^^a^ .^M^^^^ University of California • Berkeley The Theodore P. Hill Collection of Early American Mathematics Books OOlTCISEl Mercantile Arithmetic, FOR COMMERCIAL COLLEGES, AND A Hand-Book for the Counting-Room, CONTAINING ALL THE MORE USEFUL AND PRACTICAL CALCU- LATIONS OF EVERY-DAY APPLICATION, EXPLAINED ON SCIENTIFIC PRINCIPLES. HENRY A. FABER, AUTHOR OF "THE STATISTICAL ACCOUNT-BOOK' AND <'FABER'S manual." THIRD EDITION.— REVISED. CINCINNATI, O. : Queen City Commercial College, N. W. Cor. Fifth and Walnut Streets. ATLANTA, GA. : Moore's Southern Business University. 1880. Entered, according to Act of Congress, in the year 1876, BY HENRY A. FABER, In the Office of the Librarian of Congress, at Washington. A// rights reserved. PREFACE. This treatise has been prepared with special refer- ence to the wants of students of commercial colleges. All questions which tend to perplex the learner, with little or no practical utility, have been carefully ex- chided. Rules have been almost altogether omitted. The student must see the operation, and, having seen it, his judgment will enable him to deduce some method of solution for himself The subjects are treated of in the order of their simplicity and utility. They are so independent of each other, however, that the teacher may introduce them in whatever order his judgment may dictate. Numerous exercises will be found on short methods of calculation. In fact, every topic which admits of more than one form of solution, has been treated by the shortest practical method. The Author. Cincinnati, 1880. CONTENTS: TABLE OF MONEYS, WEIGHTS, MEASURES, etc.. Page 6. ARITHMETIC. Chapter. 1. Introduction . . . Page. . 15 2 Notation and . . , Numeration 17 3. Addition . 19 4. Subtraction . . . . 25 5. Multiplication . . .28 6. Division .36 7- Easy Fractions . . .43 8. Decimals .53 9. Short Methods . . . OF Multiplication and Division . . . .55 10. Percentage .... .61 11. Bills-Invoices . . . .67 12. Long Division , . . 78 13. Time .82 14. Simple Interest , .83 15. Compound , . . . . 90 16. Annual .... . 91 17. Partial Payments . .92 Page. . .94 Chapter. 18. Bank Discount 19. True Discount . . . .99 20. DiscT. Interest Bear- ing Notes . . 100 21 Complex Percentage 103 22. Time Tables .... 114 23. Average 118 24. Ratio 138 25. Proportion 139 26. Partnership .... 143 27. Joint Stock Co . . . 146 28. Compound Numbers 148 29. Foreign Exch. . . 151 30. Importing 157 31. Farming 165 32. Lumber Measure . 169 33. Fractions 171 34. Duodecimals . . . 190 35. CoMP. Proportion . 193 36. Gauging 195 INDEX, 197 to 200 inclusive. MONEYS, WEIGHTS AND MEASURES- MONEYS. Federal Money. — The unit of our money is the dollar. Accounts are kept in dollars and cents. The coins are, the double-eagle, eagle, half-eagle, quarter-eagle, and dollar; the trade dollar, half-dollar, quarter dollar, dime, and half-dime; the three, five, two, and one cent pieces. Federal money being decimal currency, ten of a lower denomination make one of a higher: 10 mills = 1 cent, 10 cents =r 1 dime, 10 dimes = 1 dollar, 10 dollars -= 1 eagle. Signs: m, mills; ^, cents; $, dollars; E, eagle. British Money. — The unit of British money is the pound sterling. Accounts are kept in pounds, shillings and pence (farthings are written as fractions of a penny): 4 farthings = 1 penny, 12 pence = 1 shilling, 20 shillings = 1 pound. Signs: d, pence; s, shilling; £, pound. The British coins are, the penny, the shilling, the crown, the sovereign, and the guinea. The value of the crown is 5 shillings; the sovereign, 20 shillings; the guinea, 21 shillings. German Money. — The unit of the money of the German Empire is the rixmark. 10 pennies (pfennige) = 1 silver- gropchen (silbergroschen) ; 10 silver-groschen = 1 rixmark (reichsmark). Signs: d, pennies; sg, silver-groschen; Rm, rixmark. French Money. — The unit of French money is the ifranc. 10 centimes = 1 decime; 10 decimes =^ 1 franc. Signs: c, centimes; d, decimes; fc, francs. MONEYS, WEIGHTS AND MEASURES. WEIGHTS. Mint or Troy Weight, used at the mint, and by jewel- ers: 24 grains = 1 pennyweight, 20 pennyweights = 1 ounce, 12 ounces = 1 lb. Signs, gr. grain, pwt. pennyweight, oz. ounce. Apothecaries' Weight.— Used in compounding medi- cines : 20 grains = 1 scruple, 3 scruples = 1 drachm, 8 drachms = 1 ounce, 12 ounces = 1 lb. Sighs, gr. grain, 9 scruple, 3 drachm, g ounce, R) pound. (1 lb.= 5760 gr.) Commercial Weight, used by Grocers, Druggists, Hard- ware dealers, etc : 16 ounces == 1 pound, 2000 pounds = 1 tun. Signs, oz. ounces, lbs. pounds, cwt. hundreds, T tuns. (1 lb.= 7000 gr.) Hay is weighed by Commercial Weight Avoirdupois — Old Commercial Weight of the U. States 16 drachms = 1 ounce, 16 ounces = 1 lb., 28 lbs. = 1 quarter 4 quarters = 1 hundred, 20 cwt. ^= 1 tun. Pig Iron (chill mold), Iron Ore, Bituminous Coal, and Hemp are weighed by avoirdupois weight. The Avoirdupois weiight is the Commercial weight of Grea BHtain. Metric Weight.— The unit of weights of the metric sys tem is the Gram. The Greek prefixes (deka, 10; Jiedo, 100; kilo, 1000) form the denominations above tlie unit. The Latin prefixes [deci, 10; centi, 100; milliy 1000) form the denomina- tions below the unit. 1 kilogram = 10 hectogram = lOOfdeca- grams = 1000 grams. 1 milligram = -^-^ centigram = -^ decigram = y^Vff g^^^* The weight of a gram is equal to 15.432 grains of Troy weight. Signs: DG., deckagram; HG., hectogram; KG., kilogram; G., gram; dg., decigram; eg., centigram; mg., milligram. . Note. — The oz. of the Mint and Apoth. weights are the same, viz: 480 grains. The oz. of the Com'l and Avoid. •weights are the same, viz: 437 J grains. MONEYS, WEIGHTS AND MEASURES. Weights op Produce per Bushel, according to usage in Cincinnati, and as fixed by statute in Ohio: Psage. lbs. Apples, dried 26 Barley 48 Barley malt, weight of bags included 34 Beans 60 Bran 20 Bran shorts 25 Broom-corn 30 Buckwheat 60 Coal, bituminous 80 cannel 70 Charcoal 30 Coke 32 Castor beans 46 Corn, shelled 66 in ear.. ..68 and 70 Hair, plastering 8 wet 16 Hominy 60 Lime, slacked 51 Malt Meal, corn 60 Middlings 40 Oats.... 32 Onions 66 Onion sets 23 Stat. Usage. Stat, lbs. lbs. lbs. 25 Peaches, dried 38 33 48 Peas 60 60 green 24 Plaster and hair 118 Peanuts, roasted 22 Potatoes, Irish 60 60 sweet 60 Rye..... 66 66 Rye malt, wt. of bags included 40 Salt 66 Seed, clover 60 60 timothy 46 45 flax 66 66 hemp 44 44 orchard grass.... 14 Hungarian grass 50 60 blue grass 14 millet 60 50 canary 60 sorghum 46 Ship stuff 40 i Shorts 30 32 ; Turnips 60 j Wheat 60 60 I Water, distilled 77.6274 60 60 Weight of a Cubic Foot oj' Cast iron 460.56 Wrought iron 486.65 Steel 489.8 Copper 565 Lead 708.76 Brass 537.75 Tin 456 White pine... 29.56 Loose earth or sand 95 Common soil ; 124 Strong soil 127 Clay 135 lbs. Yellow pine 33.81 White oak 35.2 Live oak , 70 Salt water (sea) 64.3 Freshwater 62.5 Air 07629 Steam 03689 Clay 135 Sand 113 Cork 16 TaJlow 59 Brick 119 MONEYS, WEIGHTS AND MEASURES. MEASUEES. Linear Measure is applied in measuring length and dis- tance: 12 inches = 1 foot, 3 feet = 1 yard, 5^ yds. = 1 rod, perch or pole, 40 rods = 1 furlong, 8 furlongs or 320 rods --■- 1 mile. Sign-H, in. inches, /i5. feet, yd. yard, rd. rod, fur. furlong, mi. mile. Furlongs are seldom used. 5280 ft. = 1 mile. 1 palm = 3 inches, 1 hand = 4 inches, 1 span = 9 inches, 1 meter = 3.28 feet. Scripture Long Measure. — A. digit = .912 inches, a palm = 3.648 inches, a span = 10.944, a cubit = 1 foot 9.888 inches, Si fathom = 7 feet 3.552 inches. Jewish Long Measure. — A cubit = 1.824 feet, a Sabbath dojy^s journey = 3648 feet, a mile =-. 7296 feet, a day^s journey = 175104 feet, or 33 miles 864 feet. Cloth Measure. — Cloth is measured by the yard and frac- tional parts of a yard, as half, quarter, eighth, sixteenth, etc. The yard contains 3 feet, or 36 inches. Marine Measure. — Used at sea: 6 feet = 1 fathom, 120 fathonis = 1 cable length, 880 fathoms = 1 mile. Metric Long Measure. — The unit of Long or Lineal Measure of the metric system is tlie Meter (whence the name. Metric.) The Greek prefixes, deka, etc. ; and the Latin, ded, etc., form the other denominations the same as by the gram. The meter is equal to 39.3685 inches of our linear measure. The signs of the meter and denominations above are written with capitals : M. for meter, KM. for kilometer ; those of th< denominations below the meter, with small letters — dm., deci- meter, etc. Surveyors' Measure. — 7y%% inches — 1 link, 25 links = 1 rod or pole, 4 poles or 100 links = 1 chain, 80 chains ^= ] mile, 10 sq. chains = 1 acre, 640 acres or 6400 sq. chains = 1 sq. mile or section of land. A square rod contains 272^ sq. feet. An acre contains 4356C sq. feet. MONEYS, WEIGHTS AND MEASURES. Circular Measure. — Uf^ed in reckoning latitude, longi- tude, etc., and in trigonometrical calculation: 60 seconds = 1 minute, 60 minutes = 1 degree, 30 degrees = 1 sign,* 12 signs = 1 circle. Signs. — ^^ seconds, ^ minute, ° degrees, S. sign, C. circle. 35° 3^ 2^^ would read thirty-five degrees, three minuteSj and two Measure of Time. — 60 seconds = 1 minute, 60 minutes = 1 hour, 24 hours = 1 day, 7 days = 1 week, 30 days = 1 lunar month, 365 days = 1 year, 12 months = 1 year. Square Measure is used for measuring surfaces: 144 sq. in. = 1 square foot, 9 square feet ^= 1 square yard, 30J square yards = 1 square rod, 160 square rods = 1 acre, 640 acres = 1 square mile. Signs, sq.ft., sq. yds., sq. rds., A., M. Metric Square Measure. — The unit of measure for large surfaces is the Are, from whicli are derived the Hectare and Centare. For smaller surfaces tlie denominations are tlie same as for measures of length, with the addition of the word square. Are = 100 sq. meters, or 1 sq. dekameter, or 119.6 sq. yds. Cubic Measure is applied to solids, and comprises length, breadth, and thickness, or depth. A cubic foot contains 1728 inches, that is 12 times 12 times 12 inches; a cubic yard con- tains 27 feet, or 3 times 3 times 3 feet. Metric Cubic Measure. — The Stere may be called the imit for cubic measure. It is equal to a cubic meter or 1.308 yards. Wood Measure. — Wood is sold by the cord, which should contain 128 cubic feet, closely piled, and 138 feet if stowed in a boat or barge. A pile of wood measuring 8 feet long, 4 feet wide and 4 feet high, contains a cord. 10 MONEYS, WEIGHTS AND MEASURES. Stone Measure is used for measuring masonry, which is sometimes paid for by the foot, but usually by the perch, 24| or 25 cubic feet 1 perch; the former for private, the latter for public contracts, as railroad or government work. A wall 16J feet long, IJ feet thick and 1 foot high contains a perch. Bricklayers' Measure. — ^The common dimensions of a brick are 8 inches long, 4 inches broad, and 2 inches thick. There are 21 bricks in a cubic foot of wall, including mortar. A wall 8 in. or 1 brick in thickness contains 14 bricks to the sq. ft. of surface. 12 " IJ '' " " 21 '' " " ** 2g a 2 " " " 28 " ** " " Dry Measure.— Used for measuring grain, fruit, etc.: 8 quarts = 1 peck, 4 pecks = 1 bushel. Signs, qt. quart, 'pk. peck, hu, bushel. Note.— The bushel is a cylindrical vessel, 8 inches deep and 1834 diameter, inside, and contains 2150.42 cu. in. Coal Measure. — Coal is usually sold by the bushel, which should contain 2688 cubic inches. Liquid Measure. — For measuring all liquids, except milk, beer and ale : 4 gills = 1 pint, 2 pints = 1 quart, 4 quarts =1 gallon. Barrels, tierces, etc., are no longer used as measures of capacity ; they are all gauged and reckoned by gallons. Remark.— The gallon contains 231 cubic inches. Ale or Beer Measure. — The gallon contains 282 cubic inches, and the number of pints or quarts in a gallon the same as in Liquid Measure. Metric Measure of Capacity. — The lAter is the unit of measure for capacity, and is equal to a cubic decimeter or 1.0567 quarts of United States liquid measure. MISCELLANEOUS. 11 MISCELLANEOUS. Effects of CoaL — Small coal produces about f the effect of large coal of the same species. CharcoaL — The best quality is made from oak, maple, beecli and chestnut. Wood will furnish, when properly burnt, about 16 per cent, of charcoal. A bushel of charcoal from hard wood weighs about 30, from pine, about 29 lbs. Coke. — A bushel of the best coke weighs 32 lbs. Coal fur- nishes from 60 to 70 per cent, of coke by weight. Kelative Heating Power of Different Kinds op Fuel, according to Weight. Charcoal 100, mineral or stone coal 82^, dry wood 48 J. Hence, if a tun of charcoal cost no more than a tun of mineral coal, the former would be the cheaper fuel by 21 per cent. Kelative Heating Power of Different Kinds op Wood, according to Measure. Shell-bark Hickory Pignut, . . White Oak, . White Ash, . Dog Wood, . Scrub Oak, . Witch Hazel, Appletree, Red Oak, . . White Beech, Black Walnut, Black Birch, 100 93 81 77 78 73 72 70 69 65 65 63 Yellow Oak, Hard Maple, White Elm, . Red C'edar, . Wild Cherry, Yellow Pine, Soft Maple, . Chestnut, . . Yellow Pophir, Butternut, . White Birch, White Pine, . 60 60 58 56 55 54 54 52 52 51 48 42 Digging. — 23 cubic feet of sand, or 18 cubic feet of earth, or 17 cubic feet of clay make a tun. 18 cubic feet of gravel, or earth, before digging, make 27 cubic feet when dug. 12 ' MISCELLANEOUS. Gas. — 1.43 cubic feet of gas per hour give a light equal to that of a candle; 1.96 cubic feet equal 4 candles ; 3 cubic feet equal 10 candles. Horse Power in machinery is reckoned at 33,000 lbs. raised one foot in a minute, but the ordinary work of a horse is only 22,500 lbs. per minute for 8 ho^rs. Strength of a Man. — The mean effect of the power of a man, unaided by a machine, is the rais- ing 70 lbs. 1 foot high in a second for 10 hours a day=i of the power of the horse. Note. — Two men working at a windlass at right angles to each other, can raise 70 lbs. more easily than one man can 30 lbs. A foot soldier travels 70 yards, making 90 steps in one minute, common time. In quick time, 86 yards, making 110 steps. In double quick, 109 yards, making 140 steps. Average weight of men, 150 lbs. each. Five men can stand in a s]Dace of 1 square yard. A man without a load travels on a level ground 8-| hours a day, at the rate of 3.7 miles an hour, or 31J miles a day. He can carry 111 lbs. 11 miles in a day. A porter going short distances and returning un- loaded, can carry 135 lbs. 7 miles a day. He can carry in a wheelbarrow 150 lbs. 10 miles a day. The muscles of the human jaw exert a force of 534 lbs. Hay. — 10 cubic yards of meadow hay weigh a tun. When the hay is taken out of old, or the lower part of large stacks, 8 to 9 cubic yards will make a tun. 10 to 12 cubic yards of clover, when dry, weigh a tun. Hills in an Acre. — 3 feet apart, there are 4840 hills in an acre. PAPER — SUNDRIES. 13 PAPER. SIZES OF PAPER MADE BY MACHINERY, FLAT PAPER. Letter, 10X16 Com'l Letter, . . . 11X17 Packet, 12X19 Foolscap, 13X16 Cap, 14X17 Crown, 15X19 Demy, 16X^1 Folio, 17X22 Check Folio, .... 17X^4 Tax Duplicate, Medium, Royal, . . Super Royal, Elephant, . Imperial, Columbier, Atlas, . . . Antiquarian, . 17X30 . 18X23 . 19X24 . 20X28 . 23X28 . 23X31 . 23X34 . 26X33 . 31X53 FOLDED PAPER. DESIGNATIONS OF SHEETS ACCORDING TO FOLDS OF PAPER. Folio. — A sheet folded in two leaves. Quarto. — A sheet folded in four leaves. Octavo. — Or 8vo, a sheet folded in eight leaves. Duodecimo. — Or 12mo*, a sheet folded in twelve leaves. 24 sheets = 1 quire, 20 quires = 1 ream, 2 reams = 1 bundle. Book-binders count from 16 to 20 sheets to a quire in binding account books. WRAPPING PAPER. Wrapping paper is sold by the bundle, which are generally ^hort count. The full cou7it reams contain 20 qrs. of 24 sheets each. SUNDRIES. 12 articles = 1 dozen. 12 dozen = 1 gross. 12 gross = 1 great gross. 20 articles = 1 score. 1 barrel = 200 lbs. 14 lbs. of flour = 1 stone. 14 stones of flour = 1 bbl. 1 bbl. of flour =196 lbs. 1 barrel = 31 M gallons. 1 hogshead = 2 bbls. ♦ The size of this book is 12mo. 14 THERMOMETERS. THERMOMETERS. The Celsius or Centigrade thermometer has the zero at the freezing point of water, and the distance between that and the boiling point of water divided into 100 degrees, — hence the name Centigrade. The Beaumur thermometer has the zero at the freezing point, and 80° between that and the boiling point of water. The Fahrenheit thermometer has the zero at 32° below the freezing point of water, and has 180° between freezing and boiling point of water. To convert degrees of Centigrade into degrees of Fahrenheit, multiply the degrees of Centigrade loy 9, divide the product by 5, and add 32 to the quo- tient, the answer will be degrees of Fahrenheit. To convert degrees of Eeaumur into degrees of Fahrenheit, multiply the degrees of Eeaumur by 9, divide the product by 4, and add 32 to the quotient, the answer will be degrees of Fahrenheit. To convert degrees of Fahrenheit into Centigrade, subtract 32 from the degrees of Fahrenheit, multiply the remainder by- 5, and divide the j^roduct by 9. To convert degrees of Fahrenheit into Eeaumur, subtract 32 from the degrees of Fahrenheit, multipl}^ the remainder by 4, and divide the product by 9. To convert Centigrade into Eeaumur, multiply the degrees of Centigrade by 4 and divide the product by 5, To convert Reaumur into Centigrade, multiply the degrees of Eeaumur by 5 and divide the product by 4. The sum of the degrees of Centigrade and Eeau- mur plus 32 will give the degrees of Fahrenheit. THE CONCISE MEROAKTILE ARITHMETIC. I. INTRODUCTION. AEITHMETICAL DEFINITIONS. Article 1. Arithmetic is the science of numbers. Art. 2. The theory of Arithmetic treats of the properties and relations of numbers. Art. 3. The practice of Arithmetic shows the application -of number to business, the mechanics' art, etc. Art. 4. Quantity is any thing that can be in- creased or diminished. Art. 5. Notation is the art of representing num- bers by figures. Art. 6. Numeration is the art of reading figures when arranged to represent numbers. Art. .7. The four fundamental rules of Arithme- tic are : Addition, Subtraction, Multiplication, and Division. Art. 8. Addition is the art of uniting two or more numbers into one. The result obtained by adding is called Amount or Sum. 16 ARITHMETICAL DEFINITIOJ^S. Art. 9. Subtraction is the method of finding the difference between two numbers. The result ob- tained is called, Remainder, Art. 10. Multiplication is the process of taking one number as many times as there are units in another. The result obtained is called, Product. Art. 11. Division is the method of ascertaining how many times a given number is contained in another. The result obtained is called, Quotient. Art. 12. Percentage is the method of reckoning by hundredths. Art. 13. A Fraction is a part or a number of parts of a whole. Art. 14. Interest is a percentage allowed for the use of capital. Art. 15. Bank Discount is a percentage deducted from capital loaned for the use of such capital. Art. 16. True Discount is the difference between the present worth of a note and the amount for which it is drawn. Art. 17. Proportion is an expression of equal ratios. Art. 18. Arithmetical Signs. + X 8 the sign of equality. 8 the sign of addition. 8 the sign of subtraction. s the sign of multiplication. 8 the sign of division. s the decimal sign. s the sign of proportion. s the sign of percentage. NOTATION AND NUMERATION. 17 11. NOTATION AND NUMERATION. Article 1. Notation is the art of representing numbers by symbols, called figures or digits. There are ten of these figures : 0123456789 nought one two three four five six seven eight nine The first is also called zero^ or cipher. Art. 2. When a larger number than nine is to be represented, two or more figures are used. Art. 3. Numeration is the method of reading these figures when arranged to represent numbers. Eor this purpose they are usually divided into pe- riods from the right, COMMON METHOD. Art. 4. According to the Common or French method of numeration, the first period on the right contains units, tens, and hundreds. 12 3. hundreds tens units. The second period contains units, tens and hundreds 0^ thousands ; the third, units, tens and hundreds of millions; the fourth, billions; the fifth, trillions; the sixth, quadrillions; the seventh, quintillions ; the eighth, sextillions ; the ninth, septillions ; the tenth, octillions; the eleventh, nonillions ; the twelfth, de- cillions. The higher denominations are formed by prefix- ing to decillions the Latin words, uno, duo, tre, qua- tuor, quin, sex, septen, octo, noven. ENGLISH METHOD. Art. 5. According to the English method, the first six orders have the same names and signification as 2 18 NOTATION AND NUMERATION. those of the French. Every period, however, con- sists of six orders. The second period is million; the higher denominations are named the same as by the Common method, but have different signifi- cations. Remark. — It will be noticed that each period, according to tlie Common method^ is one thousand times the preceding one, and according to the English method one million times. Hence, according to the Common method, a billion is a thousand millions, and according to the English a billion is a million millions. 26,839,506,720,052,005 according to the Common method would read : Twenty-six quadrillions, eight hundred and thirty-nine trillions, five hundred and six billions, seven hundred and twenty millions, fifty-two thousand and five. The same, according to the English method, would be pointed off thus: 26839,506720,052005, and read, twenty-six thousand eight hundred and thirty-nine billions, f^YQ hundred and six thousand seven hundred and twenty-millions, fifty-two thou- sand and five. EOMAISr NOTATIOISr. Art. 6. In Roman notation numbers are repre- sented by letters, as follows : I, one ; V, five ; X, ten; L, fifty; C, one hundred; D, five hundred; M, one thousand. A line over a letter increases its value one thousand times: thus, I> denotes ^yq hundred thousand. A letter of less value placed before one of greater value diminished the latter the amount of the value of the former: thus, -CM denotes nine hundred. MCMLYDXEYII reads, one million nine hun- dred fiiPty-five thousand, five hundred and forty- seven. MDCCCLXXYI = 1876. ADDITION. 19 III. ADDITION. Art. 1. The process of uniting two or more num- bers into one is called Addition. Art. 2. The result obtained is called sum^ amount ^ totals OY footing. Art. 3. The sign +, when placed between two numbers, indicates that they are to be added together. Note. — This book being a Mercantile Arithmetic, it is thought best to omit short examples in addition, as the parties using the same are supposed to be acquainted with the funda- mental rules of the science, but need to acquire accuracy and rapidity. The principles will, therefore, be stated, and such hints given, which, when put into practice, will enable the learner to add up rapidly and correctly. Art. 4. It is necessary, in performing the opera- tions in addition, to place units under units, hundreds under hundreds, etc. EXAMPLES. To add forty, three hundred and seventy-two, one thousand eight hundred and sixty-seven, and eight hundred and ninety-five, they should be arranged as follows : 40 372 1867 895 3174 ^. We commence the process by Adding the right hand or unit column, beginning with the lower figure, thus : 5 and 7 are 12, and 2 are 14, that being 4 units and 1 teen. The unit (4) is placed under the unit column as the result, and the teen (1) is added to the second or teens column. Next the teens are to be added, thus : 1 (the 1 tee q- obtained by adding the unit column) and 9 are 10, and 6 are 16, and 7 are 23, and 4 are 27; namely, 27 teens or 7 teens and 2 hundred. 20 ADDITION. The 7 teens are placed under the teens column as the result, and the 2 hundred are added to the third or hundreds column, and the hundr-ed column is added in the same way, resulting in 21 as the answer; namely, 21 hundred, or one hundred and 2 th(msand. The one is placed under the hujidred column as the result, and 2 thousand is-added to the 1 thousand in tlie ex- ample, resulting in three thousand, which 3 is placed under the thousand column as the answer — the whole footing will now read (commencing at the left) three thousand one hundred and seventy-four (3174). Add the following: 2365 92245 925683 7629548 18293 28392 968542 9832965 8769 67268 768656 7629824 2965 63629 329871^ 4567897,^ 3276C 24432^ 123456 7632851 Art. 5. In order to acquire rapidity, the learner should, from the beginning, avoid counting by their fingers, but should familiarize themselves with the catch figures. The catch figure is the unit figure of the result of adding two units together; thus: When- ever 5 and 6 are added together, the unit figure in the result (11) is one ; whenever 6 and 9 are com- bined, the unit figure is 5. Class exercises on giving the catch figure and on applying it will be found both interesting and profitable. The exercise may be conducted as follows: Where 7 and 9 are combined, the unit figure is — ? (6). 17 and 9 are — ? (26). 37 and 9 — ? 67 and 9 — ? 27 and 9 — ? .The teacher putting the questions, and the scholars in concert Responding by giving the answer. 5. 13965+6725+68349+76587+9825+99542= * ^ 6. 2592+18596+9382+6732+95876+29326= 7. 8549+8329+6784+7376+92542+93586= , 8. 3576+7654+3295+7628+27654+7629= 9. 3733+9258+8975+9268+9327+7652= 10. 6686+8259+9762+3876+8585+7895= 11. 2936+9286+7654+6832+9257+6873= * The answers will be found at the close of the chapter (page 24). ADDITION. 21 Art. 6. The process of adding Federal Money differs from the foregoing only in the use of the dollar ($ ) and decimal ( . ) signs. 12. $568.32+, $965+, $985.20^<^ $— ? Note. — In adding federal money the learner must "be care- ful to place the decimal points (the sign separating the dollars and cents) of the amounts of money to be added under each other, thus: $568.32 965. 985.20 13. $375.15+ $950.+ $876.51+ $7.57+ $987.56+ $781.+ $659.16+ $286.56+.56+ $185.20r= / 14. $878.10+ $758.+ $238.68+ $875.+ $658.99+ /$878.+ $751.87+ $2.85+ $286+289.54= 15. $751.+ $518.91+ $361.98+ $678.10+ $777.67 + $765.+ $958.+ $392.51+ $682.19+ $775.20= 16. $868.19+ $18.+ $85.88+ $567.50+ $678.96+ $879.+ $759.15+ $894.26+ $824.18+ $982.56= 17. $781.59+ $759.10+ $899.99+ $569.+ $569.78 + $656.71+ $871.+ $326.50+ $98.27+ $976.58= 18. $798.15+ $7.76+ $786.56+ $437.+ $788.15+ $788.88+ $935.62+ $92.52+ $768.92= 19. $889.+ $878.99+ $878.95+ $898.10+ $897.+ $987.54+ $651.25+ $329.77+ $628.95+ $628.92= 20. $18146+$71.25+$641.04+$4501+$87700= 21. $1770.03+$1006.01+ $364.01+$5442.99= 22. $2310.00+$1068.24+$26107.18+$2136.18= 23. $109.79+ $999.99+ $666.56+ $449.99= 24. $777.00+$7999.00+ $6666.00+$6730.15= 25. A merchant has 29 pieces of silk in 1 package, 35 in another, 79 in a third. In the first, there are 1497 yards, in the second, 2173, in the third, 4130. How many pieces, and how many yards in all? 26. $1.23+283+$685.04+$123.45+$78= 22 ADDITION. 27. 31465+2316532+107+3790+465321+36545G3+ 107653+23650+1007+30o72+503102+21063 is how much ? 28.18230+476+41034+9875+65432+5678+12090+ 9387+8276+565 + 13654+443z=z:how much? Ans. Sum of 27 and 28, 73440G5. .J^J) 29. 46853 + 9654+45679 + 9837+18708+7967+485 >yS^78963+84989+12345+7069+8090+7483+96748==? TAKING TWO AND THREE FIGURES AT A TIME. To enable scholars to grasp two and three figures at a time, and carry them up as one, they might be exercised on the blackboard in such sums as the following: 136377436 146739213698 954186987 782163846673 2 1 3 4 1 4 1 3 6 .2 1 2 Such exercises ought to be of frequent occurrence and scholars encouraged to answer in concert. The answers should be given instantaneously, naming only the unit figure, as shown in the column below: 8456 I ^ After writing on the right of the first column the 1345 / figures produced by pairing, the teacher may lead the 156^1^ class in adding, thus: 17 and 3? 