CASE 7. To find the least common multiple of two or more numbers. Rule. When there are only two numbers-Divide one of the given numbers by the greatest common measure of the two numbers, and multiply the other number by the quotient. When there are more than two numbers-Divide any two or more of the given numbers by their greatest common divisor. Use their quotients in their places, find again the greatest common divisor of two or more of these numbers, divide as before, and so continue the divisions until there are not two numbers that have a common measure; then multiply the common measures and the remaining numbers together, and the product will be the least common multiple. N. B. When one of the given numbers is a multiple of the other, or of the rest, it is evidently the least common multiple. When any of the less numbers are contained in a greater, they may be rejected from the calculation. The least common multiple can frequently be found by taking in succession the multiples of the greatest number, and using the lowest multiple that contains the other or the rest. Example 1. To find the least common multiple of 12 and 16. 4 is the greatest common measure. 124 X 16 48 the least common multiple. Example 2. To find the least common multiple of 9, 8, 7, 6, 4. 4) 9 8 7 6 4 4 × 3 × 2 × 3 × 7504 the least common multiple. CASE 8. To reduce fractions of different denominators to similar fractions, or fractions having the same denominator. Rule 1. Multiply both terms of every fraction by the denominators of the other fractions. Rule 2. Find the least common multiple of the denominators, and multiply both terms of each fraction by the quotient produced by dividing the common multiple by the denominator of that fraction. Observe, that when the same denominator occurs twice or more it may be used only once in finding the least common multiple. 3. Reduce, %, and 7 to similar fractions. 11 CASE 9. To find the value of a fractional quantity. Rule. Consider the numerator of the fraction as so many integers, and divide it by the denominator, reducing it into the lower denominations, as in Simple Division of quantities. 3. d., s. 14 31 § d. d., 3 d., 9 d. 16 lb., 12 lb. 71⁄2 oz., 14 oz. 10 drs. 4. 2 qr. 13 n., 3 qr. 23 n., 960 yds. This is all that is usually necessary to be expressed, but the work may be expressed thus: £20 s. × 5 ÷ 11 = 9 s. 1 d. CASE 10. To reduce a simple or compound quantity into the fraction of another quantity. Rule. Reduce if necessary the two quantities into the lowest denomination contained in either; then make the number of the quantity to be reduced, the numerator, and the other number the denominator, of the fraction required. Ex. To reduce 5 s. 9 d. to the fraction of a pound sterling. Ex. 1. Reduce 2 s. 6 d., 7 s. 10 d., and 9 s. 7 d. to fractions of a £. Ex. 2. Reduce 2 qr., 3 qr. 7 lb., and 1 qr. 24 lb. to fractions of a cwt. PRODUCTS. Ex. 1. £, . £, 77. £. Ex. 2. cwt., 13 cwt., 13 cwt. CASE 11. To reduce the fraction of one integer into the fraction of another integer. Rule. Express the integer of which the fraction is given as either the fractional part or the multiple of the integer in which the fraction is to be found; then prefix to it the given fraction in the form of a compound fraction, and find its value as a simple fraction. Ex. To reduce of a shilling into the fraction of a £, penny. and a Here, as 1 shilling is 1-20 th of a £, and 12 times 1 penny, 3-8 ths of a shilling are 3-8 ths of 1-20 th of a pound, or 3-8 ths of 12 times 1 penny. Ex. 1. Reduce s., s., and s. to the fraction of a penny. 2. Reduces., TT s., and 43 s. to the fraction of a £. FRACTIONAL ADDITION. Rule. Reduce the given fractions, if they require it, to similar fractions, or fractions having the same denominator; then add the numerators together for the numerator of the fraction of the amount, having for its denominator the common denominator. Observe. If the fractions are mixed with whole numbers, find the amount of the fractions, and if the amount is equal to or greater than unity, reduce the fraction to its equivalent whole or mixed number, and add the integers to the given whole numbers. If the given fractions are either of different integral quantities, or if an integer that can be separated into lower denominations, the values of the given fractions may be used for addition. Ex. 2. Add 4 £, 61 £, and 7% £, together. * In the addition of two aliquot parts it is to be noticed, that the numerator of their sum is the amount of their denominators; and the denominator, is the product of the same. |