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KEY TO A SCHOOL ARITHIVIETIC 
FOR INDIAN SCHOOLS 
By hall, STEVENS AND SIMS 



MACMILLAN AND CO » Limited 

LONDON JDOMBAV CALCUTTA * 
MCLDOURNL 

THE MACMILLAN COMPANY 

\ORfw UQSTOS CHICAGO 
DALI AS bAI FRANCISCO 

THE MACMILLAN CO OF CANADA, Lto- 


TORO TO 


KEY 

TO 

HALL AHD STEVEHS’ 

SCHOOL ARITHMETIC 


ADAPTED FOR USE IN INDIAN SCHOOLS 

REV ANDREW SIMS, BA. 

liONDON 3IISSIOSART SOCIETl’S IVSTITCTION, BHO^S AMl’tTB, CALCCTTA 


MACMILLAN AND CO , LIMITED 
ST MARTIN’S STREET, LONDON 


1914 



roPliJaHT 


<3L\SG0>V PPINTED AT THr TpMVErSm rRE<!S 
B\ ROBERT UACLFno«E ASD CO, LTD 



KEY TO SCHOOL ARTTmiETIC 

FOR INDIAN SCHOOLS 


EXAMPLES I b Page 8 

13-16 See Art 8 17 10000=62'>xlG 18-29 See Art 13 

30 (i) 567 =7 + 7 X 80 (n) 9036=9x1000+9 x4 


EXAMPLES I c Page 11 


871 

19 119)103072 
847 

23 rent 


607 

20 233)141431 
1631 


21 207)177399 
1179 
1449 


289 

22 471 ) 136160 
4196 
4280 
41 rcra 


1198 

23 593)710580 
1175 
9828 
4910 
166 rem 


7120 

24. 123)876513 



783 reni 


MISCELLANEOUS EXAMPLES I Page 13 

4 £17875 -£2500 x 5 7 (7126 - 3780)-257 

9 Sura of the two numbers = first no + second no , 

=(sccond no +190)+ficcond no, 
=twice second no +195 , 
tw ce second nuraber+199 = 947 , 
whence second number =196, and fii-st numbci = 391 



2 MISCELLANEOUS EXAMPLES I [CHAP 

16 If the cart cost 1 share of £62, each horse cost 6 shares of £52 , 

2 horses and cart cost 13 shares of £52 , 
i e 13 shares are worth £52, giving cost of each horse as £24 

20 Note 9998 = 10000 - 2 , see Ait 10 

22 If we divide 987654321 by 164609, 

we get 6000 as quotient and 321 remainder , 

987654000 will divide exactly by 164609 , 
hence 987654000+164609 or 987818609 is req^ number 


27 (43120-55) X 7 30 See Art 9 

34 Note 999=1000-1 , see Art 10 

42 (76428 -53)-611 = 126 45 Half of (£85 - £39) 

49 A’s score = B’s score x 2 , C”s score =jB’s score x 4 , 

total score of A, B and C'=5’s scoie x 7 = 168 , 
hence jffs score =24, A’s8core=48, U’s score =96 
Total score of remaining batsmen =255- 168-15 


51 In skeleton form the two divisions are 


rem 

rem 


61=7x8+5 

52 In skeleton form the division is 



rem aUj 
rem bj 


61=95+0, whence 5=6, a=7 


Dn isor | 529565 | Quotient 

(0 

246|6 

( 11 ) 

222|5 

(ill) 

542 


The gaps (i), (ii) and (iii) are 
filled by the nos 5049, 2244, 
1683 respectively Each of these 
IS some multiple of the divisor, 
which must be greater than 642 


Bytiial, 5049= 9x561, 2244 =4 x 561, 1683 =3 x 561 
Thus divisor=561, and quotient=943 



COMPOUND QUANTITIES 


3 


u] 


EXAMPLES n a. Page 25 

50 For e^ ery £l I get Eg 16 , 

foi £37^ I get Es 15x37^=562Es 60 cents. 


EXAMPLES n d Page 31 

11 (n) 23Es 26 cents X 10=2325 cents X 10= 23250 cents 

=232Es 50 cents 
23 Es 25 cents x 100 = 2325 cents x 100 = 232500 cents 

=Eg 2325 

23 Es 25 cents x 1000 =2325 cents x 100 = 2325000 cents 

=B8 23250 

£ fi d 


41 

5 2A 

8‘ 

(fl) 


330 

1 8 

(i) Tlie sum of (a) and (i) gives (ii) 

371 

G lOl 

00 

( 1 ) and ( 11 ) gnes ( 111 ) 

701 

8 6^ 

(111) 


£ 

s d 

& s d 

& s d 

37 

13 7 

14 23 2 7 

15 27 6 4 


29 

79 

122 


12 1 203 

12 1 553 

3|122 


16 llrf 

46 

Ic? 40 8(f 


29 

168 

732 


145 

20 451 

20 [204 
10 

20 [772 

4s 38 12s 


22 11s 

158 

244 


87 

237 

854 


203 

£1827 

£3332 


£1095 




4 


COMPOUND QUANTITIES 


[chap 


s d 


Bs a p 


Bs a p 

5 2 


17 82 10 3 

18 

173 11 9 

177 


141 


193 

61177 


4|l41 


4|579 

29 

6d 

35 

3p 

144 

885 


1410 

2123 

20f9i4 


16 1445 


16 2267 

45 

14s 

90 

5a 

141 

708 


1128 


193 

1062 


282 


1351 

£8187 


Bs 11652 


578 

Bs 33530 


cents 
19 4020 

m 

4020 

28140 

28140 

711640 cents 
=7116 Es 40 cents 


cents 
20 7655 

307 

22965 

53585 

2350085 cents 
=23500 Bs 85 cents 


24 The former dividend = Tlie latter dividend = 

Bs a Bs a 


5 3 

4 6 

6500 

5250 

16 19500 

8115750 

1218 12 a 

1968 

32500 

21000 

Bs 33718 

Bs 22968 


Bs a 
33718 12 
22968 12 

the total dividend= 

Bs 56687 8 a 



COMPOUND QUANTITIES 


n] 


EXAMPLES II e Page 34 

Bs n Bs a 

7 (99 15-10)-(99 15-12) 


cents 

12. 96 ) 864480 ( 9005 cents 
480 

90 Bs Scents 
Bs 

18 407 ) 8271 (Bs 20 

131 

16 

2096 (5a 

61 

12 

732(lp 

325 p =Be 1 11 a Ip 
Bs 20 5 a Ip, 

with lein'' Be 1 11a Ip 


£ s d 

9 56)^ 14 8(£7 

17 
20 

354(6j 

18 
12 

224 (4(f 


21 35 6c7 = 7 sixpences 

£4 05 6cf =161 ,, 


retj'’ no =-14^=23 


23 Be 1 8a =3 half-iupees , Bs 148 8a =297 half-iupces 

req"* no =-S-gi=qq 


28 Division is hut a short method foi successive suhtiactions 
Bs 7 10a 3p =1467p , Bs 100=19200p 
1467)19200(13 
4530 

129 p =10a 9p 
1 3 times, with rem*' 10 a 9 p 

30 No of 6hares=total distributed— shate dividend 

Bs 29683 4a =474932a., Bs 4 10a =74a 
req’* no = AX| . p . g .=6418 



6 COMPOUND QUANTITIES [CHAP 


EXAMPLES U f Page 35 


18 See Art 23, Ex 2 


Bs a p 
25 6 1 3 

81a 

12 

975 p 


Be a p 
483 0 9 

16 

7728 a 
U 

92745 p 


975 ) 92746 ) 95 
4995 

rera' 120= 10 a 


95 payments, with lein' 10 a 


26 Bs 600 =60000 cents , Be 2 55 cents =255 cents 

265 ) 60000 ( 235 
900 
1360 
75 cents 

235 days, with lem*^ 75 cents 

31 £1+1« +ld=253rf , £43 4s 6rf= 10373d 

req^ no 

32 The sum of the 4 profits=Bs 14917 15 a 

average annual profit=??^— 

35 A’a share =B’s share +Bs 13 
twice J3’s 8haie=Es (67 - 13)=E8 64 

Thus B gets Bs 27 and A Ea 40 

36 -ffs 8hare=(7’s ahare+£l5 , A’s share=C’8 8haio+£l6+£40 

three tunes <7's share + £16 +£65 =£265, 

ze C’b Bhare=^ ^ - ^^~'^°) =£66 

37 18Rs 20cent3=1820cBnts, 17B8 25 cents =1725 cents, 

16 Bs. 46 cents=1545 cents 

By addition, twice (A’s+B’s+C’s) 8hare=6090 cents , 

A, B, and C together get 2545 cents , 
but A and B together get 1820 cents , 

Ogets 726 centa=7Es 26 cents 
B’s share=(1645 - 725) cents=820 cents=8 Bs 20 cents, 

and -4'ashare=(1820 - 820)cents=1000cents=Es 10 



II ] COMPOUND QUANTITIES 7 

38 The total tax =(850 X 11) pence =£38 19s 2rf 

39 The no of pounds in his income=the no of times that lie? is 

contained in £45 16s 8rf, or 11000 pence Hence the req^ 
income =£1000 


40 Cost price of each=^^-|^-^=3s 


Sale 


20 

£2 2s 6cf 


12 


=3s 


profit = 2c? 


41 Profit per yd =6p But the total profit was Ps 2 4a =432 p 

no of yds of cloth bought and sold =-^-§A= 72 

42 The cost of 29 mds atl5Ps 80 cents =458 Bs 20 cents 

„ 16 „ 23 Bs 45 cents = 375 Bs 20 cents 

„ 45 mds =833 Es 40 cents 

„ 1 md = 18 Bs 52 cents 

43 As in Ex 42, it will be found that 1 lb of the mixture costs 2s l|c? 

Hence gain per pound =2|c? 

44 The carriage of 1 yd =3|ef , carriage of 1 ton=£200 

no of yds to the ton = " ° = 12800 

45 Total takings for 4 months =Bs 1555 14 a 

takings for remaining 8 months=Es 5000— Es 1555 14 a 

=Es 3444 2 a 

j Es 3444 2 a „ „ 

req® average = 5 =Es 430 8 a 3p 


EXAMPLES n g Page 39 

53 There will be 240 intervals between the 1"* and 241*'^ post 

no ofKm=^f^=6 

54 The tram runs 36 mi m 120 half-minutes 

no of yds in half a minute= =528 


ISO 



8 


COMPOUND QUANTIllliS 


[CHAP 


68 2026 yds 2 ft 
18 

3 [36 ft 
12 

2026 

16208 

36480 yds 
24 


72960 

145920 

1760 1 8755 20] 497 mi 
17152 
13120 


Tlie daily riin=(2026 yds 2 ft ) x 18 x 24 
=497 mi 800 yds 


800 


cn A j , 67 rai 500 yds 

59 Avei age advance per day = si — - — 


Km 

2 

m 

177 

31 

65“ 

31 

2 

177 

67 

487 

60 In 45 


31 

=2 mi 300 yds 


mi yds 
31)67 500(2 
62 

6 

1760 

31)9300(300 


Eem'=67 Km 600 m —67 Km 487 m 
=13 m 


re\ olution gives off 44 ft 6 in of rope , hence depth of shaft 
=44 ft 6 in X 30 =435 yds 


EXAMPLES H h Page 41 

25 1 ton=20 cwt =(20 x 4) qr =(20 x 4 x 28) lbs = 2240 lbs 

1 tonne=1000 Kg =(1000 x 2y lbs =2000 lbs +■* of 1000 lbs 

=2200 lbs 

38 One bottle of each lequires 1^ seers 

2 iiids 1 seei =162 half-seeis , 1^ seeis=3 half seers 
req** no of bottles = =64 



COJIPOUNl) QUANTITIES 


9 


ft ID 

4 3 

40 Total length =(4 ft 3 in x48)=68vds § 

Weiglit=(3x 68) oz = 204 oz = 12 lbs. 12 oz. 34 0 

204 0=68 yds 

4L Weight of rope=(2i x 280) lbs =(560+140) lbs =700 lbs 

=6 cwt 1 qr 

Adding eight of load we get total wetght=2 tons 16 cm t 1 qr 


EXAMPLES n k. Page 46 

34 Area of the 16 flower beds 

=50 sq yds 5 sq ft x 16 
=808 sq jds 8 sq ft 

Area of lawn is obtained by subtracting the 
areas of the beds and of the paths from 
the w hole area 

sq yds sq ft 
1300 0 

803 ' 8 

241 1 

2o0 0 

41 (i) 1 ac.=4 r =(4x40) sq p =160 sq p =(160x30|^) sq jds 

=(4800+40) sq yds 
=4840 sq yds 

(ii) 1 sq ini =(1760x1760) sq yds =3097600 sq yds 

=(3097600- 4840) ac 
=640 ac 

55 6 ac Or 25 p =985 p , 215 ac 1 r 35 p =34475 p 

req'* no 

EXAMPLES n 1 Page 52 

3 See Art 30 

11 Each truck can jcs (42 x 8) coh ft , or 330 ciih ft 

112 cub 5 ds =(112x27) cub ft,oi 3024 cub ft 
req'* no of trucks ="j^-j^ =9 

13 Since 7 jds =21 ft , lol of tinio=(21 x 1 x 1) cub ft =21 cub ft 
its Meight=(04x21) lbs =1344 Ills =12 ewt 




10 


SIGNS AND SYJIBOLS EQUATIONS [CHAPS 

14. Actual volume of ■ttater=(5x4x3) cub ft =60 cub ft 
its ueigbt= (60x1000) oz =3750 lbs 
Estimate =1^ tons=l ton 10 cwt =3360 lbs , 
so that I ovei estimate by 390 lbs 

15 The \olume of the cube=the volume the iiatei is raised 
„ „ = 18 X 18 X 18 = 5832 cub in 

The area of the base of the vessel =72x27 =1944 sq in 
the no of inches the watei is raised 3 

EXAMPLES n m Page 53 

9 The daily no of atrokes=(l +2+3+4+ +12)x2 

Feby 1912 has 29 days 
no of strokea=78 x 2 X 29=4524 

11 On the last day of 1910 the no of the paper was 5030 

365)5030(13 

1380 

285 

So the paper w ould then hav e been published over 13 years But 
theie iveie two leap yeais iii toese 13 jears the paper 
would then have been published 13 years and 283 days, 
7 e on and fiom Maicli 24tb, 1897 

12 Had the person been born on Dec 31st, 1891, his age viould have 

been 19 years 

These 19 years=(19 x 365+4) dayB=6939 days 
his age was 6939-24=6915 days 

EXAMPLES m a Page 56 

46 1 lb of tea costs 30if , 

1 lb of cofifee costs 18rf , 


X lbs of tea cost 30r pence 
y lbs of coffee cost 18y pence 



11 


II , HI ] SIGNS AND SYMBOLS EQUATIONS 
52 4r+2/>+J-r=4 3+2 1+9-0=12+2+9=23 
54. or|7— y+22— 7 j =3 3 7 - 5 + 22 — 7 9=127 — 68='59 
55 93-r=-3/-8^=93-3=-3 15=93-9-75-8=1 

59 — +5-g2o-2:=i-?+5 0 1- 1— ^=12+0-7=5 

X q o i ^ 

EXAMPLES in f Page 69 

3 Let X be the number, then 4(2a+3)=32 

4 Let V bo one part, then 60- ns the other pait 

Hence 3^-100=200- 8(60- 1 ) 

5 Let Six bo U’s share, then B’a Silmie is £(» +8), and A b share is 

£(t+84 15) 

Hence a+(r+8)+(i+8+13)=67 

6 Let £.r be A’s shnic, then Fa shite is £2r, and (?’s is £(2? -4) 

Hence a.+2.r+(2c— 4)=00 

7 Let A ha^e £t, then B has £(i?+10), and (7 has £3.r 

Hence i+(r+10)+3a:=8D 

8 Let A ha^ e £r, then B has £(i+37), and C has £(2 a + 11) 

hence j +(r+37)+(2a.+ll)=I88 

9 Let r be number of shillings, then 67 - t=number of sixpences 

X shillings +(67 — r) sixpences =£2 13s Od , 
or, reducing to sixpences, 21+67—1 =107 

10 Let A ha\ e Bs v, oi 4r four-anna pieces, 

then B has Rs (12-r)=(48— 4i) four-anna pieces 
Hence 4a;+5=7(48-4s -5) 

11 Let B hate v shillings, then A has 3a. shillings 

hence 3a’— 10=2(a;+10) 



12 PBIME NUMBERS [CHAP 

12 Let one have ^ shillings, then the othei has (30- ^) shillings 
hence a — 6=J of (30— 1+9), 

and, multiplying by 2, 2(a;-6)=39- v 


EXAMPLES IV a Page 76 

1-13 See Art 78 In E\ 13 the uoik is shortened by “casting out 
nines,” as explained in Ait 12 

14 “ Casting out nines ” from the last example (sec Art 1 2), e get 

12, 3, 9, 0, 11, 2 Since final rem is 2, the missing digit 
must be 7 

15 From Art 77, we have in the last example, 

sum of odd digita=5+8+6+2=21, 
sum of qnen oven digits =4+ 7+8 =19 , 
missing even digit must be 2 


17-34 

See Alt 82 



32 5 

7245 

33 2119206 

9 

1449 

9j 

9603 


^[161 

11 

1067 


23 


97 


No =5x9x7x23, No =2x9x11 x97, 

=33x5x7x23 =2x33> 11x97 

34 12 1 249984 
12 1 20832 
8 1 1736 
7 1 217 
31 

No =12x12x8x7x31, 

=23x3x23x3x23x7x31, 

=23x33x7x31 

40 From Art 78, the test foi 8 is satisfied, 

and ve get 712=8 x 89=2^ 89, since 89 is prime 



IV] PRIME NUMBERS 


13 


43 TakiDg the pnmes m order, we find that the tests for 2, 3, 5 are 
not satisfied 'We then try 7, 11, 13, 17, 19, 23, and find 
667=23 X 29 Since 29 is pnme, this is the req** result 


44 As in no 43, we find that 2, 3, 5, 7 are not factors, but that 11 is 
a factor, giving 10681 = 11 x 971 We then find that 971 will 
not divide by any of the primes 13, 17, , 37 , 971 is a 

pnme (see Art 82, Ex 3), and req** result=ll x971 

47 100a+10&+c This IS divisible by 2 or 5, when c is so divisible 


49 


100Z+10m+7i 

lOOm+lOa+i 


j- are any two such numbers 


Their difference is 99Z - 99m — 9n, which is obviously divisible by 9 


EXAMPLES IV b Page 78 


18-27 See Art 84, Ex 1, 2 

23 Since 10082=2 x 5041, and 5041 is odd, 

2 occurs to an odd power , 10082 is not a square number 


25 Here 3136 =8 x 8 x 49, 

=82x7*, 

sq root of 3136=8x7=56 

27 Here 7056 = 12 x 12 x 49, 

=122x7*, 

sq root of 7056=12 x 7 =84 

28 392=4 X 49 X 2=(22 x 7*) x 2 , req"* factor=2 

36 Here 46656=12x12x12x27, 

=123x3*, 

cube root of 46656=12x3=36 


8 1 3136 
8 (392 
49 

12 17056 

12(588 

49 


121 46656 

12 (3888 

12(321 

27 


37 


39 


Here 91125=9x9x9x125, 

=93 x 53, 

cube root of 91125=9x5=45 


9 91125 
9 1 10125 
9 (1125 
125 


The no must contain 13*, or 169, as a factor, and every other 
factor must be twice repeated Thus we have to select from 
169 X 22, 169 X 32 , , and of these 169 x 3*, or 1521 is the only 

one between 1000 and 2000 



14 


HIGHEST COMMON FACTOR 


[chap 


EXAMPLES rV d Tage 82 


117 

221 

1 2 1 

203 

319 

1 3 2 

559 

817 

104 

117 


116 

203 


616 

559 

13 

104 

8 3 

87 

116 

1 

43 

258 


104 


87 

87 



258 

HCF=13 


29 

HCI 

=43 


HCF=29 


See Art 91 

5 1 

255 

391 

1 6 2 

329 

799 

644=4x161, 

136 

255 


282 

638 

532=4x133 

119 

136 

1 

' 47 

141 

1 

161 

133 

4 j 

119 

119 



141 


133 

112 



171 

' HCF =47 


4 28 

21 

3 HCF=17 




7 

— 








HCF =4x7=28 


7 See Alt 91 


5 

527 

1147 


465 

1054 


2j^ 

93 


31 

93 


HCF =31 


8 See Art 91 


2 

623 

833 


420 

623 

29 

203 

203 

sofilo 

7 


HCF =7 


9 See Ai fc 91 
348=4x87, 
1024=4x256 


1 

87 

256 


82 

174 


5 

82 


At tins stage w c see 
that 87 and 256 are 
piime to each other , 
HCF req®=4 


See Art 91 

11 1 

3451 

9367 

1702= 2 x 851 

2465 

6902 

1998=2x9x111 2 

986 

2465 

7 

851 

111 

3 

986 

1972 



r77 
■■ — 

111 



493 


2 

74 


HCF 

=493 



37 






HCF =37x2=74 


12 See Alt 91 


1379 

2401 

1022 

1379 

357 

1022 

308 


49 

308 


294 


2[n 


7 

HCF 

=7 



mGHi:ST COMMON TACTOR. 


15 


IV] 

13 See Alt 01 


14 See Alt 01 


15 Sec Alt 91 


4 

4199 

5083 

1 10520= 4x10 x 263 3 

13547 

17081 


3530 

4109 

11 128931 203 ( 

1^02 

13547 


21003 

881 

I l2803j 1 

0[2945 

3534 


221 

884 

HCF =263 

589 

3534 

] 

acF = 

=221 


[TCF = 

589 


16 

1 


See Art 91 


17 


20077 

31279 

10002 

20077 

5 1 10075 

9] 10G02 

2015 

1178 

1178 

837 

9[837 

341 

3|93 

31 

341 


19 


21 . 


24 


See Art 91 
13503 =9x1507 
7110'i40ll507 
1 10549 1 
HCF=1507 


18 See Alt 91 
10984=4x2x2123, 
5404=4x1351 
1 


11 


2123 

1351 

1351 

1.351 

4 772 


103 



nCF=31 
See Art 01 

10905 = 3 x 5x11x103, 

04890 = 2 x 5 x 3 x 3 x 721 
103172117 
1 721 1 
HCr =103x3x5 
= 1545 

183=01x3, 22 058=2x47x7, 

793 = 01x13, 940 =2 x 47x10, 

970=01x10, 1128=2x47x12, 

HCF=G1 HCF =2x47=04 


HCF =193x4=772 


20 94248 =2’ 3’ 7 11 17, 
504900 = 2’ 6’ 3’ 11 17, 
HCF =2’ 3« 11 17 
=0732 


23 


See Arts 91 nnd 92 
422.37=0x4093, 
75582=9x2x4199, 
8892=0x4x247 
TVe now find the H C F of 
4093, 4199, 247 

117 


1 

4003 

4199 


41 M 

247 


2|491 

1720 


247 

1720 


We obtain for tliii 247 , 
req"* H C F =9 x 247 
= 2223 


403=31x13, 
744=31x0x4, 
1023=31x3x11 , 
HCF=31 

25 3150=9x3x2’, 

20244 = 9 x 3 x 972, 
90225=9x3x3075 

Since 2’, 972, and 3075 have 
no factor common to all 
thice, it follows that 
HCF =27 

26 Tlio req** number must 

divide the nurabcis 
(14490-0) and (31 530-0) 
without lemainder 



16 


LEAST COMMON MULTIPLE 


[chap 


31 2 


2 

385 

525 


280 

385 

3 

(i) 105 

(11)140 


3x35 

105 



35 


Starting from the remainder 35, ne 
can build up the work step by step till 
we arrive at the two numbers required 


Tiiiio aySn or lUO* musu euuai uuo — 

must equal the remainder (u) . whence we get 140 Then 
2 X 140+105 gives 385, and 385 x 1 + 140 gives 525 

32 Tlie required number will divide the difference between any twc 

of the three numbers and leave no remainder 
163599-142408 = 11191, 
and 166402-153599=12803, 

hence req^ no is the H CF of 11191 and 12803 

33 On)) 40051 ((iv) The gaps are filled as follows 

^ ^ („) (i) =1731-294=1437, 

(n) =4005-173 =3832 

(i) Now 1 437 (or 479 x 3) and 3832 (or 479 X 8 

294 aie multiples of the req* divisor 

Hence divi8or=479, and quotient=83 


EXAMPLES IV f Page 86 

1-7 See Art 97 8 2« 6d =5 sixpences , 10s 6d =21 sixpences , 

LCM of 5 and 21=105 

11 The lowest number which each of the four given numbers will 
divide mthout lemainde}, is then LCM 
This will be found to be 720 , hence (720+9) is req^ number 

14 Beduce to thieepences See Ex 8 

15 The answer must be the least number^ which will contain 105, 112, 

126, and 168 miles an exact number of times 

16 As Ex 15 

17 The req* time in secs must divide exactly by each of the numbers 

252, 308, and 198, since by then, A, B, and (7 must each have 
completed an exact number of rounds 

they will nesrt be all together in the LCM of 2 'j 2 secs, 
308 secs , and 198 secs 

18 From Art 97, LCM or 11781 

A II9 



V] 


REDUCTION AND ADDITION OF FRACTIONS 


17 


12 


16 


20 


27 


3003 3x1001 
2503 11x228 
3036“ 11x276’ 

391 391 

408“3x8xl7 
17x23 


EXAIVEPLES V d Page 93 

3x5x21 3xSx?x3 3 

4 

13 


315 3x105 

420“3xl40“3x5x28“3xAx?x4 
231 3x77 3x11x7 3xttx?xl 


3x11x91 8x14x7x13 
14x12x19 19 
’^14x19x23 23 

(a) 


3x8x14 
_23 
24 

28-29 See Art 107, E\ 1 
572 4x11x48 4 

1287 9x11x13 9 


(o) At tills stage we see neitliei 3 noi 8 
di\ides 391 Hence, if the fiactioii admits 
of 1 eduction, 17 must be a factoi of 391 
Bj’ division we obtain the other factoi 23 


34 


35 


954 9x8x3x83 3 
2544“8x3x8x88 8 


EXAMPLES V g Page 98 

47 LCD of 39, 65, 15=3x5x13=195, 


exp" 


5+21-13 13 18x1 


195 195 13x15 15 

48 LCD of 35, 21, 15=3x6x7=105 , 


o\p"= 


36-20+49 65 8x13 13 


105 105 5 x 21 21 

49 LCD of 12, 36, 27=4x33=108, 

99-15-8 76 4x19 19 


exp"= 


108 


108 4 x 27 27 


EXAMPLES Y li Page 100 

17 Exp"=i+^'?-+^i-S-+T^+-l-^=i+r;^+^ 

=i+t:^+5=i+*+^i=2^ 

18 Exp"=9+VV-+TV+-H-+fJ=9^^fT7Tp+'-i 

19 Exp"=9i:A+i=_L2_+i =1 +i=J 

‘ 13X0~O X3Xfl~^ 

K H S I 


B 



18 


SUBTRACTION OF FRACTIONS 


20 Exp"=3^^--V=3^-*=3*-*=3 

21 Bxp”=-|---g^-y\= 6 0 

_ «r n 1 — _1 

22 Exp"=lTT“T-s-“;rT-iiTx6xf 'sr 

t 11 1 _1JL 1=1 

^11X6X7 '3^^ 3 0 3T 


[chap 


EXAMPLES V k Page 101 


38 3 -(2^-f)=3 -2f +^=i+^-=-SV-= 1 1-5 

39 3^-li=2j-i=2+-V^=l+£^^^=P^ 

2-Tt=lHr=lH 

fii-st exp" 16 greatei, by 

41 2^--|+t‘’^= 2+-!^^^^=2-^=2* , -J niu8t be added 


42 io-(-V-J.)=10-V-+4-=10-33+|=7+>?-3 
=:7+a-CjJA.=7_Y Hence leq"* diff 


43 


(i) 6f-a?=ly5-, 

or -6f+«=-li^, 

r=6f-— ly‘^, 
or t=4^ 


(ll) t+1^ — yyjj— l'^ , 

1 *1 -I 7 _ 1 

^“■n+rrir F» 


01 T=tS 


— T®3 


EXAMPLES V 1 Page 103 

11 Exp"=3H+3t-l^-4^=l+Q-^-^)-(3 ^-s) 

=l+iT-i^=l+i-^=l 

12 Exp"=8+(i+A)-(^+^)=8+i-H- 

=8+i-:^r=8H=83 

13 Exp"=*-i+^^+^x = ao-|±^=^=^4^ 

14 Exp"=l -(^-^)+(^_|)=l_^+^=l _^+.i 

17 aH+*)-^=f-^+i^= = + l^=2 

18 (8A-3|)-(l0;ft:-6i4)=8yV-3t-10:ft-+5^-4^ 



v] ADDITION AND SUBTRACOTON OF FRACTIONS 
19 - Tir) ~ (^ To 7 “ \-r) == 2-5^ - ■3*^ - 1 + 4 r 


19 


=1 +(1-7 +^'0 - ('jV+'J'St)- 1 + c >- 7 X 1 1 
= 1 + ijV- -.V = ^ + 3X’f X7 “ 


8 r + l 4 1 . + 8 


1 OS 


20 Eeq** anioimt= Rs (2-^+ U) - 1?“? (3^+ iV) 

= R‘’ (2t V+I J -3J -••j’j)=Re (-77+ 2 i’;) 

=Ro (2 - J)=Re ■? oi 8 a 

21 Req" amount= £10- £(2j +4-fV+ \i) 

= £4-£(i+i-V+l-fr)=£l-£;-i=£3=A=.£3 5. 3rf 

22 Req'* fraction = (I - “ l*u ” } ) = rV 

\\c get fust suni=-\-of Rs 240=Rs 60, 
second flum=-j'|7 of Rs 210=Rs 72, 
thud sum=J of Rs 210=Rs 48, 
fouith aunisE-'j- of Rs 210=Rs 70 

23 A o\^es B 4^ 7^d in all Hua to the ncnicst pennj is 4s 8ff 

A paja B Qs , li psjs A 4d 

24. Length of iod=(l05-+402+38j -6l) in ^ 

==(81 +38j “'>!) in =113^' ill =11 1 in to nearest inch 

25 Residue = (l - ' ~ -J - -ji - )= i' r of pi opei tj 

-j’- of piopeitj =Rs 1200, and uholc piopeil} =5^4400 

26 ? + l+*+*+-V=l^ ^ 

“the lest" conRiBt of J of 370000, oi 68333 


27 » 2 ±-‘+ 2 f ‘.2 

{ a+b)xb+(a+h)xa — 2a b 
~ ‘ ah 

ab+b-+a-+ob~2 ab 
~ ah 

a=+y 

ab 

If rt=2 and 6=3, 

a+6 , a+6 _ 
a b 

_ r> 4. r _o oi 

,, o'+J' 2'+3» 

-55 0— 


( \ ^ 4. ^ 

a-V> ab^'^a^b 
a-b—tt+b 


a-b- 


=0 


If 0=2 and 6=3, 

,, a-b 1,1 

5^-55;+5»6 
=tV“'^“‘3V = 9 



20 


MULTIPLICATION OF FRACTIONS 


ir 

[chap 


EXAMPLES V n Page 109 

15 The daughters shaie is (l --g-), or of the estate 

req^ \'ilue=^ of £5650= £2118 15s 

16 (l - 01 of the distance leinains 

req** no of Kni 896=280 

17 I posbess of Es 3,00,000 or Ea. 1,75,000 

leq^amt =Bs 1,75,000-Es 25,000=Es 1,50,000 

18 (1 - Vs) or of the amount lemains 
Since j®g- of 1620 tons =450 tons, 

leq® tinie=A^ da}s=5 days 

19 The senioi partner gets T*r of £10043 or £3652, 

„ second „ „ of £10043 oi £2739, 

„ third „ „ xV of £10043 oi £1826 

Tlie leseive fund =£(10043 - 3652 - 2739- 1826) = £1826 

20 (l - ff) or A of the cargo remained good. Now A of 12000 =7500, 

and reqd profit=Es 2yAj:^o=,j{g 


36 

37 

38 

39 


EXAMPLES V p Page 113 

A majoi’s pay=f of 18s =16s , 
hence a captain s paj =5^ of 16s = 11s Gd 


ffs age=-i of C’s= J of 15 yrs =20 yrs , 

A’s iige= A of E’t.=A of 20 > rs =25 ^ i-s , 

M’s age=(25+20+15) yrs =60 yrs 
In Dec. 1905 its ialue=^ of Es l,20,000=Es 1 00,000 
In Dec. 1906 „ =w of Es 1,00, 000 =Eb 83,333 5a 4p 

Since the rule is ft too short, 

the apparent length of 1 ft is reallj ft , 

” » 24 ft „ VV of 24 ft , or 22 ft , 

i-r ” ” H- of 20 ft, 01 18V ft 

Had each foot of the lule been ft too long, 

a leal length of l-jL ^ -would have appealed to he 1 ft 


1ft 
22 ft 

18V ft 


i-Tft 

fj of 22 ft, 
01 20rT ft 
■H of 18V ft , 
or 16H ft 




99 


9) 



V] DIVISION OF FRACTIONS 2i 

40 Total amount cut off=-^+-j\- of , 

fraction of stick left is 

41 Total distance ho of , 

distance be walks =-^ 

42 The first has of 485 , or 14s , 

the second has of 345 , or 165 , 
and the thud lias of ISs , ol 45 , 
whence the fourth has (48 — 14 — 16 — 4) s , or 14s 


EXAMPLES V q Page 116 


25 Let X be the req** fi iction, then 4^ x ^=4V, or a x Multi- 
plying both sides by - 3 *^, we ha\e x-^, or 


r= 


_ 1 ! 


ra 


27 Keq'* no of timc3=16— (4^ of ■ 3 ^)=n— -t=15x 3=10 

28 Weha^o 103 -( 2 ^x 2 f)=-'>/— Y-=^=-XxV =2 


(i) mi =1 Kin , 
l^xfmi =5. Km , 
or 1 mi = 1 -^ Km 
(iv) 39^ in =1 m , 
hence 1 in =(l— 39J) m 


29 (i) T mi =1 Kin , (n) From (1) 

1 mi = I-} Km , 

1 rai = 1-|- X 1000 ni = 1600 m 

(v) 39 J in =1 m , 
or 39^ in =100 cm , 

whence 1 in =|^ cm =2JJ cm 

30 (11) 1 cwt =112 lbs =112xY®y Kg =50jf Kg 

(vi) 1 litre =-J pints , (vii) 1 gram =3^^ of 2^ lbs 

1 K1 =^ X 1000 pints X X 7000 grs 

=I^gals =15^^ grs 

=218J gals 


36 32 

36in=^n>=3-jB 



22 


COMBINED PROCESSES 


[CHAP 


EXAMPLES V r Page 119 

4 Exp”=3V+tx?=3l + U=4+W-=4xV 

5 Exp"=-^x£-+J=8^+J=9^ 

6 Exp"=3}+(-j-x-J)=3i+x“jf=3j:ff 

13 Exp"=-V>«7— l=8-i=7J- 

14 Exp’>=(5|)x^=2^ 15 Exp-=4i+(,V)=6TV 

22 Exp»=(4+ W)-(2+ A-^)=4|^-(l + ) 

= 4^— lx^g-=^ X -J-f = 3 

23 Bxp»=(i^ X -V-)-(2^) X (3})=\’ X V X A X V=21i 

24 Exp»=(A|^)(ii^)-(^M^)=J-^x^x^-=l^^ 

25 Exp”=(A^)-(2j)Xx'V=lfx}Xx\-=-^ 

26 Exp“=^+(4- of -V|)+(xT Tj)~ 6 + ?+f 

27 Exp® =(x®jrX -}§■)+ 2^ -(-^x o-^)“'2'+2^-i=22 

28 Exp® =(J/ X £)-( V- X ^)- H=24- 10 = - =24 - 11 -g .=12 

29 Exp"=2^+(^xi3<’-x.J)-2HT^=2j+2-2’+x'T 

=2+.;+*-^=2+^F^=2^ " 

30 Exp® = (xj- X I-) - (xV X X tV)= f ffT+TT ~ ~ :!T 

31 Eeq® 'value=(lj-— ■§-)x(7yTj-)— (ly'iy— Tij-) 

~ F X W'~ ItV = ¥ X -tV' X yt = SJ-J 

32 Eeq® value=2j+(6^ x l9^)-(2’-x 3^ -1^^) 

=2H(^xf)-0xJfxlf) 

=2j-+10-7J=5+^-l=4+ia4^=4lJ 

33 Eeq'* value =[7i of (9l-5jV)]- [(2^ of 9’ )- 5^^] 

=[7iof4f]-[:S^xJ/-5*]“ 

=(i^0f V-)-0/-xii!-)+5^l3- (See Alt 61) 
=(Hx31)-(^x19)+5tV 
~'^(31“19)+5^-j- (See Art 55) 

=i^Xl2+5TV=12B+5TV=17U 



V] 


COMPLEX FRACTIONS EQUATIONS 


23 


EXAMPLES V s Page 121 
31—1^ 1+1^^ 


22 Fraction= |^r - ‘ - -t ^ =i^— ! y-z=.^ 

3 ^ x 2 -^- V -^-2 V ' » » 6 

23 Fraction=(Ax :-ff)-(-¥-’<iV)=TF’< - =21- 

24 Fi’action=(-^x-Y-)— f)=“—14=J 

25 Fi'action=(-^ of "’f)"‘(2+^lr'^)= 

_ Ux2 21ix3 , 

27 Fraction =-i2i£-=i^= b x ^ x 2^=5 

TT+tV 7F 

28 Fraction=i “ - ^7t'^^ - = °+P - ±i 

(5+1+U)>«12 8+3+1* 

29 Exp-^vv of (V X trV)-(-v- X u)=Tmmmh=^ 

30 Exp" =-j7_— X ■j*^)=-j.'^— ^’■^= 1 - 

T- 

31 I - X’Vr-4)x45 lj-xI5 

* nA < *n'» ol *• /^ ^ i «-l\. 


2H 115-2-J 2HX45 (1H-2-5)x15 

32 TT in\_O + -Hf--if)x10’> ,ni 

^ i + l+^ ^ a + -H^)xlO'i 

_ T I S + 7 O+OO-rt 3 >, 70 igS V 70 _100 

7T-XTTrT = lrT 


30+21+15 


EXAMPLES V t Page 122 
11 Multiply by 6 , then 3a?+2r=30 
13 Multiply by 4 , then 4a:=i?+24 
18 Multiply by 6 , then 2(r+l)+3(^- 1)=24 

21 Let z be the number , then r+^=20 

o 

22 Let a be the number , then a'=20— y 



24- 


FRACTIONS OF CONCRETE QUANTITIES [CHAP 


23 Let ^ be the numbei , then 3 “5=2 

24 fr=f, 25 

26 Let ■> and 50-a. be the two paits , then 50- v=^ 

27 Let i and a: +6 be the two numbei s , then ^+(r+6)=28 

28 Lot J have r rupees and B (20-a:) rupees , then (20- t)+g=r 

29 Equation becomes s=— Multiply by 2a , then 7a=42 

2 % 

on /XT. / 1 - 3a? 6x4x7 3r, 

30 (1) Equation becomes -^= — — , or -^=12 

Multiply by 2 , then 3a? =24 
(11) Equation becomes ^ =7, 01 12t=7 

Divide by 12 , then 

(ill) Equation becomes ^=17 Multiply by 7 , then t=119 

EXAMPLES VI a' Page 126 

8 E\p"=Es 4 J^x-t\=Es ^x-^=Rs ■^=8d 3p 

9 Evp»=Rs 6-1- X H=Es x ^=Es 3 

10 Exp”=81^ xfij-=-S^-^8 XxV=-t¥s =^1 15? 7rf 

11 E\p"=Es 4 Jx-]--- 5 =Es -^ x y‘^=Es •/-^=9a 4p 

12 E\p"=6 tons X 3j*y=ia tona-f6 tons x 

Now 6 tonsx;j*^=6x20x-j*-g- cwt =13f cwt , 

Again A cwt qrs =2f qrs , 

and ■? qi =A>^ Ibg =24 lbs 

Exp” = 18 tons 13 cwt 2 qrs 24 lbs 

13 Exp” = 103s x^V=H^s =£1 2 s Zd 

14 Exp”=24 ft X 1-jV= 24 ft x^=-^0- ft =9 yds 1 ft 8 m 

15 Evp”=EBl2x^-^=Ee V=Belf=Eel 6a 

16 E\p”=18 mds x2/^=]8 mds x^=ip^ nids =39 mds 30 srs 



m] fractions of concrlte quantities 


25 


17 

18 


20 

21 

22 

23 


24 

25 

26 

27 

28 

29 

30 


Since 5j=G--3, exp’"=I?s.ll 7a 8p x(6-|-) 

=Rsll 7a 8px6-Esll 7a 8px-J- 
=Es 68 14a -E<; 2 13a Up =Es 66 Od Ip 


•5 , 

-1±L- 

* +J 

b b _ ~ e 

19 


1 

19 

pd 

II 

s + i. 

=l+i- 

Es 

a 

P 

tons 

CTrt 

qrs 

lbs 

oz 


3 

7 

4 (n) 


1 

7 

3 

0 

0 (a) 




2 






_5 


6 

14 

8 


6 

18 

3 

0 

0 

1 

1 

11 

8=^ of (a) 



6 

3 

21 

0=J of fa) 

I 

u 


6 

_n=iof(«) 



3 

1 

24 

_8=’ of (a) 


T 

1 

3 


7 

9 

0 

17 

8 


Exp® =£4 3* 8rfx-j-y, ■we dmde £4 3s 8d b} 16 and muUi- 
ply tbe result b> 23 

Since 3^=4--!-, e\p"=Es 9 6a 8p x(4~J) 

=Ee 9 6a 8p x4— Es9 6 a 8p —8 

=Ee 37 10 a 8p — Ee 1 2 a lOp =Es 36 7 a 10 p 

Exp®=28Sp — p =2U'J p =1 ac 1 i ll^p 

Exp"«Es 24 11a x5-^=Rs 123 7a +1^24 Ha xf- 
Now Es.24 llti x-jV=(-iV of Es 24 11 a )x7 
=Es 2 Oa 11 p x7=Rs 14 6a 3p 
exp"=Es 123 7a +Rs 14 6a 5p =Es 137 13a 5p 

Exp® =2 rads 30 SIS 4 cliks xi=1764 cliks x-y 
=980 chks =1 rad 21 srs 4 cliks 


Eeq*' xalue=(-2- of 7+™ of 11) half-annas 
=(3+3’-) half annas=3a 3p 

Eeq'* xalue=(| of 126-^xY sixpences =(28 +12) sixpences 

=£1 

Eeq"* value=(-^ of 90-{- of 192) in =(126-120) in =6 in 

Eeq"* value = of 1760 --jV of 220) yds =(330- 100) yds 

=230 yds 

Eeq* \alue=( of 63+^f of 55) p 

=(A^+i.?i)p=Eel 9a 5p 

Eeq* value=(-J-f- of 1980-‘-^‘V of 34) ft ^=2886 ft =962 yds 



RATIO 


26 


31 

32 

33 


22 

26 

27 


31 

32 
S3 


H 


35 


[chap 


Eeq-* value=(x®T of 112+-^of 52+f of 12) lbs =(40 + 12+32)lhs 

=3 qis 

Req^ vaUie=(^- of x*t 236+^- of 504+-^ of 352) p 
=(384+384 + 192) p =Es 5 

Eeq* -value =(-^ of tV of 120—-^ of X of 24+x\ of 462) pence 
=(12-18+126) pence=10s 


EXAllIFLES VI b Page 128 


T- . 23ia 93 . 

riaction = '5~ — =-i — oT=-r 
31 a 4 X 31 * 


n. -p I. 4/6 cm - 

24 Fraction== 7 -; ■=+ 

644 cm ** 


„ , 323 sq ft 1 

Fraction=,;s — 

68 X 9 sq ft 17 


17x19 

■x4x9 


■?-of37f8 -jlof^*'- /3x75\ /4xl35\ 

^ofl35s iofi^ Ux2/ V 9x1 / 


3x75x9 
^5x2x4x135 ^ 


Fraction=:.-??^A-= 


— r 1 


2x1760 yds 


See Art 137 We have 


4d 


=-M Also 1 ft _ 16 ft _ , 

7 yds 2 ft ~23 ft ~ 


a: lbs _ 5ja _ r 21 
28 lbs “73^a ’ 28”^ ’ 


Fiom Alt 137 

■whence x 28=2 

A sa-vres -f of Rs 2000, oi Rs 800 pei annum 
B saves of Rs 1440, oi Rs 540 per annum 
j . Rs 800 • „ 

leq^ ratio==— =X7;=--^ 

^ Rs 540 “ ‘ 

Rem'=-jL of -g. of £ 15 of a guinea 
~(tV ^ of 300— of 21 ) shillings 
shilIings=-^^5 

req* fraction=^^ ^ =^3X1=—^^^ =-§- 
i3 9s 69s 45x69 135 





rr.RcrxTAGrb 


27 


EXAIEPLES VI c. Page 131 
7 Since G % ai'c l.lleu^ tiec'!, 91 % are ttco<5 of otlici kinds , 
leq'’ imiiiljci =-j" 5 {y of 350=329 

9 Let i=tlie into pei cent , then ^=~, and •> =5 

t 100 20' 

10 Lct'^ic the icq'* ])ciccntage He pa\ cs Bo 2400 , 

i«d v=32 

100 7>00 

12 Let r lie the icq* niimher, then of i=1603 , 
hcnct r = 1603 X = 4560 


15 If Rs r be the icq'* pnee, then — of i=930| , 

TOO 

r=030ix-/^=14892 

0}- 

16 Since G-J % aie lost, 93^ % are left lit foi scimcc 

If I be oiigin.il nnnibci, ’ =13132, 

TfVO 

7=13132x^=14070 

17 If C.7 be the intonic, then the deficit is 

But tins IS the diflcicncc between the expenses and the income 
or i'(4837i-7), ^^= 18 ) 7 ^-^, i+|^^=4837i 

01 ^*-=4837i, whence 7 =4837£ x^^=4500 

19 Tlie Welsh boys=(100-45— 20— 13)%=22% of icq'* niimbci 

oth ion 

If this bo r, then ^4 , r=44x^=200 

20 He tiikcs off 3a 9 p If r ho the leq'* .ainoiint per cent , 

100~Re 1 15a 3p’ TOT"37T»p 



28 


PRACTICr 


[CHAP 


EXAMPLES VI e Pa«e 134. 


8 Re 1 5a 4p =Rs 
10 £4 1?3rf=£4T;V 
12 R ^2 14a =R6(3-J) 


9 £3 25 Gd =£31; 

11 Be 1 12a =E‘. 2-Ec J 
14 10a 8p =Ee(l-’,) 


15 


17 


8a =A of Be 1 
2a =J of 8a 


5a 4p = 


19 


5« =}of£l 
Is 3rf =} of 5^ 


22 


25 


8 a =1- of Be 1 
2a = J of 8 t 
1 a = 1- of 2 a 


10s = I of £1 
2? =1 of 10* 
6d =} of 2s 
2d =J of Gd 
=l of 2d 


Bis. 

a 

p 

Bs 

» P 

425 

0 

0 

16 9G0 

0 0 

212 

8 

0 

2 a = J of Be 1 j 

120 

0 0 

■iS 

2 

0 

3p=4of2a |_ 

15 

0 0 

Bs 

a 

p 


R-* 

a p 

324 

0 

0 

18 

3'>7 

0 0 

1108 

0 

0 



2 

1 

0 

0 


714 

0 0 




5 a 4 p =: 1, of Re 1 

119 

0 0 




1 a 4p =Jof5a 4p 

29 

12 0 

£ 


d 


£ 

s d 

464 

0 

0 

20 

425 

0 0 



4 



3 

1836 

0 

0 


1275 

0 0 

116 

0 

0 

10* =1 of£l 

212 

10 0 

28 

0 

0 

1* 8tf =j,of 10s 

33 

8 4 

Es 

n 

P 


Rs 

a p 

723 

0 

0 

24 

335 

0 0 



3 



3 

2169 

0 

0 


1063 

0 0 

361 

8 

0 

8a of Re 1 

177 

8 0 

90 

6 

0 

5a 4p =5 of Be 1 

118 

5 4 

45 

3 

_0 




£ 

s 

d 


£ 

s d 

632 

0 

0 

26 

6113 

0 0 

316 

0 

0 

5s =l of £1 

1526 

5 0 

63 

4 

0 

2* of £1 

611 

G 0 

15 

16 

0 

I cf = 1 of 2s 

305 

13 0 

5 

5 

4 

irf=;iT of 1* 

23 

9 5 

1 

6 

4 

irf =i”of irf 

12 

14 8i 



VI] 


MISCELLANFOUS EXAMPLES H 


29 


27 


29 



Bs 

a 

P 



Bs 

a 

P 


427 

0 

0 

28 


288 

0 

0 




6 





3 


2562 

0 

0 


1 

864 

0 

o' 

1 =3 of Be 1 

142 

5 

4 

8 a = 1 of Be 1 

144 

0 

0 

1 =}of 5d 4p 

35 

9 

4 

4 a = * of 8 .1 


72 

0 

0 

i =^ofla 4p 

2 

3 

i 

3 p — of 4 3 , : 

4 

8 

0 


£ 

5 

d 


( 

fi 

s 

d 


6184 

0 

0 

30 

1260 

0 

0 




5 

5s =Jof £1 

315 

0 

0 


30920 

0 

0 

2s 6d = ^ of 5s 

157 

10 

0 

05 =} of £1 

1546 

0 

0 

Is 3d=i^of2s 6d 


78 

15 

0 

25 of £1 

618 

8 

0 

2id=iofls 3rf 


13 

2 

6 

Zd =^f 25 

77 

6 

0 







MISOELLAlSrEOUS EXAMPLES U Page 135 
6 He sells ^ lbs , be adds ^ - 112^ lbs 

10 If z be the req^ no , then Bs 24-^ x | + Bs 26 xV ^ f 

=B8 25xa:+Bs 15 


11 Let V be the req'' number , then -TVr=36084 , 

ir=36084 X 25=AAa|Aoa=902100 

14 The total wages in the leq^ time must be an exact multiple both 

of 105 6a and I 65 4d, le of 126(/ and 196rf The L CM. 
of 126«f and 196(^, or (14xl4x9)rf, will therefore be the 
total wages earned in the req'' time 

This w ill tale — weeks, or 9 weeks to earn 

15 Tram travels 35 x 1760 x 3 ft in 60 x 60 secs 


16 If Bs r be req^ sum, then ^+^+750=r 
^ o 25 


5s =l of £1 
Is Zd of 55 

= 5 ^ of Is 3d 


£ 

1746 

5 

0 

d 

0 

14 

24444 

0 

0 

436 

10 

0 

109 

2 

6 

54 

11 

3 


17 



30 


MISCnLLANFOUS EXAMPIiFS IT 


[PACr 


20 


22 

26 


29 

33 

34 


The tiain t^l^cls «> IISI vIb >n 1 inin 

the number of spaces, eicli (.quil to GC jd*’, lu-wwi o\ei ui 
1 min sv III be JL' J ^ oi 2 1 If ut the Ix-r/iiinm of the viinvfe 
the iraicllcr » s opposite a post, be \\ lU count that jiO-'l .is 1 , and 
svill aftorwaids^ count 21 iiioio posts in 1^*14 \ds In my 
othei case he mil (.otint2I 
Thus 25 IS the greatest iimnbei and 2 1 tbt least 
(The student should diaw a diagram ) 


From Ai t 97, L C M , 


If 0100=r 


2C0B 

</» 


(i) 2371G0=2' 5* r- 11= 

(n) G082'ji='l- n'’ 11 Thus (i)nnd (in) nepoi feet hipiates. 

(in) 571536 = 2* 7= 

The sq root of (i) is 2= 5 7 11, of (n) is 2' 3= 7 
Hence H C F =2= 7 m 28 , l^C M =2 8^ 5 7 11 or tl5^ 

21 posts ha\o 20 sjnci-s of 11 jels tath 1tet«/en tliuin , 
the train travels <11x20 )ds oi * nil jk'I 1 min , 

Its speed is 30 tni jici hoiii 

Since y- nn =1 Km , „ x 1760 ads or 1100 \ds —1000 ni , 

100 jds 111 ^Ol III (iie/fiv).^ 

Since the length iiid bicadth iiiiist di\idi 72 niuWT cxiictlv, boUi 
length ,nid bicadth must be equal to tho jU»t 1 of 72 tiiid 07, 
If 1 ft 

tuca rcq'*=(3y3) sq ft , i c 1 sq >d 


35 Bj the method of Ai t 82 no find f 

tho nHmber=27 3= 7 3I=(2 31)x(3'’ 7) x£'' -02x03x04 

37 (1) Eap'> = 2:-+(H)+53+(I';)=:j;+(l-/f)+''/-»0“A) 

lV)=9+ 1 : + /i =03 


4< 

^ 39 11 (Qns=^l X 2240)lb3 = si . 

