To reduce fractions of different denominators to equivalent fractions having a common denominator. * MULTIPLY each numerator by all the denominators except its own, for the new numerators: and multiply all the denominators together for a common denominator. Note. It is evident, that in this and several other operations, when any of the proposed quantities are integers, or mixed numbers, or compound fractions, they must first be reduced, by their proper rules, to the form of simple fractions. * This is evidently no more than multiplying each numerator and its denominator by the same quantity, and consequently the value of the fraction is not altered. It is in many cases not only useful, but easy, to reduce fractions to their least common denominator. The rule is this: Find the least common multiple of all the denominators, and it will be the common denominator required. Then divide the common denominator by the denominator of each fraction, and multiply the quotient by the numerator-the several products will be the numerators; which are to be placed respectively over the common denominator for the answer. To find the least common multiple proceed thus: Divide by any numbers that will divide two or more of the given numbers without a remainder, and set the quotients, together with the undivided numbers, in a line beneath. Divide the second line, as before, and so on, until there are no two numbers, beginning with the lowest numbers, and only primes need be used, that can be divided; then the continued product of the divisors, quotients, and undivided numbers, will give the multiple required. Example 2 × 3 × 4 = 24 the common denominator. Therefore the equivalent fractions are 1, 1, and 1. Or the whole operation of multiplying may often be performed mentally, only 6 8 setting down the results and given fractions thus, 1, 3, 1, = 14, 19, 19, = 12 12 12, by abbreviation. 2. Reduce and § to fractions of a common denominator. 3. Reduce }, }, and to a common denominator. 4. Reduce §, 23, and 4 to a common denominator. Note. 1. When the denominators of two given fractions have a common measure, let them be divided by it; then multiply the terms of each given fraction by the quotient arising from the other's denominator. Ex. and, by multiplying the former by 7 and the latter by 5. 2. When the less denominator of two fractions exactly divides the greater, multiply the terms of that which has the less denominator by the quotient. Ex. and and, by multiplying the terms of the former by 2. 2 = 3. When more than two fractions are proposed, it is sometimes convenient, first to reduce two of them to a common denominator; then these and a third; and so on till they be all reduced to their least common denominator. Ex. and and } = and and } = 1; and ; and . CASE VII. To reduce complex fractions to single ones. REDUCE the two parts both to simple fractions; then multiply the numerator of each by the denominator of the other; which is in fact only increasing each part by equal multiplications, which makes no difference in the value of the whole. 323 17 17 2 And 24 = Also = = 34 = X Hence 48, 45, 48, 38, 38, 33 are the fractions required. It is of great importance that the student should be made familiar with this rule, both on account of the facility which it gives in actual reductions, and especially in the reductions that occur in algebraic fractions and equations. CASE VIII. To find the value of a fraction in parts of the integer. MULTIPLY the integer by the numerator, and divide the product by the denominator, by compound multiplication and division, if the integer be a compound quantity. Or, if it be a single integer, multiply the numerator by the parts in the next inferior denomination, and divide the product by the denominator. Then, if any thing remains, multiply it by the parts in the next inferior denomination, and divide by the denominator, as before; and so on as far as necessary; so shall the quotients, placed in order, be the value of the fraction required *. To reduce a fraction from one denomination to another. + CONSIDER how many of the less denomination make one of the greater; then multiply the numerator by that number, if the reduction be to a less name, but multiply the denominator, if to a greater. The numerator of a fraction being considered as a remainder, in division, and the denominator as a divisor, this rule is of the same nature as compound division, or the valuation of remainders in the rule of three, before explained. This is the same as the rule of reduction in whole numbers from one denomination to another. ADDITION OF VULGAR FRACTIONS. IF the fractions have a common denominator; add all the numerators together, then place the sum over the common denominator, and that will be the sum of the fractions required. *If the proposed fractions have not a common denominator, they must be reduced to one. Also compound fractions must be reduced to simple ones, and fractions of different denominations to those of the same denomination. Then add the numerators, as before. As to the mixed numbers, they may either be reduced to improper fractions, and so added with the others; or else the fractional parts only added, and the integers united afterwards. 1. To add and together. EXAMPLES. Here+}=13, the answer. * Before fractions are reduced to a common denominator, they are quite dissimilar, as much as shillings and pence are, and therefore cannot be incorporated with one another any more than these can. But when they are reduced to a common denominator, and made parts of the same thing, their sum, or difference, may then be as properly expressed by the sum or difference of the numerators, as the sum or difference of any two quantities whatever, by the sum or difference of their individuals. Whence the reason of the rule is manifest, both for addition and subtraction. Note 1. When several fractions are to be collected, it is commonly best first to add two of them together that most easily reduce to a common denominator; then add their sum and a third, and so on. Note 2. Taking any two fractions whatever, and 35, for example, after reducing them to a common denominator, we judge whether they are equal or unequal, by observing whether the products 35 × 11, and 7 × 55, which constitute the new numerators, are equal or unequal. If, therefore, we have two equal products 35 × 11 =7 × 55, we may compose from them two equal fractions, as 35=7, or %={• If, then, we take two equal fractions, such as 7 and 35, we shall have 35 x 11 = 7 × 55; taking from each of these 7 x 11, there will remain (35 −7) × 11 = (55 — 11) × 7, whence 35-7 we have =71, or 2=7. 55 11 In like manner, if the terms of were respectively added to those of 35, we should have 35+ 7 = 18 = 55+ 11 Or, generally, if = 11. a a ± c a = b d it may in a similar way be shown, that b Hence, when two fractions are of equal value, the fraction formed by taking the sum or the difference of their numerators respectively, and of their denominators respectively, is a fraction equal in value to each of the original fractions. This proposition will be found useful in the doctrine of proportions. 2. To add and together. 3+3=18+3=3313, the answer. 3. To add and 7 and of together. +7+ } of { = {+} + { = + % + 1 = 7 = 83. 4. To add and together. 5. To add and together. Ans. 12. Ans. 1. Ans. of }, and 9? 10. What is the sum of 3 of a pound and § of a shilling? Ans. 125s or 13s 10d 23q. 11. What is the sum of of a shilling and of a penny? Ans. 13d or 7d 1q. 12. What is the sum of 4 of a pound, and of a shilling, and of a penny? Ans. 3139 SUBTRACTION OF VULGAR FRACTIONS. PREPARE the fractions the same as for addition, when necessary; then subtract the one numerator from the other, and set the remainder over the common denominator, for the difference of the fractions sought. EXAMPLES. and . Here ៖ 1. To find the difference between 8. What is the difference between 2 of 5 of a pound, and of a shilling? Ans. 2037 or 17 8s 113d. 2100 MULTIPLICATION OF VULGAR FRACTIONS. * REDUCE mixed numbers, if there be any, to equivalent fractions; then multiply all the numerators together for a numerator, and all the denominators together for a denominator, which will give the product required. * Multiplication of any thing by a fraction, implies the taking some part or parts of the thing; it may therefore be truly expressed by a compound fraction; which is resolved by multiplying together the numerators and the denominators. Note. A fraction is best multiplied by an integer, by dividing the denominator by it; but if it will not exactly divide, then multiply the numerator by it. |