30 and 1? 41 and 9456 > ^^ 47 and 7? 54 and 1? 65 and 6? 81 and 6? 96 }5 and 2? 108 and 11? 8998 \n It will be observed that the tens produced in forming 1898 j the pairs were not named. The same course should be 1 A7Q r ^ pursued in the class, as the learner is unconscious of 1684 1 making as great an effort as he really does. 7893 ) ' When the ten is omitted by mistake, attention should 1453 )p be called to it by giving the full number, as 15 or 11 ;} 1763 / instead of 5 or 1. Qft7A \ ^ "^^^ other columns should be added without the aid 7897 ^ °^ ^^^ marginal figures. 2586 j ^ After thorough drill in this, the class should be 8529^- taught to take three figures and feven four as rapidly 1438 / as one. ADDITION. 23 30. Find the sum of 8934, 16749, 809, 67549, 98697, 746839, 1498, 829555, 9218967, 8347912, 968000, 74685. Total of the preceding two, 20815046. Foot up the following columns : 31 32 33 34 35 31645 3454 4213 1565 3654 98760 2136 6314 3657 1095 3G875 1364 2316 5437 9014 57893 4633 1369 3457 6789 14567 9897 9306 1234 9687 34564 7879 6039 3421 5764 46387 2164 8109 6789 1567 93178 4163 9876 1746 9139 78163 4569 6789 3456 1456 64518 5496 4567 1378 2345 17514 6428 5679 5932 5432 45678 8297 3263 4567 6542 21364 9287 9457 1657 1395 7198 7928 1459 6574 3642 3165 9872 1455 5638 1365 4124 8729 •9375 4932 2315 1345 9314 5976 1397 9365 3146 3162 7639 9765 3510 4165 2136 7938 3765 1096 3216 9364 3959 1456 3765 36. Add together the following numbers: 313, 2109, 6785, 2736, 798, 987, 21363, 316, 4934, 2178, 1009, 396, 298, 2753, 607, 3145, 213, 6709, 6093, 190, 2130, 2160, 716, 213, 9876, 45678, 2137, 2198, 9039, 6789, 3097, 4684, 2136, 2178, 5672, 1987, 6789. Answers promiscuously arranged: 95368, 77823, 120272, 115098, 667465, 88937, 171411. The Teacher should not permit his scholars to divide these col- umns when adding, nor should he allow them to resort to the aid of strokes or practice counting on their fingers. 37 38 39 40 41 3286 2467 34564 46321 3614 6713 109 12345 13632 1364 3654 3178 65435 14567 5436 176 145 87654 53678 7835 3976 6178 34564 86367 4678 6345 4156 13682 85432 8793 9823 7532 75671 36457 701 6023 9890 86317 21836 9804 1367 6821 24328 17354 1306 8965 9854 98713 63542 717 8632 3821 21345 78163 2103 1034 5843 1286 82645 6397 6312 1936 78654 34685 1096 4593 7136 19876 31768 2130 3687 9876 93643 65314 3107 5006 2863 6356 68231 167 7164 123 78397 64037 2109 1763 7436 21602 34685 3678 2139 1567 71346 35962 2176 8236 2563 28653 21363 5432 7860 8432 17648 78636 2137 3613 1345 82351 19854 28639 109 8736 21368 80145 1765 1756 8654 78631 87654 371 6386 1263 17639 12345 71031 9890 1345 82360 78654 1463 8243 3093 45671 12345 3168 Answers: 42838. 48213, 217166, 274993, 162504, 45063, 37293080, 25668, 275966, 3116208, 185140, 378363, 434870, 136751, 126362, 1300099, 181217, 171411, 88937, 77823, 115098, 120272, 66746, 20380194, 57436, 7158925, 1325672, $2518.52, $5617.03, $6557.68, $6660.56, $5109.27, $6508.52, $5403.86, $7668.47, $31- 621.60, $2226.33, $43.00, $22172.15, 7800, 2500, 2450, 21621200, 1243883. SUBTRACTION. 25 IV. SUBTRACTION. Art. 1. The process of taking a lesser number or quantity from a greater of the same kind or denom- ination is called Subtraction. Art. 2. The result obtained is called difference^ re- mainder, or excess. Art. 3. The sign of subtraction is — , and is called minus. 8 — 2 reads eight minus two. Art. 4. 1. Find the difference between 786 and 323. Solution. — We place the smaller number under the larger one, units under units and hundreds under hun- ^86 dreds, etc., and proceed to subtract from the right to the left, viz : 3 from 6 leaves 3 ; next we subtract the teens : 2 from 8 leaves 6, this we place in the teens place , 3 323 ^ ii'Uiii o leiives u, liii» we pjaue in Lue Lceii.-s pju-Lc , o AQO (hundred) from 7 (hundred) leaves 4 (hundred), which ^"*^ is placed in the hundreds place. The answer (Remain- der) is 463, 2. 23964— 12853 3. 2986— 258 4. 6972325— 4232323 5. 6896542— 84312 6. 276289995 — 16278585 7. 32987632 — 11976412 Art. 5. 1. Find the difference between 5354 and 897. Solution. — In this case we find that the several j ^^ JJ' ^^ numbers in the 5it6^?-a/m7i(i* are greater than those of 2*3'5*4 the minuend.^ We can not take 7 from 4, so we take o q ^• one from the teens in the 7mn. and add it to the units, ^ 9 ' making 14 — 7 from 14 leaves 7. Having taken one 1 4 5 7 from the teens, leaves 4. 9 from four we can not take, so we again take one of the next figure in the min. (3) and add it to the teens, making again 14. 9 from 14 leaves 5. 8 from 2 (1 having been taken from the 3 to add to the teens) we can not take, so we proceed to take 1 from the thou- sands, which makes 10 (hundred) +2 (hundred) gives us 12 less 8 makes 4. One having been taken from the 2 (thousand) leaves 1. Answer, 1457. ^Subtrahend, the number to be taken from the minuend. fMinuend, the greater number, from which the lesser is to be subtracted. 2G SUBTRACTION. 2. 2345678— 689829 4. 6123546 — 5261862 6. 3254298 — 1185169 3. 621129 — 509826 5. 921654 — 629827 7. 325627 — 124939 Art. 6. In order to ascertain the difference be- tween the sum of two columns, subtraction may be formally dispensed with by adding the largest column first, and by adding in the difference thus : $286.98 385.46 928.54 326.28 Solution.— Having obtained the sum of the larger column, $1927.26, we proceed to add up the smaller, viz: 2 -I- 6 + 2 are 9 and 7 (to make the result ^he same as the units in the larger column) are 16 1 (carried from the result of the units) + 4 + 3 + 4 = 12, the unit fig- ure of this result being the same as that of the dimes in the larger column, O is the difference 1 (carried from the dimes column) + 64-9 + 5 = 21 + 6( to make the difference) = 27 2 + 6 +7 + 8 =» 23; -r 9 (to make the difference) = 32 3+7+2 + 3 = 15 + 4 to make the difference = 19. Thus giving as the difference, $496.07. . $385.42 279.35 766.42 $1927.26 $1927.26 Proof: S385.42 279.35 766.42 $1431.19 $1927.26 sum of large column. $1431.19 sum of small column. $496.07 difference. Find the difference, by addition, of the following: 2. $628.93 542.69 392.75 826.37 978.62 126.58 $258.72 385.98 726.18 195.42 329.54 3. $3852.19 6829.16 9325.18 2762.29 3218.75 $625.28 398.75 285.32 975.68 932.85 $3495.94 Ans.: 463, 11111, 2728, 2740002, 6812230, 2600- 11410, 21011220, 1457, $496.07, $1600.10, $22769.69, 101303, $25170.26, $17796.82, 200688, 2069129, 291- 827, 1655849, 861684. '•■•This method was suggested to the author by E. P. Goodiiough, Esq. THE COMPLEMENT 27 THE COMPLEMENT. Taking the Complement, or " making change," is the process of subtracting a lesser number from a ''round sum." It is emploj^ed, as the second term indicates, in making change or finding the sum to be paid back to the payer out of the amount handed by him in pa^^ment. The complete number is always the sum of one or more of the denomina- tions of coin or currency — $1, $5, $10, 50c., 25c., etc. It will be found that the complement of the teens is always in the SOs, the complete number being $1 ; the payment to be made, lie. — complement, 89; the payment to be 19c. — complement, 81. The complement of the 20s in the 70s ; that of the 308 in the 60s; of the 40s in the 50s; of the 50s in the 408; of the 60s in the 308; of the 708 in the 20s, etc. It will be found to be a very profitable class-drill, to conduct an exercise on making change in the following way : Teacher. The complete number being $3, what is the com- plement out of a payment of $1.50? (The class calls out the complement, $1.50.) The drill is conducted with enthusiasm for some time on the same complete number without naming it again, naming a different payment, thus: The complete number being $5, payment $3.25, complement — ? pavment $1.85? $1.75? $3.55? $4.50? 50c.? 75c.? 85c.? $3.60, etc. The students should be required to give the denomination of the answer, whether in dollars, cents, etc. In a short time the students will find it an advantage to subtract from the left to the right instead of the reverse, by taking the $, calling $5.00 $4,910. We do not think it advisable to require the student to thus subtract from the left, but his attention may be called to the practicability, and if he find it of advantage, he should use it If the habit is once acquired, it will facili- tate the taking of the complement materially. We have con- ducted a class exercise in schools where it had never been taught, and in the -course of a half hour the complement was given by the entire class insianter. 28 MULTIPLICATION. V. MULTIPLICATION. Art. 1. Multiplication is a short method of adding. X is the sign. 3x6 = 18, reads, three times six equals eighteen. MULTIPLICATION TABLE. IX 1- 1 2X 1 = 2 3X 1= 3 4X 1= 4 IX ^ = 2 2X 2- 4 3X 2=:. 6 4X ^= 8 IX 3=3 3 2X 3 = 6 3X 3= 9 4X 3=r 12 IX 4 = 4 2X 4 = 8 3X 4^ 12 4X 4= 16 IX 5 = 5 2X 5 = 10 3y 5= 15 4X 5= 20 IX 6 = 6 2X 6:.:. 12 3X 6==. 18 4X 6= 24 IX 7^ 7 2X 7... 14 3X 7= 21 4X 7= 28 IX 8:= 8 2X 8=. 16 3X 8= 24 4X 8= 32 IX- 9 = 9 2X 9:- 18 3X 9= 27 4X 9= 36 1X10 = 10 2X10 = 20 3X10= 30 4X10= 40 1X11- 11 2X11 = 22 3X11= 33 4X11= 44 1X12 = 12 2X12 = 24 3X12= 36 4X12= 48 6X 1 = 5 6X 1 = 6 7X 1= 7 8X 1= 8 5X 2 = 10 6X 2:z= 12 7X 2==:= 14 8X 2r= 16 6X 3 = 15 6X 3 = 18 7X 3= 21 8X 3= 24 5X 4 = 20 6X 4 = 24 7X 4r=:. 28 8X 4= 32 5X 5 = 25 6X 5 = 30 7X 5= 35 8X 5= 40 6X 6 = 30 6X 6 = 36 7X 6= 42 8X 6=:=: 48 6X 7 = 35 6X 7 = 42 7X 7r= 49 8X 7== 56 5X 8 = 40 6X 8 = 48 7X 8r:r: 56 8X 8= 64 5X 9 = 45 6X 9^ 54 7X 9= 63 8X 9= 72 5X10 = 50 6X10 = 60 7X10= 70 8X10= 80 5X11- 55 6X11 = 66 7X11= 77 8X11— 88 5X12 = 60 6X12 = 72 7X12= 84 8 X 12 = 96 9X 1 = 9 10X1 = 10 11 X 1= 11 12 X 1= 12 9X ^--= 18 lOX 2:= 20 11 X 2= 22 12 X 2= 24 9X 3 = 27 10 X 3=. 30 11 X 3= 33 12 X 3= 36 9X 4 = 36 lOX ^ = 40 11 X 4=. 44 12 X 4= 48 9X 5=. 45 lOX 5:= 50 11 X ^^ 55 12 X 5= 60 9X 6 = 54 lOX 6 = 60 11 X ^= 66 12 X ^= 72 9X 7 = 63 lOX 7=- 70 11 X 7=^ 77 12 X 7=.. 84 9X 8 = 72 10 X 8== 80 11 X ^= 88 12 X 8= 96 9X 9- 81 lOX 9^ 90 11 X 9= 99 12 X 9rrrl08 9X10 = 90 10X10 = 100 11 X 10 =110 12 X 10 == 120 9X11 = 99 10X11 = 110 11 X 11 = 121 12X11 =132 9X12 = 108 10 X 12 = 120 11X12 = 132 12 X 12 = 144 MULTIPLICATION. 29 Write the multiplication table as follows: 2 times 1 or once 2 is 2. 2 times 2 are 4. 2 times 3 or 3 times 2 are 6. 2 times 4 or 4 times 2 are 8. Continue this to 12. 1. To find the sum of 123 + 123+123, we would enter the three amounts as in addition, and add for the result. In multiplication we write 123 as in the margin, and 123 say, 3 times 3 are 9 ; put 9 in the unit's place. 3 Three times 2 are 0; put 6 in the ten's place. Three times 1 are 3; which put in the hundred's place. ggg The result is 369, as it would have been by addition. TERMS. Art. 2. The number 123 is called the multipli cand, the number 3 the multiplier^ and 369 the pro- duct. The multiplicand and multiplier are also called factors. 2. To find the product of 1496 by 7. Here wo say 7 times 6 are 42 ; write 2 under the 7. 1496 Then 7 times 9 are 63, and the 4 we carried make 67 ; n write 7 and carry 6. 7 times 4 are 28 and 6 are 34; write 4 and carry 3. 7 times 1 are 7 and 3 are 10. 10472 Ans. 10472. 3. 2146X2= 4292 4. 21007X 5=* 3178X3= 9534 31497X 6 = 4167X4=16668 17843X 7= 5189X5=* 41679X 8= 7864X6= 98765 X 9= 2875X7= 73149X12= Total, 123748 Total, 2519023 *The pupil will fiU the blanks. 80 MULTIPLICATION. Observe to point off the cents in the products of the following: 6. $10.78X 9=* .. $21.37X 7=* 117. 49X 8 = 317. OOX 9= 671.49X10= 857.37X11 = 1096.49X12= 117. 07X 6 = 307. 49X 7= 678.39X11 = 467.28X12= 999. 99X 9 = Total, $33246.36 Total, $25021.08 7. 2785X357. We have here three multipliers — seven, fifty, and three hundred. 2785X7= 19495 19495 2785X5 tens= 13925 tens, or 139250 2785X3 hundreds= 8355 hundreds, or 835500 Total products, This operation might be contracted by arranging the figures as in the margin, and writing the first figure of the products of the units in the unit's place and the others to the left of it; the first figure of the product of the tens in the ten's place, or under its own multiplier, 5; and the product of the hundreds in the hundred's place. 994245 2785 357 19495 13925 8355 994245 8. 3170 X 178= 564260 6184X1794=* 3867X3784= 2896X6789= 7109X9998= 71075782 2345X3979 = 6789X2164 = 1578X 753=^5^ 9409x6781 = 2783X4679= 8976X7659= 68747184 3968X6483= 7689X2197= 6784X7898= Total product, 141049961 Total product, 242956813 Note. — Either factor may be used as a multiplier in the above :-The puj.il will fill the blanks. MULTIPLICATION. 31 9. 420001000 109608 10. 109608 420001 3360008 2520006 3780009 420001 109608 219216 438432 Product, 46035469608 Product, 46035469608000 The multiplier of the ton's place in the first opera tion being 0, we passed it, and multiplied by the 6 hundreds. In the second operation we passed the ten's, hundred's, and thousand's places for the same reason. NoTE.-^-If the learner will simply observe to write the first figure of each product under its own multiplier, he will have no difficulty in multiplying where there are ciphers. For instance, the first figure of the product by 2, in the second example, is immediately under the 2. 22. 11. 12346X 30010= 370503460 7684X 10900=* 6787X 3009= 4967X 6007= 29836769 5896X900707= 7649X 66080:= 2000X 7010=* 3160X10096= 2178X90909= 197999802 1009X90910= 21678X21006= 31784X 7009= Total, 6320532304 Total, 1013793476 Art. 3. To multiply by 10, 100, 1000, etc., we have only to annex as many ciphers ( ) to the multipli- cand as there are in the multijDlier: 35X10=350 35 Explanation. — Multiplying by the 1 and the 0, we say 10 time Sis ; write under 5, then 1 time 5 is 5, 1 time 3 IS 3=350. 350 Note. — One time the number is simply, the number repeated, so we may as well annex the cipher to the original, as above. * The pupil will fill the blanks. 32 MULTIPLICATION. 165X10 =1650 25X20 =500 165X100 =16500 25X200 =5000 165X1000 = 165000 25x2000=50000 2. 374X10 = 3. 749X2000 = 268X100 = 836X16000= 189x1000 = 341X21000= 267 X 10000= 876 X 92000 = Total, 2889540 Total, 102627000 PRINCIPLES OF MULTIPLICATION. Art. 4. When two numbers are to be multiplied together, we use for the multiplier that which will produce least figures in the operation. This will be accomplished by selecting the smaller number, ex- cept where there are many ciphers, as in Ex. 3. Art. 5. If a number of articles and the price of one article be multiplied together, the product will be the price of all at the same rate. If the price of one be in cents^ the price of all will be in cents. If in dollars, the price of all will be in dollars* Note. — Cents are easily converted into dollars, by inserting the separating point. Those on the left will be dollars, the others cents. Art. 6. The number of articles contained in any box, bale, package, etc., multiplied with the number of boxes, bales, etc., each containing a like number, will give the whole number of articles in all. Art. 7. The interest, discount, premium, commis- sion of one dollar multiplied with the number of dol- lars, will give the interest premium, etc., of the whole number of dollars. MULTIPLICATION. 33 Art. b. Aii}^ number multiplied by itself, is said to be squared or raised to the second power — any number multiplied by itself, and that number again multiplied by the first, is said to be cubed or raised to the third power. Tllus.— 2X2=4, or 2d power of 2, or 2^. 2X2X2=8, or 3d power of 2, or 2\ 2X2X2X2=16, or 4th power of 2, or '2^ Art. 9. Feet multiplied by feet, yards multiplied by yaras, etc., produce square feet., square yards^ etc. Art. 10. Any number of feet multiplied by the number of inches in one foot, will give the number of inches in all the feet. Pounds multiplied by the number of ounces in 1 pound, will give the number of ounces in all the pounds. Art. 11. Halves, thirds, fourths, multi"plied on whole numbers, produce halves, thirds, and fourths. 1. What is the price of 37 bushels of corn at 37 cents per bushel? 2. What should I pay for 357 yards of broadcloth at $2.75 per yard? 3. Find the cost of 325 acres of land at $57 per acre? Total, $19522.44 4. In 320 bales of cotton there are 460 lbs. each, how many in all? Ans, 147200 lbs. 5. In 557 pieces of muslin there are 35 yards each, how many in all? Ans. 19495yds. 6. A ship laden with flour has 7950 barrels on board, and in each barrel there are 196 lbs., how many pounds in all? Ans. 1558200 lbs. * The pupil will fill the blanks. 34 MULTIPLICATION. 7. The premium on a dollar is .03 or 3 cents, how much on $149? * 8. At 6 cents on the dollar, how much in- terest should be received on 11750? Note. — Six cents on the dollar, is the same as 6 per cent 9. At 8 per cent, premium, how much should be paid on $3764? Total, $410.59 10. Find the square of the following numbers : 37 = 1369 376 * 570= 324900 219 10960 109 . Total, 120447869 Total, 201218 11. 103'! is how much? 107^ is how much? 19'* is how much? Total, 1365973 12. How many square feet are in a room measuring 15 feet long, and 14 feet wide? * 13. How many square feet in a board 16 feet long, with an average breadth of 2 ft? Total, 242 feet. 14. How many feet are in 573 yards? Ans. 1719. 15. How many inches in 573 yards? Ans. 20628. 16. How many rods are in 374 acres? Ans. 59840. 17. If 146 is multiplied by f, what is the product? Ans. 438. * The pupil will fill the blanks. t The small figures are called indices. They are used to in- dicate to what power the numbers are to be raised. See Art. 8. 'tiity MULTIPLICATION. 35 Art. 12. — Practical Questions: 1. Find the price of 87 bushels of wheat at 84 cents a bushel. 2. If I pay 25 cents a cwt. for freight, what should I pay on 2B07 cwt. ? 3. .What will 35 acres of land cost at $25 an acre? 4. How many yds. of muslin in 6 cases, each case containing 20 pieces, and each piece 35 yds. ; and what will be the cost of the whole at 12 cents a yard? 5. 250 boards 12 feet long and 1 foot broad were sold at 2 cents a foot, what was the cost? 6. A floor measures 25 feet long and 23 broad, how many square feet does it contain? 7. A merchant failing in business can pay only 37 cents on the dollar, how much will the creditor receive to whom he is indebted $7587? 8. How many quarts are in 25 bushels? 9. In a day there are 24 hours, how many seconds are there? 10. How many pints are there in 17 bushels 2 pecks? 11. How many inches are there in 3 yards 2 feet? 12. In a ream of paper, how many sheets are there ? 13. A commission merchant receives 2 %, ov 2 cents on the dollar, how much should he receive on $1425? 14. At $3.75 a dozen, what will 7 dozen of chisels cost? Answers to the above : $576.75, $73.08, $875, 480, 132, 800, 86400, 1120, $504, $68, 575, $2807.19, 4200, $28.50. $26.25, $19.50. 36 DIVISION. VI. DIVISION. Art. 1. Division is the method of calcuhition used to separate a number into equal parts. Art. 2. The sign is -h. When placed between two numbers it indicates that the one on the left is to be divided by the one on the right. 6-r-3, reads six divided by three. Art. 3. Division is also indicated as follows: 3)6. Which indicates that 6 is to be divided by 3. |. This is a Common Fraction, and indicates that 6 is to be divided by 3 also. — — — This is also a fraction, and indicates that the f • fraction \ is to be divided by the fraction f . .5, .05 are called Decimals^ and signify that the first is divided by 10, and the second by 100. Note. — The separating point between dollars and cents is a decimal sign, and indicates that the figures on the right are so many hundredths of a dollar, $4.25 is $4y^^^. TERMS. Art. 4. The number by which we divide is called the divisor. The number to be divided is the dividend. The number produced by dividing is the quotient The number left, the remainder. dividend, divisor 3)167S4 quotient 5594 — 2 remainder. DIVISION. 37 DIVISION TABLE. The pupil can make a division table of the multiplication table, by reciting it as follows : 3 times 2 are 6, 3 in 6, 2 times; 3 times 3 are 9, 3 in 9, 3 times. To divide 346 by 2. Explanation 1. — Commencing at the lefy we say 2 in 3, 1 time and 1 left; write the 1 (time) under the 3. 2^346 2. Carrying the 1 that was left, we suppose it to ^ stand before the 4, which will raise that number to 14; X*]^ then 2 in 14 7 times ; write 7 under the 4. 4. Then 2 in 6, 3 times. Ans. 173. Remark. — Until he becomes familiar with the process, the learner might write the remainders in small figures, as in the fol- lowing example. When he has mastered the operation, he can dispense with them. 2. To divide 13076837617 by 8. 8)1 3 ^0 27 36 ^8 3 37 ^6 1 ^7 r 6 3 4 6 4 7"^2— 1 Explanation. — Commencing at the left, we say, 8 in 13, 1 time and 5 left; place 5 before the next figure, then 8 in 50, 6 times and 2 left; place 6 below, and 2 before the 7 and so on. Divide the following : Quotients. Rem. 3. 134615379--2 67307689—1 21637298452-^3 7212432817—1 59368217755-4-4 14842054438—3 1416823687949--5 283364737589—4 Rem, Rem, 4. 13645217--6 5 5. 361745731--10 1 23176841--7 2 213764952-f-ll 5 47896739--8 3 178961521--12 1 89765432-^9 8 345678900--12 Total quotients, = 21546207 =99327785 Note. — The value of the remainder may be expressed in a frac- tional form: 168-^9 = 18|; which signifies that 9 is contained in 168 eie;hteen and six-ninths times. 38 DIVISION. *6. Divide $4537.25 between 7 persons, Ans. Each person will have $648. 17f. Omit the remainders in the following : 7. $21372.00^3 8. $67849132.87^- 8 13744.00--8 16493178.00-- 7 73176.35--5* 23610934.10-f- 9 14537.07--9 12310987.47-f-ll Total quotients, $25092.50, $14, 579,927.67. When flour is $1 a barrel, the loaf will weigh 8 times as much as when it is $8 a barrel: 8X9=72 oz. at $1 a barrel. At $6, it will weigh only -J as much. '^g- = 12 oz. 9. If 6 men do a piece of work in 11 days, how long will it take 4 men to do it? 10. If 27 men in three days, do a piece of work, how long should it take 25 men ? 11. The interest on $367 for 60 days at 6 %, is S3. 67; what should be. the interest on $1687.25 for the same length of time, and at the same rate per cent.? 12. In a square foot there are 144 square inches, how many square inches are there in a room 15 feet 6 inches by 18 feet 6 inches? Art. 5. The quotient of a number divided by 2 is the ^ (one-half) of it, divided by 3 it is |, by 4 the { ; hence to find the i, ^, |, etc., of a number, we have only to divide by 2, 3, 4, etc. 1. 1 of 3716=12381 2. i of 34161143764= 1 Answers: 4880163394f, 4737147654f, 1570^^. 1874. 7057961, 2880348786, 146|, 1621 3^6^^ 306O, 1238|, 341f, $648.17-f, $7124, $1718, $14635.27, $1615.23, $16.87, $8481141.61, $2356168.28, $2623437.12, $1119- 180.68. * Remove the decimal point before dividing, and replace it in the quotient DIVISION. 39 Note. — The learner will observe that we did not. divide by the fractions in the preceding exercises; on the contrary, we multi- plied by them. Omit the fractious in the answers of questions in the following r^'oups: 3. 9| times $14567.85 is how much ? Ans. $135966.60. 4. $ 345.78X37^ 5. $4563.28X45^, 16^, 18^ Total, $24897.37^. Total, $3624130.43. 6. If I pay 12-| cents on the dollar for a loan of money, how much should I pay for the use of $4527 ? Note. — On 4527 dollars I would pay 4527 times as much as on 1 dollar — 4527 times 12^ cents; or 12^ times 4527=56587^ cents; or $565.88. What should T pay on the following amounts at the rates specified ? 7. $3146@2A, 6A 8. $71684.25@8i, 7^, 5^ 1567@3i, 5^ 89647.87@U, 2i, 3^ 7864@ei, 74 79943.57® {, 6i, 7-1 Total, $1474.04. Total, $33274.96. 9. If a steamboat be worth $155367, what would J be worth?!? 1? I? ^? 4? I? i? J^? Total, $297108.56. Art. 6. To divide by 10, we cut off the right hand figure, then the figures on the left will be the quo- tient, and those on the right the remainder. 1. Divide 25 by 10. Operation— 2 \ 5, or 2.5=2f^. 2. Divide 6498 by iOO. Ans. 64 1 98, or 64.98^=643%^ Note. — We divide by 104, 1000, etc., in the same way, only, instead of cutting off one figure, we cut off as many figures as there are ciphers — for 100, two figures; for 1000, three figures, etc. Answers : $135966.60, $12966.75, $1223.50, $11707, 125, $565.88, 364|9., 2|, $78.65, $204.49, $52.23, $80.96, $491.50, $566.21, $5973.687, $5376.318, $3942.633, $1344.718, $2241.196, $3137.675, $199,878, $5063.092, $5995.767, $307791, $206108.14, $74533.56, $84420.68, $330072.16, $636567.75, $582183.44, $73166.24, $297- 108.56. 40 DIVISION. Note. — If the 36498 had been dollars, then the answer would have been $864.98 ; or 364 dollars, 98 cents. 3. 43645-^ 10 4. $168938--- 10 and by 100 71987-^100 678476-f-lOO and " 10 81674^100 396889-f-lOO and "1000 21362-v-lOO 798755-f-lOOand " 10 Total, 6114.73 $185444.369 Art. 7. When there are cents, the division may be performed by removing the decimal jDoint toward the left. To divide by 10 we remove it one figure, to divide by 100 we remove it two figures, by 1000 three figures, etc. $55.10 ^ 10=$5.510, 3167.56 -f-100 = $l. 67,56. Note. — The value of each and all of the figures decreases ten- fold for every figure the decimal point is removed to the left. The $5 of first example become 50 cents, and the 10 cents become 10 mills or 1 cent; making the answer 5 dollars 51 cents ; not 5 dollars 510 cents. The second answer is 1 dollar 67 cents 5y6^ths mills. Divide the following, omitting the remainders: 5. % 457.87- 1677.45- 10 6. $473.04^-1000 and 100 100 15.17.--- 10 and 100 6109.88---1000 16.57-- 100 and 10 14999.99-- 100 106.07-^- 100 and 1000 Total answers, 218 dols. 66 cts. 9 dols. 85 cts. 9 mills. 7. Divide the following sums of money by 100: $645, $1678.25, $87493.57, $16453.27, $1998.38, $643.- 24, $2168, $4137.54. Total answer, $1152.16,9 Art. 8. It often happens that there are not as many figures to cut off, as there are ciphers in the divisor. In such cases we annex ciphers to the left of the dividend to make up the number. Divide $5. by 100. Ans. .05. Explanation. — This is the same as removing the decimal point two places to the left, as above. The $5 had the decimal point DIVISION. 41 on the right of the 5, it is now two places farther to the left, and therefore is divided by 100. The cipher in this case, as else- where, possesses no value. 8. $ 5-^ 10 = .5 9. $ .03-^ 10 3-^ 100 = .03 .02-4-100 4^1000 = .004 .14-^100 50-f-lOOO = .05 3.16-^100 457H-1000 = .457 Ans. $0.0362 10. Divide the following sums by 100 : 3 cents, 33 cents, $3.33, $33.33, $333.33, $3333.33. Total, $37.03,68 Art. 9. To divide by 20, 300, 5000, etc., we point off as many figures in the dividend as there are ciphers in the divisor, and divide by the 2, 3, 5, etc. The figures pointed off will form part of the re- mainder. 1. Divide 317745 by 500. 5 1 00)3177 i 45 635-245 635 %U 2. 467831-f- 20=23391i-J 3. 716849-^700= 716893-^300=2389^^3 897653-^-900 = 417368-v-500=834|g8 49673-4- 80 = Total quotients, 2083.58 Note. — To divide dollars and cents, first reduce the dollars to cents or mills. 4. Divide $36147.59 by 500. 500)36147159 5 | 00)361475 | 90 "T2'29^§ ~72295~^o^ miiis or, $72.29 |gg or, $72.29,5 In the last solution the answer is given in dollars, cents, and mills— 72 dollars, 29 cents and 5 mills. To reduce cents to mills we have only to annex a cipher to the right of the cents, as in this case. 42 DIVISION. The answers to the following are required in dol- lars, cents, and mills, omitting the remainders: 5. $13764.75---50= - 6. $16789.37-- 80 = 73968.23-^60= 67859.67-^900= 37437.18--90= 54168.23^700 = 18964.20-^80= 78910.00-^-600 = 7. Divide the following sums by 20, and give the answers as above: $1367.25, $3143.57, $2345.87, $34.57, $45670.44. 