T, =(’M^-^jr^)mds =200,', mds 
''42^ Niiiiibei =(L t' M of 15, 35, 42)+C 
4|’'^eq'> \aluc4=Rs 33j x =Es ^ =11** 20x17 



13S] 


MISCELLANKODS EXAMPLES II 


31 


. 46 Eeq^ nunibet = = 1 ^ 

47. Each fraction is gicatci than ^ , 

then sum is gi eater than A x 4 or 2 
Foi the second pait of the question, A\e find hy method of 
Alt 113 the two fiactions to he 

Ecq'> difreience=-].Y- 


49 Fraction = 


-J- of 20 -tV of 21 


<U 


_/r _s\v-— 2 _ 1 — 1 
— V f fi-j s—rr 


52 Cost at pit’s mouth —5 a x 230 =Bs 71 14 a 

Cost at pit’s mouth +caiiiage foi GO miles =E9 143 12 a 

c.ariiage foi 60 ini =Es 143 12 a — Es 71 14 a 

=Es 71 14 a foi 230 mds , 

, , , 1150a 

hence c linage per md per mile 

1150x12 
~ 230x00 P 

54 See Art 130 

55 Walking to and iiding fiom he takes 33 lira 
Elding to and iiding fiom he takes 2^ lirs 

hj tiding one nay instead of valknig ho saies IJ Ins, 
and by iiding both Ma>s instead of walking he would save 2 A his 
he would walk both wajs in (2A+2A) 01 5 his 

57 ( 1 ) 1 yd 2 ft 3 in =03 in req** ialue=G3 in x 

=225 111 =6 }ds 9 in 

( 11 ) Eeq^ value=Es 8 7a 4p x(l0- ’-) 

=Es 84 8 a 4p -Ee 1 O'h lip 

58 The first remainder is - 5 , the second is ^ of ^ or A 

if I had Es % at fust, (l — ■?, ---J- of 5 - ^ of J)i?=100 

59 £3 7a lOArf =1629 halfpence , £1=480 halfpence ^ 

The least sum of money whose i alue can he reckoned exactly in 
gold 07 or in sovereigns, will bo the LCM of 1629 half- 
pence and 480 halfpence This is (160x543x3) halfpence, 
• 01 (160 X 1629) halfpence, and would lequire 160 or of 'gold 



MISCELLANEOUS EXAMPLES H 


32 


[page 


60 ileq**diff £3-T3rof 17s6rf =(T?of 120--j\-of 35)sixpences 
=(12j-7^) 8i^iences=2s 6c? 

63 If a gallei > ticket cost % rupees, then 

S0j,+30(i+2)+20(a;+3)=16x20 


64. (ii) Fraction = 


by 36 


1 7^+6^ 


Multiply numerator and denominator 


66 


/ \ -NT * 84x12 84x12 x 8 

(i) No of metrea=-gp-= — - - =25^ 

(ii)Noofc.t=28^=.^=5V 


112x6 


9x8. 


(ill) 9 gala =(9x8) pints litres =41 1 hties 
(iv) 2 tons 4 cwt =2^ tons=2^x 2240 lbs 

=?iip5Kg=S240Ks 

67 (!) Fmctioii-l^^iV ( 11 ) Fmotion=’l|2— 

68 (i) Exp'>=7+il^+i^^iJ =8^ +1 i^=91-2=96 

(n) Exp»= 1.8^3.+ £r^V-^=i + J=-! 

69 1 lit =13 pts =(13x2) tumblers teacups, 01 OJ- teacups 

It 

6 teacups can be exactly Oiled Now ^ teacup is left 
This btie or •} litre 

70 40 Km per hr =40 x A, or 25 mi per hi , 1 e 6 mi n 12 nun 

71 3 ac 2 r 20 p =3^ ac 1 sq mi =640 ar 

Req'‘ no =(^ x 7 X 640)— 

72 1 Kg costs 51i£. 

20 

1 lb costs and 1 cvrt costs ■ x 112 
20 x2i- ons,oi 


73 Exp»=15. 

^ 20 15 ^ - 4 


20x2^ 



0 


140] ailSCELLANEODS EXAMPLES II 33 


74. Let 9r miles be the distince by sea, then 3i miles and v miles aie 
the distances by rail and coach respectnely , 

9t+3i.+a:=520 , whence t= 40 
Cost by coach =9 p x 40=360 p , cost by rail =4 p x 120=480 p 
Cost by sea=2p x 360=720 p , 

total cost=(360 + 480 + 720)p =1560p , 

hence a\erage cost per mile for whole ]oiiiney=i^^p=3p 


75. 55 Kg =(55x2^) lbs =121 lbs. cask holds (121-11) lbs or 
110 lbs of water, i e or 11 gals of water 

76 2 tons on 1 ac =2 X 2240 lbs on 4840 sq } ds 


This gives 


2x2240 
2 X 2240 


Kg on 4840x36- sq in 


xlOOOgm on 


4840x36= 


( 30 ^)= 

2 X 2240 X 1000 x .39^ x 39 J 


sq m 


21x4840 x 36 x 36 


gm on 1 sq ni 


This fnetion — f 2 ^ 2 2 .toxioo oxai s xai b _ lox tbxi ooxscxsE 

xuia - 11X1840XSX8X30X30 11X484X4X4 

to nearest integer 


77. TV+T\ + TV=-i + 1 +-! Of S = 1 + ! Of I + 1 ol {X of -J ) 

For second part of question we have of 25 = J-£- 
we have to multiply £33 25 8rf bj or 5yV 
Eeq^ value =£33 2s 8d x5^ 

= £33 2s 6d x[5-l-J-+i of-J-+J of (J-of ])] 

' =£165 13s 4d -J-£8 5s 8d +£4 2s lOrf -f-£2 Is 5d =£180 3s 3a’ 


78 50 Km =50 X ■§■ mi =31 1- mi 

the slower train is l-J- mi behind in 1 hr , 
1 e it w ill be 5 mi behind in 4 hrs 


EXAMPLES YU a Page 143 
3 1 sheep will cost £18 x -J- 4 1 man w ill take 9 days x 10 

7 1 cwt should be cained 30 mi x7, 10 cwt should be earned 
30 mi X 

9 I pay (1)^ 12 x for 1 lb , (11) Es 12 x A for 6 lbs , 

(ill) Es 12 X ^ for X lbs 
C 


K H,S.L 



34 UNITARY METHOD (CHAP 

10 1 lb costs Es 15 X - , 3 / lbs cost Es 15 x f 

11 £68 will be eained m 12 nks X-J-? 

13 In 100 days 125 men x ] ai e wanted 

14 35 homes mil eat it in 14 days x 

17 At 24 Km an hour I should take 16 hi-s x-^^ 

20 5 loads would bring 15^ tonsx 

21 In 43 wks I should save Es 62^^ x 

SC 

23 (u) a men would take p hrs x - 

* 1 * 

24 (i) Eeq^ diatance= 13^ x f nii (ii) Eeq** distance=m miles x - 

25 In 17 min it will tiavel 30 mi x^-A In i min it will tia\c] 

30mix^ 

26 Eeq* speed=(i) i ? fS' *50 per hr (u) nil x GO per In 

27 (i) Eeq^ distance=(90x 1760 x 3) ft x-gA— pei ggc 
(ii) Eeq** speed=(7M x 1760) yds x^\j^ per min 

28 Eeq^time=(i) ^ mm x^, (ii) a min x^ 

35 It loses 133 mm x — ^ 37 Req^ time=55 montlis x 

40 Eeqo time=42 days x ■ ^Y-i °7 o o 

EXAMPLES vn b Page 146 
1 The uliole= 1-=^ of A= » ©f Es 030 

3 The whole=Y=^ of r=i ot 54 in =63 in 

4 (f “ir^) 01 tV of jouiney =6 mi , of of 6 mi 

=34 mi 

5 f- of me =f of I- of of me = 5- of ■£■ of Es 3080=Es 1100 

6 and 7 As No 5 8 Eeq"* pait=^;jX 5^^ 9 As No 8 

10 1 Km =f mi and costs £ of Es 2400, oi Bs 1500 

11 1 nil =f Km and costs f of Es 311f, or Es 498 

12 1 lb = j‘y Kg and costs of 44ef , oi Is 8cf 




13 1 11) co'^ts y]--- of 1 10 f 01 

] Kjr, 01 2l 11)’> costs 1 Jrf x2}, oi 2jf? 

14 1 lit ={ pt** ®= J- of J gnl , and costs ^ of J of AOd , oi SJef 

19 "NVe base to find ^^luLh is the gicatei, OG Kin pci hi , 

oi GO nil an hr 

Since GO nn =60 x ^ Km =96 Km , tho3 me approv'^ equal 

20 A speed of 119 iiii in 2’, In's s=a speed of 110 x J Kin in j hi-s 

-=a ppced of 110 x ” Kin x-2 in 1 hi 

23 OGa bill 36 in , GO a bu\ — in oi — ni , 

30J 30 

hence Rs 30 10a or 190n buj tn x *{%'*, i r 8 nieties 

24 flja pel 1 ni =84 n pci ^.3ds=^.i per|^3ds, 

7 c 8 a pel I 3 tl gam on 1 \ d is ’ 1 , and gam 

on 48 jds IS lie 1 Ba 

26 Coin l,ind=(l- oi of acroigo 

the wholc=] 1 = V ®f i”r= V '’f 21c iuc*=408 icrcs 

27 (l-i‘'5))0i ] J- of batUboti=7J8 men , 

81C men me { J x of bittahon, 01 J of battalion , 
liciico req'* fiaction = J 

28 (1 - ^ -iV)i iV of ]on»ncy=8 nil , vliolo joninoj =128nii 

29 Mastei’’s share =(] ~ - J - J), or ’ ^ of ^cssel 

\^ hole \ c-scl^ JJt) = ? " of J-J and is vvoilh "-J{ of IJs BS'iSj, 
orRs 27037 8 a 

30 (1 - tV “ ). fS of me. = R r. 4927-,‘'3^ . 

toUl income = D-JJ = 3 '; of ,rcr=J" «f R« '1027 W 

= Rs =-4 — =RsC 033 12a 


EXAhIPLES VIII g Page 163 


Km 

27 See Art 0 90 00 

13 26 
73 30 
0 74 

0 50 = 500 in 


3dH 

29 See Ait 5 IW 

14 2 
174 
35 

64 9 

erior=0 1 jd =3 6 m 



J 


36 DECIMALS [ORAP 

30 Penmeter=(4 64+6 02+6 7) in =16 36 in 

req^ error=0 06 in , shewing 16 4 in to he a nearer estimate 

33 1 93 - 24= -(2 4-1 93)= -0 47 

36 Exp'‘=599-6=-(6-599)=-001 

40 Transposing t=16 821 -(4 04+11 4+ 0 291)=1 09 

43 3 92 

14 

0876 
5 61 
00003 
14037 

25 81 =26 correct to nearest unit 


£ZA]y[FL£S vm k Page 168 
7 To multiply 86 54 bj 2 37, add the results of (i), (ii) and (iii) 
17 1 yd =2 54 cm x 36= 91 44 cm =0 9144 m 
26 16 x 2 5 =40, 16 x 025=(18 x 2 5)-100»04 

Also 160 X 0 025=16 X 2 5 X 10-100=4 
31 2 Km 375 m =2 375 Km , 1 Km 800 m =1 8 Km 

2375 

18 

2375 Area=4275sD Km 
19000 

4 275 


32 

35 


37 


39. 


41 


Area=(46)2 sq in =21 16 sq in , 

total piessure=(21 16x15) lbs =317 4 lbs 


(i) 31416 (ii) 
28 

62832 
25 1328 

87 9648 


3 1416 (ill) 
450 

125664 

157-08 

1413 72 


1416x15x40 
=31416x600 
=31416 x 6=188496 
te 1885 ft 


1C 87 9 in ze 1414 ft 


Reqo w‘=(0 28 x 1000 x 13 5) gm =(28 X 135) gm =3780 gm 
5 cu m =(6 X 1000 X 1000) cu cm 
This weighs (5000x1 15) Kg, or (60x115) Kg, or 5750 Kg 
Value = 0 0875 X 40 = 3 5 



DECIMALS 


37 


VIII ] 

42 125 

32 

3*75 4 x 2 375 =95 

0 250 

4-000 

43 Talue=0 04 x 0 125 x 200=8 x 0 125=1 

44. Value=l 625 xO 16 x 0 5=1 625 xO 08=0 13 


EXAMPLES Vin L Page 171 

4 8 

20 Each sliare=£ g-=£06=6 tenths of £l = 12f 
21-49 See Art 167 

52 Circumference yds =122.1i2Ain =5^,^ in =90 09 in 
This=90 1 in (to nearest tenth) and crior=0-01 in 
8 6 

57 -g- =0 95555 . =0 9556 correct to 4 decimal places 


65 I 


9046 

66 

• 5 

349 968 

135 j 

r 

18092 

38o ■ 

11 

^9 9936 


13 

0 2010 

0'0670 


. 7 

6 3630 
0 9090 


guing 0 067 giving 0 909 


EXAMPLES Vm m Page 174 


19 


30 26 3 026 
89 8 9 


=034 


302 6 30 26x10 30 26 ..... 

8 9 89-10 ~ 89 ^100-34 


034 
8-9)3 026 
356 


Also 


3026 _ 30 26-10 30 26 
890 89x10 ~ 89 


100=0-0034 


22 (iv) r=4 8x-^=48x-f^=8 

214 

23 5 65)1210 

800 
2350 
90 


remainder = 90c 



38 


DECIMALS 


[chap 


29 Exp’>=72x-|4=72x-J=108 


21 

31 Exp -0 34x0 25' 


0017x4 0004 


0 34 


002 


=02 


0 08571 0 008571 
25W=-26^-°°®^ 


40 (i) diameter=y^^=15 8 in 


0 00334 


2 5603 ) 0 0085710 
89010 
12201 

15 78 
31416) 49 6 

18184 0 
2 47600 
27688 


, 1760x3ft 3520x36in 

(ii) diameter- gQQ ^ ^ jqqq ^ g ^^^g 


3 52 X 3 in 
’ 0 2618“' 


10 56 


in =40 34 in 


41 _1?L=1§2=45JLj 

0 42 4 2 


=46Ja-l- 

48 


0 2618 

rem'’=0 1 in 


42 (ill) 


37 8241 0 378241 
293 2 93 

=0 1290«-22^=0 1290^=-P 
rem’'=0 0271 


37 8241 3 78241 
90 7 ' 907 

=0 4172- fiM 32 =0 4172^= = 

rem'=00022 


40 336 
2 618 ) 105 60 
8800 
0460 
16060 


01290 


2 93 ) 0^78241 

2664 

271 

0 417 

9 07)3^41 
1544 
6371 
22 


EXAMPLES Vm n. Page 179 


0478 

30. 23)110 
180 
190 


H=0 478^=0 478 



1\ ] DLCIMAL RCDUCTION OF COaiPOUND QUANTITIES 39 


48 0 03125 =0-03’ 

49 0 071875=0 071x,=i^^=/5/jy-->jVxj- 


5» «|=5^-TVir=oi2. <■") pIi- 


EXAMPLES IX a Page 181 


23 

12> 

iP 

24. 

12. 

iP 

25 

12 

1 

iP 


16 1 

8 75 a. 


10 1 

7 75 a. 


16 

8 125 a 



0 546875 Ec 



8 484375 Es 




5 

> 5078126 

26 

12 

10 5f/ 

27 

12 

7 5rf 

28 

12 

L 

4rf 


20 

13 875» 


20 

16 625» 


20 

16 3333? 



11 694£ 



08J1£ 





8 767£ 

29 

12; 

8 25ff 

30 

121 

5 5tf 

31 

12! 

4-25rf 


20 1 

10 C875t 


20 

7 45833? 


20 

.1 35416? 



2 5344£ 



5 3729£ 




0 1677£” 

32 

16 

j 1 2 cliks 

33 

16 

Hchk 

34 



31 

1 5 ft 


40 

|U75f!rs 


20 

5 875 cot 


1 


5 5 jds 



' 5 34375mds 



4 29375 big 

1760j 

10 

0 5 








1 

lio 

0 05 


0 003126 un 


35 4 025 qi 

20 3 0G25 cwt 
5 2 153125 tons 


36 12 
16 


9p 
9 75 a 


5 19 G00375 Ea 
5 3922875 

~0 784375 of Es 25 


0 430625 of 5 tons 



40 


DECIMALIZATION OF MONEY 


[chap 


examples IX b Page 183 


13 

Bs 3 21875 

14 

Bs 5 1875 

15 

Bs9 876 


35i 



30a 



140a 


60p 







16 

Bs 2 078125 

17, 

Bs 11 5625 

18 

Bs 0 40625 


1 25n 



9-0 a 



65a 


30p 






6^ 

19 

£463 

20 

£5 72 

21 

£14823 


12 6s 


14 4$ 


16 46s 


7 2rf 


4id 



5 32d 

22 

£9 89 

23 

£0634 

24 

£11 047 


17 & 


12 68s 



0 94s 


9 6(2 


816c? 


1128rf 


2 4/ 


064/ 



112/ 

35 

Be 0 28125 

26 

Bs 1 578125 

27 

Bs 3 11875 


2 



4 



5 


Be 0 5625 


Bs 63125 

Bs 15 59376 


90-1 



50a 



95a, 








60p 

28 

Be 0 53125 

29 

Bs 

7 304 

30 

Be 078 


10 



8 



9 


o 31 2o 

Es58 432 


Bs 7-02 


50a 



6912 a 



032a 




10 944 p 



3 84p 



Bs 58 6a 11 p 


Bs 7 Oa 3p 

11 

12| 45tf 

32 

12 

4d 

33 

12 

8cf 


2o|l5 375‘f 


20 

9 33333s 


20 

17 66666s 


4 3 79875£ 


8 

7 46666£ 


40 

i 15 88333£ 


094218 of £4 


0 93333 of £8 


0 39708 of £40 



DECIMALIZATION OF MONEY 


41 


34 12 9 75c? 35 

20 7 8125s 

11 3g0625£ 
0 008 

0 091125 of £125 


37 0895 day 
21 ^ hours 
28 8 mm 
48 secs 

40 09375 
7 

6 5625 tons 
11 M cwt 
lO qr 

43 4 |3 25 r 

08125 ac 
04 

0 325 

46 £3 072 
1 44s 

5 28c? 
112 / 

49 £7 778 
15 56s 

6 72c? 

2 88 / 


1 8875 mds 
35 5 srs 
8 0 chks 


2 03125 mi 
=2 03^ mi 
=2^ mi 


41 9 9375 of £3 3s 
=9^ of £3 36 
=(10-^) of £3 3s 


36 6 4^ big 
9 5 cot 
OO chk 


39 0 0425md. 
W 

0 425 md 
17 0 STS 

42 16 | 12 75a 
Ee 0 79687 
=Ee 0 7969 


44 7 5 25d 

52 9 75 wks 
5 0 1875 yrs 
0 0375 of 5 yrs 

47 £0 71 
14 ^ 

2 ^ 

16/ 


45 £4 325 
6 5s 


48 £0 099872 
=£01 
= 25 


50 £0754 
15 08s 
0 96 c? 
3 84/ 


51-54 See Arts 178, 179 


For 57-60, see Art 179 

Thus in Ex 57 we require 0 017 x 640, or 0 17 x 64, and in Ex 58 
we require 0 2190 x 1760, or 2 19 x 176 
For Ex 59, see Ex 54, and for Ex 60, see Ex 53 



42 


DECIMALIZATION OF MONEY [CHAP IX 


EXAMPLES IX d Page 188 


25 

26 


24/=26 mils , 1/ =^inil8=(l+-^^-) mils=£0 001 X (1+-^) 

ByE. 25. l/=a>001x|.*^=^!i|^-i0001042 
1 mil = £0 001 req« difiF = £0 000042 


37 


12 half-cents=Ee 0 06 


0 96a 
1152p 

ieq« diff =12-11 52=048=-/^ 


o 

T 


38 Bs 2 624 
5 

Bs 13 12 


Es 1312 
1 92a 
U04p 


EXAMPLES X a Page 193 
38 (i) Absolute eiror=£(37 5 - 37 482)=£0 018 
Belative ^=000048 

Percentage euor If % denote the req^ rate per cent 
then Y^=0 00048 and a;=0048 

38 (ii)-44 As No 38 (i) 


EXAMPLES X b Page 195 


1 4104 

62 

000 

17 

21 89 


62 931 

99 


5 0 285 

71 

0 833 

33 

3 256 

14 

4 375 

18 


2 135 

64 

0 04 

02 

4 50 


5 89 

85 

6 03 


0016 


3 682 

82 

0 000 


3 999 

81 


3 5 08 

19 

166 

66 

0 03 

33 

6 78 

18 


7 26 

09 

306 

9 

0 

00 

332 

99 


4 0 000 

71 

0 008 

5 

0 000 

08 

0 009 

29 


8 416 

82 

01 

00 

415 

82 



IJHAP X ] CONTRACTED ADDITION AND SUBTRACTION 43 


37 

26 

10 0210 

52 

982 

8 

0 352 

94 

1020 

06 

0 384 


19 


0 

to 

07 

1000 

16 




12 Expressing each item xn 
millions^ ive have 

0 971 
8 792 
0 093 
17 828 

27 684 

giving as answer 27,700,000 

14 (i) 

=0 08+0-0064+0 00051 
(ii) r»=0 00004 

15 0 2 

004 This result differs 

0-00 8 from J oi 025 by 
0-00 16 0 00008, Inch proves 
Q-QO 032 the second part of the 
0 24 992 question 


11 Expiessing eicb item xn 
millions, we have 
4 683 
0 807 
17 493 
0094 

23 077 

gii mg as answ er 23,100,000 


13 Expiessing each item xn 
millions, we have 
£ 

65 409 
30 814 
15 659 
28 458 
8 474 
0 460 
0 936 
0 675 

150 885 

givnng as ansvi er £150,900,000 


Es 35087 

17 3614 

18 £3 7292 

23 68 

28 7 95 

0 7917 

8 33 

06 3 

12 7273 

663 94 

17 64 

17 2482 

104682 

392 5 89 



Es 1047 


19 Es2 43 

7149 2 
0 421 1 

10 000 [3 
Es 10 


Es 5 486 

12 

3 625 


888 

8 

10000 

lo 

Es 10 




44 


CONTRACTED 


[CHAP 


EXAMPLES X c Page 198 


5 62 

3 27 

2 070 

28 

4 73 

3 9 43 

57 

5 25 

15 86 


281 

1 

4717 

9 

112 

4 

49 

1 

188 

7 

39 

3 

02 

1 

47 

2 

1837 

7 

332 

3 

4953 

8 


4 003 

oU 

00 

00 

00 


013 


175 

4116 

7 

3 

0 

0 

0 


5 02 

81 

317 

6 702 84 
0 

0541 

15 08 

4 

3514 

2 

50 

3 

2 81 

1 

35 

1 

7 

0 

15 93* 

8 

38 02 

3 


078 

95 

909 

8 357 

82 

2376 

9 028 

947 

8643 

710 

6 

716 

6 

2 31 

5 

07 

1 

107 

3 

17 

3 

717 

7 

25 

0 

01 

1 



02 

1 

00 

1 



8 50 

0 

250 

0 


46 

20 

8136 

369 

6 

4 

6 

1 

4 


2 

375 

8 


11 . 


30 


7 

0928 12 0 

8631 


4 28 

9 602 

28 

4 7 

7 

1 

4 

4 

— 

® 8 

1 


40 

0404 

606 

14 0 561j 

023 

5 97001 

240 

2 

2805 

1 

2 

4 

504 

9 

242 

6 

039] 

3 



3 3491 

3 


1414 

21 

1 41421 

1414 

2 

565 

7 

014 

1 

005 

6 


3 

1999 

9 



MULTIPLICATION 


45 


16 

2400 

89 

4 237 

17 

1732 

05 

1 73205 

18 

80 


9 603 

6 


1 732 

1 


4^ 


480 

2 


1212 

4 


24 


072 

0 


052 

0 


0 


•016 

8 


003 

5 


0 


10 172 

6 


000 

1 


50 9 





3 000 

1 



19. 

00204 

76 

20 

41 

0208 

21 

59 



2 406 



7 305 




0409 

5 


28 7 

1 


Ira 


0081 

9 


12 

3 


4 


0001 

2 



2 


183 


00492 

6 


2^' 

6 




23 05731 


26 0 7601 


22 0 786 66 
421 

3146 6 
157 3 
7 9 

3 311 8 


0 563 26 
3 277 

1689 8 
112 6 
39 4 
3 9 

4 

1846 1 


79 364 
6024 

476 2 
1 5 
3 

478 *0 millions 
giving answer as 478,000,000. 


3 0807 


0 526 

333 

8 2828 

4 210 

66 

105 

27 

42 

10 

1 

05 


42 


1 

4 359 

51 

0314 

16 

31831 

942 

5 

31 

4 

25 

1 


9 

0 999 

9 


35 

5 80079 

8 

2 

5 

5 


29 As in Ex 28 we require 
4085x746 to the nearest 
unit 
746 0 
4085 

2984 
59 7 
3 ^ 

3047 4 thousands 
giving answer as 3,047,000 



46 


CONTRACTED MULTIPLICATION 


[C3HAP 


30 As m Bx 28 we require 31 As in Ex 28 tro require 
45 28 X 6 402 to «Ac nearest 80 460 x 5 073 to the nearest 

iimt iinti 

45 28 
6402 

271 7 
18 1 
1 

289 9 hundred thousands 
giving answei as 29,000,000 gi\ ing answer as 408,000,000 


80 

460 


5 073 

402 

3 

5 

6 


1 2 

408 

1 1 millions 


32 As in E\ 28 we require 33 391 3701 


5 807023 )< 43 07 to the neat - 7 52 

est unit 27T 6 

5 807023 19 7 

4 3 07 8 

232 3 " 296 1 

17 4 
4 

2o0| 1 millions 
giving answer as 250,000,000 


34 2 54 

36 

91 44 
I 4 W 
914 4 
365 8 
64 0 
_4 _b 

1348 8 cm 


45 25 

36 2 22 6 

5 

7031 

226 25 

15 58 2 

1 6093 

0 7 

220 3 

2 

135 7 

2 -0 

1565 IKg 

1 


364 1 Km 



37 224 35 

4 595 

897 4 
112 2 
20 2 
^ J, 

1030 9 hundred thousands 
giving answer as £103,100,000 


38 29 72897 

4 247 
118 9 
5 9 
1 2 
1 

126 1 hundred-thousands 
giving ansnei as £12,600,000 



X] 


CONTRACTED DIVISION 


47 


EXAMPLES X d Page 202 



818 

17 66 


03311 

1. 

1057)8 6439 

2 4 301)75 96 

3 

8 212)2 7184 


187 

32 95 


2548 


81 

2 84 


85 


1 

26 




0 02988 

9 617 


53 682: 

4 

2184)0 06527 

5 2 9332)28 210 

6 

2 9364)157 634 


2159 

1811 


10 814 


193 

61 


2005 


19 

22 


243 



2 


9 


047 

82 56 


3 406 

7 

241)1132 

8 8 425)695 57 

9 

43507)148188 


168 

21 67 


17667 



472 


264 



51 


3 


19802 

0 024 


0 051 

10 

21979)4 3524 

11. 5 87)01410 

12 

8 04)04162 


2 15450 

236 


142 


17639 

2 


62 


53 





_1290 

3 316 


0 318 

13 

13408)1 7300 

14 3 3166)110000 

15 

3142)1 


3892 

10502 


~574 


1210 

553 


260 


4 

221 


9 



22 




1732 

2 601 


0 625 

16 

17321)3 

17 31762)8 2626 

18 

3 081)1 9278 


12679 

19102 


792 


554 



176 


34 

14 


22 



48 


CONTRACTED 


[CHAP 


0 0981 

19 9135)0 89700 
7485 
177 
86 

901 

22. 2 03)18302 
32 
11 


443 

26 7816) 346423 5 
3378 
252 


92 

28 114)105 6 
6 


561 

32 (i) 5 791)3252 7 
357 2 
97 
39 


24 9 

20 2 971) 73 98 
14 56 
2 68 
1 

1132 

23 7 309) 8276 
967 
236 
17 
2 

418 

26 2164)90598 2 
403 
187 
16 

6082 6 
29 216)1 3138 6 4 
178 5 
67 4 
14 2 
13 


150 

(ii) 2 980) 447 2 
149 2 
2 


00142 
21 3 080)0 04381 
1301 
69 


1121 

24 31897) 3578 3 
388 6 
69 6 

27 

8120 

27. 8 0637) 65 4812 
9716 
1652 
39 

4 506 

30 1 2481) 5 6248 
63240 
835 
86 

0 06648 
19215) 12775 
1246 

22 

12 

1 


(ui) 3176) 445 8 


31 20 1 15 5s 

12 775 £ 192 Rs 15 cents=Rs 192 15 

£006648=15 96c? to the nearest hundredth of Ic? 


128 2 
12 






DITI'JIOX 

49 


1 132 


0863 


33 

(i) 5 28)603266 

(n) 6O«3)52SO0 



802 


4136 



274 


486 



10 





0023 

400 

2011 

1 609 

34 

(i) 3-93701)0 1000000 

(ii) 3-9370)70 200 

(ill) 3 9370 ) 6 3300 


21 

2398 

460 

2 3990 


1 

5747 

G2 

368 




23 

14 


1 4142 


69163 RcG 9163 = 1?‘!G917 to 

35 

141421)200000 

39579 

•2011 

36 

the nearest cent 

32194 I' 3|rf=£0O656. 

hence 6-91 / X 0 6 iG 


597 

31 


2 ">8 

gi\-in 

1 ® 

"a V ^ f * 

g £4 lOf V 


277 27 


605 337 

1338 5 

37 

6-2321)1728-00 

38 

15025)1000-000 

39 2 2433)3431 7 


4^53 


93 300 

1203 2 


45 33 


8 330 

86 3 


171 


837 

19 2 




80 

1 3 




11 



£C63 337, giving £065 1 1# 2<7 
to the neatest penny 


18034 _ 30090 

40 (i) 4 197CS ) 75-9133 (u) 3 89619 ) 117- 2365 
33 9367 3511 

3533 3 

195 
~28 


2 331 

(in) 5 G3G7 ) U 3799 
31065 
2881 


G3 

7 


K II S I 


D 



50 


MISCELLANEOUS EXAMPLES 


[chap 


EXAMPLES X e Page 205 

[To save space the position of the mnltipliei is sometimes moved 
The number of decimal figuies to be retained in each case must be 
decided as in Ai t 199, Ex 1 and 2 ] 


1 0 23 94 

2 18 60 2 

2 31 

01457 

47 9 

186 0 

7 2 

74 4 

2 

9 3 

55 3 

1 3 



2 71 0 

55 3 

1356 

27 7 

2 71 0 

3 9 

81 3 

086 9 

13 6 


1 6 


3 67 6 


36 73 

0 

0 073 

2 57 

1 

11 

0 

2 68 

1 

0 2367 

~53 

6 

8 

0 

1 

6 


1 

0^ 

3 


097 

13 


1872 

97 

13 

77 

70 

6 

80 


19 

181 

82 


2 423 

3 63 

64 

72 

73 

3 

64 



4 40 

55 


7 24 36 

8 

0 394 

7 31 

10 

219 

12 

09 

7 

897)959 

^(10' 

62 

9 

1 


5 2143 4 
3 721 

64 30 2 
15 00 4 
42 9 
2 1 

8 532 ) 79 75 6 ( 9 34 
2 96 8 
40 8 
5 7 


8 078 

59 


3 0103 

2^ 

is 


8 

4628 ) 2 ?6 

6(05 


5 


6 4485 

2 


1473 

4485 

2 

17 94 

1 

314 

0 

13 

4 

98153)6606 

J(67 


717 5 


30 4 
1 0 


0 

42 


80 

37 

8 

3 

_4 

41 

2 


5 312 

206 

0 

12 

4 


4 


1 

218 

9 



ON CONTRACTEO WORK 


51 


65 

471 

10398 


47 

1 

96 


59 


05 

68 

07 


J10 326 

680 

70 

20 

42 

1 

36 


41 

702 

89 


11 3214 

00 

1 01359 

3214 

0 

32 

1 

9 

6 

1 

6 


3 

9 47 ) 3257 

_6(344 


41G 6 
37 8 


12 238 

3 901 

714 
214 2 
2 

4 83 ) 928 4 (192 
445 4 
10 7 
1 0 


15 1 2327)2 5721(2 086 
1067 
81 
9 


13 6 081 75 14 . 348 15 1 2327)2 5721(2 086 

5 7002 19 1067 

30408 8 502)6^(1317 

4257 2 1592 

1 2 — ^ 

^^'^) 69 732 4 (2 011 ^ 

398 — 

li 

16 

16 F'B — In order to obtain a result correct to 4 decimal figures 

(? e , 5 significant figures) vre must decimalize £2 Is l|a in 
full Tins gives £2 05625 

48632 

2 0,5,6,2Jj ) 10 00000 
1 77500 
13000 
663 
46 

17 4 8 632 18 . 2 2046) 10 0000 ( 4536 

- 400 11816 

19452 8 793 

1^5 

25 21 ) 19472 2 5 ( 772 40 
1825 2 
605 
101 



52 


C0NTRAC3TED WORK 


19 1 414 

21 

20 1 260 

01 21 

0 00731 

1414 


1260 

01 

0 00731 

1414 

21 

1260 

01 

000005 

565 

68 

252 

00 


14 

14 

75 

60 

0 00005 

5 

66 


1 

0 00731 


14 

1587 

62 

ooo’5oo 


1 

1260 

m 


1999 

84 

1587 

62 


1414 

21 

317 

52 


1999 

84 

95 

26 


799 

94 




20 

00 

2 000 

42 


8 

00 





40 





_2 




2 828 

20 





[chap 


22 0) 0003160 
0003160 

0003180 

0000820 

0 000820 
0 000820 


9 

5 2 

3 

2 

53 

6 

6 

6 (=6'*) 

0 000009 

9(=a2) 0000002 

59(*o6) 0000000 


000065 

(u) 398 ) 0002590 
220 


1 

22735 

11 

1 

122 

7 

12 

3 

1 

2 

136 

2 

51034 

681 

0 

13 

6 


4 

695 

0 


24 


12491 

168 

954 

000 

55 

23 

789 

624 

6 

3379 

1 

62 

5 

506 

9 

6 87 

1 

118 

2 



13 

4 



1 

4 



6 87)4019 

0(5 


584 0 


34 4 


ol oolbsi-* 



53 


X.] APPROXIMATE ADDITION AND SUBTRACTION 


3 28 

OR 

26. 10 93 

6 

27 Using itsulb of 


3 2809 

1 

0 936 

and dividing 

984 

2 

109 36 


2 47 

65 

6 

981 

2 


26 

2 

32 

8 

4 84 )n 959 


2 

06 

5 

2 279 

10 76 

2 

119 59 

5 

343 


2540! 

2 

M 

5 08 
1270 


101 

6 

6451 

6 

2 54 

12903 

2 

3 225 

8 

258 

0 

16 387 

0 


29 GO nil .in hi =8B ft pei Rec, 

hence niultiplioi is ^=1 4667 


3 280 

8 

2 2046 

6 561 

6 

656 

2 

1.3 

1 

1 

9 

7 232 

8 


EXAMPLES X f. Page 211 


1 .'5 42 ± "OOS 

4 06± 005 
3 60 A 005 

ISOSdb 015 


4. 10-037 ±001 
0973 ± 001 

2001j: 001 

10011 ± 003 


7 42 3 ±05 
5 26 ± 005 
942 ± 0005 

48 502 ± 0555 

10 2 308 ± 0005 
3 00 2± 00 05 
5 3ib± 001 
5, 304 ±-0005 


2 2030 ± 5 
1450 ± 5 
3380 ± 5 
940 ± 5 

7800 ±20 

5 4 016± 0005 

0 101 ± 0005 
2 802 ± 0005 
3000± 0005 

10 009 ± 002 

8 4 680± 0005 
2 073 ± 0005 

2 607 ± 001 


11 7 000 ± 0005 
4 4 20± 001 

2 580± 0015 


3 6 07±005 

8 36± 005 
7 40± 005 

21 83 ± 015 


6 7 63±005 
5 0G± 005 

2 57± 01 


9 18 009 ± 0005 
n 900 ^0005 

6109± 001 


12 (±005x10) cm 


13 Error for each 
buck IB witliin 
± 05 Ib 


0 006 ± 0015 



54 


APPROXIMATE MULTIPLICATION 


[CHAP 


14 (j) Superioi liinit=4 75 x 6 35, 
inferior limit=4 65 X 6 25 


16 


20 


25 


26 


(ii) Siipei lor liniit=4 8x64, 
infeiior limit=4 6x62 


15 (i) Limits are 5 35x425 and 525x415 

605x505 and 5 95 x 4 95 
30 x 22 and 26x20 
430x500 and 450 x 520 


00 

( 111 ) 

(iv) 

58 

^ 

174 
4 06 

2146 


17 


732 
6 04 

43 92 
2928 

442128 


18 


50 2 
4 08 

200 8 
4 016 

204 816 


19 


0 0617 
33 5 

1851 

1851 

3085 

206695 


21 


Circumfeience=431 x(3 1416 ± 00005) ft 
=(1354 0296 ± 02155) ft 
= 1354 05115 ft, or 1354 00805 ft 
=1354 0 ft collect to nearest tentli 


4 26 
0 508 

2130 

3408 

216408 

22 Superior Iimit=314 165 x 14 7075 lbs =4620 6 lbs 
Infeiior limit=314 155 x 14 7065 lbs =4620 1 lbs 

23 408 1 24 Greatest weight 

=62 45 X 0 855 X 15 lbs =800 92 lbs 

Least weight 

=62 35 X 0 845 X 16 lbs =790 28 lbs 


204 

8 

1 

213 


0 ^ 

"o 


Highest cost=(Es 3500 x -fj^) x (80 x -J-^) 

Lowest cost=(Es 3500x-j!^)x(80XT'yi5-) 

Expiessing the product in the form 0 827 x 6 435, we got 


0827 
6 435 

4 962 
3308 

2481 

4135 

5 321745 


This apparently is coirect to thiee significant 
figuies 

If, however, we take for the first partial pio 
duct its stiict lalue, mz 

(0 827 ± 0005) x 6= 4 962 dt 003 = 4 965 or 4 959, 
we see that the ‘6’ is doubtful This means the 
‘2’ m the answ'er, and, of couise, all figures 
following it aie doubtful Tims two figures in 
the answer are correct, which illustrMcs the 
principle ' 



X.] APPROXIMATE DlVISIOir 55 


27 


In fii-sb part of question the error in 754 ni is 0 5 m. 

Since this error is multiplied or divided by the respective multi- 
pliers or divisors, we get for answers 
(i) 1 Hm , (ii) 1 cm , (ill) 1 Km , (iv) 1 mm 
In second part of question, using Art 190, 
relative error in (i) 

All — 0 V-lOO ^05 

« j> w ~764-100~- ■!4 

In the same wav the relative errors in (in) and (iv') may each he 
shewn equal to 


003612 

29 1728)62425 

j^Qgg,j After the ‘2’ in the quotient the figures 

=. are doubtful 

2170 

4420 

Erom note in Art 205, 

Upper liniit=-4-^f|^ =0 0361259 
Lower limit= ° ° =0 0361253 

These two last results are, of course, the actual limits 
The first result is a rough approMniation, sufficient to prevent 
serious error "We 'see that it is correct, howevei, to four 
significant figures 

For second part of question we have 

1 cu, in weighs 0'03612 lb , or (0 03612 x 7000) grs , 
or (36 12 X 7) gis , or 252 84 grs 
Upper limit gives (36 1259 x 7) grs , or 252 8813 grs 
Lower limit gives (36 1253 x 7) grs , or 252 8771 grs 

This shews the req^ weight is 252*8 grs , which cannot be obtained 
from the datum It suffices, however, to give a result tiue 
to the nearest grain 


453 

30 224)101600 
1200 
800 
128 


31 (i) 1760 X 3 X 12 in = 1609000 mm 
25 4 

6336)160900 ~ 

34180 

25fi0 


(u) 1609 
1609 

1609 

9654 

14481 

2588881 



56 


DECIMALS 


[CHAP X 


4 46 

32 (i) 3 74)1668 
1720 


14 91 

(u) 3107) 46 328 

15 258 
2830 0 

337 


2325 

(ill) 4070 ) 9462 800 
1322 8 
101 80 
20 40 


227000 

(iv) 3 72) 843700 i 
997 i 

2530 ‘ 

V 

c 

2 32 

33 (i) 3 1416 ) 7 2800 
996g0 
54320 


In eveiy case we see tlie pimciple holds 
good e {T , in (i) thei e ai e 3 figures in divisor 
At the 3"* division the multiple of the ‘3’ 
must come vcitically under tne ‘8’ of the 
dividend, and here the quotient begins to he 
uncei tain Thus the two pieceding figures of 
quotient aie certain So foi (ii), (iii) and (iv) 

7920 

(ii) 3 1416 ) 249000 
290880 


81360 


34 Superioi limit=^||5^=5005 secs 

Infeiioi limit= 3 secs 

35 Sp gi 

^ ® 62 425 62 425 


EXAMPLES XI a Page 217 


1 Ratio 


2 Ratio = 


70xl2rf 


3 Ratio= 


255 As 


^ 5 6 Eatio=^T4 


46qrs " "'‘"°=3r|7 

7 Ratio=|i{i2H 8 Ratio =i?20+40+^ 

561 tons ° (320+120+35) p 

9 Peicentage=^^xlOO 13 PeicenUge=|^xlOO 

14 Peicentage=|^xlOO 16 Value=Es 3 4 a x 2 jL 

17 Viilue=Rs 8 24 x 3;t 18 Vahie=Rs 2 219 x 5 84 


£19 25 



57 


CIIAl* XI ] OF COaiPOUND QUANTITIES 


Es 2 130 

4 36 

8 520 


630 

0 

127 

8 

Es 9 286 

8 


E<3 384 464x0 008351 


Es 384 

46 


8351 21 

~^75 

~7 

115 

3 

010 

2 23 


4 

Es 3 210 

6 


Value =E 8 565 x 0 04 
=Es 5 65 x4 

Value =E 8 4 725 x 4 6 
=2 363x9 


24 Vnlue=12 896x6 25 25 Valuc=£0 17133x 5 5 

= £0 85665x1 1 


26 As No 23 


27 Value =£7 24x3 375 
=£7 24x3^ 


28 

31 

34 


Value = £0 8151 2 X 4 75 
= £0 81*112x43 
X 224 + 28 + 25 
100 ~ 560 

As No 33 



29 ’ ='"'‘3 

30 


100 1081 

32 

1 1231 ft 

33 

100“ 151 ft 

35 

» 61 14 

36 

1000“361420 


a. 43 2 
100 98 473 

r _ 4710^ 
100 896431 
1 183750 

100 5250000 


37 

38 

39 

40 


It M oultl 8 a% 0 (62 - 50) % of gi oss rc\ cnuc 
The three p.iiliurs get J, icspcctncly of total pionts, 
IcaMUg -J- for the funds Of this last sliaic, the E P gets 5 , 
and the EPF 


97075-55442 
07075 


For potatoes, JL=?8230L: 035321 
' ’100 982301 


ForflaVj^= 

For wheat, Foi hailey, ^ = 

’ 100 337rf 100 25-j» 77 


EXAMPLES XI b Page 221 

1 Value = 227 sit Es (lO - ■?,) 2 Value = 309 at Es (3 - -^r) 

3 Es 417 5 IS \aluc at Ee 1 , then 8 a , 2a , Op 
4. £160 25 IS lalue at £l , then 5? , 1 1 3ff 
5 £437 6 IS value at £l , then 10^ , 7? , 7rf , 3W 

G E.S 5 0205 IS \ €aluo at Ec 1 , then 8 a , 4 a , 4 ]) 

7 Es 16 308 IS value at Eo 1 , then 8 a , 4a , 2 a , 1 a 4 p 

8 Es 1153 5 IS \alue at Ee 1 , 6hen8n,2n 

9 Es 500 76875 is value at Ec 1 , then 5 a 4 p 



58 


DECiaiALS 


[chap 


10 Hs 23 Tib IS value at Be 1 , then Bs 2, 8 a , 1 a 

11 Aiea=2 60125 bigs , Be 1 2a 8p =Bs 1-J- Then multiply 

12 Aiea=345 594 ac , £2 2s 6c? =£2 Then multiply 

13 ■\Veight=16844446 tons, £1 12s 6rf=£l|-=£(l+i+-|-) Then 

multiply , 

14 £2 916 666 15 £2 283133 

J.3 12 

37 916 7 
2 qis =\ cwt 1 458 3 
1 qi =J cwt 729 2 
4 lbs qr 104 2 

£40 208 4 


4 566 7 
2 qts =1 gal 1 141 7 

1 pt =1 qt 285 5 
£6 564 8 


16 £3 666 66 

2 fui mi 916 7 

40 yds of J mi 083 3 

£1000 0 


17 £1712 5 

7 

11 987 5 

1 ft yd 570 8 

6 in ft 285 4 

3 in =|ft 142 7 

£12 986 4 


18 B s 42870 833 3 19 B s 2073 375 

8a =l of Be 1121435 416 6 6 a 4p =J,ofBe 1 1 691 125 

5a 4p =5 of Be 1 [14290277 7 6a4p=JofEel 691125 

Bs 35725 694 3 la=jgofBel | 129 585 9 

Bs 1511 835 9 


20 £1710 725 21 £7 714 28 

10s =1 of £1 855 362 5 1 370 M 

2s 6c? =1 of £1 213 840 6 7 714 3 

10c?=Jof 2s 6c? 71280 2 2 314 3 

ic? of 10c? 3 564 0 540 0 

£1144 047 3 6 2 

2 

£10575 0 


22 £2 070 

833 

1 

99 

20 708 

3 

18 637 

5 

1863 

8 

£41 209 

6 


23 £1191 666 
30 412 5 

35 750 0 
476 6 
11 9 
2 4 

6 

£36 241 5 


24 Es27 48 


5 


137 40 


24 73 

2 

Bs 162 13 

2 


25 Bs5 029 

7 

2 6094 

10059 

4 

3 017 

8 

45 

2 

2 

0 

Bs 13124 

4 



xr] PRACTICE DECDIALIZED 59 

26 Es 382 828125 X 36 621875 
25 

3 6621875 

8 

7 

8 
5 
8 
2 
7 
9 

4 

Es 

28 1232 

n 

13552 

2 a ^ of Ee 1 154 

la 4p of Be 1 1Q2 10a 8p 

Es 13808 10a 8 p 

29 Ifc loses 8 26666' in 6 hrs 

„ 1 37777' m 1 hr 

Again, 17 days 9 his 45 min =417 75 hrs 

1 37 7777 min 
41 7 75 

55111 1 
13 77 8 
9 64 4 
96 4 
6 9 

575 56 6 min 


EXAMPLES Xn a Page 224 

1 They do * c ^ I 

2 (7—-^), or -jV ■will be filled in 1 imn- 

Tliey do or i in 1 hour 


30 (17)»=4913 


4 913 


8 788 


39 304 


3 439 

1 

393 

0 

39 

3 

43 175 

4 



Es 3828 281 


11484843 
2296 968 
229-696 
7 656 
382 
306 
26 

1 

Es 14019 883 


3 



60 WORK AND TIME [CHAP 

4 They do (^+^+^) ml day 5 They do or 1 bom 

6 (^ - ^), or IS filled m 1 mm 

7 (^+ ^ or IS filled m 1 hour 

8 (- -iY 01 IS filled m 1 mm 
\m nr mn 

9 G does or 'A' ™ ^ 

10 They do (^+^), or m 1 day 

11 They do (^+^+^), or ^ 1 ^^7 

12 A and S do in 15 days , A does ^ in 20 days Hence B 

does (-V-^) m 1 day 

13 A and 5 do ^ in 2 days, and A, B, and C do ^ in 6 ^ days 

A and 5 do -jL m 1 day, and A, B, and C do in 1 day 

C (or A) does (y^-^) m 1 day, and B does [■jV-CyS’-Tt)] 
in 1 day 

14 A and 5 do ^ in 22 days, or ^ in 1 day A does -j-gxTo ^ 

15 A and i? do J in 1 day , B and O' do J’ in 1 day , C and do J- 

in 1 day , 

A does \) m 1 day, or l day, and cams T^-of 72d , 

B » n tV» 1 » y^of72rf, 

O' „ Ki+J-j).. 1 » *»1 » » yVof72rf 

16 (i) (^-^) IS filled in 1 hour , (ii) (- 5 .+ J-- is emptied in 1 hour 

t 

EXAMPLES Xn h Page 227 

[■W B —60 nu per hr = R por sec =88 ft per sec 

1 nil per hr ft per sec , and 1 ft per sec mi per hr ] 

9 528 yds in 6 mm , or 5280 yds m 1 houi 

10 Ho rides 22 ft m 1 sec 11 He rides -j-g-xO ft in 1 sec 

12 35 X ft of tram pass m 1 sec ISi 44 x nules per hour 



61 


xn] TniE AND DISTANCE 


14 

15 

16 

17 

18 

19 

20 

21 

22 


23 


24 


It trnels 4')X-g^ ft in 1 sec., hence it tmels (150+180) ft 

aoxrq 
C X s s' 


Let r ft be length of Micluct then (r+252) ft aie tn^el’sed in 
21 sec But 30x^x21 ft aic traieraed m 21 secs hence 

r+2o2=30xf5x21 


Bite of approach =8 mi per hr and the^ meet in 4 hrs, 14 mi 
from A 


Bate of approach=10 mi per hr thej meet in 2^ hrs, 11^ mi 
from /Ts starting place 

Bate of approach =1^ nil per hr A o\ ei takes B in 6 hrs 
Bate of approach =8 mi poi hr hence they meet at 1 45 p ni 
At9a.m thei arc 33 nii apart rate of approach =11 mi pel hi 
the} meet at 12 noon, 12 milc.s fi om A 
Bate of approach =88 ft pei 1 sec , the} pass in secs 


Bate of approach (25-11 J)ft per 1 see , 
01 f-g'XlS’ ft pel 1 sec. 

(344+100x60 

time rcq'>=^— secs 

B.i(c of appixjach=74 nii per hour 
(170+150+1000) ids =1320 }ds mi 

time req’‘=^^X;^j hours 


Bate of approach = 18 ini pei hour, or I8x^.^x25 ft per 25 secs 
If T ft be length of passenger train, then T+lii = 18x-5^ x25 
ar=0G4 ft , and, since 45 nii per hr =GG ft pei sec., 
time req'*=-S4’-£+i^iiL3. secs 


EXAMPLES XII c Fai;e 229 

1 Average height 

=(5ft 10 in +5 ft 7^ in +5 ft 4 in +4 ft lOt in +4 ft 2 in )~-5 

2 Aierage tonnage in 1904=120185—20 

A\erage tonnage in 1906=(126185+107G4x2)— 28 

3 Ai pi ofits for half-year 

=[(£410 8s Cd)x4+£437 2s +£489 4s]-6 



62 


AVERAGES 


[chap 


4 Value req®=[2s 4tf xl2+2s 6d xl0+2s 9c? x8]— 30 

5 Aveiage from Jan to June 

=(£89 6s +£91 +£93 16s +£96 2s 6rf +£98+£99 17s 6d)-G 

Average from July to Dec 

=(£99 X 12 -sum of prices from Jan to June)— 6 

6 90 6 x 8 - 92 x7 

7 Distance =(88 +147 +220 +352 +686 +4840) yds in 10 min 

8 Average age req'‘={16 4 x 125+17 7 x 35}— (125+35) 

> 3514 , 3514 60, , 

10 Day’s run =-g^ mi , sea speed =gj-^^x^ knots 

Rest of passage =1000 mi , when speed=J-^^x-^ knots 

11 Eamfallfor Apiil, May and June, 1907= (1 90+0 94+1 72)=4 56in 

Average for April, 1901-1906=(1 90-0 77)=1 13 in 
„ M:ay, „ =(0 94+146)=2 40in 

„ June, „ =(1 72+1 78)= 3 50 in 


average for Apiil, May and June, 1901-1906=7 03 in 
rainfall foi April, May and June, 1901-1906=(7 03 x 6) in 
req** average foi the 3 months for 7 years 




_ 7 01X0+4 Sft 

7X3 ~ 

2 23 in 




6 mi 

pel hr 

or 176 yds 

per min 







yds 


yds 

In the nist min 

the spaces traversed are (i) 

0 

and (ii) 176 x 1 


second 

99 99 

99 

176x1 

99 

176x2 


thud 

99 99 

99 

176x2 

99 

176x3 

9 ) 

fouith 

99 99 

99 

176x3 

99 

176x4 

1 * 

fifth 

99 99 

99 

176x4 

99 

176x5 


total distance (i)=1760 yds , and (ii)=2640 jds 
Average of these totals =J—°-°-+" ° ” =2200 jds 
From the formula s=4x 176x5®= 2200 yds 

EXAMPLES Xn d Page 232 

1 (i) The laige hand gains 15 min spaces in -g^x 15 niin 

[Alt 219, E\ 1] 

00 ), „ 45 „ „ ^ X 45 min 

2 (i) The laige hand gains 45 min spaces in -1^x45 mm 

00 >1 „ (45-30) „ „ -®-g-x(45-30) min 



XII ] 
3 


MISCELLANEOUS EXAMPLES 


63 


The large hand gams (30 — 15) niin spaces in ■^x(30— 15) min 
„ „ (30+15) „ „ S-2 x(30+15) „ 

4 Tlie large hand gams (25 - 15) min sjiaces m “-§• x (25 - 15) min 

„ „ (25+15) „ „ ^Ox(25 + 15) „ 

5 Tlie large hand gams (35 - 12) min spaces in ^ x (35 — 12) min 

„ „ (35 + 12) „ „ X (35 +12) mm 

6 In ^2 X (10-1) mm and x (10+1) min after 2 o’clock 

7 -5-2 X (25 -3) mm or g-J x(25+3) mm after 5 o’clock 

8 A gams 2 yds while B runs 9 , J gains 60 >ds while B runs 

9 X 25 3 ds 

9 Suppose r lbs at 10^ a are mived with y lbs at 7a , 

then + »■+’/) 

10 Suppose rg ils at 14^^ are mixed with y gals at 17^5, 

then 14Ji + 17^=16-,(r+»/) 

11 Let j SI'S at 10 a be mixed with (18- •>) si-s at 11 a , 

then 10;c+ll(18-a;)=190 

12 THiile A walks 4 10 j ds , /? w nlks 420 ) ds 

„ „ C „ 413 3 ds 

B can giie O' 7 yds in 420 , 

7e, B „ C'7xl-;-g „ 420 x-J-?-^, 

or B „ G 2 „ 120 

13 Pcasonmg as in c\ 12, we find 

UcangncC' 16 3dh in 1728 , 

te, B „ U 16 X * „ 1728 x \ 

or 27 „ C 10 „ 1080 

14. Ecasonmg as in ex 12, we find 

B can gi\e C 5 3 ds m 90 , 

1 c , B ,, U 5 X ,, 90 X , 


or 27 


C 


15 


H „ 150 

A scores 100 avhile 2? scores 80, 

95 


27 „ 100 „ C .. 

