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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. — To multiply a decimal 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 places enough in the number, annex ciphers. Thus,
12.5; and 1.7 X 100 170. Proof. — The proof is the same as in multiplication of simple numbers.
EXAMPLES FOR PRACTICE.
1.25 X 10 =
3. Multiply 18.07 by .007.
Ans. .12649. 4. Multiply 18.46 by 1.007.
Ans. 18.58922. 5. Multiply .00076 by .0015.
Ans. .00000114. 6. Multiply 11.37 by 100.
Ans. 1137. 7. Multiply 47.01 by .047.
Ans. 2.20947. 8. Multiply .0701 by .0067.
Ans. .00046967. 9. Multiply 47 by .47.
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. 14. Multiply one million by one millionth. 15. Multiply one hundred by fourteen ten thousandths.
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.
185. 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 thirty, and three hundredths.
Ans. 12012.12012. 20. What cost 46lb. of tea at $ 1.125 per pound ?
Ans. $ 51.75. 21. What cost 17.125 tons 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.
186. Ex. 1. Divide 45.625 by 12.5.
1 2.5) 45.6 2 5 ( 3.6 5
We divide as in whole numbers, and since the divisor and quotient are the 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
45625, and 12.5 = 125 =
Then, 45.626 : 125 = 3.65, Ans., as before.
6 2 5
45 6 250
Ex. 2. Divide 175 by 2.5.
We divide as in whole numbers, and since we 2.5) .1 75 (.07 have but one figure in the quotient, we place a 175
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
16. Then 10 : X 1 =
170 .07, Ans., as before. Hence the following
1000 and 2.5
1 75 100
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.
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.
EXAMPLES FOR PRACTICE.
3. Divide 183.375 by 489.
Ans. .375. 4. Divide 67.8632 by 32.8.
Ans. 2.069. 5. Divide 67.56785 by .035.
Ans. 1930.51. 6. Divide .567891 by 8.2.
Ans. .069255. 7. Divide .1728 by 10.
Ans. .01728. 8. Divide 13.50192 by 1.38.
Ans. 9.784. 9. Divide 783.5 by 6.25.
Ans. 125.36. 10. Divide 983 by 6.6.
Ans. 148.939+ 11. Divide 172.8 by 1.2.
Ans. 12. Divide 1728 by .12.
Ans. 13. Divide .1728 by .12.
Ans. 14. Divide 1.728 by 12. ,
Ans. 15. Divide 17.28 by 1.2.
Ans. 16. Divide 1728 by .0012.
Ans. 17. Divide .001728 by 12.
Ans. 18. Divide 116.31 by 1000.
Ans. .11631. 19. Divide one hundred forty-seven, and eight hundred twenty-eight thousandths by nine, and seren tenths.
Ans. 15.24. 20. Divide seventy-five, and sixteen hundredths by five, and forty-two thousand eight hundred one hundred thousandths.
186. 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.
187. To reduce a common fraction to a decimal. Ex. 1. Reduce & to a decimal.
Since we cannot divide the nu8) 5.0 ( 6 tenths.
merator, 5, by 8, we reduce it to 48
tenths by annexing a cipher, and
then dividing, we obtain 6 tenths 8) 20 ( 2 hundredths. and a remainder of 2 tenths. Re16
ducing this remainder to hundredths by annexing a cipher, and dividing,
we obtain 2 hundredths and a re8) 40 ( 5 thousandths.
mainder of 4 hundredths, which be40
ing reduced to thousandths by anAns. .625. nexing a cipher, and then dividing Or thus: 8 ) 5.0 0 0
again, gives a quotient of 5 thou
sandths. The sum of the several .6 2 5
quotients, .625, is the answer. 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 ity lowest terms. Thus, 1636
RULE. Annex ciphers to the numerator, and divide by the denominator. Point off in the quotient as many decimal places as there have been ciphers annexed.
EXAMPLES FOR PRACTICE. 2. Reduced to a decimal.
Ans. .75. 3. Reduce š to a decimal.
Ans. .875. 4. What decimal fraction is equal to 16? Ans. .4375. 5. Reduce to a decimal.
Ans. .235294+. 6. Reduce A to a decimal.
Ans. .363636+. 7. Reduce to a decimal.
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 exart 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
187. How do you reduce a common fraction to a decimal? How can you prove the answer correct? The rule for reducing a common fraction to a decimal ?
ceded by another decimal, the whole is called a mixed repetend, and the part not repeating is called the finite part. To mark a repetend, a dot (.) is placed over the first and last of the repeating figures. Thus, the answer to example sixth, .36, is a repetend; and the answer to example seventh, .416, is a mixed repetend, of which the figure 6 is the repetend, and the figures 41 the finite part.
To change an infinite decimal to an equivalent common fraction, we write the repetend for the numerator, and as many nines as the repetend hus figures for the denominator. Thus, 36 = t; and the mixed repo tend, 416 = 418 = 888 =
100 A decimal other than a repetend is changed to the form of a common fraction, simply by writing the denominator under the given numerator. (Art. 176.) Thus, .75 = To
Tomo = qto. 8. Reduce .875 to a common fraction. 9. Change .4375 to the form of a common fraction. 10. Change 72 to a common fraction. 11. Change .i35 to a common fraction. 12. What common fraction is equivalent to.23562 ?
Ans. 33337 13. Change .093 to an equivalent common fraction.
Ans. ir Ans.
188. To reduce a denominate number to a decimal of a higher denomination. Ex. 1. Reduce 8s. 6d. 3far. to the decimal of a pound.
Ans. .428125. We commence with the 3far., which we re4/ 3.0 0
duce to hundredths, by annexing two ciphers;
and then, to reduce these to the decimal of a 1 2 6.7 5 0 0
penny, we divide by 4far., since there will be * as 2018.5 6 2 5 0 0
many hundredths of a penny as of a farthing,
and obtain .75d. Annexing this decimal to the .4 2 8 1 2 5 6d., we divide by 12d., since there will be it as
many shillings as pence; and then the 8s. and this quotient by 20s., since there will be to as many pounds as shillings, and obtain .428125£ for the answer.
RULE. · Divide the lowest denomination, annexing ciphers if necessary, by that number which will reduce it to one of the next higher denomination. Then divide as before, and so continue dividing till the decimal is of the denominatian required.
187. What is an infinite decimal ? A repetend? A mixed repetend ? How is an infinite decimal changed to the form of a common fraction ? 188. The rule for reducing a denominate number to a decimal of a higher denomination ?