Ans. 99.999. Ans. 72.927. 10. From 100 take .001. 12. From three hundred sixty-five take forty-seven ten thousandths. Ans. 364.9953. 13. From three hundred fifty-seven thousand take twentyeight and four thousand nine ten millionths. 14. From .875 take .4. 15. From .3125 take .125. 20. From .666 take .041. Ans. 356971.9995991. MULTIPLICATION OF DECIMALS. ART. 185. Ex. 1. Multiply 18.72 by 7.1. OPERATION. 18.7 2 7.1 1872 13104 132.9 12 Ans. .475 Ans. .1875 Ans. .51. Ans. 1.9. Ans. 5.4375. Ans. 7.875. Ans. .625. Ans. 132.912. We multiply as in whole numbers, and point off on the right of the product as many figures for decimals as there are decimal figures in the multiplicand and multiplier counted together. = The reason for pointing off decimals in the product as above will be seen, if we convert the multiplicand and multiplier into common fractions, and multiply them together. Thus, 18.72 187 = 1872; and 7.17173. Then 1872 × 78=132812 = 132100 = 132.912, Ans., the same as in the operation. Ex. 2. Multiply 5.12 by .012. OPERATION. 5.12 .012 1024 512 1000 Since the number of figures in the product is not equal to the number of decimals in the multiplicand and multiplier, we supply the deficiency by placing a cipher on the left hand. The reason of this process will appear, if we perform the question thus: 5.12=5% = 18, and .012180. Then = 12 X 10 100000 = .06144, Ans., the same as before. Hence we deduce the following .0 6 14 4 Ans. QUESTIONS. 512 Art. 185. In multiplication of decimals how do you point off the product? Will you give the reason for it? When the number of figures in the product is not equal to the number of decimals in the multiplicand and multiplier, what must be done? RULE.-Multiply as in whole numbers, and point off as many figures for decimals, in the product, as there are decimals in the multiplicand and multiplier. If there be not so many figures in the product as there are decimal places in the multiplicand and multiplier, supply the deficiency by prefixing ciphers. NOTE. When a decimal number is to be multiplied by 10, 100, 1000, &c., remove the decimal point as many places to the right as there are ciphers in the multiplier; and if there be not figures enough in the number, annex ciphers. Thus, 1.25 X 10 = 12.5; and 1.7 × 100: : 170. Proof. The proof is the same as in multiplication of simple numbers. EXAMPLES FOR PRACTICE. 3. Multiply 18.07 by .007. Ans. .12649. Ans. 18.58922. Ans. .00000114. Ans. 1137. Ans. 2.20947. Ans. .00046967. Ans. 22.09. 10. Multiply eighty-seven thousandths by fifteen millionths. Ans. .000001305. 11. Multiply one hundred seven thousand, and fifteen ten thousandths by one hundred seven ten thousandths. Ans. 1144.90001605. 12. Multiply ninety-seven ten thousandths by four hundred, and sixty-seven hundredths. Ans. 3.886499. 13. Multiply ninety-six thousandths by ninety-six hundred thousandths. Ans. .00009216. Ans. 1. 14. Multiply one million by one millionth. Ans. .14. 16. Multiply one hundred one thousandths by ten thousand one hundred one hundred thousandths. Ans. .01020201. 17. Multiply one thousand fifty, and seven ten thousandths by three hundred five hundred thousandths. Ans. 3.202502135. 18. Multiply two million by seven tenths. Ans. 1400000. QUESTIONS. What is the rule for multiplication of decimals? What is the proof? How do you multiply a decimal by 10, 100, 1000, &c. ? 19. Multiply four hundred, and four thousandths by thirtyand three hundredths. Ans. 12012.12012. 20. What cost 46lb. of tea at $1.125 per pound? Ans. $51.75. 21. What cost 17.125 tous of hay at $18.875 per ton? Ans. $323.234375. 22. What cost 181b. of sugar at $0.125 per pound? Ans. $2.25. 