Multiply b3’ 10 in (1) and by 8 m (ii) 

Thus while 27 scoies 800(‘^ 

lU „ 95x8 01 760 

A can gi\e C240 in 1000, 


(0 

(lO 



MISCELLANEOUS 


[chap 


G4 


16 A scores 500, while B scores 450, and C scores 432 

B can give G 18 in 450 , 
le, B „ „ 450X:^, 

01 5 „ C 20 „ 500 

17 A fires 28 shots to Si's 12, and therefore kills 7 while B misses 6 

18 A walks 100 yds while B walks 95 yds 

B „ 85 J yds while (7 walks 100 yds 

A „ 100x85i „ B „ 95x85| „ C „ 100x95, 

te, A walks 85| yds while C walks 95 yds 

C gams 9i in 95, or 19 in 190, or 1 in 10 on A 
thus C gains 176 yds in 1760 yds 

19 Bate of approach =8|- or mi per hi 

— or 242 yds per min 

they meet in or 3 min 
Since 3| mi per hr or 110 yds per min , 

the first man in 3 min walks 110 X 3 or 330 yds , 
and the second man in 3 min walks (726-330) oi 396 yds 


20 In 1 sec A runs ^ yds , while B runs ^ yds 

A runs ^ x 10^ yds , while B runs ^ x lOi yds , 

or 4 „ “ 100 „ B „ 93| „ 

req'* start is 6j yds 


6“ 15 

21 While B runs Ijd, A runs yds and C runs yds 

A runs y x x 1760^ yds while O' runs x x 1760^ yds 

or A „ 1760 „ C „ 1680 „ 

C runs 1680 yds in 5 mins 15 secs oi 315 secs 

te, C „ 1680 X ^ „ 315 x -fuj-f secs 

or C „ 1760 „ 330 secs or 5j mins 

22 Befoie they meet A runs 250 yds while B runs 200 yds , 

after „ A „ 300 „ B „ 300x5^ yds 

7 e , when A reaches the starting pt B has run 240 j'ds 
But when they meet, B was 250 yds from the starting point , 

A gels in when B is 10 yds away 



XII ] 


EXAMPLES 


65 


23 The leak admits 24/^ tons in 1 hr 

when the pumps are going, (24^^- 12) tons enter in 


25 


26 


27 


1 hr 


ic, 


te. 


she must sail 44 mi in 

12 , « 


92 


hrs 


92 

12 * 


44 X 


92 


'‘Tl 

or 6 miles in 1 hour 


24. A gained 1 jd in 40 >ds, oi 11 3ds in 440 jds 

length of com so is 440 3 ds , lun h3 A in 00 secs , and JB runs 
429 3 ds. in 90 secs 

First rate of approach =88 3 da m 10 secs = 18 1111 per houi 
actu il speed of train =(18 +4) nii , 1 c , 22 nii per hour 
Second rate of nppioach=88 3d<< in 9 secs =20 nii pei hour 
second man walks at (22 — 20) mi , i c , tit 2 nii per liour 
A has walked 3^ mi when met by B 

A has walked for G3 mins and B has w alkcd S4 mi 
in (63+3) min , 1 e , 1 mile in 12 niin , or 6 mi per hour 
The problem is the same as if B waited 11 hrs and A waited 6 lira , 
or „ B „ 6 „ A „ Ohre 

That 18, ]ust as if B started at 9 nii per hr, when A had walked 
20 miles at 4 mi per hr 
Now, Kite of approach is 5 mi per hour 

B overtakes A in 4 hours, tc, 9 houis after A started, and 
36 miles from London 

EXAMPLES XII e Page 235 

Let A ho a; and B (r+3) 3eai’s of age , 

then ?£^5(r+3)_ 

3 6 

Let A, By C be s, (r+9), (r-G) 3'ears of age lespcctively , 
then 

4 ^ 5 ^2 ’ 

whence 15r+16(i;+9)+10(a?-6)=740 and r=16 
Let silk cost 6r rupees and linen v 1 upces per yaid , 
then Or X 23+ ax 50=2114 , whence ’'=1J 
Suppose there aio j lbs at Ee ^ and (200 - r) lbs at Rs ^ , 
then -;r+g^(200-a)=190, 7a+9(200-a)=1520and a:=140 

K n s I E 



66 


MISCELLANEOUS 


[chap 


6 Let rolls be a; yds and (®+25) yds in length 

5jar+5(«+25)=2015 , 
le, ll«+10{a;+25)=4030 , whence a:=180 

7 Let 1 egg cost a; pence then 30- 16a: = 12a;— 5 

i&v 12^7 

8 Let a: be number of oranges then -jg — jg-=162 

76a; - 481;= 9720 , whence a;= 360 

9 Let a: be number required then ^=^^+6, 
i e, 66t=60a;+330 , whence r=65 

10 Let them meet in sa hours then 4(a;— 2j)+3a;=60 , 
ae, 4a:-10+3«=60, whence a;=10 

11 Let X miles be distance required He takes — hours to ivalk to 

B and back, at 4 miles an hour Walking to B at 3§ miles an 
ss 

hour, he takes ^ hours, and back to A at 4^ miles an hour, he 

iC 3^ 

takes hours The diffeience in time is 3J min , i e , hrs , 

4 ~3j‘^4i ^ 2" 7 9 “l8 


12 


63i;=36i:+28a;-7 , whence a; =7 
A gams Ij miles while walking 4^ miles, 


A 


15 


41 

6 X , I c , 15 miles 


Time=xr lira ==31 hrs 

44 * 


77 

13 Let distance be v miles at 3^ miles per hour he takes hours , 

at 3^ miles an hour he takes ^ houis , the latter' time is 
2 mins , or hour, longer than the formei 

— 4 --L— ^ 2a; 1 _3a, 

3J‘‘'30~3^ 7 ■*■30"' 10’ 

60a'+7=63r, whence a:=2J 

14 Following the method of Ex 13, we obtain 

^ JLj-i. 2a;_4A 1 

3^'"3f’*'30’ 7 ~15'^30* 

60A=56a;+7 , whence r=l| miles 



EXAMPLES 


XII ] 


67 


16 20 miles diff for tbe trains =80 miles distance between tbe towns, 

^ » H =4 „ ,, 

I.C , 14^ „ ,, = 4 X 14i ,, „ 

giving 58 miles , 

16 If A rides for ^ hours before he is oveitaken, then B iides foi 

(r-li) hours 

8r=10(T-l+), whence «=7J, whence distance=60 miles 
Eoi second pait of question, if ^=req'* number of his aftei noon, 
8i=10('r-' li)±5 , [see E\ 3, p 235] 
whence 11=10 or 6, accoiding as is ahead of oi behind A 

17 Let V be leq** number of hours aftei 9am, 

then 3Ja:+2^^±5=35 (uppei sign gives first time, lower sign 
gives second time), whence ^=5 oi 63 

18 If V hours IS time required, then 12r+9a;+ 18=60 
+18 must be changed to —18 

19 C has -jnj- of 4 mi , t e -Jr mi , start of A 

Now A in walking 6 nil gains 1 mi on O' in 1 hr 
A Irai „ jrmi „ (7 „ hr 

Again A has wxr of 5 mi , 1 e nii , stait of B, 
and B in walking 6 mi gains 1 nii on A in 1 hr 

1) of 6 mi g vins nii on A in hr 

or B „ 3J mi „ nil „ A „ 36 min 


20 


Let T miles be distance up stream 
My speed up stream is (4^ — 14) nii , ? c , 3 mi pei houi 
„ down „ (4^+1^) mi , „ 6 mi „ 


my time up stream is ^ his and down stream is hrs 

O U 

x t?— 2 

3 +— whence *=5 


Alternative Solution 

Up stream the rate of row’ing is 4^-14, 01 3 miles per hour 

.. „ 4i+4or6 „ „ 

If I^?id leturned to the stirting point, the last 2 miles would 
have taken 20 minutes, and the w hole time would have been 
24 hours, two thirds of which must have been occupied by 
the up stream journey, since the rates of rowing up and down 
are as 1 to 2 

Thus the distance up stream =2-x4x3=5 miles 

Kirs I jg2 “ 



68 


miscellaneous examples [chap XII 


21 Let him change his speed t his aftei the stait 
9i+15(7 — whence i;=5 


22 

23 


Suppose he mixes ^ seeis of milk with 7/ seeis of water 
Then 4i(^+^)=5a,, whence ^=9^ and -=j 


Suppose V gallons of -water are added 
then (18+14+-t:)xl6i=18>cl8Hl4x21, 
(32 + t)x33=18 x 37 + 14 x 42, 

whence i;=6 


24 Let A run bOl yds m 1 min 
then B runs 45l yds in 1 min 
Suppose A gets level with jB in a- min 

50Ir=4oI(ar+5) , 

50a:=45(^+5), whence v=45 

25 Suppose he swims at v miles an hour in still water 
In 1 hr he swims (a;+ 1 J) mi down stream, 

„ I hr „ (■c-l^)mi upstream, 

a;+l^=5(a7-lj), whence t=s2 

26 Let time req'' he v hours after 10 30 a m 

At this time A is 25(r+3^) miles from Biistol, 

„ B„20(T+2i) „ 

,, 0 „ 220 — SOa? „ „ 

(220 - 30a:) - 25(a;+3i)=25(a?+3V) -20 (t+ 2J) , 
le 220 - 30i;=50(t:+3*)-20(a;+2i), 
220-30^=50a:+176 - 20^-45 , 
whence giving 12 noon and 125 mi from Bristol 


MISCELLANEOUS EXAMPLES III Page 238 

1 Wheat per cwt costs 28s 5d x^^=28s Sd x-^=6s 8rf 

« 248 5ef X -^5=245 5rf x~-=6s lOc? 

H ITs llrf X 17s llrf x-;^=6s 5rf 

2 Let Es i he the ai eiage subsciiption , 

then the “othei two" give Es (v+2) and Es (t+2i) respectively 
llr=2x9+a+2+»’+2i, whence r=2^, 
the two persons giie Es 4 8a and Es 5" 



69 


PAGF 2J8 ] MISCELLANliOUS EXAMPLES III 


3 Ta\ IS £1 on 5 lbs 

tax IS £13,184,767 on 6 x 13184767 x— j* ^ tons, 

„ £13,181,767 „ tons, ic, on 29 130 tons 

4 3 j fr per 1 Kp per 2 204 lbs 

2 204 lbs - pence pei 1 lb 

= o - flV* 3 i pence pel 1 lb =15rf pci 1 lb 


5 He Moiketl U lioui-s poi ^\cek Had he voikcd foi ri2Meclxsat 
7 pice pci hoiii , he m ouid lia\ e cai ned (4 1 x 7 x 52) pice 

But be lost I'i dajs’ ^\oIk, j c (8x7x 15) pice 

Ins not oai mugs \\ ei c (4 1 x 7 x 52 - 8 x 7 x 15) pice 
liih gain on foi nici tout i itt of Us 3j pci a\ cck 
=(44 x7x52-8x7x 15) pite - 52 x 240 pice 
=8{1 1 X 7 X 26 - 7 X 15 - 52 X 30} picc =8{7(1 1 x 26-15) - 1560} pico 
=8(7x271 - 1500} pict=8 x 337 picc=Bb 12 2a 


G 


Req> \aliic=58 135823 

O « U 6 *1 • « 

=58 135323 

Ihis fraction lies bctMtcn 4 and 5 in aahic Hence, to obtain a 
fiiitd icsull collect to tico dccmial figuies, vc imiltipiy coiicct 
to f/ircc dcciinal figuies, and thus scciiic the diMsion being 
correct to (\co decimal places 


58 134 

833 

5 950917 

290670 

1 

52 322 

2 

2 906 

8 

52 

3 

2 

3 


4 

345 963 



252 DC 
1 36766 ) 3 15 063 
72 431 
1018 
1313 
83 


7 S4 875=£l , 945=£1, S15=£?5 

‘li 

Now £^=£l9=£3 0709 =£3 l<r Gd 

8 Erior=£0 000405 x317=£0157=3y Ijtf 



70 


MISCELLANEOUS 


[PAGE 


9 


10 


Hi 3 avci age speed mi m 1 lir ^ 

_Ji® ft in 1 sec =- *s — ft 1 s®o =17 ft in 1 sot 
On leturning liis speed ^^as §: of 25 mi in 2t hrs , 

01 1 mi in - — X GO mms , oi 8 mi in mins , 

-g-XSOj 


t e 28 mi in mins , j c in 178 mins 

No of gallons pei hoad=— (neglecting unneccssar} Ggs) 
=1 = =29 8=30, to ne.ucst whole numbci 

Ko\enue pei head=£l 5 -‘,’|j'y‘ (neglecting iinncccssaiy figures) 

= £0 302 = 61 


11 


12 


1 003G ' 
27 

■jO 

33 

200 7318 


70 25G] 

3 

3 0109 

8 

3011 

0 

274 3000 

1 


a-®-10 9ir+9 4j;-102 

= 113-10 9x112+9 4x11-102 
= 1331 - 1318 9+ 103 4 - 102 
= 121 +14 

=13 5 


18 nil =18 X 17G0 jdb * metres , 

tunc req-i m secs = =2JJ^ 

= 14482^gmng4hr8 limn 22 J secs 


13 B runs yds , while 0 luiis jds , while A runs 1 yd 
B „ 3^^^ X 300 yds , while 0 1 uns 300 j ds 

•0 11 A®", 01 293’ ydb, „ C „ 300 ads 

14 34 nil =34 X 17G0 x 3 ft no of hnoeks in 1 hi = '’^ . ’‘V„” . 0 . x? ^ 

whence no of knocks in t min = ~ - l ■ “ =34 

15 Income this yeai =Ils ^^ =B 3 084 , 

07 I V 109 

Income last year =Ils ^ =11® 879 , 

Income next yeai=Es 1089, and tax = 6534 p =E8 34 OiuGp 

16 Total cost = Es 34 X 10 y 5 + Es 2j‘’„ x 40 

=Es G75 + Es 92 8a =Eb 7G7 8a 
he sales Ex 392 8a 



EXAMPLES m 


71 


23n] 


17 


18 


19 


1 423 
3 

056 

2 6231 

42 691 

68 

2 846 

11 

853 

83 

28 

46 

4 

27 


14 

46 424 

49 


6 799 
64395) 43 784 
5147 
640 
61 

quotient=680 con ect to 2 places of 
decimals 


Let 1 man do the work in m days and 1 boy m h days 
5_1 , 2_J[_ 

nib 12’ 6 12“m 12 14~84’ 

■whence jn=70, and 6=168 , since a boj does -j-J-g-, 
•while a man does his fraction of a man’s '‘'• 0 Tk=j^/ 

(32 X 7J&) strokes co\ er (1760 x 3 x 12) m , 

, , , 1760x3x12 

1 stroke co>ers — " T r -= — m , 

oZ X 

1 V in , t c 261 in 


20 

21 


4 of Es 23 4a 9p +4757p +Bs 17 6875 

=Es,10 5a. 8p +Bb 24 12.i 5p +Es 17 11a. 
=Es 52 13a Ip 


Is per 1 oz IS equii to £1 per 2P oz , oi 25 fr pei Ij lb , 
/‘’5 \ 

or ^ fr per 1 lb , or x 0 002204^ fi pei gram, 

or ^Yjx2 204^fr pei Kg, or 4403fr pei Kg 


22 The company -nould pay £4000 in £20000, oi £^ in £1 , 

the bank -would iccciic £(4 of 5000), le, £1000, and A would 
pay the bank £4000 

A would also lose of £4000, claimed by linn 

he IS out of pocket to the extent of (£4000+4 £4000), 

or £7200 

After the case is taken into court, the company can pay £4000 
in £18000, or £^ in £l , 

A has to pay the bank £(4 of 5000) 

He has already lost £2000, owing to his claim being reduced, and 
on this he loses 4 of £2000 , 



72 


MISCELLANEOUS 


[page 


he IS ont of poohet to the extent of 

£(2CXX)+f of 5000+^ of 2000) 

ze, £74444, 

he IS £(74444-7200) more out of pocket, 

Z 6 j ,, £2444 M II 

23. High Tido 


Bozilognc London Bridge 

Corl 

atl0 3(Fr) ,J12 33(Fi) 

^ ^ ‘^HlSSojEng) 

f333(Fi ) 

at-! 3 23 (Eng) 

(2 58 (Iiibh) 

31 1 gms cost £4 18< lOcf , 

4 941 

66 

216gms „ £4 94166x|yi 

9883 

216 

3 


494 

0 


296 

5 


311)10 674 
1344 

0 1 


1000 =£3 8s 8rf 
G7 


25 


26 


27 


5 

Let V be the number of years req^ In r 5 rs he will ha\ c sa^ oi 
£(3000 +300i) Aifter r j IS (00 -1-30) j re remain Duiing 
that time he spends £300 per mnum 

3000+30ai =300(90 - 1 - 30), 
whence i=2o He must noik till he is 55 


From Example 2 (111), p 227, tnc see 

, j sum of lengths in feet 

time req»= --j — - — j 

%elocity of appioacb in ft i>ei sec. 


(210+ 318) X 3 528 x 3 x 60 

° (25- i 7)«a STTST-'I®® 


1 metre sells for 0 3 fr , 


0914 m 
1C 1 3d 
500 yds 


„ (03x0914)fr 

ji O 3X0 014 
>> ® S6 34 

n O 1X0 014XB00 
II " SS 34 > 


or 500 3 db „ £4 e^. 01 £5 410, 01 £5 8s 3d, 

Now 500 yds cost lOOOrf , 01 £4 3s 4<f , 
profit IS £1 4s lie/ 



73 


240] 


EXAilPLES in 


28 Witli no season ticket 

on 3 days of the ■week he pays Is lOcf x 3, or 5s Gtf , 
on 3 other dajs he pays 2s x 3, oi 6s 
Since he tra'vels for 47 weeks it costs him lls 6rf x 47, or £27 Os 6£? 
"With a season ticket it costs him £20 8s +3s x47 , 

„ „ „ £20 8s +£7 Is , 

„ » » £27 9s , 

and answer is “No , loses 8s 6rf” 


29 Cliarge=Es of 2000+^ of 1535o) 

+Ils 383 75 


30 


=Es 100 
=Es483 12 a 

No of First class passengers = 356926 x 120 


Second class 


2d 
£472793 
' lid 


=472793 X 240 x^, 


01 472793 x 48 x4 

If Second class were abolished, the receipts -n ould be 
(356926 X 120 + 472793 X 48 x 4) X liff , 
or £(356926 x 120 + 472793 x 48 x 4) x 
or 

or £ J .. P - “ . o 


.XS40 
-j ^gmo7 r B 


or £835046 2s 

But the receipts originally weie £(356926+472793), or £829719 
£5327 2s IS gained 


31 Total n eight of fi\ e bars =(2075 + 5) x 5 gm 

=10377 5 gin , oi 10372 5 gm 

Also ■weight of foui bars(3008± 5)+(2092± 5)+(2014± 5) 

+(1093 ± 5)=8209 gm , oi 8205 gm , 
whence limits are 2 1725 Kg , or 2 1635 Kg 


32 Each year a population of 1000 become 1009 1 , 
tCf „ ,, „ 1 ,, I'OOOl 

Hence, if r be population 3 years hence, 

ar=77793 x (1 0091)3=77793 x 1 0091 X 1 0091 x 1 0091 
=78500 X 1 0091 X 1 0091 =79215 x 1 0091 
=79925=79900 to nearest hundred 



MISCELLANEOUS EXAMPLES HI [PAGE 5W1 


74 

33 Suppose I travel a miles by steamer, 

then I travel (132 - v) miles by train 

' The total cost is rx ^ shillings+(132 -• a?) x shillings 

whence i;=90 

20 12 ’ 

34 Dunng the 60 miles A rows 4 stretches, i c , 24 rai in hrs 

„ 60 ,, B „ 4 „ 24 „ ^hra 

Q 

A then rows the last 2 mi in ^ hrs 

A and B row 50 miles in hrs , or 9^ hrs 

Dunng the 50 miles C lows 22 mi in 4 hours, 

„ „ B „ 20 mi „ 4 „ 

also G „ 5jmi „ 1 „ , 

and B „ 2f mi „ 4 „ 

C and B row 50 miles in 93 hours , 

G and D w in 

After B has leached the winning post, 

A rows for hrs or -j-Jtr 

A rows ^ X 5| miles , i c , — 7^ o** ^64 yds 

35 Q and B nde 2 nn in 12 min P walks 2 mi in 30 min , 

R waits 18 min Dunng tins 18 min Q wmlks (iSX'j^) mi , 
or 1-^ mi 

Let X miles be distance between meeting places 

Q takes ^ houis to walk this 
4 

P and R take ^ hours to ride this 

But R waited 18 ram , oi between Q’a dcpaiturc and Ps 

arrival , 

£C A 18 

whence r=2, giving 4 mi from start 

Tune of reaching second meeting place 

=time of riding 2 mi -f-time of walking 2 mi 
=12 nun +30 min =42 nun (fiom the start) 



PART 11. 


EXAMPLES Xm a Page 245 

IS 1,96P0(140 14 9,04,81(309 16 50,12,64 ( 708 

24^96 609r5481 1408rTl264 


17 81,90,25(905 

1805 r~ 9025 

21 1,00,6000 ( 1003 

2003 1 6009 


20 10,70,48,04 ( 3702 

67 n70 
7402\ 14801 

24 11,57,42,84,41(34021 

64 Y~2 67 
6802\ 14284 

6804l\ 68041 


26 1,74,34,56,16(132 

23^74 


202 


5 34 
10 


0) 

( 2 ) 


At stage (1) x'O ha^c found the firat 
i'no figutes in the answci 
This shew ■? that 

1 7 134 501 6 = (1 3000)2+ 534561 6 
Sim'>" at stage (2) w e see that 
17434501 0 = (13200)2+ 105610 


27 


8587 


18,40,42,09(4297 

184041 

001 00 
601 09 


EXAMPLES xm. b Page 246 


12 0,00,13 ,98,76 ( 00374 
07 \ 498 
7441 2976 

K II s I 


13 


r 


9,5048,89(3 083 
008 r 5048 
0163 \ 18489 



76 


SQUARE ROOT 


[OHAP 


15 0,00pi p9pl fiO ( 0 01 2G3 

22 rso 

24g\ I'ini 
2523\ 7CG9 

18 0,70,20 80,1G(2G0-01 

40 \276 

520oK 20 8010 


17 27,04,41,00,10(52 004 

102 ’V 04 
10J00\ 410010 

20 5,35,40,90(2 31 

13 \l SO 


4C1 


0 1 ' 


IBIOG 

Hcncu itq* imswpr=0‘01849C 

EXAMPLES Xm c Page 249 
11 0,^40(0 3794 12. "30,00(15 3102 


07 \ 540 
749\ 7im 
350 


15 2^( 17*0203 

3102 \ 10000 
JIOIOV 3100 00 
13109 


20 ijr> 

303 ^ 10 f>Q 
3001 ^ 1 5100 
30Gsn>. 201100 
' 8100 

21 




811 


,3^>>JiO 
"" 29 


28 


32 


31 


33 00000(0237 

43 \ l(.0 
4G7\ 3000 
4,7,1 \ 311 (09 
\ 47 


34 0'0j,G0( 01697 

28 \ ~2GO 

lOoy Tooo 
3787\ 27900 
370,l\ 1291(10 
\ 253 



SURDS 


77 


xnr] 


t'V=0,^8421(0 606 

36 3A=3, 

1206 \ 8421 

27 

12,1,2 \ 1185(98 

345 \ 

\ 94 

3601 \ 


3,5,0,2 


EXAMPLES Xm d Page 251 
NB N^=l 4142136, ^/3=l 7320508, 
>^=2 2360680, ^^=2 4494897, 
^/7=2 6467513, -<,^=3 3166248 


2_ ^ 


JS ^/3lc3 3 

2n/3“2n/ 3 A^a G 


25 


„ 2 2 \/2 2^/2 ^/2 
n/8“-^''n/2“ 4-2 

100 •JTi 100<yn 331 66248 

X - 


\fu 4 \fn >Jn 4x11 

,0 1 

^ ->® %/0 12 


4x11 


11 Frvct" : 


12 Fract" = 


2+‘n/3 2-n/ 3 4-a 

1 \/3+l_\/3+l '\/3+l 


Va-i Va+i 3-1 2 

-n/O-I -s^ + 1 5-1 

14 ^™t.=i^x^=’=|^=2-^/3 
\/3+i A^a-i 3-1 


15 Fract” = 


24 7+\/g 24(7+n/5)_6(7+<v/6) 

'7_^/6^7+V5 49-25 11 

16 Fract” = ^x5|±^=^±^^=ll + Ds/5 

V5-2 V5+2 5-4 

17 33,17,76(576 =24® 

107(817 


18 1 ,08,24,32,16 ( 10404 = 1022 

46 r 824 



78 


MISCELLANEOUS EXAMPLES 


[chap 


19 0,00p0p6,76,62pi ( 0 002601 
46 ^76 
520l\ 5201 


20 M 6860 (7 32 
143^68 
1462\ 3960 
146,4 \ 10360 (70 
\ 112 


21 17,23,58(4151 

81^123 
825 \ 42 58 
830l\ 1 3300 
83,0,2\ 4999(60 


=(0051)2 


7 3270 ( 2 706 
47 YIm 

64p\ 370 4^** root is 2 71 


«,51,60(6443 
124 \ 5 51 
1284\ 55 60 
128,8\ 424 


EXAMPLES Xm e Page 253 

1 Diagonal =\/(T6F+24(P) yds =V(26921 +57600) yds =289 yds 

2 Side=-s/l372-88‘' ft =V(137+88)(137-88) ft =«/225x49 ft 

=(15x7) ft 

3 Side=Vl92000- 156 cubits=438 cubits 

4 Side=\(439 ac 33 p =V2125758 25 yds =1457 9 yds 

5 Side='N/450x577 m =^259650 m =509 5 m 
Diagonal =*^259650 x 2 m =^519300 ni =720 6 m 

6 (i) We get a;2=54 x 1944, or t 2=(9 x 6) x (6 x 9 x 36) , 

a;2=92x62x62, and r=9x6x6 
(u) We get a;2=65xl625, oi t 2=(5 x 13) X (13x125), 
a;2=132x5^, and J?=13x52 

1692 X 2303 9x4x47x7x7x47 


(in)Weget r2=1692x2303= 

32 x 22 x 472 x 72 .. 3x2x47x7 


100x100 

~ 1002 ’ 100 
2 sq big 5 sq cot =3600 sq yds , side =60 yds 

cost=(2a 3p)xG0x4=(2a +3p)x240=Es -S^+GOa 

=Bs 33 12 a 



IN SQUARE ROOT 


79 


XIII ] 


8 

9 

10 


11 

12 


13 

li. 

15 

16 
17 


Area=160 ac ac = i gq mi ^ Bide=^ mile , 
■whence poMnieter=2 miles, and time req^=30 nun 
Let V ft and 3r ft be its bicadth and length respectively , 
then 3ixi=(17 x 320- 10)^=12217 5, 

•vi hence = 4070 5 and ■> = 63 8 


No of acres = 


£98’3 


£175^ 


90 

'I6’ 


No of sq yds ^ 

sido= '* — 1 °y- " ■ yds , whence pDriinetei=660 yds, and cost 
= £104 10s 

No of slab 3 =®^i 5 r=i 44 , 

=18 ft 


arca=144xl8- sq in, and side 


10ac=100 8q chains sido=10 chains 

2 sidcs+2 diagonals=(20+20\^) chains or 20(H-V2) chains 
or 440(1 + >/2) y ils 
Now 3 nil in 1 hi =88 yds m 1 ram 

time taken as mins =6(1 + 1 414) rams 

= 12 rains neaily 

Area=i x 160x88 sq yds =6600 sq yds , 
side of sq =>/6(300 yds. =81 24 yds 

(I) A =~r^=3 1416x100 sq cm =31416Bq cm 

(II) side of sq =<\/314 16 cm =17 72 cm 

( I ) Arca=i\^28x6xlOxl24=J«/4x7xl44x0=Jx2xl2x\/35 

=6>/M=6x6 916=35 49 sq in 

(II) Aiea=J*y30x lOx lOx 10=J\/lOO-x3=i^V3=ilAS£Bq in 
r=\/ j =V3i 8309=5 64 cm 


We have 8 xtPx 07854= 10000, 

10000 asr -.fni 

® - F x o 7 8 6 A = r Ja-ez = * s'! 


whence <Z=3989 cm 


18 Velocity = 332 4 x s/l + 0 00366 x 20 = 332 4 x Vl 0732 
=332 4x1 03595=3439 ni per sec. 



80 


■RECTANGULAR 


[CHAP 


EXAMPLES XIV a Page 257 


18 

19 

20 
21 

22 

24 

26 

27 

28 

29 

30 

31 


Cost=(65^x45X9)a =(19xl4)a =K8 16 10a 
Cost=(?^ij^xl8)rf=(20x27x2)rf=£4 10s 


Cost=(7 2 X 5 6 X 0 75) Es =29 Rs 70 cents 
Cost=(8 5x46x9)rf=3519ii=£l 9s 4rf 

Area=(20x8 26) sq ch =165 sq ch =16i ac 
Area=(3 123)-* sq big =9 765625 sq big 
=9 sq big 15 sq cot 5 sq chk 
Aiea=(V)' sq cb =(21 75)2 gq cb =473 0625 sq ch 
=47 30625 ac =47 ac 12 gun 4 a 
Area=66aq big Length =19^ big 
G6 

req"* breadth= big =3 big 6 cot 4 ft 

Area=37 6607 ac =376 607 sq ch Bieadth=16 06 ch 
rcq'' length = '’ -Yf ch =23 45 ch 
Aiea=(^f)® sq big =1^%- sq big , 
cost=Es IQl-x l-j*^=Es 16 6 a 6p 

(i) Area=(805 x 74) sq in =^Tnnr =5 957 Ha 

(ii) Area =5 957 Ha =(6 957 x 2 5) ac =14 8925 ac 


32 Cost=£121x-^'[^^=£^=£31f 

For second part of question ive have 2^ ac cost £12 10s 

33 Area=^^ big =16 5 big , l)readth=-y^ big =2 big 4 cot 


34 Area=(^ of 26 16 x 26 16) sq ch 

= 570 288 sq ch , 
cost = £2 12s 6of X 57 0288 
=£2 616666 x 57 0288 
= £149 225 

35 No req'>=— 1^^=1029 36 

5X5 


57 028 

8 

2 G10666 

114-017 

C 

34 217 

3 

570 

3 

342 

1 

34 

2 

3 

4 


3 

140 225 

2 


Cost=^a X 


202x144 


=810 a 


8x4 



XIV ] 


AREAS 


81 


37 Widlli=16 ft 3 in =l‘)'i in , 

38 


, 1638x7x6 

]oiigbi>= in =204 in 


If width IS X 111 , then, e\pi casing aied in *»q in , wc lia\o 
288x18x1 = 30x18x144, whence i=15 

39 On trial it will be found that length of tile must he parallel to 

width of hill If r bo no of tilts req**, 

then tx4j x33=(38 ft ll in )xl2x(20 ft 2|iii)xl2, 

, 167^x2421^ 

whence r= — — ^-*=122x57 

3^x4^ 

, . £1031x12 

cost of tiles pel dor =— rso — = j 

40 9 111 width di\ ides 30 ft 9 in c\acll\ 11 times, 

and (61 ft 6in)-15 = 01?. , 

(41 x'>2)tuifs ate wanted, of which 41 ninst be ditided 

XT ^ 10-,x24xlll , 

41. No req''= — ^7; — ^ — =019 111118 
IGx 

Wolme = .md *°‘>^lr=30l, 

lu 10 

shewing that 16 in dmdos .1 width of 24 ft exactly , 
we have 18x30^=18x30+18 xA=540-l 9, 
shew ing 0 must be dn idod 

42 Area =(34 x 28 - 30 x 24) sq > ds 

43 Cost=lfa.x (20x18-21x14) 

44 Ai ca of footpath = (83 x 47 — 76 x 39) sq ft =976 fiq ft 

=07G X 144 sq m 

Area of out tile =36 sq in , ic"'* 

Aiea of each bed = (26 x 27) sq ft 

Rs 2£x 67^x04 


ieq° no =-li<'j’‘^,'>-U-=3004 


45 

46 


Cost for footw aj = • 


cost foi icmaindci =Hs 2| x 


total cost=Ks = 

47 Cost of carpet = A a = 1 2 12 a 

Cost of stain =( -.aiZi ») x f " a =41 a 

total cost=1273H =R8 79 9 a 



82 


AREAS 


[chap 

48 Area of tiled portion=(48 x36— 36 x 24) sq ft =864 sq ft 

=864xl44sq in 
Area of one tile=(9 x8) sq m , 
req-'Ho of tiles 1728 

Area of boarded portion =(36 x 24) sq ft =(36 x 24 X 144) sq in 
Aiea of one board=(72x6) sq in , 
req"* no of boards = ^ = 288 

49 Since corridor is 2 yds wide and surrounds the court, 

its length =54 yds and its bieadth=34 yds 

Since each path is 2 yds wide and is within the court, 

it leaves 48 yds for total length and 28 yds for total breadth of 
“remainder”, 

cost of grass = (3 x 48 x 28)s = £201 12s 
Also cost of paths and coriidor, at 15s pei sq i/d , 

=(54 X 34 -48 X 28) x 15s =£(1836 - 1344) x J =£369 
total co3t=£570 12s 


1 AT e j 20 X 15 

1 Noofyds=^j^ 


EXAMPLES XIV h Page 263 

2 Length =^1^^;^ yds 


3 Width ft 

14’5 
16 

5 No of strips=2j=7^ , 8 stiips aie required 


2ix3 

. r, . T, 9 26x18 

4 C»t=E,jX-5j^ 


Also §ott]ie last stiip ’mil be wasted 


12 

6 No of strips=^=4-* , 5 stiips are requiied 


Cost=Es 2^xA>yA Waste=’.8tiip=^of(l5x2A)aq ft 

? e 6 , length=6x 18 ft =36 yds 
Cost=5-?s x36=£9 18s Waste=’J^ of (I8x2-J) sq ft 



WALLS OF A ROOM 


S3 


XIV ] 


8 

9 

10 

11 

12 

13 

14 


15 

16 
17 


18 

19 


(i) Aiea=2xl22x{l63+15l} sq ft =25^x32 sq ft=816 8q ft 

(ii) Area=2x X {20| +16x^'r} s'! =31A x36-, sq ft 

=1155 sq ft 

„ , „ 18 .2xlOx{122+lli^ „ 18x2x10x24 
Cost=Es jgX ^ yS L.=E8 ^ =Es 60 


Cost of =Es 47 4a , 

cost of ceiling=iA^iAx4^ a =Es 5 10 a , 

(i) Surface=2(6x4+6x3J +4’X3^) sq ft 

(ii) Suiface=2(3} x2^+3^-x l-j-+2j xl-J) sq ft 

= (-IS + -T + -V ) sq ft 

Cost= ^ ^ ^ q ^ X 2 a =2(4’ +3+2 x3)x2a 

° ” (caucolhiig 9) 

Aiea of walls=2(l8-2-+ll-J)x 132 =810 sq ft ,1 giving 
area of each piccc=12x3x i-t'= 81 sq ft, / pieces 

Aica of •walls=2(l6x*^+13T^)xll J =676 sq ft 
area to bo papctcd=620'8q ft and area of picco 
= 12x3xf-2=63 8q ft 

Hence no of pieces=i^^+2=9§-j+2, ic 12 pieces are icquired 

Area to be papered=(4 x 18 x 11 — 90)=702 sq ft 
Area of each piece =2^: x 12 x 3=81 sq ft 
No of picces=-^Si2 + l=93 , 1 c 10, and cost =£3 

Cost=[2x(l6-2-+10^)xll-7x3-2x4x6-4x3}]x-J xOrf 
=(594 - 21-40-14)<f =£2 3« 3d 

Let X ft be the height , tlien area to bo papcied 
=2x30xa:=60a sq ft 

Area of each picce=2A x 12 x 3=90 sq ft , costing 5« , 

35s buys 630 sq ft , 60a:=630, ■whence ^•=10^ 

No req^=2(5ixl^+3xlg-)+5|-x3=(2x V + 3)Hq ft 
= ~^W 4 -- * a=Bel 14a 


1 giving 
J Es 52 14 a 



84 


AREAS 


f 


[CHAP 


20 

21 


1 


2 

3 

4 

5 

6 

7 

8 


9 

10 

11 


No ofpieces=^H 3 ^Tr^ = 126H-, 127 

Hence co3t=(127 x 4 35) Es =552 45 Bs 

Total 11 ea ■would cost 2 x (194 + 14j) x 11 J x 
= 2 X 8 X 34i X ll|rf = 6165£f 
Again £21 6s 5d =6117<7 , 

leq® no of sq ft =-SAiLi:^Al_li=i 3 i 


EXAMPLES XIV c Page 265 


(i) Aiea= ° = ft =533 sq ft 

(ii) Cost=(8 25 x 6xl4)rf =74 25rf=6s 2i<f 
(ill) Cost =l<f x-5-§-3-=4s 11^ (neailj'), 

Gs -4s lljrf, or Is 3d, is saied 


(i) Breadth=^^=14 08m 

(ii) Breadtli=39 375 x 1408 =46 ft 2 in 




9 2x393 
36 


jds =10 1 yds 


Cost=£3 5 X *° -f uboo ~ ” ^ 24=£(16 093 x 6 25)=£100 58125 
Ai’ea to be iKiinted 2x45(565+4 35)-18=9xl0 — 18=72 sq in 
Cost=THM^^I‘^=387 5rf=£l 12s Sid 
1 sq in =1 sq mi , 244 sq in =(2 44x640) at = 1561 6 ac 

640 ac =(1 6)-sq cm , 2000 Ha =2000 x 24 x^-—^= 20 sq cm 

174 ft =210 in , 15| ft = 189 in Of these H C F =21 in 
Since the factors of 21 are 1, 3, 7, 21, ue see that the areas of the 
squaie tiles must be 1= 3-, 7=, 21- sq in respectn ely 
Thus no of \pays=4, and the different numbera of tiles are 

310X189 310X189 310X189 310X180 

1X1 > JXJ > -X7 > 31X31 

20i X 15 sq ft cost Es 16^ , 

(20i+15)xl04x2sq ft costEs — 41 2a 

zU^X 15 

Area={40x8+(40+S)xl0x2-2x7x4-5lx2}=1213sq ft 
and cost=-^a xi-y-=E9 37 14a 6p 

Aiea=4x25x3 21 sq ch =40125sq di 



XIV ] 


MISCELLANEOUS EXAMPLES 
90 61 


85 


ni =11 73 m 


12 Heiglit=- 

^ X 15 45 

13 Area=Q x (34 72 +27 08) x 20 5] sq m =633 sq m 

14 (i X [13 46+11 54] X 6 2) sq cb cost £38 75, oi (25 x 31) sq ch cost 

£387 5 , 

1 ,£387 5x10 - , _ 

10 sq ch cost — Trp — — , oi 1 ac costs £5 
'■ 25 X 31 ’ 

15 0) A=(3 6)“X0 7854=12 96x07854 = 10 18 sq m 

/NO * (43)2x07854x4 1849x31416 -o .n r t 

(ii) Co3t='‘ — ^ a = g a =Es 40 5 a 5p 

16 Area=0 7854 x (54= - 462)=0 7854 x (54 + 46) x (54 - 46) 

=0 7854 X 800 sq ft 

17 Aiea=07854x(2102-190")=07854x(210+190)(210-190)sq ft 

= 0 7 B E y ,8 .o_q^ gq yds 

18 Total cost=co3t of icctangle+cost of semi-ciicuUr veiandah 

_r 243 X 13^ X 40 ^ (13A)2 X 0 7 854 X 48 ''1 ^ 

L 9 2x9 xJ 

=(1485+381 7044) a =Ks 116 10 a 9p 

19 Aioa=(110x55) -^ds ac 

Cost=256rf xl^^x3i = 1120rf=£4 13s 4rf 

20 Side='s/l600 x 2i=\/^0 x 9=20 x 3 = 60 j ds 

Aieaof path=(602-042)=(60+51)(60-54)=114x6 sq yds 
Cost=Es 171 

21. 100 sq mi =1 sq in , 640x]00ac =1 sq in , 

Tvhenco 931000 ac =-^’- sq in, oi 14 5sq in 

22 Total area=3 x 2 3, tc 6 9 sq era 

Unshaded aiea=(4 x ■2:^ x V -)> or 2 6 sq cm 
and =(2 6 X 25) sq ft , 

shaded area= 4 3 sq cm and =(4 3x25)sq ft 

23 If sides are v yds and 2x yds , then diagonal = » x n/ 6 yds 

But 2.1^=4840 and 5:*= 484x5, 

5a;2s=484x 25 , ■whence a;\/5= 22x5 =110 yds 



86 


VOLUMES 


24 Let bieadth=a; yds and length 2t yds , 

then 2a;^=2244 5, whence t=33 5 

Peiimeter=;6r=6 X 33 5 , 

cost=(6 X 33 5 X 13 25) Es =(201 x 13 25) Es =2663 25 Es 

25 Let be no of ft in the side of total area 

Then area of path in sq ft =a:®— (a — 16)^=4x4840x9 , 
te 32a — 256=4x4840x9 
Divide by 32 and r-8 = 605 x 9, giving a=6453 
, (5453)2 29735209 

Whence ac =-435^ ac =682 6 ac 

26 Let side of larger be a yds , a*- 712=5x4840, 

whence t2=29241, and a =171 

27 Incoirect measurement of length=15 ft 

Correct „ „ =^x 15 ft =14 of 15 ft 

error in length = of 16 ft 
Similarly, error in breadth of 121- ft 
Error in area = incorrect area— con ect area 
=(15 X 12*- 15 X 14x 12-1- X 14) sq ft =15 x 12*{l -(|^)*} 
=29^ sq ft=30sq ft (nearly) 

28 Total cost = cost of land -J- cost of road 4* cost of houses 

= £(53 X 352 X 3:^)+£(^^ x^)+ £87700 
Eeceipts from sale =£ [352 (53- 9) x 51^] 

Subtracting (2) from (1), we have 

Net cnHf.= -R[fi7'7m.j.jlfi^(B3X07+B-.ldXl 1 

=£ [87700 4- 352(?-^gT+aJ^*p:!?)] 
=£[87700-17 6 X ]413]=£(87700 - 24868 8) 


6 

7 

8 

9 


EXAMPLES XIV d Page 270 

Let d metres be depth, then rfxO 65=2 6 , whence d=4. 
If d cm be depth, then 560 x 250 x d=21000 x 1000 
Weight=(i|x^xff X4801bs)=150 lbs 


Eeq^ no =42x 


3 6 


2156 



87 


XIV ] CfUBES AND CUBOIDS 


11 

12 

13 


Cost=l4x4:xl6x7a , where ns depth in feet 
Since Es21=336a, Hxtx16x7=336 

Let T feet be depth, then 9x=4\ x 2|^ x 1 J 

Let A sq yds be the area Expressing the weight in lbs , we ha\ e 


042x9Ax710 

12 


2240 


14. Volume of rainfall per big =1600 x 9 x ^ cn ft 

This weighs mds , or mds 

15 Volume of ice= AfiJJlxa x cu ft 


total w eight 


, 4840X0X5 
2X12 


57i * 


,142^p.'-928tol.s 


16 No of gallons per ac =volume in cu ft x 6| 

=4840 X 9 X f^-x AOfiX4sx2i=,27225 x galls 

17 'Weight=vol in cu ft x 710 lbs 

=224x9x-; 


1 V X 1 0 ^ _ A.xjip. awt 

8X12^112®”*^ 1C ® 


18 144x^cu in =30 lb , 

78 x 42x-?-cu in =-;2Pi^x78x42x~lbs =(5x7x7x13) lbs 


30 X 3 X G X 


144X3 

13^ 


19 No req^=* 


12 


3 44 „ 3 
4 12 12 


=120x72 


20 Let A sq ft be area , expressing the volume in cu in , 

5Ax 1728 X j-^^x-^= 486, whence A=12 

21 Let I ft be length req'*, then 

, .. 2JX 1^X14 ,, ?Xl20 e.. l. 7 in 

1 cwt =-2 — ! cu ft=Ts — cu ffci whence t=12 

11 18x144 ’ 


22 Vol of iainfall=3800xl760xl760x3x3x-j^ cu ft 
=aj^ X 10 X [38 X 176 X 176 X 9] 

=250 X 10 X [Aa2LL^«iX3. x 1760 X 3] cu ft 
=250xl0x[2006 4xl760 x 3]ou ft 
Now exp" in brackets=2006 4 mi =(66818 x 3) mi 
=(670x3) mi (nearly) 



88 


VOLUMES 


[OHAP 


EXAMPLES XIV. e Page 273 

jV B From Art 247 it follows that 

Weight=Volumex Specific Gravity. 

1 Wt =(0 3 X 1002 X 7 14) gms =21 42 Kg 

2 (i) Wt =(2 5)®x 1 2 gms =15 625 x 1 2 gms 
(ii) Wt =36 X (0 8)’ X 1 2 gms =23 04 x 1 2 gms 

3 Wt = ^4i X 3 J X 2J X ton8=2 6 tons 

4 Wt =(12 X |-f- X ^ X 62^ X 0 85) lbs =(1500 x 0 85) lbs 

5 Wt = ^4840 X 9 X 2J X ton8=121 x 3 x 125 x 0 08 tons 

=(363 X 10) tons 

6 Wt =(445x24x2x055) Kg =(1068x1 1) Kg 

=(117 48 x^) lbs 

7 Let depth be v dm , then 25 x 1 6 x a; x 13 6 =540, 

whence ^=l = 10cms 

8 (35 X 6 X 25000) litres pass in 1 hour 

^35 X 6 X 25000 x lit pass in 2j mins , 

t e (350 X 625) litres pass in 2^ mins 

9 Let depth be i; dm , then 17 5x8x r=^x3500, 

whence a. =5 =50 cms 

10 Wt =[(16 8 X 1 25) X (0 5 x 8 8)] Kg =[(2‘1 x 10) x (1 X 4 4)] Kg 

= 92 4 Kg =(92 4 X ^-) lbs = 203 28 lbs 

11 Capacity =(9 - l)x (7 -l)x (6 -l)cu in [Art 246 (ii)] 

12 Volumeofmaterial=(13xnx9-12xl0x8)cu in [Art 246(iu)] 

13 Capacity =13x9x7 cu in 

Vol of material = (14x10x8-13x9x7) cu in 

14 Eeq* no of gallons=50 x 3 x 15 x 6 x 6|^, 

50x3xl5x6x6i 60x27 ,,, , 

hence time= a mms = — ^ — mins =llj hours 

15 No of gals in lOmins =4xl-|-x(4xl760x3)x6^x^ 

=4400 x 25 



MISCELLANEOUS EXAMPLES 


89 


XIV] 


16 


17 


Since 1 ac =^-57^ sq m , if r dm be depth req^ 

o 

X 10 X 10 X rj cu dm covei 1 ac But 1000000 litres cover it, 


(100)2 

24 


X 10 X 10 X a =1000000 , i=2 5dra=25cm 


Let a? be the fmction req**, then, expiessing weight in lbs , 
rx2240 = 3^-x2^x2x622. 


whence a?=-|^5- 
Again, internal surface 

=2(&rf+W)+6Z (where ^^=2§^ft, i=2j ft, tf=2ft) 
=2(23X2+35 x 2)+25X35=33^ sq ft 
External sui face = surface enlarged by thickness of wood corre- 
sjionding to intcinal surface -i-sui face on which Iid should rest 
This last = 2 X external length x thickness + 2 x intei nal breadth 
X thickness 
external surface 


=2(3 X 2^+4 x2j-)+3x 4+2x4 X ^+ 2 x 25 x-J- 
=42j+2;=44y8q ft 


cost=(44{+35i’‘,)x9p =795X9p 

=(80-^)x9p =80x9p -3p =Es 3 11 a 9p 


18 Gip icity= ^2i^ X 2^ X 2^ cu ft =13 cu ft 

No of Ihs of wntei= 13x02^x19 = 247 X 624 = 15437f 

19 The water will use as if its volume had been inci eased by an 

amount equal to the volume of the lectangulai block Hence 
if /i be the height in feet 

2l-xl-J-xA=ll xlx , A=A ft =4^ in 


20 Vol of each brick =-V cu ft Vol of water displaced by each 

The height A to which one buck will raise water is given by 
5 x 4 xA=- 52 j-, whence A=-5-j-^ 

Again, height of water at first =^^=ll- ft , 
height left to he covered =2^ ft , 

hence if a: be req"* no of bucks whence ^=1106 



90 


VOLUMES 


[chap XIV 


21 Expressing the volume in c c and weight in gms , if s be sp gr , 
we have 280xl95x2xs=85l760, whence s=7 8 


22 


23 


24 


25 


26 


Area represented = (2 5 x 1 5) x 10^ x S’’ sq ft =25 x 1 5 x 9 sq ft 
no of blocks ueed= ^^ ^ ^ =27000 

TT i- i.1 j H 27000 x^x^x-5-x621x08 

Hence no of cartloads req“= — = 

^ 2240 x 2 5 

= ¥ 33°^ = ^0 roughly 

_ ^ 27000xAxix-}x62^-x08 
Cost= 2 m " 

=£11 4s 6d 

The river discharges in a year 

220000 X 60 X 24 X 365 cu ft out of x 4082 x 640 x 4840 X 9 ou ft 

220000 x 60 x 24 x 365 




te 


■J^x 4082 x 64 x 484 x9 
l ooooxaoxsas q 

3~1X408SXS2X3 


cu ft out of 100 cu ft 


% o*’ HfififF % or 39% 


„ 400000x30 
Since cu 

64 


ft of 


Let s sq ft be area of reservoir 
water are wanted per day, 

20 X a?=. * - 0 . o .. oogx . ? 3 o ^ 134 (July-Dee = 184 days) , 
whence a?=800x 120x 184 sq ft =17664000 sq ft =406 ac 
Let y sq ft be area of catchment Since 10 in of rainfall are 
sufiBcient for 6 months, y X cu ft =vol of reservoir, 

yx 14=20x17664000, whence y=423936000sq ft orl5sq mi 

Annual discharge =(30000000 x 100 x 1 ^) cu m 
If V be the req'*no of days, since a; days =a:x 24x60x60 mins, 
170 X 60 X 60 X 24 xa;= 30000000 x 100 x 1;J , 
whence a;=- ^ - g gg - ^ ^ =255 

(1400 X 4 X 1760 X 3) cu ft of water pass in 1 hr 
(1400 x 4x 1760 xSxoj) gals „ „ 1 hr 

1400X 4X1760X 3 x-^x 24x365 gals „ „ 1 year, 

1400X4X1 7flOX3xaSX24X306X23 , 

4 x 7000 x 2240 tons of matter pass in 

1 year 

This erp° — oooxa^cBXBii 


tons=2374586 tons 



CHAP X\ ] GRAPH OF A STRAIGHT LINE 


91 


27 "We have ynW ™ =Z x(0 002)2x0 78")4 , 

^~ d ~ 6osiTr c “ » this is the length of 9 Kg of copper , 
length of 7 5 Kg x ooo - aV^ r c =265 m 


EXAMPLES XV. a Page 279 


16 The points, m the ordei gii en, 
are represented by A, B, C, D, 
E, F, G, H in Fig 1 By 
measurement the distance of 
any one of them from the 
origin IS said to be 5 units 

Also if P(x, y) stands for any 
one of the points, 
OP2=a'=+y2 

The value of 3?-\-y- is found 
b} calculation to be 25 in 
each case Tlius OP=5 



Fio 1 


EXAMPLES XV b Page 281 


1 The points are shewn in Fig 2 , 
the first senes, in the given 
order, is A, B, C, D, E, and 
the second senes P, Q, R, S, T 
Tlie coordinates of the inter- 
section of these lines arc seen 
to be 5 and 8 , hence the 
point IS (5, 8) 



K H s I 


G 


iJO 2 


92 


GRAPH OP A STRAIGHT LINE 


[chap 


3 Substituting the given values foi x 
in turn in the equation, the cor- 
responding values of y are found 
to be 10, 12, 16, 6, 0 Hence the 
points required are (0, 10), (1, 12), 
fs, 16), (-2, 6), (-5, 0) These 
are the points A, B, C, D, E plotted 
in Pig 3, and aie seen to he on 
a line cutting OX at E and OY 
at A 



Fio s 


5 (i), (ii), (ill) The three lines are parallel, one passing through 

the origin, and the othera making intercepts -4 and 6 on 
the y axis 

6 As in Ex 2, it will he seen that each graph is a line through the 

origin 

7 These points aie plotted in the solution of XV a, 16 Since they 

aie equidistant from O, the curve is a circle 



FlO 4 


aTl\rn OF A STRAIGHT LINK 93 

Tlie>50 art* the pttinLs A, B, C, 0. E plottetl m Tig 4 Tliey Iio on 
the hnc cntling OX at E and OY at A 

Siniihih f'H tin. second oqnition wc ohtaiii the points (0, 8), 
(1, «i), (2, t), O, 2) ( >, 0) 'I hcAc arc the points P, Q, R, 8, E 
They ho on n lino cutting OX it E nntl OY at P 

Tims the 1 cquiicd cooi'dinatca of the point of intoi section are 4, 0 

2 Suhstitiiling tho gi\cn viliics foi r, the coi responthiig ^ allies of v 
in the"lirs)fc equation me 17, 13, 9, 5, 1, -•!, in the t'Ccoml 
equation -9, -0*, -1, -1^, 1, Thc> mil be found to 
lie on t\v<i straiglu lines iiitci'^ectiiig at tho }}Oint (2, 1) 



3 The lines are shewn as (i), (u) in Fig 5 
On (i) at P when rs - y is seen to be 3 , 

(ii) It Q when i/rsrlC, i is i-con to be 1 3 nearly 








)4 GRAPHS [chap 

4 Substituting the given values for x in turn in the equation, ive 

obtain y= -4, —3, -2, -1,0, 1 
Plotting the points, "we see they be on a straight line 
When y=3, a;=25, and when -1;= -15, 3/= -5 

5 Let us take of an inch as unit foi y and one inch as unit for x , 

then the graph of 11^+6 will be aa in 3?ig 6, in which the 
line has been drawn by joining the points (0, 6), (2, 28) 

Now we see that t =1 8 at the point P, and here y=2Q, nearly 
Again w=20 at the point Q, and r=OR=128, approximately 
In obtaining this last result we observe that OR is greater 
than 1 2 and less than 1 3, and we mentally divide the tenth 
in which R falls into ten equal parts (i c into hundredths of 
the umt) and judge as neaily as possible how many of these 
hundredths are to be added to 1 2 





( 

1 

1 





L 

ij 


J_L 

U 


j 



jj 

B 

11 

11 


i 





EH 

MB 






1 


1 

■ 


— 


— 

^ “ 

II 

! 



