REDUCTION OF FRACTIONS. Case 1st.—To reduce fractions to their least terms; or to express them by the least whole numbers possible. Find the greatest common measure of the numerator and denominator, and divide both the terms of the fraction thereby, and the quotients will be the terms required, which will express the given fraction in its lowest terms. Therefore 3 is the greatest common measure. Though the rule already given is general for all fractions whatever, yet it will not be always necessary to find the greatest common measure, as many fractions may be reduced to their lowest terms by inspection, as in the first six examples; and whenever the sum of the digits of any term of a fraction, or any number whatever, can be divided by 3 or by 9, the whole term or number may be divided by 3 or 9. When both terms are even numbers, they may be each divided by 2; and if the sum of the digits of an even number can be divided by 3 or 9, this number may be also divided by 6 or 18. When the terms of a fraction, or any number, ends with a cipher or 5, the whole can be divided by 5; and if the sum of the digits of any number ending with a cipher or 5 can be divided by 3 or 9, this num ber may be also divided by 15 or 45. When the two last digits of any number can be evenly divided by 4, the whole can be divided by 4; and when the 3 last digits can be evenly dividedby 8, the whole may be divided by 8. By the proper application of this information, much labour, may be often saved.* Here the sum of the digits in each term of the fraction, is 9, therefore, the total of each term may be divided by 9; which being done, the fraction becomes:###, the terms of which not falling under any of the prescribed conditions, we may reasonably conclude the fraction is reduced to its lowest terms. The first rule, by the common measure, always reduces the fraction to its lowest terms; but when the terms of a fraction are large, and the divisors not such as have been specified, it is sometimes doubtful whether the fraction is in its lowest terms or not; it will be, therefore, advantageous to carry the reduction by the second rule, as far as it will go, and then apply the first rule to the result. Crample. Here we instantly perceive that 6 is a common measure, and dividing the terms of the fraction thereby, it becomes *; proceeding then by the first rule, we discover that 143 is the greatest common measure, and that # is the fraction in its lowest terms. Let the following fractions be reduced to their lowest terms each respectively. - * For a farther elucidation of those curious properties in numbers, and for a demonstration, I refer the reader to John Walker's Philosophy of Arithmetic, pages 25 to 29: 2. Case 2d.—To reduce a mixed number to its equivalent improper fraction. Multiply the integer by the denominator of the fraction, and to the product add the numerator, and the denominator being placed under this sum, will give the fraction required. Crample. Let 54 be reduced to its equivalent improper fraction. 54 7 Answer . . . . ** Reduce the following mixed numbers to their equivalent improper fractions. Case 3d.—To reduce an improper fraction to its equivalent whole or mixed number. Divide the numerator by the denominator for the integral part, and place the remainder, if any, over the numerator for the fractional part, and it will form the whole or mixed number required. (trample. Let 397 be reduced to its equivalent whole or mixed Reduce the following improper fractions to their equivalent whole or mixed numbers. Case 4th.--To reduce a whole number to an equivalent fraction, having a given denominator. Multiply the whole number by the given denominator, and under the product thereof, place the same denominator, and it will form the fraction required. If there is no given denominator, place an unit as a deno minator. &rample. Reduce the following whole numbers to the form of frac tions, with the denominators opposite said numbers, respectively. - - Wh. Num, Denom. - Wh, N. Denom, 1. 5 ... .. 8 . l l ... ... 3. 9 ... .. 7 4. 16 . . . . 9 5. 11 ... .. 6 6, 19 . . . . . — 7. 15 . . . . 3 8. 17 . . . . 4 9. 13 . . . . . 9 10, 31 . . . . 13 Case 5th.--To reduce a compound fraction, to an equivalent simple one. Multiply all the numerators together for a new numerator, and all the denominators together for a new denominator, and the produets will form the fraction required, If a part of the compound fraction or quantity be a whole or mixed number, it must be reduced to an improper fraction; and when it can be done, any numerator and denominator of the given fractions may be divided by the same number, and the quotients used instead of themselves; also when any two contrary terms are the same, they may be wholly expunged. Let the following compound fractions be reduced to their equivalent simple fractions or value, each set respectively. Case 6th-To reduce fractions of different denominators, to equivalent fractions, having one common denominator. Multiply each numerator continually into all the denominators, but its own, for the numerators, and all the denominators together, for a common denominator. But if the least common denominator is required, find the least common multiple of all the given denominators, which will be the least common denominator required ; divide this common denominator severally, by each denominator of the given fractions, and the several ‘quotients multiplied by the numerators of the given fractions, will be the respective Inumerators. &ramples. Let #, + and # be reduced Let },3,4,&be reduced tofracto equivalent fractions, having tions equivalent thereto hav When in any set of fractions, one of the denominators is such, that each of the others may be evenly divided thereinto, it is evident that this denominator is the least required. Let }, +, and # be reduced Let {, }, + and or, be reduced to fractions equivalent there- to fractions equivalent thereto, to, having the least common having the least common deInnon denominator, mominator. |