UC-NRLF F THE VERS1TY OF< BY T. M. HUNTER, RECTOR TO THE ASSOCIATION FOR THE REVIVAL '.OF SACRED MUSIC IN SCOTLAND. HUNTER'S Elements of VOCAL MUSIC : An Introduction to the Art of Beading Music at Sight. 6d. %* This work has been prepared with much care, and is the result of long practical experience in teaching. It is adapted to all ages and classes, and will be found considerably to lighten the labour of both teacher and pupil. The Exercises are printed in the Standard Notation, and the Notes are named as in the Original Sol-fa System. CONTENTS. Musical Scales. Exercises in Time. Syncopation. The Chromatic Scale. Transposition of Scale. The Minor Scale. Part Singing. Explanation of Musical Terms. HUNTER'S SCHOOL SONGS Por Junior Classes, With Preface by Rev. JAMES CURRIE, M.A., Principal of the Church of Scotland Training College, Edinburgh, and Author of the u Elements of Musical Analysis," etc. %* These "SONGS" consist of Verses composed expressly for this work, or carefully selected from approved sources. In the Songs for Junior Classes the TUNES are- Simple, and princi- pally arranged for two voices ; while in those for Advanced Classes they are more complex in arrangement, of greater length, and mostly written for three voices. The Melodies, which are either original or adaptations from the great Composers, are printed in the STANDARD NOTATION. The entire Series is intended as a Musical Manual for the Singing Class or the Family Circle. FIRST SERIES, containing 60 Songs,- price 4d. Morning Song. The Fairy Queen. The Merry Month. (Hound for 3 Voices.) Boyhood. Friendship. (Bound for 3 Voices.) The Fairies 1 Dance. Herald of Spring. Charming littleValley. Bright beams the Morning. What you've to Do. (Round for 3 Voices.} The Swing. Edinburgh: OLIVER AND BOYD. London: SIMPKIN, MARSHALL, AND Co. Hunter's School Songs with Music. FIRST SERIES (for Junior Classes) Continued. Little Things. To the Praise of Call John the Boat- All Hail! gentle Truth. (Round for man. (Round for 3 Spring. 3 Voices.) Voices.) Fairy Light. The Kite. The Brook. [here. Cuckoo. (Catch.) Now the Sun sinks in The Summer now is The Bugle Horn. the West. (Round Evening Sun. Come,0 Come Away ! for 3 Voices.) The Bell doth Toll. (Round for 3 Voices.) Before all Lands. (Round for 3 Voices.) The River. Music. (Round for 3 Joyous Spring-time. Come, Come Here! Voices.) (Round for 3 Voices.) (Catch for 4 Voices.) The Seasons. Winter Song. The Cricket. Like a May -day. The Snowdrop. Play is Done. (Round (Round for 3 Voices.) The Little Spring. for 3 Voices.) Hark ! How the Morning Call. Sunshine. Lark. (Round for 3 The School Bell. Early to Bed. (Round Voices.) Oft by the Deep Blue for 3 Voice*.) The Little Busy Bee. Sea. Come, May ! thou love- The Sea. Winter's Departure. ly Linge/er. Ever Blooming. England and Her Softly, gently flow our (Round for 3 Voices.) Queen. Days. The Daisy. Boat Song. Evening. Birds are Singing. Try Again. Come, Brothers. (Round for 4 Voices.) Holiday Song. (Round for 3 Voices.) The Moon. Evening. (Canon for Boat Song. The Little Lark. 4 Voices.) SECOND SERIES, containing 63 Songs, price 4d. W.-lconiP, bright and Patriotism. To the Cuckoo. sunny Spring. The Village Green. Dew-drops. Sunmu-r Mu ruing. Sister, Wake. (Round The Bugle. Autumn. for 6 Voices.) [ing. The Child's May-day Little Jack Ilorner. Children, join in Sing- Song. (Rou>> To the Rainbow. The Reaper's Song. In the Harvest Morn I am Merry. (Hound . MLJ is rising. so cheering. for 3 Voices.) The Bells. (Round for Be kind to each Billy and M.-. 3 Voices.) other. The Valley. Sleep, my Baby. He that would live. The Monkey. Daybreak. (Round for 3 Voices.) Music Kvprywhpro. Tin; Woodcutter's Shall we, oppressed Hear the Big Clock. Night Song. with Sadness. (Round for 3 Voices.) Sing we together. Youth's Desires. Mountain Boy's Song. (Round for 4 Voices.) Come from Toil. [Continued at end of BooTc. PRACTICAL ARITHMETIC SENIOR CLASSES. HENRY G. C. SMITH, TEACHER OF ARITHMETIC AND MATHEMATICS, GEORGE HERIOT'S HOSPITAL. SIXTH EDITION. EDINBURGH : OLIVER AND BOYD, TWEEDDALE COURT. LONDON : SIMPKIN, MARSHALL, AND CO. 1871. Trice 2s. bound. Answers to Ditto 6d. E2UCATIOH IlfiB, Now ready^ a New Edition of PRACTICAL ARITHMETIC FOR JUNIOR CLASSES. BY HENRY G. C. SMITH. Price 6d. Answers to Ditto, 6d. C. I EDINBURGH : TED BY OLIVER AND BOYD, TWEEDDALE COURT. **~nrr- fv ~ THIS Manual, which is a Sequel to Practical Arithmetic for Junior Classes, is intended for the use of those who have mastered the Fundamental Rules in Simple and Compound Numbers. Considerable space, in accordance with the importance of the subject, has been devoted to the explanation of Fractions ; and the other branches also have been illustrated with a view to practical instruction. The Exercises, which are copious and original, have been constructed to combine interest with utility. They are arranged in distinct Sections, and are accompanied with Illustrative Processes. As the work is essentially prac- tical, the explanatory remarks in elucidation of the various processes are more of an applicate than an abstract character. The Answers to the Exercises hi this Manual are published in a separate form. CONTENTS. Page Arithmetical Tables 7 Prime and Composite Numbers 12 Vulgar Fractions 19 Decimal Fractions 51 Continued Fractions 76 Practice 77 Allowances on Goods 90 Simple Proportion 92 Compound Proportion . . 103 Statistics 108 Commission and Brokerage Ill Insurance 114 Interest 115 Discount 124 Equation of Payments 127 Stocks 128 Profit and Loss 131 Distributive Proportion 137 Alligation 141 Barter 144 Chain Kule 145 Exchange 147 Involution 154 Evolution 157 Scales of Notation 168 Duodecimals 171 Series 175 Compound Interest 180 Miscellaneous Exercises 186 ARITHMETICAL TABLES, Z. MONEY. MONEY OF ACCOUNT. 4 farthings = 1 penny d, 12 d. =1 shilling s. 20 s. =1 pound . d. for denarius: s. for solidus: for libra. DECIMAL DIVISION OF 1. 10 mils, m. =1 cent c. 10 c. =1 Horin fl. 10 fl. =1 pound . COINS IN CIRCULATION. GOLD. Sovereign, 1 ; Half-sav. 10s. 1869 sovereigns are coined out of 40 lb troy of sterling gold. SILVER. Crown, 5s. ; Hf.-cr. 2s. 6d. ; Florin, 2s.; Shilling, IB.; Sixpenc*,6A.', Groat, 4d.; Thrcepencc t 3d. 66 shillings are coined out of 1 Ib. troy of sterling silver. BRONZE. Penny, Id.; Halfpenny, fyl.\ Farthing, Jd. In 100 parts of the bronze metal for these coins, there are : 95, r; 4, tin; and 1, zinc. To 1 Ib. avoir, there are: Of pennies, 48; of halfpennies, 80 ; of farthings, 160. OBSOLETE COINS. Tester, Gd.; Dollar, 4/6; Noble, 6/8; Seven Shilling piece; Angel, 10/; Half- guinea, 10/6 ; Mark, 13/4 ; Pistole, 16/ ; Guinea,21/; Carolus,23/; Jacobus,25/; Moidore, 27/ ; Joannes, 36/. The denominations of Scots Money are one-twelfth of the value of the cor- responding names in sterling: thus, 1 Scots = 20d. sterling. lg. = Id. * Also, 1 merk, * = 13Jd. II. WEIGHT. The Act 5 Geo. IV. cap. 74, which established the IMPERIAL WEIGHTS AND MEASURES, came into operation on 1st Jan. 1826. By the Act 18 and 19 Vic. cap. 72, the Imperial Standard of Weight is the Pound Avoirdupois, deposited in the Exchequer at Westminster, and of which copies are placed in the Mint, the Royal Society of London, Greenwich Observatory, and the Palace at Westminster. AVOIRDUPOIS WEIGHT. Avoir. Wt. is the general weight of commerce. 1 Ib. avoir. = 7000 grains. 16 drams, dr. = 1 ounce oz. 16 oz. = 1 pound Ib. 28 Ib. = 1 quarter qr. 4qr.orll21b.=: Ihundredwt. cwt. 20 cwt. = 1 ton T. Also, 14 Ib. = 1 stone In London, a stone of butcher-meat = 8 Ib. In Liverpool, &c., 100 Ib. = 1 cental. TROY WEIGHT. Troy Wt. is used in weighing the precious metals and in philoso- phical experiments. 24 grains, gr. = 1 penny wt.dwt. 20 dwt. = 1 ounce oz. 12 oz.or5760gr.= 1 pound Ib. At the Mint, the ounce is divided into 1000th s. The fineness of gold is estimated in carats. Pure gold is said to be 24 carats fine. Sterling gold, of which every 24 parts contain 2 of alloy, is 22 carats fine. ARITHMETICAL TABLES. The fineness of silver is estimated in oz. and dwt. Sterling silver, of which 1 Ib. contains 18 dwt. of alloy, is 11 oz. 2 dwt. fine. 151$ Diamond carats = 1 oz. troy, which is also = 600 Pearl grains. APOTHECARIES' WEIGHT. 20 grains, gr. = 1 scruple ^ SQ =1 drachm 3 83 =1 ounce ^ In the above, the ^ is the ounce Troy of 480 grains ; but in the NEW SYSTEM of weights adopted by the General Med- ical Council in October 1862, the Q and 3 have been abolished, and the is the ounce Avoirdupois of 437 J grains. III. LENGTH. By the Act 18 and 19 Vic. cap. 72, the Imperial Standard Measure of Length is the Yard, deposited in the Exchequer at Westminster, and of which copies are placed beside those of the Standard of Weight. LINEAL MEASURE. 12 lines, I. =1 inch in. 12 in. = 1 foot ft. 3 ft. =1 yard yd. 5 yd. = 1 pole po. 40 po. = 1 furlong fu. 8 fu. or 1760 yd. = 1 mile ml. Also Length of 3 barleycorns = 1 in . Breadth of 4 barleycorns=l digit = f in. ft. Palm = 3 Hand = 4 Span = 9 Cubit Step Pace Fathom = 6 ft. -J GEOGRAPHICAL MEASURE. 6076 ft. nearly = 1 geog. ml. 60 geog. ml. = 1 degree of the Earth's circumf. 21600 geog. ml. = the Earth's circ. Also 3 geog. ml. = 1 league SURVEYORS' MEASURE. n. = 1 link Ik. 100 Ik. or 66 ft. = 1 chain 80 ch. := 1 mile ch. ml. 4 nl. or 9 in. 4 qr. CLOTH MEASURE. = 1 nail nl. = 1 quarter qr. = 1 yard yd. Flemish ell =3 qr. I English ell = 5 qr Scotch //=37in. (French = 6qr. IV. SURFACE. SQUARE MEASURE. This Table is formed by squaring the corresponding denominations in Lineal Measure. 144 sq. in. = 1 sq. foot sq.ft. 9 sq. ft. = 1 sq. yard sq.yd. 30 sq. yd. = 1 sq. pole sq.po 40 sq. po. = 1 rood ro. 4 ro. or 4840s. yd. =1 acre ac. 640 ac. =1 sq. mile sq. ml. Also 100 sq. ft. = 1 square of flooring 36 sq. yd. = 1 rood of building SURVEYORS' MEASURE. 10,000 sq. Ik. = 1 sq. chain 10 sq. ch. =1 acre V. SOLIDITY. CUBIC MEASURE. This Table is formed by cubing the corresponding denominations in Lineal Measure. 1728 cub. in. = 1 cub. ft. 27 cub. ft. = 1 cub. yd. Also 5 cub. ft. = 1 barrel bulk B. B. 8 B. B. =1 ton measurement 40 cub. ft. of rough timber = 1 load 50 cub. ft. of hewn timber = 1 load 216 cub. ft. = 1 cubic fathom ARITHMETICAL TABLES. VI. CAPACITY. According to the Act 5 Geo. IV. cap. 74, the Imperial Standard Measure of Capacity is the Gallon, which contains 10 Ib. avoir, of dis- tilled water weighed in air at the temperature of 62 Fahrenheit, the Barometer being at 30 in. The Standard Measure is deposited in the Exchequer at Westminster. MEASURE OF CAPACITY. pt. qt gal. fc qr. 4 gills gi. 1 pint 2 pt. =1 quart 4 qt. =1 gallon 2 gal. = 1 peck 4 pk. = 1 bushel 8 bu. =1 quarter Also Pottle = 2 qt I Coomb = 4 bu. Strike = 2 bu. | Load = 5 qr. Last = 10 qr. The Imperial Gallon = 277-274 cub. in , is the highest measure for liquids. The weight of an Imperial Bushel of wheat varies from 56 Ib. to 64 Ib. : by the Tithe Commutation Act of England it is taken at 60 Ib. The following" were abolished by the Act 5 Geo. IV. cap. M. Cub. in. Imperial Wine Gallon =231= '8331109 Gallon Ale Gallon =282 = 1-0170446 Winchester Bushel Heaped Measure, usod for coals, &c., was abolished by the Act 5 and 6 Guliel. IV. cap. 63, which enacted that after 1st Jan. 1836, " all Coals, Slack, Culm, and Cannel of every Descrip- tion, shall be sold by Weight and not ." The bushel was = 1 Winchester bushel -|- 1 quart = 2217 02 cub. in., but when heaped in the form of a cone = 2815-486 cub. in. 3 heaped bushels = 1 sack 12 sacks = 1 chaldron When the terms Hogshead, Pipe, &c. are used, it is merely as the names of r I =2150-42= -9694472 Bu. * The Weights and Measures of the United States of America are the same as those used in Great Britain, with the exception of the Measures of Capacity, which continue to be the subdivisions and multiples of the Winchester Bush- el f r dry goods, and of the Wino Gal- lon for liquids. casks of wine, &c., and not as meas- ures, for the contents must always be expressed in Imperial Gallons. When the names Puncheon, Tierce, are applied to casks of sugar, molasses, &c., their gross and net weight* must be stated. APOTHECARIES' FLUID MEASURE. 60 minims 11^=1 fluid drachm f 3 8f3 =1 fluid ounce fg 20 f g =1 pint O 8 O =1 S allon C O for Octarius ; C for Congius. 1 f of distilled water weighs 1 oz. VII. INCLINATION. ANGULAR MEASURE. 60 seconds " =1 minute 60' 90' 4 L or 360 30 = 1 degree = 1 right angle L =. 1 circle Also = 1 sign of the zodiac VIII. TIME. MEASURE OP TIME. 60 seconds, sec. = 1 minute min. 60 min. = 1 hour ho. 24 ho. = 1 day da. 7 da. =1 week wk. 4 wk. = 1 common month co.mo. 365 da. 1 or 52 w. J-zr 1 common year co. ye. Id. J 365 da. 6 ho. = 1 Julian year Ju. yr. 366 da. = 1 leap year The year is divided into 12 cal- endar months : January February March April May June Da. 31 28 31 30 31 30 July August September October November December Da. 31 31 30 31 30 31 In leap year, February has 29 days. * See Allowancet on Good*. A2 10 ARITHMETICAL TABLES. QUARTERLY TERMS IN ENGLAND. Lady Day March 25 Midsummer June 5 Michaelmas Day Sep. 29 Christmas Dec. 25 Easter Day, on which the Movable Feasts depend, is the first Sunday after the Paschal Full Moon, which hap- pens on March 21, or next after it. When the Full Moon is on a Sunday, Easter Day is on the next Sunday. QUARTERLY TERMS IN SCOTLAND. Candlemas Feb. 2 Whitsunday May 15 Lammas Aug. 1 Martinmas NOT. 11 The Sidereal Day is = 23 ho. 56 min. 4-09 sec. It is the true time of the earth's revolution on its axis, or the interval between two successive me- ridian transits of the same star. A sidereal clock is always kept in an as- tronomical observatory. The Apparent Solar Day is the inter- val between two successive meridian transits of the sun's centre. This day varies in length. The difference be- tween Apparent Solar Time as shown by a sundial, and Mean Solar Time as indicated by a well-regulated clock, is termed Equation of Time. The Mean Solar Day of 24 hours is used for the purposes of civil life. Astronomers in using the Mean Solar Day begin at 12 o'clock noon, and reckon the hours onward to 24. The Astronomical agrees with the Civil Reckoning from noon to midnight; but from midnight to noon the former is a day behind, thus : CiTil Time. Astron. Time. Sep. 10 : 7 p.m. = Sep. 10 : 7 ho. Sep. 11 : 11 a.m. = Sep. 10 : 23 ho, Since the sun apparently describes a circle or 360 in 24 hours, 15 of longitude correspond to 1 hour of mean solar time; thus the time at a place in 45 E. long, is 3 hours before that of Greenwich, while in 60 W. long, it is 4 hours behind it. The Periodical Month or sidereal rev- olution of the moon is = 27 da. 7 ho. 43 min. 11-5 sec. It is the time of the moon's revolution round the earth, or the interval in which the moon re- turns to the same place in the heavens. The Lunar Month or synodical rev- olution of the moon is = 29 da. 12 ho. 44 min. 2'87 sec. It is the interval be- ween new moon and new moon, or between two successive conjunctions of the sun and moon. The Jews use a year of 12 lunar months of 29 or 30 days each ; and to make it somewhat correspondent to the solar year, intercalate a month of 29 days, 7 times in a cycle of 19 years. The Mohammedans use a year of l! lunar months or 354 days, and add a day to the year 11 times in 30 years. The Sidereal Year is = 365 da. 6 ho. 9 min. 9*6 sec. It is the time of the earth's revolution round the sun. The Solar or Tropical Year is = 365 da. 5 ho. 48 min. 49'7 sec. = 365*24224 days. It is the interval between two successive passages of the sun through the vernal equinox. The solar year regulates the seasons, and is there- fore the proper standard for regulating the civil year. Julius Csesar adopted a nominal year of 365 da. 6 ho. In the Julian Calendar, every year whose number is divisible by 4 contains 366 days. The Julian Calendar was introduced 45 B.C. Its error is = 365'25 da. 365'24224 da. = -00776 da. p yr., or 3-104 da. in 400 years. In the 16th century an error of 12 days had accumulated, but as it was determined to reckon merely from 325 A. D. the year of the Coun- cil of Nice Gregory XIII. ordered ten days to be omitted in October 1582. In the Gregorian Calendar, every year whose number is divisible by 4 is a leap year, except when divisible by 100 and not by 400 ; thus, while 1600 is a leap year, 1700, 1800, and 1900, are common years. 400 years in the Gregorian Calendar or New Style (N. S.) are thus 3 days shorter than 400 years in the Julian Calendar or Old Style (O. S.). The error of the Gre- gorian Calendar in 400 years is there- fore 3-104 da. 3 da. = -104 da., or 00026 da. %> yr. N. S. was introduced into the British Empire in Septem- ber 1752. O. S. is still used by the Greek Church. The difference be- tween O. S. and N. S. is progressive. In the 16th and 17th centuries it was ten days; in the 18th, eleven; in the 19th, twelve. MEMORANDA. Sack of Flour or Meal = 280 Ib. Barrel >/ = 196 * Quire of Paper = 24 sheets Ream // = 20 quires Bale " = 10 reams Roll of Parchment = 60 skins Pack of Wool = 240 Ib. Long hundred = 120 Gross = 144 ARITHMETICAL TABLES. 11 METRIC SYSTEM OF WEIGHTS AND MEASURES. The Use of the Metric System was rendered permissive in the United Kingdom by the Act 27 and 28 Vic. cap. 117. When the metre was first definitively introduced in France in 1799, it was adopted as the ten-millionth part of the Quadrant from the N. Pole to the Equator, but subsequent calculations have, however, shown that it is not pre- cisely so. MEASURES OF LENGTH. Metre. Inches. Millimetre = -uol = -03937079 Centimetre = -01 = -3937079 Decimetre = -1 = 3'937079 METKE =1- = 39-37079 Metre Yard*. Dekametre = 10 = 10-93633 Hectometre = 100= 109-3633 Kilometre = 1000= 1093-633 Myriametre = 10000 = 10936*33 MEASURES OF SURFACE. 8q. Metres. Sq. Yards. Centiare = 1 = 1 19603326 ARE = 100 = 119-603326 Sq. Metret. Acres. Dekare = 1000 = -2471143 Hectare = 10000 = 2-471143 MEASURES OF CAPACITY. Cub. Mtr-. Pint. Centilitre = -UUuul = '0176077 D.-cilitre = -0001 = -176O77 LITRE = -001 = T76077 Cub Metre. Gallons. Dekalitre = -01 = 2-20027 Hectolitre = -1 = 22-0097 Kilolitre = 1- = 220*097 Milligram = -U01 = "oil m = -01 = 'I Decigram = -1 = 1-.V, (greater than) the highest number in the series ; thus we finish by eliding 49, which is <-{-> 7. (3) Find the primes from 100 to 150. 101, 103,4^ 107, 109,434-, 113,4*5-,4i?-, 119, 121, 423-, 495; 127,1297 131, 433; 435; 137, 139, 444, -H3; 44^44?-, 149. * The sign <-h, for " divisible %," was introduced by Mr Barlow of Woolwich Academy in 1811. f Eratosthenes, curator of the Alexandrian Library, died B. c. 194. PRIME AND COMPOSITE NUMBERS. 13 Elide every third number after 105, the first number -H 3 .. fifth n , f 105, , 5. * seventh n 105, // >, 7. eleventh 121, n n. All the composites are now elided, for 149 is < (less than) the square of the next prime, 13. Find the primes below 1000, giving those in each hundred as a separate exercise. 2, PRIME FACTORS. THE primes that make up a composite number are termed its Prime Factors; thus, 2, 2, 2, 3, 7, are the prime factors of 168. Resolve the following into prime factors. 2 2 KM _ (2X2X2X3X7 i||- tor2'X3X7 2 2 091 5 2X2X3X7X11 ||- tor 2*X3X7X11 2 T2 3 231 3 21 7 77 ~ 11 1. 6 6. 42 11. 98 16. 143 21. 245 26. 624 2. 12 7. 55 12. 100 17. 154 22. 264 27. 1188 3. 15 8. 66 13. 105 18. 165 23. 275 28. 1331 4. 21 9. 70 14. 110 19. 192 24. 343 29. 1452 5. 30 10. 75 15. 125 20. 242 25. 539 30. 1584 Resolve the following into prime factors, and combine them into sets of three factors each, not greater than 12. 8 1 0^^2 X 3 * X 5 : 3'X3*X'-'X5 Itis easier to obtain the 91810=9x9X10 =9X9X10 three factors thus:- -go =9xlo 31. 225 35. 495 39. 405 43. 704 47. 240 51. 960 32. 315 36. 616 40. 448 44. 756 48. 486 52. 1152 33. 392 37. 968 41. 504 45. 792 49. 729 53. 1296 34. 441 38. 1089 42. G72 46. 1056 50. 768 54. 1728 3. GREATEST COMMON MEASURE. A NUMBER which divides another without leaving a remainder is termed a Measure or Factor of that number ; thus, 8 is a measure of 24. A number which divides two or more num- bers without leaving a remainder is termed a Common Measure of those numbers ; thus, 6 is a common measure of 24 and 36. 14 PRIME AND COMPOSITE NUMBERS. 3 The greatest number which divides two or more numbers without leaving a remainder is termed their Greatest Common Measure (G. C. M.) ; thus, 12 is the G. c. M. of 24 and 36 ; 6 the G. c. M. of 24, 36, and 54. Numbers whose G. c. M. is 1 are prime to each other. Com- posite numbers may be prime to each other ; thus, 25 is prime to 36. (1) Find the G. c. M. of 78 and 300. I. By Prime Factors. 78 = 2 X 3 X 13 ; 300 = 2* X 3 X 5* G. c. M. = 2 X 3 = 6 Since 2 and 3 are the only factors common to 78 and 300, 2X3 or 6 is the o. C. M. of 78 and 300. II. By Division. 78)300(3 1 234 66)78(1 2 66 12)66(5 6 G. c. M. 60 G. c. M.~6)12(2 12 6 is a common measure of 78 and 300. For, 6 measures 2X6 or 12; 5X12 or 60; 60 + 6 or 66; 66 + 12 or 78 ; 3X78 or 234; and 234 -j- 66 or 300. No number ^> 6 is a common measure of 78 and 300. Since every measure of 78 measures 3 X 78 or 234, every common meas- ure of 78 and 300 measures 234 and 300. Now if a number is con- tained a certain number of times in 234 and another number of times in 300, the difference between the quotients is an integer, which is the number of times the number is contained exactly in 300 234. Every common measure of 78 and 300 therefore meas- ures 300 234 or 66. Since it measures 78 and 66, it measures also 78 66 or 12 ; hence also 5 X 12 or 60 ; and finally 66 60 or 6. No common measure of 78 and 300 can therefore be > 6 ; but 6 is a common measure of 78 and 300 ; .. (hence) 6 is their G. C. M. Find G. C. M. of 78 66 300 234 12 12 66 60 6 1.48,78 2. 56, 98 3.121,143 4. 342, 665 5. 448, 784 6. 203, 261 7. 375, 525 8.841, 899 9.961, 1178 10. 1243, 1469 11.1001,1287 12.1131,2639 13:9889,986 14.1792,1832 15. 1850, 1517 16. 1792, 1847 17. 3927, 5049 18. 1287, 1551 19. 1537, 1802 20. 3056, 3629 21.1261,22116 22.3243,37976 23.31484,109268 24.82739,57693 25.10759,20405 26.714285,857142 27.49593,43902 28.17641,22243 3. PRIME AND COMPOSITE NUMBERS. 15 (2) Find G. c. M. of 42, 56, and 49. 42)56(1 14)49(3 42 42 14)42(3 G. c. M. 7)14(2 42 14 Every c. M. of 42 and 56 is a measure of their G. c. M., 14 ; .*. every c. M. of 14 and 49 is a c. M. of 42, 56, and 49 ; but 7 is the o. c. M. of 14 and 49 ; .-. 7 is G. c. M. of 42, 56, and 49. (3) Find G. c. M. of 192, 56, 44, 128, 94. Take any two numbers, as 44 and 94 ; 2 is their G. c. M. The o. c. M. required cannot therefore be > 2. Now 2 measures the numbers 192, 56, 128; .'. 2 is the G. c. M. required. Suppose we had selected 56 and 128, their G. c. M. is not a measure of all the rest. G. c. M. of 8 and 44 is 4. G. c. M. of 4 and 94 is 2, which measures 192, and 2 is the o. c. M. required. To abridge the process, it is expedient to select at first two num- bers whose G. c. M. is among the least of the mutual G. c. measures. (4) Find G. C. M. of 27, 216, 48, 105, and 405. 3 is G. c. M. of 27 and 48 ; 3 is a M. of 216, 105, and 405. .-. 3 is the G. C. M. of 27, 48, 216, 105, and 405. Find G. C. M. of 29. 45, 27, 54 30. 90, 84, 81 31. 56, 84, 63 32. 24, 36, 48, 216 33. 32, 40, 64, 108 34. 72, 84, 66, 176 35. 198, 495, 209, 660 36. 146,730,365,219 37. 924, 378, 612, 246 38. Find the Greatest Common Divisor of 12460 and 10769. 39. Find the greatest number cancelling 1859 and 3003. 40. Find the length of the greatest line exactly measuring the sides of an enclosure 216 yd. long and 111 broad. 41. Find the greatest measure of capacity contained exactly in two measures containing respectively 6 gal. 7 pt. and 8 gal. 6 pt. 42. What is the greatest sum of money contained exactly in 2 " 9 1 and 2 " 3 " 11? 43. Find the greatest sum of money contained exactly in 34 77 and 70 "12 2. 44. George, James, and John, wish to spend 2/6, 1/10, and 3/5^, on the same kind of squibs. Find the price of the dearest squib they can purclin 45. Two apprentices carry 1147 and 961 ivory balls respectively from the workshop to the showroom. The balls are carried in baskets of equal contents, which are filled and emptied several times. How many balls are in a basketful ? 16 PRIME AND COMPOSITE NUMBERS. 46. Two frigates having the same number of guns fire a number of rounds. The one has fired 608, and the other 1102 shots. How many guns has each ? 47. The Nemesis and Mceander frigates having the same number of guns greater than 36, fire a number of rounds. The one has fired 352 and the other 484 shots. How many guns has each ? And why is the limitation " greater than 36" necessary? 48. Two opposition coaches, which have run full during the season for the same number of days, have had 4807 and 3971 pas- sengers respectively. How many days has the season lasted, and how many passengers does the one contain more than the other ? LEAST COMMON MULTIPLE. A NUMBER which contains another an exact number of times is termed a Multiple of that number ; thus 48 is a Multiple of 8. Measure and Multiple are correlative terms : 7 is a measure of 14, 14 is a multiple of 7. A number containing two or more numbers an exact number of times is termed a Common Multiple of those numbers ; thus 48 is a Common Multiple of 4, 6, and 8. The least number containing two or more numbers an exact number of times is termed their Least Common Multiple (L. c. M.) ; thus 24 is L. c. M. of 4, 6, and 8. When two or more numbers are prime to each other, their L. c. M. is their product ; thus L. c. M. of 3, 5, 7, and 11, is 3X5X7XH. (1) Find L. c. M. of 15 and 21. 15 = 3 X 5; 21 = 3 X 7. L. c. M. 105 = 3 X 7 X 5. Every common multiple of 15 and 21 must contain 3, 5, and 7. But 3 X 5 X 7 is the least number containing 3, 5, and 7; .-. 3 X 5 X 7 is L. c. M. of 15 and 21. L. c. M. of two numbers = Product - G. c. M. Thus, of 15 and 21; G. c. M.=3; Product = (3 X 5) X (3X7). ^ In finding the L. c. M. of 2 numbers it is thus easier to di- vide one of the numbers by their G. c. M., and multiply the quotient by the other number. PRIME AND COMPOSITE NUMBERS. 17 (2) Find L. c. M. of 224 and 256. G. c. M. = 32. L. c. M. = v* 4 X 256 = 7 X 256 = 1792. (3) Find L. c. M. of 384 and 564. G. C. M. = 12. L. c. M. = 32 X 564 = 18048. Find L. c. M. of 1. 27, 36 2. 42, 56 3. 35, 49 4. 72, 48 5. 52, 78 6. 34, 51 7. 144, 180 8. 216, 225 10. 200, 250 11. 224,343 9. 196, 343 12. 324, 360 13. 420, 798 14. 225, 375 15. 234, 390 (4) Find L. c. M. of 16, 18, 21, 24, 30, 32, 36. i. 2 2 2 3 -tft *% 21, 24, 30, 32, 36 21, 12, 15, 16, 18 21, 6, 15, 8, 9 21, -B-, 15, 4, 9 7, 5, 4, We elide 16 and 18, which are respectively measures of 32 and 36. We divide by the prime 2 so Ions as it is contained in more than one number. Since 2 is not contained in 21, we continue to write 21 until we divide by the next prime. In the 4th line we elide 3, a measure of 9. The factors contained in the numbers in addition to 2, 2, 2, 3, are 7, 5, 4, 3 ; and as these are prime to each other, L. o. M. = 2X2X2X3X7X5X4X3 = 10080. IF. 12 | jfr 4fr, 21, 24, 80, 32, 36 7, j 5, 8, 3 Since the factors of 12 are contained in one or other of the num- bers, we may divide by 12, and find the other factors contained in the numbers. 12 is not a measure of 21, but on dividing 21 by 3, the o. c. M. of 12 and 21, we obtain 7. Similarly we divide 30 and 32 respec- tively by 6 and 4. The facl factors contained in the numbers besides those of 12, are 7, 5, 8, and 3 ; and as these are prime to each other, L. c. M. = 12X7X5X8X3= 10080. In the First Method we divide by a prime so long as it is contained in two or more numbers. In the Second, we divide by a composite number whose factors are contained in one or other of the numbers ; and when any number is not a multiple of the divisor, we divide it by their G. c. M. 18 PRIME AND COMPOSITE NUMBERS. (5) Find L. c. M. of 21, 24, 25, 27, 28. 2 21, 24, 25, 27, 28 21, 12, 25, 27, 14 3 21, 6, 25, 27, 7, 2, 25, 9 L. c. M. 25 X 9 = 37800. ii. 12 | 21, 24, 25, 27, 28 7, 2, 25, 9, 7 L. c. M. = 12X7X 2 X 25 X 9 = 37800. 16. 4, 6, 10, 12 17.8,12,15,18 18.12,16,18,20 19.12,16,18,27 20. 10, 6, 15, 12 21.12,15,20,40 22.12,28,35,21 23. 32, 36, 49, 56, 42 24. 20, 24, 25, 27, 45 25.28,30,32,36,42 26.35,40,42,49,28 27.8,14,18,21,32,28 28. 24,27,28,32;36,56 29. 15,21,24,27,28,35 30. 25,32,63,40,35,56,80 31.30,36,32,48,40,54,63 32. 30,33,36,42,48,63,55 33.27,36,45,54,63,72,81 34.35,45,56,63,40,72,28 35.15,21,33,24,35,40,77 36.56,40,24,88,55,21,33 37. Find the least number containing the 9 digits. 38. Find the shortest distance that three rods of 8 ft. 3 yd. and 4 yd. will exactly measure. 39. Find the content of the smallest vessel that may be exactly filled by using a gallon, a 10 pint, or a 12 pint measure. 40. A rides at 10 miles an hour, B drives at 6 miles an hour, and C walks at 3 miles an hour. Find the shortest distance they may all traverse in an exact number of hours. 41. Tom, Dick, and Jack, agree to spend the same sura in pur- chasing fire-wheels at the rate of ld., 4d., and 2d. respectively. What is the smallest sum they can expend? 42. What are the prime factors of L. c. M. of 12, 35, 28, and of 21, 15,20? 43. Mention the prime factors of L. c. M. of 12, 28, 35, 21, 55, and of 15, 33, 20, 77, 44. (6) Find the least number which, when separately di- vided by 2, 3, 4, 5, 6, always leaves the remainder 1. Least Number <-h 2, 3, 4, 5, 6, is L. c. M. of 2, 3, 4, 5, 6=60; 60+1 or 61 is the number required. 44. Find the least number which, when separately divided by 2, 3 ...... 7, leaves the remainder 1. (7) Find the least number which, when separately di- vided by 2, 3 ...... 8, leaves the remainders 1, 2 ...... 7 respectively. L. c. M. = 840 ; 840 lor 839 is the number required. 45. Find the least number which, when separately divided by 2, 3 ...... 9, leaves the remainders 1. 2 ...... 8 respectively. 19 VULGAR FRACTIONS. IK a unit is divided into ' - ! - ' - ' three equal parts, and two i 1 of them are taken, the parts thus taken form ' - * - ! - ! - ' tico-thirds of one unit. 2 If two units as a whole are divided into three equal parts, and one taken, the part thus taken is one-third of two units. That which we have obtained by either method is written, f . It is termed a VULGAR FRACTION, of which 2 is the Nu- mmifnr and 3 the Denominator. The Denominator of a Vulgar Fraction indicates the number of equal parts into which a unit is divided ; and the Numera- tor, the number taken. Or, the Numerator indicates the number of units, and the Denominator the number of equal parts into which these units, considered as a whole, are di- vided, and of which one is to be taken. If 2 units are eacli di- vided into 5 equal parts, 1 we obtain 10 fifths. The integer 2 is thus reduced to the fractional form, 1 3 . (1) Reduce 3 to an equivalent fraction with denominator 7. s=y 1. 1!^ luce 9 to equivalent fractions with denominators 4, 8, 7. 2. ,. 33 .. " " " 3, 5, 8. 3. .. 29 " 11,13,20. 4. * 37 * > 12, 14, 15. 5. A baker divides 12 rolls into 4 equal parts each. How many fourths has he ? 6. Into how many eighths of a yard can a draper cut 17 yards of cloth ? Suppose we have two units 1 2 s and three-fifths of a unit, by dividing each of the units into fifths, and adding in the three-fifths, we obtain thirteen-fifths. 23 = _ = 13 ~5 5 " ~S ' Every number which, like 2|, is thus made up of an integer and a fraction, is termed a MIXED NUMBER. 20 VULGAR FRACTIONS. Reduce the following mixed numbers to a fractional form : 13. 928^3 14. 15. T3" ~~~ 13 1. 7J 4. 8 T 6 T 7. 90 I 7 10. 2. 11| 5. 8 T \ 8. 79 ?& 11. 3. 13* 6. 15jf 9. 23 1 1 12. 16. How many eighths of a yard are in 7 yd. ? 17. How many twelfths of a penny are in 8 T 5 5 d. ? 1 8. How many sixteenths of a yard has a draper sold who has disposed of 9 T 3 5 yd. ? If we take the ' ! ! ! : ! ! ' ^ ! ! fraction y, we find we can make up two units, with three-fourths over. V = 2f. Every fraction which, like y , has its numerator > its de- nominator, is > 1, and is termed an IMPROPER FRACTION. Every fraction which, like , has its numerator = its de- nominator is = 1, and is termed an IMPROPER FRACTION. Every fraction which, like f , has its numerator < its de- nominator, is < 1, and is termed a PROPER FRACTION. Reduce the following improper fractions to whole or mixed numbers : (1) VV = 14. (2) >A = 7 T V (3) i| = 1. 1. 86 2. H 3. * T V 4. 5. 6. 8. 4300 9. *ff* 10. ij?a 1L 2500 12. 2^f2 13. 15. 3 T 16. A grocer, who has sold 89 quarter-pounds of tea, wishes to know how many Ib. he has sold. 17. A draper, who has sold 117 sixteenths of a yard, is asked how many yards he has sold. 18. The average length of a year, according to the Gregorian Calendar, is ' 4 4 fi 9 - days. Express this as a mixed number. If we take any fraction, as f, all fractions with the denomina- tor 7, having the numerator < 5, as ^, |, &c., are < 4; and all with the numerator 5, having the denominator > 7, as f, |, &c., are also < f . By diminishing f VULGAR FRACTIONS. 21 8. the numerator, or increasing the denominator, we thus dimin- ish the value of a fraction. Again, all fractions with the ' - - - . - 1 - : _ denominator 7, having the nume- ? rator > 5, as f, ^ &c., are > f ; ..... and all with the numerator 5, but ' the denominator < 7, as J, , &c., are also > 4- By increasing the ' - ! - ! - ! - - * numerator, or diminishing the de- I nominator, we thus increase the value of a fraction. (1) Mention 4 fractions with denominator 12, next > T A i A Ai i 8 a' A" (2) Mention 3 fractions with numerator 5, next < T 5 T . A i**> A TV 1. Mention 4 fractions with numerator 9, next < T ' T . 2- - 3 ...... 6, > T V 3. 6 " denominator 9, > $. 4. " 5 ...... 10, , < T ',. 5. 3 " " numerator 13, > < jf. 6. 3 denominator 17, < if. We may multiply -^ by 2, by - ' - ' - ' - - - doubling the number of the parts ; f I I thus, $ X 2 = $. We may multiply -| by 2, by ^ -- : doubling the magnitude of the s parts; thus, f X 2 = f . By multiplying the numerator or dividing the denominator, we thus multiply the value of a fraction. Multiply the following fractions by integers : 12 X 2 .{), 12, a common factor of 24 and 60, is contained 2 times in 4. To tor by 2, a), J.Z, a COI11H1UI1 lildUi Ui if* aim uv, ia WJJIKMUWU 24. To multijjly ^ by 24 we may therefore multiply the numera- 2, and divide the denominator by 12. 1. i X 5 5. A X 3 9. A X 12 2. w X 4 6. if X 4 10. /* X 14 3. TT X 4 7. U X 7 11. H X 27 4. w X 8 / 8. H X 16 12. X 15 9. 22 VULGAR FRACTIONS. 13. Seven purchasers each buy g peck of meal. How many pecks have been bought? 14. 1 Ib. troy = -} f 1 Ib. avoir. How many Ib. avoir. = 15. Find the number of degrees = 25 grades, of which each = i 9 o deg. 1O We may divide f by 4, by taking one-fourth of the number of parts ; thus, | -f- 4 = f . We may divide f by 2, by taking as many parts of half the magnitude; thus, | -=- 2 = T 3 o- T 3 o (MfTiWUUfC ) UlUO, ^ -7- AI TO* By dividing the numerator or multiplying the denominator, we thus divide the value of a fraction. Divide the following fractions by integers : 3X6 In (3), 3, a common factor of 18 and 15, is contained 6 times in 18. To divide * by 18, we may therefore divide the numerator by 3, and multiply the denominator by 6. 