ber thereof, they may both be expunged, and the other members continually multiplied (as by the rule) will produce the fraction required in lower terms. 6. Reduce of of to a simple fraction. Thus 2×5 4X7 =185 Ans. Ans. 1=14 7. Reduce of of of to a simple fraction. CASE VI. To reduce fractions of different denominations to equivalent fractions having a common denominator. RULE I. 1. Reduce all fractions to simple terms. 2. Multiply each numerator into all the denominators except its own, for a new numerator; and all the denominators into each other continually for a common denominator; this written under the several new numerators will give the fractions required. EXAMPLES. 1. Reduce to equivalent fractions, having a common denominator. 1 + 2 + 2=24 common denominator. Ans. 1975 888 and 123 to a common denominator. 6. Reduce and of 1 to a common denominator. Ans. 1 2 3 8 768 3456 The foregoing is a general Rule for reducing fractions to common denominator; but as it will save much labour to keep the fractions in the lowest terms possible, the following Rule is much preferable. RULE II For reducing fractions to the least common denominator. (By Rule, page 155) find the least common multiple of all the denominators of the given fractions, and it will be the common denominator required, in which divide each particular denominator, and multiply the quotient by its own numerator for a new numerator, and the new numerators being placed over the common denominator, will express the fractions required in their lowest terms. EXAMPLES 1. Reduce and to their least common denominator. 4)2 4 8 2)2 1 2 1 1 1 4×28 the least com. denominator. 82x1 4 the 1st. numerator. 84×3-6 the 2d. numerator. 8÷8x5=5 the 3d. numerator. These numbers placed over the denominator, give the answer equal in value, and in much lower terms than the general Rule, which would produce 12 11 18 2. Reduce and to their least common denomina, Ans. # # # 9 24 2 16 16 3. Reduce and to their least common denomiAns. 17 18 nator. 4. Reduce and to their least common denominator. Ans. 1 18 & CASE VII. To Reduce the fraction of one denomination to the fraction of another, retaining the same value. RULE. Reduce the given fraction to such a compound one, as will express the value of the given fraction, by comparing it with all the denominations between it and that denomination you would reduce it to; lastly, reduce this compound fraction to a single one, by Case V. EXAMPLES. 1. Reduce of a penny to the fraction of a pound. By comparing it, it becomes of 2 of 2 of a pound 2. Reduce 5 X 1 X I 6x 12 x20 5 1440 Ans. of a pound to the fraction of a penny, Compared thus, 40 of 20 of 1d. Then 5 X 20 X 12 3. Reduce of a farthing to the fraction of a shilling. Ans. s, 4. Reduce of a shilling to the fraction of a pound. Ans. T 9 5. Reduce of a pwt. to the fraction of a pound troy. Ans. 1336 6. Reduce of a pound avoirdupois to the fraction of a cwt. Ans. 1 cwt. 7. What part of a pound avoirdupois is of a cwt. Compounded thus of of 38}}j=3 Ans. 8. What part of an hour is of a week. Ans. 18=2 9. Reduce of a pint to the fraction of a hhd. Ans. 10. Reduce of a pound to the fraction of a guinea. Compounded thus, 4 of 20 of s. 4 Ans. 11. Express 5 furlongs in the fraction of a mile. Thus 5 of 11 Ans. 12. Reduce of an English crown, at 6s. 8d. to the fraction of a guinea at 28s. Ans. of a guinea. CASE VIII. To find the value of a fraction in the known parts of the integer, as of coin, weight, measure, &c. RULE. Multiply the numerator by the parts in the next inferior denomination, and divide the product by the denominator; and if any thing remains, multiply it by the next inferior denomination, and divide by the denominator as before, and so on as far as necessary, and the quotient will be the answer. NOTE. This and the following Case are the same with Problems II and III. pages 75 and 76; but for the scholar's exercise, I shall give a few more examples in each. EXAMPLES. 1. What is the value of 211 of a pound? 2. Find the value of of a cwt. Ans. 88. 9fd Ans. 3qrs. 3lb. 1oz. 124dr. 3. Find the value of 3 of 3s. 6d. 5. How much is of a hhd. of wine? 6. What is the value of 11⁄2 of a dollar? Ans. 7oz. 10dr. 7. What is the value of of a guinea? Ans. 5s. 71d. Ans. 18s. 8. Required the value of 17 of a pound apothecaries. Ans. 2oz. 3grs. Ans. £4 13s. 54d. of of of a hhd. of wine? Ans. 15gals. 3qts. CASE IX. To reduce any given quantity to the fraction of any greater denomination of the same kind. [See the Rule in Problem III. Page 75.] EXAMPLES FOR EXERCISE. 1. Reduce 12 lb. 3 oz. to the fraction of a cwt. Ans. 195 1792 2. Reduce 13 cwt. 3 qrs. 20 lb. to the fraction of a tun. Ans. 33 Ans. 3. Reduce 16s. to the fraction of a guinea. 4. Reduce 1 bhd. 49 gals. of wine to the fraction of a tun. Ans. 5. What part of 4 cwt. 1 qr. 24 lb. is 3 cwt. 3 qrs. 17 lb. 8 oz. Ans. ADDITION OF VULGAR FRACTIONS, RULE. REDUCE compound fractions to single ones; mixed numbers to improper fractions; and all of them to their least common denominator (by Case VI Rule II.) then the sum of the numerators written over the common denominator will be the sum of the fractions required. EXAMPLES. together. 1. Add 51 and of Then 51 reduced to their least common denominator by Case VI. Rule II. will become Then 132+18+14—1oa—630 or 65 Answer. 32 1 14 13 |