duct by 4 we add 1, since the product of the two rejected figures, 15, by 4, approximates to 1 hundred, which would require 1 to be carried; and so on, it being sufficient to increase the partial product only by such a number as approximates most nearly to that which would have been carried, provided the two rejected figures next to the figure of the multiplicand had been retained. 2. Multiply 325.701428 by .7218393, retaining only three places of decimals in the product. Ans. 235.104. 3. Multiply 56.7534016 by 5.376928, retaining only five places of decimals in the product. 4. Multiply 843.7527 by 8634.175, retaining only the integers in the product. Ans. 7285109. DIVISION OF DECIMALS. 274. Ex. 1. Divide 1.728 by 1.2. Ans. 1.44. opERatios. We divide as in whole numbers, 1.2 ) 1.7 2 8 ( 1.4 4 Ans. and, since the divisor and quotient 1 2 are the two factors, which, being mul tiplied together, produce the dividend, 5 2 we point off two decimal figures in the 4 8 quotient, to make the number in the two factors equal to the number in the product or dividend. The reason for pointing off will also be seen by performing the example with the decimals in the form of common fractions. Thus, 1.728 = lTWir = \m: and 1-2 = h\ = \l Then Hot + « = tttt X ft = = \H = hVff = 1-44, Ans. as before. 2. Divide 36.6947 by 589. Ans. .0623. Operation. "\Ve divide as in whole 589)3 6.6 947 (.0 623 Ans. numbers, and since we have 3 5 3 4 but three figures in the quo —— tient, we place a cipher be 1 3 D 4 fore them, and thus make the 117 8 decimal places in the divisor . - (• n and quotient equal to those \ Ly7 of the dividend. 1 ' b i ■piig reason for prefixing the cipher will appear more obvious by solvin"- the example in the form of common fractions. Thus, 36.6947 = 36^7 = »l and 589 = if*. Then « + *f* = JWof X A = tfffifo = = .0628, Ans. as before. Hence the following Rule. — Divide as in whole numbers, and point off as many figures in the quotient as the number of decimal places in the dividend exceeds the number in the divisor; but if there are not as many, supply the deficiency by prefixing ciphers. Note 1. — When the decimal places in the divisor exceed those in the dividend, make them equal by annexing ciphers to the dividend, and the quotient ,will be a whole number. Note 2.— When there is a remainder after dividing the dividend, ciphers (nay be annexed, and the division continued; the ciphers thus annexed being regarded as decimals of the dividend; and to indicate in any case that the division does not terminate, the sign plus (-f-) can be used. Proof. — The proof is the same as in division of whole numbers. Examples. 3. Divide 780.516 by 2.43. Ans. 321.2. 4. Divide 7.25406 by 9.57. Ans. .758. 5. Divide .21318 by .38. 6. Divide 7.2091365 by .5201. Ans. 13.861+. 7. Divide 56.8554756 by .0759. Ans. 749.084. 8. Divide 119109094.835 by 38123.45. Ans. 3124.3. 9. Divide 1191090.94835 by 3812345. 10. Divide 11910909483.5 by 38.12345. 11. Divide 11.9109094835 by 381234.5. 12. Divide 1191.09094835 by 3.812345. 13. Divide 11910909483.5 by .3812345. 14. Divide 1.19109094835 by 3.812345. 15. Divide .119109094835 by .3812345. 16. Divide 30614.4 by .9567. Ans. 32000. 17. Divide .306144 by 9567. Ans. .000032. 18. Divide four thousand three hundred twenty-two,and four thousand five hundred seventy-three ten-thousandths by eight thousand,and nine thousandths. Ans. .5403-)-. 19. How many yards of calico at $0.0775 per yard can be purchased for $ 10.85? 20. What costs 1 acre of woodland when 19.65 acres are sold for $ 982.50? Ans. $ 50. 21. Divide three hundred twenty-three thousand seven hundred sixty-five by five millionths. Ans. 64753000000. CONTRACTIONS IN DIVISION OF DECIMALS. 275. To divide a decimal by 10, 100, 1000, &c. Remove the decimal point as many places to the left as there are ciphers in the divisor, and if there be not figures enough in the number, prefix ciphers. Thus, 2.15 -r- 10 = .215; and 1.9 -r- 100 = .019. Examples. 1. Divide 31.675 by 10. 