■ 





II 

II 








■ 

1 


1 




s 









1 

■I 

1 





1 


■ 


1 




a 










T 








1 






■ 










EH 








1 


£3 




1 


















1 


m 




1 


















1 


n 




■ 




















m 




■ 


















■ 


i 




1 





- 













1 


i 




1 









T 









1 


■ 




■ 





m 




II 









1 


1 




■ 





m 




II 









1 


m 




1 




1 





la 

1 





IT 


IE 

1 


m 



Li 

1 




il 




B 

■I 





- 

“t 


-± 

■ 


w, 



It 

_L 




|e 


IB 


M 



1 



1 

■1 

II 

■1 

IC 


1 



III 

II 




1 

E 

n 


n. 






1 

II 

II 

III 

II 


L 



III 

II 




II 


r 

r 

It 



IQ 

r 

r 

r 

rr 


rr 

TT 

T 

r 


r 

ri 

1 

1 

1 

j 



Fia 0 


EXAMPLES XV d. Page 289 

6 Measure time horizontally and population vertically 

Take 0 1 of an inch as unit in each case , also it will be con- 
venient if we begin measuring abscissa, at 1830, and oidinates 
at 20 

The graph is given in Pig 7 , it will be seen that it passes exactly 
through the extreme points and lies evenly among the others 
The populations in 1848 and 1875, at the points A and B respec- 
tively, will be found to be 27 8 millions and 45 3 millions 



96 GRAPHS [chap 

7 A convenient scale is 0 6" horizontally to each month, and 0 1" 
vertically to each degiee The curve is shewn on half this 
scale in Fig 8 It should be remembeied that the curie 
need not necessarily pass through all the points plotted, but 
only as near to them as possible (Art 259) 



8 Take 0 1" and 0 4" to represent the units of circumference and 

radius respectively The points correroonding to the given 
values of C and r will be found to he on a straight line 
The lequired values may be read off fiom the graph 

9 The requiied values are given by the abscissae of P and Q (Fig 9) 



10 The points are plotted in Fig 10, and are seen to he nearly on 
a stiaight line By tiial it is found that the line through 
the 1** and 4*’’ points lies most evenly amongst the remaimng 
points 


XV] 


pkactical applications 


97 


Assuming the equation to be y=ax-\-h and, substituting the 
coordinates of the 1“ and 4‘'* points, ne find a = 6 =tt 
H ence the equation is lly=3r+35 
Putting y=ll 5 m this, we have r=30 5 , 

• • ^—10 t • y— 5 9c • 





[chap 


98 GRAPHS 

13 The graph is shewn in Fig 12 The abscissi® of P and Q aie 
17 and 36 5 respectively Hence the required prices aie 
Rs 17 and Rs 36 5 


Fio 12 


14 


The graph is shewn on a small scale in Fig 13 The student 
^ould draw his own on double the scales here employed 



rio 13 


L5 Take one-tenth of an inch as unit in each case 

Following the method of the Example in Art 260, it will bo 
found that a straight line passes accurately through the 
2“^, 5*‘', and points, and lies evenly among the rest 

Assume y=ax+h, and, substituting the coordinates of the first 
and last points, we find a=0 5, 6= -3 
Thus the required equation is ?/=0 5r— 3 
Fiom the graph, uheny=16, a?=36, 
x=71, y=32 5 

From the equation, when a:=164, w=79 


XVI ] BATIO 99 

16 See solution of £\ 11 

17 Take a scale of 10 ycais to the inch horizontally and 10000 

population to the inch veitically It will also be convenient 
to take for the oiigin the point (1810, 20) As in the 
Example of Art 259, it will be found that the giaph of P 
is a cm VC passing accuiately thiough the 2"^, 4*'', 6“‘, and 
V*’' points, and lying evenly among the rest 
Tlic graph of Q, as in the Example of Art 260, is a straight line 
passing accurately thiough the 1“, S"*, and T'** points, and 
lying evenly among the othera 

They intersect at the point (1883, 45), which gives the requited 
year 

18 A scale of 05" horizontally to each year of Expectation, and 

0 2" vertically to each y ear of Age should be employed, the 
origin being the point (34, 6) It will bo found on plotting 
the points that between the ages 10 and 22 the giaph is 
appioximatcly rcpicscntcd by .i straight line Eead on the 
auscissa: of those points on this line whose ordinates aie 
12 and 20 


EXAMPLES XVI a Page 291 


9 


Since the same rate per ac is chaigcd. 


ratio = 


£33 8s Ad 20 
£55 2s 9rf~33 


10 


Let d nil bs the total distance and x and y mi pei hr then rates , 


then 49r=rf, and 8Ay=d , 


49t= 847/, and -= 


84 _ 1 ^ 
49~ 7 


4 


11 Eatio of speeds =^r“ = g 


12 


Eatio = 


n 0 

IIP 

PCS 

lOS 


20 


13 


The first takes 6| hrs , the second 6 hrs , 


as in No 10, ratio of Epeeds= 


£_8 

6i~9 


14 The now fraction is -J ^ Eut ■^= ^ 

This shews since 

We now see 16 must be added to the consequent 



100 

15 


RATIO 


[chap 


17 

18 


Sinco 8 > 5, we should have to add more to the numerator than 
to the denominator in order to get a fraction which =-|- 
But in this case the quantities added are equal, shewing 

8-|"€K 8 

If 5(8+a)=7(5+a), whence a=2J 


0 ) 


5481 
62 43’ 


(n) 


653x1728 02 
62 43x16 02 


( 111 ) 


4 05x1728 02 
62 43 X 16 02 


Let Vi cu ft be volume of iron, Vj cu ft volume of steel , weight 
or Vj = Vi X 7 2 X 62 43 lbs , weight of V 2 = Vg x 8 0 x 62 43 lbs 

Since these ate equal, Vi x 7 2 x 62 43= Vg x 8 0 x 62 43 


y!_ 80 _W 

Va 72“ 9 


19 Br = 


100X1700X30 

1 in =100 mi , 1^ in =180 mi Smi’>' 830 mi =8 3 in 

20 4 5 in =50 mi , hence 1 in =i§A mi , z c scale is 1 in to 11^ mi , 

IIF=-— i 

if9.x 1760x36 

We find 6 6 in =73 3 mi , and 3 9 in =43 3 mi 


EXAMPLES XVI b Page 298 

1 -6 See Art 268, Ex 1 and 2 7-9 See Art 268, Ex 4 

11 -13 See Art 268, “ T/iree quantities a, h, c, etc ” 

tt 6 6“ 

^“a’ ^5-17. See Art 268, “Three quantities, etc” 

18 ~=^, 

X air 

19 -25 As in Alt 268, Ex 3, put x for missing term 
26 Eatio=7«;^=0 061 

4 0 1 



PROPORTION 


101 


XVI ] 


lib 5 2240 lbs 6x2240 

1 Kg “11 ’ 1 Kg “ 11 ’ 

hence 1 ton=—— ^^^ of 1 Kg =1018 Kg 


28 

29 


2920 yds =f^ x 2920 m =-a^ 

1 ht ^7 1 K1 _7000 

1 pt 4’ Igall 8x4’ 


x40 m =2667 m 
1 K1 =^of Igall =219gaUs 


30 From Art 264, Ex 2, vre have 

wb of 1 cu ft of cast iron 
sp gi of iron wt of 1 cii ft of •water 
sp gr of copper” w t of 1 cu ft of copper 
wt of 1 cu ft of water 
wt of 1 cu ft of cast iron 
“ wt of 1 cu ft of copper 

if 70 lbs be weight required, ^ , 
whence w=4498 


EXAMPLES XVI c Page 301 


1 

(i) Es 18 x-^ , 

(ii)-(iv), as No (i) 



2 

Es 55 x |^ 

3 

74 mi x-|^^ 

4 

7 in x|4f 

5 

26 daysx-|-|- 

6 

180 hr x|f 

7 

21 min x-|^ 

8 


9 

110 yds x^ 

10 

22^ knots X ^ 

11 

3^ mm x-|^ 


12 (i)£56x^ 

> 0 ^) ^ 

13 

(i) 57|^mi x^. 

(u) 

192 mins x , 

(ill) 28 

ini x-l^ 

14. 

Es52xf-f 

15 

96Kgx|^ 

16 

27tf 


17 Esl302x^ 18 20cwtxAl 

15 5 17^ 

19 Time without stoppages =270 min , 

aveiage running speed=246x mi 

Time with stoppages =288 mm , 

average speed = 246 x-^y mi 



102 


PROPORTION 


[chap 


20 2362 ozx^ 21 2145 m 22 4712 m 

23 259x^ 24. 220x|| 25 6|ozx^ 

26 404mi x|-perlir,ze 404nn x^x-|^ per 48 mm 

27 Scale is 126720 in to 1 in , or mi to 1 in , 

t e 2 mi to 1 in , givmg 3 8 in and 5 1 mi foi other answeis 

28 118J mi to 7i in = 15 8 mi to 1 in 

•p ■p 1 1 - ■ - 

~16 8X1760X8X1B 1001088 

6 9 in =6 9 X 15 8 mi =109 mi (nearly) 

29 256 6 litres X -If =710 litres=7l0 Kg 

30 420 lit =420 Kg =i^x 420 lbs =924 lbs Time=66 min x 


EXAMPLES XVI d Page 304 


1 

Es 141-Jx-fl 

2 

£l404^^Xi^ 3 

£49|^X^ 

4. 

3276 lbs x|^ 

5 

39 mi X — 

5 

6 

65a x?i 

3i 

7 

100 min X ^ 

8 

27x^ 9 

33xi45 

107 6 nn.. S 0 00 


3} 

2Zi 

101 

.47 6 — 136 3 

10 

29 

45 srs X — ^ 
16iV 

11 

75 mi X ^ 

6^ 

12 


13 

1 cwt x^ 

li 

14 

41- 

200 lbs X ^ 

15 

Es28fxfe 

16 

TJn 1 18 00 

17 

*^1520 

18 

Es 125 x^ 

19 

^ 17 0 2 


20 Assets =-|^ 

of Es 1250 , loss=Es 


21. (i) Ta'?=-j--^- on every lie 1 of gross income , 
tax=^-|— of gloss income 

(ii) Net income = gloss income- tax =][§^.§ of gross income 



PROPORTION 


103 


x\t] 

(lu) Ta\= j-iJ-j- on evoiy Re 1 of net income , 

of net income 

22 Fiom No 21, net incomo=Rs 16633 x \ V® 

23 Piom No 21, net incomes: Es 6100 x ^ ,T ^ v 

ite 1 

=Es 6100X},*; 

24 Fiom No 21, gioss income =Es 59,J x3-J-" 

25 From No 21, t!i\=Es 2150’ x-j-J-y 

26 Valiic=i“itex=jV =£2035x12 27 Rates = £5676 = x|g 

28 His net income nonld haic been 187 p pci Ro 1 of gioss income 

instead of 188p pci Re 1 of gioss income 

Hence net income icquiicd = Rs 6264 x [-g~ 

29 Total ta\ is Is Orf in the £ , gioss income = £584 x£^^ 

30 If total income is £», then — - 120)=17}^ , whence •» =580-, 
Net income = total income ta\ 

31 Let nij uncnincd incnnie bo £i , 

then TrJp0- of 1174+ Jg- of ■» =70^ , whence i=5285 

ko9 91 1 

32 4 7 in X;:^=-^ in , 701 nii to 4 7 in =15 nii to 1 in 

<0^ 6 

n 1? 3 = 1 

1 5X1700X 1X1!, 0504 00 

33 Rs 2450 x JyV =!*« 53iC=i3!.=Rs 3524 neai ly 

34 2’ 3- hrs X ?. a=180 niin 

35 Rs4345xf^5 36 Rs 1500x'*^=Rs 

58J-g- 

37 78 6 gms x V 

38 Net income = of £1253 

•■“0 saved out of eveiy £ of net income , 

' of of £1253 IS left This= £1 



PROPORTION 


104 


[CHAP 


39 Since first section of 18 740 Km cost £3704, nt tlie same rate 

the whole railway will cost ^3704, 

or £a_r,^Tj<^pj:^£ 3 6 ^XjL^8 6 J^£4^|^^jg500Q 

40 1 yd =23 lbs , or II Kg , 1 metre=V x x®t 

41 "We have 11344 x 8 pmts=l ac , or 11344 x 8 x f lit =|- Ha , 

1 Ha litres 

7^3 

42 1 cn in of gold weighs 0 58 oz x -8^ , 

■ . 3X8 - ^— cu in weigh 8 7 oz , z e 0 78 cu in weigh 8 7 oz 

90X0 68 O' o 


J 

EXAMPLES XVI e Page 314 

7 Take one tenth of an inch as the unit in each case Supjmse 

y rupees aie equivalent to v shillings Then, as m E\ , 

Art 277, Hence, measming rupees verti- 

cally and shillings hoiizontallj% the requiied graph is at once 
obtained by joining the origin to the point whose cooidinates 
are 21 and 16 The abscissa coi responding to oidinate 13 is 
17 The ordinate coiiesponding to abscissa 80 is 61 
Thus 13 iupees=17 shillings and 80 Bhillings=61 rupees 

8 Take scales of an inch to 1 gallon along the aMS of x, and one 

tenth of an inch to 1 litre along the a\is of y As in Ex 7, 
y=^a;, which is the equation of a line through the ongin 
Join the origin to the point (4 4, 20), and the graph is 
obtained Tlie ordinate corresponding to an abscissa 2i is 
seen to be 114, the abscissa concsponding to an ordinate 
20 9 IS seen to be 4 6 
Hence 2J gallons =11 4 litres, 
and 20 9 litres = 4 6 gallons 

9 Take scales of an inch to the yard along the axis of r, and an inch 

to a metre along OY ffoin the point whose coordinates are 
(6 01", 6 50") to the origin The ordinate corresponding to 
an abscissa 2 22" is seen to be 2 03", and therefore 
2 22 yard8=20 3 metres 
Hence 22 2 yards=20 3 meties 



XVI ] TREfliTED GRAPHICALLY 105 

10 This 18 most easily done on a very large scale Take 0 1" hon- 

ronLilly to each gnin, and 1 O’ veitically to each gram Since 
161 grains=l 17 grams, ne liaie 905 grains=5 85 grams, 
and the graph will be the line joining the origin to the point 
(9-0o", 0 85") 

The abscissa corresponding to ordinate 3 5 is 54 1 
The ordinate corresponding to abscissa 30 9 is 2 0 
Hence 3 5 grams=54 1 grains, 

and 30 9 grains=2 0 grams , 

1 C 3 09 grains=0 2 gram 

11 Take scales of an inch to 1 minute along the axis of r, and an 

inch to 1000 jards along the axis of y Plot the points, and 
it will be found they are not in a straight line llierefoie 
the distances arc not propoitional to the times 
Por the second part of tne question, join the oiigin to the point 
(2, 700) and produce this line Read off the ordinate corre 
spending to abscissa G^, and w e get for req"* distance 2300 yds 

12 See solution to XV d 8 

14 Draw the graph of y = r, 

and read off the > allies of y coi responding to ;f=40, 64, 78 

15 Cf XV d 12 Plotting the points as in this example, we find 

they do not he on a straight line Hence the premium is 
not directly proportional to the age 
For the second part of the question, join the ongin to the point 
(35, 2 8), and read off the ^alue5 of y corresponding to 
a;=25, 30, 40, 45, on this line 

16 Take a scale of an inch to a j ear along the axis of x, and an inch 

to 100 increcMc, along the axis of y 

Regarding the “beginning of 1905” as the “end of 1904,” take 
the point (’04, 0) as origin The increases for 1905, 1906, 
eta, are 200, 420, 663, 928, respectively 
Plotting the points (’04, 0), (’05, 200), (’06. 420), (’07, 662), 
(’OS, 928), we see they do not lie on a straight line hence 
the increases are not pioportional to the times 

17 Take 01" veitically to represent 1 penny and 1" hoiizontally to 

represent 1 hour Let y shillings be the wages for x hours , 
then the graph is that of the equation a; 

The required values can easily be read off 



COMPOUND PROPORTION 


106 


[chap 


18 The rela^lon between the actual marks x and the raised marks y 

IS clearly which repiesents a st line through the 

origin ^ 

The graph is shewn in Fig 14, the line being drawn through the 
origin and (68, 100) 

The required marks aie given by the ordinates of P and Q, and 
are 90 and 72 respectively 



EXAMPLES XVI f Page 317 
As in Art 279, Ex 1, leq® area=9G sq ft x6X “ 

As in Alt 279, Ex 1 req^ weight =2i mds x 3k x-| 
96mix^xi 4 Es22Jx-=x^ 

3jtonsx-kx-|x.5 6 225 X 2240 lbs X 




2400 7X21' 


20 wks X k^^x-? 
9 Ra9x-5-xk-£ 

11 Es736x^x^ 


8 Esl500x^x-3f 
10 12 men X x -k 
12 9 people X x 


CX)!kIPOUin) PROPORTION 


107 


XVI ] 


10 - ^Tir 3 

13 7menx;^X:j. 

15 65 ^ "rnr ^ ^ 

17 Es.o0xgx|| 

19 £4275x^4xS-xiA 

3x112x530 


21 (0 1 HP X 

(ni) 1 H P X 


33000 * 

440> 10x100x3 
33000 


7^ 20 

14 32menX|^x-^ 

16 24 da}sx|-lxf^ 

18 8.ksx^x|^ 

20 120 pagesx^-^x-j^ 

/ ^ TTT> 51x112x100x6 
(ii) 1 H P X — 


33000 


22 1500 men X 2-5 X J A 23 35 da\sx{-^x C- 

24. iSosx^xfl 25 40H.P x:-;-^x4^ 

26 S^hrsxjxf 27 30 mi. x x K-^- 

28 40 English navvies are paid le x 30 x 40, z e £420 for the vorli, 
50 Belgian vrorkmen , 5«x40x56, le £560 „ „ 

shewing the English naiwr is more profitable 
The cost required = £C000 x *-2-g- 


EXAMPLES ZVL g Page 321 
L-15 See Art 2S1 

16 Let sides be 5r, Ir and 8r jd® , 

then 5x-t-7j:— 12r=270, whence r=135 

17 Let the m.arks for the questions be 2 Om, 3 2m, 4 Sm, 

then 2-0m-i-3 2m-i-4 8’7J = 150, whence n«=15 

18 1^, 4, 2f are proportional to 3, 8, 5 , lime is -fe 

X 3 

To find percentages, we have 

19 Let ^ and ^ metres be lengths, then ^+^"^§=289 and r=30G 

20, Let 2^x, 3j? and 3|-:c feet be sides, 

then 2Jx-*-3r J-3j-T=33S , whence x=40 

H 


KHS I 



108 PROPORTIONAL DIVISION [CHAP 

T 

21 The shares of the piofits are of £165^, of £165^, 

and J-J- of £165^ 

22 See No 21 

23 Contiibutions are as 3 4 5, 

r 3 

whence percentages aie given by j^=22> 

G gets of Es 15,000 

24 They furnish respectively of Es 51,000 

Foi second part of question 

25 The creditors leceive ° lespectively of £126 

26 Shares are respectively of 480o? 

27 Amount req ^ of 300 x 123 274 grs 

=V-of 100x123 274 grs 

of 12327 4 grs =33900 35 grs 

28 Weight req'*=^— ^ of 3750 gm 

29 1 cu m = 1000 litres and weighs 1295 gms 

Hence weight of oxygen= of 1295 gms =298 3 gms 

30 Weight req^=-^^ of 207 4 Kg 

31 If shares aie £7v, £3r and £2a;, then 7a7+3a;+2a?=72 6 

32 Suppose C gets Es x, then B gets Es ^ and A gets Es - x — , 

3v 15? ” 8 2 

a+-^+-j^=20§, whence r=6 

33 Since 5 2=35 14, and 7 13=14 26, 

A’s share E’s share <7’s share =35 14 26 
If shares aie Es 35i, Es 14a? and Es 263?, 
then 35» +143+26r=12^ , whence v=^ 

34 Since 4s 5s 4rf =3 4=21 28, 

and 8s 9rf 7s 6rf =7 6=28 24, 
the shares may be taken as £21 a?, £28r and £243? , 
21r+28i+24r=146, 
and r=2 



XVI ] PROPORTIONAL DIYISION 109 

35 3 - 1 - 20 + 3 =26 7 + 56+ 2=65 

Tin=(:Ar+^)=tlm of total weight 
Coppei =(:-^+|-«)='-i^ 

Zinc=(^+-5^)=-^^ „ „ 

36 The shares are -S- " Jj*- -■ respectively of £384 

37 See No 36 It should he divided in proportion of 

50 x 6 85 x 4, or 15 17 

38 See No 36 Thev <5lioiild be di% ided in proportion of 

60x3 40x5 125x2, oi 18 20 25 

39 and C arc m the firm foi 12, 8, and 6 months respectively 
Rs 15,577 8 a should be dn ided in proportion of 

40 X 12 3x8 50 X 6, or 40 2 25 

40 A,B, and C are in partnership for 3, 3, and 2^ years lespectively 

Bs 9660 must be diindcd in proportion of 

8x3 10x3 6x2i, or 8 10 5 

41 Suppose Vi, Vj the \olumes, Sj and s, the specific graaaties , 

then by note at end of question, 

Wg Vg $2 

giving 6 oz and 9 oz. 

42 From note in No 41 , ivcights are as 3x7 2x4 5x2, 

1 c as 21 8 10, gii mg 42 lbs , 16 lbs and 20 lbs. 

43 The shares are asjx-j- -J-x^ 

7 c as ^ J • 3 ^^, or as 2 3 10 

44 The number of coins in each group is jointly proportional to the 

total \aluc of the group directly, and to the value of each 
coin (in the group) inversely because the gt eater the total 
value of the group, the more coins vre must have in it , but 
supposing the \alue of each com is increased from a ciown 
to a sovereign, there will be fewer coins in the gioup 
no of coins in the difierent groups vnll be as 
f Aj eras 7 8 16 

45 Tlie volumes are proportional to weights taken directly and to 

specific graiities taken inversely 

volumes are as ? c as 7 5 4 

Hence for second part of question we get 1 4 lit , 1 0 lit , 0 8 lit 



110 


PROPORTION AND VARIATION 


[CHAP 


EXAMPLES XVI li Page 327 
1 2 4 in x ^ 2 If a, in be distance req^ then ^=3 6 

26® 

3 40 sq cm X 

*7 7 ^ 63 ^ 

4 (i) 31 4 cm x^ , (ii) 314 sq cm x^, 5 80x^,sq yds 

6 132x|Ssq mi =132x14® sq mi =2587 2 sq mi 

2 5 - ^ 

7 (i) ]^ft , (ii) 145 x|^ [From Art 283, let 8=lfi 

Now when t=l, s=lQ 1 1=16 1, and we get s=16 1«® ] 


1 3® 

8 3 28 lbs Xp^ and 3 28 lbs Xp 

19 12 12® 

9 Breadtli=6cra x— Thickness =4 cm x-g- Wt =1 4Kg x-gg- 

2 5® 5® 162 6 

10 Weight of first =1 30 lbs x-g 3 -=l SOx^^ lbs =— gj- lbs 


Weight of second =1 3 lbs 


7®_446 9 
^^ 2 ®” 8 


lbs 


11 


Let N be no of levolutions lequired Since the no of revolutions 
in a given time vanes inversely as diameter of the wheel, 


N ^12 
130 20’ 


whence N=78 


[The second wheel need not be taken into account ] 


12 Volume req®= 1728 x-iAjZ^ cu in 

13 The first is paid 28s for 48 hrs and does 48 x 6, i e 240 units of work 

The second,, 25s „ 54 hrs „ 54x4,?c216 „ „ 

we have to compare the costs for 1 unit, viz and 
The latter is the smaller, giving ‘ the second man ’ for answer 

14 9d X ^ X = lOd and a small fraction 



XVI ] PROPORTION AND VARIATION 111 

15 Let B put in Us. ^ 

A lends at tlie rate of [Bs 23230 x 4+Rs 27000 x 8 ] foi 1 montb 
B lends at the rate of [Hs rx 7+Rs (r— 3000)>'5] for 1 month 
(SeeEx.X'^T g, No 36) 

7r+5(i-3000)=23250 x 4 +27000 x 8 , whence 7=27000 

16 If W he weight reqS then 

from which W= ^L^=0 01338 gm 

For second part of question, 

difference is 000183 m 001355, 

43 \ 

17 If h be height rcq"*, then p=g . whence /t= 2 j 

18 VTc have or Substituting the given values for 

lb o“ 4 

X in turn in this equation, we find the corresponding values 
of 3 ^ to be 0, 1, 1, 4, 9, ^ 

If the given values of r wexe negative, the resulting values of y 
would be positive (see Art 54), and would be 0, 1, 4, 

9, ^ The graph is shewn in Fig 15, on a reduced scale 



Fio 15 


6 7X 



112 MISCELLANEOUS EXAMPLES IV [PAGE 

19 If we substitute the given values for C in turn in the equation, 

the corresponding values of S are found to be 0 1", 0 4", 0 9', 
16", 2 5", 3 6', 4 9', 6 4" Taking a scale of 2 in along the 
axis of a to 1 in circumference, and 1 in along the axis 
of y to 1 ton, and plotting the points, we obtain a graph 
resembling the right-hand half of that in Ex 18 (Fig 15) 

20 On completing the table, we shall find it convenient to take a 

scale of 01" to a mile hoiizontally and 0 1" to 0 125 lb 
vertically After plotting the points, we shall obtain a 
graph resembling the right-hand half of that in Ex 18 
(Fig 16) 


MISCELLANEOUS EXAMPLES IV Page 331 


A 


1 (i) Multiply numeratoi and denominator of first fiaction by 45 

' - - 13 A 


It becomes 


133 


Operate on second fraction in the same way with 16, 


and it becomes 


1 8 


1C 


18 

133 


106-36* 

(ii) Treating first fraction as in (i), we get 

OA-IS 03S _17 16 

P 6 0 + 7 9 1 8 A 6 0 0 


40 mi per hr =40 x ^ ft per sec 
=68^ ft per sec 
33 knot8=^ y« - V^ ft per sec 
=55^ ft per sec 


(See p 60 of Key. Note on 
Ex XII b) 


4 f mi -40Q0=-£>L i -7^o^ 9y^xi3 gg 

5 Total length of a side = 35 x 12 in 

No of tiles that can be placed 1^=52^ , 

52 complete tiles must be used, leaving a 2 in space at each end 
As this holds good for all sides of the courtyard, we have a square 
piece of paving consisting of 52 x 52, or 2704 tiles 
Uncovered space = area of courtyard -area of tiling 
=t(35 X 12)8-(52 X 8)2]=(420«-416S)=(420 -1-416)(420-416) 
=836x4=3344 sq in 



MlisCKLLANLOUb LXAMPLLis IV 


113 


201 875 


7 Value = JRs 13 6875 x 20 4375 


1 1 S6875 


204 875 


01 462 

5 

12 202 

5 

1050 

0 

113 

4 

010 

2 


Bfl 2b0 422<C 


8 i X-vV+TWT^ V ^ ~ f 

9 1 cw U costs £ I’s X {4 + t ^ -^5 It js sold foi £^VV 112 , 

the profit on 1 c\vt. = £(-l^x 112- Ij x 14“lr^TV)=j£’lryj 
■a hence profit on 2 cwt 24 lbs =£3^'- H* 

10 £15 5=(25 20x 15 5) fi , oj 3D0 6 fr 

11 Amount req^=£}-|44n'==~^ II*' 1^ 

12 If X be no of d ij s i (hj*', then --/g- x r = 1 2] , 
whence x— 113y\, giMiig fm result 11 1 da>s 

C 

13 13 % of 3 inds 30 sn - AV of 150 Sis = SIS =19’ SI'S 

^+ffH=?7T + :-f= = y-^-=‘^288447 

15 Arca=-3j*;,‘’^4’o— =12 8 ac , » c 13 ac. (to nearest acie) 

Monej req'* =£17 U Gd x6l+£21 lOs Grf xG\ = £39x6\ 

” = £253 lOs 

20, 5Voha\e 2ac2i 7 12 po _ (2x4x 1 0+2x40 +7 12)po 
3-’ ,ic ~ S’- X 1 X 40 po 

_i0^_07O7 

“^60 " 

17. Ans =(16i '' lOlG-05) Ivg = 10510 8 Kg 

18 A nins (28 x 5) ^ ds hilc C runs ( 16 x 3) yds in 1 5 secs , 

tc A „ 140 ^ds „ B „ 138 jds, 

n A „ 700 }ds „ B „ 138x42-^, or 690 yds 


41 cu ft of tc ik weigli 2240 lbs , 

1 „ „ ■"il^lbs, >c 54 Iba 


91 



114 


MISCELLANEOUS EXAMPLES IV 


[page 


„191 6281 „191 

20 Valuer £63», x 0 628125 = ^ i000“ ^T" ^ 


5025 

‘8000 


= £l^=£39 19s 9|d 


21 

22 

23 


Let A, B. C be the tbiee numbers Using notation of Ait 97, 
we have A B C=}nX mX p)C=(nnp X) xX®= 29172 x 17® 

Let ^ annas and («+5) annas pei score be req’’ prices 
Then 25i+25(i;+5)=625 , whence t=10, 

5 cwt 3 qi 20 lbs =664 lbs , 2 cwt 16 lbs =240 lbs , 

value=£44rT^x 


24 If A pays B Es 90, E paj s C Es 90 and C pays A Es 90, then they 
are in ]ust the same position as beioie, foi each pa>s and 
receives Es 90 After this, howei er, A owes B Es 35, B owes 
C EeO, and A ones G Es 23 They can theiefoie settle 
accounts by A paying B Es 35 and C Es 23 


E 

25 Area=35-^ big , expondituie=Es 440 xSg'^+Es 3808^ 

=Es (1320+ 371i-+38089-)=Es 5500 

26 4 men and 9 boys do ^ of n oik m 1 day , 


8 

9) 

18 

R 

99 ^ 

99 

1 » 

(1) 

Again 3 

99 

6 

7 

99 1 ^ 

99 

1 » 


9 

99 

18 

99 T 

99 

1 » 

(2) 


From (1) and (2) wo see that 1 man does of work in 1 day , 

tel man does of noik in 1 day and would do the nliole 
in 12 days 

Again, since 3 men and 6 boys do in 1 day, 

it follows that 6 boys do A) ™ 1 ^ay, 
and 1 boy would do the whole in 18 days 
Otherwise Let a man take v days and a boy ^ days, 
tneii a man does - in I day and a boy 1 in 1 day , 



MISCELLANEOUS EXAMPLES IV 


115 


Multiplying (1) bj 2 , niiiltipljmg (2) by 3 and subtracting, 
■we get -=;^, and ^ =12 

T 12 

Substituting tlio \aluo of r in (1), wo find y=18 

27 E\p"=31 21325+3 20876 - 20 5375 - 13 632625 

=34 122 -34170125 

=0 251875 =0 251 J 

28 Let tlio cost of tho Army be i pei cent of the total 

X 91 710000 91 7 , , V - 

^ 100“ 183 392264 183 6^"®'^* 0499, 

r=49 9=60 (to ncaiest integei) 

Similarly for tlio Navj , 

r 29,520,000 295 , , . 

iOO=t8t59^=l8TO<”“"'5>”®'®' • 
t= 16 1 = 16 (to nojirest iiitogci) 

Also for tho CimI Seniccs, 

— 23,500,000 235 i,\_q joo 

100 “183,592,264 1836^ ’ 

r=12 8=13 (to nearest integer) 

29 Second man runs 5214 yds in 2133 secs , 

21 JJX 170 O in 1 hr, 
or 5 mi in 1 hr 

30 Let the thud class faic bo x shillings, 

then tho first-class faro is (r+H) Bhilhnga , 
32o(r+li)+l250z=300x 20 

DiMdo by 25, then 13(r+ii)+50i=12x20 , whence a;=3^ 


(i) Exp"= 


xl^+il 

^ •ya* OA 


1^1+2^’' 12 '34. 

_ go+n ■ 1 1 _a 

30 + J0“34 4 

^ ^ tV+iV-^ 48 1+0+8 

32 It was seen tw ice in tho first 13 secs , three times in the first 
26 secs , and so on , 

„ , , total niimbei of secs . - 

req'* no of times hi 



116 MISCELLANEOUS EXAMPLES IV [PAGL 


33 


35 

36 


37 

38 


39 


40 

41 

42 

43 


Let each man earn ^ annas, each boj y annas pei d.ij 
Then „ 6r „ „ 6^^ „ poi week, 

Ca. X 8+6?/ X 4=30x16, \ 
and 6ix.')+6yx3=19^ xl6 / 


These become on i eduction, 

12.r+6y=120,\ 
and 10i+6y=10J,j 


gixing 7=8, y=4 


From Ex 21, p 306, gross iental= of £7101 14^ 6ef 
1 sox oz =Aii^4jP gis , 

pure gold in 1 sox =;-j of 


See Arts 82 and 88 


G 


6897000 X 56 X 16 oz were produced poi 187000x4840 sq yds , 


oaoTxrnxia „ 

16TX4840 




„ 1 sq jd , 


7 c Y—.y OZ , 01 68 oz., \xcio produced per 1 sq jd 


Given price is £;^ per metres , 


i e 
te 
xe 


X 25 40 ft i)or - A" — moti cs , 
permctic, 
per metre. 


i c fr , 01 5 14 ft pci metro 


Exp" = £ ^ tW"” + 0875 X 0 58 - £0 9625 X 1 439 

=£(1 2658 +0 9786-1 3850) 

= £0859 = 17s 2d 


Total weight of 1 sox = ’-j of weight of pui c gold in it , 

1 sox weighs 113 X ] ^ gr , 

and Ya X ^ ^ 3 x 7000, or nearly 57 sox s w cigli 7000 grs , or 1 lb Av 
4060 yds remain , total no of men w anted during 2“'* year 
IS 1000 X 4-8-S-{j-, gix ing 363 extra men 


H 

Trke scales of an inch to 5 lbs boiizontalh and an inch to 5 Kg 
vertically As in Art 277, Example, ?/=-’,{ 7 Join the point 
whose coordinates aio (31, 11) to the origin The inquired 
values may bo read ofl fiom the gi aph 



117 


334 ] MISC15LLANK0US EXA^IPLES IV 

oi r1 

44. Totil dobts=£71j2 ITcncc he owes of £110 3s Zd to A, 

of £110 3s 3d fo D, and ^ of £410 3s 3d to C 
i 15j ’7 Vij 

45 Let r bo the no of cnncei ts, then oi=21 y 2^ , 

whence r=8 4, gn ing 9 as niswcr 

46 Snppf'se x persons wci e admitted w ithoiit psj inent 

Tlicn (17563 - r) paid Gd , tnd of these 12 15 pstd Is extra , 
(1556S-r)y 6+1215x12=101610 (receipts in pence), 
and x-=1123 

47 71 ac Ir 15 po =(71+0 25+009175 

71343 75 
2G25 

112 687 5 
12806 2 
1426 9 
356 7 

£187 277 3 giving £187 Gs 7rf 

48 ^?o of men ncccsFni^ to do the leqniied woik is 7x2x3, or 42, 

wo must icpi.icc 12 of tlicso b> bojs , 
hence ^ of 12, or 30 bo^s .iic leqimed 

I 

49 If A sq in be am req’’, then o\pi cssing the \ oliimo in cu in , 

8x36xA=3xl728, and A=18sq in 

50 Take a acalo of 0 4 " to 1 i iipcc hoii/onkallj , then each 0 1 ' repie- 

stnts 4.1, .and in cstimati coiitct to the nc.aiest la can 
easilj bo m.ide A hc.aIo of 0 5" to 1 ri \ ci tic Rhould bo 
taken If j/ms tost -» lupees, is in Ait 277, Ex,implc, since 
Rs 02 8.a =Rs ’ ^ and 1 iiid =40 sis , y=-x'WV‘'’ This may 
be Miittcn »/=’?», 01 ?/=2j» Take foi oiigin the point 
(16, 37) and join it to the point (32, 7J) bj a sti.iight line 
From this gtaph tlic icqiiiicd i.iliias may be icid of! 



118 MISCELLANEOUS EXAMPLES IV [PAGE 


51 3 men oi 7 women will take 64 days to do a piece of woik twice 

as great 

Now 7 men and 6 women take 7 times as long as 49 men and 
35 women , 

i c 7 men and 5 women take 7 times as long as 49 men and 
15 men , 

1 e 7 men and 5 women take 7 times as long as 64 men , 
ee 7 men and 5 women take 64 daysx or 21 days 

N B 35 IS the L C M of 5 and 7 • 

35 women can be replaced by an exact no of men 

52 15 oz cost llja , 16 o? cost }-^ of llja 

53 Let s ft be length of tiain, v ft per sec be its speed 

From Art 214, fi+132x3=i>x7J, (1) 

s=nx3 (2) 

Subtracting (2) from (1), 396=vx4i, and i;=88 ft per sec, 
giving 60 mi per houi 

54 (Draw a diagram ) 

Coat in annas=8x(25^xl9^-2W x 16H-)+ V x2^1rXl6H■ 

=8 X 25i X 19i^+ ( V— 8) X 21 } X 16H- 
=8xAlx4^-^Xi3lxi5A 
=ii^_<LjX=3664 
cost=Es 229 


56 


(i) Eap®= 


i V 4 V 

4+2t ^ 


J 


(ii) Exp”= 


S — 13 
TT TTS 


1 — 1 1 


TT”T!r 


' 3 1 

■ TTT^ 

■ m 


57 (Draw diagram ) Ai ea of path =(82 x 75 - 74 x 67) sq ft , 

which reduces to 132 sq yds 4 sq ft 

58 A does in 8 days what B does in 15 days and what C does in 

12 days , 

B’s time=A’s time x -1^=12 wks x-J/i- 

59 Take scales of 1" to 02 Kg hoiizontally, and 02" to 1 lb verti- 

cally As in Art 277, Example, ori/=-^v Take 

for origin the point (1 9, 27) Join it to tho point (3 8, 64) 
From this giaph the requited values may be read oil 



119 


336] mSCELLANEOUS EXAMPLES IV 


60 27 mds 5 srs =1085 sis 

Cost=209a X III X ^^^=4800 a =Es 300 


K 


61 22 cwt 3 qrs 21 lbs =22 9375 cwt 
£12 16s Si =£1283333 


22 937 
1 


5 

2 8333333 


229 375 
45 875 
18 350 
688 
68 
6 


£294 364 


1 

8 

9 

7 

1 

6 


NB It 18 easier to multiply by the 
money, on account of the 'S’s’ 
recurring 


giving £294 7s 3^ 


62 

63 


Exp"= 


1 +^—^ 


x6§-= 


co-Ko-s 1 . 

■flO+20-32 


1 4.1 — 8 

4,41 po 01,5 (21000004 
41 

4,2,0,o\ 00 01 5 


48 

7 


64 Using notation of Ex 2, page 330, we get 

3(x+y)=27 and 9(r-y)=27 
These give a:+y=9 and x-v=S 
Adding, we find x=6 , whence y=3 
Answer for hours 


65 1^ pints in 1 mm =3 gallons in 16 min -A x60 gals m 60 min 
=-^x60x;5-2^1it ml hour, 

Fssol ^ 


66 His time must be greater m the ratio 1760 1710 
(See Arts 269 and 270 ) 

time reqmred=r3^ X mm =35^ mm =3 mm 31| secs 



120 


MISCELLANEOUS EXAMPLES IV 


[page 


L 


67 Let a seer cost % annas, then -^x6+a=35, and r=3f 


68 


69 

70 


Required sum is £183 3s 10«? x 

=£183 692 x 0 730208 
= £134 2s 8e? 


183 602 

0730208 

128 684 

4 

5 610 

8 

36 

7 

1 

4 

£134 133 

3 


(1600 X 9) sq ft cost Ra 165 , 

(80 X 1760 X 3) sq ft cost Es 165 x ‘* YbVo°xir o* 4840 

1320 yds 01 I nil =18| in , \i hence 1 nn =25 in , 

1 sq mi =625 sq in , giving second ans\soi 


71 (i) Pioduct<:4x0 9, ^c 3 6 

(ii) Quotient > ** , le 44 

(ill) From Art 226, p 246, we see that fust figiiio in n/0 324 is 0 5 

72 Total evpendituio=Rs 2 13a 6p xl22=Es 346 15a 

Expenditure in Sept =Rs 3 5n x 30= Rs 99 6a 

„ Oct =Rs28a x31=E5 77 8a 

„ Nov =Ro 1 15n x 30= Rs 68 2a 

expendituie in Dec =Es 111 15a 
This will give Es 3 10 a as answei 


M 

73 (24 25 X 16 8) sq ch =(2 425 x 16 8) ac , 

ient=£5 775 x 2 425 x 16 8= £(14 00438 x 16 8)=£235 2735 

74 13 men and 15 hoys take 16 times as long to hoe i piece of 

ground as 13x16 men and 15x16 bojs take, oi as 13x16 
men a7id 15x9 men take, oi ns (208+135) men, ?c 343 
men take 


req« no of days = 7 x x x x 16 = 10 

Aliter If we woik with a fi action of a man, 
since 1 hoy=y^ of a man, 


we have 13 men and 15 hoys=(l 3 +i{-^A) men , ? e 21^ 
of days i equired = ^7 x y x ^ x = 10 


no 


men 



121 


&37] inSCELLANEOUS EXAl^IPLES IV 

75 Let A, B nnd C'get Es i, Es ar and Es 4t respectively , 

r^-3^+4r=78322l r=981j^ 

A gets Es 981 9 a , etc. 

76 SeeEx 2,p 227 Velocity of appioach=(60+40)Km ,2 c lOOKm 
Sum of lengths =175 tn , 

time req‘*=- ^ ^ oVi^o b o ® O’OOnS hrs , or 6 3 secs 

For second part of question, we get \elocity of approach 
(60-40) Km, , j c 20 Km , giMng time req** as 31 5 secs 

77 Let X ft be thickness of beam 

9r2 X 32=3); X 112, whence r =:15 ft 

78 Take scales of 1 in to 10 oranges horizontally, and 1 in to 20 a 

^el■tlcall 3 Since Es 3 2a =50 a , as in Ait 277, Evample, 

2/=^»r, or y=]-Si 

Join the point whoso cooi’dinates are (13, 10) to the origin 
From this graph we can read off the lequued values 


N 


79 1 cu ft of waters (30 48)* gms , 

62 426 lbs =(30 48)3 X 0 001 Kg , 


1 Kg = 


62426 

(3048)3 


lbs, xe 


62426 

28320 


lbs , xe 


2 2 lbs 


80 "Wt =1 6x5-^ x2jxl-r^ oz oz =30 75 oz 

32 8 10 SCO 

As a simple test, take block to be (5x3xl|^) cu in , it then 
weighs 30 oz 

81 (See Ex 2, p 330 ) Let t miles per hour be rate of stream and 

y miles the distance row ed 

Then y= 7(8 — r) Also y=5(8+r) , 

7(8-x)=6(8+3:), whence r=lj 



122 


MISCELLANEOUS EXAMPLES IV 


[page 


82 Let each man construct x yards pei day 
Then, if y yards be length of path, 

3«x2=|-1760, (1) 

and 18a =^-1760 (2) 

Adding (1) and (2), 24a=^, and 6a:=|| 

Substituting this value for 6a: in (1), 

■wo have — 1760, and y=1760x20, ic 20 miles 

iaO O 

83 Let I ft be length of train and v ft pei sec be velocity of train 

Since 2 miles an houi=^ ft per sec, and 4 miles an 
hour=|4 > velocity of approach of first person 

and train=»--||^ of second person and train=»--|4 

From Ex 2, p 227, we see time req'*= — ^ — length 

> r t 1 velocity of approach 

or length = velocity of approach x time req* , 

f=(®-|^)x9, also Z=(®— ■|^)xl0, 

10(®--||-)=9(®--|^), and »=AS^ft per sec, 

or 22 mi per hour 

84 The equation to OP is y=-lA^a?, 

ory=1414r Cf Art 275 

O 

85 (3+15+105) times, z e 123 times 93=105 — 15+3 

86 Let d ft be the required depth Then weight of water 

=1000 X 24 X 15 X d oz , and also=9000 x 10 x 16 oz 
Since these are equal, we find cf=4 ft 

87 Timereq«=23x7xf^x^days, ^e 138 days 

88 Let 1 seer costa annas At higher price 1 seei costs (t+ 1^) annas , 

15| chk cost ^^(®+lJ) annas , 

^(®+lJ)='i:, and fl7=38|a 



123 


337] JIISCELL4NE0DS EXASIPLES IT 

89 For second part of question, total hcigbt of the four men 

=(172± 005) X 4 in =(6 88± 02) in 
Total height of the tliree=(l 8+1 G5+1 38i 00'ix3)m 

=(5-03 ± 01')) in 

Difference =(1 85 ± *035) m , gi\ ing for limits 1 815 ni , and 1 885 m 

90 (i) Area=[(U4)2-(8G)2]x0 7854sq ft =43 97 sq ft 

=44 sq ft 

(ii) Weight=G 7 X [(7 3)- - (4 7)-] x 0 7854 gm = 1G4 18 gra 

= 1G4 gm 


P 


91 Let X runs be avenge requued 
Then 12 x24+3v= 15x30, and 3r=54 

92 (i) Co‘!t=Il3 j 11*1 x87+14a 6p x2x87 

=(Rs5 11a +Rel 13a)x87 = Rs7 Si x87=RsG')2 Sa 
(ii) 8250 at Es 7 14 a per 1000= 8] at Ea 7 14 a eich=etc 
(ill) (49y5-x 1) mi =5(G0 -i^)x ’ mi =25 nii — mi 
=25 mi —495 yds =24 nii 1205 3 ds 


93 Cost=[2x(5l+4)x3+5ix4]x8a =79x8a =Es 39 8a 

94 (3 5 y 7 4) sq im = 1 2535 tp 3'\^ sq nil 

12535,r^ 

1 sq in =— — gq mi 
3 1 X / 4 

Now < O'OOl . •Tr"inr-(3 5 x 7 4) < 0 0001 

the aquaie root of this is 001 
This shew s -szSto neglected 

1 sq in =~ ~ - ^^”^ - sq nii =483 9 sq mi , ? c 484 sq mi 
scale IS 1 in to 22 mi , since 484 =22- 


95 Let him row x mi per hour on still water 
Then 21=3 >(j:— 2), and a;=8 

time required=-^-hrs., ie 2hr8 
K H s T “ I 



MISCELLANEOUS EXAMPLES IV [PAGE 339 
"I I M I I I I I M M Tl I I I I I l~n I Ll-Lt^ 


124 



96 See Fig 16, which is di-awn on half the given scale The line is 
such that each ordmate is numerically eight times the coire- 
sponding abscissa ' '^Also the distance tra\elled at 8 mi per 
hr in any number of horns is 8 times that number of hours 
Hence the oidinate of any point on the line gives the distance 
travelled in the time indicated by the abscissa of that point 

The graph of A’s motion is the line OPH 

Since B starts at 2 p m at 12 mi per hr and has theiefore ^one 
12 mis at 3 p m , his graph will be the line obtained by join- 
ing the points (2, 0) and (3, 12) Similarly (P& is obtained 
by joining (3, 0) and (4, 16) 

The time and place of JTs overtaking A are given by the cooids 
of the point common to their graphs, viv W, and aie 6 pm 
and 48 nils fiom O 

To find at what times A and B are 8 mi apait, we must discover 


PERCENTAGES 


125 


CHAP wnr] 


for w bat abscissT* tbe vertical distance between the two graphs 
icpiescnts 8 iiii Tins will be the Cissc foi tbe abscissa; 
maiking 4 pm and 8pm (Tlic latter is not shewn in tbe 
diagram ) 

The rest of tbe solution will be obMous from tbe hguic 


EXAMPLES XVn a Page 341 
land 2 See Art 287, Ex 1 

^ oi. 

3 Let T be rate per cent , then -—=^ 4-6 As No 3 

100 25 

7-12 See Alt 287, Ex 2 13-16 As No 3 

17 Let r be the rate per cent , then t=14 0 

18-21 As No 17 22 Eeq-* value =7^ of £573 =£22 920 

23-24 As No 22 25 Ecq-* \alue=^ of £58*=£2 499 

26 Ecq^\alue=T-jTrof £150675=£6 0^ Crf 

27 Eeq** ^^0** 4=£4G 253 

28 (i) Eeq\;iluo=x5u of £«302 729=£9 08187 

(ii) Ecq’' >albesf of ^1572 11s lOrf of £1672 11s lOrf 

=-jVV,of ^786 6s IW =T^ of £8649 5s W 
=£8649254 

29 Eeq** commission = of Es 3750^. ** 

30 Error = ^ of length = -j-J-j- of length 

« 

Ifa?bereq'*peicentage, ■j^=TJT’ a =0 694 

31 Loss=n41 in 7 j 00 or m 100, giMiig AJ-0-, oi 15 2 % 

32 Increase is 268 on 4680, or od 0 7 % 

EXAMPLES XVn h Page 344 

r 

JVjS In some of these examples ? is used for tbe quantity 
to be found 

1 If Es r be Ins income, of Es r is left , -^r^^Es 1860 

luu 100 


s 



126 


PERCENTAGES 


[chap 


2 Req^ population of 

3 PiomAit 289, x is the present population , 

a; 62,130 

4. From Art 289, first rem'^=Y^ of 145600 , 

of ^ of 145600 

5 »=-3^of7a 6p 6 From Art 289. -5^ of r= 140 (pence) 
7 First remainder=-f of x , from Art 289, of ^=95880 


8 of *=532 


9 As m 8, ^=74000 


10 Frojn Art 289, a year ago the fund stood at of Ks a? , 

it -will now stand at of of Bs a; , 
of of Hs r=Es 167584 

11 Let a represent 182 460, then a - , 

100 

, 5ja_94io 94 6 x 182 460 

”100°“ 100 100 

12 Increase =Es 177280 on Es 4780320 , i e J 'y »U> x 100 on 100 

gmng3 7l% ‘ “ ’ 

13 Applying the piinciple of Art 289, we have 

giving 24% 

14 ^ ~ B790000 

lOQ lifikoeooe 

* 

15 Yield of olives per ac =-4-g- j ^- g - g - ° - lbs =35 lbs , 

currants „ =-^ Vi r AVo°” =2083 lbs , 
figs „ 11,3 ^1154 

If percentage of land occupied by olives is *, 

Similarly percentage of land occupied by currants 

= lAAOft 0X100 o 

6 bo 3 100 " j 


11 

» 



PROFIT AND LOSS 


137 


XVII ] 

16 Total imports =£523075163 
As in No 15 vv 0 obtain 

percentage for Biitisb possession3=3-A«^_^gA^-^J-Aa.=rl.g.2^==2l 
„ US 

„ France 

„ other countries =^^J^iJ}-fJAA=£-=J-^A=42. 

17 As in No 15 e obtain 

percentage for 18G8=— “-f* J ‘■|fl-AA= r^^=67 , 

,1 fifiQ = *-f ni oBXi 00 — s- 'isx _ gg 

It *laUo 5 

ic*rn__ K '■a4«oxioo_fi4ri'’ _fir. 

„ lo/u imrrzro 

•c. 1 , j . total expenses x 100 

For whole period percentage = r-r 

^ ° total earnings 

— lA=JLiAi_«JLiA<> = iffgo-t _ gg 
— -TTl^ ^4 oo 

18 As m No 15 we obtain 

percentage incicasc for England and "Wales 

„ „ Scotland ==i^V‘t^=¥(r7-=U 

19 As in No 15 wo obtain 

percentage for England and Wahs=-^Y^ygy^—=:A^^=13 , 

.Q»nflnn>l _fl0POOOXlOO_0«»O_lK 

„ ocouana TTTirirDa! rrs — 

Tmlnrirl _4P_OOOJ<100_«PSO_T1 

„ ireianu iTTTtntV TTI — U 


EXAMPLES XVn c Page 347 
1-5 See Art 291, E\ 1 6-9 See Ait 291, Ex 2 

10 Profit=(Ee 1 2a 9p -15a)=3a 9p on 15a , 

req'^ poiccntige=-^^^x 100=25 

11 Loss = 18s on (£3 2s +18s), or 18* on £4 , 

18? 

rcq'> peicentage=^ x 100=22^ 

12 CP-4%ofCP=SP, or-i'';^ofCP=SP, 

*5 P ~iTnr 1^® 75=Bs 72 

13 SP=CP+14%ofCP, or & P =1-^ of CP = }-^ of 100a 



128 


PROFIT AND LOSS 


[chap 


14 CP-12i%ofCP=SP, ^ofCP=Es2#, 

CP=Es2fXgp=Es3 

15 (i) Profit=£3 7s \d on £40, 

req^ percentage = x 100=-^^^ =8 4 % gain 
(ii) Loss=£3 10s ftd on £25 15s , 

req** percentage = ^|25° 15^ ^ ^00=^-|^ = 13 7 % loss 
(ill) Profit=£9 13s Oifl? on £77 4s Qd , 

req-* percentage x 100= = 12 5 % gam 

16 5|rf per lb =664rf per cwt , £2 7s llrf =675c? 

gain per cent x 100=12 

17 CP per 100=-f4x 100a , 

SP per 100= of J4xl00a =Es 8 12 a 

4s lOlrf 

18 CP for 130 lbs =32s erf , gain pei tent = 327 "e^^ 100=15 

19 110 knives cost Es 100, and are sold for Bs 121 , 

gain per cent =21 

20 For 25 eggs, CP =I5a , and SP =18a , 

gain per cent =-j^ of 100=20 
92'?' 