23. What cost 375.25bu. of salt at $0.62 per bushel? Ans. $232.655. DIVISION OF DECIMALS. ART. 186. Ex. 1. Divide 45.625 by 12.5. Ans. 3.65. OPERATION. We divide as in whole numbers, and 12.5) 4 5.6 2 5 (3.65 since the divisor and quotient are the 375 812 750 625 625 two factors, which, being multiplied together, produce the dividend, we point off two decimal figures in the quotient, to make the number in the two factors equal to the product or dividend. The reason for pointing off will also be seen by performing the question with the decimals in the form of common fractions. Thus, 45.625 = 625 45625, and 12.5 = 121: 451000 = 45625 1000 before. 1000 10 125 = $250 = Ex. 2. Divide .175 by 2.5. OPERATION. 2.5).175 (.07 175 We divide as in whole numbers, and since we have but one figure in the quotient, we place a cipher before it, which removes it to the place of hundredths, and thus makes the decimal places in the divisor and quotient equal to those of the dividend. The reason for prefixing the cipher will appear more obvious by solving the question with the decimals in the form of common fractions. Thus, .175=100%, and 2.5 = 21% = 25. Then 1% ÷ 25 = 25000 τόσ = .07, Ans., as before. Hence QUESTIONS. Art. 186. In division of decimals how do you point off the quotient? What is the reason for it? If the decimal places of the divisor and quotient are not equal to the dividend, what must be done? RULE. Divide as in whole numbers, and point off as many decimals in the quotient as the decimals in the dividend exceed those of 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. 1 NOTE 2. When there is a remainder after dividing the dividend, ciphers may be annexed, and the division continued, the ciphers thus annexed being regarded as decimals of the dividend; to indicate in any case that the division does not terminate, the sign plus (+) can be used. NOTE 3. - When a decimal number is to be divided 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 1.25 10.125; and 1.7 100= .017. Proof. The proof is the same as in division of simple numbers. 19. Divide one hundred forty-seven, and eight hundred twenty-eight thousandths by nine and seven tenths. Ans. 15.24. 20. Divide seventy-five and sixteen hundredths by five, and forty-two thousand eight hundred one hundred thousandths. Ans. 13.846+. QUESTIONS. What is the rule for division of decimals? What is note 1 ? Note 2? Note 3? What is the proof? 21. Divide six hundred seventy-eight thousand seven hundred sixty-seven millionths, by three hundred twenty-eight thousandths Ans. 2.069+. REDUCTION OF DECIMALS. ART. 187. To reduce a common fraction to a decimal. Ex. 1. Reduce § to a decimal. OPERATION. 8) 5.0 (6 tenths. 8) 20 (2 hundredths. Ans. .625. Since we cannot divide the numerator, 5, by 8, we reduce it to tenths by annexing a cipher, and then dividing, we obtain 6 tenths and a remainder of 2 tenths. Reducing this remainder to hundredths by annexing a cipher, and dividing, we obtain 2 hundredths and a remainder of 4 hundredths, which being reduced to thousandths by annexing a cipher, and then dividing again, gives a quotient of 5 thousandths. The sum of the several quotients, .625, is the answer. 8) 40 (5 thousandths. Or thus: 8) 5.0 0 0 .625 To prove that .625 is equal to §, we change it to the form of a common fraction, by writing its denominator (Art. 176), and reduce it to its lowest terms. Thus, 1000 = ğ, Ans. Hence the following nator. 625 RULE. Annex ciphers to the numerator, and divide by the denomiPoint off in the quotient as many decimal places, as there have been ciphers annexed. to a decimal. Ans. .875. Ans. .4375. Ans. .235294+. Ans. .363636+. Ans. .416666+. NOTE. In reducing a common fraction to a decimal, when the denominator contains other prime factors than 2 and 5, there cannot be an exact division of the numerator; but, on continuing the division, some figure or figures of the quotient will be continually repeated. A decimal, of which there is a continual repetition of the same figure or figures, is called an infinite or circulating decimal. The figures that repeat are called repetends. When the repetend is pre QUESTIONS. Art 187. How do you reduce a common fraction to a decimal? How can you prove the answer correct? What is the rule for reducing a common fraction to a decimal? |