# Full text of "Arithmetic for advanced classes; being a continuation of Trotter's lessons in arithmetic for junior classes: containing vulgar and decimal fractions; simple and compound proportion, with their application; simple and compound interest, involution and evolution, etc"

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C-NRLF B 2b2 7MT LIBRARY OF THE Jniversity of California. Received %yUy^^ • ^^9 9 • tccession No.^ y ^"M^ • Cla^s No, are now ready. In the PRIMER the vowel sounds are presented in an easy and natural manner, being in every case exemplified by real words rather than by arbitrary syllables, and arranged m rhymmg groups. The lessons are composed of sentences woven mto narratives, and hieroglyphic lessons have been introduced for the purpose of making the work of revisal more varied and interesting. In the FIRST STANDARD the narrative form has been pre- served throughout, and the lessons, while inciidentally supplying considerable information, are mainly intended to enable the child to overcome the mechanical difficulties of reading. They have therefore been made as light and attractive as possible; many elliptical, and, as a new feature, several alliterative and hiero- glyphic, lessons have been constructed. Easy lessons are also given in Script for the reading and writing of Manuscript. In the SECOND STANDARD a variety of interesting matter has been simplified by the syllabification of difficult words and the grouping together of common affixes. A novel feature is the introduction of lessons on the Tenses of Verbs. Useful information is imparted on common objects and animals, with lessons inculcating duty and honour. In Dictation a large pro- portion of the matter is shown in Script ; while the Exercises Oliver and Boyd's New Code Olass-Books. appended to these, direct increased attention to the subjects presented, and furnish plenty of school- work. In the THIRD STANDARD, as the child will now have acquired considerable fluency in easy reading, a varied selection has been made from authors that have long been favourites with the young. In the Dictation all the difficulties in spelling monosyllables and easy dissyllables have been anticipated, and the Exercises, which are partly in Script, have been constructed so as to foster the habit of observing words and their distinctions. 11. GEOGRAPHY. Three little works have been prepared by INIr W. Lawsox, F.Il.G.S., St Mark's College, Chelsea; Author of " Geography of the British Empire," etc. 1. The GEOGRAPHICAL PRIMER will be found adapted to the requirements of Standard IV. The meaning of a Map is clearly explained ; an outline is given of the Chief Divisions of the World ; while the numerous facts have been selected and arranged to suit the age of the pupils. 2. The GEOGRAPHY OF ENGLAND meets the requirements of Standard V., and is intended to succeed the " Geographical Primer." The style and su!)ject are a little in advance, and there is some attempt to show the dependence of one part of the geography upon another. A Chapter on the principal Railways will be found to meet the increasing desire for information on this subject. 3. ELEMENTS OF PHYSICAL GEOGRAPHY. This work has been written as a " Specific Subject," with special reference to the New Code. The language and illustrations are simple, and suited to the capacity of pupils of from ten to fourteen years of age. III. ARITHMETIC. This subject has been undertaken by Mr Alex. Teotter, Teacher of Mathematics, etc., Edinburgh ; Author of " Arith- metic for Advanced Classes," etc. Part I. embraces Standards 1 and 2. „ II. „ „ 3 and 4. Part III. [in preparation) will embrace Standards 5 and 6. [Continued at end of Book. LESSONS IN ARITHMETIC FOR WITH TABLES OF MONEY, WEIGHTS, AND MEASURES, ACCORDINO TO THE IMPERIAL STANDARDS. By JAMES TROTTER, LATE OP THE SCOTTISH NAVAL AND MILITARY ACADEMY, Author of "A Complete System of Arithmetic," etc. EDINBURGH: OLIVER AND BOYD, TWEEDDALE COURT. LONDON : SIMPKIN, MARSHALL. AND CO. Price 6d., or 8d. cloth. Advanced Arithmetic, in Continuation of this Work, 6d., or 8d. cloth. Also, strongly bound together in leather. Is. 3d. Answers to both Works, 6d. each. 1872. vt T7 SCHOOL-BOOKS BY JAMES TROTTER, LATE OF TUB SCOTTISH NATAL ANI> MILITARY ACADEMY. LESSONS in ARITHMETIC for Junior Classes. 6d. A CoMPLBTB System of ARITHMETIC, Theoretical and Practical. 3s. Teotteb's Edition oi MUTTON'S BOOK-KEEPING. 2s. A Complete System of MENSURATION, by Ingram & Trotter. 2*. Ingram and Trotter's EUCLID; containing the Elements of Plan© Geometry and Trigonometry Is. 6d. Ingram's Concise System of MATHEMATICS. Revised by Mi Trotter. 4s. 6d. Trotter's LOGARITHMS and PRACTICAL MATHEMATICS. 3a. Ingram and Tbotteb's Elements of ALGEBRA. 38. PRINTED BY OLIVER AND BOYD, EDINBURGH. ADVERTISEMENT TO THE ENLARGED EDITION. The following little Work was originally composed for the use of the Author's Junior Classes. It was afterwards submitted to the public, in the hope that it would be found worthy of an introduction to Public Schools and Academies, and that, from the number and variety of the Exercises, it might prove a useful auxiliary to Governesses and Families. This hope having been fully realized, the present Edition has been subjected to a careful revision, and enlarged by the introduction of simple illustrations of the various rules and of a considerable number of Practical Exercises ; at the end of the work also, are given Exercises on that system of Decimal Coinage which, in course of time, is most likely to be adopted in this country. These additions have been made by the Author's son, Mr Trotter, Teacher of Mathematics, &c., who has recently pre- pared a Continuation of this Work for Advanced Classes. MULTIPLICATION TABLE. 2 times 4 times 6 times 8 times 10 times 12 times 2 are 4 2 are 8 2 are 12 2arel6 2 are 20 2 are 24 3 ... 6 3 ... 12 3 ... 18 3 ... 24 3 ... 30 3 ... 36 4 ... 8 4 ... 16 4 ... 24 4 ... 32 4 ... 40 4 ... 48 5 ... 10 5 ... 20 5 ... 30 5 ... 40 5 ... 50 5 ... 60 6 ... 12 6 ... 24 6 ... 36 6 ... 48 6 ... 60 6 ... 72 7 ... 14 7 ... 28 7 ... 42 7 ... 56 7 ... 70 7 ... 84 8 ... 16 8 ... 32 8 ... 48 8 ... 64 8 ... 80 8 ... 96 9 ... 18 9 ... 36 9 ... 54 9 ... 72 9 ... 90 9 ... 108 10 ... 20 10 ... 40 10 ... 60 10 ... 80 10 ... 100 10 ... 120 11 ... 22 11 ... 44 11 ... 66 11 ... 88 11 ... 110 11 ... 132 12 ... 24 12 ... 48 12 ... 72 12 ... 96 12 ... 120 12 ... 144 3 times 5 times 7 times 9 times 11 times 20 times 2 are 6 2 are 10 2 are 14 2 are 18 2 are 22 2 are 40 3... 9 3 ... 15 3 ... 21 3... 27 3 ... 33 3 ... 60 4 ... 12 4 ... 20 4 ... 28 4... 36 4 ... 44 4 ... 80 5 ... 15 5 ... 25 5 ... 35 5... 45 5 ... 55 5 ... 100 6 ... 18 6 ... 30 6 ... 42 6... 54 6 ... 66 6 ... 120 7 ... 21 7 ... 35 7 ... 49 7 ... 63 7 ... 77 7 ... 140 8 ... 24 8 ... 40 8 ...56 8... 72 8 ... 88 8 ... 160 9 ... 27 9 ... 45 9 ... 63 9... 81 9 ... 99 9 ... 180 10 ... 30 10 ... 50 10 ... 70 10... 90 10 ... 110 10 ... 200 11 ... 33 11 ... 55 11 ... 77 11... 99 11 ... 121 11 ... 220 12 ... 36 12 ... 60 12 ... 84 12 ...108 12 ... 132 12 ... 240 EXPLANATION OF ARITHMETICAL TERMS AND SIGNS. Number is either a unit, or consists of a collection of units ; being the name given to our conception of things considered as one or many. Abstract numbers. When we consider numbers in their general nature, without referring them to any particular subject, they are then called abstract; as, 3, 7, 10, &c. Concrete or applicate numbers. When we consider number not in its general nature, but as applied to certain })articular things, as, two pounds, three pence, &c., it is termed concrete or applicate. A WHOLE number cousists of one or more units. A FRACTION consists of one or more parts of unity. A mixed number consists of a whole number and a fraction. A COMPOUND number cousists of several applicate num- bers joined together in one expression ; as, £4, 6s. 8d. An even number is that which can be divided into two equal whole numbers. An odd number is that which cannot be divided into two equal whole numbers. A PRIME number is that which can only be divided by itself and unity, without a remainder; and numbers are said to be prime to each other when no number but unity will divide both without a remainder. A SQUARE NUMBER is the product of any number by itself. A CUBE NUMBER is the product of a number and its square. A COMPOSITE NUMBER is that produced by multiplying two or more numbers together; thus 28 = 4X7 is a composite number, and 4 and 7 are called its component parts. An ALIQUOT PART is a number which is contained in a greater an exact number of times ; thus 4 is an aliquot part of 16, but not of 17, as it is contained exactly 4 times in the former, and in the latter 4 times and 1 over. An integer is any whole number; as, a pound, a mile, &c., or, 1, 2, 4, 6, 9, &c. Minuend is the greater number in Subtraction. buBTRAHEND is the Icss number. 4 ARITHMETICAL TERMS AND SIGNS. MuLHPLicAND in Multiplication is the number to be multiplied or repeated. Multiplier is the number by which we multiply, or which expresses how often the multiplicand is to be repeated. Product is the sum or result of the operation in Mul- tiplication. Factors. The multiplicand and multiplier are called factors of the product. Divisor in Division is the number by which we divide. Dividend is the number to be divided. Quotient is the number which shows how often the divisor is contained in the dividend, or the result of the operation. Denomination in applicate numbers is the name of the subject to which the number is applied; as, pounds, shil- lings, yards, miles, &c. Numerator is the upper number of a fraction, and shows how many parts of unity are expressed by the fraction. Denominator is the under number of a fraction, and shows into how many parts the unit is divided, A COMMON measure is any number that will divide two or more numbers without a remainder, and their greatest common measure is the greatest number that will do so thus 2 is a common measure of 12 and 18, and 6 is their greatest common measure. A COMMON multiple of two or more numbers is any number that contains each of them an exact number of times, and the least number that will do so is called their least COMMON MULTIPLE ; thus 48 is a common multiple of 12, 6, and 4, and 12 is their least common multiple. = {equal to) denotes equality; thus 21s. = 1 guinea. -f- Cptiis) addition ; thus 6 4-4=10. — {minus) subtraction; thus 7 — 3= 4. X {multiplied hy) multiplication; thus 4X3 = 12. -7- {divided hy) division; thus 18 ~- 6= 3. : {is to) : : {as) are signs used in proportion to denote an equality of ratios; thus 4 : 6 : : 8 : 12 denote that the ratio of 8 to 12 is the same as that of 4 to 6, and is read, 4 is to 6 as 8 is to 12. i represents & farthing ^ or the quarter of any thing. i a halfpenny, or the AaZ/of any thing. f three farthings, or three quarters of any thing. AEIIHMETiCAL TABLES. ADDITION AND SUBTRACTION TABLE. 1 2 3 4 5 6 7 8 9 11 3 4 5 6 7 8 9 10 11 12 4 5 6 7 8 9 5 6 7 8 9 10 6 7 8 9 10 11 12 13 14 15 7 8 9 10 11 l2 13 14 l5 16 8 9 10 11 12 13 14 15 16 17 9 10 11 12 13 11 15 15 17 18 10 11 12 13 14 15 16 17 1! 19 11 12 13 14 15 16 17 18 19 20 2l 22 23 24 25 26 27 28 29 30 31 12 13 14 15 16 17 18 19 20 !i 22 23 24 25 26 27 28 29 30 31 32 33 34 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 17:18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 59 40 1 2 3 4 5 2 3 4 5 6 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 ?1 32 33 18|19 19|20 2021 21j22 2223 23:24 24 25 25^26 26 27 20 21 22 23 24 25 26 27 28 29 30 3l 32 33 34 35 36 37 38 39 6 7 8 9 7 8 9 10 lOill 1112 12 13 13 14 10 11 12 13 14 15 16 17 18 19 20 11 12 13 14 15 16 11 18 19 20 21 12 13 14 15 16 17 18 19 20 21 22 13 14 15 16 I7 18 19 20 21 22 23 14 l5 16 17 18 19 20 21 22 23 24 15 16 17 18 19 20 21 22 23 24 25 18 19 20 21 22 23 24 25 26 17 18 19 20 21 22 23 24 25 26 27 18 19 20 21 22 23 24 25 26 27 28 19 20 21 22 23 24 25 26 27 28 29 20 21 22 23 24 25 26 27 28 29 30 27 28 29 30 31 32 33 34 35 36 37 28 29 30 31 32 33 34 35 36 37 38 Note. Before commencing Arithmetic it is absolutely neces- sary that the pupil should commit to memory that part of the preceding table which is cut off by a double line. The remaining part should likewise be learned as soon as possible. The same remark applies to the Multiplication and Division Table on the next page, as well as to all the tables which follow. Indeed the earlier that a child begins to learn the Arithmetical Tables, the more lasting will the impression be upon the mind, and his pro- gress in Arithmetic afterwards will be easy and unobstructed. ARITHMETICAL TABLES. X) ^ s § o o CO o t o CT) "^ 00 00 ■o" CO o 00 co" CO t CO o t o" o" CO t o CO CO 05 00 cO~ t 1 co" o CO 00 CO CO CO C5 t oo" CO CO" CO CO i o 00 CO CO CO CO i 8 § CO CO cr> CO CO (N CO o CO oc (N 00 §! CO c^ C<l c^ 00 CD CO 1 CO OS 00 l> o CO CO CO" CO CO kO t CO CO CO CO 05 t CO CO CO CO i o 00 CO 2 CO lO CO CO CO CO o CO 00 o CO OC' QO cT CO CO CO CO CO CO CO CO CO CO l> CO t CO CO t CO 00 l> CO "<# CO Ci Co" o 00 Oi Ci CO CO oo" CO t co~ o 05 X; £L CO CO To 1 CO CD 00 00 CO o CO to CO -^ o 'f CO CO CO CO CO o CO CO CO FT o -^ CO o CO 05 tH t CO s — ( o X) CO ^ ^ -* s CO 00 s -ob- o o CO CO CO CD 00 CO o 00 CO 05 cd" 00 o" co" t CD CO 2 00 co~ CO t To CO 00 ?^ 05 o 05 !>■ CO xO co" CO o o CO t CO CO o C5 X CO CO 00 00 C5 o CO 2 CO CO 00 CO CD CO CO CO 2 00 o CO CO CO CO CO g 1 o § ^ o 00 CO o o 00 05 05 00 00 t>. GO CO 05 X o 05 00 05 CO t o CO CO CO 00 CO CO § 2 CO o 00 GO CO CO CO CO 00 o CD CO CO CO s 00 o 05 CO 05 CO o GO o kO 05 s CO o CO CD g g 00 o 05 c o to TO 00 OS CO CO c ^ 00 CO s^ CO o -^ o CO CO CO 00 -^ CO CO CO CO ^ CO 00 GO CO CO kO o CO CO kO 00 CO CO ^ to CO 00 o CO CO CO 00 00 tH 05 o CO CO CO CO CO CO CO o CO CO CO CO *>* CO T— ( CO CO ^ o CO 00 00 05 o CO '"' (M CO Tt< lO CD r> 00 CJ5 o ^ CO CO 05 o CO Note. In using the preceding table for a Division one, we take the numbers in the left-hand column out of the numbers in the same horizontal line, and the number of times each is con- tained will be found either in the top or bottom line. ARITHMETICAL TABLES. STERLING MONEY. 4 farthings qi's. =■ 1 penny d. 12 pence =: 1 shilling 5. 5 shillings =. 1 crown cr. 20sMm„gs ={Lrrefg^''£ 21 shillings = 1 guinea G. TROY WEIGHT. 24 grains ^r.= 1 penny weiglit (fitf^ 20 dwt. = 1 ounce oz. 12 ounces = 1 pound lb. Gold, Silver, and Jewels, are weighed by Troy Weight. APOTHECARIES' WEIGHT.* 20 grains gr. = 1 scruple ^ 3 scruples = 1 dram ^ 8 drams = 1 ounce 5 12 ounces = 1 pound lb. Used only for medical prescriptions. AVOIRDUPOIS WEIGHT. 16 drams cZr.^i 1 ounce oz. 16 ounces = 1 pound lb. 28 pounds = 1 quarter qr. 4 quarters = 1 hundred wt. cvA. 20 hund wt.= 1 ton T. 112 lbs. = 1 cwt. 7000 grains = 1 lb. avoird. 14 lb. = 1 stone This table is used for all articles, except Gold, Silver, and Jewels. LINEAL MEASURE. 12 lines li. 12 inches 3 feet 5i yards 40 poles 8 furlongs reO yards = 1 inch 171. = 1 foot ft. =. 1 yard i/d. = 1 pole po. = 1 furlong fur. = 1 mile ml. = 1 mile 2 yds. or 6 feet = 1 fathom 2 i feet = a military pace 4 inches = 1 hand 1 1 foot = 1 cubit 22" yds. or 66 ft. = 1 chain; and as the chain contains 100 links, each link is = 7*92 inches, and 80 chains = 1 mile. CLOTH MEASURE. 2} inches = 1 nail nl. 4 nails = 1 quarter qr. 4 quarters = 1 yard yd. 8 quarters = 1 Flemish ell Fl. e. 5 quarters = 1 English ell En. «. 6 quarters =: 1 French ell Fr. e. 37 inches = 1 Scotch ell Sc. e. GEOGRAPHICAL MEASURE. 6076 feet nearly = 1 geo. mile 3 miles = 1 league le. 20 leagues = 1 degree dpg. or" 360 degrees =. the earth's cir- cumference SQUARE, OR LAND MEASURE. 144 square inches = 1 square foot 9 sq. feet = 1 square yard 30 J sq. yards ^ 1 pole or perck 40 perches = 1 rood ro. 4 roods =1 acre ac. 640 acres = 1 sq. mile 36 sq. yards = 1 rood of building 100 sq. feet =1 square of flooring 10 sq. chains, or ) ^ 100,000 sq. links | — -^ acie CUBIC, OR SOLID MEASURE. 1728 cubic inches = 1 cubic foot 27 cubic feet = 1 cubic yard 40 cubic feet of) rough, or 50 of V= 1 load lo. hewn timber J 42 cubic feet = 1 ton shipping 6 cubic feet = 1 barrel bulk MEASURE OF CAPACITY. 2 pints pt. = 1 quart qt. 4 quarts = 1 gallon 2 gallons = 1 peck 4 pecks = 1 bushel 8 bushels = 1 quarter pk. bu. qr. ANGULAR MEASURE. 60 seconds " =1 minute ' 60 minutes = 1 degree ** 30 degrees = 1 sign 5. 12 signs = 1 circle cira. • In the British Pharmacopoeia (1864), the whue the lb. avoir, of 7000 grains, and the oz. 1 . Troy of 4f10 prains has been abolished, »oir. of 437^ grains, have been adopted. ARITHMETICAL TABLES. APOTHECARIES' FLUID MEASURE * 60 minims min. = 1 drachm Jl. drm. 8 drachms = 1 ounce /f. oz. 20 ounces = 1 pint O. 8 pints = 1 gallon C. HAY AND STRAW WEIGHT. 36 lbs. avoir. = 1 truss of straw 56 lbs. = 1 truss of old hay 60 lbs. = 1 truss of new hay 36 trusses = 1 load Hay sold befween the begirning of June and the end of August, of that gear's growth, is reckoned new. TIME. 60 seconds sec. = 1 minute mi. 60 minutes = 1 hour ho. 24 hours =. 1 day da. 7 days = 1 week we. 4 weeks = 1 co. month mo. 365 days, or 52 ] weeks and 1 >-= 1 co. year t/e. day j 365J days = 1 Julian year 366 days = 1 leap year The year is divided into 12 cal endar months, viz. : QUARTERLY TERMS. In England. Lady-day, March 25. Midsummer, June 24. Michaelmas, September 29. Christmas, December 25. In Scotland. Candlemas, February 2. Whitsunday, May 15. Lammas, August 1. Martinmas, November 11. FLOUR & BREAD WEIGHT. A peck-loaf = 17 lb. 6 oz. avoird. A half-peck do. = 8 11 — A quarter-loaf = 4 5J — - A peck of flour is 14-44 lb,, or 14J lbs. nearly, and a bushel 57J lbs. very nearly. Five bushels make a sack, which ought to weigh 288-8 lbs. avoirdupois. The number of days in each month may be easily remembered from the following lines : Thirty days hath September, April, June, and November; All the rest have thirty-one, Excepting February alone, Which hath but 28 days clear, And 29 in each leap year. 365 days 5 hours 48 min. 50 sec. = 1 solar or tropical year. January 31 days. February 28 — March 31 — April 30 — May 31 — June 30 — July 31 days. August 31 — Septem.30 — October 31 — Novem. 30 — Decern. 31 — MISCELLANEOUS TABLE. 24 sheets 20 quires 10 reams 12 articles 20 articles 12 dozen 12 gross 120 articles 500 bricks 1000 tiles : 1 quire of paper : 1 ream : 1 bale : 1 dozen : 1 score : 1 gross r 1 great gross = 1 great hundred : 1 load = 1 load 500 herrings = 500 red do. = 10(X) sprats = 60 herrings = 100 lbs. avoir.: 56 lbs. = 64 lbs. = 256 lbs. = 112 lbs. r 19i cwt. = : 1 barrel : 1 cade : 1 cade :1 keg : 1 bari. gunpowder : 1 firkin of butter : 1 firkin of soap : 1 baiTel of soap : 1 barrel of raisins : 1 foddftr of lead ♦ AocordiiMi to the British Pharmacopoeia (18C4>. ARITHMETICAL TABLES. Farthings. Pence. qrs d. d. s. d. 4= = 1 12=1 6. .. 1; 13.. .1 1 6. .. It 14.. .1 2 7. .. 1: 15.. .1 3 8. .. 2 16.. .1 4 9. .. 2J 17.. .1 5 10. .. 2% 18.. .1 6 11. .. 2| 19.. .1 7 12. .. 3 20.. .1 8 13. .. 3i 21.. .1 9 14. . 3$ 22.. .1 10 15. . 3| 23.. .1 11 IG. 17. 18. 19. . 4 24... 2 25.. .2 1 26.. .2 2 27.. .2 3 20. 21. . 5 . 5i 28.. .2 4 29.. .2 5 22. . 5 J 30.. .2 6 23. . 61 31.. .2 7 24. . 6 32.. .2 8 25. . 6J 33.. .2 9 26. . ^ 34.. .2 10 27. • 4 35.. .2 11 28. . 7 36.. .3 29. • 7} 37.. .3 1 30. • 7} 38.. .3 2 31. • n 39.. .3 3 82. . 8 40... 3 4 33. • 8i 41.. .3 5 34.. . 8} 42.. .3 6 35.. • n 43.. .3 7 36.. . 9 44.. .3 8 37.. • H 45... 3 9 38.. . 9f 46.. .3 10 39.. • n 47. ..3 11 40.. .10 48... 4 41. •lOi 49.. .4 1 42. •lOJ 50... 4 2 43. .io| 51. ..4 3 44 .11 52... 4 4 45. .llj 53... 4 5 46. .11} 54... 4 6 47. .lll 55.. .4 7 48. .12 56.. .4 8 MONEY TABLE. d. s. 57=4 58.. .4 59.. .4 60... 5 61.. .5 62.. .5 63..,5 64.. .5 65.. .5 66.. .5 67.. .5 68.. .5 69.. .5 70.. .5 71.. .5 72.. .6 73... 6 74.. .6 75.. .6 76.. .6 77.. .6 78.. .6 79.. .6 80... 6 81.. .6 82.. .6 83... 6 84.. .7 85.. .7 86.. .7 87.. .7 88.. .7 89.. .7 90.. .7 91.. .7 92.. .7 93... 7 94... 7 95.. .7 96.. .8 97.. .8 98.. 8 99... 8 100.. .8 101.. .8 Shillings. 1 . d. sk. £ s. 9 20= =1 10 21. ..1 1 11 22. ..1 2 23. ..1 3 1 24. ..1 4 2 25. ..1 5 3 26. ..1 6 4 27. ..1 7 5 28. ..1 8 6 29. ..1 9 7 30. ..110 8 31. ..1 11 9 32. ..1 12 10 33. ..1 13 11 34. .1 14 35. ..1 15 1 36. .116 2 37. .1 17 3 38. .1 18 4 39. .1 19 5 40. .2 6 41. .2 1 7 42. .2 2 8 43. .2 3 9 44. .2 4 10 45. .2 5 11 46. .2 6 47. .2 7 1 48. .2 8 2 49. .2 9 3 50. .2 10 4 51. .2 11 5 52. .2 12 6 53. .2 13 7 54.. .2 14 8 55. .2 15 9 56. .2 16 10 57. .2 17 11 58. .2 18 59. .2 19 1 60. .3 2 61. .3 1 3 62. .3 2 4 63. .3 3 5 64. .3 4 sh. 65= 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85., 86., 87., 88., 89., 90., 91., 92.. 93.. 94.. 95.. 96.. 97.. 98.. 99.. 100.. 101.. 102.. 103.. 104.. 105.. 106.. 107.. 108.. 109., £ =3 .3 .3 ..3 ,.3 9 .3 10 ,.3 11 .3 12 ,.3 13 .3 14 .3 15 .3 16 .3 17 .3 18 .3 19 .4 .4 1 .4 .4 .4 .4 .4 .4 .4 4 .4 10 .4 11 .4 12 .4 13 .4 14 .4 15 .4 16 .4 17 .4 18 .4 19 .5 10 ARITHMETICAL TABLES. NUMERATION TABLE. Units 9 Tens 98 Hundreds 987 Thousands 9; 876 Tens of Thousands 98; 765 Hundreds of Thousands 987 ; 654 Millions 9 ; 876 ; 643 Tens of Millions 98; 765; 432 Hundreds of Millions 987 : 654 ; 321 Billions 9; 876; 543; 210 Tens of Billions 98; 765; 432; 109 Hundreds of Billions 987; 654; 321; 098 Trillions 9; 876; 543; 210; 987 EOMAN NOTATION. The Eomans used the following letters only for numbers, viz. I one, V five, X ten, L fifty, C a hundred, D or Iq five hundred, and M or CI^ a thousand. Any letter followed by another of equal or less value denoted the sum of their separate values ; thus III three, LXXVl seventy-six. Any letter followed by another of greater value denoted the liflerence of their separate values ; thus XL forty, XC ninety. Every 3 annexed to Iq, and every C and 3 joined to CIq, increased the value ten times ; thus 1q3 five thousand, CCIq;;) ten thousand. A line drawn over a letter denoted that its simple value was increased a thousand times ; thus X ten thousand, XL forty thousand. 1 or 1 XVII or 17 II .. 2 XVIII .. 18 III .. 3 XIX .. 19 IV .. 4 XX .. 20 V .. 5 XXI .. 21 VI .. 6 XXII .. 22 VII .. 7 XXIII .. 23 VIII .. 8 XXIV .. 24 IX .. 9 XXV .. 25 X .. 10 XXVI .. 26 XI .. 11 XXVII .. 27 XII .. 12 XXVIII .. 28 XIII .. 13 XXIX .. 29 XIV .. 14 XXX .. 30 XV .. 15 XL .. 40 XVI .. 16 L .. 50 LX or 60 LXX 70 LXXX 80 XC 90 C 100 CI, &c. 101 CC, &c. 200 CCCC or CD .. 400 lO or D 500 l0CorDC,&c. .. 600 loCCCC, DCCCC, or CM 900 CIo or M 1000 CIoCorMC,&c... 1100 MM or II, &c. .. 2000 loo 0^ ^» «^c. .. 5000 lODO or L, &c. .. 50,000 ARITHMETIC. Arithmetic, as a science, explains the propei'ties of num- bers, and as an art, the methods of computing by them. The fundamental rules are. Numeration, Notation, Ad- dition, Subtraction, Multiplication, and Division. The characters by which all numbers are expressed are, 1, one or unit; 2, two; 3, three; 4, four; 5, Jive; 6, six, 7, seven; 8, eight; 9, nine; 0, cipher or nought. NUMERATION Is the art of reading a number expressed in figures. Trillions. Billions. Millions. Thousands. Units. 604; 450; 360; 412; 474. Read or write in words the following : 24079 — Twenty-four thousand and seventy-nine. 79_97_18— 24— 81— 67— 76— 35— 67— 26— 53— 91 — 19 —48—101—208—84—110—802 — 111 — 109—119 — 125 — 152—319—913—301—310—4617—4107—4170—28410— 20814—5106—74125—47010—2097431—501746-730087— 1730086—9704010—21070—20202020—5170409 — 2017101 — 74107 — 1074010 — 29654301 — 102030401 — 157301074 —748017018—547207542—63710073001—54872193543270. NOTATION Is the art of expressing any given number in figures. Express in fig^ires the following : Five thousand and sixty-four — 5064. Seventy-four — ninety-six — one hundred and one — one hundi-ed and ten — one hundred and eleven — two hundred and eight — one hundred and eighteen — one hundred and thirty-one — one hundred and thirteen — seven hundred and eight — nine hundred and eighty — two thousand, three hun- dred and twenty-one — nine thousand and seven — twenty- one thousand and ten — one hundred and fifty thousand and five — six millions, forty thousand and thirty — eighty- niue millions, one hundred and forty thousand and twenty- six — seven hundred billions, ten millions, eleven thousand, one hundred and one — four hundred and one millions, seventy thousand and seventeen — eight trillions, twenty billions, sixty-nine millions, four thousand and sixty-three. B 12 SIMPLE ADDITION Is the method of finding a number equal to several num- bers taken togetlier. The number found is called the sum or amount. EXERCISES ON THE ADDITION TABLE. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 2 3 4 5 6 7 8 9 4 5 6 7 2 3 4 5 6 7 8 9 3 4 5 6 2 3 4 5 6 7 8 9 2 3 4 5 2 3 4 5 6 7 8 9 1 2 3 4 2 3 4 5 6 7 8 9 4 1 2 3 2 3 4 5 6 7 8 9 2 5 1 2 2 3 4 5 6 7 8 9 3 2 5 7 7 14 21 28 35 42 49 56 63 19 22 26 34 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 7 6 9 5 4 9 7 7 6 3 6 2 6 2 4 3 3 5 2 2 1 3 4 4 4 2 1 7 3 5 9 8 4 1 7 2 8 8 8 5 7 2 8 4 6 6 4 3 4 1 4 1 2 7 9 3 3 9 5 7 5 2 6 8 6 6 5 1 3 6 1 4 1 6 8 4 1 2 3 8 7 4 1 5 8 2 6 2 7 9 2 6 1 7 5 5 2 3 4 7 2 1 3 8 2 8 8 8 9 9 8 6 3 2 3 3 2 4 9 4 4 4 7 6 3 3 9 4 1 4 4 3 5 1 5 7 5 3 8 2 6 7 2 1 5 4 4 6 2 6 1 2 6 2 7 4 3 1 2 6 2 5 7 3 7 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 7 4 7 8 1 7 4 7 9 9 6 3 5 7 9 2 2 9 7 9 2 2 9 7 2 7 8 2 2 5 6 3 6 6 6 1 9 6 5 8 4 1 1 9 4 1 1 5 2 3 8 9 3 6 6 8 7 4 8 6 8 8 4 1 4 4 8 8 3 8 2 4 7 5 7 3 7 2 3 8 2 7 7 8 4 4 2 9 3 2 9 5 1 8 7 1 5 2 7 2 8 6 8 1 1 5 4 3 4 2 3 6 9 4 2 2 9 3 4 3 6 2 9 6 1 8 7 8 5 8 6 8 6 7 5 4 1 5 2 6 2 8 4 8 7 5 3 9 2 7 5 3 2 1 1 1 5 6 3 4 1 7 4 3 9 3 8 1 8 8 6 4 3 7 6 2 9 8 9 6 SIMPLE ADDITION. 13 Example. Add together 847, 478, 19, and 951. Ans. 229 Solution. Arrange the numbers as in the margin ; adding the units' or right-hand column, 1 and 9 are 10 and 8 are 18 and 7 are 25 ; write down 5 and carry 2 to the second column : 2 and 5 are 7 and 1 are 8 and 7 are 15 and 4 are 19 ; write down 9 and carry 1 to the third column : 1 and 9 are 10 and 4 are 14 and 8 are 22 ; write down 22, and the answer is 2295. The work may be checked by adding the columns down^ wards. 847 478 19 951 2295 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 234 754 869 649 214 314 987 374 215 118 982 475 698 495 421 431 879 743 152 181 342 638 986 218 638 209 798 437 521 811 758 863 213 821 863 920 654 865 634 214 875 921 542 637 759 516 465 685 463 579 426 192 121 736 975 651 546 856 346 798 11. 12. 13. 14. 15. 16. 17. 18. 7486 2146 4816 5411 2222 8888 5555 4848 4867 6412 6184 2196 3333 9999 6666 5959 2194 1093 7298 3482 4444 nil 7777 6767 1942 3901 8735 9876 5555 2222 8888 7676 7368 2473 4567 3846 ^m^ 3333 9999 9595 3687 . 3742 8912 2198 7771 4444 1111 8484 19. 20. 21. 22. 23. 24. 25. 26. 1234 7921 4869 1276 1874 2764 4872 6729 6678 1297 5728 2761 7481 6427 3481 6278 9012 3808 6372 3849 2310 3818 5834 6483 3456 8076 7184 4598 1046 2984 6287 7321 7890 6487 8296 6623 9875 4629 7821 1234 1234 7923 9543 6312 6793 9273 1234 6678 27. 28. 29. 30. 31. 32. 33. 34. 9847 2146 4121 1214 2009 .9817 9002 7189 6438 6148 1246 6421 9002 1789 2009 9871 5279 6437 3459 9543 4716 2138 6174 8312 7346 2977 9528 8259 6174 4817 4716 7184 8978 3888 6473 3746 8136 7864 6318 4687 6438 8436 8987 7898 2198 3189 8912 9813 14 SIMPLE ADDITION. 35. 36. 37. 38. 39. 40. 41. 8729 4816 7286 9112 9876 5469 5726 7298 3729 3465 2968 2187 3874 6275 4165 5412 2187 4627 4632 5286 3874 2189 8046 7129 3729 2893 9684 9873 3145 4208 1408 8463 3984 5836 3521 8729 9807 1076 2198 5726 2194 1234 42. 8768 7543 2189 9138 4672 8279 43. 44. 45. 46. 47. 48. 49. 50. 7284 4869 2790 4286 5216 2149 1876 2168 4563 4964 4623 6384 2615 9186 3848 8614 3629 5208 2347 2198 3842 3456 2193 2196 9245 2080 5867 5486 2876 7289 1984 5483 5483 1897 3867 2173 3184 9738 4876 3146 2196 7986 9218 4817 7296 4865 3842 8965 51. 52. 53. 54. 55. 56. 57. 58. 9726 6295 4872 2138 4965 9876 3097 2974 8643 4368 2198 5483 3846 6298 9808 9084 5273 7348 8169 9654 3876 6786 4097 9840 2736 2763 2367 3672 6723 7236 1876 9489 1894 9653 6539 4875 9864 2198 8965 1284 9867 2198 4963 2186 4372 7234 8629 5814 3095 1986 9631 8472 3729 2139 6243 8145 59. 60. 61. 62. 63. 64. 65. 66. 7486 2981 7298 2184 4763 4863 3456 9871 2193 1892 8917 4218 8769 6348 7890 2179 4728 3720 2347 5763 2986 2176 1234 5046 2089 4175 5486 3698 4863 8472 5678 6804 9082 5176 3847 7296 3648 2784 9012 8470 4754 2347 9176 4738 2198 5168 3456 1986 7538 6129 7153 7219 1927 3615 7891 3459 67. 63. 69. 70. 71. 72. 73. 74. 1874 2112 7411 2119 86 834 2174 2487 7486 331 1721 9112 186 4747 4187 87 8291 2897 862 7486 5681 6363 83 7428 9182 987 48 3849 2196 5995 3189 83 3748 729 5496 486 987 559 5 4876 1876 8297 7486 42 91 87 1765 42 741 54 59 3798 8746 63 6789 9899 3496 7289 876 7983 3904 7298 49 114 SIMPLE ADDITION. 15 75. 76. 77. 78. 79. 80. 87469 18734 89846 19876 98846 11421 84697 81423 72844 28674 21174 21896 33442 47884 51168 54869 38965 69847 21756 58337 27489 96843 56897 38176 67498 21486 98472 21876 21984 47897 39846 68742 21224 48638 51478 38189 27485 89638 18769 88768 31894 49898 58744 48621 97652 21777 98499 98974 81. 82. 83. 84. 85. 86. 74985 71279 84120 98797 98724 34563 12345 48694 21796 38468 84786 78908 67890 38848 69845 21896 86749 42809 90876 97120 38471 54868 87498 90786 65217 17208 18769 98976 98863 36094 71489 80967 48684 48698 97377 27158 38594 74689 18769 38489 98776 38646 48684 98467 38478 89765 19864 64583 87. 88. 89. 90. 91. 92. 47216 90804 49899 48899 87748 47189 86143 79048 98765 37744 51123 98765 31487 21886 34775 44768 17648 38486 21879 66477 21984 89443 48679 34896 39842 38896 56348 34886 15015 69847 23876 59769 84237 29876 27987 97849 54875 27998 73486 54869 92764 38488 16846 54889 54997 12345 89898 21776- 93. 94. 95. 96. 97. 98. 94863 42174 8989 49864 97867 8765 8639 7148 98798 644 1008 219 86394 837 654 6449 976 38480 7563 1896 94568 21786 54890 10846 75638 61784 28 38486 12789 8973 6387 4721 2875 9876 76 87997 29846 12 38486 54868 38147 738 4875 38469 94783 987 21898 84778 99. 7368+ 8451+5184+6372+ 3147+1763 + 2189. 100. 5436+ 2195+7964+6830+ 8347+5146+ 798. 101. 73847+85487+3486+5763+84695+3146+ 495. b2 16 SIMPLE ADDITION. 102. 103. 104. 105. 106. 714816 187621 876548 971028 918765 148617 317849 721473 876980 187659 548389 948647 374869 487694 876591 821864 218698 968768 527389 243876 217784 384869 486842 938765 438762 548987 198768 172986 387659 387624 987786 2147-29 348697 647548 876554 489754 987486 374898 475486 554433 457986 579864 548694 754864 765432 107. 108. 109. 110. 111. 314579 869457 304756 274816 908076 145793 694578 475630 748163 807069 457931 945786 756309 481634 760908 579314 457869 563098 816345 219374 894632 578694 630987 123456 475432 946328 786945 789063 234567 173849 463284 123456 890637 345678 948386 632846 789012 637890 876543 872198 328466 345678 378906 765432 749865 778998 901234 906378 654321 384976 112. 113. 114. 115. 116. 548637 493128 795846 497864 998776 486378 931284 598467 864794 887769 863789 312849 218694 468479 776698 637890 128493 580308 684947 433821 749087 740086 984678 218624 388466 490876 409648 394867 374186 218968 471874 218408 973842 987384 478149 548643 184820 298765 219864 941798 896847 123456 458738 718698 217486 376849 654321 219986 398748 989999 729287 987489 487219 216847 874865 117.47563+74298+98254+214865+652193+381964 +300892+476983+396847+734682. 118. 2 14736 + 637240 + 509984 +998447 + 219863 + 863214+792186+197235+748692+897628. 119. 742869+38475+8476+317286 + 863217 + 9846 +72354+748693+7486+95476+4721864. SIMPLE ADDITION. 17 120. 121. 122. 123. 124. 786904 5744 217846 849784 216 72189 186473 3868 84 47386 2891 862 9 7698 472 749863 7648 778466 377669 8 2847 97448 47 4886 67489 47283 189654 3848 847334 738789 898647 347219 543896 27 48 98 386 8965 7846 2748 749 4 47 987654 64 7864 786473 889764 87654 736 987654 876 74869 78 876489 125. 7486 + 7489 + 9846 + 3748 + 5634+7486+ 9847 +5329+4675+3869+9873+8469+4683. 126. 5276+8943+9486+3114+98760+3456+72894 +729+89657+3846+47836+7584+48765. 127. 74486+311472+68476+38169+744869 + 1870 +542138+216746+9876+521869+31468. 128.7486957+75312984+9104763+7238641+521437 +43879654+9876+34819+9896543+47869847. 129. The population of London, in 1851, was 2,362,236 ; of Dublin, 258,369 ; of Edinburgh and Leith, 191,221 ; of Glasgow, 329,097; of Liverpool, 375,955; of Bir- mingham, 232,841 ; of Manchester, 316,213 ; of Bristol, 137,328 ; and of Leeds, 172,270 : required the amount of the whole. 130. Bought a house for £3150 ; what should it be sold for to gain £275? 131. The number of wrecks and collisions on or near the coasts of the United Kingdom in 1852 was 1015 ; in 1853, 832; in 1854, 987; in 1855, 1141; and in 1856, 1153 : find the whole number during these five years. 132. The total number of British Cavalry who joined the Allied Army in the Crimean Campaign, was 4819 ; Artillery, 7032 ; Sappers and Miners, 403 ; and Infantry, 43,726 : how many men joined in all? 133. In 1856, the passengers conveyed by Kail in Scotland were. First Class, 1,664,005; Second Class. 1,952,240; Third Class, 9,476,226; Mixed, 4767: find the total number. c 18 SIMPLE ADDITION. 134. In the same year the receipts were, First Class. £232,130; Second Class,£171,588; Third Class,£436,564; Mixed, £14,892 : required the whole sum. 135. Find the sum of twenty- seven thousand, eight hundred and forty-nine — thirty-eight thousand, live hundred and forty-six — eight thousand and nine — twelve thousand, nine hundred and sixty-three — five thousand and forty — five hundred and seventy-eight thousand and forty-six — nineteen thousand and sixty — twenty-seven thousand, eight hundred and forty-seven. 