21 If it cost X annas, then ^^=92 a, and r=100a 

120x 

22 If it cost a; rupees, then ’•=E8 12i 

23 If £« be C P , then and a:=£16 

24 If Be a; be C P , then ^g=65j, and t=Es 72^ 

25 150 yds at 8 a 9p per yd are worth Es 82 Oa 6p=Bs 8203125 
If Es a; be C P , ^^=82 03125, and a;=Es 65 625 

26 9rf per doz =15rf per score if x pence be C P , 

-j^=15 and r=12^rf 

27 Let X pence be C P per lb , then and x=2d , 

cost per ton =2rf x2240=£18 13s 4rf 


CP=Es2fx^=Es3 



Pl^OFIT AND LOf>5 


129 


WII ] 


28 Lot£» be CP per gro'ss, then ^ = £24 

His S P must be llg of £2 1, oi £28 16? 

29 Let i sbillings be C P per half cw t , then = 105, and t = 1 1 2? , 

C P pel lb =2* 


EXAMPLES XVII d Page 350 
1 Ecq'> S P = Vi" of B<; 301 2 Bcq<* b P of 19s 

3 Beq-* S P = of Be 1 9a 9 p 
A Bcqo S P of 2irf x 112=£1 5? 

5 SincoEs81=iVffofCP. Bs 90= of C P = of C P . 

hence loss is 20 % 

6 105a=^^ofCP, 93 a =-|^-^ofCP,giMng7% loss 

7 43a=l^-ofCP, 38a=^x||of CP^^VofCP.and 

loss is 5 % 

90i 

8 Let z annas bo rcq** pnco, thui 100=99, and r=ll0a =Bs G 14a 

9 If he bought at -> annas pci store, he pud lOi. annas 

He icccivcd '^xl80a J-g-g-x 107. = 180x-|f , iv hence T=30a 

10 2|rf per lb =308</ per cut , 308rf =x(pir of CP , 

hence £1 11? Gcf, oi 378rf =^^x of CP=igg of CP, 
gi\ing8% proht 

11 She sells 108 oranges foi 90a, and since hoi loss is 10%, then 

CP IS 100a If she sold thoiii at 10 foi 12a, she would 
icccnc 129 Ga , giving a piolit of 29 G % 

[iVZ? The number of oianqes sold docs not aOcct the gfiin oi loss 
pel cent 108 is a convenient number betause the fust S P 
lb then 90 a , uliich at once giies 100 a as the CP] 

12 24a of CP , 21a =]-5^x-=] of CP, oi of CP, 

giving 12% piofit 

13 Let £i be cost puce, then ^^=49 8, and 'i:=£415 



130 


PROFIT AND LOSS 


[chap 


20 V 

14 Let Es a, be coat price, then 5, and a.=EB 37 5 

(See note to no 13 ) 

30^ 

15 Let £x be cost of carnage, then jqq= 10> a'Od a;=£33j 
(See note to no 13 ) 

20 1* 

16 Let X rupees be piinie cost, then and a;=E& 46| 

(See note to no 13 ) 

22a 

17 Let V pence be C P , then 2^=44, and r=200cZ , 

of C P =176o? =14s 8rf (See note to no 13 ) 

22 

18 Let % shillings be CP per doz, then and x=i0s , 

S P ieq'*s= of 40s =48s (See note to no 13 ) 

19 See Art 293 It costs G Ea 600 x x oi Es 693 

20 See Alt 293 I must pay £3 2 X — x oi £4 10s 


21 

23 

24 

25 

26 


27 


See Ai t 293 He pays Es 2000 x x or Es 2100 
Let X shillings be A's cost, then x x fj^x i^=121, or t= 100 
He got 40<f X ]-•§-§• X I X or 57c? 

C P to dealer=^^of Es 160=Es 200, profit=Es 50, i e 25 % 

Shopkeepei pays of first price, and sells at of fiist puce , 
his gain IS xiAr of C P on outlay of fjrv of C P , or 10 % 

Ahte) If article first cost £100, dealer paid £110 and sold at £121 
profit IS £11 on £110, oi 10 % 


Let X be piofit pei cent Thud S P of first S P 

A1.0 tM S P o£ ig „( S P , 


120 ^ 100+a;_157i 
100 100 ^ 100 “ 100 ’’® 
whence r=5 


inn,. 157^x100x100x100 
100x120x125 ' 


28 Let5niaker%, then £6x^x^-^tf x^=£9 9. 


ino_. 0 oxiooxiooxi no i 
LUU+X- 6XlS6Xlfo > ^=20 



XVII ] mSCELLANEOUS EXAMPLES ON PERCENTAGES 131 
29 


(i) Req"* price=7jrf =8s 3d 

120 . 110 .112; 

8 


(n) Req'> prico=25^ 


Torj iin n^l 

(ill) If £ar bo cost rcq^ , thon £xx^Xy^x^^"=£14 85 , 


100 100. 100 

*~1‘^®'^ ’^120^110 ^1121’ 01 «10 


(iv) If r be profit pei cent , then of C P =^7:^ x x 


120 110 mj 
100^ 100 100 ’ 


and 100+jr=148^, or x==48\ 

100 100x100x100’ 


EXAMPLES XVn e Page 353 

1 Lot X bo rcq'i population , adding tho bn tbs and subtracting 

11^ 

tbo deaths, =85600, and a^=80000 

1 liT 41 

bii tbs 8800, dcatbs=^»3200 

2 Total CP =(8 x2j+20x 20)« =600s Total SP =28x215 = 672s 

gjiin IS 72s on 600s outlay, or 125 on lOOs , giving 12 % 

3 Suppose 4 lbs of cbicoi> arc nii\ed vntb 17 lbs of cofiee 
TotalCP =(4x3 + 17x24)a =420a Total SP =21 x25=525a 

gain 18 105 a on 420 a , oi — a on 100 a, gn mg 25 % 

4 Capital icq"=Es 10000 X ’-5-^ x x J-^x l-5^=Es 20736 

5 7 eggs are sold foi 4 x l-J-g-a , 7 e 5 aie sold foi 4 a 

6 Let Rs X be the whole aint req’’ Then Rs is di\ ided in the 

propoition of Rs 12500 Rs 8500 ^ 

Now Rs 12500 +Es 8500=Es 21000, 

12500 , 40i 8500 , 40^ , „ 

M000°^lW=^°^100+^°°’ ^=Re393/ 8a 

■IJL o 

7 Let £a; be rent , then ^ - 25 of 1150=77^ of 1150 , 

lUU 100 

XI hence T=£n4j- 

8 Lot x% of the n hole pass Now 95 % of boj s and 60 % of gii Is pass , 

jf* 05 no 

J^of 2300=^ of 2000+^ of 500, and t=88 



132 mSOELLANEOUS EXAMPLES ON PERCENTAGES I'CHAP 


9 Suppose A pays £100 , lie leceives £105, 

and D pays £105 x x J-g^, 
te D pays £157i , A would gain 57^ % 

10 Suppose each article costs lOOrf Then C P is 300rf and S P is 400c? 
Profit=100d on 300c?, or 33 Jo? on 100c?, or 33 J % 


11 He actually sells at £21 — £1 6s 3c? , oi £19 13s 9c? 

Now £21=-^-^ of CP , 

£19 13s 9c? , or £l9iJ=|iBx ^ of C P of C P , 
profit=55 % 


12 Let a? % be gain req** j of Es 36000 =Es 12000 , 

'3 of Es 36000=Es 14400 , reni>‘=Es 9600 

ra 9600.?^ of 36000 

Cancelling we have 9600+18000+96(100+ r)=39609, and a;=25 

120ar 

13 Let £t be cost of goods, then i^s;25, and Ts=20f 

gross profit £(25-20f)=£4J Now 10 % of £25=£2j , 
net profit=£(4J-2j)=£l^ 

14 Let a; lbs of first kind be mixed uith y lbs of second kind 

Expressing puces in annas we have 

105 o: S 

25(r+y)=j^(14r+24y), 3a.=2y, and -=g 

15 Let V gals of water be added to y gals of milk Expiessing 

puces in pence, since 2c? per pint=16c? per quart, we have 

140 r n 

16(iJ+y)=^ of 13y , 80r=lly, and 


16 Value of total wool produced in N Z ="§-§-5-5- of £8593000 
Value of wool expoited fiom N Z =£4041000 
Since amount is proportional to value, if v be req^ percentage, 
X _ 4041000 

10*^ of 8593000 


,_' 1041Xg 0f60 .-.pn 4 041XS0fB0 

8603X180E6''-‘"''~8 603X1 80B6 


XlOO 


III X 100 (coutiacting)=0 661 x 100=66 %. 


and 



SVll] MISCELLANEOUS EXAMPLES ON PERCENTAGES 133 


17 Let £x be toUl ^alue of Canada’s fai m products, 

90ar 

then :j^=131 millions and 7:=14 6 millions 
Total imports to Bntain a= ° x 13 1 millions 

=-|^xl3 1 millions=71 7 millions 

18 1=^ of CP =£3, CP=£15 and SP=-i^of £15=£18 

If £3; be advertised price, then -^=18, and t=£24 


19 Let V seers of water be added to 1 seer of milk , 
(T+l)x3a =-J-g^xlx2^a, and 1 =^ 

20, Let T rupees be cost price. 


lOoX . 110 , 95» , -n ann 

wo 100’ 


2L Ps 20 coriesponds to an increase of 4 %, 

of CP =Rs 20, and C P =Rs 500 
22 Let T sbillings be price per quarter at whicb be bought, 

then of 2803!+^ of 320r+i^^ of lOOr- 7003.=total gam 

This reduces to Go^x and =£100 19s 7rf expiessed m shillings 
63jr=2019/;, and t= 30§, le £1 10s lOif 


23 Let V pence be C P per cwt , then of 4 x 112 

a;=266<f =£1 2s 2d 


24. Let X lbs of first be mixed m ith y lbs of second 
Then 94(a;+y)=^^‘{633;+78^}, le T=y 

25 Let Es x be the puce requiied , then 8x+22x^=^^x3150, 

and t=Rs 180 ^ 

26 Water in first mixture=(2 x 25+3 x 15) parts out of 500, ? c 19 % 
■water in second mivture=(^\^ of 5+|-) parts out of 5^,te 23% , 

spirit in second mixture =(100— 23) %,te 77 % 

27 1 Kg or T lbs cost 3x<f , 1 lb cost Since it sold for 2d 

r profit IS ^d on outlay of ^^-d, giving 28 6*% as profit he 

thought he made 

Again 1 Kg or 1000 gm cost 3 5d , 1 gm cost 0 0035tf , and 

453 6 gms cost (453 6 x 0 0035)c? , i e 1 5876c? , 
gam IS (2-1 5876)c? on outlay of 1 5876c?, grvmg %> 

ze 26% 





134 SIMPLE INTEREST [CHAP 

28 Cost of materials and manufecture=iT5Tr of £3750 

Remaining expense3=£(387 +940+136 +200+-j-5-5- of 3750) 
=£17005 

profit = £(3750- ^ of 3750- 1700 5) 

=£(j^ of 3750 -1700 5) = £737 
In second part of question, gioss sales = of £3750, oi £5250 , 
cost of matciials and manufacture =-j^'’ji- of £6250 
Remaining expenses=£(387+1880+136+ 200+-j-J-{5- of 5250) 
=£2655 5, 

profit=£(5250— of 5250-2655 5) 

=£(t^ of 5250 - 2655 5)= £757 , 
increase in annual profit=£(757-737)=£20 


EXAMPLES XVIII a Page 358 
1-15 See Alt 298, £s 1 and 2 


16 


19 


£3712 

^ 

74 24 
18 56 

92 80 
3 | 

27840 
34 80 

31320 
=£313 4s 


£ 

£ 

£ 

1 457 

17 51 27 

18 6 41 

s•^ 

3 

n 

13 71 

15381 

1923 

% 


10025 

5484 1 

46143 

208325 

6 855 

76 905 

2i 

61 695 

538 335 

41 6650 

=£61 13s llrf 

=£538 6s 8c? 

2 6040 


20 Intel est= 


44 2690 
=£44 5s 6rf 
£701 11s Irfx6ix44 


£761 


100 

11s IrfxS 


of 2784) 


21 


10 

=£228 9s Ad 

Interest=£4 08121 x 1 x 4 

=£8162=£8" 3s 2d. 



XVIII 


SIMPLE INTEREST 


136 


22 


2i 

25 

26 

27 

28 

29 

1 

2 

3 

4 

5 

6 

7 


Pun 

Amt 


. =£1 567 

25 

62 69 
14105 


8 76 795 

3 

9 599 

4 

156 725 


£166 324 



4 V 7 


23 


^ £244tLx3Jx3 
100 

=£ Y:f^=£27 459 
=£27 9s 2d 
Amt =£271 10s 10c? 


I=Esl8 4290625x22x2|=Rs 18 4290625 x6=Es 110 9a 3p 
Amt =Es 1953 7a 9p 

I=Rs 11 47171875 x5jx2i=Ra 1147171875x12 
=Rs 137e60625=Rs 137 10a 6p Amt =Rs 1284 13 a 3p 


I=£]2 4x^x3l=£-V=£l0 Cs 8d 
I=Rs 5 484 xix4i=Rs 5 484x9 =Rs 12 34 
I=Rs 4375x2 x4=Es ll 3 =Rs 11 10a 8p 
Amt =Rs 449 2a 8p 

I =Rs 62 80 X 2J^ X A = Rs 70 65 Amt = 6350 Rs 65 cents 


EXAMPLES XVIII b Page 361 
146 days =5 year , 

I=Rs 7 50x gx3|=Es 11 25, and Amt =E8 761 4 a 
292 days = I year , 

I=£l8 35x^xli=£l8 35, and Amt =£1853 7s 
219 days= i| year , 

I=Iis 16 875x^x6=Rs 60 75, and Amt =Rs 1748 4a 

73 days =-^ year, I=£2 52J-x^ x3=£l 6125=£1 10s 3rf, 
and Amt =£253 11s lit? 

I""~' T65x 2°^ ~ 8 , Amt = £1702 16s 

4s 3c? . 

Amt =£869 8s 5c? 

Time =146 days=| yeai , 

6 118, and Amt =Et 431 10a 



136 


SIMPLE INTEREST 


[chap 


8 Time =196 days , 

Es 219x196x10 
365 x3 


=Ss 3 92, and Amt %Bs 222 14 a 9 p 

j Es 3 7375x3x35 Es 394 5375 
5x8 “ 40 


9 Time=219days=2yeai , 1= 40 

=Es 98634376, and Ajnt=Es385 9a 9p 

10 Time=146 days=g year , 

I=iI®OiJi|ii55=£10 736, and Amt =£791 10s 9rf 

11 I=Es316x^x5=?l^^^=Esl428 =Ee 1 6a 9p 

12 I=Es5 47x^Sjj^xL=Es 5 47xf^=Es 2 0231 =Es 2 Oa 3p 

13 I=Es70x^x4=5iiiill^=Es65 9726 =Es65 15a6p 

14 I=Es 708xmx|=52-^^=Es76716 =Es 7 lOa 9p 


15 I = Es 965 90625 x ~ x =Es 9 6590625 x 


=f=Es 103736=Es 10 6 b 


16 Time=65days, I=£41J 
and Amt =£415 18s lOd 


I=£413x^x4=^?^=£2 9419 , 


17 Time=80days, I=£5 10x^x3=^^=£3 3534 , 

and Amt =£513 7s l<f 

18 Time=101days, I=£2450x^Xg=^^^i^=£23 7281 , 

and Amt =£2473 14s 7d 

19 Time=212 days , I=£]36875x|^x^=’ ^^ ^ ^ 

^ £2756^6625 ^^37 7^^^ ^ =£1406 10s 3rf 

20 Time= 108 days , 

I=£402 5375xl|x^=£4 025375xJ|x| 

=£4 025375 x^=—- ^|P^ - =£4 9131 , 

and Amt =£407 9s 



2C7IU] 
21 


SIMPLE INTEREST 


137 


Time=146 days=-|- year , I=Rs 12 50x-|x-^=Rs 1875, 
and Amt =Rs 1268 12 a 

22 From July 5 to Nov 18 is 136 days , 

on Nov 18 he receives £204 13s 2d 
From Nov 18 to Feb 20 is 94 days , 

on Feb 20 he receives £301 18s 8d 

23 From May 1 to Dec 31 is 244 days, Aug 1 to Dec 31, 152 days , 


I=Rs 3 50 X ^ X J^+Rs 2 70 X ^ X 




Rs 170 8 . Rs 82 08 
73 

Rs 948 3 


82 08\ 15 
73 j^4“ 


FT 

Rs 252 88 
73 


1 5 


15 


73 


=Rs 12 9904 


I owe Rs 632 15 a 9p 


EXAMPLES XVin c Page 363 
[ih. theie examples the sj/nibols P, I, r, n are used as rd Art 298 ] 
1-8 See Art 302, Ex 1 9-12 See Art 302, Ex 2 

13 P+P2i^=5430i, or P=^>«^S=£5400 


1609 


IF Qivoi—o'jK o_ given Amt 527 4375 

14 3^x24-8 75, P _-^___=__=£485 


15 t5<6=24. 


given Amt 204 8 » 

^“Amt of £1-1024“*^^“ 


16 P+Px^=254*V. 

17 Int on £P=(Int on £l)x P, 
24 9 


P=254t^x^=£236J 
152 

1125~^ la, 9 


18 As in 17, P=^-^^=£237 (to nearest £) 



138 


SIMPLE INTEREST 


[chap 


19 As in 17, P=4^ =?®lii^^=£1076 (to nearest £) 

T^X-I IbXO 
100 

20 p+pxi^=327ll7, P=327117x— £310.16s 


200 


P=2000x^=£1778 (to nearest £) 

^2sO 


100 

21 P+Px 5^=2000, 

22-26 See Art 303, Ex 1 

27-30 Find interest by subtiacting Principal fiom Amount , tben 
see Art 303, Ex 1 

31 If n be req* fraction of a yeai, 

sjvmlnj ^i_.028yi=10saay» 

Int for 1 year 250x-j-gTj- 

4740 


32 As in 31, 


23 7 


1572 85x11 

1572 85 Xj^ 

4740 x 73 346 02 , 

33 Aam31,»=^^y.a-gx366days 
= ^\^5^28 days 


x365 days 


34r37 See Art 304 38-40 Find inteiest and see Art 304 

.=4i 


12A 


41 ,=__JiZ55^±_= 

Int at 1 per ce^^ 

42 As in No 41, } = - 

437ixfxx^ 

43 If Principal is £100, then Interest is £200, and, as in 41, 

200 


Int at 1 per cent 650 x ^ x 
11? 

-=4 




100 X 25 X - j^ Q ^ 


=8 


44 As in 41, r= 


50 


564x2iXy^ 141 


=|2?=356 



XVIII ] 


DISCOUNT 


139 


45 As in 41, r= 


100 


3550 X ^ X 


■=S-67e 


71 


46 P+PxM^=2519 9 


P=|?? of 2519 9=£2057 
245 


47 (i) Annual interest is x 52, t e £6 5 

Px-|^=6 5, or P=650xYt-=£236 7s 


(ii) Annual interest =£^^ 


mr 

pyMiLl_3^ P_3^_£55 6, 


48 As in Alt 303, Ex 1, «= 


305x-j-^ 
100 


yra 


= 30 5^ - 5 X 365 days =240 days 


12 904 

49 As m Art 303, Ex 1, »=rTrr ^ yrs =0 43 }rs =157 duys 

1000 X -j-jj-jy 

This gives Jan 9th 

60. Erom Jan 1 to Aug 8=219 days=-f yr Fiom Art 303, 

36 


P= 


fi ^TxnF 


= £1500 


EXAMPLES XVni d Page 368 

1 Disc =R 3 2020x-J-x4=Rs 20 3 a 3p 

2 Disc =Rs 63 31 x -j?- x =Rs 92 5 a 3 p 

3 Disc =R3 694203126xi-ixA=Rs 08677x33=Rs 28 634 

=Rs 28 10a 3p 

4 Disc =Rs 5 31375 x-l-x 3=Rs 6 6a (146 days=^ j'ear ) 

5 Disc =Rs 3 67921875 x x i=Rs 6 518 =Rs 5 8 a 3 p 

(219 d.ivs=-^ yr ) 

6 Disc =Rs 8 90 x^x7=Rs 51 9166 =Es5114a9p 

K II S I K 



140 


DISCOUNT 


[CHAP 


7 

8 

9 

10 


Disc =Rs 7 2990G25 x = x ]^=Es 12 166 =Rs 12 2 a 9 p 
Disc =Rs 5 312G5625 x x 3=Rs 14 G09 =Rs 14 9 a 9p 

Disc==£7G2xiPx4=^-|^=£l4 697=£l4 13s llrf 

ODU /O 

87 „ £0 4107 x 261 £1071927 

Disc x£2 0535x^x3= ^3 ^3 

=£1468 =£1 9s 4c? 


11 10%=-j-V Neglecting 3s 8d for percenUge pui poses, we have 

sum ieq'’=£25 13s 8c? --^^of £26 10s =£23 2s 8c? 

12 As in 11, sum req'*=£l4 3s 2d of £14=£13 9s 2c? 

13 As in 11, sura req^=£41 11s of £41 10s =£39 9s 6c? 

14 As in 11, sum req^ = £38 4s 8c? --jljj- of £38 =£37 5s 8c? 

15 5%=-3r If Rs r be amt req*’, —^=26 359 and a,=Rs 27 75 

16 10% = i\^ If £« be amt req^, — = 16 65 and a:=£18 5 


EXAMPLES XVIII e Page 370 

1 Time=73 days=^ year, Disc =Rs 603 x-^-x-|-=Rb 3 Oa 3p 

2 Time=219 days=^ year, Disc =Es 9 5275x4x5 


3 Time =99 days, 

^BsJOl 752 
73 

4 Time =83 days, 

Rs 242 028 


=Rs 28 9 a 3p 

D«C.E3381xSx4.Sl»^£iS5 
obo 73 


=Rs 41335 =Il8 4 2a 3p 

Disc =Rs486x — x3=^^-?5|2iM? 

ooD to 


73 


=Rs 3 315=Rs 3 5a 


5 Time=61 days , Disc =Es 27 6325 x x s= 


61 5 Rs 1685 5826 


365 ‘ 


=Rs ll545=Rs 11 8a 9 p 


146 



XA'iir 3 


BILLS OF EXCHANGE 


141 


« TA T, o,- l'»4 !> Rs244 09 

6 Tunc = 1 Til du** , Disc =1?*? 3 1 < x-j^^X 2 = s-j 

=Rs 3 i-137=Rs 3 3a Gp , Gisli ieq'*=Es 317-Iis 3 5n Gp 

7 Time=202da33=^ joai , Disc. = £10 312x j X'5-=£34 8166 

=•■£34 1G< 4rf , Cash rcq'*= £1934 4« -£34 16s 4rf 


8 Timc=83da%s, 

£4270 9891 
IIG 


^ ...83. .7 £7 3311x681 

Disc — ..i36 <f>i)J X X 2 “ 


=£29253 =£29 6* Irf , 


Gusli ieq‘>= £3676 11s- £29 3s Id 


9 Tmie-=121 da>s , 

£G31 7917 


121 7 £0 745917x847 

Disc =£, 4591 r X 3 ^ X g= ^ 


=£8 655=£8 13s Irf , 


73 

Casli ieq'*=£745 1 8s 4rf-£8 13s Itf 

10 Tiinc=- 11 dava , Disc = £10 31642 x x 6 
£1 031 G42 X 492 £607 5089 


11 Fioin Art 298, 1 = 


- =£C%3=£G 19s Iff , 

hi hi 

Cash icq'>= £1031 12s 10ff-£G 19s Iff 
Pxrxn 


100 


Mhcic I IS in tins case the discount , 


lOOxI 100x303 „ 

^ Py« 605xi 


1 on V o or. 

12 As in No 11, r=ig-^i^'=2i % 


,n , , -KT ,, lOOxI lOOxGiV 1 

13 rrom formula in No 11, r=-„ =-??? — 3'> =9 months 

’ Pxr 175x5 * ^ 


14. As in No 13, months 

15 Off a hill of £100 he t ikcs £4 , he elm ges £4 foi the loan of 

£9G for 10 months , tc £4 x ]-5 foi the loan of £9G foi 1 j eai , 
hence rate icq** =5 % 

16 As in 15, lie elm ges £2^ foi the loan of £97^ foi 3 months, 

t e £10 foi the loan of £97^ foi 12 months , 

hence rate icq'’=1026% 



142 


COMPOUND INTEREST 


[CHAP 


EXAMPLES XVIII f Page 372 


B In this set of examples, 

A„ denotes Amount in n years , 

P„ „ the Principal at the beginning of the ti® yem , 

In » the year’s interest 


Bs 

1 Pi 200 
Ii _10 

Pa 210 
L 10 5 

As 220 5 
Bs 

4 Pi 3500 
ii _m 
Ps 3675 
lo 183 75 

Pa 3858 75 
la 192 9376 

Aa 4051 6875 


£ 

7 Pi 225 
Ii 9 


Pa 234 
la 9 36 


10 


P 3 

243 36 

Pi 

225 

Cl 

18 36 


£ 

p, 

3546 

II 

1773 

Ps 

3723 3 

Is 

186 165 


P 3 

3909 465 


I 3 

195 473 

25 

P 4 

4104 938 

25 

Pi 

3546 


Cl 

558 938 



Bs 

2 Pj 2500 
Ii JOO 

P, 2600 
i; J04 

Aa 2704 
Bs 

5 Pi 760 
Ii J5 
Pa 765 
la 15 3 

Pa 780 3 
la 15 606 
P 4 795 906 
I 4 15 918 

A 4 811 824 


Bs 

3 Pi 3760 
Ii 112 5 


Pa 3862 5 
la 115 875 


A 2 

3978 375 



Bs 


P, 

3300 


Ii 

165 


Ps 

3465 


Is 

17325 


P3 

3638 25 


I3 

181 912 

5 

P# 

3820 162 

5 

Ii 

191 008 

12 

A4 

4011 171 



8 Pi 

£ 



£ 


425 


9 Pi 

725 


Ii 

17 


Ii 

2175 



442 


Ps 

746 75 



17 63 


Is 

22 402 

5 


45968 


Ps 

769 152 

5 

I3 

18 387 

2 

Is 

23 074 


P 4 

478 067 

2 

P 4 

792 227 

07 

Pi 

425 


I 4 

23 766 

81 

Cl 

53 067 1 


Ps 

815 993 

8 




Pi 

725 





Cl 

90 994 


11 Pi 

£ 

345 75 


12 P, 

£ 

472 9 


Ii 

13 83 


Ii 

14 187 


Ps 

369 58 


Ps 

487087 


I 2 

14 383 

2 

Is 

14 612 


A, 

373 963 


Ps 

501 699 

61 




Is 

15 050 





Aa 

516 751 




\A'ni ] 


COMrOUXD INTLRFST 


143 


£ 

13 Pi 124T711 
I, G21Pj !f?') 

P, i30*j69s|0‘) 
I; O'! 291 I 03 

A. U71 104) 


!?«? 

14. P, 473 89 
I, 18 034 1 0 

P, 492 SOI 'O 
Ij 19 712 ho 

Pj 512'.1 g!1() 
I- 20 300 )04 

A3 533-017 ( 


E«! 

15 P, 3000 
I, 123 

P, 5123 
I3 128 125 1 

P 3 3253125 
I 3 131 328 )1 

Aj 3384 453) 


Es 

16 P, 4300 

T 

' 1123 

Pi 

Ic {‘V 

Uo 

A, 1690 628 1 


4091 23 

no G3 

11728!12 


18 


P| 

I. 


rr^n 

Iris 

Pj 7033 73 


E«= 
67-iO 
270 
33 73 


20 


^3 

P. 

Il 

P^ 

la 

P3 

I3 

^3 


Uic 

33 208 

73 


7371 IGS 

73 

flfiJT 

291 81G 

73 


30 833 

81 


7702 871 



£ 



1001 233 


/ln?7 

48 030 

99 

l-TiW 

4 003 



1033 273 

07 

/inTT 

49 39S 

19 


4133 

18 


1707 004 

14 

fr?.iT 

31 210 

13 

l?Jc 

4 207 

31 


1702 482 



17 P, 
I, 


.{r 


rytt 

Co 


E-! 
4750 
142 5 
23 73 




4910 25 


117 487 

5 

24 381 

25 

5088 318 

75 

132 049 

36 

25 411 

59 

52GG410 



£ 


19 P, 

Ii 


r-ioTj 

Uoff 





5010 373 
200 GG3 


23-082 1 


3242 320 

87 

209 692 

83 

20 211 

60 

5478 225 



£ 

21 P, 2000 

I, 100 

P„ 2100 
i; _203 
Paj 2203 
Is} 53 123 

As} 2200 125 
P, 2000 

C r 260 125 



U4 


COaMPOUND INTEREST 


[OHAP 



£ 



£ 


Pi 

6620 


23 P, 

3600 


II 

224 8 


Ii 

90 


p, 

l; 

584^8 


Pa 

3690 


233 792 


la 

92 25 


pi 

6078 692 


Pa 

3782 25 


hi 

121 571 1 

8 

la 

94 556 

25 

Aa} 

62001631 

8 

Psj 

3876806 

25 

Pi 

6620 


Isi 

48 460 

] 

Cl 

580 164 

1 

Asj 

3925 266 

13 




Pi 

3600 



Cl 325 266 




£ 




Rs 


Pi 


8467 725 


25 P, 


504 687 

|5 

If 

find 

84 577 

25 

T 

/vfin 

20187 

5 

*1 

tiiff 

21 144 

11 


IzriiJ 

2 521)44 

Pa 

(ife 

8563 446 

56 

Pa 


527 398 

44 

la 

85 634 
21 408 

46 

®1 

la 

(t 

21 095 
2 636 

04 

99 

Pai 

& 

8670 489 

63 

P.’i 

551 131 

37 

laj 

21 676 
6 419 

22 

laj 

{± 

6 511 
2 755 

31 

TO 

Aji 


8697 584 

'o 

Aa} 

659 398 

34 

Pi 


8457 725 


Pi 

504 687 

5_ 

01 


239860 


cr 


64 711 



*"=’ etc, denote the Princ,pals> 


26 P, 

1“ hs-I^yr’s I (^) 


£ 

320 75 
6 415 


2”<'Iialf.yr’s 

S-^halfyr's 

Aij 


327 165 


6 543 

_3_ 

333 7081 

3 

6 674 

16 

340 382 

1 


27 P, 

1*» half-yi >s I (jTp) 


£ 

5206 

1302 


Pu 

2«'Jl)alf.yr's I (^) 
P, 

3"«balf-yr’8 

Paj 

4“' half -yr ’6 I (jL) 
Aa 


533 82 


13 345 

5 

547 165 

5 

13 679 

14 

560 844" 

64 

14021 

11 

574 866 




XVIll ] 


COMPOUND INIEREST 


145 


R<! 


28 

P, 

16000 


1*' half-j 1 *8 

M 

rhf 
fhi _ 

320 

40 



Pii 

16300 


2"^lialf-3i « 

M 

.-iofl ^ 

327 2 
40 0 



p, 

1672S 1 


3’^ half -3 r ’•> 


iSu 
.chu _ 

314 562 
41S2C 

• 1 

1 2 


Aij 

1 

7101 182 




Efe 


30 

P, 


250 


1* qnailoi’8 

I 

{± 

2 5 

_0625 



Pii 


253 125 


2''’qinit«i-’8 

I 

f JoIT 

2 511 
0 032 

25 

81 


Pq 


256 290 

06 

qtiartoi s 

I 

it 

2 562 
0 640 

80 

72 


Pij 


250 492 

67 

4*’’ qiiai tei s 

I 

Xih^ 

2 504 
0 619 

02 

73 


A, 


262 716 



£ 

31 (n) P, 100 
l-^qu.itcisl 1 21 

PH 101 25 

2'>'»quaiter'hl 

Pji 102 510 (i2 

S'-quartci^Bl jgo 29 

Pi 3 101707 on 

4 quarici<5i 

At 105 095 

cficctn c annii il rate 
== 6-095% 


E*! 


29 Pi 

100000 


i*‘iiiif3i’si 

1000 

250 


Pi} 

101250 


2“'* Iialf -3 r *9 I 1 

KToV 

1012 5 
253 125 


Pa 

102515 625 


3"* half- 3 1 ’si 

lTu5 

1025 166 
256 289 

25 

06 

P-’! 

103797 070 

31 

4**’ half 31 ’si 

1037 970 
259 492 

70 

07 

Ps 

105094 533 

68 

3''’half-M ’s I ■f'' 

' lion 

1050 945 
262 736 1 

31 

33 

All 

106108 215 

r 


£ 

31 (0 P, 100 

D'kilfMM _5 

Pli 105 
bI 5 25 

P, 110 25 

cfltcln c mnuiil into 
= 10 250% 

£ 


31 (in) P, 


100 


1’* q\Mi Ici-’b I 

0 5 
0125 


Pil 


100 625 


2”^ qinrtci 's I 

/ 2?iC 

IhO® 

0 50.1 
0125 

12 

78 

Pli 


101 253 

90 

3’’* qiiarlci’s I 

/:'/rc 

\nhn 

0 506 
0126 

27 

57 

Pu 


101 886 

74 

4" quai tcPs I 

hh 

0 509 
0127 

43 

36 

A, 


102 524 


cffcrtne 

annual i-ate 



= 2 524% 



116 


COMPOUND INTLREST 


[CILVP 


EJC&IVIPLES XVm g Page 376 


1 

Q 


/VZi A,„ P„, I« l)a\ 0 the ‘?ame nieinings as 111 Ex Xnil f 
A..=:Es 387 x(l 03)2=Es 387x 1 0fi09=Es 110 5683 


Ai=Rs 873')x(l 04)3 
= ^-5 873 j X 1 124864 
=Bs982 569 


873 3 
1124864 


873 5 


87 35 


17 470 


3 494 

0 

60S 

8 

62 

4 

3 

£ 

982 568 

7 


3 Ai= £053 775 X (105)3 
=£953 775x1 157625 
= £1104113 


933 775 

1 157623 


933 776 
93 377 

1 

5 

47 688 

8 

6 676 

4 

572 

3 

18 

1 

4 

8 

1104 112 

9 


A p given A mt _ £296 _ £296 _ 
Amt ofil (103)’~106()9”*' 


c p_ given Amt £340 95 £340 95 

Amt of £1 (103)3 ”1092727“*'^^^ 


6 


p_ gi\cnAm t £3244 9_£3244 9 
Amt of £1 (1 04)* ~ 1 0816 


=£3000 


7 p_ gi^cn Amt £5343112 £5343112 
Amt of £1 (1 04)1 1 124864 


p_ gncnAmt £3920162 £3820162 

Ant of £1 (1 05)3 1 157625 

9 D_g »cnAmt £275 05 £375 95 £275 95 
Amt of £1 (1-025)«‘“(1 0506)=“ 1 1038 



XVIII ] 


COMPOUNn IXPERFST 


147 


10 


11 


12 


_ gn on Amt R«; 300 
“Amt of Re 1“ (1-0 1)3 
Rs 300 


1 124864 
=Ro 2CG 009 


2m fiOO 


1 12,4,8,6,4)300 


p_ given Amt _R‘! 8 jO 
“ Amt of Re 1 ~ (1 *0.-))- 1 1025 

p_gnenAiiit Rs 2000 
Amt of Re I “ (1-03)3' 

R« 2000 
1-092727 
=R-» 1830 283 


R<;850 _ 

i^ = Rs n09/5 


750272 

7535 4 

7862 

1113 

101 


1830 233 

1 0 , 0 , 2 J ^7 )2000 

0072730 

330914 

3096 

911 


I 

i*> c._gi'cnAmt Rs 00-27 875 R^ 6027 875 „ 
Amt of Re 1 “ (I olfSy 1 076400*25 


14, P= 


(I 0375)- 

gncn Amt _ Rs 3025 265025 Rs 3925 265G25 
Amt of Re 1 “(1-025)3 x 1 0125 ~ 1 076890025 X 1 0125 
R« 3925 200 „ 

' 1-090351 758 


15 A=:Rs 250x1 04x1 08=Rs 2808 

16 From Ai t 31 1, the Amt in 2 jcai-s = P x (1 05)' , 

licncc Amt in 2l yi-s = P x (1 05)- x 1 025 
For second pirt of question 
P— Amt 

“Amt of Re 1 


Rs 22050 
^(1 -05)- X (1-025) 
Rs 22050 
■ 1 1025x1-025 
Rs 22050 


1 13006 
=Rs 19512 


19512 

11^0,0,6)22050 0 
10749 4 
678 9 
13 9 
26 
3 


17 P= 


given Amt Rs 2811 90G25 


Amt of Re 1 1 01 x 1 04 x 


525 Rs 2811 90025 „ 
105 1 12476 



148 


COMPOUND INTEREST 


[CHAP 


18 Aa= P X (1 035)2= 1 07123 (approx ) 

19 Cl =A3 -P=Px(105)*-P=P{(105)2-1}=Px 0 16763 (approx), 

we have £50=Px0 15763, and P=£50 -0 16763=£317 

20 Pop" for 1891=^^=33600, pop" foi 1881=^^=30000 

21 Eeq'* population = 347865 x 1 02 x 1 025 x 1 03 to nearest hundred 

=3478 65 X 1 0769 to nearest unit 
= 3746 

Hence answer =374600 

22 Cl on £!=£(! 04)3- £1 =£0124864 

SI on £1 =0 04 X 3=£0 12 , diff for £l =£0 004864 
Hence diff for £3125 = £3125 x 0 004864 =£15 2 


23 C I on £1 = £(1 04)« - £1 = £0 169869 
SI on £l=0 04x4=£016, 

diff for £1 =£0 009869, and diff for £1000 =£9 859 

24 C I on £1 =£(1 05)3 _ £i = £o 157625 

SI on £l=£005x3=£0 15 , diff for £1=£0 007626 
Hence diff for £2400=£2400 x 0 007626=£18 3 


25 C I on £1 = £(1 0375)3 _ £l = £o 07640626 

SI on £1 = £ 0375 x 2 = £0 075 , diff foi £l = £0 00140626 

Hence diff for £2960 =£2960 x 0 00140625 =£4 163 


26 C I on £1 = £(1 02)« - £1 = £0 08243216 

SI on £1 =0 02 X 4=£0 08 , diff foi £l =0 00243216 
Hence diff for £7545 5 =£7545 5x0 00243216 
=£7 5455 x 2 43216=£18 352 


27 Prom Note, Ex 2, p 376, 

diff on £P 


P= 


£0 6 


diff on £l 0 0025 


=£240 


“ •^■"27,P.^^=£3125 29 Asm27,P-^^-H,37M) 
^ ^“27,P=5^.E.,000 

31 The C I IS reckoned on Es (26333 3125 - 13000), 

1 e Es 13333 3125 Amt =Es 13333 3125 x (1 035)® 

= Rs 13333 3125 X 1 071226 =Es 14282 977 



XVIII ] 


COMl’OUXD INTLllEST 


149 


Rs 

P, 4000 
Ij JOO 

P, 4200 
In 210 

p^ 4 no 

220 5 

Pj 4630 h 
I 5 231 fi2‘) 

Pc 4862025 

Rs 
P, 200 
I, 10 

A, 210 
Subtract _20 
P, 160 
i; _8 

A, 168 
Subtract ’>0 

5 118 
I 3 5 0 

Ai 123 0 
Subli.ict '50 


total saMng‘>= 


I, 

A, 

Subtract 


73-0 
3 69^ 

77 505 
50 

27 095 


Rs 

P. 1000 
p; 4200 
P 4 4410 
Pc 4630 5 
Pfl 4863 025 

22102 525 


ic R*= 22103 


34 Its at the end of each year 

/ ]2’'\ 

~V ~T^/ aaluo at the 

beginning of the jeai 
This fraction i educes to J 
Vilue icq‘'=Rs 24000 x(5j)' 

=Rs 24000 X 14068 


I c Rs 27 9 a 6 p 


EXAMPLES XVIII h Page 381 

2 See Ex, Alt 315 

3 Since £(100-121), or £87^, is cash payment foi a sum of £100, 

after discount it 12f, per cent has been deducted, we haxc 
y 7 35 

g^=l^> ''hicli becomes 7/=^ % This is the giapli foimed 

bv joining the origin to the point (40, 35) A convenient 
scale IS 0 1" to 1 shilling along each axis The lequiied 
values can casil'v be read off 



150 STOCKS [chap 

4 From Aifc 316 we see the graph \iill he lineal 

Measure time horizontally (1 inch to 10 years), and Amount 
lertically (1 inch to £40) beginning at £260 
The first graph is the line joining the points (6, 260) and ^15, 350) 
The second graph is the line joining the points (5, 330) and 
(20, 420) In each of these lines the ordinate of any point 
gives the Amount foi the number of yeais gi\en by the 
corresponding abscissa 

Again, these graphs intersect at a point where r=:25, ■?/=450 
Thus each Principal with its Intel est amounts to £450 in 
25 years 

When a,=0 theie is no Interest, thus the Piincipals will be 
obtained by reading oflT the values of the inteicepts made by 
the two giaphs on the ^-axis These are £200 and £300 
respectively 

Note To obtain the result y=200 it will be necessary to 
continue the ^-axis downwards sufficiently fax to shew this 
ordinate 

EXAMPLES XIX a Page 384 

1-6 See Alt 324, Ev 1 7-12 See Art 324, Ex 2 

13 Bs 100 stock IS worth Bs 95 cash , 

Bs 2880 stock sells for Bs x 2880 cash 

14 Bs 3500 stock costs Bs 3220 cash , 

Es 100 stock costs Bs 100 cash 

15 £285 stock costs £228 cash , 

£100 stock costs £5^ X 100 cash 

16 £100 Consols is worth £83^ cash , 

£855 Consols is worth £j^ x 855 cash 

17 £102J cash buys £100 stock , 

£1000 cash buys x 1000 cash 

18 £560 stock costs £480 cash , 

£100 stock costs £^-§^ X 100 cash 

19 Es 100 stock sells foi Es 94^ cash , 

Es 2912 stock sells for Es ^ x 2912, or Es (94 5 x 29 12) cash 



xrx] 


.STOCKS 


101 


20 H*? 100 sinck co>t‘« lla 1 J.'il n>»li , 

ll-> 'i7S2 slock to..lK Hh Y 'i7S2, oi Hs {^u 82 y IS*! .">) casli 

1 U(i 

21 Ps 100 vtock costs IJs 100* «isli , 

1{«,3’V72 slock cn-.ts ll« / 3"w2, oi Ps (37 72 y 100 O) cash 

lUU 

22 On Bs If-O stock the Ids'!} j.: ^ 

on R= 760 stock the lo'S is Bs 780 

23 On £100 “tod the piolit is £3| , 


on £G 70 stocl the piofil is £.~‘,xG>0 


'io>» 


24 Ps 131 1 cash hnjs stock nft< iwanls sold foi Rs 133 cash , 

1 n 

lienee nq‘ proceeds r= Ps x7 >0 

I JI 4 

25 £107 cash hn.is stock aftcnnuls sold foi £102 cish , 

hrnce req* c\sh — £{ x 175 
2G On £100 stock the loss is ■t.'it , 

* nl 

on £ JWO fctock the loss la £^ y 2700 

27 On an outhj of Ps 1 12^ the gain ik lls (117 - 1 12i), 01 Ps,.ii , 

mj^ gam=P« ,-[*-7 y 11871 —Ps '47 x ■ '-C-— Ps '-i' 

■-Rs37«i 

28 On Rs 100 stock the profit is Rs (107^ - 101 i), 01 Rs 7 ( , 

7? 

on Rs 3720 hlO'k the gam ib Rs ^^jX3720=-Rs ^*O,jX.1720 

= Rs 213 14a Op 

29 Am in No 28 we get icqMovs— £, x 1C42=£70 18s 1(7 


EXALIPLES XIX b Pngo 386 
1-3 SSceArt 325, 1'x 1 4-6 See Ait .12.7, L\ 2 

7 1'* slock inronic:^Rs 3 y l-{ -r-Rs 337 G73 , 

2 " ' stock income --- Rs 1 1 / I-I J =-Rs 3 10 1 73 , 

2”* stock gives the grcatci income h} Rs 10 8a 

8 Req'>inc=-£2y /’.Y y 9 Req"inc 1 of -Csj x 



152 


STOCKS 


[chap 


10 Req''inc=Rs2^x^^=Rs276 x 786=R-32l615=Rs21 9a 9p 

11 Req-> me =R 3 4^ X -Wt? =Rs ^^=Rs 113 839 =Rs 113 13 a 6 p 

12 Req0mc=£2ix|g=£^=£l5 1975=£l5 3s llrf 

13 Req* amt of stock =E9 100 x^ 

297 


zMV Wv 

14 Req^ amt of stock =Rs 100 x-^^ 15 Req*sum=Rs llOxgj 

16 Req«sum=£l00x^ 17 Req** sum=£115ix^ 

^ ^ „665 X 8SX2 X 240_„ 

18 Req'»sum~£l41iXg^^a^-£ 4x5x11x226 ^ ° 

19 Req'iamt s=£l00x^^ 20 Req'*sum=Rs 104x-S^ 

21 As m No 18, req« amt = £132^ x = £^^22iil = £2414^ 

22 £90 cash buys an income of £4| , 

£100 cask buys an income of £4kx-^^, giving 6 % 


23 As in No 22, £100 cash buys an income of £3^ -Hr* 8" ? % 

24 As in No 22, £100 cask buys an income of £5^x^^, oi £Vj?-, 

giving 3 79 % 

25 As in No 22, £100 cask buys an income of or 

giving 4 44 % 

TOO 

26 As in No 22, £100 cask buys an income of £4x or £f4i, 

giving 3 35% “^5 

27 1"‘ stock income from £100= £3 =£3 529 , 

2“* stock income from £100= £4 x £4 166 

2"* stock IS better by £0 64, or 0 64 % 


28 1*‘ stock income from £100=£4 x ■5-§^=£3 174 , 
2»^ stock income fiom £100=£3^ x ^§|-=£3 240 

2“* stock is better by £0 07, or 0 07 % 

29 1*‘ stock income from £100=£6 x ^^=£3 671 , 
2<^ stock income from £100=£6 X ^^=£3 448 

l‘‘ stock IS better by £0 12, or by 0 12 % 



STOCKS 


163 


XIX ] 

30 1“‘ stock income from Es 100=Es 4 x ^2 'TT*' > 

2°^ stock income fiom Es 100=Es x ] =Es 

2 "’’ stock IS the bettei 
Foi second p<ii t of question, 

diff of incomes on Es 100 invested =Es 

diif of incomes on Es 7020 invested =Es -fS-®- x = Es 10 

31 Let Es X he sum iniested by each, the income of the first 

=Es 3 X ^ , the income of the second=Es 4 x • 

l"-^=2, and ^=1512 

32 Let Es ^ be the leq^ amt of stock id’s income=Es 3i x , 

E’s income =Es.2ixY^» ^--^^-^^=100, and r=Es 10000 


100 100 


33 If Es X be the req*' price, then Es x cash pioduces Es 4 income , 

4 

Es 100 cash pioduces Es -X 100 income, 

-X 100=5, and a=80 
r 

34 As in No 33, — xl00=2i', and r=110 

A\ 

35 As in No 33, — ■xl00=6, and r=75 

X 

36 Let Es x be icq'^ puce, then income fioni Es IGOO invested is 

IGOO 


3xM^ 

X 


3x-^^^=50, and a. =96 

X ’ 


9400(1 

37 As in No 36, 5^x =1200, and a:=110 

9OKO 

38 As in No 36, 5^x-^ — =95J, and r=136j 

39 If £r be req"* price, &x cash produces £6 income , 

£100 cash produces £-xl00 income Tins, after income tax 


230 G 

has been paid, becomes — x ^ x 100 But this 13 £5 , 
230 6 



154 


STOCKS 


[OHAP 


40 4 45 IS peicentage on £100 cash , 

- £21 7s Qd, or £21375 is peicentage on £100 x 
or £480 (to neaiest £) 

41 If £?? be req^ amount, income from it is £2| x ^ , 

2|x^=60, and a =£2007 (to nearest £) 


EXAMPLES XIX c Page 392 

1 Rs 100 stock sells foi Rs (l05j-^), oi Rs 1051 cash , 

hence leq** ca8h=R8 6785 x 

2 Rs 100 stock costs me Rs (00^+^), or Rs 91 cash , 

hence req^ piice=Rs 2750 x-j"^ 

3 Rs (lOl-f +-^), or Rs 101-’ cash buys Rs 100 stock , 

Rs 3552^ buys Rs 3552^ X stock 

4 Rs (93 J - }■), or Rs 93|- cash is realised on Rs 100 stock , 

req** amount of stock=Rs 100 

5 Rs 100 stock costs Rs (125|-1-|-), or Rs 125’ cash , 

■JQKJ. 

reqo cost=Rs 5782xi^=Rs 7256 410 

6 Buying price=9l|+^, oi 9ll, selling price =90^- -J-, or 90 

loss on Rs 100 stock =Rs l-^ , hence req^ loss=Rs I'jX 

7 Gam on buying and selling Rs 100 atock=Rs (108 - 105), oi Rs 3 
Half year’s dividend on Rs 100 stock =Rs l^ , 

total gain on Rs 100 stock =Rs 4| 

Hence on outlay of Rs 105, gam is Rs 4| 

» )] Rs 3500, gain is Rs 4^ x YtP^j oi,Rs 158 5a 3p 

8 Gam on buying and selling Rs 100 stock=Rs (93J-91i)=Es 2 
1 quaiter’s dividend on Rs 100 stock=Re g , 

total gam on Rs 100 stock =Rs 2| , 
hence on outlay of Rs 91|-, gam is Rs 2|, 

„ Rs 4380, „ Rs 2§ x or Rs 126 



BROKERAGE 


155 


XIX ] 


9 

10 

11 

12 


13 


14 

15 


16 


17 


18 


Rs 100 stock costs Rs 96| cash , 

4200 

leq'* incoine=Rs 2| x -qqi-i oi Rs 120 

Rs 100 stock costs Rs 1 04 cash , Rs 3| income costs Rs 104 cash , 

104 

hence req^ sura=Rs 400x oi Rs 12800 

£100 stock costs £l03j\ cash , £4 income costs £103^ cash , 

1 d 1031 226875 n^nnn 

hence req“ sum= £2/5 X -j-*"=£' , or £/090 


£100 stock costs £103} cash , 

£3 X 5-|-g- IS net income from £103} c.ish , 


1 d 103^ n2070000 

hence req® sum = £ oOO x — - - = £- 




113 


01 £18319 


Let £i he leq"* price, then £(i +|)cash piocluces £4| income , 

43 ' 43 

hence £100 cash produces £100 x income , £100 x j =5, 

l+J I'T" 4 

or 100x4|=5(a4-i) , hence i+4 = 95, and i=94| 

Rs 100 stock can he bought foi Rs (lOOx )+hrokeiage , 

1 e Rs 92} + brokerage, or Rs 92g 


Income from first stock =:Rs 3x60=Rs 180 
Sale of fiist stock realises Rs 60x91, or Rs 5460 cash 
Hence income from second stock=Rs 4 x Y-A” 13® » 
req^ loss=Rs (180-168), oi Rs 12 
Income from first stock =Rs 3ix55=Rs 192} 

Sale of first stock realises Rs 91 } x 55 cash 

nil X 55 

Hence income from second 8tock=Rs 4x =Ri3 198 , 

req^ change=Rs (198-192}) , te Rs 5} gam 


Income from first stock =Rs 620 

Sale of first stock icalises Rs 125 xi^ p;^, oi Rs 15500 
Hence income from second stock =Rs 3x4-''7jY~=®s 500 , 
req** change is Rs (620 - 500) , i e Rs 120 loss 
£1365 cash buys £100 x 4-^p- stock, or £1500 stock 
£1000 of the stock realises £93} x -AV £*135 cash 

The remaining £500 stock realises £85 x cash, or £425 cash , 
reqo loss=£(1365-935-425), or £5 
L 


K II s I 



156 


BROKERAGE 


[OHAP 


19 £4340 cash buys £100 x stocb, or £4000 stock , 

my first income is £3^ x 40, or £140 
The £4000 stock realises £110x40, oi £4400 cash 
£4400 cash then buys an income of £5^ x or £200 , 
req^ ineiease=£(200-140)=£60 

20 £26180 cash buys £100 X £28000 stock. 

his fiist income is £3 x 280, or £840 
£14000 of this stock realises £92| x 140, or £12915 cash, which 
purchases in the 4 p c 's an income of £4 x £532 577 

The remaining £14000 stock brings in £^ of 840, or £420 income , 
req"* difference =£(532 577 + 420 - 840), oi £112 577 increase 

21 £500 R’ Stock at 95 costs £475 , 

biokerage at on £475 =£2 7s Gd 

22 £500 Stock at 128^ costs £642 lOs , 

biokerage at % on £642 lOs =£3 4s 3d 
£500 N T D Stock at lllf costs £558 15s , 
brokerage at ^ % on £558 15s =£2 lOs llrf 

