7. What is the difference 11. From 1 take 1 hunbetween 999 and ninety-nine dredth. Rem. 0.99. hundredths? Rem. 998.01. 12. From.1000 take 1 thou 8. From *0.9173 subtract sandth. 0.2134. 13. How many are 71.01.9. From 742 take 195.127. 19.71? Rem. 546.873. 14. How many are 100% 10. From 9.005 take 8.728. 0.01? DIVISION OF DECIMALS. 22 Since any ANALYSIS. 125. 1. If 14.25 lb. of butter be divided into 3 equal shares, how many pounds will there be in each? Here we wish to divide 14.25 into two factors, one of 8)14.25(4.75 12 which shall be 3, and the other such a number as, multiplied by 3, (101) will produce 14.25. We first seek how many times 3 in 14, and find it 4 times, and 2 units over.' The 2 21 units being 20 tenths, we join them to the 2 tenths, making 22 tenths, and, dividing these by 3, the quotient is 0.7, and 15 0.1 over: but 0.1 being 0.10,(116) we join the 1 to the 5 15 hundredths, making 0.15, and dividing by 3, the quotient is 5 hundredths. The whole quotient then is 4.75 lb. Το prove that this is the true quotient, we multiply it by the divisor, 3, (4.75X3=14.25) and reproduce the dividend. dividend may be regarded as the product of the divisor and quotient taken as factors, (101) and since the product must have as many decimal places as are contained in both the factors,(121) it follows, that the number of decimal places in the divisor and quotient, counted together, must be just equal to the number of decimal places in the dividend. 126. 2. If 18 bushels of wheat be divided equally among 4 men, how much will each receive? 4)18(4.5 bu. Here we find that 18 bushels will give each man 4 bushels, 16 and that there will be 2 bushels left. We now add a cipher to the 2, which multiplying it by 10,(91) reduces it to 20 tenths, and dividing 20 tenths by 4, the quotient is 0.5; 20 each man will, therefore, receive 4.5 bushels. tion may be continued, and in pointing the result, the ciphers annexed are to be regarded as decimals belonging to the dividend. 127. 3. What is the quotient of 0.0084 by 0.42? Omitting the ciphers, we find 42 in 84 just 2 times; 0.42)0.0084 0.02 bu since there are 4 places of decimals in the divi84 Ans. dend, and only 2 in the divisor, there must be 2 places also in the quotient; we therefore place a čipher at the left of the 2 in the quotient, between it and the separatrix, to make up the deficiency. We see by this example, that if a quantity be divided by a decimal, the quotient will be larger than the dividend. Hence by RULE. 128. Write down the divisor and dividend, and divide as in whole numbers. Point off as many places for decimals from the right hand of the quotient, as the decimal places in the dividend exceed those in the divisor. NOTE 1.-If there are not so many figures in the quotient as the number of decimal places required, supply the deficiency by prefixing ciphers. 2.-Should the decimal places in the divisor exceed those in the dividend, make them equal by annexing ciphers to the latter. 3. Whenever there is a remainder after division, by annexing ciphers to it, one or more additional figures may be obtained in the quotient.(126.) QUESTIONS FOR PRACTICE. 4. In 68.43 hours, how ma- Let the pupil point the folny times 1.5 hours? lowing answers according to 1.5)68.43(45,62 Ans. the rule. 60 9. What is the quotient of 4263 by 2.5? Ans. 17052. 84 10. What is the quotient of 75 4.2 by 36? Ans. 116+ 11. What is the quotient of 93 3298 by 7.54? Ans, 437+ 90 12. What is the quotient of 43 by 5.4? Ans, 45, 30 NOTE.-- When the quotient is 30 not complete, it is denoted by pla cing the sign +- after it, in which 5. Divide 1 by 0.5. case more quotient figures may be Quot. 2.* obtained by annexing ciphers to the remainder 6. Divide 1 by 2. Quot. 0.5.* 13. u=how many? 7. Divide 7.02 by 0.18. Quot. 39. 