136. A merchant has £1275 in the bank ; his goods are worth £2760 ; his household furniture, £565 ; and debts owing to him, £674 : how much is he worth ? 137. What quantity of tea was consumed in the United Kingdom in 1856, England having consumed 47,986,635 lbs. ; Scotland, 6,583,233 lbs. ; and Ireland, 8,708,344 lbs. ? 138. In 1856, the Emigrants to Canada consisted of 5555 English; 3872 Scotch ; 4357 Irish ; 3136 Prussians , 2806 Norwegians; 1249 Germans; 823 Belgians; 260 Swiss, and 381 Italians, French, &c. ; find the whole number. 139. Two travellers start from the same place and travel in opposite directions, the one travels 75 miles the first day, 63 the second, and 45 the third ; while the other travels 65 miles the first day, 180 the second, and 378 the third : how far distant will they then be from each other ? 140. In 1856, the quantity of cofi*ee consumed in Eng- land was 33,019,884 lb. ; in Scotland, 1,197,6851b. ; and in Ireland, 778,385 lb. : what quantity was consumed in the United Kingdom ? 141. In 1856, the tonnage of registered ships in the British Empire was in England, 3,461,031 tons ; in Scotland, 592,974 ; in Ireland, 250,455 ; in Jersey, Man, &c., 62,496 ; and in the Colonies, 949,780 tons : find the amount of tonnage. 142. A merchant owes to A £597, to B £694, to C £748, to D £899, to E £1045, and to F £1303 j how much does he owe in all? 19 SIMPLE SUBTRACTION Is the method of taking a less number from a greater. The greater number is called the minuend^ the less, the subtrahendj and the number found, the remainder or difference. Ex. From 7986734 Take 2463212 Diflf. 5523522 Ex. 607482678 minuend. 5140346 subtrahend. 602342332 remainder. 217486973489 105342341056 46798765483 23214342352 85179684729 23123461304 600796857439 342526125 10008694758 3242745 85069857497 3042354443 Ans. 9890771. Ex. From 10574363 take 683592. Sol. 2 from 3 leaves 1, write down 1 ; 9 from 6 we cannot, but 9 from ten leayes 1 and 6 are 7 ; write down 7 ; having borrowed ten, carry one to 5 is 6 from 3 we cannot, but 6 from ten leaves 4 and 3 are 7 ; write down 7 and carry one to 3 is 4 from 4, &c. The work may be checked by adding the lower nurnber and remainder together, or by subtracting the remainder from the upper number. From Take Difference Proof Proof 10574363 683592 9890771 10574363 683692 217634821643 90000000000 124368412781 47654321809 47386743841 31728698748 10. 11. 12. 987214638475 1 47869386481 63112141763 298765428969 j 18976248656 32197648763 13. 14. 15. 804765786935 30241704862 47214127004 276548674876 18702930409 21807163047 16. 17. 18. 172876548734 20468754874 53748688714 89658714968 9876547185 31765948976 20 SIMPLE SUBTRACTION. 374869040735 9876524698 10074021004 21047386943 734861047 987654897 22. 23. 24. 734869548647 20417386984 15473846731 27486009829 1763047098 7348209872 25. 111473869875 9174869989 26. 27. 21765483642 1 60000472986 9176254961 | 73864786 28. 29. 30. 300712684734 10203040506 60708090104 987000487 1020304050 6070809017 31. 32. 33. 10000473698 1 34072986410 20172345604 784629 1 29738047306 1073647298 70047216384 17047386473 40100721647 1976006548 8721738462 1700876109 87. 38. 39. 21734007201 21738400078 40072173867 9172073167 4764873091 74169081 40. 41. 42. 100002402503 600043216753 100000643289 76543209 67429768 854989 43. From 44. From 45. From 46. From 47. From 48. From 49. From 50. From 51. Take 52. Take 53. Take 54. Take 55. Take 748163486 take 79106474 and 549876. 2104738400 take 219846736 and 2173844. 2174863 take 478654+312842+176348. 548629+748634take318467+21986+73894 2198641+200473take54876+78698+9846. 8047- 5278- 9873- 7048+5734 take 2174+3846- ■8497. 9176+8796 take 8976+7421+1121. •7894+2198 take 4987+8746+1471. 2173+4173+9876from 78469+2174+8459. 74867382176983 from 4879684721674974. 58217384698746 from 5763847218698481. 91047384687690 from 9476347869485203. 20734076948763 from 9846738479894210. SIMPLE SUBTRACTION. 21 56. The battle of Waterloo was fought in 1815, and the battle of the Alma in 1854 ; how many years elapsed between them ? 57. A merchant owed £2476, but has paid £1587 ; how much does he still owe ? 58. A man born in 1775, died in 1858; what was his age? 59. Napoleon I. born in 1769, died in 1821 ; what was his age ? 60. A man was 98 years old in 1858 ; when was he born ? 61. Americawas discovered in 1492; howlongis it since? 62. A piece of cloth contained 1074 yards ; 274 yards were sold to one person and 123 yards to another ; how many yards remained ? 63. From Edinburgh to York by rail is 209 miles, and to London 413 miles ; how far distant is York from London ? 64. A ship sails from London to Sydney, a distance of 13,640 miles ; after sailing 7684 miles, how far has she still to sail ? 65. What number added to 354896 will make 432678? 66. The sum of two numbers is 4789246, and the less is 849758 ; what is the greater ? 67. Howlong is it since the invention of printing in 1430? 68. In 1856, the number of Post-office Orders issued in the United Kingdom was 6,178,982 ; the number issued in England and Ireland was 5,693,459 : how many were issued in Scotland ? 69. The receipts from passengers and goods by rail in Scotland amounted to £2,319,217 in 1856, and from goods alone £1,464,041 ; find the receipts from passen- gers alone. 70. How long is it since the invention of gunpowder in 1400? 71. B was born when A was 27 years old ; what age is A when B is 51, and how old is B when A is 76 ? 72. A merchant owed to A £7486, but has paid him £4736; to B, £5746, but has paid him £3721; to C, £10,844, but has paid him £7483 ; to D, £5748, but has paid him £4106 ; to E, £5120, but has paid him £3980; and to F, £11,111, but has paid him £9879 ; how much does he owe to each, and how much in all ? 22 SIMPLE MULTIPLICATION Is a short method of finding the sum of any given num- ber when repeated as many times as there are units in another given number. The number to be repeated is called the multiplicand^ the other number, the multiplier^ and the result is called the product. The two given numbers are also called factors of the product. Case L When multiplier does not exceed 12. Ex. Multiply 5974587 by 8. Ans. 47796696. Sol. 8 times 7 are 56, write Multiplicand 5974587 down 6 and carry 5 ; 8 times 8 Multiplier 8 are 64 and 5 are 69, write down Product 47796696 9 and carry 6 ; 8 times 5 are 40 = and 6 are 46, &c. 1. 384607592176 X 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. 2. 597260875486 X 2, 4, 6, 8, 10, 12, 11, 9, 7, 5, 3. These Exercises may all be checked by Addition. Case II. When the multiplier is found in the table. Ex. Multiply 74867384 by 14. Ans. 1048143376. 74867384 X 14 = 2 X 7 Ex. Mult. 748673 by 20 20 149734768 prod, by 2 7 1048143376 prod, by 14 14973460 1. 748674869 2. 530472937 3. 374216487 4. 796548737 5. 975318642 6. 759386154 7. 649587596 8. 927635849 9. 123456789 10. 987654321 X16, X15, X22, X33, X25, X35, X36, X48, X55, X64, 18, 24. 21, 32. 30, 28. 42, 45. 36, 49. 27, 44. 40, 42. 54, 56. 60, 63. m, 70. 11.219703842 12.504382796 13. 846593742 14. 142857142 15.846153846 16. 952380952 17.543207159 18.791364857 19. 517369428 20. 629752837 X72, 77, 81. X84, 88, 90. X96, 99,110. X96, 81, 63. X80, 96, 77. X 81,121,144. X 99,132,121. X 84,110,100. X56, 54,132. X 45,121, 81. SIMPLE MULTIPLICATION. '23 Case III. When the multiplier is not found in the table, and does not exceed 156, or 12 X 12 + 12. Ex. 74238476X26=5X6+1 5 371192380= 5 times 5 1855961900 = 25 >^ 74238476= 1 " Ex. 67584937X38=6X6+2 6 405509622 = 6 times 6 2433057732 = 36 »* 135169874 = 2 ,' 2568227606 = 38 times 1930200376 = 26 times 1. 674295386 X 17, 23, 26, 29, 31, 34, 37, 43, 46. 2. 965830295 X 38, 47, 62, 58, 62, 68, 74, 79, 83. 3. 534869738 X 39, 59, 69, 75, 87, 93, 103, 105, 115. 4. 275963849 X 19, 38, 47, 59, 74, 87, 95, 137, 149. Case IV. When the multiplier exceeds 156. Ex. 3210421765X235 235 16052108825= 5 times 9631265296 = 30 « 6420843530 =200 « 754449114775=235 times 1. 74863847 X 2. 43958172 X 8. 79586216 X 4. 31596857 X 5. 74951084 X 6. 16847593 X 7. 39416809 X 8. 20537958 X 9. 53104009 X 10. 69073854 X 11. 90768300 X 12. 71765184 X 364, 729. 513, 624. 734, 856. 807, 965. 760, 398. 976, 304. 854, 930. 216, 648. 729, 356. 457, 390. 278, 936. 548, 690. 25. 51948673X7040908. 26. 94076803X4667890. 27. 72584692X1234667. 1. My income is £29 per week ; what is it per annum ? 2. 87 parishes are each assessed £37 ; what is the whole assessment ? Ex. 48769486X407500 407500 24384743000 341386402 195077944 19873565545000 13. 5976843 X 2798, 6005. 14. 3179648 X 4035, 3907. 15. 5271809 X 4576, 7689. 16. 6485937 X 3090, 7406. 17. 7268369 X 6480, 4729. 18. 5184736X2751, 6043. 19. 4958674X1234, 6678. 20. 6396274X9560, 8009. 21. 7261587X8154, 6700. 22. 8430957 X 8900, 3007. 23. 9376864X7461, 6893. 24. 1069769 X 9876, 4500. 28. 40769864 X 70049000. 29. 36947582 X 84000960. 30. 52749683 X 90004396. 24 SIMPLE MULTIPLICATION. 3. How many sheaves are in a field containing 327G Bhocks, each 12 sheaves ? 4. How many miles does a ship sail in 17 days at the rate of 169 miles a-day ? 5. How many hours are there in a year ? 6. How often does the seconds hand of a watch re- volve in a day and in a year ? 7. A railway train travels at the rate of 35 miles an hour ; how many miles does it travel in 56 hours ? 8. A ship's cargo consists of 435 boxes, each contain- ing 598 apples ; find the number of apples. 9. How many letters are there in a volume of 436 pages, each page 39 lines, and each line 52 letters ? 10. Sound moves at the rate of 1142 feet in a second; how many feet will it move in 75 seconds ? 11. A peal of thunder is heard 35 seconds after seeing the flash of lightning ; how far distant is the cloud ? 12. A train consists of 13 carriages having each 3 com- partments, each containing 12 seats ; how many passen- gers would find seats ? 13. What is the value of an estate, containing 7564 acres at £56 per acre ? 14. A ship's crew of 375 men is provisioned for 115 days, now each man is to receive 16 ounces a-day; how many ounces have they in all ? 15. A ship after sailing 37 hours at the rate of 7 miles an hour, encounters a storm, which drives her back during 7 hours at the rate of 12 miles an hour ; she then sails at her original rate during 5 hours ; how many miles will she now be upon her voyage ? 16. How many shots does a fleet of 3 ships of 72 guns each, 5 of 91 and 7 of 42, fire in 93 rounds ? 17. How many soldiers are there in 12 regiments of 9 companies each, and each company consisting of 95 men? 18. How much powder does a sixteen gun-battery of 18 pounders expend in 18 hours, if each gun is discharged 22 times iji an hour, the charge for an 18 pounder being 6 lbs. ? 25 Divisor. Dividend. 7)47286492 6755213^ 7 Quot. SIMPLE DIVISION Is the method of finding how often one number is con- tained in another. The number we divide by is called the divhor, the num- ber to be divided, the dividend^ and the result, the quotienU Case I. When the divisor does not exceed 12. Ex. Divide 47286492 by 7. Sol. 7 is not contained in 4, but 7 in 47 is 6 and 5 over; place 5 before 2, then 7 in 52 is 7 and 3 over ; 7 in 38 is 5 and 3 over, &c. The work is proved by multiply- ing the quotient by the divisor and adding in the remainder, 1. 7298763408 ~ 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. 2. 5487219876 — 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. 3. 9846798764 -^ 12, 6, 2, 3, 11, 9, 10, 8, 5, 4, 7. Case II. When the divisor is found in the table. Ex. Divide 74263849 by 14. 2 )74263849 ^14=2X7 7 )37131924 — 1 quot. by 2 5304560— 4 ... 14 47286492 Proof. 20 4X2+1=8+1=9 rem. 1. 34867896-M5, 16, 18. 13. 47654876- 2. 48678963-^20, 21, 24. 14. 76548764 ■ 3. 86789634-^-25, 27, 28. 15. 98765432 - 4. 21750486-H30, 32, 33. 16. 23457219 - 5. 30975219-^35, 36, 40. 17. 34807280- 6. 93048765-r42, 44, 45. 18. 48702083 - 7. 12345678-^48, 50, 54. 19. 54621487 - 8. 23456789-^56, 60, 63. 20. 18765486- 9. 34567890-^64, 66, 70. 21. 33144777 - 10. 51146784-^72, 77, 80. 22. 11847654- 11. 38712967-1-81, 84, 88. 23. 28048694 ■ 12. 76921783+90, 96, 99. 24. 78648769 • Ex. 275473 - 2,0) 27547,3 13773^§ Quot. 100,108,110. 120, 121, 132. 144, 121, 108. 72, 81, 50. 54, 96, 44. 108, 88, 77. 132, 81, 56. 144, 54, 48. 121, 36, 32. 120, 64, ^Q. 110, 72, 42. 108, 99, 35. c2 26 SIMPLE DIVISION. Case III. When the divisor is not contained in the table. Ex. Divide 48769847 by 7486. Divisor. Dividend. 7486)48769847( 7486 X 6 = 44916 Ans. 6514f 043, Quotient. 6ol4f£|? 7486 Divisor. 38538 ... X5 = 37430 11084 ... Xl = 7486 35987 ... X4 = 29944 Kemainder. 6043 39084 52112 26056 45598 6043 Rem. 48769847 Dividend. 1.77486694- 2.54809678- 3.48096785- 4.57486786- 5.38492136- 6.48675846- 7.21486483- 8.54862187- 9.48765486- 10.30846298- 11.74869548- 12.34112118- 13.21476548- 14.58643876- 15.79864879- 16.54867384- 17.79847684- 18.54867486- ■23, 31, 43. ■26, 37, 47. •29, 39, 51. •17, 19, 13. •52, 53, 57. ■61, 75, 69. •73, 74, 78. •79, 82, 83. ■85, 86, 87. ■89, 91, 95. ■97, 92, 98. ■93, 74, 94. ■784, 842. •542, 876. ■325, 498. ■173, 156. •139, 147. •163, 184. 73846548 65482173 87460094 86754800 4- 38476700 -^ 48216784 -^ 25. 7384698700 26. 4869873846 27. 7298740000 28. 3216504000 29. 2190874860 30. 5486384766 31. 4768754867 32. 7321987645 33. 5419738473 34. 2176548698 35. 1876487693 36. 3175486987 217, 298. 248, 263. 376, 483. 800, 900.' 600, 390. 740, 500. - 17640. - 47687. - 87000. - 36500. - 17000. -37480. , - 176487. - 279864. - 548637. - 248765. - 764869. - 987654. 1. A product is 4822150080, and one of the factors 704 ; what is the other? 2. My yearly income is £364 ; what is that per week ? 3. Great Britain and Ireland contain a population of 27,675,780, and their surface is 121,385 square miles; how many inhabitants is that on an average to the square mile? 4. France contains a population of 35,700,000, at the rate of 175 to the square mile ; how many square miles is the surface of France ? SIMPLE DIVISION. 27 ft. If a floor 40 feet long require 1280 stones, each a foot square, to pave it; what is its length? 6. The number of letters in a volume containing 746 pages is 1,846,350 ; how many letters are in a page? 7. An assessment for the poor of £5616 is raised from 48 parishes ; how much is levied from each parish ? 8. If a pigeon fly at the rate of 56 miles an hour, what time would it take between Edinburgh and the Cape of Good Hope, a distance of 5544 miles ? 9. Divide £6725 equally among 25 men. 10. In how many days will a ship accomplish a voy- age of 4473 miles, sailing 213 miles in a day? 11. How many loaves, each weighing 69 ounces, can be made from 16,491 ounces of flour? 12. The circumference of a wheel is 13 feet ; how often does it revolve on a road 68,640 feet long ? 13. A tax of £7791 is to be levied from 53 parishes; how much must each pay ? 14. Divide 343 oranges equally among 7 boys. 15. How many carriages, each containing 36 passen- gers, would be required to convey 648 persons ? 16. A gentleman's income is £6205 per annum ; how much is it per day ? 17. One man alone can build a wall in 378 hours ; in how many hours would 7 men do the same ? 18. 7 regiments, consisting of 716 men each, are to be reduced into 4 others of equal strength ; how many men will be in each new regiment ? 19. How often can 375 be subtracted from 744375? 20. 15,855 ounces of beef are divided among 755 sol- diers ; what is the weight of each man's ration ? 21. How many dozens of wine are in 64 pipes, each containing 756 bottles ? 22. A product is 2632938, and one of the factors 246 ; what is the other factor ? 23. A ship sails 5712 miles in 28 days ; how many miles is this on an average per day ? 24. The circumference of the Earth is 25,000 miles nearly ; how long would a person take to travel this dis- tance at the rate of 40 miles per hour ? 28 SUPPLEMENT TO MULTIPLICATION AND DIVISION. I. When the multiplier contains a fraction. Ex. 6487536 X 8f 5 )19462608 = 3 times. 38925213- = I times. 51900288" = 8 times. 557928091 Product. 1.7486948X 2.5721987X 3.7121846X 4. 5987648 X 5.3842198X 6. 4876529 X 7.7214867X 8.4962184X 9. 4763148 X 10.2147634X 11.9847693X 12.6478796X 13. 5463784 X 14. 8754964 X 15. 8075084 X 41 63, 8i. _, 71,10^. 41, n, 6|. 84,121,2^. 37A, 46^. 65^1,304^. 113^\,312if 416^V 549if 1791, 484^'^. 44U, 574j. 41 1, 59 A- 84A, 93,\. 108ii,275A. II. When the divisor con- tains a fraction. Ex. 487654-^-31 30 487654 _5 5 16 f 2)2438270= prod, by 5 (8 )1219135 1523911 Quotient. l.4765847-^- 2.5862190-^ 3. 4948645-^ 4. 5482169-:- 5.7928465^ 6. 5786478-f- 7.87486736^ 8. 57638469. 9.78621475- 10. 86275846- 11.51840963- 12.78219865- 13.21973465 14.34758694 15.97986089 Q 3 K 6 m 3 As Q6 "T25 ^85 . - 12i. 13i, 14f, 155. 244|. 18J. 56i^ 94i«. 104,11 -^ 17 631 84i — 71 6 ^58^^, 644H. ^ 251^,5124^. -^7364 3674^. EXERCISES ON THE PRECEDING RULES. 1. In 1856, the number of seamen registered in Eng- land was 156,913; in Scotland, 29,987; in Ireland, 13,403; in Jersey, Man, &c. 5424 ; and in the Colonies, 62,032 : lind the whole number. 2. In 1851, the population of the South-eastern Counties of Scotland was : Linlithgow, 30,590 ; Edin- burgh, 259,493 ; Haddington, 36,363 ; Berwick 36,165; Peebles, 10,804 ; and Selkirk, 9802 : find the sum. 3. In 1856, the number of births registered in Scot- land was 52,301 males and 49,447 females ; and the number of deaths was 29,417 males and 29,039 females : find the excess of births over deaths in that year. EXERCISES. 29 4. How many passengers are in a train consisting ot 4 first class carriages, each containing 18 persons ; 3 second class containing 30 each, and 2 third class con- taining 40 each ? 5. In 1856, the number of marriages in the 33 coun- ties of Scotland was 20,487 ; what was the average number in each ? 6. The British Army at the battle of the Alma was composed as follows : — Light Division, 5454 men; 1st Division, 4711; 2d, 4222 ; 3d, 3794; 4th, 4419 ; Cavalry, 1100; Artillery, 2700; Sappers and Miners, 400; how many men were engaged in all ? 7. At the same battle, the loss of the British amounted to 2196 killed and wounded; how many effective men remained ? 8. How many yards are in 15 pieces of cloth, each containing 56 yards ? 9. Mercury's distance from the Sun is 36,793,000 miles, Mars' distance is 108,031,000 miles greater than Mer- cury's, and Neptune's is 2,710,114,000 miles greater than Mars' ; find the distances of Mars and Neptune from the Sun. 10. In 1856, the number of Post-ofiice Orders issued in Ireland was 461,723; the number in Scotland was 23,800 more than in Ireland, and the number in England exceeded that in Scotland and Ireland together by 4,284,490 : how many were issued in Scotland, in Eng- land, and in the United Kingdom ? 11. How many times is Mount Blanc, 15,732 feet in height, higher than Arthur Seat, which is 820 feet high ? 12. At the battle of the Alma, the Fusilier Guards lost 11 officers and 170 non-commissioned officers and men killed and wounded ; the Grenadiers, 3 officers and 126 men ; and the Coldstreams, 3 officers and 27 men : at Inkerman, the Fusiliers lost 9 officers and 169 men ; the Grenadiers, 9 officers and 223 men ; and the Cold- streams, 13 officers and 178 men. How many of the Guards fell at Inkerman more than at Alma? 13. A pear-tree one year produced 14,861 pears, aver- aging 11 to the pound ; how many lbs. were produced? ^4- In how many days will a boy read through the D 30 EXERCISES. Bible, which contains 31,173 verses, if he reads 39 verses daily? 15. How often does the hammer of a clock strike in a day and in a year y 16. One female can cut out 300 gross of blanks for steel pens in a day ; how many will she cut out in a year of 313 days? 17. A steel pen manufactory sends out 180,000,000 pens yearly ; how many boxes, each containing a gross or 12 dozen, would they fill ? 18. A gentleman has 3 farms containing 675 acres ; the first and second together contain 490 acres; and the second and third 425 acres . how many acres are in each farm ? 19. The gallant Sir John Moore fell at the battle of Corunna in 1809, at the age of 48 ; in what year was he born? 20. The Sun's diameter is 882,000 miles ; how many times is it greater than the Earth's diameter, which is 7920 miles ? 21. Divide 1584d. among 3 girls and 5 boys, giving each girl twice the number which a boy gets. Sol. Since each girl gets 2 boys' shares, 3 girls have 2X3 = 6 boys' shares ; the number of boys' shares is therefore 6 + 5 = 11. Hence each boy gets 1584 4- 11 = 144d., and each girl 144 X 2 = 288d. 22. How much grain will a farm of IG fields, each 29 acres, produce, if one acre produces 9 quarters of grain ? 23. A gentleman gave £484 to two charities, and to one he left 3 times as much as to the other ; what did he leave to each ? 24. Several volumes contain 10,192 pages ; in how many days would a person read the whole, reading 4 hours a day and 7 pages an hour ? 25. In a church there are 12 windows ; in the lower sash there are 12 panes and in the upper 18 ; how many panes of glass are there in all ? 26. A gentleman has 4 farms, containing 240, 375, 408, and 425 acres respectively, and he wishes to divide them into as many others of equal size ; how many acres will there be in each farm ? EXERCISES. 31 27. In one class there are 150 boys, in another 145, in a third 140, in a fourth 135, and in a fifth 130; how many are there on an average in each ? 28. The sum of two numbers is 2779, and their difi'erence is 293 ; what are the numbers ? Sol. 2779 2779 Add 293 Subtract 293 2 ) 3072 2 ) 2486 1536 is the greater. 1243 is the less. 29. At an election, the successful candidate had a ma- jority of 84 votes out of 572 votes ; how many had each of the two candidates ? 80. The loss of the French and Sardinians at the battle of the Tchernaya or of Traktir Bridge, amounted to 1792 men killed and wounded; the French loss was 1292 more than the Sardinian : what was the loss of each? 31. Divide 204 apples among 4 girls and 5 boys, giving each girl 3 times as many as a boy. 32. Galileo died in 1642, and Newton in 1725; how long is it since each of these events, and how many years elapsed between them ? 33. A gentleman dying, left £45,000 ; to his widow he bequeathed ^ of his estate, and the remainder was to be equally divided among his 4 children ; how much did he leave to each ? 34. A ship at sea fires a gun, the report of which is heard 12| seconds after seeing the flash ; how far distant is the ship, sound moving at the rate of 1142 feet per second? 35. Two casks of wine contain together 151 gallons, and one contains 31 gallons more than the other; how many gallons does each contain ? 36. What number being divided by 337 gives 9472 for the quotient, and 108 for the remainder ? 37. Divide £416 among 6 men and 8 women, giving each man 4 times as much as a woman. 38. Two brothers being asked their ages, said that the sum of their ages was 63, and that the difference of their ages was 9 ; find their ages. Ex.Red.l9718farth.to£. 4 )19718 Farth. 12 ) 4929 ^ Pence. 2,0) 41,0s. 9|d. £20, 10s. 9§d. 32 COMPOUND NUMBERS I. STERLING MONEY. REDUCTION Is the method of bringing numbers from one denomina- tion to another without altering their value. To bring higher to lower denominations multiply. To bring lower to higher denominations divide. Ex. Red. £20, 10s. 9id. to farth Mult, by 20 and add 10s. 410 Shillings. Mult, by 12 and add 9d. 4929 Pence. Mult, by 4 and add 2f. 19718 Farthings. 1. Reduce£35, 17s. 4id.; £28,lls.ll|d.; £40, 10s. lOf d.; €200. 10s. 8fd.; £574, 19s. 11^.; £409, 17s. 4id. ; £147', 10s. lOid. ; and £105, 2s. 4id. to farthings. 2. Reduce £470, 10s. O^d. ; £270, lis. 6d. ; £672, 18s. 9^.; £486, 12s. l^d. ; 12s. 4id. ; 17s. S^d. ; lis. lid. £700, 10s. 2d. ; £21, 15s. S^d. to halfpence. 3. Reduce £87, 19s. lO^d. ; £11, lis. lid. ; £50, 19s. 6d. £47, 15s. 9id. ; £400, 10s. O^d. ; £290, 16s. 4d. ; £403, lis. ll^d.; 16s. 8id. ; lis. S^d. ; 13s. 4^d. ; £43, 12s. 4d. and 15s. 8^d. to halfpence and farthings. 4. Reduce 4786; 3040; 7098; 48769; 73846; 4098 7214 ; and 38463 farthings to pence, shillings, and pounds 5. Reduce 4876; 7487; 3562; 1749; 3689; 2177 5848 ; 7216 ; 111111 ; 33333 halfpence to d. s. and £. 6. Reduce 78469 ; 738467; 87698; 714086 farthings: 48763; 21764; 50487; 140715 halfpence: 729374; 89214; 47865; and 571640 pence to sovereigns. 7. Reduce £2716, 2s. 2id. ; £4176, 12s. S^d. ; £3108, 14s. 7id. ; £176, Os. 2id. ; £417, Os. O^d. ; £49, 17s. 6d. ; and £2010, 10s. 6|d. to halfpence and farthings. 8. Reduce 41763 ; 58462 ; 71209 ; 17268 ; 38467 ; 84762; 47219; and 876213 farthings to sovereigns. REDUCTION. 33 Ex. Red. £2475 to crowns and guineas. 2. £2475 X 20 £2475 20 5 )49500 s. Ans. 9900 cr. f 3)49500 s. 17 )16500 Ans. 2357 gu. 3 s. 9. Reduce £7485; £3876; £4921; £3817; and £3760 to crowns and guineas. 10. Reduce 17486; 887; 2130; 2491; 2168; and 7430 guineas to pounds. Ex. Red. £21, 17s. 6d. to sixpences. Ans. 875 sixd. £21, 17 s. 6d. Proof. 20 2 )875 sixd. 437 s. 2,0)4Vs. 6d. 2 (sixd. in Is.) £21, 17s. 6d . Ans. 875 sixd. c===; 11. Reduce £121 ; £45, 7s. 6d. ; £56, 18s. 6d. ; £79, 18s. ; £84, 5s. 6d. ; and £99, 19s. 6d. to sixpences. 12. Reduce 448; 977; 2163; 3729; 4125; and 5763 sixpences to shil. and pounds. 13. How many half-crowns in £42, 7s. 6d. ; £54, 12s. 6d. ; £67, 15s. ;"'and in £99, 17s. 6d. ? 14. Reduce 528 ; 1254; 3453; 4869; 5871; and 7459 half-crowns to pounds, &c. 15. How many threepences are in £49, 7s. ; £54, 6s. 3d. ; £72, 19s. 6d. ; and in £84, 14s. 9d. ? 16. Reduce 1748; 2153; 3785; 5142; 6897; and 7455 threepences to shillings and pounds. 17. How many florins are in £170 ; £144 ; 6743 far- things ; 1786 pence ; and in 436 shillings ? 18. Reduce 43 guineas ; 77 gu. 8s. l^d, ; £37, 2s. 6d. ; 93 gu. 2s. 4»d. ; 78 gu. 18s. 9|d. ; and 18s. 9|d. to farth. 19. Find the sum of £18, 19s. 4^d.+5 crowns+17 half- crowns-f-234 florins-}-! 7 guineas, in farthings and pounds. 20. How many pounds will a man save yearly, by lay- ing aside 5s. 9M. weekly ? 21. How many penny stamps may be obtained for £49, 17s. 7d.? 34 COMPOUND ADDITION. Example. Sol. The sum of the farthings is 10 = £381 17 s. 6id. 2^d., write down ^d. and carry 2 to the 148 12 9^ pence. The sum of the pence is 44 = 412 16 7| 3s. 8d., write down 8d. and carry 3 to the 319 11 ll| shil. The sum of the units column of the 470 19 9f shil. is 28, write down 8s. and carry 2 to £1733 18 8^ the tens of the shil. : the sum is 7 ten shil. ===== pieces = £3 and 1 ten shil. piece, write down 1 and carry 3 to the pounds. The sum of the pounds is £1733, and the whole answer is £1733, 18s. 8^d. — The results in Compound numbers may be checked as in Simple numbers. 1. 2. 3. 4. £ s. d. £ s. d. £ s. d. £ s. d. 24 11 4 J 31 17 lU 27 15 1\ 14 19 111 16 18 9i 13 14 lOi 72 18 10 12 16 Si 61 10 2| 42 16 9.f 36 11 5i 29 11 5A 32 17 lU 24 12 4i 63 10 4^ 18 18 8£ 45 16 3i 56 18 Hi 41 17 lOi 15 14 10 96 11 41 49 19 111 68 16 Si 15 14 9 69 13 7i 94 13 7 86 15 51 19 17 lOi 12 16 10 17 11 lU 74 11 91 91 19 6i 14 18 n 16 15 5i 47 16 10 51 14 lU 29 15 6i 18 18 Hi 51 10 4i 18 13 2\ 31 17 111 81 17 lOi 15 8 111 81 10 9| 9. 10. 11. 12. 17 18 HI 42 18 9i 29 10 8^ 14 12 Si 16 13 7i 36 17 2| 34 8 11^ 17 13 61. 19 12 41 34 16 3| 76 7 7i 98 19 2 20 11 3| 43 12 111 82 11 10| 84 10 6| 31 17 Si 45 19 101 49 17 71 18 11 111 17 16 111 53 11 4| 63 13 91 19 10 81 13. 14. 15. 16. 45 17 SI 40 21 51 S 3| 82 12 llf 49 16 4| 38 19 11 64 19 111 4M8 71 38 18 71 51 15 5| 73 17 10^ 72 16 9 64 4 llf 86 16 7 84 16 91 S8 11 4f 39 17 6i 13 14 101 18 18 11| 22 15 101 83 11 4f 89 19 6 17 14 10| 16 18 7i COMPOUND ADDITION. 35 17. 18. 19. 20. £ s. d. £ s. d. £ s. d. £ s. d. 274 13 lOf 426 16 4J 410 10 lOf 329 19 111 476 12 8J 246 13 3^ 104 17 4f 293 18 11 567 11 4| 642 17 8f 816 11 \\\ 932 17 6i 658 13 Ik 351 9 lU 681 13 4 456 16 10| 549 18 7i 513 12 Al 168 12 lOJ 564 13 3i 721 16 111 135 11 lU 473 2 U 645 17 6 213 19 lOf 497 18 101 734 3 0^ 897 13 4i 132 15 7i 974 19 9i 347 11 111 978 19 9^ 21. 22. 23. 24. 476 11 llf 847 13 6| 984 17 m 484 15 4| 725 13 4i 472 19 2i 845 15 0| 846 13 8| 870 11 9| 756 14 llf 756 13 21- 725 16 10^ 708 17 lOi 793 18 5A 384 11 5i 857 11 Ak 534 16 3i 904 15 81 479 18 lOi 583 3 9i 729 19 111 405 12 lOi 721 19 111 879 16 8| 297 18 4| 762 16 81 562 13 8i 405 17 Ik 972 15 3| 636 17 9^ 629 16 9i 896 19 111 25. 26. 27. 28. 325 14 71 420 12 7J 508 19 71 874 13 6 256 15 81 500 13 8^ 850 11 2 847 16 8i 719 11 61 721 16 6| 793 16 71 856 17 101 971 13 lOi 217 18 lU 918 13 41 865 11 21 47*2 16 8i 172 16 4| 981 17 10 832 18 6 749 18 111 901 17 21 974 18 Q^ 823 7 4| 426 19 9^ 847 19 111 953 12 111 748 16 3 273 17 10^ 487 15 lOi 947 13 3^ 784 17 4| 29. 30. 31. 32. 219 13 111 549 16 2 J Ill 11 111 204 8 0| 192 16 5^ 495 17 10 222 17 10^ 420 -3 n\ 921 17 4 954 11 2| 363 18 2| 569 18 21 476 13 &l 867 18 31 746 12 91 931 13 11 764 18 lOi 678 19 3i 805 13 7| 139 15 4| 647 11 11 786 12 9 508 19 61 721 4 8^ 513 17 101 954 13 81 741 10 41 801 12 7 315 19 8^ 987 16 10 417 11 U 971 19 101 33. £473, 18s. 10|d.+£972, lis. 4id.+£987, 19s. llid. +£852, 17s. 9id. + £112, 15s. 61d. + £521, 14s. 8|d. -f £846, 13s. 7|d. + £613, 12s. 9^d. + £716, 17s. ll|d. 36 COMPOUND ADDITION. 34. 35. 36. 37. £ s. d. £ s. d. £ s. d. £ 8. d. 540 13 llf 321 17 8i 463 18 9 897 11 81 419 19 9 123 16 3i 364 19 lOi 879 12 71 914 13 41 231 15 4i 643 11 2i 798 13 4| 411 12 2 213 18 11 634 17 10 789 16 llf 114 8 6| 312 16 8J 346 18 9i 978 4 2 701 13 4 132 11 2i 436 12 71 987 11 4| 158 17 llf 474 13 6 183 16 8 363 15 71 815 16 8 744 11 3f 831 15 41 633 18 9f 38. 39. 40. 41. 948 12 71 574 11 lOf 530 11 4f 876 11 llf 489 13 61 457 16 41 504 17 111 768 19 lOl 894 19 91 739 19 9 876 19 91 687 14 41 276 15 41 395 11 10| 743 12 lOf 527 13 8 568 17 llf 953 14 41 549 13 61 498 19 llf 729 19 9 476 15 11| 985 19 llf 409 17 lOf 276 11 111 729 13 8i 859 18 10| 490 17 101 467 13 41 297 16 41 764 13 81 385 13 4f 42. 43. 44. 45. 210 11 5i 741 18 11 116 17 41 901 18 71 101 13 61 387 16 5f 161 13 5f 910 13 41 354 16 21 862 10 101 600 17 21 864 12 111 726 15 111 629 17 61 560 13 81 648 17 6f 367 18 10| 748 15 5f 74 11 3f 899 11 11 481 19 111 796 17 llf 9 8 6 988 15 7^ 816 15 81 869 19 10 408 12 llf 749 18 llf 964 17 lOf 176 17 8f 780 16 91 548 17 61 489 19 111 298 11 41 473 18 111 721 16 21 984 12 61 476 10 6f 729 16 71 387 19 9i 571 18 llf 80 16 101 780 13 31 807 12 111 709 11 41 89 13 61 10 16 71 840 15 llf 742 18 91 476 13 4- 47. 48 19 41 480 13 61 408 15 91 72 11 41 568 17 111 367 13 2f 673 18 81 469 12 lOf 576 19 8f 864 13 4j 61 7: 48. 500 11 llf 499 9 " 72 18 308 15 38 12 380 11 596 12 65 13 962 18 298 14 llf If 49. 344 18 101 436 19 111 87 13 " 728 12 864 11 111 86 13 41 987 12 49 14 986 17 806 4 91 61 2i COMPOUND ADDITION. 37 50. A owes to B £743, lis. 6id., to C £325, 4s. 8|d., to D £750, 19s. lOfd., to E £113, lis. lljd., to F £1041, 13s. 8fd., to G £89, 16s. 8d., to H £1430, 153. ll|d., and to I £740, 16s. lOd. ; how much does he owe in all ? 51. A paid to B £675, 13s. 7id., to C £298, 16s. lOid., to D £749, 13s. 7id., to E £97, 18s. 6|d., to F £987, 13k. Hid., to a £75, 13s. 8|d., to H £1279, 17s. 4fd., and to I £684, 13s. ll^d. ; how much did he pay in all? 52. A person collected in January £744, lis. 8fd., in February £896, 17s. lO^d., in March £472, 17s. 4id., in April £583, 16s. llfd., in May £739, 17s. 6fd., in June £1096, 13s. 8id., in July £578, 12s. 8^d., in August £1374, 18s. 5jd.,in September £458, lis. llid.,in October £735, 13s. 4id., in November £2179, 16s. 4id., and in December £532, lis. Ijd. ; how much did he collect? 53. I received from A £736, 15s. lid., from B £874, 13s. 8fd., from C £879, 17s. lOid., from D £84, Us. 2|d., from E £98, 17s. lOfd., from F £921, 16s. llid., from G £1093, 10s. 4^d., and from H £729, 8s. lid. ; how much did I receive in all ? 54. A owes me £274, lis. lOid., B £89, 13s. 7d., C £74, lis. lid., D £96, 18s. 9^d., E £65, lis. 2id., F £418, 4s. 6fd., G £173, 13s. 4id., H £748, 17s. 6fd., K £847, 13s. 4id., and I have in the bank £7486, 17s. ll|d. ; how much am I worth? 55. A housekeeper's account was, for beef, &c., £4, 2s. 7id. ; tea and coffee, 21s. 3d. ; sugar, 17s. 7|d. ; potatoes, 5s. 6|d. ; butter, lis. l^d.', fruit, 21s. 3^d. ; and bread, 43s. 7d. ; find the amount. 56. A corn merchant laid out on wheat, £597, lis. 6^d. ; on barley, £409, 17s. 4|d. ; and on oats, £347, 9s. llfd. : what should he sell the whole for to gain £79, 18s. 11 ^d. ? 57. A gentleman left to his widow, £7692, 17s. 4id. ; to each of his two sons, £3000, 17s. 6d. ; to each of his four daughters, £2559, 18s. 7^d. ; and to his other rela- tives, £4975, 8s. 4id. : how much was this in all? 58. A gentleman owes his tailor, £23, 14s. 8^d. ; his bootmaker, £14, 7s. 3jd. ; his grocer, £48, 17s. 7|d. ; his baker, £35, 16s. 8^d. ; his house-rent is £115, 17s. 6d. : how much must he draw from the bank to pay these sums ? 38 COMPOUND SUBTRACTION Ex. From £429 17s. Sjd. Take 145 12 3i Diff. £284 5 5^ Ex. From £501 15s. 6^d. Take 250 4 Gj Diff. £251 11 Oi 1. 2. 3. 4. £ ». d. £ 5. d. £ s. d. £ 5. d. 146 12 7i 247 16 lOf 375 15 6i 508 6 7| 73 5 2^ 184 5 3^ 183 10 4^ 348 3 7i Ex.From£742,15s.8id.take£653,17s.9id.Ans.£88,17s.l0|d. From £742 15s. S^d. Take 653 17 9| Diff. £88 17 lo| Sol. 2f. from If. we cannot, but 2f. from 4f. (or Id.) is 2f. and If. is |d.; write down |d., and carry 1 to 9d. is lOd. from 8d. we cannot, but lOd. from 12d. (or Is.) is 2d. and 8d. are lOd. ; write do^Am lOd., and carry 1 to 17s. is 18s. from 15s. we cannot, but 18s. from 20s. (or £1) is 28. and 15s. are 17s. ; write down 17s., and carry 1 to £653 is £654 from £742 are £88 ; write down £88. 