23 £800 Consols at 91 costs £728 , brokerage at ^ % on £800 =£1 
£650 E» Stock at IIOJ realises £719 17s 6d , 

biokerage at -^ % on £719 17s 6d =£3 12s 

24 £2375 ly Stock at 126 realises £2968 16s , 

brokerage at -J- % on £2968 16s =£14 16s lid 
£4000 India Stock at 97^ costs £3900 , 
biokerage at ^ % on £4000= £5 

25 £3600 Consols at 89§ realises £3217 10s , 

brokerage at ^ % on £3600= £4 10s 
£2700 B? Stock at 1191 costs £3226 lOi , 

brokerage at J-% on £3226 10s =£16 2s 8d 

26 £400 N T Stock at 103£ costs £415 , 

brokerage at } % on £4]5=£2 Is 6d 



157 


XIX ] SHARES 

27 (i) £300 India Stock at 93 costs £279 , 

Inokerage at % on £300=7« Ct/ 

(ii) £400 Trans\ aal Stock at 95^ costs £382 , 
brokerage at ■|- % on £400=£l 
(ill) £500 Mexican Stock at 95 costs £475 , 
brokerage at on £475 =£2 7s 6d 

EXAMPLES XIX d Page 396 

I Ro 383x500 2 Rs(200- ^tV) x60 

3 Cost of £l shares = (£2 17s Gtf +44rf)x44=£l27 6s Gd 
Cost of £5 shares = (£6 6s 3tf +9£f )x80=£508 

4 Rs(15+2^)x50 5 (£2+£|-9rf)x250 

6 Rs 353-(Rs 15+Rs 2^+4a ) 

7 (i) Rs 4i X 35 <ii) R« 61 X -jW 47 

(ill) (Rs71x}-S^+8a)x75 

8 Req'' dn idend = Rs 100 x 20 x -j-3^ x — Rs 58 7 a 

9 (i) On £5 the dividend is of £3, oi £-i^, 

req"* percentage=lg-2.x-j^, or £6 
(ii) On £13^ the dividend is of £9 , 

100 8 

req** percentage=j^x of £9, or £5 Gs 8d 

(ill) On £36 the dividend is 17s x 2 , 

req® percentage = 17s x2x-yt/*-j or £4 14s 5d 
£375 

10 (i) Req® no of shares (ii) Req® costprice=£7ix20 
(ill) Req® dividend =£ 53 - x-5^y- 

II ( 1 ) Price (with brokerage) per share=£103 + ls 3d =£10l-§-, 

req® no of shares=£60>^— lOj-^ 

( 11 ) Req® cost=£10j-3-x 36 ( 111 ) Req® dividend=:^ of £6x 50 



158 


STOCKS AND SHARES 


[chap 


12 On £100 of first stock he realises £96^ Tf £v be the puce of 

each share, with £96^ cash, he buys shaies IIis income 

901 ^ 105 

IS now -64 X , and tins by the question w x 2| 

t:=£1335 


EXAMPLES XIX e Page 397 

1 I buy £5520 x oi £6000 Consols , 

£6000 Consols must sell foi £5730 cash, 

01 £100 „ „ £95^ cash 

- ^ ^ 1200 „13200 

2 Req^ income =£2| x ■= £ 14 465 

3 Income on investing Rs 100 in 1*‘ stock=Rs x -}-^-^=Rs 2 96053 , 

„ ), Rsl00„2“'> „ =Rs2jxJ^“-=Rs2 94118, 

the first IS more piofitable 
For second part of question, we have 

diff for Rs 100 invested* Re 0 01935 , 
req'' diff =Re 0 01035 x 32 30=Re 0 625 

fiyOf) 

4 Let Rs % be leq^ puce , then inconie*Rs 23 x - — j (allouinE foi 

biokeiage), 

6200 .1 2? X 6200 

’^'^““176 — ’ alienee r*9G| 

5 Let £i be the leq^ price, then income from £100 cash 

IS £7 X (allowing for brokerage) , 

7x;^^=4, whence x+^=!^^ and t=1743 

6 Income = Rs 3ix-^J-^-® 

Foi second part of question 

on Rs 90 outlay, the loss is Rs (90- 87), or Rs 3 , 


on Rs 7500 


Rs3x^ 


7 Let Rs r be 1 eq^ price Then Rs 115 cash buys Rs 44 x — income 

44x115 

— =5, and r=1034 

The half-yeaily dividend =l of Rs 5 x "^Y/=Rs 51 



159 


XIX ] STOCKS Al^D SHARKS 


8 Let £ 1 1 bo his capital Income fi om !•* stock =£2| x ^ , 

income from 2“'* stock =£3ixr^ 

“ 105 

i 2 1 n/Sji , 3ix3t\ j-r 

total incomc=-e( 

7 * 1 1 

Now 5=yr of 'll , ? c Ills income =5= of liis capital, 

O oJ Om 

01 3 J % of hi3 capital 

9 Ecq-* cost=£59|x2 25=£(60-l)x2 25=£(135 - 0 281) 

=£134 719 

Brokcn{TC=-^jj- of £131 719=£0 674 total cost=£l35 393 


10 


11 


12 


Total sum due to broker=i£994 19e +£2 11s =£997 10s 

,997 5 
■99”75' 


req** ,amt of stock=£l00x^^“ = £l000 


% 183 

Let Rs a bo the i cq** sum Income is Rs G x ^ x 
and this is Rs 1200 

®^ril^Tl=^200, and 28800 

%■ 230 

If £x be the amt of 3 p c. stock, income is £3 x =75; x 

and this=£l423 240 

Hence x= £15400 This loalises £78? x 154, which is inaestcd in 
4 pc. stock Allowing foi biokenge and income ta\, it 

11 t 783x154 230 Xiro-m 

jields an income of £4x — 10^75" ^940’ £453 o49, or 

£453 11 ^ the req'’ change IS ^10 ICs gam 


13 Income from the 3 p c ’8=Rs 3 x x ] =Rs 146 14 a 

„ „ „ =Rs3tx''j!y^9=R^i66 4a 

the second was the moie adaantageous by Rs 9 6a 

14 £4376 GW R stock at 124 costs £7426 4r 10<f 
Brokerage at ^ % on £5426 4s lOrf = £27 2s 8rf 


15 Suppose he invests Rs a 

^ 7* 

Income fi om 3 p c 's =3 x i x , income fi om 4 p c ’s= 4 x ^ x , 


37 


4i 


2x97J-^2xl25 


=510, and a =16250 



160 STOCKS AND SHARES [CHAP 


16 Let Es ^ be invested in cicli stocL 

Income fiom 6 p c '8=0 x , income fiom 9 p c ’s=0 x ^ , 

^-^-220. and a-472!K) 

17 Let£^ be amt ncccss.n> , lonmin nftci pajing Icgacj duty 

income from 4 p c ’a (alloii iiig for income tax) 


4ix 


0» ..230 

“"*-^10x1031 ’^210* 
9r 230 


10xl03Jt 210 


x^=C0, md t=1G00 


18 Amt of 3pc. 8lotb = 100xA^=£l0000 

£5000 stock loalisca £02 x 50, oi £1600 If £i bo req'* jicrcentagc, 
then £4600 bungs in an income of -Lr v oi £100» 

Since the remaining £5000 stock brings in an income of £150, 

-no bale 150+ 100 js= 300+50, and i<=D% 

19 Lot V shillings bo the ixite per £ 

•c /> i I. 1 .D-ii 4500 20 — j" 

From fust stock income = £2 Jx-|^x — j^y— 

£4560 stock icabscs £85 x 45 6 , 

f 1.1 . F5xl5G 20- r 

income fiom second slock = 1 x — j— \ —=5^ 

4x85x 15 6 20- r 2^x 1560 20-3 „„„ 

— m — ""-ir-'Too- »^~o-=2oo, 

or S0-r /lx8 -. X 4r^ _ 2l x I5n0\ _ „ _ 

20 V 114 “ ioo~7 ’ 

or ^’(13G-n4)=20 0, 

(20-7)x 22=118 and 3 = 1 

AlUer Income from 2» pc's=£2i:x45 C=£114 (%vitboHt tax) 
£4560 stock realises £85 y 15 G, oi £ ISTO 
This brings in an income of £4 x YrVi ^2130 (n ilhoiit tax) , 
difrcicnce of incomes (mihoiit tax)=£22 , 
hence the tax on £22 is (£22- £20 18s), or 225, 
and tax is I5 in the £ 



STOCKS AND SHARES 


161 


XIX ] 


20 First income=Rs 8 x 50=R'5 400 

The 50 shaies realise Bs 9000, and tins buys shares at Rs 50 
These shaies produce an income of Rs x x 

01 Rs 450 reqa difTerence^Rs 50 gam 


21 In case, Rs 115^ cash produces Rs 5^ income , 

r. inn T> El 100 _ 1100 

t c Rs 100 „ „ Rs 5i X 01 Rs income 

In 2"'* case, Rs 7^ cash pioduces Rs 3^ x-j^^ income , 

■D lAn r> oi 10 100 „ 700 

? «? Rs 100 „ „ Rs 3h X X — or Rs income 

” - 100 7^ lo5 

rates of interest are as VA** f or as 155 147 

Alitev The incomes fiom Re 1 iniested in each company are 
proportional to the respective rates of interest 

gi 1 j 

Re 1 in\ ested in fimt company pioduces Re oi Re income 

34" 


Re 1 


second 


” 77V 115 ” 


nites of interest aie as xtt, 01 as 145 147 


22 The first income (fiee of taTc)=£8 x 38 4=£307 4^ 

On £3840 stock he realises £187^ x 38 4 

His second income (free of ta\)=£4+ x ——^^^=£300 

“ lUo 

Hence diffeience (free of ta\)=£7 4s 

tax on £7 45 =£7 45 -£6 I85 =65 , 
whence tax=10rf in the £ 


23 His fiist income =Rs 2| x ” • x }-0-a=Rs 2291 666 
On selling out he lealises Rs 80000 x ^ 

His second income =Rs 2|^ x ' " = Rs 2632 978 

difierence=Rs 341 312=Rs 341 5 a 


24 Let £a be spent on the Stock 


^ 19 

Net income from this =£3 x x ^ 

3x1x19 52x783x20 2610x13 

975x20 * 8x19x3 “ 19 

Adding l5 01 £0 05 foi Contract Note Stamp, 

reqo answer=£1785 8394 =£1785 16s 9rf 


= £1785 7894 



162 MISCELLANEOUS EXAMPLES Y [PAGE 

25 He realised Rs (]00+150)xl00, or Rs 25000 Each Rs 10 share 

cost Rs X 10 , he bought ^25000—^^, oi 3000 shares 

26 £500 Canada Stock costs £104|^ x 6, or £521^ 

Thus £478| is left to be invested in N Z Stock 

This buys £100 x^^, or £489 770, or £489 15* 5cl NZ Stock 
y/i 

27 £20000 Jamaica Stock costs £101^ x 200, or £20225 
£10000 Cardiff Stock coats £97 x 100, or £9700 

£20075 remains This buys £100 x 

or £20642 Rnstol Stock 

Net income on £50000 invested 

=£(3i X 200+ 3 X 100 + 3 x 206 42) X ^ 

=£(700 + 300+ 619 26)x^=£l619 26xi|=£l538 297 
average income per cent =£— - °| - ° ---=£3 08 

28 Rs 40000 stock costs him Rs 65 x 400, or Rs 26000 
He owes the bank 4 % of of Rs 26000, oi Rs 416 
He leceives Rs 2^ x400, oi Rs 900 dividend 

By sale of Rs 40000 stock, he realises Rs 70 x 400, or Rs 28000 
his piofit IS Rs (28000+900-26000-416), or Rs 2484 
Now Rs 2484 is profit on an outlay of Rs 26000, 
giving Rs 5^* profit on Rs 100 outlay, or 9 6 % 


MISCELLANEOUS EXAMPLES V Page 401 

A 


1 

2 


2 2046 lbs =1 Kg , 1 lb =^ 3 ^ Kg , 

1 ton or 2240 lbs =1016 Kg 


10 tons 13 cwt 74 lbs =(l0+^+^)tons=10H^ tons 

reqa cost=£ll X 10 ^=£X X (l 0 +i|A)=£^l||ll| 

-£ll|.+ £f^=£ll 2 q. 3 ^ X51=£ll 13s 4d+16s ll|<f 
=£12 9s 3cf to nearest penny 



MISCELLANEODS EXAMPLES V 


163 


401] 

3 Let Rs X be the pai tnei-’s shaie of the piofits 

Then 0 24 x r=7800, and r=32500 , req** fiaction=-Jf f§-=-| 

4. £5 Ms Crf=117is, 2« 

, " 117^ , 470 , 

area of ioom=-^ sq jds sq 3 'd 8 =470 sq ft, 

470 

and req'* breadth ft =20 ft 

5 As in Alt 292, E\ 1, leq^ S P =-V/ 80=Rs 96 

SC 

6 Let Bs T be the i eq‘‘ amount of stock His income is Rs 3| x 

5|^=Rs 84-jV, and t=Rs 2250 


B 

7 Cost for 20 3 ds = £2 11« 5rf =£2i-J^ 

cost foi 1 yd =-fjf of £2]^-^ , 

f 339 “t“ 

no of3ds in peumetei =-^ =^fv=2640, 

V of £2.7TJ‘ 

and no of yds in side=660 , req"* acieage=-5-^^®^®-2.=90 

8 Total CP =240a , total SP=250a 

(i) actual gam =10 a , (ii) percentage gain =-215^ x 100=4-J- 

9 112 lbs of coffee cost 168$ , 52 lbs of coffee, or 12 lbs of tea 

cost 168$ X , 

22 lbs of tea cost, or 572 lbs of sugai cost o ^^3$ 

1 X 1a 

req** weight of a cask of sugar costing 42$ =572 lbs x -24^= 168 lbs 

10 Let the req** area =$• sq cm , then 1800xa=350x0'02 , 

whence ■c=T^inr 1 tiW *^9 =0 39 8q mm 

11 Convenient scales are 01" to Itf hoiiaontally, and 01" to 1 lb 

leiticilly Since 1 cut =112 lbs, and £1 6s 8rf =320c?, as 
111 Alt 277, Example, y= or y=-^% Join the origin 
to the point (20, 7), and the graph is obtained The required 
values may then be read off 



164 MISCELLANEOUS EXAMPLES V [PAGE 

12 From Art 313, 

the amount in 3 yis =Es 1756^1 +^) 1756x1 0225® 


Rs Rs Es 


1756 

1022 

1795 510 

6 

10225 

1835 908 

98 

10225 

1766 

1795 51 


1835 90S 

98 

3512 

35 910 

2 

36 718 

18 

3 512 

3 591 

02 

3 671 

82 

878 

897 

76 

917 

95 

1795 610 

1835 908 

98 

1877 216 

93 


req^ Cl =Rs 1877 3a 6p -Rs 1756=Rs 121 3a 6p 


Tj df .. Rs20 11a6p Rs 20^4 /m» 

13 Req" fiaction=— H=T 7^=0 17 

^ Rs 121 14 a Rg i2if 100 


14 9 0707(3 0117596 

9 = 3011760 

601 

602l\ 10600 
60227\ 457900 
60234 \ 36911 
\ 5749 
\ 373 


15 Interest = 1^ a x 52 on 240 a 
=78 on 240 
“ 7 % ^100 on 100 

“32U 


16 


Req"* fiaction = 


16ixl2Jsq ft 
21= X 18^ sq ft 



17 (8+10+12), or 30 shovelfuls are thrown in per 1 min , 

the first man would take 90 mm x , the second, 90 min x-j-g-, 

the thud, 90 imn x , or 337-^ nun , 270 min , and 225 min 
lespeetively ,ie 5^ hours, 4^ hours, and 32 hours respectively 


18 2“^SP=l«tSP x|56t^= 1"‘SP x|?=^of SR 

xvs Ola 81 9 

Also 1»‘ S P =135 % of C P , 2“>> S P =-V- of 135 % of C P , 
or 2"^ S P =165 % of C P , hence req"* gain =65 % 

(See Art 292 ) 



402] 


MISCELLANEOUS EXAMPLES V 


165 


19 Volume of each cu ft , 

17^0 

■weight of each brick x 145 lbs 

Hence, if x be leq** no of biick^, x 146 xa=5 x 2240, 

, 6 x 2240x1728 2240x128 / i\ 

"= 9x4ix T xi45 =-9l^=^^°Q 

20 In vtalking 4^ miles, B can gi\e mile stait 

„ 1 mile, B ,, A. ^ y) ,, 

or „ 1 yd , i? „ ^ ytl „ 

,, 1350 yds, B „ j4 ^ of 1350yds, or 150 yds start 

21 Lot X shillings be leq^ C P , the marked price = of x shillings, 

and S P of of ^ shillings Also £8 4fi 8d =164^s 

05xl30x^ tajo j 494x100*' looi a loc Atf 
100x100 ”1®43, ^“3x953030“^®®-” ^ 

22 2 cwt 26 lbs =250 lbs , req"* cost=250 x 0 4536 x 4 75 fr 

=1000x0 1134 x 4 75 fi =113 4 x 4 75 fr =539 fi (nearly) 


23 1"^ dividend=Rs 4 x •Sy^^=Rs 360 

Rs 9000 stock realises Rs 112ix Wir* 10126 , 

2“'* dividend=Rs6jx^^^=Rs 330, giving Rs 30 loss 

24 Let Rs x be one pait, then Rs (1130— .v) is the other pait 

of 1130, «,d ^-678 , 

hence we obtain Rs 678 at 7 %, and Rs 452 at 2 % 


26 Req** ratio = 


Rs 269 8a Rs 269^ 7 

Rs500 8a Rs 500f 13 



166 MISCELLANEOUS EXAMPLES V [PAGE 


27 £38 6 

47 


1544 


440 yds =i mi 

270 2 
1814 2 

9 65 


220 vds =i- mi 

4 825 


55 yds of 220 yds 

1206 

25 

55 yds =4 of 220 yds 

1206 

25 

6 yds =■?! of 65 yds 

109 

66 


1831 197 

16 


giving £1831 4a 


28 Let Es ^ be req^ puce 
Then 

30i + ^ X 20=^ of 5000, 
and a?=150 


(2 men with 1 boy do \ of the work in 1 day, 

U man „ 2 boys „ ^ „ „ 1 „ 

f 4 men with 2 boys do -f of the woik in 1 day, 

\l man „ 2 ,, ,, ,, 1 ,, 

by subtraction, 3 men do -y, or of the work in 1 day , 
that IS, 1 man does of the work in 1 day 
1 boy does -b-- ^ or of the woik in 1 day 


Since their daily rates of work are proportional to their weekly 

■lAr 

rates, the wages of a boy must be Es 7 x^=Es 4 

ITT 

30 Let X be the original population of each part , 

195a;=39390xl00, and a:=20200 


F 

31 30 eggs are bought for 25 a and sold for 36 a , 

11 a is gained on an outlay of 25 a , 

44a „ „ lOOa, oi 44% 


32 His estimate of 30 in is leally 12 in, of 1 in is leally ^in 
of 1 sq in IS leally f^x,}^ sq in Now 2 ac 3 r =11 r , ’ 

his estimate of 11 1 is really ^x-|^xllr=12 5ier=313ac 



MISCELLANEOUS EXAMPLES V 


167 


403] 

33 His first income =Ks 2^ x i-5-A^=Bs 275 
The stock realises Rs 10000 

If he in\ ests Rs o in the 4 pei cent stock, his mcome is Rs 4 x 

4xj^=273, and a.=9625, 
he retains Rs (10000 — 9625) or Rs 375 
oggX 7^ 

34. Time=73d'i3S Int€rest=£3Ax-j^x^ 

=£.1xJ3!^x^=£-}^=£2 725=£2 14s 6<f 
Amount =£389 6s Qd +S2 14s 6'^f=£392 Is 21 ? 

35 Cutting the 17 in edge from the 18 in edge, the 9 in edge fiom 

the 30 in edge, lie gets 3 pieces out of each sheet, and 
requires 12 sheets E ich cheet has an aica of (18 x 30) sq in , 
01 3| sq ft 

Hence 12 sheets co'^t 2a x 3^ x 12, oi 90a The area of the 36 pieces 
IS sq ft X 36, and then a alue is 3 a x - >(■ 36, oi 
114|a His profit is therefore 23|a on 90a outlay, oi 27f % 

36 Let the man’s daily wage he 7r annas, the •mfe’s 4r annas, 

the son’s 3 t annas, then 7rx2-*-4tx3^+3.Tx4=80, and 
T=2a , the man’s wage is 14a, the woman’s 8a, the 
hoy’s 6 a. 


G 


37 103 tons 5 cMi; 3 qrs 

=(l03+^+r^)tons 
=103 2875 tons 
req^ price 

=£5 423x103 2875 
=£560 335 =£560 6s 8rf 


103-2S7 

5 

a 425 

51C 437 

5 

41313 

0 

2 065 

7 

31C 

4 

560 334 

G 


Volume of casting = 


loa g8-rxs240 _ ioas s-sxgs 
488 Cl 

= 8 0 = 0 . s ^4Y4 1 cu fj. 

C 1 


38 Volumeof siood=(26xl9xl8— 25x18x17) cu in 

=18(26x19-25x17) cu in =18x69 cu in=\8^8^cu ft , 

weight of ho''c=-Jj^®g‘’-x 40 lbs =i3-r- lbs =28| lbs 



[PAGE 


168 MISCELLANEOUS EXAMPLES V 


39 


40 

41 


Let each woman do 2a: units of work pei day, 
then each man does 3a; „ „ 

50 men working for 80 days do 3a: x 50 x 80, or 12000a: units, 
and the cost is 5^s x 50x80, oi £1100 
Again, 20 men and 30 women do (3a: x 20+ 2a: x 30), 
or 120a: units per day , 
they would do 12000r units in 100 days, 
and the cost would bo (6^s x 20+2|s x 30) x 100, or £962 10s , 
the leq® difference IS £1100— £962 10s, or £137 10s less 

Let £P be leq"! sum , then from Art 313, Px(-^^)®=308^, 

P=308^ X (^)>'=ao 


Cost Price=Rs 38 , Selling Pnce=Rs 38 x 

If Rs a; be the “ maiked” piice, then Selling Piice=Rs , 
95i: -- 105 , „ 

100=^® ""lOO’ 


42 Suppose ho invests Rs v in the first stock, and Rs 23100 - r in 

the second His income from the fiist 8tock=Rs 3 x , 

from the second=Rs 4^x — , 

“ 1»35 

a:=8100, giving Rs 8100 at 81, and Rs 15000 at 135 


43 12345 


1 2345 


12345 


2469 

0 

370 

35 

49 

38 

6 

17 

15239 

12345 

90 

15239 


3047 

98 

457 

20 

60 

96 

7 

62 

18813 

66 


giving 1 8814 


H 


44 


Exp"= 


(6?+4*)x12 


_ Ba-13 0 10 

eo+C6 ~o so 

— ^3 9 _ 10 
T3 6 leTT 


— -I T - 8 

vb uir 


— i 

e 


21-194 

93+87 



404] 


MISCELLANEOUS EXAMPLES V 


169 


45 100 yds -weigh 14 08 lbs , 


or ioo. m 
100 

1 ni 


0 0 0 s s ° ’ 

14 08X1 OP 


14 08X1 00_ liS>Ll_0 9 OP OQ ho 

0 oossxiod* 2 0 0 I or /D gm 


46 Let X gallons of the first, and (84 -iT) gallons of the second be 

taken 

Water in fiist+water in 8econd='water in mixture , 
^+^( 84 -ar)= A of 84, and r=24 , 
req^ answer is 24 gallons of fiist, with 60 gallons of second 

47 Since 24 hours=:86400 secs, and 2^ min =135 secs, the min 

hand moves over (86400+135), or 86535 sec spaces in 86400 
actual secs , 

it indicates 1 sec of time in % actual secs , 

„ 1 hr „ •|-845“6 1 , his 

When it shews 5 p m on Wednesday, it has moved over 
53 hour-spaces since noon on Monday , 
correct timc=53 x =53 x -g^ ^rs 

This reduces to 52 hrs 55 mm 2 secs after 12 noon on Monday, 
giving 4 hrs 55 min 2 secs after 12 noon on Wednesday 

48 If profits are Rs r, since 40 % of Rs v is shared equally by them, 

we need only considei the leniaining 60 % of Rs t, oi Rs — 

5 


This IS divided in the ratio of 13500 7500, or 9 5 , 

e 


their shares of this aie ^ of Rs and A of Rs — resp'>^ 

id. Q ' id R 


14 

— of of ^+450, and a.=Rs 2625 

14 5 14 5 


49 Discount=Rs 9 8a on Rs 190, te Rs 9^ on Rs 190, or 5 % 

60 E\penses=;^ of £41647310=^ of £41647310 

=£6073566 =£6074000 (to nearest thousand) 

For second part of question, 

the old rateable value of £1 is now £l x 
The cost to collect £1 x ’-gg- is now £-VV ^ ■H'O' > 

ji >1 ^ ^ 

Thi8=363rf, giving 3s Ojrf 


9) 



170 


MISCELLANEOUS EXAMPLES V 


[page 


51 216 eggs ai e bought foi ^ x 216 annas, or 180 a 
If X eggs are broken, 

then 216 - ^ ai c sold foi J-g- x (21 6 - r) annas 
Since a profit of 20% is made, x(216-a;)=180x J-g-g , 

and a:=36 

52 Let £P be the roq'* sum, then fioin Ai t 313, 

£800 £800 


P= 


(1 04)< 1 16986 
=£683 8 =£684 to ncaicst £ 


C83 8 


11.69,80)80000 
0803 
450 
99 

53 Space left uncoveied IS (32x26 -29^ y 22^), or 136] sq ft 
Since (32 x 25), or 800 sq ft cost £28 to co\ cr. 

Amount saicd=£28x^^=£l]-gY=£4 16? 4f;cf 

54 A, B and 0 togetbci fill I of a cat t in 1 inin 

A and 0 „ tV » 1 > 

1 1 
13 ^ ” ” 

B fills (I - Vr)j -V of a call in 1 ni'n 

01 ^ 


A and B 


^ ” (a 131,)’ " 20 

A „ or tV 


Rs 7 8 a should bo divided in the pi opoi lion of :iV * 
01 as 4 5 6, yl getting Rs 2, /? Rs 2 8n , <7 Rs 3 


J 


55 (i) 13s 3W X480=(13? 4rf -irf)x480 = (£= -7-;-5y)x480 

= £320- £1 =£319 

(ii) 24x25x26 = 6x4x25x26=6x100x26=16600 

03 1 

(ill) Incomo=^ of Rs 12000=^ of Rs 12000=Rs 300 






600 100 400’ 


shcMing the fust inicstiiicnt giics the largoi income 



171 


406] MISCELLANEOUS EXAMPLES Y 

56 7 men and 4 boys do f the work of (14 men and 8 boys), 

oi (14 men and 3 men), oi 17 men in the same time , 

7 men and 4 boys -will take 4 times as long 
to do a piece of woik twice as gicat , 

req^ time=17 days x x 4= 12 days 

^ •'17 men •' 

4P 

57 If £P be the Principal, the Interest =£j^ , 

4P 

leq^ tiine=£P— yrs =25 years 


58 


Let d feet bo the req"* depth , 100 gals = cii ft , 


9xrf= 


or d= 

0 2 3 ’ 


100 
BO OT 


® (nearly) 


59 


Let £.t be price of 3 %, and let £2A be invested 

Tlie income from 1'‘ stock is £4 x from 2'"^ stock is £3 x 

, ^ , nr 2A 4A . 3A lOA 

and the total income is £5 x H — =rr^ 

100 90 % 100 

Divide by A, and 


_4 3_£ 

90*^a“100’ X 100"90’ a~90’ 


and r=54 


60 B runs 7 5 nii per hr , or 1 mi in 01 8 niin , 
in G min B inns | nii , 01 1320 yds 

Q 

In 6 min A runs 1 mi x 01 1408 yds , 
req^ distance =(1408 -1320), or 88 jds 


K 


61 Let X be req"* number, 111 a year he sends 12r rupees, or 
£iacxT!3!^, 

12jcx-J^= 100, whence a;=125 


62 Eeq'* weight=7 207 gm x 2 3 x 2 3 x lOOO 
=5 29x7 207 Kg 
=38 Kg 


7 

207 
5 29 

OT 

0 

1 

4 


_6 

OT 

•0 


K n s I 


M 


K.l> 



[page 


172 MISCELLANEOUS EXAMPLES V 


63 The two kinds of eggs aie worth 135 at (2s 3rf +ls 9c?), 
or 135 dt 4s T, scoie=4s x6|=£l 7s 
220 lbs of lard at Sjc? =110 lbs of lard at Is 4^c? Combining 
this with the buttei, we get 110 lbs at (Is 4^c? +ls 7^o?), 
or 110 lbs at 3s a lb =£16 10s 
106 cakes at 11s 3c? a do? =318 cakes at 3s 9c? a do? 

Combining these with the biscuits, 

we get 318 ai tides at (3s 9c? +4s 3c?) a do? , 
or 318 at 8s a do? =318x-j®^s =^ 318s =£10 12s 
Total=£l 7s +£16 10s +£10 12s =£28 9s 
Discount=10c? X 28-j^=(280+4^)£? =£1 3s 4W, 
giving £28 9s — £l 3s 4ic?, or £27 5s 3Jc? 
for the discounted bill 


64 Suppose I hold Rs v in the 3 pei cents and Es (7200 — x) in the 


*1? 

5 per cents , the incomes fiom these are Rs 


7200-1! 


7200 


and Rs 5 X — — Since the total income =Rs 3\ x 

„ 3,+5(7200-r)=11x720H 

and 1 =6000 

I hold Rs 6000 at 3 %, and Rs 1200 at 5 % 

65 In the mi\tuie of 9 gallons, 

'ilcohol=(-j=^ of 3+^^ of 5) gals =4^ gals , giving 50 % 

Igi 

66 Lot his wages be r i upees per week His weekly savings= ^ x 

87-4 100 

1 upees, and his weekly expenses = i;Xy^ rupees , 

87i 110 

his increased weekly expenses = i x x i upees 
Since his inci eased weekly wage is (a7+lj) rupees, 
he saves + li - -c x x rupees pei week 

52(^+U-^x^)=52a!xJA^-13 

Divide by 52, and 

3 V 1 

giving r+--_=-__ Multiply by 80, and we get 
80t+120 — 777!=10r— 20, whence a;=20 



406] 


MISCELLANEOUS EXAMPLES V 


173 


L 


67 

68 


(i) E\p"= 


45 + 2V{r 


= 1 


_3_+^ 
(ii) E'p"=^^-^ 




16 0 _ 
OXT T _ 1 

6020 123 

770 


. . _ . j „ 60 niin 

(0 Eeq-i speed=2 im 


=3 mi per houi 


(ii) Eeq^ speed=3 mi in 1 lii =(3x1760x3) ft in (60x60) secs 
= ft , or 4 4 in 1 sec 

60X60 ’ 

After going half a mile, 30 min remain in which he must walk 
2^ mi he must ti alk 5 mi in 60 miu , oi 1 houi 


69 Let the atch cost Rs t , he sells it foi Rs i x , but this is the 

1094 

same as Rs 115 X rx-j^=115x-j5|/^, and r=100 


70 Let P,o bo the population 10 years ago, P the population to day , 
then P= P,o X Also P= 22000000 x =23100000 , 

PioX 1-^8^=21100000 and P,o=23100000xf“-^=21000000 
Req^ incrcase= P - P,o=23100000 - 21000000 = 2100000 


71 4 qts weigh 10 lbs , 1 qt weighs 2 5 lbs , and 1 lit weighs 1 Kg , 

01 2 205 lbs 

1 qt 2 5 lbs 2 >00 500 10 , , . 

1 lit “ 2 205 lbs - 2205 “441 8 8 

and 8 8 qts =10 lit 

If y quarts are equnalent to x litres, then jq or 

Take 0 5" to 1 liti e hon/ontally, and 1" to 1 quart vertically 
As in Alt 277, E\iiraple, if -we ]oin the origin to the point 
(10, 8 8), the giaph is cmtained and the icquired values may 
bo read off 

72 Total cost=6441000 fl x 1000 

=£531 8 X 1000=£532000 (nearly) 

Since 110 Km =110x g Km , 
cost pel mile=£532000x— ^=£f^ x 1000 

=£lA£l_o X 1000=£7 7 X 1000 

" = dB8000 (nearly) 


6318 

1211 )644100 

38G0 

m 

106 



174 


mSCELLANEOUS EXAMPLES V 
M 


73 Eeq‘‘ int =Es 8740X 


S 2 0 ^ 4 C 
3 6 5 ^100 

=Es 8740xff§xf5% 


=Es 


1 730S S 
73 


[page 

237 057 
73) 1730=) 2 

650 

39 


=Es 237 057=Ea 237 la 
74 Let him hold £a? stock , his income (after deduction of income tax) 


IS 


a?x2+ 19 
100 "^ 20 ’ 


■c X 2^ X 19 
100x20 


= 639 ^, 


19 X a: 10792 10792x40 

or .A =-H7r-) and x= 77; =£22720 


20x40 20 


19 


75 


Area of largei =10 x 1 01 x 1 01 ac =10 201 ac. 
diff of aiea=0 201 ac =972 sq yds 7 56 sq ft 


484 
2 01 

968 
4 84 

972 84 sq yds 
9 

7 56 sq ft 

76 1 sq mi , or 640 ac =625 sq in , or 30 ac =AA^^^ sq in , 

req* length of 8 ide= m x •JZva. x 1 73205 in 

— in =5 41 in (nearly) 

O 

77 Let X rupees be pnme cost , then, as in Ait 293, 

y V 1 20 y 1 as V 140_iql „_ 105X100X100XI00 _ pi 

^^TTnr’^T?nr’'TDTr-J^«*8> ana x— 8x120x123x140 “”4 

78 Net profits+balance-£5000=£26613 8s 6(7 
Thus £26613 8s Gd is paid on a capital of £500000, 


or £: 


26613 8s 6(7 
5000 


£100 


This gi\es a dividend of 5%, and the remainder, £1613 8s 6(7, 
IS earned foiward 


N 

79 1 cm per 1 sec =19 8 Km per 1980000 secs 

Now 1980000 secs =— yrs =-k»- 8 . oo yrs 

00X24X305'' 6260 •' 

=3 8 yrs 


3 76 


525,6 )19800 

4032 

353 



MISCKLLANEOUS EXAMPLES V 


175 


80 Let rcq‘* pi ice be i fi-iiic'i 

Tlicn t X 100=(2 75 X 80 +0 5 X 20) X 

=(220+]90)x 1-^ , 

, _ •‘j-oji-iio —A m f, 

^ 1 0 DM 0 0 * **1 I* 

81 Tlio Intel Hill aoIuiiio of the box is (17 -2)®, oi 15’ cii in , 

15® 

l')’’cu in would contAih oio worth £421{^, oi £-^ . 

hence 2* cn in „ „ „ £l 

But the 01 e in the bo\ is ictuallj woith £(42]|-78j), or £7®, 
the intci mil \ohinic of the box nnist be 2®x 7® cii in , 

01 1 1® ell III , 

and tho thickness of the umttiinl=i(17 -14) in , or li in 

82 Re 1 pci da3 =Rs 365 pci nuiiuin , 

1 eq'* pci centagc= 100 t ^ " DTr=" i" i 

83 TIic second cichst iidos 1700 3 ds 111 200 sees , 

„ „ 17G0x’-g^, 01 171G3ds in 105 secs 

But tho fii-st iidcs 1700 3 ds in 195 sets , 
leq' st<ijts=(1700-171C) >ds =11 3ds 

84 DiiT of the two intcicbts=intcicst on fust itai’s intci'cst , 

Rs 115 "la. -Rs 112 8n , 01 Rs 2jtj is intcicst on Rs 112 8a , 

01 Ri „ „ Rs 100 


Thus req'* inte=2l and req'' Piiiicipal=Rs 112A x-^=Rs 4500 


85 £11 18t If/=£12-1^ 8(/=£l2--,Vt>f 

£2 12ji Cr/= £3-7s Or/ = £1-J of £.1 

86 As in Alt 302, En 2, wt haio 


O' Px]-}f^-^=r»18 5G2',, 
P=45485G25x.} 


4'>48562 6 
14G 


45185G25 
18I0-f2‘4)0 
2723137') 0 

146567 ) 664000 12) 0 { 4530 0 
77822 
1038 
Ml 
10 



176 


MISCELLANEOUS EXAMPLES V 


[page 


87 (a-)’cu in weigliOGaS lb or^lb 

(|)“ weighs 1 lb 
53 

(|)®x40, or ^cu in weigh 40 lbs 
length of edge=A m =2^ in 

88 The engine must lun a number of feet equal to the LCM of 

13o ft , 275 ft , and 1320 ft mile) 

03 03 52 1 1 

This LCM =23 33 53 11 ft = ■ „ mi 


23 33 53 11 
' 2® 5 11 3 


17G0 3 
mi =11J mi 


89 


The fiist kind costs hiin of 20*1 , or 16 a pci lb , 
Izo 


the second 


100 

112 ^ 


of 27 a , 01 24 a pci lb 


Selling at 23 a per lb , his gam on a mixture of 2 lbs is 
46 a -(16+24) a, or 6 a 

on an outlay of 40a he uill gain On , giving 15 % 


90 His income = of £19250= £577-? 

An inciease of 10 % on this =£57^ 

Each £100 stock sold icaliscs £09 (xisli 
This buys stock it 101 and brings in 
an income of £4x^'y’^, 01 ££^ 

increase of income pci £100 stock sold=££-^-£3=£-3-J- 
Hence an increase of £57| in the income icquiics the sale of 
£100x£-;x57^, 01 £7150 stock 


91 Suppose ?/ a& are reaped bj v men Then, as in Ait 277, E\ , 
Taking scales of 0 1" to 1 aa lioi i7ontally, and 0 1" 
to 1 man vertically, if we join the oiigin to the point 
(24, 29), we obtain a giapb flora which we can read on the 
required values 


92 Let req^ cost be % shillings Then, fiom Ait 293, wo have 

, v=I-Oi^-3-OiI=Aoo=55| 

req'* cost=Hs 56 



MISCELLAiNLOUS EXAMPLES V 


177 


400 1 


P 


93 Ecq* intcrcst=R'’ x ~ y 2378 

_T>, 

■*' ioo'"ie 3 
=R<!l 21 R 3 
=R'! 12 3 a 


2378 
187 

2378 
1002 4 
lG(i 4f> 

30.5) nib bG(l2183 
70G 
GC8 
T03G 
IIG 
G 


94 


95 


36 in cost 59rf , or 

— 5—. in co'st X 2'> 2 fr , 
and 1 ni cost ft , 

o, 1 m cost ' n>o>> lo . s : f, ^ 

«p 4 

or G 77 fi 


5 0x0 7^30 37 
21 

_4 13 x 30 37 
24 

_1G2 3031 
“ 21 


20 3217 


=G 774 


1000 fills = 1000 X 0 16 ui ft =160 tu ft , 

160 x 60 x 21 cu ft aic djsclnigod III ,i da\ 

If 1 IS rcq"* no of dnsb, then 160 x 00 x 24 xt = 300x300x20, 
and r=7{ I) 


96 The outl 1} on 120 cfi"s=:Rs 3 12 i +Re 14a =Rs 3 Since 20 

nio broken, the «!cllinfi |iiuc of 100 innst be Rb 5+Re 1 14a , 
I c Rs 6 11 i , 01 110 a Mcisuie 10 inn.i'j to the inch hori- 
xontalR, and 10 eggs to the ineh acituiilh The equation of 

thegnph Mill he oi Tom the point (11, 10) 

to the oi ifiin, and fi oni this graph i c id ofl the i cqiiii cd a allies 

97 32 sq jds 1 nj ft =289 sq ft =17- sq ft , length=!l7 ft 

aic.t to 1)0 ])aintcd 

=(17y 11^x4+17 X 17-7^ x3’, -51, X 3;i) sq ft 
=(17y 1G*+17X 17 -23-20) «q ft =1026 sq ft 
ll 17 

Hence cost=“,j a x 102G=-g-y57a =Rs 30 4 a 6p 


98 


Let £t be icq"* gross incunic Then net income=t 
Kom £513 16s 6cf =£513-,'^=0-Y(/~ 


-cx 


jB 

160 


X 


eai _ gorra 
_*o 4 0 " » 


-Of- ixinpyj^o 
40XBSXej4 


= £620 



178 CONTINUED FRACTIONS [CHAP 

99 Let V cu cm be tbe volume of each pellet , 
then 58 V=22x 1 xO 7854 , 

V=:. * x . o ..."- 8 s 0 8-5 416=54 2 cu mm 

58 SO 

100 Let L c\\ t be the load when AP=a; ft (BP=(10 - r) ft ) 

Then, flora Art 283, L= When x=5, L=12, 

and ue find ^=300, 

To find the req^ loads, we put ar=l ft, 2 ft, 3 ft, 4 ft succes- 
siveh, and find L=33J cwt, 18| cwt, 14^ cwt, 12j cwt 
respectively 


1 

4 

7 

10 

13 


EXAMPLES XX a Page 411 


riact“=- 




Exp»=7- 


Fract"=- 


2+j 
1 

2+'J 
2 

8-V 

±1 

6+iri 

3 


2 Fiact"=- 


5 Piact“= 


8 Fract”= 


6 

4 . 1 0 
T-rj 

11 

3+1^ 

.-It*. 


3 Exp’‘=2+ 

6 Fract"= 

I- 

9 Exp''=5- 


1+i 

10 


f •§* 

2 

4-1 

2 +-®- 


Tir 


11 Fract"=-— ^xlO| 12 Fract"= 

% ” T"? ^ 


2 + 


Exp^=.__^=,_3 

lA 


3f 


14 Exp”=2-- 


8 - 


lA 




EXAMPLES XX b Page 415 


1 

2 

3 


Tlie first twoconveigentsaiefand^ 3'^convergent=L§i±==-, 

® 3X1+1 4^ 

4“ convergent= f^f±i =ff , 5‘» conveigent=fff|.^-« 


The first two convergents aie x and V' The others aie 

13>=a + 3 SOXl + 13 4SX3+ 3 D 166X6 + 42 

4X2+1’ 0X1+4 ’ 13X3+9’ 48X6+13" 

The first two con\ eigents aie -x and %■ The others are 

- y 3 + 1 7X1 + 2 . 9X4 + 7 43X2 + 0 0'X6+4 3 

6X3 + 2’ 17X1 + 6’ 22X4+17’ lo''6X2 + 22 ’ 232X6 + 106 



X3L] 

4 


CONVERGENTS 


As in Art 336, Tre have 

3 
o 


1 

79 

291 

6 

25 

54 


1 

4 


Tbisgiies ^ ^ 


_i i_ 1. 

2+ 0+ 4 


The first ti^o conierg** aie ^ and -j- , 


the 3”* and 4‘*' are 
5 As in Art 336, vre have 


1X2 — 1 3X6 + 1 


4xe+j> iixfl+4 


625 6 

431 3 
10 10 


Thisgiies^^^^ 


The first two converg** aie ^ and -ft 
the 3"* and 4‘*‘ aie 


lJXo+6> 46X1+13 


6 As in Art 336, we have 


This gives 1 + ^ 


2 233 313 1 

10 73 80 1 

3 3 7 2 *=‘ ■" -*2-^1- 

1 ■ 

The fiist two com erg** are -J- and 2 » 
the 3*^ and 4‘» are 
As in Art 336, v.e haie 
4 349 1128:3 
4 25 8113 

1 6’6 

The first two converg** are t and -V* i 
the 3"> and 4‘- aie 

As in Art 336, we ha\ e 
jl 
5 
1 


1 1 1 . 
lO-r 2+ 3 


This gives 3+^ ^ ^ I 


7 

1000 

1139 

6 

27 

139 

3 

3 

4 



1 


This gives 1+^ i 


The first two converg** are -I- and -5^ , 
the 3"* and 4«- aie 


9 As in Art 336, w e hav e 


1414 1 1 


11000 
172 
32 
2 , 

The first two converg** are -j- and -r , 
the 3*^ and 4**> are 


414 2 
70 2 
6|3 


Tins gives 1+^ ^ ^ ^ i 1 


179 



180 CONTINUED FRACTIONS [CHAP 


10 As in Alt 336, we have 


1 

393 

1000 

5 

179 

214 

1 

4 

35 


1 

3 


Tins gives ^ i ^ ^ ^ 1 


The first two converg** aie ^ and -{ , 
the r and 4‘- are 


11 


As in Art 336, we have 


3 

3029 

10000 

6 

290 

911 

2 

32 

43 

10 

10 

11 



1 


This gives 


_i 1 1 1 i_ 1 L 

3+ a+ 3+ 0+ 1+ 2+ 1+ 10 


The first two conveig‘* aie j and -jV , 


the 3"* and 4“* are 


3X3+1 1 OXfl+3 

10X3+3> jaXO+10 


12 See Alt 339, E\ 2, and Ait 340 
The 4“ and fia’ Converge* are and 

the error in taking for the C F is less than 

aUo Si^lOu^SS 

13 1 metre— 39 37 in =1 0936 yds Exjness this as a CF 
"We obtain aT for the 4*'' Conveig^ 

1 metie=^yds (ncailj), or 32 incties=35 yds (neaili) 


14 One div“ of the first = div"* of tlie second 

Expressing this as a C F , we obtain j ” as the Conxeig*- 
one div" of the fiist=^ J div"‘ of the second (nearly), 
or 31 div”’ of the first=40 diV” of the second (neaily) 


15 See Art 339, E\ 2, and Ait 340 The 6“' and 7“‘ Conveig** of 
1 41421 are Ag- and Of these, is greater, and -f-Jg- 

IS less, than the C F Since, then, the C F lies betw'een 
them in value, it follows that the dificience of gg- and the 
C F 18 less than gg _ or . \ 

70 100’ 70X100 


16 


See Alt 339, E\ 2, and Art 340 

The 4‘'' and 5‘'' Converg'* arc f-gandJ-Bg 

the 0 F differs fiom AAl by less than — ; — k; , or 

llo ssXlOXllS’ 

Thus IS the req*® approximation 


1 


3680 



CONVLRGENTR 


181 


XX] 


17 


19 


Tlio •1'*’ ind .'>*’> Coii\ejg^aie md pd“t Since the CF lies 
between these in \'ilnc, it follow ■; that the CF diifcis from 
■34 1 j 5 less th.in -rr-T^* rrvrs-ri which is less than 
0 001 

Isow IS ncaier the C F in \a1nc than J-J 
the error in this cise is ilso le^s than 0 001 

1 fr = — =sj!« 

C^-0 *3 

The 3"* and 4**" conieig^* of o® iie and 

As in No 17 it niaj be shewn that 1 fr differs from 

01 fiom by a quantity Itss than 7 - 7 — f > or , 
foi IT) fr the diff is less than If 


20 lies between 5 and G Put \^=5 + r 

(o + r) —26, 01 10r+jr=l 
^ 1^1 1 1 
^ 10+ > 10-*- 10+ 10 + 

The 3*^ and 1‘'' com erg-* arc and J-o£'5'i 
rroin Alt 310, it foUow*s that diffciN fioni 

b) a fraction l(ss than :T r o,oxiM¥ oT : F iFv r 6 ' 3 0 - i 

21 Considci the deciiinl poition 0242218 It icduccs to a CF 

whose first and third coniei gents aio ] and thus shewing 
8 daja in 33 jcaic to bo nioic correct than 1 daj in 4 } is 
Nov 8 dai s in 33* i rs =0 21242 1 day in 1 y 1 

eiior for 1 yr =(0 242121-0 212218) d.iy =0000206 daa 
1 da> would bo the erioi for — —A — — yrs , 01 4854 3 is 

• OOOO^OO*'^ 


EXAMPLES XX c Page 418 


1 (i) 27’i3 

31 

8259 

^ 30,1 29 
8652 20 
Subt Ttiiini ^9 
864883 
1 C 8649 


(») 


Subt 


ic 


32 7 
16 8 

.549 .36 
31 

1648 08 
78 48 

1726 56 
69 

1723 87 
1726 



182 approxlmation [ciur 


2 Eeq'’ ai ea=(62 3)- x 

=3881 29Xff 
=12193 46 sq m 


0 31830089 
3 (i) 3 14169C25 ) I 000000000 
57022200 
26I0G 278 
073037 
~ 31009 *' 
2780 
272 


3881 20 

;u 

lien 87 
I 501 17 

12108 34 
Subi Tsririj 4 88 
12103 46 


iV/? From Alt 313, 
-=314159265 


(") i-W-ir+roTr+miini) ("*) I?cq'' dmmcloi 
=0 3333 -(001 +0 005 +0 00002) j,5 

=0 31831 “ r 

=291 80x0 3183 m 


4 (1) 


R1 

3724 

f 110 3 
oi J8 8 
40731 


=93 m (nciih) 

(u) 5 gT 41 

,J- 40 78 

614 01 


SOI 

■S5 

0 3183 

IF 

> 

2 

9 

2 

3 


1 

92 

1-9 


5 


6 


giving 1073 ,m1s 
I b luaj bo eisili ahoiMi 
id” 

(i) 3600 

Subtinct 300 

3280 

Add -V ’>214 
1117 1 
gning 3337 lu 


gi\jng 61 1 jds 
that 1 +’>’( 5 r“ V 

\ iIm 

00 B^OO (=5 nnlci) 

Subtnrl 880 

7020 

Add Jc 120 71 
8015 71 
gi\ ing S04G m 


22 jds =20 in , 

1760 3 ds =1600 m 
But ncojds =1760x0 01430 ni 
= 1600 3 m 

Error=(l6093-1600) m =93 m 


on 

30 

i 

1 

on 

30 

040 

or 

01 

S(> 

ICOO 

32 



APPROXIMATION 


183 


XX] 

Foi second part of question, we have 

1 mile=80chains=80(20+-i^+-7V)®' =(l 600 + 8 +l) ni =1609m 

7 (i) 5 ft 3 in =63 in req^ eqmvalent=63(^^+-j^) cm 

=(157 5+2 52) cm =1 60 m , 


(ii) req** equi*=39 375(i^+-j-^) cm = 

(9844+1 57) cm =100m 

14 


353 55 

Add i~o'(r 014 


14 

A-dd. 1 0 0 0 6 00014 


353 55 

■A.dd. j'ij 0 '0 0 00007 


14142 

1 41421 


494 97 

=V2 

Add TO' o' 

4 95 


Add 1 0 0 u 0 

05 


Add 2 ou i) 0 

00 



499 97 


=500 m 


(i) -^=020000 

00 

(u) ^=014286 

71 

8 

o 

o 

II 

00 

i=0 02040 

82 

^=000800 

00 

i=0 00291 

55 

^=000160 

00 

^=0 00041 

65 

p=0 00032 

00 

^=0 00005 

95 

CO 

o 

8 

o 

o 

II 

40 

^0=0 00000 

85 

i=0 00001 

28 

^^=0 00000 

12 

^=0 00000 

26 

i=0 00000 

02 

1 

O 

o 

o 

o 

o 

o 

Ii 

-^la 

05 

i= 0 00000 

00 

^0=0 00000 

00 

0 16666 
ginng 0166667 

OT 

025000 

00 




184 


APPROXIMATION 


[CITAP \X 


1 

2^ 

1 

2 4' 
1 

2 1 G’ 
1 

2 4 6 8 ' 
1 

2 4 0 a 10' 
1 

2 4 0 8 10 12 ' 
1 

2 4 G 8 10 12 ir 
1 

2 1 6 8 10 12 11 lO' 
1 

2 4 G 8 10 12 14 10 18° 


;0 500000 

00 

:0 125000 

00 

:0 020833 

33 

0 002G04 

17 

0 0002C0 

42 

0 000021 

70 

0000001 

35 

0000000 

10 

0000000 

00 

0C18721 

27 


gning 0C48721 


11 - 1 - :^0 313 m 33 

1 

--^=0 33333333 -0'' 9) -011111111-0 =0012310 Gi 

;ri7-=001231’iG7-(3xr.) =000111322 -5 = 0-000823 01 
5 o’ 

~=0 0041 1322 -(Ox 7) =0 00015725 -7 =0 000003 32 

7 o* 

jj-L.=000013725-(9xO) =0 00005031-9 = 0 000003 G3 
-^,=000000305-11 =0 000000 51 

jgi^=O000005G5-(9xn)=O00000Ofi3-13=OO00000 03 

o^ic^ 

giMiig 0 316574 



CHAP XXI ] 


LOGAHITHMS 


185 


EXAMPLES XXI 1> Page 425 
9 log^=loga-log(6c)=loga-log6-loge 

10 log ^=log a- - log (itf*) =log a- - log 6 - log <? 

=2 log a— log 6-3 logo 

11 log ^^=log(a-6^)-logc®=loga-+log6“ -logo® 

=Moga+5 log 6- 5 logo 

12 log — - p — =log = log - log a^= log + log - log 

0sjct^ 

=ilog6+ilogc-21oga 

13 log36=log(22 32)=log2=+log3==21og2+2log3 

14 logj-Jg*logg3l^=logl-log(2= 3=)=-2log2-31og3 

15 log•/G48=log^/2^=log(25 32 ) = } log 2+2 log 3 

16 IogN/Mx4'2«=log(v^^ ■yF)=log(2^ 3^ 3')=log(2^ 

=log(2’ 3 '^) = Hog 2+^ log 3 

17 log ( = 3 ^ - rVff) = log 49 = log 7= = 2 log 7 

18 log =^+logVT"-=log(H V¥)=log =,=log2-log3 

19 We have log PR" = log A, or log P + 7i log R = log A , 

7ilogR=logA-logP, and w=— ^ 

—d? 