9. 44 20 2. 3. 4. -T- 2 *H2 *- 6 4 9 5. 6. 7. 5 3 8 6 10. 11. 12. |f 30 12 21 11. 13. What part of a mile does a stream flow tp minute which flows 2* 5 mile in 7 min. ? 14. 12 oz. troy = ^1 Ib. avoir. What part of 1 Ib. avoir, is 1 oz. troy? 15. 64 squares of a draught-board occupy f sq. ft. What does one square occupy ? Having given any fraction, as f , I by taking one-third of the num- \ i L_ ber of parts each three times as I large, we have the same fraction expressed, as f . By dividing the numerator by 3, we divide the fraction by 3 ; and by di- viding the denominator by 3, we multiply the fraction by 3, and thus the fraction is unaltered in value. By dividing the numerator and the denominator of a fraction by the same number, the value of the fraction remains unaltered. The fraction f , when expressed as |, is said to be in its LOWEST TERMS. A fraction is in its lowest terms when its numerator and denominator are prime to each other. 1 VULGAR FRACTIONS. 23 11* Reduce the following fractions to their lowest terms : In I., we divide the numerator and denominator successively by the common factors 4, 7, 4, selected by inspection. In II., we at once divide the numerator and denominator by their O. C. M., 112. The product of the factors 4, 7, 4, used in I., is = G. c, M., 112. (2) J. By inspection, || = iif = > f (3) Ufi. I- IS 2. 4. ff 5. ^ 6- Ill 7 8- im 9- Uff !0- iiil H 12. O. C.M. 74 13. Hit 14. 4f*f 15- Mft 16- /T% 17. UH 18- iM 19. HIJ 20. *jJSS 21. 22. 5 1 1 TTJUUT 10212 TTJT^ 23. ttitf 24- IBH5 = H- Of^ 3927 *** 3U4"^?y 26. !iJ 27. 28. 29. 30. ii|44 31. 32. 33. 3 3 u% ] 34. jffi 35. i$ 36. AVA 37. AVA 38. 39. 40. JH 4.9 285714 ^^" W^-5V5 43. 44. 45 t 857142 46. imii 47 616384 48. HHH 12* Having given any fraction, as J, by taking twice as many parts of half the magnitude, we have the same fraction expressed, as T 8 o By multiplying the nu- merator by 2, we multiply the fraction by 2; and by multiplying the denominator by 2, we divide the fraction by 2 ; and thus by multiplying the numerator and the denominator of a frac- tion by the same number, the value of the fraction remains unaltered. Take any two fractions, f, {. Of the Common Multiples of 4 and 6, let us take 24, | = Jf , and f = . The fractions have thus been reduced to equivalent fractions with a Common Denominator. 12 is the L. c. M. of 4 and 6 ; j = T \ and f = JJ. The fractions have thus been reduced to equivalent fractions with the Least Common Denominator. 24 VULGAR FRACTIONS. |0 1. 2. 4 7 8 9 11 12 Reduce the following to equivalent fractions ha least common denominator (L. C. D.) : ;i) i, i, TV, it- L. c. D. = L. c. M. of 3, 6, 12, 16 == 48. Since the denominator of is contained 16 F - M - times in 48, we multiply its numerator by 1 6. 16 Similarly, we divide the L. c. D. by the deno- minator of each fraction, and multiply the 8 corresponding numerator by the quotient. ^ The number, showing how often the denominator 4 of a fraction is contained in the L. c. D., is that by which the numerator of that fraction is g multiplied, and may be termed the FRACTIONAL MULTIPLIER (F. M.) 4, f, 1 13- 4> ?, t, , f 4, 1 ving the 2 _ 32 3 "~ 48 5 .__ 40 .6 : "" 48 7 _ 28 12 48 15 _ 45 16 ~~ 48 3 -f *, J, 1 14- TT, A> A) A 33, -e* f, {, 1, T\ I 5 ' t) T\, 4, H, -1, iJ j, *, 4 H 17- M f A? If iV f7 7 11 1Q 11 11 11 11 11 11 , "5", TT, T3 1O * T2", ^4", T, 4", "615, Tl" t, T\, , i8 19. $, {, ii, 44, ft, 11 7 17 19 11 90 4 24 124 624 3124 J ^4~J "32, ^"ff ^ U< T) 2"S? T2"5"J "62"5"J "3"T2 5" 23572 91 3711973 T) TTJ TT, T, TT ^ i ' ^^J Uo, *2> 2O? 24"> 2"ff 359 36 99 555 555 TJ t> TT> 12> "33 *" t? IT) Tf7, TfT, 2TJ -5? f3 3 3 3 9Q 11 15 17 35 31 23 > 4~5 T) fi?J ^36" l ' T2^J 1"6, T) "3~6^ "3"2"J ^T (2) *, T 9 T, TV L. c. D. =7 X 11 X 13 = 1001. When the denominators are prime to each n x ^ other, the F. M. of a fraction is the product of the denominators of the other fractions : thus, the F. M. of f is 11 X 13, which is = 7 X 13 7 X 11 X 13 6 _858 7 ~1001 iT~~Iooi 4 308 7 7XH (3) *, A, T\, 41 - L. c. D. = 7 X 13 X 19 X 23 = 39767. Although the denominators are prime to * each other, yet it is often convenient, when there are four or more fractions, to obtain the F.JM. by dividing the L. c. D. by each 3959 denominator in succession : thus, the F. M. ofM-^fiRI 7X13X19X23 ,. . 2093 13 "~ 1001 3 _ 17043 7 "~ 39767 J__ 3059 13 "~ 39767 7 _ 14651 19 39767 12 _ 20748 23 ~ 39767 ui 7 is ou5i ? which is = 13 X 19 X 23. 1729 VULGAR FRACTIONS. 25 12. 25. 1, 1. 31. i i i, 1, T'T 26. 1, f, 3 T 32. 1 1! ?', " 27. I, *. 4, A 33. 1, T, ii, T 9 F 28. i, 1, T? T\ 34. f *, 1, i? 29. i |, T \, T 4 T 35. f, *! f, TV 30. i, i T^J TT 36. fi i, f, T 8 T, 1 con ipar in if, respectively = J, }, J. Order of Magnitude, f , j|, J. By reducing the fractions to a common denominator, we at once discover the order of magnitude. (2) i I, I, iV * Complements, J, i, i, T \y, f Order of Magnitude, r %, |, J, f , f . That which, when added to a proper fraction, makes up unity, is termed its COMPLEMENT. Of a number of fractious, that which has the least complement is the greatest fraction. (3) T V, A) A* respectively = ^, ~, ~. Order of Magnitude, T V, A> rr- (4) f y , H, , respectively = |, 7 -&, f , f . Order of Magnitude, ft, |}, f, |f . Of the series of fractions T 2 3 , 5 3 5 , /f, 5 4 g , arranged in order of magnitude, let any two, as 5 a ff and / 4 , be taken. Reducing tlK-in to the common denominator 20 X 34, we have the nu- merators respectively == 3 X 34 and 20 X 5, of which the for- mer is the greater. When fractions are arranged in order of B 26 VULGAR FRACTIONS. Iflt magnitude, the product of the numerator of any fraction by the denominator of the next less is > the. product of the de- nominator of the former fraction by the numerator of the lat- ter; thus, in the series { |, ||, |i, ; 15 X 26 > 16 X 23, 23 X 36 > 26 X 31, 31 X 6 > 36 X 5. 1. i,f,S 3. I,' 1 1' I 5312 1?2") T) TU A 2 5 1 O. VV. -^Vi ^ 1 1 7 33' 6 8211 7 5 -3-) 7-5-5 T2-) ) 10. *, it, H, i*, I 11. if, if, i}, is 19 9 15 19 16 12 1 ^' TO) 1F> 0) TT) TS 13. 14. , T , A A 16. A, A. A A, ** n3 4 5 216 ' T7> ** 2^> TT> "5T> 18. T V, ) A 3T> Let us ADD the fractions ^, J, and |-. ' By taking any line as the unit, we place the lines representing , |, f , in a line, and thus obtain their sum. To express the value of the sum, or to add the fractions arithmetically, we reduce them to equi- valent fractions .^ having a common <<<< . . . . i . denominator, and 1 add the numerators of the equivalents, nators, the L. c. D. is generally taken. I 1 Of common denomi- (1) i + I + I- I A 1 = A i 4. i *=ii f f = 2 A (2) i + i + + * + A = Otherwise 4- l 5. 9-4+A-f A-hA 13. + + + TW + H + 5 % 14. T ' T + A + A + A + T'A + 3* 15. H + + A + AV + TT + , VULGAR FRACTIONS. 27 (3) If +11 =2- In the diagram, having rep- resented 1 and If by lines, we place the integers together, and tlion the fractions in the same line with them, and thus obtain the sum whose value is 2 + == I 1 1 1 I +1 1 2 3 i In adding Mixed Numbers we need not reduce them to improper fractions. (4) tf + 7f = 19 234 + 896 4 + * 16 + 429 17. 6i+7|+8J+9$ 19. 20. 21.9J+10|+11f+5U The work may be abridged by combining, in the process of adding, those fractions whose denominators are either the same or have a common measure, as in the following examples : (5) & + A + TV + i + f + 1 = 1 11 (6) 4 T A ""24" 2 I 1 1 _ 6 + U TT "T "5T 51 7 2 3" 81-4+*+* W-A+I+A+A 28 VULGAR FRACTIONS. 14* (7) A student spends I of the day in teaching, T V in at- tending classes, ^ in study, / in recreation and meals, and -; in miscellaneous reading. What part of the day is he thus occupied ? - - - 4 - T6- 37. i of a pole is in sand, and T 4 5 of it in water. What part of the pole is thus below the level of the water ? 38. In an Allied Camp, of the soldiers are natives of England, T 2 5 of Scotland, T y ff of Ireland, and 550 f Wales. What part of the camp is under British colours ? 39. Of the chairs in the University of Edinburgh, 3 9 5 of the num- ber was founded in the nineteenth century, in the eighteenth, and ^ in the seventeenth. What part of the whole was founded in these centuries ? 40. In 1685, the regular infantry, and the regular cavalry of England, were respectively T | 5 and T to of the militia. What part were they together of the militia ? 41 . Of the prismatic spectrum, red occupies J, orange 5 3 o, and yellow T 2 S . What part of the whole do these three colours occupy ? 42. What part of a piece of cloth has a draper sold, who has cut ffi 3 G> S 5 2, &, and ftofit? 43. A treasurer has expended fcfc, ? 7 g , i, 7 5 , and ,*, of a given sum. What part of the whole has he laid out ? 44. In 1853, of the number of freshmen belonging to Cambridge iff belonged to Trinity College, 3 5 2 3 g to St John's College, & to Gonvilleand Caius College, and 5 | B to Queens' College. What part of the whole did these form ? " 45. Of the water of the Dead Sea, T || B is muriate of lime, ; V? muriate of magnesia, &*>** muriate of soda, 3^ sulphate of lime. What part of the whole are these ingredients ? 15* ^ Let us SUBTRACT a frac- tion, as -|, from an integer, - - - : _ ' ....... as 2. We dimmish one of 1 | 2 the units by |; thus, | | = |. This, with the other unit, makes the whole remainder If. In subtracting a proper frac- tion from an integer, we find the complement of the fraction, and diminish the integer by 1. (1) 18- = 1. 18 2. 10 3. 9 - T 4. 11 5. 8- 6. 23 - VULGAR FRACTIONS. 29 15. Let us subtract f from f . By - T 8 3 reducing the fractions to a com- '....... mon denominator, we find f= T \, ?$ T 9 2 1 = T 9 j 5 an( l by taking T 8 5 from T 9 ^ we obtain the remainder T ' T . W f f - 21 * IT* 7- J J 15. 2 9 36" ~~~ ft 23. a 15 8. 1-i 16. 1 1 TT ~ T\ 24. B i\V 9. 17. l-l 25. fl ~" I, 10. | | 18. if T 26. TT T3 11. | | 19. T 9 T- f 27. if -if 12. it - T 7 * 20. ~ T 9 7 28. 19 11 ^l T^ 13. T * T T 21. 1 3 T* 1 9 29. tt-*A u - ii ir 22. H- 1 7 30. i! H In subtracting a mixed -2 numuLr, as ^5-, iruin dii- other, as 3, we find the 1 2 3 difference first between -T 5 o the fractions, and then '- 1 i t i t i i J between the whole num- f 3 5 1 s * bers. Thus, J or -fa from 3 Ol * T 8 TJ leaves T 3 5 , and 3 2=1, So, 3|-2i=3 2 + - ~ 5 1 Io~ l r = 2 + ^^ = 2^ 31. 3i 2J 35. 18| 10J 39. 17i| 13 1 2 32. 7^ 5^ 36. 17f 10 T \ 40. 18if 17 sW 33. 17f 13 T % 37. 23|i 19JJ- 41. 29 T\V 9- r vv 34. 6| 3 ,\- 38. 16iJ 14JJ 42. m 2 T v 5 Let us take 1| from 3i. Since we -i cannot subtract or from or f , we reduce one of the 3 units to sixths. i 2 1 \ or | diminished by is thus == . ~6 2 1 1. So, 3i 1 2 l-l~ 61 2 1 1 6 X In subtracting the sixths, instead of taking 4 from 9, we may subtract 4 from '), ;ind add in 3. The prac- tical advantage of this method is illustrated in A, in which we take = 11 If B Units. Sixths 3 3 1 4 1 5 30 VULGAR FRACTIONS. 15 the lower numerator from the common denominator, and add * in the upper numerator. In B, we may consider the mzto as units of a lower name, of which six make up a higher unit. The solution is then obtained as in Compound Subtraction. 44. 16i 13| 45. 14f 9|i 46. 16* 10f 47. 14* 13 48. 15H 4 49. 13J 11$ 51. 13i - 52. 6*- 53. 181* 50. 8| - 4if 54. 23{.| 22 jf (5) | of a pole is below the level of a pond, ^ of it is in the water. How much of it is in the ground ? 55. g of a pole is above the bottom of a pool, and T \ is in the pool. What part of it is above the level of the water ? 56. A retail draper who has bought f of a piece of cloth, sells i of the piece. What part of it has he over ? 57. f of a common is laid out as bleaching- ground. What part of it is over ? 58. A sailor has spent T 9 3 of Ms life at sea. What part was spent before he went to sea ? 59. A person succeeding to a legacy left by an ancestor or de- scendant in the direct line, pays T o f the value as duty. What part is over ? 60. Of the prismatic spectrum, the blue, indigo, and violet rays together occupy , and the blue and indigo together T 5 g . What part does the violet occupy ? 61. The number of pear and apple trees in an orchard is f of that of the whole, and that of the pear trees is 5 7 5 . What part of the whole is the number of apple trees ? 62. Of a consignment of guano from Saldanha Bay, f g consisted of carbonate of lime and phosphates of lime and magnesia, and of the phosphates. What part of it was carbonate of lime ? 63. In 1857, the number of parliamentary electors in Scotland was ? y ? of the whole number in Great Britain. What part of the whole number was the number in England and Wales ? 2- i- -i- 4- 1 + I - A + i 5- i + 8+T 9 5 -l* 6. f ~ VULGAR FRACTIONS. 31 (2) A traveller has gone of a journey on foot, T \ on horseback, by rail, and the rest by coach. What part has he gone by coach ? * _i_ i\ i /20 + 24 + 45\ is T-*; J- -^ 180 ) 13. * of a pole is blue, 5 red, and the rest white. What part of it is white ? 14. A student has in three weeks read respectively 5 5 T , f , and of the First Book of the JEneid. What part of it has he yet to read ? 1">. A soldier while in the army had spent | of his life in the United Kiii'_rdom, 5 5 7 in Canada, T ' a in Gibraltar, in India, and 5 ' 7 in the Crimea. What part of his life had he spent before rafisti 16. having used T 8 , 5 7 ^, and of an ingot of gold, wishes tci know what part still remains. 17. Of the whole time spent by Professor Piazzi Smyth in the LOmical Ivxpedition to Teneriffe in 1856, T 2 g was spent in the lowlands of TencrifFe, T Y 7 at Guajara, and T 2 T 6 , at Alta Vista. What part was spent in the voyage? 18. Of the component elements of albumen, i is carbon, T 5 hydrogen, and 5 , nitrogen. What part of it does the remainder, consisting of oxygen, phosphorus, &c., constitute ? 19. Of the whole number of Jehoshaphat's " men of valour " in Judali and Benjamin, the three divisions of Judah were respec- tively Jg, 5 7 S , and 2 V What part belonged to Benjamin? 20. Of the black and mulatto population of Cuba in 1850, the free mulottoes were 5 3 T yi the free blacks s ViV an( ^ * ne mu latto slaves 5? | 5 . What part was the number of black slaves ? 21. Of the annual salaries of the principal, depute, and assist- ant clerks of the Court of Session, 5 deputes receive T | n each, and 9 assistants T 1 3 each. What part does each of the 4 principals receive ? In Mri/m-LYiNG a fraction by another, as by f, we consider that since th<> nultiplier is of 2, the product will be of 2 *. Nn-.v '2 X $ = 5, and the required product is = | -^ :$ = ,"-, which is thus = X f . In multiplying fractions together, the product of the numer- 32 VULGAR FRACTIONS. ators becomes the numerator of the product, and the product of the denominators the denominator of the product. In multiplying by an integer we repeat the multiplicand as many times as there are units in the multiplier ; in multiplying by a fraction we take that part of the multiplicand which is denoted by the multiplier. | X | may be expressed as f of , or -J of -, which being the fraction of a fraction is termed a COMPOUND FRACTION, in contradistinction to a SIMPLE FRACTION, as f. A Com- pound Fraction is reduced to the form of a simple one by mul- tiplying the numerators arid the denominators, as in Multipli- cation of Fractions ; thus, f of | = A. We may consider - of -f | either as i of 2 X 4 or, as < in the diagram, we may di- i 4 s is vide | into three equal parts, and take two of them. Si- milarly, we may take f of , A T s either as -| of 4 X f, or, as in the diagram, we may divide intone equal parts and take/owr of them. 4 /IN 8 V 1 7 V _ 6 ft 1 2 3 \ / "S r\ *Tw 9 ^ 10 4"3> ?!>' 5 Since 2 is a common factor of 8 and 10, we CANCEL these numbers, and write the number of times the factor is con- tained in each. By thus cancelling any numerator with any denominator with which it has a common factor, we obtain the product in its lowest terms. (f)\ e y 7 "^"s/" 7 " i W TTT * T8 .35- X ^. T-S. 5 3 The numerator of the product is = 1 X 1. Unity takes the place of a numerator or a denominator cancelled with any of its multiples. X _ 29- v 31 _ 1. I X * 9. I X 5| 17 Qi V Q 3 -U. U T A 0-6? 2- * X A 10. * X 7f 1 Q 72 V 2 O O TIT ^ 2~TT 3- % X A 11. f X 161- 19. 19i X 16| 4- | X i$ 12. I X 18i 20. 231 X 3i| 5. 4^ X ff 13. 1 T \ X 91 173 v is'* ^* - 1 ^ X T^ f 6- 4 X 14. 2A X 4 22. |J X H- 7. A x if 15. | X 64 23. 4A X 17,1 8- it X if 16. A X 7| 24 * a 3 X 3 l ~- 17. VULGAR FRACTIONS. (4) * X 5| X 4 T T =| X ^ X = Vi 33 25. i X 2| X -- 26. J X 3i X 1 27. 3| X X 28. 2J X 4 X 3f 29. 3 T ' T X T V X 5J 30. 3f X & X 61 31. | X i 32. 6| X 33. -J X 4 34. T T X 1 = 10*. X 8i fr X 6* X 6| r X 41 35. 7J X * X i4 36. 81 X If X * Reduce the following Compound to Simple Fractions : -a- (5) i of * of A of =i_X f X * X = T V 2 5 (6) 1-ofi 37. 4 of of J 38. f off of if 40. f of f of J4 41. 4 of J of 8} 42. f of 44 of 4 of 4 j 43. f of 2J of a of 64 43. 44 45. | of 2J of f c Of T 7 g- Of S. 1 1 n f i 3 '* 01 T 3 4 X f X ^ X J| = 11 46. 4$ of, % of 4 J of 47. 4 of | of | of 4 of T% of 7 48. J of 64 of A of if of 5 49. ff of 16f of A f7 TT 51. 52. 53. 54. of T 4 T of - Of i Of ^ - T of8iof8^ - of T 4 ? of6561 (7) If a train runs of a mile in a minute ; how many miles will it run in f of 431 min. ? ml. 29 -4- 3 87- 29 "5" s^ ~5 /^ T ^"~ Q^ ^N n_ f\ "2" ~~~ "4" ^^ 4 ^'^^* -3- 2 55. A soldier was in hospital 5 \ of the time he served in India, which was 5 e , of his life. What part of his life was he in hos- pital? 56. A sailor's share of prize-money is 5 7 5 of a midshipman's, whose share is 2 -\ of a lieutenant's. What part of a lieutenant's share does a sailor get? 57. Jack, who gets } of a plum-pudding, gives of his share to Tom, who gives 1 ; of his to Harry. What part of the plum-pudding i I arry get ? B 2 34 VULGAR FRACTIONS. 17* 58. A schoolboy prepares his lessons at home in of the time he plays, which amounts to T % of | of a day. During what part of a day does he prepare his lessons ? 59. On the Geelong and Melbourne Railway, the fare per mile by the third class is ^ of that by the second, which is | of that by the first, which is 3|d. Find the fare per mile by the first. 60. Find the receipts of a railway for a week which amount to $fi of 6384. 61. The* number of registrars employed in the Census of 1851 was T yy 7 of that of the enumerators, of whom there were 38740. Find the number of registrars. 62. If a train runs a mile in f of 3f min., in what time will it run -fo of 23 miles ? 63. 24 flagstaffs are placed on a road at the distance of of 73 J yards between each. How many yards are between the first and the last. The number of spaces between a number of objects placed in a line is one less than the number of objects. 18. In DIVIDING a fraction by another, as f by f , we consider that since the divisor is of 2, the quotient obtained by dividing by f is 3 times as large as that obtained by dividing by 2. Now | 2 = |, and the required quotient is = f X 3 = f f -r" I thus produces the same result as f X f In dividing a fraction by another, we invert the divisor, and proceed as in Multiplication of Fractions. A fraction inverted is the RECIPROCAL of the original fraction ; thus, f is the re- ciprocal of f . The product of a fraction by its reciprocal is = unity. To divide f by , we may, *. \ : \ i L ." as ill the first diagram, ac- I | 1 | cording to the previous explanation, take one-half of f , which is |, and by taking three parts each = |, we obtain |. Expressing I and | in the same 1 f denominator as T 9 F and T \ re- ' .....i... spectively, we see in the second 13 T 9 s 1 diagram that if we take 8 twelfths as the unit. 9 twelfths con- tain 9 of those parts of which the unit contains 8. T \ is thus f of T\> or I is the quotient obtained by dividing T \ b*y -^ or is | -T- f. (i) i ~ T ' T = * x v = = i&. (2) ^-7{f= X = VULGAR FRACTIONS. 35 18. 1. 1 - 1 9. 4 _ - A 17. 19| - - f 2. 1 - - 1 10. - - AV 18. 17A , 3 3 3. * ~ - 1 11. t? - - if 19. 41-7- 4. * - 12. If ~ - ti 20. 11* 5. IT " - if 13. 5| - - .{..I. 21. 2 '7 _ 6. - - ** 14. 7* - - i J 22. 3 ii - - 4 7. ii- _ 2 15. 3f - - 1 23. 14| - 8. TT - - 3T 16. 6 * - - if 24. 2i" (3) J -5- T 7 r of 3| =' "*" v ] i v -8- ^- X 10 = W- (4) Aof4j-r-f = 25. |} - I of 10* ||i -r- J of 25$ X 26. 27. 28. 29. 30. 0^ 2| -1- 7 f of 12| 33. f of IA -T- Jl of | 1 J * of 3| 34. | of H- A of 4 of 7i 35. A of iii -T- f of 1 f of || -M of A We may write the quotient | -f- f in the following form : 17 v" 6 " 5. X 5 31. ^ of j 32. 33. 34. 35. 36. H- wliich the dividend becomes the numerator, and the divisor the denominator of a COMPLEX FRACTION. A Complex Fraction has a fraction in either its numerator or 3 2 5- B 3 ninator, or in both of them: thus, -i, , -^, 13, are T ' 4" *^ 'I" coni]lcx fractions. The reduction of Complex to Simple Frac- tions is similar to the Division of Fractions. Reduce the following Complex to Simple Fractions : (5) f = A- We have multiplied the numerator and the denominator of the fraction by 4, the denominator of the numerator. So, when either the numerator or the denominator is an integer, we multiply the numerator and the denominator by the de- nominator of the fractional term. (c) -i- = * -;- -T \y/ 13 o X TT [9 = *l 36 VULGAR FRACTIONS. quotient may evidently be obtained by multiplying es of the complex fraction for the numerator, and 18. The extremes means for the denominator. the the (7) _ 12* 5 35- . 2 1 = 14. W- 11 J 3 We may cancel either of the extremes with either of the means. As the numerator and the denominator will likely he expressed in the lowest terms, we thus cancel the first with the third, as 35 with 77, and the second with the fourth, as 4 with 6. 37. 6 8* 42. 3 T 47. $ 38. 7 43. 7 48. S "9 Ttf 15" 39. 9 12| 44. if 49. 19* 28 T 7 -g 11 O s 40. 134 45. 50. T? 41. 7 46. it 51. 24| (8) tof_3j | of 34 , 13 x -25-5 13X7 _ 91 6X20 120- (9) 52. 53. 5 5 5 *- 4 ^ 3 3 T ff T 8 T |of3i 54 tVofiai 3 x. 5 8 f 6} 56. 1. 1 *of9* 5 *of!3* ^^ 1 f T 00. -J 7_ 57. JL T T TJ ii | i VULGAR FRACTIONS. 37 18* O 1 ) How many pieces, each 30 yards, are contained in 114} yards? 14 114{ + 30| =^ X -4= || = 3jJ pieces. 3 58. If a piece of cloth is 29f yards in length, and a remnant 1 {j yard ; how many times is the former as long as the latter? 59. How many squares, each $ sq. inch, are contained in 132 sq. inches ? 60. How many postage- stamps, containing $f sq. in., are in a sheet of 172* sq. in.? 61. How many times can a measure of | pint be filled out of a yessel containing 63 1 pints ? 62. How many times will a coin 2 inches in circumference turn round in traversing 30 inches ? 63. Mercury is 13f times as heavy as water, and gold is 19 1 times. How many times is gold as heavy as mercury ? 64. The pellicle from which goldbeaters' skin is made is 3^0 inch thick, while gold leaf is 335^05 m thick. How many times is the former as thick as the latter? 65. The largest scale of the Ordnance Survey Maps is lineally T?*72 f that of nature, and the smallest is g 3 joo- How many times is the farmer as large as the latter? 66. The mass of the Earth is 3 55 'j 5T , and that of Jupiter is y^ of that of the Sun. How many times is the mass of Jupiter as great as that of the Earth ? 67. A book of 240 leaves without boards is ji inch thick, and another of 180 leaves without boards is T 7 5 inch thick. How many times is the paper of the form eras thick as that of the latter ? (12) How many men are in a regiment of which T % = 255 men ? 255 -r fi; = 255 X V = 85 men - The regiment is evidently = '3 of T * 5 of the regiment, but T J ff of the regiment = 255; 'hence the number in the regiment = '3 of 255 = 850. 68. Find the length of a pole of which f = 18 ft. 69. The Pylades war steamer, having 2 1 guns, has T S 3 the num- ber which the Princess Royal war steamer has. Find the number of guns in the latter. Find the distance from London to Kurrachee, that from the head of the Red Sea to Kurrachee, which is 1700 miles, being T y 3 of it. 38 VULGAR FRACTIONS. 18* (13) Of a pole, T ^ is painted white, ^ green, JJ red, and the remainder which is 5 ft. is painted black. Find the length of the pole. T\r T KG i ii = GO == "e o 2 1 i = J; 5 ft. -S- J = 10 ft. 71. Of the area of the five great lakes, Lakes Erie and Ontario together contain , Michigan and Huron together f , while Lake Superior contains 32000 sq. miles. How many square miles do they in all contain ? 72. Of an army $ is English, 5 7 ? Scotch, T $ ff Welsh, and the remainder numbers 4796 Irish. How many are there in all ? 73. Of the distance from Edinburgh to London by rail, via Car- lisle, that from Edinburgh to Carlisle is , from Carlisle to Preston < 9 ii of 3J of 11 H of* of 10. ?i -> A O f 24 (3) ( + *- *) 155 99 40 -f 120 114 34 180 - 180 45' From f i we are required to subtract |- -f- f diminished by f . By subtracting A -f- 1 we obtain a remainder too little by f . By adding f to this remainder we therefore obtain the re- quired result. When " " is placed before a parenthesis, we change the " + " an d " " signs of the enclosed quantities respective- ly to " " and " +," and add or subtract as indicated by the changed signs ; thus : 12. H + A- (H 4- 13. (4) t "r.f 1 8 ^- T! = 15. 16. ( 18. *_=.* l+S VULGAR FRACTIONS. 41 19 The following show the difference in value produced by changing the place of the parenthesis : 19. 20. 21. - T of x X !?-- 2O In REDUCING the fraction of a quantity to a lower name than that in which it is given, we multiply the fraction by the num- ber of times the former is contained in the latter ; thus, in reducing / T foot to the fraction of an inch, we multiply the numerator by 12, and obtain f inch, which is = f of ^ foot. (1) Reduce ^V oz. troy to the fraction of a grain. 2. 3. 4. & 7. 9. 1 X 20 OZ. rry 9 ,1. a , s. __ cr. . , s. , hfd. V T cr sixd. T cwt. Ib. av oz. av. 9 ^' 10. .T^lb. tr oz. tr. 11. T VV ac po. 12. ^,da ho. 13. ,jscwt Ib. ^ 15. T ^tf ml yd. 16. T y T bu gal. 17. T fcfu yd. 18. -gVo ho min. In reducing the fraction of a quantity to a higher name than that in which it is given, we divide the fraction by the num- ber of times the former contains the latter ; thus, in reducing T \ f/rain to the fraction of a Ib. avoir., we multiply the denom- inator by 7000, and obtain 1^33 Ib. avoir., which is = TT UTy of 7000 gr. (2) Reduce 4d. to the fraction of a crown. 7 X 10. 20. 21. i . s. 22. $s gu. 23. jib T. 24. |f in yd. 42 VULGAR FRACTIONS. 31. -f min da. 32. ; da co. yr. 33. if qt qr. 34. | cub. in cub. yd. 35. |f po ac. 36. ijf gr lb. av. 2O 25 - If sec ............ no - 26. -I gal ............. bu. 27. 4| yd ............. ml. 28. |f oz. tr ......... lb. tr. 29. if pt ............. gal. 30. fl pk ............ qr. In reducing the fraction of a quantity to a name which is neither a measure nor a multiple of the name in which the fraction is given, we both multiply and divide as in the fol- lowing example : lb. av. to the fraction of a lb. troy. (3) Reduce 720 In multiplying by 7000, we reduce the fraction of a lb. av. to that of a grain, which, when divided by 5760, becomes that of a lb. troy. 42. !!. oz. tr oz. av. 43. fflk ft. 37. |fl cr. 38. -Hgu . 39. Jnl ft. 40. f E. E yd. 45. ^ co. mo co. yr. 41. T 4ihjlb. av lb.tr. 46. ifgeog. ml Imp.ml. In reducing a compound quantity to the fraction of a simple or a compound quantity, we proceed as follows : (4) Reduce 1 * 2 * 7 to the fraction of 1 * 13 // 5. 1*2*7 = 271d. 1*13*5 = 401d. 1*2*7 = ift of 1*13*5. Having reduced the quantities to the same name, we find that since 1 2 7 contains 271 pence, of which 1 ,, 13 // 5 contains 401, the former is JJ of the latter. 47. 11/6 1 48. 2/2* 1 49. 2ft. 8 in 1 yd. 50. 3ro. 15 po 1 ac. 51. 6fu. 15 po 1 ml. 52. 6oz. 3dwt. ...1 lb. tr. 53. 4/4 13/8 54. 7/8J 13/3J 55. l//15//3 3//13//9 56. 3 oz. 4 dwt 57. 3 fu. 44 yd 58. 2qr. 3nl 59. 2ro. 14 po 60. 7bu.3pk 61. 7 ho. 12mm. 62. 4 da. 17 ho 63. 2230 X 64. 66 32 X 23 /x .. 2 lb. 6 oz. 3ml. 3yd.lqr. 3ac. 1 ro. Iqr.Sbu. ..3da.4ho. lwk.3da. 360 ..90 VULGAR FRACTIONS. 43 2O (5) Reduce s. to the fraction of % ., or find what part f s. isof4?. . 3 5 X 20 _ a _ 3 X 27 TStf 100 X 10 f , Q ,, OI * 4. (2). 65. T \ 6G. f s 67. T 3 34- ^ai 7T5 - = 5//2i. 24 2 Otherwise: j- 1 = 7 v of 25 96 {g 2 25//0//0 77. 44 s. 84 T 3 A ^' av< 91. *y y ml. 78. J4 s. 85 -|^ CWt. 92. 4f ac. 79. J4J . 86 19 T 1 93. *y- y oz. tr. 80. 44 . 87 . T y T lb. tr. 94. T v- 5- bu. 81. Jf cr. 88 It J d - 95. || pk. 82. 4f4 gu. 89 . 4J sq. yd. 96. T V Ik. 83. fl. 90 . 44 cub. yd. 97. A q f Ju. yr. (7) Find the value of J of 9 T \ acres ac. ac. ac. ro. 10 po. Jof9 T ac. = JX W = 3 ,Y =: 7, 7 , j^ =^_ 10 = 23 j ; ao. ro. j)0. 12 3 7 A ac. = 7 // // 234. 44 VULGAR FRACTIONS. OQ ac. ro. 10 po. ac. ac. ro. po. Otherwise: x '*'=i* 4 * = W 9 9 A = 9)64 7 // // 98. f of 5} cr. ^ofSfhf.cr. 102. T \of3fu. 12 po. 100. | of 3 // 7*6 103. | of 2 ac. 3 ro. 104. 1 of 3| s. 105. $of2ho.34min. 106. | of 3 wk. 6 da. 107. Express a Russian Archine, which is \ of a yard, as the fraction of a mile. 108. Express the height of Ben Macdhui, which is \\\ of a mile, in feet. 109. Schiehallion, where Maskelyne made a series of observations on the Density of the Earth, is nearly | of a mile high. Express its height in feet. 110. Harton Coalpit, where Airy conducted a series of observa- tions on the Density of the Earth, is f \ of a mile deep. Express its depth in fathoms. 111. The velocity of sound is 5 7 5 of a mile ^ sec. Express it in ft. 112. Express 5 dwt. 9 gr., the weight of a guinea, as the frac- tion of 1 Ib. troy. 113. In an estate of 3173 acres 20 poles, the roads occupy 66 ac. 1 ro. 8 po. What part of the estate is occupied by roads ? 114. The distance traversed by an express train in T 5 5 hour is run by a goods' train in | of 1 J hour. What fraction is the for- mer time of the latter ? 115. The National Subscription, promoted by Cromwell in aid of the Waldenses, amounted to 38097 7 3, of which Cromwell gave 2000. Express the latter as the fraction of the former. 116. In November 1855, the Patriotic Fund amounted to 1,296,282 4 7, of which Glasgow subscribed 44,943 1 10. What part was the Glasgow subscription of the whole ? 117. Express 58|? yards, the depth of an Artesian well, as the fraction of another which is /^ of a mile deep. 118. What fraction is an oz. avoir, of an oz. troy? 119. Reduce a grain to the fraction of a dram avoir. 120. Express a Ib. troy in avoir, weight. 121. In Scotland, during June 1856, the mean weight of vapour in a cubic foot of air was 3 T 7 o grains. Express this as the fraction of 1 Ib. avoir. 122. In Scotland, during April 1856, the mean weight of vapour in a cubic foot of air was T o55s Ib. avoir. Express this in grains. 123. Mont Blanc is 15780 feet above the level of the sea, and VULGAR FRACTIONS. 45 2ODhawalagiri is 5 g 7 g' 5 miles. Express the height of the former as the fraction of that of the latter. 124. A degree of longitude on the parallel of Greenwich is nearly = of a degree of the Equator, which is = 60 X 6076 ft. Find the number of Imperial miles in the former. Find the sum of f ac., } of 3f ro., and } of 16} po. I. II. ac. ro. po. ac. | ac. =0*2 // 16 J of 3| ro. = // 2 // 36} } of 16} po. = // 4} 1*1* 17} f }j c.=lac. Iro. 17}po. In adding fractions expressed in different names, we may, as in I., find the value of the fractions, and then proceed as in Compound Addition; or, as in II., we may reduce the frac- tions to the same name, and having added them, find the value of their sum. 7. T. + | cwt. + \ qr. lo3fro.=|fro.= |f= 2. A . + f fl. + } s. 3. I ac. + 2 j} ro. + 3 po. 9. f f.+7f cwt.+lj|qr.+20flb. 4. f ml. + T 3 T fu. + T * T po. 10. , 5. f Ib. + 1} oz. + 2} dwt. 11. T ' ft. + | yd. G - T6u -+i s -+!i of Vf the sea, is r>j s s 9 4 miles, and that of Aconcagua in the Andes 1 ) feet greater than 4 miles. Reduce the latter to the fraction of the fori; 50. The attraction of gravity at the Equator is less than that at the Poles by 5 5 on account of centrifugal force, and ^^ on ac- count of the earth's oblateness. Find the sum of these fractions, and give a fraction with the numerator 1, to which the sum is nearly equal. DECIMAL FEACTIONS. IN Integers we employ the decimal notation, by which the places ascending from right to left have respectively ^ the local value of units, tens, hundreds, thousands, &c. Fractions in which the decimal notation is employed are termed DEC- IMAL FRACTIONS. In Decimal Fractions, the places de- scending from left to right have respectively the local value of t< i> if ix, hundredths, thousandths, &c. Thus, in 4'235, the point is placed to the right of the units' place, and the inures to the right of the point represent 2 tenths, 3 hun- ths, 5 thousandths; '235 denotes T % + T ^ + isW == +^ + 5 = ||; and 4-235 = 4 T m- Similarly, -0379 I _ 300 + 70 + 9 379 denotes ^ -f- TBVV T TSUSTJ "'ioooo 10000' 52 DECIMAL FRACTIONS. <26 A Decimal Fraction may be expressed in the form of a vul- gar fraction, having the figures of the decimal as the numerator, arid 10, or a power of 10, as 100, 1000, &c., as the denom- inator. The number of figures in the decimal is = the number of ciphers annexed to " 1 " in the denominator of the vulgar fraction. Ciphers annexed to a decimal do not alter its value ; thus, 36 = -360 = -3600, for T %% = ^ = T 3_e_o_o_. Express the following decimals in the form of vulgar frac- tions : (1) -1341 = T V&V (2) '00739 = . 1. -3 2. -27 3. -167 4. -231 (3) -005 = 13. -8 14. -125 15. -3125 16. -15625 5. 6. 7. 4153 8827 32471 98347 9. 10. 11. 12. -009 -0007 -000093 -000107 (4) '0848 = T jjfr = 17. -032 18. -004 19. -0625 20. -7168 21. -0425 22. -46875 23. -00256 24. -000375 Write the following fractions in the form of decimals : (5) T Vo = *71. (6) T ^-o- = '003. 25. 26. 27. 28. 29. 30. 31. 71ER50 33. 34. 35. 307 ToooooS" 27* By Amoving the decimal point of a number one place towards the right, we increase the value of the number tenfold ; thus, 34 X 10 = 3-4; -07 X 10 = -7. By moving the decimal point of a number one place towards the left, we diminish the value of the number tenfold: thus, 7'13 10 = -713: 79 -i- 10 = -079. In multiplying a decimal by a power of 10, we move the point as many places towards the right as there are ciphers in the multiplier; and in dividing by a power of 10, we move it as many places towards the left as there are ciphers in the divisor. (1) Multiply and Divide -00347 by 1000. 00347 X 1000 = 3-47 00347 -f- 1000 = -00000347. 27. DECIMAL FRACTIONS. (2) Multiply 3-219 by 10000. 3-219 X 10000 = 32190. (3) Divide 7830 by 100000. 7830 -r- 100000 = -0783. 53 1. 2. 3. 4. 5. 6. 0369 X 1000 J17<3 X 100 42839 X 10000 3-216 X 1000 7-23 X 10000 15-9 X 10000 7. 8. 9. 10. 11. 12. 