8. Divide $34567.25 by 10, 12, 20, 100, 30, 50, 70, and 90. Total, $11132.84,6. 9. Divide $367897.87 by 100 and the quotient bv 10, 20, 30, 40, 50, 60, 70, 80, 90. Total, $4719.74,4. " 10. Divide $17654.37 by 100 and the quotient by 3, 10, 7, 40, 30, 50, 70, 90 and 80. Total, $298.78. 11. Divide $314937 by 100 and multiply the quo- tient by 7, then divide the quotient by 30, 60, 40, 12, 9, 80, 90. Total, $26117.89,9. 12. Find the sum of |-, i, J^, ^\, ^\, 5^, J^, of the yj^th part of $6739.45. $61.21,3. 13. The ^% 3L, J^, ^1^, ^1^, ^-i^th of $16894 39 divided by 100, is how much? Total, $41.38,8. Art. 10. — Practical Questions: 1. At 16 cents a bushel for coal, how many bushels can be purchased for 127 feet of lumber at $3.75 a hundred? 2. Three persons invest $6000 in business. The first, $3000; the second, $2000; and the third, $1000; and their gains are $2400; what was each man's share? Answers: $494.16,9, $2161.11,8, $11132.84,6, $2628.- 53,3, $4719.74,4, $26117.89,9, $298.78, $41.38,8, $61.- 21,3 ; 29f , $1200, $800, $400. EASY FRACTIONS. 43 VII. EASY FRACTIONS.* Art. 1. A fraction is a part or number of parts of any thing considered as a whole. Fractions are of two kinds, common and decimal. A common frac- tion is written with two numbers, called terms, having a line between them, as |-; a decimal fraction with one number, having a period at the left, as .5 (five- tenths). Art. 2. A common fraction indicates division, the upper number being the dividend and the lower the divisor. In treating of fractions, the dividend is called the numerator and the divisor the denominator. The denominator indicates the number of parts into which the whole is divided, and the numerator the number of such ^^arts under consideration. Art. 3. Value of a Fraction. — The lowest value of a fraction is expressed by the figure 1 for a nu- merator, and the highest value a number as great as the denominator less l.f \ represents the lowest value of fractions of the denomination of ninths, while -I represents the highest value of that denomi- nation.J *This chapter is introduced for the benefit of that large class of scholars who leave school before completing the study of Arithmetic. The subject of fractions is treated of at length in the latter part of this book. TThis does not apply to improper fractions, which, as the name indicates, are not strictly fractions. X 1. Since this is the case, it is evident that fractions decrease in value as their denominators increase, the numerators remain- ing the same. \ is less than J, J than i, \ than J. 2. It is also evident that the value of a fraction depends on the relation of the numerator to the denominator, or, in other words, the number of times the numerator is contained in the denominator, f is equal to |, because the numerator 3 is con- tained in its denominator, 6, the same number of times that the numerator 4 is contained in the denominator 8. 44 EASY FRACTIONS. When ii number is divided into two parts, each part is called a half; into 3 parts, each 'part is called a third; into 4 parts, each part is called a fourth; into 5, a fifth; into 12, a twelfth; into 18, an eighteenth; into 25, a twenty-fifth; into 100, a hundredth; into 476, a /oi^?' hundred and severity-sixth part. 1. When a number is divided into 10 parts, what is each part called? Into 11? Into 20? Into 33? Into 45 ? Into 97 ? Into 62 ? 2. When divided into 31, what? Into 69? Into 103? Into 364? Into 155? Into 1000? Into 3144? 3. Whicli is greater, f or f ? Ans. f. Reason. — Because it will take less to make it a whole num- ber. The first fraction requires J to make it a whole number, while this one requires only J. 4. Which is greater, -i or -f ? | or |? f or f ? fori? |or|? XorA? H ov^? |or|? Since the value of a fraction depends upon the re- lation of the numerator to the denominator, [note 2, page 43,] both terms may be multiplied or divided by the same number without altering its value. 1X2^4 ^^^ 2-.2_l 4X2 8 4-- 2 2 Now, I and ^ possess the same value as f, because their respective numerators are contained the same number of times in their denominators. 5. Change f to twentieths. 3X5 15 Explanation. — By multiplying the 4 by 5, we 4\/5 20 change the denominator to twentieths; and by mul- tiplying the numerator by the same number we preserve the original value. 6. Change f to Sths; ^ to 12th8; 4- to 20th8; |- to 14ths ; f to 12ths ; f to 18ths ; f to 30ths. A-n«5WPr<5' 75 487 8 1276 6 4 4 15 2 4 3 18J 30' 4- EASY FRACTIONS. 45 Art. 4. To reduce a fraction to its lowest terms is to divide the numerator and denominator by such a number or numbers as will do so without a remain- der. When the terms can not be exactly divided by any number greater than 1, the fraction is in its sim- plest form. 1. Eeduce ^, -j^j, ^, ^, -^, -i^, to their lowest terms. When a single number will not reduce the frac- tion, other numbers may be used, as below. , 2. Eeduce -g^frr ^^ ^^® lowest terms. 5)9MT(ll)Tik(TiT- 3. Eeduce to their lowest terms, -^^j -^^^ ^|^, and Teir- Fractions may be Proper^ Improper^ Simple^ Com- pound or Complex. We shall treat of only the three former at present. A proper fraction is one whose numerator is less than its denominator, as ^, An improper fraction is one whose numerator is equal to or greater than its denominator, as f, ^. Art. 5. A simple fraction is a single fraction, and may be proper or improper, as f, f . Art. 6. When a whole number and fraction ap- pear together, they are called a mixed number, as 5|. Art. 7. Improper fractions may he changed to whole or mixed numbers by dividing the numerator by the de- nominator.'^ ^Answers: 3^, ^, ff, ^-^, ^% 2f, ^, i -L,'^, TT) Tj TT- *This is simply acting on the principle that the numerator is the dividend and the denominator the divisor. 46 EASY FRACTIONS. To change ^ to a mixed number. 5)13 Explanation. — There are 5 fifths in one whole num- 23 ber; in 13 fifths there are as many Is as the number of times 5 is contained in 13, which is two times, with 3 fifths over, making 2|. 1. Change |, f , f, -^, ^-^-, %4_ to whole or mixed numbers. Art. 8. To change whole or mixed numbers to im- proper fractions is an operation the reverse of the last, which scarcely needs explanation. 1. Change 9|- to an improper fraction. 9f Explanation. — In 1 wliole number there are 5 fifths; _5 in 9 there are 9 times 5 or 5 times 9 fifths, to which we i9 add 4 fifths, and we have -Y-. 2. ^Change the following mixed numbers to im- proper fractions : 3|, 9^ 8f, 5f , 41f , 97i 16|. Art. 9. To multiply a fraction by a whole number is simply to multiply the numerator without alter- ing the denominator, or to divide the denominator without altering the numerator. To multiply -^ by 6. 12 -11-^^ ^"^ ^2- Eeason. — Assuming that 7 is a whole number, multiplying it by 6 gives 42 ; but since it is not a whole number, but twelfths^ the 42 is f|=3r«2 or Z\. By decreasing the denominator, the fraction is in- creased (as it takes fewer of the small parts to make^ a whole number) ; hence, the 7 represents halves in- stead of twelfths. |-=3-i-. Answers: H, 1^, If, 3, 30^, 69^ ^, ^, -^S ¥. ¥. ^, ^, ¥. 3i, h ¥, 69,^. * The learner should prove the accuracy of his work by last article. EASY FRACTIONS. 47 1. f X7-? 3. ^9^x12 = ? 5. i|xll = ? 2. IX9-? 4. ifx6^? 6. 1^X12 = ? Art. 10. To multiply a whole number by a fraction^ we multiply the numerator without altering the de- nominator. 1. Multiply 25 by |. 25X3 fourths=75 fourths, or -"^, which, changed to a mixed number, [Art. 7] =18f. 2. 35xf = ? 3. 134X2^--? 4. 16xA = ? Art. 11. To multiply a mixed number by a ichole number. Multiply 7f by 9. 7S Explanation. — 3 fourths multiplied by 9—27 fourths, 9 or 6f ; and the 7 multiplied by nine=6o, plus the 6=69, QQ3 making the product 69|. Or thus : 7| A 3^1X9 = ^^ = 691 1.181x5=? 2. 29fx8 = ? 3.83^x7 = ? Art. 12. To multiply a whole member by a mixed number. 1. Multiply 29 by 8|. 29 8f Explanation.— Multiplying 29 by 2 thirds, we have ,qi 58 thirds, or 19J, which we write in the first line. Then 232^ 29X8^232, which, added to 19J=251J. 25ii Or thus : 29X¥=^f^=251 J. 2. 15x3|=:? 3. 12X12^ = ? 4. 14xl7f=? Answers: ^,>f, lOf, 4if, 9^, 10||, 28, 6^, 4f, 69f, 60, 150, 6f, 93|, 238, 5831, 20, 251i 150, 246|, 18f, 53|, 15|, 72^, 5f . The Teacher will find it important to require the learner to preserve the process, as he will be apt to adopt clumsy methods of solution. 48 EASY FRACTIONS. To multiply a fraction by a fraction. 3. Multiply! by f Assuming the numerator 5 to be a whole number, -|X5=-^^; but 5 is not a whole number, but 5 sixths; lience -^^ is 6 times too much. • j^- divided by 6=i|, or f . [Note 1, page 43.] Art. 13. Hence, to multiply a fraction by a frac- tion, we multiply the numerators together for a new nu- merator , and the denominators for a new denominator. |X|-=Yfj which, reduced to its lowest terms =|-.* 2. fXf-? . 3. ^X|-? 4. iXli-? Art. 14. To midtiply a mixed number by a fraction or a mixed number. 1. Multiply 15| by -^. l^=:-^f, which, multiplied by -jSj^^ W ^i^ l^^- 2. Multiply 8f by 16f . 83^3^ and 16|=^. ^x^=^fF=145f|==145f 3. 12iXl6t---? 4. 14fx-i^-? 5. 18i-Xl2i = ? Art. 15. To divide a whole number by a fraction or a mixed number. 1. Divide 315 by f, or, in other words, find how often I is contained in 315. Solution. — Before we can measure 315 by fourths, we must change it to fourths. In 1 there are 4 fourths ; in 315 there Answers: fi, f, |i, 14^, 146|, 208^, 12f 231^, |, 2^^ *It will be observed tliat to multiply by a fraction does not increase the multiplicand, as in whole numbers; but, on the contrary, decreases it, the f being less than J. To account for this, it is only necessary to remember that a whole number is reduced to the denomination of a fraction by being multiplied by it. 6X1^18 fifths, or 3f. Much more is a fraction reduced in value if multiplied by a fraction. From this we readily infer, 2. That to divide by a fraction increases the dividend. EASY FRACTIONS. 49 are 315 times 4 or 1260 fourths, which, divided by 3=430. Hence, f is contained in 315 420 times. Operation. 315 or ^^Xi-=-T--=^^0 4 3 j 1260 420 . 2. 320-v-|-=? 4. 541-f-|=? 3. 27-- 1 = ? 5. 684--^=? 6. 987- 7. 136- 8. Divide 25 by Sf Operation. 25X2 l)alves=-\o and 5|X2=Y. 50^11 4A, or ¥XA-=W=^4^|=43^. Art. 16. Hence, to divide by a fraction, we multi- ply by the denominator and divide by the numera- tor, or invert the divisor and proceed as in multipli- cation. 1. 157-f- 3i-=? 4. 345-- 6f=-?. 7. 195-f-16|=? 2. 22-f-12|=? 5. 39---15i=? 8. 39--124rr=? 3. 16--16f=.? 6. 79-f-37^=? 9. 87--3l|^ ? To divide one fraction by another. 10. Divide I by f. Operation. fx|-=i|=i^. Explanation. — By inverting the divisor, we obtain |§, the terms of which, being divided by 2, give ^. 11. t2j^|=? 14. 31i--f =? 17. f -^li=? 12. T%-^i- ? 15. 31f-^= ? 18. T%^3i= ? 13- li=f=? 16. 13|-|=? 19. 5^6f=? Art. 17. To divide when either divisor or dividend is a mixed number and the other term a whole number^ both terms may be reduced to the same denomina- tion. [Art. 15.] I. Divide 34571 by 13. Answers : 32|, 265f , 2434|, 19740, 2280, 151|, 24f , 1 1 iLO. JL9 Q«2 9 Qf?3 IfiXJLO 9 7 1 IO4.J.6 119 64 511 2-25 9 8 517 ^3 2 2 998 •^■^ir> *^¥6> ^1^) ^TO"? ^6 4> Zi T? ^T25- 50 EASY FRACTIONS. 3457J Explanation. — The dividend containing the 4 fraction of J, both terms are reduced to fourths, 62)13829(565 ^^^ division performed as in whole numbers. 104 '-The result shows that the divisor is contained • oAo ^^ ^^1^ dividend 265 times, with a remainder of Q19 49 fourths [Art. 2, Principles of Division, page ^^ 46], or 26511 times. 269 49 The same by short division, 13)3457 J Explanation.— 13 is contained in 3457 265 "25549 times, with a remainder of 12, which, reduced ^^ to fourths, including the J of the dividend, is 49 fourths. 13 not being contained in this an even number of times, the denominator is increased 13 times, (which is the same as to decrease the numerator,) which gives the same fraction as by long division, |f . 2. 1398| : 56 — - ? 5. 1255| : 350—? 3. 2564 : 7 — ? 6. 7961:421—? 4. 1939 -- 8^=? 7. 467|-- 12=? Art. 18. To subtract a fraction from another of the same denomination is simply to subtract the less nu- merator from the greater. 1. From ^ take -^. Q SI 2_5 — 9 ^' ¥2 42 — ' 4 i_6 5_ ? ^•22 22 • Art 19. To subtract a fraction or mixed numoer from a whole number. 1. From 9 take 3f. Answers: 275|f, 24^, 36^^^, 3|4^, 1^1 232i|, 38f^, h h h iV> h h h h A. A^ tV. EASY FRACTIONS. ' 51 The following formula will render the operation simple : Whole number. Fourths EXPLANATION. — Arranging the less un- , ^* der the greater, we find we can not take 3 fourths from fourths; so a whole number . "T "T or 1 is added to botli terms. In 1 there ^ ^j are 4 fourths, from which we take 3 fourths, ^'* ? and we have a remainder of 1 fourth. To the 3 add 1 and we have 4, which, subtracted from 9, leaves 5, giving for the answer 5|. 2. 13— 41=.= ? 5. 11-2 i=? 8. 52— 27i^? 3. 15— b\=l 6. 7— |=r? 9. 13—121:^? 4. 29-121= ? 7. 14—1^= ? 10. 89—75^= ? Art. 20. To subtract one fraction from another of a different denomination, it will be necessary first to reduce both to the same or a common denominator. 1. From I take f By Art. 3, |- can be changed to 56ths by multi- plying both terms by 7, and ^ can be changed to 56ths by multiplying both terms by 8, giving -|| and 1^, the difference between which is -^-^^ the answer. It will be observed that the multipliers used in this case were the denominators, 7 and 8, which, multiplied together, give a common denominator^ and multiplied into the numerators of each other give the new numerators. Operation: |-^=ff-4|_^. 2. From f take \ . 5. |— 1= ? 3. From | take |. ' 6. 6^ — fz=z? 9. 4. From | take ^. Art. 21. To add fractions of the same denomination, the numerators only are added, and the sum reduced to a mixed number or its lowest terms Answers: |, 61 9f, 41, 24f 8|, 8|, 16|, 13f, 13f, 5 53. 23 5. _3_2 1 23 1 5 3 193 1 T¥: *^4' 3Tj 8' 193^' 6"' T2"' ¥' "2 8"? ~E' ^^t' T2* 52 EASY FRACTIONS. 1. Add l+f+l+f 3 Explanation.— Here the four numbers are added 6 together, making 21 eighths, which, reduced to a mixed 5 number, are equal to 2f. 2- iV+A+A+A+A+i^+A=? Art. 22. To add fractions of different denominations ^ they should first be reduced to a common denomi- nator, as in subtraction. X_I3.: — ? i.4_3. 4.16 10 12^^11 When three or more fractions of diiferent denomi- nations are to be added together, they may be re- duced to a common denominator by multiplying all the denominators together, as above, and then by multiplying each numerator by all the denomina- tors except its own.* 1. Find the sum of ^-+1 +|. 2 x4x 6^=^48r:r:(7ommon denominator, 1 X 4X 6=24=:i^zr5i^ numerator. 3x2x6=36=AS'econd numerator 5 X 2 X 4=40= Third numerator. 100=Sum of numerators. Hence, 0^=2^=2,^. The J in the example was multiplied by 24, giving |f ; the f by 12, giving ff ; and the | by 3, giving ff. 2. f +|=? 5. |+t+|-=? 8. 2f+ I+I-? 3. i+i=? 6. ^+i+i = ? 9. 5i+6i+| = ? 4. A+I-? '^. A+i+A=? 10. |+2|+^=? Answers : 2fT 3^, UfJ^ ^J^, H, 4X, 12, 3^^, H, 05 12? 2' ^24- *This is simply multiplying both terms by the same number. DECIMALS. 53 VIII. DECIMALS. Art. 1. A Decimal Fraction expresses its value in one term, and is known from a whole number by its having a period called a decimal point at the left. .5 is a decimal. The value of a decimal is more easily ascertaine( than that of a common fraction ; while operations in decimals are performed with nearly the same ease as those in whole numbers. Art. 2. The numerator only, of a decimal frac- tion is written, the denominator always being 10 or some power of 10. A decimal composed of one figure as .5, will have 10 for a denominator. One composed of two figures, as .75, will have 100 for a denominator, etc.; hence to find the denominator, we have only to write as many ciphers as decimals, and annex a 1 to the left. Remark. — Ciphers on the extreme right of a decimal possess no value. .500 is the same as .5. The value of .073 is how much, expressed as a common fraction ? Arts. jj|^ What is the fractional value of the following: .23, .007, .013, .75, .12, .11, .4, .42, .710, .0076, 10.7? Art. 3. The remoyal of the decimal point one place to the right or left, increases or diminishes the fraction 10 times. 31.67X10=316.7, and 316.7-^-10=31.67. ADDITION AND SUBTRACTION. Art. 4. Operations in Addition and Subtraction of decimals are performed in the same way as those in addition and subtraction of simple numbers. The pupil who has calculated Federal money, is already 54 DECIMALS. acquainted with these operations. He has only to observe that units be placed under units, etc., or what is still more simple, to place the decimal points directly under each other, and proceed as in simple numbers. 1.07 +.001 + 37.045 + 10.06+.0007 would be done thus: 1.07 .001 37.045 10.06 .0007 48.1767 1. Find the sum of .007+31.06+ .1009+3 00.07=* 710.34+2.406+67.709+ .0006 = 314.60+.0006+ .0027+ .001 = 714.06+ .003+ 8.007+ 800.= 2. Total, 2748.3678 123.006+.18532+.0185+1672.3+1865.01=* 184.003+.0185+11.10+18639.01+1657.003= 0.005+2683.17+2.95+6892.02+.0031= Total, 33729.80242 3. Find the difference between 107.06 and .213. 617.07—41.7106=* 107.06 10.06— .9092= .213 illSt ?S:_ Total, 725.7592 4. 21.80— .0503=* 364.2—128 9= 6.295—2.654= Total, 260.6907 Note. — For Muhiplication, Division, and Reduction of De- cinaals, see chapter XXXIII, pages 173-176. ':-- The pupil will fill the blanks. 'SHORT METHODS. 55 IX. SHORT METHODS. PROPERTIES OF NUMBERS. Art. 1. Numbers ending with 5 or are divisible by 5 without a remainder. Art. 2. If the two right hand figures of a number are divisible by 4 without a remainder the whole number is divisible by 4. Art. 3. If the three right hand figures of a num- ber are divisible by 8 without a remainder the whole number is divisible by 8. Art. 4. If the sum of the figures of any number be divisible by 3 or 9 the'whole number will be di^ visible by-B or 9. SHORT METHODS OF MULTIPLYING. Art. 5. An even part of 10, 100, 1000, etc., can be multiplied mentally by division. ALIQUOT PARTS OF 100. ALIQUOT PARTS OF 1000. 50 =i 14|=4 333i= 1 331=1 121=^ 250=^ 25=1 10 =-1, 166i=^ 20=4 81= J. 125=1 ^n = h H = i\ 83i = J^ To multiply by a part of 100, we suppose two ciphers to stand at the right of the number and di- vide by the part. To multiply by a part of 1000, we suppose three figures, etc. 1. To multiply 176 by 12^. Operation 8)17600 Arts. 2200 56 SHORT METHODS OF MULTIPLYING. 2. To multiply 379 by 250. Operation 379000 ^^^^^^ ^^^^ 4 3. To multiply $49.75 by 125. Operation 4975000 ^^, ^^, ^ — ^ =621875 cents, ® or S6218.75 Ans. Note. — Only a few answers will be given in the following, as the pupil can prove the accuracy of his calculations by multi- plying in the ordinary way. 4. $140 X 12^=$1750 7. 949 X333J= 5. 3767X 81= 8. 179 X 2i= 6. 9987 X 25 = 9. 769 X 3|= 10. 675 yards @ 37-J- cents. Operation 675 at a dollar =$675. 00 at 25c. =\ 168775 at 121c. =1 84.371 Ans. 2537121^ 11. 715X621 cents. 14. 9876 X $2. 18| cents. 12. 947X871 15. 719X 3.621 13. 194Xl8f 16. 965X 4.37^ I^OTE. — The multiplier of the 12th Ex. wants only 12J cents, or J of being a dollar; so we find the cost of 947 at a dollar, and take off ^ of it. Other examples may be solved in the same way. Art. 6. To find the cost when there are fractions in both factors : 18f lbs. @ 12i cents. Operation, 18f lbs. @ a dollar : = $18.75 at 121 cents =1 or $2.34 Ans. 2. 37|- lbs. @ 18f cents. Operation, 37i @ a dollar irr : $37.50 at 121:== lor 4.687 " 6^ = 1 ^' 2.34 3 Ans. $7.03 SHORT METHODS OF MULTIPLYING. 57 1. To multiply 424 by 97 : Operation. 424X100 = 42400 424X 3= 1272 41128 Art. 7. When the multiplier wants from 1 to 12 of being 100, 200, 3000, etc., the work can be con- tracted by multiplying by one of these, and subtract- ing as many times the multiplicand as the multiplier is short of them. Art. 8. When the multiplier is 29, 39, 49, etc., we multiply by the next higher number and subtract the multiplicand. 1. To multiply 176 by 59 : Operation. 176X60 = 10560 2. 671 X 39 176 3. 59X689 TT—- 4 89X784 ^""^^^ 5. 167 X 29 Art. 9. To square numbers that end with 9. What is the square of 29? Operation. 29 29 841 ExPLAx\AT[oN. — Writing 1 for the first figure of the product, we add 1 to the ten's place of the multiplier, and multiply the sum on the multiplicand less 1: 3X'^8^84, with the 1 annexed =841. Find the square of the following numbers mentally : 99, 59, 119, 79, 19, 69, 129, 89. Art. 10. To square any number of 9*s instantane- ously, and without multiplying. Commencing at the left, we write as many 9's less one, as the number to be squared, an 8, as many O's as 9's, and a 1. The square of 9999999 is 99999980000001. 58 SHORT METHODS OF MULTIPLYING. Remark. — The square of any number of 3*s will be the ^ of the square of the 9's. Art. 11. To 7nultiply by 375, 625, 750, or 875, we first multiply by 125, (Art. 5, Ex. 3,) and that pro- duct by 3, 5, 6, or 7. 1649X625 = ^-^l^^=??i^=1030625 o o Art. 12. To square numbers under 135, ending with 5 The first two figures on the right of the product will always be 25 ; and to find the others, we add 1 to the ten's place and multiply it on the ten's and hundred's places above. To square 115 : Operation. 11 12 13225 Art. 13. To square a number containing a half, aH 12^, we multiply the whole number by the next higher number and add a fourth. 8J squared =8X9+^=72^. Art. 14. To multiply hy numbers of two figures containing the figure 1. 1. Multiply 346 by 15. Operation. 346 1730 5190 2. Multiply 346 by 51. Operation. 346 1730 17646 These operations might be performed mentally. Taking the 2nd, for instance, we say one time 6=6 ; then 5 times 6=30 and 4 of the upper line =34; 5 times 4=20 and 3 of the upper line and 3=26 j 5 times 3=15 and 2 = 17. Product 17646. SHORT METHODS OF DIVmiNG. 59 SHORT METHODS OF DIVIDING. Art. 15. To divide by a number composed of two or more factors j"^ as 96. which is comiDOsed of the factors 12 and 8, or G48 which is composed of 9X8X9. This operation is performed by using the factors instead of the whole number. 1. Divide $7854 by 32. Operation. 4)7854 8)1963—2 How the true remainder is found : 245 — 3 The first remainder is 2 dollars, because it is left from the dollars that were divided. The second re- mainder is four times as great as if it were from the first line, because every figure of the second line is four times as great as if it stood in the first line. Four times 3 = 12, and the 2 of the first remainder equal 14, the true remainder. Ans. 245||. 2. Divide 6371 by 336. Operation. 6)6371 Remark. — The true remainder of this example ^ is found by multiplying the last remainder by 7, g^ \p^\ 4 to make it of the same value as if it were from the line above, and that by 6, to make it of the 18-^7 same value as if it were from the upper line: 7X7X6=294, to which, add 6X4+5 or 29. The true remainder is 323. Ans. 18|||. 3. Divide 1463 by 28 7. 4571--441 4. " 7714 '* 72 8. 1987--378 5. '* 1943 '' 49 9. 9843--720 6. " 8765 ** 343 10. 1456-f-729 Answers : 103f , 25^1 , *5^, IfH. 52i, 39f|, lOif, * Such numbers are called eomposUe. When a number can not be di- vided into factors, it is called a prime number. 60 SHORT METHODS OF DIVmiNG. DIVISION BY CANCELLATION. Art. 16. To cancel, signifies to blot out or make void. Division by cancellation is performed by writing the terms in fractional form, and dividing them by any number that will do so without a remainder. 1. To divide 1463 by 28. 209 4 Explanation. — The first terms were canceled by dividing by 7, leaving ^J^, which finished, is 52J. 2. 3465-!- 35 5. 1962-^ 22 3. 2763-- 81 6. 6876^-152 4. 6545-^245 7. 5436-^-144 Art. 17. To divide by aliquot parts of 100, 1000, etc. This process is the reverse of that under Art. 62, page 69. 1. To divide 7654 by 25. Operation. 76.54 306,16=306^% Note 1. — The decimal value of the remainder is always obtained first by this process. If we divide the 16 by the multiplier, it will give the true remainder or 4. 2. If the aliquot part is of 100. there will be two places; if 1000, there will be three places of decimals. 2. To divide 19765 by 125. Operation 19.765 8 158.120 158. ^\ 3. 17630^33i, 74910-r-12^, 87396^-125, 824--2i. or 158. y\A^ PERCENTAGE. 61 X. SIMPLE PERCENTAGE. Art. 1. Percentage embraces all those calcula- tions in which 100 is made the basis of comparison. Art. 2. Fer cent, signifies by the hundred. 6 per cent, signifies 6 for every 100. ^ is the sign, and is written thus : 6 ^ , which reads, six per cent. The 6 is called the rate. Art. 3. One per cent, of a number is that number divided by 100. One per cent, of $320 is $3.20. To find any other percentage of a number, we multiply one per cent of it by the rate. Art. 4. Percentage may be divided into simple and complex. Art. 5. Simple Percentage embraces all those calculations in which both the principal and the rate are known and is applied to Premium, Discount, Exchange, Taxes, Commission, Brokerage, Insurance, Insolvency, Loss and Grain, etc. 1. Find 6% of 758. 7.58=1 per cent. 6 45.48=6 per cent. 2. 3% of 215=6.45. 3. 9% of $756= 4. 4% of 788= 5. Ifo of $179= Note 1. — To find 1^ of the following, remove the dec. point two places to the left, and point ofi^ four figures in the product. 2. Give your answers in dollars and cents. If you have 6 mills or more, add a cent to the cents; if less than 5, omit them, 6. $768.15@6%= 7. $566.75@13 % = 8. $196.55@5%= 9. $789.15@33J% = Note. — When the percentage of sums under $10 is required, it will be more simple first to multiply by the rate and then divide by 100. 10. 25^ of $0.97= 11. 50 % of $0.17= 12. 65% of 0.08= 13. 33J% of 0.76= 62 PERCENTAGE. Art. 6. Premium is a percentage to be added; discount, a percentage to be subtracted from the face or par value of a bill, note, etc. Art. 7. The method of settling accounts between persons in distant places by draft is called Exchange, If between persons in the same country it is called Iti- land or Domestic Excliange. Art. 8. Exchange may be par, or it may be at a 'premium or at a discount. 1. What will be the cost of a draft ©n New Orleans for $3200 at 1% discount? 2. What amount will pay for a bill of exchange on New York for $1860 at f % premium? 3. Ax ^% discount, what should I receive for New York exchange calling for $728 ? 4. At ^% premium, what will a bank pay for a draft on Chicago for $276 ? Art. 9. Commission or brokerage is the percent- age charged by a commission merchant, factor, agent, or broker, for transactii^g business for another. Art. 10, Commission is usually reckoned on the whole amount of sale, purchase, or collection. 1. At 2^ per cent., what is the commission on $17640? 2. A merchant sells goods for another to the amount of $4371.87, what is his commission at 5 per cent ? 3. A broker receives \ per cent, for selling $2500 worth of merchandise for a commission merchant, what was the amount of his brokerage? 4. A of New Orleans buys sugar for B of Cincin- nati, to the amount of $7100, what is the amount of his commission at 1^ per cent. ? Answers: $441.00, $218.59, $6.25, $106.50, $3192.00, $1866.98, $727.09, 276.69. PERCENTAGE. 63 Art. 11. Insurance is a guarantee against the loss of property by fire or the dangers of transporta- tion. The amount paid for guarantee is called the Pre- mium. It is a certain percentage on the estimated value of the property insured. This percentage or rate is large or small, in proportion to the risk. 1. How much should be paid to insure a bouse valued at $1674, premium li%, and policy $1.50? 