5. 6. 7. 8. 427 13 4i 450 10 44 296 16 34 609 13 64 298 16 lOi 276 13 B| 109 11 114 379 14 7f 9. 10. 11. 12. 825 11 6 742 17 84 408 18 10 504 16 74 296 13 6i 486 13 94 298 19 9| 329 16 8f 13. 14. 15. 16. 742 13 84 200 804 10 14 476 17 7f 427 12 n 99 12 04 721 13 6 298 17 11 17. 18. 19. 20. 547 10 976 15 4 705 44 325 6 04 238 11 Of 762 18 K 396 17 2^ 298 13 04 21. 22. 28. 24. 610 17 3| 542 16 04 726 17 104 789 11 5| 496 16 lU 486 16 04 498 17 11 499 11 74 25. 26. 27. 28. 271 13 0^ 542 16 04 980 15 6 832 17 64 148 17 0| 347 19 111 890 15 6| 328 18 7f 29. 30. 31. 32. 532 16 7 424 19 10 173 16 14 410 12 44 325 11 9| 248 19 104 99 17 Of 147 13 74 COMPOUND SUBTKACTION. 39 83. 34. 35. 3a. £ s. d. £ s. d. £ s, d. £ s. d. 901 18 0| 386 16 4 409 11 H 251 13 4^ 496 18 U 293 17 2| 359 11 6 151 17 9| 109 13 10 96 13 lOi 38. 499 17 3i 399 17 61 39. 256 11 2i 193 4 7i 40. 704 14 4 407 14 9i 41. 42. 43. 44. 275 12 ^ 326 9 3 533 4 0^ 214 17 11 186 11 7i 263 18 H 353 7 6f 142 17 iH 45. 46. 47. 48. 973 2^ 841 1 0^ 711 5 2 817 739 3 If 418 1 n 117 5 lU 718 03 49. £748, 13s. 6id.— £589, 15s. 8|cl. 50. £721, 15s. 8d. —£629, 13s. lUd. 51. £721, 17s. 6|d.+£853, 13s. l|d.— £684, 13s. 6id. +£789, 17s. lid. 52. £987, 2s. lid.+£305, 2s. Hid.— £896, 12s. 8id +£296, 17s. 9id. 53. A merchant bought goods for £578, 15s. 6^d., and sold them for £642, 8s. 7Jd. ; what did he gain ? 54. Borrowed 500 guineas and paid £125, 17s. 4id. at one time and £298, 14s. 5id. at another ; what is still due? 55. The receipts of a railway one year amounted to £48,984, 17s. 8id. ; and the year followmg to £50,492, 2s. 3d. ; find the increase. 56. A housekeeper went to market with £5 ; she paid for beef 17s. 6id. ; mutton, 12s. 7id. ; fish, 7s. ^^L.', tea, 6s. 5d. ; coffee, 2s. 3^d. ; sugar, 7s. l^d. ; vegetables, 7s. 3|d. and sundries, 4s. l|d. ; with what sum did she return ? 57. A owed to B £748, 16s. 7id., but has paid £398, 17s. 6^d. ; to C £1000, but has paid £899, 17s. 4id. ; to D £470, lis. 4id., but has paid £381, 13s. 4£d. ; to E £721, 18s. 7|d., but has paid £643, lis. 9id. ; to F £896, 13s. 2id., but has paid £799, 17s. l|d. ; how much does he still owe to each and in all? 58. A merchant has in cash £7328, 17s. 11 Jd., goods worth £12,748, 16s. lOd., furniture £574, 18s. lljd. ; 40 COMPOUND SUBTRACTION A owes him £112, 17s. 6id., B £327, 18s. 7Ad., C £486 13s. 8|d., D £89, 16s. lO^d. and E £136, 18s. 8id. ; at the same time he owes to F £574, 18s. 11 ^d., to G £324, lis. 7|d., to H £723, 18s. 6d., to I £327, 17s. 4|d., and to K £587, 10s. 3fd. : how much is he worth? 59. A gentleman's yearly income is £500, his household expenses £294, 13s. 7:|d., rent £54, 13s. 6d., taxes £20, lis. 8id., servants' wages £25, 17s. lid., tradesmen's ac- counts £52, lis. 7fd., and incidental expenses £24, 17s. ll^d. ; how much does he save yearly? 60. Three ponies cost £35, 15s. 6d ; the first and second cost £26, 10s. 4d. and the second and third £30, 3s. 9d. ; find the price of each. 61. A, B, and C contributed £109, 18s. l^d. to a charity ; A and B contributed £61, 3s. 3^d. and A's contribution was £21, 10s. lO^d. less than C's: how much did each contribute? 62. A bankrupt's debts amount to £19,728, 15s. 7jd. and his effects to £12,899, 17s. 8|d. ; how much is he de- ficient ? 63. A gentleman dying left £17,584, 17s. 6d. ; to his widow he left £3756, l8s. 9d. ; to each of his three sons, £2573, 7s. 6d, ; to each of his two daughters, £2000, 14s. 3d. ; and the remainder to his other relatives : how much was this ? 64. A bankrupt owes to A, £329, 10s. 7^d. ; to B, £748, 17s. ll^d. ; to C, £876, 17s. lOjd. ; to D, £1783, 17s. Hid.; to E, £578, 19s. 3|d. ; to F, £1047, 18s. 6jd. ; and to Gr, £1270, 8s. 8^d. : at the same time he has in cash, £520, 17s. S^d. ; in bills, £325, 16s. lO^d. ; goods valued at £984, 17s. 6d. ; H owes him £44, 16s. 7id.; I,£72,lls.7|d.; K, £84, 13s. 4id. ; and L, £105, 17s. llfd. How much will his creditors lose by him? 65. A tax of £975, 17s. lO^d. is raised from 5 towns ; the first town pays £190, 14s. 8^d. ; the second, £204, 15s. 7|d.; the third, £199, 17s. 8f d. ; and the fourth, £219, 15s. 3id. : how much does the fifth town pay? 66. A merchant laid out £756, 18s. 9^d. on wheat, bar- ley, and oats ; the sum laid out on wheat and barley was £437, 6s. 2d., and on barley and oats, £540, 12s. l^d. : how much was laid out on each ? 41 COMPOUND MULTIPLICATION. Case I. When the multiplier is not greater than 12. Ex. Multiply £8, 17s. 9id.by 9. Ans. £79, 19s. lUd. Sol. 9 times If. = 9f. or 2Jd. ; write down £8 17s. 9id, Jd., and carry 2d. 9 times 9d. are 81d. and 9 2d. are 83d. or 6s. lid. ; write down lid., and £79 19 \i^ carry 6s. 9 times 7s. are 63s. and 6s. are = 69s. ; write down 9s. and carry 6 9 times 1 are 9 and 6 are 15 ten s. pieces = £7, 10s. ; write down 1 before 9s. and carry £7. 9 times £8 are £72 and £7 are £79. 1. Multiply £27, 17s. 8jd.by2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. 2. Multiply£36,15s.ll|d. by 7,5,2,9, 6,11,4,10, 8, 12,3. Case II. When the multiplier does not exceed 156. Ex. Multiply £3, 16s. 7id. by 16 and by 26. 2. £3 16s. 7id.X26=5x5-f 1 1. £3 16s. 7jd.Xl6 = 4X4 | _5 4 19 3 0| = 5 times. 15 6 5 = 4 times. 4 5 95" 3 15 16 1 J = 25 // £61 5 8 =16 times. 7i= 1 '/ d. £99 11 8i = 26 times. £ 5. £ 5. d. 1.27 11 4iX 14, 18, 20 13.42 18 6iX 17, 23, 26 2.31 17 2|X 24, 27, 30 14.36 15 lliX 29, 31, 34 3.19 18 6iX 32, 25, 36 15.25 13 lOfX 38, 43, 39 4.34 14 lliX 42, 45, 48 16.38 11 OiX 47, 59, 68 5.28 19 lOiX 49, 54, 65 17.45 12 8IX 67, 69, 74 6.74 13 8|X 56, 60, 64 18.56 18 6iX 75, 76, 79 7.16 19 lliX 63, 66, 72 19.73 16 4|X 83, 87, 86 8.35 OfX 70, 77, 81 20.84 19 llfX 89, 93, 98 9.54 19 OiX 80, 90, 96 21.91 13 4iX 94,103,107 10.47 13 111X108,121,144 22.94 15 2iXll5, 95,106 11.45 15 6|X110, 99,132 23.96 17 81X117,127,139 12.39 17 10|X 84, 77, 81 24.99 19 11|X148,126,137 42 COMPOUND MULTIPLICATION. Case III. When the multiplier exceeds 156. Ex. Multiply £2, 13s. 4|d. by 536. Ans. £1431, Os. 2d £2 13s. 4|d.X6 = £16 Os. 4id.= 6 times. 10 26 13 11^ X3= 80 1 lOi = 30 times. 10 £266 19 7 X5= 1334 17 11 = 500 times. £1431 2 =536 times. 1. 2. 3. 4. £ s. d. 5 18 11| X 583, 742 6 5 2^X 879, 986 3 13 4i X 1004, 2963 4 17 9| X 3125, 7518 £ *. d, 5. 7 19 lOfX 7384, 6472 6. 10 11 lliX 5809, 4365 7. 13 16 8iX 9416,10738 8. 16 17 101X27580,74087 Find the price of, 1. 27 cwt. of sugar at £3, 16s. 6d. per cwt. 2. 38 tons of steel at £25, 17s. lid. per ton. 3. 45 quarters of wheat at £3, 6s. 7^d. per quarter. 4. 67 dozen Madeira at £6, 8s. llf d. per dozen. 5. 53 gallons whisky at 12s. 2f d. per gallon. 6. 86 acres of turnips at £17, 16s. 8d. per acre. 7. 57 cwt. butter at £4, 5s. 8Jd. per cwt. 8. 67 lbs. tea at 7s. ll^d. per lb. 9. 58 acres of grass at £7, 8s. 7id. per acre. 10. 75 cwt. Carolina rice at £2, 15s. 7^d. per cwt. 11. 46 sugar loaves, each 17^ lbs., at ll^d. per lb. 12. 17 boxes pimento, each 87 lbs. at lljd. per lb. 13. 19 cwt. potashes at £1, 7s. llfd. per cwt. 14. 116 cwt. tallow at £2, 10s. 7id. per cwt. 15. 73 gallons rum at 18s. O^d. per gallon. 16. 59 ounces of gold at £3, 17s. ll|d. per oz. 17. 153 bushels malt at 7s. A^d. per bushel. 18. The daily pay of a foot soldier is Is. Id. ; how much is this yearly ? 19. A farm of 379 acres is rented at £3, lOs. 7^d. per acre ; how much is the whole rent ? 20. A merchant bought 25 pieces of cloth, each con- taining 20 yards at £1, 2s. t^d. a-yard, and sold the whole for £612, 10s. ; what was his gain? COMPOUND MULTIPLICATION. 43 21. If the weekly forage of a horse be 14s. 6^d. ; what sura will be required to keep a regiment of 750 horses for a year ? 22. The rent of a house is £1, 10s. 6Jd. per week ; how much is that in the year ? 23. How much will a farmer pay for cutting down his crop, if he employs 53 reapers for 3 weeks at 2s. llfd. each per day ? 24. If an hospital contains 80 boys, and each on an average costs Is. 3^d. a-day for food and clothing; how much will each, and also the whole, cost in the year ? 25. How much will a tax on property of £8746 yearly value amount to, at 2s. 2f d. per pound ? 26. A clerk's salary is £2, 17s. 9d. a-week ; how much is it yearly ? 27. Find the price of 7 pieces of cloth, each 45 yards, at £1, 2s. 7^d. per yard. 2i». The pay of an Ensign in the Foot Guards is 5s. 6d. per day ; what is it yearly ? 29. A bankrupt owes his creditors £4876, and pays them 8s. 6^d. per pound ; how much does he pay in all? 30. How much does the pay of a regiment of 895 men amount to in a year, at the rate of Is. l^d. to each man per day ? 31. Find the value of a lac of rupees, that is 100,000, at Is. llfd. each. 32. How much will a farmer receive for a field of wheat containing 16 acres, if each acre produces 7^ quarters, and the price of wheat is £2, 16s. 7d. per quarter? 33. A farmer has a field of potatoes containing 1000 drills ; now if each drill produces 19 bushels, how much will he receive for each drill, and also for the whole, at the rate of 4s. 7Jd. per bushel ? 84. A butcher purchases 4 oxen for 49 guineas, and sells the beef, which amounted to 165 stones, at 6s. llfd. per stone, and he gets besides £2, 3s. 5|d. for the hide, &c. of each ; what is his net gain ? 35. The weekly receipts of a railway are £1768, 17s. 8id. ; how much is this per annum ? 36. Find the value of 17 tons of coal at 15s. 3d. per ton. 44 COMPOUND MULTIPLICATION. 37. What should 7 chests of tea, each containing 74 lb., cost, at 3s. lOd. per lb. ? 38. An hospital contains 165 boys, and each requires for food and clothing Is. 2Jd. a-day; the governor's salary is £368, 7s. 6d. yearly, and 4 teachers have each £182, 14s. 8d. yearly ; the porters' and servants' wages and board amount to £215, lis. 6d. per annum ; and the treasurer's salary amounts to £400 yearly : what is the annual income of the hospital, supposing the yearly sur- plus to be £597, 18s. 9d. ? COMPOUND DIVISION. Case I. When the divisor does not exceed 12. Ex. Divide £27, 13s. 7id. by 5. Ans. £5, 10s. SJd. Sol. 5 in £27 is 5 times and £2 over; £2 :40s. and 13s. are 53s. 5 in 53s. is 10 times and 3s. over; 3s. = 36d. and 7d. are 43d. 5 in 43d. is 8 times and 3d. over; 3d. = 12f. and If. are 13f. 5 in 13f. is 2 times and 3 over. 5 )£27 13s. 7|d . £5 10 8i I 5 Proof £27 13 7i 1. Divide £35, 17s. 8id. by 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. 2. // 74 15 7i by 3, 5, 7, 9, 11, 2, 4, 6, 8, 10, 12. Case II. When the divisor is found in the table. Ex. Divide £30, lis. 8id. by 18. Ans. £1, 13s.ll|d.^\. 2 )£30 lis. Sj d. -M8 = 2 X 9 9)15 5 10^- 1 quot. by 2. £ 1 13 11 1- 1 " by 18. 1X2 + 1 = 2 + 1=3 rem. £ s. d. 9.612 16 3i-r 80, 84, 90 10.714 13 10^+ 70,121,144 11.896 11 2i+ 20, 42, 81 12.968 17 9|-f- 88,108,132 13.742 16 8i+ 63, 72, 81 14.845 17 9^+120, 44, 56 15.874 5 6f+ 36, 88,108 16.997 19 1U-+121, 132, 144 1. 374 16 2. 456 18 d. 3.354 11 lOf+32,35,40 4. 729 19 9i+30, 36, 44 6.847 17 llf-f-45,48,49 6. 783 11 9 ■ 7. 874 14 10*- 8. 956 15 4| 16,21,25 ■24, 27, 28 ■54, 55, 56 ■60, 63, 64 •50, 66, 72 COMPOUND DIVISION. 45 Case III. When the divisor is not contained in the table. Ex. Divide £27, 13s. 7id. by 17. Ans. £l,12s.6id.-rV 17)£27, 13s. lid. ( £1 12s. Gfd.^^'y 17X1 = 10 6 10 3=4 times. Mult, by 20 4 213 s. .26 1 = 16 " 17 X 1= 17 1 12 6i=l . 43 |Rem. 17X2= 34 £27 13 7i Proof. Mult, by 17X6 = Mult, by 17X3 = £ s. 476 13 764 17 1387 11 17 i r t: \' 9 12 115 d. 102 13 _4 54 f. 51 3 rem. d. 8^-7-13,19,23 8i-rl7,26,29 4i-^37, 46, 58 4. 7480 16 lOi-f-67, 78, 89 5.5387 14 11^47,58,69 6. 6892 17 4i-^53, 57, 59 7.9673 8 lU-^73,75,79 a 2723 4 U-f-83,91,95 £ s. 9.9873 18 10.3725 17 11.4876 11 12.5983 13 13.6473 12 571, 219, 432, 729, 843 343 563 984 d. llf-f- 6 -^ 41-^8463,1729 14.7893 17 10i-^- 2748, 9740 15.8739 16 11^^5490,6753 16.9378 11 71-^4386,9897 Case IV. To divide one sum of money by another. Ex. How often does £99, 8s. 5id. contain £1, lis. 6|d. £1, lis. 6|d. ) £99, 8s. 5Jd. 20 31s. 12 378 d. 4 1515 f. 20 1988 s. 12 23861 d. 4 )95445 f. (63 times. 9090 4545 4545 46 COMPOUND DIVISION. £ 8. d. £ s. d. £ s. d. £ s. d. 1. 113 12 6 - - 2 10 6 9. 402 2 - -4 3 9i 2. 51 7 3 - - 12 2| 10. 103 14 0|- -0 18 4i 3. 630 7 8^ - 1 14 6^ 11. 308 15 0- -2 7 6 4. 248 17 3^ -72 2^ 12.9093 12 - -2 12 1\ 5. 484 19 4^ -13 17 H 13. 201 9 9^- -3 9 5f 6.2855 7 o^ -27 14 4 14.4349 1 n- -6 3 8| 7. 2673 1 6 - - 7 13 n 15.4539 8 4 - -0 18 7i 8.8866 0^ - 1 5 Oi 16.4574 15 3^- -2 11 2^ 17. How many moidores, each 27s., are in £149, 17s. sterling ? 18. How many francs, each 9|d., are equal to £9, 15s. ? 19. How many books at 3s. 6d. are equal in value to 868 at Is. 9d.? 20. £lj 3s. 4d. was distributed among a number of boys, each received Is. 8d. ; how many were there? 21. How many pounds Irish, each 21s. 8d., are equal to 1404 pounds Scotch, each Is. 8d. ? 22. The railway fares of a certain number of passengers amounted to £26, 12s. 6d. ; the fare of each was 35s. 6d. : how many were there ? SUPPLEMENT TO COMPOUND MULTIPLICATION AND DIYISION. Ex. £4 17s. e^d. X 4g 4| 3)9 15 1 =2 times. 3 5 19 10 Oii 2 £22 15 21 J = 4§ times. Ex. £27, 10s. lljd. -r4|. 4| )£27 108.11id. 5^ 5 22 [2 )137 14 8^ =5 tunes. 1 1 1)68 17 4'g -l \b_ £6 5 2^/22 £ 5. d. £ 8. d. 1. 4 8 7iX 41, 5f 1. 270 18 llf- - 4i, 5^ 2. 6 17 10| X 6^, H 2. 384 13 lOi- - 9i, 114 8. 7 14 IH X 7^, H 3.487 11 5^- " 9f, 10| 4. 8 19 7i X 8f, 9f 4. 592 13 n- -llf, 13| 6. 9 12 1\ X 111, 12t\ 5. 756 17 lu- -16i, 29^ 6. 12 14 7i X 16y% 27A 6. 847 11 9^- -47|, 59i 7. 17 18 11|X 19f, 26^ 7. 967 11 H- -84t?^,97t*, 6. 24 11 7^ X 46 ,V 52^ 8. 989 17 lU- r8U\,89^\ COMPOUND DIVISION. 47 1. Divide £746, lis. 6d. equally among 48 men. 2. If 38 cwt. sugar cost £108, 16s. 8d. ; what is that per cwt.? 3. If 32 quarters of wheat cost £110, lis. 6d. ; what is that per quarter ? 4. A gentleman spends £960 a-year ; what is that a- week, and a-day ? 5. A gentleman's income is £1000 ; what should his daily expenses be to save £340 a-year ? 6. A labourer earns 15s. 7^d. per week, but he must save £12 a-year for house-rent and clothes; how much may he spend per week ? 7. If 46i lbs. tea cost £18, 17s. 6id. ; what is that per lb. ? 8. If27| gallons Cognacbrandycost £33, 17s. 6d.;what is that per gallon ? 9. If 34 men gain £1360 in a year; what does each gain per week, and per day ? 10. Divide £255, 17s. 6d. among 7 marines and 75 sailors, giving each marine twice as much as a sailor. 11. A farm of 156 acres is let for £375, 18s. 3d. ; what is that per acre ? 12. A joint-stock company consists of 527 shares, and the capital is £500,000 ; what is the value of a share ? 13. A merchant bought 6 pieces of cloth, each 56 yards, for £308, 12s. 6d., and sold it at 19s. llfd. per yard; how much did he gain upon the whole, and per yard ? 14. How much cloth at 15s. 6i d. per yard can be bouglit for£95, lis. 7id.? 15. How much wine at £2, 2s. 6d. per dozen can be purchased. for £297, 10s.? 16. In £59, 17s., how many crowns, half-crowns, and florins, and of each an equal number? Ans. 126 of each. Sc*L. 1 crown = 60d. £59, 17s. 1 h.-cr. = 30 _20 1 florin =_24 1197 s. ll4d. )14364d . 126 of each. 17. How many guineas, half-gumeas, crowns and florins, and of each an equal number, are in £25, 6d. ? 48 COMPOUND DIVISION. 18. How many gallons of brandy can be bought for £625, 19s. 6d. at 36s. 6d. the gallon? 19. The revenues of an hospital amount to £1807, 8s. yearly ; how many boys wiU it maintain, if each costs £18, 16s. 6^d. ? 20. A gentleman distributed £19, 14s. 6d. among some poor people, giving each 10s. ll^d. ; how many poor were there ? 21. Divide £73, 7s. 44 d. among 3 men, 5 women, and 10 boys, giving each man twice a woman's share, and each woman 3 times a boy's share. 22. If a man gains 2s. 6d. a-day, and spends Is. lO^d.; how many days must he labour to pay a debt of £11, 7s. 6d. ? 23. A farmer, who employed 49 reapers for 4 weeks to cut down his crop, paid them in aU £158, 12s. 9d. ; how much was that to each reaper, and what was the daily wages of each ? 24. A merchant pays for gas £12, 17s. 6d. yearly, at the rate of 10s. per 1000 cubic feet; how many cubic feet does he consume in the year ? 25. The wages of an equal number of men, women, and children amounted to £24, 7s. 6d. ; each man earned Is. 6d., each woman Is., and each child 9d. : how many were there of each ? 26. A house and its furniture are worth £3750, 16s. 8d., but the house is worth 7 times as much as the furniture ; what is the value of each ? 27. If 1000 muskets are worth £3333, 6s. 8d. ; what is the price of each ? 28. A corn merchant lays out £581, 17s. on equal quantities of wheat at 42s. per quarter, barley at 36s. 6d. per quarter, and oats at 29s. 3d. per quarter ; what quantity of each did he buy ? 29. How many gallons of ale at 3s. 6d. a-gallon should be exchanged for 75 gallons brandy at 38s. 6d. per gallon ? 30. A person spends £8, 12s. 6d. weekly ; what must his daily income be that in 12 years he may lay by £312 r 49 BILLS OF PARCELS. Mr James Scott 14 gallons malt aqua 13 rum 12 hollands 9 brandy 15 dozen port wine 16 sherry Edinburgh, Jan. 2, 1858. Bought of William Oliver^ @ 15/6 £ .. 18/6 .. 24/6 .. 55/6 .. 47/6 .. 36/6 Mr Andreio Turnhull Bought of John Smart & Co, 27J yards superfine black cloth @ 21/8. ..£ 17| blue do. .. 23/6... 15| olive do. ..14/9... 23» mixt do. ..17/10.. 343 bl. cassimere .. 6/4i... 3n .drab do. ..5/9^... £ Mr John Williamson Bought of J. & W, Allan, 17 reams large thick post HP. . @41/7.... ..£ 23 small do. do. .. 32/9.... 13 foolscap ruled .. 20/3.... 16 coloured yellow .. 25/8i... 18 do. green ..24/111. 21 marbled .. 19/11... £ Mr John Anderson Bought of William Tod, 13j lbs. green tea @ 6/61.... ..£ 17^ hyson skin .. 5/3^... 26 1 souchong .. 4/lU... 191 pekoe .. 5/8^... 27 raw sugar .. 6^... 35 refined do. .. 8.... £ 50 BILLS OF PARCELS. Mr William Brown 56 cwt. raw sugar 29 boxes oranges 5 lemons 150 sugar loaves ea. IS^lbs 52^ cwt. of molasses A chest of black tea, ST^lbs. Bouglit of Drysdale & Co. @ 50/8 £ .. 34/lU .. 19/4i .. 8|p.lb. .. 17/6 p. cwt. .. 4/3jp.lb... Mr George Thomson Bought of David Wright, 54i yds. superfineBrussels carpet @4/10|..£ 71 fine do. do. ..3/9 67f superfine English do. .. 2/11 J.. 294 fine do. 17^ floor-cloth 15J.. V crumb-cloth do. ■2/M . 5/71... . 8/9i... Mr David Simpson 52 quarters wheat 47 barley 39 oats 17 pease 19 beans 117 stones hay Bought of Richard Davidson, @46/6 £ .. 43/5., .. 27/8. .. 45/3. .. 46/8. 9i.. Miss Murray 14^ yds. pink sarcenet 17f green silk 25 printed calico 23 J ..... Norwich crape 19 gingham 24| do. striped 27| silk velvet Bought of Thomas Watson, @3/7i £ .. 4/2i .. l/2i .. 3/2i .. llf •• 10^ .. 14/8^ 61 II. WEIGHTS AND MEASTJKES. KEDUCTION. Ex. Eed. 8472 grs. to ib. f 4)8472 grs. (6)2118 16 )2118 2,0)35,3 dwt. 12)17 oz. 13 dwt. lib. 5 oz. 13 dwt Ex. Red. 3 lb. 4 oz. 5 dwt. to gi'S. Mult. by 12 and add 4 oz. 40 oz. Mult, by 20 and add 5 dwt. 805 dwt Mult. by 24 19320 grs. 1. Reduce 27 lbs. ; 14 lbs. 10 oz. 13 dwts. ; 57 lbs. 8 oz. 12 dwts. 16 grs. ; 82 lbs. 3 oz. 15 dwts. 20 grs. troy re- spectively to grains. 2. Reduce 27653 dwts. ; 476890 grs. ; 478670 oz. ; 72586 grs. ; 5147G0 dwts. ; and 738469 grs. to pounds. 3. Reduce 29 lbs. 2 oz. 3 drs. 1 scr. 18 grs. ; 18 lbs. 4drs, ; 461bs. 4oz. 2 scrs. ; and 205 lbs. 15 grs. to grains apothecaries' weight. 4. Reduce 4968 drs. ; 72190 scrs. ; 618764 grs. ; 5489 oz. ; 73864 drs. ; and 892164 grs. to pounds. 5. Reduce 24 tons ; 6 tons, 3 cwt. 2 qrs. 14 lbs. 12 oz. 12 drs. ; 15 cwt. 27 lbs. 14 drs. ; 27 lbs. 13 oz. 15 drs. to drams avoirdupois. 6. Reduce 21704736 drs.; 41876 lbs.; 219864 oz. , 518764 lbs. ; 21983 qrs. ; 714867846 drs. to tons. 7. Reduce 3 fur. 34 po. 3 yds. 2 ft. ; 17 miles, 2 fur. 28 po. 2 yds. 9 in. ; and 81 mis. 1 fur. 26 po. 3 yds. 2 ft. 6 in. to inches. 8. Reduce 71846 yds. ; 4189628 inches ; 4596327 ft. ; 8476 po. ; 51486973 in. ; and 7184896 ft. to miles. 9. Reduce 4 yds. 2 qrs. 1 nl. ; 24 yds. 3 nls. ; 25 Eng. ells, 3 qrs. 2 nls. ; 53 Fie. ells, 2 qrs. 3 nls. ; 56 yds. 3 qrs. 3 nls. to nails. 10. Reduce41764 nls. ; 5174 qrs. ; 318769 inches ; 49864 nls.; 217384 inches; and 8172144 nls. to yards and English ells. 11. Reduce 27 ac. 3 ro. 16 per. ; 84 ac. 2 ro. 24 per. 28 yds. ; 108 ac. 1 ro. 36 per. 25 yds. 5 ft. 84 in. to square inches. 52 REDUCTION. 12. Reducel47847684sq.inches; 218764yds.; 5189764 ft. ; 31874 per. ; and 84726084 inches to acres. 13. Reduce 48 cub. yds. ; 403 cub. yds. 21 ft. 908 in. , 700 tons of ship. ; 672 loads of hewn timber ; and 876 do. of rough, to cubic inches. 14. Reduce 17486936 cu. in.; 784693 cu. ft.; 874869684 cu. in. ; and 784627 cu. ft. to cubic yards and tons of shipping. 15. Reduce 8 qrs. 3 bu. 2 pk. 1 ga. ; 208 qrs. 7 bu. 3 pk. 1 ga. 3 qts. ; 409 bu. 2 pk. 1 ga. 3 qts. 1 pt. to pints. 16. Reduce 7486984 pts. ; 87634 pks. ; 918764 gals. ; 8176 bu. ; 514876 pts. ; and 784673 qts. to quarters. 17. Reduce 208 galls. 3 qts. 1 pt. ; 476 galls. 2 qts. ; and 749 galls. 1 qt. 1 pt. to pints. 18. Reduce 74869 pints ; 586476 pints ; 3486 qts. ; and 79040 pints to gallons. 19. Reduce 42 signs 39° 36' ; 81^ 15° 49' 59'^ ; 208s 20° 56' 28''; and 315^ 19° 34' 38'' to seconds. 20. Reduce 718460"; 87654'; 374°; 8178640"; 71860'; 7186940"; and 7184° to signs and circles. 21. Reduce 36 co. ye. 219 da. 18ho. 15 min. 27 sec. ; 380 CO. ye. 219 da. 23 ho. 29 min. 36 sec. ; and 7184 Julian years to seconds. 22. Reduce 71847630 sec. ; 48196219 min. ; 81468 ho. ; 31817640 sec. ; and 7187210 min. to com. and Jul. years. 23. How long would it require to count 800 millions of sovereigns, at the rate of 120 in a minute? 24. How many seconds have elapsed since the birth of Christ or in 1858 Jul. years? 25. The distance of Jupiter from the sun is 494,513,000 miles ; express this in feet. 26. Saturn revolves round the sun in 10,756 days, 5 ho. 16 m. 32 sec. ; how many seconds is this ? 27. In Scotland there are 29,167 square miles j how many acres does it contain ? 28. The polar axis of the Earth is 41,706,360 feet ; express this in miles, &c. 29. Light travels at the rate of 192,000 miles per sec. ; in what time will it travel between the Sun and Saturn, the distance being 906,643,000 miles ? 53 COMPOUND ADDITION. TROY WEIGHT. 2. lbs. oz. dwt. gr. lbs. oz. dwt. gr. lbs. oz. dwt. gr. 27 10 11 18 31 10 19 23 67 8 13 22 36 11 16 17 53 8 17 19 76 9 18 20 41 8 18 23 35 4 15 20 18 11 15 23 34 9 10 12 20 3 4 6 81 4 12 14 26 5 8 19 25 9 16 17 15 3 13 13 37 11 16 15 48 11 19 21 56 11 4 8 47 10 18 22 84 10 16 18 9 10 8 17 APOTHECARIES' WEIGHT. 4. 5. 6. lbs. oz. dr. Bcr. oz. dr. sc. gr- lbs. oz. dr. Bcr. 18 9 4 1 11 7 2 19 56 10 7 2 17 8 7 2 10 4 1 15 84 11 5 1 26 5 4 8 3 2 18 96 10 6 2 62 11 7 2 7 5 1 17 31 8 4 1 25 10 4 1 9 4 2 15 28 9 7 2 64 11 2 2 11 6 1 18 86 11 7 2 AVOIRDUPOIS WEIGHT. 7 8 s . ton. cwt. qrs. lbs. cwt. qrs. lbs. oz. qrs. lbs. oz. dr. 74 18 2 25 27 2 18 14 14 14 14 14 87 16 3 27 31 3 26 15 27 26 15 15 65 13 1 20 46 1 24 13 38 22 13 12 29 11 18 49 2 27 11 47 11 12 13 94 17 3 26 37 24 12 31 18 15 11 38 14 2 19 84 3 16 10 72 27 13 15 45 19 1 27 93 1 27 15 29 22 10 4 LINEAL MEASURE. 10. 11 12. mis. fu. po. yds. fu. po. yds. ft. po. yds. ft. in. 45 3 27 4 17 18 2 1 39 '2 1 7 76 7 39 5 25 36 1 2 45 4 2 11 64 5 29 3 64 31 3 1 53 3 1 10 85 6 34 2 45 39 5 2 32 5 2 8 58 7 26 4 74 26 4 25 4 2 y 69 3 37 2 55 34 5 2 74 5 2 11 73 5 33 4 79 18 1 1 43 2 1 9 54 COMPOUND ADDITION. CLOTH MEASURE. 13 14 15. yds. qrs. nls. in. En.ell qrs. nls. in. Fr.ell. qrs. nls. in. 27 3 3 2 45 3 2 1 48 5 3 2 73 1 1 36 4 3 2 76 3 2 48 2 75 2 1 1 51 4 1 1 86 2 1 1 64 4 3 2 36 5 3 2 74 2 3 2 38 4 1 2 71 4 1 39 1 2 76 3 3 1 67 1 2 76 1 3 2 69 4 2 2 84 5 3 2 SQUARE OR LAND MEASURE. 16. 17. u I ac. ro. pe. yds. ro. pe. yds. ft. pe. yds. ft. in. 36 3 39 30 14 27 18 4 27 29 8 67 45 2 33 24 25 11 19 8 31 30 4 96 72 3 27 29 32 17 21 4 27 27 1 99 85 1 36 30 36 38 29 7 38 11 7 84 96 3 38 27 48 28 26 6 29 26 3 47 71 2 31 25 86 39 30 8 36 30 8 98 78 3 39 30 74 36 28 7 39 30 8 99 MEASURE OF CAPACITY. 19 20 21. qrs. bu. pe. Ra. bu. pe. ga. qt. pe. ga. qt. pt. 56 7 3 56 2 1 2 74 1 3 74 6 2 39 1 3 38 1 2 45 3 1 47 3 1 2 84 1 3 63 7 3 76 1 1 1 48 1 2 34 6 2 79 3 3 56 1 3 47 7 3 34 1 1 75 2 38 5 2 49 3 1 3 63 1 3 TIME. 22. 23. 24 co.ye. da. ho. m. da. ho. m. sec. da. ho. m. sec. 28 96 18 15 84 12 56 59 55 18 54 11 38 27 16 53 96 18 41 36 73 19 49 56 45 99 23 59 72 23 59 59 85 16 46 55 76 84 11 52 67 22 48 45 98 17 41 36 35 219 16 18 36 21 18 47 87 22 28 39 74 361 15 55 79 20 29 55 76 20 18 49 84 278 11 46 93 23 51 42 65 23 54 51 COMPOUND ADDITION. 55 25. A corn merchant bought 208 qrs. 3 bu. 1 pk. of barley ; 336 qrs. 2 pk. of wheat ; 236 qrs. 4 bu. of oats ; 125 qrs. 1 bu. 3 pks. of rye ; 86 qrs. 1 bu. 1 pk. of pease ; and 79 qrs. 2 bu. 2 pk. of beans : how many quarters did he buy ? 26. The distance from A to B is 2 ml. 1 fu. 30 po. 5 yds. ; from B to C, 7 fu. 4 yds ; from C to D, 1 ml. 25 po. ; and from D to E, 3 ml. 1 fu. 2 yd. : find the distance from A to E. 27. A clothier, at various times, bought 28 yds. 2 qrs. 1 nl. of cloth ; 37 yds. 2 qrs. ; 47 yds. 1 nl. ; 37 yds, 1 qr. 2 nl. ; and 67 yds. 2 qrs. 2 nl. : how much did he buy in all ? 28. London is in latitude 51° 30^ 48'' N., and Sydney is in lat. 33° 51^ 40'^ S. ; what is the diflference ? COMPOUND SUBTRACTION. TROY WEIGHT. 1. 2. 3. lb. oz. dwt. gr. lb. oz. dwt. gr. lb. oz. dwt. gr. 96 10 13 14 36 5 17 21 82 2 1 16 47 10 19 21 19 7 18 23 79 11 14 20 apothecaries' weight. 4. 5. 6. lb. oz. dr. scr. lb. oz. dr. Bcr. oz. dr. scr. pr. 39 2 3 1 52 7 7 41 6 2 18 29 7 5 2 46 8 7 1 34 7 1 19 84 65 avoirdupois weight. 7. 8. 9. ton. cwt. qrs. lb. cwt. qrs. lb. oz. qrs. lb. oz. dr. 84 13 2 11 46 1 23 12 17 21 11 10 69 14 3 25 29 2 22 15 9 22 13 14 10. mis. fu. po. yds. ^ 3 22 4 3 25 5 lineal measure. 11. fa. po. yds. ft. 35 33 4 1 17 36 4 2 po. 39 19 12. yds. ft. in, 2' 2 8 4 2 10 56 COMPOUND SUBTRACTION. CLOTH MEASURE. 13. 14. 15. yds. qrs. Ills. in. E.ell. qrs. nls. in. Fr.ell. qrs. nls. in. 72 2 1 1 93 4 2 1 81 4 2 1 66 2 1 2 63 4 3 2 41 5 3 2 SQUARE OR LAND MEASURE. 17. ae. ro. pe. yds. ro. pe. yds. ft. pe. yds. ft. in. 65 2 31 21 53 18 27 3 25 21 4 10 36 3 31 24 19 28 29 5 8 29 4 11 MEASURE OF CAPACITY. 19. 20. 21. qrs. bu. pe. pa. bu. pe. ga. qt. pe. ga. qt. pt. 21 3 2 54 2 1 2 18 3 18 4 3 1 49 2 1 3 9 1 3 1 22. co.ye. da. ho. m. 41 138 14 25 18 147 16 46 TIME. 23. da. ho. 90 13 43 21 40 18 25 46 m. sec. da. 24. ho. m. BBC. 70 19 15 55 31 22 15 59 25. The latitude of Edinburgh is 55° 57' 23'' N., and the latitude of Pekin is 39° 54' 13'' N. ; find the difference. 26. Mars revolves round the sun in 686 da. 23 ho. 30 m. 41 sec, and Venus in 224 da. 16 ho. 49 ra. 10 sec. ; find the difference. 27. Three farms contain 4536 ac. 3 ro. 25 per. ; the 1st and 2d contain 3327 ac. 30 per., and the 1st and 3d 2752 ac. 15 per. ; what is the size of each? 28. A merchant bought 756 qrs. 3 bu. 2 pk., and sold to A 208 qrs. 3 bu. 1 pk., and to B 315 qrs. 2 bu. 2 pk. ; what quantity has he left ? 29. A piece of silk measured 43 yds. 2 qrs. 1 nl. 1 in. ; after 27 yds. 3 qrs. 2 nl. 2 in. have been sold: how much remains ? 30. A traveller arrived at a railway station at 26 min. and 32 sec. past 1 o'clock, and found that the train did not start until a quarter past 2 o'clock ; how long had he to wait ? 57 COMPOUND MULTIPLICATION. 1. 17 lb. 8 oz. 15 dwt. 21 grs. X25, 27, 29, 47 2. 32 lb. 11 oz. 17 dwt. 19 grs. X 16, 24, 31, 38 3.25 lb. 4 oz. 3 drs. 2 scr. 18 grs. X 18, 22, 34, 43 4. 47 lb. 9 oz. 7 drs. 1 scr. 13 grs. X20, 28, 37, 41 6. 36 ton. 14 cwt. 2 qrs. 21 lb. 1 1 oz. 13 drs. X 30, 32, 39, 46 6.43ton. 16cwt. 3qrs. 261b. 13oz. 15 drs.X33, 35, 47, 51 7.21 mis. 5 fu. 29 po. 2 yds. 1 ft. 11 in. X36, 40, 52, 57 8.37 mis. 7 fu. 36 po. 3 yds. 2 ft. 8 in. X42, 45, 53, 58 9. 56 yds. 3 qrs. 2 nls. 2 inches X54, 56, 67, 71 10. 73 E. ells, 4 qrs. 3 nls. 1 inch X60, 64, 68, 17 11.47 Fr. ells, 5 qrs. 2 nls. 2 inches X^^, 70, 69, 73 12.38 Fl. ells, 2 qrs. 1 nl. 1 inch X72, 77, 19, 13 13. 56 ac. 2 ro. 31 pe. 23 yds. 5 ft. 47 in. X44, 48, 59, 61 U.87 ac. 3 ro. 39 pe. 27 yds. 8 ft. 126 in. X49, 50, 62, 65 15.43 qrs. 3 bu. 2 pe. 1 gal. 3 qts. 1 pint X80, 81, 75, 78 16.76 qrs. 4 bu. 1 pe. gal. 2 qts. 1 pint X84, 88, 79, 82 17. 34 CO. ye. 156 da. 21 ho. 56 min. 57 sec. X90, 96, 86, 98 18. 71 Ju. ye. 213 da. 19 ho. 42 min. 49 sec. X 108, 121, 107 19. A sovereign weighs 5 dwt. 3 grs. nearly ; find the weight of 1000 sovereigns. 20. What is the weight of 35 brass guns, each weigh- ing 6 cwt. 3 qrs. 9 lbs. ? 21. How far will a postman travel in a year, if he walks 9 mis. 3 fu. 8 po. 5 yds. daily ? 22. How much grain will a farm of 25 fields, each 12 acres, produce at the rate of 8 qrs. 7 bu. 2 pk. 1 gal. per acre ? 23. A cubic foot of water weighs 2 qrs. 6 lbs. 8 oz. ; what weight of water is there in a cistern whose content is 72 cubic feet ? 24. How much cloth would be required to make coats for a regiment of 875 soldiers, allowing 3 yds. 1 qr. 1 nl. to each ? 25. A cartload of coal weighs 19 cwt. 2 qr. 18 lb. ; how much will 37 cartloads weigh? 26. Find the content of 17 farms of 14 fields, each con- taining 9 ac. 3 ro. 19 per. 4 yds. 58 COMPOUND DIVISION. 1.387 lb. 4 oz. 13 dwts. 18 grs. 2. 496 lb. 11 oz. 19 dwts. 22 grs. 3.576 lb. 10 oz. 4 drs. 2 scr. 18 grs. - 4.765 lb. 8 oz. 7 drs. 1 scr. 12 grs. - 5.876ton.l5cwt..3qr.201b.l3oz.l2dr.- 6.987ton. 18cwt.2qr.261b.l5oz.8dr.- 7. 475 mis. 7 fu. 38 po. 3 yd. 2 ft. 11 in.- 8. 754 mis. 3 fu. 25 po. 2 yd. 1 ft. 10 iii.- 9. 375 yds. 3 qrs. 2 nls. 1 inch 10. 573 E. ells, 4 qrs. 3 nls. 2 inches 11. 876 Fr. ells, 5 qrs. 2 nls. 2 inches - 12.768 Fl. ells, 2 qrs. 1 nl. 1 inch 13. 476 ac. 3 ro. 36 pe. 25 yds. 4 ft. 96 in. ■ u. 674ac. 2 ro. 24pe. 28yds. 5ft. 102 in.- 15. 987 qrs. 7 bu. 3 pe. 1 gal. 2 qts. 1 pint- le. 879 qrs. 4 bu. 2 pe. gal. 3 qts. 1 pint- 17. 578 CO. ye. 134 da. 15 ho. 44 m. 58 sec- 18. 488 Ju.ye. 341 da. 21ho. 56m. 58 sec- 16, 18, •15,14, ■20, 22, 30, 42, •28, 25, 27, 32, ■33, 40, •35, 36, •44, 45, •48, 49, 50,54, -55, 56, -60, 63, -64, 70, -66, 72, -77; 80, -81,84, -88, 90, 23, 17, 31, 39, 37, 38, 43, 46, 53, 52, 98, 19 29 26 34 39 41 47 51 57 87 117 273, 181 371,811 713, 645 298, 364 756, 643 209,316 369, 691 19. 14 hhds. Jamaica sugar weigh 234 cwt. 2 qr. 14 lb. ; find the weight of each. 20. How many canisters, each containing 1 qr. 7 lb., can be filled from 37 cwt. 21 lb. ? 21. 133 bars of silver weigh 156 lb. 3 oz. 17 dwt. 2 grs. ; w^hat is the weight of each ? 22. Find the circumference of a wheel which revolves 5267 times on a road 8 mis. 7 fu. 32 po. 5 yds. long. 23. An estate contains 5837 ac. 2 ro. 29 per. ; into how many farms, each containing 32 ac. 3 ro. 37 per., may it be divided ? 24. A spring yields 72 gallons of water an hour, and supplies 675 families ; how much may each family use daily ? 25. How many steps, each 2| feet, will a man take in walking 9 miles ? 26. In 2 cwt. 2 qr. 5 lb. 4 oz. 8 drs. ; how many parcels of 4 oz. 5 drs., 5 oz. 6 drs., 7 oz. 8 drs., and 8 oz. 5 drs., and of each an equal number ? 59 MISCELLANEOUS EXERCISES. 1. A was born in 1805, and B 20 years after; when was B born, and what are their present ages ? 2. A general, commanding an army of 45,550 men, fought a battle, in which 5217 were killed, 11,781 wounded, and 518 amissing; he likewise threw 2157 into one garrison, and 1786 into another; how many effective men remained under his command in the field ? 3. What number being divided by 374 will give 8647369 for the quotient, and 76 for the remainder? 4. The product is 78469468, and one of the factors 4876 ; what is the other? 5. Two persons start from the same place, and travel the one 35 miles, and the other 42 miles a-day; how far will they be distant from one another at the end of 44 days if they both travel the same way, and how far if they travel in opposite directions ? 6. A person, after paying to A £71, to B £84, to C £121, to D £118, to E £217, and to F £196, has still remaining £254 ; how much had he at first ? 7. In leap year how many days in each of the 12 calendar months, and what is their sum ? 8. How many days from March 3d to November 1 9th ? 9. How many days from April 1st to December 29th? 10. A man was born in the year 1821, when will he be 85 years of age ? 11. A man was bom in 1815, what was his age in 1858 ? 12. A boy can point 1 6,000 pins in an hour, how many at that rate will 16 boys point in a year of 365 days, if they work ten hours each day ? 13. If the population of the globe is taken at one billion, how many die yearly, if we suppose a generation to last 36 years? 14. At a game of cricket A, B, and C score 112 runs, A and B score 79 runs and B and C 70 runs ; how many did each score ? 15. The Iliad contains 15683 lines, and the -^neid 9882 lines, now if a boy reads 112 lines daily; in how many days will he finish them ? 16. A merchant lodged in the bank on Monday £744, lis. 7^d., and drew out on Tuesday £579, 18s. 6|d. ; lodged on Wednesday £1054, 17s. 8d., drew on Thursday £873, 198. 60 MISCELLANEOUS EXERCISES. 9id. ; lodged on Friday £1786, 13s. 10|d., and drew out on Saturday £1297, 13s. llfd. ; how much remained on Tuesday, Thursday, and Saturday after drawing? 17. If a yard of cloth costs £1 , 2s. 6^d., what cost 85 yards ? 18. If 74 yards of cloth cost £84, 17s. 6d., what cost lyard? 19. If 25 yards cost £24, 5s. lOd., what cost 5 yards? 