250. To find the least common multiple of two or more fractions. Ex. 1. What is the least common multiple of,, and 216? OPERATION. 3 3 33 2, 16, 216 = 2, 8, 18. Least common multiple of the numerators = 33 = 4 Ans. 84. Least com 84 mon multiple 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 33 81 as the least number that can be exactly divided by the given fractions. RULE. Reduce the fractions, if 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 common 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,, and § ? 3. Find the least number that 315, 77, and 51 will divide without a remainder. Ans. 152. 4. What is the least common multiple of 3, 9, and 1? 5. What is the smallest sum of money with which I could purchase a number of sheep at $2 each, a number of calves at $4 each, and a number of yearlings at $93 each? and how many of each could I purchase with this money? Ans. $112; 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 5 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 174 days; A 480m., B 640m., and C 720m. 7. How many times the least common multiple of 3, 43, and 5, is the least whole number that 32, 43, and 5 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, of a mile, and 7 of a gallon. 252. REDUCTION OF DENOMINATE FRACTIONS. 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 de nomination to a lower. Ex. 1. Reduce of a pound to a fraction of a penny. make a shilling, there will be 12 times as many pence as shillings, 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 Toʊ of a pound to the fraction of a farthing. 3. Reduce o of a pound troy to the fraction of a grain. 4. Reduce of a pound, apothecaries' weight, to the fraction of a scruple. 5. Reduce of a cwt. to the fraction of an ounce. 6. Reduce of a ton to the fraction of a pound. 7. What part of an inch is 8. What part of an inch is of an ell English? 'ggʊ of a mile? 9. Reduce 35816 of a league to the fraction of an inch. 10. Reduce 509550 of an acre to the fraction of an inch. 11. Reduce T2 of a tun of wine measure to the fraction of a quart. 12. What part of a pint is of a bushel? 13. What part of a minute is 14. Reduce 320 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 of a penny to a fraction of a pound. Since 12 pence make a shilling, there will be as many shillings as pence, ors.; and since 20s. make a pound, there will be as many pounds as shillings, or £. Ans. RULE. Divide 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? 3. What part of a pound is 4. Reduce of a scruple to the fraction of a pound. 5. Reduce weight. of an ounce to the fraction of a hundred 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 of an inch to the fraction of a mile. 9. Reduce 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 of an ounce? Ans. 255. To find the value of a fraction in whole numbers of lower denominations. Ex. 1. What is the value of of a £. Ans. 5s. 5d. 1far. = 1s. = = = 55S.; and since 12d., of a shilling is of ffd.=55d.; and, since 1d. 4far., of a penny of 4far. ffar.1far. Therefore, £. 5s. 5d. 1 far. This is equivalent to multiplying the numerator of the fraction by the numbers required to reduce it to successive lower denominations, beginning with the highest, and dividing each product by the denominator, as in the operation. 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 of a shilling? 3. What is the value of 7 of a guinea, at 28 shillings? 4. What is the value of of a cwt.? Ans. 3 d. Ans. 21s. 9d. 14far. Ans. 7oz. 17dr. 5. What is the value of of a lb. avoirdupois? 眚 6. What is the value of § of a lb. troy? Ans. 10oz. 13pwt. 8gr. 7. What is the value of of a lb. apothecaries' weight? Ans. 33 53 19 12 gr. 8. What is the value of § of an ell English? 9. What is the value of 1 of a mile? Ans. 2qr. 3na. Clin. 14. What is the value of of a hhd. of wine? 256. To find the value of whole numbers in a fraction of a higher denomination. Ex. 1. What part of a £. are 5s. 5d. 19far.? RULE. £. as the answer. 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. EXAMPLES. 2. Reduce 3d. to the fraction of a shilling. Ans. 3. Reduce 21s. 9d. 14far. to the fraction of a guinea. Ans. 7. |