273 100 5236 1000 367 10000 72-3 100 98-475 -f- 1000 8-375 -H 10000 f ^ re duce a vulgar fraction, as |, to a decimal, we must multiply the numerator and the denominator by such a num- ber as will produce a power of 10 in the denominator. Since 1000 is the lowest power of 10 which contains 8, we multiply the numerator and the denominator of by - < V > -, which is = 125. 1 = 1^=^^55: -375, Now, 3X125 = 3 X 1J ir- = 3 -H 5 | ience tne figures of the decimal are obtained by annexing ciphers to the numerator of the vulgar fraction and dividing by the denominator. The number of places in the decimal is = the number of annexed ciphers. AYlien we can readily find how often the lowest power of 10, which is a multiple of the denominator, contains it, we multiply the numerator by the quotient; thus, Since the prime factors of 10 are 2 and 5, no number con- taining any other prime factor will exactly divide a power of 10. Hence, those Vulgar Fractions only whose denominators in the lowest terms of the fraction have no other prime factor than 2 or 5, produce TERMINATE DECIMALS. Express the following vulgar fractions as decimals : (1) * = "75. 1. i 2. i 4- t 5- I G. I !) T J T = -056. 125 )| or T J ? = TSU3 = 056. 1) ,fe = T fo Of ; \ = -0075. 7. A 13. 9 19. 1 3 25. AV 8. it 14. *V 20. Wj 26. 183 6^^ 9. T V 15. r2ir 21. iVff 27. 329 on 10. A 16. 7 Yl 22. 7 28. 233 23S 11- *i 17. TVlT 23. T^TT 29. 1 KIT 12. A 18. 1 1 T5T5 24. if 30. *15 54 DECIMAL FRACTIONS. 29 In the ADDITION of Decimals, we place tenths under tenths, hundredths under hundredths, &c., and thus add figures hav- ing the same local value. (1) 67-37 + -1883 + -0965 + 6-314 + 77-4006. 67-37 We carry as in integers; thus, for 14 ten thou- -1883 sands, we write 4 in the ten thousandths' place, -0965 and carry 1 to the thousandths' column. Simi- 014 larly with the thousandths and the hundredths. For 13 tenths, we write 3 in the tenths' place, and 77'4 ( carry 1 to the units' column. 151-3694 1. -30103 + -47712 + -60206 + -69897 2. -096 + -0096 + 96-0096 + -96 3. 7-0096 + -314 + -326 + 81*093 + 325-73 4. -7146 4. -003 + 94-216 + -314 + 95-279 5. 93-423 + -875 + -329 + 4-326 + 57-916 6. 373-912 + 37-3912 + 3739-12 + 3-73912 7. 247-35 + 9-168 + -709 + 82-361 + 18-017 8. -73 + -0073 + -073 + -00073 + -000073 9. .716 + -00716 + 716-0716 + -0000716 (2) Add J, J, and T 5 g by Vulgar and Decimal Fractions. H jL = -3125 = m = 1-9375 10. i + I + A + 11. i + l + l + 12. ' 3O In the SUBTRACTION of Decimals, we find the difference between figures of the same local value. (1) .59 _ -043. By taking 3 thousandths from 10 thousandths, we '59 obtain 7, which we write in the thousandths' place. .Q43 We proceed as in integers, taking 5 from 9, or 4 from 8, c. '547 1. -5475 -4212 2. -875 -525 3. -275 - -198 4. 5-25 3-875 5. 3-125 1-9375 6. 8-425 5-3875 7. 1-25 -175 8. 2-834 2-786 9. 3-245 1-2375 10. 1-1 -0009 11. 8-75 7-00009 12. 9-03 -90003 DECIMAL FRACTIONS. 55 3O. (2) Subtract - 13. J - | 14. J} - V from J by Vulgar and Decimal Fractions. ^ = U = "44 JM = TV = -4375 ,fc, = -0025 16- I - | 17. if - 18. is In the MULTIPLICATION of Decimals we proceed as in in- tegers, and point off as many decimal places in the product as there are together in the multiplicand and the multiplier. (1) Multiply -347 by 2-3. 347 X 2-3 = jfo x *s 7 o B i '7QQ1 1041 "7981 In working by vulgar fractions, we see that the number of ciphers in tin- denominator of the product is = the sum of the numbers of hers in the denominators of the factors ; so, the number of il places in the product is = the sum of the numbers of the factors. *53 (2) Multiply -53 by -0047. -0047 371 212 002491 74213 TOO 519-491 5-09 67000 3563 3054 _ 341030 -5.3 X -0017 = T Vu X T sVro = '002491. (3) Multiply -74213 by 700. Since one factor contains five decimal places, and the other ends in two ciphers, we point off three places in the product. (4) Multiply 5-09 by 67000. Since one factor contains two decimal places, and the other rnds in three ciphers, we annex one cipher to the product. 1. 5-27X4-83 2. -430 x 2-1!) 89X-76 4. 2-38x3-47 5. 5-G2X-213 6. -278X-547 7. 5-27 X -00483 8. -0436X '00219 !). 18-9X-000076 10. -238X-0347 11. -0562X-0000213 12. -00278 X '000547 13. 52-7X48300 14. 4-36X219000 15. -189X7600 16. -00238X347000 17. -00562X21300 18. 27800X '000547 31. 56 DECIMAL FRACTIONS. 19. 98-7654 X '983427 22. -007639 X 763900 20. -123456 X '654321 23. 87'6591 X 684000 21. 5-78934 X '000763 24. -000009 X '000983 25. 100 X '01 X '001 X -0001 X 1000 26. 300 X '003 X '0003 X 3000 X '00003 27. 5000 X 500 X '0007 X '035 X '00005 28. -003 X '03 X '3 X '0003 X 30000 Find the following products by Vulgar and Decimal Fractions : 29. | X A X 2j 32. 4 X A X 30. 31. X i X X X 33. 34. | X 2f X X X 32* In the DIVISION of Decimals we divide as in integers, and point the quotient so that it may^ contain as many decimal places as are in the dividend, diminished by the number in the divisor. (1) Divide 228-75 by 30-5; and 6-4 by 25-6. 30-5)228-75(7-5 25'6)6-400(-25 2135 512 1525 1280 1525 1280 = T V X X = 4 = <25 < H A X In dividing 228-75 by 30-5, since there are two places in the di- vidend and one in the divisor, we point off one in the quotient. In dividing 6'4 by 25'6, since we use three places in the dividend and one in the divisor, we point off two in the quotient. The following examples illustrate various modifications of the general rule : (2) Divide 48-97 by -59 ; and 292-3 by 3-95. 59)48-97(83 3-95)292-30(74 472 2765 177 1580 177 1580 (3) Divide~ 7 68625 by 91500; and 32-1 by 128400. 91500)-68625(-0000075 128400)32-100(-00025 6405 2568 4575 15420 4575 6420 Since in dividing -68625 by 915 we would have -00075, by in- creasing the divisor 100 times we diminish the quotient as many times, and thus obtain -0000075. Similarly, in dividing 32-1 by 128400, the number of decimal places in the quotient is = the sum of the number of decimal places used in the dividend, and of the number of annexed ciphers in the divisor. DECIMAL FRACTIONS. 57 32. Divide 2230-1 by -769 ; and 1400 by -00224. 769)2230-1(2900 1538 6921 6921 00224)1400-00(625000 1344 560 448 TT20 1120 In dividing 2230*1 by 769 we would have the quotient 2*9. By diminishing the divisor 1000 times we increase the quotient as many times, and thus obtain 2900. Similarly, in dividing 1400 by 00224, we annex as many ciphers to the quotient as there are dec- imal places in the divisor, diminished by the number of decimal places used in the dividend. We may often find it of advantage to reduce the divisor to an integer, and move the decimal point in the dividend as many places towards the right as we do in the divisor. According to this method, the examples in (2) and (4) would be expressed in the following manner : 1897( 395)29230( 769)2230100( 224)140000000( 1. 1-7503-7-7-61 2. 40-3858 -r- 6'34 3. 39-538 -T- -53 4. 392-37-7-31-9 5. 110-i'Jf> 1-53 6. 5-2441 -7- 22-9 7. 1750-3 -f- -0761 8. 4038-58 -T- -0634 9. 3953-8 -r- -053 10. 39237 -7- -319 11. 1109-25 -7- -0153 12. 524-41 -7- -0229 13. 175-03 H- 76100 14. -403858 -r- 63400 15. -39538-7-5300 16. -39237-7-3190 17. -110925-7-153000 18. -52441-7-22900 19. -0156366 -7- -0042 20. -03486 -r- 4-98 21. -378816 -7- 5-919 22. 20973-6 -7- -8739 23. 9110-64-7-2900 24. 7-127577 -7- 1-0053 Find the following quotients by Vulgar and Decimal Fractions : 25. f -Mi 27. 1)4-7- 10J I 29. 44 -7- A 26. 7i -f- TS 28. 44-4-2$ I 30. 3j -7- 12f In an INTEKMINATE DECIMAL, one figure or a series of figures * continuously recurs. The figures which recur form a Period. AVln-n the decimal contains the recurring period only, it is termed a Pure Interminate, as -333, &c., written -3 ; '036036, &c., written '036, AVI i en the decimal contains a terminate as well as an interminate part, it is termed a Mixetf Interminate, as -1666, &c., written -16 ; -159090, &c., written -1590. When the period contains one figure, the decimal is called a Repeater ; but when more than one, it is called a Circulator. c 2 58 DECIMAL FRACTIONS. 33* rURE INTERMINATE. MIXED INTERMINATE. Pure Repeater as... -3 Mixed Repeater as... -16 Pure Circulator // ...-036 Mixed Circulator....// ...-1590 A vulgar fraction whose denominator in the lowest terms of the fraction contains neither of the prime factors 2 or 5, pro- duces a pure interminate ; thus, = *3 ; y = '428571. A vulgar fraction whose denominator in the lowest terms of the fraction contains 2 or 5, and one or more of the other primes, produces a mixed interminate ; thus, = -16 ; 14^ = -5236. Express the following vulgar fractions as decimals : (1) f = -857142. By annexing ciphers to 6 and dividing by 7, we find that the quo- tient consists of a period of six figures. (2) ,' T = -3i8. The interminate part of the decimal begins at the second place, and consists of a period of two figures. (3) T V = -05882352941 17647. When the numerator is unity, and the T ' T = -05882 T 6 7 denominator such a prime as will produce e_ _ -35294 * a considerable number of figures in the pe- riod, we may work as follows : By taking out "J the decimal, say to 5 places, we obtain T ' 7 i . = -05882 T 6 7 , which, multiplied by 6, gives the decimal for T 6 7 . Proceeding similarly with the other final vulgar fractions, as in the subjoined process, we have T ' 7 = -05882352941 17 6470588 T y By ^examming where the figures begin to recur, we obtain a period of sixteen figures as above. 1- * 2. 3. 4. 6. 7. 8. 9. 10. 34. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. if li Express the following interminate decimals as vulgar fractions (1) '185. 1000 X '185 = 185-185 1 X '185 = -185 Therefore, 999 X '185 = 185 And, 485 = iff = / T t In reducing a pure interminate to the form of a vulgar frac- tion, we take the period as the numerator, and write " 9 " as often in the denominator as there are figures in the period. 34. i. 2. 3. 4. (2) (3) 4 64 07 962 (4) DECIMAL FRACTIONS. 48i = f|i = Jf 076923 = 7 769? r?S53 3 5 = TT' 5. 135 9. 296 6. 288 10. 023i 7. 259 11. 00369 8. 48 1 12. 02439 38i 1000 X 68i :=: 681-81 10 X 681 = 6-8i re, 990 X 68i = 675 13. 14. 15. 16. 59 428571 153846 000407 047619 And, -681 = f}$ = if. In reducing a mixed interminate to the form of a vulgar fraction, we take for the numerator the difference between the ml numbers, which respectively contain the figures of the decimal and those of its terminate part ; and for the denomina- tor we write " 9 " as often as there are figures in the period, and annex as many ciphers as there are figures in the terminate part. ,_. .1->'-M 1234 12 naa - . en The following method may also be employed in reducing a mixed interminate to the form of a vulgar fraction : 681 = -6 + -081 = of -81 35. .!..-. ...*. 594 + 81 T5 1 f ' I! To 1 555 990 "so = 55' 17. 16 21. -7045 25. -0054 29. 00962 18. 116 22. -0045 26. -0916 30. 000216 19. 0138 23. -0054 27. -0916 31. 5142857 20. 416 24. -0054 28. -0916 32. 1076923 (10 3 -f -8i + -037 + -375. Since the terminate decimal '375 occupies three places, the interminate part of the sum *333 333333 MS at the fourth place. The periods, con- ic of 1, 2, and 3 figures respectively, are 037 181818 037037 extended G places beyond the terminate dec- 375 imal. Mini as they then recur in the same rela- tive order, the period in the sum thus consists 1-563552188 of (J places, which is the L. c. M. of 1, 2, 3. In extending periods to as many places as are denoted by the L. c. M. of the number of places in each, we are said to make the periods siinilar. 60 DECIMAL FRACTIONS. 35 I* 1 the Addition of Interminate Decimals, having extended the Interminates to the longest terminate part, we make the periods similar and then find the sum. (2) -3 + 4 + -7 = 1-5. 3 _|_ 4 _|_ 7 14. Since, by extending the decimals a place to the right, we would obtain the same sum, we add in 1, and thus obtain the sum = 1-5. (3) 4-962 + -416 + 5-076923. 4-96 296296 As the periods have been made similar, we '41 666666 first add the columns at the beginning of the 507 692307 similar periods to find the number to be car- ried to the last column. 10*45 655270 (4) -3 + 4 + -5 + -6 = 2. 6 = 18, so with the carrying figure the sum is When we obtain 9 as a repeater, we write and carry 1. 1. -5 + -1 +'6 + -3 2. -2 + -8 + -7 + '4 + -6 3. -09 + 45 + -27 + -54 4. -36 + -18 + '63+-8i 5. -962 + -26i+-i62+-i85 6. 370-1--259 + -636+-407 7. -509 + -037 + -75 8. -216+-216+-2i6+-2l6 9. -037 + '503 + -142857 12. r 4- WW 4~ 4~ ' 36. In Subtracting an Interminate Decimal from another, we make the periods similar, and then find the difference. 91142857 3J962962 5 179894 275 j.96|296 078 703 030 - -0300 Having found that we carry 1 from the be- ginning of the period, we take 3 from 7, &c. (2) -275 1962. In subtracting an Interminate from a Ter- minate, instead of carrying frorn the beginning of the period, we may subtract each of the figures in the inter mina'te from 9 ; thus, having obtained 703 by taking 296 from 999, we carry 1 to 6 in the subtrahend. 1. -16 -07 2. -216 -1583 3. -243 -074 4. -076923 -0375 5. -234 -1672 6. -285714 '-0093 7. -306 -009 8. -003 -0003 9. 10. 11. 12. DECIMAL FRACTIONS. 61 I n Multiplying an Interminate Decimal by a Terminate, we proceed in the following manner : 7623 (1) -7623 X 27-5. In multiplying by 5, we carry 3 from the be- ginning of the period ; similarly, in multiplying by 7, we carry 5 ; and in multiplying by 2, we carry 1. We then extend the periods, and find the sum. 27-5 38118 533663 1524752 1083 X 4 216 X 7 32 X 9 142857 X 5 5. -962 X 11 6. -753 X 64 7. 8-46 X 846 8. 7-27 X 72 20-96534 9. 3-09 X 37 10. -037 X 23 0033 X 606 09756 X 250 11. 12. In Dividing an Tnterminate Decimal by a Terminate, we extend the dividend until the quotient recurs. (2) -148 -f- 12. 12)048148148 012345679 13. -857142 -r- 6 14. -03523 ---26 15. -0231-7-308 16.57-18-7-37 17. 24-106 -7- 32 18. 33-3 -f- 271 In Multiplying or Dividing by an Interminate Decimal, we reduce it to a vulgar fraction. (3) -076923 X '285714. 285714=3- -076923 7) -153846 021978 (4) -536 -7- -5. 5 = J -536 9 5)4-8272 9654 We may sometimes reduce both the multiplier and the multipli- cand, or both the dividend and the divisor, to vulgar fractions. 19. -27 X '3 20. -037 X '027 21. -02439 X '00369 22. 2-25037 X '4i8 23. 10 -3 24. 23 -7- 2-09 38. If wo wish to have 3-14159265358979, &c. correct to 8 dec- imal places, we take 3-14159265; but if we desire to carry it 25. -00369 -7- -00271 26. -02439 -T- -17073 27. ^ X '03 28. A- X A 29. -00813-7-^ 30. & -7- If 62 DECIMAL FRACTIONS. 33t 4 places merely, it will be more accurate to write 3'1418 than 3-1415, for the fifth decimal place being above 5, the for- mer is nearer to the true decimal than the latter, and is thus a nearer APPROXIMATION. (1) Give approximations to -8450980400+ from 9 places to 1 place successively. 845098040+ ; -84509804+ ; -8450980+ ; -845098+ 84510; -8451; '845+5 '85; -8+. By affixing " + " we mean that the true value of the decimal is > the approximation ; and by affixing " ", that the former is < the latter. If -8450980400 had been terminate, we would have written 84509804 merely. But if we had written only eight places in the approximate decimal, it would seem as if we knew not the next two. Give approximations to the following from 9 places to 1 place successively. 1. -0413926852 2. -3010299957 3. -4771212547+ 4. -6020599913+ 5. -6989700043+ 6. -7781512504 (2) Find the sum -428571 + '39024, to 6 decimal places. "VVe extend the decimals to 7 places, and '1285711 I finding the sum of the 7th column, we add in 3Q024.3Q-I- the carriage to the 6th, and thus obtain the OJU OJ ~T" sum correct to 6 places. '818815+ (3) Find the sum 1-05 + -571428 + -83 + -39024, to 4 decimal places. To obtain the last figure as the nearest ap- 1 '055556 proximation, it is often necessary to extend the 57 1 429 decimals two places beyond the number re- *8333 quired. The sum of the 5th column, increased by the carriage from the 6th, being nearer 20 than 10, we carry 2 to the 4th column. 2 '8506 7. 7-27 + 9-2916 + 8-36 to 3 pi. 8. -036 + -036 + -036 to 4 pi. 9. -02439 + -003 + 3-1416- to 4 pi. 10. -91908- + -72428- + -72607+ to 5 pi. !2- A + Y+ A^to 4 pi. (4) From -12195 subtract -OG93 to 5 places. 05264+ (5) From -142857 subtract -00813 to 5 places. DECIMAL FRACTIONS. 63 38* T Iie extra fi ure in the Gth place of the re- mainder being > 5, we increase the figure in the 5th place by 1. 012987013 002710027+ 01027699 20. 1-041393 -698970+ to 5 pi. 21. 1 -41497+ 1-32222- to 4 pi. 22. 7r ^_ _| T to5pl. r to 5 pi. 142857+ 008130+ 13473 (6) From -012987 subtract -00271 to 8 places. Since the extra figure is>5, we cancel the carriage. 15. -7 -72916 to 4 pi. 1G. '259 -0027ito5pl. 17. -962 -90 to 4 pi. 18. -0625 -0416 to 3 pi. 23.^ 7 10. -2 -0083 to 3 pi. 39. . * n CONTRACTED MULTIPLICATION we obtain a product which is correct to a certain number of places. If we wish to find the product of the terminate decimals 5*2467 and 4-2635 tv four decimal places merely, it is evident that the fig- ures to the right of the line in A are unnec<-s>;iry. In B, we commence the first line by multiplying the figure in the fourth place having the local value of 7 ten thousandths by 4 imits ; the second line by multiplying 6 thousandths in the third place by 2 tenths, adding in the carriage of 1 from 2X7; the third line by multiplying 4 hundredths in the second place by 6 hundredths, adding in the carriage of 4 from 6X6, &c. Since the first column on the right has the local value often thousandths, there are thus four decimal places in the product, as required. To insure accuracy in the last decimal place of the approxi- mate product, we work for one place more than what is re- quired. To accommodate the eye, we invert the multiplier, and put its units' place under the place in the multiplicand whose local value is the same as that of the last decimal place for which we are working. (1) Multiply 5-2467 by 4-2635 to 4 places. Working for five places, we invert the multiplier, and put the figure in the units' I'ljK'o of the multiplier under the fifth place of the multiplicand. In ridding, AVC carry 1 from the last coin A 5-2467 4-2635 B 5-2467 4-2635 26 157 3148 10493 209868 22-3693 2335 401 02 4 209868 10493 3148 157 26 0545 . 22-3692 5-24670 Inv. (5362-4) 2098680 104934 31480 1574 262 22-3693 64 DECIMAL FRACTIONS. . . .. 39. (2) Multiply -02439 by '037 to ^ ( 730730-) 8 places. 731707 Working for nine places, we place the 170732 inverted multiplier so that its units' place -~~ may be under the ninth of the multiplicand. ^ We carry 1 from the last column, and as 170 there are five significant figures in the pro- duct, we prefix three ciphers. "00090334 1. 4-5625 X 3-375 to 5 pi. 2. 5-7563 X 3-996 to 3 pi. 3. 69-235 X 2-525 to 3 pi. 4. 14-36738 X 30-61725 to 5 pi. 5. -0842367X52-6739 to 6 pi. 6. -74216 X -8237 to 5 pi. 7. 4-02439 X '5027 to 5 pi. 8. 5-857142 X 8'09 to 5 pi. Let us find the product of the approximate factors 324*1674+ and 2-12967 +. The former may stand for any number be- tween 324-16735 and 324-16745; and the latter for any be- tween 2-129665 and 2-129675. Since the product of the least values 324-16735 and 2-129665 = 690-367859, and that of the greatest values 324-16745 and 2-129675 = 690-371314, the product of the approximate factors can therefore be guaranteed to two decimal places only, as 690*37+. As the factors in the ac- 324-1674* companying process are ap- 2-12967 * proximate, we see that of the nine decimal places in o the product seven are inde- terminate. The number of 1 Q 291 determinate places is=9 7 ; soon 9 = 5 + 4, the sum of the -? 5 ~ numbers of decimal places in ******* 2691718 450044* 75066* 0088* * * * ******* the factors; 7, corresponding '" to the number of figures in the factor 324-1 674+, is = 3 + 4, the sum of the numbers of integral and decimal places in that factor. By cancelling the number of decimal places in the fac- tor having the greater number of figures, we have 9 7 = 5 3. The number of determinate places is = the number of decimal places in the factor having the fewer figures diminished by the number of integral places in the other. (3) Find the product of 31-7436 by 31-7436 76321+ to as many places as can be Inv - (12367*) depended on. 222205 Since the number of decimals in the factor 19046 having the fewer figures is = 5, and the num- 952 her of integral places in the other is = 2, the 63 number of determinate places is = 5 2 = 3. We therefore work for 4 places. 24-227 DECIMAL FRACTIONS. 65 39. Find the following products to as many places as can be depended on : 9. 2-183+ X -00704. flgr -00704, whose significant figures extend over three places, has in all Jive decimal places. The other factor contains one in- tegral place. The number of determinate places = 5 1. 10. -000732 X 2-8+. ^T In -732+ X 2-8 +, the number of reliable places would = 1. Since we have -000732+ as a factor, we remove the point three places to the left, and thus increase the number of reliable places. 11. -23 X 7142-3. 43T The number of decimal places in the factor having the fewer figures being less by 2 than the number of integral places in the other, we cannot depend on the last 2 integral places of the pro- duct, and thus can give it in hundreds only. 12. 1-375 X -2304. When one of the factors is terminate, the number of determinate places is = the number of decimal places in the approximate fac- tor, diminished by the number of integral places in the terminate. 13. 17-69235+ X 2-00976 17. -7854- X '0036712 18. -052+ X 12345- 19. -275 X 3-2463 20. 2-005 X -00017 14. 1G-;UG7 X 8-3146 15. 3-247 X -00603 16. 3-1416- X -007009 4O In CONTRACTED DIVISION, we obtain a quotient which is cor- rect to a certain number of places. (1) Let us divide 74-0625 by -3147, of which both are terminate, so as to obtain three decimal places in the quotient. By inspection, we find '3,1,4,7,0,0) 74'0625 ( 235'343 + by dividing 7-1 by -3, that 6294 there will be three integral 11122 places in the quotient. We q, * * thus require six figures in ^ the quotient. Annexing as 16815 many ciphers to the divisor 15735 as make it contain six fig- ~~i' Turn ures, we find the first figure in the quotient, and then elide a figure from the divi- ~T36 it each successive step. 1 9 / need not write the ci- L * in the first two partial 10 products. 9 66 DECIMAL FRACTIONS. 'K \ .A-IOKAA / .c\f\c\AC\c.Q I ,5 ) '012500 ( 0< 210 1 o^ 26 25 4O (2) Divide '0125 by 30*725, to obtain seven decimal place? in the quotient. We find that as there will be three prefixed ciphers in the dividend, the number of figures necessary to make up the required number in the quotient will be 7 3, or four. We commence to . divide by 3072, and elide a figure at each successive step. To obtain a certain number of figures in the quotient of two terminate decimals, we begin the division by having as many figures in the divisor as are = the number of required decimal places increased by the number of integral places in the quo- tient, or diminished by the number of prefixed ciphers in it. We then continue to elide a figure from the divisor at each successive step until it is exhausted. Find the quotients of the following numbers having ter- minate decimals : 1. 6-75 -f- 3-25 to 4 pi. 2. 10- -f- 4-75 to 3 pi. 3. 20-6 3-3125 to 3 pi. 4. 6-23475 -r -04875 to 3 pi. 5. 4-12189 -^ -04763 to 2 pi. 6. -004365 -H -71215 to 5 pi. 7. -0007 -h 3-125 to 6 pi. 8. -00034625-^631 -247 to 10 pi. When either the divisor or the dividend, or both, are ap- proximate, we can depend on only a certain number of places in the quotient, as may be seen in the following examples : (3) 2-5 -f- -0773. 0773)2-500(32-3+ 2319 181 155 ~26 23 (4) -0031416--- -674. 67)-00314|16(-0047 268 46 47 (5) 6-143-^-007354. 00735|4) 6-143 (835-+ 5883 260 221 39 37 (6) -007316+ -H 7-4. 7-4)-007316 (-000989 666 656 592 64 67 DECIMAL FRACTIONS. 67 4O* I R ft N cases, we first find the initial figure in the quotient and point it. When the divisor is approximate, and the dividend has more determinate places than are in the divisor, as in (3) and (4), we begin to elide the figures in the divisor after the first par- tial product. When, as in (5), the dividend is approximate, and the divisor can produce more determinate places than are in the dividend, as many figures only of the divisor must be D as will make the first partial product contain no more than are in the dividend. But when, as in (6), the divisor is terminate, and has its significant figures extending over fewer places than the number of the determinate in the dividend, we carry on the division in the ordinary way till the dividend is exhausted, and then commence the contraction. - "525 10. -00313 7-4 11. 1-0367 -7- -94364 1-J. 12-3 -8738 13. 2-575 -;- '234 14. 10- -7- -5236 15. 5-2673 + -06731 16. 2-0167+ -7- -733 17. 1-0035 -0417 + 18. 10- 21-63 19. 1- 2-302585093 20. 4- -+ 2-167 + 21. -1 -000767 + 22. 72-1 -00312 23. 10- -h -000763 24. -007635 -f- 7-142 + 25. -073167 -h 2-25 26. 1- 12-56637 + 27. 42-75 ~ -00077 28. 630- -f- -0739 29. -0125 -i- 71-23 30. 10- ~ 2-718281828 + In i:i;ir< IN*; a simple quantity to the decimal of another in a higher name, we annex ciphers to the number of units in the r, and divide by the number which shows how often a unit of the lower name is contained in one of the higher. (1) Reduce 9d. to the decimal of I/. d. 12)9-00 We thus change T 9 a to a decimal. - - * i OS. In reducing a compound quantity to the decimal of a simple quantity, we reduce the number in the lowest name to the dec- imal of"the next higher, to which we prefix the integer in the latter, and so proceed till we obtain the decimal of the required (2) Reduce 4 Ib. 7 oz. 15 dwt. to the decimal of 40 Ib. The accompanying process is equiv- t ,~ ^; alent to the following: ISdwt. = *!<*. = '75 o* z. 7-75 oz. = Lu> Ib. = -64583 Ib. 40 ! 4'64583 Ib. 1 2 3 11). 4^u_8_3 = -] 1614583 of 40 Ib. '11614583 68 DECIMAL FRACTIONS. 41. When the quantities are expressed in mixed numbers con- taining vulgar fractions or decimals, we proceed as follows : (3) Reduce 4| min. to the decimal of 15*2 hours. 60 ) 4-75 min. 15-2) -07916 ho. (-0052083. We may sometimes cancel thus : 4-75 _ -25 25 60 X 15'2 60 X '8 48 ~ In reducing a compound quantity to the decimal of another, we find the vulgar fraction, which shows what part the former is of the latter, and reduce it to a decimal. (4) Reduce 2 //I I// 8 to the decimal of 5//7*7i- By the method of 20., No. (4);2//ll//8 = if If of 5//7//7i 4 f|f= -48006194+ Otherwise: By reducing 11/8 to the decimal of 1, and pre- fixing 2, we obtain 2'583, and are thus said to have re- duced 2//ll//8 to the decimal of 1. Similarly, 5*7*74 reduced to the decimal of 1 =5-38125. 2-583 -^ 5-38125 = -48006194+ 1. 8d Is. 2. 15cwt 1 T. 3. 30 in 1 yd. 4. 7/6 1. 5.13/44 1. 6. 5/6 1. 7. 8 oz. 3dwt lib. troy. 8. 3fu. 10 po 1ml. 9. 2ro. 30 po 1 ac. 10. 3qr. 15|lb 10 cwt. 11. 3bu. 3ipk 5qr. 12. 6 ho. 9^ min 3 da. 13. 2/8J 5/3J 14.7/81 15/3 15. 6/7 18/9 16. 3oz. 5dwt lib. 3 oz. 17. 2bu. 3pk 5bu 1 pk. 18. 2ft. Sin 3yd. 2ft. 19. 5fu. 8 po 7fu. 20ipo. 20. 5min.l6isec..3ho.l5min. 21. 2327 / 37 // 90. 22. 5 cwt. 3qr 2T. 10 cwt. 23. 3 da. 101 ho... .3 wk. 4 da. 24. 6f min 7ho.30min. 25. From Delhi to Bombay the direct distance is 720 miles ; and from Delhi to Madras, 1080 miles. Reduce the former to the decimal of the latter. 26. Westminster Hall is 270 feet long and 75 feet broad. Re- duce the latter to the decimal of the former. 27. Reduce a sidereal day, which is = 23 ho. 56 min. 4'09 sec., to the decimal of a solar day of 24 hours. 28. Reduce the sidereal day of Jupiter, which is = 9 ho. 55 min. DECIMAL FRACTIONS. 69 41. 50 sec., to the decimal of the Earth's sidereal day, which is 23 ho. 56 min. 4-09 sec. 29. Reduce a solar year, which is =. 365 da. 5 ho. 48 min. 49-7 sec., to the decimal of a sidereal year, which is = 365 da. 6 ho. 9 min. 9*6 sec. 30. Express the height of the Peak of Teneriffe, which is = )2 feet, as the decimal of a mile. 31. Express 3 " 17 " 10 , the value of 1 Ib. troy of sterling gold, in the decimal of 1 . 32. The Danube is 1630 miles long, and from the source of the >uri to the mouth of the Mississippi the distance is 4000 miles. uce the former to the decimal of the latter. 33. Reduce the weight of a Cologne mark, which is =, 3608 grains, to the decimal of 1 Ib. troy and of 1 Ib. avoir. 42* In finding the value of a decimal of a unit, we multiply the decimal by the number of times the given unit contains the next lower unit, and so on as far as may be required. (1) Find the value of '7895. 7895 20 7895 = s. 15-7900 12 d.9-48 4 f. 1-92 (2) Find the value of '583 oz. ' troy. oz. -583 20 dwt. 11-6 24 gr. 16- By multplying the intermmate decimals, we obtain 583 oz. = 11 dwt. 16 gr. The following examples afford additional illustration of finding the values of decimals : (3) Find the value of 2'75 of 5'45 acres. 5-45 2-75 (4) Find 5 cwt 5 cwt. 3 28 4 the value of 2'425 of 3 qr. 16 Ib. qr. 16 Ib. = 660 Ib. 2-425 660 14550^ 14550 2725 3815 1090 ac. 14-9875 4 ro. 3-9500 40 1600-5 57 qr. 4 Ib. 14 cwt. 1 qr. 4'51b. po. 38-00 70 DECIMAL FRACTIONS. 42. 1- 2. -975 3. 2-875 s. 4. -4375 gu. 5. 1-05416 6. -7302083 7. -275 Ib. av. 8. -16 oz. tr. 9. 3-142857 cwt. 10. -583 hour - 22. 5-24 of 3 11. 7-0625 ac. 12. 2-0945 cub. ft 13. -55 of 4-204 ac. 14. 2-75 of -04yd. 15. -003 of 3-6 ml. 16. 4-125 of 243 ac. 17. -075 of 3 bu. 2 pk. 18. 3-0916ofllb.4oz.lOdwt. 19. -325 of 7 ho. 24 min. 20. -432 of 5 cwt. 2 qr. 24 Ib. 21. -037 of 15-201 yd. 23. -725 of 7-76 bu. 24. 3-425 of 4-003 cwt. (5) Find the value of -0025 ac. -f 3-45 ro. + -0076 ac. + -009 po. ro. po. ac. 0025 ac. = // 0-4 = '0025 3-45 ro. = 3 //18- = '8625 0076 ac. = // 1-216 = '0076 009 po. = // 0-009 3 //19-625 = -87265625 25. 2-003 ml. + -275 ml. + 1050 yd. + -025 ml. 26. 3-3 -5 s. + -075 cr. 285714 guin. 27. -425 ho. + -003 min. -275 ho. + -925 min. 28. Express the hectolitre, = -343901 qr., in bu. and pk. 29. Express the Linlithgow wheat boll, = -499128 qr., in bu. and pk. 30. Express in grains troy, a weight -00024 Ib. avoir, heavier than a kilogram, which is 2*20462 Ib. avoir. 31. From Paris to Berlin by railway is a distance of 1308 kilo- metres, of which each is = 1093*63 yards. Express the distance in miles and yards. 32. Mercury revolves round the Sun in 87-9692580 days. Ex- press the period of revolution in days, hours, and minutes. 33. Express in avoir, wt. the weight of a Prussian pound, which is -46771 of 2-20486 Ib. avoir. 34. Find the length in inches of the Greek foot, = * * of the Roman foot, which was '97075 foot. 35. Find the weight of 3| cubic feet of water at 62-455 Ib. avoir, per cub. ft. 36. The radius of a circle is = -1591549 of its circumference, which contains 360. Find the angle whose arc is = the radius. DECIMAL FRACTION'S. 71 43. ^' e cal1 the tent h of a Pound Sterling, a florin. In extend- ing the decimal division of the Pound, it was proposed to call the hundredth a " cent," and the thousandth a " mil." 1 florin = -1 = 2s. 1 cent = -01 = 2|d. 1 mil = -001 = |ff. 1 shilling = ^ florin; 1 farthing = -JJ or 1^ mil. To express a sum of money as the decimal of 1, we may work as in 41. ; but to do it mentally, let us consider the following analysis : s. f. fl. m. 14/10i = 14 + 41 = 7 + 42iJ 14/10} = -7427083. For the first place, we take half the number of the shillings. For the second and third places, we express the pence and far- things as farthings, and increase the number by 1 if it is > 24. For t\\& fourth and///'/// places, we multiply the number in the second and third, or when the number is > 25, its excess above 25, by 4, and add 1 for every 24. For the sixth and seventh multiply tho number in the fourth and fifth, or when the number is > -Jf>, 50, or 75, its respective excess above 25, 50, or 75, by 4, and add 1 for every 24. When the number of shillings is odd, we work for the next r even number of shillings, and add 5 to the second place ; thus, 15/10J = 14/10J + 1/ = '7427083 + '05 = -7927083. Jd. =: -0010416, any sum of money expressed in the deci- mal of 1 contains no more than six terminate places. When there are more than six places the seventh is interminate, being either : I). Reduce the following sums of money to the decimal of 1. (1) 17/5 1 = -8739583. 8. d. 8. d. 8. d. 8. d. 1. 12 // 6 7. 14 * 5J 13. 18 "10f 19. 7 // Ox 2. 18 // 8. 16 // 34 14. 12 // 6i 20. 13 6f 3. 4 // 9. 12 * 4 15. 8 " ? f 21. 14 ,2J 4. 2 // 7 10. 6 // 4j 16. 3 22. 19 ' 8* 5. 12 // 10 11. 8 // 3 17. 9 // 8i 23. 3 6. 14 // 8 12. 18 //1U 18. 15 x/ 7 i 24. 19 //11J To -urn of money approximately to three decimal j;i, or in florins, cents, and mils, we adopt the prin- ciple of approximate decimals (see 38.), by increasing the 72 DECIMAL FRACTIONS. 43,number of farthings by 1 when it is > 12, or more than half- way up to 24, and by 2 when it is > 36, or nearer to 48 than to 24; thus, 16/4J = '8197916 = -819jf, being nearer to 820 than to '819, is approximately = '820. (2) Reduce 16/7 to three places of the decimal of 1. 16/71 = -831. Reduce the sums of money, Nos. 1 to 24, approximately to three places. Being familiar in 42. with the common method of finding the value of the decimal of 1, we may now consider the fol- lowing plan : Let us find the value of '9238. -9,238 By pointing off the first place, we 952 obtain the number of florins, '9238=18/5i|f. Now, since 96 farthings = 1 florin, we must multiply by 96. But as 96 = 100 4, we put 4 times the multiplicand two places to the right, and then subtract. The number made up of the first two places on the left is the number of farthings. (3) Find the value of -7145. -7,145 7145 = 14/3HI- 25. -125 26. -225 27. -375 28. -975 29. -3125 30. -7625 31. -9875 32. -5375 33. -4236 34. -5168 35. -8274 36. -4537 13,92 37. -7219 38. -8437 39. -2914 40. -3853 To obtain the value of the decimal of 1 to the nearest far- thing without a fraction, we proceed as follows : Let us find the value of '7287. We consider it approxi- mately = -729, which is = 14 s. + 29 mils. Since 25 mils = 24 f., we subtract 1 from 29, and obtain 29 mils = 28 f. nearly, and -729 approximately = 14/7. To obtain the number of shillings, we divide the number of cents in the first two places by 5, the number of mils being = the remainder with the figure in the third place annexed ; thus, 883 = 17s. + 33 mils ; '824 = 16s. + 24 mils. When the second figure is < 5, we may obtain the number of shillings by doubling the figure in the first place. In reducing the number of mils to farthings, we adopt the principle of approximate decimals, and subtract 1 when the number is > 12, or more than half-way up to 25, and 2 when > 37 or nearer to 50 than to 25. DECIMAL FRACTIONS. 73 43. W Express -768 to the nearest farthing. -768 = 15s. + 18m. = 15/4. Valuate the decimals, Nos. 33 to 40, to the nearest farthing. fiT The pupil may now construct a table, showing the correct and the approximate decimals of 1 from d. to I/, so that by men- tally inserting the decimal for the number of shillings, the decimal of any sum may be obtained. MISCELLANEOUS EXERCISES IN DECIMAL FRACTIONS. 