20 WehaAe log^=logV, 01 log7r+3logtf-log6=IogV, 

V hence logrf=-J(logV+logG — logr) 



186 


LOGARITHMS 


[chap 


EXAMPLES XXI c Page 429 
4-11 See Alt 368 

12 4 5703-5=t- ^^^ fi . " - P -aJ=T3141 

13 ^(3 8123)=^(6+3 8123)=T6354 

14 i(T 5632)=-i(r 1264)=i(5+4 1264)=1 8253 
16 i(2 1305)=-^ (6 3915)=-^(8 + 2 3915)=2 5979 

EXAMPLES XXI d Page 432 


1 log 283 =1 4518 

diff for 4= 6 

log 17 6 =12455 

difF for 2= 5 

log product = 6984 

2 log 8 03 = 9047 

diflF for 4= 2 

log 189 =I 2765 
diflf for 3= 7 

logproduct= 1821 

3 log 470 =2 6721 

dilF for 8= 7 

log 6 39 = 8055 

log product = 3 4783 

4 logs 7 = 5682 
log 8 9 = 9494 
log 023 = 2 3617 

log pioduct=I 8793 

6 log 31 9 =15038 

log 151= 1790 
log 9 7 = 9868 

log pioduct=2 6696 

6 log 43 =16335 

log 8 07 =_ 9069 

log 0392= 2 5933 
log product = 1337 


antilog 698 = 4989 
diflF foi 4= 5 

antilog 6984=4 994 

antilog 182 = 1 521 
diff for 1= 0 

antilog 1821 = 1 521 

antilog 3 478 =3006 
diff for 3= 2 

antilog 3 4783 = 3008 

antilogT879 = 7668 
diff for 3= 6 

antilog! 8793= 7673 

antilog 2 669 =466 7 
diff for 6= 6 

antilog 2 6696=467 3 

antilog 133 =1358 
diff for 7 = 2 

antilog 1337=1 360 



LOGABITJBIS 


187 


XKI J 


Numerator 

7 log 17 3=1 2380 

log numerator =1 2380 
12380 

subtract 2 409') ' 
log fraction =2 7686 

Numerator 

8 log 487 = 2 6875 

log numeratoi =2 6875 

2 6875 
subtract 3 8060 

log fraction =2 8815 

Numerator 

9 log 2 17 = 3365 

diff foi 9= ^ 

logminiorator= 3383 

_3383 

subtract 7 9 )29 

log fraction =0 3854 

Numerator 

10 log 0125 =2 0969 

diff for 4= H 

log numeratoi =2 0983 

2 0983 
subtract 0G133 

log fraction =3 4850 

Numerator 

11 log 2 38 =03766 
log 3 90 =0 5911 

dilf for 1= 1 

log numerator =0 9678 

0 9678 
subtract » 0 6839 

log fraction =0 2839 

KBS I 


Denominator 
log 294 0=2 4683 
diff foi 8= ^ 

log denominator =2 4695 

antilog 2 768 = 05861 
diff for 5= 7 

antilog 2 7686= 05868 

Denominator 
log 6390 =3 8056 
diff foi 8= 5 

log denominatoi =3 8060 

antilogl 881 = 07603 

diff foi 6= 2 

antilog 2 8816 = 07612 

Denominator 
log 897 =T 9528 
diff foi 3= ] 

logdononiinato] =T 9529 

antilog 385 =2 427 
diff foi 4= 2 

antilog 3854=2 429 

Denominator 

log 410 =0 6128 
diff foi 5= 5 

log denominatoi =0 6133 
antilog 3 4850=0 003055 

Denominator 

log 4 83 =0 6839 

log denominatoi = 0 6839 

antilog 0 283 =1919 
diff foi 9= 4 

aiitilog0 2839=l 923 



188 LOGARITHMS [CHAP 


Numerator 

Denominator 

12 

log 14 7=1 1673 



diff foi 2= 6 



log 38 0 = 1 5798 

log 387 =2 5877 


dift foi 5= 6 

diff foi 9= 10 


lognumeiator=2 7483 

log denoniinatoi =2 5887 


2 7483 

antilog0159 =1442 


subtiact 2 5887 

diff foi 6= 2 


01596 

antilogO 1596=1 444 


Numerator 

Denominator 

13 

log 925 =29661 



difF foi 9= 4 



log 159 =0 2014 

log 74 0 =18692 


diff foi 7= 20 

diff for 3= 2 


log numerator =3 1699 

log denoniinatoi = 1 8694 


31699 

antilog 1 300 =19 95 


subtract 1 8694 

diff for 5= 2 


log fraction = 1 3005 

antilog 1 3005= 19 97 

14 

Let ^= V5 1, 01 (5 1)^ Then log i=^log 5 1=^(0 7076) 


=03538 

=log(2 258) 

15 

Let ^='^11, or (11)^ Then log ^=-J logll=-J(l 0414) 


=0 3471 

=log(2 224) 

16 

Let r=( 097)* Then log t= 

=4 log (097)= 2 9868x4 


=5 9472= 

log( 00008855) 

17 

Let «=^/l015, or (10 15)* 

Then log r=| log (10 15) 


II 

CD 

O 

o 

11 

0 2516=logl784 

18 

[153 76=153 8 coriect to foui significant figures ] 


Numerator 

Denominator 


log 153 8=2 1869 

log 276 =2 4409 


log 0137=2 1367 

log 0038=3 5798 


log numerator=0 3236 

log denominator=0 0207 


0 3236 



subtract 0 0207 

antilog 0 3029 = 2 008 


log fraction =0 3029 




XXI] 


LOGARITHMS 


189 


19, [3302 7=3303 coirect to foui bignificant figuics ] 


Numerator 

log 3303 = 3 5180 
log 14 3 = 1 1553 

log ninnci‘ator=4 0742 
4 0742 
subtract 4 29^ 
log fraction =8 3785 


Denominator, 
log 0'iGl=2 7490 
log 387 =T 5877 
log 0001= 3 0590 
log denominator =4 2957 
antilogS 3785 = 2 391 x 10’ 


20 Let t=5® 

Tlicn log ^ =8 log 5 =0 0990 x 8 = 5 5920=log (3 90S x 10®) 

21 Let r=ll® 

1 hen log i = G log 1 1 = 1 041 4 X 6 = G 2484 =log (1 772 x 10®) 
22. Lot r=7' 

Tlien log t= 7 log 7=0 8451 x 7=6 9157 =log (8 235 x 10®) 

23 Let j?=13® 

Tlien log^=5logl3=l 1139x6=56695=log(3*711xl0®) 

24 Ix!t r=A/82058, or (82 558)^ Then log t = ’ log (82 558) 

= > log (82 5G)= ’ (1 91G8)=0 0389= log (4 354) 


25 Let i=17®x29® 

Tlien log'i=31ogl7+2log29=l 2304x3+1 4024x2 
=0 6100= log (4 130x10^) 

26 Let i=(2 301)® 

nicn log 1=5 log (2 301)=0 3619 x 5=1 8005=log(G 449x10) 

27 Lot t=( 089)'' Then log t= 4 log (•039)=4-(2 9494) 

= l(5 7976)=I±^pLP=T 3997= log (2 510x10-1) 

5® X 19® 

28 Let T= — — Tlicn loga =3log5+2logl9— 5log6 

= 0 0990 X 3 + 1 2788 x 2 - 0 7782 x 5 = 0 7030 =log 5 802 

29 Let or (3 47)-^ 

Tlien loga;= -41og3 47=0 5103x(-4) 

= -(21C12)=-3+(3-21612) [See Art 307] 

=3 8388=log(G 900x 10-3) 



LOGARITHMS 


190 


[chap 


30 Let «=fiaction Tlien, correcting denom^ to four significant 

figuies, 

logT=i(log 01367+log 0296-log 8735) 

=4(2 1358+2 4713-2 9412) 

=i(7 6659)=+(8+l 6659)=4 8329=log(GS06x 10-^) 

31 Let i=fraction Then log^=J(log 678+ log 9 01 -log 0234) 

=-i(i 8312 + 9547 - 2 3692)=i(2 4167)=l 2084 
= log (15 85), giving 16foi ansiici 


EXAIffPLES XXI e Page 435 

From Art 376, logA=log P+«log R 

log A = log 370 + 26 log 1 04 = 2 5682 + 25 X 0 01 70 
=2 9932= log 984 5, gmng Es 985 
A=Es 250 x(l 0i)«x(l 025)* 

log A =log 250 + 5 log 1 05 + 4 log 1 025 
= 2 3979 + 5 x 0 0212 + 4 x 0 0107 
=2 5467= log 352 2, gmng Es.352 
Fiom Alt 376, it follous that 

log P = log A - « log R = log 3000 - 15 log 1 035 = 3 4771 - 15(0 01491 
=3 2'-.36=log 1793, gmng £1793 ' * 

FiomAit 376. ^ - Log A - log P lo g 3000 -log 1130 
log R Jog 1 05 

— 3. 4T_7 1-10RT1 0 4J40 or> 

0 0.1^ -^j-57Y-^=20}eai’s 

If n be the leq'' no of years, then, as in 4, 

26532-24314 02218 
=13, 


n = log ^50 -log 270 
log 1 04 


0 017 


0 0170 

son will be (8+13), or 21 years old 
Let N be the req** population Then N =4459000 x (0 9477)* , 
log N=log^4459000+ 31og(0 9477)=6 6493+3 x T 9766 
=6 5791= log (3794000), gmng 3794000 
If ^ be the mean pioportional, then t*=2 87 x 30 08 
21ogr=log287+log 30 08=0 4579+1 4782=1 9361 
log r =09680, and t=9290 ’ 


If y be the thud proportional, then 

’ *^ 0 0238 * 

and logy=21og7 805-log(0 0238) 

=1 7848-2 3766=3 4082 , 

3^=2660 



LOGAEITHMS 


191 


XXI] 

„ ^ 25 2 x 289 3 6786 x 8= 

® (3^y 31= ’ 

log/=log578 6 + 2log8-21og31=2 7624+1 8062 - 2 9828 
=15858, and /=38 53 

9 If w kilograms be req** weiglit, w=0 00776 x 540 x 36 x 22 
log w=log( 00776)+log o40+log 36 +log 22 
=3 8899+2 7324+1 5563+1 3424=3 5210 
15=3319 Kg 

10 From formula we get 

log i =log T+ ^ “ logi7)=log 3 142 +i(log 150 - log 981) 
=0 4972+^(2 1761 - 2 9917)=0 4972+^(1 1844) 

=0 4973+4 (2 +11 844)=0 4972+1 5922 = 0 0894 , 
i=l 228=1 23 secs (neail)) 

VJ 

11 Wehave ;•==—, ]ogr=J(log3+logV-log4-log;r) 

Jog r= J(log 3 +log 248 6 - log 4 - log 3 1 42) 

= J (0 4771 +2 3956 - 0 6021 - 0 4972) 

=i(l 7734)=0 5911 , 
r=3 9 cm 

12 As in No 11, log i = J(log 3 +log V — log 4 - log s-) 

log (log 3 +log(18 2)= —log 4 - log tt) 

= KO 4771 + 3 y 1 2601 - 0 6021 - 0 4972) 

= J(31581)=1 0527 , 
r=ll 29 cm anddiam^=22 58 cm 

13 If T be rcq"* lalue, then log5;=logm+2logw— log2 

log r= log 9 17 + 2 log 17 64 — log 2 
=0 9624 +2x1 2465 - 0 3010 
=31544, and t=1427 

14. We have logF=logm+2logu— log 9 r— logr 

=log 24 7+2 log 60 — log 32 19 - log8 4 
= 1 3927 +2 X 1 7782 - 1 5077 -0 9243 
=2 5171 F=329 

15 Weha^e 21og'!;=log5f+log} — log289 

=logf|i|+log 4000-log 289 
= log 32 2 - log 6280+log 4000 - log 289 
= 1 ^79 - 37226+3 6021 -2 4609 =2 9265 , 
log ®=i(2 9265)= 14632 and «=0 2905 



192 


LOGARITHMS 


[CHAP 


If ^= vxSx ~ 60 * + 

or log v=log 2 -hlog 3 142+]og 400C^ 2905) —2 log 60 

=0 3010 + 0 4972+ 3 6021 - 1 4632 - 3 5564 
= 1 3807 Hence a:=24 (neat ly) 


16 Let W Ills be the req^ weight of watei 

Then W= 62 3 lbs x of vol of roller in cu ft ) 

4 m-l 
=^23x~ 

log W=log02 3+log4+logsr+2logr+]ogZ -logl728-log5 
=log 62 3 + log 4+ log 3 1 42 + 2 log 13 + log 36 

-log 1728 -log 5 

= 1 7945 + 0 6021+0 4972 +2 2278 + 1 5563 


ri 7945) 
0 6021 
0 4972 
2 2278 
Ll 5563. 

6 6779 


-3 2375 - 0 6990 


_ /3 2375) _ 2-4^4 
[06990/ ~ ^ 


3 9365 
W=551 lbs 


17 Let A cm be the req^ height (See Ait 284, III) 

log /i= J(3 log 14 2 - log 2)= J(3 4569 - 0 3010) 

= 5(3 1559) =10520, /£=113cni 

18 Let A cm be the req'' height 

For the 3 dl pot we haio and 

* (12 0)“ 1 pint ’ 

3 log /i=3 log 12 6+log 1 76+logO 3=3 3012+0 2455+1 4771 

=3 0238 , logA=l 0079, and A =10 2 cm 

For the 5 dl pot we shall have 

3 log A=31og 12 6+log 1 76+logO 5=3 3012+0 2455+T 6990 

=3 2457, log A=1 0819, and A=121 cm 

For the 1 lit pot we shall haie 

31ogA=3log 12 6+log 1 76+logl=3 3012+0 2455 

=3 5467 , log A=1 1822, and A=15 2 cm 

19 We have from Ait 376, 7 t=— g^~!PSP , 

logR 

„_ loglOQ-^og50 38 2-1 7023_0 2977_-o 

log 1 025 0 0107 0 0107 



LOG \MTIIMS 


193 


XXI ] » 

20 "Wo ha\o <i‘— Q 7 ^-, A\lieio V is \oIunie of cjlindei 

2logt/i=log V - log(0 7854)- log /i 

=log - log(0 7854) - log 15 3 
= log 2o G - log 1,1 6 - log(0 7854) - log 1 5 3 
=1 4082 - 1 1335 -T 80^51 - 1 1847 
=T 1949, log <7=1 5974, tind <7=03958 cm 

21 From foimuli in No 20, 

log A = log V — log(0 7854) — 2 log <7 

=l 0 g - ggg - -log(0 7654)- 2 log(03) 

=1og(450x 10'')-lo_g8 S8-log(0 7854)-2 log( 03) 

= 5 0532 - 0 9484 - 1 8951 - 4 95 12 
= 7 8555 , A = 71 G90000 cm = 7 1 C900 m 

22 Lot t=N''(l<7)>xH— L logt=4(5log(.it7)+logH-logL), 

01 log r=A{ri(log3+log<7)4-log H -log L} 

=i{0(log3+ 125)+log38- log 17G0} 

=^{5 X (0 4771 +0 G284)+ 1 5798 - 3 2 155} 

=1 { 5 5275 + 1 5798 - 3 2 155 } 

=lO 8018)=! 9.309, 

x=85 29 gallons 

23 Lot V in bo icq*' picssine Thou t =(} 39^)’''x30 in fust case 

log p= 15(log 130 - log 147)+ log 30 
=15(2 1139-21071)+! 1771 
=15y(T 94G0)+1 4771=1 1990+1 1771=0 0701 
1=4 743 in 

In second cise e get 

log r=50x 19400+1 1771=3 3300+14771=2 8071 , 
1=001.113 

24 If psq ft be icq** aica, then •/(O 020x12)-’+ 10- 

7 c i = IGn/CO G2G)-x l-'x 4-’+fG-’=16x 4^(0 620)- X 3-! +4= 
=10 X 4 X 2 \/(0 J13)-y l^+2-*=128\^4882 

log t= log 128+4 log(4 882) =2 1072+4(0 0886) 

=2 1072 +0~34+3=2 4515 , 
jr=280Rq ft 



MISCELLANEOUS EXAMPLES VI Page 448 

Most of the graphical eramples m this set depend on the methods ^ 
Jit^ 377 and 378 In many cases a reference to a diagram ‘will he 
all the solution necessary 

1 M.iik off OY=6 0' to repiesent 60 miles Through Y draw Y2 

parAllcl to OX 

Join O to the point m hose abscissa along OX represents 1 hour, 
and hose ordinate measured up from OX is 12 miles,^ The 
oidinate of an^ point on this line will measure the distance 
tiaielled tow aids Y by the first rider 
Join Y to the point whose abscissa along YZ represents 1 hour, 
and whose oidinate measured do\rn wards from YZ represents 
9 miles The resulting hne'wiU shew the distance tra.\elled 
towaids O at the end of any time (For this line all ordinates 
must be diawn to YZ ) 

The two lines cut at the point whose abscissa is 286", which on 
the given scale repiesents 2 his 52 min 
The vertical distance betw een the lines is 1 8 ' for the ahsicssa 2 O', 
1 e the ndeis are 18 miles apait after 2 lii-s 

2 Take the scale as in Ex 1 The graph representing A’s motion is 

the line joining the origin to the point (1 O', 0 6 ') iTs graph 
IS found by joining (1 5 , 0) to (2 5", 0 8') Tliese tw o lines meet 
at the point (6 0", 3 6') Hence the time of meeting is 6 0 p m 
Measuimg the diffeience between the oidinates of coirespondmg 
jjoints on the two line's, we find thatd’s graph is 05' abo\c 
Us foi the abscissa marking 3 30 p m , and that therefore A is 
3 miles ahead of A at that time Similarly i? is 3 miles 
ahc.id at 7 30 p m 

3 A convenient scale is 25 miles to the inch vertically, and 1 hour to 

the inch honzontallv The graphs are shewn on half this 
scale in Fig 17 Tfie answeis aie given by the coordinates 
of P and the abscissa of LWl, HK 



195 


PAGE 449 ] MISCELLANEOUS EXAMPLES TI 

4. Let Rs ?/ lepresent the ■sal iij' iftei ir 3 eais, Ks a, Rs b the annual 
inci ease and the initial salary 1 espectn ely 
Then we have the relation y=ar-\-h, which is the equation of a 
straight line A scale of Rs 800 to the inch veitically, and 

10 yeais to the inch hori/ontallj will be found conienient 
The line can then be drawn b^ joining the points (6, 1280) 
and (15, 2000) The initial salary is gi\en b> the intercept 

011 OY, 'and the salary foi the 21st year is gi\en by the 
ordinate coi responding to an abscissa 20 

5 CP+y\^of CP=16s 4rf, CP =14? of 16s 4rf= 14s 7c? 
Profit on each article =£10 18s 9c? —100= 2s 2^0? 

Selling Price = 14s 7c? +2s 2|e?=16s O^c? 

6 Reckoning in thousands, the total no of gallons supplied 

=16 5 X 610+ 17 9 X 730+ 13 3 x 520= 10065 + 13067 + 7956=31088 
The total population =610 +730 +520 =1860 , 

ai erage supply per head= W«nf =^6 7 


24 - 

7 Suppose he holds &x stock , his income fioni this is £ of 

19 2^ 

and after income-tax has been deducted, it — of of x , 

19-r2A , , 8908 x 200 x 20 

^of j^of r=4484rj and x=- . -- —a - - = £18880 


20x5x19 


8 A runs 3^x9 ft while B luns 4| x7 ft, 

or A „ 14x9f6 „ B „ 17x7 ft, 

or A „ 126 ft „ B „ 119 ft , 

A catches B in 126 ft when B has a start of 7 ft , 
or A „ Z?„18ft „ B , „ 1ft, 

or A „ B „ 18 X 25 yds „ B „ „ 25 yds 

A must inn 18 X 25 yds , or 450 y ds 

9 The graphs are shewn in Fig 18 A’s graph is the broken line 

XDEM, his rest of half an-hour being shew n by tlie horizontal 
line OE B’s graph is the line HKN The "solutions are 
given by the abscissce of P and of the points where KL, MN 
meet the hour-axis 



go 9 48 10 


196 


MISCELLANEOUS 


[page 



OB 


197 


4-19] EX^lMPLhS VI 

10 Sco Tig 10 OY IS of nny comcinenb length and lepiesents the 
time taken In A to iiin 120 j.nci'' In the same tune 
/} and C ^\ould lun 100 3 iids and 80 jaids lespcctuely 
ITonce the gnphs of the tin cc runnel's aie found hj joining 
O to the points nmked A, B, C in the figuic Find the point 
Q on OB ^^hn,h coi icspond** to an iihscissa 80 yaids, .ind draw 
a horizontal cutting OC, OA in R and P Ihen the distance's 
lun In C and A wlicii Zi h<is gone 80 j’aids me gnen by the 
absciss TJ of R and P 


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Flo 19 


11. I ig 50 lb dniw n on a 1 educed scale 
/I’s gi'apli IS the line OP 

Since /J lias a start of 8 3 <h R is a point on his graph 

Also /j bo its A b3 1 second , S is a jxiint on his graph 

Thus />"s graph is the line RS 
.1 bo its C b3 40 3 d*- , Q is a point on C’n graph 

FindT the point on ^s graph corresponding to 15 seconds, and 
nieasiuc TK downw uds to icpicbciit 18 3ds Then K is also 
a point on (7's graph 

Thus C\ graph is the hue QK 'Wheii pioduccd this meets the 
time axis at L. Then, since OL lopiesents 5 seconds, C must 
has e started I seconds after A 

As the graphs arc tlirec patallH linos, the speeds of the runners, 
which are incasuicd 113 the slopes of the lines, are equal and 
the race would end in'a dead heat 






198 


MISCELLANEOUS 


[page 



12 Let d ft be the req^ depth 

cu ft of water m cistern =5x4xd, 
cu ft of watei on xoof =20xl2x-j^ 
5x4xtZ=20xl2Xy'S2-, and d=2 


13 


14 


fr 25 105 =£1 , 
fr 430 50=£i|^»- 


17148 

25105) 43050 

179450 

3715 

1204 

~200 


TlieSI foi3 3rs at 5% = j-^of Ks 401-|x3=Rs 60 4a 


The amount at C I foi 3 j is at 5 % 

=Es 410|-x(l 05)3=Rs A^-lx 1 157625 
=Rs 3213 X 0 144703125 =Rs 464 931 
=Rs 464 15 a 


0 144703125 
3213 


434100 

4 

28-940 

6 

1447 

0 

434 

1 

464 931 

IT 


Cl =Rs 464 15a -Rs 401 10a =Rs 63 5a 
req« diff =Rs 63 5a -Rs 60 4a =Rs 3 1a 


449] EXAMPLES VI 199 

15 Space left uncovered =(30 x 28 - 27V x 25+) sq ft sq ft 

B S B 

This -would cost Rs 420 x .■ * ■ or Rs 69^, to carpet 

30 X 28 

Rs 69 h-, or Rs 69 6a , is saved 

16 1 lb av =7000 gr of pure gold 

Each £ contains of 123 gr of pure gold If x be req** number, 
■we have of 123 xa= 7000, ■whence 

Thus 62 sovereigns can be coined 


17 (Fig 21 ) A ’a graph is found by joining X to P Find the point 
R on this line ■which coriesponds to 4 pm Then R is a 
point on R’s graph Thus jo’s'giapli is the line joining Q 
and R The solutions can no^w be read off from the figuie 



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200 MISCELLANEOUS [PAGE 

18 Since one goes 8 miles i\hile the othei goes 4 miles, the speed of 

the first IS tuite tli.it of the second Let their speeds be 
2a miles and a miles pei horn lespectnel}, and let them meet 
X miks fiom B 

The first goes (12+ 1 ) miles in houis, the second goes 

. , 12-ri 12+a; 12 — r 

(12- r) miles in — hours , - - — = 

' ^ a 2a a 

Multiply by 2a, then 12+i=2(12+i), and t=4 

For graphical leiification, take a scale of 2 miles to the inch 
a ei tically and suppose A at the oi igin The intes ai e as 2 1 
Take as the unit of time that interval dunng which the 
fastei goes 8 miles, the slower 4 miles Bcpiesent this unit 
bi 4 in on the t axis, and along it maik 4 in , 8 in , 12 in , 
fioni the oiigiii, 1, 2, 3 lespectiiely 

The gi“ipli of the fastei will then be obtained by joining the 
point (1 , 8) to the oi igin It passes through the point , 
(1 3, 12) Now join the last point to the point (3, 0), and 
we cleaih obtain the graph for the return jouinej' of the 
fastei triieller Siniilailj the giaph of the slower traaellei 
is the line joining the point (1, 4) to the oiigin it outs the 
‘leturn’ graph of the /aster at the point (2, 8), shewing the 
men meet 8 miles fioin A i e 4 miles from B 

19 See Fig 22 A scale of 30 miles to the inch \ ei tically, and 1 In 

to the inch hoiizontally will be found iiioic coinenient than 
the one here emploj ed 

The beginning and end of his journey, being at difieient speeds, 
will be lepiesented b> different giaphs Tliese aie obtained 
by joining A to P and B to Q The abscissa of the point of 
lntLl>^ectIon of AP and BQ will gi\e the rcquiicd time 



201 


450 ] EXAMPLES VI 

20 Total increase=185000x^l^+23-)000x V6V*+325000 x2j^ 

=891 7 + 761 4+1677 =3330 1 
If ar % be the a\ ei age increase, 

the total increise=j^ of (185000+235000 + 325000) 

= r(1850+23j0+3250)=7450i , 

7450» =3330 1 and t=0 447 % 

• 

21 The total receipts foi the first four inonths=£l >"i5 Is 
The receipts for the rcninming eight months must be 

£5000-£l555 Is, oi £3444 19f , 

req** average takings =£430 12s 4f(f 


22 Since 7931520=765 x 2 * x 3^ it follows that the onl 7 pairs aie 

{765, 76> 2" 3‘} and {765 2', 765 3<}, 

becau'so, i. hen m c dii ide the numbers forming each pair by 765, 
tlie quotients are prime to each othci For if thc\ were not, 
their H C F would not bo 765 Tins maj bo seen in the case 
of such a pan us 765 2^ 3- and 76 > 2' 3^ , the H C F is not 
765 but 765 2* 3- Again t iking anotliei pan, 705 and 
705x2'*, their LCM is 765x2'*, oi 0120, not 7931520 In 
the simc way any other case mn} be dealt with 


23 Let the Sides bo 2 1 ft and 3i ft Area= 6 ?"Fq ft 

and cost=R«« 3 -^x 6 T=, x6r==24, 

and i-=141, oi i =12 , 
hence lengths are 24 ft and 30 ft lespectnelj 

24. Tlio companj paj s out of Rs 20000000 as interest on mortgage. 


If Rs » bo the gross annu il icceipts, then expenses are , 

40 ^ 

lOO’ “loo 


40 4^ 

then net annual receipts .uc t - - Trjk of Rs 20000000 



202 


MISCELLANEOUS 


[page 


tins must bs equal to the inteiest on the shareholders’ capitalj 
VIZ 6 % of Rs 40000000 

of Rs 20000000= of Rs 40000000, 
or ^-900000 =2000000, and ar=Rs ^552^222 

100 V 


the erage -weekly receipts 
_Tf aooooooo 

0X6 2 

— Tie 20000000 
312 

=Es 92948 11a 


92948 72 


812 ) 29000000 

2960 
1520 ' 
2720 
2240 
560 


25 18 eggs cost 12a , or 12 eggs cost 8a , but 12 eggs are sold for lO^a , 
gam IS 2ja on an outlay of 8a , giving ~x 100, oi 31j % 


B £17 5s 7d 

=£17 279 

1666 

=\alue of 1 ton 



40 



£691 166 

6 

= value of 40 tons 

2 cwt ='j^of 1 ton 

1727 

9 

= „ 2 cwt 

8 lbs =i^oflcwt 

061 

7 

= „ 8 lbs 

4 lbs = of 8 lbs 

030 

9 

= „ 4 lbs 


£692 987] 

1 

= value of 40 tons 2 cwt 12 lbs 


Giving £692 195 9cf 


27 Let the weights of unit volumes be 4s, 5s and 6s ounces re 
spectively, and let the substances contain 3», 4v and 5v units 
of volume lespeotively Then the respective -weights are 
4sx3«, *18x41) and 6s x 5® ounces, or 12s», 20s» and 30s® 
ounces No-w 5 lbs 13 oz =93 oz 

12sv+20s®+30s®=93, and s®=-J oz , 
hence 12s®=18 oz , 20s®=30 oz , 30s®=45 oz 
Thus the weights are 1 lb 2 oz , etc 



203 


r»i ] 


LXVMPLES VI 


28 (1 ig 23) TiKp 2ri niilc*i to the inch vcilitilh, and 1 horn to 
"the inch hoii/onUilK The fignie shewn is clmwii on half 
these St lies 

Mcisine ort AB alon;: the \tiliuil axis to lepie-cnl lit) miles and 
driM a lioii/ont.il thiough B Plot iho points whose co- 
rn dm ites lepi'csent (I hi , 26 miles), (2 hi s , 10 milts), ( } liis , 
()0 miles) ele , and join t n h sm tessn e p iii ns in the di.igirm 
Thu lionrontil thioiigh B is seen to be cut at ii point on the 
‘ 7 ’ oitliinle Hence B la rc.ichcd .it 7 p in 



29 


Tlie liorrontil through the point w hnso oixlin itc i oprcaonts 
48 nils fioin B (/t 71 inN fiom A) cuts tlic gnpli at P, 
whoso ab'.uss.v K pi c«eiit 17 pm .ippioMiinitch 


Increase of acreage between 180,i and lOCX) 
= J93770 at., and tins coiicsponds to 
4‘^tX)00(X) lbs 

1 ae cont tins ^ lbi , 

01 2 18 lbs * 

Foi second jj-irt of question, 
decrc.ase of ,ieieago=1107 ac 

decrease of cropt=^-« x 1107 lbs 
= 290700 lbs neulj 


2177 

10,3.7.7.0) 48000000 
0216 
1196 
110 

1197 V 18=67106 
206517 
10377 ) 67166 00000 
^020 
120270 
~1000S 
203 


16 









MISCELLANEOUS 


[PAGE 


30 (Fig 24 ) The graph foi the X to Y train must he obtained by 
joining the points whoso coordinates, measuied from X are 
(0, 2 33) and (35, 3 33) For at 2 33 p m the tiain is still at 
X,’ and at 3 33 p m has travelled 35 miles 



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Via 24 

31 I biglia IS rented at Es 600 
Now 1 bigha=(1600x9) sq ft 

hence (1600 X 9) sq ft are rented at Es 600 
(34 X 125) sq ft are rented at Es ” ° ” p o ' o x p ^ 

01 Es 177 083, giving for answei Es 177 1 a 3 p 

32 Let V rupees be the marked price He sells at rupees 

But he gains 15 % on an outlay of Es 72 , 

he sells at Es 72 x , 

90t 115 

hence i=Es92 

33 Since ^’s Tiages are greatei than B'b f<yr the same time, we must 

mulfply sn 10, V or ^ 

Again, since A takes less time than B foi the same piece of worl, 
we must multiply £11 lOs by (See Art 279 ) 

req'’ charge=£ll^x|^x||=£ll 



EXAMPLES VI 


205 


4rA ] 


34 For first period total Cl op 

=(13 22 1 X 15 2) iinllioii bushels 
=201 million bushels 
For second pci lod total crop 
=(17 ins X 18) million bushels 
=310 million bushels 

For first period, amount of uheit per head 
= ”4- bushch=6 bushels 
For second period, amount of Mheat per lieid 
= -7^^ bushels =8 1 bushels 


13 

1 

224 

152 

132 

2 

GG 

1 

o 

G 

200 

•9 

171 

198 

1 

8 

172 

0 

137 

5 

309 

5 


35 If the total e\pcn<!cs foi r bojs arc represented by £y, the 
annablc part may bo denoted b> £ar, and the constant part 
b\ Sfj lienee j and y satisfj a lineai equation y=a.r+6, 
V, hci e a and b arc constant quantities licnee the graph is a 
straight lino (See Art 378) 



lio 25 


As the numbers aio largo, it will be conacnient if uo begin 
measuiing ordinates at GOO, and abscissaj at 100 This 
enables ns to bring the requisite portion of the graph into 
a sni.illci compass VTicn a =105, t/= 650 , and when a?=128, 
i/=742 Thus two points P and Q are found, and the line 
PQ, 18 the required graph 

By measiiicmeiit we find that when r=115, y=G90, and that 
when v=710, r=120 Thus the required answers are £690, 
and 120 boys 


206 


MISCELLANEOUS 


[page 

36 As in E-c 35 the graph must be a straight line Hence ive need 
only join the points (12, 3840) and (16, 4320) and then read 
off the ordinate coiresponding to an abscissa 15 

A convenient scale is that of 5 patients to the inch hoiizontally 
and Es 200 to the inch veitically The veitical graduation 
may commence at Bs 3500 


37 Let the respective populations be 3a? and 8a?, then the total 
population IS 11a? , 

the total number of males is -5^ x Sar+^Vi? ^ 


01 in 11a?, or x ^ m 100, or ^ in 100, gn mg 45 % 
100 111 ? 


100 


11 


38 Internal edge=l m =100 cm , external edge =102 cm , 

volume of metal =(102^ — 100®) cu cm =61208 cu cm 
weight of metal =61208 gm x 7 =428 456 Kg 

39 Error in £ (vw ~ roVff) 

=£(100 - 96) in £100, or 4 % 


Again error=y^ of 567 f = 

40 After the first replacement, 
the first cask contains 
148 gals of wine I 
\l2 gals of water / ’ 

^ g i of cask IS wine 1 
of cask IS water / 

The 12 gallons then diawn, 
from the first cask contain 
4 of 12 gals of wine 

The amount left, 
in the first cask is 
48 gals of wine and water, 
of which -f- of 48 gals is wine 


x567f =22 68 f,or 23 f 


the second cask contains 
("12 gals of wine I 
\l8 gals of water/’ 

^ g / ■§■ of cask IS wine \ 
of cask IS watei / 

fiom the second cask contain 
•§- of 12 gals of wine 

in the second cask is 
18 gals of wine and water, 
of which ■§■ of 18 gals is wine 



452] 


EXAJIPLES VI 


207 


after the second replacement, 


the (list cask contains 
(J of 48+-§- of 12) gals of Mine, 

1 43 r gals of Mine \ 
\16J gals of ivateij ’ 


t e 


or of cask is Mine 

60 

or of c.isk IS Matei 
60 ^ 


the second cask contains 
(|- of 18+-| of 12) gals of Tvine, 
„ ri6jgals of Mine I 
\l3^ gals of MaterJ ’ 

C 1 ^*4 

I or ’ * of cask is wine 
30 

* ® l JOl 

j — or of cask is water 


The 12 gallons then draMn, 
from the first cask contain 
!-§• of 12 gals of Mine 

The amount left, 
m the first cask is < 

48 gals of wine and Mater, 
of which J S' of 48 gals is wine 

after the thiid replacement, 
the first cask contains 
(■J-S-of 48+ of 12) gals of wine, 
or 41 28 gals of m me 


from the second cask contain 
H of 12 gals of wine 


in the second cask is 

18 gals of Mine and watei, 
of which of 18 gals is wine 


the second cask contains 
(H" of 18+-’-S- of 12) gals of wine, 
or 18 72 gals of wine 


41 3 men and 4 bo} s do ^ of tli 

but 3 men „ 1 boy „ J 
3 boys do (-J-- J)i or -V 
and 8 boys „ j> or 
But 4 women and 8 boys do -J- 
4 women do (J - J), or ^ 


c M'oik in 1 day , 

„ 1 day , 

„ 1 day, 

„ 1 day 

„ 1 day , 

» 1 day, 

„ 1 day. 


giving 15 days for req’’ time 



208 


MISCELLANEOUS [PAGE 


42 £32 17s 


= £32 89 
2 

6875 

511 

657 93 

7 

164 48 

4 

3 29 

0 

32 

9 

fi 82604 

0 


43 9 big 18 cot 2 cbk — (0"t"oQ"f"iQX2o) 
=9 90625 big 
Es 1664 4 a =Es 1664 25 


req^ ground lent 
_T>s iic 

— 9 9 0 025 

=Ea 168 


IGS 


qqOG25) 1664 250 
G73 6250 
79 2500 


44 (l 24 X jYi) 

, 4 52X16X21X144 

time req®= 1 Ji x ' i l ®®“ 


^ 120^0^X3 7 1 gees 

= 100874 secs 

=28 hr 1 min 14 secs 


273x1260= 343980 
100873 8 
3 41)3^980' 

2980 

^20 

1^ 

307 

60 1 100874 
GO 1 16804 sees 
281ir 1 nun 


45 We have -§■ of time on coach =1 hr 

time on coach=A, oi 2| his 

Had he travelled this part of the joumei by tiain, ho would have 
been 1 hr less, oi 13 hrs But ho would travel the whole 

distance by tiain in 2 horns distance by coach =-g' of 
60 mi =45 mi 


46 


He has to pay back altogethei in four year's the equivalent of the 
amount of Es 344810 at 5 % C I foi that time 


This =Es 344810(1 05)* (See Ait 313) 
=Es 344810(1 21550625) 

=Es 419118 7 


344810 

0 

1 2155062 

344810 

68962 

3448 

1 

1724 

05 

172 

40 

2 

06 


06 

419118 

7 


Since he pays by annu<il instalments of Es 97240 8a , oi Es 97240 5, 
the hrst instalment paid at the end of the 5ist yen amounts 
to Es 97240 5(1 05)® in three more years at 5 % C I 
Similarly the second instalment paid at the end of the second 3 ear 
amounts to Es 97240 5(1 05)® in two more years at 5 % C I 



EVAMPLES VI 


209 


In the •%mo v,k% the thud in'^talment imoniits to Es 97240 5(1 05) 
.it the end of the tune 

With the last instalment, his payments amount to 

I 972401 5 


Bs. 97240 o (1 053 + 1 052+ 1 05 + 1 ), 
01 Es 97240 5(4 310125), 
or Es 419118 7 as above 


4 310125 

38S0C2 
29172 2 
972 4 
9 7 
1 9 


I 419118, 7 

_J> 0 IPOX-X-OOO _Bg H 
aixssos 7 04 10> 

31 /’^\2 -08/ 70410 . 

= £201 X X =£^-? ” 2 , 

0410> oaum 

req"* number of Napoleons =31 527 

48 Suppose he mixes a- lbs of the fii-st kind w ith y lbs of the second 
It costs him (3l r+5y) annas , he sells it at 4j(i+y) 

4i(r+)/)=a ]-g-gx(3lT;+5y), and multiplving b\ 100, 
450(r+y)=!l02(3H+5^), or 450i +450y=357» +610^ , 

931=601/, and -;=g=|, 

ho mixes 20 of fust kind with 31 of second 


49 Exp» IS greatei than or or ro 5 ^^, 

01 0 0000748 


Exp» IS less than or or 000008 


50 Let £r bo the buj ing price, then a- x\j^ is the selling price 

xuu 

Nott a loss of in the £ leaves £vx^^ out of £r 
- 240 

or |^-23375-|?,r, 

” (5ffi-S'-23373.or^r.23370. 
or r= = \Y.~ * ° ^ = 1 25 X 24 =£30 



210 


MISCELLANEOUS 


[page 


51 Let e shillings be the pnce of an English piece, 
and/ „ „ a Fiench piece 

Then e aiea of English piece 22^x12 5 
/~ aiea of Fiench piece ~ 18 X 9 ~3 

c=5 of /, and we must multiply the puces of the French 
pieces by 5 


On Level Uphill Downhill 

52 In miles / rate=4,I f JT’s iate=3, rate=5, 1 

pel hi \ra rate=3 jIPs rate=| of 3 /IPs rate=^ of 5 / 

_ ^ 

P leaclies C in liis, or 23 hrs , X reaches B in f hrs, 

01 2 hrs , and in 3 hr more he goes (3 x 3 ) mi , or 2 mi 

beyond B 

at the end of 23 his , X and T are 2^ mi apart 
X and Pthen go touaids each other at 3 mi an hi , and J of 
5 mi an hr lespectively , their rate of approach (see Arts 
216, 217) 19 (3+1 of 6), or 6| mi pei hr , they meet in 

|| In , 01 J hr In this time P ivill be (| of 5 ) x 01 1 J mi 

fiom C , they meet 11 miles fiom A 


53 Take a scale of 1 in to 10 yds along the axis of y 

Maik along the axis of a; the points 0 , 1 , 2 , 3 , 4 , etc, an inch 
apait 

Let 1 in along the r-a\is represent the time JJ takes to walk 
20 jds Now, Mhile B walks G yds, A \ialks 7i, hence, 

while B walks 20 yds , A walks 20 X 01 25 3 ds 

Since B has a stait of 20 yds, we shall obtain his giaph by 
]oining the points ( 0 , 20 ) and ( 1 , 40) Siniilaih A’s graph 
IS obtained by joining the points (0, 0) and (1, 25) 

( I ) AVhen B has walked 50 yds, he is (20 + 50), 01 70 yds fiom 

Me stait From the giaph A is then 62i yds fiom the stai t 
Hence A is (70 -62k), or 7^ 3 ds behind B 

( II ) When A has walked 130 yds , B will be seen from the graph 

to be 124 yds from the stait Hence A is (130-124), 01 
6 yds ahead ' 

(ill) It wnll also be seen that when B is 116 yds fiom the start he 
96 y^s°^ H must have walked (116-20), or 



211 


154 ] EXAMPLES VI 

54 Along OX measure off OA=2 6" to represent 2 hr 10 min On 

the vertical through A mark the point B, such that AB=1 0" 
Then since the man’s speed against the stieam is (4^—1^), 
? e 3, mis per hr , the first part of his ]ourne} is represented 
by the line ]oining O to (I 2", 1 5") He returns at the rate 
of (44 +H), le 6, mis per hr, and at 210 is 2 mis from 
his sorting point Hence the second pait of his motion is 
shewn by a fine drawn from B to represent 6 mis per hr 
This is found by joining B to the point whose abscissa, 
measured to the left of A, is 1 2", and whose ordinate is 4 0' 
The graphs cut at a point whose coords maik 5 nils and 1 hr 
40 rain Hence the distance rowed up stream was 5 miles 

55 Let req* height be h cm , then , 

/i.3=176x(12 4)3, and 31ogA=logl 76+3 log 12 4 
or log A= J( 2455 + 3 2802)=! 1752, and /i=14 97 cm. 

56 Its value in 1 yi =Es 27400 x 0 95 
Its value in 6 yrs =Ea 27400 x (0 95)® 

=Es 27400 x 0 735092 
=Es 20141 5 

57 Suppose a man does 5a units of ivork per day 

„ woman „ 3a „ „ „ 

„ boy „ 2a „ „ ,, 

Then 10 men in 25 days do 5a x 25 x 10, or 1250a units , 

„ 8 women „ 25 days „ 3a x 25 x 8, or 600a units , 

„ 6 boys „ 25 days „ 2a x 25 x 6, or 300a units 

Thus the work consists of (1250a+600a+300a), or 2150a units, 

and three times the \\ ork consists of (2150a x 3) units 

Again 19 men in 1 day do (5a x 19), or 95a units , 

20 women „ 1 day „ (3ax20), or 60a units, 

30 boys „ 1 day „ (2a X 30), or 60a units , 

giving a total of (95a + 60a + 60a), or 215a units 

they will do a vvoik consisting of (2150a x 3) units in 



212 


MISCELLANEOUS 


[page 


58 4 lbs of A should be mixed with 7 lbs of B , 

weight of A should be TX of weight of mixture 
But 4 cu in of A is actually mixed with 7 cu in of B, 
j weight of 1 eu in of A 151 
weight of 1 cu in of B 117 
weight of 4 cu in of A 151x4 _ 604 
weight of 7 cu in of B 117 x 7 819* 


and the weight of A actually is 
of mixtuie 

the percentage of A’s weight actually is jTrirX 100, oi 42 445 , 
but the percentage of A’s weight should be X 100, or 36 383 , 
eiror=(42 445 - 36363)%, or 6 08% 


59 


<1=0 8 in , 6=0 7 in , c=l in 

area of section in figure =(0 8 x-^) sq in =0 68 sq in 


aiea of section of haystack 

=(0 68 X 36) sq yds =(0 68 x 36 x 9) sq ft , 
and its volume =(0 68 x 36 x 9 x 30) cu ft 
Hence its weight= ° «8Xsax| | ? . () , x . i Q .^ i^=29 5 tons, 
giving 30 tons 


60 External volume=(3y x2j^xl^) cu ft cu ft 

=11302cu ft 

Since 1 gal of water occupies cu ft, 38 gals occupy 
^x38cu ft 

interval volume = cu ft =6 099 cu ft , 

whence volume of stone =(11 302 — 6 099) cu ft =5 203 cu ft 


Now 6 203 cu ft of stone weigh 874 lbs , 
and 5 203 cu ft of water weigh 6 203 X 62 3 lbs 

req<« number of times= ^^ 203 x 02 d 

— B7* 

— SXT 

=2 69=2 7 


6 

203 

G 

23 

312 

0 

10 

4 

1 

6 

CO 

0 


2 00 

32 , 4 )^ 

220 

32 



213 


455] EXAMPLES VI 

61 Let £r be the rcq'* sum Tlio Amt of £^ for 3 jts =£a x (1 03)® 

(See Alt 313) 

Tlie respective Amts of tlic £100 payments aie 

£100 X (103)=, £100 X (103) and £100 

Tims the total pajment=£.rx(l 03)=+£l00(l 03=+l 03+1) 

Similarly if lie paid £1')0 a year, the total payment would be 
£150(1 03=+ 103+1) 

rx(l 03)®+ 100(1 03=+l 03+1) =100(1 03= +103+1), 

01 (l-03)=a=60(l 03=xl 03+1) =00(1 0G09+1 03+l)=50 x 3 0909 

00x3 0100 
^ 1-03= 

_ 104 510 
1-092727 
=141 4306 
=£141 Os 

62 Let r jeare be his present ago, his total s.ilary foi 13 jears 

after he was 20 was 

Es (1000 y 3 + 1 200 + 1400 + 1600 T 1600 + 2000 + 2200 

+ 2400 + 2600 + 2800 + 3000), 

or Es 24000 The no of jt.us his salaiy is stationaiy will 
be (a — 13-20), 01 a -33* Dining this time he leceivcs 
Es 3000 ,i 3 car But Ins salary during this whole period, 
(j:-20) 3 cai'^, a\enged Es 2500 

24000 + 3000(i -33)=2500(i -20), and dividing b 3 500, 
48+6(r-33)=5(j -20) , whence r=50 

63 3425= 5= 137, 1829 = 31x01, 3245=5 11 59 
Lot X be the third gii cn nurabci 

Tlius the candidate found the L C M of 

5 11 59, 31 59 and a:, (1) 

instead of the L C hi of 

5= 137, 31 59 and % (2) 

As the L C M for (1) and (2) was the same, r must have had as 
factors 

11, which does not occui in (2) , 

0= and 137, which do not occur in (1) , 
also 2, since x is an c\ en number 


141 4306 
1 092727 ) 164 5450 
452723 
15632 
4705 
334 



214 


MISCELLANEOUS 


[page 


X must be a multiple of 2 5® 11 137, or 76350 
But 75350 IS the only 5 figure multiple of 75350 , 
the third given numbei = 75350 


64 1 inteival on the fiist=-|^ m , on the second=-5g^ , 

req" diff =(Y-^) “ 

Shoitest distance ieq‘’=L C M of ^ in and ^ in , 
or -i-w 1“ and -1^ in 
This L C M =-^ 5 ^ in = 14^ in 


65 We have 35 ft ™ “ Similarly 4 ft =\-^ m 

Since the depth is the same in each case, if x gals are contained 
in the fiibt trench, 

X 48 x 2 48 x 2 x 3x105 x 2400 

2400~^ IM’ 32x128 

3 ^105 

Now 1 gal =8 pints = 4" litres , 

req^ no of litres = 4 - ^ y . ? ° x ==81000 

V S ^ 1 S o 7 

66 Suppose B walks 1 mi. in x min When they meet, B has walked 

5^ mi in 5^x min , A has walked 3V mi in 3^ x 18 min 
These times diffei hy 3 min , 5+r=3jx 18+3, and i?=12 

B’s speed is 1 mi in 12 min , or 5 mi per hr 

67 Suppose B, instead of supplying the £500 later on, supplied 

£l00.r at the beginning Then, since the time is the same in 
each case, A’s proportion of the piofits would he 

1100+(1300+100i)+1700 281), or 

of £f6I3A-«6f3A, or j^=^(or , 

and v=3l 100i=333J, 

so that B would have supplied £333] at the beginning 

Now £333] lent for 12 months— £500 lent foi — months, 

oOO 

or 8 months Hence req'’ time is 4 months later 



r\AMrLES VI 


215 


68 15 lbs. per 1 sq in =l">x7000 grs per ^3^ 3^^ ‘’q 
TixTOOO 1 riy7000y(30 37)= 


gm per 1 sq in. 