14. =8.3 Ans, 8. Divide 0.0081892 by 21.75-16.75 0.347 Quot. 0.0236. 15. 9.31+5.09-1.758.46;3.58 =5. * These are called Reciprocals. VULGAR FRACTIONS CHANGED TO DECIMALS. ANALYSIS. 129. If we divide an apple equally between 2 boys, the part which each will receive will be $ an apple, or the quotient of 1 divided by 2; if we divide 1 apple between 3 boys, each will receive $, or the quotient of I djerided by 3. In like manner, if 3 apples be divided between 4 boys, 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 equiva. lent 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 t, which is the decimal expression for f. And again in the fraction 4, adding a cipher to 3, and dividing by. 4, the quorient 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)1.0010.04 Ans. 6. Change 44 to a decimal. 1.00 Ans. 0.7045+ 2. Change t, d, and if to 7. Change + to a decimal. equivalent decimals. Ans. 0.173+ Ans. J=0.5, 1=0.25, and I=0.75. 8. Change sio to a decimal. Ans. 0.002, 3. What is the decimal expression for is of a day? 9. Change 7t to a mixed Ans. 0.2 day. pumber. 4. Change ff of a rod to a 10. Change to a decimal decimal. 1:31. Having become familiar with the method of changing Fulgar 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 QUESTIONS FOR PRACTICE. 1. There are 3 pieces of 4. What is the product of cloth, one contains 45 yards, 24 by $? 24+0.5=12, Ans. one 3, and the other 54 yds. 5. In 28 rods how many how many yards in the yards, 55 yards being equal whole? to one rod? 43=4,5 5}=5,5, and 28X5.5=154 34=3.75 rods, Answer. 51=5.25 6. In 154 yards how many rods? Ans. 13.50=131 15=154 rds. +5,5=28 2. There are 4 boxes, each rods, Ans. of which contains 58 lb. of 7. What is the quotient of sugar; how many pounds in the whole? 12 by 1? 51-5,375. Ans. 21.5 lb. 12:=12+24, Ans. 3. A person having 17 By these examples it appears tons of hay, sold 67 tons; tiplication and increased by divis that a number is diminished by mulhow much had he left? ion, when the multiplier and divisor Ans. 10.925 tons. are fractions or decimals. FEDERAL MONEY. 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, 11, They might be read, one dollar, one dime, one cent and one mill, or, one dollar, eleyen 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 cagle and 1 dollar, or eleven dollars. Twenty-five eagles, 8 dollars, 4 dimes, 6 cents and 3 mills, may be written and read, es mills. 2} decim. 25 8. 4 6 3 258. 46 3 258. 463 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. 1. If I pay 4 dollars 62 3. F holds a note against G cents for a barrel of soap, 5 for one hundred seven dollars dollars 28 cents for a barrel and six cents, one against H of flour, and 10 dollars 8 cts, for forty-nine dollars sevenfor a barrel of pork, what do teen cts. and one against K I give for the whole? for nine dollars ninety-nine 4.62 cents; what is the amount of 5.28 the three? Ans. $166.22. 10.08 4. A man bought 24 yards Ans. $19.98-19 dollars of broadcloth for $15.50, 6 and 98 cents. yds. of lutestring for $5.25, 7 yds. of cambric for $5.25, 2. A owes B $78, C $46.27, and trimmings to the amount D$101:09, and E $28.16; of $4.12; what was the awhat is the amount of the four mount of the purchase? debts? Ans. $253,52. Ans. $30.72. MULTIPLICATION OF FEDERAL MONEY. 1 134. Rule.-—The same as for the Multiplication of Decimals.(122) QUESTIONS FOR PRACTICE. 1. What will 34 yards of 3. If a man purchase 4 cloth cost, at 37 cts. per yard? handkerchiefs at 62 cts. each, 0.37 8 yards ribbon at 17 cts. per 34 yard, and 5 yards of lace at 44 cents per yard; what is 148 the whole amount? 111 Ans. $6.04. 4. What will 47 pounds of $12.58 Ans. coffee cost, at 22 cents per 2. What will 156 yards of pound? Ans. #10.34. cloth cost, at $1.67 per yard? 5. What cost 59 dozen of Ans. $260.52 eggs, at 59 cents per dozen? |