each boy will receive of an apple, or the quotient of 3 divided by 4, and generally a Vulgar, or Common Fraction denotes the division of the numerator by the denominator.(22, 103) The fraction, for example, denotes that 1 is divided by 2, but since 1 does not contain 2, the quotient is less than 1, and must therefore be expressed in parts of unity. Now if we add a cipher to the dividend, 1, it becomes 10 tenths;(126) and 10 tenths divided by 2, the quotient is 0.5.(125) Hence the decimal 0.5 is equivalent to. Again, in the fraction, if we add a cipher to the 1, it becomes 10 tenths as before, and 10 tenths divided by 3, the quotient is 0.3, and 0.1 remains. Joining a cipher to 0.1, it becomes 0.10, and dividing again by 3, the quotient is 0.03, and thus may we go on as far as we please, getting by each additional cipher a 3 in the quotient, which is 10 times less than the preceding, as 0.333+, which is the decimal expression for . And again in the fraction, adding a cipher to 3, and dividing by 4, the quotient is 0.7, and 0.2 remain; adding a cipher to 0.2, and dividing again by 4, the quotient is 0.05;-0.75 then is the decimal expression for And generally, 130. To change Vulgar Fractions to Decimals. RULE-Annex ciphers continually to the numerator, and divide by the denominator, so long as there shall be a remainder, or until the decimal be obtained to a sufficient degree of exactness. The quotient will be the decimal required; and it must consist of as many decimal places, as the number of ciphers annexed. If the quotient does not contain so many figures, make up the deficiency by prefixing ciphers. (127) QUESTIONS FOR PRACTICE. 1. What is the decimal ex- 5. What are of a month pression for? in decimals? Ans. 0.375 mo. 25)1.00(0.04 Ans. 1.00 6. Change to a decimal. Ans. 0.7045+ 7. Change to a decimal. Ans. 0.173+ 8. Change to a decimal. Ans. 0.002. 9. Change to a mixed number. 10. Change to a decimal 131. Having become familiar with the method of changing Vulgar Fractions to Decimals, whenever fractions occur, the pupil has only to substitute for them their equivalent decimal values, and proceed as if they had been given in decimals. To illustrate this remark, take the following 132. Federal Money is the established currency of the United States. Its denominations are all in a decimal or ten-fold proportion, as exhibited in table 1, page 38. The dollar is considered the unit money, and all the lower denominations are regarded as decimal parts of a dollar. Thus the dime is 1 tenth, or 0.1 of a dollar, the cent 1 hundredth, or 0.01 of a dollar, and the mill 1 thousandth, or 0.001 of a dollar; and placing these together, dol. d. c. m. 1 1. 1 1, They might be read, one dollar, one dime, one cent and one mill, or, one dollar, eleven cents and one mill, or, one dollar, one hundred and eleven mills or thousandths. The place next to dollars, on the left, is eagles, and 11. may be read, 1 eagle and 1 dollar, or eleven dollars. Twenty-five eagles, 8 dollars, 4 dimes, 6 cents and 3 mills, may be written and read, decim. 25 8. 4 6 3 258. 46 3 258. 463 Eag. ∞.dol. Hence any sum in Federal Money may be regarded as a decimal, or mixed number, and may be managed in all respects as such. Federal Money is usually denoted by the character, $, placed before the figures; and in reading it, dollars, cents and mills are the only denominations usually mentioned. ADDITION OF FEDERAL MONEY. 133. RULE. The same as for the Addition of Decimals. (118) QUESTIONS FOR PRACTICE. 3. F holds a note against G for one hundred seven dollars and six cents, one against H for forty-nine dollars seventeen cts. and one against K for nine dollars ninety-nine cents; what is the amount of the three? Ans. $166.22. 4. A man bought 24 yards of broadcloth for $15.50, 6 yds. of lutestring for $5.25, 7 yds. of cambric for $5.25, · and trimmings to the amount of $4.12; what was the amount of the purchase? Ans. $30.72. MULTIPLICATION OF FEDERAL MONEY. 1 134. RULE.-The same as for the Multiplication of Decimals.(122) 6. At 16 cents a pound, what will 18 pounds of butter cost? what will 27 lbs. cost? 7. What is the cost of 126 bushels of rye, at 62 cents a bushel? Ans. $78.75. 8. What cost 87 bushels of oats at 33 cts. per bushel? at 41 cents? at 37 cents? at 25 cents? 9. If a person spend 61 cents a day, how much will that be a year? Ans. $22.814. 10. What cost 63 yards of calico, at a quarter of a dollar a yard? Ans. $15.75. 11. What cost 1758 pounds of tea at $1.15 per pound? Ans. $2021.70. SUBTRACTION OF FEDERAL MONEY. 135. RULE. The same as for the Subtraction of Decimals.(124) QUESTIONS FOR PRACTICE. 1. A man bought a pair of oxen for $76, and sold them again for $81.75; how much did he gain? Ans. $5.75. 2. Take 1 mill from $100, what remains? 3. I bought 5 yds. of cloth at $5 a yard, and paid six 5 dollar bills; who must receive change, and how much? 4. A man bought 100 lbs. of wool at 33 cents a pound, and sold the whole for $31.494 how much did he lose? 5. A person having $200, lost 2 dimes of it; how much had he left? 6. A person bought 24 yds. of cloth at $1.50 per yard, and paid $26.55, how much remains unpaid? Ans. $9.45. 7. I bought 6 yards of cloth at 76 cents a yard, and gave a 5 dollar bill; how much change must I receive? 8. How much must be added to 83 cents to make it $5? DIVISION OF FEDERAL MONEY. 136. RULE. The same as for the Division of Decimals. (128) QUESTIONS FOR PRACTICE. 1. If 24 lb. of tea cost $7.92 what is that a pound? Ans. $0.33. 2. If 125 bushels of wheat cost $100.25, what is it a bushel? 3. Six men, in company, uy 27 bush. of salt, at $1.67 a bushel, what did each man pay, and what was each's share of the salt? Ans. $7.515, and his share 4 bush. 4. If $1268 be divided equally among 15 men, what will each receive? Ans. $84.53. 5. A man dies leaving an | $78, how much is that an estate of $35000; the demands acre? against the estate are $1254. 65; the remainder, after deducting a legacy of $3075, is divided equally among his 6 sons; what is each son's share? Ans. $5111.725. 6. If 12 acres of land cost 7. Divide $7 between 9 men,what is each man's share? Ans. $0.777. 8. $12-how much? Ans. $0.006. 9. $81928+5=how much? MISCELLANEOUS QUESTIONS. 1. From 2 take 0.16289. Ans. 1.83711. 2. At 12 cents a pound, what will 87 lb. of butter cost? Ans. $10.874. 3. If a person spend $100 a year, how much is that a day? Ans. $0.273. 4. How much sugar at 12 cents a pound can be bought for $15.50? Ans. 124 lb. 5. A owes B $15.58, and is to pay him in rye at 67 cents a bushel, how much rye will be required to pay the debt? Ans. 23.25 bu. 6. If buttons be 9 cents a dozen, what are they a piece? Ans. $0.0075. 7. The President of the United States receives $25000 a year; how much is that a day? Ans. $68.493. 8. A man buys a chest of tea weighing 40 lb. for $35; at what price per pound must he sell it to gain $10 on the whole? Ans. $1.25. 9. If 6s. make one dollar, how many dollars in 45s.? $7.50. |