I''jx7000 1 Ti y 7000 y (30 37)= 

' 15 43 (30 37)' 13 4,3 

13 y 7000 X (30 37)2 _ 

Tr>-43-^000— p«r 1 sq m 

isx7oqo>;(3qi")s„ 

■ l-.j3xl.l6!)x ld0 ^g pc. 100,q cn, 

103X15JOOOG9 ^ 

= Y513 Pe> 100 ‘"1 

=103 X 1 0043 Kg per 100 sq cm 
= 103 17 Kg pel 100 sq cm 
cxcess=5 17 in 100, 01 5 5/' 


69 Letlii-.in«.onicforl90C»be €.1 , then Ins income foi 1901=£.rx-^, 

intl Ihcrt'foic Ins income foi 1902=-&rxy*^x-j’’^ Hence his 
e\penditure foi 1002 is ry I’jj x -j'j; y 

’ ^ o’s' ^ iV ^ 4 = r+50 and 1 =£4000 

70 Tx-t jcq* pneo pei gillon bo £r Outlav=10i y 200=£100 
£ 100 , amounts to £100 y(l 03)* («!Co Art 313) 

175r=100x(l 03)', 

j _ 100 X (1 03)< 100 > (1 1023)*’ 100 y 1 21 35 4x1 2155 
-ina r_ ^ 

=£0 0916=135 llrf 

71 In 1 c« cm of tlie mixture, there sre 07 cii cm of sulphuric 

acid and 0 3 cu cm of w ater Tins u eiglis 

(07x1 842-1-0 3 y 1) gni , or 1 5894 gm 
Hence 1 cu cm of mixture weighs (1) 1 5894 gm , (2) 1 615 gra , 

1 gm occupies (1) cu cm (2) ^Yc cu cm , 

thus loss of xolumc=(Y7- V 7T “roTo) rsViTT 

= KsaTvV^ cu cm 1117-^00 cm 
=— cu cm in 1 cu cm 

=1^? P®** I or 1 % 



216 


MISCELLANEOUS 


[page 


72 


73 


74 


Smce the course is 1760x3 yds, oi 5280 yds, the second man 
Avill have run ^ of 5280 yds, or 4950 yds , the thud will 
have run =-§ of 5280 yds , or 3630 yds Now 
4950 = 528x10 - 330, and 3630=528x7-66, 
giving as answei 330 yds and 66 yds respectively, from the 
etaiting point 

His net income=Es 1650 - Ks 140=Es 1410 Es 1410 is interest 

on Bs 28000, giving Bs pei Bs 100, or 6 04 % 

For second part of question, net income 

=Bs 1410-:^ of Bs 1550=Bs 1371 35 
lUU 

Now Bs 1371 26 interest on Bs 28000=Bs ^ 1 %^ -- interest on 
Bs 100 

27ow giving 490% 


234 mi coat £1 642, 234 x 1760 x 0 9144 m cost 1 642 x25 17 fr , 

or 234x1 76 x 09144 Km cost 1642x25 17 c , 


1 Km costs j hl 


lognumerator =log 164 2+log25 17=2 2153+1 4009=3 6162, 
log denominator = log 234 + log 1 76 + log 0 9144 

=2 3692+0 2455 +1 9611 =2 5758 , 


log fraction=3 6162 - 2 5758= 1 0404, 
and fraction = 10 97 (centimes per Km ) 

Aliter As above we get 1 Km costs oj 4 xi Tdx ' o ~ D i T r ® 


75 


This=^ 


4132 1 > 


370 58 

= 10 97 (centimes per Km ) 


2517 

2 34 

1G42 

176 

2517 

2 34 

1610-2 

1638 

100 6S 

14 04 

5 03 

4 11 84 

4132 91 

0 9144 

37659 ) 413291(10 97 

3 70 65 
411 

36701 

164 

280S 

16 


376 68 


Take a scale of 01" to 5 mm lioiizontally and 01' to 2 miles 
vertically Assume P is at the oiigin Join the origin to 
the point (60, 40), and we obtain the giaph of the motor car 
Tlie tram goes at the rate of 1 mile per minute Join the 
points (15, 0) and (25, 10), and produce this line as far as the 
point (50, 35) (Tins will be the graph of the tram as far as 



456 ] 


EXAI^IPLES VI 


217 


the fiist stopping place) Join the points (50, 35) and 
(55, 33) by a homontal line (This will be the train’s graph 
while waiting at the station ) Again ]oin the points (55, 35) 
and (100, 80) (Tins will be the giaph of the tiain aftei 
leaving the station ) 

(i) It will be seen that when both are in motion, the tiain and 

motor-car are together 30 miles from P and 40 miles from P 

(ii) At the point on each graph where the oidinate is 150, the 
abscissae aie 170 and 225 respectively Thus the leq** length 
of time is (225 - 170) min oi 55 min 


76. After the first replacement, 

there are (fl—h) gals of spirit out of the a gals of the mixture , 

the quantity of spirit is of the whole mixture, or 
of a gals ® ® 

Also, since the w.ater and spiiit are supposed to be mixed 
vmformly, the amount of spirit in any quantity of the mixtme 

18 of that quantity 

when h gals are tahen out of the mixture, 

of h gals of spiiit are lemoved. 

spirit left of of 6^ gals 

of 

a ' a \ a J 


[If the student finds any difficulty in the above solution, let him 
put a=5 and 5=2, and then work through the question 
with these values ] 


77 A, B and C together do -{fV of the work in 1 day , 

they do ■aV xlO, or -J- of the woik in 10 days, and of the 
work remains to be done 
Now C does of the work in 1 day 
After A’s withdrawal he works for 20 days, 

and after B’s withdrawal he woiks for 96(1-1-7), ^^8 days, 

he woiks for (20-1-128), 01 148 days, and does or -g- of the 
work 

Thus B, aftei A’s withdiawal, must have done (•&“•§) of the 
work, 01 of the w ork m 20 da) s , 
he would do the whole work alone in 20 x 6, or 120 days 



MISCELLANEOUS 


218 

78 Let req^ sum be £P and » the rate per cent 

The Amt on £P at C I foi 2 yeais at i per cent 

= p(l+j^)' (See Alt 313) 

the Cl on £P for 2 yeais at i per cent = P^l *^1^) “ ^ 

_2Pr P?g 
~100'’‘l00^ 


Now the SI on £P for 2 years at » per cent = 


2Pr 

100 


2P> Pi^ 2Pi P;2 

diff of Cl and SI for 2 

= 13s 4d , or £5 

Similarly, 

/ r V 

the Cl on £P foi 3 3’'eais at 1 per cent = P( 1 + 1^] ~ P 

_3P; 3P?g Pr3 
100'*'100*'^1D(P 


Now SI on £P foi 3 years at » per cent = 


3Pr 

100 


a.ff .f 01 «.asi t„r3,ea.,.^+2g+^-|« 

_3P?2 Pr3 
1002 1003 


We have to solve 


P>2 _2 
1002 3’ 

3P;^. 01 

1002’^10CP~^^ 

Substituting from (1) in (2) we get 

2 Pr® Pr® 


= £2 Os 4o?,or£2,j\j 




1003 eo 


Divide (3) by (1), then and r=2 5 

Substituting in (1) this value for r, 

and P=£10663=£]066 13s 4d 



EXiVMPLEvS VI 


219 


«7j 


79 


80 


n/25+V125 
=>725 + 11 1803399 
=n/36 1803399 
=6015 


^25,(11 1803399 
2> 

jOO 30,18,0339,9(6 015 

17000 1201 HlSOS 

12015' 



700000 

89191 

22110 

2070 


60239 

164 


Let Bs 7t nnil Bs 5i be tbo ie'?pective capitals 

Tlie cltlei now has Bs 7vx(l + ’i), oi Bs lOi, the Tount^et 
Bs (5 j?— 707) 


51-707 43 

lOt 100’ 


oi 50a - 7070 =43r , 


81 


a?=1010 Hence the elder has Bs 7070, the joungei Es 5050 


Beq'* weight=(1034 x 820 x 08xll 35) gin 
=(aC 1280 X 9 OS) gm 
=7848 Kg 


SOI 


28 
9 08 


7778 5 
09 1 

7847 0 


82 


£1 17s 5V 


14 lb', =1 cwt 
7 lbs =^ of 14 lbs 
3^ lbs =J of 7 lbs 
2 lbs = I of 14 lbs 


= £1 872 91GCC 
10,1 


187 291 

60 

5 618 

75 

234 

11 

117 

05 

058 

52 

031 

41 

193 353 

53 


=£101 7s Id 


=ialuc of 1 cnt 


=\aluc of 100 cwt 
= „ 3 cn t 

= „ 14 lbs 

= „ 7 lbs 

= „ SJ lbs 

= „ 2 lbs 

= total lalue 


83 Take 12 o’clock at the ongin, and 0 2" to 1 min horizontally 
Take 1" to 5 miles a erticaPy Plot the points gi\ cn m the table 
and ]om tbcin succcssnch bj striiglit lines It will be seen 
that the speed is gieatcst between 5 and 10 miles flora tlie 
stait, since between these two points the giapb has its 
greatest dope or gradic7it It tlion traiels 5 miles in 5 min , 
or 60 miles pci hom 

84. (20-5) lolls cost Bs 90 -Bs 70, oi 15 rolls cost Es 20 
Hence 5 lolls cost Bs -,,9- 
But gramophone ind 5 lolls cost Bs 70 , 
gianiophono costs Es 63’, 

gramophone and 50 tolls cost Es G3j+Es 66*;, oi Bs 130 
P 


Kits I 



220 


MISCELLANEOUS 


[PAOr 


85 (i) 2204 lbs =1 Kg, or 2204x16 o/ =1 Kg 

1 oz Kg Kg =0 0283575 Kg 

=28 3575 gm 

But 1 0 / =-”/ gm =28 3333 gm 

erroi is (28 3575 - 28 3333) m 28 3333, or 0 0242 in 28 3333 , 
req'^ percentage =';‘* 9 *JV 3 * 3 *-= 0 085, or 009% 


(ii) 1 Ib Kg =04537205 Kg coriectlj 

1 lb =tV Kg =0 4345455 Kg lougbly 
As in (1), we find eiror is 0 000825 in 0 4537205 

leq* percentage is WaVlrzP/* o* 0 18 % 


86 


A produces ||| lb 


B 

139 

” 794^ 


C 

12 „ 

=0 197 lb 

0 

112 „ 

» 632 “ 

=omib 

E 

>1 118 “ 

=m =“”2"' 


C produces tbe most butter per gallon, then follow in order of 
meiit D, B, A, E 


87 


Convei ting to cm and gm , 

tVnrlcnPSSs - . 

req tmcaness - 1 q u o x i o o xTTT 


cm = 


1-47000 
. J 0 0 U 4 


cm =7 4 mm 


88 The respective jafes of inteiest aie as 1 2 3 

Then the total respective intciests are as 1 4 9, giving respec- 
tively rVi TT 1 T of 2450 


89 In every hour the minute hand tiaieises 60 minute spaces, while 
the hour-hand tiaveises 2i of such spaces Thus in every 
60 min of time the minute hand m ill gain 37f minute spaces 
on the houi-htvnd , it will theiefore gam 15 minute spaces in 

15 X 5^1 01 15^ min of time past noon 

The hands will first be togethei after 1 o’clock when the minute 

f*f\ 

hand has gamed 2i niin on the hour-hand, te in 2^ x -pjr, or 
2^ min past one - 



EXAMPLES VI 


221 


45S] 


90 Since tliey nil aJtn their ^jiceds proportionally, there nill he no 
difltience in the rc^peUv c positions of the iiinncrs at the end 
of the dilTeitnt racc^ vhateiei thc»c ’ipccch may he Hence in 
oiii c.ilciil.ition w e need tahe no account of the ratio 7 8 9 
Now A runs 100 >ds while B luns 90 yds , 
and A „ 429 } ds „ C „ 440 j ds , 

A runs 429x100 jds, while B runs 90x429 ids , and C runs 
440x100 yds , 

B runs 90 x 4 29 yd<5 while C i uns 440 j ds , 

B luns 4x96y 4 29, or 1047 30 jds while C runs 1700 , 

C can heat B 11 2 04 j ds 


91 Cost of digging per sq ft of suiface= 

Costofiubble „ „ =r^ 

Cost of gravel „ „ =-}■*- x ^Sd = 

Cost of rolling „ „ = 

total cost per sq ft. of suiface=(f, + ’ + ’{ + ii)'^= 5^ 
total cost for the whole smface=^gff x880x3xSG 
= 143200rf =£005 


92 Vslueof total coinage=£l919881') 

■o . f 11 £18598000 

Percentage of gold =j;^^^^xl00 

= 90 8 

Tj * r 1 ill 0490 

Percentage of s.lv ei = x 100 

=2 00 

Percentage of bron7e= ^^ - ^|^Y> - x 100 
=0 52 


9C8 

192)18538 
1308 
156 
2 66 

1920)5102 ( 
1265 
“ll3 
0 52 

1920 ) 1003 3 
433 
49 


93 Cash Tcceiv cd on sale 


-£( 




of 130o) 


== £(JUU2aj:+ JL5^l_a)== £0.^^= o = £2180 

Stock bought at 272i=£5i^^jH12=£“If2i^=£800 , 

total brokerage =5-1 t; of £ (850 + 1300 + 800) =£=^ 
= £J^=£3 13s 9£/ 



222 


MISCELTANEOUS [PAGE 


94: 


Let Es r be his outlay on each siyoid his total outlay is 
Es 2000r, and, since he gams 15 %, Ins total receipts would 
be Es 20001 But he sells each svioid at Es 12^, 

and there are 2000) swords 

2000^ X T^= ^ 2000 X 12^, 


« 9BXSOO_OX2 07Xl 0_0__p- inli 
and ^—100X10X2000X115 •'^'^lo 


Now xVtt of 2000, or 700 swords, prove worthless 

his total receipts are Es (12|;^ x 1300), or Es 16818| 
But his total outlay is Es (l0}-g-x2000), or Es 21375 , 
his Ioss=Es (21375 -16818|)=Es 4556 4 a 


95 By calculation we obtain the following table 


Take scales of 1" to 10 years hori- 
zontally and 1" to 1000 population 
lertically Take the point (1850, 

5000) as origin Plot the points 
as in Alt 259, Ex It will bo 
seen that a smooth curve can be 
drawn tliiough them, and the 
required results may then be read 
off 

96 See Art 313 

Last July's payment of Es 1000 will amount to 
Es 1000 X (1 025)® by next January 
This year’s January payment of Es 1000 will amount to 
Es 1000 X (I 025)® by next Januaiy 
This year’s July payment of Es 1000 will amount to 
Es 1000 x(l 025) by next January 
Next January’s payment will be Es 1000 
total payment due next Januaiy 

=Es 1000x(l 025)®-bEs 1000x(l 025)®+Es 1000x(l 025)+Es 1000 
=Es 1076 8906 -bBs 1050 625 -1-Es 1025 + Es 1000 
=Es 4152 5156=Es 4152 8a 3 p 


Year 

1850 

1860 

1870 

1880 

1890 

1900 

Pop" 

5000 

5400 

5940 

6702 

7641 

8787 


5000 

Add 87. ^ 
5400 

Add 10 7. ^ 
5940 

Add 10 7. 594 
„ 37, 178 2 
6702 2 
Add 10 7, 670 22 
„ 47, 268 088 

7640 5 
Add 107. 764 Ot 
„ 57. 382 025 




223 


459] EXAMPLES VI 

97 Let rcq^ distnnco bo i Km 

From A to B I ude (t - 3) Kia , and take hrs , 

I walk I Km , and take i, oi J hrs 

Fiom B to A I iidc r Km , and take — hrs 

15 

r-j , 1_ 4i-3 l_i 

16^ ■‘‘6 15’ 63 "^6“ 15’ 

jc 10(4i-3)+105 =42x, A\heiice i=37 5 Km 


98 Tho L C M of 22 1 and 3G4 = 291 2 

In 2912 sec? aftei staiting xl and B am\e for the fiist time at 
tho staiting point In this time ^1 will have gone round 
13 times and B 8 times , A pabsca B 5 times 


1031 

485 

1 031485 

1-031 

49 

30 

95 

1 

03 


41 


8 

1063' 

96 


1063 

96 

1 031485 

1 003 

96 

31 

92 

1 

06 


43 


9 

1097 



100 Ecq''aiea=127 35x98 27 sq m 


127 35x98 27 
(0 9144)-' 


sq 3db 


127 35x98 27 

■- - - no 

(0 9144)- X 4840 


12 735x0 9827 
0 83013 x 4 84 


12 733 
082 

7 

0 830 
4 81 

13 

114G1 

5 

3 344 

5 

1018 

8 

bG8 

0 

25 

5 

33 

4 

8 

0 

4 04C 

8 

12 514 

7 


3 05 

40,4,7 ) 12514 
373 



224 


MISCELLANEOUS 


[page 


101 1"* yeai'’s dividend = 


Es 3x10000 _Es 2000 
105 7 


. , , , Es 2000 , Es 3 X 2000 Es 2000 

2"^ years dividend = — ^ ! — iot xI — ~ — 

Es 2000x110 
7 X 107 

Es 2000x110 , Es 3x2000x110 




3"* year’s dividend = 


7x107 
Es 2000x110 


108x7x107 
3 \ Es 2000x110x111 


■ 7 X 107 

Es 2035000 
^ 6741 




108/ 7x107x108 

=Es 301 14a 3p 


102 


By investing Ee 1 the respective inteiests leceived aie Ee 037, 
Ee 036, Ee 030, Ee 046, and Ee 042 

the stocks arranged in relative oidei of merit as profitable 
investments stand thus — Bank of Madias, Bank of Bengal, 
Govt Paper (3), Govt Paper (3|), and Govt Consols 
By investing Es 97 3 the net inteiest received is Es 3J x 
by investing a lac of lupecs the net inteiest xeceived is 
Es ^ 7-0 X § X 3503 7 a 


103 


Since 27 gals =216 pints, there aie 2154 pints left after 1*‘ day 
216 pints are i educed to 2154, or 1 pint to pints 


Thus the multiplying ratio (see Aits 269-271) is HIM foi each 
day 216 

after 20 days 216 X ^ or 216x^||i^ pints remain 


104 


Eeq"* no = 16 X 


60 secs 
10 secs 


X 


5 teeth _16 x 60 x 5 
24 teeth ~ 10x24' 


105 Suppose tfi inches fell in 1899, in 1900, etc 

Then rfi+cfj+rfs =74 94, le (24 98x3), 

c? 4+«^6 +c?g =88 86, 1 c (29 62 x 3), 
-di+ds = 4 80, 

d^ — dQ= 636, 

-d, = 7 47, 

<^2 — C?3 = 0 17 


0 ) 

( 2 ) 

(3) 

(4) 

(5) 

( 6 ) 



460] 


EXAMPLES VI 


225 


106 


Adding (2), (3) and (4), 3^/,= 100 2 and rf,=33 34 in 

Subst for rfj in (4) Me find «/^=26'98 in 

Subst for </, in (3) mc find rf4=2854 in 

Subst foi in (S) Me find f/>=25 87 iii 

Subst foi dn in (C) mc find <73=25 70 in 

Subst for da and d^ in (1) m c find d^ =23 37 in 

Let req"* length be r yds 

Aiunsi.’vds •wbile5iiius(r-15)3ds and <7xuns(i-29) yds (I) 


Agiiii B runs i jds Mliile 77 luns (t — 15) yds 


( 2 ) 


From (1), 


ZJ’s 1 -ite T - 1 *) 


C”s rite a - 29 
f-lo r 


from (2), 


^’s l“ltC 


C’s rate i - 15 
Multiply by (r - 29) x ( i - 15) , 


c— 29 X — 15 

(j:-16)x(i - 15)=i(» -29), oi r»-30i + 22 >= i=- 29r 
Subtract i-froin each side, then — 30r+225= - 29i, and » =225 

107 Pcq« value=— (see E\ \vi d , No 21) 


£2000 x 9 x 220 £600 


229x240 8 

108 The ‘remaindei ’=l- 5 -p=i 

3 5 I » 


'=£75 


his total profit=:^ of of of ^ 


m 

loS 


20 

100 


1^ 

15 


— l-=r^s=15i gning 164 % piofit 
300 200^300 600 100 * 

109 Let £r be his oipiUl at first 7 inontbs’ inteiest at 3|% is 

100 ’ capital -{-interest=£^T+ 

•n** 

He then intests £^v, after six months he diuMs a dnidend 
320 

of £l of ^ of and lealises by sale £^^^ of 

320 lOo 320 


2 109 

the total return =£^5 


3 

39? 


106^ 


3 

^ 109 ’^ 320 ’'"^ 109 


39? N 


net gam on £i=£^, givmg 1| % gam 



226 


MISCELLANEOUS 


[page 


110 A puts in 126 oxen foi 3 months, which costs him as much as 

126 X 3 oxen foi 1 month Siimlaily the cost to JS and C is 
that of 162x5 oxen and 72x12 oxen lespectively foi 
1 month 

Thus then expenses are as 126 X 3 162 x 5 72x12, oi as 7 16 16 

Thus A pays ^ of Es 570, B pays J-s of Rs 570, C pays -if of 
Es 570 (see Ait 281) 

111 Let X be the no of children in tow'n, out of 100 in town and 

countiy, then 100 — r is the numbei in the countiy 
The total cost is [69^ + 39(100 - 1 )] pence it is also 60 x 100 pence 
69t+39(100-^)=6000, and t=70 
Thus leq^ percentages aie 70 and 30 

112 The bookseller contributed Es 8000 for 366 days 
His partnei contiibuted Es 11500 for 108 days 

then shaies aie as 8000x366 11500x108, ze as 488 207 
Thus the bookseller leceives of Es 1654 
=488xEs 2 3799=Rs 1161 6a. 

His paitnei receives ^-9- of Es 1654 

= 207 X Es 2 3799 = Es 492 10 a 

113 Let be leq*" puce , then the fiist bicycle leahsed > 

100 

the second bicycle cost realised 

The thud bicycle was bought with this and the addition of 
£5 165 3c? , or £5tv , 

65 ..I'SOr , _ 65x6017.65x21 

100 “ lOT xloO 100^4 ~ 

Hence j^+333=10|-g, oi ^=63^, and r=£l7i 


65x60i 7 65x21 

100x100 ■’'100x4 


Hence i^+333=10|-g, oi ^=63^, and r=£l7i 


114 1 m =3 28 ft , 1 sq m =(3 28)® sq ft 

Hence 1 aie=(3 28)® x 100 sq ft 
and 1 Ha =(3 28)® x 100 x 100 sq ft 

- 328X3 JS 
~ 4840XP 

= 4JJlaA8 ac -lliAl 
006X0 “®~ 6446 

=2470ae nearly 


2 4C98 
5445 ) 13448 
25380 
3800 

43 



EXAMPLES VI 


227 


4G1 ] 


115 Ti no ai (M = 4 X 3 14159 x (4000)® sq mi 

Approx .uea=4x 3 142857143 X (4000)® iq mi [-®t;S.= 3 142857143] 
DifTorenco = 4 x 4000® x 0 001267143 sq mi 
=4x4®x 1267 143 sq mi 
=64x1267 143 sq mi 
=81097 sq mi (moic) 

116 Tlie companv lias to pay back altogctlici the Amt of Rs 1658775 

at4%CI 


12G7 

143 

6 

4 

76028 

4 

5068 

6 

81097 

0 


Tins =Rs 1658775 x(l 01)^ (see Ait 313) 
=Rs 1658775 x(l 0816)® 

=Rs 1658775x1 16985856 
=Rs 1940532 


1058775 

1 

16085856 

1658776 


IGWrl 

5 

00326 

50 

14028 

08 

1327 

02 

82 

94 

13 

26 


83 


ii 


1D-J05321 13 


Since tliej pay by annual instalments of Rs 156976, tbe first 
instalment paid at the end of the fiist \ear amounts to 
Rs 456976 x (1 01)® in 3 moie years at 4 % C I 
Sirailaily the second iiisUlment paid at the end of the second 
jeat amounts to Rs 45697Gx(l 04)®in2moieyeaisnt4%CI 
In the same nay tbe tliiid instalment amouutb to Rs 456976 x 1 04 
at tbe end of tbe time 
With tbe last insUlmcut tbe payments 
amount to 

Rs 456976(1 04®+l 04®+l 04+1), 

01 Rs 456976(1 124864+1 0816+1 04+1), 

01 Rs 436976 X 4 246 164, 
or Rs 1940532, ns aboi e 


466076 

4 

246464 

1827004 

913*)3 

2 

18279 

0 

2741 

8 

182 

8 

27 

4 

1 

8 

1940532 

0 


117 At tbe intc of 3®, mi pci In , they row 308 ft in 63 secs Since 

tlic next gun is fiicrl ivbcn they ha\e lOMcd foi 60 secs, it 
follows that the icpoit takes 3 secs to leacb them But in 
3 secs , sound ti a\ els 1 100 x 3 ft , oi 3300 ft 

req"* distance =(308 + 3300) ft, oi 3608 ft 

118 Tboii times foi the mIioIo 3 ouinoy aic ns 42 56, oi ns 3 4 , 

tboir speeds aie iis 4 3 (see Ait 271), and they meet at a 
point uinch dnidcs the nbolo distance in tbe latio of 4 3 , 
z,e they meet at ^ of tbe distance fioni Liicrpoo], and in if of 
42 min or 24 nun 



228 


MISCELLAWEOUS 


[PAGi, 


Take 4 in vertically to lepiesent tlie distance between Liverpool 
and Manchester Mark the successive inch divisions along 
the y-dxis fiom the oiigin inclusive, 0, 1, 2, 3, 4 For time 
take 1 in to represent 10 rain along the r-axis Assume 
Liverpool to be at the origin Join the origin to the point 
(42, 4), to obtain the graph of the first train The line 
joining the points (0, 4) and (56, 0) will be the giaph of the 
second tiain They cut at the point i\hose abscissa n, 24 


119 Let each man do m units of work, and each boy h units of woik 
per daj' 

9 men and 6 boys in 2 days do (18m+12&) units , 

5 men and 7 boys in 3 days do (15m+2l6) units 
But these sets of units are equal , 

18m+126=15j»+2l6, or m=2b 
the work consists of 18»i+126, or 546+126, or 666 units 
Again 2 men and 5 boys do (2m+56) units in 1 day 
Now 2»i+56=66+56= 116 since 116 units are done in 1 day, 
666 units will take 6 days 


120 

At first 

After 1*‘ 
operation 


After 2"'* 
operation 


1“ Jar contains 
8 pints of brandy 

7 pints of brandy 


2°'’ Jai contains 


8 pints of water 
'8 pints of"! 

water, 1 /f watei, 

1 pint of brandy 

biandy,J 


} 


'(7+^) pints of' 


'(8 -t) pints') 

f (7+^) pints' 

brandy. 

- • 

of water, 1 

1 of water. 

f pint of 


(1-^) pints j 

I pint 

. water 


L of brandy,; 

1. of brandy , 


121 (l)7r=3|=^ 

^.b76xl=6V6x(HA-+T^) 

= 57 6 

144 

36 

327 

18 327=18 33 to two places of decimals 
(2) TT houis=3 hours, 8 minutes, 30 seconds =3 hours, 8i minutes 
=32^=3^ hours=5^=3 1416 houis 
7r=(i)31416=(u)3il^ 



4G2] 


EVAaiPLES VI 


229 


1 

78 

876 

4 48 

Since ester nal dinmctci = 7 5 in , 


50 

Weight of hollow splicio 
=0 5236(7 53 - 73 ) X 4 48 02 
=0 5236(421 875-343) x 4 48 oz 

31 

6 

55 

31 

353 

36 

0 5236 

=0 5236(78 875)x 4 48 0 ? 

= 185 02 . 

17G 

7 

1 

66 

07 

06 

21 


185 

02 


123 Let n tini d-cliss tickot cost r pence, then a second cUss ticket 
costs pence, and a fiist chiss ticket costs 1 ^ x pence 

Now16j 10Arf=202irf f+U«+13xlA«=202A, 

or i+3r+5r=405, and t=45 
the tickets ai o 1 3 x 1^ x 45rf ,01 9« 4 W , etc , 

and the rates pei mile are 01 2 Arf , etc 

45 


124 


Tlie req'* number must bo odd, since it is a prime 


It ends in 9, giving 


or 


7, 

99 

or 

99 

6 , 

99 

or 

99 

3, 

99 


/ 189, 279, 969, 459\ 
\819, 729, G39, 649 / 
1167, 257, 347\ 
\G17, 527, 437 / 
fl45, 295\ 

\415, 325/ 



as possibilities. 






79 


( 1 ) 

( 2 ) 

(3) 

(4) 


Now every number in 

99 » 


39 


99 


(1) IS a multiple of 9,' 

(3) H j> 5, ■ 

(4) „ „ 3, 


and cannot be pnme 


From ( 2 ), w c find 527 = 17 x 31, 

and 437=19x23 

257 and 347, although both piimc, viill not satisfy the conditions 
We aie thus left mth 167 and 617, which are both prime 


125 


He rows at 4 miles an lioui on still water 

„ li X 4, or GJ miles an lioui dow n stieam , 
the stieam flows .it Ij miles per hour 
He rows up 6 tic.am at (4-1^,), or Z-, miles pei hour 

req"* tinio=^^+^^ hrs =| hr 



MISCELLANEOUS 


230 


[PAGL 


126 He has to pay back altogefchei the amount of Bs 25220 foi 3 yeais 
at 5 % C I This =Rs 25220 x (1 05)® (See Ait 313 ) 
Suppose Bs P IS the yeaily instalment Then Rs P paid at the 
end of the first yeai amounts to Rs P x (1 05)® m 2 more yeai s 
at 5 % C I 

Similarly Rs P paid at the end of the second year amounts to 
Rs PX105 


"With the last instalment, the payments 
settle the debt, 

and P(l05®+105+l)=25220x(105)® 
P X 3 1525 = 25220 X 1 157626, 


and P= 


36230X1 1BT63B 
3 1526 
2S22X^ 6306 
1 261 


=Rs 9261 


4 

6303 

262 

2 

9261 

0 

2315 

3 

92 

6 

9 

3 

126.1)11678 

2(9261 


3292 


12 


127 


Suppose a lbs aie allowed fiee to each passenger 
When they shaie the luggage 2a lbs are not paid for, 

and (345 -2t) lbs cost them (Rs 3 2a +Rb 5), or 130a , 

ee 345 - 2 ^ 1 ^^ cost 10 a 
13 

When one of them owns all the luggage, x lbs are not paid for, 
and (345 - u) lbs cost him Rs 11 4a , or 180a , 

t e cost 10 a 

?^-=-4-^, or 18(345 -2a)= 13(345 -r), and a;=76 


128 Let I be the leq^ numbei Expressing cost and receipts in 
annas, and equating them, we get 

122r=28r+4800, and r=51^ 

Thus the least possible no of copies is 52 

To verify graphically, take scales of 50 rupees to the inch 
hoiizontally, and 10 photogravures to the inch veitically 
Cost of pioducing 40 photogravuies=Rs 370 

Join the points (300, 0) and (370, 40), and we obtain the graph 
for the cost of pioduction of any number of copies Again 
40 copies are sold foi Rs 305 

Join the oiigin to the point (305, 40), and the graph for the 
‘ sales ’ is obtained 

Tliese giaphs will be found to meet at a point whose ordinate 
lies oetween 51 and 62 in value 



4C2] 


EXAMPLES VI 


231 


129 When ho t>irns bach, to the st.irting point, 10’ has gone He 

•irn\c3 thcie in 10' nioie Since the lest of the party will 
reach tlie station 1 Jioiii aftci thej staited, he has now to 
walk 3 miles in 40', oi at the rate of 4^ miles ,in houi 
Toaerif> this icsult giaphicalh, take 3 in horizontally to lepie- 
sent 1 hour, and 1 in \ ertically to represent 1 mile 
Join the oiigin to the point (1, 3), and we obtain the graph foi 
the majoiiU of the mit^ Ne'tt, 3010 the points (J, ^), (J, 0) 
and (1, 3) siiccessn cl-v , and we haic the gi”iph for tte ‘one’ 
tourist after he lea\ cs the 1 cst of the party The line joining 
the two last-mentioned points rcpicsents a rate of 3 miles in 
3 of an hour, or 4^ miles an hoiii 

130 X miles per houi =52S0i feel per 3G00 seconds 

'•>2807 44 r 

^ 3000 P®'* » 


131 


30 

45i 


but by the rule we ha\e 


so that on ~ ft 
«iU 


132 


we haie an enor of ft , 

1C on 44 „ „ 1, 

1C on 100 „ „ Vr=? 1=2 27, 

1C 2 % to the ncaiest integer 

Maaimuni weight of soicicign= 123 47447 grams 
Minimum „ „ =122 7 „ 

TJie heap of sovereigns weighs within the limits 384 ± OxlSOgi 
gieatcst no of sovs = = = 564 9 , 

greatest no of sov s = ')G4 
The least no of sows = 1 o^ ' ^f ~ =050 07 , 

least no of sois =060 

Let 'ix be the no of teeth in tlie fii'st w heel 
then 6i is „ „ second „ 

5r „ „ thud „ 

and 4i „ „ fouitli „ 

Now LCM of 7r, 6r, 5r, 4r is 420r 

when 420r teeth Inave passed on each wheel, each will have 
made an exact numbei of revolutions, and thej will be 
simiiltaneoublj in thou original positions 

the first must reiohc IrOi^ qq etc, etc 
7x 



232 


MISCELLANEOUS 


[page 


133 Rg (6760 - 6500), oi Rs 260 is the interest on Rs 6500 foi 1 yr 

Rs or Rs 4 „ „ Rs 100 „ 

Thus if Rs P he the req** sum, 

P(104)=0500 (Alt 313) 

P=^^=R3 6250 

II 

134 Let req^ Icngtli ho x ft Cost per sq ft 

11 1 * 25 20 

Tlien toUl cost='Cx6x-^5 =ix0x -^x^^-fr 
But total cost =787 4x 1 53 X 1 5fr 

1 1 9ri 9f) 

rx0x^x=^=787 4xl53xl6, 


'nhence r= 


7874x1 53x1 5x9x20 7874x17 


Gxl»x25 20 


07 


1338 58 
' 07 


= 1912 ft =637 jds 


135 In tlie first year piont=Rs (35,40,000 - 25, 10, 000)=R8 1 0,30,000 
In the second jeai rcccipts=^ of Rs 35,40,000=Rs 34,16,100 , 

„ „ expenses=i^' of Rs 25,10,100=Rs 25,22,550 , 

„ „ pioGt=RB 8,93,550 

Difierence of tlio two irais’ profits=Rs 1,36,450 , 


136450 1364 5 


ieq« pciccntage=100x^-zj^= 


1030000 103 


:132 


136 Tnteiest iecencd=^ of £9000+^ of £7200 (1) 

Gish realised hy sale=i^ of £9000+^^ of £7200 (2) 

Total cash ohtained=(l)+(2)=^^ of £9000+^- of £7200 
Hence req"* gain=^^^-l) of £9000+(^j^-l) of £7200 
=^|of £7200-^^^of £9000=£(I^-^)=£116 13^ 4tf 



463] 


EXAirPLES VI 


233 


137 \^r0090-V0 091 

=10044 - 0 3017 
=0 7027=0 703 


1,00,90 (10014 
200,4 r 009000 
I 931 
0,09,1(0 30160 
GOlPlOOO 
0026 39900 

3744 


138 Let 2n ^^orl.n^cn be engaged on the Gist house, 
and n „ „ second „ 

Suppose each is paid Be :> pet lioui 

Then the cost pci day of the first house 

“I 

=Rs 10rx2n+Rs 2x-^x2w=R8 267Jr, 

and the cost per day of the second house=lOrx«=Rs lOnr 
Again, the first house takes 4 months, oi 4 x 30 daj's, 
and the hccond „ 7 „ or 7 x 30 „ , 

26n E X 4 X 30 + 10« r x 7 x 30= 17400 
Dmdcby 30, then 104?iE+70nr=580, or 

cost of fii-st houso=Rs 26n'i x 4 x 30=Rs 10400 , 
cost of second hou''o=Rs lOiw x 7 x 30=Rs 7(X30 


139 Let Vj be the volume of water, Vg that of the other liquid 
Then w oighs V, kilograms, and Vo w eighs Vo x 1 340 Kg 
The volume of the mixtuie is (Vj+Vo), 
and this weighs (Vi+Vo)x 1 270 Kg 
(Vi+Vo)xl270=Vi+V2Xl340, or V,x027=V,x0 07 , 
V, 7 

^ =^ Thus are water, 5 -|- other liquid 


140 Let req** distance be e miles, which the fiist train takes 50 min 

to travel Its rate in miles pei min is therefoie — , and at 

- 50 ’ 

12 20 it has travelled 20 — miles Similarly at 12 20 the 

second tiain has traiclled 15 miles and the third 10 — 
I 45 40 

milAn 


20r 15r 
50 45 


lOr 

40 


= ^GS, 


2a. , T 295 - , 

or ■]5"+3+4="“g~> a=3/* miles 



234 


MISCELLANEOUS 


[page 


Penmeter of room=2(a+6) ft 

Area of •wall8=2c(a+6) sq ft 
Width of paper=rf ft 

no of yds of paper req‘‘= — ^ 


„ ^ , 2c(a+b) n 

Cost of paper=— ' x rupees 


nc(a+b) 

288d 


142 Since 1 md =40 srs , after the first replacement, 

-g- md IS milk Aftei the second, of -g-, or (I-)® md is milk 
Similarly, aftei the 3'“ replacement, (g)® md is milk 
Now suppose 1 md cost x annas at first 
Then (f-)® md is sold for r annas , 

? c 1 md IS sold for (f-)* a, or a annas 
gam on v annas=(^|- 3 r- 1 ) annas=^-S-5a: annas , 
gain pel cent =^t.J^^s=49 27 


143 (i) Correcting to four significant figuies, 

the fraction becomes 

87 3S?10 684 

Numerator Denominator 

log 52 45 =1 7197 

log 378 4 =2 5780 log 87 32=1 9411 

log 0 0209 = 2 3201 log 0 584 = 1 7664 

log mimerator = 2 61 78 log denormnato} = 1 7075 

log fraction =2 6178 - 1 7075=0 9103, 
and antilog 0 9103 = 8 134 

(ii) Denote the expression by r, then 

log r = 2 log 1 03 + 4 log 1 025 + log 1 05 
= 0 0256 + 0 0428 + 0 0212 = 0 0896, 
and antilog 0 0896 = 1 229 

144 See Art 311 and the note to Ex , Art 313 From this we get 

leq^ Amount= £500 x (1 03)® x (1 025)* x 1 05 

= £500x1 229=£6145=£614 10s 



4G4] 


EXAMPLES VI 


235 


145 TVntez in nn\tiire=(-j'\Jjj- of 1 gal +-j-^ of 3 gals +1 gal ) 

— (■rVsr+ TVir+ - ) =T<f<r -A mi\tuTe=4i gals 


82 


Now gals out of 4^ gals out of 100, oi I 85 % 


146. 712 lbs 11 cigb nioic than 1728 cu in 
1 lb weighs inoie than ^^‘2 cu in 
If V shot go to the lb , 

then their >olume eu 111 , 

^1728 x 0 x 82 
^> 712^ 

04 

or r > JLI=A 2 <«£i± 2 :, or 23732 

^5«-XJ HIBO’ 

6(1 

Similarly, 

1 lb weighs less than ^rVs” cm m > 
and the volume ^ < ~~ cu in , 

238GG 
r V 

Tlius 1 lies betivecn 2373 and 2387 


172Sx6x64=CC3552 
3 141-)9x 89=279 60151 
2373 2 
2 79602 ) 6635 52 
1043 48 
204 67 
8 95 
66 

864 x 512=442368 
314159x1)9=185 35381 
2386 6 
185354) 4423 68 
716 60 
160 64 
12 2 6 
124 


147 2 oz. of blucstonc and 1 } or of lime go to 1 gal or 10 lbs of 
watei , 

or 2 07 of bluestone and 07 .. of lime go to IGO or of watei , 

2 gm of bluestono and 1^ gm of lime go to ICO gni of water, 
or Vin>” giM of blucstonc and VVu* of lime go to 1000 c c 
of w ater , 

i c 12 5 gm of blucstone and 7 81 gm of lime go to 1 litre of 
watei 


148 Let ? =no of rupees lent 

Then 3 (1 04)< - ^(l 08)-=:650 , 

le t (1 1G98585G-1 1GG4)=G50, 
ie tx0 003468")G=G50 
r= 187939 

Principal =Rs 187900 to the ncaiest hundred 
Q 


K ns I 



MISCELLANEOUS 


[page 


236 

149 Let one man’s woik in one day lepiesent a unit of woik 
Since 150 men in 25 days complete i of the work, 

150 X 25 X 4, le 15000, units of \i oik aie needed foi the whole 
100 men in 60 days complete -f of the woik 
of the woik, ? e 5250 units of work, remain 
But 130 -(25 +60 +10), te 35, days only remain 
1 e 150, men ire required 


150 Let X Bs =cost of goods 
Then ^ of stock sells foi Es 


=Es 


rx 140x3 
~100x4 
105r 
100 


The remaining of the stock sells foi Es 


V X 140 X 60 
100 X 100 X 4 


=Es?^ 

100 


whole stock sella for Es 
gam =26 % 


126t 

100 


151 Suppose the man lows r mi an hour on still water, and that the 
stream flows at y miles per hr 

Then he rows up stieam at (v—y) mi per hr, and down stieam 
at(i+y)mi per hi 

In 18 nun , or hi , he ions stieam He lows 

this distance donn stream at {^+y) mi pei hi , and therefore 

takes hrs 

10(a-+y) 

He rows the last 1^ mi down stieam in his 

^+ 1 / 

the total time between his passing and overtaking the bottle 


= (~ 
VIO 


3(r-y) , U 


v+yr 


01 6^+15' hrs 


10(i+y) v+yj I0{v+y) 


But this IS the time taken by the bottle to float down 1^ mi , 
1 e hrs 

y 

or {6v+l5)y=l5{x+y), ic 6vy=l5v 
Divide by x, and y =2J 



466] EXAJIPLES VI 237 

» „ , 6000 x(103yox 003 

152 Substituting the given values a= (i o3)io_l > 

log numeiatoi = log 5000 + 10 log 1 03 + log 0 03 

=3 6990 +01280 + 2 4771=2 3041 , 
nunioi'atoi =antilog 2 3041 =201 4 
Noting that lOlogl 03 = 0 1280, and antilogO 1280=1 343, 

-u e see that denoniinatoi =1 343 - 1 =0 343 
a=£§^=£587 2 

153 (6^x5000)cu cm aie dischaiged per 1 min (see Ait 247, Ex 3), 

1000 cu cm , 01 1 litie aie discharged pei — miii , 

‘ioooo 

?c 30000 lit aio dischaiged pel -- — ^min, oi I663 min 

154 Suppose the clients deposit Rs x The inteiest fiom the prefei- 

4 7 St? 

ence stock =Rs x g, fiom the mortgage3=Rs x ^ He 

pays Rs ~ x ^ as interest on deposits 

Multiply by 1200, and 16i + I5t-24r = 12000000 , 

■whence ^= 17,14,286 to iieaiest Re 


155 Suppose thcie aie t cu cm of the Fust ind y cu cm of the 

second liquid Then then weights aie tx0 68 gni , and 
V X 1 04 gm , 1 espcctively 

Siinilaily the weight of the mi\tuie=(t x082+yx082) gm 
a.x0 08+yxl 04=t+082+yx0 82, and 
Hence 1?=: J J-, and of the whole nii\tuie 

^ = X 100 % = 61 1 % , and y = /^ of 100 % = 38 9 % 

156 Since 1 5 sq in is inci cased to 1 sq ft , 01 144 sq in , 

1 sq in „ 01 96 sq in , 

1 in „ in , 01 9 8 in 

Hence 4 in , 9 8 x 4, or 39 2 in , 

and 6 in „ 9 8 x 6, 01 58 8 in 

the least dimensions are 59 in and 40 in 



238 


MISCELLANEOUS 


[page 


157 


Let a unit of ■woik=woik of one boy in one hour 
A man boys 


{(I4xt)+13}x3x2 

5 


no of units of work to be done 


=^x^= 45 units 


If for this work there are x men, 

theie must be 27 - ^ boys 
Then ix^+(27-a?)=45 

7r+108-4ar=180, 

3t=72, 

0^=24 

So theie are 24 men and 3 boys 

10 boys are replaced by that no of men 


158 Let 
Then 


i?=no of 
120i?_ 

100 


Kms per hour travelled by motor cai 


« « » 
75 75 m 

X 

100 

75 750 25 

X 123 ;’^ 120 ’ 


tram 


9000=*7500 + 25t, 
25a:=1500, 

i ;«=60 


159 Let Rs x be the letail price per gross Since a loyalty of 10% 
IS paid and discount of 40 % is allowed, oi ^ is the 

net receipt per gross by the manufacturer 
But this allows a profit of 25 % on Rs 170 , 

X 125 

2=170Xj^, whence ^=Rs 425 per gross 
puce per article =Rs 2 15a 3p 

For second part of question, the net receipts on each article aie 
of Rs 3 But each article costs 

req-* proat=Re f^=5^a 


144 



4G6] EXAMPLES VI 239 

160 The line for the Litin niaihi is found by joining the points 

(31, 30) and (163, 120), and foi the Gicek niaiks hy joining 
(56, 40) and (IGl, 100) 

The most convenient scale is that of 10 niaiks to the inch for 
each case, beginning at 60 for the unreduced niaiks and nt 30 
foi the 1 educed inaiks 
The liguie will however be large 

The two lines cut at a point whoso cooids aie approximately 
(74, 60) Hence an actual niaik of 74, either in Latin or 
Greek, appeals as 60 when leduccd 

161 Since 15 nii pei hr =22 ft per sec, 

m 50 mill he IS ncai er the fort by 60 x 22 x 60 ft 
But the intcival between the l"*^ and the 52'“’ lepoit is 51 niin , 
sound takes 1 mm to travel 50x22x60 ft, 
giving a speed of 1100 ft poi sec 


162 The volume of water (lowing thiough the pipe per sec is that of 
a cylindci whoso b<iso aiea is tt 6® in, or it | sq ft, and 

whoso height is 1 ft , giving ^ cu ft 

If the 1 cq'' time bo v lira , then in this time f j x t x 60 x 60 j cu ft 
aio dischaigcd ^ 

But this must equal a cuboid of watei whoso base area is 
(20 X 48 10 X 9) vq ft , and whose depth is iV ft 

tXt:x60 x 60=20 x 4840 x 9x:^ 

4 12 


eoy-i p 10X0X4 
' 3141 s J vooxoox 


g * !!.- g4g —OR RR 

12 3 141E0X3 D 4S1T7 


163 Income at fiist=Rs 

_ 1402 3X7X2 
-Xl 9 3 

=Rs 525 

„ 14625x98x4 
2"-» incoino=Rs 

_ 1 402 rXnflX4X2 

iiexiPB 
=Rr 525 

Thus there is no change in income 



240 


MISCELLANEOUS 


[page 


164 


Cosfc=Rs 


£lOM_7s 
Is 31 


247290 

"15H 

» i. 7 s n nxi B 

rrs 


_ 2 8 1 7 7 6 

TTf 


=Ra 15,516 3a 9 p 


165 Fast biain >• 

Times of departure 12 0 12 3212 3812 4412 5012 56 

A I * * * * * * * I I I I B 

Times of departure 12 5612 3912 2212 5 

< Slow train 

The above diagi'am shews tliat if the line weie double, the trains 
would pass each othei between the 2“'^ and 3"^ stations from 
B , also the times foi the trains at these stations shew the 
lequiied station is the 2'“* out from B 


166 


Eveiy 62 3 lbs weight of the man displaces 1 cu ft of iiatei , 

hence leq** volume cu ft, oi 2 25cu ft 

For the second pait of the question, if ^ be the req"* number 
the total volume occupied by the men =2 25 x -p cu ft 


But this IS of 360 x 40 x 54 eu ft , 

2 25a;=^ of 360 x 40 x 5J, or r= ^ =360 x 8 x 11 

=31680, giving 31700 to neaiest bundled 


167 The volume of water passing the budge pei hour 

IS 180 X 36 X 17500 cu dec , giving I- *U> . ^i . ji . x . i 7 b gy 
pel ram , or 1890000 lities pei nun 

168 Cost foi the ordinary burners 

34 

=33 a X X 96 X 13 X 26 (6 months=26 weeks) 

=■ 1 . 3 X T X YuV = 3747 744-1 

Amt saved =0 33 of 3747 744 a =Rs 77 4 a 9p 

169 Amt of guaranteed stock and preference stock 

=£17904062+£11925808=£29829870 
Half year’s dividend on this at 5 % 

=-jV of £29829870 =£745746 76 



467] 


EXAJIPLES VI 


241 


If e'^actlj £40000 is earned o%er, the anioant for the dividend 
on ordinary stock would be 

£1492156 5875 - [£40000 +£745746 75], 
or £1492156 5875 - £785746 75, 
or £706409 8375 

On this supposition the leq** percentage per half jear would be 
£706409 8375x100 £7064 

£35538259 ’ 3554 

This IS gi eater than 1| and less than 2, and nearer to the lattei 
value But if 2 % is taken, the sum earned o\er will clearly 
be not as much as £40000 Thus to satisfy the conditions of 
the question, we must take 1| % 

Now 1 J % on £35538259 

=(2-i) of £355382 69 
= £710765 18 -£88845 6475 

=£621919 5325 (interest on ordinary stock) (1) 

The inteiest on the other stock =£745746 75 (2) 

Adding (1) and (2) we obtain for total interest 
£13676662825 

amount carried over=£1492156 5875 - £1367666 2825 
=£124490 3 =£124490 bs 


170 From Art 284, iii , 

Volume of statuette iy 5^ _ 125 
Volume of model ~ 48® ~ 24® ”13824 


Supposing the statuette were made of plaster, it would weigh 
of 150 lbs Since, howevei, siher is 12 9 times as 
heavy as plastei, it w eigbs of 150 x 12 9 lbs , 


or 


1 S 5X50X4 3 
1 «3C 


lbs , or 17 5 lbs 


171 36 in IS correct length of unit, 36 18 is incorrect length of unit 
Now the greater the measure of the unit, the smaller the number 
of units in any distance Hence we have 

correct measurement of side _36 18 1'005 
incorrect measurement of side 36 ~ 1 
correct measurement of area _ 005y_l 010025 

r incorrect measurement of area ~ \ 1 / 1 

Now 1 010025 > 1 by 0 010025, i e by 1 0025 % 



242 


MISCELLANEOUS 


[PAGE 


172 Let Vi, V 2 be the volumes of the spheres 
„ Si, So „ sp gravities „ 

» Wi, W 2 „ weights „ 

„ 1 1 , 1 2 >j radii „ 


Then from Ait 247, E\ 1, we see that 
V, j,® rA, W« 


3 ‘ 

'2 '2 
j X 289 


Substituting given values, 
17 > 2 ® \17/ r, 17 


173 Their invested capitals are in the pi opoition 6 4 3 
A uses wliole capital for 4 months, 
and half „ 8 „ 

B and C each use whole capital for 12 months 
then shales of the piofits are in the proportion 
(6x4)+(3x8) 4x12 3x12, 
le 48 48 36, 
te 4 4 3 

So A has of Es 2024 =R 3 736 


174 The inciease in fiist population is 45682x008, in the second, 
25408 x 0 045 , the deciease in the thud is 18960x0 1 If v 
be the req** average inciease pei cent, this actual inciease is 

(45682+25408+18960)x~, or 90050x3^ 

luu 100 

90050 X j^=45682 x 0 08+25408 X 0 045 - 18960 x 0 1, 

01 9005a=365456+114336-1896, 
whence =3 2 


175 Since 2s 8cf per cwt =£2 13s 4£f per ton, 

each ton of oie costs him £8 16s 8tf +£2 13s id, or £1U 

But only of this 13 coppei, of which only 92% can be 
extiacted and sold Hence if £v be leq** price, 

0^ t'uw of (soo Art 292) , 

^=-¥-xHvXJ50^xJ^o=£ 62 10s 



243 


46S] 


EX^VMPLKS VI 


176 Suppose the thickness of a halfpennj is nn , 

then the thickiiest. of a peniiv is I 23 r in 
\ oluine of halfpcnns =0 78'j4 x 1- x r cii in , 
and \olunicof ponnj =0 78 1 1 x (1 23)" x 1 23.'* cii in 

07854xl-xj I 1 


nq^ friction = 


0 7854 y (1 23)' X 123 a (123)’ 1861 


=034 


177 Marked price = nominal cost pi ice X (see Art 289) 
But hccaii''e of the fniiidulent balance, 

no minal cost p rice _ 1 + iV _ 16 
actual cost piice 1 “ 15 ’ 

iinminal cost pncc=actual cost price x } 2 
Hence niaikcd price = actual cost piiccx } J^x 
=actual cost price x 
and req'’gain = 12% 


178 

I 

II 

Ill 


S wine, 


9 V me, 


4 wine, 


8 w atcr 


13 iratei 

. 


28 watei 


In II there is room for 8 more gallons 
after fllling up II fioni 111 thcic arc 

f (‘l+i'j of 8) Mint, 1 /lO wine, ) 
1(13+ of 8) w itei,J ’ (22 watci / 

In I there is looiii for 16 moio g-illons 
after filling up I fioiii IT thcic aic 

f(8+ ’,2 of 16) wine, ) fl3 wine, 1 
1(8+ y; of 16) water,/ * ^ ll9 watci / 

179 A 1 ea of cnclosui e = (1 0 x 4840 v 9) sq ft 

length of its sidc=^/(46400 x 9) ft 
=660 ft 

length of Bticct=(4x690) ft 
and its w idtli is 30 ft 
no of paMng stones ieq^=4y 690 x 30 X A 

= 110400 





244 


mSCELLANEOUS 


[page 


180 Let =pieaent value of bill , 

then r^^=513 

fi 1 ixsoo 

* Ttri 

=506^ 

=£506 5 a 

total assets=£l422 lOs 4d 
and debts =£2134 lOf 

he pays in the pound £lgi|i| 

=£666 
=13s 4d 


181 The decimal evidently terminates when the denominator is 
(i) any power of 10, (ii) any pow ei of 2, (iii) any power of 5, 
(iv) a product of any po\iei of 2 ana any poiiei of 5 
And (i) IS included in (iv) 

But, after the fraction has been reduced to its lowest terms, 
with any of the othei nos, 3, 6, 7, 9, 11, etc, as denominator, 
or as a factor in the denominator, the decimal does not 
terminate 

where 8=2^ the decimal terminates 


182 


(i) A +B+C do tV of the woik in 1 day 

(ii) B+C+D „ -tV „ „ 

(ill) C+D+A „ „ „ 

(iv) D+A+B „ „ „ 

combining (i) and (ii), A 


18 20 2 9 10 180 
(0 II ^“■®=I8“24'"3'F8""72 


A+B-2D=^+~= 


2+5 


But(iv) A+B+D=^ 


180 ‘ 72 9 4 5 2 9 4 5 2 


321=^- 


3 9 9 4 5 2 
40-21 


3 9 4 5 2 


— 1 9 
“iTTSTT 

D does of the work in 1 day 



469] 


EXAMPLES VI 


245 


A+B-^-C+J) do iV+ ulfg - “ 1 

— ISO+IB 


_ 1 <10 
-• 3 :rro 

A+iS+C'+jD do the work in - fs^ - days, 
1 c m 16-j^V days 

183 Let Bs v=niaiked piice of article 

957? 

Then Rs j^=piice obtained from customei 

BatRs712ix|= „ „ „ 

93 c ^,-1 4 
100“ '^^‘^3’ 

19j;^ 1425 4 
20 2 ^3’ 

19 c= 19000, 
r=1000 

niaiked pnce=Bs 1000 

184 1'* income = Rs 4-41^ x § 

=Rs 1575 
„ 44100 X 98k X 5 


2“^ income=Rs 


98x110^^ 


=Rs ^ ^ ” iJfJJLKO- 

98X2X1773 

=Rs 2000 

increase in income=Rs (2000-1575) 
=Rs425 

185 The losei runs 9iV yds in 1 sec 

he runs 100 yds in secs , 

7 c 11 secs 

The winner takes 1 sec less 

he runs 100 yds in 10 secs 

100 yds in 10 secs = ^i 7 eoxio” P®* 
=20^ „ „ 
100 yds in 11 secs = ~ ff ' ^ox ' i ' i° 


99 


99 



246 


IIISCELLINEOUS [PAGE 


186 


"Wlion li Ins onco completed tlio coulee A is not half way lound 
So ZJ, while going lound the second time, will catch up A 
Let 1 =110 of secs from start befoie they aio together again 


In rsecs A goes 


-1- of the comse 
224 


B goes once lound and of the course, 


t e 


224 + r 
224 ' 


But in V secs B goes ^ of the couree 
88 

1 224 + a 

224 ’ 

224 1 =19712 +88r, 
136J, = 19712, 
a = 144J^ secs 


87 


Intel est paid=Rs ~ ” x i o o ' ^ ~ ~ 5 10 a 

Interest obtained=Rs 5 10a +R& 2 13a =Rs 8 7a. 
Let a = late of interest obtained 


Then 


250 X 3 X ^ 
4x100 

15r_135 
8 16’ 


v=S.=4X% 


188 Original value=Rs 54321 

Le.s3 5% 2716 05 

.Aftei 1 yi =51604 95 
Less 5 % 2580 2476 

After 2 yis =49024 7025 
Lest. 5 % 2451 235125 

Aftei 3 yis =40573 467375 
Less 5 % 2328 673368 

Aftei 4 yis =44244 794 
Less 5 % 2212 2397 

After 5 yrs =42032 5543 
Less 5 % 2101 6277 

After 6 yis =39930 9266 
t e value to the neaiest rupee=Rs 39931 



4G9] 


FAAMPLES VI 


247 


189 Ijot i=no of iiiimitc 5>pnco'! coieicd by lioui band bcfoie bands 
coincide 

Tlicn 12 . 1=110 of minute spaces co\eitd bj minute liand 
12.1 -C0=i, 
lli=G0, 


_ r O _ r 


r= 


1 J 


»1T 


So tint bauds shotild coincide aftei G5i t nuns 
Rut tbcj coincide aflei 00 nuns of concct tunc 
in CG niiiiB. of cm i cct time clock is nun slow 

ij ^ “4 ,, ,, „ 


nun sloiv 

= 1112? 


w 


190 


■R bile D poos ‘140 i ds , C goes 12 1 1 ds 

n „ ids 

I * 4 (I t 4J1JL'' - < r,lo 

' » n I I o~ ’ 


V 

V) 


= n2;.iLA vds 


’T> 4 B j 

/? wins 1y 7J-4-J-'j 3 ds 


01 Afioow nil iTFii AT TUI L iNri''m inros IT nonriiT MAti rnosr and co ltu