2. Divide 916.05 by 100. 3. Divide 7.0461 by 100000. 4. Divide 70.461 by 100000. 5. Divide 704.61 by 100000. 6. Divide 7046.1 by 100000. 7. Divide 70460 by 100000. 8. Divide .70400 by 100000. 9. Divide 196.5 by 1000000. Ans. .0001965. 10. If $3500 are paid for 1000 yards of broadcloth, what is it a yard? Ans. $ 3.50. 11. When $ 1025 are paid for 40 boxes of sugar, each containing 250 pounds, what is the cost of 1 pound? Ans. $0.1 Of 276° When the divisor contains many decimal places, and only a certain number of decimals are required to be retained in the quotient, the work may be contracted as follows:— First consider how many figures, in all, it is necessary for t,he quotient to contain. I'hen, by using the same number of figures from the left of the divisor, find the first figure of the quotient, and, instead of bringing down a neio figure from the dividend, or annexing a cipher to the remainder, reject a figure on th» right of the divisor at each successive division, and make thi other figures a divisor. In multiplying such a divisor by the quotient figure, observe to add to the product the number nearest to that which woidd have been carried if no figures had been rejected. 1. Divide 695.57270875 by 52.35775, and retain in the quotient three places of decimals. FIRST OPERATION. product by 1, -4- 1. 17 19952 0 325 Ans. 13.285. By inspection, it is evident that the first quotient figure will be of the order of tens, and therefore the quotient will contain two places of whole numbers; and as there are to be three places of decimals, it must contain five figures. Hence, we divide at first by five figures of the given divisor, counting them from the left toward the right, thus using the 52.357 and rejecting the figures, 75, on the right. In multiplying each contracted divisor by its quotient figure we increase the product by having regard to rejected figures, as in contracted multiplication of decimals (Art. 273). The nature and extent of the contraction will be seen by comparison with the common method as shown in the second operation, in which the vertical line cuts off the figures not required. Note. — When the given divisor does not contain as many figures as are required in the quotient, we must begin the division in the usual way, and continue till the deficiency is made up, after which begin the contraction. 2. Divide 4327.56284563 by 873.469, and retain five decimal places in the quotient. Ans. 4.95445. 3. Divide 252070.52Q751 by 591.57, and terminate the operation with four decimal places in the quotient. Ans. 426.1043. 149 2 1958 04087 210 4. Divide 70.23 by 7.9863, and retain in the answer four decimals. 5. Divide 12193263.1112635269 by 123456789, and let the quotient contain as many decimal places, plus one, as there will be integers in it. Ans. 9876.54321 REDUCTION OF DECIMALS. 277. To reduce a decimal to a common fraction. Ex. 1. Reduce .125 to its equivalent common fraction. Ans. OPERATION. .125 = rtfife = Mj = A = 4 Ans. Erasing the decimal point and supplying the denominator, which is understood, we have -fflfg, which reduced to its lowest terms equals \, the answer required. Rule. — Erase the decimal point, and write under the numerator its decimal denominator, and reduce the fraction to its lowest teiins. Examples. 2. Reduce .875 to a common fraction. Ans. {. 3. Reduce .9375 to a common fraction. Ans. 4. What common fraction is equivalent to .08125? Ans. r^j. 5. Change .00075 to the form of a common fraction. 6. Express 31.75 by an integer and a common fraction. Ans. 31f. 7. Express 96.024 by an integer and a common fraction. Ans. 96TJr. 8. Express 163.04 by an integer and a common fraction. 9. Express 1001.4375 by an integer and a common fraction. Ans. 1001J6. 10. Express 1457.222 by an integer and a common fraction. 11. Express 19678.36 by an integer and a common fraction. 12. Express 9163.8755 by an integer and a common fraction. Ans. 9163£J§£. 278. To reduce a common fraction to a decimal. Ex. 1. Reduce J to a decimal. Ans. .375. |