2. At 2\ % premium, wh^it should I pay on $6710 worth of goods, including policy at $2.50? 3. At 4 ^ premium and policy $1.75, what should I pay on a freight of furniture worth $2200 ? Art. 12. The term Stocks signifies shares in incor- porated companies, while Bonds represent government and municipal securities, and mortgaged securities of corporations. 1. What is the value of 10 shares of railroad stock, at 5 % premium, par value per share being $50 ? 2. Bought $1500 of bonds at 1.02 %, what did I pay for them ? 3. Sold $500 in railroad stock at a premium of 5 ^, and received sugar at 10 cents per pound, how many pounds did I receive? 4. Purchased 50 shares of Pacific railroad stock at a discount of 2^ %, par value being $100, what did they cost me? 5. Sold $5000 in bonds at a premium of 2J ^, what did I gain after paying a broker \ % for selling ? 6. Bought $2600 of canal stock at 3 % below par, and sold it at 4 ^ above par ; how much did I pay, and what did I gain by both transactions? Answers: $26.61, $170.25, $89.75, $525.00, $1530.00, 5250, $4875.00, $112.50, $182.00, $2522.00. 64 PERCENTAGE. Art. 13. Tax is a sum imposed or levied upon society to defray its expenses. Art. 14. Foil tax is a specific sum assessed on male citizens. Art. 15. Duty is a tax levied by the general gov- ernment upon imported goods. Art. 16. Specific tax is a fixed sum levied upon specific things without regard to value. Art. 17. A general levy is an assessment upon projDerty according to its value. 1. The general levy of a county was as follows: 8 m ^^ for school fund, 4 m ^ for specific purposes, Q m % ^^^ sinking fund ; how much tax will I have to pay on $750 personal property and on $2800 real estate ? Solution. — We first add the several rates together to ascer- tain the whole assessment, viz : 8 -[- 4 -|- 6 = 18 m ^, or 1.8. 1^8_ c/^_ 1750 _|_ $2800 r= $3550 taxable property. lj«o % of $3550 z=z $63.90. . 2. Under the same levy what tax would a farmer have to pay on $256 personal property and $8200 real estate, including poll tax for three male persons @ $2 a person, and a special tax of $5 for a piano and $3 for a gold watch ? 3. What would be the duty on an invoice of im- ported silks, costing $560 in gold (including dutiable charges), at 35 ^ less 10 % ? Note. — The 85 cfo is to be reckoned on $560, and the 10 ^ to be taken from the result. It will be observed that 35 (fo less 10 (fo is not 25 ^, but 31 J ^. Answers: $176.40, $63.90, $166.21. * One m. per cent, is one mill on one hundred dollars. PERCENTAGE. 65 Art. 18. Marking Goods. — This is done by select- ing samples of each kind or quality of goods, and putting on them a private nfark, indicating the cost price, the selling price, or both. Every house has got its own peculiar marks, which generally consist of the letters of some word or phrase, instead of figures. For instance, if we take the word importance, we will have a letter for every figure, and can readily substitute the former for the latter : 1 2 3 4 5 6 7 g-i 9 J m p o r t a n c e When a figure is to be repeated, an additional letter is need. Take <7 in this case. The selling price is commonly found by adding to the cost price a certain amount per cent, to cover the freight and other charges, and allow a remunera- tive profit. To facilitate labor, some merchants make this per- centage an even part of 100, and add the same part of the cost price to itself 1. To add 12^ per cent, to the cost price of goods at $1.20 per yard. 12J being J of 100, add J of 120 (15c.) to itself, which will make the selling price %\.Zb. 2. To add 25 % to books at S4.80 per doz. 25 being J of 100, add } of 480 (120c.) to itself, which makes tlie selling price %Q.0() per doz., or 50cts. each. 3. To 30c. add 20 % profit. 4. $1.20 " ^ % charges and 20% profit. 5. 1.75 " 2\% freight and 10% " 6. Add 14 % to $1.75 7. Add 15 % to $0.16 8. <^ 12% to 0.87 9. " 25% to 0.05 10. " 53^ to 1.67 11. " 18^ to 3.16 Answers: $1.35, $6.00, $1.50, $1.97, $2.00, $0.97, $0.18, $2.56, $0.06, $0.36, $371.00, $3.73. 66 PERCENTAGE. Art. 19. MISCELLANEOUS EXEKCISES. 1. Eeceived a consignment of tea, which I sold for S1678, how much should I return the consignor, after deducting charges $150, and commission 2^ %? 2. Sold a consignment of cloths for $6750, with the assistance of a broker, who charged me ^ ^ ; what amount of money did I make, com. 2| % ? 3. A commission merchant sells goods for his prin- cipal to the amount of $3000, and charges 21^ com., what does he make by the operation, after Spaying a broker J^^ for his services in effecting sales? 4. Insured | of a steamboat worth $18000, at 1| % premiura : what did it cost me ? 5. How much should an insurance company pay to an insured who hold a policy for $2000 on his dwelling, the damages being estimated at 68%. 6. What is the amount of loss on a policy insuring mdse., $1200; fixtures, $300; building, $4000; dam- age on building, 37-1-%; mdse. saved, $300; bal. of mdse. damaged, 75%; loss on fixtures, 83J%. 7. A merchant holds three policies of insurance, as follows: iEtna, $1000 on leaf tobacco, $1500 on cigars, $100 on fixtures, and $200 on retail stock; Aurora, $2000 on leaf tobacco; American, $1500 on leaf tobacco and $1500 on cigars. Damage on leaf tobacco, $3000; cigars, $2500; retail stock, $300; fixtures, $200. What must each company pay? 8. After reserving 5% commission on sales, amount- ing to $520,75. how much should I return to my principal ? What is the commission on the following amounts: 9. $364.15@3^%= 10. $36.21@li% = Answers: $67.50, $202.50, $1360, $2425, $2216.67, $1333.33, $2250, $494.71, $12.75, $4.91, $60.36, $83.29, $0.54, $5.80, $135.49, $4.30, $1486.05, $118.12,5. BILLS — INVOICES. 67 XL BILLS -INVOICES. WK/n goods are sold, it is the duty of the mer- chant, or one of his clerks, to make out a statement of the quantity, kind, and price, of each article, for the satisfaction of the purchaser, and to enter at the foot of such statement the whole amount of the pur- chase, with the payment received, if any, or the terms of settlement. If the goods are bought to sell again, this statement is commonly called an Invoice^ otherwise it is called a Bill^ especially by the pur- chaser. A bill or invoice is sometimes delivered to the buyer at the time of purchase, but it is usually sent with the goods, or if the buyer resides at a distance, by mail. An invoice should specify the place and date of sale, the names of buyer and seller, a description of the goods, the prices of boxes, etc., used for packing, and in some kinds of business, the terms of sale. When goods are received, the quality and quantity are compared with the invoice, and the selling prices made out from it, after which it is filed away or pasted in a book for future reference. An Account is a statement of goods sold at differ- ent periods of time. Accounts are taken from the ledger, and often contain items in favor, as well as against the buyer. Finding the cost of a number of articles at a certain price, and placing the amount opposite, is called, in bill-making, extending; adding the col- umns, footing up. In making out bills, the three requisites are rapidity, legibility and accuracy. The principal is accuracy. t)B BILLS — INVOICES. Remabk. — The bills that, come in are usually called invoices \^ DRY GOODS. New Orleans, March 4, J 876. Mr. W. A. Dickey, Bought of Charles Shannon. , 26 pieces Calico, 825 yds @ 14c %JlS^tS^ 2 " " 120 '' @ 9c /a. ?0 12 " Twilled Muslin, 340 yds @ 10c ^ J ^ 3 cases @ 75c \ 'X^ Received in payment his note at 90 days. w /A J ^^5^ Charles Sni'tmoN, * Per H. U. Cincinnati, July 31, 1876. Mr. R. Nelson, Bought of Haseltine, Macfarland & Co 60 12 15 Braid Bonnets, @$0.622 ^ 68 6 Swiss Straw Bonnets, '.' 1.25 70 4 7 Braid " *' 1.50 80 2 7 " ** " 3.00 86 2 7" " " 3.75 6 Pes. No. 1 Tafft. Ribbon, " 15 5 " " 2 " " " 28 3 " " 4 " " " 48 2 " " 6 " " " 75 1 " " 12 " " " 1.10 3 " Bonnet Ribbon, " 2.00 2 " " " " 2.50 3 1 Box Ruches, !!!!!!.7.*.*.*.*.!.'. " 1^50 425 \ u (( ^ a 2.25 210 \ Doz. BunchesFlowersi.*.*!.*.'.'.'.*.*.'.'!!.'.' '' 18.00 I " " Feathers, "36.00 1 Po. Black Silk, 20 yds " 87^ S80.47 , Note. — The numbers in the column on the left are those marked on the boxes and packages. BILLS — INVOICES. 69 y GROCERIES AND LIQUORS. Cincinnati, Sept. 1, 1875. Mr James O'Shaughnessy, Bought of King & Daly. I Hhd. N. 0. Sugar, ;ifg8 ^^1080 lbs @$0.07 % 4 Bris. jST. 0. Molasses, *|l 169 gals " 35 1 Trs. Rice, 7?§ 630 lbs ;.. " 4 20 Bags Rio Coffee, 3200 lbs " 11 2 Half chests Black Tea, '50-1^72 lbs '' 25 100-28 3 " " Yng. Hyson do. 150 lbs " 50 1 " ' Imperial do. 60 lbs " 40 2 " ** Gunpowder do. 110 lbs '' 60 1 " " Colong Blk. Tea, 45 lbs " 40 6 Doz. ground Cinnamon, *' 40 6 '* " Allspice, " 40 6 " '' Pepper, " 40 4 " " Mustard, " 75 1 Box 5 lump Tobacco, i|J 108 lbs " 25 1 " pound lump do. ^\l 124 lbs " 20 1 " Va. pound do. ^|g 120 lbs " 35 1 " Slump do. i|gl25 1bs " 22 20 Brls. Rect. Whisky, 800 gals " 17 4 " Ginger Wine, 160 gals " 60 i Cask French Brandy, 40 gals " 4.00 I " Port Wine, 45 gals " 2.00 10 Brls. Bourbon Whisky, 405 gals " 1.00 \ Brl. Holland Gin, 20 gals... . " 1.50 $1761.45 * Gross Weight — weight of hogshead, etc., and contents. t Tare or weight of bag, box, etc. Ten per cent, is usually deducted for sugar. I Gallons in each barrel. ** Net Weight — weight of goods In hogsheads, etc. 70 BILLS — INVOICES. Messrs. Gaff & Baldwin, Cincinnati, June 8, 1876. Bought of Straight, Deming & Co 100 Boxes cheese, ^^oo 3590, @ $ .08 $ 30 Firkins Butter, 336o 2820, " .15 100 Boxes* $20.00 Starch, 4810, '' .05 100 " $25.00 Star Candles, 4000,... " .20 20 bbls.^ $25.00 Lard Oil, 810 gals., '' .85 50 " Mess Pork, " 16.00 10 Tierces S. C. C. Hams, 335o 300O, " .11 30 Kegs Lard, ijjg 1334, '' .12J 15 bbls. Mess Beef, " 15.00 Com. for purchasing, $1521.75, " 2^ fo Drayages, 16.00 Insurance on $5000.00, 59.88 QUEENSWARE. $4152.87 Philadelphia, May 17, 1858. Mr. W. Anderson, To Samuel Asburj & Co. W. A. [c] 23 Crates Queensware, per ship 140 @ 163. Lancaster, as per invoice ren- dered, £115 I65. 4d^bU.U Exchange 10 fo prem., 51.41 Ins. 2J % @ 5^ per £, 14.33 $579.88 Int. 47 days, from Mar. 31 t^ May 17, 1858, 3.76 Cash, $588.64 Duties, etc., on the above, Invoice, £115 16s 4d Com. 2^fo 2 llsUd £118 145 Sd @ $4.84 per £, is $574.57, duties 24 fo $137.90 Custom, House Fees, 1.00 Freight, £10 17 3 @ $4.80, 52.14 Drayage, 9.50 Cash, $200.54 • Price of empty boxes and barrels. BILLS — INVOICES. 71 Cincinnati, December 1, 1876. Mr. Newton Thompson, Germantown, Ohio, Bought of William Andersoa WA [C] 3 doz. Edged Plates, @$0.40 $ 115 10 " " " " 50 5 " CC " " 50 4 " " Dishes, ea. $1.75, 2.25, $2.75.t 1 " " Bakers, ea. $L50, 2.00, $2.50. 1 " " Beaded Nappies, each $1.75, 2.25. i " " Chambers, " 3.50 2 " '' Bowls, " 80 24 " '' " '< 60 3 " '* '' '' 50 " " Pitchers, " 3.50 " colored '' ea. $2.50, 4.00. 4 " '' Bowls, *' 87^ 5 " " " " 65 6 " " " " 55 9 Sets CCTeas, " 20 36 " Painted Teas, '' 20 Crate, " 1.00 3, CA'PS^i $56.73 HATS, CA^S,^AND FURS. Messrs. D. W. Fairchild & Co., t^ u^ c r.^ ^ l r^ -it, ' Bought of Frost & Griffia tl328 ) 1336 } 3c. 18 doz. 2 x B. C. Shanghai, @ $13.00 1337 J {^|3 I 2c. 6 " Fr. felt Hungarian, " 21.00 . 1491 Ic. 6 ''^ B. C, '' 18.00 6 cases 75 cents each, cooperage 12^ cents each, drayage 37-^ cts., 5.63 $473.63 ♦ " CC " signifies cream color. " Teas, cups and saucers, t Three qualities. J Numbers on cases. 72 BILLS — INVOICES. Cincinnati, Sept. 16, 1876. Messrs. Lewis Evans k Co., Bought of D. W. Fairchild. ^ doz. Men's black cass. Hungarian, @ 21.00 $10.50 1 u a u u li u 27.00 51 " " «' « " 33.00 i " " " " '' 24.00 J " *' " broad brim wool, " 14.00 I " " " wove Senate, " 12.00 i " " " cashmerette, " 15.00 ^ " Boy's " wool Hungarian,. " 7.00 J " *' caps assorted, " 12.00 i " " " " " 9.75 i '' " cloth caps,. " 9.00 i " children's fancy caps, " 8.00 i " " '' " " 13.00 j\j " men's cloth caps, *' 14.00 j\ " boy's " " " 10.00 1 Case at 75 and 1 at 50 cents,.... $264.63 BOOKS. Philadelphia, Oct. 9, 1876. Messrs. Applegate & Co., Bought of Childs & Peterson. 300 Kane's Arctic Explorations, cloth, $4.22 100 " " " sh., 5.07 50 " '' " hlf. antique 7.19 50 " " ^ " full " 8.45 50 Bouvier's Law Dictionary, 2 vols., 8.45 30 " Institutes, 4 '' 12.67 400 Shepherd's Constitutional Text Book, .63 18 Cases and drayage, 24.17 $3633.77 Exchange 1 ^, $36.33 Insurance, 44.50 Freight, 85.41 $3800,01 BILLS — INVOICES. 73 HAKDWARE. Cincinnati, Messrs. A. C. Morris & Co., Bought of Tyler 6 Mouse hole Anvils ^1. ]. 13. 0. 3. 22. 0. 2. 12. 1. 2. 12. 1. 3. 16. 1. 1. 14. 873 lbs. @ 141 1 Case best cast steel, assorted,. IX5 liXi 4. 0. 26. 474 lbs. @ 17^ Pocket Knives: , Nos. 1212 1518 22 29 32 37 Doz. 3 3 12 3 6 8 Price, 4 s. 4s. 6d. 8s. 4s.9dl0s. 3s.9d. £11 5s. 9d. @ $5.00 5 doz. narrow Butts, each, 3 in. .85 " 3^ "$1.15 10.00 Less 10 fo discount, 1.00 9.00 Less Extra 12 ^, 1.08 Package and Dray age, June 3, 1876. Davidson & Co 40 82.95 $207.35 56.44 Mr. James Brov^rn, Cincinnati, 7.92 3.25 67.61 $274.96 July 3, 1876. 10 Kegs lOd. nails,. 6 " '^' 5 2 4 3 6 8d. 6d. " 4d. 'V 8d. fence nails, 8d. brads, lOd. finishing nails, Amount forwarded Bought of Guiou & Kizer @$3.50 3.75 4.25 4.75 3.75 3.75 4.50 $141.50 Long weight, cwL qrs. 74 BILLS — INVOICES. Amount brought forward, 1141.50 4 Kegs 8d nails, @ $4.75 2 Doz. No. 1 tinned bottom coffee mills, " 12.00 o " '' 2 '' " " " " 5 00 3 u u 3 u u a u u I'j-^Q 2 " Britannia hoppers, " 7.00 4 " Brass '* " 4.00 6 " Iron " " 3.00 4 " No. 8 X Janus faced locks, " 13.00 2 '' '' ^' a " '' " " 15.00 $335.00 TAILORING. Richmond, Va., Feb. 3, 1876. Mr. Michael Tracey, To James Humble. Sep. 9, For making and trimming one overcoat, $12.00 " 1 silk velvet vest, 10.00 " ^ doz. pair socks, @ $2.37^ 1.19 Feb. 5 1876. Settled, ^23.19 James Humble. FARMING. Lexington, Ky., Oct. 29, 1876. Mr. W. R. Henderson, Bought of R. S. Wharton. Jan. 3, 3 Durham heifers, @ $25.00 1 Bay 2 yr. colt, " 50.00 Apr. 11, 140^ Bushels corn, " .50 5^ Doz. chickens, " ^'^^h Aug. 9, 123^ lbs. butter, " .18| Cr. $225.97 Apr. 4, By 25 Hogs, 3147 lbs. @ $0.3^ " IPlow, $15.00. Oct. 30, " Cash for balance, R. S. Wharton. BILLS — INVOICES. 75 ~^^^ EXERCISES FOR PEN AND PAPER. The pupil should provide himself with bill paper, and make out bills from the following transactions, using his own name as clerk or principal, as he refers. 1. Apr. 3, 1876, Sold to Mrs. E. Nelson, 22 yds. black silk @ $1.25,12 yds. black silk velvet @ $4.87, 15 yds. linen @ 75 cts.,'47 yds. W. flannel @ 621 cts., to be charged to account of Eichard Nelson. / 2. Sold to Mr. H. Schnicke, 1 overcoat @ $17.50, ^ doz. shirts @ $32.50, .J doz. pocket hkfs. @ $5.75, ^ doz. pairs socks @ $3.75. Eeceived paj-ruent in cash, A. B. (clerk). Feb. 3, 1876. 7^. June 9, 1876. Sold to Cyrus Wright on order /of A. J. Eice, 2 doz. Gillot's pens @ 12^ cts, 1 box F. envelopes @ $1.50, .} doz. penholders @ 50 cts., 1 copy Bjn'on @ $1.50, 2 copies Shakespeare's 2)lays @ $2.50. Indorsed on order. Note. — This transfiction is a very common one, and should be thoroughly understood by the learner. It supposes us to be in- debted to A. J. Rice, who, in his turn, is indebted to C. Wright, in whose favor, he (Rice) draws the following order on us: " Cincinnati, June 8, 1876. Messrs. Nelson, Kizer & Co. will please let C. Wright or bearer liave goods to the amount of Twenty dollars, and charge to ray account. $20.00 A. J. EicE.'^ The amount of the bill being less than that of the ordur, Wright is permitted to keep the latter, after we write across the back of it, " June 9, paid $8.50, N., K. &Co.^' 76 BILLS — INVOICES. The amount, $8.50, is then charged on our books to Eice. A better way is, take Eice's order, charge him with the $20.00, and give Wright a due bill for the balance, $11.50. 4. May 13, 1876. Sold to H. J. Minor, Louisville, 3 chests Congou tea, marked H. J. M.— 21, 22, 23. No. 21, 102 lbs. gross, tare 21 lbs. ; No. 22, 103 lbs. gross, tare 22 lbs. ; No. 23, 99 lbs. gross, tare 20 lbs., @ 75 cts. 5. Sept 9, 1876. Sold to Eobert O'Brien 125 yds. carpet @ $1.12, .10 pieces Irish linen, 198 yards @ *26 cts., 6 pieces muslin, 71 yds. @ 12^ cts., 5 pieces French merino, 175 yds. @ 87 cts., I doz. silk um- brellas @ $51.50, 12 pieces black silk velvet, 250 yds., @ $3.25. Paid drayage $1.00, and insurance @ 1^%. 6. Jan. 9, 1876. Sold to Andrew Spence, Pittsburgh, 6 hhds. sugar, and shipped same on steamboat Bos- tona, Miles, master. The hhds. were marked and numbered ''A. S., 5, 6, 7, 8, 9, 10.'' No. 5 weighed 1424 lbs., tare 27 lbs. ; No. 6, 1573 lbs., tare 31 lbs. ; No. 7, 1397 lbs., tare 35 lbs.; No. 8, 1576 lbs., tare 37 lbs. ; No. 9, 1498 lbs., tare 30 lbs. ; No. 10, 1675 lbs., tare 36 lbs. @ 12^ cts. Paid drayage, $2.50, insur- ance 1^ %. 7. Dec. 3, 1876. Sold to Mrs. Sophia Dodd, 20 yds. I muslin @ 15 cts., 18 yds. French merino @ 87^ cts., 1 silk bonnet @ $5. Eec'd in payment B. E. Gooley's due bill, W. Dodd's favor for $30, and gave our due bill for balance of note unpaid. BILLS— INVOICES. 77 EXERCISES IN MAKING EXTENSIONS. The following exercises can all be solved by the short methods explained in another part of this work: They are designed for the blackboard. 67i@ 157 '' 216 '' 917i@ 119f " 175 '' 143J ^' . 216f " 116J '' 718^ ^' 1. Find the cost of 15 yards muslin @ 12^ cents, 22 yards silk @ 821 cents, 45 yards ticking @ 25 cents, 150 yards satfn @ $2.25, 45 yards calico @ 8^ cents, 125 yards M. Delaine @ 27 cents, 121 yards French merino @ 62^ cents, 6 pieces sheetings, 197 yards @ 33J cents, 12 pieces shirtings, 376^ yards, @ 12^ cents, 15 gross spools, $2.25. 2. Find the cost of 2 cases assorted cassi meres, 175 yards, @ $1.75, 8 pieces blue cloths, 216 yards, *917J at a dollar =$917.60 $917.50 1 800 @ 5.17= $4136.00 6 2 @, 6.17 deducted 10.34 ;0.83i 4197 @ $1.25 6191 @$3.20 0.121 464* '^ 7.621 116/:. '' 3.37^ 1.87^ 119f ^' 1.45 26^ " 1.82^ 6.55* 3671® 1.35 197 @ 3.13 1.50 4881 ^' 1.65 682# " 1.25 2.17 771J- ^' 2.121- -j-798 " 5.17 6.55 1671 " 2.50 677^ *' 4.87^ 1.75 719f " 1.63 35 '^ 0.35 1.37i 711i " 1.35 115i " 3.121 ^' 0.18| 0.45 125 ^' 0.65 417 6505.00 $4125.66 458.75 =J of the cost at a dollar. 45.87 J==-j-^ of the cost at 60 cts. 6009.62J 78 -LONG DIVISION. XII. LONG DIVISION. Art. 1. The previous operations in division have been performed mentally^ the learner writing only the quotient. This is called Short Division, and is to be preferred when the divisor is a small number, or caa be reduced to a small number, as 200, 12000, which by pointing off the ciphers are reduced to 2 and 12 (Art. 9, p. 41.) But when the divisor is 19, 23, 79, 536, etc., the operation would be too difficult and tedious to perform mentally. In such case the greater part of the process has to be written^ and is known by the name o? Long Division. Example 1.— To divide 3147 by 6. SHORT method. 6)3147 ~524| long method. dividend, quotient. 6)3147(524^ Explanations. — 1. To perform this operation OA ^ we say, 6 in 31, 6 times, and write 5 in the quo- tient and multiply it on 6, which makes 30. This 14 we write under the 31. 'i 2 2. We now subtract the 30 from the 31 as we would perform an operation in subtraction. The 27 remainder is 1. Instead of supposing this 1 oj. to stand before the 4 in the dividend, we bring down the 4 to it, which makes 14. 3 3. Six in 14, 2 times, we write 2 in the quo- tient, multiply it by 6 and write the product underneath. 4. Subtracting this 12 from the 14 we find a remainder of 2, to which we annex 7, brought down from the dividend, making the number 27 5. Six in 27, 4 times, we write 4 in the quotient, multiply it by 6, and write the product 24 underneath. 6. Subtracting as before we find a remainder of 3, under which we write the divisor 6, making |. LONG DIVISION. , 79 2. To divide 834716 by 723. 723) 834716(1154||| 723 ' This is the product of 1X723. 2 Tiie remainder after subtracting 723 '1117 from 834, with 7 brought down. » 723 3 The product of 1X723. 4 The remainder of 1117— 723, and 1 * 3941 brought down ^ 3615 5 Tlie product of 723X5. -- — -^ 6 The remainder of 3941—3615, and 6 3266 brought down. ' 2892 "^ The product of 723X4. -^ 8 The last remainder=||| Note. — The pupil should put a dot under each figure brought down, to prevent its being taken twice. 3. To divide 67314968 by 163000. 163|000)67314|968(412i|fgg| 652 * 211 163 "484 326 158 Remarks. — 1. Instead of using the whole divisor in finding a quotient figure, it will generally do to use only the first one or two figures. For instance, in Ex. 2, say 7 in 8, instead of 723 in 834; and in Ex. 3, say 16 in 67, instead of 163 in 673. 2. The products should never exceed the numbers above them; (number 3 should not exceed number 2;) if they do, a smaller number should be put in the quotient. 3. For every figure brought down from the dividend, there should be one in the quotient. Where the divisor is not contained in the small dividends, as the 310 in the 68 (Art. 2, Ex. 1), a cipher should be written in the quotient, and another figure taken down. 4. The divisor can not be contained more than 9 times in the new dividends. * These three figures are a part of the remainder, as shown in the quotient. 80 LONG DIVISION. Art. 2. To divide dollars and cents, we reduce them to cents, then our quotient or answer will be cents, which are easily converted into dollars, by in- serting the decimal jDoint. 1. To divide $3168.20 by 310. 310)316820(1022 or $10.22 310 "682 620 ~620 620 2. Divide 9765837 '' 65. Ans. 150243||. 3. 1763-^76= 4. 3167-^-119= 7964--87='- 71438-^-320 = 89737--98 = 67898^764= 77168-^19= 78637^-892= Total remainders, 138 5.* $10000--7109 = 7185--1990= 67416--- 144r= 3784-- 642= Total remainders, 958 6. $140.98-^-671 = 730.45~-126 = 164.87-^-144= 1710.14-f-166 = Total quotients, $479.06 Total quotients, $17.44 PRINCIPLES OF DIVISION. Art. 3. If we divide the price of a number of things of equal value by the number, we obtain the price of one. Art. 4. The quotient will always be in the same name with the dividend or number to be divided. If the dividend be dollars, the quotient will be dol- lars ; if it be rods, the quotient will be rods. * Reduce these to cents before dividing: $10000=1000000 cents CSee Note page 38,) «ind omit the remainders. EXERCISES IN MULTIPLICATION AND DIVISION. 81 EXERCISES IN MULTIPLICATION AND DIVISION. 1. If 23 yds. of muslin cost S3. 45, what will one yard cost?* 2. If 117 men can do a piece of work in 48 days, how long will it take three times that number to do it? 3. How many men can do a piece of work in 5 days, that took 10 men 25 days? 4. If a case hold 29 pieces of muslin, how many will it take to hold 7250 pieces ? 5. If 15 men can do a certain piece of work in 75 days, how long will it take 1 man to do it ? 6. If 7 dozen silver spoons cost $35.35, what will 3 dozen cost? Note. — Find the cost of 1 dozen, then the cost of 3. 7. If two-sevenths of a ship cost $14602, what will the seven-sevenths, or the whole ship cost? METHODS OF PROOF. Art. 5. Division and Multiplication being con- verse operations, the one is' proved by the other. DIVISION. PROOF. 88)3715(97 97=quotient. 342 38==divisor. 295 776 266 291 "29 rem. 3686+ the rem. 29=3715=dividend MULTIPLICATION. PROOF. multiplier, prodact. multiplicand. 465 25)11625(465 25 100 2325 ' 162 930 150 11625 l25 125 6 82 TIME. XIII. TIME. TO RECKON TIME. Art. 1. Business men usually reckon 30 days to the month; but when a note is given at one, two or three months, it falls due on the same day of the month it was given, plus the days of grace. Some notes and bonds draw interest from date. When such is the case, the time is computed as fol- lows : 1. What is the difference of time between January 3, 1878, and February 9, 1879 ? 3rrs. mos. days. Operation. 1879 2 9 1878 1 3 Am. 116 ExPiiANATiON. — We call January the first month, February the second, etc. 2. What is the difference in time between April 3, 1878, and January 1, 1879? yrs. mos. days. Operation. 1879 1 1 1878 4 3 8 28 Find the difference of time between the following dates : January 1, 1874, and April 2, 1876 = October 9, 1871, and Jan. 1, 1875 = June 23, 1875, and Dec. 9, 1878 = Total, 8 yrs. 11 mo. 9 ds. SIMPLE INTEREST. 83 XI \. SIMPLE INTEREST. Art. 1. Interest is a percentage charged for the use of capital. It is regulated by the year or month. 6 per cent, (per annum) signifies* szo: dollars on every hundred dollars for a year.* Interest may be divided into simple and compound. Simple Interest is percentage on capital alone. Compound Interest is interest reckoned on both capital and interest. TERMS. The terms are Principal, Rate, time and Amount. The Principal is the sum or capital loaned. The Bate is the percentage charged. The Amount is the sum of principal and interest. Note. — The legal interest of the United States is 6 per cent. Vi hen no per cent, is named in this book, 6 per cent, is understood. 2. Mills are omitted in the answers. Art. 2. The interest on any sum of dollars for 60 days, is equal to as many cents as there are dollars.f The int. on $100 for 60 days is 100 cents or $1.00. u u 1250 " '' " 1250 •' ^' 12.50. Find the interest on the following: 1. $1749 for 60 ds. 2. $1009 for 60 ds. 785 ^' '^ " 719 '^ " '' 9000 " " ^' 5000 '^ " " Total, $115.34 Total, $67.28 * It is customary to reckon interest for all rates at 6 per cent., and afterward to increase or diminish as necessary. See Art. 6. t Since the interest on $100 for 360 days is $6 (Art. 2), for 60 days, it is one-sixth as much or $1.00; but $1 is 100 cents or as many cents as there are dollars in the princioal. S4 SIMPLE INTEREST. Art. 3. To find the interest for any number of daySy we take that part of the interest at 60 days, that the number of days is of 60. To find the interest of $120 for 30, 20, 15, 12, or 10 days. The interest on $120 for 60 days, is $1.20. For 30 days it is J of $1.20, or 60 cents. For 20 days it is | of 1.20, or 40 cents. For 15 days it is i of 1.20, or 30 cents. For 12 days it is | of 1.20, or 24 cents. For 10 days it is I of 1.20, or 20 cents. Reason. — Since 30 days is J of 60, the interest for 30 days will be J of that for 60 days; 20, 15, 12, and 10, are also equal parts of 60. Note. — When the days are not even parts of 60, we divide them into even parts. For 18, we take 16 and 8 ; for 27, take 12 and 16; for 37, take 30, 6, and 2; for 110, take 60, 30, and 20. 2. Find the interest on $211 for 93 days. 2 11 =int. for 60 days. The student, after some 1 1 055:= a a 30 " practice, should not lose time f -j QP- (t t< q n by writing the divisors, or 2^ lUO— ^ ^j^g jj^^,g ^^ ^Yie right, as in «3 27 =s *' " 93 *' *^^^ example. TABLE. ALIQUOT OR EVEN PARTS OF SIXTY DATS. To be committed to Memory by the Pupil. 30 days =i 12 days= I 5 days=y'^ 2 days=3'^ 90 '< i^ 10 " ' 4. '^ 1 1 *' J ip; u — 1 f\ a — 1 ^ '* — 1 10 — :j D — yu ^ — TT(5 ds. ds. 3. $797.00 for 10=$1. 33 4. $1000 for 2= * Ititerest is seldom reckoned on cents. If less than 60, reject them, otherwise add a dollar to the dollars. Simple interest. 85 $1000 71 61 190 days. ,00 for 27= .97 for 47= .80 for 45= .27 for 16 = 6. days. ^1799.14 for 98= 387.66 for 67 = 199.44 for 41 = 450.22 for 29= Total, $6.03 Total, $35.75 days. 7. $719.99 for 11 = 55.18 for 9 = 88.17 for 69= 466.00 for 78= $1997.00 for 7.88 for 17.97 for days. 13= 54= 35 = 10.00 for 120= Total, $8.47 days. $1000.00 for 97 = 650.00 for 67 = 10.70 for 13= 127.57 for 51 = 368.17 for 118= 718.57 for 125= Total, $4.71 10. days. $1999.20 for 23= 361.74 for 18= 78.93 for 23 = 1467.20 for 34= 7100.18 for 77- 29.00 for 99 = Total, $46.76 Total, $108.96 Art. 4. To find the interest for years and months. In a year there are 6 sixty days ; therefore we mul- tiply the interest for 60 days by six times the num- ber of years, and as there are half as many sixty days as months, we multiply the interest for 60 days by half the number of months. Kecapitulation. — Consider the dollars cents ^ and multiply by 6 times the number of years plus half the number of months j and for the days take aliquot parts as before. 