123, 124. DECIMALS. 8. Multiply 0.026 by 0:003. - 12. Twenty-five x 0.25 'Prod. 0.000078. | are how many ? 9. Multiply 125 by 0.008. 13. Seven + 117% 1.024 Prod. 1. =how many ? 10. Multiply 25.238 by 12.17.1 14. 128.75 + 144.25 x Prod. 307.14646. 0.06=16.38 Ans. 11. Multiply 5 thousand by 1 15. 0.004 + 0.0004 X 5 thousandths. Prod. 25. 0.00002=0.000000088 Ans. SUBTRACTION OF DECIMALS. ANALYSIS. 123. 1. What is the difference between 43.25 rods and 22.5 rods? We write down the nuinbers as for Addition, with the 43.25 largest uppermost. As there are no hundredths in the 22.5 subtrahend, we bring down the 5 hundredths. Pro ceeding to the 10ths, we are unable to take 0.5 from 0.2; Ans. 20.75 rods. we therefore borrow a unit from the 3 units, which be ing 10 tenths, we join 10 to the 2. making. 12 tenths ; from which we take 5 tenths, and write the remainder, 7 tenths, in the place of tenths below the line. The rest of the operation must be obvious. 2. From 24 hours take 18.75 hours, what remains ? Here, as we cannot take 5 from nothing. we borrow 24. 0.10 from the 4 units, or 400 hundredths; then taking 5 18.75 ( 0.05) from 0.10, the remainder is 0.05. The 400 huse dredths has now become 390 hundredths, or 39 tepths, or Ans. 5.25 h. 3.9; then 0.7 from 0.9 leaves 0.2, and so on. RULE. 124. Write down the numbers as in Addition of Decimals, observing to place the largest number uppermost. Beginning at the right, subtract as in Simple Subtraction, (99) and place the decimal point in the remainder directly under those in the given numbers. NOTE 1. When the numbers are all properly written, and the results correctly pointed, the decimal points will all fall in one vertical column, or directly under one another, both in Subtraction and Addition. NOTE 2-In numbers giren for Addition or Subtraction, the decimal places may all be made equal by annexing ciphers to a part of them,(116) without altering their value, and then all the decimals will express similar parts of a unit, or be of the same denomination QUESTIONS FOR PRACTICE. 3. A person bought 27.63 lb. | 7. What is the difference of cinnamon, and sold 19.814 between 999 and ninety-nine Ib. how much had he left ? hundredths ? Rem. 998.01. 27.63 8. From 0.9173 subtract 19.814 0.2134. 9. From 742 take 195.127. Ans. 7.816 lb. 10. From 9.005 take 8.728. 4. From 468.742 rods, take 11. From 1 take 1 hundredth. 76.4815 rods. Rem. 0.99. Rem. 392.2605. 12. From 1000 take 1 thou5. From 9 ft. take 0.9 ft. i sandth. what remains ? 13. How many are 71.01-Ans. 8.) ft. | 19.71 ? 6. From 2.73 take 1.9185. 14. How many are 100 Rem. 0.8115. 0.01 ? DIVISION OF DECIMALS. 12 21 15 15 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 3) 14.25 ( 4.75 which shall be 3, and the other such a number as, mul tiplied 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 units being 20 tenths, we join them to the tenths, making 22 tenths, and, dividing these by 3, the quotient is 0.7, and 0.1 over: but 0.1 being 0.10, (116) we join the I to the 5 hundredths, making 0.15, and dividing by 3, the quotient is 5 hundredths. The whole quotient then is 4.75 lb. To prove that this is the true quotient, we multiply it by the divisor, 3, (4.75*3=14.25), and reprocuce the dividend. Since any 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, ihat 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? Here we find that 18 bushels will give each man 4 4) 18 ( 4.5 bu. bushels, and that there will be 2 bushels left. We now 16 add a cipher to the 2, which multiplying it by 10, (91) reduces it to tenths, and dividing 20 tenths by 4, the quotient is 0.5; each man will, therefore, receive 4.5 bushels. Hence by annesing ciphers to the remainder of a division, the operation may be continued, and in pointing the result, the ciphers annexed are to be regarded as decimals belonging to the dividend. 20 127. 3. What is the quotient of 0.0084 by 0.42 ? Omitting the ciphers, we find 42 in 84 just 2 O. 42 )0.0084 (0.02 Ans. times; but since there are 4 places of decimalo 84 in the dividend, and only 2 in the divisor, there must be 2 places also in the quotient: we there. fore place a cipher at the left of the 2 in the quotient, between it and the separatrix, to make un the deficiency. We see by this example that if a quantity be divided by a decimal, the quotient will be larger than the dividend. 