1. Find the price of 30 Parian statuettes @ 1-775 each. 2. In January 1856, the number of days during which rain fell in Scotland was 13, and the amount which fell was 2*38 inches. Find the daily average for each of the 13 days. 3. How many ac. ro. and po. are in a park containing -08 of 155-1875 acres? 4. If 31-75 poles are feued for 2-38125, how much is it per pole ? 5. Find the sum of -3125, -4375s., and -75d. 6. In March 1856, in Edinburgh, the thermometer at the highest was 51' 1, and at the lowest 29*4. Find the difference or range. 7. Find the value of -00375 Ib. troy of sterling gold @ 31710 8. Of 100 parts of matter in locust beans, sugar and gum form 61*10, other vegetable matter forms 31*55, and moisture 5. Of how many parts does the remainder, which is mineral matter, consist ? 9. The distance from Paris to Leipsic by railway is 1225 kilo- metres, each 1093*63 yards. Express it in miles. 10. Of the manure of dissolved bones "1571 of its weight is organic matter. Find the weight of organic matter in 80 tons of manure. 1 1 . Express the sum, T g of 4| + J + j i of 5 7 S + 5 3 5 , as a decimal. 12. In February 1856, at Sandwick, Orkney, the barometer at the highest was 30-543 inches, and at the lowest 28*843 inches. Find the difference or range. 13. The following rents are drawn from a property : mansion, 150-15; farm, 470*475 ; parks, 80*875 ; feus, 7 '625. Express the total in , s., d. 14. Find the price of 14 cwt. 3 qr. 14 Ib. rice @ '625 jg> cwt. 15. The time of Jupiter's rotation on his axis is 9 ho. 55 min. 50 sec., and the period of his revolution round the sun is 4332-5848 days. Reduce the former to the decimal of the latter. 16. A line in a diagram in a book published in the sixteenth century, which now measures 6*83 inches, has shrunk to * of its original length ; find what it had been. D 74 DECIMAL FRACTIONS. , 17. A cubic inch of pure water weighs 252-458 grains, find the weight of a cylindrical inch which is '7854 of a cubic inch. 18. A gallon of pure water weighs 10 Ib. avoir. ; and a cubic inch, 252-458 grains. From these data, find the content of a gallon. 19. The period of the revolution of the Earth round the Sun, measured sidereally, is 365-2563612 days, and that of Mars is 686*97964580 days. Reduce the latter to the decimal of the former. 20. The height of the Peak of Mulhacen in Spain, formerly es- timated at 3555 metres, has been found to be -156 kilometre less. Find its height in feet at 39*37079 inches ^ metre. 21. A gallon of pure water weighs 10 Ib. avoir., find the weight in oz. of a pint of whey of which the Specific Gravity is 1-019. ^gT When \ve mention the Specific Gravity (s. G.) of a substance, we show how many times it is as heavy as pure water ; thus, the s. G. of lead being 11'35, any volume of lead is 11-35 times the weight of the same volume of water whose s. G. is 1. 22. Find the weight of 12 gallons of olive oil, of which the s. G. is -915. 23. Find the content of a block of granite 5*5 ft. long, 3-2 ft. broad, and 1*6 ft. deep. 24. A metre is = 39*37079 inches. Reduce an inch to the deci- mal of a metre. 25. What decimal of the whole time necessary to burn a ton of coals continuously at the same rate is that required to burn 2-20486 Ib. ? 26. Divide 31-4 among 6 men and 11 youths, giving a youth 525 of a man's share. 27. The weight of a cubic foot of pure water is 999-278 oz. avoir., find the weight in Ib. avoir, of the air in a room 12 '5 ft. high, 16-25 ft. long, and 10*4 ft. broad, air being 815 times as light as water. 28. In March 1856, the weight of vapour in a cubic foot of air in Edinburgh was 2-24 grains. Find in the decimal of a Ib. avoir, the weight of vapour in the atmosphere of a room 12 ft. in height, length, and breadth, supposing that there was no fire and that the window was open. 29. Reduce \ of ^ of ^ to a decimal. 30. Express the sum, f of If + f of f| + '2, as a decimal. 31. Reduce f guinea to the decimal of 1. 32. Express the sum of S J 7 and 5 J T as a decimal. 33. The Polar and Equatorial Diameters of the Earth are re- spectively 41,707,620, and 41,847,426 feet. Express each decimally in miles. 34. Find the number of miles in the Meridional Circumference of the Earth, supposing that it contains 40,000,000 metres, each 39-37079 inches. DECIMAL FRACTIONS. 75 Gravity of Hydrogen, that of air being 1, is 069, while that of air as compared with water is -0012. Express the relative weight of Hydrogen as compared with water. ggT Water is the standard for solids and liquids, and air for gases. 36. The s. a. of carbonic acid gas, that of air being 1, is 1-524. L6 relative weight of carbonic acid gas as compared with T. 37. Reduce an oz. avoir, to the decimal of an oz. troy. 38. Keduce a Ib. troy to the decimal of a Ib. avoir. 39. A Winchester bushel is = -9694472 Imperial bushel. Ex- press an Imperial bushel as the decimal of the former. 40. A zinc bar, which at 32 Fahrenheit measures 1 inch, at 212 measures 1 -003 inch ; find the length of a bar of the same metal at . which at 32 measures 2-25 inches. 41. What decimal multiplied by i of T 9 5 produces fi? 42. Divide 1-3125 equally among a number of almsmen, giving each -375 ilorin. What is the number ? 43. What quantity of sugar @ -025 ip Ib., will cost 19'575 Hod 14. Divide the sum of -075 and -0075 by the difference of 7-5 and 45. The yard measure made by Bird in 1758 was 36-00023 inches long. How many times would this measure be contained in a mile. 46. In 1825, the Stirling jug or pint measure was measured in Edinburgh, and found to contain 104-2034 cubic inches. Reduce this to the decimal of an Imperial gallon. 47. On the floor of a room 10 ft. 8 in. long and 8'25 ft. broad, dust has accumulated to the depth of -075 inch. Express the volume of dust in the room as the decimal of a cubic foot. 48. The maximum delivery of a reservoir is 567-07 cubic ft. of r ^ minute, and its minimum delivery 516*66 cubic ft. Find the number of gallons, each 277-274 cub. in., delivered on an equal average in 24 hours. 49. The mean diameter of the Earth is 7912-409 miles. Find the surface of a sphere of the same diameter, found by multiplying the square of the diameter by 3-1416. 50. Find the content of a sphere of the same diameter as the earth, found by multiplying the cube of the diameter by -5236. 76 2 ^ / CONTINUED FKACTIONS. lF we take a vulgar fraction, as Jf , and divide the numerator and the denominator by the numerator, we obtain ^^ = ^-^ Similarly, T ft = ^, and = ^ . We have thus J$J L i __ 1_ __ 1_ In the last form, we observe that sffi 3 * 3 L_ every numerator is unity. A com- 2 L_ plex fraction, in which every numer- 3 ^' ator is unity, and every denominator includes the succeeding parts of the fraction, is termed a CON- TINUED FRACTION. In the foregoing process, we have obtained the continued fractions : l - 1 . 1 . 1 These fractious are re- 3 ' 31' 3 i_ 3 i_ spectively = ^ 4, T \, 2 2 I 2 L- and i- We find that 3 ^' * we have reproduced the original fraction J|i. As the other fractions continually ap- proach to it in value, they are termed Convex-gents. The con- vergents are alternately greater and less than the original fraction. (1) Find the convergents to |J. The practical method of finding the con- vergents is to proceed as in finding the G. c. M. of 121 and 415 (see 3.). We may write the quotients 3, 2, 3, 17, i 1 __j ;.L- .I/L - .C_-_.L dace the in a column, and opposite the first we p unity in the Numerators' column, and Quot. 3 2 3 17 Num. 1 2 7 121 Den. 3 7 24 415 first quotient 3 in the Denominators'. In the second line the numerator is = 2 X 1, and the denomina- tor is = 2 X 3 + 1. In the third, the numerator is = 3 X 2 -}- 1, and the denomina- tor is = 3 X 7 + 3. In the fourth, the numerator and the denominator of the original fraction are reproduced. The convergents are, % f, 2 7 ? , |||. (2) Find the first three convergents to 3-14159. By proceeding as in finding the G. c. M. of 14159 and 100000, we obtain the first three quotients, 7, 15, 1. The convergents are, Quot. 7 15 1 Num. 1 15 16 j, and 3 T y,; or, y, f|, f *|. Find the convergents to the following fractions : ! Jft- I 2. fltf . | 3. T V*. | 4. ff Den. 7 106 113 CONTINUED FRACTIONS. 77 4*5* 5. Find the convergents to yVsV ^" We first reduce the fraction to its lowest terms. But whether we do so or not, the fraction is reproduced in its lowest terms. 6. Find the fifth convergent to -7854. 7. Find the third convergent to '5236. 8. The Specific Gravity of oxygen is T y 5 ' T of that of carbonic acid gas. Give the fourth convergent to this fraction. 6 days. Give six convergents. 12. The solar year is = 365*24224 days. Give the fourth con- vergent. 13. A metre is = 39*37079 inches. Find the fourth convergent to the fraction which a yard is of a metre. 14. A Scotch acre is = 1-261183 Imperial acre. Find five con- vergents to the fraction which an Imperial is of a Scotch acre. PKACTICE. > PRACTICE is the method of computing by means of Aliquot Parts. A number contained an exact number of times in another is an aliquot part of it : thus, 7 is an aliquot part of 21 ; 10/ of 1; 6/8 of 1; 14 Ib. of 1 cwt. ALIQUOT PARTS. 10 //O 6//8 5//0 4//0 3 // 4 s. d. 2//6 2//0 0*8 0//6 46. (l^Find the price of 794 yards of silk @ 2/6 V yd. 794 = price of 794 yd. @ 1 99 // 5 = @ 2/( d. 6 4 3 2 H 78 PRACTICE. s. d. 1. 8462 @ 10 a. *0 7. 8472 @ s. 1* a. 8 13. 7342 @3 u. *4 2. 7926 .. 6 *8 8. 7904 .. 5* 14. 9836 .. 5 *0 3! 8248 .. 2 *6 9. 8463 .. 4* 15. 9246 .. *8 4. 7923 .. 4 '/O 10. 9527 .. 3* 4 16. 9372 .. *6 5. 7686 .. 3 *4 11. 3513 .. 2* 6 17. 7236 .. *8 6. 7968 .. 2 *0 12. 6723 .. 1* 8 18. 8943 .. 6 *8 (2) What cost 7689 oranges @ lid. and @ fd. each? 7689s. . . t VOJl. 3 4 zoOo7 H i I/ 961*11 . @lid. 12 2,0 5766f 48*l*li 48,0*6 d. i. 19. 7268 @ 6 23. 8464 @ li 27. 6847 @ 20. 8379 .. 4 24. 7932 .. 1 28. 8467 .. 21. 3848 .. 3 25. 7233 .. li 29. 6593 .. 22. 5766 .. 2 26. 7923 .. i 30. 7892 .. 9 (1) Give two aliquot parts which make up 7d. and 5^d. d. d. 711 ( 6 = iof I/ r, i ( 4 = of I/ 74^. - ^ +1 , = ? of 6 d. 5id.=| +1 _ I Qf a | 1. Find two aliquot parts which compose the following rates : 3|d. ; 7d. ; 7Jd. ; 4jd. ; 6fd. ; 6jd. (2) Give two aliquot parts which make up 8/4 and 12/6. OH _ f 5/ = i of 1 - 2 fi _ ( 10/ = J of 1 - t +3/4= j of 1 -| +2/6 = \ of 10/ 2. Find two aliquot parts which compose the following rates : 7/6; 3/9; 5/10; 6/3; 2/11; 4/8. (3) Find the aliquot parts which, when respectively sub- tracted from I/ and 1, leave 9d and 16/8. 9d _f I/ ' \ 3d. = i of I/ 16/8- ~ 3/4 = i of 1. 3. Find aliquot parts which, when respectively subtracted from I/ or 1, leave the following rates : lOid.; 9d.; Hid.; 17/6; 13/4; 15/. (4) Find three aliquot parts which make up 8^d. and 15/7*. PRACTICE. 79 47. f 6 = i of I/ Jd. = 4+2 = of6d (+> = ^of2d ( 10/=iofl 15/7^ = -^+ 5/=i.oflO/ ( +7id. = 4 of 5/ 4. Find three aliquot parts which compose the following rates : 9jil.; ?id.; 8|d. ; 7/8J; 11/10J; 13/9. 48. (1) Find the price of 4671 loaves @ 4|d. and @ 9d. J 1 I I/ 4,1. Jd. 2,0 4671s. . @ I/ d. 3 ' 114671s. . . @ 1//0 I/ 1| 1167* & . .. 0//3 . . @4d. 194* 71 .. 1 97* 3f .. i 2,0)350,3 // 3 . @ // 9 175//3//3 )184,8//lli @ 4fd. 49. 92*8*111 (2) Find the price of 846 yards of cotton @ SJd. and ^@T d. 846s. . . @ I/ 846d. . . @ Id. 4 i )u-> . @ 4d. I 1 r I/ 105//9 ... 11 i. d. 6768 . . . @ 8d. O1 1 1 Jtrl 2,0)38,7*9 '7 "9 . @ 5id. J t i 12 Zll^ . . .. 5U 5B556}7T7@ 7f d- 2,0)54,6//4 27//6//41 33 @ d. 9. 6723 @ d. 2i 17. 6874 @ 2| 3. 4673 .. 4. 8423 .. 8. 87 74 41 5 31 81 94 10. 7247 .. 11. 3475 .. 12. 4672 .. 13. 2435 .. 14. 6724 .. 15. 7233 .. 16. 9894 .. 61 3J li 61 6i 84 11 18. 8674 .. 52 19. 7683 .. Jfc 20. 8267 ..mt 21. 8956 4p)f 22. 5732 Jr 9 * 23. 746MFHi 24. 8722 .. 7 (i) Find the ?> 1/I'-' J price of 423 yards of clotli* 1/10, and *i > *! 423 .@l s d 423s. . . @ 1^0 2/ 1 42// 6..@2//0 J 5 i ji 211// 6 . .. 0//6 2/ 3//10//6 .. O// 2 : 7 2 t ? 6d 26// 5i .. 0//OS 38// 5^/6 @ 1" 10 2 0)66 O// Hi @ lx/t >l 33^111 80 PRACTICE. 49. (2) Find the price of 846 cwt. of rice @ 7/9}, and @ 11/7$. 2/6 3|d. i I 1 2/6 846 . . @ 1 s> d . d. 6 1J I 211*10 . @5*0 105*15 . ..2*6 13* 4*4J ..0*3| 330* 9*4J @ 7*9} 846s. 11 9306 . 423 . 105//9 _ s. d. @11*0 . 0*6 2,0)983,4*9 @ll*7a 491*14*9 (3) Find the amount of 793 railway fares @ 16/8, and @6/9. 5O 793 . . . @ 1* 0*0 1793s. ..I/ 3/4 i 1 132* 3*4 .. 0* 3*4 6 g J 660*16*8 @ 0*16*8 d. 4758 . . @ 6*0 9 i 594*9 . .. 0*9 2,0)5352*9 @ 6*9 267*12*9 s. d. s. d. s. d. 1. 4567 ( 1* U 17. 798 ( S 10* 6 33. 893 @ 16* 8 2. 3283 .. 1* 7i 18. 742 .. 4* 8 34. 979 .. 17* 6 3. 5687 .. 1* 6| 19. 467 .. 5* 3 35. 894 .. 18* 4 4. 8672 .. 1*10J 20. 923 .. 1*10 36. 897 .. 18* 9 5. 937 . 15* 21. 916 .. 4* 6 37. 374 .. 8* 9 6. 423 . 13* 4 22. 743 .. 3* li 38. 968 .. 4* 4i n t . 341 3// Q o* u 23. 123 .. 2* 4 39. 763 .. 9* 6 8. 876 . 12* 6 24. 732 .. 2* 9| 40. 423 .. 13* 54 9. 827 . 11* 8 25. 428 .. 3* 8 41. 346 .. 9* 9| 10. 729 Q A 26. 293 .. 5* 6 42. 729 .. 18* 74 11. 873 . 4* 2 27. 468 .. 2* 3 43. 777 .. 19* 24 12. 798 . 5*10 28. 736 .. 10* 8 44. 947 .. 4* 8 13. 149 . 10*10 29. 716 .. 1* 54 45. 589 .. 5* 74 14. 824 . 11* 3 30. 637 .. 2* 8 46. 346 .. 5* 5 15. 899 6* 3 31. 468 .. 17* 4 47. 777 .. 9* 9 16. 243 . 2*11 32. 823 .. 7* 84 48. 732 .. 7*10i (1) Find the price of 783 qrs. of wheat @ j f^^g ^ qr ' 783.. . @l 783 ... @ 1 3 5 2349 . . . @3 00 113915 @5" O'.-O ton 1 391-10 .. 0"100 12/6 i 5|| 489 7-6 .. 0-12.6 l / 3 i r 10/i 48189 .. 0" 13 342512"6 @4" 7-6 2789" 89@311"3 PRACTICE. 81 5O ( 2 ) ^d the price of 379 quarters of barley @ 2//3//5 ^ qr. 379s. 51. 43 \& j./ 1137 > 8. d. d. 1516 f @ 43//0 4 i I/ II 126//4 ... . , 0//4 a H * I/ II 47//4' , . , \j> i O//H 2,0)1 647,0*8 J . . . 43//5i 823//10//8 8. d. s. d. s. d. 1. 916 @ 4//16//0 11. 985 @ 7//11//8 21. 896 @ 7//19// 2. 169 .. 3//15//0 12. 946 .. 7// 8//9 22. 846 .. 6// 8// 4 3. 843 .. 2//13//4 13. 853 .. 10//16//8 23. 859 .. 2//12// 6 2 .. D'/ 3//4 14. 976 .. 10//12//6 24. 987 .. 4// 7// 6 5. 847 .. 3//12//0 15. 793 .. 12//13//4 25. 739 .. 4//11// 8 6. 974 .. 3// 7//6 16. 847 .. 11//13//9 26. 463 .. 4// 1//10 7. 874 .. 5//16//8 17. 569 .. 6//13//4 27. 568 .. Iff I// 6f 8. 734 .. 5" 8//4 18. 279 .. 4//15//0 28. 984 .. 9// 2// Si 9. 986 .. 6//15//0 1 ( .>. 947 .. 5//18//4 29. 719 .. 25// 9// 8J 10. 793 .. 7//17//6 20. 539 .. 4//18//8 30. 346 .. 27//15// 6f It is often convenient to employ the FLORIN as the unit of computation. (1) Find the price of 489 tons of coal@ 14/, and @ 1*3 ^ ton. 489 fl. . . @ 2/ 489 fl. . . . @ 2/ 111 342,3 fl. . . @ 14/ 244 //Is. 342^6 5379 562,3 fl. Is. . @ 23/ 562*7 The most convenient method of reducing a sum expressed in and a. to fl., is to annex half the number of s. to the number of ; thus, 3 " 4 = 32 fl. (2) Find the price of 878 cwt. of sugar @ j 2 //15//4 878 fl. . @ 2/ 28 7024 1756 d I I II 24584 . . @2"160 8 |}| Fl. || 292*1 4 .. 0" 0"8 2429,1"0"8 @2*154 2429"2-8 D 2 JF1. 878 fl. . . 17 @2/ .. 0* 14^0 0-8 6146 1 1 878 J | 2921"4 1521,8"l,/4 @1 14/-8 1521-17"4 82 PRACTICE. 51. 1. 794 2. 798 3. 823 4. 697 5. 796 6. 267 7. 937 8. 469 9. 835 10. 974 11. 826 12. 563 52. (*) Find the P rice of 749 TT cwt - @ n / 8 V cwt 2 ! 13. 943 @1//16 25. 763 @1//14// 8 6 1 14. 879 .. 1//18 26. 269 .. 0//17// 4 8 15. 937 .. 2//12 27. 263 .. 1//13// 6 14 16. 893 3// 4 28. 798 .. 0//16// 6 16 17. 828 '.'. 7// 8 29. 839 .. 0//12// 4 12 18. 726 .. 5//14 30. 346 .. 2//15// 8 18 19. 699 .. 3// 5 31. 876 .. 0//16// 3 7 20. 893 6// 7 32. 732 .. 0//17// 9 11 21. 467 ;; 7// 9 33. 356 A "1 A . O 19 22. 796 .. 9*11 34. 797 !.' 1//15//10 13 23. 876 .. 8//17 35. 798 .. 2//18// 2 17 24. 539 .. 11//13 36. 529 .. 3//11//10 io/ 1/8 749 374//10//0 62// 8//4 T 6 T of 11/8 = Q// 6//4 T \ 437// 4//8J T 6 T s. d. 11//8 _6 ll)70//6 6//4J (2) Find the price of 292 T \ Ib. @ ll/5 & Ib. IO/ 1/3 1 io/ 1/3 292// 8// 9 146// 4// 18// 5// 3// 0//1 6*4 Since ^ of 1 = 8/9, the price of 292 T T g Ib. @ 1 is 2928"9. 167//10//1 In the method of (1), we first find the price of the integral num- ber 749 cwt., by taking the parts which make up the rate 11/8 ; and then add in the price of the fractional number, T \ cwt. In the method of (2), we first find the price of the mixed number 292 T 7 8 Ib., at the unit of computation 1, and then take the parts whicli make up the given rate. The first method is of more general application than the second, which is only conveniently applied when the denom- inator divides the unit of computation without producing a fraction. 1.216i@13-/ 4 2. 547|.. 16// 8 3.899^.. 9// 6 4. 447 f .. 5// 9 5.967^.. 6//10 6. 793| .. 17// 1 7. 468| .. 16// 6 8. 794f .. 19// 6^ s. d. 9. 235| @0//18// 4 10. 324-^ .. 2// 3// 8 11. 829| .. I// 6// 3 12.247 T V. 1"13" 4 13. 794 T %..2// 5//10 14. 823 T V.. I// 9// 4 15. 299|..0// 8*114 16. 834 T \ .. 0//17// 9 s. d. 17.273 T \@1 //3 //9 18.347f ..0//17// 5J 19. 423 ..0//17//1H 20. 342^..0// 5// 34 21.827| ..0//19// 2^ 22.286* ..0//12// 9^ 23. 999 T \ .. 1//13// 54 24.889^ ..2//14// 71 PRACTICE. 53. 83 ALIQUOT PARTS. qr. i = 14 lb. i = 71b. f = 41b. i = 3i lb. These Aliquot parts are given as examples. The pupil having a thorough knowledge of the Arithmetical Tables can easily find ali- ' f the various denominations in WEIGHTS AND MEASURES. (1) Find the price of 7 cwt, 2 qr. 7f lb. i 2qr. 1 qr. IGlb. 14 lb. I = i To 2ro. Iro. 32 po. 16 po. 8 * 6*8, ! 8*6*8 & cwt. price of 1 cwt. 7ft. ill., ilb. f 1 T* 1 cwt. 2qr. 71b. 7 cwt. qr. price of 7 * / ... 0//2/ ... 0*0/ ... 0*0/ ... 0*0/ lb. /O '0 ^7 'OJ '04 58* 4* 0* 0* O// CO / 8^, ^. 4 J 7 / 9O 1 3 ' Z 3> T* 63 * 1 * 4, T 9 T price of 7 * 2 * 7-f (2) Find the rent of 353 ac. 2 ro. 10 po. @ 2*7*6 & ac. 353 . . . rent of 353 ac. @ 1 2 706 ) ;ac. ro. po. j 1 88* 5 ( i . . 353 // * (c 2*7*6 2/6 5/ 44* 2* 6J 2ro. i 1 ac. 1* 3* q .... * 2 * . lOpo. i 2ro. 0* J ... 0*0*10 . . 839 // 14 // 2i \ . . 353 // 2 // 10 @ 2 // 7 // 6 the number in the name in which the price of a unit is is small, as 7 cwt. in (1), we find its price by multiplication, ;md then take parts for the numbers in the lower names. But when the number is large, as 353 acres in (2), we may find its price liy taking p.-irts for the rate, and then finding the price of the num- bers in the lower names as in (1). cwt 1. 1:5 qr. I). " 2// 14 ^vcwt. @ 1*17* 4 9. yd. nl. ' 2 ^yd. @ 0*15* 4 '/ 3*14 .. 1* 19* 8 10. 19* 3/ ' 3 .. 1* 3* 4 * 2*16 .. 1* 1* 10 11. 227* 2/ 1 9/y 7 fi 1. 122 / 3* 9 J .. 2* I// 71 12. 313* 3* 3| q i o clwt trr. gv oz '15*12 6. 17//11//15 @ -0* .. 1* 8 13. ac. 17 / ro. '2* po. 20 @ 6*10* fhvt. gr. ^vlr 14. 43 / '3// 35 .. 4*16* 8 7. !> '15*23 @46// 6 15. 365 / '!// 19 .. 8*13* 4 K \\' '14^/22; .. .T7'/ 2* 6 16. 49 3// 37i .. 3* 7*11 84 PRACTICE. 53. qr. bit. pk. ^ qr. 17. 7 // 4* 2 @ 2 // 8 *0 18. 11*7* 2 .. 2//16// 8 19. 0*7* 3 .. 3// 5// 4 20. 301*5* 1}.. 3// 3// 8 gal. pt. gi. ^ gal. 21. 3//5// 2 @ 0* 8* 22. 0//7// 3 .. 0*16* 23. 125 * 4// 1 .. 1*10* 3 24. 73 //5// 2f.. 1//12'/ 6 (3) Find the price of 195 cwt. 2 qr. 11 Ib. @ 4// 13//4 ^ cwt. 1 V cwt. = 5/ V qr. = 1/3^7 Ib. = 2|d. V Ib. Since 195 cwt. 2 qr. 11 Ib. 195*11*1 U = 195 cwt. -f- 2 qr. -f 7 Ib. 5 -f- 4 Ib., the price at 1 aa* cwt. is = 195 + 2 X 51 + 1/3 + 4 X 2jd. = 195*11 /llf. 6/8 1 | Having thus found the price Q19//1'v/1 of 195 cwt. 2qr. lllb. @l ^ cwt., we proceed to find it at the required rate. (4) Find the price of 14 yd. 1 qr. 2 nl. @ 6/4 : I/ ^ yd. = 3d. ^ qr. = f d. ^ nl. The price of 14 yd. 1 qr. 2 nl. @ I/ yd. is = 14/ + 3d. + 2 X Jd. = 977'/19// 65// 3//l yd. 1 ^ T. = I/ ^ cwt. = 3d. ^ qr. 1 ^ ac. =. 5/ ^ ro. r= 1 ^oz. tr.ml/ ^ dwt.m 5/ ^ oz. tr.=3d. ^ dwt.= T. cwt. qr. # T. 25. 6 // 13 // 3 @ 5 // 7 // 6 26. 73//19// 1 .. 0//13//4 27. 17 // 3// 2 .. 4* 2*6 cwt. qr. Ib. $> cwt. 28.23* 3//14@l // 8*4 29. 13 // I// 21 .. 2//10//0 30. 19 // 2//11 .. 0//14//6 Ib. oz. dwt. |v Ib. 31. 3* 7//11 @5*11*8 32. 43 * 5//17 .. 10//13//4 33. 37 // 9 // 7 .. 3 * 17 // 4 oz. dwt. gr. fv oz. 34. 7* 13*17 1*16*3 35. 6//17// 9 .. 0//17//3 36. 3// 5//13 .. 0*7*6 l/^ gal. = ac. ro. po. ^ ac. 37. 13*2*30@2* 3* 6 38. 14 '/I// 27 .. 3// 3* 4 39. 37//3//11 .. 5//10* 8 yd. qr. nl. ^ yd. 40. 7*2* 3@0*17* 4 41. 8*3* 1 .. 2* 2* 6 42. 23//2// 2 .. 5// 6 // 8 qr. bu. pk. ty qr. 43. 7//3// 2@1* 3// 1^ 44. 6//5// 3 .. 4//10* 45. 15//3// 1 .. 3// 5* 10 gal. pt. gi. & gal. 46. 5*3*1@0*16* 47. 17 //I// 2 .. 0//17// 4 48. 163*0*3.. 2* 2* PRACTICE. 85 The following special methods are useful in A\oirdupois Weight, 11 ^ cwt. = 5/3 sp qr. = 1/35 & 1 Ib. = 2id. ^lb. Is. ^ cwt. = 3d. ^ qr. = fd. ^ 7 Ib. = f farthing ^ Ib. (5) Find the price of 4 cwt. 2 (6) Find the price of 16 cwt. qr. 5 Ib. @ 6"6s. ^ cwt. 3 qr. 15 Ib. @ 4/6 per cwt. 4 X 1"1"0 = 4" 4 16/ + 3 X 3d. = 0"16" 9 2 X 0"5"3 = 0"10" 6 2X|d.-j-ff.= 0" 0" 5 X 0"0"2i = 0" 0llj 4 "1 5} 6 28"12" cwt. qr. Ib. 49. 3 3 " 12 50. 27 1 w 18 3 7" cwt. qr. Ib. ! 7"7 <$> cwt. 51. 6 " 2 " 14 @ 8/6 ip cwt. 5"5 ... 52. 7 " 1 20 .. 16/3 ... We may now obtain a method for finding the price of 1 Ib. when that of 1 cwt. is expressed in and s. (7) Find the price qp> Ib. @ 7 "5s. per cwt. 7 " 5s. = 7 " 7s. 2s. 7 x 2id. 2 X ?f- = l/3 } W Ib. Having given the price of 1 cwt. in . and s., to find that of 1 Ib., we multiply 2 id. by the number of ., and ff. by the differ- ence between the number of . and s. ; and increase or diminish the former product by the latter, according as the number of s. is > or < than that of . Find the price ^ Ib. at the following rates <$> cwt. : 53. 8"lls. | 54. 6"10s. | 55. 9"2s. | 56. llls. Id. ^ Ib. = 2/4 $* qr. = 9/4 ^ cwt. "b. = 7d. {p- qr. = 2/4 ^ cwt. (9) What cost 13 cwt, (8) Find the price of 3 cwt, Iqr. 13lb. s. d. 3 X 9 " 4 1 X 2 4 13X0^1 @ 5$d. ^ Ib. s. d. = 1" 8" = 0" 2" 4 Q /; J,, J per Ib. ? s. d. 2 X 9 " 4 3X2-4 1"11 5 5} 710J 717 1 8" 4"lli : 0" : ^ 1* 16^ @2fd. s. d. L8" 8 7" 5" 8 13 cwt. qr. Ib. 57. 3 2 " 5 @ 2d. . 68. 17 . 3 " 19 .. 9;d. 59. 9 cwt. @ 7 id. 60. 26 cwt. .. 4|d. 86 PRACTICE. Since the numbers 12 and 20 are employed in the Money of Account, we may easily obtain methods for finding the prices of 1 2 and 20 articles with some of their multiples, when that of a unit is given, which may be convenient in MENTAL COMPUTATION. In finding the price of One Dozen, every penny in the price of the unit becomes a shilling. When the price of the unit is below 1/8, that of the dozen is below 1. (1) 12 @ 2d. = (2) 12 .. 3*0. == 2/ 3/3 (3) 12 @ (4) 12 .. = 4^12,6 Find the price of one dozen at the following rates ^ unit : 1. 50. 3. 10iO. 5. 1/3 7. 2/8 2. 7d. 4. 9^0. 6. 1/7 f 8. 3/5$ In finding the price of Two Dozens, every penny in the price of the unit becomes & florin. When the price of the unit is below 10d., that of the dozen is below 1. (5) 24 @ 4d. == 8/ (6) 24 .. 5|d. = 11/6 (7) 24 @ 1/5 = l14s. (8) 24 .. 2/3$ = 2146 Find the price of two dozens at the following rates ^ unit : 9. 10. 3d. 90. 11. 12. 7fO. 13 14. 1/6 15. 16. 3/7 5/8| In finding the price of Four Dozens, every farthing in the price of the unit becomes a shilling. When the price of the unit is below 5d., that of the four dozens is below 1. (9) 48 @ 3d. = 12/ (10) 48 @ 1/5 J = 39s. Find the price of four dozens at the following rates ^ unit : 17. 20. 19. 1^0. 21. 70. I 23. l/6i 18. 40. 20. 3*0. 22. lOjd. | 24. 1/lOf In finding the price of Eight Dozens, every farthing in the price of the unit becomes a florin. When the price of the unit is below 2d., that of the eight dozens is below 1. (11) 96 @ 20. = 16/. (12) 96 @ 7d. = 218s. Find the price of eight dozens at the following rates ^ unit : 25. 2iO.; 26. l0.; 27. 5f 0. ; 28. l/6f. In finding the price of Any Number of Dozens, every penny in the price of the unit becomes as many shillings as there are dozens. (13) 84 @ 40. = l8s. (14) 144 @ 7|0. = 4-13s. Find the price of the following : 29. 72 @ 3d. | 30. 108 @ 5|d. | 31. 144 @ 6|d. | 32. 60 @ 8*0. In finding the price of One Score, every skill itif/ in the price of the unit becomes one pound, and every penny becomes 1/8. PRACTICE. 87 54-. (15) 20 @ 3/6 = 3"10s. (16) 20 @ 7/5^ = 79*2. Find the price of one score at the following rates ^ unit : 33. 7/; 34. 5/3; 35. 7*d.; 36. 2/4$. In finding the price of Two Hundred and Forty units, every penny in the price of the unit becomes a pound. (17) 240 @ 5d. = 5. (18) 240 @ 1/2 1 = 14" 15s. Find the price of 240 units at the following rates yp unit : 37. 8d.; 38. 7sd.; 39. 1/11 1; 40. 5/7. In finding the price of One Hundred units, every penny in the price of the unit becomes 8/4, and every shilling becomes 5. (19) 100 @ 4d. = 1"13"4. (20) 100 @ 5d. = 2510. Find the price of 100 units at the following rates ^ unit: 41. 7d.; 42. 9{d. ; 43.2/3; 44.19/11. 55. MISCELLANEOUS EXERCISES IN PRACTICE. (1) A bankrupt whose debts are 3075 offers a composition of 1 1/3 ^ . How much does he pay ? 3075 10/ 153710 192 3 < 1 10/ (2) Find the weight of 124| bushels of wheat @ 63 Ib. sp bushel, cwt. qr. Ib. 1 cwt. 56 Ib. 124 > 1 14 62 7 / " 21 / 3 " 2 I 1 cwt. ^ bu., we 69 3 23| 1 Living found the weight of 124| bushels I take aliquot parts for 63 Ib., or 2 qr. 7 Ib. 1 . Find the price of 288 dressing-glasses @ 7/9 each. : nd the value of 840 stones of hay @ Tjd. each. 3. I'ind the price of 6 T. 15 cwt. oat manure @ 85^9 fj.1. 4. Wh.it does a chemical manufacturer receive for 5 1. Ib cwt. 2 qr. of sulphate of ammonia @ 1910s. # T. ? 5. Fin.l the price of 17 cwt. 3 qr. 14 Ib. of marine salt 2/6 .V bankrupt whose debts are 2016 offers a composition of 14/;;3 -a, . Find his effects. 7. How mudi is got for a silver epergne, weighing 7 Ib. 3 oz. 10 dv. 1 i mi-hand @ -V ^ oz. ? 88 PRACTICE. 8. What does an ensign receive in 365 days @ 5/3 ^ day ? 9. A French sub-lieutenant receives 1350 francs ^ annum. To how much sterling is this equal, reckoning a franc at 5 l s ? 10. Express in sterling the annual salary of a field-marshal of France, which is = 30,000 francs. 11. Find the value of 300 Austrian florins @ 2/0 each. 12. Find the value of 325 Rhenish florins @ 1/8 each. 13. Find the value of 360 Prussian dollars @ 2/1 Of each. 14. What is the value of a lac of 100,000 rupees @ 1/1 Oj each ? 15. What is the value in sterling of 5000 rubles @ 3/l each ? 16. To what sum in sterling are 1600 West Indian pistoles, each 16/, equivalent? 17. On Oct. 16, 1854, the stock of tea in London amounted to 47,522,000 Ib. Find the duty @ 2/1 ^ Ib. 18. A newspaper sold at 3^d. has a circulation of 3500. How much is received for each issue ? 19. Find the weight of 331 qr. 3 bu. of wheat @ 62 Ib. & bushel. 20. Find the weight of 692 qr. 5 bu. of oats @ 42 Ib. ^ bu. 21. Find the weight of 242 qr. 7 bu. of barley @ 54 Ib. ^ bu. 22. What is the weight of 1248 bu. of wheat @ 2 qr. 4 Ib. # bu. ? 23. Find the weight of 720| bu. of barley @ 1 qr. 26 Ib. y bu. 24. What is the weight of 200 bu. of oats @ 1 qr. 16 Ib. y bu. ? 25. Find the import duty on 14 cwt. 2 qr. 14 Ib. prunes @ 7/ ^cwt. 26. Find the import duty on 16 cwt. 3 qr. 21 Ib. Berlin wool @ 6d. y Ib. 27. Find the amount of excise duty charged in England in 1855 on 83,221,004 Ib. of hops @ 2d. y Ib. 28. Find the amount of excise duty charged in the United King- dom in 1855 on 166,776,234 Ib. of paper @ ld. ^-Ib. 29. Find the whole pay of 34- majors of Dragoon Guards and Dragoons in the British Army for 31 days, @ 19/3 each ^ day. 30. What did a writer's clerk whose income was 110^- annum, pay for income tax in 1855, at the rate of lld. ^ ? 31. What did a minister, whose stipend in 1856 was 326"10"5, pay for income tax @ 1/4 ^ ? 32. A bankrupt whose debts are 30,000 pays 8/3 T r 2 y . How much does he pay? 33. In 1852, 590,767 oz. of gold coin were exported from the United Kingdom. Find the value @ 3"17/'10| y oz. 34. Reckoning the ducat at 4/2J, find the value refused by Shyloclc, when he says : " If every ducat in six thousand ducats Were in six parts, and every part a ducat, I would not draw them, I would have my bond." PRACTICE. 89 55. (3) Find the gross rental of the following 5 farms : nc. ro. po. s. I. 263 "0 38 @ 1"11 6 i 414 II. 457 " 39 .. 1 5 571 III. 49 " 3 5 .. 2 5 112 IV. 156 " 2 32 .. 1"15 274 V. 146 1 39 .. 1-13 4 244 1616 6 1* 35. Find the amount of a minister's stipend: 30 qr. 7 bu. 5 7 5 pk. barley @ 39/6 y qr. ; 12 qr. 2 bu. 3 5 7 5 pk. oats @ 24/10J f qr. ; 40 bolls oatmeal @ 18/10 ; and 48610|. 36. In the Edinburgh grain market, on 52 Wednesdays ending Oct. 22, 1856: 42,915 qr. wheat were sold at an average price of 73/7 ; 42,206 qr. barley @ 42/1 ; 44,558 qr. oats @ 31/5. Find the amount. 37. Find the value of the average annual agricultural produce of a parish : 1386 qr. wheat @ 65/1 ; 1350 qr. barley @ 40/6 ; 2314 qr. oats @ 29/9; 82,500 stones hay @ 7$d.; 204 acres turnips @ 12'-26; 204$ acres potatoes @ 13"! "8. 38. Find the rental of an estate containing 4 farms : 375 ac. 2 ro. 30 po. @ 3"12"6 y ac.; 432 ac. 1 ro. 20 po. @ 3"2"6 v ac. ; 280 ac. 3 ro. 25 po. @ 2126 y ac. ; 413 ac. ro. 15 po. @ 217 y ac. (4) Edinburgh, Mrs Jones, Sept. 14, 1857. Bot of Adam Coburg, General Draper, s. d. s. d, 5 pieces, each 46 yd. merino @ 4." 3 8 .. .. 80 yd. cotton .. " 4 3^ .. .. 54yd. linen .. 29 f yd. 48 12 86 Mr James White, Grocer, Perth, To Price & Co., Wholesale Merchants, Glasgow. s. d. 1857. June 26. Sept. 11. In writing out the following Accounts, supply Names and Dak*. 39. 286 loaves @ 7*d. ; 140 loaves @ 6Jd. ; 89 fancy ^loaves @ 8d. ; 147 doz. biscuit 3d. per doz.; 176 Ib. flour @ 2Jd- 5 chests congou, each 2 qr.l lib. @ 3"8 fib. 3 hhd. sugar, each 13 cwt. 2 qr. .. 394 .. cwt. 3 cwt. 1 qr. 14 Ib. coffee .. 49"6 .. cwt. 14 cwt. 2 qr. 3 Ib. cheese .. 0"5.. ID. 61 79 8 37 ;i86 8 13 7 5 14 4 Of 8! 90 PRACTICE. 55. 40. 648 silk mantles @ 14/10|; 420 richly trimmed mantles @ 45/; 600 yd. satin @ 8/10 J; 252 silk velvet mantles @ 71/9 ; 140 Paisley shawls @ 47/6 ; 246 foreign shawls @ 66/8. 41. 900 yd. moleskin @ 1/2J; 500 yd. plaiding @ 1/4; 250 yd. flannel @ 1/5; 600 yd. gingham @ 4f d. ; 1800 yd. unbleached cotton @ 3d. ; 200 yd. twilled linen @ 1/5 ; 80 yd. pilot cloth @ 6/5; 200 yd. pack sheeting @ 5d. 42. 348 squares of Windsor soap @ 5d. q> square; 440 doz. squares of honey soap @ 10/6 ^ doz. ; 200 bottles marrow oil @ 1 If d. ; 288 pints castor oil@ 1/2 ; 350 pots polishing paste @ 5|d. ; 1 cwt. 2 qr. 7 Ib. starch @ 6d. f Ib. 43. 740| Ib. coffee, No. I., @ I/; 370 Ib. coffee, No. II., @ 1/2 ; 561J Ib. coffee, No. III., @ 1/4; 311 Ib. coffee, No. IV., @ 1/8. 44. 4965 qr. wheat @ 41/4; 236 qr. barley @ 39/2; 483| qr. oats @ 26/1 ; 146^ qr. beans @ 39/5. 45. 14 pieces, each 37 yd. @ 10/5 ^ yd.; 11 pieces, each 53 yd. @ 12/4 f yd. ; 19 pieces, each 44| yd. @ 13/8 ^ yd. ; 23 pieces, each 59 yd., @ 16/7J ? yd. 46. 124 qr. 7 bu. wheat @ 55/5 f qr. ; 88 qr. 4 bu. barley @ 45/3 w qr. ; 138 qr. 3 bu. oats @ 23/8 y qr. ; 181 qr. 5 bu. beans @ 40/8 ^ qr. 47. 6 chests congou, each 2 qr. 17 Ib. @ 3/9 f Ib. ; 13 hhd. brown sugar, each 13 cwt. 1 qr. 18 Ib. @ 36/4 ^ cwt. ; 3 casks molasses, each 7 cwt. 2 qr. 14 Ib. @ 13/9 ^ cwt. 48. 14 cwt. 2 qr. 14 Ib. Cheshire cheese @ 50/ y cwt. ; 17 cwt. 3 qr. 14 Ib. Wiltshire @ 40/ $* cwt. ; 23 cwt. 1 qr. 18 Ib. Gouda @ 28/^cwt. ; 15 cwt. 2 qr. 13 Ib. American @ 35/ ^ cwt.; 27 cwt. 3 qr. 16 Ib. Carlow butter @ 77/ f cwt.; 39 cwt. 1 qr. 14 Ib. Waterford @ 72/ ; 47 cwt. 2 qr. 20 Ib. Dutch @ 84/ # cwt. ; 23 cwt. 2 qr. 7 Ib. Limerick @ 66/8 ^ cwt. 56. ALLOWANCES ON GOODS. IN selling goods by weight, an ALLOWANCE is made for the box or package containing them. The weight of any commodity, with that of the box or package containing it, is termed Gross Weight ; the weight of the box, Tare; and the weight of the commodity, Net Weight. If a chest of tea weighs 80 Ib., and the empty chest 16 Ib., the Gross Weight is 80 Ib., the Tare 16 Ib., and the Net Weight 64 Ib. Draft is an allowance given to a retailer to enable him to tarn the scale in selling a commodity in small quantities. Gross Weight. Tare. cwt. qr. lb. qr. lb. // 3 //10 // 17 // 3 // 7 // 16 // 3 // 5 // 14 // 3 // 4 // 15 3 // //26 2 // 6 // 2 // 6 ALLOWANCES ON GOODS. 91 56. A wholesale merchant in selling a chest of tea may deduct 1 lb. The Commercial Allowances Tret and Cloff are now obsolete Cloff was similar to Draft. Tret was an allowance given on goods liable to waste. (1) Find the net weight of 4 chests of tea, of which the gross weight and tare are respectively as follow : I. II. III. IV. 2 // 2 // 20 Net Weight. 1. How much honey is sold, when in placing a jug weighing 7f oz. in one scale, weights amounting to 3 lb. 3f oz. are placed in the other ? 2. A railway truck weighing IT. 16 cwt. 3 qr., when loaded witli wool, weighs 8 T. 11 cwt. 1 qr. What is the weight of the wool? 3. A two-horse cart, weighing 13 cwt. 2 qr. 21 lb., when loaded with compost, weighs on the machine of a toll-bar 2 T. 1 cwt. 1 qr. 7 lb. What is the weight of the compost? 4. Find the net weight of a barrel of flour; gross weight, 1 cwt. 3 qr. 10 lb.; tare, 12 lb. 5. Find the net weight of 12 drums of Turkey figs ; gross weight, 24 lb. 8 oz. ; tare, f lb. each. Find the net weight of 3 tierces of coffee, of which the gross wt. 2 qr. 9 lb. each, and the tare 2 qr. 17 lb. each. 7. Find the weight of coal brought up by a train of 20 trucks depot, the average of each truck being 10 T. 17 cwt. 2 qr. gross, and 3 T. cwt. 3 qr. tare. 8. Find the net weight of 3 hogsheads of sugar, of which the gross weight and the tare are as follow: I. 13 cwt. 2 qr. 14 lb. gross; tare, 1 cwt. 1 qr. II. 12 cwt. 1 qr. 13 lb. gross f tare, 1 cwt. 20 lb. III. 13 cwt. 1 qr. 20 lb. gross; tare, 1 cwt. 1 qr. 7 lb. Find the net weight of 3 tierces of coffee, of which the gross ht is respectively 5 cwt. 2 qr. 13 lb. ; 4 cwt. 1 qr. 12 lb.; 6 cwt. qr. 17 lb. ; and the average tare 2 qr. 7 lb. $> tierce. 92 ALLOWANCES ON GOODS. (2) Find the net weight of 9 bales of wool, each 3 cwt. 3 qr. 14 Ib. gross ; draft, 2 Ib. ^ bale ; tare, 16 Ib. V cwt. cwt. qr. Ib. 3 // 3 // 14 Gross wt. of 1 bale // // 2 Draft " 12 Draft Suttle 9 16 Ib. 4 1 cwt. 34 // 2 // 24 * 9 bales 4 // 3 // 23} Tare " 29 // 3 // Net weight 10. Find the net weight of 231 cwt. 2 qr. 3 Ib. gross; tare, 14 Ib. w cwt. 11. Find the net weight of 200 cwt. 1 qr. 4 Ib. gross; tare, 20 Ib. f cwt. 12. Find the net weight of 8 chests, each 1 cwt. 2 qr. 14 Ib. gross ; tare, 1 6 Ib. ^ cwt. 13. Find the net weight of 29 chests, each 1 cwt. 1 qr. 7 Ib. gross ; tare, 12 Ib. ^ cwt. 14. Find the net weight of five half- chests of tea, tare being 20 Ib. y cwt., and gross weight respectively, I qr. 19 Ib. ; 1 qr. 18 Ib. ; 1 qr. 20 Ib. ; 1 qr. 21 Ib. ; 1 qr. 16 Ib. 15. Find the net weight of 4 chests tea, weighing respectively 75 Ib., 84 Ib., 63 Ib., 83 Ib. ; draft, 1 Ib. ^ chest; tare, respectively 13 Ib., 17 Ib., 14 Ib., 15 Ib. 16. Find the net weight of 20 casks madder, average gross weight of each cask being 15 cwt. 2 qr. 14 Ib. ; draft, 5 Ib. ^ cask ; tare, 17 Ib. ^ cwt. SIMPLE PEOPOBTION. IN comparing two numbers, by finding how many times the one is as large as the other, the quotient obtained expresses the relation or RATIO of the dividend to the divisor ; thus, the ratio of 16 to 8 is y 5 ; of 14 to 5, V 5 of 2 to 9, f . In expressing the ratio of two numbers, as of 16 and 8, we write it thus, 16:8. The first term, 16, is called the Ante- cedent, and the second, 8, the Consequent. Four numbers are said to be Proportional when the ratio of the first to the second is equal to the ratio of the third to the fourth. On examining the four numbers, 14, 8, 35, 20, we find V = f & or 14 : 8 = 35 : 20, and say 14 is to 8 as 35 is to 20, which we write as follows 14 : 8 : : 35 : 20. SIMPLE PROPORTION. 