1. To find the interest of $120 for 1 year 4 months and 20 days. 86 SIMPLE INTEREST. 1.20 Explanation. — The interest for 60 days is 120 cents; S for 1 year and 4 months it is 8 times 120 or 960 cents; and for 20 days it is J of 120, or 40 cents, making the 9.60 sum $10.00 — the interest required, 40 Arts. $To7oO 2. Find the interest of ^240 for 3 years 4 months and 10 days. Ans. $48,40. 3. What is the interest of $1467.45 for 2 years 6 months and 17 days. Ans. $224.21. Find the interest of the following: yrs. mos. days. 4. $321.00 for 2 3 15.* $44.14 1767.00 for 7 4 21. 783.66 897.25 for 3 i 6 27. 192.41 898.57 for 2 - 7 ^ 25.t 148.09 716.27 for 2 1 9= 90. .57 810.98 for 1 6 7= 73.94 50.00 for 3 7 . 18= 10.90 8.00 for 9 •■^./ 27 4.48 yrs. mo» days. 5. $3140.79 for ^ y 7 7 = 795.17 for 2 1 1 = 3.90 for 3 5 15 = 1057.57 for 1 11 11 = "tCotal, $526.01 yrs. mos. dfivs. 6. $2674.57 for 1 8 21 = 7143.45 for 2 1 18 = 1742.67 for 1 9 13= - 2100.00 for 2 1 1 = 4109.85 for 1 6 17 - Total, $2022.35 *Find the interest for 2 years 4 months, and deduct the int. foi 16 'days. t Call this 2 years 8 months, and deduct the int for 5 days. SIMPLE INTEREST. 87 yrs. mos. days. 7. $7856.00 for 1 1 29 = 677.19 for 3 3 3= 287.17 for 1 7 16 = 97.19 for 5 10 14= 10.10 for 1 3 19= $743.95 yrs. mos. davs. $57.87 for 2 6 14= 120.14 for 7 7 7 = 340.00 for 9 1 24= 1657.00 for 1 3 24= 769.75 for 2 3 18= $487.40 Art. 5. Having the interest at 6 per cent, to find the \rMrest at any other rate. This is done by taking aliquot parts of 6, and in- creasing or diminishing the interest, as the rate is more or less than 6 per cent. At 2 per cent, the interest is I of that at 6 per cent. At 4 " " it is I less than at 6 per cent. At 8 " ^' it is J more than at 6 per cent. . At 10 " '' it is ig« of that at 6% ; so we have only to move the decimal point in the Q% interest one phice to the right, and divide by 6. For 15 %^ we move the decimal point in the same way, and di- vide by 4 ; and for 20 % by 3. 1. Let the interest at 6 % be $240. At 2 % it will be I as much, or $ 80 " 8 % " " " I more, or 320 " 10 % " '* '' 10 times i of $240, or 400 88 SIMPLE INTEREST. 2. To find 14 %, 7i %, 8J %, and 10| % of $350 for 60 days. 4)3.50 int. at 6 % ^^^f^ 3 ^f;- f^^ \ ^^ .87,5" " \\% '.14 5 '' " i?^ 84723~~ " " 71 % $3.50 int. at "6 % $3.50 int. at 6 ^ 1.16.6 '^ " 2 % 1.75 '' " 3 % »19.4 " *' J % .87 5 " "^H % $4.86 8J% $6.13 10^% yrs. mos. days. 3. $798.18 for 6 1 6 @ 9% =$438.10 1000.00 for 4 2 4 @ 7%= 292.44 yrs. mos. days. 4. $340 for 2 2 20 @ 2i% = 600 for 3 4 15 @ 6i% = 850 for 1 2 12@8|% = Total, $237.22 5. Find the interest of mos. days. $617.18 for 3 18 @ 15% = 460.74 for 2 5 @ 18% = 765.12 for 8 16® 20% = Total, $151.55 6. Find the interest on the following at 10% $710 for 92 days= 7. $496 for 91 day8 = 1978 for 27 " = 671 for 86 " = 8889 for 128 " = 100 for 104 '' = 75 for 117 " = 269 for 73 " = Total, $351.47 Total, $36.91 •Art. 6. It is customary for bankers to lend money^ and discount b} the month instead of by tne ytar. SIMPLE INTEREST. 89 This percentage is easily converted into 6% interest, and the work performed with as much ease as before : 1 % per month is 12% per year, or 2 times 6% \\% " " is 18% ^' ^ '' or 3 " 6% 2 % " " is 24% '' '' or 4 " 6% 1. Find the interest of the following : $65.00 for 80 days® 2 % per month = 40.00 '' 33 '' @\\% '' " = 190.00 ** 63 " @2 % " *' = • 700.00 ''93 " @3 % *' " = Total, $77.20 Art. 5. The work, when computing interest, can often be abbreviated. Sometimes advantage may be taken of the aliquots of hundred ; at other times it will be of advantage to transpose the terms and consider the days as dollars and the dollars as days; or the rate (if it is some other rate than 6%) may be reduced mentally to 6%. For instance, in the second question in the last group, the $40 may be considered $120, and transposing the term and the 33 multiplied by 2, mak- ing 66c the answer. It will materially abridge the operation and expedite the labor, if the learner will observe to avoid the use of all lines, figures or marks that are not absolutely necessary. As, for instance, when using aliquot parts, to write only the results of division, as shown in the following example: Interest of $321 for 2 years, 1 month and 8.21 22 days at 10% per annum. "40^25 Explanation. — Mentally it is found that there are 1.07 Vl\ 60 days in 2 years and 1 month, to multiply by ^07 which we divide by 8. The division by 6 and the multiplication by 10 were performed simultaneously, 41.302 giving $68,836 or $68.84 as the answer. ~7:r7:7?. 68.806 90 COMPOUND INTEREST. XY. COMPOUND INTERJEST. Art. 1. In Compound Interest the interest is con verted into principal ev^ery quarter, half year or year. Capital is thus more rapidly increased, than by simple interest. Any person acquainted with the principles of simple interest will readily understand how to com- pute compound interest. 1. What is the compound interest of $1000 for 1^ years at 6 %, payable semi-annually (half-yearly)? The interest of SIOOO for 6 mo8.= S30.00 Add the principal, 1000.00 Amount for 6 mos. $103oToO Interest on $1030 for 6 m^os., 30.90 Amount for 1 year, $1060.90 Interest on $1060.90 for 6 mos., 31.827 Amount for 18 mos., $1092.727 Principal, 1000.00 Compound interest for 1-J years, $92.73 2. Find the compound interest and amount of $1865 for 3 years, 3 months, at 8 ^, payable tri-^monthl}^. 3. What is the compound interest and amount of $486 for 4 years, at 10 ^, payable annually? 4. What is the compound interest and amount of $672 for 4 years, at 6 % per annum ? Answers: $1092.73, $848.38, $2412.56, $711.55, $92.73, $176.38, $547.56, $225.55. Kemark. — At 6 per cent, money will double itself in 11 years, 10 months and 21 days. At 5 per cent., in 14 years, 2 months, and 15 days. At 3 per cent., in 23 years, 5 months, and lOJ days. ANNUAL INTEREST. 91 XVI. ANNUAL INTEREST. Annual Interest is the term applied to interest on a note that is drawn with the clause ^' interest payable annually." When this interest is not paid at the end of the year, it draws simple interest till paid. 1. A note for $300 at 3 years, 6% interest, pay- able annually, had nothing paid on it at maturity. How much was due? Int. on $300 at 6% =$18 = int. for 1 year. 3 $54 = int. for 3 years. Int. on $18 for 2 years, 2.16 " '^ 1 year, 1.08 Principal, 300.00 $357.24 amount. 2. What is the amount of a note, at the end of 4 years for $368, for 2 years, 8% interest, payable annually, that had nothing paid on it until settle- ment? Note. — When a note is overdue interest is calculated up to the date of maturity, as in Ex. 1, and simple interest is calcu- lated on the amount from maturity till paid. 3. What should I pay at maturity to redeem my note for $800, payable 3 years after date with 10% interest, payable annually, nothing having been paid at maturity? 4. A note for $720, at 4 years, 6% interest, pay- able annuall}^, had nothing paid at maturity. How much was due? 5. What is the amount of a note, at the end of 3 years, for $1268, payable 2 years after date, bearing S% interest, payable annually, no payments having been made until settlement? Answers : $497.91, $1064, $738.84, $1597.32, $357.24, $908.35, $497.92. 92 PARTIAL PAYMENTS. XVII. PARTIAL PAYMENTS. Art. 1. Notes, bonds, etc., drawing interest, are sometimes paid by installments, and the amounts thus paid, indorsed on them. The legal rule for computing interest on installments, may be expressed thus : Apply the payment to the discharge of the inter- est, and if there is a remainder, subtract it from the debt. When the payment is less than the interest due, it is not applied to the discharge of the interest or debt, but is indorsed on the note until the install- ments exceed the interest; then the sum of the payments are computed as below. 1. $576. Cincinnati, Oct. 9, 1875^ On demand, 1 promise to pay Eobert Ingles, or order, five hundred and seventy-six dollars, with interest, value rec'd. Samuel Dunning. On the note are the following indorsements; RecM Dec. 16, 1875, $100 " Feb. 28, 1876, 3 '^ July 27, " 150 Required the amount due Sept. 3, 1878. yi mos. ds. From 1875 12 16 Take 1875 10 9 Difference, 2 7, or 67 days. Amount of note, $576.00 Interest on $576 for 67 days, 6.43 Total amount due, $582.43 PARTIAL PAYMENTS. 93 Total amount due, $582.43 Installment, to be subtracted, 100.00 Balance due, $482.43 The second payment is less than the interest due, and no calculation is required. From Dec. 16th, 1875, to July 27th, 1876, is 7 mos., 11 days. Balance due, $482.43 Interest for 7 mos. 11 days, 17.75 Amount due, 500.18 Amount of payments, 153.00 Balance due, $347.18 From July 27 to Sept. 3d, is 38 days. Balance, 347.18 Interest for 38 days, 2.19 Amount due Sept. 3, 1876, $349.37 2. $650. Boston, June 3, 1868. For value rec'd, I promise to pay on demand t • H. Crooks, or order, six hundred and fifty dollars, with interest at 6 % per annum. Indorsements. J . i . JJAVIS. Jan. 6, 1870, $95 Oct. 13, 1870, 350 Jun. 3, 1875, 12 Sept. 7, 1877, paid the balance, how m.uch was it? Ans. $405.92. 3. On a note drawn Sept. 3, 1877, for $650 with legal interest, there are the following indorsements : Oct. 4, $100 Nov. 3, 2 Dec. 19, 210 Apr. 3, 1878. the balance ; how much was it? Ans. $354.32. M BANK DISCOUNT. XVIII. BANK DISCOUNT. Art. 1. Discounting notes consists in buying them at less than their nominal value, or the amount for which they are drawn. The difference between the nominal value and the price paid is called discount. There are two kinds of discount : True Discount, which is interest paid in advance on the present value of a note, and Bank Discount, which is interest paid in advance on the face of the note. The latter resembles compound interest, as it is interest on both interest and principal.* When a note is discounted in bank, the interest of the note for the time it has to run, and at the banker's rates, is deducted from the sum called for by the note. This species of discount is therefore reckoned in the same way as interest. 1. How much discount should be deducted from a note of $500 at 90 day^ 2 $5.00= int. for 60 days. 2.50= " '' 30 ^' .25= '^ " 3 '' (grace) Ans. $7.75 2. $1500, Columbus, Jan. 8, 1879. Sixty days after date, I promise to pay Messrs. M'Ewen and Banfill one thousand ^ve hundred dol- lars, value received. William Dodd. Eequired the discount at 6 ^. Ans. $15.75. * The present worth of a note drawn for $100, payable in a year at 6 per cent., is $94.34, and the interest is $5.66; that is, the principal and interest together, are equal to $100, or the face of the note; so when a banker discounts from the face of a note, he discounts on both principal and interest. BANK DISCOUNT. 95 Bankers prefer lending money on short time, and by the day, instead of by the month. Notes are usually drawn for 30, 60, or 90 days ; and interest 18 always charged on the days of grace, 1. What is the bank discount on a note of $120 at 60 days, at ^ % per month? Ans. $1.26. 2. Find the discount on a note of $575.75 at 90 days, at the same rate. Ans. $8.92. 3. What is the bank discount on a note of $450 for 60 days at 2 % per month? Ans. $18.90. Remark. — The discount on $450 at 2 per cent, per month, is the same as the discount of 4 times $450, qf $1800 at 6 per cent per annum. 4. How much money should be paid by a banker who discounts a note of $350 at 30 days, at 1^ % per month?- Ans. $344.22. 5. What will be the proceeds of a note drawn for $670 at 60 days, at 2 % per month? Ans. $641.86. 6. At 1^ % per month, how much proceeds should be recovered on a note of $1749.57, drawn at 90 days? Ans. $1668.12. 7. Find the discount on a note of $1678.25, drawn at 90 days at IJ % per month ? 8. At 2i % per month, what is the discount on a note of $688 at 90 days? 9. At If % per month, what will be the proceeds of a note drawn for $6784, at 60 days? Answers to the foregoing: $47.99, $65.03, $6534.69. 10. Find the discount on the following: $1310.00 for 60 daj^s @ 2 % per mo. 746.87 " 90 " *' 1-J " ^' *< 219.56 " 30 *' " 1 1867.25 '^ 20 " " 2^ " ** 1367.00 ^' 15 " " 3 " " $152.57 96 BANK DISCOUNT. . $1673 for 30 days (a>, 1 6789 '' 3 nios. u 2 1987 " 9 mos. a 1 6745 " 10 days a n % per mo. $693.21 Find the amount of proceeds of the following: 12. $3768 for 10 days @ 4 % per mo. 1767 " 15 " " 8 '' '' " 8767 ^' 6 " '* 1^ " " '^ $14165.43 13. $167.39 978.00 897.87 for 2 mos. U Q U u 3 (( @ ^ % per mo. '' 20 '' " an. << 25 '^ " " $1900.25 Note. — As a review exercise the pupil might cal- culate interest on cents as well as dollars. Art. 2. Bankers frequently discount notes that are partly matured ; when such is the case, the following table will assist the accountant in computing the dis- count: A TABLE Showing tho number of days from any day in one month, to the same day in any other month, throughout the year. MONTHS. i X5 u a ft < ^ S >> ft 1 > o 1 January, 365 334 30fi 275 245 214 184 153 122 92 61 31 31 365 337 306 276 245 215 184 153 123 92 62 59 28 365 334 304 273 243 212 181 151 120 90 90 59 31 3'i5 335 304 274 243 212 182 151 121 120 89 61 30 365 334 304 273 242 212 181 151 151 120 92 61 31 365 335 304 273 243 212 182 181 150 122 91 61 30 365 334 303 273 242 212 212 181 153 122 92 61 31 365 334 304 273 243 243 212 184 163 123 92 62 31 365 335 304 ♦274 273 242 214 183 153 122 92 61 30 365 334 304 304 273 245 214 184 153 123 92 61 31 365 335 334 February, ^m March, 275 April, ?-W May, ■;ii4 183 July, 153 August, ..-. }n September, 91 October, 61 November, 30 December,...: 365 BANK DISCOUNT. 97 Use of the Table.^To find the time from Feb. 13 to March 23, in the following example : In the left hand column we find February in the second line, and running the eye along till we come under " March,'^ we find the number 28; hence from Feb. 13 to March 13 is 28 days ; to March 23d, will there- fore be 10 days more, or 38 days. The discount will be reckoned for 93 days minus 38 days, equal 55 days. 1. A note drawn on Feb. 13, 1878, for $900, at 90 days, was discounted on March the 23d, at 2 ^ per month, how much was paid by the borrower? Ans. $867. ' 2. What proceeds should be paid on a note of $346 at 90 days, drawn on Nov. 3d and discounted on Dec. the 7th, at IJ % per month? ^riS. $335.79. 3. A note of $689, made Sept. 9, payable- in 60 days, was discounted on Oct. 5th, at 2 % per month, what was the discount? Ans. $16.99. Note. — If the decimals be carried out to three or four places, the cents may differ slightly from the following totals. (4.) Face of Date. Time. When Rate of Disc't. Note. • Disc'td. $167.50 Jan. 3, 1879 60 days Feb. 7, 2 ^ per mo. 9876.00 Feb. 7, " 90 " Mar. 12, 2^ fo " " 789.00 Jun. 18, '* 30 " July 3,4% " " 1897.00 Feb. 21, " 90 " Apr. 1, 1^ fc '' Total, $555.24 (5.) 1676.37 Apr. 3, 1879 90 days May 9, 2 % per mo. 679.39 Mar. 9, " 30 " Apr. 3, 2^ fo " " 7168.00 Jun. 13, " 60 " July 9, 1^ fo '' " 816.37 Aug, 12, " 30 " Sep. 6, 21 fo " " Total, $167.74 7 98 BANK DISCOUNT. (6.) *No^te°.^ Date. Time. ^isc'td. Rate of Disc't. Discount 2676.00 Jan. 9, 1879 90 days Feb. 1, li fo per mo. 7187.00 Feb. 3, " 60 " Mar. 13, \\ ; what was the rate per cent. ? 3. With an investment of $4944, what rate per cent, will bring $618 ? Art. 5. To find the Eate of Income for a given investment obtained at a discount or for a premium. 1. If I invest in 6 ^ interest-bearing bonds, pay- ing 10 % premium, what per cent, of income will I receive ? Solution. — If I buy at 10 cfo premium, I pay $110 for a bond of $100. The investment of $110 brings me but $6— hence: exlOp^jO^ lip 11 >' /" 106 COMPLEX PERCENTAGE. 2. Bought 5-20 (6 %) bonds @ 12 ^ premium ; whate rate per cent, income will I receive, including J % brokerage? Note. — The premium and brokerage added will give what was paid above the par value. 2. Bought railroad bonds bearing 8 ^ interest at a discount of 4 % ; what will be the rate per cent, of my income ? Art. 6. To find the Cost Per Cent, of interest- bearing stocks in order to make a certain per cent, income. 1. What should I pay on the dollar of a 6 ^ bear- ing bond to make an income of 10 % ? Solution.— A $100 6 cfo bond will pay $6, and $6 is 10 ^ of $60. Hence, I must buy 6 ^ bearing bonds @ 60 cents on the dollar to make 10 ^. FORMULA : —^ — = 60. 10 2. At what rate should I buy a 7^ ^ bearing bond to make an iacome of 10 ^ on my investment? 3. How much should I pay on the dollar of an 8 ^ bond to make an income of 12 ^ ? Art. 7. To find the Cost of an investment bought at a premium or discount and sold at a premium or discount, when the gain and rates of premium and discount are known. 1. Bought stock at 4 % discount and sold it at 6 % premium, gaining thereby $200 ; what was the amount of my investment ? Solution. — Buying at 4 ^ discount and selling at 6 ^ pre- mium gives a profit of 10 9^. FORMULA : ^^0X100^ $2000, face of bonds. 10 COMPLEX PERCENTAGE. 107 4 9^ of $2000=: $80; that taken from the par value gives $1920 as the investment. 2. Bought stock at 10 % discount and sold it at 2^ ^ premium, realizing $87-^; what was the par value of the stock, and how much did I invest ? 3. Bought stock at 2 ^ premium and sold it at 2J % discount, thereby losing $450 ; what was the par value, and how much did I invest? Art. 8. To ascertain the Time a note has to run, the discount, etc., being known, so that a certain rate will be equal to the interest at another certain rate. 1. How long will a note, discounted at 20 % per annum, have to run to make 22J % interest per an- num. Solution. — The discount at 20 ^ per annum is equal to the interest @ 25 9^ per annum (Proof: Interest on $100 for 1 yr. @ 20 ^ ^ $20, giving $80 as the proceeds, 25 ^ of which will be $20). Hence, in 1 year, or 360 days, a discount of 20 % will equal the interest at 25 %. To ascertain in how many days a discount would bring 22J 0/^ interest, use the following — 360 days X 25% X 2+ , . , FORMULA : ^—^ ^^—-^ — ^ which 22^%X5 8 ;^ stands canceled: ^^0 X 25 X ^i ^^OO days. % Proof.— The discount for 200 days on $100(^^20^^ ---=$11.11, giving as proceeds $88.89, the interest on which for 200 days @ 22.]% is $11.11. The numbers used in the formula are 360, the number of days in a year; 25, the rate of fnterest per year; 22 J, the rate of interest for the time desired; 2 J, the difference between the rate of discount and the rate of interest for the desired time; and 5, the difference between the rate of discount per year and the rate of interest per year. 2. How long will a note, discounted at the rate of 108 COMPLEX PERCENTAGE. 20 ^ per annum, have to run to make 24 % interest per annum? . 3. To make 11 J ^ interest per annum, how long would a note, discounted at 10 ^ per annum, have to run ? Art. 9. To ascertain the Eate of Gain on articles which, by being bought at a certain lower rate, will produce a certain higher rate of gain. 1. If an article be bought at 10 ^ less, and the rate of gain thereby increased 15 %, what would be my rate of gain ? Solution. — By buying at 10 cfo less, we pay 90 cents on the dollar. Hence, by buying the cost is 90 cents on the dollar. 10 cents gained on 90 cents is a gain oflli^. \i \\\ cents gain on the dollar in the cost will make 15 cents on the dollar in the gain, the entire gain per cent., the following formula will work out the selling price : 15X100 15X9X100 ,^. formula: — = — — — =135. 11-^ 100 $1.35 being the amount an investment of $1.00 will produce. Hence, we gain 35 cents on the dollar, or 35 %. Note. — The example may be worked by proportion, thus: Hi : 15 -i^ 100 : 135. For \\\ is to 15 as 100 is to 135; or, the gain per cent, buying, is to the gain per cent, selling, as the cost price, is to the selling price. 2. If an article cost me 12 J ^ less, my rate of gain was increased 16 %] what was my rate of gain? 3. What will be my rate of gain, if I buy an article ' at 10 % less, and thereby gain 16f % more. Answers: 50 ^, 12 %, 36 ^ ; 400 days, 300 days, 200 days ; $10200, $10000, $700, $630, $1920 ; 66f cts., 73^,60 cts.; Si%,H%,^^%; 121^,7^^,7%; S872, $735, $2454.32, $2400; 24^^, 8 ^, 10^,22J%. COMPLEX PERCENTAGE. 109 Art. 10. To find the amount /or icMch a note may he draivn to realize a certain sum after being discounted. 1. Kequired the face of a 90-day note which will realize $275.23, after being discounted at 2% per month. [nterest on $1 for 93 days at 2% per month = .062. Proceeds of SI --^ $1,000 — .062 = .938. Since there are as many dollars in the principal as the proceeds of $1 is contained times in. the proceeds given, $275,230-^.938 will give the principal required, $293.42 +. Proof.— Interest on $293.42 for 93 days at 2% per month =^ 18.19 -4-, which, subtracted from $293.42, leaves $275.23, the proceeds. 2. The proceeds are $212.60, time 63 days, rate 1^% per month ; required the principal. ^ 3. What principal will realize $120 proceeds in 6 months at 10% per annum? 4. The time is three months, rate 10% per an., pro- ceeds $168.97; what is the principal? 5. The rate is 12% per annum, proceeds $693.75, time 4 months; required the principal. Art. 11. To find the rate per cent., when the prin- cipal, interest, and time are given. 1. The principal is $300, time 60 days, interest $5; required the rate. Interest on $300 for 60 days at Q % = $3. At 1 ^ =rz .50. It is obvious that the rate will be as great as the number of times 1 ^ is contained in the interest given. Hence, $5.00 -f- 50 = the rate, 10 ^. Proof.— Interest on $300 for 60 days at 10^ =$5. Answers: $293.42, $219.52, 10 %, $173.30, $126.32, $722.66. *The learner can prove his work by computiag interest on the principal found. 110 COMPLEX PERCENTAGE. 2. The principal is $396.15, time 13 months 9 days, interest $26.34,3 ; required the rate. 3. What is the rate per cent, on $144 for 5 days, when the interest is 24 cents? 4. Eequired the rate on $250 for 60 days, when the interest is $3.50. 5. The principal is $820, time 30 days, interest 86.15 ; what is the rate? Art. 12. To find the Time, when the principal^ rate per cent and interest are given. Grace being allowed only on notes and drafts, where neither is named it is not reckoned. 1. The principal is $1440, rate 10 ^ per annum, interest $37.50 ; required the time. Interest on $1440 for 1 day at 10 % = 40 cents. Since there are as many days as the interest for 1 day is contained times in the interest given, $37.50 -^ 40 =: 93f , or 94 days. Proof. — Interest on $1440 for 94 days at 10 % per annum = $37.60.* 2. The principal is $1674, rate 2 ^ per month, in- terest $59.87; required the time. 3. In what time will a note for $600, at 6 ^ per annum, draw $27.50 interest? 4. A note for $375 drew $21 interest at 6 ^ per annum ; how long did it require to do it? 5. A merchant wishes to know the time it will take a balance of $917.50 to make $60.80, with interest at 10%. Answers: 8|-%, 12%, 6^ 11 months 6 days, 54 days, 108 days, 239 days, 94 days, 275 days. 9%. -'Interest is never reckoned on the fraction of a day, hence the difference. COMPLEX PERCENTAGE. Ill Art. 13. MISCELLANEOUS EXEKCISES. 1. What is the gain per cent, on goods bought at SI. 20 and sold at $1.35? 135 = 8elling price. 120=cost price. 15= actual gain. j^nd*y^^^(5C.=gain on one cent. On 100 cents there will be 100 times J/^j or i-520_o = i2_6_ or 12^. Note. — If the gain is an even part of the first cost take the same part of 100. This is the reverse of operation 1st in last Article. In the present Ex. 15 is J of 120, therefore the gain per cent, is \ of 100 or 12^. FIRST COST. SELLING PRICE. GAIN PER CENT 2. $2.00 $3.00 3. 1.25 1.50 4. 0.75 1.00 5. 0.10 0.12^ Total, 1281 What was the first cost of the goods marked a» follows? 9. 3.75 @ 25 % loss. 10. 0.87^ '' 12» % " 11. 0.12^ " 50 % " 6. 115.87 @ 12^ % gain. 7. 14.54 '^ 3^ % " 8. 00.87 " 16| % " 12. A bill of $1687. 75, had been reduced 10%, what was the original amount? 13. A carpenter puts in an estimate at 25 % off the bill of prices, and another puts one at 10 % off the first; how much per cent, off the bill was his discount ? Answers : 12^ %, 50 %, 20 %, 33J %, 25 %, $102.97, $10.90, $0.74, $5, $0.25, $1, 1875.28, 32^ %. 112 COMPLEX PERCENTAGE. 14. Bought a bbl. of apples for $1.75, and sold it for $2.25 ; what did I gain per cent? 15. Sold 25 bbls. potatoes for $39 ; how much did I gain per cent, if they cost me $1.25 per barrel? 16. Bought 150 bbls. of flour @ $5.25, paid for drayage $7.50, and porterage $1 ; at what per barrel should I sell it to gain 15 %. 17. Bought 15 horses at $125 each and sold the lot for $3500 ; what was my gain per cent, after paying $25 for their feed ? 18. Sold a safe which cost me $80 for $75; what • was my loss per cent. ? 19. Bought a bill of goods for $350, paid freight $15.20 ; insurance, $5 ; drayage, $3 ; and sold them for $425. What was my actual gain, and what my gain per cent. ? 20. Bought Henry Ullhorn's note for $750 at a discount of 15 ^ ; what did 1 pay for it? 21. Sold Henry Hazin's note of $320 for $300; what was the rate of discount? 22. A jeweler, whose business capital is $10000, makes 100 % on his goods, and takes in on an aver- age $20 a day. A grocer, whose capital is $1000, profits 15 %, and takes in $35 a day. The jeweler's- expenses being $1000, and the grocer's $300 per year; what does each one gain % on the capital invested ? 23. A pork merchant receives a quantity of pork to be sold on commission, at 2J ^; or he may have the whole on his own account at 7-^ cents per pound ; should he sell on commission, or buy, supposing he can get 8|^ cents a pound? Answers: 28f %, $6.10, 84^ ^, 6i ^, 14 %, $51.80, $637.50, 6^ %, $6500, lllf %, buy. COMPLEX PERCENTAGE. 113 24. A huckster commences business on $50, turns his money every 3 days, making 2 cents on every 10, how much does he make in the year, provided he spends $15 a month for rent, and puts out his gains at 6 % interest at the end of every month ? 25. A bookkeeper who receives a salary of $1500 a year, and loans his emplo3^er $2000 at 10 ^, is offered a fourth of the profits on $8000 for five years, for his capital, influence and services; would he gain or lose by accepting the offer, the profits of the business be- ing 20 % per annum ? 26. The assignee of an insolvent debtor reports to the court that preferred claims (which must be paid in full before the general creditors are entitled to a dividend) have been proven to the amount of $386 ; other claims, $40630 ; that he has realized from collections and sales, $8650. The costs of court to date are $8.50; the fee for assignee's counsel, $50 ; assignee's commission on the cash reported, d% ; auctioneer's commission, 2% on $3260 sales. Give the assignee's and auctioneer's commissions, the per cent of dividend (without a frac- tion) that can be paid to the general creditors, and the balance of cash that will remain in the hands of the assignee after paying costs, fees, commissions, preferred claims, and dividend to general creditors. 27. The final report of the assignee in the above case, shows that all the property of the assignor has been re- duced to cash ; that there is in his hands $10680. The unpaid costs are $18.60 ; assignee's commission on the money reported, less the balance on hand at last report, 5%; assignee's attorney's fee, $200; sundry expenses of the trust, $196.92. What is the assignee's commission at this settlement, and how much will the general credr itors receive on the dollar ? Answers: $65.20, $432.50, 18 %, $394.40, $412.78, 28 %, $514.28, Lose $6500, $1048.05. 