In most cases the common denominator can be found by inspection. Ex. 1. Reduce and to equivalent fractions having the same denominator? Here,, and 3=2; therefore and Ans. In this operation we multiply the terms of each fraction by the same number; therefore we do not alter the value of any of the fractions used. The multiplier for the terms of any fraction will always be the quotient obtained by dividing the least common multiple by the denominator of that fraction. In the fractions and 3, 6 is the least common multiple of 2 and 3; then 62-3, the multiplier for ; and 6÷÷3=2, the multiplier for . Then 1x3 , and 1x2 3X2 , as before. Ex. 2. Reduce 1, 1, 8, 11, fs, to others having a common denominator? 7 4)2 4 8 16 18 90 2)21 2 4 18 144 99 Here we write the denominators in a line, and find their least common multiple is 144, which is the denominator of their equivalent fractions. We find the multipliers thus:-144-2, the denominator of the first, 72, the numerator of the equivalent to; again, 1444 the least common multiple. =36. For the third, we say 144 1404 1 1 2 9 Then 4×2×2×9=144, 8-18, and 18 x 5-90, the numerator of the equivalent to §; and so on. Reduce the following to their equivalents having the same denominator : : 89. To express one quantity as the fraction of another of the same kind. RULE.-10. Write the former as numerator, the latter as denominator. 2o. If they are of different denominations, they must be reduced to the same denomination. It is always proper to reduce fractions as simple as possible in the answer to their lowest terms-improper fractions to mixed numbers, &c. Ex. 1. What fraction of 18s. is 6s. ? Here } Ans. Ex. 2. What fraction of 2s. is 21d.? Reduce to the fraction of a pound— 1. 17s. 6d., 15s. 9d., 1s. 10d., 15s. Reduce to the fraction of a shilling — 3. 9d., 7 d., 2 d., 41d., 34d. Reduce to the fraction of a cwt. 4. 2qr. 14lb., 3qr. 14lb., 1qr. 7lb., 2qr. 211b. Reduce to the fraction of an acre 5. 2r. 20p., 3r. 10p., 1r. 30p., 3r. 25p. ADDITION. CASE I. 90. When the fractions have a common denominator. RULE.—Add the numerators together and under their sum write the common denominator. Ex. 1. Add 3 and 4? Here we say 2+3=5, and write 4 the common denominator under it; 2=1 Ans. cause 1+1+1, and 4=1+1; we have 3+4=1+1+1+1+1 =1; compare this with the illustration. The fractions to be added should always be written as in 88, Ex. 2. 1. 1+ 1+ 1, 3+ 4+ 3, 3+ $+ $+ 4 2. ++, }}+}+í, 3+ §+ 3+ } 3. At&tú+fï, 13+}+18+13 4. 23+23+}}+{} CASE II. 91. When the fractions to be added have not a common denominator. RULE.-10. Reduce them to others equal in value, and having a common denominator. 2o. Add these others as in CASE I. Ex. 1. To add and ? We bring them to equivalent ones having 11 Ans. the same denominator, and then add these equivalents. The reason is evident from CASE I., and 88, = of a pound 10s.; of a pound 25s. £1 5s., or £11. = 15s; then 15+10= Ex. 5. 1+ + 1, 3+ 4+ §,3+}+ § 6. ++ 4, 3+11+ §, ŝtí+1 7. §+16+32, 11+ 8+18, §+3+ 8+18 8. 72+13+18+13, 8+ 3+38+ 9. 13+1+18+21, 3+ 2+ 3+ 8 10. &+ + 8+18, 18+11+13+} 92. To add mixed numbers. RULE.-Add the fractional parts by the rules given above, and carry the whole number of the sum to the addition of the whole 93. When the fractions have a common denominator. RULE. Take the numerator of the less from the numerator of the greater, and under the remainder write the common denominator. 94. When the fractions have not a common denominator. RULE. Bring them to equivalent ones having a common denominator, then proceed as in CASE I. Ex. 1. Find the difference between and ? Here, 1, and 4; then by 93,4 REASON. Since of a pound 5s., and of a pound = 4s. ; we have 5s. a crown= 3d.; then - 4s. = 1s. of a pound, as above. Also, of 1s. 3d., and of a crown 12d.; then 1s. 3d.-1s. RULE.-10. Subtract just as in pence and farthings. 2o. Take care to borrow one when the fraction of the smaller number is greater than the other fraction: this 1 must always be considered as divided into the number of units expressed by the common denominator. Ex. 1. From 33 take 1§? From 33-2+1 1g=1 Ans Here, as we cannot take § from , we borrow 1 from the 3, and call it ; then we say 5 from 8, and 3 remain; 3 and 3 are 6, that is, §; or, 5 from 11, and 6 remain; we proceed in the same manner in every case. It is necessary to observe that in subtracting pence and farthings we always say the 1 penny borrowed is 4 farthings, whereas in the subtraction of mixed numbers, we suppose the unit borrowed divided into parts the same kind as expressed by the fractions. Ex. 5. 17—91, 163–74, 54—25 6. 117-14, 10}s.—1gs., 127s.—3s. 7. 10211-1015, 947—19, 71)—} ANS. 71,817, 33 1011, 8118., 838s. 9147, 75, 70% MULTIPLICATION. 96. To multiply fractions. RULE.-Multiply the numerators together for the numerator of the product, and the denominators for its denominator. |