and the fifth had what was left. How much did each 66 3d had 4336 1600 of 100=25 of 100-11 294. 336 1680 of 100-115 of 100=11 11 3. 336 4th had 1785 of 100-255 of 100-12 7 34 4336 5th had 7735 of 100-105 of 100=53 15 4 14336 VULGAR FRACTIONS REDUCED TO DECIMALS. 95. To change a vulgar fraction into an equivalent decimal fraction. Let us endeavor to change into an equivalent decimal fraction. This fraction is the same as of a unit; and as 10 tenths make a unit, the fraction is the same as 3 of 4o of a tenth, 3 tenths+ of a tenth. Again, of a tenth is the same as % of 10 of one hundredth,—7 hundredths+ of one hundredth. But of one hundredth is the same as of 10 of one thousandth, 5 thousandths. Therefore of a unit 3 tenths, 7 hundredths, and 5 thousandths, or as usually written, -0.375. Hence we deduce this RULE. Annex a cipher to the numerator, and then divide by the denominator. If the dividend will not contain the divisor, write O in the quotient and annex another cipher, and then divide; to the remainder annex another cipher, and again divide by the denominator; and so continue to do until there is no remainder, or until as many decimal figures have been obtained as may be desired. The quotient will be the decima. fraction required. NOTE. It will be seen that this rule bears a close analogy to rule under ART. 91, as it ought; since the values of the successive figures in a decimal fraction decrease in a tenfold ratio. EXAMPLES 1. What decimal fraction is equivalent to? 16)100(0.0625 40 32 80 80 2. Wha decimal is equivalent to ? Ans. 0.05555, &c. 3. What decimal is equivalent to? 4. What decimal is equivalent to? 5. What decimal is equivalent to ? 6 What decimal is equivalent to +? Ans. 0.05. Ans. 0.04. Ans. 0.3333, &c. Ans. 0.142857, &c. 7 What decimal is equivalent to ? Ans. 0-0909, &c. 8. What decimal is equivalent to ? Ans. 0·076923, &c. .9. What decimal is equivalent to ? Ans. 0.0588235, &c. 10. Change into an equivalent decimal. Ans. 0.75, into an equivalent decimal. 11. Change 16. Change into an equivalent decimal. Ans. 0.875. 17. Change 18. Change 19. Change into an equivalent decimal. Ans. 0.95. into an equivalent decimal. Ans. 0.98. into an equivalent decimal. Ans. 0.928571428, &c. In the foregoing process of converting a vulgar fraction into an equivalent decimal fraction, we continue to annex ciphers to the remainders, and to divide by the denominator of the vulgar fraction; hence, whenever we obtain a remainder like one that has previously occurred, then the decimal figures will commence a repetition. And as no remainder can exceed or equal the divisor or denominator of the vulgar fraction, the whole number of different remainders cannot exceed the number of units in the decominator less one; consequently, when the decimal figures do not terminate, they must recur in periods whose number of places cannot exceed the number of units less one in the denominator of the equivalent vulgar fraction. Decimals which recur in this way, are called repeiends. When the period begins with the first decimal figure, i is called a simple repetend. But when other decimal figures occur before the period commences, it is called a compound repetend. A repetend is distinguished from ordinary decimals by a period or dot placed over the first and last figure of the circulating period. 96. The following vulgar fractions give simple repor ends: +=0·3. +=0.i. =0·09. +=0·076923. =0·0588235294117647. =0·052631578947368421. =0·047619. =0·0434782608695652173913. 97. The following ones give compound repetends: =0.16. =0.0714285. t=0.06. =0·05. *=0·045. =0·0416. 98. Those simple repetends, which have as many terms, less one, as there are units in the denominator, we shall call perfect repetends. The following are some of the perfect repetends: +=0·142857. =0·0588235294117647. =0.05263157894736842i. =0·0434782608695652173913. 20-034482758620689655172413793i. NOTE.-For some interesting properties of repetends, see Higher Arithmetic. REDUCTION OF DENOMINATE DECIMALS. 99. A denominate decimal is a decimal fraction of a unit of a particular kind. Thus, 0·45 of a £, is a denominate decimal, since the unit is £1; for the same reason, 0.25 of a foot is a denominate decimal, the unit being 1 foot. What is a denominate decimal? Give some examples. CASE I. To reduce denominate numbers of different denominations to a decimal of a given denomination. Let it be required to reduce 15s. 6d. 3far. to the decimal of a £. I. 3far.=4d.=0·75d. II. 6d. 3far. is therefore the same as 675d.; if we divide this by 12, it will become 6.75 =0.5625s. 12 |