20. What cost 93 cwt. of sugar at £2, 16s. 8^d. per cwt.? 21. What cost 1 lb. of tea at £96, lis. S^d. for 275 lb. ? 22. How many letters in a book of 21 volumes, each 840 pages, each page 48 lines, and each line 41 letters? 23. If a mason gains 18s. 6d. per week, and lays up 2s. T^d. per week ; how much does he spend, and how much does he lay up in a year ? 24. How many revolutions does a wheel, which is 2 J yards in circumference, make in 3^ miles ? 25. A traveller walks 25 miles a-day, after travelling 75 miles, another follows him at the rate of 30 miles a-day ; in how many days will the second overtake the first ? 26. If a man's wages are 21s. per week, how much may he spend weekly to save £13, 13s. a-year? 27. A farm of 96 acres is let for £96, 16s. 6^d., what is that per acre ? 28. Gained £274, 19s. 8id., but afterwards lost £189, 19s. llfd. ; what is my net gain? 29. How much will a labourer earn in 219 days at 2s. l^d. per day ? 30. A labourer earns £35, 17s. lO^d. a-year, how much is that per week ? 31. 16 men purchased a lottery ticket for £25, which turned out a prize of £3150; how much of the ticket did each pay, and how much did each receive of the prize ? 32. A merchant has in cash £2385, 17s. llfd., in bills £12,748, 16s. 6d., tea valued at £748, 16s. llfd., raw sugar £289, 17s. 6|d., refined sugar £112, 17s. S^d., whisky £348, 17s. lOd., rum £240, lis. 7id., brandy £497, lis. 7|d., gin £241, lis. 7^d., wines £1298, 3s. 4|d., porter £84, lis. ll^d., ale £73, 16s. 9d., in other articles £876, 13s. 9^d., and debts owing to him £2381, lis. lid.; at the same time ho owes to A £481, 17s. llfd., to B £973, 16s. 7^^ ^o C £876, 16s. lOd., to D £584, 16s. 4|d., to E £683, 13s. ^d., to F £297, 16s. lO^d., and in bills £7348, 16s. 7fd.; what is his net worth ? 33. In £23, 2s., how many shillings, sixpences, and four- pences, and of each an equal number ? MISCELLANEOUS EXERCISES. 61 34. "Wliat quantity of tea at 3s. 9§d. per lb. should be exchanged for 728 lbs. of sugar at 6^d. per lb. ? 35. A took to market with him £148, 17s. lOfd., and he there received from B £741, lis. lO^d., from C £629, 168. S^d., from D £946, lis. 6d., from E £493, 16s. llfd., from F £748, 16s. 9id., from G £387, 10s. 6fd., and from H £876, lis. 7^d. ; but in coming home he was robbed of £2587, lis. 8f d. : how much did he bring home with him? 36. A person paid for a feu to build a house £1276, 17s. 6|d. ; the mason's bill amounted to £1485, 178. 3f d., the joiner's to £487, 16s. 9fd., the plasterer's to £184, 19s. 9id., the slater's to the same, the painter's to £120, lis. 7fd., the plumber's to £56, lis. lO^d., besides other charges to £37, lis. 9^d. ; now he wants to sell it so as to gain £470, lis. 9^d. : how much does he expect for it ? 37. A person gains £1, 5s. 7jd. per week, and spends 19s. 8^d. per week ; how much does he save in the year ? 38. A person gains £1, 2s. 7^d. per week, and spends £45, 17s. Ifd. in the year; how much does he save in the week? 39. Divide 91 lb. 7 oz. 11 dr. of tea among 12 men and 24 women, giving each man | of the share of a woman. 40. A merchant began business with a capital of £950, 17s. 6d. ; at the end of the year he had in cash £350, lis. 8id., in bills £256, 17s. 8^d., in goods £850, lis. 2^d., and debts owing to him £572, lis. 7fd. ; at the same time he owed in bills £381, 17s. 2Ad., to A £340, 18s. 7^d., to B £120, lis. 4fd., to C £49, Hs. 6fd., to D £36, 17s. 8^., to E £49, lis. 2^d., and to sundries £134, 18s. 6d. : whether has he gained or lost, and how much ? 41. Bought 24 pieces of cloth, each containing 30 yards, for £840, 17s. 6d., and sold 400 yards at £1, 4s. 3d. per yard ; how must I sell the remainder per yard to gain £84, 2 s. 6d. upon the whole ? 42. Bought 480 yards of cloth for £560, 6s. 8d., but 120 yards being damaged, I am obliged to sell them at a loss of £20, 13s. 4d. ; how must I sell the remainder per yard so as to gain £60, 16s. 8d. upon the whole, and what did the damaged part sell at per yard ? 43. A merchant clears by his trade £1 590, 17s. 6f d. yearly ; his household expenses amount to £580, 17s. 7|d., house rent £120, lis. 9^, taxes £45, 17s. 8|d., shop rent £140, lis. 9id., taxes £56, 17s. 8|d., servants' wages £175, lis. llfd., tradesmen's accounts £170, lis. 9|d., and incidental expenses £49, 17s. 8|<l. ; what is his net gain ? 62 MISCELLANEOUS EXERCISES. 44. What is the value of 12 1 gallons of rum at 18s. 4d. per gallon ? 45. Bought sugar at £3, 16s. 6d. per cwt., how much was that per lb. ? 46. After paying at one time £847, lis. 8|d., at another £650, 10s. 4id., at a third £549, 16s. T^d., at a fourth £729, 18s. 4|d., at a fifth £1084, 19s. 8id., and at a sixth £1578, 15s. 5id., there remained due £2196, 17s. lOfd.; what was the original debt ? 47. What is the stock of a banking company, which con- sists of 154 shares, each £578, 17s. 8^d. ? 48. Divide £17, 12s. lid. among 3 men, 4 women, and 5 children, giving each man 2 times the share of a woman, and each woman 3 times the share of a child. 49. A gentleman gave £10, 10s. to pay for his lodging from 1st May till 10th July, at Is. ll^d. per night; what change should be returned to him ? 50. The longitude of New York is 74° 0' 3" W., and of Cal- cutta, 88° 20' 27" E. ; find the difference. 51. Discounted six bills; the first amounted to £340, 17s. 8^d., the second to £473, lis. 9|d., the third to £576, 17s. 8|d., the fourth to £605, 15s. 4|d., the fifth to £680, 17s. 2|d., and the sixth to £720, Is. 9^d. ; the discount upon each was respectively £7, 19s. lOf d. ; £9, 12s. ll^d.; £11, 18s.8|d.; £12, 16s. 8id.; £12, 19s. 9|d. ; and£13, 5s. ll|d. : what was the net proceeds of each, and of the whole ? 52. A gentleman's yearly income is £1560, 16s. 8^d. ; how much may he spend monthly, weekly, and daily, to save £500 a-year ? 53. A gentleman gave his daughter for her fortune an es- critoire, containing 12 drawers, each drawer was divided into 18 compartments, in each of which was £24, 17s. 6fd. ; what was the daughter's fortune? 54. Divide £1120, 10s. 6d. among 10 men and 3 boys, giving each boy only ^ of a man's share. 55. How many guineas, half-guineas, crowns, and florins, and of each an equal number, are contained in £36, lis. 6d. ? 56. Divide 886 ac. 3 ro. 25 per. of land among A, B, and C, giving A 32 ac. 2 ro. 35 per. less than C, and B 98 ac. 3 ro. 15 per. more than C. 57. The freights received for a voyage were, from A £127, 6s. 8H, from B £141, lis. 7fd., from C £174, 17s. lO^d., from D £84, lis. 9|d., from E £79, 12s. 4id., from F £112, 13s. 6id., and from G £14, lis. 2Jd.; how much was the whole freight ? MISCELLANEOUS EXERCISES. 63 5^3. A merchant bought 87 yards of blue cloth for £1, 2s. 2^d. per yard; how must he sell it per yard to gam £12, 13s. 6d. on the whole? 59. The rent of a shop, including taxes, is £95, 19s. 7Jd, a-year ; how much is tliat weekly and daily ? 60. Bought 9 pieces of cloth, each 35 yards, for £164, 12 s. 9^d., and sold 108 yards at lis. 2d. per yard; how must 1 sell the remainder per yard to gain £47, 8s. 7fd. in all? 61. In 20 lbs. 11 oz. 14 drs. of sugar, how many packages, containing 2 lb., 9 oz., and 4 oz., and of each an equal Qumber ? 62. A common consists of 440 ac. 2 ro. 20 per. ; into how many fields, each containing 5 ac. 3 ro. 20 per. can it be divided ? 63. Bought 44 pipes of wine for £2640, and gained by selling them as much as 11 pipes cost me ; what was a pipe of it sold for ? 64. A ship's company took a prize of £17,240, lis. 9d. ; the captain got i of the whole, the 2 lieutenants got each y*g of the remainder, the 3 midshipmen got each ^^g of what was left, and the remainder was equally divided among i crew of 218 men ; what was the share of each ? 65. Divide £172, 18s. 3d. among 4 men, 7 women, and 13 children, givmg each man 3 times the share of a woman, ami each woman 5 times the share of a child. 66. A father divides his estate among his 3 sons ; the eldest gets £6000, the second | of the eldest, and the third f of the second ; what was the value of the estate, and the shares of the two younger sons ? 67. A bankrupt who owed his creditors £7856, paid them only £3250, 12s. 6d. ; what was that per pound? 68. A and B gain joinjtly £56, 17s. llfd., A and C £48, 17s. lO^d., and B and C £60, lis. 8Jd. ; what is the whole gain, and the share of each ? 69. Received 147 yards of cloth at 14s. 6d. per yard in exchange for 441 lbs. of tea ; find the price of the tea per lb. 70. An equal number of men, women, girls, and boys, are employed at a manufactory, each man receives Is. 6d. per day, each woman Is. 3d., each girl 7^d., and each I now the sum required to pay their daily -^ ^ £15, 63. l^d. : how many of each are empla 71. A merchant purchased 245 yards o^Sid^at 10^ T^dUi per yard, now 20 yards became worthlesl^feena >being damj^ ^s -. aged ; he sold the remainder for 18s. 9d. v- what did he gaintt a^ i 72. 14 lbs. of tea at 3s. lOd. per lb., 16 lbs. at 4s. 2d^^ 64 MISCELLANEOUS EXERCISES. lbs. at 4s. 6d., and 35 lbs. at 5s. 3d. are mixed together; what should it be sold for per lb. ? 73. How many packages of coffee, containing respectively 2 lb., 1 lb., I lb., and ^ lb., and of each an equal number, can be made from 16 cwt. 10 lbs.? 74. A prize of £2982, 14s. 2d. is divided among a captain, 2 lieutenants, 3 ensigns, and 120 soldiers; the captain is to have 5 shares, each lieutenant 4 shares, each ensign 2 shares, and each soldier one share : how much should each receive ? 75. A father left to his eldest son 4500 guineas more than he left to his second son, to the second 12500 crowns more than to his third son, and to the third he left 9000 guineas ; find each son's portion. 76. Divide £786, 13s. 6^. among 3 persons, giving the first £140, 16s. lOd. more than the second, and the second £90, 18s. lOd. more than the third. 77. A merchant bought 145 gallons of whisky at iSs. 6d. a gallon ; how many gallons of water must he add to it, that he may gain £7, 12s. 6d., and reduce the price to 12s. 6d. per gallon ? 78. £5, 19s. 2d. is to be divided among 3 classes of poor people, there are 8 in the first class, 9 in the second, and 10 in the third; the share of the first class is to be 1^ time that of the second, and the second twice the tmra ; find the share of each class. 79. The weekly wages of A and B are £3, 7s. 9d. ; of A and C £3, 12s. 3d. ; of B and C £3, 13s. : what are the daily wages of each ? 80. Mercury revolves round the sun in 87 da. 23 ho. 15 min. 44 sec. ; Venus in 224 da. 16 ho. 49min. 10 sec. ; Mars in 686 da. 23 ho. 30 min. 41 sec. ; Jupiter in 4332 da. 14 ho. 2 min. 8 sec. ; and Saturn in 10,756 da. 5 ho. 16 min. 32 sec. : how many revolutions has each of these planets performed in 1858 solar years? DECIMAL COINAGE. In anticipation of a Decimal Coinage being introduced into this country, the system most likely to be adopted is shown in the following DECIMAL COINAGE. 65 TABLE OP DECIMAL MONET. 1 mil (m.) = ^To'oTj = li^- 10 mils == 1 cent (c.) = £t^o = 2|d. 100 mils = 10 cents = 1 florin (fl.) = £t'o = 2s. 1000 mils = 100 cents = 10 florins = £1 = 20s. 6d. = 25m. = 2c. 5m. ; Is. = 50m. = 5c.; 2s. 6d. = 125m.= Ifl. 2c. 5m. ; 5s. = 250m. = 2fl. 5c. ; 1 Os. =500m. = 5fl., etc. The pound sterling, which is now divided into 960 parts, would thus be divided into 1000 parts, and calculations in money would be performed as in the Simple Rules, by placing a point after the pounds, and making the florins occupy the Jlrst place after the point, the cents the second, and the mils the third place ; thus : — £24, 2fl. 7c. 5m. would be written decimally, £24-275 £36, 7c. £48, 6fl. 5m. 1. £25, 2. 57 8. 90 4. 76 5fl. 8 3 1 Express decimally, 6. £27, 6. 30 7. 17 8. 5fl. 3c. 9 2 4 4 9. 10. 11. 12. £36-070 £48-605 £150, 5fl. 8c. 4m. 490 2 5 708 5 910 4 2 13. £20-450 14. 36-050 Read in £'s, florins, etc. 15. £47-825 I 17. £99-005 | 19. £210-065 16. 90-605 18. 100-725 20. 102-708 ADDITION AND SUBTRACTION. 1. £8, 2fl. 5c. 4- £7, £24-133 = £24, Ifl. 3c. 3 5m. + £8, 3fl. 8m. Ans. £24-133 = £24, ifl. 3c. 3m. Ex. 2. £85, 3fl 5m. — £58, 4fl 3c . 6m. Ans. £26-869. Sol. Writ cimally unde proceed as in and in the n e the amounts de- r each other, then Simple Numbers, jsult place a point Ex. Sum 1. £8-250 7-575 8-308 , £24-133 Ex Dii . 2. £85-305 58-436 af. £26-869 below thft ot 1. £ fl. c. 5 4 3 6 7 8 6 9 5 9 10 7 1 15 m. 2 4 5 2 9 £ 14 18 24 30 45 53 fl. 5 8 2 7 2. c. 9 9 4 2 m. 6 5 7 6 9 5 £ 75 67 50 39 30 24 3. fl. c. m. 8 2 9 5 6 7 6 6 2 5 8 66 £ 150 127 217 363 460 604 DECIMAL COINAGE. £ 234 342 423 432 243 324 £ 100 140 104 110 101 104 6. fl. 6 2 2 2 7. 8. 9. 185 4 2 5 197 4 6 567 8 2 3 67 5 9 179 3 5 7 498 9 3 5 10. 11. 12. 759 6 3 2 842 1 6 975 2 699 7 6 5 483 2 6 586 6 3 5 MULTIPLICATION AND DIVISION. Ex. 1. £75, 5fl. 5m. X 42. Ans.£3171-210 = £3171,2fl. Ic. Ex. 2. £185, 2c. 5m. — 25. Ans. £7*401 = £7, 4fl. Im. Ex. 1. £75-505 42 151010 302020 Product, £3171-210 Ex.2, 25 J f 5| £185-025 (51 37-005 Quotient, £7-401 Sol. Multiply or divide the amount, expressed decimally, by the multiplier or divisor, and point off three figures from tiia right of the result. 5. £145,5fl.2c.5m. X26,31 6. £415, 2c. 4m. X 79, 85 7. £154, 4fl. 8m. X 101,163 8. £514, 3fl. 8c. 9m. X 695, 2045 13. £359, 4fl. 5c. 5m. -H 29, 37 14. 641 2 3 2 -^61,73 15. 783 1 I 1. £75, 3fi. 5c. X 25, 45 2. £63, 7fl. 6c. 5m. X 16,63 3. £97, 6c. 8m. X 77, 96 4. £84, 7fl. 5m. X 63, 81 9. £184, 2fl. 7c. 5m. -f- 25, 45 10. 127 5 7 5 -^63,8l 11. 129 7 4 5 -^35,55 12. 126 5 8 8 -T-33,77 8 -r- 122, 131 16. 3083 8 8 5 -i- 365, 355 17. How much will a man's wages amount to in a year, at £1, Ifl. 2c. 5m. per week ? 18. Divide £68, 3fl. 5c. 5m. among 3 men, 5 women, and 7 children, giving each woman twice the share of a child, and each man thrice the share of a woman. 19. A man gains Ifl. 2c. 5m. in a day, and spends Ifl. per day ; how many days must he work to pay a debt of £9*375 ? THE END. EDINBUEGH : PRINTED BY OLIVER AND BOTD. ARITHMETIC f FOR ADVANCED CLASSES; BEING A CONTINUATION OF TROTTER'S LESSONS IN ARITHMETIC FOR JUNIOR CLASSES : CONTAINING VULGAR AND DECIMAL FRACTIONS; SIMPLE AND COMPOUND PROPORTION, WITH THEIR APPLICATION ; SIMPLE AND COM- POUND INTEREST, INVOLUTION AND EVOLUTION, ETC. By ALEXANDER TROTTER, TEACHER OP MATHEMATICS, ETC., IN EDINBURGH, Author of " A Key to Trotter's Complete System of Arithmetic," etc feta mtUn. EDINBURGH : OLIVER AND BOYD, TWEEDDALE COURT., LONDON : SIMPKIN, MARSHALL, AND CO. 1872. SCHOOL-BOOKS BY JAMES TROTTER, LATE OF THE SCOTTISH NAVAL AXD MILITARY ACADEMY. LESSONS in ARITHMETIC for Junior Classes. 6d. A Complete System of ARITHMETIC, Theoretical and Practical. 3s. Teotteb's Edition of HUTTON'S BOOK-KEEPING. 2s. A Complete System of MENSURATION, by Ingram & Trotter. 2s. Ingram and Trotter's EUCLID ; containing the Elements of Plane Geometry and Trigonometry. Is. 6d. Ingram's Concise System of MATHEMATICS. Revised by Mr Trotter. 48. 6d. Trotter's LOGARITHMS and PRACTICAL MATHEMATICS. Ss, Ingram and Trotter's Elements of ALGEBRA. 38. EmNBURGH : OLIVER AND BOTD, TWERDDALK COURT. ADVERTISEMENT. This work is designed for those Pupils who have thoroughly mastered the Simple and Compound Kules ; and it has been tlie Author's aim to adopt as simple language as possible in the explanatory remarks. Each subject is accompanied by an example fully worked out and minutely explained, and has been treated as amply and carefully as its importance demanded. The Exercises, which are all new, are numerous and practical ; and Answers to them are published in a separate form. CONTENTS. Page Tables of Money, Weights, and Measures, 3 Greatest Common Measure, 5 Least Common Multiple, ib. Vulgar Fractions, 6 Miscellaneous Exercises in Vulgar Fractions, 11 Ratios and Proportion, 13 Simple Proportion, ' 14 Compound Proportion, 21 Practice, 25 Miscellaneous Exercises, 28 Decimal Fractions, 30 Interminate Decimals, c{4 Miscellaneous Exercises on Decimals, 38 Commercial Allowances, 39 Commission and Brokerage, 40 Simple Interest, 42 Discount, 47 Insurance, 49 Stocks, 61 Equation of Payments, 53 Distributive Proportion, . 64 Simple Fellowship, ib. Compound Fellowship, 66 Profit and Loss, 57 Exchange, 60 Duodecimals, 62 Involution, .63 Evolution, 64 Extraction of the Square Root, . •. . ib. Cube Root, . . V 66 Compound Interest, ,...,.,,.. 68 Miscellaneous Questions, 70 Decimal Coinage, 73 TABLES OF MONEY, WEIGHTS, AND MEASUEES. MONEY. qrs. d. 4 1 8. 48 12 1 £ 960 240 20 1 TROY WEIGHT. Grs. Dwt. 24 1 Oz. 1 480 20 1 1 1 Lb. 5760 240 1 12 1 1 Gold, silver, and jewels are weighed by Troy Weight. APOTHECAR lES' WEIG] HT. Gr. Scr. 20 1 Dr. 60 3 1 Oz. 1 480 24 8 1 1 Lb. 5760 288 96 12 1 1 Used only for medical prescriptions. AVOIRDUPOIS WEIGHT. Dr. 1 Oz. 161 1 Lb. 2561 16 1 Qr. 71681 448 28 1 Cwt. 28672 1 1792 1121 4 1 To. 573440 1 35840 22401 80 20 1 LINEAL MEASURE. 7000 grs. = 1 lb. avoir. ; 14 lb. = 1 stone. This table is used for all articles, except gold, silver, and jewels. In. 1 Ft. 12 1 Yd. 36 3 1 Pol. 198 1 16^ 5J1 1 Fur.I 7920 660 220 1 40 1 1 Ml. 63360 5280 1760 1 320 8 1 1 4 inches = a hand ; 6 feet, or 2 yds. = a fathom; 3 miles = a league. CLOTH MEASURE. In. 1 NL 1 2i 1 1 Qr. 1 9 4 1 1 Yd. 36 1 16 4 1 1 3 qrs. = 1 Flemish ell : 5 qrs. = 1 English ell ; 6 qrs. = 1 French ell; 4 qrs. 1 inch, or 37 in. = 1 Scotch ell. SQUARE OR LAND MEASURE. Sq. in. Sq. ft. 144 1 Sq.yd.l 1296 9 1 1 Per.| 39204 272i 30i| 1 Ro 1 1568160 1 10890 | 1210 1 40 1 jAc. 6272640 143560 1 4840 1 160 1 4 M 36 sq. yds. = a rood of building, and 100 square feet = a square of flooring. Land is measured by a chain 66 feet in length, divided into 100 links, eacli = 792 inches. 10,000 square links = 1 square chain, and 100,000 sq. links, or 10 square chains, = 1 acre. I I 1 I I 4 TABLES OF MONEY, WEIGHTS, AND MEASURES. CUBIC OR SOLID MEASURE. ANGULAR MEASURE. 1728 cubic inches = 1 cubic foot, and 27 cubic feet = 1 cubic yard ; 40 cubic feet of rough, or 50 cubic feet of hewn timber, = a load ; 42 cubic feet = a ton of shipping ; 5 cubic feet = H barrel bulk. MEASURE OF CAPACITY. Pts. Qt. 1 - 2 1 Gal. ^ 8 4 1 Pk. 16 8 2 1 Ba. 1 64 32 8 4 1 1 Qr. 512 1 256 1 64 32 8 1 1 3600J 60 |_ l_|_Circ^ 1296000 I 21600 j 360 | 1 TIME. Min. Ho. 1 60 1 1 Da. 1440 24 1 1 Co. Ye. 1 525600 8760 1 365 1 1 60 sees. = 1 min. ; 7 da. = 1 wk. ; 4 wks. = 1 CO. mo. ; 52 wks. and 1 da. = 1 CO. ye.; 365J da. = 1 Julian ye.; 366 da. = 1 leap ye. ; 365 da. 5 ho. 48 m. 50 sec. = 1 solar or tropical year. FLOUR & BREAD WEIGHT. A peck-loaf = 17 lb. 6 oz. avoird. A half-peck do. = 8 11 — A quarter-loaf =4 5^ — A peck of flour is 14-44 lb., or 14J lb. nearly, and a bushel 57| lb. very nearly. Five bushels =: a sack, which ought to weigh 288-8 lb. avoirdupois. HAY AND STRAW WEIGHT. 36 lb. avoir. = 1 truss of straw 56 lb. = 1 truss of old hay 60 lb. = 1 truss of new hay 36 trusses = 1 load Hay sold between the beginning of June and the end of August, of that year's growth, is reckoned new. QUARTERLY TERMS. In England. Lady-day, . March 25. Midsummer, . June 24. Michaelmas, . September 29. Christmas, . December 25. In Scotland. Candlemas, . February 2. Whitsunday, . May 15. Lammas, . August 1. Martinmas, . November 11, MISCELLANEOUS TABLE. 24 sheets 20 quires 10 reams 12 articles 20 articles 12 dozen 12 gross 120 articles 500 bricks 1000 tiles : 1 quire of paper : 1 ream : 1 bale : 1 dozen : 1 score ; 1 gross : 1 great gross 1 great hundred 1 load 1 load 500 herrings : 500 red do. : 1000 sprats 60 herrings ; 100 lb. avoir. : 56 lb. 64 1b. 256 lb. 112 lb. 19J cwt. : 1 barrel : 1 cade : 1 cade :lkeg : 1 barrel gunpowder : 1 firkin of butter : 1 firkin of soap = 1 barrel of soap = 1 barrel of raisins = 1 fodder of lead AKITHMETIC THE GREATEST COMMON MEASURE. The greatest common measure or divisor of two or more numbers is the greatest number which divides them without any remainder. Ex. Find the G. C. M. of 201 and 469. Ans. 67. Solution. Divide the greater number 201) 469 ( 2 (469) by the less (201), and the last divi- 402 sor (201) by the remainder (67) continu- ~67) 201 ( 3 ally imtil there is no remainder; the last 201 divisor (67) is the greatest common meas- ure of the two numbers. The G. C. M. of three numbers is obtained by finding that of two of them, and afterwards that of the result and the third number. Find the G. C. M. of, 1. 126 & 777 2. 584, 803 3. 2449, 2573 5727 & 7802 5824, 13376 1557, 2249 7. 16531, 31659 a 3247, 4393 9. 42039, 23701 THE LEAST COMMON MULTIPLE. The least common multiple of several numbers is the least number which contains each of them an exact num- ber of times. Ex. Find the least common multiple of 4, 6, 10, 18, and 30. Sol. Arrange the numbers after each other in one line ; divide by 2 as often as any of the numbers will divide by 2, then by 3 in the same way, again by 5, and so on by all the prime num- bers ; the continued product of all the divisors (2X2X3X3X5) is the least common multiple of the numbers. Find the L. C. M. of, 1. 7, 12, 14, 15, 24 I 6. 27, 32, 36, 72, 108, 144 2. 8, 16, 20, 24, 36 | 6. 12, 15, 32, 60, 64, 120 3. 4, 10, 14, 21, 28 I 7. 8, 11, 104, 52, 88, 143 4. 8, 16, 14, 10, 35 ' 8. 11, 26, 34, 52, 68, 187 Ans. 180. 2 4, 6, 10, 18, 30 2 2, 3, 5, 9, 15 3 1, 3, 5, 9, 15 3 1, 1, 5, 3, 5 5 1, 1, 5, 1, o 1, 1, 1, 1, 1 6 VULGAR FRACTIONS. A FRACTION consists of one or more parts of unity, and is expressed by two numbers, the one placed above the other with a line between them ; thus, ^- ^^^^^'^'^l' ' ' 9 Denommator. The lower number is called the denominator, and shows into how many equal parts the unit is divided ; the upper number is called the numerator, and shows how many of those equal parts have been taken to make up the fraction — the two together are called tlie terms of the fraction. A fraction also indicates an unperformed division ; thufc ~ signifies 14 divided by 9. A proper fraction is one whose numerator is less than its denominator, as |, |, -j^. An improper fraction is one whose numerator is equal to or greater than its denominator, as ^, |, Y. A mixed number consists of a whole number (or inte- ger) and a fraction, as 3y\, 39^. A compound fraction is a fraction of a fraction, as f off, i of If of J. A complex fraction is one which has a fraction for its numerator or denominator, or both, as 3, =1 =• ^ r 9 4^ An integer has one for its denominator, as 12 = K*. A fraction is multiplied by multiplying its numerator or by dividing its denominator, and is divided by dividing its numerator or multiplying its denominator. The value of a fraction is not altered by multiplying or dividing its terms by the same number. REDUCTION OF VULGAR FRACTIONS. Case I. To reduce a fraction to its lowest terms. Ex. Reduce |^| to its lowest terms. Ans. -Jf. Sol. 1. Divide the terms of the fraction I 144^ __ 48-^4 _ 12 (144 and ]56) by those numbers which I 156-^3 62^4 13 measure them exactly (3 and 4), until no number can bo found that does so, the fraction is then reduced to its lowest terms (||). Sol. 2. Find the G. C. M. (12) of the terms of the frac- tion, and divide them by it for the lowest terms {\ |) of the „ , 111 • 12 12 fraction. ^^ '. .^ = 75 as before. VULGAR FRACTIONS. Reduce to their lowest terms, Case II. To reduce a mixed number to an improper fraction. Ex. Reduce 5y\ to an improper fraction. Ans. -f j. Sol. Multiply the integer (5) by I 5 e =5X11+6 __.55-H_ 61 the denominator (1 1), and to the I ' ' n n u product (55) add the numerator (6), then under the sum (61) place the denominator (11) for the fraction (f {). Reduce to improper fractions, 4f; 6f ; 9^; 12^?; 15^^; 17^^; 25^^; 33ii; 45^V; 57A; 113/^; 237^VV; 69^^^; 147 VA; 178^Vt; 273V^Vt. Case III. To reduce an improper fraction to a whole »r mixed number. Ex. Reduce 4V *o ^ mixed number. Ans. 13y\. Sol. Divide the numerator (157) I ,5, .^„ . .« .« , bythedenominator (12),andtothe I ^5 — 10/ — 1^ — 10^2 quotient (13) annex the remainder (1) with the denomina- tor below it (j'5) for the fraction (13 j',). Reduce to whole or mixed numbers. Case IV. To reduce a compound fraction to a simple one. Ex. Reduce J of 1 J of $ to a simple fraction. Ans. |. Sol. 1. Multiply all the I , ^f 1^ of * — «of « of ^— «- —^ numerators together and I ^ ^^ ^^^^ 5 — 4 ot gOi g — ^^g—^ all the denominators together, and reduce the resulting frac- tion (yVs) to its lowest terms (|). Sol. 2. Strike out all those factors which j 5 4 5 are common to the numerators and deno- J ^^ ^ ^^ o ^^ 9 minators, and proceed with the numbers I ^ ^ " that are left, as in Sol. 1, for the fraction in its lowest terms. Note. Mixed numbers must be reduced to improper fractiona before multiplying. a2 8 VULGAR FRACTIONS. Reduce to simple fractions, 5 of 4 of A; T\of|f ofj; f of I of 15; A of 3^^ of lof^V; Aof7f ofl3i; ^Vofl2iof3i; vVof5|ofl3i; ^\ of 12^ of A ; if of 6| of 3/^ ; 4 of 8 J of 5. Case V. To reduce fractions having different deno- minators to others of equal value having the least com- mon denominator. Ex. Reduce f of ^, f, i, 1^ to their least common denominator. Sol I ^of« - - 1«-^ 3 4 ,,_1><126,3X63 4X28 12X36 ■o J 1^^^-" 4 10' -^7—5^4^ 5' 7 —2X126' 4X63' 9X28 7X36 Keduce the compound to simple and the mixed numbers to impro- per fractions ; find the L. C. M. (252) of the denominators for the required one, and divide it by each of the denominators ; then multiply the quotients ^126, 63, 28, and 36) by the respective numerators (1, 3, 4, 12) for the required numerators. The fractions must all be in their lowest terms before pro- ceeding as directed. Reduce to their least common denominator, *T5 TT ^^ 2TJ "^T) TIT 01 7f5? T2? TT> S 01 -ff-j 8 4 17 19 111 aT> T3'? 77? 7F' T^"S" 13 16 6 7 '7 •j-g-j ^U) -s^ty? TT? T's^i" 126 189 112 432 252' 252' 252' 252 *■' Si ?' -g^' T3' TTT 94 2rwf3 11 13 ^* TT' TT 01 T¥? 2T? 7 ^^ ^B" 4 13 17/^f6 3 19 nf 69 17 ! 10. Case VI. To reauce tractions irom one denomination to another without altering their value. Ex. 1. Reduce £f to the fraction of a penny, i. e. from a higher to a lower name. Ans. ^-^d. Sol. Multiply the numera- I ^5_ 5x240 ^ =— d =— d forby the number of the lower I 9 9 * 9 ' 3 * name contained in the higher (240), and reduce the fraction to its lowest terms. Ex. 2. Reduce ^ lb. to the fraction of a cwt., i, e. from a lower to a higher name. Ans. -^^-^ cwt. Sol. Multiply the denomi- I iii f _^j. _Lpwf nator by the number of the I 9 ^l^- — 9xii2^'^^-— 252^^^' lower name contained in the higher (112.) VULGAR FRACTIONS. 9 Ex. 3. Reduce £ J to the fraction of a guinea. Ans. f gu. ., -3 3X20 3X20 5 feOL. £4 = 5 S. = ^^ gu. = ^ gu. 1. Red. I qr., fd., y\s., |cr., & 14s. 7|d. to fractions of a pound 2. >f £/y, |d., ^|s., fgu., &/i of3s.3d. // n a farthing 3. ff I lb., /y to., j\ oz., y-V (l^M & 1 cwt. 21 lb. »/ 3 cwt. 4. n ? ml., I fu., j\ yd., I ft., & 4 ft. 7 ^ in. >- >^ sl pole 5. ff £|, for., 13s. 4d., ^of 7s,6d.,& fofSh.cr. " aguinea 6. If y*g Co, ye., { I mi., f sec, & 5 h. 48 m. 50 s. >• a day 7. ». fE.E.,i?FrE.,|Fl.E.,fSc.E.,&3(ir.3nl.'/ a yard Case VIL To find the value of a fraction in units of lower names. Ex. 1. Findthevalueof f ml. Ans. 4fu.17p.4yd.10in. Here, 5 ml. fu. po. &c. -^ 9 = 4 fu. 17 po. 4 yd. 10 in. Sol. Divide the numerator, as so many of the given name, by the denomuiator, as in Compound Division. Ex. 2. Find the value of f of 7s. 6d, Ans. 5s. Sol. 7s. 6d. X 2 = 15s. and 15s. -h 3 = 5s. Find the values of, 1- ^tV fs., |d., A cr., ^%gii., and | of 16s. 8d. 2. ^^ cwt., if qr., f lb., I ml., i^ yd., f of 2 ml. 50 yds. 3. *E.E., .{-^Fl.E., fac, fro., Js.per., -}f of 4ac. 3ro. 4. Txto.jAac, T-«y0f3qr. 3b., |ml., /-j- lb.tr., ^-^of 18s.9id. 5. » of 5s.6id., I of t\ of 3f gu., if co. ye., -,\ Jul. ye. 6. Joff ofS^cr., jf of|iofl8jbu., fofaS.ye,, f of2i|of4j-'' ADDITION OF VULGAR FRACTIONS. Ex.1. 5 + |of4|+A=| + | + ,^ = ^iy^±^ Sol. Reduce the fractions to their least common deno- minator by Case V., then add the numerators, and below the sum (509) place the denominator (315). Ex. 2. £| + /;^s. + *d. = 7s. 6d. + 2id. + Jd. J = 7s.8fd.|. Sol. Find the values of the fractions by Case VII., and add as in Compound Addition. 10 3.3i VULGAR FRACTIONS. 9 T^ T^ _L 6 I T T6 T^ T2 5. 4i + 5 6.7^ "6 13. 7 _i_ 6 T-^ nr -^^ 8-3f + 74 + 6f + 9f 10. 6^of/^+l|+^3^ofl 11. l«ofi|+fof/^+lU 12. 2^of4| + ^\of|+|of| 13. £f + |s. + |d. 15. 16. 17. 18. ^ cwt. + f qr. + 41 lb. ^\ton+^\cwt.+ T%lb. T^oac+TT^o. + Sfper. 19. £f + * gu. + f cr. + 1 of 8s. lOid. 20. ^3^ qr. + f bu. + -j^ pk. + ^^ of 9 qr. 4 bu. 21- tV ye. + ^\ da. + ^V ho. + ^^V of 85 da. 1 ho. SUBTRACTION OF VULGAR FRACTIONS. Ex. 1. I of 2 - - = |-i| = ^«A = ^7^ Sol. Prepare the fractions as in Addition, and subtract the numerators. ■t 3 1 ^' T TT 3. 14f— 6i 6. 6- 6.15- T -7A 8.42 — 2441 13. 8f oflf-^AoflyV 14. 4-,VofH — 2iof I 9. 10^-2^ 10. f of4^— lofJI 11. ^«^ofH-T\of|| 12. 7iof4f-6iofA 15. 134of H — ^ofl^ 55 Sol. Find the values of the fractions by Case VII., and subtract them as in Compound Subtraction. 17. 18. 19. 20. -4qr. Tfgu- TuCWt, ^\ lb. — I oz. tr. 21. A ml. — 17^\ po. ^E. E. i§Fr.E.-^Vyd. .j^ ac. — t of 2 ro. 25. TVofigye. — 4of^iye. MULTIPLICATION OF VULGAR FRACTIONS. Ex. T\XilfX4i = H?^g = |; orvVXi|fX4^ = |X|Xi = f. Sol. 1. Multiply all the numerators (4 X 165x41) to- gether for the numerator (27060), and all the denominators VULGAR FRACTIONS. 11 (11 X 164 X 25) together for the denominator (45100) ; then reduce the fraction (||i|§) to its lowest terms (|). Sol. 2. Strike out all the factors that are common to the numerators and denominators, and proceed with those num- bers that are left as in Sol. 1. 7. 54 X I of 6| 8. 98 X I of ^\ 9-TVof^\X T\of6| 10. 8^of 14f X 7§of6vV ii-(12i + 6|)X(4i-2f) 12. (21 3_14^V)X (6^3^+71) } ll^d., Is. 9|d., & 2s. 6|d. p. yd. 14. // 174ilb.&212^Vlb.@10id.,ls.7TVd.,&3s.4id.p.lb. 1. f X f X 1 2, t\ X 1 X f 3. 4- X 6 "8 X 7 1X5 4. H X A X 4 5. 27 X 2 of 6 6. 37 X tS of 3 3 T6 L3. Val H yd .&16f yd DIVISION OF VULGAK FRACTIONS. T?Y -♦ nf 5 -1- 1 3 1 O _L. 1 3 1 O V 1 4 2 O shx. T or ^ -r- x:f — ¥t ^^ t^? — 21 X tt — -g-^- SoL. Invert the divisor (if), and proceed as in Multiplication. of 4 T6 . 25-^ 37-^ off of^l 7. 42 -^- 8.56^ Vofil 9. ■ '^ " 10. nf 2 7 of if 63 s of '— 7 01 23- 2T\of|4 ii-(6f + 7i)^(7i-2|) I2.(8t-4i)-f-(4j + 2|) l6|,21f,24A,26xV,&32H 14. G72cwt. 3qr. 14|lb.-Mlf , 12|, 15|, 18/^, 23/,, & 27^^ 13. £375, 6s. lOJ^d. ■ 14#, MISCELLANEOUS EXERCISES IN VULGAK FRACTIONS. 1. Find the sum, difference, product, and quotient of ^\of6f andf of|f 2. A can do a piece of work in 8 days which B can do in 9 days ; what part will they do together in 1 day ? 3. What number added to 4 of 5^ gives 14|? 4. What part of 3f is | of f ? 5. What number is that ^ of which is equal to 30 ? 6. A and B can do a piece of work in 8 days which A alone can do in 12 days ; what part of it can each do alone in one day ? 12 VULGAR FRACTIONS. 7. A gentleman having ^ of a ship, worth £3115, pur- chases another person's share which is ^ of f of it ; what part has he now, and what is its value ? 8. Wliat number divided by ^\ of 7^ gives 240 ? 9. How many chests of tea, each containing 124f lbs., can be filled from 73 cwt. 2 qrs. 1^ lb. ? 10. What number is that j*-,- of which is equal to 25 ? 11. What part of 5 guineas is f of £3 ? 12. A farmer went to market with £2| ; he received there £73^\, £89^\, and £49/^: with what sum did he return ? 13. What number multiplied by J of 7| gives | of 15f ? 14. Divide £4125 among 4 men, 6 women, and 12 chil- dren, giving each woman | of a man's share, and each child -f of a woman's share. 15. Two persons, by trading, gained a certain sum, the first lodged f of the capital, and received £200 as his share of the gain ; what was the whole gain, and the second's share ? 16. What numl taken there remains 35 ? 17. Two places are 72 miles distant from each other : A starts from the one at the rate of 12f mis. an hour, and at the same time B starts from the other, to meet A, at the rate of 18j miles in 2 hours ; when and where will they meet ? 18. A gentleman's income is j\ of || of £7560, and he spends | of f of it ; how much does he lay up ? 19. A person spends ^ of his money + £2, and has left :^ of it -f- £3 ; what sum had he at first ? 20. What number is that from which ^ of it being taken there remains 40 ? ' 21. I of the trees in an orchard are pear trees, Jf are apple trees, and there are 50 cherry trees ; what is the number of trees ? 22. A man's present age is 65 years, 5 years since his son's age was f of his ; what is the son's present age ? 23. A cistern can be filled by two pipes in 24 and 25 minutes respectively, and can be emptied by a third in 32 min. ; what part of it will be filled in 12 min., the three pipes being all open ? VULGAR FRACTIONS. 13 24. A person has f of a ship worth £4200, and he sells ^ of his share ; what part has he left, and what is its value ? 25. A and B can do a piece of work in 6 days, A and C the same in 8 days, and B and C in 12 days ; what part could the three together perform in 5 days ? 26. A ship and its cargo are together worth £23750, and the cargo is 5^ times more valuable than the ship ; find the value of each. 27. Simplify (14I + 6VV — 2^V) X ^f-S^V 28. A father left ^% of his estate to one son, and the remainder to another ; the difference of their fortunes was £750 : what was the estate worth ? 29. Divide £2000 among A, B, and C, giving A | of the whole, B | of A's share, and C the rest ; find also what fraction of the whole C receives. 