93 Since V = it, -H-*? = |ff, and U X 20 = 8 X 35. When four numbers are proportional, the Product of the Means is = the Product of the Extremes. According to the arithmetical interpretation of Definition of Proportionality in Euclid (Book V. Def. 5), four numbers are pro- portional when the first, or a multiple of the first, contains the second as often as the third, or a like multiple of the third, contains the fourth. Let us take 8, 2, 28, 7 ; 8 contains 2 four times, and 28 contains 7 four times ; hence 8 : 2 : : 28 : 7. Again, take 27, 48, 63, 112 ; sixteen times 27 = nine times 48, and sixteen times 63 = nine times 112 ; hence 27 : 48 : : 63 : 112. In SIMPLE PROPORTION, we are required to find a number to which a given number may have a given ratio. t Find a number to which 56 may have the ratio of 24 to 63. Let x be the required number, then 24 : 63 : : 56 : x ; and since the product of the means is = the product of the ex- tremes, 24 times the required number, or 24# = 63 X 56, therefore the required number, x = ^ = 147. The fourth term in a proportion is termed the Fourth Pro- portional to the other three. We have seen it is obtained by multiplying the second term by the third, and dividing the product by the first. (1) Find the fourth proportional to 21, 30, and 28. 21 : 30 : : 28 : x 10 4 , =^=40. -*" We may cancel the common factors of the first term with those of the second or the third. Find fourth proportionals to the following numbers : 1. 6, 14, 12 2. 8, 24, 5 3. 7, 18, 21 4. 3-6, 4-2, 6-6 5. -27, 11-7, 2-1 6. 15-3, 2-89, -171 (2) Find the fourth proportional to 5j, 9$, and 5J : 9| : : i : x V : V : : * : * 2 14 & T * 94 SIMPLE PROPORTION. 57. Find the fourth proportionals to the following numbers : 7. 3i, 5J, 8,V 9. i, |, A 8. 3f, 6f, 1'Ji 10. i, 7|, When there are three numbers, of which the first is to the second as the second is to the third, the third is termed the Third Proportional to the first and the second, and the second is the Mean Proportional between the first and the third. (3) Find the third proportional to 9 and ll. 9 : 11J : : llj : x 5 rf 45v 45 * Q 45 X-4&- 2 2 5 1 4 , - * X , * - 4X4X9- ^ - 14 T 14-jjg. is the third proportional to 9 and llj, and 11 J is a mean proportional between 9 and 14 T ] 7 . Find the third proportionals to 11. 9, 15 I 13. 5, 12. 49, 56 ! 14. 2-7 58. (!) K 27 cwt. sugar cost 51, what cost 63 cwt. ? cwt. 27 ... . . 51 63 ... . . x cwt. cwt. 27 : 63 : : 51 : x 7 17 -33- > C ~^~ 4?11Q -3- The greater the quantity of sugar, the greater will be the price. Since we multiply the third term by the second and divide by the first, in order to obtain the fourth term greater than the third, the second must be greater than the first. Having stated 51 in the third term, we place 63 cwt. in the second and 27 cwt. in the first. Having stated the number which is of the same kind, or is homogeneous to that which is required, we place the greater, or the less of the other two homogeneous numbers, in the second term, according as the fourth term should be greater or less than the third. The following method may sometimes be adopted : Since 27 cwt. cost 51 1 cwt. costs } and 63 cwt. cost je^LXJL 1 = 119. SIMPLE PROPORTION. 95 58. (2) If 39 men can do a work in 168 days, in how many days can 72 men do it? men. days. 39 168 72 x men. men. days. days. 7'2 : 35, and his assets 321*6*3. How much can he pay ^ 1? Debts. Assets. 635*10*5 321*6*3 1 x 535 // 10 // 5 : 1 : : 321 // 6 // 3 : x Here we say as 535//105 of debt is to 1 of debt, so is 32163 of assets to x of assets. We may however state and work as fol- lows : 535*10*5 : 321*6*3 : : 1 : x _20 20 20 S . 10710 s. 6426 s. _JL2 12 128525 d. 77115 d. 25705 15423 20 25705)308460(12 s. 308460 When all the terms are homogeneous, we can state the proportion in two ways. 31. A tenant whose rent is 53"6"8 pays a tax of l"13/>4. Find the tax on a rent of 36. 32. Find the rent of a tenant who pays 13/9 of poor's-rates, at the rate of 5Jd. f . SIMPLE PROPORTION. 99 33. A bankrupt's debts are 525" 10^6, and his assets 375*7, 6. How much can he pay f ? bankrupt whose assets are 3420 pays a composition of 9/6 ^ . Find the amount of his debts. ;>."). The tax paid on an income for the year ending April 5, 1856, was 19"4, at the rate of 1/4 ^ . Find the income. 36. A clerk, after paying 2/<2"l of income-tax for the year end- ing April 5, 1854, found that he had 9817//11 over. What was the rate # ? 37. If the shadow of a staff 3 ft. 7 in. high measures 4 ft. 9 in., find the height of a steeple whose shadow is 158 ft. 4 in. 38. A farmer inadvertently used stone weights of 26 Ib. 8 oz. each for 28 Ib. What would 2 T. 13 cwt. of grain appear by these rhts to be ? 39. A merchant used weights of 27 Ib. 12 oz. instead of 28 Ib. Find the true weight, which would appear 1 cwt. more by the false weights. (7) If a person can walk 8 miles in 2| hours, how far can he walk in 3^ hours ? 84 miles ..... 2i hours x miles ..... 3^ hours ho. ho. ml. ml. 2i : 3J : : 84 : x . 40. If 26{| yards of cloth cost 8f, what cost 111| yards? 41. If 39 S 3 8 cwt. of rice cost 18^, how much rice may be had for3i|? 42. If 93| yards of damask cost J of 45f $, what cost 113 T ', yards ? 43. If 54 men can do a work in 29| days, how many men will . do it in 35iJ days? 44. For every 5J miles that A walks, B goes 4J miles. ^Ho long will B take to traverse a distance walked by A in 6 hours? V train at the rate of 25| miles an hour traverses a distance in 3i hours. In what time will one at the rate of 24 miles an . A mile in * of 2| hours. In whattime can he walk J of 1 { $ mile at the same rate ? 47. If ft of a vessel is worth 1393, what is the value of | of f. of the vessel ? 100 SIMPLE PROPORTION. 58* (**) If 2 '45 cwt. cost 22'75, how many cwt. may be had for 11 -7? cwt. cwt. 22-75 : 11-7 : : 2-45 : x 1-75 -9 49 "35 J_ 1 "5 5)6-3 1-26 cwt. We have cancelled the first and the second terms by 13, the first and the third terms by 5. By multiplying the first and the third terms by 100, we clear the decimal points, and obtain two numbers which when cancelled by 7 are 5 and 7. 48. If 4-06 cwt. of rice cost 3*480, how much rice may be bought for 7-625? 49. A wall whose height is 9*1875 ft., casts a shadow of 10*5 ft. Find the length of the shadow of a steeple 93 -8 ft. high. 50. A bar of cast-iron, whose Specific Gravity is 7*207, weighs 80 Ib. Find the weight of a bar of cast-brass of the same size, whose s. G. is 8-100. 51. A jar of honey, whose s. o. is 1-450, weighs 4| Ib. Find the weight of olive-oil, whose s. a. is -908, contained in the same jar. 52. A block of Parian marble, whose s. G. is 2*560, weighs 2J tons. Find the weight of a block of Carrara marble of the same size, whose s. G. is 2*716. We now give some MISCELLANEOUS EXERCISES, which include several important Applications of Proportion. 53. After paying 7d. qp* as income-tax for the year ending April 5, 1854, a gentleman had 971"161 over, on what had the tax been charged ? T 1 n 0*7 0/49 // 5 : 97116//1 : : 1 : x 54. A person paid 1 1 ^d. ^ as income-tax for the year ending April 5, 1856, and had 104"147 of net proceeds. Find his income. 55. The ratio of the diameter to the circumference of a circle was given by Peter Metius as 113 : 355. Find the circumference of a fly-wheel 10 ft. in diameter. 56. A cistern can be filled by a pipe running 3| gallons y minute in 54 minutes ; in what time can it be filled by another running 4 gallons ^ minute ? 57. If 300 labourers can make an embankment in 48 days, in how many more days will 60 fewer do it? SIMPLE PROPORTION. 101 58. 77 tailors can execute an order of regimental clothing in 30 days ; how many more must be engaged to fulfil the order 8 days sooner ? 59. If 33 masons can build a wall in 47 days ; and if, after work- ing 11 days, 15 leave ; in how many days after the 15 leave will it be finished ? <6T 33 masons csmjinisJi the wall in 47 11 or 36 days. Since 15 masons have left, 18 remain. masons. days. Hence, 18 : 33 : : 36 : x = the number of days after the 15 have left. 60. If 17 men can do a work in 89 days ; and if, after working 33 days, 3 men leave ; in how many days in all will the work be done ? 61. If 64 men can perform a work in 57 days ; and if, after work- ing for 12 days, notice is sent to finish the work 9 days before the stipulated time ; how many additional men must be engaged ? 62. If 3 men can do as much as 4 youths; and if 13 men can do a work in 9 days ; in what time can 12 men and 8 youths do it? youths. men. ZT 4 : 8 : : 3 : x = 6 6 + 12 = 18 men. men. days. 18 : 13 : : 9 : x 63. If 4 men can do as much as 7 youths ; and if 15 men can do a work in 16 days; in what time can 16 men and 14 youths doit? 64. Find the Horse Power of an engine which can raise 5 tons of coals per hour from a pit whose depth is 66 fathoms. %3T The labour necessary to raise 1 lb. through 1 foot is termed the Unit of Work (U. W.) Watt found that a horse could do 33,000 units of work ^ minute. 1 H. P. = 33,000 U. W. 5 tons = 11200 lb. 66 fathoms = 396 ft. 11200 X 396 = 4435200 U. W. ^ ho. 6,0) 443520,0 73920 U. W.

hour. When and where do they meet ? 7."). From Carlisle to Preston is 90 miles. A train leaves Car- lisle at 12 "15 a. m., at 40 miles f hour, and Preston at 2 a. m., at 36 miles f hour? When and where do they meet? T Find where the Carlisle train is when the Preston train starts, and then proceed as in the other examples. COMPOUND PBOPOBTION. 59. "WE have seen that the ratio of one number to another may be expressed by a fraction, of which the antecedent is the numer- ator and the consequent the denominator. Thus, the ratio of 4 to 5 is = , and the ratio of 6 to 7 is f . Since the com- pound fraction | of -f is = $, we say that the ratio of 24 to 35 is compounded of the ratios of 4 to 5 and of 6 to 7. Hence, if one number is to another in the ratio of 24 to 35, it is in the ratio compounded of the ratios of 4 to 5 and of 6 to 7. Thus, since 24 : 35 : : 48 : 70, the ratio of 48 to 70 is compounded of the ratios of 4 to 5 and of 6 to 7. We write these numbers in the following form : * if}:: 48: 70. 45 = 4 of f = i|. 48 X 5 X 7 __ 4 X 6 X 70 70 X 5 X 7 ~~ 5 X 7 X 70' and 48 X 5 X 7 = 4 X 6 X 70. ; he Product of the Means is = the Product of the Extremes. In COMPOUND PROPORTION we find a number to which a given number may have a ratio compounded of two or more ratios. Find a number to which 72 may have a ratio compounded of the ratios 4 : 5 and 6 : 7. 104 COMPOUND PROPORTION. 59* Let x be the number, yO . / Then since the product of the extremes is = the product of the means, 4X6X^ = 72X5X7 The required consequent is = its antecedent X the other con- sequents -=- the other antecedents. (1) If 4 horses plough 45 acres in 10 days, in what time will 6 horses plough 81 acres? Before stating, we may write the terms in two rows. This me- thod is particularly useful in writing down a question to dictation. 4 horses, 45 acres, 10 days 6 horses, 81 acres, x days. Horses 6 : 4\ Acres 45 : 81 j : 27,0)324,0 (12 We follow the same method as in Simple Proportion ; thus 6 horses will take a less number of days than 4 horses ; hence 6 : 4. Again, 81 acres will require a greater number of days than 45 acres ; hence 45 : 81. We thus consider each pair of terms separately in refer- ence to the required number. \Ve may work every question by resolving it into questions in Simple Proportion. The foregoing question may be resolved as follows : I. If 4 horses plough 45 acres in 10 days, in what time will 6 horses plough the same number of acres ? Horses. Days. 6 : 4 : : 10 : II. Now if 6 horses can plough 45 acres in ^y- days, in what time will the same number of horses plough 81 acres? Acres. Days. 45 : 81 : : LXJ : x - "x* 81 _ 12 dilys . COMPOUND PROPORTION. 105 59. (2) If 21 reapers cut 3 ac. 3 ro. of corn in 4| days, in what time will 24 reapers of the same strength cut 16 ac. 1 ro. ? 21 reapers, 15 roods, \ 3 days 24 reapers, 65 roods, x days. Reapers 24 : 21 Roods 15 : 65 13 ) } : : = VW = 17jft days. 5 In (1) the number of days is in the inverse ratio of the number of -, and in the direct ratio of the number of acres. In (2) the number of days is in the inverse ratio of the number of reapers, and in the direct ratio of the number of roods. 'nay illustrate (1) as follows: 4 horses plough 45 acres in 10 days 1 horse ploughs do. in 10 X 4 days 6 horses plough do. in ^~ days do. plough 1 acre in ^| days do. plough 81 acres in ~~^ days. Similarly we may illustrate (2) or any other exercise in Com- iiid Proportion. 1 . If 3 families of 6 persons each consume 28 loaves in a week, how many will 9 families of 5 persons each consume in the same time? ^ This Question, and others similar ^ to it, may easily be worked by one statement in Simple Proportion. 2. A housekeeper having used 6 pots of jelly with 14 loaves each 12 slices, wishes to know how many will be used with 8 h 7 slices? :;. If 13 bushels of oats serve 3 horses for 11 days, how many bushels will serve 7 horses for 12 days? 1 f G boys are boarded for 10 months for 270, for what ought i:; Ix-vs to he boarded for 7 months? 5. If 8 labourers earn 148 in 12 days, what will 17 labourer mi in 5 days? , - r t If 22,500 types are used in setting up 12 pages each 25 lines, how many types will be required in setting up 17 pages typr- nnd breadth each 31 lines? 106 COMPOUND PROPORTION. 59. 7. A family may live for 3 months in the country for 24" 10, what will be required to maintain them in town for 9 months, sup- posing 3 in the country to be equivalent to 4 in town ? 8. If a traveller walks 140 miles in 8 days walking 7 hours a- day, how many miles may he accomplish in 12 days walking 8 hours a-day ? 9. If 3 tailors make 5 vests in 1 1 hours, in what time will 1 1 tailors make 15 vests ? 10. If 64 yards of carpet, 3 qr. wide, cover the floor of 4 equal 'rooms ; how many yards of carpet, 1 yd. wide, will cover 3 of them ? 11. If the 4 Ib. loaf costs 8d. when wheat is @ G4/ ^ qr., find the weight of the penny loaf when wheat is @ 56/. 12. A bootmaker who employs 15 men fulfils an order of 25 dozen pairs of Wellington boots in 4 weeks, in what time may he accomplish an order of 45 pairs by employing 3 additional men ? 13. If 24 cakes can be made out of 3/ worth of oatmeal when meal is @ 18d. ^ pk., how many cakes can be made out of 10/3^ worth when meal is @ 1 3d. ? 14. Captain Basil Hall, in computing the time in which Sir Walter Scott might execute the MS. of Kenilwortli, introduces the following: if 120 pages of 777 letters each may be written in 10 days, in what time would 3 volumes of 320 pages of 864 letters each be written ? 15. A railway company charges 18/ for the carnage of 9 cwt. 40 miles, (1) What should be charged for carrying 10 cwt. 54 miles? (2) What weight should be carried 27 miles for 54/? (3) How far should 3 cwt. be carried for 15/? 16. If 7 compositors set up 15 sheets in 6 days, (1) In how many days will 21 compositors set up 30 sheets? (2) How many sheets will 27 compositors set up in 14 days ? (3) How many compositors will set up 25 sheets in 7 days ? 17. If 36 labourers clear 513 yards for a railway in 6 days, (1) How many will clear 3800 yd. in 10 days? (2) How many yd. will be cleared by 156 labourers in 18 days ? (3) In how many days will 16 labourers clear 190 yd. ? 18. If 4 masons build 27 yards of wall in 5 days working 9 hours a-day, in how many days will 32 masons build 81 yards of a similar wall working 10 hours a-day? 19. If 12 boys are boarded 10 months for 498, find the board of 18 boys for 9 months, supposing that the cost of boarding 4 of the former = that of 3 of the latter. 20. If 5 is sufficient to maintain 8 labourers for a fortnight COMPOUND PROPORTION. 107 59* when corn is at 28/ qp> qr., how much will be required to maintain 6 labourers 29 days when corn is at 32/ ^ qr. ? 21. If 20 men, of whom the average strength is f of an ordinary man's strength, can load 81 trucks in 8 hours; in what time will 32 men, of the average strength of ^ of an ordinary man's strength, load 63 trucks ? 22. If 7 labourers mow 50 acres in 9 days of 8 hours each, (1) IIow many acres will 14 labourers mow in 3 days of 6 ho. ? (2) How many labourers will mow 25 acres in 18 days of 7 ho. ? (3) In how many days of 9 hours each will 14 labourers mow icres? (4) By working how many hours a-day will 20 labourers mow 500 acres in 2 1 days ? 23. 8 men can dig a trench 200 yards long, 2 ft. broad, and 6 ft. deep, in 12 days, (1) How many will dig another 160 yd. long, 3 ft. broad, 5 ft. deep, in 6 days ? (2) What length of trench will 7 men dig in 11 days, suppos- ing it 4 ft. broad and 7 ft. deep ? (3) What breadth of trench will 6 men dig in 8 days, sup- posing it 50 yd. long and 6 ft. deep ? (4) What depth of trench will- 12 men dig in 15 days, sup- posing it 50 yd. long and 4 ft. broad ? 24 If 17 men cat 33/ worth of bread in a week when the 4 Ib. h at Bd., what value of bread will 9 men eat in 2 weeks when the 2 UK loafisat4Jd.? 2 - 1 f a family by using 2 gas-burners ?i hours a-day pay 1-5 - when gas is @ 10/ F 1000 cub. ft., what will a family Mirners 4 hours a-day pay p quarter when gas is @ 7/6 v . Tnfcldles, of which 8 weigh 1 Ib., serve 4 winter even- ings from 5 to 11 P. M. ; how many candles, of which 6 weigh 1 Ib., will serve 3 spring evenings from 7 to 11 P. M.? 27. If 330 slices, A inch thick, are obtained from 12 rounds of beef, how many similar rounds will supply 495 slices, 1 inch thick , 28 If the part representing land cut out of a map of a countiy / l( )sn miles in extent weigh 384 grains; find the extent ot drawn on the scale of 56' on that of 100 sq. ml. to the sq. hound run while the hare runs 420 yards ? 108 COMPOUND PROPORTION. 59* 30. If the horse Flying Dutchman takes 10 strides while the horse Nonsuch takes 9, but if 6 strides of the former are equal to 5 of the latter, what distance will the latter run while the former runs 1200 yards? 31. If 6 bars of metal, 2 ft. long, 6 in. broad, and 3J in. thick, weigh 126 Ib. ; find the weight of 7 bars 3 ft. long, 41 in. broad, and 3 in. thick. 32. The weight of 35 cubic inches of gold, of which the Specific Gravity is 19'258, is 355-270 oz. troy; find the weight of 49 cubic inches of silver, of which the Specific Gravity is 10-474. 33. A slab of granite containing 3, s s cub. ft. weighs 541 Ib., find the weight of a piece of pumice stone containing If cub. ft., the s. o. of the former being to that of the latter as 175 to 61. 34. A contractor having engaged to lay ten miles of railway in 150 days, finds that 90 men have finished 3 miles in 80 days ; how many additional men must be engaged to finish it within the time ? 35. A stabler lays in 80 bushels of oats to feed 15 horses for 16 days ; at the end of 4 days he receives other 5 horses ; how many additional bushels will be required for the given time ? 36. The diameter of the Sun is 882,000 miles. His apparent diameter as seen from the Earth is 32' 1-8". Find the apparent diameter of a globe of fire as large as the Solar System, 5,700,000,000 miles in diameter, viewed at the distance of the nearest fixed star 206,265 times as distant as the Sun. COMPUTATIONS made at a certain rate per hundred (per cen- tum) are termed PER-CENTAGES. Per-centages are used in Commercial Arithmetic in finding Commission, Interest, &c. They are often employed in ques- tions of Statistics. STATISTICS. .STATISTICS treats of the numerical data of any subject. Thus, if we examine the number of persons who pay Income Tax, the amount annually paid, &c., we are said to inquire into the Statistics of the Income Tax. Again, if a Table gives the amount of Tea annually imported and consumed in Great Britain, with the amount of duty paid, &c., it is said to furnish the Statistics of the Tea Trade. The Statistics of a country treats of its population, rev- enue, and general resources. STATISTICS. 109 6Ot (1) Of 93,498 births registered in Scotland in 1855, 47,872 were males. Find the per-centage. 93498 : 47872 : : 100 : x = 51-201 per cent. 1. Find the per-centage of alloy in sterling gold, of which 1 Ib. troy contains 1 oz. alloy. 2. In 1851, of 335,966 emigrants from the United Kingdom, 257,372 were Irish. How much per cent, was the latter number of the whole ? 3. In 1855, the produce of silver in the United Kingdom amounted to 561,300 oz., of which 4947 were from Scotland. Find the per- centage that the latter number was of the whole. (2) The number of poor relieved in Scotland for the year " ending 14th May 1848 was 100,961 ; for 1849, 106,434 ; and for 1850, 101,454. Find the increase per cent, from 1848 to 1849, and the decrease percent, from 1849 to 1850. 106434 100961 100961 : 5473 : : 100 : x = 54-2091 per cent, of increase. 106434 101454 106434 : 4980 : : 100 : x = 46*7896 per cent, of decrease. 4. The number of letters delivered in the United Kingdom in the year preceding Dec. 5, 1839, when penny postage was gener- ally introduced, was 82,470,596; and in 1840, 168,768,344. Find the increase per cent. In 1854, the number of letters delivered in the United King- dom was 443,649,301 ; and in 1855, 456,216,176. Find the increase per cent. . 6. The total number of railway tickets issued in the United Kingdom in 1850 was 66,840,175; and in 1851, the year of the Great Exhibition, 78,969,623. Find the increase per cent. 7. The population of Ireland in 1841 was 8,175,124; andin!851, '.52,385. Find the decrease per cent. (3) A sample of bone manure was found to contain 15-83 per cent, of sulphate of lime. Find the weight of the sulphate in 12 tons of manure. Tons. 100 : 15-83 : : 12 : x T. x = '1583 X 12 = 1'9 110 STATISTICS. 6 Ot When the rate per cent, contains an approximate decimal, the result can be obtained to a certain number of decimals only (see 39). In some cases the required result is necessarily a whole number. 8. The number of representatives in the House of Commons is 658. Of this number, or even of 654, which was for some years the number of representatives, the per-centage for Scotland is 8'1. Find the number of the Scottish representatives. 9. A sample of turnip manure was found to contain 20'5 per cent, of sulphate of lime. Find the weight of the sulphate in 20 tons of manure. 10. The Queen's Remembrancer in Scotland has a salary of 1250 ^ annum. Find the salary of his chief clerk, which is 44 per cent, of his own. (4) The Estimate for the Science and Art Department in Scotland, for the year ending 31st March 1856, was 1763. Find the estimate for the succeeding year, which gave an increase of 3*165 per cent. 100 : 103-165 : : 1763 : x = 1818*16 x 1-03165 X 1763 = 1818-8 11. In 1855, the number of marriages registered in Scotland was 19,639. In 1856, the increase was at the rate of 4-318 y cent. Find the number of marriages in the latter year. 12. The population of Scotland in 1841 was 2,620,184. Find the population in 1851, which had increased at the rate of 10*2496 per cent. 13. The population of England and Wales in 1841 was 15,914,148. Find the population in 1851, which had increased at the rate of 12-65202 per cent. (5) In 1855, the per-centage of deaths, amounting to 62,154, was 2-06884 of the estimated population. Find the estimated population. 2-06884 : 100 : : 62154 : x = 3004300. 3,004,300 is the reliable number obtained from the given number of decimal places. Had we taken the rate per cent, as 2-07, we would have obtained 3,000,000 merely. To obtain 3,004,290, the correctly estimated population, we require 6 decimals in the rate per cent. In statistical computations we can reproduce all the places of whole numbers only when a sufficient number of decimals in the per-centage is given. 14. In 1856, when the number of acres in Scotland on which wheat was cultivated was 70,522 more than in 1855, the increase STATISTICS. 1 1 1 6O.WAS at the rate of 36'8646 per cent. Find the number on which wheat was cultivated in 1855. 15. In Scotland, during the year ending May 14, 1855, the de- crease in the number of registered poor was 3217 from the former t As the decrease was at the rate of 3-09992 per cent., find the number relieved during the year ending May 14, 1854. (6) ^December 1856, the number of deaths in London was 14, GIG. This was an increase of 2*482 per cent, over the number of deaths in December 1855, in which the num- ber showed a decrease of 17-408 per cent, from that in December 1854. Find the number of deaths in December 1855 and in December 1854. 100 + 2-482 = 102-482 : 100 : : 14616 : x = 14262 100 17-408 = 82-592 : 100 : : 14262 : x = 17268 16. In 1851, the population of the United Kingdom, which was 27,674,352, had increased from 1841 at the rate of 73-361818 per cent. Find the population in 1841. 17. In 1812, the census of China in the seventeenth year of Kiaking amounted to 362 millions. This gave an increase of 8'7 per cent, since 1792, when a statement was made to Lord Macart- ney in the fifty-seventh year of Kienlung. Find the census in 18. In 1856, the number of deaths in England and Wales was 391,369; the decrease per cent, was 8-18103 from the previous year ; find the number in 1855. COMMISSION AND BBOKEBAGE. 61* COMMISSION is a per-centage allowed to an agent for buying or selling goods. / jt BROKERAGE is a per-centage allowed to a broker lor trans- ferring the right of property, or for assisting in the sale or purchase of goods. , A merchant often allows a per-centage to a customer when he pays goods in Ready Money. This allowance, termed JCOUNT, must be distinguished from Bank Discount (see 64), in whose calculation the element of time is introduced. 1 Express the following per-centages as allowances 40; 33i; 25; 20; 12$; 5, percent. 112 COMMISSION AND BROKERAGE. 61* 2. Express the following per-centages as allowances ^' s. : 25 per cent. = ^ = J. J of Is. = 3d. 50; 33; 16f ; 12 J per cent. 3. Express the following allowances as per-centages : 7/6 sp- = = }. | of 100 = 37^ per cent. 10/; 5/; 2/6; I/; 8d. ; 6d. ^ . 6d.; 4d. ; 3d.; 2d. ; IJd.; Id. ^ 5. (1) Find the commission on 578*10'/6} @ 2} per cent.(/ ) 100 : 578*10*6} : : 2} : x We therefore multiply the sum by the rate per cent., and divide by 100. 578*10* 6} 578-526 2* 2} 8)1735*11* 6} 8)1735578 216*18*ll}f 216947 1157* 1* 1157052 13,73*19*11}} 13-7401 = 13*14*91 20 14,79 13*14*91 ,% 12 9,59 4 (2) Find the brokerage on 347*12//6 @ ^ / , and @ 8/4 c / . 8) 347*12*6 ,43* 9~^0| 5/ i 20 3/4 i 8,G9 8/8 i 347*12* 6 86*18* 1 57*18* 9 1,44*16*10- 12 20 8,28 p6 A __!?. M5 =,*& = & 1T62 4 a COMMISSION AND BROKERAGE. 113 61 Find the commission on : 4. 1260 . . . @ 5 % 5. 1274*17*8 .... 4 / 6. 375* 7*6 .... 2 7o 7. 840*11*6 . . 5 / 8. 375*15 .... @ 31 c 9. 509*10*6 4| c 10. 846*17*3 44 e 11. 723*11*6 4/6 c 12. 8467*10*6 13. 3176*13*4 Find the brokerage on : 14. 5260*12*6 . . @2/8/ 15. 324* 3*4 . . .. 7/3 / 16. A commission agent sells goods to the amount of 53610.. Find his commission @ 2i %. 17. A broker sells 50 shares of the Bank of Scotland, each 196. Find his brokerage @ i / . 18. A traveller for a sugar-house transacted business in a pro- vincial town to the following amount : Raw sugar, 620 ; crushed sugar, 547"10; refined sugar, 320/45; molasses, 200126. Find his commission @ 3 / . 19. An agent is allowed 5 / f r selling goods and guarantee- ing the debts to his employer. His sales in a year amount to 15,375" 106, and his losses to 375"42. Find his income. 20. An agent is allowed 5 % for selling goods and guaranteeing the debts. His sales amount to 13,756108; his bad debts to 200" 15 ; and his doubtful debts, amounting to 500" 16, are valued @ 12/6 tf- . Find his probable income. 21. An agent is allowed 5f / for sales and risk of debts. Sales amount to 15,246"10; debts, amounting to 609"15, are valued @ 10/6 q? . Find his probable income. 22. An invoice, containing an account of goods purchased, is sent by an agent to his employer. The price of goods is 409" 12"6; charges for packing, &c.,7"12"9; commission on the whole, @ 2* / . Find the amount of the invoice. 23. An agent sent to his employer in St Vincent's an account of ' the sales of 56 tierces of sugar, each 8 cwt. 3 qr. 16 Ib. average net weight, @ 62/^ cwt.; deducting commission @ 2$/ j duty, 15/ V cwt. ; freight, &c., 180"12"9. Find the net proceeds. 24. An agent, who is offered a commission of 5J / on amount of sales with risk of bad debts, or a commission of 3f / on amount of sales without any risk, accepts the former. The sales amount to 8500, and the bad debts to 147 "15. How much has he gained or lost by his choice ? 114 INSUEANCE. 62^ NSURANCE * s a contrac t by which a company engages to in- demnify the value of property against loss. The owner, whose property is insured, pays to the Insur- ance Company a certain per-centage or Premium on the sum insured, on which a Government Duty also is chargeable. The deed of contract between the Insurance Company and the owner of the property is termed the Policy of Insurance. (1) Find the expense of insuring a cargo valued at 525* 12//6; premium, 2 guineas /o j duty, 3/ / 5 commission to agent for effecting the insurance, | c / - fcl 525*12*6 4- 525*12*6 600 @ 3/ 7 = 18/ 2 2,62*16*3 90 When the sum insured is not a multiple of 100, 12,56 the duty is charged on 1051* 5*0 52*11*3 11,03*16*3 12 the next greater multiple. 20 6,75 ,76 4 12 T 9,15 Premium, . 11* 0*9 Commission 2*12 * 6| Duty, . . 0*18 * 14*11*3] 1. Find the premium on insuring an hospital for 3400 @ 3/6 / . 2. Find the premium on insuring farm stock for 530 @ 2/6 %. 3. Find the expense of insuring household property to the amount of 469" 10 @ 1/6 / ; duty, 3/ %. 4. What was paid for insuring a house for 750 @ 2/6 % ; duty, 5. What was paid for insuring a cargo for 1250 @ 1"17"6 %; duty, 2//o? 6. An agent insures a cargo for 1370 @ 3 guineas % ; duty, 4/ / ; commission oil the sum insured @ / . What is the total expense ? 7. A house factor insures four houses for 560, 940, 420, and 780 respectively, @ 1/6 / ; duty, 3/ %. Find the expense. 8. Insured 3250 on a ship @ 3 % ; duty, 4/ / ; commission, \ /o- Find the expense. INSURANCE. U5 62. 9. An agent insures 4530 on a cargo 4 guineas % duty 4/ % ; commission, / . Find the expense. 10. A ship, worth 5500, had a cargo worth 2670. All the expenses connected with insuring the ship and the cargo to their full value amounted to 4"18 / . How much was paid? (2) Find what sum must be insured on property worth 3846, so that, in case of total loss, the whole, including the expense of insurance, may be recovered. The ex- pense is premium, 3 gum. / ; comm?, 1%; duty,4/ / . 3// 3 10 4 100 3*17 = 96//3 : 100 : : 3846 : 4000 The expense of insuring 4000 @ 317 8 | = 154. The net sum thus recovered = 4000 154 = 3846. By insuring 4000 over all the expenses. 11. What sum must be insured to cover 1530, the expense of insuring being 4"7"6 "/.? 12. How much must be insured to cover 3890; premium, 2 guin. /. ; commn ,"/,; duty, 3/ /o ? 1.;. I I>\v much must be insured to cover 5005; premium, 3"! "/.; commn, J | ; duty, 4//? 14. What sum must be insured to cover 429, all the expenses connected with the insurance being 2 "10 / ? 15. A cargo is worth 2442, and the expense of insuring it amounts to 217"6/ . What must be insured to cover the value? INTEBEST. 63JNTEREST is a per-centage charged for the loan of money. The money lent is termed the Principal, and the sum of the Principal and the Interest is termed the Amount. (1) Find the interest on 280//13//6 for 1 year @ 3J % & annum. 100 : 280//13//6 : : 3* : x 280*13*6 X 3 ~~ The Interest on a sum for 1 year = Principal X Rate % .1- 100. For conciseness, we may use the Initials in a formula. 116 INTEREST. 63. i = P.XJ* 100 280*13*6 280-675 3 -035 140* 6//9 1403375 842* 0//6 842025 9,82// 7//3 9-823625 20 16,47 12 9*16*5J JJ = 9*16*5! nearly. 5,67 _4 2,68 Find the interest for 1 year on 1. 320 @ 3/ 2. 647//15//6 .. 4 % 3. 802*11*6 @ 3i / p* ann. 4. 772*16*9 .. 4} % .. (2) Find the Int. on 567//5//6 for 7 yr. @ 4J. % V ann. Prin. 100 : 567//5//6 ) ~. 1 Int Yr. 1 : 7 } : : 4 ^ : * 567//5//6 X 7 X 4J 100 Int. on a sum for a number of years = Principal X NO of Years X Rate % -f- 100. y P X Y X R 100 567// 5//6 567-275 7 -315 3970//18//6 1701825 4J 8509125 1985* 9//3 178-691625 15883*14*0 178,69* 3//3 20 13,83 178//13//9| { | = 178*13*10 nearly. 12 9,99 4 INTEREST. Find the Int. on 8. 564,13H **,o .. o .. .. 3 v / 9 361,14 fi 2"17,6.. 4 .. .. 2J% 10.' 874*18*8 .'.' 8 '.'. '.'. 2j/. (3) Find the Int. on 321,15,4' for 2 yr. 5 mo. @ 3 J l 5. 750 for 7 \ 6. 216, 4,6 .. 5 } .:* :ii': 321,15,4^ 2 4 mo. 1 yr. 1 mo. I 4mo. 643,10,9 107, 5,U 321-769 2. 12)1608845_ 134070~ 643538 - 7// 2332824 100) 2527-226 w o^ii^ 25-27226 = 25,5,5^ Find the Int. on 11. 374,17,3 for5mo.@3r/ 14.876,14,6^..2y.3m.@27 8 12. 769,13,3 .. 8 13. 467, 2//4J.. 5 .. 47 ... 8 15. 723//16,3|..3y.llm...3i% 16. 846,12,6 ..2y.7m...57 (4) Find the Int. on220,4//7 from April 1 to Sept. 11, (Qf * /o Da. , 4 Prin. 100 : 220,4,7 ) Da. 365 : 163 f J ___ 220//4*7X 163X4, 36500 > or, with a more convenient divisor, __ 220//4//7 X . 31 Of) 163 Int. on a sum for a number of days = Principal X N 2J %, and to July 16 @ 2 / . On examining the process in (8), we see that 200X1020 = 200X102x2x5 73000 73000 = Interest on 200 for 102 da. @ 5 /.. As rates of interest may be reduced to 5 / , we May 1 1 may consider the following plan on which Interest 12 . Tables used in some banks have been constructed. 13 f Let a sum be deposited on May 11, when Int. is 54 ' at 3 / . By writing J or '6 opposite May 12, add- ing -6 continuously till the rate changes, say on May 15, to 2 / , and then adding jr- or '5 continuously, 17 . we can at once see how many days @five / will pro- 18 . duce the required interest. Int. on 200 from May 11 to May 15 @ 3/ , and to May 18 @ 2J /, 200 X (4 X 6 -f- 3 X 5) _ 200 X 39 __ 200 X 3-9 X 10 73000 ~" 73000 ~" 73000 Int. on 200 @ 5 / for 3'9 days as given in the table. (9) What Principal will produce 210 of Interest in 5 years @4'/ ? Prin. . x _ ^>4L = 1050. 1-2 1-8 2-4 2-9 3-4 3-9 For Years : P = g For Days: P = 52 . What principal will produce 384 of Interest in 6 years @ 4/ ? 53. What principal will produce 153 of Interest in 4J years @ 4A / ? 54. Find the principal of which the Interest for 50 days @ 4 / is 14" 12. (10) What Principal will amount to 1260 in 5 years Int. on 100 for 5 yr. @ 4/ Amount of 100 12~12 6 : : 100", = 124 INTEREST. 