OPERATIOX. = common nu coloo col clay colo 36, least common multiple of the numerators, 36 of 4 48, new denominator. (merator. 6 of 5 45, new denominator, Ans. 40, new denominator. We find the least common multiple of all the numerators, which is 36, for the common numerator; and to obtain the several new denominators we take such a part of the given denominators, respectively, as the common numerator, 36, is of each given numerator. Thus, both terms of each fraction being proportionably increased, its value is not changed. Rule.— Find the least common multiple of the given numerators for a common numerator. Take, for the new denominator of each fraction, respectively, such a part of its given denominator as the common numerator is of its given numerator. Note. — Compound fractions, or whole and mixed numbers, must be reduced to simple fractions, and all to their lowest terms, before finding the common numerator. EXAMPLES 2. Reduce , 4, 4, and $ to other fractions of equal value having a common numerator. Ans. 34, 34, 35, 31. 3. Change š, 24, and 14 to fractions having a common numerator. 4. A can travel round a certain island, which is 50 miles in circumference, in 445 days, B in 6 days, and C in 6 days. If they all set out from the same point, and travel round the island the same way, in how many days will they all meet at the point from which they started, and how many times will each have gone round the island ? Ans. They will meet in 320 days; A will have gone round the island 75 times ; B, 50 times; and C, 48 times. GREATEST COMMON DIVISOR OF FRACTIONS. 247. The greatest common divisor of two or more fractions is the greatest number that will divide each of them, and give a whole number for the quotient. 248. To find the greatest common divisor of two or more fractions. OPERATION. 100 240 4 } 45 Ex. 1. What is the greatest common divisor of 15, 25, and 51 Ans. 5 45, 23, 51 = 15, 29, = 12, , Greatest common divisor of the numerators = 4 | Greatest comLeast common denominator of the fractions required. Having reduced the fractions to equivalent fractions having the least common denominator, we find the greatest common divisor of the numerators 12, 100, and 240 to be 4. Now, since the 12, 100, and 240 represent forty-fifths, their greatest common divisor is not 4, a whole number, but 4 forty-fifths; therefore we write the 4 over the least common denominator, 45, and have as the answer. RULE. — Reduce the fractions, if necessary, to their least common denominator. The greatest common divisor of the numerators, written over the least common denominator, will give the greatest common divisor required. EXAMPLES. 2. What is the greatest common divisor of , $, s, and ? Ans. ts: 3. What is the greatest common divisor of 123, 91, and 81? 4. What is the greatest common divisor of છે, , and ? ? 3 Ans. et 5. What is the greatest common divisor of 34, 570, and 215? 6. A farmer has 33 bushels of corn, 67} bushels of rye, 707 bushels of wheat. He wishes to put this grain, without mixing, into the largest bags, each of which shall contain the same quantity. Required the number of bags and the quantity each will contain. Ans. The capacity of each bag, 33 bushels; and the number of bags, 51. 7. I have three fields; the first contains 7311 acres, the second 8841 acres, the third 13911 acres. Required the largest-sized house-lots of the same extent into which the three fields can be divided, and also the number of lots. Ans. Size of each lot, 74, acres ; number of lots, 41. LEAST COMMON MULTIPLE OF FRACTIONS. 249. The least common multiple of two or more fractions is the least number that can be divided by each of them, and give a whole number for the quotient. OPERATION - 843 mon = 4 Or, 250. To find the least common multiple of two or more fractions. Ex. 1. What is the least common multiple of , f6, and 216? Ans. 84 , 16, 216 = , , 1%. Least com Least common multiple of the numerators = 33 multiGreatest common divisor of denominators 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. — Recluce 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. 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 4, s, and ? 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 ļ, 4, 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 $ 44 each, and a number of yearlings at $ 9f 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 3} 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 Ans. 44 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 171 days; A 480m., B 640m., and C 720m. 7. How many times the least common multiple of 31, 43, and 51, is the least whole number that 31, 43, and 54 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. 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. To reduce a denominate fraction from a higher denomination to a lower. Ex. 1. Reduce oto of a pound to a fraction of a penny. Ans. d. 253. OPERATION. or 1 x 20 20 20 X 12 240 3 d. d. Ans. 640 640 8 Since 20s. make a pound. Or, d. Ans. there will be 20 times as 640 8 many shillings as pounds, 82 8 62475.; and since 12d. make a shilling, there will be 12 times as many pence as shillings, or 248d. gd. 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 tato of a pound to the fraction of a farthing. 3. Reduce gobo of a pound troy to the fraction of a grain. 4. Reduce qu'on of a pound, apothecaries' weight, to the fraction of a scruple. 5. Reduce gero of a cwt. to the fraction of an ounce. 6. Reduce goo 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 trosso of a mile ? 9. Reduce 380g of a league to the fraction of an inch. 10. Reduce 25094500 of an acre to the fraction of an inch. 11. Reduce 11'sz of a tun of wine measure to the fraction of a quart. 12. What part of a pint is zo of a bushel ? 13. What part of a minute is totoso 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 of a penny to a fraction of a pound. Ans. ato 3 OPERATION. 3 3 3 S. ; £. = 96 96 x 20 1920 1 £. Ans. 640 3 1 Since 12 pence make Or, £. Ans. 8 X 12 x 20 640 a shilling, there will be 12 as many shillings as 4 pence, or 65. ; and since 20s. make a pound, there will be zby as many pounds as shillings, or ało £. 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 ? 5. Reduce it of an ounce to the fraction of a hundredweight. 6. Reduce of a pound to the fraction of a ton. 7. Reduce ț 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. |