30. What number multiplied by } of J, and the pro- duct divided by ^-^j^ of 5i, will give for the quotient -j't of 64 of 4? RATIOS AND PROPORTION. In comparing two numbers of the same kind, their ratio or relation to one another is found by dividing the first ' by the second ; thus, the ratio of 4 to 2, generally writ- ten 4 : 2, is 4-^2 = 2; of 3 mis. to 6 mis. is 3-^-6 = 1. The first number is called the antecedent, and the second the consequent ; the two together are called the terms of the ratio. Proportion consists in the equality of ratios; thus, since 4:2 = 8:4, the numbers 4, 2, 8, and 4, consti- tute a proportion : they are generally written 4 : 2 : : 8 : 4, and are read as 4 is to 2 so is 8 to 4. In every proportion the product of the 1st and 4th terms (or of the extremes as they are called) is equal to the product of the 2d and 3d terms (or of the means) ; thus in the proportion 4 : 2 : : 8 : 4 we have 4 X 4=2 X 8. Hence the first three terms of a proportion being given, the 4th is found by dividing the product of the 2d and 3d terms by the 1st. £144 14 SIMPLE PROPORTION or the RULE of THREE. When three terms of a proportion are given, the object of this rule is to find its 4th or last term. Of the three given numbers, two are always of the same kind, and the remaining one is of the same kind as that which is required. Ex. 1. If 16 men earn £32 in a week; what sum will 72 men earn in the same time ? Ans. £144. Sol. 1. Place that term which is of Men 16 : 72 : : £32 the same kind as the answer is to be, for the third or right-hand term (£32). 2. Consider from the nature of the question whether the answer is to be greater or less than the term written down : if greater (as in this Ex.), place the greater of the two remaining terms (72) in the middle, and the other on the left (16) ; but if less, place the less of the two like terms in the middle, and the other upon the left. 3. When none of the terms is compound (as in this Ex.), multiply the 2d and 3d terms together (72 X 32), and divide the product (2304) by the 1st or left hand- term (16) for the answer, in the same name as the 3d or right-hand term (£'s). 1. If 25 yds. of velvet cost £30; what should 760 yds. of the same cost ? 2. If 14 cwt. of sugar cost £42 ; what should be paid for 207 cwt. ? 3. A train runs at the rate of 73 mis. in 3 hours; in how many hours will it run 438 mis. ? 4. A person spends £500 yearly ; how much will he spend in 146 days ? 5. If 7 men do a piece of work in 36 days ; in how many days will 9 men do the same ? 6. What cost 162 copies of a book, when 171 copies cost £19? Ex. 2. If 6 cwt. 3 qrs. of tea cost £170, 2s. ; what should 27 cwt. 3 qrs. of the same cost ? Ans. £699, 6s. Sol. State the question as before. Reduce the 3d term to the lowest name in it (shil.), and the 1st and 2d terms to the lowest name in either (qrs.) ; then multiply the 2d and 3d SIMPLE PROPORTION. 15 6 cwt. 3 qr. : 27 cwt. 3 qr. : : £170, 2s. _4 _4 __20 27qrs. Ill qrs. 3402s. 3402 27 )377622 2,0 )1398,6 s. £699, 6s. terms togethef (111 X 3402), and di- vide the product (377622) by the 1st term (27) for the answer, in that name to which the 3d term was re- duced (shil.). 7. What cost 4 yds. 3 qrs. 2nls. of cloth, when 15 yds. 2 qrs. cost £8, 3s. 4|d. ? 8. If 13 cwt. 14 lbs. of coffee cost £131, 13s. 9d. ; how much may be bought for £13, 3s. 4Jd. ? 9. If a person walks 14 mis. 2fu. 28 po. in 4 ho. 6min. 40 sec. ; how far will he walk in 9 ho. 3 min. 20 sec. ? 10. How many yds. of linen at 3s. 6d. a-yd. should b( given for 136 yds. of muslin at 2s. 7^d. a-yd. ? 11. If 4 yds. of cloth cost 84s. 4d. ; what will 27 yds. 2 qrs. cost? 12. Find the value of 2 qrs. 3 pks. of wheat, when 36 qrs. 2 bu. 2 pks. cost £76, 5s. lid. Ex. 3. If 14 lbs. of tobacco cost 73s. 6d. ; what cost 10 lbs. of the same ? Ans. £2, 12s. 6d. Obs. When the first and ^;4 lb. : ^0 lb. : : 73s. 6d. either of the other terms can i ^ - ^.^ . be divided without remain- j P ^ ^9^a. der by the same number, the j 126 quotients may be used in 5 place ofthe original numbers. | £2, 12s. 6d. = 630d. 13. A courier travels 176 miles in 4 days; how far will he travel in 15 days? 14. How much sugar may be bought for £95, at the rate of 28 lbs. for 14s. 3d. ? 15. A man's wages are £37, 2s. 6d. a -year ; what should he receive for 219 days' service? 16. If the 8d. loaf weigh 4 lbs. 5i oz. ; what should the shilling loaf weigh ? 17. How many pairs of stockings, at 14s. 6d. per doz, pairs, may be bought for £30, 19s. 10 Jd. ? 18. Find the value of 1 cwt. of sugar, when 3 cwt. 14 lbs. cost £10, 18s. 9d. b2 16 SIMPLE PROPORTION. 19. What cost 5 pieces of silver, each 3 lbs. 4 cz. 12 dwt., at 5s. 9d. per oz. ? 20. If the quartern loaf costs lOjd. when wheat is at £3, 10s. per qr. ; what should it cost when wheat is at £2, 3s. 4d. per qr. ? 21. A bankrupt's effects an;iount to £3528, and he compounds with his creditors for 12s. 3d. per £1 ; what is the amount of his debts ? 22. If 16 men consume £10 worth of beef when the price is 7-^d. per lb. ; what value of beef will they consume in the same time when the price is lO^d. per lb. ?* 23. Sound moves at the rate of 1142 ft. in a second, and the report of a gun is heard 14| sec. after seeing the flash ; how far distant is the gun ? 24. How many paces of a man, each 2i ft., are equal to 150 steps of a horse, each 2f ft. ? 25. A bankrupt's debts amount to £7428, and he com- pounds with his creditors for 10s. 9|d. per £1 ; find the amount of his effects. 26. Find the value of 8 cheeses, each 26 J lbs., at T^d. per lb. 27. At what time between 6 and 7 o'clock are the hour and minute hands of a watch exactly together? Sol. (11 : 12 : : 6 hours.) 28. Bought 17 yds. 2 qrs. of cloth for £16, 2s. 3id. ; what should 4 yds. 3 qrs. be sold for to gain £2, 5s. 2|d. on the whole ? 29. If 36 gallons of whisky, worth 17s. 6d. a-gallon, be mixed with 4 gallons of water ; what should be the price of a gallon of the mixture ? 30. A person pays £65, 6d. for income-tax, at the rate of Is. 4d. per £1 ; what is his income? 31. Required the circumference of a circle whose di- ameter is 22035 mis., the ratio of the diameter to the circumference of a circle being as 113 is to 355. Sol. (113 : 355 : : 22035 mis.) 32. A pound troy of standard gold is coined into £46, 14s. 6d. ; find the weight of a sovereign. * When the same term is twice mentioned in a question, that term must he altogether excluded. SIMPLE PROPORTION. 17 33. The ratio of standard to pure gold being 22 to 24; what is the value of an ounce of pure gold ? 34. A garrison of 3300 men have provisions for 12 months; how long would the same provisions serve 4950 men ? 35. A 16 gun-battery discharges 1760 cwt. of shot in a certain time ; how much will an 18 gun-battery discharge in the same time ? 36. The chain of QQ ft. for measuring land is divided into 100 links ; what is the length of a wall measuring 1760 links? 37. What is the commission on £477, 2s. 6d., at £27^ per £100? 38. A pound troy of standard silver is coined into 66s. ; find the weight of half-a- crown, and of a florin. 39. The ratio of standard to pure silver is 37 to 40 ; what is the value of a lb. of pure silver? 40. From a garrison of 2000 men with provisions for 9 months, 500 are sent out ; how long will the provi- sions serve the remaining men ? 41. If 250 men dig a trench in 5 da. 5 ho., working 12 hours a-day ; in how many days would they do tlie same, working 1 1 hours a-day ? 42. If 49 men do a piece of work in 3f days ; in how many days will 48 men do the same ? 43. A garrison being besieged, has 49 days' provi- sions, at the rate of 15 oz. a-day for each man ; how long will they be able to hold out if each receives lOJ oz. a-day ? 44. Required the charge for 12125 cubic feet of gas- at 5s. lOd. per 1000 cubic feet. 45. The rent of a farm of 350 ac. 3 ro. 20 per. is £1710, 10s. 3|d. ; what should be the rent of another of equal quality, containing 525 ac. 1 ro. 20 per. ? 46. If £14, 8s. be the interest on £360 for a year ; wliat sum will gain £33, 12s. in the same time and at the same rate per cent. ? 47. A garrison of 2500 men, with provisions for 7 months at the rate of 21 oz. a-day for each, is reinforced by 1000 men ; how many ounces a-day must each be allowed that the provisions may last that time ? and if c 18 SIMPLE PROPORTION. each receives the full allowance, hov/ long will the pro- visions serve? 48. Find the value of 5 bars of steel, each weighing 4 cwt. 3 qrs. 14 lbs., at £12, 14s. 4d. for 10 cwt. 3 qrs. 21 lbs. 49. In what time would 6 battalions of foot, each 375 ft. in length, march through a town If mile long, at the rate of 75 paces of 2| ft. per minute? 50. A piece of work can be done by 45 men in 13 days ; now at the end of 6 days, 10 men leave : in how many days will the remaining men finish the work ? 51. What is the price of 6f Fr. ells, at £66, lis. for 72|Eng. ells? 52. What is the price of 7 pieces of cloth, each 16§ yds., at £3, 4s. 9d. for 3|- Scotch ells? 53. A bankrupt's debts amount to £4020, and his assets to £3266, 5s. ; how much will this afford his creditors per £, and how much will A lose, whose claim is £560, 13s. ? 54. A gentleman's income is £3867, 15s. per annum; his expenses amount to £1050, and he wishes to save £500 : how much may he spend between Whitsunday and Martinmas ? 55. How many yards at 4s. l^d. are equal in value to 1231 yards at 12s. 7id. per yd. ? 56. If 4 to. 5 cwt. 14 lbs. of lead cost £50, 17s. 6d. ; what should be given for 20 to. 11 cwt. 49 lbs. of the same ? 57. A cubic foot of chalk weighs 2784 oz., and a cubic foot of basalt 2860 oz. ; how many cubic feet of the former are equal in weight to 7830 cubic feet of the latter? 58. A column of chalk weighs 20 cwt. 2 qrs. 24 lbs. ; re- quired the weight of a column of basalt of the same dimensions. 59. How much wheat can be bought for £101, 5s. when 7 qr. 4 bu. 3 pks. cost £20, 10s. 0|d. ? 60. What should be paid for 102 qrs. 3 bu. 2 pks. of oats, at the rate of £5, 4s. ll^d. for 4 qrs. 2 bu. 2 pks. ? 61. How much water must be mixed with 250 gallons of "whisky, at 14s. 6d. per gal. to reduce the price tc 12s. 6d. per gal.? SIMPLE PROPORTION. 19 62. How much water must be mixed with whisky at 15s. a-gal. to fill a cask of 360 gals., so that a gallon of the mixture may be worth 13s. 4d. ? 63. If the rent of 4J acres be £7, 13s. ; wliat will be the rent of 5^ acres ? 64. An express train runs 58 mis. in 1 h. 30 m. with two stoppages of 3 minutes each ; in what time will it run 435 miles with 5 stoppages of 4 minutes each ? 65. What quantity of linen at 2s. 6d. a-yard should be exchanged for 5 dozen pairs of shoes at lis. a-pair? 66. A and B barter, A has oats at 24s. per qr., which he rates at 27s. 6d. to B for sugar at 75s. per cwt. ; at Avhat should B rate his sugar to be even with A, and how many cwts. should he give for 175 qrs. of oats? 67. If 78 qrs. 5 bu. of barley be given for 53 qrs. 1 bu. of wheat at 64s. 9d. per qr. ; what is the barley valued at per qr. ? 68. A grocer mixes 56 lbs. of tea at 4s. per lb. with 44 lbs. at 5s. ; how should he sell 11 lbs. of the mixture to gain £5, Is. lOd. on the whole? 69. At what time after 2 o'clock are the hour and min- ute hands of a watch exactly together ? 70. Find the diameter of the earth, whose circumfer- ence is 24850 miles nearly. 71. B gives to C 12 gallons of brandy at 37s. 6d. per gal. and £14, 12s. 6d., and receives from him tea at 4s. 6d. per lb. and 7 cwt. 2 qrs. of sugar at £3, 5s. per cwt. ; what quantity of tea did B receive ? 72. Find the weight of 600000 sovereigns, 1869 sove- reigns weighing 40 lbs. troy. 73. Lent a friend £455 for 6 months ; how long should he lend me £630 to return the favour ? 74. If f cwt. cost £5i ; what should | cwt. cost ? 75. If ^\ of a ship be worth £420 ; what should | of it cost? 76. What velocity will a falling body have at the end of 7i sec, if it acquires a velocity of*168|- ft. in 5i sec? 77. A hare starts 140 yards before a greyhound, but while the hare runs 5 yds. the dog runs 7 ; how far must the do^r run to catch the hare ? 20 SIMPLE PROPORTION. 78. If -/y of an estate be worth £500 ; what is the vaUie of f of it? 79. If 30 horses be maintained for 5 montlis on a cer- tain value of oats when the price is 22s. 6d. per qr. ; how many horses may be fed for the same sum and time when oats are at 25s. per qr. ? 80. A person, after paying income-tax at Is. 4d. per £, has remaining £665 ; required his income. 81. How long would a cannon-ball with a velocity of 2000 ft. per second take in passing from the earth to the moon, a distance of 237630 miles ? 82. The distance of Jupiter from the sun is 494513000 mis. ; what is the length of its orbit, supposing it circular ? 83. The same planet performs its revolution round the sun in 4332-| days ; what is his mean motion in 365^: days? 84. Jf a tower 150 ft. 4 in. high cast a shadow of 181 ft. 1 in. : what length of a shadow wiU a pole 38 ft. 6 m. high cast at the same time ? 85. How many revolutions will a coach-wheel 3^ ft. ii? diameter make in 4 miles ? 86. The weight of an 18-pounder iron gun being 41 cwt. 2 qrs., and the weight of a 12-pounder 33 cwt. 2 qrs. ; how many 12-pounders will be equal in weight to 469 18-pounders? 87. What should be paid for 15 cwt. 1 qr. 14J lbs. of lead, when 14 cwt. 3 qrs. 16 lbs. cost £17, 7s. 6d. ? 88. A person whose annual income is £650, spends £15, 2s. 6d. a-week for the first 20 weeks ; what should his daily expenses be during the rest of the year, to save 89. Bought 7 pieces of cloth, each containing 61 yds., for £424, 6s. 7^d. ; what should 241 yds. of the same be sold for to gain £5, 6s. 9d. on the whole? 90. A can do a piece of work in 9 days which B can do in 12 days ; in how many days would they be able to finish the work, working together? 91. A cistern has two spouts, by one of which it can be filled in 3 months, and tfy the other it can be emp- tied in 8 months ; in what tune will it be full, supposing it empty and both spouts running? SIMPLE PROPORTION. 21 92. A can do a piece of work in 6 days, which B can do in 8 days ; after A has been working 2 days, B comes to help him ; in what time will they finish the work to- gether ? 93. A starts from a certain place at the rate of 5 miles an hour ; after 2 hours, B starts from the same place at the rate of 6^ miles an hour ; when will B overtake A, and how far will each have travelled ? 94. A grocer uses a weight of 15|oz. instead of the pound avoirdupois ; how much does he cheat his customers by selling 365 such pounds ? 96. A cistern, containing 399 gallons, is emptied in a certain time by a pipe which discharges 4| gals, per minute, and another is emptied in the same time by a pipe which discharges 7-§- gals, per minute ; how many gallons does the last cistern contain? 96. A wine merchant uses a measure containing 1224 cub. in. instead of 131^ c. in. ; of how many gallons, each 277 J c. in., does he defraud the public by selling 739 J such measures ? COMPOUND PROPORTION. When a question requires for its solution two or more statements of Simple Proportion, the method of finding the answer by one operation is called Compound Pro- portion. Ex. •!. If 30 men eat £9 worth of bread in 12 days, when the price of the loaf is 8d. ; what value will 64 men eat in 10 days, when the loaf is at 6d. ? Ans. £12. Sol. 1. Place upon the right hand that term which is of the same kind as the answer is to be (£9). 2. Take from the question two terms that are like one another (30 men and 64 men) and state til em, without any reference to the Men, 30 : 64 : : £9 Days, 12 : 10 Price, 8 : 6 2880 : 3840 : : £9 9 2880 )34560 £12 worth other similar terms, as in Simple Proportion ; in the same way, take other two similar terms (12 da. and 10 da.), and state them as in Simple Proportion below the last pair, and proceed thus till all the terms are stated. 22 COMPOUND PROPORTION. 3. Multiply all the left-hand terms together, and also the middle terms, then work out the resulting Proportion (2880 : 3840 : : £9) as in Simple Proportion for the an- swer (£12). When some of the terms are Compound, they must be re- duced as in Simple Proportion ; the work may be greatly abridged by cancelling. Ex. 2. If 14 persons spend £5, 5s. in 10 days ; how long will £42 serve 16 persons? Ans. 70 da. Sol. State the question as in Ex. 1, and reduce £5, OS. and £42 to sh. Arrange the mid- dle and the right- hand terms, with the sign of Multi- plication between £ 5, OS. £42 105s. : 840s. : : 10 da. Persons 16 : 14 8 7 $0XUX^O 1X7X10 ^0X10 70 da. them above a line, and the left-hand terms below it ; then cancel the upper and under numbers as much as possible, as in fractions, and divide the product of the remaining numbers above the line by the product of those below for the answer. 1. If 45 men cut down 120 acres of grass in 7 days ; how many acres will 84 men cut down in 10 days ? 2. If 300 soldiers consume 4 barrels of flour in 10 days; how many soldiers will 12 barrels serve for 15 days? 3. If 48 yds. of cloth, 4 quarters wide, cost £24, 12s. ; what should be paid for 36 yds. of the same, 6 quarters wide ? 4. What is the interest on £383, 5s. for 325 days at 4^ per cent, per annum ? 5. If 30 men consume £7 worth of bread in 10 days, when the price of the loaf is 8d. ; what value of bread will 40 men consume in 15 days, when the loaf is at 7id. ? 6. If 30 men can do a piece of work in 12 days of 10 hours each ; in how many days of 8 hours each will 45 men do a piece of work 6 times as large ? 7. If 63 cwt. be carried 42 mis. for £3, 10s., when tho rate of carriage is ^d. per mile per cwt. ; what distance should 142 cwt. be carried for £8, 17s. 6d., when the rate is Id. per mile per cwt. ? COMPOUND PROPORTION. 23 8. At 2^ per cent, per annum £375 was lent, and it now amounts to £431, 5s. ; how long has it been lent ? 9. The pound weight of standard gold is coined into £46, 14s. 6d. (22 carats in 24 being pure gold) ; what is the value of 3 ounces of pure gold ? 10. If 3 men or 5 boys do a piece of work in 8 days of 10 hours each ; in how many days of 9 hours each would 4 men and 10 boys do a piece of work 3 times as large ? 11. If 40 masons build a wall 56 yds. long in 10 days of lOj hours each ; how many hours a-day must 60 ma- sons work to build a wall 120 yds. long in 20 days? 12. If 120 men can dig a trench 150 yds. long, 4 yds. wide, and 2 deep, in 7^ days of 10 hours each ; what length of a trench, 5 yds. wide and 3 deep, will 200 men dig in 15 days of 12 hours each? 13. If 14 horses plough 112 acres in 40 days ; how many horses would plough 64 acres in 16 days ? 14. If 30 men earn £80, 14s. in 15 days ; how many men will earn £107, 12s. in 12 days ? 15. A traveller completes a journey of 240 miles in 3 days of 12^ hours ; in how many days will he complete a journey of 360 miles, travelling 9 hours a-day? 16. If the 8d. loaf weighs 3 lbs. 4 oz. when wheat is at 64s. per qr. ; what should the shilling loaf weigh when wheat is at 72s. per qr. ? 17. Required the avoirdupois weight of 600000000 sovereigns, there being 1869 sovereigns in 40 lbs. troy, and 7000 grains in a pound avoirdupois. 18. A certain value of bread is sufficient to serve 3200 men for 44 days when the loaf is at 9d., allowing each man 16 oz. a-day ; how many men will 7 times the value serve for 112 days, at 20 ounces each per day, when the loaf is at lid.? 19. If 135000 bricks, 8 in. long, 3^ in. broad, and 2f in. thick, be required to build the walls of a magazine ; how many bricks, 14 in. long, 4 in. broad^ and 3 in. thick, would be sufficient for the same ? 20. If 7 compositors set up a volun^e of 12 sheets in 21 days of 12 hours each ; how many would be required to set up 3 volumes of 10 sheets in 35 days of 9 hours each? 21 COMPOUND PROPORTION. 21. 35 masons build 48 yds. of a wall which is to be 192 yds. long in 12 days of 12 hours each ; how many additional masons will be required to finish the wall in 18 days of 10 hours each? 22. If 15 men build a wall, 80 ft. long, 3§ ft. thick, and 9 ft. high, in 27 days of 10 hours each ; in how many days of 12 hours each will 25 men build a wall 100 ft. long, 2 J ft. thick, and 8 ft. high? 23. A garrison of 4050 men, with provisions for 5 months at the rate of 32 oz. a-day for each, is reinforced by 750 men, and cannot be relieved for 8 months ; how many oz. a-day must each man be allowed that the pro- visions may last that time ? 24. The cost of papering a room with paper 3 qrs. wide, at 3f d. a-yard, is £2, 3s. 9d. ; what would be the cost if the paper were 1^ yd. wide, and the price 4|d. a-yard? 25. A block of marble, 5 ft long, 4 wide, and 1 ft. 3 in. thick, weighs 39 cwt. 2 qrs. 8 lbs. 13 oz. ; what is the weight of another block, 8 ft. long, Ah wide, and 2 ft. 4 in. thick? 26. In what time will the interest of £750, r2s. 6d. be sufficient to pay a debt of £112, lis. 10^ d. at 4 per cent. per annum ? 27. Find the interest of £1418, Os. 6d. for 375 days, at 3^ per cent, per annum. 28. If the 6d. loaf weighs 3 lb. 4^ oz. when the wheat is at 56s. a-quarter ; what is the price of wheat per qr. when the 8d. loaf weighs 3 lb. 13| oz. ? 29. If 18 men working 9 hours a-day, or 36 boys work- ing 6 hours a-day, can do a piece of work in 8 days ; in how many days would 10 men and 24 boys do a piece of work 5 times as large, all working 8 hours a-day ? 30. A contractor engages to construct 3 J mis. of a road in 90 days, and for this purpose he employs 120 men, who work 8 hours a-day, but after 60 days, he finds they have only accomplished 2 mis. of the road ; how many additional men must he employ to finish the work in the appointed time, the men working 9 hours a-day ? J] 25 PRACTICE -^^^sss^ Is an expeditious method of finding the values of goods by means of aliquot parts. A less number is said to be an aliquot part of a greater when the less is contained an exact number of times in the greater ; thus 3 is an aliquot part of 24, which contains it exactly 8 times ; so also is 2 s. 3d. of 18s., which contains it exactly 8 times. TABLE OF ALIQUOT PARTS. lOs. = £i lid. = i^of 6d. = is. 6s. 8d.= £i. Id. = £^i^ = ^\s. 5s. — loflOs.— £i |d. =iof6d. = ^Vs. 4s. = £i 7 1 , 3s. 4d.— ^ of 10s. — £i id. = £^1^ = ^Vs- 2s.6d.= iofl0s. =£i AVOIRDUPOIS WEIGHT. 2s. —i of 10s. — £Vo 10cwt.= i ton ls.8d.= iof6s.8d.=£V^ 5 cwt. = J ton Is. 4d.= £^V 4 cwt. = \ ton Is. 3d.— J of 10s. — £Ve 2 cwt. = -^-Q ton Is. =- £^V 2 qr. = i. cwt 6d. = is. =£^^^ 161b. = 1 cwt 4d. = |s. 14 lb. = ^ qr. = ^ cwt 3d. = is. 71b. =^qr. = TVcwt 2d. = is. 41b. = 1 qr. This table should be extended by the pupil. Case I. When the price is an aliquot part of £1, Is., or Id. Ex. Find the price of 2744 yds. at 4d. and 3s. 4d. per yd. 4d. = is.) 2744s. = val. at Is. 2,0) 914s. 8d. val. at 4d. Va.at4d.=£45, 14s.8d. Sol. 1. Since 4d.=r^s. the value at 4d. = ^ of the va. at 1 s. ; now 2744 yd. at Is. =2744s., hence ^ of 2744s. = 914s. 8d., or £45, 14s. 8d. is the value at 4d. Sol. 2. Since 3s. 4d. =£^, the va. at 3s. 4d. = I of the va. 3s. 4d. = £i)£2744va. at £1 £457, 6s.8d. va. at3s.4d. at £1 ; now the va. at £1 is £2744, hence \ of £2744 = £457; 6s.8d. =va. at 3s.4d. Find the values of, 1. 7459 oz. at id., fd., Id., Ijd., 2d., 3d., 4d.,&6d.per oz. c2 26 PRACTICE. 2. 1786 yds. at Is., Is. 3d., ls.4d., Is. 8d., 2s., 2s. 6d., 3s. 4d., and 6s. 8d. per yd. 3. 3457 lbs. at 10s., 5s., 4s., 3s. 4d., Is. 8d., Is. 4d., 6d., and 4d. per lb. Case II. When the price is not an aliquot part of £1, Is., or Id. Ex. Find the value of 575 lbs. at 3s. 9d. per lb. 2s. 6d. =£i)£ 575 va. at £ 1 £71, 17s. 6d.va.at2s.6d. ls.3d.=:riof 2s.6 d. 35, 18s. 9d . >> Is. 3d. 3s. 9d. Sol. 3s.9d.= 2s.6d. + ls.3d.; now the va. at 2s. 6d. by Case I. is £71, 17s. 6d., £107, 16s. 3d. and since Is. 3d. = ^ of 2s. 6d., the va. at Is. 3d. = i of the va. at 2s. 6d. = ^ of £71, 17s. 6d., or £35, 18s. 9d. The sum of the values at 2s. 6d. and Is. 3d. = the value at 3s. 9d. Note. When there are £'s in the price, multiply the quantity by them, and take aliquot parts for the s. and d. Find the values of, 4. 375 at 4s. 4d., 4s. 8d., 5s.6d., 6s.3d., 7s. 4d., 8s. 4d., 10s. lOd., and 13s. 4d. each. 5. 692 at 12s. 4id., 13s. 9id., 14s. 4d., 15s. 7id., 18s. lOd., 19s. 5d., 19s. 7id., and 19s. lOd. 6. 1476 at Hd., 2id., 3f d., 4id., 6|d., 7id., 8id. & lOid. 7. 297 at £1, 14s. 8d., £1, 18s. lOd., £2, 5s. 7d., £3, 15s. lUd., £5,17s.l0d., £6,16s.6d., £7,lls.9d.&£9,18s.4id. 8. 379 at £2, 2s. 2d., £3, 3s. 3d., £7, 7s. 7d., £11, lis. Case III. When the price consists of £ and s. only. Ex. Find the value of 493 cwt. at 39s. per cwt. Sol. Multiply the quantity (493) by 493 half the number of shillings (19|) ; then 19 J double the right-hand figure of the pro- ^^7^ /?q/.i r. duct (3^) for shillings (7s.), and the ybid^_£Jbi, /s. rest C961) are £'s. Find the values of, 9. 1476 at 2s., 3s., 7s., 9s., lis., 13s., 17s., 19s., 29s., 43s., and 47s. 10. 1729 at 16s., 18s., 21s., 37s., 95s., £5, 5s., £5, lis., £16,4s., and£17, lis. PRACTICE. 27 Case IV. When the quantity contains a fraction. Ex. Find the value of 579| yds. at 21s. 3d. per yd. Sol. 1. Is. 3d. = £j\ ) £579 value at £1 36, 3s. 9d. va. at Is. 3d. I of 21s. 3d. = 15s. ll^d . Val. of 5791 yds. = £615, 19s. 8id. Sol. 2. 579| X 4 Is. 3d. = £j\ ) £2319 value at £1 144, 18s. 9d . va. at Is. 3d. 4 ) £ 2463, 18s. 9 d. va. of 4 X 579| £615, 19s. 8jd. va. as before. Find the values of, 11. 476| and 549| at 14s. 8d., 15s. 5d., 17s. 8d., & 19s. 6d. 12. 375^ and 742^ at 13s. 9d., 12s. 8d., 14s.lld., & 17s.6d. 13. 1146f&1763/Tat26s.4d.,37s.9d.,46s.7id.,&76s.4id. 14. 2986f and 4863,-V, at £4, 17s. 5d., £6, 14s. 7id., £7, 2s. 2id., and £10, lis. 9d. Case V. When the quantity is compound, and the price of one of the highest name is given. Ex. Find the price of 4 cwt. 3 qrs. 14 lbs. at £3, 16s. 7|d. per cwt. 2qr. = ^cwt.)£3 16 Sol. Find the va- lue of the quantity of the highest name given, then take ali- quot parts for the lower names, and add. Thus, 2 qr. = ^ cwt., va. of 2 qr. = i of £3, 16s. 7id. ; 1 qr. = ^ of 2 qr., va. of 1 qr. 15 6 1 18 lqr.=^of2qr. 19 141b.=iqr. 9 7^ va. of 1 cwt. 6 va. of 4 cwt. 3i '' 2qr. Ifi'- Iqr. 6||" 141b. £18 13 6JJva.of4cwt. 3 qr. 14 lb. ^of£1,18s. 8|d.; and 141b. ='iqr.,"va. of 14 lb. = | of 19s. Ifd.^. The sum of the separate values gives the whole value. Find the values of, 15. 3 cwt. 3 qrs. and 6 cwt. 2 qrs. 21 lbs. at £4, 5s. 4d. and £4, 7s. 6d. per cwt. 16. 7 cwt. 1 qr. 16 lbs. and 21 cwt. 2 qrs. 18 lbs. at £5 16s. 8d. and £6, 17s. 8d. per cwt. "« PRACTICE. 17. 12 lbs. 3 oz. 15 dwt. and 27 lbs. 5 oz. 6 dwt. at £3, 3s. 4d. and £5, 16s. 8d. per lb. 18. 17 yd. 3 qr. 2 nl. and 37 yd. 2 qr. 3 nl. at £1, 3s. 6d. and £1, 17s. 6d. per yd. 19. 27 ac. 3 ro. 12 per. and 47 ac. 2 ro. 25 per. at £4, lis. 8d. and £5, 15s. per acre. 20. 33 qr. 3 bu. 3 pk. and 67 qr. 1 bu. 2 pk. at £2, 2s. 4d. and £2, 7s. 4d. per qr. 21. 14 bu. 3 pk. 1 gal. and 17 bu. 2 pk. 1 gal. at £2, 4s. and £2, 12s. per qr. 22. 45 ac. 3 ro. 24 per. and 67 ac. 1 ro. 14 per. at £5, 13s. 4d. and £6, 8s. 4d. per ac. MISCELLANEOUS EXERCISES. 1. Find tlie price of 116 cwt. of sugar at £3, 14s. 8d. and £4, lis. 9d. per cwt. 2. Kequired the value of 1147| yds. of cloth at £1, 2s. 7id. and£l,4s. 8id. per yd. 3. 'What should be given for 18 cwt. 3 qr. 21 lbs. and 27 cwt. 2 qr. 21^ lbs. of tea, at £25, 7s. 6d. and £27, 19s. 8d. per cwt. ? 4. A bankrupt's debts amount to £1250, and he com- pounds with his creditors for lis. 10|d. per £1 ; find his effects. 5. How much sterling money is equal to 1000 francs, each 9fd. sterling? 6. Find the cost of digging a ditch, the solid content of which is 6753 cubic yds., at Is. 8|d. per cubic yd. 7. The daily pay of a foot-soldier is Is. Id. ; how much does it take to pay a regiment of 750 men for a year at that rate ? 8. The annual cost of the Police of Paris amounts to 5335295 francs of 9|d. each ; express this in sterling money. 9. How much sterling money is equal to 2000 rupees, each2s. l^d.? 10. Required the income-tax on £975, 17s. 6d. at Is. 4d. per £1. 11. The annual rent of a parish is £36750, and a tax is PRACTICE. . 29 assessed for the poor at 2s. IJd. per £ ; how much will it amount to ? 12. In 1856, the expenses of the British Postal Service were £1660229, and the net revenue was £1207725; express each of these in francs and rupees of Is. lOJd. 13. Sold 273 qrs. 5 bu. of wheat at £2, 15s. per qr. ; 159 qrs. 3 bu. of barley at 42s. 8d. per qr. ; and 79 qrs. 6 bu. of oats at 22s. 7d. per qr, ; find what sum was received in all. 14. Find the value of 14 cwt. 3 qrs. 16 lbs. of tobacco at £23, 7s. lOd. per cwt. 15. A farm, containing 675 ac. 3 ro. 24 per., is let at £l, 17s. 6d. per acre; what is the whole rent? 16. Edinburgh, May 15th, 1871. James Drummond, Esq., bought of Robert Hunter, 163| lbs. tea at 3s. 9d. per lb., 167i lbs. sugar at 7id., 147i lbs. coflfee at Is. 8d., 1 1 cheeses, each 56 lbs., at 8Jd. per lb. ; what is the whole value ? 17. Leith, April 17th, 187 1 . Alexander Clark bought of Scott and Co., 179J doz. sherry at 27s. 6d., 185J doz. port at 36s., 163 gals, aqua at 17s. 6d., 17 gals, brandy at 36s. 6d., and 21 doz. claret at 47s. 6d. ; what is the whole value ? 18. The number of sovereigns coined in 1855 was 8448482, each weighing 5 dwt. 3y\ grs. ; required the whole weight. 19. The number of shillings coined in 1855 was 1368499, each weighing 3 dwt. 15y\ grs. ; required the whole weight. 20- What is the value of 3 casks ot molasses, each 7 cwt. 3qr. 3iib.,at 12s. 7|d. per cwt., and duty 4s. 2d. per cwt.? 21. A gentleman has 3 farms ; the first contains 450 ac. 3 ro. 24 per., and is let at £1, 13s. 4d. per ac. ; the second contains 564 ac. 1 ro. 36 per., and is let at £1, 16s. per ac. ; and the third contains 635 ac. 2 ro. 16 per., and is let at £2, Is. 8d, per ac. ; the taxes which he pays upon each are respectively 5s., 6s. 8d., and 3s. 4d. per ac. : what is the full rent of all his farms, the amount or taxes which he pays, and his net income ? 22. A bankrupt owes £7580, and he can pay 15s. 7id. per £ ; what are his effects worth ? 30 DFXIMAL FRACTIONS. A FRACTION which has unity with one or more ciphers after it for its denominator is called a Decimal fraction ; as, -j%, T^^TJ- Such fractions are expressed without their denominators by pointing off, from the right of the numerators, as many figures as there are ciphers in the denominators; thus, -^^ ^§§, g-%Vtr ^^^ written '4, 1'69, •412. "^^^len the number of figures in the numerators is less than the number of ciphers, the deficiency is made up by prefixing ciphers to the numerators ; thus, -j^^ = -04, TTi%%V^ = -00251. Ciphers on the right of a decimal do not alter the value; thus, -040 = ^^g^, = ^^^ = -04. The fii'st figure after the point indicates tenths ; the second figure, hundredths ; the third, thousandths, and so on ; that is, in decimals, as in integers, the value in- creases in a tenfold ratio from right to left. Decimals are therefore operated upon in the same way as integers, due attention being paid to the placing of the point. 80 ) 7-0000 «^o=-0875 REDUCTION OF DECIMALS. Case I. To reduce a vulgar fraction to a decimal. Ex. Reduce ■^''■^ to a decimal. Ans. -0875. So L. Divide the numerator (7 ) by the denom- inator (80), annexing ciphers to the numerator until the division terminates or repeats ; then point off as many figures from the right of the quotient (875) as there were ciphers annexed (4), and make up the defi- ciency (if any) in the quotient by prefixing ciphers to it. When the division terminates, the result is called a Finite decimal ; if not, it is called a Repeating or Cir- culating decimal, according as one or several figures recur, and a dot is placed above the repeater, or above the first and last figures of the circulate ; as, -|, = '6 ; 1= -142857 (seep. 34). Reduce to decimals, 1 1 3 i 13 27 231 19 511 17 23 197 n-nA 1001 O 1 19 173 153 11 183 214 363 279 87_ '^- ¥7? TaT' 6^5? TTT-^J 2T5) T3T5) SJ-^T^ TdTTJ -g^TsJ T3'6 25- «14 7 6 25 6 36 54 121 143 15 .?,111 *• T! ^J -ST? TJ T2 6' 115") 2lT^> TTbOJ TTeXy? ^¥7^J -^^'4^ ^ T 6 4^' DECIMAL FRACTIONS. 31 Case II. To reduce a finite decimal to a vulgar fraction. Ex. Reduce '0375 to a vulgar fraction. Ans. -g^, Sol. Write the given decimal (-0375) as the | Tofeao = /o numerator of the fraction, and for the denominator write unity ^ with as many ciphers after it as there are figures in the decimal (4) ; then reduce the fraction dlg^o) t^ ^^ \ow- est terms (g^^). Reduce to vulgar fractions, 1.-5; -25; -75; -625; -3125; -03125; •18725r096875; -000575 2. -48; -364; -4248; -0672; -4152; -04096; -03136; '00048 3. -525; 6-0425; -00675; 8-0864; -0001875; 1-04264; 2-18575 Case III. To reduce a compound quantity to the decimal of a higher name. Ex. Reduce 4 cwt. 3 qr. 21 lb., or 553 lb., to the dec- imal of a ton. Ans. -246875 ton. Sol. Reduce the given | 5531b. = ^V?^ to- = •246875 ton. quantity (553 lb.) to the fraction of the required name (by Case YI., p. 8), and again (by Case I. of dec.) reduce the fraction (/2V5) to a decimal (-246875). 1. Reduce 2s. 6d., 3s. 9d., 53. 6d., 12s. 8id., 15s 9f d., and lO^d., to the decimal of £1. 2. Reduce 3 qr. 211b., 2 cwt. 141b., 4 cwt. 17^^., 141b., 2 qr. 141b., and 981b., to the dec. of 1 ton. 3. Reduce 2 ml. 7fu., 4 ml. 16 po., 37 po. 5^ yd., 7fu. 34 po., 8 ml. 16 po., and 374 yd., to the dec. of 9 ml. 4. Reduce 2 ac. 3 ro., 3 ac. 24 per., 2 ro. >21 yd., 17 per. 151 yd., 1 ac. 363 yd., and 363 yd., to the dec. of 3 ac. 5. Reduce 4 oz. 1 dwt., 19 dwt. 15 gr., 21 gr., 11 oz. 15 gr., 10 oz. 10 dwt., and 1 dwt., to the dec. of 1 lb. tr. 6. Reduce 14s. 7d., 16s. lid., 3 cr., 4 half-cr., f of 14s. 5id., and 3^d., to the dec. of 1 guinea. 