63* 55. Find the principal which in 4 A yrs. @ 4 / will amount to 962. 56. What principal will amount to 1017" 15 in 6 yrs. @ 3 / ? 57. What principal lent from March 10 to May 22 @ 5 % will amount to 712 "9"5? (11) At what rate must 730 be lent for 95 days to amount to739//10? Prin. 730 : 100 ) Jj nt ; ft , SGSOOXO* . Da. 95 : 365 } ' ' *' 9 " 10 : x = 736x95 = 5 ' For r,: R = 58. At what rate must 424 be lent for 2 yrs. to produce 26" 10 of Interest ? 59. At what rate must 255" 10 be lent from April 1 to June 20, to produce 2 "16 of Interest? (12) Lent 1825 @ 3/ OJ when will 10// 13 of Int. be due? Prin. 1825: 100 \ J^' 36500 x 213 71 ~ Int. 3 :lO//13f : : 365 : x== i825xeo = 71 For Years: Y = For Days: D = 60. How long must 670 be lent to produce 134 @ 5 %. 61. How long must 91 "5 be lent to produce 2 of Int. @ 5 %? 62. Lent 511 on Jan. 1, 1856, @ 4J/ , when will it amount to 517*13? 64 ' DISCOUNT. DISCOUNT is a per-centage charged for the payment of money before it is due. 200. London, March 15, 1858. Three months after date, I promise to pay to Mr William Jones, or order, Two hundred pounds for value re- ceived. James Brown. II. 200. I London, March 15, 1858. Three months after ^ date pay to me or order. Two hundred pounds for value | received. To Mr James Brown. ^ William Jones. DISCOUNT. 125 64* No ' I ' is the form of a Promissory Note, in which Mr James Brown promises to pay 200 in 3 months after the given date. No. II. is the form of an Inland Sill, drawn by Mr William Jones and sent to Mr James Brown, who on accepting it writes his name across the bill, and becomes bound to pay 200 in 3 months after the given date. If Mr Jones who holds the note or the bill cashes it at the bank before it is due, as on April 19, the bank charges discount for ad- vancing the money. The value of a bill when it is discounted is termed its Present Value. The value of a bill when it becomes due is termed its Future Value. We may compare the Present Value and the Future Value to Ready Money and Credit Price. Goods which may be had on credit for a certain sum may be bought for less ready money. The Credit Price is the Present Value of the goods increased by Interest ; the Keady Money is the Future Value diminished by Discount. The Bank or Common Discount is the Interest on the Future Value of the bill. The True Discount is the Interest on the Present Value of the bill. The Present Value lent out when the bill is discounted amounts to the Future Value when the bill becomes due. The True Discount is the difference between the Future Value and the Present Value. Common Discount (C. D.) = Int. on Future Value (F. V.) e Discount (T. D.) = Int. on Present Value (P Hence, C. D. - T. D. = Int. on (F. V. - P. V.) But, F. V. P. V. = T. D. Hence, C. D. T. D. = Int. on T. D. The difference between the Common and the True Discount on a bill is = the Interest on the True Discount. In Great Britain and Ireland, Three Days of Grace are given on all bills except those drawn " at sight," which are payable on presentation. When a bill, running for a number of months, and dated on the 31st of a month, becomes due in a month having fewer than 31 days, it is nominally due on the last day of the month, and legally due on the third of next month. Find the Common and the True Discount on a bill for 200 drawn March 15, 1858, at 3 months; discounted April 19, @ 4 / . Nominally due, June 15 From April 19 to June 18 = 60 days. Legally due, June 18 Amount or Future Value . Common Discount, or Int. on 200 I . . I// 6//3 for 60 days @4/, .... Not Proceeds 126 DISCOUNT. 64. Int. on 100 for 60 days Future Value. Present Value. 1004 : 200 : : 100 : x = 198*13*10j|fJ| If we wish the answer correct within a farthing, we may express the fraction decimally, and use contracted division. 100-6575 : 200 : : 100 : x = 198'693 = 198*1310J. Amount or Future Value 200 True Net Proceeds or Present Value . True Discount I// 6// Proof. True Discount l Ib. so as to gain 33 / a ? 28. Find the prime cost of coffee $ cwt. sold @ 1/10 ^ Ib. with a profit of 10 / e . 29. What was the prime cost of goods sold for 26"5 with a loss of 12/ ? 30. Bought 7 cwt. 3 qr. Java rice for 4"105. How must it be sold y Ib. to gain 20 | ? 31. Find the prime cost of a work of 10 vols, sold @ 10/6 ^ vol. with a profit of 16 , . 32. A contractor gains 16 / by performing a piece of work for 233"! 9"5. What is his outlay for workmanship and materials? 33. A paper merchant bought 100 reams of foolscap, and sold 50 reams @ 1"5, with a gain of ll / ; 25 reams @ l>-8 ; and the remainder, being damaged, @ 17/8. Find the total prime cost, and the gain or loss / . 34. Find the weekly outlay of the proprietor of an omnibus who receives on an average 315"3 every lawful day, and thus clears 75 %. 35. At what price must cloth bought @ 5/6 sp- yd. be rated so as to allow 4 /<, discount for ready money and gain 9 T ' T e /o by the money received ? 36. Suppose a bootmaker pays on an average 6/4 for the leather and furnishings of a pair of boots, and 6/4 for the workmanship ; what must he charge his customer so as to allow him a discount of 5 e !o, and gain 50 / by the money received ? (9) Sold goods for 225//10 with a gain of 12} % What would have been gained or lost 7 by selling them for 187*10? 100 225//10 : 187*10 : : 112} : x = 93} ~6j Loss % (10) Sold a bale of leather for 14*14, and gained 17| c / . How should it have been sold to have gained 18 / ? 117| : 118 : : 14/14 : x r= 14*15 S. P. PROFIT AND LOSS. 135 67. 37 AjDookseller having bought two copies of the seventh edition of the Encyclopaedia Britannica at the same price, sold one @ 25 with a profit of 9j' T "/.. How much did he gain |. by selling the other @2710? 38. A merchant of Lyons by selling silk @ 10 francs # metre gained 20 "/.. What did he lose % by selling silk of the same prime cost @ 8 francs ^ metre? 39. Lost 36 / by selling cloves @ 8d. ^ Ib. What would have been gained or lost / by selling them @ 1 d. ^ oz. ? 40. Gained 13 / by selling paper @ 9/5 y ream. What was lost / by selling paper. of the same value @ 8/3 y ream? 41. Sold a bale of leather for 15, and lost 25 e /.. How should it have been sold to have gained 33 / ? 42. Sold pencils at the rate of 3 for 2d., and gained 33 / . What would have been gained or lost / by selling them @ 5d. q? doz. ? 43. A bootmaker by selling boots @ 24/ ^ pair gains 50 / . What must he have charged to have given a discount of 5 / , and to have gained 78& / ? (11) Find the prime cost and selling price of goods sold with a gain of 32 / e , and of 16/'17/'4 in all. 32 : 100 : : 16*17*4 : x = 52*14*2 P. C. 16*17*4 Gain. 69*11*6 S. P. 44. Sold goods with a loss of 20 / , and lost 57"6"8 by the transaction. What was the prime cost? 45. Find the selling price of goods by which there was a loss of 2 / or of 54" 10 by the whole transaction. 46. What does a draper receive for 39 yd. of cloth which he sells with a gain of 2/ $x yd. and of 26 / ? 47. Sold cheese with a gain of 2^d. $> Ib. or of 62 %. At what was it bought and sold ^ cwt. ? 48. Sold 39 casks of cod-liver oil, each containing 52 gallons, with a loss of 1 / , and of 8107i on the transaction. What was the prime cost $>- gallon ? 49. Find the original outlay of a publisher who sold 2000 copies of a guide-book, with a gain of 6d. ^ copy and of 25 %. 50. Find the outlay of a publisher who sells 500 prints of an en- graving with a gain of 5/6 f print and of 35 ^f I . (12) How much sugar, bought @ 2*13*8 V> cwt. was sold @ 5d. W Ib., with a total loss of 3*18*9 < 136 PROFIT AND LOSS. 67. 2*13*8 V cwt. = 5fd. P. C. & Ib. 3*18*9 5_d. S. P. * J78 S< id. Loss. * 945^ 3)3780 3780f. "I2601b. = llcwt. Iqr. (13) How many prints of an engraving must a picture - dealer sell @ l//ll//6, so that he may gain 5l / on an outlay of 250 ? 100 : 151J : : 250 : x = 378 S. P. 1*11*6 378 31 s. 7560 s. 63 sixd. )15120 sixd. (240 prints. 51. Bought a cargo of oranges @ 12/6 ^ chest, and sold it with a gain of 30 / , and of 18" 15 in all. How many chests were in the cargo ? 52. How many yd. of cloth bought @ 13/2J ^yd. must a draper sell @ 16/6 to gain 3"19"6? 53. What quantity of butter bought @ 2138 y cwt. must be sold @ ?id. ^ Ib. to clear 418 ? 54. Bought haddocks @ 3/4 y long hundred (120). How many must be sold at 7d. ^ dozen to gain 12/6? 55. How much sugar bought @ 42/ ^ cwt. must be sold @ 6d. ^lb. to gain 20 in all? 56. Bought 10 cwt. of sugar @ 44/ ^ cwt., and sold it at 4^d. ^ Ib. . How much tea bought @ 3/1 y Ib. must be sold @. 4/4 ^ Ib. to cover the loss on the sugar ? 57. Sold iron @ 5 "6 y ton, with a profit of 6 / , and of 21 106 in all. What quantity was sold ? 58. A drysalter purchases goods @ 58/4 f cwt., and by retailing them gains 2 "17 "6, being at the rate of 4 / . What quantity was sold ? 59. A grocer buys sugar @ 37/4 $> cwt., and by selling it @ 62 ^ % profit gains 5"5"5. What quantity does he sell ? 60. Bought a cargo of oranges @ 15/ sp- chest, and sold one-half of them @ 19/6 ^ chest, and the other with a loss of 10 %, but gained 27"76 on the whole. How many chests were bought? (14) Bought goods for 53, and sold them for 75, with one year's credit. What was gained / ? Let us first find the Present Value of 75, reckoning the rate of interest here and in all the following examples at Five per cent. PROFIT AND LOSS. 137 105 = Future Value of 100 in 1 yr. @ 5 / . 105 : 100 : : 75 : x = 71* P. V. of S.P. The question is now reduced to the following : Bought goods for 53, and sold them for 7 If / ; what was gained / ? 53 : 71f : : 100 : x = 134f S. P. Gain / = 34f f . These two statements may be united as follows : 105 : 1001 . . 1on . 53 : 75 f ' ' 10 ' x (15) How must cloth, bought @ 6/9 ^yd., with 3 months' credit, be sold so as to gain 5 / , and allow 9 months' credit. F. V. of 100 @ 5 / for 3 mo. and 9 mo. = 101J and 103J. 10H : 103| ) s. a. s. a. 100 : 105 | : : 6*9 : x = 7*3 T V S. P. 61. Bought goods for 59, and sold them for 89 with one years credit ; what was gained / ? G2. What was lost by selling 288 yards of cloth for 182 8, bought 6 months ago @ 12/6 ^ yd. ? 63. Bought goods for 70, and sold them for 70 guineas with twelve months' credit ; what was gained or lost % ? G4. I low must goods be sold to gain 5 %, and give 9 months' credit, bought the same day for 81 with 3 months' credit? 65. What is gained or lost % by selling goods @ 47*13 "4 y cwt. bought 6 months ago @ 8/ ^ lb. ? 66. What is lost % by selling goods with 6 months' credit, bought 6 months ago for the same money ? 68. DISTRIBUTIVE PROPORTION. IN DISTRIBUTIVE PROPORTION we divide or distribute a given number into parts which have a given ratio to each other. (I) Divide 376//5 of gain among three partners in an ad- venture whose risks are respectively 225, 150, and 250. 225 150 250 625 225 376//5 : x = 135* 9 625 625 150 250 37G//5 376//5 : x = : x = 90 v 6 150//10 376* 6 138 DISTRIBUTIVE PROPORTION. 68 ^ ie sum ^ ^ e r i s ks 625. As the whole risk is to each risk, 50 is the sum to be divided to the share of each. The sum is thus divided into parts proportional to 225, 150, and 250, which may be cancelled by their common factor 25. The following method is often convenient : 225 150 250 * = 9 X 15*1 = 135* 9 6 6 X 15*1 = 90* 6 10 10 X 15"! = 150*10 25)376"5(15*1 376* 5 (2) A sum of 1000 was bequeathed to four relations, and by an inadvertency in the will, it was stated that they were to receive J, , J, and of the sum respectively. How much should each receive according to the spirit of the will? 6 6 X 66*13*4 = 400 4 4X 66*13*4= 266*13*4 3 3 X 66*13*4 = 200 2 2 X 66*13*4 = 133* 6*8 I5)1000(66*13*4 1000 We divide 1000 in the mutual ratios of , J, , . The sum of these fractions = { | is greater than unity. T ' 5 is therefore one- fifteenth of the sum. Dividing 1000 by 15, we multiply by 6, 4, 3, 2, successively to obtain the respective shares. 1. Divide 84 into parts having the mutual ratios of 2, 3, 7. 2. Divide 1200 into parts having the mutual ratios of 11, 12, 13, 14. 3. Divide a line 4 feet long into parts having the ratios of the first four odd members. 4. Divide 100 into parts having the ratios of the cubes of the first three numbers. 5. Divide 390 into parts having the ratios of , , J. 6. Divide 1331 into parts having the ratios of the reciprocals of the first three even numbers. 7. Apportion a house tax of 618"8 among 3 joint proprietors, who pay in the proportion of the annual values of their properties, which are 30, 40, and 60 respectively. 8. A vessel is divided into 64 equal shares, of which A, B, C, D, have 6 shares each; E, 12; F, 16; Gr, 4 ; and H the remainder. Find their respective shares in sustaining a joint loss of 158" 10" 1. 9. Divide a profit of 689 among 3 partners, of whom the first owns T 2 3 of the joint stock and the second T 5 3 . 10. A, B, C, D, invest 450, 230, 190, and 110 respectively DISTRIBUTIVE PROPORTION. 139 * n a speculation. Find their respective liabilities in a joint loss of 31312. 11. Three partners respectively claim ^, {|, and , s of the gain of an adventure amounting to 1260. Give to each a proportionate share. 12. Divide 5 guineas among George, James, and Henry, who respectively claim , *, and , so that they may have proportionate shares. 13. An analysis of the manure of dissolved bones gives the following results for every 100 parts: Water, 13*97; Organic Matter, 15-71 ; Soluble Phosphates, 21'63 ; Insoluble Phosphates, 11-43; Sulphate of Lime, 15'83; Sulphuric Acid, 15'63; AJj^jline Salts, 1-10; Silica, &c., the remainder. Find the weight of each in a ton of dissolved bones. 14. Oil of vitriol (HO, S0 3 ) contains by weight, 1 of Hydrogen, 32 of Oxygen, and 32 of Sulphur. Find the weight of each in a gallon of oil of vitriol which weighs 18| Ib. (3) D, E, and F, gain 564 : D's capital of 300 has been in trade for 6 months ; E's, which is 400, for 3 mo. ; Fs, which is 500, for 2 mo. Find the share of each. D, 300X6=1800,9 9x28//4=253//16 E, 400X3=12006 6x28//4= 169// 4 F, 500X2=100015 5x28//4= 141 20)564_ 564 28//4 The use of 300 in trade for 6 mo. is equivalent to that of 6 times 300 for 1 mo. Similarly, 400 for 3 mo. is equivalent to 3 times 400 for 1 mo. ; and 500 for 2 mo. to 2 times 500 for 1 mo. Taking the time of 1 month alike for D, E, F, we see that the shares are proportional to 1800, 1200, and 1000. (4) A commences trade with 3000 : in 3 months B joins him with 4000 ; at the end of the next 2 months A takes out 1000 ; in 1 mo. after C joins them with 2000, and B adds 1500; in 2 mo. after C takes out 500: at the end of 12 months they divide 2760 of gam. What is the share of each ? A has 3000 in trade for 5 mo., and 2000 for 7 mo. B // 4000 " 3 // and 5500 // 6 C // 2000 " 2 // and 1500 // 4 * . ( 3000 X 5 = 1 5000 A { 2000 X 7 = 14000 R (4000 X 3 = 12000 1 45000 B ! 5500 X 6 = 33000 f 4 ' 140 DISTRIBUTIVE PROPORTION. 68. c | 2000 x 2 = 4000) 10000 \ 1500 X 4 = 6000 f 10 ' 00 29 X 32'/17//lH = 952//17//l^ 45 X 32*17*1 J$ = 1478*11*5? J 10 X 32//17//lf = 328*11*58$ 84)2760 2760* 0//0 32*17*1$ * 15. In a copartnery, A's capital of 400 has continued for 9 mo. ; B's of 350 for 8 mo. ; C's of 600 for 2 mo. Divide 570 of gain among them. 16. Three cattle-dealers rent a field of 9 acres @ 5 ^ acre: A puts in 6 cows for 2 months ; B, 9 cows for 1 mo. ; C, 12 cows for 3 mo. How much does each pay ? , , 17. At the end of 12 months, D, E, F, having a joint capital of 6000, find that they have lost 625. D's capital of 2500 has been in trade for 12 mo., E's of 1500 for 8 mo., and F's for 4 mo. What is the loss of each? 18. A and B enter into partnership, the former with 1800, the latter with 900 : in 8 months B adds 300 to his capital. Divide a profit of 840 between them at the end of 12 months. 19. A has 300 in trade for 7 months, when B joins him with 400. At the end of the next 3 months C joins them with 300. Divide 549 of gain among them after 18 months' trade. 20. A, B, and C, enter into partnership on Jan. 1, 1856, with a capital of 1000 each. On April 30, B withdraws 400, and C makes up the sum. On Aug. 28, A withdraws 200, and C makes up the sum. On balancing their books for the year they find they have a gain of 365. What is the share of each ? 21. Three graziers rent a field from May 11 to October 19, 1857, for 43. A agrees to pay 13 for grazing 12 oxen; B, 18 for 18 oxen; and C the remainder for 20 oxen. To how many days is each grazier entitled ; and if the oxen go into the field in the order A, B, C, on what days do B's and C's severally enter? 38/, and 17 @ 50/. 5. On Feb. 6, 1856, the following quantities ol wheat were sold at the six highest prices in the Edinburgh Grain Market: 8 quarters @ 96/; 4 @ 84/; 21 @ 78/; 13 @ 76/; 1 @ 75/; 2 @ 74/. Find the average price ^ qr. as deduced from these prices and quantities. In Alligation Alternate, we find the proportional quantities of ingredients of given prices which will produce a compound of a given price. (2) Mix spirits @ 8/3, 7/9, 6/6, and 8/4 ^ gal. so that the compound may be worth 8/ ^ gal. 142 ALLIGATION. I. d. gal. f 78, 3 X 78 = 234 qr \ 93 -, 4 X 93 = 372 99_l is x 99 = 1782 1 100 ! _3 X 100 = 300 28 28) 2688 96 n. d. gal. f 78 -i 4 X 78 = 312 n J 93-] 3 X 93 = 279 J0 1 99 J 3 X 99 = 297 [ 100 J 18 X 100 = 1800 28 28)2688 96 We express the prices in the same name. To obtain a compound at 96d. we must mix two ingredients, of which the one is dearer and the other cheaper than the com- pound. We may, as in Method L, connect 78d. with 99d., and 93d. with lOOd. The act of thus connecting or binding the prices together is the reason why the rule is termed Alligation. If spirits worth 99d. qp gal. are sold @ 96d. there will be a loss of 3d., and if spirits worth 78d. are sold @ 96d. there will be a gain of 18d. Since 18 X 3d. = 3 X 18d., the loss on 18 gallons worth 99d. will balance the gain on 3 gallons worth 78d. We therefore write the difference between 96 and 78 or 18 opposite its alternate number 99 ; and the difference between 99 and 96 or 3 opposite its alternate number 78. We proceed similarly with 93d. and lOOd. In Method II. we may connect 99d. with 93d. and 78d. with lOOd. When the differences between the price of the compound and that of a dearer and of a cheaper ingredient connected together are equal) we may take any equal quantity of each of the latter ; thus, instead of 3, 3, 4, 18, we may take a:, a, 4, 18, where x may be any quantity. 6. Find the proportional quantities of sugar @ 5d. and 8d. that must be sold to make the average price 7d. ^ Ib. 7. What proportional quantities of potatoes @ 2/, 3/, and 3/6 # bushel must be sold to make the average price 2/9 y bushel ? 8. Mix tea @ 4/6, 4/2, 3/4, and 3/9 y Ib., so that the compound may be worth 3/1 1 y Ib. 9. What proportional quantities of wine @ 15/, 12/, 18/, 19/, and 21 1 y gal. must be sold to make the average price 16/ ^ gal. ? (3) What quantities of tea @ 5/3, 4/5, and 2/9, must be mixed with 21 Ib. @ 6/1, to make the whole worth 5/ mr\, TU Q 69. f33-j 3X3 = Ib 9 33d. 53 1 ALLIGATION. 143 3X3 = Ib 9 33d. 13 X 3 = 39 .. 53d. 60 -, X = 39 .. 53d. OU 1 63-J 27 X 3 = 81 .. 63d. 173 J 7 . . . . . 21 .. 73d. [33-, -, 13 + 3= 16 X fi = 9|f @ 33d. W|S H =13Xfi= 8 A .. 53d. -J 27 = 27Xfi = 16}| .. 63d. [73- J 27 +7 = 34 . . . . 21 .. 73d. Having found the proportional quantities as formerly, we multi- ply them by the ratio of the given quantity to its corresponding proportional quantity. Similarly, when the quantity of the compound is given, we mul- tiply the proportional quantities by the ratio of the given quantity to the sum of the proportional quantities. 10. How much wheat @ 42/ and 56/ must be sold with 13 qr. of wheat @ GO/ to make the average price 50/ y qr. ? 11. How much sugar @ lOd. and! Id. must be mixed with 9 Ib. of 7d. sugar to make the whole worth 8d. ? 1 2. How many gallons of water must be mixed with 63 gallons of spirits @ 8/ so that the prime cost may be 7/ f gal. ? ^* We alligate Sj with 0. Or we may solve this by proportion, s. s. gal. pal. 7 : 8 : : 63 : 72. .-. Number of gal. of water = 72 63. 13. How many gallons of water must be mixed with 47j gallons of spirits @ 6/3 to make the prime cost 5/ ^ gal. ? 14. How many gallons of each kind of wine @ 15/3, 16/4, 17/2, and 18/1, must be sold to make the average price of 154 gallons 17/fgal.? 15. The Specific Gravity of an alloy of gold and copper is 16-65, while that of gold is 19-2, and that of copper 9. Find the weight of gold and copper in 144 oz. of the alloy. 16. A crown made of gold and silver weighs 150 oz. and displaces 13-824 cub. in. of water. Had it been gold it would have displaced 12-96 cub. in. of water, and had it been silver it would have displaced 23-04 cub. in. Find the weight of gold and silver in the crown. &F This question is founded on the story of Archimedes and Hiero. Hiero had given a goldsmith a certain quantity of gold to make a crown. In course of time, the artificer presented a crown of the same weight as that of the quantity of gold ; but as Hiero suspected a fraud, he requested Archimedes to discover if any baser metal had been alloyed with the gold. Archimedes considered that if the crown contained any metal lighter than gold, it would be larger than a pure gold crown of the same weight. Having o tained a mass of pure gold and of the other metal, each of the same weight as the crown, he found the quantity of water which each ol the three displaced, and from these data discovered the proportion of each metal in the crown. 144 7O. BAETEE. IN BARTER, two parties mutually give goods of equal value in exchange. (1) Exchanged 164 Ib. of tea @ 4/8 W Ib. for coffee @ 1/7 ^ Ib. How many Ib. of coffee were received? x Ib. of coffee @ 1/7 = 164 Ib. of tea @ 4/8. (2) In return for 146 qr. wheat @ 70/ ^ qr., an agent re- ceived Wilts cheese @ 88/ ^ cwt., and Dunlop cheese @ GO/ ^ cwt., obtaining 6 cwt. of Wilts for every 5 of Dunlop. How many cwt. of each were received ? 6 X 8*8 = 528 5 X GO = 300 828s. = the price of 1 parcel of both kinds of cheese in the given proportional quantities. x parcels @ 828s. = 146 qr. @ 70/ = 10220s. o - 146 X 70 - 10220 1971 T^orpplc x ~828 ~T* -- lz aST parcels. Each parcel contains 6 cwt. of Wilts and 5 cwt. of Dunlop. 6 X 12^0-V = 7 4 A cwt. = 74//0// 6|J Wilts. 5 X 12^V = 6 HSf cwt - = 61//2//24 5 W Dunlop. Proof 4 74 A cwt ' @ 88/ = roof ^ 61 ^ cwt @ 6Q/ = 3702 6 . s> 10220s. 1. How many yd. of cloth @ 2/3 are worth 54 Ib. of tea @ 4/1? 2. What is the price ^ yd. of cloth, of which 200 yd. are worth 2 cwt. 2 qr. 25 Ib. @ 93/4 y cwt. ? 3. How many gallons of brandy @ 24/6 ^ gal. are worth 35 doz. loaves of refined sugar, each 16 Ib. @ 70/ ^ cwt. ? 4. Exchanged a tierce of sugar weighing 8 cwt. 3 qr. 14 Ib. for 31 cwt. qr. 7 Ib. rice @ 18/ y cwt. Find the price of the sugar ^lb. 5. How many yd. of linen cambric @ 5/6 must be given in ex- change for 15 dozen pairs of boots @ 18/ y pair, and 13 dozen pairs of shoes @ 8/ y pair ? BARTER. 145 7O* 6. ^ baker, who has run an account with a grocer for 12 Ib. tea @ 4/2, 60 Ib. sugar @ 6d., 3 Ib. coffee @ 1/8, and 13 drums of sultana raisins, each 20 Ib., @ lid. f Ib., has a contra-account of 23 dozen loaves @ 7d. ^ loaf. How many loaves @ 8d. will settle the account ? 7. A dairyman, who has supplied a baker with 90 pints of milk @ 2d., 13 pints of cream @ 10d., and 80 Ib. of butter @ 10d., agrees to take an equal number of loaves @ 7d. and 7d. How many of each does he get ? 8. Exchanged 28 Ib. of tea @ 4/2 for coffee, and got 5 Ib. of coffee for 2 Ib. of tea. How many Ib. of coffee were got, and what was its price ^ Ib. ? 9. In return for 80 qr. barley @ 56/ ^ qr., of the value was re- ceived in bone-dust @ 88 & ton, and the rest in money. How much money and how many tons of bone-dust were received ? 10. In return for 165 cwt. flour @ 15/ f cwt., an agent received 3 chests of tea, each 81 Ib., @ 4/4 ^ Ib., and 8 doz. loaves of re- fined sugar, each 19| Ib. What was sugar ^ Ib. ? 1 1. In return for 14 cwt. 2 qr. 20 Ib. Glo'ster cheese @ 77/f cwt. ; beef @ 8d. y Ib., and mutton @ 7d. f Ib., were received in the ratio of 7 Ib. of beef for every 3 Ib. of mutton. How much of each was received ? 12. Exchanged 6 cwt. 2 qr. 3 Ib. salmon @ 1/6 y Ib., 20 tur- bots @ 4/2, 16 dozen haddocks @ 4/6 & doz., and 15 pints of shrimps @ 6d., for 2 cows @ 9"13 each, 160 Ib. beef @ 7d., 240 Ib. pork @ 5d., and 80 pairs of fowls @ 3/9 ^ pair. How many Ib. of mutton @ 7d. must be given for the balance ? 71. CHAIN EULE. (1) IF 5 pheasants are worth 4 grouse; 5 grouse, 8 par- tridges ; 2 partridges, 5 snipes ; how many snipes may be had for 10 pheasants ? x snipes = 10 pheasants 5 pheasants = 4 grouse 5 grouse = 8 partrid; 2 partridges = 5 snipes Having arranged the pairs of equal values or equations, so that numbers of the same name are on different sides, wo examine the equations as follows . 146 CHAIN RULE. Snipes. Snipes. 1 partridge = f \ S[ ouse I- ?_XJJ<1 or 5 pheasants ) 2X5 or 8 5 P ge dgeS }= ^ 1 Peasant = j 1 grouse = |f 10 pheasants = ^ We see then that the number of snipes = 10 pheasants is obtained by dividing the product of the numbers on one side by the product of those on the other. N' 32 Imperial bushels or Linlithgow wheat firlots, and 16 LHnlithow wheat firlots = 11 Linlithgow barley firlots? 3. How many Scotch acrek= 100 Irish acres, 121 Irish acres = 196 Imperial "Stores, and 126\Imperial acres = 100 Scotch acres ? 4. 8 Scotch miles = 9 Imperial miles; 14 Imperial miles =11 Irish miles. How many Irish miles = 112 Scotch miles? rfgT The mutual ratiosTirtue preceding examples are convenient approximations. 5. 2 quarts of plums are worth 3 of pears ; 6 of pears = 5 of apples ; 8 of apples cost 2/4. Find the price of 3 quarts of plums. 6. In 1855, the mutual ratios of the weights of bales of cotton imported at Liverpool from the following places were as follow : 2 from Bombay = 3 from Egypt ; 9 Brazil = 4 United States ; 7 Brazil = 5 Egypt ; 7 Calcutta = 5 Madras ; 14 United States = 15 Madras. How many from Calcutta were = 50 from Bombay? CHAIN RULE. 147 7. By examining the average weight of the bales of cotton im- ported at Liverpool in 1843, the following were obtained : 55 from Egypt = 69 from W. Indies ; 35 from Alabama = 43 from the Upland U. States, from which 207 = 350 from Egypt; 91 from Alabama = 215 from Brazil, from which 27 = 13 from E. Indies. Ilbw many from W. Indies = 165 from E. Indies ? 8. From t^c Imperial averages for the week ending 30th April IS^jt^aj^eared that the price of 39 quarters of barley = that of 73 of oats ; 68 of barley = 73 of beans ; 27 of beans = 28 of pease ; 39 of wheat = 58 of rye, of which 153 = 143 of pease. How many quarters of wheat = 638 of oats ? 9. 4 talents were =#75 Ib. avoir., and each talent contained 3000 shekels. Find thd weight of a shekel in oz. avoir. 10. 273 quarters of irheat = 638 of oats, of which 73 = 39 of barley, sold @ 42/7 ^ quarter. Find the price of 1 quarter of wheat. 11. By a comparison of the apothecaries' grains of different countries, it was founfx that 17 German = 20 British; 85 German = 86 NeapolitaTTpS7 Spanish = 45 Austrian; and 185 Spanish = 172 Neapolitan. How many British = 90 Austrian? 12. A mile = 8|0 chains = 63360 inches; a chain = 100 links. How many inched are in a link ? 13. 176 Ib. troy = 144 Ib. avoir., each 7000 grains, of which 3608 = 1 Cologne mark. How many Cologne marks = 451 Ib. troy? 14. If a *aeire = 39-37079 in. be taken as 40)0 o 0>000 of the earth ' s circumference, how many miles are in the earth's circumference? 15. 4 nautical miles = a German mile; the earth's circumfer- ence contains 5400 German miles = 40,000,000 metres. How many feet are in a nautical mile ? 72. EXCHANGE. EXCHANGE is the method of changing the money of one country into that of another. The Par of Exchange is the real comparative value of the money of two countries, estimated by the weight and fineness of the coins. , f ., The Course of Exchange is the comparative value ot t money of two countries, which fluctuates according to tne circumstances of commerce. In Exchange, 1 is generally adopted as the unit ;of '^PJ" Thus the rar of exchange with France is 25 francs 22 centii ! When 1 is tSe unit, the equivalent i* foreign money vane, 148 EXCHANGE. in the course of exchange ; thus, 1 may be exchanged at one time for 24 fr. 30 c., and at another for 25 fr. 50 c. When a foreign coin is taken as the unit, the equivalent in sterling varies in the course of exchange ; thus, while the par with Naples is 39f d. ^ ducat, the exchange may at one time be 38d., and at another 40d. ^ ducat. CANADA. Accounts are kept in , s. D. Currency, of which 1, being taken as = 4 dollars of the Nominal value of 4/6 each, is = 18/ sterling. Hence the nominal par is 100 cur- rency = 90 sterling. But as the real average value of the dollar is 4/2, 1 currency = 16/8 sterling, and the real par is 108 currency = 90 sterling. The Nominal Par is taken as the standard, and a Premium is added to show the course of exchange. At a premium of 8 7 0) 108 currency=90 sterling. WEST INDIES. The old currencies are now superseded by sterling. Of the foreign coins in circulation, the principal are the dollar = 4/2, and the doubloon = 3//4. (1) How much sterling is = 327 currency, at a premium of9 c /o? B>j the Chain Rule. l09 r - e 327 9 e O * or- Ster ' * = 327Curr. 109 . td27 . . 190 . x or x = ^^ = 300 Ster. iuy (2) How much currency is = 8460 sterling, at a premium of91/o? *on terl c n Q 8 icn "? cy ' f Curr. z=8460Ster. 90 : 8460 : : 1091 : * or j S ter.90=109JCurr. = 10293 1. How much currency will an emigrant to Canada receive for 135"76 sterling, at a prem. of 8 / ? 2. An emigrant on arriving at Toronto changes 6 crowns, 7 hf.-crowns, 37 shillings, and 5 sixpences sterling, to currency, @ the rate of 15d. for I/ sterling. How much currency does he receive ? 3. How much sterling is = 324"2 3 currency, remitted from Montreal, at a premium of 8 % ? 4. An agent at Quebec wishes to remit to his employer in Lon- don 489"12"ld. To how much sterling will this be equal at a premium of 8 J / ? 5. How many dollars @ 4/2, or how many doubloons @ 3 "4, must a Jamaica merchant receive from his correspondent at Cuba, who Is due 320 ? EXCHANGE. 149 6. An agent changes 3000 dollars to sterling @ 4/2 ^ dollar, at Kingston, Jamaica, on embarking for Halifax, Nova Scotia, and on arriving changes the sterling to currency @ 8 / premium. How much currency does he receive ? UNITED STATES. Accounts are kept in dollars and cents. 10 cents* = 1 dime ; 10 dimes= I dollar* ($) ; 10 dollars = 1 eagle. The par of exchange, deduced from the gold coins, is $1 = 4/l nearly; from the silver coins, $1 = 4/2 J nearly. In custom-house valuations, $1 = 4/2. The nominal par of exchange is $1 = 4/6 ; hence, $40 = 9, or $100 = 22//10. We take the nominal par as the standard, and add a premium to $100 ; thus, at a premium of 9} / , $109 = 22*10. (3) How much sterling is in $1 11-55, at a premium of 9-| /o? ft ft 109-125 : 111-55 : : 22*10 : x = 23. (4) How many $ are in 2560, at a premium of 9 % ? ft 22*10 : 2560 : : 109 : x = $12401-77}. 7. How much sterling is in i$3390, remitted from New York, at a premium of 8 / ? 8. How much sterling is in ft994'25, remitted from Philadelphia, at a premium of 9& / ? 9. How many ft are received at Boston for 738, at a premium of 9 - - / ? 10. How many ft are in 7659, received at New Orleans, at a premium of 10 / ? The following Illustrative Processes may suffice for the rest of the Exercises : (5) Change 999//12 to florins at Vienna, at the rate of 'lO florins 50 kreuzers ^ 1. fl kr 1 ? 12. How many francs must be remitted from Brussels to pay a bill of 987146, @ 25 fr. 10 c. $* 1 ? 13. How much sterling must be remitted to Paris to settle an account of 9900 francs, @ 24 fr. 75 c. f 1 ? 14. Plow much sterling must be sent to Antwerp to be equivalent to 25663 fr. 75 c., @ 24 fr. 50 c. ^ 1 ? HOLLAND. 5 cents = 1 stiver ; 20 stivers or 100 cents = 1 florin or guilder = 1/8. Par of exchange, 12 florins = 1. 15. How many florins must be paid at Amsterdam in order to liquidate a debt of 1500"8 ; exch. 12 fl. 6 c. f 1 ? 16. A merchant at Gouda consigns cheese to the amount of 8993 florins to an agent in Scotland. How much sterling must the latter remit @ 1 1 fl. 50 c. ^ 1 ? SWITZERLAND. 10 rappen = 1 batz ; 10 batzen = 1 franc = 1/2 nearly. The French coinage is also used. 17. A London jeweller remits 701"126 to a watchmaker in Geneva @ 25 fr. 30 c. f 1. How many francs does the latter receive ? 18. A merchant of Geneva, on coming to Berne, changes 518 French to Swiss francs @ 148 French for 100 Swiss francs. How many Swiss francs does he receive ? AUSTRIA. 4 pfennings = 1 Icreuzer; 20 kreuzers = 1 zwanziger; 3 zwanzigers or 60 kreuzers = 1 florin = 2/0 nearly. Par of exchange, 9 fl. 50 kr. = 1. EXCHANGE. 151 72* 19 - How many florins will be received at Vienna for 786 "14"6, exch. 10 fl. 30 kr. ^ 1? 20. The sum of 19868 fl. is remitted from Augsburg in Bavaria, where the Austrian coinage is used. Find the value in sterling @ 10 fl. 45 kr. ^ 1. SOUTHERN GERMANY. 4 pfennings=l Jcreuzer ; 60 kreuzers = 1 florin = 1/8 nearly. Par of exchange, 120 fl. = 10. 21. How many florins will be received at Frankfort-on-the- Maine for 767, exch. 119J? 22. An agent at Munich remitted 2241 florins. Find the value in sterling, exch. 124|. PRUSSIA; HANOVER, &c. 12 pfennings = 1 groschen; 30 groschen = 1 thaler = 2/10f nearly. Par of exchange, 6 thai. 27 gr. = 1. 23. How many thalers may be had at Dantzic for 72615"6, exch. 6 th. 20 gr. ? 24. How much sterling is = 36473 th. 20 gr. remitted from Hanover, exch. 6 th. 10 gr.? Bremen. 6 schwaren = 1 grote ; 72 grotes = 1 rixdollar = 3/3i nearly. Par of exchange, 609 r. d. = 100. 25. How many r. d. may be had at Bremen for 575, exch. 608 ? 26. How much sterling is = 1517 r. d. 36 gr. remitted from Oldenburg, where the Bremen coinage is used, exch. 607 ? Hamburg; Lubec. l2 pfennings = 1 schilling ; 16 schil- lin __ i mar / Cf Money is distinguished into Banco and Currency. Banco is used in Hamburg in exchanges in whole- sale transactions and in Bank business. Currency consists of coins in circulation; the marks current of Hamburg, and Lubec are, from the latter, termed marks Lub. The agio or difference between banco and currency varies from 20 to Zbl Par of exchange, 13 ink. 10J sch, banco = 1 , 1 mark banco = 1/51 nearly; 1 mark current = 1/2* nearly. 27. How many mk. banco are = 876 ; 8, exch 13 mk * scl. ? 28. How much sterling will be received m London for 27783 mk. banco, remitted through the Bank of Hamburg, exch. 1, 12 2 9 ! C A merchant pays 6461 mk. cur, into the Bank of Hamburg How much banco is entered on the books, agio being 24i fci equivalent, agio 21 / ? 152 EXCHANGE. DENMARK. 16 shillings = 1 mark ; 6 marks = 1 Rigsbank dollar = 2/2 nearly. Par of exchange, 9 R. d. 10 sk. = 1. NORWAY. 