OPERATION 3 33 81 mon 250. To find the least common multiple of two or more fractions. Ex. 1. What is the least common multiple of 4, 66, and 216? Ans. 81 , 166, 216 = Least comLeast common multiple of the numerators 33 multiGreatest common divisor of denominators 4 (ple required. Having reduced the fractions to their simplest form, we find the least common multiple of the numerators, 3, 3, and 33, to be 33. Now, since the 3, 3, and 33 are, from the nature of a fraction, dividends, of which their respective denominators, 4, 8, and 16, are the divisors (Art. 216), the least common multiple of the fractions is not 33, a whole number, but so many fractional parts of the greatest common divisor of the denominators. This common divisor we find to be 4, which, written as the denominator of the 33, gives 3 81 as the least number that can be exactly divided by the given fractions. RULE. - Reiluce the fractions, ?f necessary, to their lowest terms. Then find the least common multiple of the numerators, which, written over the greatest common divisor of the denominators, will give the least common multiple required. Or, Reduce the fractions, if necessary, to their least common denominator. Then find the least common multiple of the numerators, and write it over the least cominon denominator. NOTE. The least whole number that will contain two or more fractions an exact whole number of times, is the least common multiple of their numerators. EXAMPLES. 2. What is the least common multiple of , S, and $ ? Ans. 44 = 24. 3. Find the least number that 316, 73, and 54 will divide without a remainder. Ans. 154 4. What is the least common multiple of ğ, , and 10? 5. What is the smallest sum of money with which I could purchase a number of sheep at $ 21 each, a number of calves at $ 41 each, and a number of yearlings at $9 each ? and how many of each could I purchase with this money? Ans. $ 1124; 50 sheep: 25 calves ; 12 yearlings. 6. There is a certain island 80 miles in circumference. A, B, and C agree to travel round it. A can walk 31 miles in an hour, B 4 miles, and C 54 miles. They start from the same point and travel round the same way, and continue their travelling 8 hours a day, until they shall all meet at the point from which they started. In how many days will they all meet, and how far will each have travelled ? Ans. In 17} days; A 480m., B 640m., and C 720m. 7. How many times the least common multiple of 33, 43, and 54, is the least whole number that 34, 43, and 51 will exactly divide. DENOMINATE FRACTION. 251. A DENOMINATE Fraction is one in which the unit of the fraction is a denomination of a compound number ; as, of a pound, g of a mile, and 7 of a gallon. REDUCTION OF DENOMINATE FRACTIONS. 252. REDUCTION of denominate fractions is the process of changing fractions from the unit of one denomination to that of another, without altering their value. 253. To reduce a denominate fraction from a higher enomination to a lower. Ex. 1. Reduce oto of a pound to a fraction of a penny. Ans. g d. OPERATION. Or, or 20 1 x 20 20 20 X 12 240 3 d. d. Ans. 640 Since 20s. make a pound, d. Ans. 640 there will be 20 times as 8 82 8 many shillings as pounds, 62475.; and since 12d. make a shilling, there will be 12 times as many pence as shillings, or 347d. = d. RULE. Multiply the given fraction by the same numbers that would be employed in the reduction of whole numbers to the lower denomination required. EXAMPLES. 2. Reduce tzoo of a pound to the fraction of a farthing. 3. Reduce gobo of a pound troy to the fraction of a grain. 4. Reduce zice of a pound, apothecaries' weight, to the fraction of a scruple. 5. Reduce goto of a cwt. to the fraction of an ounce. 6. Reduce gooo of a ton to the fraction of a pound. 7. What part of an inch is zig of an ell English ? 8. What part of an inch is TToggo of a mile ? 9. Reduce 780167 of a league to the fraction of an inch. 10. Reduce z5096550 of an acre to the fraction of an inch. 11. Reduce TI'm of a tun of wine measure to the fraction of a quart. 