7. Reduce £14, 5s. lOd., and £16, 17s. 6d., to the dec. of 20 guineas. Case IV. To find the value of a dec. in lower names. Ex. Find the value of 1-375 ac. Ans. 1 ac. 1 ro. 20 per. Sol. Multiply the given decimal (-375) by the number of the next lower name contained in that given (4), and point off, from the right hand of the product, as 1-375 ac. X 4 1-500 ro. X 40 20-000 per. .?2 DECIMAL FRACTIONS. many figures as are in the decimal (3) ; again, mnltiply tho decimal part of the last product (-500) by the number of the next lower name contained in this last name (40), and point off as before ; proceed in the same way as far as necessary. The figures on the left of the points are integers of the re- spective names. Fii>d the values of, 1. £-4625; -37258.; -6875 gu.; '3425 or.; £-7725; -9258. 2. -3775 cwt.; -4275 ton; -68725 or.; -3975 cwt. ; •4375 ton; •4651b. 3. -8975 ml. ; -3875 ac. ; -496725 da. ; -8975 co. ye. ; •1875 bu. ; -3875 ro. 4. -4725 Eng. E. ; -45875 yd. ; -2875° ; -9875 Ju. ye. ; £8-6775; 9*725 gu. 5. -242242 da.; £-8794^ 8-746 ton; 10-12125 ac. ; 11-6874 ml.; 14-72 yd. Case V. To reduce shillings and pence to the deci- mal of £1 mentally. Ex. 1. Reduce 4s. 10|d. to the dec. of £1 mentally. Ans. £-24375. Sol. Divide the shillings by 2 for the first figure of the decimal (-2) ; the farthings in the pence and farthings (42), increased by their 24th part (l^f or If), gives the next two figures (43) ; then to the third figure annex the remainder (f ), reduced to a decimal (75), for the value (£-24375). Ex. 2. Reduce 15s. 8id. to the decimal of £1. Ans. £'784/^, or £-784375. When the shillings are odd, 50 must be added to the far- things, increased as above. Reduce to the decimal of £1 mentally, 1. 7s. 6d. ; 10s. 7id. ; 14s. 8id. ; 15s. 9d. ; 19s. 8id. ; 12s. 7id; 18s. 9id. 2. lis. 8d.; 128. 4d.; 13s. 5d. ; 14s. lljd. ; 16s. 8ad.; £1, 17s. 9d.; £3, 18s. 7d. Case VI. To value the decimal of £1 mentally. Ex. Value £-6354 mentally. Ans. 12s. S^d. Sol. Divide the first two figures (63) of the dec. by 5 for shillings (12) ; to the remainder, if any, annex the thin! DECIMAL FRACTIONS. 33 figure (5), and tins number (35) diminished by its 25tli part (1) gives farthings (34), which must be reduced to pence (8^d.) Note. Since three places of the decimal only are required, call the third figure one more than it is, when the fourth is 5 or upwards, otherwise reject it. Find mentally the values of, £•972; £-496; £'896; £1-8914; •873; -785; -999; 6-4377; 1. £-525; 2. -675; £11-7364 22-8976 Sum Diff. 7-96752 -0478 8-01532 7-91972 ADDITION AND SUBTEACTION OF FINITE DECIMALS. Ex. Fmd the sum and difference of 7-96752 and -0478. Sol. Arrange the numbers so that the decimal points may be directly under each other, and proceed as in integers ; then place the point in the result directly under the other points. Find the sum of, 1. -25, 4-675, -00475, 84, 96-23725 2. -0046, 217, -284, -0478, 3-44756 3. -728921, -00043, 211-2, 86-114875 4. -00867, 1-432, 247, -00083, -674 5. 9-732, -048076, -1234, 6-7289, -214, -7 6. -6498, -37293, 311-4, 21-72, -00875 7. -046, 36-479, 2-101, -111, -04789 8. -3596, -24798, -35, 37, -00705 9. £375, 16s. 6d., £331,14s.9d., £375, 18s.9|d., £157,7s.8id., £97 4s. 6d. 5s. 9d. 10.47-98625— 13-97846; 7-0423— -4789; 28-23546— 16^479258 11.29-46537—21-57698; 8-2458— -0034; -6728934— -00628575 12.1-4386- -004289; -04657- '00827 ; -4798231- -46991482 13. 17 cwt. 2 qr. 14 lb.— 14 cwt. 3 qr. 21 lb. ; 29 qr. 5 bu. 2 pk. — 23 qr. 7 bu. 3 pk. MULTIPLICATION OF FINITE DECIMALS. Ex. Multiply 4-7025 by -0025. Ans. -01175625. Sol. Multiply the factors together, as in in- tegers, disregarding the point ; then, from the right hand of the product, point off as many figures as there are decimal places in both fac- tors (8), making up the deficiency (if any) by prefixing ciphers (1) to the product. 4-7025 ♦0025 235125 9405 •01175625 d2 34 DECIMAL FRACTIONS. Note. A decimal is multiplied by 10, 100, 1000, &c. by re- moving the point one, two^ three, «&:c. places to the right; an, 2-134 X 10000 = 21340. 1.4^6275x4-63, 5-75, -046, -824, 86-4, -005, 1000, 10001 2. •00796x37-8, 42-89, 7-824, -046, -3724, 1-738, 100, 10101 3.27-8372x8-434,96-429,-00429,-7426,3-14152,-001,1001 4. Val.27yd. 3qr . 2nl., at 2/6, 4/9|,23/4^,27/10|,31 /I li p.yd. DIVISION OF FINITE DECIMALS. Ex. Divide 3-146 by 42-8. Ans. -073505 nearly. Sol. Make the decimal places in 42800 )3146-000000 both numbers alike by annexing ci- -073505 phers (2) to that which has the least number ; then divide as in integers, and to the remainder (if any) annex ciphers to carry on the division : the number of ciphers last annexed (6) is the number of decimal places in the answer (-073505). Note. A decimal is divided by 10, 100, 1000, &c. by removing the point one, two, three, &c. places towards the left; as, 2-134 -f- 100 = -02134. 1. 382-8825-7-55, 1-36, 31-5,-325, -1309, -00119, 2-75, 81-9, 100 2. 24-22728-r-l-15, 85-5, -805, -0368, 103-5, 369-6, -01539, 1000 3. -0483923^-20-35,475-6, 8-584,-03157,-002552,l-221, -0019721 4. £4779, 17s. 8jd.-M7-5, 4-375, 281-25, 687-5, -03125, -1875 INTERMINATE DECIMALS. In reducing vulgar fractions to decimals, when one or more figures of the quotient recur, the result is called an Interminate decimal. Decimals consisting of one or more figures which re- cur, are called pure repeating or circulating decimals ; as, -3, -142857. Decimals consisting of a non-recurring and a recurring part are called mixed repeating or circulating decimals ; as, '46, '3796 : the non-recurring parts (4 and 37) are generally called the finite parts of the decimals (-46 and '37960 INTERMINATE DECIMALS. 35 REDUCTION OF INTEEMINATE DECIMALS. Case I. To reduce a pure repeater or circulate to a vulgar fraction. Ex. Reduce '75 to a vulgar fraction. Ans. ^| or -§f. Sol. Write the given decimal for the numerator, and be- low it, for the denominator, place as many nines (2) as there are decimal places; then reduce the fraction (||) to its lowest terms (y). Reduce to vulgar fractions, 1.-3, -6, -4, -9, -27, -36, -396, -594, -185, -259, -063, -0072 2. -571428, -153846, -190476, -171, -428571, -095238 Case II. To reduce a mixed recurring decimal to a vulgar fraction. Ex. Reduce '0476 to a vulgar fraction. -MII&. -g-goo -5900' 2^T5 Sol. From the given decimal (-0476), considered as an integer, subtract the finite part (04) for the numerator, be- low which, for the denominator, place as many nines as there are figures in the circulating part (2), with as many ciphers annexed to them as there are figures in the finite part (2) ; then reduce the fraction (/9V0) to its lowest terms l^^^^^^). Reduce to vulgar fractions, 1. -472, -583, -16, -027, -QliS, -4891, -6381, -729693 2. -146, -2359, -0026, -06563, -11818, -4585, -68472 Case III. To make circulates similar. Ex. Make -03, -775, and -76034 similar. Sol. Extend each decimal as many places beyond the longest finite part (2), as is indi- cated by the L. C. M. of the number of places in the several circles (6). Make the following circulates similar. •03333333 -77575757 •76034034 1. -143, ^037, -014 2. -3742, -0089, ^476 3. -1769, -2456, 'im 4. -402, '179, •0423 5. -117, -1146, -0894 6. -2486, -1175, -0624 7. 1^729, •2684, -0046, 2-478 8. 4-3, 8-729, •4673, -OOl 9. -00101, -26770, -1421, 3-428 E 36 INTERMINATE DECIMALS. ADDITION AND SUBTRACTION OF INTERMINATE DECIMALS. Ex. 1. Find the sum and difference of 37-143 and 29-8736 Sol. Since there are only repeaters given, extend them one place beyond the longest finite part, and carry or borrow at 9 on the right hand; the right-hand figure of the result is a repeater or 0. 37-1433 29-8736 Sum 67-017 DifF. 7-2696 Ex. 2. Find the sum and difference of 71-857 and 43-97642. Sol. Make the circulates similar, and add the carriage (if any) from the left-hand column of the circles to the right-hand figure of the under circle, before adding or subtracting. 71-85757575 43-97642642 Sum 115-83400218 Diff. 27-88114933 After the right-hand figure of the result has been obtained, proceed as in Finite decimals. Find the sum of, 1. 1-6, 23-052, -583, 6-3, -05i I 3. -7, -245, -006, -043, -i 2.7-8, 8-964, -729i, 14-0561 4. -72, -345, -854, '00625 5. 9-76, 8-427, -0864, -75, 8-4572 6. -1341, -672, -1487, -05, 6-4576 7. 11-i, 1-63, -9625, -03, 7-1167 O 2 5 3 4 6 or\f\ 3 9. £12, 14s. 5d., £27, 16s. 7id., £33, 16s. 6^d., £17, lis. 8d., and lOfd. 10.29-146— 13-17, 16-72—4-6583, 57-6854-14-976851 11.17-36—3-143, 21-867—7-863, 41-6872-19-478623 12. 17 ml. 3 f. 4 yds.— 6 ml. 39 po. 5 yds., 18 cwt. 3 qr. 161b.— 7 cwt. 181b. MULTIPLICATION OF INTERMINATE DECIMALS. Ex. Multiply 42-375 by -037. Sol. When the multiplicand is a repeater, and the multiplier a finite decimal, multiply as in integers, but add 1 to the right-hand figure of the product for every 9 in it; then extend the several products the same length and add them. 42-375 •037 296628 1271266 1-567895 INTERMINATE DECIMALS. 37 7-14672 '69 6432054 42880360 Ex. 2. Multiply 7-14672 by -69. Sol. When the multiplicand is a circulate and the multiplier a finite decimal, multiply as in integers, and add the carriage from the left of the circle to the right-hand figure of the circle ; then extend the several products the same length and add them. 4-9312414 When the multiplier is interminate, or when both are in- terminate, reduce the multiplier to a vulgar fraction, then multiply by its numerator and divide by its denominator. In dividing, instead of ciphers, the repeating or circulating figures must be annexed in their order to carry on the division: thus, 6-285714 X -3 = 5-2857142 X ^ = 1-761904. 1.5-7963 X 4-8, 5-76, 8-942, 7-63, 11-5, -0042, -176, -0087 2.8-1426 X 7-3, 6-84, 9-428, 3-76, 1-51, -43, -0074, -4875 8. 6-8726 X -4, 6-6, 7-2, -043, -069, -0428, '0072, -1456 4.7-8763 X -6, -63, -171, -069, -487, 3-i48, 6-709, 8-472 DIVISION OF INTERMINATE DECIMALS. Ex. 1. Divide 26-80583 and 21-18729 by 8. Sol. When the dividend only is inter- minate, divide as in finite decimals, and in carrying on the quotient, instead of ciphers, annex the repeating or circulating figures in their order, until a repeater or circulate is obtained in the quotient. 8 )26-80583333 3-35072916 8)21-18729729 1-64841216 Ex. 2. Divide 6-7283 by G'6. Here 6'6z=z6^=^z=y> Sol. When the divisor or both are inter- minate, reduce the divisor (6-6) to a vulgar fraction (6 §), then multiply by its denomi- nator (3), and divide by its numerator (20) for the quotient, which may be carried on as in Ex. 1. 1. 7-416-^-125, 187-5. 43-75, -52, -427, -729 2. -2307692 -^ -65, 4-16, -975, -0208, 4-*63, 7-83 8. 14-i42857 -r- '84, -583, 68-4, 7-23, 81-6, 9-63 6-7283 3 20 )20-18498 1-009249 38 MISCELLANEOUS EXERCISES ON DECIMAL FRACTIONS. 1. Find the values of £-31416; -142857 ton; -12645 ml. ; -846283 lb. troy; -1728 yd. ; and '78646 day. 2. From the sum of 7-6, 8-24, 9*1583, subtract 4-72, and multiply the remainder by 21-24. 3. Find the value -875 of 17s. 6d. + -143 guinea — -725 of 8s. 6d. 4. Express 1456*725 cub. ft. in French steres, each . 35*31658 cubic feet. 5. Find by decimals the value of 14 cwt. 3 qrs. 14 lbs. of coffee at £9, 6s. 8d. per cwt. 6. The ratio of the diameter of a circle to its circum- ference is 1 : 3*14159 ; required the mean equatorial cir- cumference of the earth, the diameter being 7925*626 mis. 7. Find the diameter of the planet Jupiter, its circum- erence being 273318*33 miles. 8. Reduce £4, 13s. 4d. to the dec. of £5, 12s. 6d. 9. The gold eagle of the United States weighs 270 grains, 21*875 carats fine; what is its value in sterling money, 46*725 sovereigns weighing a lb. troy, 22 carats fine? 10. A and B can do apiece of work in 4*8 days, A and C the same in 4*4 days, and B and C in 5*45 days ; find by decimals in what time the three together could do it, and also separately. 11. Find the value of -1875 ton + -375 qr. + -8751b., and reduce the result to the decimal of 3 cwt. 3 qrs. 12. The whole area of France is 131017513*33242 acres; find the area in hectares, each 2*473614 acres. 13. Multiply -06723 by -00401, and divide 301*5 by -00045. ^ 14. A French metre is 3*2808992 imperial feet ; how many imperial feet are there in a quadrant of the me- ridian, or in 100000565*268 metres ? 15. What is the hourly motion of the earth, whose distance from the sun is 95 millions of miles and period of revolution 365 days ? 16. An ounce of tea costs 2-8125d. ; what should be paid for 6 583 lbs. of the same? EXERCISES ON DECIMAL FRACTIONS. 39 17. Find the value of 14-8745 ml. — 3*427 mile. 18. £4, 6-625S. X 27-45 and 993-714285-^234-2. 19. Keduce -025 of 4 ml. 3 fu. 19-264 pole to the dec, of 2 ml. 5 fu. 20. Keduce 2-275 of 4i gu. to the dec. of £20, 9s. 6d. 21. Find the values of -073 of 8s. 6d. and -785 of 15s.9d. 22. Reduce '2 of |- of 4 cwt. 3 qrs.23 lbs. to the dec. of 31cwt. 2qrs. 24 lbs. 23. Divide £1050, 3|d. among 5 men, 8 women, and 16 boys, giving each woman -5 of I of a man's share, and each boy -5 of 1-25 of a woman's share. 24. Divide £1125 among A, B, and C, giving A 4- of 2-625 of the whole— £45; B 1-025 of what A receives -|- £75, and C the remainder. COMMERCIAL ALLOWANCES. The Gross weight of goods is their whole weight, including the weight of the cask, barrel, &c., which contains them. Tare is the weight of the cask, barrel, &c., which con- tains the goods, or it is an allowance made for it. That which remains after deducting the tare is called the Tare Suttle. Tret is an allowance of ^V? or 4 lbs. on every 104 lbs. of the Tare Suttle for waste. — Draft is also an allowance sometimes made, and is deducted before the Tare. The Net weight is what remains after all the allow- ances have been deducted. Questions under this head may be solved by Practice. Ex. Find the net weight of 5 hhds. sugar, each 10 cwt. 3 qrs. 4 lbs., allowing tare 14 lbs. per cwt., tret 4 lbs. per 104 lbs., and draft 4 lbs. per hhd. 10 cwt. 3 qr. 4 lb. gross weight of each hhd. 4 draft on 1 hhd. 10 3 0X5 141b.= icwt.)53 3 6 2 24^ tare. 4lb.p.l04lb.=^'^)47 3i tare suttle. 1 3 6H tret. 45 24 1 7 net weight. 40 COMMERCIAL ALLOWANCES. Find the net weight of, 1. 48 cwt. 3 qr. 14 lb., tare 16 lb. per cwt. & tret as usual. 2. 79 cwt. 2 qr. 21 lb., tare 7 lb. // // // 3. 147 cwt. 1 qr. 7 lb., tare 81b. // // // 4. 984 cwt. 3 qr. 16 lb., tare 14 lb. // // ^ 5. 748 cwt. 2 qr. 12 lb., tare 12 lb. // // // 6. 896 cwt. 1 qr. 241b., tare 18 lb. // // // 7. 4 chests of tea, each 12 cwt. 3 qrs. 14 lbs., allowing tare 15 lbs. per cwt., tret as usual, and draft 3 lbs. per chest. 8. 7 casks of sugar, each 14 cwt. 2 qrs. 18 lbs., allowing tare 14 lbs. per cwt., tret as usual, and draft 4 lbs. per cask, and the value of the net weight at 6id. per lb. 9. 3 hhds. tobacco ; the first, 8 cwt. 3 qrs. 14 lbs., tare 3 qrs. 7 lbs. ; the second, 9 cwt. 2 qrs. 7 lbs., tare 2 qrs. 19 lbs. ; and the third, 10 cwt. 1 qr. 16 lbs., tare 3 qrs. 7 lbs., allowing draft 7 lbs. per hhd., and tret as usual. 10. Bought 2709 lbs. of coffee, and was allowed 1 lb. gratis to the score; what did I pay for it at 16id. per lb.? 11. How much pure silver in a mass weighing 42 lbs., allowing 15 dwt. of alloy in each lb. of the mass? 12. Purchased 4 bags of rice, each 265 lbs., and was allowed 3 lbs. gratis to every 50 lbs. ; what did it cost at 3id. per lb. ? COMMISSION AND BROKERAGE. Commission is an allowance of so much per cent, paid to one person for transacting the business of another. Brokerage is a smaller allowance of the same nature. Ex. 1. Find the commission on £475, 12s. 6d., at £li per cent. Ans. £7, 2s. S^d. Sol. Multiplythesum(£475,12s.6d.) | £475 12 6 by the rate per cent. (1^), and divide 1^ the product (£713, 8s. 9d.) by 100. 100)713 8 9 ^ Com. £7 2 "81 When the rate is guineas, add :^\ of the sum to itself, then multiply by tlie rate as £'s, and divide by 100 when the rate is shillings, &c., take aliquot parts of £1, and divide by 100. Sol. 1. 5^0) £322 10 16 2 338 12 6 6 1,00)8,46 11 Com. at 2^ p. c, £8 9 3 3f COMMISSION AND BROKEKAGE. 41 Ex. 2. Find the commission on £322, 10s., at 2^ gu. and 8s. 9d. per cent. Sol. 2. 5s.r=£|) £322 10 80 12 6 2s.6d.=:^of5s.= 40 6 3 ls.3d.=iof2s.6d.=20_3__l| 100 )141 1 10^ Com.at8s.9d.p.c.,£l 8 2^:} 1. Find the com. on £148, 2s. 6d., at 2,^, 3, 3i, and 3| per cent. 2. Find the brok. on £152, 10s., at ^, f, -j*^, andl^p.cent. 3. Find the com. on £500, at 2s. 3d., 3s. 9d., 4s. 3d., and 16s. 9d. per cent. 4. Find the brok. on £4216, 5s., at 3s., 2s. 6d., 3s. 4d., and 4s. per cent. 5. Find the com. on £2850, at 1|. gu.. If gu., l-^ gu., and 2^ gu. per cent. 6. Find the brok. on £375, 17s. 6d., at 3s. 9d., 5s. 3d., lis. 3d., and 13s. 4d. per cent. 7. An agent sells for his employer goods to the amount of £1260, 17s. 6d. ; the expenses attending the sale amount to £14, 2s. 6d. : what is his commission at 3^ per cent. ? 8. A banker discounts bills to the amount of £1252, 10s. ; what is his commission at ^ per cent. ? 9. An agent charges 3| per cent, for commission and risk of bad debts ; his sales during the year amount to £9275, 15s., and his losses to £150 : required his net income. 10. A broker is authorized to purchase £1120, 10s. of 3 per cent, stock ; what is his brokerage at \ per cent. ? 11. An agent's annual sales amount to £12783, 13s. 4d. ; his bad debts, valued at 12s. 6d. per £1, amount to £360, and his losses to £150 : what is his income, if he is al- lowed 5 per cent, for commission and guarantee ? 12. A factor collects the half-yearly rents of 3 farms ; the annual rent of the 1st is £1250 ; of the 2d, £775 ; and of the 3d, £840; the charge for repairs on each is f, 1^, and 2 per cent. : what sum will he remit to the landlord, his factorage being 3} per cent, upon the rental ? 42 INTEREST. Interest is the money paid for the loan of money. The money lent is called the principal^ and the sura of the principal and its interest is called the amount. Interest is divided into simple and compound. SIMPLE^ INTEREST. Case I. To find the interest for any number of years. Ex. Required the interest of £375, 2s. 6d. for 4 years, at 3i per cent. Ans. £50, 0%. 4d. Sol. Multiply the principal by £375 2 6 the rate per cent. (3J), and by the number of years (4) ; and divide the product (£5001, 13s. 4d.) by 100. £1250 8 I 100) £50,01 13 Int. £50 3 4 4 4 Note. For months take parts of a year, or multiply by them and divide by 12. 1. Find the interest of £1345, 10s. for 1 year, at 5, 5|, 5^, 4^, 4|, and 3| per cent. 2. Find the amount of £575, 13s. 4d. for 1 year, at 2^, 3, 3J, 4i, 5, and 5f per cent. 3. Find the interest of £1200, 13s. 4d. for 5 years and G years, at 2i, 3J, and 4i per cent. 4. Find the amount of £150, 17s. 6d. for 6f yrs. and 7^ yrs., at 2i, 3, and 5 per cent. 5. Required the interest of £1244, 15s. for 5 ye. 7 mo. and 4 ye. 5 mo., at 3i, 4, and 5f per cent. 6. Borrowed £750, at 2J per cent. ; what sum will be required to discharge the loan at the end of 5 ye. 8 mo. ? Case II. To find the interest for any number of days. Ex. Find the interest of £675, 10s. for 195 days, at 4 per cent. Sol. Multiply the princi- pal by twice the rate per cent. (8), and by the number of days (195) ; and divide the product by 73000 (i. e. 2 X 365 X 100). Ans. £14, 8s. 8id. |f f £675 10 8 5404 195 73,000) £1053780 £14 _0 _ INTEREST. 43 7. Required the interest of £677, 10s. 7icl. for 198 (lays and 364 days, at 2^, 3^, and 4i per cent. 8. Find the amount of £1436, 14s. 7id. for 1 ye. 95 da. and 2 ye. 5 da., at 2^, 3^, and 4 J per cent. To divide by 73000. Taking Ex. 2— To the pounds ^) £1053780 of the dividend add ^ of itself, yV) 351260 j'^ of this third, and ^^^ of the t*^) 35126 last; then point off 5 figures 3513 of a decimal from the right £14^ gs. 8^d. = £14-43679 hand of the sum. This gives the interest too much, by about Jd., for every £10 of interest. 9. Required the interest of £483, 12s. 6d. from Mid- summer to Christmas, and from Christmas to Midsum- mer, at 2^, 4, and 5 per cent. 10. Lodged in the bank £748, 5s. on March 1st, find the amount of this on Dec. 31st, interest at 2i per cent. 11. Borrowed £1051, 4s. on April 4th ; ^ at 3^, and the rest at 3^ per cent. ; what sum should be returned on Nov. 11th? 12. Find the interest of £210, 12s. 8id. for 275 days, at 4J and 4^ per cent. ; and of £445, 18s. 2d. for 252 days, at 2^ and 3i per cent. 13. Find the amount of £4482, 4s. from Feb. 23d to June 13th, at 3-^, 4-f, and 5^ per cent. 14. What is the interest of £1005, 17s. 7d. from Jan. 1st to May 16th, and from April 7th to Dec. 15th, at 3|, 3^, and" 4^ per cent. ? 15. Borrowed £340, 19s. 5d. on Lammas-day at 4 per cent., and £361, 7s. at 4-i- per cent, on Michaelmas-day ; what sum will discharge the whole loan on Whitsunday following ? Case III. To find the interest when a debt is dis- charged by partial payments at short intervals of time. Ex. Borrowed £750 on Jan. 1st, of which £200 was paid on March 4th, £250 on May 15th, and the balance on August 1st ; what was then due, interest at 2^ per cent. ? Ans. £307, 10«. 44 INTEREST. Obs. J rom Jan. 1 st to March 4th is 62 days, Mar. 4th to May 15th is 72 days, and from May 15th to Aug. 1st is 78 days. The balances are multiplied by the respective days. Jan. 1. Borrowed £750X62=46500 Mar. 4. Paid 200 Balance 550X72=39600 May 15. Paid 250 Balance 300X78=23400 £109500 (Mult, by 2^X2=) 5 Divide by 78000) £547500 £7, 10s. Balance as above 300 Aug. 1. Amount due £307, 10s. 16. A bill of £960 was due Jan. 4th, of which £300 was paid on April 9th, £260 on July 11th, £200 on Nov. 11th, and the balance on Dec. 31st ; what was then due, interest at 3 per cent. ? 17. Lodged in the bank £1500 on May 13th, and drew £300 on July 11th, £400 on Oct. 1st, £350 on Dec. 14th, £200 on March 29th, and the balance, along with the interest, on June 9th ; what was then drawn, interest at 2J per cent. ? 18. Borrowed £1050 on March 9th, of which ^ was paid on June 14th, £200 on Sept. 23d, -f of the remain- der on Nov. 29th, and the balance on Feb. 11th; what was then paid, interest at 3^ per cent. ? 19. Lent, Jan. 15th, £1320, and received -25 of it on March 31st, £400 on June 18th, -5 of the remainder on Aug. 19th, £100 on Dec. 30th, and the balance on March 1st ; what was then received, interest at 3| per cent. ? Case IV. To calculate interest on Accounts-Current. An Account-Current is a statement of the mercantile transactions of two persons when immediate payments are not made. It is written on two pages marked Dr. and Cr. ; the Dr. or left-hand side containing all sums paid by the person furnishing the account ; the Cr. or right-hand side those 2:>a^c? to him. Ex. Required the interest on the following Account- Current between March 2d and Nov. 11th, interest at 3i per cent. INTEREST. 45 Dr. Wilson & Co. in Account-Current with Murray & Co. Or, Mar. 2. To Cash, £550 June 30. " Do. .320 Sept. 15. >^ Do. .470 Nov. 11. » Int. . 1 16 2| £ 1341 16 2f Nov. 11. ToBal. £166 16 2i Dr. da. Prod. Mar. 2. £550X254=139700 June 30. 320X134= 42880 Sept. 15. 470X 57= 26790 209370 April 14. ByCash, £400 May 17. >' Do. .350 July 31. >' Do. . 425 Nov. 11. /' Bal. . 166 16 2| £1341 16 2| Cr. da. Prod.* April 14. £400X211= 84400 May 17. 350X178= 62300 July 31. 425X103= 43775 190475 1 90475 £ 18895 X 7 73000) £132265 Interest =£1, 16s. 2|d. Sol. Multiply each sum by the number of days between the date opposite to it and the last date (Nov. 11). Add the Dr. and Cr. products separately, and multiply the sum of each by twice its respective rate per cent. ; subtract their products, and divide by 73000 for the interest which is entered on the Dr. or Cr. side of the Account, " To or By Interest ;" as the Dr. or Cr. products, after multiplying by twice the rates, is the greater. Then add each side of the Account, and write the difference on the less side, with the words *' To or By Balance." 20. Find the interest, at 3 per cent., on the following Account-Current from March 2d to Oct. 28th, 1871. Dr. J. Brown in Account-Current with A. Anderson, Cr, To Cash, . . Do.. . » Do.. . £750 . 240 . 560 April 17. By Cash, . . £640 May 30. " Do. . . .550 July 18. - Do. . . .310 Mar. 2. June 3. Aug. 30. 21. Required the interest on the following Account- Current, at 4 per cent., to April 8th, 1871 Dr. R. Scott in Account- Current with Thos. Younger, Cr. April 14. To Goods, . . £650 Aug. 12. » Cash, ... 400 Dec. 15. /' Do 700 June 11. By Goods, . . £740 Oct. 19. '/ Cash, .... 625 Feb. 14. // Do 300 4G INTEREST. 22. Eequired the interest on the following Account- Current to Oct. 15th, allowing Smith 3J per cent., and Weddell 3J per cent. Dr. Jas. Smith in Account-Current with Henry Weddell, Cr, June 3. To Cash, . . £700 Aug. 1. By Cash, . . £1050 Oct. 7. " Do. . . . .500 Dec. 12. n Do.. . . . 600 Jan. 9. " Do. . . . . 650 Mar. 15. » Do.. . . . 450 May 29. f Do. . . . . 420 July 30. '/ Do. . . . . 160 23. Required the principal and interest on the follow- ing Account-Current to Nov. 29th, allowing Harrison 4 per cent., and Cochrane 4 J per cent. Dr. H.Harrison in Account-Current with John Cochrane, Cr. Sept. 11. By Cash, . . £750 Jan. 1. '/ Do 360 May 9. '' Do 250 Sept. 20. " Do 540 July 4. To Cash, . . £600 Nov. 11. " Do 240 Mar. 2. - Do 400 July 12. /' Do 300 24. Required the principal and interest on the follow- ing Account-Current to Jan. 12th, allowing Henderson 2^ per cent., and Clark 2i per cent. Dr. Henderson & Son in Account-Current with Jn. Clark, Cr. Aug. 2. To Cash, . . £840 Dec. 11. " Goods, . . 800 April 9. " Cash, ... 700 Aug. 14. " Goods, . . 840 Oct. 17. By Cash, . . . £960 Feb. 3. // Do 750 Jmie 9. " Do 900 Nov. 1. " Goods, ... 600 The following examples are solved by Simple and Com- pound Proportion : 25. In what time will the interest of £725, 16s. 8d. at 2.^ per cent, pay a debt of £81, 13s. Ud.? 26. How long must £532, 5s. lOd. be lent to amount to £548, 4s. 4d. at 3 per cent, per annum ? 27. What sum will amount to £476, 9s. 5id. in 224 days, at 3| per cent, per annum ? 28. In what time will £1825 amount to £1840, 18s. 6d. at 3J per cent. ? 29. In what time will any sum of money double itself at 2 J, 2f , 3, 3^, 3|, and 4 per cent. ? and in what time would any sum of money treole itself at each of these rates ? 30. At what rates per cent, will any sum of money double itself in 7^, 10, 11^, 12^ 15, 16, 20, and 25 years? 31. How long must £1125 be lent to amount to £1188. 15s. at 4 per cent. ? DISCOUNT. Discount is an allowance granted for discharging a debt before the period allowed for payment has expired. The p7^esent value of a debt due at the end of a certain time is that sum tlie amount of which for the given time is equal to the sum due at the end of that time : Thus, the present value of £105, due 2 years hence, at 2a per cent., is £100; for the amount of £100, for 2 years, at 2^ per cent., is £105 ; and the discount allowed for present payment is £105 — £100 = £5. Ex. Find the present value of £913, 10s., due 6 months hence, at 3 per cent., and also the discount. Ans. £900 and £13, 10s. Sol. The interest on £100 for 6 months, at 3 per cent., is 3 X A = £1 i, and the amount of £100 for that time is £101 ,. Then, by Proportion, 101^ : 100 : : £913, 10s. : £900 present value, and £913, 10s. — £900 = £13, 10s. is the discount, or by Proportion £101^ : £1^ : : £913, 10s.: £13, 10s., discount as before. 1. What is the present value of £1158, lis. 6d., due 4 months hence, at 2^- per cent. ? 2. Wliat is the discount upon £345, Is. lO^d., due in 9 months, at 3 per cent. ? 3. What sum will amount to £285, 4s. 4d. in 3 years, at 3 per cent. ? 4. A debt of £188, 12s. Sjd. is to be paid; £47, 16s. 4d. in 2 months, £89, 8s. G^d. in 3 months, and the rest in 4 months ; wliat discount should be allowed for pre- sent payment of the whole, interest at 4 per cent. ? 5. Required the present value of £527, lOs. 4d., due 219 days hence, at 3j per cent. 6. What is the difference between the interest of £608, 10s. 6d. for 146 days, and the discount upon it due in 146 days, interest at 2i per cent. ? 7. Bought goods to the amount of £2400 ; ^ due 1 month lience, ^ due 2 months hence, i due 4 months hence, and the rest due 6 months hence ; what sum will be sufficient to pay the whole now, interest at 3 per cent. ? 8. Required the discount on £373, 16s. IJd,, due 3, 4, and 6 months hence, at 4 per cent. 48 discoujST. 9. 1 am offered a discount of £40 for present payment of £640 worth of goods to be paid 3 months hence ; at what rate per cent, is the offer made? 10. The present value of £436, 5s. 4id. due a certain time hence, is £420, 10s. ; required the time, interest at 2i per cent. In discounting Bills, bankers find the interest on the amount for the time which the bill has to run for the discount ; the difference between this discount and the amount is called the net j)roceeds. In this country three days, called Days of Grace^ more than the term of the bill are allowed. Ex. Find the net proceeds of a bill of £572, 10s., dated April 8th, at 3 months, and discounted June 3d, at 3i per cent. Here, 3 months from April £572, 10s. 8th is July 8th, and adding 3 38 'lays of grace, the bill is due on July 11th; again from June 3d to July nth is 38 days. Then the interest on the amount (£572, 10s.) for 38 days, at 3^ per cent., viz. £2, Is. 9d., is the discount, and the net proceeds is found by subtracting £2, Is. 9d. from £572, 10s. The interest is calculated to the nearest penny. 21755, Os. 73,000)£ 152285, Qs. Discount = £2, Is. 9d Amount = £572, 10s. Od. Net proceeds =£570, 8s. 3d. L £572, 10s. the nearest penny. Find the net proceeds of the following biUs : Amount. Date. Term. Discounted. Rate. 11. £672, 12s. Jan. 4. 4 mo. Mar. 5. 2^ per cent. 12. 743,11s. Feb. 9. 6 // May 11. 3 // z^' 13. 897.15s. Mar. 11. 5 // June 14. 31 // // 14. 983, 4s. May 12. 7 // Aug. 17. 3|: // // 15. 1260, 14s. June 15. 3 V July 26. 4 // // 16. 1340, 17s. Oct. 14. 5 // Dec. 26. 4i // // 17. 1572, 8s. Apr. 10. 8 // Sept. 30. 4| // ^ 18. 2183, 16s. Dec. 30. 7 // Mar. 1. 2| // // Discounts on merchants' bills are generally calculated the same way as m Commission. 49 INSURANCE Is a contract by which an individual or company, in con- sideration of a certain allowance called premium, agrees to repay the owners of the goods, or other property in- sured, any loss or damage which they may have sus- tained to the amount stated in the written agreement or Policy of Insurance. The policy of insurance in this country must be written on Stamped Paper, the amount of which is called Policy-duty, and is always charged upon exact hundreds ; thus, if the sum insured be £510 or £570, the duty is charged on £600. The calculations are made the same way as in Commis- sion and Brokerage. Ex. 1. Find the insurance on £310 at 3s. 6d. per cent., and policy-duty 2s. 6d. per cent. 3s. 6d. per cent, on £310 = £0, 10s. lO.^d. 2s. 6d. " » on 400 = 10 Sum required for insuring £310 = £1, Os. 10|d. 1. What must be paid for insuring £920 at 3s. 6d., 4s., OS. 6d., 6s. 3d., 12s. 3d., and 13s. 4d. per cent.? 2. What is the premium for insuring property to the amount of £3530 at 2i, 3i, 4^, 5^^^ H gu- and 3^ gu. per cent. ? 3. What must be paid for insuring £4350 at £4^, £2, 2s. lOd., £3, Is. 6d., 2igu., 3^ gu., 3| gu. per cent., and policy 3s. per cent. ? 4. What is the expense of insuring £12500 on the ship Isabella from Leith to Calcutta, at 2i gu. per cent., policy 2s. 6d. per cent., and commission \ per cent. ? 5. Insured £12520 on a ship at 5 gu. per cent., and policy 3s. per cent. ; she received damage to the extent of £3250 ; what sum will be recovered, allowing If per cent, discount on the loss ? * 6. Insured £14350 on the ship Ohio from Leith to New Orleans at 4^ gu. per cent., policy-duty 2s. 6d. per cent., and commission i per cent. ; she received damage tc * To find the sum recovered, from the amount of the damage, subtract the premium and other charges. 50 INSURANCE. the amount of £2580 ; hoAv much will be recovered, al- lowing 2^ per cent, discount on the damage ? 7. Insured £6750 on a ship at 7^ gu. per cent., £10050 on the cargo at 3| gu. per cent, and £500, the net freight at 5 gu. per cent. ; the policy-duty was | per cent, and commission i per cent. ; required the whole expense of insurance. Ex. 2. What sum must be insured to recover £7700 at 2^ gu. per cent., policy 5s. per cent., and commission 17s. 6d. per cent., in case of total loss? Sol. From I £100— (£2,12s.6d.+5s.+17s.6d.)=£96,5s. £100 subtract | and £96, 5s. : £100 : : £7700 : : £8000 sum. the rate and other charges; then state, as the remainder (£96, 5s.) is to £100, so is the given sum (£7700) to the sum to be insured (£8000). How much must be insured to recover in case of total loss, 8. £2365 at 5gu. per cent., and policy 3s. per cent.? 9. £4459 at If gu. per cent., and policy 5s. per cent.? 10. £1384, 10s. on a single voyage at 6|- gu. per cent., policy 5s. per cent., and commission f per cent. ? 11. What must be insured on a ship worth £6750, and the value of the cargo £15954, to cover the whole value; premium 8 gu. per cent., policy 5s. per cent., and com- mission ^ per cent. ; 3^ per cent, to be returned if the ship sailed with convoy, which she did ? Ex. 3. How much must be insured on a voyage out and home to cover £9120, 5s. at 3| gu. per cent., policy 5s. per cent., and commission -^^ per cent. ? Here £100— (£3, 18s.9d.-f 5s.