FRACTIONAL REDUCTION. CASE 1. To reduce an improper fraction to its equivalent whole or mixed number. Rule. Divide the numerator by the denominator; the quotient will be the equivalent whole number, and if there is a remainder, annex it as the numerator of a fraction having the given denomi nator. Example. To reduce 537 to a mixed number. 5.37 = 537 ÷ 25 = 2113 4 Ex. 1. Reduce 32, 25, and 112 to equivalent whole numbers. 2. Reduce 25 33, and 617 to equivalent mixed numbers. 79 10001 3. Reduce 32, 4, and 1980' to equivalent mixed 179 numbers. 382 Products. Ex. 1. 4, 5, 28. 181, 16, 26. Ex. 2. 34, 6, 56. CASE 2. To reduce a mixed number to an improper fraction. Multiply the whole number by the denominator of the fraction, and add in the numerator; then make this amount the numerator to the required fraction, having for its denominator the denominator of the given fraction. Example. To reduce 73 to an improper fraction. Ex. 1. Reduce 43, 52, and 11 to improper fractions. 2. Reduce 7, 2311, and 5413 to improper fractions. CASE 3. To reduce a complex fraction to its simple terms. Rule. When only one of the terms contains a fraction, multiply both terms by the denominator of the annexed fraction.* When both terms contain fractions, multiply them by the product of the denominators of the annexed fractions, or by their least multiple. For the multiplication of a fraction by a whole number see Obs. 3. Fractional Multiplication. The least multiple is the smallest number or product in which the given numbers are contained. CASE 4. To reduce a compound fraction to its equivalent simple fraction. Rule. Mixed numbers and whole numbers being expressed as improper fractions, and complex fractions being reduced to simple fractions, multiply all the numerators together for a new numerator, and all the denominators together for a new denominator. Observe, that if the same number occurs as both a numerator and a denominator, it may be cancelled; and if one numerator and one denominator will divide by the same number, each may be divided, and its quotient used in its place. N. B. Only one numerator and one denominator may be taken at a time, but the cancelment or division may be repeated as often as possible. EXAMPLE. To reduce of 24 of § of 9 to a simple fraction. The mixed number and the whole number being expressed fractionally, and the numbers being arranged for multiplication; beginning with the 1 st numerator 3, we find no denominator that will divide it, and therefore we pass on to the 14, which, with the 7, divides by 7, cancelling the 7 in the denominator, and making the 14 into 2;-this 2 divides with 8, cancelling the former and making the latter 4; and then the two fives cancel one upon the other; there then remains 4 in the denominator which will not divide with either the 3 or the 9, and therefore compounding these latter numbers we get 27-4 ths or 63. N. B. We have here placed a 1 above or below the numbers to be cancelled, but this is not necessary in the performance of these calculations. EXERCISES. Reduce of of 11 to a simple fraction. Product 5 Reduce of 1 of of 34 to a simple fraction. Reduce of of 15 of 4 of 85 to a simple fraction. 49 69 644 CASE 5. To find the greatest common measure to two numbers. Rule. Divide the greater number by the less; then if there is a remainder make it the divisor of another operation, in which the preceding divisor is made the dividend, and thus continue the calculation until there is not any remainder, and the last divisor will be the greatest common measure. EXAMPLE. Find the greatest common measure to 612 and 828. 612) 828 (1 612 216) 612 (2 180) 216 (1 36) 180 ( 5 180 The greatest common measure is 36, being the greatest number that will divide both 612 and 828. EXERCISES. Ex. 1. Find the greatest common measure to 171 and 180. 2. Find the greatest common measure to 4536 and 4872. 3. Find the greatest common measure to 1785, 3255, and 7140. Ex. 1. 9. PRODUCTS. Ex. 2. 168. Ex. 3. 105. N. B. The principle of this rule may be easily explained from the above example; for as 36 is the exact measure of 180, it must also exactly measure 36 and 180, or 216; again, as 36 measures 216 and 180, it must measure twice 216 and 180, or 612; lastly, as 36 measures 216 and 612, it must measure the sum, or 829; hence 36 is the common measure of 612 and 828. CASE 6. To reduce a fraction to its lowest terms. Rule. Divide both terms of the fraction by their greatest common measure. Note 1. Instead of finding the greatest common. measure, divide the numerator and denominator by any number that will exactly divide them; repeat the calculation, if possible, with the new fraction produced, and so continue the operation until the fraction is reduced into its lowest terms. Note 2. Numbers ending in 0 are divisible by 10, and in two ciphers by 100, &c. Numbers ending in 5 are divisible by 5. If the sum of the figure is divisible by 9 or 3, they are divisible by those numbers. Even numbers are divisible by 2. If the units and tens are divisible by 4 the numbers are divisible by 4. If the units, tens, and hundreds are divisible by 8, the numbers are divisible by 8. |