114 XXII. TIME TABLE FOB COMPUTING INTEREST AND AVERAGE. Number of days from \st of January to any other day of the year. In lenp-years^ add 1 to the days after 28th of February. o 1 1 2 1 3 2 4 3 5 4 6 5 7 6 8 7 9 8 10 9 11 10 12 11 13 12 14 13 15 14 16 15 17 16 18 17 19 18 20 19 21 20 22 21 23 22 24 23 25 24 26 25 27 26 28 27 29 28 30 29 31 130 ~9(T 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 U4 115 116 117 118 119 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 111181 2 3 4 5 6 1 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 2121243 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 244i 245 246 247' 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268j 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 3001 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 TIME TABLE 115 FOR COMPUTING INTEREST AND AVERAGE. Nvmber of days from 1st of Juty to any other day of the year. In leap-years^ add 1 t* the days after 2Sth of February. p r (w p f c o 1 i u 1 1 a* p > p a o p •-i 1 Z ^ o ^ ? s p ^f 31 M 92 123 153 1 184 215 243 274 304 335 1 2 1 32 63 93 124 154 2 185 216 244 275 305 336 2 3 2 33 64 94 125 155 3 186 217 245 276 306 337 3 4 3 34 65 95 126 156 4 187 218 246 277 307 338 4 5 4 35 66 96 127 157 5 188 219 247 278 308 339 5 6 5 36 67 97 128 158 6 189 220 248 279 309 340 6 7 6 37 68 98 129 159 7 190 221 249 280 310 341 7 8 7 38 69 99 130 160 8 191 222 250 281 311 342 8 9 8 39 70 100 131 161 9 192 223 251 282 312 343 9 10 9 40 71 101 132 162 10 193 224 252 283 313 344 10 11 10 41 72 102 133 163 11 194 225 253 284 314 345 11 12 11 42 73 103 134 164 12 195 226 254 285 315 346 12 13 12 43 74 104 135 165 13 196 227 255 286 316 347 13 14 13 44 75 105 136 166 14 197 228 256 287 317 o48 14 15 14 45 76 106 137 167 15 198 229 257 288 318 349 15 16 15 46 77 107 138 168 16 199 230 258 289 319 350 16 17 16 47 78 108 139 169 17 200 231 259 290 320 351 17 18 17 48 79 109 140 170 18 201 232 260 291 321 352 18 19 18 49 80 110 141 171 19 202 233 261 292 322 353 19 20 19 50 81 111 142 172 20 203 234 262 293 323 354 20 21 20 51 82 112 143 173 21 204 235 263 294 324 355 21 22 21 52 83 113 144 174 22 205 236 264 295 325 356 22 23 22 53 84 114 145 175 23 206 237 265 296 326 357 23 24 23 54 85 115 146 176 24 207 238 266 297 327 358 24 25 24 55 86 116 147 177 25 208 239 267 298 328 359 25 26 25 56 87 117 148 178 26 209 240 268 299 329 360 26 27 26 57 88 118 149 179 27 210 241 269 300 330 361 27 28 27 58 89 119 150 180 28 211 242 270 301 331 362 28 29 28 59 90 120 151 181 29 212 271 302 332 363 29 30 29 60 91 121 152 182 30 213 272 303 333 364 30 31 30j 61 122 183 31 214 273 334 31 116 PAST-TIME TABLE FOR COMPUTING INTEREST AND AVERAGE. Number of days frovi Jannary 1st to any day of the previous year, pmrs add one day before February 28th. In leap' 1 o i 1 1 1 > g e-i c 1 1 o 1 o B a- 1 o 1 p- "T 365 334 306 275 245 214 ~1 184 153 122 92 61 31 1 2 364 333 305 274 244 213 2 183 152 121 91 60 30 2 3 363 332 304 273 243 212 3 182 151 120 90 59 29 3 4 362 331 303 272 242 211 4 181 150 119 89 58 28 4 5 361 330 302 271 241 210 5 180 149 118 88 57 27 5 6 360 329 301 270 240 209 6 179 148 117 87 56 26 6 7 359 328 300 269 239 208 7 178 147 116 86 55 25 7 8 358 327 299 268 238 207 8 177 146 115 85 54 24 8 9 357 326 298 267 237 206 9 176 145 114 84 53 23 9 10 356 325 297 266 236 205 10 175 144 113 83 52 22 10 11 355 324 296 265 235 204 11 174 143 112 82 51 21 11 12 354 323 295 264 234 203 12 173 142 111 81 50 20 12 13 353 322 294 263 233 202 13 172 141 110 80 49 19 13 14 352 321 293 262 232 201 14 171 140 109 79 48 18 14 15 351 320 292 261 231 200 15 170 139 108 78 47 17 15 16 350 319 291 260 230 199 16 169 138 107 77 46 16 16 17 349 318 290 259 229 198 17 168 137 106 76 45 15 17 18 348 317 289 258 228 197 18 167 136 105 75 44 14 18 19 347 316 288 257 227 196 19 166 135 104 74 43 13 19 20 346 315 287 256 226 195 20 165 134 103 73 42 12 20 21 345 314 286 255 225 194 21 164 133 102 72 41 11 21 22 344 313 285 254 224 193 22 163 132 101 71 40 10 22 23 343 312 284 253 223 192 23 162 131 100 70 39 9 23 24 342 311 283 252 222 191 24 161 130 99 69 38 8 24 25 341 310 282 251 221 190 25 160 129 98 68 37 7 25 26 340 309 281 250 220 189 26 159 128 97 67 36 6 26 27 339 308 280 249 219 188 27 158 127 96 66 35 5 27 28 338 307 279 248 218 187 28 157 126 95 65 34 4 28 29 337 278 247 217 186 29 156 125 94 64 33 3 29 30 336 277 246 216 185 30 155 124 93 63 32 2 30 31 335 276 1 1 215 31 154 123 62 1 31 PAST-TIME TABLE FOR COMPUTING INTEREST AND AVERAGE. Number of days from July \st to any day in the past year. 117 o e^ > w o !2; Q o tH >^ K >■ ss: t_ o S s a at a (6 1 B 1 o o B 1 o a a 1 1 p. I ^ c a J 1 365 334 303 273 242 212 1 181 150 122 91 61 30 T 2 364 333 302 272 241 211 2 180 149 121 90 60 29 2 3 363 332 301 271 240 210 3 179 148 120 89 59 28 3 4 362 331 300 270 239 209 4 178 147 119 88 58 27 4 5 361 330 299 269 238 208 5 177 146 118 87 57 26 5 6 360 329 298 268 237 207 6 176 145 117 86 56 25 6 7 359 328 297 267 236 206 7 175 144 116 85 55 24 7 8 358 327 296 266 235 205 8 174 143 115 84 54 23 8 9 357 326 295 265 234 204 9 173 142 114 83 53 22 9 10 356 325 294 264 233 203 10 172 141 113 82 52 21 10 11 355 324 293 263 232 202 11 171 140 112 81 51 20 11 12 354 323 292 262 231 201 12 170 139 111 80 50 19 12 13 353 322 291 261 230 200 13 169 138 110 79 49 18 13 14 352 321 290 260 229 199 14 168 137 109 78 48 17 14 15 351 320 289 259 228 198 15 167 136 108 77 47 16 15 16 350 319 288 258 227 197 16 166 135 107 76 46 15 16 17 349 318 287 257 226 196 17 165 134 106 75 45 14 17 18 348 317 286 256 225 195 18 164 133 105 74 44 13 18 19 347 316 285 255 224 194 19 163 132 104 73 43 12 19 20 346 315 284 254 223 193 20 162 131 103 72 42 11 20 21 345 314 283 253 222 192 21 161 130 102 71 41 10 21 22 344 313 282 252 221 191 22 160 129 101 70 40 9 22 23 343 312 281 251 220 190 23 159 128 100 69 39 8 23 24 342 311 280 250 219 189 24 158 127 99 68 38 7 24 25 341 310 279 249 218 188 25 157 126 98 67 37 6 25 26 340 309 278 248 217 187 26 156 125 97 66 36 5 26 27 339 308 277 247 216 186 27 155 124 96 65 35 4 27 28 338 307 276 246 215 185 28 154 123 95 64 34 3 28 29 337 306 275 245 214 184 29 153 94 63 33 2 29 30 336 305 274 244 213 183 30 152 93 62 32 1 30 31 335 304 243 182 31 151 92 31 31 118 AVERAGE. XXIII. AVERAGE. Art. 1. When several payments have to be made at one time, or when one bill has to be paid v^^ith several notes of different lengths of time, an average has to be sought, the process of finding which is called Average, or Equation of Payments. 1. A merchant sells a bill of goods amounting to «4000, to be paid as follows : $400 in 30 days, $600 in (lO days, $1000 in 90 days, and the balance in 4 mos., or 120 days, what would be a mean or average time of payment for the whole? A credit of $400 for 30 ds., is the same as a credit on $1 for 12000 ds- » 600 " 60 " " " " " 1 " 36000 " " 1000 '' 90 " " « " " 1 " 90000 " " 2000 " 120 " " " " « 1 "240000 " 4000 378000 On one dollar there is a credit for 378000 days. On $4000, there is a credit for ^U^^=^H ^^J^- That is, the $4000 might be paid in 94^ days^ or on the 95th day, without either party sustaining loss by interest. 2. A merchant sells goods to the amount of $1700, $500 payable in 60 days, $300 payable in 90 days, and $900 payable in 30 days, what is the average time of payment of the whole? 3. Sold a bill of goods, amounting to $700, J of which is payable in 90 days, J in 4 mos., and ^ in 6 mos.; required the average time of payment. Answers: 49 days, 143 days. AVERAGE. 119 Art. 2. To find the average date of purchase. 1. Purchased goods as follows, what was the average date of purchase? Dec. 31, a bill of $300, Jan. 3, a bill of $100, Jan. 9, a bill of $200, Jan. 18, a bill of $800, Jan. 23, a bill* of $500. ' Ans. Jan. 15th. Remark. — If the amounts abo"ve were equal, and the intervals also equal, the average date of purchase would be on Jan. 9th: because it is midway between the first and last dates. ExPL. — The first was due at the time of purchase; the second, 3 days after; the third, 9 days after, etc. 300 X = 100 X 3= 300 200 X 9= 1800 800X18 = 14400 500X23=11500 1900 )28O00 14i|, or 15 days after Dec. 31, the date of first purchase, which brings the time up to Jan. 15 th. If these debts had been contracted on a credit of three months, a note dated Jan. 15 would be given to settle the bill. 2. What is the average date of purchase of the fol- lowing ? Jan. 1, Mdse., $360, Feb. 6, Mdse., $325, March 8, Mdse., $180, April 3, Mdse., $65, May 13, Mdse., $275, June 8, Mdse., $70. Am, Jan. 15, March 3, Feb. 26. 120 AVERAGE. §1000.00 3500.00 9734.00 976.50 1037.00 3. The following goods were sold on a credit of hO days : JSTew York, Apr. 3, 1876. Mr. James Callen, Bought of Eobt. Boggs. Jan. 1, Invoice of Coffee, ., ^' 6, '* *^ Sugar,' . Mar, 9, <' '' Sunds., . Apr. 3, " '' 1T6247750 Kequired the average date of purchase, or date of note. 4. Philadelphia, Dec. 3, 1859. Mr. Henry Higgins, Bought of James Kiel, Sept. 3, Invoice of Calicoes, .... §3150.00 ^' 19, " ^' Muslins, 1174.00 '^ 20, " " Silks, 3500.00 Oct. 19, '' " Sundries, .... 1743.00 $9567700 Required the date of maturity of a 3 months' note, grace included. Find the equated time of payment for the follow- ing: 5. 6. Apr. 3, $167.25* May 7, $674.40 - 9, 374.00 Jun. 7, 168.37 ^' 19, 176.00 " 10, 370.20 - 20, 371.00 '^ 15, 167.00 « 25, • 197.87 " 19, 679.60 '' 30, 300.00 July 23, 679.45 May 9, 150.57 • Aug. 18, 993.18 ^' 23, 720.18 " 19, 875.57 Answers : Feb. 22, April 30, July 11, Dec . 23. ^When the cents are under 50, reject tliem, otherwise add a dollar to the dollars. AVERAGE. 121 Art. 3. Whe7i goods are pvrchased at different dates and on different lengths of credit. 1. Purchased the following bills of merchandise; required the average date of maturity, or the equated time of payment for all : Apr. 3, a bill of $250 on 3 months' credit. a g a u jgy a g u u May 1, « " 250 '' 4 " " Jun. 9, " " 320 ''2 " . " If we substitute the date of maturity of each of these bills for the date of purchase^ and arrange them in the order of time, we shall have a problem in all respects similar to those under last Art. The first bill falls due July 3d,* the second Oct. 9th, the third Sept. 7th, the fourth Aug. 9th. Ar- ranged in the order of time, they appear thus : July 3, $250 Aug. 9, 320X37 = 11840 Sept. 7, 250X66 = 16500 Oct. 9, 157X98=15386 977 43726(44T[|4 or 45 days. 3908 4646 3908 9TT Hence the date of maturity is 45 days after July 3d, or on Aug. 17th ; from which time till the day of settlement, interest is due on the whole amount. * Days of grace are not allowed on invoices. 122 AVERAGE. When some of the purchases are at cash price. 2. What is the equated time of payment for the fol- lowing: Jan. 1, 8600 on 3 mos., Feb. 3, 1670 at cash price, Mar. 3, $950 on 6 mos., May 3, S550 for cash? The first payment falls due on April 1st, the second, being for cash, was due at the time of purchase, Feb. 3d, the third Sept. 3d, and the last May 3d. Ar- ranging the dates and amounts in the order of time, «8 before, we have Feb. 3, $670 X 00 Apr. 1, 600 X 57 34200 May 3, 550 X 89 48950 Sept. 3, 950X212 201400 2770 284550(103 days nearly, or \lth May, Note. — The amount due on Sept. Sd, is $2770 plus the interest on that amount, from May 17th. From May 17th. to Sept. .3d. = 109 days. Interest on $2770 for 109 days = $50.82, which, added to $2770 = |2820.:32. Find the equated time of payment of the following; 3. July 1, $675 on 3 mos. ; 13th, $619.54 on 2 mos. ; 19th, $147.67 at cash rates ; 23d,- $678.44 on 5 mos. 4. Sept. 3d, $937.15 on 30 days' credit; 9th, $897.78 on 90 days' credit; 17th, $619.18 at cash prices; Oct. 3d, $777 on 60 days. Eequired the amount due on each of the following on July 1st. 5. Jin. 9th, $678.44 @ 60 days; 20th, $419.88 at cash price; 29th, $789.14 at 3 mos. Answers: October 17, Nov. 2, May 17, $1919.85, $4708.49. AVERAGE. 123 6. April 9th, $1678 on 3 mos. ; June 18, $1000 at cash prices; 21st, $879.55 on 60 days; 23d, $371.19 cash ; 20th, $785.25 cash. A?is. $4708.49. Art. 4. APPLICATION TO ACCOUNT SALES. An Account Sales is a detailed statement of goods sold by a commission merchant, on account of the party who sent them. The person or party who sends goods to another to be sold for himself, is called tho consignory the per- son to whom they are sent, the consignee, and the goods sent, the consignment. COMMISSION HOUSE OF STRAIGHT, DEMING & CO. Shipment 18. No. 7828. Sales for account of Messrs. Gaff & Baldwin. By sundries, Jun. 4, T. B. Colgan & Co. @ 60 days, 8 hhds Sugar. 1095 1020 1100 1120 1080 1240 8965 1200 1110 896 8069 @7/g $600.13 Jun. 6, G. Newton & Co.,® 60 days, 10 hhds. Sugar. 1080 1040 • 1090 1340 1120 1020 1240 1100 11440 1200 1210 1144 10296 @ 6| $707.85 124 AVERAGE. Sales for account of Messrs. Gaff & Baldwin — continued. Jun. 10, B. Yilgers k Co., @ 60 days,20 hhds. Sugar. 1060 1240 1210 1110 1180 1005 1055 1285 1240 1100 1185 1210 1300 1325 • 1010 1140 1120 1205 23185 1205 1000 2318 20867 ©6^5 1343.31 $2651.29 Charges, Jun. 1, Fd cash st'mrLandis for fr't,$ 87.18 '' *' Drayage, 9.50 " 14, Insurance, 4.63 ** '' Storage, 9.50 ** " Commission and guarantee, 132.56 243.37 Netproc'ds due by equa^n^ Aug, 13, $2407.92 E. O. E. Cincinnati, June 14th, 1876. Straight, Deming & Co., per F. Jelke. Recapitulation of answers of Art. 1-4 inclusive : Jan. 15, Feb. 22. April 30, May 17, July 11, Aug. 7, Aug. 17, Oct. 17, Nov. 2, Dec. 23. 45 days, 49 days, 95 days, 143 days, $1919.85, $4708.49. AVERAGE. 125 Art. 5. When payments are made before a note or bill is due, to find how long after maturity it should run, to balance the interest on the advanced payments. 1. A merchant holds a note of $500 at 6 months. Three months before it is due, he receives $100, and one month before it is due, he receives $300, how long should he allow the balance to run, to equal the interest on the advance? The int. on $100 for 3 mo8.=int. on $1 for 300 mos. " " 300 u 1 u _ a u I u 300 " 600 mos. Hence, the interest on the advanced payments is equal to the interest of $1 for 6^0 months ; that is, a balance of $1 should have run 600 months, but the balance due on the note is $100; therefore, it should run fg§ months =6 mos. Proof. — The int. of the $100 that is to run 6 mos. =$3. The int. of $100 (the first pay't,) for the 3 mos.=$1.50 " " 300 (the sec. pay' t,) " "1 ^^ = 1.50 Total interest on advance =$3.00 2. A note of $600 given on Jan. 3, 1876, payable in 6 months. 4 months before it was due, $100 was paid on it, and 3 months before it was due, $200 was paid; how long in equity should the balance run? 3. A merchant owes $700 due 8 months from the time he contracted the bill ; 5 months afterward, he pays $200, and 2 months after that, $300, how long should the balance remain unpaid? Answers : 3 mo. 10 days, 4 mo. 15 days, 4 mo. 12 days. 126 AVERAGE. 4. If I borrow $600 from A at one time, and $500 at another, each for 4 months, how long should I lend him $1000 to return the favor? Ans. 4 months 12 days. APPLICATION TO ACCOUNTS CURRENT. Art. 6. In applying equation to accounts current, both the debit and credit sides of an account have to be averaged, for which reason, the operation is usually called Compound Equation. Dr. J. Doe in acct with R. Rob. Cr. $ c. * c. 1876. 1876. July 3 To Mdse., 1000 00 Aug. 1 By Cash, 500 00 (( 7 il u 500 00 u 12 u u 500 00 Aug. 18 U (( 250 00 Eequired the amount due on Aug. 18th, supposing the sales to have been made at cash rates. We first find the average of the debit side, because it contains the earlier. date. The credit side is aver- aged from the same date. July B, 1000 X 7, 500 X 4= 2000 Aug. 18, 250X46 = 11500 1750 )135O0(7j||, or 8 days from 1225 July 3. "T25 Aug. 1, 500X29 = 14500 « 12, 500X40=20000 1000 )34500, or 35 days from July 3. AVERAGE. 127 The debit side averages July 11th, that being the mean time of purchase. The credit side shows that the payments average Aug. 7th ; hence the whole debt was due from July 11th to Aug. 7th (27 days), for which, interest should be charged; and the bal- ance, $750, was due from Aug. 7th to Aug. 18th, for which interest should also be charged. Interest on $1750 for 27 days =$7.88 '* « 750 " 11 " 1.38 $9.26 Which added to the debit side of Doe's account^ gives a balance against him of $759.26. Art. 7. APPLIED TO STATEMENTS. When settlement is made by note, an average date of payment is found by dividing the difference of the prod- ucts by the balance of debt, and counting backward or forward from the assumed date. 34500—13500=21000 days, which divided by 750 gives 28 days, to be counted backward from July 3d, giving June bth. Explanation. — In the preceding pperation we assumed that the whole debt was due on July 3d, making the sales subject to a discount equal to that on $1 for 13500 days (favor of buyer), and the payments to a discount of 34500 days (against buyer), showing a balance against him of 21000 days on $1, or 28 days on his debt of $750. To be against him we must count backward, for to count for- ward would give him longer time to pay the note. Proof.— Interest on $750 from June 5th to Aug. 18th (74 days), $9.25. The difference of 1 cent is caused by the fraction of a day reckoned as a full day, on preceding page. 128 AVERAGE. H g o 88 8 8 888 8 § B §3 g g ^^8 o iS o ^ lO r-l r- (N CO o 05 CO %" g O CM O ^ ^ g ^ ■ ■- 1 T3 ;n .-. £ -' ^ :S; :« oo a 0) : ;i. : 3 -o O tr • :j 0/ 3 i^lo "■" c:. 30 : :^ :: :-S i 0," : 2 c !cV^ll X 1 3 K« •5f||.| oo |.« 1 4i -T -.1 r= "^ '^ '^ -a 0/ s C-- 7 i o -:: ^i 5 .- « » o 2 ■■^lii Ss.KO" ^ »-- ^ . .... 03 - - - - = = - - - o oo» c; o cor:-+i i ■M (N 1— 0^ ri (N r- I^Tr; ^ ^ >> S a>3i -^i - rr^'S ---s ^ <: ^. -5 d88 8i5 ^%S 88 8 88 88 8£ s lO^ C^'O lO'f-f O— ' O OO iOtH oc o ^^55 5|3:3 ^g Sg § §5 ^S ^ -«3 ll • o n r-t ^5 ^:^^ Ti^ ^ — • O CO -+ -* 1 C 'O o (N ^: ro 1 CO • • - : :>>:-- : cu : ' a : -^ i ft = bO : 0'^ £ i £ = oil!: SJ58 o C M o -J a- a> •-2 Is a» o Ml 1 1 8 CO 5 1 .o r-""^ --■* -•»" -- - -^ x" w -* = «> g^3 ^^'a ^^ °° S'^ *3 •* ^ . o CD -< t^ -; ^ s- ;-. >> r^ (3 1 A 2-i" .<1> ^1/ 5 :: •* " i. - 3 - - 1 1 AVERAGE. 129 Dr. A. Mills. Ck. 1876. $ c. 1870. $ c. Feb. 18 To Mdse, 600 00 Apr. 3 By Cash, 600 00 (( 28 (( (( 700 00 a 12 (( ' 11 400 00 Mar. 17 li u 800 00 Eequired the amount due July Ist; also, the equated time at which a note would have been dated. Art. 8. APPLICATION TO STORAGE. Storage is usually charged for by the box, barrel, etc., for the month. Eeceived July Eeceived " 1, 4, bbls. days, products, 200 X3= 600 300 500X2=1000 Delivered " Balance ^' Eeceived " Eeceived '^ 6, 10, 16, 100 400 X4=1600 300 700X6=4200 200 900X2=1800 Delivered ' ' Balance '< Eeceived ^' 18, 21, 600 300 X3= 900 500 Balance on hand 800 30)10100 336f The products divided by the number of days in a month, give the number of barrels chargeable for a month. Answers : $1229.20, 337 days. 9 180 AVERAGE. Art. 9. GENERAL EXERCISES. 1. A bill of $1000 is to be paid in five equal install- ments, at 3, 4, 5, 6, and 7 months, what time should be allowed, if the individual will pay it all at once? 2. The following bills of goods have been pur- chased at different periods ; required the average time of payment, allowing 30 days credit on each : , $1347 on Jan. 1, $167 on Feb. 3, $1794 on Feb. 8, «6783 on Feb. 10, $1076 on Feb. 19, $319, on Mar. 6, and $1674 on April 9. 3. What should be the date of a 60 day note for the following bills: $168 purchased on April 6, $3196 Apr. 9, $1668 May 3, $6847 June 1? 4. When would a 90 day note fall due, given for the following bills: $673 on June 3, $710 on July 6, $415 on July 9, $678 on Aug. 3? 5. What is the storage of the following account, closed Aug. 4, at 5 cents per bbl. per month? Apr. 3, received 167 July 1, delivered 200 " 9, '' 145 " 3, " 150 " 17, " 450 " 7, " 190 May 18, " 198 6. At 5 cents per barrel, what will the storage of the following amount to on Oct. 24? July 9, received 167 Oct. 1 delivered 125 Aug. 5, '' 378 " 3 " 500 ^* 9, " 780 " 19 " 450 *• 31 '• 178 (Answers on next page.) AVERAGE. 131 7. How much was due on the following account on Sept. 13, 1859; bills sold on 60 days time:^ Dec. 8, 1858, $1676, Jan. 9, 1859, $1675, Feb. 14, paid $500, Apr. 16, paid $1000. No credit being allowed on the following bills, required the balance due at date of last purchase jv paj'nient. Debit side. Credit side. 8. Apr. 9, $600 May 13, $700 ^' 9, 500 " 15, 100 " 15, 700 Balance due May 15, 9 June 3, $365 " 19, 784 July 18, 594 Balance due July 18, 1858. 10. Dec. 7, $1874 1859. Jan 3, 1678 '' 21, 712.53 June 20, $300 *' 29, 500 " 30, 100 1859. Jan. 7, $1000 April 14, 900 What will be average date of maturity of the ollowing, and what will be the balance due Mar. 7? 11. Jan. 1, $1673 on 3 mos. credit, Mar. 4, $1000 " 9, 740 on 2 " " *' 7, 500 " 29, 500 on 4 ^' " Answers: March 16th, May 13th, Oct. 6th, 5 mo., $141.80, $168.96, $846.23, $1009.93, $1921.65, $2426.41, May 8th, $1398.40. * Some reckon every fraction of a day as a whole day; others only fractions that are over J^. 132 METHODS OF AVERAGING. METHODS OF AVERAGING. There are two methods of averaging known to ac- countants, viz.: the Interest Method and the Product Method. By the first interest is fully computed upon every item up to the day of settlement. By the Prod- uct Method the time is usually reckoned from the date of the first item, and multiplied into the va- rious amounts. From this arises discount. A new method, now introduced for the first time, is -a modification of the latter, which we shall call the Interest- Product Method. By it the time is reck- oned from the last item of the account, or the day of settlement, which results in giving the interest direct without further calculation, except to divide by 6000. The Time Tables on pages -114 and 115 are used for the Discount-Prodtict Method, and those on pages 116 and 117 for the Interest-Product Method. Pr. • 12. C. A. Walworth. Or. 1876. 1876. Jan. 1, To Mdse, $300.00 Feb. 9, By Mdse, $200.00 Mar. 3, " '' 500.00 23, '' " 100.00 300X 200X39^7800 500=61=30500 Dr. products. 100X53=5300 13100 Cr. products. 800 300 13100 300 500)17400 Bed. due 500 34f or 35 days from Jan. 1 or Feb. 5. Explanation. — Assuming both purchases and sales to be due on January 1, W. would be entitled to a discount on his purchases equal to that on $1 for 30500 days, and I would be entitled to a discount on my purchases equal to that on $1 for 13100 days, making a difference of 17400 days in W.'s favor on the balance, $500. The discount on $1 for 17400 days is equal to the discount on $500 for 17400 days--500=34i days from January 1, to be reckoned forward, which will give Feb- ruary 5. Had the discount been against him, it would have' ■ehown that the balance was past due on January 1, which AVERAGE. 135 would indicate that the time would be counted backward frona that date. Should the balance, instead of the equated time, be required, the difference between tlie Dr. and Cr. products may be divided by 6000 to find the discount on the balance ta the day of settlement at the rate of 6 per cent per annum. Any amount being 100 per cent of itself, when multi{)lied l)y a num- ber of days, will be 100 per cent per day. Six per cent per an- num being the l-6000th part of 100 per cent per day ; In-nce by dividing the difference as aforesaid, we have the discc.unt on the balance up to the assumed day of settlement at tiie rate named. 17400 : 6000 = $2.90 discount. This process we shall name the Discount- Product Method. Dr. 13.- J. C. HiNTZ. Or. 1869. 1869. July 3, To Mdse. $1000.00 Aug. 1, By Cash, Sooo.oa 7, " " 500.00 u 13^ u 500.00 Au.l8, " " 250.00 Assuming the day of. settlement to be July 1, we have. lOOOX 2= 2000 Days. 500*X31 =15500 Days. 500X 6= 3000 " 500X43=21500 " 250X48=12000 1000 37000 Cr. products. 1750 17000 " 17000 Dr. products. 1000 20000 Difference of do. $750 Balance due. 750)20000(26 ds. Explanation. — The sum of the credit prod- 150 ucts being greater than that of the debit prod- ~5()0 ucts, shows that the discount is in my favor. 450 Hence, in order to sett'e on tlie assumed date, -Tq his payments would be at a discount- of 27 days for the balance; but as it would be im- possible to settle on a past date, we will have to charge him in- terest from 27 days prior to July 1 (June 4) to the real day of settlement, whatever that may be, or else take his note, dated June 4, bearing interest from date. Say the date of the settle- ment is January 1, 1870. Interest on $750 from June 4, 1869^ to January 1, 1870 (211 davs^ is $26.37, which, added ta $750=$776.37, balance due with interest, January 1, 1870. This result, however, may be ascertained directly by the In- ter est- Product Method, as shown by the same example on page 134. 1 34 AVERAGE. INTEREST-PKODUCT METHOD. Art. 10. To ascertain the balance due on the day of settlement the Inter est- Pro duct Method may be employed. By it the actual date of settlement is used instead of an assumed date, as by the Discount- Product Method. 14. Find the balance due January 1st, 1870, of the account of J. (7. Hintz, page 133, by the Interest-Prod- uct Method. 1869 Jnlv 3. ..$1000X182= 182000 Jul> 7... 500X178= 89000 Aug. 18... 250X136= 34000 Au^. 1 $500X153=76500 Aug. 13 500X141=70500 Dr. Interest-Product, 305000 Cr. " " 147000 Cr. Interest-Product, 147000 60)1580.00 $ 26.33 Interest due January 1, 1870. 750.00 Balance of account. $776.33 Balance, incUiding interest. Explanation. — A purchase made on .July 3 — terms cash — should pay interest up to the date of settlement. That being, in tliis case, Jan. 1, 1870 the debtor is to be charged with inter- est up to that date, viz.: for 182 days. The interest on $1000 for 182 days is equal to the interest on S182,000 for one day, (or on $1 for 182000 days.) The same with every item on the Dr. side, making the Dr. Interest-Product equal to the interest on $305,000 for one day. The day of settlement being Janu- arv 1, 1870, Hintz paid $500 on August 1, 1869, 153 davs be- fore the day of settlement, and $500 on August 13, 1869, 141 days before the day of settlement, and is therefore entitled to a credit for interest on the respective amounts. The Cr. Interest- Products, 147000, being equal to the interest on $147,000 for one day. The amount on which Hintz is entitled to interest for one day being $147,000, and the amount upon which he is chargeable with interest for one day being $305,000, he is chargeable with interest on $158,000 for one day more than the amount upon which he is entitled to receive interest. The interest on $158,000 AVERAGE. 135 for 60 days at the rate of 6 per cent per annum is $1580.00 — tliat being 1 per cent of tlie amount — 1 day is l-60th of 60 days;, hence, by dividing $1580.00 by 60 we have the interest for 1 day on the balance of the products, viz.: $26.oo. The advantage of this method over the Discount- Product Method is obvious. By this nothino; is as- sumed. Interest is actually reckoned from the date of the first item of account to the chiy of settlement^ and the accrued interest obtained without further calculation. Should the timehe required it is readily found by dividing the difference of the product by the balance of the account. By the Interest-Prod- uct Method the interest is simply charged to the side of the account on which is the greater pi*oduct, irrespective of the bahmce of the account; whereas^ by the Discount-Product Method the time has to be reckoned backward or forw^ard from the date ob- tained, and the interest computed and applied after- ward. 15-16. Find the balance due July 1st, 1877, by the Interest- Product Method, of the accounts of B. H, Langdale and Edw. Witte, page 137. 