+6s.3d.) = £95,10s.; hence by Comp. Proportion / £95^ : £1 00 : : £9120, 5s. : £10000 sum. I 95^: 100 How much must be insured on a voyage out and home to cover, 12. £223729 at 4igu. p. c, pol. 5s. p. c, & com. 8s. 6d. p. c. ? 13. £145924 // 3|-gu. // ^ 3s. // // lOs. 14. £580326 // 5^gu. // '^ 5s. /^ // 13s.6d. '// i5.£157323// 7^gu. // // 6s. // // 5s.6d. // INSURANCE. 51 16. Insured 250 hlicls. sugar, at £24 per lihd., from Ja- maica to Leitli, at 10 gu. per cent. ; policy-duty 5s. per cent., and commission | per cent. ; to return 5 per cent, if the ship sailed with convoy and arrived, which she did . on her arrival, however, it was found that 200 hhds. only were shipped : required the sum due to the insurers. Note. The insurers charge ^ per cent, on the value of the goods not shipped, in returning the premium upon them. 17. Insured 350 chests of tea, at £12, 10s. per chest, from Canton to Leith, at 9i gu. per cent. ; policy-duty 5s. per cent., and commission 5s. lOd. per cent. ; to re- turn 4 per cent, if the ship sailed with convoy and arrived, which she did : on her arrival it was found that only 300 chests were shipped, and these were so much damaged that they sold only for £11, 10s. per chest; whereas, had they been undamaged, they would have brought £13, 16s. : how much is due by the underwriters ? STOCKS. Stock is the name given to the money borrowed by government to defray the expenses of the nation ; it is also the term applied to the capital of any bank, rail- way, or trading company. AVhen £100 of stock is sold for £100 sterling, the price of stock is said to be at par ; the price of stock, however, is continually fluctuating. When we see the 3 per cents, quoted at 93, it signifies that £93 sterling is the selling price of £100 stock, and that £3 is the annual dividend on £100 stock, or £93 sterling. Stock is bought and sold through the agency of brokers, who charge usually i per cent, on the amount of the stock for their trouble. The following examples illustrate the several cases which are met with in stocks : Ex. 1. How much 3 per cent, stock at 93 can be purchased for £3131, 7s. 9d.? Here £93 : £3131, 7s. 9d. : : £100 : £3367, Is. 8d. stock. Ex. 2. How much will be received by selling £2150 Bank stock (7 per cent.) at £220^, and brokerage ^ per cent. ? Here £220^ — | = £220 sum received for £100 stock. Hence £100 : £2150 : : £220 : £4730 sum received. 52 STOCKS. Ex. 3. What rate per cent, is derived from the 3 per cents, at £96? £96 : £100 sterling : : £3 : £3i per cent. Ex. 4. How much must be invested in Russian 5 per cents. at 104i to produce an annual income of £300, allowing ^ per cent, for brokerage? Here 104^ + 5 = 104|. Hence £5 : £300 : : £104| : £6262, 10s. sum to be invested. Ex. 5. At what rate should money be invested in the 4 per cents, to yield 3^ per cent, interest? Sol. 3^ : 4 : : £100 : £114f per cent. 1. How mucli stock can be purchased for £68728, Os. O^d. in the 3 per cents, at 91^, 91 J. 92, 92 1, 93, 93-», 93i, and 93j per cent. ? 2. How much sterling money will be required to pur- chase £5750, 3 per cent, stock at 92|, 92^, 92i, 93, 93^, and 93^ per cent., including brokerage ^ per cent.? (Here £100 stock will cost | more tlian the prices given.) 3. Find the yearly income derived from investing £6012, 7s. O^d. in the 4 per cents, at 84, 84i, 85, 85^, 88, and 92^. 4. What rate per cent, is derived from the Russian 4^ per cents, at 95, 95j, 95i, 96, 96f , and 99 ? 5. How much sterling must be invested in the 4 per jcnts. at 83^ to produce an annual income of £252, 10s.? 6. At what rate should money be invested in Bank stock to produce 3^, 3f , 4, 4i, 4^, and 5 per cent. ? 7. What is the price of India stock (lOi per cent.), when £4752 can purchase £1728 stock? 8. What is the price of the 3 per cents., when £3412, lOs. invested in them produces £105 per annum? 9. If £8932 be invested in the 3j per cents, at 101^, and sold out at 102| ; what difference will it make in my income to reinvest the proceeds in Bank stock at 220? 10. When the 3 per cents, are at 93, India stock at 230, and Bank stock at 208 ; which is the preferable investment, including brokerage -| per cent. ? 11. Invested £3196 in the 3 per cents, at 94, and was obliged to sell at 92-| ; what was the whole loss ? 12. Invested £5194 in Danish 3 per cents, at 53, and sold out so as to gain £294 ; at what price was it sold ? 13. Invested £3570 in Bank stock at 212|, and sold out at 228^ ; what is gained, allowing -| per cent, for brokera<2fe ? STOCKS. 5.3 14. How mucli is derived annually by investing £5590 in India stock at 215 per cent. ? 15. A has £2400 in 3 per cents. ; how much must he invest in 3^ per cents, at 84 to have an income of £350 V 16. At what rate must money be invested in Russian 4i per cents, to yield 3| per cent. ? 17. If £2261 be invested in 3 per cents, at 84, and sold out at 85 J ; what difference will it make in my income to reinvest the proceeds in Dutch 4 per cents, at 95? 18. How much 3 per cents, at 99 f must be sold out to pay a debt of £931 ? 19. A father leaves his son ^ of his fortune in 3 per cent, stock, i in the 3| per cents., and the remainder £2100, in 4 per cent, stock ; what is his annual income ? 20. Invested £3683 in Russian 4 J- per cents, at 95}, and sold out so as to gain £43-5 ; at what price was it sold ? EQUATION OF PAYMENTS Is the method of finding the time when two or more debts due at different periods may be discharged at one payment without loss to either party. Ex. Find the time for discharging at one payment £300 due in 3 mo., £200 in 4i mo., and £400 due in 6 mo. Sol. Multiply each sum by its 3 X 300 = 900 respective time (3 X 300, &c.) ; then 4^- X 200 = 900 divide the sum of the products 6 X 400 = 2400 (4200) by the sum of the debts (900). 9^00 ) 4200 Ans. 4§ mo. Find the time for discharging at one payment, 1. £40 due in 3 mo., £45 in 4 mo., and £55 in 6 mo. 2. £110 due in 72 da., £140 in 84 da., £200 in 96 da., and £240 in 108 days. s. £300 due in 210 da., £420 in 340, £500 in 365 da. 4. A debt, -i of which is due in 6 mo., | in 7 mo., ^ in 9 mo., and the remainder in 10 months. 5. A debt, i of which is due on Christmas-day, ^ on Whitsunday, -J on Nov. 11, and the rest on Jam 1. 6. A debt, -^ of which is due on March 14th, and \ on the 14tli of each succeeding month. 54 EQUATION OF PAYMENTS. The following exercises may be solved in a similar mamier: What is the average price per qr. of, • 7. 40 qrs. wheat at 60s. 6d. per quarter, 20 qrs. at 65s., 30 qrs. at 75s. 6d., and 60 qrs. at 80s. ? 8. 12 qrs. barley at 42s. per qr., 18 at 45s., 20 at 39s., 24 at 36s. 6d., 30 at 45s., and 36 qrs. at 43s. 4d. ? 9. 10 qrs. oats at 25s. per qr., 20 qrs. at 23s. 6d., 25 qrs. at 26s., 30 qrs. at 26s. 6d., and 45 qrs. at 27s. ? 10. A wine-merchant mixes 5 gals, sherry at 28s. per gal., Avith 8 gals, at 30s. 9d., 10 at 36s., 12 at 42s. 8d., and 16 at 42s. 6d. ; what is the average price per gal. ? 11. A grocer mixes 12 lbs. tea at 3s. 4d. with 42 lbs. at 3s. 8d., 25 lbs. at 4s., 28 lbs. at 4s. 3d., and 30 lbs. at 4s. 6d. ; what should the selling price per lb. of the mix- ture be to gain £4, lis. 4d. upon the whole ? 12. 8 oz. of gold, 24 carats fine, are melted with 16 oz. 23 carats fine, 18 oz. 21-| carats fine, and 20 oz. 18 carats fine ; what i^ the average fineness of the mixture per oz. ? DISTRIBUTIVE PROPORTION Is the method of dividing a number into parts propor- tional to as many given numbers. This rule is employed to divide the gain or loss of a company in proportion to the shares or stocks of eacli partner, and is then termed Felloivship or Partnership^ which is either Simple or Compound: Simple Fellowship, -when each partner's gain is proportional to his stock only ; Compound Fellowship, when each partner's gain is proportional to his stock and the time of its being employed. SIMPLE FELLOWSHIP. Ex. Three merchants, A, B, and C, in company, gain £420; A\s stock is £500, B's £400, and C's £300: re- quired eacli man's share of the gain. Sol. Add the ^^00 + £400 + £300 = £1200 stocks; then state ^1200 : £500 : : £420 : £175 A's share. asthesum(£1200) 1200: 400:: 420: 140 B's >• IS to each part- 1200 : 300 : : 420 : 105 C's .> ner's stock, so is Whole gain, £420 the gain (£420) to each partner's gain. DISTRIBUTIVE PROPORTION. 55 1. Divide 7020 into 3 parts proportional (1) to tlie niim- . bers 55, 65, and 75, and (2) to the numbers 3, 4, and 5. 2. Three merchants, X, Y, and Z, gain by trade £225 ; X's stock is £425, Y's £350, and Z's £225 : find each man's share of the gain. 3. A bankrupt owes £1470 ; A's claim is £350, 17s. od., B's £415, 8s. 9d., C's £420, 16s. 3d., and D's the rest ; his effects amount to £980: what will each receive? 4. Divide 4428 into 3 parts proportional (1) to the num- bers 2, 3, and 7, and (2) to the fractions a, i, and i. 5. A tax of £2997 is to be raised from 4 towns ; the number of inhabitants in each is respectively 2100, 2400, 3000, and 3600 : how much should each town pay ? 6. Three graziers, L, M, and N, rent a park for £104, 7s. 6d. ; L puts in 12 oxen and 8 horses, M 8 oxen and 12 horses, and N 10 oxen and 10 horses ; how much ought each to pay, if 2 oxen eat as much as 3 horses ? 7. Gunpowder consists of 74-8 parts of nitre, 13*3 of charcoal, and 11*9 of sulphur ; how much of each will be required to make 214 cwt. 32 lbs. of gunpowder? 8. Gun-metal consists of 100 parts of copper and 11 of tin ; how much of each of these is there in a brass-gun which weighs 19 cwt. 3 qrs. 8 lbs. ? 9. Divide £1620, 15s. among D, F, and E, giving D 6 as often as F 5, and E 4 as often as F 5. 10. A pound troy of sterling silver consists of 37 parts of pure silver and 3 parts of alloy, and is coined into 66s. ; how much of each is in 231s. ? 11. A pound troy of sterling gold consists of 22 carats of pure gold and 2 carats of alloy, and is coined into £46-725 : what quantity of each is there in £700, 17s. 6d. ? 12. 37 ac. 3 ro. 2 per. of ground is to be divided among three persons. A, B, and C, in proportion to their estates ; A's being worth £560 a-year, B's £640, and C's £700 : what part should each receive ? 13. Four companies of 60, 56, 52, and 36 men, require to furnish 51 men daily for a particular duty, in proportion to their strength ; how many must each furnish ? 14. A gentleman hired a carriage for 40 miles for £5, 5s. ; at the 10th milestone he admits 3 others, and at the 15th milestone other two : what should each pay? 56 DISTRIBUTIVE PROPORTION. COMPOUND FELLOWSHIP. Ex. A and B enter into partnership ; A contributes £500 for 5 months, and B £630 for 10 months ; they gained £572 : what share of the gain should each receive? Sol. Mult, eachpart- 500 X 5 = 2500 ner's stock by the 630 X 10 = 6300 time it continues, Sum of prod. =. ggQQ then State as the gSOO : 2500 : : £572 : £162, 10s. = A. sum of the products 8800 : 6300 : : 572 : 409, 10s. = 13. and 6300), so is the whole gain (572) to each partner's gain. 15. L and M enter into partnership ; L advances £525 for 6 months, and M £375 for 8 months ; they gain £221, 8s. : what is the share of each? 16. A, B, and C, engage in trade ; A's stock of £1200 continues for 8 months, B's of £1575 for 10 months, and C's of £1455 for 12 months ; they gain £998, 18s. : what is each man's share ? 17. X, Y, and Z, rent a grass-park for £39, 19s. ; X put8 in 12 oxen for 4 months, Y 15 for 6 months, and Z 18 foi 8 months : what part of the rent should each pay? 18. E, F, and G, enter into company; E advances £1200 at the first, after 3 mo. F advances £1400, and after 5 mo. G advances £1600; the whole gain during 12 mo. was £573 : required each man's share. 19. A and B engage in trade for 12 months ; A advan- ces at first £1200, and after 7 mo. withdraws £500 ; and B advances at first £800, and after 5 mo. £400 more ; they gain £2004, 15s. : how much of it belongs to each ? 20. The wages of A and B for 4 weeks amount to £12, 19s. ; A works 9 hours a-day for f of the time, and 10.^ ho. a-day for the rest of the time ; while B is idle one day a-week, and works 11 ho. a-day the rest of the week : how much of the wages should each receive ? 21. A, B, and C, rent a grass-park for 14 months at a rent of £97 ; A put in 20 oxen, and paid £30 ; B put in 25 oxen, and paid £40 ; and C put in 36 oxen, and paid the remainder : how long should each hold the park ? Note. Each party's proportional is here found by dividing the sum which he paid by the number of oxen he put into the park ,* thus, A's proper, is here 30 -^ 20 = 1^, &c. DISTRIBUTIVE PROFORTIOX. 57 22. Four graziers rented a field for 9 mo. at a rent of £80 ; A pat in 120 sheep, and paid £14 ; B 30 oxen, and paid £24; C 180 sheep, and paid £18 ; and D 36 oxen^ and paid the remainder : liow long should each man re- tain the field, if 5 sheep eat as much as an ox ? 23. A common, consisting of 506 ac. 23 per., is to be divided among 4 persons. A, B, C, and D, whose estates on which their claims are founded are respectively £8000, £7500, £6400, and £6000 yearly, while the value of the land allotted to each is 64s., 60s., 50s., and 48s. per acre : what quantity of the land should each receive ? 24. Tlie gain of three merchants was £802, 10s., of which A's share was £360, B's £262, 10s., and C\s the remain- der ; now A's stock of £4000 continued 6 mo., B's 5 mo., and C's 4 mo. : what was the stock of each ? PKOFIT AND LOSS Is that branch of Arithmetic which treats of the gains and losses of merchants, and which enables them to fix the prices of their goods so as to gain or lose so much per cent, upon them. The price at which goods are bought is called the prime cost, tlmt at which they are sold the selling pt^ice ; when the selling price is greater than the prime cost^ the difference is called gain, otherwise it is called loss. The calculations are made by means of the Compound Rules, Practice and Simple & Compound Proportion. Ex. 1. Bought tea at £21 per cwt., and sold it at 4s. lO^d. per lb. ; what was the gain or loss per cwt. and per lb. ? Obs. The S. P. being greater than the P. C. per cwt., the difference (£6, 6s.) is the gain per cwt., from which the S. P. of 112 lbs. at 4/101 — £27,6s. Prime cost of do. =21 Gain per cwt. = £6, 6s. £6, 6s. H- 112 = 1/1^ gain per lb. gain per lb. is found by dividing by 112. Ex. 2. Bought tea at 3/9 per lb. ; at what price should it be sold per lb. to gain 10 per cent.? Obs. £100 worth is sold for £110; hence £100 : £110 : : 3/9 : 4/1^ S. P. per lb. Ex. 3. Bought tea at 3/9 per lb., and sold it at 4/1^ ; what was the gain per cent. ? Obs. 4/1 1 — 3/9=4^d. is the gain on 8/9; hence 3/9 : 4id. : : £100 : £10 the gam per cent. 58 PROFIT AND LOSS. Ex. 4. Gained 10 per cent, bv selling tea at 4/1^ per lb. ; what was the P. C. per lb. ? Sol. £110 : £100 : : 4/1^ : 3/9 P. C. per lb. Ex. 5. Bought sugar at 37/6 per cwt. ; at what price should it be sold to lose 8 per cent. ? Obs. £100 worth is sold for £92 ; hence £100 : £92 : : 37/6 : 34/6 S. P. per cwt. Ex. 6. Gained 7^ per cent, by selling coffee at 1/9^ per lb. ; what is gained or lost per cent, by selling it at 1/10 per lb. ? Sol. 1/9| : 1/10 : : 107^ : 110, & 1 10— 100=£10 per cent. gain. Ex. 7. Bought goods at 15/3 and 4 months' credit, interest at 5 per cent. ; at what rate should they be sold to gain 6 per cent., and allow a discount of 4 per cent. ? For the 4 months' cred. 101§ : 100 : : 183d. : 16/6| S. P. " the gain ... 100 : 106 " the discount . . 96 : 100 1. Bought 3 cwt. 3 qrs. 14 lbs. of tea at 3s. 9d. per lb. and sold it at £23, lis. 4d. per cwt. ; what was the gain per lb., per cwt., and on the whole? 2. Bought 4 casks sugar, each 3 cwt. 2 qrs. 21 lbs., at od. per lb. ; what should the whole be sold for to gain 14s. per cwt ? 3. Sold 143 yds. at 10s. 3d. per yd. and gained £13, 8s. l^d. ; what was the P. C. of 1 yd. and of the whole ? 4. Bought muslin at Is. 4^d. per yd. ; how should it be soh to gain 7^ per cent. ? 5. Bought soap at 4|d. per lb., and sold it at 5|d. ; how many lbs. must be sold to gain 13s. 9d. ? 6. Sold 326 dozen wine at 31s. 6d. per doz. and gained £24, 9s. ; what was the P. C. of 1 doz. and of the whole ? 7. How much per cent, is gained by selling Is. worth of goods for Is. l^d. ? 8. Bought goods at £3, 6s. 8d. ; how should they be rated to gain 4 p. cent., and allow the purchaser a discount of 5 p. cent.? 9. Bought sago at 70s. per cwt., and sold it at 73s. 6d. ; what was the gain per cent. ? 10. Gained 3^ per cent, by selling 126 yds. of cambric for £48, 16s. 6d. ; what was the P. C. per yd. and of the whole? 11. Bought linen at 3s. 2d. per English ell, and sold it at the same per yd. ; what was the gain or loss per cent. ? 12. At what price should a yard of gingham which cost Is. 5d. a-yard, be sold to gain 15s. 8|d. on 151 yds.? 13. Lost 4* per cent, by selling goods at £72, lis. 3d.; what was their prime cost ? 14. Gained 10 per cent, by selling coffee at £10, 10s. lOd. per cwt. ; what was the prime cost per cwt. and per ton? PROFIT AND LOSS. 59 15 Bought 4 cwt. 3 qrs. 21 lbs. of raisins at 98s. per owt. ; how much per cent, was gained by selling the whole for £28, 2s. lO^d., the expenses of the sale being 16s. 5|d.? 16. Grained 3^ per cent, by selling tea at 5s. 3d. per lb. ; what was gained or lost per cent, by selling it at 5s. per lb. ? 17. The prime cost of a book is 6s. 8d., the expense of sell- ing is 3 per cent., and the gain is 12 per cent. ; what is the selling price of 40 copies of the book ? 18. Lost 3 J per cent, by selling butter at 16s. 3|d. per stone ; what was gained or lost per cent, by selling it at 1 s. 4d. per lb.? 19.. By selling 5 apples for 2d., 3 per cent, is gained; what is gained or lost per cent, by selling 18 for 6d. ? 20. How much per cent, is 2s. 6d. profit per £1 ? 21. Bought 50 reams of paper at 18s. 6d. per ream; 3 per cent, was lost in selling : what was the whole loss ? 22. A merchant bought 252 gallons of wine at 35s. 6d., but J of it being damaged, he sells it at a loss of 2^ per cent. ; how must he rate the remainder per dozen to gain 5 per cent, on the whole ? 23. Bought 4 casks of brandy, each 126 gallons, at 5s. 3d. a -bottle : now each cask leaked a gallon ; how should the re- inainder be rated per gallon to gain 10 per cent, and allow a discount of 4 per cent. ? 24. Bought a horse for £40, and sold it for £45, and 3 mo. credit, interest at 5 per cent. ; required the gain. 25. Purchased 108 yds. of cloth at 18s. 9d. a-yd., but being damaged, I am willing to lose 5 per cent, in selling it ; for how much must a yard and also the whole be sold ? 26. A buys goods to the amount of £2025, and sells them to B for £2250, who in turn disposes of them to C at a profit of 4 per cent. ; how much per cent, above their prime cost did C pay for them ? 27. Bought 350 qrs. of wheat at £2, 12s. 6d. per qr., and sold f of them at a profit of 7^ per cent., and the rest at a loss of 2^ per cent. ; what was gained upon the whole? 28. Purchased 4350 yds. of linen at 2s. 7^d. per yd., and field i of the whole at 2s. 8^d., | at 2s. 9d., and the remain- der at 10 per cent, profit; requu-ed the price of the remain- der per yd., and the gain upon the whole. 29. By selling an article for £43, 10s. I lost 3 J per cent., and recovered the loss by selling another for £19, 10s. ; what was the gain per cent, on the second article ? 30. Bought sugar at 70s. per cwt. ; how must I sell it per cwt. to gain 5 per cent., and allow the purchaser a discount of 4 per cent, and 4 months' credit, interest at 6 per cent. ? 60 EXCHANGE Is the method of valuing the money of one country in that of another, according to a certain rate. The intrinsic value of the money of one country com- pared with that of another is called the Par of Exchange, and is determined by the weight and fineness of their coins. The Com'se of Exchange at any time is the value of a fixed sum of the money of one country estimated in that of another : from various circumstances this is contin- ually fluctuating. In some countries, money is distin- guished into Banco and Currency, or into Specie and Paper money, — the former being more valuable than the latter by a certain rate per cent., which is called agio, discount or premium. TABLES OF FOREIGN MONEYS. France. — 100 centimes=10 decimes=l franc=9^d. ster. nearly. Par of exch. with London in gold, 25 francs 22 cents for £1 ster. ; in silver, 25 francs 57 cents for £1 ster. Holland and Belgium.— 100 cents = 20 stivers = 1 florir = Is. 8d, Par of exch. with London, 12 fl. 9 cents for £1. Hamburg. — 192 pfennings — 16 schillings = 1 mark. 3 marks or 48 schillings = 1 rixdoUar of exchange. Par of exch. with London, 13 marks 10^ sch. for £1 ster. Money is here divided into banco and currency ; the agio fluctuates between 20 and 25 per cent. Accounts are kept in currency, and exchanges are made in banco. Portugal. — 1000 reas = l milrea=:57p. ster.; 400 reas = 1 crusado, and 1000000 reas = 1 conto = £239, lis. 8d. ster. The discount on paper money is about 24 per cent. ; ex- change money is ^ in paper. Russia. — 100 copecs=: 1 silver ruble = 37 Jd. ster 1 paper ruble = 10| ster. nearly. Turkey. — 40 paras = 1 piastres 2 |d. ster. Par of exch. with London, 100 piastres for £1 ster. North America and West Indies. — £100 ster. at par == £llli currency, or £100currency= £90 ster. In Jamaica, £1661 currency = £100 ster. United States. — 100 cents=10 dimes=1 dol.=4s. 6d. ster. The par of exch. with London was originally 4| dol. for £1 Bter. ; this value being now too small, a variable premium of 9 or 10 per cent, is added to the par value. EXCHANGE. 61 East Indies. — 192 pice = 16 annas = 1 sicca rupee = 2 s. ster. nearly. 116 current rupees =100 sicca rupees; 100000 rupees = a lac, and 10 million rupees =: a crore. The Calculations of Exchange are made by means of Proportion or Practice. Ex. How much sterling money is equal to 750 copecs, exchange at 3s. 2id. per ruble ? Sol. 100 copecs : 750 copecs : : 3s. 2^d : £1, 4s. Ofd. ster. 1. How much sterling money in 11619 francs 30 cents, and in 21126 fr., exch. at 25 fr. 20 cents, and at 25 fr. 15 cents per £1 ster. ? 2. In £420, 17s. 6d. and £580, 13s. 4d., how much French money, exch. at 25 fr. 20 cts., and 25 fr. 16 cts. per £1 ster. ? 3. How much sterling money in 2145 marcs 15 sch. and in 5845 marcs 2 sch., exch. at 13 marcs 10 sch. and 13 marcs 8 sch. per £1 ster. ? 4. In 456325 reas, and in 874625 reas, how much sterling, exch. at 56d. and at 57|d. per milrea ? 5. In £212, 17s. 6d. and in £318, 2s. 6d., how much money of Holland, exch. at 12 fl. 8 cts. and at 12 fl. 9 cts. per £1 ster.? 6. How much Turkish money in £124, 5s. and in £340, 7s. 6d., exch. at 100 piastres, and at 103^ piastres per £1 ster.? 7. How much sterlhig money in 100 rubles 50 copecs, and in 1825 rubles 25 copecs, exch. at lOd. and 10|d. per ruble? 8. How much Hamburg currency in £360, and in £756, 13s. 4d., exch. at 13 marcs 8 sch. banco per £1 ster., agio 20 per cent., and at 13mar. 1 sch. banco per £1 ster., agio 2 5 p. cent.? 9. How much sterling in 435 rupees 9 annas, and in 750 rup. 5 an. 8 pice, exch. at 2s. and 2s. 4d. per rupee? 10. How much United States cun-ency in £250, 12s. 6d., and in £742, 17s. 6d., exch. at 4| dollars per £1 ster., pre- mium 8 and 10 per cent. ? rr: How much sterling in £364, and in £1008 Canadian currency, exch. at 112 and 112^ per cent.? 12. How many current rupees in £376, 5s. and in £980, exch. at 2s. and 2s. O^d. per sicca rupee? 13. How many rupees in 3452 dollars 80 cents, and in 5179 dol. 20 cents, exch. at -415 dol. and '416 dol. per rupee? 14. How much Hamburg currency in 706 francs 70 cents, and in 869 francs 50 cents, exch. at 100 marcs banco for 185 francs, agio 20 and 25 per cent. ? 15. In 11880 milreas current, and in 10560 milreas current, how much ster. at 56d. per milrea, agio on paper money 20 per cent., and at 57}d. per mil., agio on paper money 24 p. cent.? 62 DUODECIMALS Is a method employed for multiplying feet and inches, &c. by feet and inches, &c. A foot is divided into 12 inches, an inch into 12 parts or primes, and a prime into 12 seconds. Ex. Multiply 6 ft. 3 in. 4 pts. by 7 ft. 2 in. 5 pts. Sol. Arrange the numbers so that ft. may be below ft., in. below in., &c. Multiply by the ft. (7) in the multi- plier as in Compound Multiplica- tion ; in the same way, multiply by the inches (2), but write the pro- duct one place nearer to the right hand ; again, multiply by the parts (5), and write the pro- duct one place nearer to the right hand than the last ; then add the separate products, carrying at 12. The answer is 45 s. ft., 2 twelfths of a s. ft., six 144ths of a s. ft. {i. e. 6 s. in.), and eight 144ths of a s. in. ; now 2 twelfths = twenty-four 144ths ; hence the answer may be written 45 s. ft. (24 + 6 y^^) s. in. = 45 s. f. 30 y?^ s. in. 6 ft. 3 in. 4 pts. 7 2 5 43 11 4 12 6 8 31 4 8 45 2 6 8 ft. in. pt. ft. in. pt. ft. in. pt. ft. in. pt. 1. 7 4 6 X 4 6 5 7 6 4 2. 10 5 3 X 3 5 4 3 10 5 3. 11 6 9 X 2 2 4 3 4 8 6 11 9 4. 12 8 4 X 3 6 9 7 3 6 9 10 3 5. 15 9 10 X 2 6 3 3 9 6 8 9 4 6. 16 11 2 X 3 9 6 6 8 9 10 11 3 7. 18 9 8 X 4 7 6 7 9 3 12 11 9 8. 21 10 7 X 6 3 5 9 5 4 11 3 6 9. 48 4 9 X 7 10 11 10 10 5 11 8 9 10. 56 3 6 X 14 6 4 15 9 2 17 6 6 11. 78 6 4 X 21 4 6 25 7 9 32 8 4 12. 99 11 8 X 36 10 3 49 11 9 54 7 6 Note. The area of a board is found by multiplying the length by the breadth, and the cubic content by multiplying the length, breadth, and thickness together. 13. Find the area of a board 4 feet 7 in. broad and 18 feet 9 in. long. U. Find the area of a floor 12ft. 6 in. 4 pts. by 18 ft. 6m. 3pts. 15. Find the area of a wall 17 ft. 4 in. 6 pts. long and 10 ft. 6 in. high. 16. Find the content of a cistern 7 ft. 4 in. long, 6 ft. 6 in. DUODECIMALS. 63 deep, and 3 ft. 9 in. wide, and the number of gallons it would contain, each 277J c. in. 17. Find the cubic content of a block of marble 3 ft. 4 in. long, 2 ft. 10 in. wide, and 1 ft. 8 in. thick. 18. What is the length of a floor containing 44 s. yd. 96 s. in., whose breadth is 17 ft. 6 in. ? 19. What length of carpet f wide will cover a floor 22 ft. G in. long and 18 ft. 4 in. broad? 20. How much paper will be required to cover the walls of a room 27 ft. 8 in. long, 20 ft. 3 in. broad, and 12 ft. 6 in. high? 21. How many gallons of water must be run off from a cistern 8 ft. 6 in. long, 4 ft. 3 in. broad, and 6 ft. 8 in. deep, to make the surface sink a foot ? 22. The paving of a court-yard cost £13, 4s. at 5s. 6d. per sq. yard ; how broad is it, its length being 36 ft. ? INVOLUTION. When a number is multiplied by itself any number of times, the process is called Involution, or the raising of Powers. The original number is called the root^ and the products powers of the root. Powers are often indicated by writing the number once, and a small figure (called the index or ex- ponent of the power) a little to the right above the number, denoting how many times the number is to be taken as a factor. Thus, 5 2=5x5 = 25, is the second power or square of 5. 53 =5 X 5x5=1 25, is the third power or cube of 5. 5 7 = 5X5X5X5X5X5X5=78125, is the 7th power of 5. r ^ \K 3X3X3X3X3 243 ,, ^ „,, - _ (A)'= 11X11X11X11X1 1 = 161051 = t^^ ^^*^ P^^^^ ^^ I'l- It may be noticed that 5^=53 X 5*= 125 X 625 = 78125 as above, i. e. the sum of the indices of powers of the same number, is the index of their product. 1. Find the cubes of 21, 33, 44, 67,89,11-9, l-25,& 1-075. 2. Raise 24«, 75^, l-05«, 2-15«, •025r and 1-025^ ^-^=£=25^ 3.Raise(f)«, (|)«, i^sy, {^%y, GV)^ and(^)^: .^:^ 4. The side of a square is 11 feet; finS its- area. (ll<*=::r area in s. feet.) X* ' [ i 5. The side of a cube is 8 feet; find its content. {S^S' " ^ content in c. feet.) c^sr ^ 64 INVOLUTION. 6. The side of a square court-yard is 22 ft. 6 in. ; what is its area ? 7. The side of a cubic cistern is 6 ft. 3 in. ; what is its content ? 8. A cubic foot of quartz weighs 2640 oz. ; required the weight of a piece 4^ in. in the side. 9. A cub. ft. of chalk weighs 2784 oz. ; find the weight of a column 4 ft. 6 in. in the side. 10. A cub. ft. of water weighs 1000 oz. ; what weight of water does a cubic cistern contain, whose side is 4 ft. ? 11. How many dice, i in. in the side, can be cut from a cubic piece of ivory 6 in. in the side ? 12. How many squares, 3 in. in the side, can be cut from a square piece of pasteboard, whose side is 1 ft. 6 in. ? EVOLUTION Is the method of extracting the root of a given power. The square root is the method of extracting the second root of a given number, or of finding a number which, when raised to the second power, produces the given number; thus, the square root of 169 = 13, for IS'* = The cube root is the method of extracting the third root of a given number, or of finding a number which, when raised to the third power, produces the given num- ber]^ thus, the cube root of 1331 is 11, for 11 ^ = 1331 ; The sign V placed before a number indicates that the square root of the number is to be taken ; %/ placed before a number indicates that the cube root of the number is to be taken. EXTRACTION OF THE SQUARE ROOT. Ex.1. Extract the square root of 9177-64. Ans. 95-8. 91,77,-64(95-8 root. 81 185 1077 5 925 Sol. 1. Divide the given num- ber into periods of two figures each, beginning at the units^ figure. 2. Find the greatest square number in the first period (81), place its root (9) on the right 1908 15264 15264 of the given number, and subtract its square (81) from the EVOLUTION. 65 first period (91) ; then to the remainder (10) annex the next period (77) for a resolvend (1077). 3. Write the double of the figure in the root for a partial divisor (18), and find how often it is contained in the resolv- end (1077), omitting its right-hand figure (7); place the number of times (5) after the last figure of the root, and after the partial divisor (18), for a complete divisor (185); then multiply the complete divisor (185) by the figure last placed in the root (5), and subtract the product (925) from the re- solvend (1077) : to the remainder (152) annex the next period (64) for a new resolvend (15264). 4. To the last complete divisor (185) add its right-hand figure (5) for a new partial divisor, and so proceed until all the periods are brought down. Note. When there is a remainder, after bringing down the last period, the root may be carried on decimally, by annexing periods of two ciphers each to the remainder. The square root of a fraction is found by taking the square roots of its terms, if they are exact squares ; if not, the frac- tion must be reduced to its equivalent decimal and its square root taken. Extract the square roots of, 1. 5184, 6889,9801, 14884, 17161, 22201, 297025,&958441 2.1100401,1279161,3786916,4008004,14356521,60481729 3. 9862-4761, 99-980001,56-725, 597-184, 674-85,&948-625 4.127-3,2479-6, 118-63, 2459-147, 4-8, 5-4245, & 121-45 Note. The repeating figures of the decimals must be annexed in their order, in periods of two figures each. 5.2, 3, 5, 7, 11, 12, 13, 17. and 19, each to 6 places of dec. 6-tVt, t%V nh 3^1, Hff, Hi, 14^, 17Vt, and 24f. 7. 102030201, 10020210201, and 9018027018009. Ex. 2. Find a mean proportional between 9 and 25. Ans. 15. Sol. V9 X 25 = V225 = 15, for 9 : 15 : : 15 : 25. 8. Find a mean proportional between 7 and 28, 15 and 135, 24 and 96, 18 and 288, 44 and 396, 19 and 46. Note 1. The side of a square equal to any given area is the square root of that area. 2. Circles are to each other as the squares of their diameters. 3. In a right-angled triangle, the square of the hypothenuse, or side opposite the right angle, is equal to the sum of the squares of the other two sides. GQ EVOLUTION. 9. Find the side of a square to contain 756 s. yds. 10. A gentleman's estate contains 4851 ac. 1 per., and ho wishes another of equal area in the form of a square ; required its side. 11. An army of 58564 men is to be formed into a square ; how many men will the front contain ? 12. The diameter of a circular pond is 540 ft. ; what is the diameter of another 5 times as large ? 13. Two ships sail from the same port, the one due east 180 mis., and the other due south 230 mis. ; what is the dis- tance between them ? 14. A wall is 83 ft. high ; what length of line will reach from the top to a point 67 feet from its base ? 15. The wages of a certain number of men amounted to £561, 2s. 6d. at 2s. 6d. per day ; they wrought as many days as there were men employed ; what was the number of men ? 16. A ladder 84 ft. long reaches from the edge of a ditch, 40 ft. wide, to the top of a wall on the opposite side of the ditch; what is the height of the wall? 17. A room is 48 ft. long, 36 ft. broad, and 16 ft. high ; what is the length of each of the diagonals, and also the diagonal of the contained space ? 18. What is the length and breadth of a parallelogram 4 times as long as it is broad, whose area is 3 ac. ? 19. 79524 trees 16 ft. distant are planted in a square plan- tation ; what is the length of the side ? 20. A room as broad as it is high, and 32 ft. 6 in. long, contains 8937 c. ft. 1 254 c. in. ; find the height. 21. Arrange 24964 soldiers so that the number of men in rank may be 4 times the number in file. 22. The paving of a square enclosure cost £36, 9d. at 9d. per square yard ; find the length of its side. EXTRACTION OF THE CUBE EOOT. Ex. Extract the cube root of 12-812904. Ans. 2*34. Sol. 1. Divide the given number into pe- riods of 3 figures each, beginning at the units^ figure. 