12. What part of a pint is são of a bushel ? 13. What part of a minute is azotogo of a year? 14. Reduce zoo of a hundred-weight to a fraction of an ounce. 254. To reduce a denominate fraction from a lower denomination to a higher. Ex. 1. Reduce g of a penny to a fraction of a pound. Ans. ato Or, 1 Since 12 pence make £. Ans. 8 x 12 x 20 640 a shilling, there will be is as many shillings as 4 pence, or os. ; and since 20s. make a pound, there will be zo as many pounds as shillings, or ało £. Ans. RULE. — Diride the fraction by the same numbers that would be employed in the reduction of whole numbers to a higher denomination. EXAMPLES. 2. What part of a pound is of a farthing? 5. Reduce 6 of an ounce to the fraction of a hundredweight. 6. Reduce of a pound to the fraction of a ton. 7. Reduce { of an inch to the fraction of an ell English. 8. Reduce 4 of an inch to the fraction of a mile. 9. Reduce į of an inch to the fraction of a league. 10. Reduce of an inch to the fraction of an acre. 11. Reduce { of a quart to the fraction of a tun, wine measure. 12. Reduce of a pint to the fraction of a bushel. 13. Reduce $ of a minute to the fraction of a year (3651 days). 14. What part of a hundred-weight is į of an ounce ? Ans. Fero 255. To find the value of a fraction in whole numbers of lower denominations. Ex. 1. What is the value of 1 of a £. Ans. 5s. 5d. 1. far. OPERATION 5s. 5 d. 3 £. 20 Since 1£. 20s, l of a £. is 55 of 20s. ils. 56 s.; and since ls. 12d., 1 of a shilling is of 12d. 1 2 ffd.=5,6d.; and, since 1d. 4far., * of a penny = Ú of 4far. 11) 60 d. (5 d. = {ffar. = 18 far. Therefore, £. 5 5 5s. 5d. 1 far. This is equivalent to multiplying the numerator of the fraction by the numbers required 4 to reduce it to successive lower de11) 2 0 far. (1far. nominations, beginning with the high est, and dividing each product by 11 the denominator, as in the operation. far. Ans. 5s. 5d. 13far. RULE. Multiply the numerator of the given fraction by the number required to reduce it to the next lower denomination, and divide the product by the denominator. Then, if there is a remainder, proceed as before, until it is reduced to the denomination required. EXAMPLES. 2. What is the value of jg of a shilling ? Ans. 3 d. 3. What is the value of 7 of a guinea, at 28 shillings? Ans. 21s. 9d. 1ffar. 4. What is the value of 11 of a cwt. ? 5. What is the value of of a lb. avoirdupois? Ans. 7oz. 17dr. 6. What is the value of g of a lb. troy? Ans. 10oz. 13pwt. 8gr. 7. What is the value of Az of a lb. apothecaries' weight? Ans. 33 53 19 12 15gr. 8. What is the value of g of an ell English ? Ans. 2qr. 3na. Clin. 9. What is the value of 1} of a mile ? Ans. 6fur. 30rd. 12ft. 8 in. 10. What is the value of of a furlong? Ans. 35rd. Oft. 2in. 11. What is the value of 15 of an acre ? Ans. 2R. 6rd. 4yd. 5ft. 127 in. 12. What is the value of it of a rod ? 13. What is the value of i'y of a cord ? Ans. 9ft. 1462 * in. 14. What is the value of g of a hhd. of wine ? Ans. 6gal. 2qt. 1 pt. 045gi. 15. What is the value of 3 of a hhd. of beer? Ans. 42gal. 16. What is the value of }} of a year (365} days)? Ans. 174d. 16h. 26m. 5. sec. 17. What is the value of 73 14 of a dollar? 4 OPERATION. fi£. Ans. 256. To find the value of whole numbers in a fraction of a higher denomination. Ex. 1. What part of a £. are 5s. 5d. 1 i far. ? We reduce the 3s. 5s. 5d. 14 far. 2880 5d. 1 Iftar. to elevenths 1£. 10560 of a farthing, the low est denomination in the question, for the numerator of the required fraction, and 1 £. to the same denomination for the denominator. We then have the fraction 3. £. as the answer. 11 RULE. — Reduce the given numbers to the lowest denomination mentioned in either of them. Then write the number which is the fractional part for the numerator, and the other number for the denominator, of the required fraction. 28.80 £. 10 560 EXAMPLES. Ans. It Ans. |