17-18. Find the balance due January 1st, 1877, by the same method, of the account of N. J. Jones, page 137. Also of G. A. Walworth's account, page 132. 19. July 1st, 1873. K. H. Langdale's account is a& follows : 1873 1873 Jan'y 3, To Mdse, $300.00 Feb. 6, By Cash, $ 100.00 " 4, " " 250.00 May 3, " " 1,000.00 Feb. 9, " « 730.00 June 3, " " 160.00 May 6, " " 800.00 20. Charge him with the interest up to July 1^ 1873, close his account and bring down the balance. 136 AVERAGE. Charge him with goods bought since the day of last settlement, as follows: July 3, $500; Aug. 6, $100; Nov. 8, $100. Credit him with cash, paid as follows: Aug. 30, $800; Dec. 1, $600. Find balance due, with interest, January 1, 1874. COMPOUND METHOD. Art. 11. Find the balance due^ July 1, 1874, of the following : 21. Theodore Baur. 1874 1874 Jan. 12, To Cash, $ 500.00 Jan. 3, By Mdse, 90 ds,...$1000.00 Feb. 5, " Acc'p 60ds.... 120.00 June 30, ** Mar. 8, " Mdse, 600.00 * June 3, " Note, 3 mos... 100.00 $1220.00 60 ds,... 150.00 $1150.00 Arranged according to the dates when the items are due. Dr. Jan. 12 Apr. 9 Mar. 8 Sept. 6 Int. $500X170 12()X 83 500XU5 luox Dis. 67 Cr. Apr. 3 Aug. 29 Dis. $1000X 150X59 Int. 89 Dr Cr. 85000 99()0 57500 6700 89000 95700 60)656.10 $10.93 int.Dr 8850 Dr. Ride, of ac. $1220.00 Cr. ♦' ♦* 1150 00 Difference 70.00 Interest Dr 10.93 Balance due $80.93 ExpiiANATiON.— It will be seen that both the Discount and In- terest-Product Methods are em- ployed in tills solution. B is Dr. for the interest on the three first items of the Dr. side of his ac- count, because it was due before the day of settlement, (July 1, '74), and he is credited with the discount on the fourth item, that being due (Sept. 6, 74) 67 days after the settlement. He is cred- ited with the interest on the $1000 paid 89 days before the set- tlement, and cUai'^ed with (he discount on the $150 due (Aug. 29t h,) 59 days after the day of set- tlement; he is therefore eh a ijre- able with the interest on $1618.10 for one day, less the interest on $957.00 for that time, viz: $10.93. COMPOUND METHOD. 137 22-24. Find the balance due January 1, 1877, hy the Compound Method, of the accounts of Langdale and Witte, Ex. 25 and 27, also the balance due July 1, 1876, of the account of N. J. Jones, Ex. 26. 25. E. H. Langdale. 1»76 1876 July 3, ToMdse, 6mos , 560.87 July 1, By Balance, 127.15 15, '* " 3 " 149.50 30, " Accept. 60 ds, 300.00 Aug. 21, " •* 3 " 2000.00 Aug 29, " Cash, 460.00 Sep. 18, " " Cash, 396.40 Oct. 20, *' Note, 3 mos , lUOU.OO Oct. 15, '* •* 175.20 31. ♦' Cash, IdO.UO 21 " " " 425.16 31, " Mdse Ret., 250.00 27, " Cash, 100.00 Nov. 30, " Cash, 450.00 31, *• Mdse,3mos , 506.18 30, ** Balance, 2144.77 Nov. 28, « " 4 ♦♦ 30, " " 4 •* 1, To Bal.. 197.45 321.16 4^31.92. 4831.92 Dec. 2144.77 26. N. 1. Jones. 1876 1876 Jau. 1, To Balance, 650.00 Jan. 8, By Mdse, 3 mos. 160.00 Feb. 3, " Cash, 245.00 15, * - 6 - 710.87 15, ♦' Note, 60 ds, 416.87 • Feb. 14 ' u 2 " 910.14 Mar. 18, " Accept. 30 ds. 1000.00 Apr. 16, ' " Cash, 1000.00 June 4, " " 60 " 750.14 June 8, ' " 4 mos, 9e above ratios would be expressed thus: 1:2 and 2:1, and would be read one is to two and two is to one. French mathematicians divide the first term by the second; English the second by the first. The English method is used here, 3 : 6 will equal |- or 2, ^ : |-=-=::r|. Art. 2. Numbers or quantities of different de- nominlations/ can not have a ratio. lYe can not compare 3 trees with 5 books. But.if the numbers are capable of being reduced to the same denomina- tion, they can be compared ; for we can say 3 feet is to 2 inches, as it is the same as to say, 36 inches is to 2 inches.. Each number is called a term of the ratio. The first term is called antecedent; the second, conse- quent. The value of a ratio depends upon the relative size of its terms. Every ratio may be formed into a fraction by mak- ing the consequent the numerator and the antecedent the denominator, thus: 4:8 = | = 2; and 8:4 = | PROPORTION. 139 XXV. PROPORTION. Art. 1. Two ratios may be equal to each other. 2 : 4, = 4 : 8. 2 bears the same relation to 4 that 4 does to 8. Art. 2. When ratios are equal, the numbers or terms which compose them, are said to be in propor- tioUj and are written thus: 2 . 4 : : 3 : 6, and read 2 is to 4 as 3 is to 6. The first and last terms, as the 2 and 6, are called extremes, and the second and third the means. Art. 3. The same ratio may arise by comparing 4 quantities, two of which are different in denomina- tion from the other two. tuns tuns $ $ 3 : 6 : : 6 : 12. The ratio is 2. Art. 4. * If the extremes are multiplied together, the product will be equal to the product of the means, 3X12=36 6X 6=36 Hence, when any 3 terms are given, we can readily find the fourth, by dividing the product by the odd term. If we had only the three first terms of the tuns tuns $ above proportion: that is, 3 : 6 : : 6, the fourth term would be found by dividing the product of 6X6, or 36 by 3,=:12, or the fourth term as above. To apply this in practice, we have only to sup- pose the 3 tuns and 6 tuns to be coal, and the $6, the price of 3 tuns. Then 3 tuns is to 6 tuns, as the price of 3 tuns is to the price of 6 tuns. 140 PROPORTION. 2. What will 35 lbs. of sugar cost, if 7 lbs. cost 77 cents ? Statement. — 7 : 35 : : 77 is to the price of 35. lbs. lbs. cents. 7 : 35 : : 77 35 385 231 7)2695 $3785 The above operation might have been abridged by cancellation. H u ^ 77X5=3.85 ^n5. 3. If 6 lbs. of tea cost $4.75, what will 15 lbs. cost ? 4. Find the price of 37 horses, when 16 cost $1500? 5. What will 120 bbls. of potatoes cost, if 21 cost $67? 6. 42 bushels of beans cost $87.50, what will 3 bushels cost ? 7. If 610 bushels of wheat cost $1670, what will be the price of 27 ? 8. If 3 bushels 1^ pecks of beans cost $12.50, what will 5 bushels 4 qts. cost? bu. pks. bu. qts. The bushels, Statement, — 3 1^ : 5 4: : 12.50 pks., and qts. 4 4 had to be re- duced to qts.. 13 20 in order to be 8 8 of the same denomination. 108 164X12.50 Art. 176. • — = =$18.98 \ 108 iin5M?(?r5.— $3468.75, $11.87*, $6.25, $382.86, $73.92: $.185. PROPORTION. 141 9. If 27^ lbs. of butter cost $3.75, what will 16^ lbs. cost? 10. Find the price of 12^ dozen of chickens at 30 cents a pair. 11. The price of 21 tuns, 13 cwt., 3 qrs., and 15 lbs. of hemp is $1680.55, what will 15 cwt. cost? 12. What will 54 lbs. 7^ oz. of tea cost, if 15J lbs. cost $8.47? 13. If f of a ship cost $7000, what will j% cost? These fractions need not be reduced to the same denomination. 14. If 6 men do a piece of work in 7 days, how long will it take 5 men to do it ? In stating the previous question, we compared quan- tity with quantity and cost with cost. In this question there is nothing relating to cost, so we must adopt another method of making the statement. Perhaps the simplest is the following : 1. Inquire what is wanted, and put the term of that name to the right. In the question above, time is wanted, so we put the term of that name to the right. 2. Ascertain by reasoning, whether the quantity wanted will be greater or less than that given ; if less, put the smaller of the two numbers for the middle term ; if greater, put the greater of the two terms for the micMle term. In the above question, we reason, that it will take 5 men a greater time than 6 men, so we put the greater of the two terms (6) in the second place. men. men. Statement. — 5 : 6 : : 7 days. The answer is 8| dpys, or 8 days 4 hours, reckoning 10 hours to the d^y. 15. If 2 men plow a field in 3 days, how long will ^* take 3 men to do it? Answers,— i22.b0, $2.25, $7350, $58.09, $30.25, 2. 142 PROPORTION. 16. If 26 yards of linen cost $13.50, what will 10 yards cost? 17. If 8 coats can bo made from 10-1 yards of cloth, how man}^ can be made from 31^ yards? 18. If the interest of S750 for 3 years, 4 months, and 10 days be $151.25 (360 days to the year), w^hat is it for one year? 19. The interest of £100, from 3d of April to 25th February, is £6 5s. 9'?Jd., what is it per year? 20. A, E, and C are in partnership, and their gains for the year are $6757, what is each man's share, suppose A invested $1567, B $2600, and C $3798? The sum of their investments is to each man's in- vestment, as the total gains to each man's gain. 21. M invests $6500, N $1487, O $3654; in three months, it is found that their gains are $1678, w^hat IS each man's share? 22. A lends B $1000 for 13 months 10 days, how long should B lend A $8271, to return the favor. 23. If the shadow from a two foot rule be 6 in., what is the hight of the tree that throws a shadow of 75 feet ? 24. If 7 men can hipld 21 perches of masonry in a day, how many men will it require to build 156 perches in a day. 25. The shadow of a tree being 87 feet; two nails being driven in the tree 3 feet apart, show a distance on the tree of 4J feet, what is the height of the tree? 26. The net profits of a concern being $1860; A's interest is $8750, and B's interest is $8190 ; what is each man's gain ? Answers: 9, 7, 300, 58, 52, 49, $5.19, $5.90, $4.50 $45, $910, $936.89, $950, $1329.34, $936.94. PARTNERSHIP. 143 XXVI. PARTNERSHIP. Art. 1. When two or more persons associate together to carry on a business, they are said to be in partnership, and are called a firm^ house, or com- pany. The funds, propert}^, and merchandise furnished by partners for carrying on business, are called Btock or capital, and the gains are called dividends. The liabilities of a partnership or individual busi- ness are the debts, and the assets their available means, including the indebtedness of others to them. An inventory is a list or statement of those things which constitute assets. Art. 2. In keeping partnership accounts, each member of the firm should be credited with all that he brings into the concern or business, and be charged or debited with all he takes out, just the same as if he had no interest in it. Art. 3. The calculations peculiar to partnership, relate to the division of property and profits. 1. A, B, and C have been in business one year, and find they have made a net gain of $3476, which is to be divided as follows : A is to have J, B J, and C J; required the share of each. $3176^^1738, A's share; $3 4_7 6=$869=B^8 share; and $869=C'8 share. 2. X, Y, and Z purchase a tract of land for $2000; X giving $600, Y $900, and Z the remainder. In one year afterward, they sell it for $5500; required each person's share of the proceeds. 3 A, B, and C invest $2000 each. In 3 months their gross gains are $2000 ; expenses, including $250 for additional services of C, $600, what will be each man's share of the gain? 144 * PARTNERSHIP. 4. D's interest in a partnership is ■^. What is his share of a gain to the firm of $3467.18 ? 5. E, F and G own a steamboat worth $35,000, their respective shares being ^, -^, -^q. What is the profit of each after deducting $1350 expenses, from $5450. gross profits ? Art. 4. Interest on Investment. 6. H, I and J invest in partnership $3400, $2900, and $1500 respectively, and at the end of the year find a net gain of $2600. Allowing 6% on their investments, what amount is each entitled to in proportion to the capital advanced ? Interest on Investment, 8468. Net profits, $2600, minus $=468=^2132, to be divided pro rata. $2132 : 7800=.2733 gain on :ffl.OO. .2783X3400=$929;33 H's share. .2733-X2900= 792.67 I's share. .2733-X1500= 410.00 J's share. 82132.00, whole gain. The respective shares of gain may be ascertained bytlie follow- ing proportion: The whole investment is to H's investment as the wliole gain is to H's gain, thus: 87800 : 3400 : : 2132 : H's gain. 39 17 X 213 2 39)36244 #929.33 $7809 : 2900 : : 2132 : 792.67 (I's gain) 8VS00 : 1500 : : 2132 : 410.00 (J's gain) 7. K, L and M engage in partnership with a capital of $15000, to share equally, K investing $10000, L $3000, and M $2000, and L and M to receive salaries of $1500 and $1200 a year respectively ; allowing interest on their investments, which remained intact, what is each part- ner's share in a gross gain of $5700, expenses being $1950, exclusive of partner's salaries? Art. 5. Winding up a Losing Concern, 8. R, S and T, equal partners, with a capital of $30000, finding that they are losing money, agree to dissolve, and on March 4, 1874, leave the property in the hands of T to settle. At this time the effects were cash on hand PARTNERSHIP. 145 $500, merchandise $17500, bills receivable $1300, and book accounts $1000, and their liabilities were bills pay- able to the amount of $2100. On September first T reports as follows : sales of merchandise $14000, on hand $1500, cash on hand $13000, notes $300, uncollected bills $750, liabilities extinguished ; expenses $650. Of the remaining efiects T proposes to take the Mdse at a dis- count of 50% if his partners take notes and accounts at the same rate. Failmg to agree, they sell the goods at auction for $900, and T agrees to take the bills re- ceivable in payment for the collection of the unsettled bills which he thus guarantees. Required the amount coming to each, allowing T 1 % commission for settling the business? Art. 6. Average Capital, 9. U, V and W engage in business January 1, 1874, investing respectively $3000, $2000, and $1000, and agreeing to share the gains and bear the losses in the ratio of their average capital. A-pril first U draws $100, May first V draws $200, and July first W draws $100. Assuming the gains to be $1500 at the end of the year, what was each partner's share ? u. Int. on ^3000 for 12 mos. $180 $175.50 •♦ " 100 " 9 " 4.50 112.00 57.00 $175.50 V. $344.50 Int. on $2000 for 12 mos. 8J 20.00 344.50 : 175.50 : : 1500=U's share. " " 200 " 8 " 8.00 : 112 00 : : 1500= V's " : 57.00 :: 1500=W's *' W. $112.00 Int. on $1000 for 12 mos. mo.oO " " 100 " 6 " 3.00 $57.00 The question may also be solved by Products. Answers: $768.75, $1281,25, $5281,00, $1083,49, $4834.50, $1738, $860, $487,66, $764.15, $248.19, $2050, $466.66, $1650, $1375, $2475, $4100, $50. 10 146 CORPOKATIONS. XXVII. JOINT STOCK COMPANIES. Art. 1. A JoiJit Stock Company is a body of men associated together in a species of partnership, to carr}^ out some heavy undertaking requiring the in- vestment of more capital than individuals or part- nership companies commonly possess. Joint stock companies are usually incorporated by act of legis- lature, with certain privileges. Eailroads, canals, bridges, etc., are generall}' constructed bj^this species of combined interest, and many banking and insur- ance houses, scholastic institutions, etc., are owned and managed by joint stock companies. When an association of this kind is to be formed, a few leading persons make an estimate of the prob- able amount of capital required, divide it into equal shares of from $10 to $100, or $500, according to the nature of the undertaking, and issue certificates of ownership for each share. These are called certifi- cates of stocky and are transferable. Persons own- ing certificates, are called stockholders. Joint stock companies are usually managed by a president and board of directors, elected for the pur- pose, by the stockholders. When shares sell for the price named in the certi- ficate, the stock is said to be at par; if above this value, they are said to be above par; if below it, below par. Besides the stocks of companies, there are govern- ment stocks, which consist of bonds that have been issued by state officers, for the purpose of borrowing money. These draw interest at a specified rate. In dividing the profits of joint stock companies, it has been found more convenient to declare the divi- dend as so much per cent. COEPORATIONS. 147 1. What is the cost of 10 shares of railroad stock at 5 % below par, the original cost being $100 per share? Find the cost of 10 shares, at SlOO and deduct 5 ^. 2. A banking institution declares a dividend of 18 % on a capital of $376198, what amount of money should a stockholder receive, who holds 5 shares valued at $200 each? 3. I hold 15 shares (each of $100) of stock, in gas works, which have declared a dividend of 20 %, " how much am I entitled to after my gas bill of $20, is deducted ? 4. How many shares of United States stocks at 2 % above par, can I biiy for $1224, the original cost being $100 per share? 5. What amount of stock can I buy for $1687, if I am allowed 2 % commission on the amount in- vested ? The amount I am to receive is to be j§^, or ^'^ of the amount of stock purchased — not J-^ of $1687, for that would be commission on commission and invest- ment. Let the amount to be invested be represented by f{{, and to this add -'^=f J ; then we discover that $1687 is 1^ of the amount to be invested, '||'''=33.078= g?^, or my commission, which if we multiply by 50, will give us the amount to be spent, $1653.90. To prove this, find the com. on $1653.90 at 2 %. 6. A broker receives S6785, which he is desired tc invest in State stocks, how much should he invest, and allow himself 2| ^ on the investment? 7. What amount of stock can a broker buy for 16700, and allow himself J % on the investment? Answers: $180, $950, $280, $16658.35, $6619.49, 12, $1780. 148 COMPOUND NUMBERS. XXVIII. COMPOUND NUMBERS. The application of the fundamental rules to num- bers of different denominations, gallons, quarts, and pints; hundreds, quarters, and pounds^ etc., will, it is presumed, be sufficiently taught in the following examples : BRITISH MONEY. Art. 1. Tg add compound numbers. What is the amount of the following sums of Brit- ish money? Solution. — We first add the fractions, calling £. S d thcn^ bX\ farthings^ which makes 6 farthings ; these 1ft 17 4-1 ^® reduce to pence, by dividing them by 4. |=r 1Q fi ?! ^1^^ ^** ^"^ *' ^^^ ^^^ *^® ^ P®^"^ ^^ *^^® n ^7 of column of pence, which makes 20 pence; this num- 17 7 oj ber divided by 12 (the number of pence in a shil- rr TZ ^ ling)=l shilling and 8 pence. Write the 8 under Q'i) ii o-^ ^jjg pence, and add 1 to the units of the shilling's place, which makes 21 ; write 1, and add the 2 to the ten's column =3 or 31 shillings, which divided by 20=£1 and 11 shillings left. Write the latter under the shillings, and add the 1 pound to the pound's column =<£55. Ans. £55, lis. 8^d. 2. Add the following: £ 17 18 ll|+£ 14 17 2i+£ 16 14 8 = £ 17 19 0J+£ 45 llf +£111 10 21= £116 16 6 -l-£320 14 5|+£ 38 18 8 = Total, £700, O5. M. Art. 2. To subtract compound numbers. Subtract £14, 75. Q\d. from £19, 4s. M. £ 5, d. Solution. — We can not take J from nothing, 29 4 3 so we add a penny to both numbers; then sub- 14 7 64- tracting the 1 from a penny, or i, we have | ^ left. Adding Id. to the 6d., we have 7d., which A -tr* Q 3 ^® ^^^ ^^^ subtract from the 3d. above, and * ^^ ^i accordingly add Is. to both numbers; 7 from COMPOUND NUMBERS. 149 Is. 3d. or 15d., leaves 8d. Adding Is. to the shillings, we have 8s., which can liot be taken from 4s. without adding £1 to both numbers; £1 to 4s.=24s. ; 8s. from 24s.=16s. Then adding £1 to the 14, we have £15, which, taken from £19^=£4, making the answer £4 16s. 8|d. 2 Subtract the following : £ s. d. £ s. d. 17 10 Si— 14 5 3 = 119 7 6 — 17 19 5J= 500 — 20 18 8 = 176 14 7J— 129 15 7i = Total, £620, 13s. 9^^. Art. 3. To multiply compound numbers. Multiply £17, 4s. d^d. by 8. Operation. £17 4 9 J 8 £137 18. 2. After performing operations in addition, the learner will readily see how this is done. £ s. d. 17 18 8JX 7 = 120 16 6^X12= 365 7JX 9= Total, £4860, 15s. OJd Art. 4. To divide compound numbers. Divide £157, 13s. 6}^. equally between 25 per sons: 150 COMPOUND NUMBERS. Operation. 25)£157 135. 6^ (£6, 6s. l^d., or 150 £6, 65. If6?., nearly. £7= remainder. 20 153=shilHngs in £7, with 135. of the 150 [dividend added. 3=remainder in shillings. 12 42=pence in 3 shillings, and 6 pence 25 [from the dividend. 17=remainder in pence. 4 70=farthing8 in 17 pence and J. 50 20=remainder, or |g farthings. Recapitulation. — We first divided the £157 by 25; then 163 shillings by 25 ; then 42 pence by 25 ; and, lastly, the 70 farthings. £ s. d. Divide 167 18 6f by 25 = 768 14 3} by 125= 17 11 3J by 875 = Total, £12^"l75^ £ 5. d. Divide 25 18 4 by 5 76 12 8 by 4 1 15 9 by 3 162 12 6 by 30 Total, £30 75. 2d. FOREIGN EXCHANGE. 151 XXIX. "FOREIGN EXCHANGE. Art. 1. In calculating Foreign Exchange the money of one country has to be expressed in that of another. A bill drawn in New York on an English house, will be expressed in pounds, shillings, and pence, ' The relative value of moneys of different countries depends on the par of ExcJiange, and the course of Ex- change. The Par of Exchange is the comparative value of the coins of the different countries, and is fixed, while the relative purity of the coins is the same. The par of exchange between the United States and Great Britain is $4.8665 to the pound sterling. The Course of Exchange usually depends upon the relative state of indebtedness of the merchants of the different countries, and the supply of gold and silver; accordingly, the course of exchange will sometimes be above, and sometimes below par. FORM OF A FOREIGN BILL. Exchange for £1567. Cincinnati, June 3, 1876. Thirty days after sight of this first of Exchange (second and third of the same tenor and date unpaid,) pay to the order of J. H. Story, the sum of one thousand hwQ hundred and sixty-seven pounds sterling, value re- ceived, and place .to my account as advised. To William Morgan', Esq., C. H. GuiOU. Liverpool, England. Note. — Foreign Bills are generally drawn in seteoftwo, three, or four ; that is, copies of the same bill are made out and trans- mitted by different conveyances to the payee, one of which being received and accepted, or paid, the others to be void. 152 FOREIGN EXCHANGE. BKITISH OR STERLING EXCHANGE. Aet. 2. British or Sterling Money Reduced to Federal Money or United States Curi'ency. The calculations relating to sterling money have been reduced to simple operations. In the daily papers we find quoted in gold or currency the precise value of the pound sterling in dollars and cents, as in the following example, the operation of which we give below. 1. Required the value of £157 9 2 in Federal Money, when sterling exchange is quoted at 4 86 in gold, and gold at 10% or 110. By Decimals. 12)2.0 20)9.166 £157.4583 486 9447498 12596664 6298832 ^765.247338==cost in gold. By Aliquots. 486 157 9 2 3402 2430 486 s.d. 76302 6 8= =\ 162 2 6= -i 607 765.247. 76.524: =cost in gold. -10% |841.771=cost in currency. Akt. 3. To assid the learner we give ihefoUovdng Table of AlIQUOTS OF A POUND. 10 o=i 6 8=.i 5 0=i K d. 4 0= 3 4= 2 6= d. 0-tV d. 6-^ 2. Sterling at 4 87J in gold, and gold at llOJ, quired the currency for £147 6 8. re- FOREIGN EXCHANGE. 153 3. The quotation for sterling being 540 in currency, how much will buy a bill for £652 10? 4. Required the currency for the following l)ill at 3 davs sight ; 486J^, gold at 110 ; £376 4 6. 5. What will pay for a sight bill for £319 4 9, with the market at 489 in gold, and gold at 9 J premium ? 6. How much will a bill for £794 5 4, cost in cur- rency, sterling exchange being quoted at 486 i, and gold at 110? 7. Required the cost of £113 3 3 at the same quota- tions. (Ex. 6.) Art. 4. To Reduce Federal to Sterling Money. 8. How much British Money can be bought for $841.77, exchange being quoted at 486 in gold, and gold at 110? In other words, if £1 cost $4.86+10%, what sum in the same currency can be bought for S841.77? 841.77 : 4.86+.486=157.458 20 9.160 12 1.920^£157 9 2. (See Ex.1.) 9. Sterling at 4 89 in gold, and gold at 107J, what amount of a bill can be bought for $1051.35? 10. Required the amount of a bill that can be bought for $31.49, sterling quotations being 4 86f , in gold, and gold being 111^. 11. Sterling quotations being 520, in currency, what sight bills can be bought for $650 ? 12. Required the face of a sight bill that can be bought for $50, sterling quotations being 489 in gold, and gold at 10% premium. 13. Sterling at 487 J in gold, and gold at 109, what amount of a bill can be bought for $79.56? 14. Required the amount of a bill that can be bought for $47.20 currency, sterling at 487, gold at 108? 154 FOREIGN EXCHANGE. Answers: $841J7, $1709.37, $791.87, $3523.50, $2013.37, $4250.52, S605.58, £157 9 2, £200, £14 19 5, £5 16, £8 19 4, £1 10, £125, £246 13, 4. £9 5 11. GERMAN EXCHANGE. Art. 5. The money of the whole German Empire is Beichmark and pfennige. Signs: Rm. Reichmark, d. pfennig. 1 i?m.=100c?. COMPARATIVE TABLE. Rm. 1 Prussian Thaler (30,Silber- groschen,) ... =3 1 Florin, Austrian Coinage, = 2 7 Florins, S'th Qer. Currency, (Siiddeutsche Wahrung,) =12 Rm 10 Mark-Banco, (Hamburg,)= 15 100 Florins, (Holland,) . =169 5 Francs, (France,) . = 4 10 £, (British,) . . . =203 "--^ ■ 'Tede 5 Francs, (France,) . = 4 0£, (British,) . . . =203 97 Cents, (Federal Money,) = 4 Art. 6. To Reduce German to Federal Money. 1. Required the value of Rm, 1264 in United States currency when exchange is at par, and gold is quoted at 110. First Method. Second Method. 1264 Rm. A : Rm, ItU : : 97 c : a; 24.25 1 316 ■316^ ?1 5056 2212 2528 2844 $306.52 in gold. $306.52 in gold. 30.65^ 10% cost of gold. 30.65 $337.17 in currency. $337.17 in currency. Explanation — Fir^t Method. — If one Rm. is equal to 24Jc in gold, Rm. 1264 are equal to 1264 times 24^c, viz.: $305.62 in gold. Gold being quoted at 110; i.e., 10 per cent above par, we add 10 per cent to the value in gold to obtain the value in currency. The Second Method is by proportion, which see page 139. 2. The quotations for German Exchange being 98^, and for gold 110^, find the cost in currency of a bill for Rm, 1892. FOREIGN EXCHANGE. 155 3. German Exchange at 100, and gold at llOJ, what will be the cost of a bill for Em. 720 ? 4. What will be the cost of a bill for Em. 58, German exchange being quoted at 98, and gold at 112^. Art. 7. To Reduce federal to German Money. How much German Exchange can be bought for '^125.40 in currency, when the quotation is 4:96 and gold no? 24 Explanation. — If four B^n. are 2.4 equal to 96c in gold, one Rm. ia 26!4J1 2540.0(475 Bvi. equal to 24c in gold, gold being at IQ5g 10 per cent premium, one Bm. will -.Q^^ cost 10 per cent more in currency !.Zf^ or 26 4.10th cents. $125.40 in cur- rency will bring as many Bm. as 26.4 1^2^ is contained therein, viz : Bin. 475. i:j20 5. Required the amount of a bill that can be bought for $1862, exchange being quoted at 95, to Bm. 4, and gold at 112? 6. German Exchange at 96, and gold at 110, what amount of a bill can be bought for $125 ? 7. Required the amount of a bill that can be bought for $42.25 currency, exchange being quoted at 96 and gold at 111? 8. What is the face of a bill that costs $666.68 in currency, exchange 100, gold 115? Answers: Rm. 475, Rm. 473.45, Rm. 7000, Rm. 158.60, $15.98, $512.86, $198.90, $337.17, Rm. 2318.89. FRENCH EXCHANGE. Art. 8. The unit of French Money is the franc, (a silver coin equal in quality to our silver coins.) l/c.= 10 decwies, 1 (/ec.=10 centimes. The par value of the fc. is about 19|^ cents, or fcs. 512^ to $100 in gold. Art. 9. To Reduce French to Federal Money. 1. French Exchange being quoted at 510 and gold at 156 FOREIGN EXCHANGE. Ill, requii:ed the cost in United States currency of a bill for /cs. 2465. 510)2465.00(4.83- 4.83- 2040 111 4250 4.83- 4080 48.33- 1700" 483.33- 1530 S536.50 in U. S. currency. It will be noticed that the amount of the bill is divided by the quotation of the French Exchange, and the result multi- plied by the gold quotation. French Exchange may be worked by proportion. /cs.510:/cs. 2465^$111 : $536.50. 2. Kequired the cost in currency of a bill for /cs. 727.6, exchancre being quoted at 520, and gold at 110. 