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 repetends: 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. 0.052631578947368421. 200434782608695652173913. 25=0.034482758620689655172413793i. 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 £. II. 6d. 3far. is therefore the same as 675d.; if we divide this by 12, it will become III. 15s. 6d. 3far.=15-5625s.; this divided by 20, gives for the decimal sought. The work may be more concisely done, as in the following We placed the different denominations above each other, so that the smallest denomination stood at the top; we then supposed ciphers annexed to the 3 farthings, and divided by 4, since 4 farthings make one penny, and the quotient, which must be a decimal, we placed at the right of the 6d; we next divided 6·75d. with ciphers annexed, by 12, because 12 pence make one shilling, and the quotient, which is also a decimal, we placed at the right of the 15s.; finally, we divided the 15.5625s. by 20, because 20 shillings make one pound. In dividing by 20, we cut off the cipher, and then divided by 2, observing to remove the decimal point one place to the left. We therefore have this RULE. Place the different denominations above each other, so that the lowest denomination may stand at the top; commencing at the top, annex ciphers, and divide each denomination by the number expressing how many of such denomination make a unit of the next higher denomination. The last quotient will be the decimal required. Repeat this Rule. EXAMPLES. 1. Reduce £8 5s. 2d. 1qr. to the decimal of a £. UCTION OF DENOMINATE DE 8.259375 of a £. 2. Reduce 3qr. 2na. to the decimal of a yard. OPERATION. 4/2 0.875 of a yard. 3. Reduce 1ft. 4in. to the decimal of a yard. OPERATION. 12/4 31-3333, &c 0-4444, &c. of a yard. 4. Reduce 376. 4oz. 8pwt. 1gr. Troy, to the decimal of a pound. Ans. 3.36684027777, &c., of a lb. 5. Reduce 3h. 30m. 10sec. to the decimal of a day. Ans. 0-145949074074, &c., of a day. 6. Reduce £3 5s. Od. 2far. to the value of a £. head. Ans. £3.252083333, &c. 7. Reduce 28 gallons of wine to the decimal of a hogsAns. 0.4444, &c., of a hogshead. 8. Reduce 4s 6d. to the decimal of a £. Ans. £0.2270833, &c. 9. Reduce 18s. 34d. to the decimal of a £. Ans. £0.915625. 10. Reduce 3 pecks, 5 quarts and 1 pint to the decimal of a bushel. Ans. 0.921875 of a bushel. 11. Reduce 11hr. 16m. 15sec. to the decimal of a day. Ans. 0.469618055, &c., of a day. 12. Reduce 20 rods, 4 yards, 2 feet and 6 inches to the decimal of a furlong. Ans. 0-521969696, &c., of a furlong. 13. Reduce 42m. 36sec. to the decimal of an hour. Ans. 071, of an hour. 14. Reduce 30 days, 3 hours, 27 minutes, 30 seconds, to the decimal of a year of 365-24224 days. Ans. 0.0827617, &c., of a year. 15. Reduce 5hr. 48m. 49.536sec. to the decimal of a Ans. 0.24224 of a day. day? CASE II. To find the proper value of denominate decimals. Find the value of 0·778125 of a £. |