24 shillings = 1 mark ; 5 marks = 1 species dollar = 4/5 nearly. Par of exchange, 4 sp. d. 53 sk. = 1. SWEDEN. 48 shillings = 1 rixdollar banco = 1/8 nearly. Exchanges are generally effected through Hamburg. 31. How many Rigsbank dollars are in 432, remitted to Copen- hagen; exch. 9 R. d. 10 sk. ^ 1 ? 32. How many species dollars are in 1050"! 0, remitted to the Bank of Norway at Trondheim ; exch. 4 sp. doll. 53 sk. ^ 1 ? 33. How much sterling is = 5300 species dollars, remitted through a branch of the Bank of Norway at Bergen ; exch. 4 sp. doll. 50 sk. ? 1 ? 34. How much sterling is = 740 rixdollars banco, sent from Stockholm; exch. 12 r. d. 16 sk. banco y 1 ? RUSSIA. 100 copecs = 1 ruble = 3/1 . 35. A British merchant sends 867" 14^6 to an agent at St Petersburg; what does the latter receive @ 3/l ^ ruble? 36. How much sterling must be remitted to Riga to discharge a bill of 1200 R 1 ? EXCHANGE. 153 72. 42. How many Austrian lire are = 3415, remitted to Venice ; exch. 48| d. for 6 Austrian lire ? 43. How much sterling is in 2300 lire nuove, remitted from Genoa ; exch. 25 1. 30 c. y 1 ? 44. How much sterling is in 4590 Tuscan lire, remitted from Florence ; exch. 30 1. 60 c. $* 1 ? ROMAN STATES. 10 bajocchi = Ipaolo ; 10 paoli = 1 scudo or crown = 4/2 nearly. Par of exchange, 48 paoli or pauls 45. On visiting Home, an Englishman changes 37 "12 "6. How many pauls does he receive at the rate of 47 pauls ^ 1 ? 46. How much sterling is = 3697 scudi, 68 baj., remitted from Ancona, at the rate of 48 pauls y 1 ? NAPLES. 100 grani = 1 ducat = 3/3J nearly. SICILY. 600 grani = 1 oncia = 10/3 nearly. 47. A merchant in Naples receives a bill from London to the amount of 861. To how many ducats is this equal; exch. 41d. y ducat? 48. How much sterling is = 846 oncie; exch. 123d. y oncia? TURKEY. 40paras=l^'^re=2d. ; about 120piast.=l. EGYPT. 40 paras = 1 piastre=2%d. ; // 100 // =1. 49. A traveller pays an interpreter at Constantinople the sura of 500 piastres. What is the value in sterling at 120 piastres f 1 ? 50. Change 125"! to piastres at Alexandria @ 97 \ piastres f 1 . GREECE. 100 lepta = 1 drachma = 8|d. nearly. Par of exchange, 28 dr. 15 Ip. = 1. 51. Find the difference in Sterling and in Greek money between 4416 and 1317 dr. 42 Ip., exch. at par. EAST INDIES. 12 pice = 1 anna ; 16 annas = 1 rupee = 1/10*. 52. A Calcutta merchant makes a payment of a lac or 100,00( rupees. Find the amount in sterling @ 1/10J. 53. Sent to Bombay goods worth 299"163. To how many rupees is this equivalent @ 1/1 each? CHINA. 1000 le or cash =. 1 leang or tael, reckoned by the East India Company @ 6/8. 720 taels = 1000 dollars of 4/9 i nearly. 54. How much sterling is = 5400 taels, paid at Canton, reck- oning them @ 6/6 each ? 55. A merchant of Hongkong sells goods to the amount of 846 *134. How many taels does he receive @ 6/8 each? G 2 154 EXCHANGE. I NDIRECT EXCHANGES between two countries are effected through the medium of another. It is seldom that the medium is effected through more than one intermediate place. (9) How much sterling must be paid in London to pay 749 Rigsbank dollars in Copenhagen through the me- dium of Hamburg; exch. 13 mk. 6 sch. banco = 1; 200 R. d. = 300 mk. banco. x = 749 R. d. R. d. 200 = 300 mk. b. mk. b. 1 = 16 sch. sch. 214 = 1 x == 749 X 300 X 16 __. g^ 56. How many francs = 250, sent to Paris through Hamburg ; exch. 13 mk. 14 sch. banco = 1 ; 185 fr. = 100 mk. ? 57. Find the number of mk. curr. = 180, remitted through Hamburg ; exch. 13 mk. 12 sch. banco = 1 ; agio 20 / . 58. How much sterling must be remitted to Berne through Paris to be equivalent to 6325 Swiss francs ; exch. 25 fr. 30 c. *p 1 ; and 148 French = 100 Swiss francs? 59. How much sterling is = 60,180 paras; exch. between Con- stantinople and Vienna, 210 paras =r 1 florin ; between Vienna and London, 9 fl. 50 kr. = 1 ? 60. How much sterling is = 530 th. 7 gr. ; exchange between Berlin and Paris, 3 fr. 60 c. ^ 1 th. ; between Paris and London, 25 fr. 20 c. y 1? 73. INVOLUTION. INVOLUTION is the continued multiplication of a number by itself. The continued product thus obtained is termed a Power of the given number ; and the number of times the number is used as a factor denotes the Index of the power. Thus 2X2X2X2X2X2 = 64 = sixth power of 2 = 2 6 . (1) Find the seventh power of 27. 27 X 27 X 27 X 27 X 27 X 27 X 27 = 10,460,353,203 Instead of multiplying by the number successively, we may use those powers of which the sum of the indices is equal to the index of the required power ; thus, INVOLUTION. 73. 155 27X27X27= .... 19,683 .... - 2 7' 19683 X 19683 = 387,420 489 = 27X2?i - 27 387420489X27 = 10,460,353,203 = 27X27 = 27' Find the following powers : 1. 17 3 2. 32* 3. 36 4. 5. 6. 98* 99 5 101 7. 8. 9. II 7 158 149 10. 13' 11. 309 5 12. 1002* (2) Find the sixth power of T 3 r . 3- = 729 ( T 3 T )e = 11 = 1771561 13. 14. (I) 3 (A) 4 15. 16. (*) (U) a 17. 18. (f) 8 (A) 5 19. 20. (*)' (fl) 4 (3) Find the 5th power of 1-025 true to 6 decimal places (see 39.) 1-025 XI '025 = 1-050625 l-050625xl'025 = 1-076891 1-076891 X 1-050625 = 1-131408 21. 1-04* to 4 pi, 22. 1-05* .. 6 .. 23. 1-03' .. 4 .. 24. 1-025 6 to 6 pi. 25. 1-045' .. 7 .. 26. 1-035'.. 7 .. 27. 2-625' to 6 pi. 28. 3-165" .. 4 .. 29. 9-999 4 .. 4 .. 30. Find the area of a floor 19| ft. square.* 31. Find the cubic content of a die whose side is }| inch. 32. How many sq. ft. are contained in the aroura, 50 Greek ft. (each 1-01146 ft.) square? 33. How many sq. yd. are in the are, 10 metres (each 39'37079 in.) square? 34. Find how many cub. ft. are in the stere or cubic metre. 35. How many flagstones 14 in. square will be required to floor a kitchen 2 1 ft. square ? 36. Find how many cubes | inch in the side can be cut out of 7 cub. ft. 74 cub. in., allowing 3 cub. in. for waste. Circles are proportional to the SQUARES of their diameters. 37. How many times is a circle 27 ft. in diameter as large as another 15 in. in diameter? 38. The paving of a circular floor 25*6 ft. in diameter cost 9" l.,4 ; what cost the paving of a similar floor 38-4 ft. in diameter? * When we say a surface is 19| ft. square, we mean it contains 19| X 19 square ft. A surface 10 ft. square is 10 times as large as a sur- face containing 10 square ft. 156 INVOLUTION. 73* Spheres are proportional to the CUBES of their diameters. 39. TJhe weight of a metallic ball inch in diameter is -398 oz. Find the weight of another of the same metal f inch in diameter. 40. A ball inch in diameter displaces '128 oz. of water; how many oz. will another 2 in. in diameter displace ? 41. How many times is the Earth, whose mean. diameter is 7912 miles, as large as the Moon, whose diameter is 2140 miles? A body, in falling, traverses 16'1 ft. during the first second, 4 X 16* 1/2. in two seconds, and so on, the SPACES traversed being proportional to the SQUARES OF THE TIMES. 42. Through what space will a body fall in 2J seconds? 43. To what height must an aeronaut ascend so that a ball let fall from his balloon may reach the ground in the quarter of a minute ? I. The square of the sum of two numbers 9+7 is = the sum of their squares increased by 9+7 _ twice their product.* 92 _i_ 9x7 Thus, (9 + 7) 2 = 9 2 +2X9X 7 + 7 2 9X7 +7 or 16 2 =81 + 126 + 49 = 256. 9 2 +2(9X7)+7 2 Similarly, (40 + 3) 2 = 40 2 + 2 X 40 X 3 + 3* or 43 2 = 1600 + 240 + 9 = 1849. II. The square of the difference of two 9 7 numbers is = the sum of their squares dimin- 9 7 ished by twice their product.* 9^ 3X7 Thus, (9 7)-'=9 2 2X9X7 + 7' 9X7 +7 2 or 2 2 = 81 126+49 = 4. 92~ Similarly, (50 4)- = 50- 2 X 50 X 4 + 4 2 or ' 46 2 = 2500 400 + 16 = 2116. III. The product of the sum and the differ- 9 + 7 ence of two numbers is = the difference of 9 7 their squares.* 9219 O~~T" Thus, (9 + 7) X (9 7) = 9 2 7 s 9 X 7 7 2 or 16X2=81 49 = 32. cp HjT Similarly, (50 + 7) X (50 7) = 50 2 7 2 or 57 X 43 = 2500 49 = 2451. * These propositions are more conveniently remembered in their algebraic form. II. (a &)* = III. (a + &) (a b] =a INVOLUTION. 157 From III., we obtain a convenient method for obtaining the square of a number mentally* By III. (77 + 3) X (77 3) = 77* 3' Hence (77 + 3) X (77 3) + 3* = 77 2 or 80X74 + 3 3 = 77 2 Find the difference between the given number and a num- ber near it ending in 0. Take a third number, so that the dif- ference between it and the given number may be = the former difference. The square of the given number is = the product of the other two numbers increased by the square of the common difference. Thus, 93 2 = 90 X 96 + 9 = 8649. From I., we obtain a method applicable when the number to be squared ends in 5 or . By I. 75 2 = 70 2 +2 X70 X5 + 5 3 = 70X70 + 10X70+5 9 = 80X70 + 25 When the last figure is 5, the square may be found by multi- plying the number of tens by the next greater number, and then affixing 25. Similarly, (9fc) = 9 X 10 + = 90*. Square the following numbers mentally : 1. 21 2. 61 3. 33 4. 47 5. 56 6. 89 7. 74 8. 68 9. 72 10. 97 11. 82 12. 64 13. 85 14. 75 15. 35 16. 65 17. 195 18. 895 19. 395 20. 495 21. 19j 22. 22J 23. 17i 24. 25J EVOLUTION. EVOLUTION is that process by which we find a number which when multiplied a certain number of times by itself, repro- duces a given number. The number found is termed a root of the given number. SQUARE ROOT. . The SQUARE ROOT of a number, when multiplied by itself, reproduces the original number ; thus, 3 is the square root of 9, 3 = ^9 = 9* 5 8 = V 64 = 64 *' Take any number, as 43, we know that 43 = (40 + 3) = 40* + 2 X 40 X 3 + 3*. * This is sa Darvel, Ayrsh powers. tid to be the method ^^^^^^A^ lire, who has acquired some celebrity for her arithmetical 158 EVOLUTION. 74. Let us now in re- 40 producing the number determine the method 2x40+3 of finding the Square Root. 40 '+2X40X3+3 '(40+3 40 2 2X40X3+3* 2X40X3+3' Subtracting 40 2 , we leave 2 X 40 X 3 + 3 2 . Further to obtain the quotient 3, the divisor must be 2 X 40+3. No number containing 1 figure can have more than 2 fig- ures in its square. No number containing 2 figures can have more than 4 figures in its square. Since 1 1 place in a number corresponds to a period of 2 places in it square, ^efore ex- tracting the square root, we point off in periods oftwb places, commencing at units' place. (1) Find ^1849. The greatest square root in 18 ^s '4. Sub- 4 ) 18,49 (43 tracting 4 2 , we have 2, which with the next * 1 period annexed is 249. Doubling* 4, we see " that 8 in 24 is 3 times. Anne^ng;3 to 8^ we subtract 3X 83, and having no remainder, find 43 = ^1849. I , We have first subtracted 40 2 = 1600; we have then sub- tracted 2 X 40 X 3 + 3* or (2 X 40 + 3) X 3 = 249, to make up 43*. ' (2) Find ^12744$. The greatest sq. root in 12 is 3. Sub- 3 )12,74,49(357 tracting 9, and annexing the next period, 9 we have 374. Doubling 3, we see that (55 \ "374 as we have a figure to annex to 6, the 325 next figure in the quotient will be 5. 7 ^ 7 "loiq Subtracting 5 X 65, and taking down the TO Tq next period, we have 4949. Adding 5 , to the divisor, we obtain 70, the double of 35. The next fig- ure being 7, we subtract 7 X 707 or 4949 ; and thus find that 357 = ^127449. We have first subtracted 300 or 90,000. ^ Having then subtracted 2 X 300 X 50 + 50' or (2 X 300 + 50) X 50, we have now subtracted in all (300 + 50) 2 or 350*. We then subtract 2x350X7 + 7* or (2x350 + 7) X 7, fad thus complete the square of 357. In extracting the square root, no remainder can be greater than twice the root obtained. Thus, in finding the greatest square root in a number to be 8, it is evident the number is less than (8 -f- 1)* or 8* + 2 X 8 + 1. When 8* is subtracted, the remainder is therefore less than 2X8 -f- 1. or not greater than twice 8. EVOLUTION. 74. 159 Find the square root of 1. 1024 2. 4225 3. 3136 4. 137641 5. 50625 6. 401956 7. 5499025 8. 9897316 9. 10. 11. 12. 13. 14. 15. 16. 7365796 27415696 20820969 14235529 16232841 70207641 31843449 79263409 17. 80568576 18. 62473216 19. 88887184 20. 22992025 21. 56987401 22. 58415449 23. 236144689 24. 998876025 I, Find ^672-35675 to 5 decimals. )6,72-35,67,5(25-92984 2 )6,72'35,67,5(25'92984 45 4_ 272 225 4735 4581 15467 10364 51849 5103 45 509 5182 509 5182 4 272 225 4735 4581 518588 5185964 . -.50 4666 41 437 0900 5,1,8,4 4148704 22 219600 20743856 1 475744 15467 10364 5103 4666 437 415 22 21 1 In extracting the square root of a number, we need only extract as many figures as the number next greater than half the number of the required figures. In the example before us, we require 5 decimals, and as there are 2 integral places in the root, there will thus be 7 figures in all. We need only extract 4 figures, and then finish as in Contracted Division (see 40.) Let us now examine the closeness of the approximation. In comparing the first part of the root which is extracted, with the second part which is required, we must attend to local value, by adding as many ciphers to the former as will give it 7 figures, the required number in the root. When the square of the first is subtracted, the remainder is = twice the product of the first and second with the square of the second. We now merely divide this by twice the first, so that the quotient = the second with the square of the second divided by twice the first. Now the second contains 3 places, hence its square contains no more than 6 places ; and as twice the first cannot con- 160 EVOLUTION. 74* tain less than 7, the square of the second, divided by twice the first, is a proper fraction, and hence less than 1, so that the quotient is a convenient approximation to the second part of the root.* (4) Find _ 5 257851,1 168 EVOLUTION. 76* The fourth root of a number may be found by taking the square root * of its square root ; the sixth root, by taking the square root of its cube root, &c. Find the following roots by Horner's Method : 1. #2 4. #228886641 ' | 7. #21224-09008801 2. #20 5. #35806100625 8. #81-108054012001 3. #200 6. #20730-71593 9. #148035889 10. #17 11. #^i T 12. #|| of A of li. 77. SCALES OF NOTATION, IN the common notation, the local value of the figures ascends in the SCALE of TEN. We may, however, adopt other scales : In the scale of 6, " 1 " in the second place being six times the value of " 1 " in the first, " 10" represents 6, the lose of the scale. Again, " 1 " in the third place being six times the value of " 1 " in the second, " 100" represents 36, the second power of the base. " 2534 " in the scale of 6, or (2534) e , is = 4 + (3 X 6) + (5 X 6*) + (2 X 6 3 ) ; (65284). = 4+ (8 X 9) + The number of characters used in any scale is denoted by its base. In the scales of 11 and 12, we may represent 10 by D for Decem; and in the scale of 12, 11 by U for Undecim. (1) Express 451 in the scale of 6. / j r -t In dividing 451 successively by 6, we 75//1 find that 451 = (2X6* )-f (OX6)-f(3X6) I i 451 = (2031),. To reduce a number in the decimal scale to its equivalent in another scale, we divide the number successively by the base of the latter, and to the final quotient annex the succes- sive remainders. 1. Red. 666 to scale of 6 2. 315 // // 4 3. // 225 // // 7 4. Red. 313 to scale of 8 5. // 222 // // 2 6. // 1859 n // 12 (2) Express (1234) a in the decimal scale. 1234 5 7 (1234), = (1X5 3 ) + (2X5') + (3x5) + 4 = 5{(1 X 5*) + (2 X 5) + 3} + 4 194 SCALES OF NOTATION. 169 To reduce a number in any scale to its equivalent in the decimal scale, we multiply the left-hand figure by the base of the former, and add in the next figure to the right, and pro- ceed similarly till all the figures are taken in. Reduce the following to the decimal scale : 7. (423) 5 9. (3567) 8 11. (2D98),, . 5 8. (1243) 6 10. (12345) 9 12. (DU10) la (3) Express (2143) 8 in the scale of 7. To reduce a number from one _(2143) 6 7298 scale to another, of which neither n 7 42//4 is the decimal, we first reduce to gg g x/ Q the decimal, and then to the re- - ^ . quired scale. 298 = ( 604) * ' 13. Reduce (1001001), to the scale of 3. 14. Reduce (2D43)j , to the scale of 7. 15. Reduce (4U57) l , to the scale of 2. The pupil will now see that the "higher the base of the scale, the fewer figures are necessary to represent any number; but lit same number of figures is required in two scales, then the left-hand figure in the Mgher is less than that in the lower scale. (4) Reduce 23, 34, 41, to the scale of 3, add them and prove the work. 23 = (212) 3 (10122). 34 = (1021). * 41 = (1112) . gg 98 = (10122). 98 Tnd in the 4th, 3 = (10),. 16 Reduce 64, 127, 95, to the scale of 2, and find the sum. 17 Reduce 2^ 14B, 79, to the scale of 12, and find the sum. (5) Reduce 2002 and 1271 to the scale of 4, and find their difference. (23123) 4 2002 = (133102) 4 Tl 1271 = J103313)^ "731 = "723123^ lg 3 from 2 we cannot, 3 from 4 leaves 1, 1 and 2 are 3. 1 and 1 make 2, 2 from 4 leaves 2, &c. H 170 SCALES OF NOTATION. 7*7 18. Reduce 625 and 367 to the scale of 5, and find their difference. 19. Reduce 237 and 74 to the scale of 9, and find their difference. The Arithmetical Complement (A. c.) of a number in any scale is obtained by subtracting the number from the base, or the next greater power of the base. The Arithmetical Com- plement of a number is so called because its figures and those of the number together fill up the scale. In the decimal scale, A. c. of 7 = 10 7 = 3 ; A. c. of 213 =r 1000 213 = 787. In the scale of 6, A. c. of (3). = (10) 6 (3) e = (6) 6 ; A. c. of (342) e = (1000) .- (342) . = (214) .. The best method of finding A. c. is to commence at the left hand, and subtract each figure from the base diminished by one, except the right-hand figure, which we take from the base. In the scale of 8, to find A. c. of (263) 8 , we take 2 from 7, 6 from 7, and 3 from 8, and thus obtain (515) 8 . 20. In the decimal scale, find the A. c. of 43, 726, and 2817. 21. In the scale of 6, find the A. c. of (24) 6 , (253) 6 , and (1243) e . 22. In the scale of 12, find the A. c. of (24) 12 , (346) 12 , and (28DU) 12 . (6) Reduce 1691 and 127 to the scale of 12, and multiply them. 1691 = (U8U) la (I>4345) ia 127 = _(D7) la -124 11837 6D25 20292 9952 __ 214757 = (D4345) ia 7 times U = 77 = (65) 12 . Write 5 and carry 6, &c. 23. Reduce 2341 and 725 to the scale of 7, and multiply them. 24. Reduce 741 and 1286 to the scale of 6, and multiply them. 25. Reduce 198 and 241 to the scale of 12, and multiply them. (7) Reduce 753 and 29 to the scale of 7, and find tho quotient. 153 "264 224 (40), 26. Reduce 864 and 72 to the scale of 3, and find the quotient. 27. Reduce 78467 and 317 to the scale of 12, and find the quotient. 171 78. DUODECIMALS. IN Duodecimal Multiplication, we descend in the scale of twelve pom the/tf, which is adopted as the unit of computation. . ^towdfoot is divided into 12 inches or primes (') ; an inch into 12 hues, parts, or seconds (") ; a line into 12 thirds ("'), &c. Descending from the square foot in the duodecimal scale, the names are as follow : twelfth of sq.ft. (') ; 8 q. inch (") ; twelfth ofsq. in. ("') ; sq. line ("" or Iv ), & c . Let AB be a lineal foot, di- vided into 12 in. each = BH. The square BD is a .' n 80,77^,75, "30 22. In Venice the clocks strike to 24. How many strokes are made in a day ? 23. A boy gains 10 marbles on Monday, 3 more on Tuesday, and 3 more on each successive day. How many has he gained in six days ? 24. A merchant gained 90 during the first year in business, and 35 more in each successive year than the one preceding. How much has he gained in 20 years ? 25. A labourer saved Id. the first week of the year, and |d. more on each successive week. How much has he at the end of the year? 26. A body falls 16'1 ft. during the first second, thrice as far dur- ing the second, and so on. How far would a body fall in six seconds ? 27. If 20 sentinels are placed in a line at the successive distance of 40 yards; how far will a person travel who goes from the 1st to the 2d and back; from the 1st to the 3d and back; and so on till he goes from the 1st to the 20th and back : and how long will he take at the average rate of 3 miles f hour? SO. GEOMETRICAL PROGRESSION. A GEOMETRICAL PROGRESSION (G. P.) is a series of num- bers uniformly ascending or descending by a common ratio ; and is therefore appropriately termed an EQUIRATIONAL SERIES. 2, 6, 18, 54, &c., is an ascending G. P., in which the common ratio is f or 3. 1, J, J, ft, &c., is a descending G. P., in which the common ratio is \. * Let a the first term, d = the difference, S = the sum, and I = the nth term, or the to of n terms; then I orthenthterm=o(n IJo, S= H 2 JJOThe Dormer series is as follows: Term Thel? Term 1st _- 2 = 2 1st 2d = 2X3 = 6 2d 3d = 2 X 3 2 = 18 3d 4th = 2 X 3 3 = 54 4th &c. &c. &c. &c. 178 SERIES. The latter series is as follows : = 1 =1 = 1 X (I)* = I = 1 X U) 3 = i &C. cNJC. To obtain any term in a G. P., we raise the common ratio to the power whose index is less by one than the number showing the rank of the term in the series, and then multiply the power by the first term. In the 1st series, the llth term is = 2 X 3 10 = 118098. In the 2d series, the 20th term is = 1 X Q) 1 9 = ^ ATT- (1) Find the 9th term in the G. P. 7, 21, &c. Ratio = y = 3 ; 9th term = 7 X 3 8 = 45927. (2) Find the 6th term in the G. P. 2 J, 1 J, &c. Ratio = H -r- 2J = J; Gth term == 2J X (f) 5 = ! X AV = A- Find the Find the 4. 10th term in 81, 27, 9, &c. 5. 7th ., g, 3, i, &c. 6.5th /, i 1. 6th term in 4, 8, 16, &c. 2.5th 7, 28, 112, &c. 3. 9th " i, 1, 5, &c. 7. Of seven purses, the first contains 1/4; the second, 2/; the third, 3/; and so on in the same ratio. How much does the last contain ? 8. A person who found a potato imitated the example of Samuel Budgett and planted it. At the end of the first season he obtained 25 potatoes; and during each successive season the whole crop of the preceding one was planted and increased in the same ratio. Find the crop at the end of the fifth season. 9. Out of a vessel containing 10 gallons of brandy, T ' 5 was ex- tracted and replaced with water, T ! 5 of the content was again extracted and replaced with water, and so on for seven times. How much brandy is finally in the vessel ? 45T The first term is 10, the ratio ^j, and the number of terms 8. Let us find the sum of the G. P. 2, 6, 18, 54. Ratio = | = 3. 3 X Sum = (2X3)+(2X3)+(2X3*) + (2X 3*) (3 1) Sum = (2X3 4 ) 2 = 2 (3 4 1) Sum = 2 X ^ T = 80. SERIES. 179 8O Let us find the sum of the G. p. 9, If, ft, ^ Ratio = If -f- 9 = i. |XSum= (9X|)+{9xa) 2 }+{9xg) 3 } (l-i) Sum = 9-{9X(|)'}=9$l-(i)} Sum=9x^ 4 =10-. To find the sum of a G. P. we raise the ratio to the power denoted by the number of terms, divide the difference between this power and unity by the difference between the ratio and unity, and multiply the quotient by the first term.* Any term in a G. P. is a Mean Proportional, or a Geome- trical Mean between two equidistant terms; thus, in the G. P. 1C, 24, 36, 54, 81; 36=^16 X 81=s/24x54(see57& 74.) (3) Find the sum of 7, 14, 28, to 10 terms. Ratio = V 4 = 2. Sum = 7 X ^f = 7161 - (4) Find the sum of |, fa T { T , to 8 terms. Ratio = ,V -r = i- _ I \s * (?) _ 1 \s 65535 V 4 U1H = f X ~fZ.i" T X e"f f 36" * 3^ terms. 10. Find the sum of 2, 4, 8, to 12 j 13. Find the sum of 3, f , T 3 5 , to 7 1 1 . ,/ / 5, 15, 45, * 8 12. 14. 15. 16. Of seven boys, the first has 64 nuts, the second 96, and so on in the same ratio. How much have they in all ? 17. Of five brothers, the eldest has 759'-76, the second two- tliirds of this sum, and so on in the same ratio. How much have they in all ? 18. A gentleman on taking a house for twelve months ignorantly agreed to pay 1 mil as rent for the first month, 1 cent for the se- cond, 1 florin for the third, and so on in the same ratio. To what would the rent amount ? The number of terms in a descending G. P. may sometimes be infinite; thus every Interminate Decimal is an infinite descending G. P. * Let a = the first term, r = common ratio, S = the sum, I = the last of n terms ; then I or the nth term = ar -- 1 , 180 SERIES. In ;7, which is = T ^ + T fo + T ^u + &c. ad infin. (co), the ratio is T V Now a fraction when raised to a power becomes less as the index of the power becomes greater ; when therefore the index is infinite, the fraction becomes 0. Hence, Sum which is = -^ X 1 7" ^V* is= T VXJ- 1 - To * 15 - T ? g - _IP_ - 7 " 1 - I* " I** ' The sum of an Infinite descending G. P. is = the first term divided by the difference between the ratio and unity.* (5) Find the sum of T y& + TU**TO + &c - Ratio = TT fcs ; Sum = T y& ~ (1 TU W) = f See 34, No. 1. 19. Find the value of -45, or the sum of T V 5 + ToVW + &c - 20. Find the value of '037, or the sum of I 81. COMPOUND INTEREST. WHEN a sum is lent for a number of periods or terms at COM- POUND INTEREST, the Interest is added to the Principal at the end of each term, and the Amount obtained becomes the Prin- cipal for the next term. On 600 lent for 5 years @ 5 / , the Simple Interest would be 150; and the Amount, 750. But at Compound Interest the Amount would be as follows : Principal for the first year .... 600 Interest // // // ..... 30 Principal for the second year .... 630 Interest // // .... 31'5 Principal for the third year ..... 661-5 Interest // // // ..... 33-075 Principal for the fourth year .... 694*575 Interest // // // .... 34'72875 Principal for the fifth year ..... 729-30375 Interest // // '/ ..... 36-4651875 Amount for 5 years ..... 7657689375 Original Principal ...... 600 Compound Interest ..... 165-7689375 * When n is infinite, and r < 1 . r n = 0, S = j ( ^_ . COMPOUND INTEREST. JgJ 81. Exercises in Compound Interest may be performed by .s method, but a more concise plan may be obtained by considering the following : Interest on 1 for 1 year 5 / = -05 Amount on 1 // // // // // = 1-Q5 Since the Amount for any year becomes the Principal for the next, we obtain the following proportions : Principal. Amount. > & 1 : 1-05 : : 1-05 : 1-05 2 = Am*, for 2 years. 1 : 1-05* : : 1-05 : 1-05 3 = // // 3 // 1 : 1-05 3 : : 1*05 : 1-05* = * // 4 // 1 : 1-05* : : 1-05 : 1'05 5 = // // 5 // Therefore 1 : 600 : : 1-05* : 600X1 -05 *= 765-7689375 To find the AMOUNT of a given sum for a number of terms at Compound Interest, we raise the Amount of 1 for one term to the power denoted by the number of terms, and mul- tiply by the given sum. (1) Find the Amount of 450 and the Compound Interest on it for 3 years @ 4 %. Am* of 1 for 1 yr. @ 4 % = 1*04 Am 1 , of 450 for 3 yr. @ 4 % = 450'X 1 '04 s = 450 X 1-124864 = 506-189 = 506*3*9 J Comp d Int. = 506//3//9^ 450 = 56*3*9 J. In involving the Amount of 1, we take as many places in the powers as will produce the result correct to three decimal places* (see 39, 73, & 43.) (2) Find the Amount of 547*625 for 4 years @ 5 / , payable half-yearly. Am 1 , of 1 for 4 yr. 5/ V ann.=l'025 Am*, of 547-625 for 8 half years 5 % = 547-625 X 1-025 8 = 667-228 * Calculations in Compound Interest are often effected by having the amounts of 1 at the important rates tabulated for a series of years. Exercises in Compound Interest afford good illustration of the advan- tages of Logarithms. The Questions prescribed above are, however, given for such periods as enable them to be easily solved by Invoh tion. 182 COMPOUND INTEREST. Gl* Find the Amount of the following sums : 1. 600 for 2 years @3% 2. 300 * 3 // // 5/ 3. 800 // 4 // // 3/ 4. 400 // 4 // f 47o 5. 700 4 '/ // 2i/ 6. 834 // 5 // // 3% 7. G97'/15//Ofor6yrs.@2i/ a 8. 468//10//6 // 4 // // 4/ 9. 232// 7//6 * 8 // // 3% 10. 35// 3//9 // 3 // // 3|/ 11. 666'/13//4 5 // // 2J/ 12. 267//19//2 // 7 // // 4i/ 13. Find the Amount of 670 for 3 years @ 6 / , supposing the interest to become due half-yearly. 14. Find the Amount of 684 for 3 years @ 4 / , supposing the interest to be due quarterly. 15. What is the Compound Interest on 764-42 "6 for 4 years @ 5 / , due half-yearly ? 16. Find the Compound Interest on 29" 15 for 3 years @ 3 / , due quarterly. 17. Find the difference between the Simple and the Compound Interest on 750 for 3 years @ 4 %. 18. A sum of 300 is lent for one year @ 4 / ; find the difference between the Simple and the Compound Interest, due quarterly. 19. To what will a legacy of 500 left to a boy 11 years of age have accumulated at Compound Interest, on his attaining majority at 21 years of age, allowing Interest @ 5 / ? 20. A legacy of 2500 was left to a young lady in 1852 on con- dition that it should be improved at Compound Interest for a mar- riage-portion. To what will it have accumulated at her marriage in 1860, reckoning Interest @ 5 / ? We may require to find the Principal which, improved at Compound Interest, may at a future date amount to a given sum; thus, let us find a sum which in 6 years @ 3J/ will amount to 700. . Am 4 , of 1 for the given time = 1'035 6 =1-229255 Amount. Principal. ^.^ 1-229255 : 700 : : 1 : x = We work by Contracted Division (see 40.), and obtain the result 569-450, the Present Value of 700. To find the PRESENT VALUE of a given sum due in a given time at a given rate, we divide the given sum by the amount of 1 for the given time.* * Let P = Principal, A = Amount, R = Kate, n = number of years, A - P (\ 4- AV P ~ A P + ' - COMPOUND INTEREST. jg 3 alUG f 500//12 " 6 > due in 7 years 1-03 7 = 1-229874 = 407-054 = 407//1//1. 21.900duein2yrs.@47o 22. 700 * ,4 , 7 5/I 23.1200 // // 4 // // 3/ Find the Present Value of 24.1405//ll//6duein4yr.(5)4 / a 25. 105//11//3 // // 3 * 7SA. 26. 333// S//4 // // 5 // // 2--/I 27. What sum will in 3 years @ 4 / amount to 100, supposing the interest to be paid quarterly ? 3. Find the sum which, with half-yearly payments of hit- will at 6 / amount in 4 years to 253-354. 29. A merchant who has increased each year's capital by a tenth t finds that at the end of twelve years he has 3985" 16" 1. Find his original capital. 30. A sloop was bought by A, who sold it to B, by whom it was sold to C, who finally disposed of it. Each gained 30 % on his prime cost. C sold it for 6592 ; what did A pay for it? One of the most important applications of Compound Interest is in the calculation of ANNUITIES. An Annuity, as its name imports, is a sum payable yearly for a certain number of years ; an Annuity may, however, be payable at equal intervals of any duration, as half-yearly, quarterly, \;c. Suppose a person, entitled to an annuity of 30 ^ annum for 5 years, payable yearly, draws none of it till the end of the time ; to what will it have amounted, reckoning interest at 4/ ? 1 of the annuity might be lent at the first payment for 4 years, and become at Compound Interest 1'04 4 ; 1 at the second payment might be lent for 3 years, and become 1*04 3 ; 1 at the third payment might be lent for 2 years, and become 1-04 2 ; 1 at the fourth payment would in 1 year become 1-04; and to these we would add the fifth payment of 1. The Amount of an Annuity of 1 for 5 years @ 4 / is thus = 1-04*+ 1-04 3 + 1-04 2 + 1-04 + 1. The sum of this / c of\\ - />1'04-1 <.1'04-1. Geometrical Progression (see 80.) is = * 1 . 04 _ 1 = * r^ Having found the Amount of an Annnitv of 1, that of 30 J.Q^S 1 for the same time and rate = 30 X r^j To find the AMOUNT of an Annuity, we diminish the amount of 1 for the given time and rate by 1, divide the differ- 184 COMPOUND INTEREST. 81 ence by the interest of 1 for one term, and multiply the quotient by the given Annuity. (4) Find the Amount of an Annuity of 25 payable half- yearly in 4 years @ 5 / . Int. of 100 for yr.=2'5; Int. of 1 for 1 hf.yr.='025 Am 1 . oflfor8hf.yr.=l-025 8 ; Annuity for yr.=12'5 Amount of Annuity = 12-5 X -^^ss 12-5 X ^- = 12-5 X 8-736116 = 109-20145. 31. Find the amount of an annual rent of 25 for 8 years @ 5 / . 32. Find the amount of an annuity of 60 payable yearly for 6 years @3/ . 33. The Lord Justice Clerk of Scotland has an annual salary of 4500. To what would it amount in seven years @ 4 / ? 34. Find the amount of an annuity of 36 payable quarterly for 2| years @2/ . 35. A gentleman of fortune, entitled to an annual pension of 200, payable half-yearly, allows it to accumulate for 10 years. Find the amount @ 5 / . 36. A salary of 180, payable quarterly, is not drawn for 1^ years. Find the amount @ 5 / . 37. Find the amount of 4 half-yearly dividends of 2000 stock in the three per cents, reckoning interest @ 4 / . ^gr The half-yearly annuity is one-half of 3 / on 2000. Suppose a person, desirous of obtaining an annuity of 70 W annum for 10 years, wishes to know how much he must pay for it @ 3 %. 1.A91O 1 The amount of this would be 70 X 03 . The sum to be paid for the annuity would evidently be that which in 10 years would produce this amount. We would therefore re- quire to find the Present Value of the Amount by dividing it by the amount of 1 for the given time (see p. 183). 70 X 1 ' 03I < 1"" 1 -T- l'03 l = 70 X I ~T^ R I 03 To find the PRESENT VALUE of an Annuity, we diminish 1 by the Present Value of 1 for the given time and rate, divide the difference by the interest of 1 for one term, and multiply the quotient by the given Annuity. (5) Find the Present Value of an Annuity of 30//17"6 payable quarterly in 2J years @ 3i / . COMPOUND INTEREST. jgr 81. I ^.of100for :ly r. = :.875;Int.oflforl q uar.=.00875 Present Value of 1 for 11 quarters = Annuity for 1 qr. = 7'71875 = j O f Present Value of Annuity = 7-71875 X ^FoogTgn *00875 = 7-71875 X ^=7-71875 X 10-4436 = 80-6115. UnHmited ' the Annuit ^ is termed a A person wishing to obtain a perpetuity of 200 y annum is desirous of knowing the sum to be paid for it @ 5/ The amount of any sum, as 1, for an unlimited time being QO (infinite), its reciprocal, or the present value of 1, due in an unlimited time, is hence = 0. Present Value of 1 = ^ = 0. Present Value of Perpe- tuity = 200 X = = ^ = 4000. The sum of 4000 lent out @ 5 / will produce 200 in perpetuity. (6) Find the Present Value of a Perpetuity of 99//2//6 ^ annum @ 3 %. Present Value = = 3050. 38. Find the present value of an annuity of 40 payable annually for 10 years @ 4 / . 39. Find the present value of an annuity of 6210 for 3 years, payable half-yearly, @ 5/ . 40. Find the present value of a perpetuity of 2 1 0" 17 6 ^ annum 41. The Lord Justice General of Scotland has a salary of 4800 tp annum. Find the present value of this for 10 years @ 3/ . 42. A tenant, on taking a lease of a house for 7 years @ 19 ^ annum, pays the present value. Find the sum, reckoning interest @ 4 / . * Let a = Annuity, R=Eate/ , r=^ = Int. onl, n=N of years, Amount of an Annuity = a X -- - - . i __ L_ Present Value of an Annuity = a X (H-r). r a 100# Present Value of a Perpetuity = or -^ . 186 COMPOUND INTEREST. 43. A colonel of the Royal Marines on half-pay has 264" 12 6 ^ annum. Find the present value of this annual salary for 6 years @ 4%. 44. What ought to be paid for a property giving an annual rent of 1878"6, reckoning @ 4| / ? 45. What sum paid in January 1858 will produce an annuity of 50, payable half-yearly until July 1861, @ 4| %? 82. MISCELLANEOUS EXERCISES. 