PROBLEM 4. 172. To divide one decimal fraction by another: RULE. Divide as in whole numbers, and point off as many figures for decimals in the quotient as the number of decimal places in the dividend exceeds those in the divisor. (a) If the number of figures in the quotient is less than the excess of decimal places in the dividend over those of the divisor, supply the deficiency by prefixing ciphers to the quotient. Ans. .0012. 4. Divide .000744 by .62. NOTE 1. The dividend is a product, the divisor and quotient being the factors (Art. 77); hence the rule for pointing the quotient. NOTE 2. The rule for determining the place of the point in the quotient may also be explained by changing the decimals to the form of common fractions and performing the division; thus, .625.256zz÷25=ff=2.5. NOTE 3. By attending to the relative size of divisor and dividend (Art. 83), we have another mode of fixing the place of the decimal point in the quotient; thus, 25; 2.5 = by dividing the dividend by 10 (Art. 161), we have To of the preceding quotient; dividing again by 10, .25= To of the 2d quotient; dividing again by 10, To of the 3d quotient. Now dividing the divisor by 10, .25 = 10 times the 4th quotient; dividing again by 10, 10 times the 5th quotient; and so on to any extent. 25=.025 = = = 2.5 = 172. Rule for dividing decimals? What is said of ciphers in the quotient? Reason of the rule for pointing the quotient? Second explanation? Third? 5. Divide 38.7425 by .25. 6. Divide .09936 by .276. 7. Divide .000975 by .15. 8. Divide 17.472 by .48. 9. Divide 234.7744 by 62.44. 10. Divide 58.794 by 12.3. Ans. 154.97. (b) If there are more decimal places in the divisor than in the dividend, the number may be made equal by annexing one or more ciphers to the dividend. The quotient will then be a whole number; thus, 4.5.18 4.50÷.18=25. 11. Divide 3647 by .125. = Ans. 29176. (c) If there is a remainder after all the figures of the dividend have been used, the division may be continued by annexing ciphers to the dividend. Each cipher annexed becomes a decimal place in the dividend. In some examples this operation may be continued until there is no remainder, but in others there will necessarily be a remainder, however far the operation may be continued, This latter class of examples gives rise to circulating decimals; thus, .7 ÷ .9.7777, etc. Again, .8÷.11.727272, etc. In the first of these examples, the figure 7 will be repeated perpetually, and in the second example, the figures 7 and 2 will be repeated in like manner. Whenever the remainder consists of the same figure or figures as any preceding dividend, the quotient figures will begin to repeat. It may be remarked, however, that, if the divisor contains no prime factors but 2's and 5's, the divison can always be continued until there shall be no remainder; but if there is any other prime factor in the divisor, the division can never be completed unless the same other factor is in the original dividend; for a 172. What shall be done when there are more decimal places in the divisor than in the dividend? What is done when there is a remainder? The cipher annexed is what? When can the division be completed? When can it not be completed? Why? dividend is not divisible by a divisor unless it contains all the factors of the divisor; whereas annexing ciphers to the dividend introduces no prime factor into it except 2's and 5's. 14. Divide .13 by 8. 15. Divide 7.2 by .16. 16. Divide 8.7 by .25. 17. Divide 3.6 by 7.5. NOTE 4. When a decimal is not complete, we sometimes place the sign +after it, signifying that there is a remainder. In Ex. 25, divide by the factors of 300, viz. 100 and 3; i. e. move the point two places to the left and then divide by 3. 26. Divide 3.6412 by 400. 27. Divide 56.427 by 8000. 28. Divide 36.49 by 600. Ans. .009103. 29. Divide three thousand eight hundred and fifty-three hundred-thousandths by thirty-two millionths. Ans. 1204.0625. 30. Divide eighty-four and eighty-four hundredths by fortyeight thousandths. PROBLEM 5. 173. To reduce a common fraction to a decimal. Ex. 1. Reduce X 100 to a decimal fraction. 300 = 75; and 75 100.75, Ans. If a number be multiplied by any number, and the product be divided by the multiplier, the quotient will be the multiplicand 172. For what is the sign + sometimes used? (Art. 84, c). Now, in the above example, is multiplied by 100 by annexing two ciphers to the numerator; the fraction 392 is then reduced to the whole number 75, and, finally, 75 is divided by 100 by placing the decimal point before the 75; ..=.75. Hence, RULE. Annex one or more ciphers to the numerator and divide the result by the denominator, continuing the operation until there is no remainder, or as far as is desirable. Point off as many decimal places in the quotient as there are ciphers annexed to the numerator. 2. Reduce to a decimal fraction. X 1000 - 3000 =375; and 375 ÷ 1000 —.375, Ans. 9. Reduce,,, 13, 74, 75, and 7 to decimals, Every decimal fraction is a common fraction, and, if its denominator be written, it will appear as such. It may then be reduced to lower terms, or modified like any other common fraction. This proves the rule in Art. 173. 10. Reduce .48 to the form of a common fraction and then to its lowest terms. .48, Ans. 11. Reduce .125 to its lowest terms. .125==&=&=, Ans. 12. Reduce .17 to the form of a common fraction. 13. Reduce .275, .325, .00025, and .00625. Ans. 2.8284, Ans. 15. Reduce 1.5, 3.75, 8.25, 9.125, and 2.0125. 173. Rule for reducing a common fraction to a decimal? Explanation? 174. Is a decimal also a common fraction? How is this made evident? How ay the rule in Art. 173 be proved correct? PROBLEM 6. 175. To reduce whole numbers of lower denominations to the decimal of a higher denomination. Ex. 1. Reduce 2pk. 3qt. to the decimal of a bushel. RULE. Having annexed one or more ciphers to the lowest denomination, divide by the number it takes of that denomination to make one of the next higher, and annex the quotient as a decimal to that next higher; then divide the result by the number it takes of THIS denomination to make one of the NEXT higher, and so continue till it is brought to the denomination required. 2. Reduce 9s. 6d. 3qr. to the decimal of a pound. NOTE. In dividing by 20 to reduce the decimal of a pound, and in all similar examples, we may point off the 0 in the divisor, and then divide by 2, but in such a case the point in the dividend must be moved one place toward the left, for by so doing both divisor and dividend are divided by 10, and .. the quotient is unchanged (Art. 84, b). 3. Reduce 2ft. 9in. 1b. c. to the decimal of a yard. 4. Reduce 3cwt. 2qr. 201b. 8oz. to the decimal of a ton. 175. Rule for reducing the lower denominations of a compound number to the decimal of a higher denomination? Principle? Mode of dividing when the divisor is 20, 40, etc.? When the divisor is a mixed number? |