3. What will be the cost of a bill for fcs. 226.66, French Exchange 518, gold 112^? 4. French Exchange at 518 J, and gold at 115, what will be the cost of a bill for/cs. 52.5? Art. 10. To Reduce Federal to French Money, 5. The quotation being 510 for French Exchange and 111 for gold, required the amount of a bill that can be bought for $536.50 currency. The process is the reverse of the above. The propor- tion would be $111 : $536.50 : :fc8. 510 : fcs. 2465. 6. Kequired the amount of a bill that can be bought for $260 currency, French Exchange being quoted at 515, and gold at 110. 7. French Exchange at 516|, and gold at 109|, what amount of a bill can be bought for $1410 currency? 8. The quotation being 512 and 108|, what will be the face of a bill that costs $682.75? Answers: $37.57, $49.23, $153.92, $3253.15, $536.50, fcs. 367.57, fcs. 1217.27, fcs. 2465, fcs. 3221.82, fcs. 6650.82.'* $11.64. IMPORTING. 157 XXX. IMPORTING. Art. 1. Importing is the business of buying goods in a foreign, to sell in the home market. A tax, un- der the name of Duties or Customs, is imposed by gov- ernment on most imported articles of commerce. Such taxes are levied for the purpose of creating revenue to defray the expenses of government, or to protect home manufacturing and agricultural inter- ests. Duties are regulated by a scale of rates called a Tariff, and are altered according to the exigencies of the times. A high tariff signifies high rates of duties ; a low tariff, low rates of duties. In the United States, a high tariff is called for, when . the expenditure of government exceeds the revenue. In Great Britain, it is advocated when imported articles sell so cheap as to interfere with the sale of home products. The persons appointed to examine imported goods and collect duties, are called Custom House OfficerSy and their place of business, the Custom, House. Art. 2. Duties are of two kinds : ad valorem and specific. Ad valorem duties consist of a rate per cent, on the value of the goods, as stated in the invoice; specific duties, of a stated sum of money on the quantity imported, without regard to value, as $1 a gallon, $20 a ton. Certain allowances are made on goods charged with specific duties. These are draft, tare, leakage, and breakage. These allowances sometimes consist of a percentage of the weight or quantity, and some- times of a specified deduction. Tare is an allowance made for the weight of the 158 IMPORTING. box, barrel, bag, crate, etc., which contains the goods, and is usually calculated by percentage, etc., after the deduction for draft is made. Draft or tret is an allowance made for loss by weighing in small quantities, and for impurities to which some goods are subject. On 112 lbs., or less, it is 1 lb.; from 112 lbs. to 224 lbs., 2 lbs.; from 224 lbs. to 336 lbs., 3 lbs. ; from 336 lbs. to 1120 lbs., 4 lbs. ; from 1120 lbs. to 2016 lbs., 7 lbs. ; more than 2016 lbs., 9 lbs. Note. — The draft, though not stated in the question, is to be deducted before other allowances are made. Leakage is an allowance of 2 ^ on liquids, in casks, paying duties by Ihe gallon. Breakage is an allowance on bottled liquors, usually 5 ^, but on ale, beer, and porter, 10 ^. Gross Weight is the total weight of goods and box, barrel, etc. Net Weight is what remains after all deductions are made. Art. 3. Goods imported may be placed in Govern- ment warehouses and the duty paid on withdrawal therefrom, or duties may be paid at once and the goods taken by the importer. 2. Entries should be made within twenty -four hours of the arrival of the goods. 3. Each class of merchandise should be entered by itself, without regard to rate of duty, and goods de- signed for warehouse entered by the case. 4. All charges incurred before shipment at the for- eign port should be added to the invoice value ; and every invoice * is subject to duty on at least 2\ ^ IMPORTING. 159 commission. Except the fee for Consul's Certificate^ ocean insurance, and freight, which are not dutiable. 5. Fractions are omitted in reckoning duties. Half a dollar is considered $1; under that the fraction is rejected. 6. On the back of the blank form of entry is an affidavit to be made by the owner or owners of the merchandise, or some one acting for them under power of attorney, stating that the invoice produced is the only invoice received for the goods ; that the entry contains a true account of said goods; that nothing has been concealed whereby the United States may be defrauded ; that if any mistake is dis- covered in the future the affiant will make it known to the Surveyor of Customs, etc. 7. There are three separate entries on the blank — next page — which, in practice, would be made out on separate papers. 8. The first entry represents merchandise subject to both specific and ad valorem duties, and a reduction of 10 % on the rates. Art. 4. Custom House values of foreign currencies. Crown of Sweden, Norway i Pagoda of Madras 1.84 and Denmark 268 | Patacans of Uruguay Dollar, Egypt 1.0039 '^ "" '^ ' Dollar, Mexican 1.0475 Dollar, Central America.. .965 Florin of Austria 476 Florin, Southern Germany. Florin of Netherlands..... .405 Franc, France k Belgium. .193 Lira. Italv 193 Mahbul), Tripoli 8909 Mark reichs(rix),Ger.Em. .2382 Milreis of Brazil 5456 Milreisof Portugal 1.0847 Peso of Cuba 9258 Peso of Chili 9123 Peso of Venezuela 7773 Peso of Columbia : 965 Pound Sterling, Gr. Brit.. 4.8665 Piastre, Turkish 0439 Rix mark, Germany 2382 Rupee of India 4584 Ruble of Russia 7717 Tale of China 1.61 Thaler (see dollar) Yan of Japan 997 160 V,: «; 1 • 00 c4 OS rtj »o OS «c '5 "3 t- ^ a> €^ s o o o CO !>- o o o CO CO lO t-- oo «o o lO O 1-J co «d OS o «o — lO «o CO lO >< c^ cs lO '^ ¥k ^ H ^ ^ ^ O P tT •^ h4 "o 'o ^ tuo bo o .s _c i^ >, 01 3 H^ ft_ « ^ a — . ^ ■'" 1 tft - 1 00 =^ oi 'Ci t- CO €©■ €© a ^ ^• fe 1^ €©■ ^ o m o p (§) »o {■J O? «o «o pq d «o CO Tjj Tf 00 b- CO O 00 00 03 rH to OS tO vo d d — I>1 O r-i C^ CO CO oo CO oo o CO '■IH 00 00 Ed 2? eo eo oc 5 5 «rt 2: 1 ^ ■^ -^ < O a o :^ ^r::^ 1 f 1 a 2 PJ !» ^ o .r o •^ s K "5 02 •» tn ?^ tn' CQ c aj (U (JJ -^ K S -2 s o3 r c: bO S < cj-d o ^^ O h-^ o P^ >o o go go o f^ o tt- 1 \^ o ^5 /^ \ . / 1¥5> tTJCiO' lZ4gH5» Sg^HOJ 5155' 5555? ?5S3?> 9 3 6 9 8 7 6 Answers arranged promiscuously ^, J, jfyy, iHfj g 3, JL 2 3 _4 9 1 1 9 5> 25J 8) ^l> llQZ) ■&) 3J38 Art. 4. To reduce a mixed or whole number to an improper fraction. This is done in the same way that we would reduce feet and inches to inches. 1. Eeduce 5 J to an improper fraction; that iSj in 5-| how many eighths? 5X Explanation. — In every whole number there are 8 g eighths, and in 5 whole numbers there are 8 times 6, or 40 eighths, to which add 7 eighths, and the result is 4? 47 eighths. Eeduce the following numbers to improper frac- ..ions: 7|,6|, 5l|, ITg^, HS^g, 16^. Answers.— ^,^, V, 1*, 'W, ^V", Vc'- FRACTIONS. 178 Art. 5. To reduce improper fractions to whole or mixed numbers^ is an operation the reverse of the last. 1. Eeduce y to a mixed number. 47 — PSX 2. Eeduce the following to whole or mixed num- bers, and the remaining fractions to their lowesi terms : 146 456 364 161 J196 100 4160 3179 7854 llOOO g > 5 » 5 > 15 J 21» > 5 > g » 1^'E J H64 » S6 Note. — When fractions are to be reduced to their lowest term, and the learner should be unable to see what number both the denominator and the numerator can be divided by, the Greatest Common Divisor may be found by dividing the denominator by the numerator, and the numerator by the re- mainder, and the old remainder by the new remainder, etc. The last divisor — i. e., the one which can be divided into the dividend without a remainder is the Greatest Common Divisor, 130)169(1 To find the greatest common divisor of 130 -Jf^, we proceed as follows: 39)130(8 13 being the divisor, that divided 39 117 without a remainder, is the greatest com- ~TF')39C3 ^^on divisor: 130-^-13 = 10 39 'l69--13 = 13 MULTIPLICATION OF DECIMALS. Art. 6. In this rule we multiply as in whole numbers, and mark off as many places of decimals in the product as there are in the two factors. 1. 5.7X6.107 6.107 There are 3 places in this factor, 5.7 and 1 place in this " 42749 30535 34.8099 so we mark off 4 in the product. Answers: 182, 10i|, 24J, 50f, 12^ 693J, 17^3^, 129^, 9^, 47|i. 174 DIVISION OF DECIMALS. Art. 7. When the product contains fewer figures than there are decimals in the factors, we make up the number by annexing ciphers to the left. 2. 100X.0005. ^QQ^' The product contains only 3 figures 1 (^00), so we annex one more cipher to 500 make up four, the number contained or .0500 in the factors. Arts. .05 .107X.05 =* 61.04X.0007= .7103X.004 = Total, .0509192 DIVISION OF DECIMALS. Art. 8. Division of decimals is effected in the same manner as division of whole numbers, with the diflTerence in using the decimal point. 77ie divisor must contain as many 'places of decimals as the dividend, 1. To divide 34.8099 by 6.107 6.1070)34.8099(5.7 The dividend contains 4 de- 30 5350 cimals, and the divisor only 3; — 7:7,770 so we point off one in the quo- _flL_ in Multiplication. Note. — When a remainder occurs, we may annex ciphers in- definitely, and carry out the quotient to as many places as we desire. Art. 9. When the dividend does not contain as many decimals as the divisor, annex ciphers to the right of the former, until it contains the same num- ber ; the quotient will then appear in whole num- bers. If a remainder occurs, annex ciphers, and the result will be decimals. FRACTIONS. 175 2. Divide 3066 by .1783. .1783)3066.0000(17195.7 1783 12830 12481 ~^90 1783 17070 16047 1l0230 8915 13150 12481 Four ciphers have been an- nexed to the dividend, and a fifth annexed in finding the 7 of the quotient; so we point off 1 decimal. When there are not figures enough in the quotient to make up the number of decimals in the dividend, annex ciphers to the left of the former. 3. I>ivide 10.70067 by 370.4. 370 0^0.70067(.0288 7408 32926 29632 "32947 29632 Here the quotient pro- duced only three figures (288), which, with the one in the divisor makes only four decimals; so to make the number equal to the decimals in the dividend we annex a cipher to the left 314.06-~10.73 =29.2693 17600-^785.4 = 3170.09-^2.4014= 417.456-^31.145 = Total, 1385.1825 176 REDUCTION OF DECIMALS. 30.f)'40~-493.67 = 10.8739^117.406 = 6.342-^-22.973 = 1467.06-j-196.04 = Total, 7.91420 REDUCTION OF DECIMALS. Art. 10. To reduce a common fraction to a decimal^ we annex ciphers to the right of the numerator, and proceed as in division. Reduce ^ to a decimal, f to a decimal. 2)10 4)300 75 775 Eeduce ^ to a decimal. 3)100000 .3333 This quotient might be carried out in definitely. It is called a repeating decimal. Such h decimal is marked thus, .3. Its fractional value is restored by using 9 instead of 10 for the denomina- tor. 3 = 1. Reduce 4 to a decimal. 7)1000000000000 This is called a circulating 142857142857 decimal, and is marked thus, i 45?85'7 1 4 -2 8 5 7 __ 1 Its fractional vulue is restored in the same manner as that of the .3 in the preceding example. Express the following fractions decimally : 33 1 % i 1 6 1 5 4 3 '71 1 4 3 3 T7 Total, 1.1186 Total, 1.2239 FRACTIONS. 177 Art 11. To find the value of the decimal part of a denominate number^ as £0.75, $0.33 J, etc. 1. What is the value of .5 of a yard? If we wanted to know the value of 5 yards in a lower denomination, we would multiply by 4, as 5X 4=20; that is 20 quarters. The same principle ap plies in decimals. .5 4 2.0 Ans. 2 quarters. £0.345 is how much? , 345 Explanation. — The next lower denom- 2Q ination to pounds is shillings; so we multiply by 20, and point off three figures 6.900 shillings, ^o correspond with the number in the 22 factors. The next lower denomination is , pence; so we multiply by 12, and the next 10.800 pence. farthings, which we multiply by 4. The ^ answer is 6 shillings, 10 pence 3^ farthings, 3.200 farthings Note. — 1. The ciphers on the right need not be used, as they possess no value. 2. Observe that the whole numbers are not multiplied, else the shillings and pence would be reduced to farthings. Find the value of 3. .625 of a gallon. Ans. 2, 1. 4. .1425 of a year. Ans. 1, 21 3 . 5. .8323 of a £. Ans. 16, 7|. 6. .1374 of a tun. Ans. 274,12.8. 7. .0037 of a lb. Troy. Ans. 21.3. Art. 12. To reduce denominate values to decimals. Eeduce 6 shillings, 10 pence, 3i farthings to the decimal of a pound sterling. 178 DECIMALS. 1. 5) 10 4)'~372 _ — 1_ This operation is the reverse of the 12)10.8 last. Observe that the 3 in the second 2Q\ g Q line was annexed after reducing i to a I 1_ decimal; so with the 10 and 6. £ .345 2. Eeduce 3 quarters to the decimal of a yard. 3. Eeduce 6 lbs. 3 oz. to the decimal of a cwt. 4. Eeduce 12s. 6|d. to the decimal of a £. 5. Eeduce 12 lbs. to the decimal of a tun. 6. Eeduce 1 foot 3 J in. to the decimal of a yard. 7. Eeduce 16 oz. to the decimal of a tun. The pupil can prove his calculations by last Art. PRACTICAL QUESTIONS. 1. At 56 cents a pound, what will 127 lbs. 6 ounces of tea come to ? 16)60 .375 decimal part of a pound. 127.375 56 764 250 6368 75 7133.000 or $71.33 Remark. — This is not the shortest method of computing the above ; the object being merely to show the practical application of decimals. 2. At 15 for a pound sterling, what will be the value of £16 8s. lOd.? Arts. $82.21. 3. What will be the value of the following sums 9f money, at the same rate? FRACTIONS. 179 £167 10s. 3Jd., £19 2s. 6d., £10 10s. lOJd. Total,$985.91 4. At $75 per hundred (112 lb.), what will 14 cwt., 3 qrs. and 15 lbs. cost? 5. Find the cost of 3 tuns 15 cwt. of hemp, a S140 per tun r' Answers.--U116.29j $525. MULTIPLICATION OF COMMON FEACTIONS. Art. 13. A fraction is multiplied by a whole number, by simply multiplying the numerator with- out altering the denominator, fx 7=7X3, or '\^j which reduced to a mixed number, equals 5|-. Art. 14. Fractions can also be multiplied by di- viding the denominator, without altering the numer- ator. -4.X5=5)/3=|, or IJ. Multiply the. following fractions: 1. |x 5=14 4. /^X 11=2.357 2. |x 4=3| 5. j\X 9=3.316 3. |xl2=8 6. j%X 6=2.824 Art. 15. Mixed numbers are multiplied by whole nutnbers, as compound numbers are multiplied. Let it be required to multiply 4|^ by 7. Whole Nos. Eighths. Illustration^ 4 5 Explanation. — Seven times 6 tj eighths equal 35 eighths, or 4 ^_ whole numbers and 3 eighths. 32 3 Seven times 4=28 and 4 make or 321 32. Ans.S2l. ° It will not be necessary for the pupil to" write his work in so formal a manner as in this illua- tration. 180 FKACTIOXS — ADDITION AND SUBTRACTION. 2. Multiply 61 by 12. 12 813 Answers. Answers. 3. 6f X 8=54 7. 914| X 120=109760. 4. 7'i X 7 = 50| 8. 63^ X 15= 952.5 5. 8| X 6=53| 9. 127y\X 20= 2543.636 6. IJ/^X 12=13/^^3 10. 110^ X 14= 1542.333 Art. 16. To multiply fractions together, we multi- ply the numerators together for a new numerator, ^,nd the denominators together for a new denomin- ator. 1. fXf=A, orf. These operations might have been abbreviated by what is called cancellation. In the first example, for instance, f is to be multiplied by |, that is, the numerator 2, is to be multiplied by the numerator 3; but the 2 is also to be divided by 3, for f signifies that 2 is to be divided by 3 ; therefore, since the 2 is to be multiplied by 3, and divided by 3, it remains exactly the same, and the 3 of the denominator is said to cancel or make void the 3 of the numerator. In the following operations, the canceled figures will be known, by having a line drawn across them. 2 $ 2 In the second operation, the 2 of tXz==~ the numerator and the 6 of the de- nominator are uncanceled, making |, 2 $ ^ $ which reduced by dividing both by 2, ^^4^ 2^ "a e^quals J. ^ . The 2 and 6 might have been can- ^ $ 4 $ celed also, by dividing both where tX-X-X- they stood in the question, as in the ^ o 3d example, placing only 3 as a de- nominator, and 1 as a numerator. FRACTIONS. 181 1 is alwa^^s to be understood, where a number ha» been canceled. 1 Some prefer arranging the terms 1$ ^ 3 of canceling fractions as in the 15 margin, with the denominator or 8 t^ 37 divisor on the left, and the numer- ator on the right. Explanation. — The first 2 was canceled in the 18, leaving 9; the 24 and 9 were canceled by dividing both by 3, leaving 8 and 3; the 74 was canceled by the second 2. The fractions arranged in the usual order are Answers, Answers, 3. 4x I X f =i. 7. l^X 2iXy«^=3.38 4. fXiXi|X/,=i 8. 6iX |Xif=4.8 5. iXAX^,X^\= .0041 9. 87iX^f.X^^= .05 6. iXliX f X i .0265 10. ^fiX52|x 4 =5.75 Art. 17. Compound fractions are reduced to simple ones, by multiplication. Let it be required to reduce J of f of f to a simple fraction. We know by in- spection, that one-half of f is J, and that J of f is J, the answer. By multiplication ^XfXf=3^^=J. I $ $ By cancellation -X-X-=i. ^ ^ $ 4 ^ Answers. Answers^ 2. I of I of ./y=J 6. f of f of 9| =7.389 3. 4 of If of 34=. 3 7. fof J of /-J =.0068 4. I of ^f of A = .286 8. 41 of IJ of J =1.5 5. 4 of 1-1 of f = 1.75 9. |of fX fXli=.4286 10. At 11} cents a pound, what will 147^ lbs. of coffee cost? 11. What will 7^ lbs. of cheese cost, at 9i cents per lb. ? 182 FRACTIONS — DIVISION . 12. At 12J cents a pound, what will 120 lbs. of sugar cost ? 13. What will 14^ lbs. of beef cost, at 6| cents a pound ? 14. Fifteen and a half yards of muslin at 9J cents, will cost how much? Ansivers.— 71 cents, $14.80, $16.59, 98 cents, $1 .43. DIVISION OF COMMON FEACTIONS. Art. 18. Division being the reverse of multipli- cation, to divide a fraction by a whole number ^ we divide the numerator or multiply the denominator. 3)6 6 per day, that is equal to the work of one man for 3X5X8=120 hours; 15X12X7 feet is equal to 1260 cubic feet, 17 X 14 X 6 feet is equal to 1428 cubic feet. 2 men work- ing 10 hours per day is equal to 1 man working 20 hours per day. It takes 120 hours to dig 1260 cubic ieet, hence 10^ feet per hour. In 20 hours 210 feet can be dug. 1428 feet: 210 feet= 6f days or 6 days 8 hours. 2. If 6 men in 15 days dig a trench 18 feet long, 7 feet wide, and 5 feet deep, in how many days will 21 men dig a trench 125 feet long, 9 feet wide, and 4 feet deep? JL?i^. 30.61 days. 3. What is the interest of S6784 for 2 years 6 months, and 15 days, at 6 % per annum (365 days)? Statement. $6 days, 365 927 dollars, 100 6784 Arts. $1033.77 4. The interest of $1467 for 3 years, 4 moa., and 12 days, is $450.72, what is the rate per cent?* 5. The interest on $786.55 at 10 % is $176.44, what is the time? 6. The interest of a certain sum of monej^ for 4 years, 2 months, and 20 days at 6 % is $100, required the principal? ♦ The pupil can prove his own work by computing the interest by the method taught in the first part of this book. GAUGING. 195 XXXVI. GAUGING. The process of finding the capacity of barrels, etc., is callen Gauging. Art. 1. To find the capacity of a vessel in the form of a cylinder^ square the diameter in inches, multiply by the length in inches, and the product by 34, then point off four figures from the right, and you have the capacity in wine gallons. 1. Find the capacity in gallons of a cistern measur- ing 8 feet in diameter and lO feet in depth. Solution.— 8 ft. = 96 inches ; 10 ft. == 120 inches. 96 X 96 X 120 X 34 =^ 3760.1280, or 3760xVVo g^^^- Note. — To find the capacity in bbls., divide the i.umber of gal. by 31 J (the number of gal. to a bbl.). 2. Find the capacity, in gallons and barrels, of a cistern measuring 10 feet in diameter and 12 feet in depth. Art. 2. Having the head and hung diameters, to find the mean diameter add two-thirds of the difference to the head diameter. To find the capacity of a barrel or cask, ascertain the mean diameter and proceed to solve as under Art. 1. 1. A cask, having for the head and bung diameters 30 and 36, and length 40 inches, holds how many wine gallons? 30 — 36 == 6. f of 6 = 4. 4 + 30 =: 34 mean diam. 342 = 1156 X 40 X 34 = 157.2160 gallons. 2. Find the capacity of a barrel measuring 17 inches at the head, 21 inches, bung, and being 2 feet 3 inches long. 3. What is the capacity of a barrel, having the head diameter 36 inches, bung diameter 40 inches, and lenorth 46 inches? 196 GAUGING. Art. 3. Having the top and bottom diameter of a vessel in the form of a frustrum of a cone, to find the mean diameter add half of the difference to the smaller. 1. Find the capacity in gallons of a vat, in the form of a frustrum of a cone, the diameter at the top being 5 feet, and at the bottom 7 feet, and the depth 6 feet. Solution.— 5 ft. = 60 inches. 7 ft. = 84 inches. 60 fn m 84=r24, half which (12) added to 60 (the smaller diameter) =* 72 inches mean diameter. 72 X 72 X 72 (depth in inches) X 34, etc. 2. What is the capacity in gallons of a vat, the top and bottom diameters being 4 and 6 feet, and the depth 6 J feet? 3. How many gallons will a vat hold, measuring 6 feet at the top, 6^ feet at the bottom, and 7 feet in depth? Art. 4. To find the number of gallons of linseed oil in a barrel, add one-third of the number of pounds to the net weight in pounds of the barrel, and divide the sum by 10 (there are 7^ pounds of linseed oil to the gallon). 1. How many gallons of linseed oil are contained in a barrel weighing 315 pounds net? Solution.— J of 315 = 105 + 315 = 420. 420 divided by 10 = 42 gallons. 2. Find the contents in gallons of a barrel linseed oil weighing 324 pounds. 3. In a barrel of linseed oil weighing 298 pounds, are how many gallons ? Answers: 157.216 gal., 35J gal., 119.369 barrels, 3760.128 gal., 1269.0432 gal., 42 gal., 43.2 gal., 39.733 gal., 233.835 gal., 7050.24 bbls. 223.82 bbls. 954.72 gaL, 1359.74 gal, 32yV gal. iisriDEx:, Account Sales^ 123 Account Current 128 Acre, Hills in an 12 Square feet to an 8 Acres in a field 165 Adding, Rapid method of. 22 Addition, simple 19 of Compound numbers 148 of Decimals 63 of Fractions 185 Ad Valorem duties 157 Agents, see Commission 62 Aliquot Parts of 60 84 of 100 55 of 1000 55 of £ 352 Aliquots, Multiplication by 55 Annual Interest 91 Apothecaries Weight 6 Arithmetical Definitions 16 Signs 16 Average 118 applied to accounts current.. 126 applied to statements 127 applied to storage 129 Discount product method.... 133 Interest product method 134 Compound method 136 applied to account sales 123 Capital to calculate 145 Averaging, Methods of 132 Avoirdupois Weight 6 B Bank Discount 94 Bankruptcy 113 Bills of Exchange 151 BiLLs-Invoices 67 Accuracy in 67 Credit on 74 Discount oflf 73 with gross weight and tare.. 69 receipted by clerk 68 BiLL-Making, exercises on 76 Bin, to measure 166 Binders' count of paper 13 Bonds, Stocks, etc 63 Breakage 158 Bricklayers' Measure 10 British Money 5 Exchange 151 Brokerage and Commission 62 BusHELin inches 166 of produce. Weight of 7 C Cancellation, Division by 60 Centigrade Thermometer 14 Change, to make rapid 28 Charcoal 11 Circular Measure 9 Cistern, to find contents of 195 Coal Measure 10 Coke 11 Commercial Weight. 6 Commission and Brokerage 62 Complex Percentage 103 To find gain per cent 103 To find principal 104 To find rate per cent 105, 109 To find rate of income 105 To find cost per cent 106 No find cost of investment,. 106 To find the time 107, 110 To find the rate of gain 108 To find the amount 109 Miscellaneous exercises Ill 198 INDEX. Complement 27 Compound Numbers 148 Proportion 194 Interest 90 Corporations 146 Cubic or Solid Measure 9 Foot, weight of substances.. 7 CusTOM-HousE Entry 160 Values of Currencies 159 Customs 157 D Decimal Point, Power of. 63 Decimals, Addition of. 53 Subtraction of 54 Multiplication of. 178 Division of 174 Reduction of 176 Digging 11 Discount and Premium 62 Discount, Two kinds of 94 Bank 94 True 99 Discounting off Bills 73 Discounting Notes 94 Discounting lot. -Bearing Notes 100 Banker's Method 100 Broker's Method 101 Equitable Method 102 Division 36 Long 78 Method of Proof of. 81 Short 36 Principles of 80 Short Methods of. 59 of Fractions 182 of Decimals 174 by Cancellation 60 Draft or tret 158 Dry Measure 10 Duodecimals 190 Duty 64, 157 E Easy Fractions 43 Effect of Coal ii English Invoices lGl-164 Equation of Payments 118 Equation of Time 118 Exchange 62 Par of 151 Course of 151 British or Sterling 152 Foreign 151 French 155 German 154 Extensions, exercises on making 77 F Fahrenheit Thermome'^er 14 Farming 165 Federal Money 5 Foreign Exchange 151 " Invoice.*; ....: 161 «' Bill of Exchange 151 " Currencies... 159 Fractions, Common 171 Value of a 43 Easy 43 Addition of 185 Subtraction of 183 Multiplication of 179 Division of. 182 Reduction of. ... 172 Decimals, see Decimals...'?, 173 French Money ... 5 Exchange 155 G Gain and Loss 65 per cent 6{ 111 Gas Measure 12 Gauging 195 German Money 6 Exchange 154 Goods, Exercises in marking.... 65 Grain, Weight of, per bushel.. 7 Greatest Common Divisor \73 Gross Weight 168 INDEX- 199 H Hay in a Ton 12 Hills in an Acre 12 Horse Power 12 Heating Power of Fuel 11 I Importers 157 Importing 157 Introduction 15 Insolvency 113 Insurance 63 Interest 83 Annual 91 Compound 90 Simple 83 To find the true 109 To find the rate of 109 To find the time 110 To find the amount 109 on Investments 144 on cents, How to reckon 84 for days 84 for months 85 for years 85 Investments 63, 103 Invoices, British 161 to 164 and Bills 67 J Jewish Long Measure 8 Joint Stock Companies 146 L Land, To lay off a quantity of... 165 Measure ,. 9 f.EAKAGE 158 Least Common Denominator.... 186 Least Common Multiple 187 Long Division 78 Liquid Measure 10 Lumber business, calculation for 169 M Man, Strength of. 12 Marine Measure 8 Marking Goods 65 Measure, Ale and Beer 10 Bricklayers' 10 Circular 9 Cloth 8 Coal 10 Cubic or Solid 9 Dry 10 Jewish Long 8 Land 9 Linear or Long 8 Liquid 10 Marine 8 Metric, of capacity 10 Metric, Long 8 Scripture Long 8 Square or Surface 9 Stone 10 Surveyors' 8 Time 9 Wood 9 Men, Average Weight of. 12 Mercantile Order 75 Metric System of Weights and Measures, Weights 6 Long Measure 8 Square Measure 9 Cubic Measure 9 Measure of Capacity 10 Mint or Troy Weight 6 Multiplication 28 of Fractions 179 of Decimals 173 by AliqHOts 55 Method of Proof of. 81 Short Methods of. 55 Principles of. ? 32 Table 28 200 INDEX. N Net Weight, What 158 Notation and Numeration 17 English Method 17 French Mefhod 17 Roman Method 18 Notes, Discounting 94 Form of Promissory 94 Numbers, Compound 148 Properties of 55 P Paper, Sizes of 13 Partial Payments 92 Partnership, Remarks on 143 Calculations 143 Int. on Investment.,.,. 143 Average capital 144 Winding up... 144 PAST-Time Tables 116 Payments, Equation of. 118 Poll Tax 04 Percentage 61 Compfex 103 Premium and Discount 62\ Printers' Count of Paper 13 Produce, Weight of, per bushel 7 Promissory Note, Form of 94 Properties of Numbers 55 Proportion, Simple 139 Compound ,. 194 R Ratio 138 Reaumur Thermometer..... 14 Reduction of Decimals 176 Roman Notation 18 S Scripture Long Measure... 8 Short Division 36 Short Methods 55 Signs, Arithmetical 16 Simple Interest 84 Percentage 61 Proportion 139 Discount 62 Specific Duty 157 Tax 64 STATUTESof Weight of Bushel... 7 Square Measure r. , 9 Sterling Money 5 Sterling Exchange 151 Stocks, Bonds, etc 63 Stone Measure 10 Strength of Man 13 Subtraction 25 of Decimals 53 of Fractions 183 of Compound Numbers 149 T Tariff 157 Tax , 64 Thermometers 14 Time, to Reckon 83 Measure of 9 Time Table, Gse of. 97 Interest 96 and 114 Past 110 Ton, Weight per 6 Tret 158 Troy Weight 6 True Dtscouat«;rr. 99 Interest 109 W Weight per Bush§I^' Produce 7 of Cubic Foot.......;......... 7 Troy 6 Commercial 6 per Ton 6 Avoirdupois 6 Metric 6 Gross, tare :ind net. What^^isgL- Live Stock i68 ' Winding up a Losing Concern... 144 Wood Measure 9 Wrapping Paper 13 /a^}^ ; J I ~1X C}^(X, 1 / ^ "^ G} 9