1. FIND the L. c. M. of all the multiples of 3 from 6 to 27 inclusive. 2. Find the G. c. M. of 25 X 45 and 5 X 3 s . 3. What is the G. c. M. of the square of 48 and the cube of 18 ? 4. Find the L. c. M. of the first ten even numbers. 5. Reduce y^l^B to its lowest terms. 6. Arrange , ^, , f and ji, in order of magnitude. 7. Subtract the sum of $ + jj + u + 13 + it from 5 - 8. Find that number of which ({j -f ; |) is = 51 . 9. Multiply * of 2 by 2 iiL, 10. Multiply |-f | by J , and increase the product by T 5 5 of 1 1. 11. Find that number whose fifth diminished by its seventh is = 3?. 12. From the square root of '000169 subtract the square root of 00016. 13. Find the decimal which when added to the difference of 5 | 5 and -002775 produces the square of '215. 14. Subtract the cube of 1-6 from 130 times '0325. 15. Find the interest on -219 for 47 days @ 3'6 %. 16. A grocer by selling sugar @ 6d. ^ Ib. loses d. ; find his loss / . 17. From Edinburgh to Glasgow by railway is 47 ^ miles. In what time will a train traverse the line at the rate of 990 yards y min., allowing 5 S ? hour for stoppages? 18. The number of copies in the first edition of the Lay of the Last Minstrel, which was 750, was to that in the seventh as 15 to 7 1 . Find the number in the latter. - 19. From 1847 to 1857, the Revenue of the City of Edinburgh was 70,629, and the Expenditure 57,684. What per-centage was the difference or surplus of the former ? 20. A, at the rate of 4^ miles an hour, walks a distance in 3 T ' 5 hours ; in what time will B walk the same distance at the rate of 5 of 51 miles an hour ? MISCELLANEOUS EXERCISES. 187 82* 21 - Find tne square root of 10 5 316 -. 22. Find the cube root of -296. 23. Find the H. p. of an engine which can raise 4 tons of coala per hour from a pit 77 fathoms deep. 24. The centre arch in Westminster Bridge, which is 76 feet wide, is the seventh from the side, and each arch is 4 ft. narrower than the adjoining one nearer the centre. Find the width of the first arch. m 25. Divide 5 among A, B, C, D, in the mutual ratios of , , , and . 26. A sum of 1343"146 collected for a family of orphans was laid out at 6 / per annum. Find the value of a half-yearly payment. J 7 . Reduce 1 dwt. to the decimal of 1 Ib. avoir. Divide 25832 in the ratios of the squares of the reciprocals of the first four odd numbers. 29. The Admirals, the Vice Admirals, and the Rear Admirals of the British Navy are each divided into 3 classes of Red, White, and I Hue, and the classes of each rank contain the same number. The number of Admirals is 5 7 g of that of the whole, which is 99, and that of Men-Admirals is -fr. Find the number in each class of Admirals. 30 The walls of Rome erected by Aurelian have been calculated to contain 1396 hectares, each 2-47114 acres. Express the area in ^SL Thfl weight of an American dollar is 412 \ grains, of which is pure silver. Find the weight of pure silver in 100 dollars 32. If on every guinea of selling price half-a-crown is gamed ; find the gain on 1000 of buying price. 33. Victoria Bridge on the St Lawrence is within 50 yards 2 miles in length ; in what time will a train traverse it at the r f J/F^ for 4 years @ 5 /., and for 5 years @ 4 /., payable half-yeaily n ; Horticultural Society, founded by Sir Joseph Banks ,, w as remodelled in 1856; been incorporated by Royal Charter, m what year did i " lny days elapsed between the Annular Eclipse of 15th ^^^ S! Kri the cube of 11 in the scale of 3. . rom York to London is a distance of 192 miles , 188 MISCELLANEOUS EXERCISES. at ^ 1C same time from each terminus, the one from York at the rate of 40 and the other at 32 miles an hour. How far from London will they meet ? 40. F starts at 12 h at 6| miles an hour, and B at 12 h . 30. At what rate must B travel to overtake F at 2 1 ? ? 41. 3 Russian versts are = 3500 yards. Reduce a verst to the decimal of a mile. 42. Assuming the length of a glacier, described by Principal Forbes, to be 20 miles, and its annual progression 500 ft. ; how long would a block of stone take to traverse its length ? 43. Of 150 encumbered estates in Ireland, the numbers in the four provinces were respectively as 1, 2, 3, and 4. Find the num- ber in each province. 44. How many metres are in a Scotch mile, taking 1 Scotch mile = 1-123024 Imperial mile; and 1 metre = 39*37079 inches. 45. How many metres are in a Scotch mile, taking the following approximations, 8 Scotch miles = 9 Imperial miles, and 32 metres = 35 yards ? 46. If, in victualling a crew, 80 days are allowed for an outward and homeward voyage to Oporto ; 1 g 5 of this time for one to Deme- rara ; to Boston, f of that to Demerara ; to Valparaiso, Y of that to Boston. Find the time allotted for an outward and homeward voyage to Valparaiso. 47. In 1851, the population of Glasgow was 3'58866 per cent, of the population of Edinburgh more than double the latter, which was 161,648. Find the former. 48. Of an estate, the uncultivated part is 5 3 5 , the cultivated part, f , and the remainder under wood contains 65 acres. How many acres are in the whole ? 49. In 1 855, in the naval armament of France, the number of line of battle ships was f of 100; that of frigates, which was 12 less than f of the number of the line of battle ships, was f of that of the smaller vessels, and the number of steam vessels was ^ f of double the number of frigates. HJOW many were there of each? 50. What length of rails requires 873 T. 1 cwt. 1 qr. for their construction @ | cwt. ^ yard ? 51. Find the greatest depth of Lake Erie, which is | of that of Lake Huron, whose greatest depth is f of that of Lake Ontario, which is of that of Lake Michigan, whose greatest depth is {$ of that of Lake Superior, which is 990 ft. 52. In the Walcheren Expedition, out of an average force ot 40,589, there were 4212 deaths. Find the per-centage. 53. The circulation of a periodical was 38,500 ; of the whole, the MISCELLANEOUS EXERCISES. 189 of stamped copies was T 1 4. How many copies were un- stamped ? 54. Find the surface of a floor 28 ft. 7 in. long, and 15 ft. 6| in. broad. 55. Find the sum of V|| -f- S/ffs + if HI- 56. Find that number whose square root is = $ of 5J -f- 7 f * ! 57. Find the true discount on 2217/>3f for 3 months at 5 / . 58. In 1850, the states of Ohio and Tennessee, nearly of equal extent, produced 59,078,695 and 52,276,223 bushels of wheat respectively. Find the difference of their weight, reckoning the bushel at & | cwt. 59. If A pays ll|d. ^ for income-tax, what is his income when the net proceeds are 116"3"1 ? 60. If a courier traversed a distance of 400 miles in 36 hours, in what time did he traverse | of | of f of 6| miles? 61. No. 1585 of the Athenceum appeared on 13th March 1858 ; on the hypothesis that it has been regularly published once a-week, find the date of No. 1. 62. Find the price of 3 cwt. 2 qr. 13 Ib. carrots @ 16/ ^ 240 Ib. 63. Reduce a talent of 3000 shekels, each | oz. avoir., to the decimal of 1 cwt. 64. A train contains 13 trucks laden with coals; the average weight of a loaded truck is 10 T. 8 cwt. 1 qr., and that of an empty truck 3 T. 16 cwt. 2 qr. Find the weight of coals conveyed by the train. . 65. If, in the Russian tariff, the duty on Scotch herrings is 40 copecs y pood ; how much sterling is this f cwt., a ruble of 1 copecs being = 3/1 i, and a pood being = 36 Ib. avoir. ? 66. A miser collected 370 in packets of pound notes, crowns, half-crowns, florins, shillings, and sixpences. The values of five of the packets were the following fractions of the whole -.-packet of notes, W,; of crowns, ^5 of hf.-crowns, ***; ^ norms, , ; of shillings, ,V Find the number of notes, crowns, hf.-crowns, florins, shillings, and sixpences. ^ 67. Find the value of | cr. + f s. n. + s- 68. A can do a work in 7* days, B in 6f days, and C m 5| days. In what time can they do it by working together ? 69. A field contains 18 ac. 2 ro. 18 po., and another 7 ac. 3 ro. 7 po. Find the side of a square field of equal area to ^ 70. Of the two members of parliament wtm ^^^ in 1857, the number of votes polled *""*** ^ n Ythe was to.its excess above the other number as 902 tc h citation of a newspaper in the first quarter of a yea, 190 MISCELLANEOUS EXERCISES. 82 was 3200, and in the second quarter 3600. What would the cir- culation in the third quarter require to be to show the same ratio of increase ? 72. An angler, by using a single hook and a tackle of four hooks alternately for equal times during a day, caught 9i Ib. with the former, and 11 Ib. with the latter. On another day he caught 25 Ib. with the former ; what might he have taken with the latter? 73. Texas contains 274,362 square miles. Into how many lots, each 4536 acres, might it be divided? 74. The managers of a congregation buy a site of f rood for 500. How much will they pay for 5 9 5 acre ? 75. A person who has paid 642 of income-tax has 142 "15 "10 over. How much has he paid ^ ? 76. 70 masons can build a mansion in 61 days; after working for 10 days, 15 more are engaged. How many days fewer will be occupied than would otherwise have been ? 77. If 7 men can do as much as 11 youths, and if 21 youths can do a work in 13 days; in what time can 14 men and 4 youths doit? 78. A starts on a journey at the rate of 3 miles an hour, B fol- lows in | hour at the rate of 4 miles an hour. How far on will B overtake A? 79. In the household book of a ducal family we have the follow- ing entry by the steward: " Given your lordship on New Year's Day to give your grandchildren and the servants and several others, 32"6"6." Taking this as Scots money, which is one-twelfth of sterling, express the sum as the fraction of 100 sterling. 80. Find the 10th term of the series 1, 1|, 2, &c. 81. Find the 10th term of the series 1, 1, 2i, &c. 82. The population of a country in 1854 was 4,500,000, and if it has increased each year at the rate of 10 per cent, on the preceding year; find the population in 1859. 83. In an estate in Sweden, the arable land contains 200 ttmn- lands ; meadowland, 2 per cent, less than the arable ; and wood- land, 1 per cent, less than 7 times the arable. Find the area in acres of the estate, a tunnland being = 1-2312 acre. 84. From Dresden to Prague by rail is 150 kilometres, each 1093-63 yards : a train leaves Dresden at the rate of 48 kilometres q? hour, and in a quarter of an hour afterwards another leaves Prague at the rate of 40 kilometres qp hour. How many miles from Prague will they meet ? 85. Give eight convergents to the fraction which a kilometre is of a mile. MISCELLANEOUS EXERCISES. 1'Ji 82, 8 ^- Wlmt sum invested in the 3| per cents at 93 [ will produce 61 "5? 87. Deposited 500 in the National Bank on 1st September 1856, when interest was @ 2 %; "on 8th Oct. it rose to 3| / , and on 15th May to 4 / . Find the interest due on 8th June 1857. 88. At what rate must 273 be lent from 1st January to 27th May to produce 4-914 of interest? 89. Find the price of 19 cwt. 2 qr. 7 Ib. @ 1"86 y cwt. by decimals. 90. Find the value of 17-375 cwt. @ -5625 by decimals and by practice. 91. In the reign of Henry VIII., among the monasteries and religious houses whose revenues were confiscated, there were 186 belonging to the Benedictines with a revenue of 65,87 9" 14, and 173 to the Augustines with a revenue of 18,691"126. Reduce the average of one of the latter to the decimal of that of one of the former. 92. Divide the square of 390,404,646 by the square of 123,456,789, and let the quotient be carried out till it contains 3 significant figures in the decimal. 93. A capitalist who had invested 3120 sterling in stock @ 97*, gold 2500 stock @ 98, and the remainder of the stock @ 96. Find his gain or loss. 94. A labourer's wages for 30 days are 3"18"9. Find the wages for the working days in January and February 1860, new year's day being on a Sunday. 95. If 1 Ib. troy of sterling gold is worth 46f , find the weight of 3465 sovereigns and 1792 hf.-sovereigns. 96. The Brisbane Prize Fund of the Royal Scottish Society of Arts amounts to 175 in the 3 per cent. Government Consolidated An- nuities. Find the value of the fund at 90; and the value of the pria 97 Find the weight of an oaken block 2-25 ft. long, 16 inches broad, and ft of 1 '625 ft. thick; a cubic foot of water weighing 999-278 oz. avoir., and the specific gravity of oak being '925. 98 The deflection of the Earth's curvature is 8 inches mile, 32 inches for 2 miles, and so on, * a****^S . -j-ii.~ ^:^4- n - nno , "FTi-nd tlifl heicrnt or a Iignt portional to the square of the distance. J"^ above the level of the sea which is visible for 14 nautical each 6076 ft. c . . , n _ -.wiinnrv 99. 42 men, whose average strength is f of that of ****% an, can do a piece of work in 4 t days which other 5 = men can do ... 4 days. What is the average strength of one of the latte. as compared with that of an ordinary man? 100. The period of the Earth's revolution IB 365 256 days, man in 192 MISCELLANEOUS EXERCISES. that of Mercury 87 '969 days. Express by Kepler's Law the deci- mal that the distance of Mercury from the Sun is of tha.t of the Earth. 101. In a heavy gale, a flagstaff 60 ft. high snaps 28'8 ft. from the bottom, and not being wholly broken off, the top touches the ground. How far is its point of contact from the bottom ? 102. Seventeen trees are standing in a line 20 yards apart from each other ; a person walks from the first to the second and back, thence to the third and back, and so on to the end. How far will he have walked ? 103. If the value of 1 oz. troy of sterling gold { fine is 3'89375 ; find the value of 1 Ib. avoir of pure gold. 104. A lunation = 29*53 days, is the period in which the moon passes once through her phases. After a cycle of 223 lunations, known to the Chaldeans as Saros, eclipses recur in the same order and magnitude. Find the date of the eclipse in the next cycle corresponding to the solar eclipse of 28th July 1851. 105. When a body floats in a liquid the weight of the liquid dis- placed is = the weight of the floating body. The effective length of a vessel is 96 ft., the effective breadth 22 ft., and the draught of water 9 ft. Find the weight of the vessel, taking the weight of a cubic foot of water roughly at 62 Ib. 106. Find the mean discharge per second of the River Tay, sup- posing the area of its basin to be 2400 square miles, the annual fall of rain to be 30 inches, of which is lost in evaporation. THE END. PRINTED BY OLIVER AND BOYP, EDINBURGH. EDUCATIONAL WORKS PUBLISHED BY OLIVER AND BOYD, EDINBURGH; SOLD ALSO BY SIMPKIN, MARSHALL, AND CO., LONDON. *** A Specimen Copy of any work will le sent to Principals of Schools, post free, on receipt of one half the retail price, in postage stamps. English Reading, Grammar, etc. Armstrong's Eng. Composition.... P. 6 Eng. Etymology 6 Cennon's English Grammar 4 First Spelling-Book 4 Dalgleish's English Grammars 5 Gram. Analysis. 5 Eng. 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To which are added, a Comparative View of Ancient and Modern Geography, and a Table of Chronology. By ALEXANDER ERASER TYTLER, Lord Woodhouselee, formerly Professor of History in the University of Edinburgh. New Edition, with the History continued. 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 INTRODUCTION by W. K. TWEEDIE, D.D. 2s. Simpson's History of Scotland ; with an Outline of the British Constitution, and Questions for Examination at the end of 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 for Examination at the end of each Section. 3s. 6d. 14 Writing, Arithmetic, and Book-keeping. WRITING, ARITHMETIC, AND BOOK-KEEPING. 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. Practical Arithmetic for Junior Classes. By HENRY Gr. 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 of 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 not to be found." Practical Arithmetic for Senior Classes ; being a Con- tinuation of the above. By HENRY GL 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. English Journal of Education. 1 ' There are, it must be confessed, few good books on arithmetic, but this certainly appears to us to be one of them. It is evidently the production of a practical man, who desires to give his pupils a thorough knowledge of his subject. The, Rules are laid down with much precision and simplicity, and the ilk iitelligible to boys of ordinary capacity." Lessons in Arithmetic for Junior Classes, By JAMES TROTTEH. 66 pages, 6d. stiff wrapper; or 8d. 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. Answers, 6d. Each subject is also accompanied by an example fully worked out and minutely explained. The Exercises are numerous and practical. precision and simplicity, and the illustrations cannot fail to make them int< "" Writing, Arithmetic, and Book-keeping. 15 A Complete System of Arithmetic, Theoretical and Practical ; containing the Fundamental Rules, and their Application to Mercantile Computations ; Vulgar and Decimal Fractions ; Invo- lution and Evolution ; Series ; Annuities, Certain and Contingent. By Mr TROTTER. 3s. KEY, 4s. 6d. * t * All the 3400 Exercises in this work are new. They are applicable to the business 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 length 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 ALEXANDER TROTTER, Teacher of Mathematics, etc., Edinburgh. Is. KEY, 2s. Each rule is followed by 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 Application to Mercan- tile Calculations; Vulgar and Decimal Fractions; Exchanges; 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, Teacher of Mathematics, etc., in Edinburgh. Is. 6d. KEY, 2s. 6d. Each Rule is followed by an example worked out at length, and minutely explained, and by numerous practical Exercises. Button's Arithmetic and Book-keeping. 2s. 6d. Button's Book-keeping, by TROTTER. 2s. S^ts of Ruled Writing Books, Single Entry, per set, Is. 6d.; Double Entry, per set, Is. 6d. Stewart's First Lessons in Arithmetic, for Junior Classes; containing Exercises in Simple and Compound Quantities arranged so as to enable the Pupil to perform the Operations with the greatest facility and correctness. With Exercises on the Proposed Decimal Coinage. 6d. stiff wrapper. Answers, 6d. Stewart's Practical Treatise on Arithmetic, Arranged for Pupils in Classes. With Exercises on the proposed Decimal Coinage. Is. 6d. This work includes the Answers ; with Questions for Examination. KEY, 2s. Gray's Introduction to Arithmetic; with Exercises on the proposed Decimal Coinage. lOd. bound in leather. KEY, 2s. 16 Copy-Books^ Mathematics, etc. Lessons in Arithmetic for Junior Classes. By JAMES MACLAREN, Master of the Classical and Mercantile Academy, Hamilton Place, Edinburgh. 6d. stiff wrapper. The Answers are annexed to the several Exercises. Maclaren's Improved System of Practical Book- KEEPING, arranged according to Single Entry, and adapted to General Business. Exemplified in one set of Books. Is. Gd. A Set of Ruled Writing Books, expressly adapted for this work, Is. Gd. Scott's First Lessons in Arithmetic. 6d. stiff wrapper. Answers, 6d. Scott's Mental Calculation Text-book. Pupil's Copy, Gd. Teacher's Copy, Gd. Copy Books, in a Progressive Series. By R. SCOTT, late Writing-Master, Edinburgh. Each containing 24 pages. Price : Medium paper, 3d ; Post paper, 4d. Scott's Copy Lines, in a Progressive Series, 4d. each. The Principles of Gaelic Grammar ; with the Definitions, Rules, and Examples, clearly expressed in English and Gaelic; containing copious Exercises for Reading the Language, and for Parsing and Correction. By the Rev. JOHN FORBES, late Minister of Sleat. 3s. Gd. MATHEMATICS, NATURAL PHILOSOPHY, ETC. Ingram's Concise System of Mathematics, Theoretical and Practical, for Schools and Private Students. Improved by JAMES TROTTER. With 340 Woodcuts. 4s. Gd. KEY, 3s Gd. Trotter's Manual of Logarithms and Practical Mathe- MATICS, for Students, Engineers, Navigators, and Surveyors. 3s. A Complete System of Mensuration ; for Schools, Private Students, and Practical Men. By ALEX. INGRAM. Improved by JAMES TROTTER. 2s. Ingram and Trotter's Euclid. Is. 6d. Ingram and Trotter's Elements of Algebra, Theoretical and Practical, for Schools and Private Students. 3s. Music, Drawing, School Registers. 17 Introductory Book of the Sciences. By JAMES NICOL, F.R.S.E., F.G.S., Professor of Natural History in the University of Aberdeen. With 106 Woodcuts. Is. 6d. SCHOOL SONGS WITH MUSIC, By T. M. HUNTER, Director to the Association for the Revival of Sacred Music in Scotland. Elements of Vocal Music : An Introduction to the Art of Reading Music at Sight. Price 6d. %* This Work has been prepared with great care, and is the result of long practical experience in teaching. It is adapted to all ages and classes, and win be found considerably to lighten the labour of both teacher and pupil. The exercises are printed in the standard notation, and the notes are named as in the original Sol-fa System. CONTENTS. Music Scales. Exercises in Time. Syncopation. The Chro- matic Scale. Transposition of Scale. The Minor Scale. Part Singing. Kxplanation of Musical Terms. Hunter's School Songs. With Preface by Rev. JAMES CURRIE, Training College, Edinburgh. FOR JUNIOR CLASSES : 60 Songs, principally set for two voices. 4d. Second Series : 63 Songs. 4d. FOR ADVANCED CLASSES : 44 Songs, principally set for three voices. 6d. Second Series : 46 Songs. 6d. School Psalmody ; containing 58 Pieces arranged for three voices. 4d. GEOMETRICAL DRAWING-. The First Grade Practical Geometry. Intended chiefly for the use of Drawing Classes in Elementary Schools taught in connexion with the Department of Science and Art. By JOHN KENNEDY, Head Master of Dundee School of Art. 6d. 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Is. 6d. %* This work forms a Complete Course of French for Beginners, and comprehends Grammatical Exercises, with Rules; Reading Lessons, with Notes; Dictation; Exercises in Conversation; and a Vocabulary of all the Words in the Book. The Edinburgh High School French Conversation- GRAMMAR, arranged on an entirely New Plan, with Questions and Answers. Dedicated, by permission, to Professor Max Mullcr. 3s. 6d. KEY, 2s. 6d. The Edinburgh High School New Practical French READER: Being a Collection of Pieces from the best French Authors. With Questions and Notes, enabling both Master and Pupil to converse in French. 3s. 6d. The Edinburgh High School French Manual of CONVERSATION and COMMERCIAL CORRESPONDENCE. 2s. 6d. In this work, Phrases and Idiomatic Expressions which are used most frequently in the intercourse of every-day life have been carefully collectt-d. Care has been taken to avoid what is trivial and obsolete, and to introduce all the modern terms relative to railways, steamboats, and travelling in general. Ecrin Litteraire : Being a Collection of LIVELY ANEC- DOTES, JEUX DE MOTS, ENIGMAS, CHARADES, POETRY, etc., to serve as Readings, Dictation, and Recitation. 3s. 6d. Letter from PROFESSOR MAX MULLER, University of Oxford, May 1867. " MY DEAR SIR, I am very happy to find that my anticipations as. to the success of your Grammar have been fully realized. Your book does not require any longer a godfather; but if you wish me to act as such, I shall be most happy to have my name connected with your prosperous child. Yours very truly, MAX MU'LLER. " To Mcns. C. II. Schneider, Edinburgh High School." The French New Testament. The most approved PROTESTANT VERSION, and the one in general use in the FRENCH REFORMED CHURCHES. Pocket Edition, roan, gilt edges, Is. 6d. Chambaud's Fables Choisies. With a Vocabulary containing the meaning of all the Words. By SCOT and WELLS. 2s. Le Petit Fablier. With Vocabulary. For Junior Ola By G. M. GIBSOX, late Rector of the Bathgate Academy. Is. Gd. French. 19 Standard Pronouncing Dictionary of the French and ENGLISH LANGUAGES. In Two PARTS. Part I. French and English. Part II. English and French. By GABRIEL SURENNE, late Professor in the Scottish Naval and Military Academy, etc. The First Part comprehends Words in Common Use, Terms con- nected with Science and the Fine Arts, Historical, Geographical, and Biographical Names, with the Pronunciation according to the French Academy and the most eminent Lexicographers and Gram- marians. The Second Part is an ample Dictionary of English words, with the Pronunciation according to the best Authorities. The whole is preceded by a Practical and Comprehensive System of French Pronunciation. 7s. 6d., strongly bound. The Pronunciation is shown by a different spelling of the Words. 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Intended as a Class-book for the Student and a Guide to the Tourist. Map. Pronunciation marked throughout. 3s. 6d. Surenne's Pronouncing French Primer. Containing the Principles of French Pronunciation, a Vocabulary of easy and familiar Words, and a selection of Phrases. Is. 6d. stiff wrapper. Surenne's Moliere's 1'Avare : Comedie. Is. stiff wrap- per ; or Is. 6d. bound. Surenne's, Moliere's le Bourgeois Gentilhomme : Comedie. Is. stiff wrapper ; or Is. 6d. bound. 20 French. Surenne's Moliere's Le Misanthrope: Come'clie. Le MARIAGE FORCE : Comedie. Is. stiff wrapper; or Is. 6d. bound. Surenne's French Reading Instructor, Reduced to 2s. 6d. Hallard's French Grammar. 3s. 6d. KEY, 3s. Gd. Grammar of the French Language. BY AUGUSTE BELJAME, B.A., LL.B., Vice-Principal of the Paris International College. 2s. Beljame's Four Hundred Practical Exercises. Being a Sequel to Beljame's French Grammar. 2s. ** Both Books bound together, 3s. 6d. The whole -work has been composed with a. view to conversation, a great number of the Exercises being in the form of questions and ansv First French Class-book, or a Practical and Easy Method of learning the FRENCH LANGUAGE, consisting of a series of FRE\< n and ENGLISH EXERCISES, progressively and grammatically arranged. By JULES CARON, F.E.I.S., French Teacher, Edin. Is. KEY, Is. This work follows the natural mode in which a child learns to speak its own language, by repeating the same words and phrases in a great variety of forms until the pupil becomes familiar with their use. Caron's First French Beading-book: Being Easy and Interesting Lessons, progressively arranged. With a copious Vocab- ulary of the Words and Idioms in the text. Is. Caron's Principles of French Grammar. With numerous Exercises. 2s. KEY, 2s. Spectator. " May be recommended for clearness of exposition, gradual pro- gression, and a distinct exhibition to the mind through the eye by means of typo- graphical display : the last an important point where the subject admits of it." An Easy Grammar of the French Language. With EXERCISES AND DIALOGUES. By JOHN CHRISTISON, Teacher of Modern Languages. Is. 4d. KEY, 8d. Christison's Recueil de Fables et Contes Choisis, a 1'Usage de la Jeunesse. Is. 4d. Christison's Flenry's Histoire de France, Racontee a la Jeunesse. With Translations of the difficult Passages. 2s. 6d. French Extracts for Beginners, With a Vocabulary and an Introduction. By F. A. WOLSKI, Master of the Foreign Language Department in the High School of Glasgow. 2s. 6d. Wolski's New French Grammar. With Exercises. 3s. dl. Latin and Greek. 21 EDINBURGH ACADEMY CLASS-BOOKS. THE acknowledged merit of these school-books, and the high reputation of the seminary from which they emanate, almost supersede the necessity of any recommendation. The " Latin " and " Greek Rudiments " form an intro- duction to these languages at once simple, perspicuous, and comprehensive. The "Latin Rudiments" contain an Appendix, which renders the use of a separate work on Grammar quite unnecessary ; and the list of anomalous verbs in the " Greek Rudiments " is believed to be more extensive and complete than any that has yet appeared in School Grammars of the language. 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Hunter's Livy. Books XXI. to XXV. With Critical and Explanatory Notes. Reduced to 3s. Latin Prose Composition : The Construction of Clauses, with Illustrations from Cicero and Caesar; a Vocabulary containing an Explanation of every Word in the Text ; and an Index Verborum. By JOHN MASSIE, A.M. 3s. 6d. Dymock's Csesar ; with illustrative Notes, a Historical and Geographical Index, and a Map of Ancient Gaul. 4s. Dymock's Sallust; with Explanatory Footnotes and a Historical and Geographical Index. 2s. Caesar ; with Vocabulary explaining every Word in the Text, Notes, Map, and Historical Memoir. By WILLIAM M'DowALL, late Inspector of the Heriot Foundation Schools, Edinburgh. 3s. M'Dowall's Virgil ; with Memoir, Notes, and Vocabulary explaining every Word hi the Text. 3s. Neilson's Eutropius et Aurelius Victor ; with Vocabu- lary containing the meaning of every Word that occurs in the Text. Revised by WM. M'DOWALL. 2s. Lectiones Selectae : or, Select Latin Lessons in Morality, History, and Biography : for the use of Beginners. With a Vocab- ulary explaining every Word in the Text. By C. MELVILLE, late of the Grammar School, Kirkcaldy. Is. 6d. Macgowan's Lessons in Latin Reading. In Two PARTS. Part I., Improved by H. FRASER HALLE, LL.D. 2s. 17th Edition. Part II. 2s. 6d. The Two Courses furnish a complete Latin Library of Reading, Grammar, and Composition for Beginners, consisting of Lessons which advance in difficulty by easy gradations, accompanied by Exercises in English to be turned into Latin. Each volume contains a complete Dictionary adapted to itself. Latin and Greek. 23 Mair's Introduction to Latin Syntax : with Illustrations by Rev. ALEX. STEWART, LL.D. ; an English and Latin Vocabulary, for the assistance of the Pupil in translating into Latin the English Exercises on each Rule; and an Explanatory Vocabulary of Proper Names. 3s. 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This contains good translations of difficult passages, with exact information upon points of antiquities derived from the best and most modern authorities." Grammatical Exercises on the Moods, Tenses, and SYNTAX OF ATTIC GREEK. With a Vocabulary containing the meaning of every Word in the Text. On the plan of Professor Ferguson's Latin " Grammatical Exercises." By Dr FERGUSSON. 3s. 6d. KEY, 3s. 6d. %* This work is intended to follow the Greek Rudiments. Homer's Iliad Greek, from Bekker's Text. Edited by the Rev. W. VEITCH, Author of " Greek Verbs, Irregular and Defective," etc. 3s. 6d. Homer's Iliad, Books I., VI., XX., and XXIV.; with Vocabulary giving an Explanation of every Word in the Text, and a Translation of the more difficult Passages. By Dr FERGUSSON. 3s. 6d- 24 Latin and Greek. LATIN ELEMENTARY WORKS AND CLASSICS. 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Hunter's School Songs with Music, SECOND SERIES (for Junior Classes) Continued. The Golden Rule. Our Bonny Boat. Would you be loved? (Round for 3 Voices.) (Round for 4 Voices.) (Round for 3 Voices.) December Night. Welcome ! Welcome ! Song of the Bees. The Lambkin. little Stranger. Slaves to the World. Haste thee, Winter, Hark ! the little Birds (Round for 3 Voices.) haste away. are singing. (Round Queen of the Fairies' Evening Hymn. for 4 Voices.) Song. Song of the Brook. Wild Wood Flowers. Moonrise. Cradle Song. Never say Fail. Work and Play. The Change of the The Meadow Spring. The Year's last Hour Seasons. The Mariner's Song. is sounding. (Round Now we're met. Echo. (Round for 3 for 4 Voices.) (Round for 4 Voices.) Voices.) The Garment of Truth. The Rivulet. Man the Life-Boat! The Farewell. HUNTER'S SCHOOL SONGS For Advanced Classes. FIRST SERIES, containing 44 Songs, price 6d. The Lark. Ever-flowing, mighty The Mountain Shep- Lo! the Heavens are Ocean. herd Boy. breaking. Home, Sweet Home. TheTraveller'sReturn. Gather your Rosebuds. A Man's a Man for a' The Wayside Stream. Freedom's Land. that. Bright are the Glories. Humble is my little Rule Britannia. The cloud-cap t Towers. Cottage. (Pound ColdtheBlastmayblow. While Gladness hails for 3 Voices.) Sun, Moon, and Stars. the parting Year. Bright are Young Life's Oh, see the lovely Mariners, spread the golden Treasures. golden Sun ! Sail. Good Night! (Hound A Southerly Wind and LettheSmilesofYouth. for 4 Voices.) a Cloudy Sky. The Cuckoo. (Round Morning Star. (Sound for 3 Voices.) for 3 Voices.) Hark! the Bonny. Around the Winter Good Night. (Round for 3 Voices.) Fire so bright. Now Sing the girds. God Save the Queen. Come tothe Hills away! (Mound for 3 Voices.) When the rosy Morn Fair Morn is up. Come, honest Friends. appearing. Hark! 'tis the Wild (Rouncifbr 8 Voices.) How great is the Plea- Birds singing. Christmas. sure. (Round for 3 (Round for 3 Voices.) Come, conie, my Play- Voices.) The Spring breathes mate's.- \R*G Voices.) The Eagle. around us. Now Autumn rich. Sweet Spring is return- Work while you may." (Ottnon for 4 Voices.) ing. (Round for 4 Voices.) The Shipwreck. YA 043?0 Hunter's School Songs with Music. SECOND SERIES, containing 46 Songs, price 6d. Slumber, gentle In- Now we are met. The Pleasures of the (Round for 3 Voices.) \ Wood. Now the Sun, his jour- ! Beautiful Primrose, ney ending. JO Toil, from thee fant. By and By. The Quail's Call. Life is Onward. ! The Psalm of Life. I I comes every Joy. How Sweet tobe Roam- Cursed be the Wretch. ' The Skylark. ing. (Round for 3 , (Round for 3 Voices.) I Night March. Voices.) [ Let us all be up and The Sea-King's Song. The Fisherman's Cot- doing. tage. How lovely are The Lorelei. days of Spring. My Heart's in the See the Conq'ring the Highlands. Ode to Nature. 'Tis Hum - Drum. Hero comes. I love to Wander. Murmur, gentle Lyre. (Sound for 3 Voices.) . The Open Window. The Chapel. | The Wayside Well. Sweet the Pleasures. Ye high-born Spanish (Round for 3 Voices.) Lordly Gallants. L' Amour de la Patrie. The Sun is career- ing. Noblemen. Come again. Come, Follow me mer- rily. (Round for 3 Voices.) Patriotic Song. Home. Farewell to the Forest. A Fairy Song. Autumn Winds. Silent Night. Those Ev'ning Bells. Wind, gentle Ever- j Brightly the Sun is green. (Round for 3 beaming. (Round Voices.) j for 6 Voices.) Forest Song. ! In this little Island. SCHOOL, PSALMODY; Containing 58 Pieces, arranged for Three Voices. Price 4d. CONTENTS. Abbey. Bedford. Belgrave. Bethlehem. Bishopthorpe. Bur ford. Culross. Dismission. Doversdale. Doxology. Duke Street. Dundee. Dunfe rin line. Dusseldori. Erfurt. Evan. Franconia. French. Grace. Harrington. Huddersfield. Irish. Jackson's. Leipsic. London, New. Macclesfield. Manchester. Martyrdom. Mehcombe. Morven. Narenza. Newington. New St Ann's. Norwood. OldC. Old CXXIV. Sanctus. Selma. Soldau. Southwell. Stroudwater. St Ann's. St Bride. St David's. St John's. St Magnus. St Mary's. St Matthias. St Michael. St Paul's. St Stephen's. St Thomas. Tallis. Wareham. Weimar. Winchester. Wittemberg. York. *,* A Specimen Copy of any of these Books will le sent to Principals of Schools ly OLIVEB AND BOYD, on receipt of one-half the retail price in stamps. Edinburgh : OLIVER AND BOYD. London : SIMPKIN, MARSHALL, AND Co.