SECTION II. Reduction. It is often convenient, and sometimes necessary, to change the form in which a fraction or a mixed quantity is expressed. For instance, to prepare different fractions for addition or subtraction, we must always express them in other fractions, which shall have a common denominator. The process, by which the form of a fraction is changed, its value remaining the is called Reduction. same, A. To reduce an Improper Fraction to a Mixed Quantity. 1. What is the value of ? 2. What is the value of 2? ANS. 1. ANS. 3. As 5 fifths are equal to 1, 19 fifths must be equal To reduce an improper fraction to a whole or mixed quantity, Divide the numerator by the denominator, and annex the remainder, if any, to the quotient, in the form of a fraction. Reduce the following fractions to whole or mixed quantities: 5.) 17. 6.) 24. 7.) 398 8.) 177. In 1 there are 3 thirds, and in 7 there are 7 times 3, or 21 thirds; and 2 thirds added to 21 make 23 thirds. 17. Reduce b + to an improper fraction. a The integral quantity, b, reduced to a fraction, becomes a; [See Sec. I.] and the whole quantity is ab ab+, or ab+b, which expresses the same value. α Hence, to reduce a mixed number to an improper fraction, multiply the integer by the denominator, and add the numerator to the product. The sum will be the numerator of the fraction required. Reduce the following mixed numbers to improper fractions: 18) 94. a 19.) a + . 20.) x y + 10+ a bb a 21.) b++. 22.) a−x+x. 23.) aa + · α b+14 26. Reduce a+b++ to an improper fraction. We are here required to find two other fractions having the same denominator, which shall be equiva lent to and . If both terms of the first fraction, , be multiplied by the denominator of the second, it becomes ; and if both terms of the second fraction, ?, be multiplied by the denominator of the first, it becomes. 29. Reduce, and to a common denomina We multiply both terms of each fraction by the de nominators of the other fractions. 30. Reduce and to a common denominator. ANS., and. by by Hence, to reduce several fractions, having different denominators, to other fractions of equal value with a common denominator, we have the following RULE For the numerators: Multiply each numerator by all the denominators except its own. D. To reduce Fractions to their Least Terms. As small numbers are more convenient to work with than large ones, a fraction should always be kept in its least terms. For this purpose, the following general RULE will be found useful: Divide both the numerator and denominator by any quantity which will divide all the terms of both without a remainder. Experience will suggest various expeditious ways of applying this principle. Reduce the following fractions to their lowest terms The learner will remember, that the numerator is a dividend, that the denominator is a divisor, and that the value of the fraction is to be found in the quotient. The signs prefixed to the several terms of a fraction, affect those terms only; but when a sign is prefixed to the fraction itself, it affects its whole value, that is, the value of all the terms taken collectively. Without regarding the signs, we know that the value of the fraction is b. Now, by the common rules of ab α |