& 1. Reduce,, and to their least common denominator. 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, (24-8, 2-6, 2-3, and =2) and multiplying the quotients by the respective numerators, (8× 1 =8, 6 × 3 =18, 3 × 5=15, and 2×11=22) and the new numerators (8, 18, 15 and 22) written over 24, the common denominator, give, 18, and 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. 246. To reduce fractions to integers of a lower denomination, and the reverse. ANALYSIS. 1. Reduce of a pound 1. Reduce 7s. 6d. to the to shillings and pence. fraction of a pound. £x20=68. and 687s. 6d. 90d. £1-208.➡ -74s. but s.X12-43d. |=240d. then 7s. 6d.=2l. and 48d. 6d. Then £= 78. 6d. Hence 247. To reduce fractions to integers of a lower denomination. RULE.-Reduce the numerator to the next lower denomination, and divide by the denominator; if there be a remainder, reduce it still lower and divide as before: the sev-| eral quotients will be the anawer. =£. Hence, 248. To reduce integers to fractions of a higher denomina tion: RULE-Reduce the given number to the lowest denomination mentioned for a numerator, and a unit of the higher denomination to the same for a denominator of the fraction required. 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 7, 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 and 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. Ans. 131. 5. What is the sum of of a week and of a day? 8. What is the sum of 4, }+2=H+Aw.-, and ? Ans. S 2d. 14h. Ans. 251. SUBTRACTION OF FRACTIONS. ANALYSIS. 1. What is the difference between of a dollar and fo of a dollar? evidently expresses 2 tenths more than 3 tenths; then is the difference. 2. What is the difference between 2 of a yard and of a yard ? Here we cannot subtract from, for the same reason that we could not add them,(49). We therefore reduce them to a common denominator, (,) and then the difference of the numerators, (9-8-1) written over 24, the common denominator, gives for the difference of the fractions. RULE.-Prepare the fractions as for addition,(250) and then the difference of the numerators written over the common denominator will be the difference of the fractions required. |