QUESTIONS FOR PRACTICE. 2. What is the least com 3. What is the least num mon multiple of 3, 5, 8 andber which may be divided by 1. Reduce of a dollar and of a dollar to a common denominator. If each term of, the first fraction, be multiplied by 5, the denominator of the second, the becomes, and if each term of 2, the second, be multiplied by 2, the denominator of the first, becomes; then, instead ofand, we have the two equivalent fractions, and which have 10 for a common denominator. 2. Reduce, and & to a common denominator. (230), Multiplying the terms of by 24, the product of 4 and 6, the denominators of the other two fractions, becomes; again, multiply the terms of by 18, the product of 3 and 6, the denominators of the first and third fractions, becomes ; and lastly, multiplying the terms of by 12, the product of 3 and 4, the denominators of the first and second, becomes ; then instead of the fractions, and, we have the three equivalent fractions, 4, and 24, which have 12 for a common denominator. From a careful examination of the above, the reason of the following rule will be manifest. 240. To reduce fractions of different denominators to equivalent fractions having a common denominator. RULE.-Multiply all the denominators together for the common denominator, and each numerator by all the denominators except its own for the new numerators. 241. Ans. 14, 14 and 1. Ans. 금융, 공금물 and 공동률. 6. Express and of a dollar in parts of a dollar of the same magnitude. Ans. 18 and 25. TO REDUCE FRACTIONS TO THEIR LEAST COM- ANALYSIS. 1. Reduce,, §, and to their least common denomi nator. The common denominator found by the foregoing rule is a common multiple of the denominators of the given fractions, but not always the least common multiple, and consequently not always the least common denominator. The least common multiple of the denominators, 3, 4, 8 and 12 is 24 (238), which may be divided into thirds fourths, eighths and twelfths; for the new numerators we must therefore take such parts of 24 as are denoted by the given fractions; and this is done by dividing 24 by each of the denominators (248, 24-6, 24-3, and 4-2), and multiplying the quotients by the respective numerators, (8X1-8, 6x3=18, 3×5-15, and 2X11-22), and the new numerators (8, 18, 15 and 22) written over 24, the common denominator, give, 14, 1 and 2 for the new fractions, having the least possible common denominator. Hence, 242. To reduce fractions of different denominators to equivalent fractions having the least common denominators. RULE.-Reduce the several fractions to their least terms (235). Find the least common multiple of all the denominators for a common denominator. Divide the common denominator by the denominators of the several fractions, and multiply the quotients by the respective numerators, and the products will be the new numerators required. QUESTIONS FOR PRACTICE. 2. What is the least com- 3. What is the least com QUESTIONS FOR PRACTICE. 2. What part of a pound is f of a cwt.? 3X4X28 336_6 Ans. 2. What part of a cwt. is of a pound? 246. To reduce fractions to integers of a lower denomination, and the reverse.v ANALYSIS. 1. Reduce of a pound to shillings and pence. £3×20=60s. and 60s-7s.; but s.X128d., and 48d. 6d. Then £7s. 6d. Hence, 1. Reduce 7s. 6d. to the fraction of a pound. 7s. 6d. 90d. £120s. 240d.; then 7s. 6d.££. Hence, 248. To reduce integers to 247. To reduce fractions to integers of a lower denomina-fractions of a higher denomination. tion. RULE. Reduce the numer- RULE. Reduce the given ator to the next lower denom- number to the lowest denomiination, and divide by the de- nation mentioned for a numernominator; if there be a re-ator, and a unit of the higher mainder, reduce it still lower, denomination to the same for and divide as before; the sev- a denominator of the fraction eral quotients will be the an- required. swer. QUESTIONS FOR PRACTICE. 2. In of a day, how many bours? 3. In of an hour, how many minutes and seconds? 4. In & of a mile, how many rods? 5. In of an acre, how many roods and rods? 2. What part of a day are 8 hours? 3. What part of an hour are 6m. 40s. ? 4. What part of a mile are 120 rods? 5. What part of an aore are 1 rood and 30 rods? 249. ADDITION OF FRACTIONS. ANALYSIS. 1. What is the sum of 3 of a dollar and of a dollar? As both the fractions are 9ths of the same unit, the magnitude of the parts is the same in both-the number of parts, 3 and 4, may therefore be added as whole numbers, and their sum, 7, written over 9, thus, Z, expresses the sum of two given fractions. 2. What is the sum of of a yard and of a yard? As the parts denoted by the given fractions are not similar, we cannot add them by adding their numerators, 3 and 2, because the answer would be neither nor; but if we reduce them to a common denominator, becomes, and, (240). Now each fraction denotes parts of the same unit, which are of the same magnitude, namely, 24ths; their numerators, 8 and 9, may therefore be added; and their sum, 17, being written over 24 we have of a yard for the sum of 2 and 3 of a yard. 250. To add fractional quantities. RULE.-Prepare them, when necessary, by changing compound fractions to single ones (224), mixed numbers to improper fractions (218), fractions of different integers to those of the same (247,248), and the whole to a common denominator (240); and then the sum of the numerators written over the common denominator, will be the sum of the fractions required. |