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 lecimal places in the divisor exceed those in the dividend, make them equal by aunexing ciphers to the latter. 3. Whenever there is a remander 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 ? Aos. 116. + 11. What is the quotient of 3298 by 7.54 ? Ans. 437+.. 12. What is the quotient of 43 by 5.4? Ans. 45. 30 NOTE.- When the quotient is not complete, it is denoted by plac ing the sign + after it, in which 5. Divide 1 by 0.5. case more quotient figures nay be Quot. 2. * obtained by annexing ciphers to the remainder. 6. Divide 1 by 2. Quot 0.5. * 13. 46.39=how many? 7. Divide 7.02 by 0.18. 9-27 · Quot. 39 74+13-45.5 8. Divide 0.0081892 by 21.75–76.75–8.3 Ang. 0.347. Quot. 0.0236. | 15. 9.3175,09–1.75–8.46--8.58 * These are called Reriprocals. !=.5. VULGAR FRACTIONS CHANGED TO DECI MALS. 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 re divide 1 apple between 3 boys, each will receive , or the quotient of i divided by 3. In like manner, if 3 apples he divided between 4 boys. each boy will receive a 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 d, for example, denotes hat l is divided by 2, but since 1 does not contain 2, the quotient is less han 1, and must therefore be expressed in parts of unity. Now if we add a cipher to the dividend, I, it becomes 10 tenths, (126); and 10 tentos ivided by 2, the quotient is 0.5. (125) Hence the decimal 0.5 is equivElent to d. Again, in the fraction { if we add a cipher to the 1, it becomes O 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 y 3, the quotient is 0.03, and thus may we go on as far as we please, geting by each additional cipher a 3 in the quotient, which is 10 times less han the preceding, as 0.333-t, which is the decimal expression for . And again in the fraction a. adding a ciplier 10 3, and dividing by 4, the uotient is 0.7, and 0.2 reniain; adding a cipher to 0.2, and dividing again Y 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 Eivide by the denominator, so long as there shall be a remaioSer, or until the decimal be obtained to a sufficient degree of exactness. The quotient will be the decimal required; and it nust consist of as many decimal places, as the number of cibhers annexed. If the quotient does not contain so many gures, make up the deficiency by prefixing ciphers.(127). .. QUESTIONS FOR PRACTICE. 1. What is the decimal | 3. What is the decimal expression for z's ? expression for of a day? 25 ) 1.00 (0.04 Ans. Ans. 0.2 day. 1.00 4. Change 11 of a rod to a decimal. | 5. What are of a month 2. Change }, }, and 3 to in decimals ? equivalent decimals. Ans. 0.375 mo. Ans. L=0.5, 1=0.25, and 6. Change it to a deci0.75. mali Ans. 0.7045 + 7. Change to a deci- ' 9. Change it to a mixed inal. Ans. 0.173.+ number. 8. Change zdo to a deci- i 10. Change i to a deci Ans. 0.002. mal. mal. 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 QUESTIONS FOR PRACTICE. 1. There are 3 pieces of 5. In 28 rods how many cloth, one contains 41 yards, 1 yards, 54 yards being equal one 33 and the other 51 yds. to one rod ? how many yards in the | 51= 5.5 and 28 x 5.5=154 whole ? rods, Ans. 41=4.5 6. in 154 yards how ma32=3.75 ny rods? 51=5.25 154.0 Ans. 13.50=131. 5.3=154-5.5== 28 rods, 2. There are 4 boxes, Ans. each of which contains 53 7. What is the quotient lb. of sugar; how many of 12 by 1? pounds in the whole ? 12.0 | 5f=5.375. Ans. 21.5 lb. 0.5=12:-0.5=24 Ans. 3. A person having 174 By these examples it ap. tons of hay, sold 67 tons ; pears that a number is diminhow much had he left ? isbed by multiplication and inAns. 10.925 tons. creased by division, when the 4. What is the product multiplier and divisor are fractions or decimals. of 24 by į? 24 x 0.5=12 Ans. Federal Money. 132. Federal Money is the established currency of the United States. Its denominations are all in a decimal or ten-fold pro portion, as exhibited in table ). page 31. The dollar is considered the unit money, and all the lower denominations |