Machinery's Handbook, 31st Edition
6
Fractions
Examples:
2 3
15 3
2 3
17 3
1 2
18 2
1 2
19 2
2 3
1 2
5 +
= + =
9 +
= + =
5
=
9
=
To convert mixed numbers to improper fractions, multiply the whole number by the de- nominator and add the numerator to obtain the new numerator. The de nominator remains the same. For example, 2 = = = An improper fraction is converted to its mixed number form by dividing the numerator by denominator and placing the remainder over the denominator. Sometimes the fraction part can be reduced, as the second example shows. 17 8 = 17 ÷ 8 = 2 1 2 -- 2 2+1 × 2 ----------- 7 16 --- 3 16+7 × 16 ------------ = 5 2 -- 3 55 16 --- 1 8
26 16
= 26 ÷ 16 = 1 = 1 10 16 5 8
Equivalent Fractions: A fraction raised to its equivalent form (“higher terms”) by mul- tiplying numerator and denominator by the same number (that is, by multiplying by a form of 1). For example, 1 ⁄ 4 × 4 ⁄ 4 = 4 ⁄ 16 and 3 ⁄ 8 × 4 ⁄ 4 = 12 ⁄ 32 . Any integer n can be expressed as a fraction with a chosen denominator value of m by simply writing n as n /1 and multiplying by m / m . Example: To express 4 as an equivalent fraction with a denominator of 16, write 4 ⁄ 1 × 16 ⁄ 16 = 64 ⁄ 16 Reciprocal: The reciprocal of any number a other than 0 is 1/ a . (0 has no reciprocal, since 1/0 is undefined.) The reciprocal also is called the multiplicative inverse , since a × 1/ a = 1. For example, the reciprocal of 8 is 1 ⁄ 8 ; the reciprocal of 4 ⁄ 7 is 7 ⁄ 4 . Least Common Denominator: Fractions cannot be added or subtracted without a com- mon denominator. For example, 2 5 inator in the answer is the same denominator seen in the fractions. In general, a c b c a b c + = + 1 5 3 5 + = = + 2 1 5 , a simple computation, since the denom- . But fractions with different denominators cannot be added or subtracted until they are converted to equivalent forms that have common denominators. This is done by rais- ing the fractions to higher terms (as explained previously). While any common multiple serves as a common denominator, it is preferable to use the least common multiple (LCM) of the denominator, referred to as the least common denominator (LCD). For example, 36 is the LCD of 2 9 5 6 and , since the LCM of 9 and 6 is 36. Raising each fraction to its equiva- lent form having a denominator of 36 is shown:
2 9
4 4
8 36
5 6
6 6
30 36
× =
× =
and
Example: In the case of 9 11
7 10 and the LCD is the product of the denominators, 11 ×
10 = 110. Raising each fraction to its equivalent form is shown: 9 11 10 10 90 110 7 10 11 11 77 110 × × = = and
Adding and Subtracting Fractions and Mixed Numbers To Add or Subtract Common Fractions: 1) Convert each fraction to terms of the least common denominator; 2) add or subtract numerators; 3) if answer is an improper fraction, change it to a mixed number; and 4) reduce fraction part if necessary.
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