2. Find the greatest cube number in the first period (8), place its root (2) towards the right of the given number, and subtract its cube (8) from the first period (12); then to 2"-X 300 12-812,904 (2-34 1200 8 4 189) 4812 63X3 1389 y 4167 6 9) 645904 694 X ^ 158700 645904 2776 161476 EVOLUTION. 67 the remainder (4) annex the next period (812) for a resolvend (4812). 3. Write 300 times the square (4) of the figure in the root for a partial divisor (1200), and find how often it is contained in the resolvend (4812), then place the number of times (3) to the right of the figure in the root. Again to the former part of the root (2) add its double (4), to the sum (6) annex the trial figure (3), and multiply this number (63) by it (3) ; then add the product (189) to the partial divisor (1200) for a complete divisor (1389). Multiply this number (1389) by the figure last placed in the root (3), subtract the pro- duct (4167) from the resolvend (4812), and to the remainder (645) annex the next period (904) for a new resolvend (645904). 4. Place the square of the last figure in the root (9) below the last complete divisor (1389), add it (9) and the two lines above it (189 and 1389) together, and to the sum (1587) annex two ciphers for a new partial divisor (158700). 5. With this partial divisor find another figure (4), and place it in the root. To the number on the right (63), which was multiplied by the last figure of the root (3), add the double of that figure (6), annex to the sum (69) the new trial figure (4), then multiply the number thus found (694) by the trial figure (4), and add the product (2776) to the partial divisor (158700) for a complete divisor (161476), and so proceed till all the periods are brought down. Note. When there is a remainder, after bringing down the last period, the root may be carried on decimally by annexing periods of three ciphers each to the remainder. The cube root of a fraction is found by taking the cube roots of its terms when they are exact cubes ; if not, the fraction must be reduced to its equivalent decimal and its cube root taken. Extract the cube roots of, 1. 76765625, 143877824, 260917119, 485587656. 2. 997002999, 25128-011089, 143795466-919, 865-250742889. 3. 14-75, 118-62, 1-47825, 7-6, 8-36, 94-8, to 6 places of dec. 4. 2, 4, 6, 7, 9, 12, 13, 16, to 6 places of decimals. 5. 1030607060301, 27054306369020601. «• Uh mh HU, H, 51, -000000405224. Note. Similar solids are to each other as the cubes of their like dimensions. 68 EVOLUTION. 7. The side of a cubic vessel is 10 in. ; what should be the side of another to contain ^ as much ? 8. A block of granite is 6 ft. long, 6 ft. broad, and 4 ft. thick ; what are the dimensions of another 3 times as heavy? 9. A stone is 8^ ft. long, 7 ft. broad, and 5 ft. thick ; what are the dimensions of another 9 times as large, and the side of a cube equal to both ? 10. A cubic block of marble is 8 ft. in the side ; what are the length and breadth of another 3 times the weight, whose thickness is 3 ft., and length twice the breadth ? 11. The solid content of a cube is 407 ft. 1673-in. ; how many square ft. are in its surface ? 12. A vessel contains 411540 c. in., and has its sides in proportion to the numbers 3, 4, and 5 ; what are its sides? COMPOUND INTEREST. When a sum of money is put out to interest, and its amount at the end of a fixed period is considered the principal for the same period and at the same rate, the original sum is said to be improved at Compound Interest. Case I. Given the principal, rate, and time ; to find the amount and the interest. Ex. 1. Find the compound interest of £100 for 3 years at 2 per cent, per annum, the interest payable yearly. Here 2 per cent. =: j^-^ = ^\^. j5»5 ) £1 00 Principal for 1 st year. 2 Interest for 1st year, jig) 102 Principal for 2d year. 2*04 Interest for 2d year. 3^) 104-04 Principal for 3d year. 2-0808 Interest for 3d year. 106-1208 Amount at end of 3 years. 100 Principal for 1st year. £6, 2s. 5d. = £6*1208 Interest for 3 years. 1. Find the compound interest of £875 for 5 years at 2, 2|, 4, 5, 7|, and 10 per cent., the interest payable yearly. 2. Required the amount of £450, 10s. for 6 years at 2, 2^, 4, 5, 7^, and 10 per cent, per annum, compound interest. When the number of payments of interest is small, and the rate an aliquot part of 100, this method answers very well. The followiner method is suitable for all cases : COMPOUND INTEREST. 69 The amount of £1 for 4 years at 3 per cent., when the in- terest is payable yearly, is that power of the amount of £1 for 1 year (1'03) which corresponds with the number of years (4), i.e. (l-OS)* ; when the interest is payable half-yearly, the amount of £1 for half-a-year is 1-015, and for 4 years or 8 half-years it is (1-015)« ; in the same way when the interest is payable quarterly the amount of £1 for 4 years is (1 -0075) ^ «. Ex. 2. Find the compound interest of £375 for 5 years at 5 per cent., the int. payable (1) yearly, (2) half-yearly. Sol. 1. Amt. of £1 for 5 ye. at 5 p. cent.=(l-05)s=l -276282 Multiply by 375 Amt. of £375 for 5 ye. at 5 per cent. = £478*605750 Subtract 375 gQj^ 2. Compound interest of £375=£103,12s.l4d. Amt.of£lfor5ye.,i.e.l0h.-ye.at5p.c.=(l-025)^<'=£l-28008 Multiply by 375 Amt. of £375 for 10 h.-ye. at 5 p. cent, per an.=:£480-03000 Subtract 375 Compound interest of £375=£105,0s.7^d. 3. What is the compound interest of £750 for 5 years at 3 per cent., 8 years at 4 per cent., and 7 years at 3 J per cent., the interest payable yearly ? 4. What is the amount of £350 for 6 years at 2 p. c, 8 ye. at 2 J p. c, and 10 ye. at 3i p. c. compound interest, the in- terest payable yearly ? 6. What is the compound interest of £120, 10s. for 4 ye. at 2 p. c, 5 ye. at 4 p. c, and 6 ye. at 5 p. c, interest payable half-yearly ? 6. What is the amount of £240, 12s. 6d. for 2 years at 3 p. c, 3 ye. at 4 p. c, and 2f ye. at 5 p. c. compound interest, the interest payable quarterly ? 7. What is the compound interest of £375, 14s. for 3J ye. at 3 p. c, 4 ye. at 2J p. c, and 4^ ye. at 6 p. c, the interest payable three times yearly ? 8. A merchant began business with £1000, which he in- creases every half-year by J ; what will his capital be at the end of 5 J years ? Case II. To find the interest on bonds when the in- tervals between the payments are great. Ex. Lent on bond £1050 at 4 per cent., Aug. 12th, 1855; and received on Sept. 15th, 1856, £300; on Oct. 70 COMPOUND INTEREST. 20th, 1869, £350 : what was the balance due, including the interest on Dec. 15th, 1870 ? Ans. £502, 19s. 84d. Aug. 12, 1867. Lent at 4 per cent., . . . £1050 Interest on ditto for 399 days, 45-9123 Amount, 1095-9123 Sept. 15, 1868. Received in part, 300 Balance, 795-9123 Interest on ditto for 400 days, 34-8893 Amount, 830-8016 Oct. 20, 1869. Received in part, 350 Balance, 480-8016 Interest on ditto for 421 days, 22-1827 Amount, 502*9843 Dec. 15, 1870. Received in full, 502-9843 9. A bond of £975 became due on January 15th, 1866, of which was paid April 21st, 1867, £250; July 29th, 1868, £200; Oct. 16th, 1869, £300; and the balance on Dec. 17th, 1870: what was then paid, including interest at 4J per cent. ? 10. Lent on bond £1225, at 2^ per cent., on March 4th, 1864, and received £320 on June 17th, 1865 ; £250 on Aug. 7th, 1866, £300 on Nov. 12th, 1867; £200 on Jan. 13th, 1869 ; and the balance on April i9th, 1870 : what was then due, including the interest ? 11. Borrowed on bond, at 3 per cent., £875 on Jan. 4th, 1865, and paid £200 on March 7th, 1866; £150 on June 13th, 1867 ; £200 on Sept. 11th, 1868; £150 on Nov. 17th, 1869; and the balance on Jan. 4th; 1871 : what was then paid, in- cluding the interest ? 12. Borrowed, at 3 J per cent, £1500 on June 4th, 1865, of which was paid, Aug. 1st, 1866, £350; Oct. 9th, 1867, £250; Nov. 21st, 1868, £400; and the balance on Dec. 31st, 1869: what was then paid, including the interest ? MISCELLANEOUS QUESTIONS. 1. How many francs, each 9^d., are equal in value to 209 half-crowns ? 2. If whisky at 14s. 6d., 15s. 6d., 16s., and 17s. a-gallon, be mixed in equal quantities ; what should a gallon of the mixture be sold for to gain 5 per cent, and allow a discount of 6| per cent. ? 3. A cubic foot of water weighs 1000 oz. ; how many tons of water will a cistern 16 ft. 6 in. long, 15 ft. 4 in. broad, and 5 ft. 6 in. deep contain ? MISCELLANEOUS QUESTIONS. 71 4. Find the value of § gui. ; reduce 5s. 7^d. to the frac. of 9 gui., and 3 ml. 2 fur, to the frac. of 1 ml. 6 fur. 12 poles. 5. A ladder, 45 ft. long, reaches to a window 27 ft. from the ground on one side of a street, and, without moving the foot, it reaches to a window 36 ft. high on the other side ; find the breadth of the street. 6. 248 trees are planted in the breadth of a plantation at a distance of 5 ft. 4 in. from each other; what is the breadth of the plantation, allowing the same distance between the trees and the fence on both sides ? 7. If £435 gains £58, 14s. 6d. in 4 J years ; what is the rate per cent. ? 8. The side of a cubic piece of marble is 32 ft. ; find the side of a piece 7^ times as large. 9. Find the value of a rectangular piece of ground 48 ft. 4 in. by 34 ft. 6 in., at 24s. per s. ft. 10. Exchanged 19 cwt. 2 qr. 12 lb. of cofiee at £9, 6s. 8d. p. cwt. for sugar at 7^d. and tea at 4s. 6d. per lb. ; there was 5 times as much sugar as tea : how much was there of each ? 11. If 7 lb. sugar be equal to 3 of cofi^ee, and 6 of cofiee to 2^ of tea; how many lbs. tea are equal to 168 lbs. sugar? 12. A cask is f full, and after 40 gals, were run off", it was ^5 full ; how many gals, could the cask contain ? 13. If a globe 9 in. diameter weighs 27 lbs. ; what will a globe weigh whose diameter is 25 in. ? 14. Purchased 1260 lbs. tea at 4s. per lb., but J of it being damaged, 25 per cent, was lost in selling it ; the remainder was sold at 4s. 6d. per lb. : how much per cent, was gained at the latter price and on the whole ? 15. In 1854, the number of births registered in England was 324069 males and 310336 females; how many males were bom for 100 females ? 16. What fraction multiplied by the square of 1^, and the product divided by the cube root of §i§, produces 3 ? 17. Invested £10710 in new 2^ per cents at 74f ; how much must I invest in 3 per cents at 90J to produce an in- come of £500 yearly ? 18. In 1801 the population of Scotland was 1608420, and in 1851 it was 2888742 ; what was the increase per cent, during that time ? 19. What is the thickness of a solid foot of stone that is 9 ft. 4 in. lon^ and 2 ft. 6 in. broad? 20. A certain number of persons were fined 5s. 6d. each, but 3 of them having no money, each of the others had to pay Is. lOd. more than their fine ; how many persons were there ? 72 MISCELLANEOUS QUESTIONS. 21. Reduce 14s. 11 Jd. to the dec. of £5, 19s. 6d., and y\ of 2f d. to the dec. of half-a-crown. 22. In 1855 the number of births registered in Scotland was 93498, of which 47872 were males, and 45626 females ; what decimal of the whole were males and females ? 23. Find the present value of £475, 15s. due 4 years hence, at 2^ per cent, simple interest. 24. A grocer buys sugar at 5d. and 7d. per lb. and mixes them in the proportion of 3 ; 5 ; what will he gain per cent, by selling it at 7id. per lb. ? 25. A square contains exactly 2^ ac. ; find its side. 26. In the Centigrade thermometer the freezing-point is zero, and the boiling-point 100°; in Fahrenheit's the freezing- point is 32° and the boiling-point 212° : what degi*ee C. corre- sponds to 68"" F., and what degree F. corresponds to 45° C. ? 27. What is the shortest piece of cloth that shall at the same time be an exact number of yards, EngHsh ells, Flem- ish ells, and French ells ? 28. A person spends £10, 4s. 2d. in 35 days, and he saves £93, 10s. lOd. yearly; what is his income? 29. ^ of an army was killed in battle, j'jj was taken pris- oners, Y*^ died from sickness, ^'^ was in hospital, and 31375 effective men remained ; how many were there at first ? 30. A person being asked his age, answered, if to my age you add ^ and i of it, the sum will be 59 ; what was his age? 31. The corn produced by a field was found to be 200 qrs. or ^ more than what was sown ; how much was sown ? 32. Bought £126 worth of tea at 4s. 6d. per lb., some of which being damaged, I sold the remainder at 4s. 9d. per lb., which produced £106, 17s. 6d. ; what quantity was damaged? 33. A gentleman gave to three persons £78, 6s. 6d. ; the second received § of the first, and the third f of the second : what did each receive ? 34. A person bought a horse, gig, and harness for £60 ; the horse cost 7 times as much as the harness, and the gig was A the price of the horse and harness ; what was the price of each? 35. What must be the depth of a cistern which is 6 ft. 3 in. long and 4ft. 6 in. broad, to contain 481*665 gals, of water? 36. Light travels at the rate of 192000 miles per sec. ; how long does it take to travel from the sun to the earth, a dis- tance of 95 millions of miles ? 73 DECIMAL COINAGE. The pupil, having worked the Elementary Exercises in Decimal Coinage, at the end of the " Lessons in Arith- metic," and also those given under Decimal Fractions (page 30), may now solve the following questions. TABLE OF DECIMAL MONEY. lmil(m.) =£^^V^ = |4f. 10 mils = lcent(c.) =£'iio = 2|d. 100 mils = 10 cents = 1 florin (fl.) = £^V = 2s. 1000 mils = 100 cents = 10 florins = £l = 20s. Ex. 1. Reduce £12, 17s. 9d. from the present to the proposed system. Ans. £12-8875 = £12, 8 fl. 8 c. 7i m. Here, bv Case V. p. 32, £12, 17s. 9d. = £12-8875 = £12, 8fl. 8c. 7^m. Ex. 2. Reduce £7, 8 fl. 2 c. 5 m. from the proposed to the present system. Ans. £7, 16s. 6d. Here, by Case VI. p. 32, £7, 8fl. 2c. 5m. = £7-825 = £7, 16s. 6d. Reduce from the present to the proposed system, 1. 6s. 6d. 2. 7 3 3. 18 4 4. 19 7 Reduce from the proposed to the present system, 5. 14s. 2^d. 9. £2, 8s. 9d. 13. £12, 13s 4d. 6. 13 7^ 10. 4 15 8 14. 15 16 8 7. 15 8i 11. 7 6 8 15. 17 14 7 8. 17 5 12. 9 10 10 IG. 21 12 8 17. £-425 18. •675 19. •850 20. •925 21. £-763 22. ^574 23. -235 24. -075 25. £4-6375 26. 6-8125 27. 7-4025 28. 9-7875 29. £10-7750 30. 12-6666 31. 15-3333 32. 17-8166 33. A man earns £58, 7fl. 7c. 5m. per annum, his expenses are £49, 8fl. 9c. 7m. ; how much does he save ? 34. What is the value of 27oz. of silver at 2fl. 7c. 5m. per oz. ? 35. A man's wages are £1, 2fl. 7c. 5m. weekly ; how much do they amount to in a year ? 36. What is the weekly rent of a house, when the yearly rent is £65, Ifl. 4m. ? 37. If 35 quarters of oats cost £53, 3fl. 7c. 5m. ; what is the rate per quarter ? 74 DECIMAL COINAGE. 38. A bankrupt who owed £3595, paid his creditors £2786, Ifl. 2c. 5m. ; how much did he pay per £1 ? 39. If 15 gallons of whisky cost £13, Ifl. 2c. 5m.; what should he paid for a cask containing 125 gals. ? 40. Find by practice the value of 17cwt. 2qrs. 14 lbs. of sugar at £2, 4fl. 5c. per cwt. 41. A man's wages are £50, 2fl. 2c. 5m. for 146 days; how much is this per annum ? 42. What is the commission on £575, 2c. 5m. at 2 and 3J per cent, ? 43. What is the brokerage on £796, 2fl. 5c. 6m. at J, J, and § per cent. ? 44. How much should be paid for insuring £5750, 2fl. 5c. at 3 per cent., and policy Ifl. 2c. 5m. per cent. ? 45. What is the interest on £487, 7fl. 5c. for 4 years, at 2^ and 4 per cent. ? 46. Find the amt. of £896, 5fl. for 3 years, at 2 and 5 p. c. 47. Find the interest on £228, Ifl. 2c. 5m. for 198 days, at 4 and 4^ per cent. 48. What should £2851, 5fl. 6c. 2Jm. amount to in 1 year and 99 days, at 4 per cent. ? 49. What sum will amount to £251, 8fl. 7c. 5m. in 4 months, at 2^ per cent. ? 50. Divide £153, Ifl. 4m. among 4 persons, so that J the share of the first, J of that of the second, J of that of the third, and \ of that of the second may make up the same sum. 51. What is the rent of a farm of 525 ac. 3 ro. 25 per. at £3, 5fl. 2c. 8m. per acre ? 52. A bill of £919, 8fl., dated Feb. 14, at 6 months, was discounted June 13, at 3^ per cent. ; what was the net pro- ceeds, deducting commission ^ per cent. ? 53. If 7 1 per cent is gained by selling tea at £22, 5fl. 7c. 5m. per cwt. ; what is gained or lost per cent, by selling it at £22, 8fl. 9c. per cwt. ? 54. In what proportions should tea at Ifl. 2c. 5m., and 2fl. per lb. be mixed to reduce the price to Ifl. 7c. 5m. per lb. ? 55. What part of £9, 7fl. 5c. is £8, 4c. 3|m. ? 56. In what time will the interest of £437, 6fl. 7c. 5m. pay a debt of £52, 5fl. 2c. Im., at 4 per cent, per annum? KDIKBURGH : PRINTED BY OLIVER AND BOYD. 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REID'S GRAMMAR, with Analysis of Sentences, HUNTER'S SCHOOL SONGS, with Music, THE PRINCIPLES OF ENGLISH GRAMMAR; with a Series of Progressive Exercises, and a Supplementary Treatise on Analysis of Sentences. By Dr James Douglas, lately Teacher of English, Great King Street, Edinburgh. Is. 6d. DOUGLAS'S INITIATORY GRAMMAR, for Junior Classes. Printed in larger type, and containing a Supplementary Treatise on Analysis of Sentences. 6d. DOUGLAS'S PROGRESSIVE ENGLISH READER. . A New Series of English Reading-Books. The Earlier Books are illus- trated with numerous Engravings. First Book. 2d. I Third Book. Is. I Fifth Book. 2s. Second Book. 4d. | Fourth Book. Is. 6d. | Sixth Book. 2s. 6d. DOUGLAS'S SELECTIONS FOR RECITATION, with Introductory and Explanatory Notes; for Schools. Is. 6d. DOUGLAS'S SPELLING AND DICTATION EXERCISES. 144 pages, price Is. AthencBum.—" A good practical book, from which correct spelling and pronunciation may be acquired." 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From Dr Joseph Bosworth, Professor of Anglo-Saxon in the University of Oxford; Author of the Anglo-Saxon Dictionary, etc., etc. " Quite a practical work, and contains a vast quantity of important in- formation, well arranged, and brought up to the present improved state of philology. 1 have never seen so much matter brought together in so short DALGLEISH'S GRAMMATICAL ANALYSIS, with Pro- gressive Exercises. 9d. Key, 2s. DALGLEISH'S OUTLINES OF ENGLISH COMPOSI- TION, for Elementary Schools, with Exercises. 6d. Key, 4d. DALGLEISH'S INTRODUCTORY TEXT-BOOK OF ENGLISH COMPOSITION, based on Grammatical Synthesis; con- taining Sentences, Paragraphs, and Short Essays. Is. DALGLEISH'S ADVANCED TEXT-BOOK OF ENGLISH COMPOSITION, treating of Style, Prose Themes, and VersiQcation. 2s. Both Books bound together, 2s. 6d. Key, 2s. 6d. ENGLISH GRAMMAR, founded on the Philosophy of Language and the Practice of the best Authors. With Copious Exer- cises, Constructive and Analytical. By C. W. 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The sections on Descriptive and Narrative Essays have been entirely rewritten. KEY TO THE Improved Edition, including Directions for teaching the Work. 2s. 6d. HISTORY OF ENGLISH LITERATURE; with an Outline of the Origin and Growth of the English Language. Illus- trated by Extracts. For Schools and Private Students. By William Spalding, A.M., Professor of Logic, Rhetoric, and Metaphysics, in the University of St Andrews. Continued to 1870. 3s. 6d. Spectator. — " A compilation and text-book of a very superior kind. . . The volume is the best introduction to the subject we have met with." POETICAL READING-BOOK; with Aids for Grammatical Analysis, Paraphrase, and Criticism; and an Appendix on English Versification. By J. D. Morell, A.M., LL.D., Author of Grammar of the English Language, etc.; and W. Ihne, Ph.D. 2s. 6d. STUDIES IN COMPOSITION : A Text-Book for Advanced Classes. By David Pryde, M.A., Head-Master of the Edinburgh Merchant Company's Educational Institution for Young Ladies. 2S' Recently published. ENGLISH COMPOSITION FOR THE USE OF SCHOOLS. By Robert Armstrong, Madras College, St Andrews; and Thomas Armstrong, Heriot Foundation School, Edinburgh. Part I., Is. 6d. Part II., 2s. Both Parts bound together, 3s. Key, 2s. ARMSTRONG'S ENGLISH ETYMOLOGY. 2s. ARMSTRONG'S ETYMOLOGY for JUNIOR CLASSES. 4d. 8 ENGLISH READING, GRAMMAR, ETC. SELECTIONS FROM PARADISE LOST; with Notes adapted for Elementary Schools, by Rev. Robert Demaus, M.A., late of the West End Academy, Abardeen. Is. 6d. DEMAUS'S ANALYSIS OF SENTENCES. 3d. EWING'S PRINCIPLES OF ELOCUTION, improved by F. B. Calvert. A.M. 3s. 6d. Consists of numerous rules, observations, and exercises on pronuncia- tion, pauses, inflections, accent, and emphasis, accompanied with copious extracts in prose and poetry. RHETORICAL READINGS FOR SCHOOLS. By Wm. 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This is a collection, alphabetically arranged, of the principal roots^ affixes, and prefixes, with their derivatives and compounds. OLD TESTAMENT BIOGRAPHY, containing notices of the chieif persons iil Holy Scripture, in the form of Questions, with references to Scripture for the Answers. 6d. NEW TESTAMENT BIOGRAPHY, on the same Plan, 6d. FISHER'S ASSEMBLY'S SHORTER CATECHISM EX- PLAINED. 2s. Part I. Of what Man is to believe concerning God. II. Of what duty God requires of Man. GEOGRAPHY AND ASTRONOMY. GEOGKAPHY AND ASTEONOMY. Ik compiling the works on these subjects, the utmost possible care has been taken to ensure clearness and accuracy of statement. Each edition is scrupulously revised as it passes through the press, so that the worke may be confidently relied on as containing the latest information accessible at the time of publication. A COMPENDIUM of MODERN GEOGRAPHY, Political, Physical, and Mathematical: With a Chapter on the Ancient Geog- raphy of Palestine, Outlines of Astronomy and of Geology, a Glossary of Geographical Names, Descriptive and Pronouncing Tables, Questions for Examination, etc. By the Rev. Alex. Stewaet, LL.D. Carefully Eevised. With 11 Maps. 3s. 6d. SCHOOL GEOGRAPHY. By James Clyde M.A., LL.D., one of the Classical Masters of the Edinburgh Academy.. With special Chapters on Mathematical and Physical Geography, and Technological Appendix. Corrected throughout. 4s. Athmceum.—'-'''WQ have been struck with the ability and value of this ■work, which is a great advance upon previous Geographic Manuals. . . . Almost for the first time, we have here met with a School Geography that is quite a readable book, — one tliat, being intended for advanced pupils, is well adapted to make them study the subject with a degi'ee of interest they have never yet felt in it. . . . Students preparing for the recently instituted University and Civil Service Examinations will find this their best guide." DR CLYDE'S ELEMENTARY GEOGRAPHY. Corrected throughout. Is. 6d. An Appendix on Sacred Geography has now been added, which will be found amply sufficient for ordinary uses. Fresh interest has been given to many old names by the mention of quite modern facts connected with the corresponding places. AN ABSTRACT OF GENERAL GEOGRAPHY, compre- bending a more minute Description of the British Empire, and of Pales- tine or the Holy Land, etc. With numerous Exercises. For Junior Classes. By John White, F.E.I.S., late Teacher, Edinburgh. Carefully Revised. Is. ; or with Four Maps, Is. 3d. WHITE'S SYSTEM OF MODERN GEOGRAPHY; with Outlines of Astronomy and Physical Geography; comprehending au Account of the Principal Towns, Climate, Soil, Productions, Religion, Education, Government, and Population of the various Countries. With a Compendium of Sacred Geography, Pi'oblems on the Globes, Exercises, etc. Carefully Revised. 2b. 6d, ; or with Four Maps, 5s. 9d. 10 GEOGRAPHY AND ASTRONOMY. KUDIMENTS OF MODERN GEOGRAPHY. By Alex. Reid, LL.D., late Head Master of the Edinburgh Institution. "With Plates and Map of the World. Carefully Revised. Is.; or with Five Maps, Is. 3d. Enlarged hy 36 pages of extra information regarding the Counties and principal Railways of the United Kingdom, The names of places are accented, and accompanied with short descrip- tions, and occasionally with the mention of some remarkable event. To the several countries are appended notices of their physical geography, productions, government, and religion ; concluding with an outline of sacred geography, problems on the use of the globes, and directions for the construction of maps. FIRST BOOK OF GEOGRAPHY: Being an Abridgment of Dr Reid's Rudiments of Modem Geography. With an Outline of the Geography of Palestine. Carefully Revised. 6d. This work has been prepared for the use of young pupils. It is a suit- able and useful companion to Dr Reid's Introductory Atlas. DR REID'S OUTLINES OF SACRED GEOGRAPHY. 6d. This little work is a manual of Scripture Geography for young persons. It is designed to communicate such a knowledge of the places mentioned in holy writ as will enable children more cleai-ly to understand the sacred narrative. It contains references to the passages of Scripture in which ttie most remarkable places are mentioned, notes chiefly historical and descriptive, and a Map of the Holy Land iu provinces and tribes. AN INTRODUCTORY GEOGRAPHY, for Junior Pupils. By Dr James Douglas, lately Teacher of English, Great King Street, Edinburgh. Carefully Revised. 6d. DR DOUGLAS'S PROGRESSIVE GEOGRAPHY. An entirely new work, showing the recent changes on the Continent and elsewhere, and embracing much Historical and other Information. 160 pages, 1«. Carefully Revised. 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Besides a complete treatise on the science of geography, this work contains the elements of astronomy and of physical geography, and a variety of problems to be solved by the terrestrial and celestial globes. At the end is a pronouncing Vocabulary, in the form of a gazetteer, con- taining the names of all the places in the work. ELEMENTS OF ASTRONOMY; adapted for Private Instruction and Use of Schools. By lingo Reid, Member of the College of Preceptors. With 65 Wood Engravings. 3s. REID'S ELEMENTS OF PHYSICAL GEOGRAPHY; with Outlines of Geology, Mathematical Geography, and Astrox- OMY, and Questions for Examination. With numerous Illustrations, and a large coloured Physical Chart of the Globe. Is. SCHOOL ATLASES. A GENERAL ATLAS OF MODERN GEOGRAPHY; 29 Maps, Coloured. By Thomas Ewing. 7s. 6d. WHITE'S ELEMENTARY ATLAS OF MODERN GEO- GRAPHY. 4to, 10 Maps, Coloured. 2s. 6d. Contents.— 1. The W^orld; 2. Europe; 3. Asia; 4. Africa; 5. Nortli America; 6. South America; 7. Euglaud; 8. 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While the utmost attention has been paid to accuracy, the narratives have in every case been rendered as instructive and pleasing as possible, so as to relieve the study from the tediousness of a mere dry detail of facts, A CONCISE HISTORY OF ENGLAND IN EPOCHS. By J. F. CoRKRAN. With Maps and Genealogical and Chronological Tables, and comprehensive Questions to each Chapter. New Edition, with the History continued. 2s. 6d. The writer has endeavoured to convey a broad and full impression of the great Epochs, and to develop with care, but in subordination to the rest of the narrative, the growth of Law and of the Constitution. HISTORY OF ENGLAND FOR JUNIOR CLASSES; with Questions for Examination. Edited by Henry White, B.A. Trinity College, Cambridge, M.A. and Ph.D. Heidelberg. Is. 6d. Athenceum. — " A cheap and excellent history of England, admirably adapted for the use of junior classes. The various changes that have taken place in our constitution are briefly but clearly described. It is surprising how successfully the editor has not merely avoided the obscurity which generally accompanies brevity, but invested his narrative with an interest too often wanting in larger historical works. The information conveyed is thoroughly sound; and the utility of the book is much increased by the addition of examination questions at the end of each chapter." HISTORY OF GREAT BRITAIN AND IRELAND; with an Account of the pi*esent State and Resources of the United Kingdom and its Colonies. With Questions and a Map. By Dr White. 3s. Athenceum. — " A carefully compiled history for the use of schools. The writer has consulted the more recent authorities: his opinions are liberal, and on the whole just and impartial ; the succession of events is developed with clearness, and with more of that picturesque effect which so delights the young than is common in historical abstracts." HISTORY OF SCOTLAND FOR JUNIOR CLASSES; with Questions for Examination. Edited by Dr White. Is. 6d. 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With two large Maps, etc. 3s. 6d, WATTS' CATECHISM OF SCRIPTURE HISTORY, and of the Condition of the Jews from the Close of the Old Testament to the Time of Christ. With Inteoddction by W. K. Tweedib, D.D. 28. SIMPSON'S HISTORY OF SCOTLAND; with an Outline of the British Constitution, and Questions for Examination at the end ol each Section. 3s. 6d. SIMPSON'S GOLDSMITH'S HISTORY OF ENGLAND; With the Narrative brought down to the Middle of the Nineteenth Century. To which is added an Outline of the British Constitution. With Questions for Examination at the end of each Section. 3s. 6d. SIMPSON'S GOLDSMITH'S HISTORY OF GREECE. With Questions for Examination at the end of each Section. 3s. 6d. SIMPSON'S GOLDSMITH'S HISTORY OF ROME. With Questions foi Exaninatim at the end of each Section. 3s. 6d. 14 WRITING, ARITHMETIC, AND BOOK-KEEPING. WKITHTG, AKITHMETIO, AND BOOK-KEEPINa. This section will be found to contain works in extensive use in many of the best schools in the United Kingdom. The successive editions have been carefully revised and amended. ARITHMETIC ADAPTED TO THE NEW CODE, in Three Parts. By Alexander Trotter, Teacher of Mathematics, etc., Edinburgh. Parts I. and IL, embracing the first four Standards^ are now Beady. Each containing 36 pages, 2d., stifif wrapper. Answers to Parts I. and II., price 3d. each. Part III. in Preparation, PRACTICAL ARITHMETIC FOR JUNIOR CLASSES. By Henry G. C. Smith, Teacher of Arithmetic and Mathematics in George Heriot's Hospital. 64 pages, 6d. stiff wrapper. Answers, 6d. From the Rev. Philip Kelland, A.M., F.R.SS. L. & E., late Fellow oj Queens^ College, Cambridge, Professor of Mathematics in the University of Edinburgh. "I am glad to learn that Mr Smith's Manual for Junior Classes, the MS, of which I have examined, is nearly ready for publication. Trusting that the Illustrative Processes which he has exhibited may prove as efficient in other hands as they have proved in his own, I have great pleasure in recommending the work, being satisfied that a better Arithmetician and a more judicious Teacher than Mr Smith is ndt to be found." PRACTICAL ARITHMETIC FOR SENIOR CLASSES; Being a Continuation of the above. By Henry G. C. Smith. 2s. Answers, 6d. Key, 2s. 6d. *** The Exercises in both works, which are copious and original, have been constructed so as to combine interest with utility. They are accompanied by illustrative processes. LESSONS IN ARITHMETIC FOR JUNIOR CLASSES. By James Trotter. 66 pages, 6d. stiff wrapper; orSd. cloth. Answers,6d. This book was carefully revised, and enlarged by the introduction of Simple Examples of the various rules, worked out at length and fully explained, and of Practical Exercises, by the Author's son, Mr Alexander Trotter, Teacher of Mathematics, etc., Edinburgh ; and to the present edition Exercises on the proposed Decimal Coinage have been added. LESSONS IN ARITHMETIC for ADVANCED CLASSES; Being a Continuation of the Lessons in Arithmetic for Junior Classes. Containing Vulgar and Decimal Fractions; Simple and Compound Proportion, with their Applications; Simple and Compound Interest; Involution and Evolution, etc. By Alexander Trotter. New Edition, with Exercises on the proposed Decimal Coinage. 76 pages, 6d. in stiff wrapper ; or 8d. cloth Ansioers, 6d. Each subject is also accompanied by an example fully worked out and minutely explained. The Exercises are numerous and practical. WRITING, ARITHMETIC, AND BOOK-KEEPING. 15 A COMPLETE SYSTEM OF ARITHMETIC, Theoretical and Pi'actical; containing the Fundamental Rules, and their Application to Mercantile Computations; Vulgar and Decimal Fractions; Involution and Evolution; Series; Annuities, Certain and Contingent. By Mr Tbottee. 8s. Key, 4s. 6d. *^* All the 3400 Exercises in this work are new. They are applicable to the tusiness of real life, and are framed in such a way as to lead the pupil to reason on the matter. There are upwards of 200 Examples wrought out at lengtlt and minutely explained. INGRAM'S PRINCIPLES OF ARITHMETIC, and their Application to Business explained in a Popular Manner, and clearly Illustrated by Simple Rules and Numerous Examples. Remodelled and greatly Enlarged, with Exercises on the proposed Decimal Coinage. By Alkxandeb Teotteb, Teacher of Mathematics, etc., Edinburgh. Is. Key, 2s. Each rule is followed ty an example wrought out at length, and is illustrated by a great variety of practical questions applicable to business. MELROSE'S CONCISE SYSTEM OF PRACTICAL ARITHMETIC ; containing the Fundamental Rules and their Applica- tion to Mercantile Calculations; Vulgar and Decimal Fractions; Ex- changes; Involution and Evolution; Progressions; Annuities, Certain and Contingent, etc. Re-arranged, Improved, and Enlarged, with Exer- cises on the proposed Decimal Coinage. By Alexander Trotter, Teftcher of Mathematics, etc., in Edinburgh. Is. 6d. Key, 2s. 6d. Each Rule is followed hy an example worked out at length, and minutely explained, and by numerous practical Exercises. HUTTON'S ARITHMETIC AND BOOK-KEEPINa. 2s. 6d. BUTTON'S BOOK-KEEPING, by Trotter. 2s. Suts of Ruled Writing Books, — Single Entry, per set, Is. 6d. ; Double Entry, per set, Is. 6d. 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