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Algebraic Expressions - Evaluation involving Fractions and Mixed Numbers



A fraction  is a number that can represent part of a whole.
Examples : 1/4, 2/3, 5/4, 7/3
A proper fraction is simply a fraction whose numerator is less than the denominator. More precisely, the absolute value of a proper fraction is less than 1.
Examples : 1/4, 1/2, 3/4
An improper fraction or a top-heavy fraction is simply a fraction whose numerator is greater than (or equal to) the denominator. More precisely, the absolute value of an improper fraction is greater than (or equal to) 1.
Examples : 5/4, 7/2, 9/4
A mixed number is the sum of an integer and a proper fraction.
Example: 2 4/5 where 2 is the integer part and 4/5 is the fraction part.



An improper fraction can be converted to a mixed number in three steps:
  1. Divide the numerator by the denominator.
  2. The quotient (without remainder) becomes the integer part and the remainder becomes the numerator of the fraction part.
  3. The new denominator is the same as that of the original improper fraction.

Example : To convert 31/15 to a mixed number.
  1. Divide : 31 ÷ 15
  2. Quotient = 2 ; Remainder = 1
  3. So, the mixed number is 2 1/15



A mixed number can be converted to an improper fraction in three steps:
  1. Multiply the integer part by the denominator of the fraction part.
  2. Add the product to the numerator of the fraction part. The resulting sum is the numerator of the new (improper) fraction.
  3. The new denominator is the same as that of theoriginal fraction part of the mixed number.

Example : To convert 2 1/15 to an improper fraction.
  1. Multply : 2 × 15 = 30
  2. 30 + 1 = 31
  3. So, the improper fraction is 31/15



Addition and subtraction of fractions :

To add/subtract fractions with the same denominator, simply add/subtract their numerators and write down the denominator.
Example : To add 4/9 and 1/9, simply add the numerators 4 and 1. Thus, 4/9 + 1/9 = 5/9.
Example : To subtract 1/9 from 5/9, simply subtract the numerators (1 from 5). Thus, 5/9 − 1/9 = 4/9.

To add/subtract fractions with different denominators, rewrite them first as equivalent fractions with the same denominator and then add/subtract.
Example : To add 1/2 and 1/9, rewrite them as equivalent fractions : 1/2 = 9/18 and 1/9 = 2/18.
Thus, 1/2 + 1/9 = 9/18 + 2/18 = 11/18.
Example : To subtract 1/9 from 1/2, rewrite them as equivalent fractions : 1/2 = 9/18 and 1/9 = 2/18.
Thus, 1/2 − 1/9 = 9/18 − 2/18 = 7/18.



Multiplication of fractions :
To multiply fractions, multiply their numerators and denominators separately.
Example : To multiply 4/9 and 5/7, simply multiply the numerators 4 and 5 (to get 20) and then multiply the denominators 9 and 7 (to get 63).
Thus, 4/9 ×5/7 = (4 × 5)/(9 × 7) = 20/63.

A fraction may be multiplied by an integer by writing the integer as a fraction whose denominator is unity (1).
Example : To multiply 3/23 and 7, write 7 as a fraction (7/1). Thus, 3/23 × 7 = 3/23 × 7/1 = (3 × 7)/(23 × 1) = 21/23. 



Division of fractions :
The reciprocal (or multiplicative inverse) of any nonzero number x is the number 1/x. The reciprocal of the fraction p/q is the fraction q/p.
Example : The reciprocal of 7 is 1/7, and the reciprocal of 3/5 is 5/3.

Any number may be divided by a fraction by simply multiplying the number by the reciprocal of the fraction.
Example : To divide 15 by 4/7, multiply 15 by 7/4. Thus, 15 ÷ 4/7 = 15 × 7/4 = (15 × 7)/4 = 105/4.

Try the Quiz : Algebraic Expressions - Evaluation 4

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