Reciprocal and Division of Fractions

Reciprocal and division of fractions are two different methods. When the numerator and denominator of a fraction are interchanged then it is said to be it’s reciprocal. Suppose a fraction is a/b, then it’s reciprocal will be b/a. A fraction is a numerical quantity that is not a whole number. Rather it represents a part of the whole. For example, it tells how many slices of a pizza are remaining or eating of the whole pizza, such as one-half (½), three-quarters (¾)  etc. Division of fractions involve is an operation performed on fractions with multiple steps included.  Also, learn dividing fractions here.

Parts of Fraction
The fraction has two parts:

  1. Numerator
  2. Denominator.

Types of Fraction: Fractions are basically of three types, proper, improper and mixed. Learn the definitions below.

Proper Fraction: If both the numerator and denominator are positive, and the numerator is less than the denominator.

Example: 2/5, 1/3, 3/6, 7/8. 9/11, etc.

Improper Fractions: Whereas fractions having numerator greater than the denominator are called Improper fraction.

Example: 8/3, 3/2, 6/3, 11/9, etc

Mixed Fraction: When a whole number and a fraction are combined it is known as a mixed fraction.

All these details were the basics of fractions. Now let us learn reciprocal of fractions along with its division.

Reciprocal of Fractions

Reciprocal

The fraction obtained by swapping or interchanging Numerator and Denominator with each other is known as Reciprocal of the given fraction.

For example, a reciprocal of 5 is 1/5 , a reciprocal of 8/3 is 3/8.

The reciprocal of a mixed fraction can be obtained by converting it into an improper fraction and then swap the numerator and denominator.

For example, to find the reciprocal of \(\small 2\frac{1}{3}\);

  • Convert the mixed fraction into improper fraction:\(\small 2\frac{1}{3}=\frac{7}{3}\)
  • Now invert the fraction: 7/3 and 3/7, where 3/7 is called reciprocal of 7/3 or \(\small 2\frac{1}{3}\).

Note: The product of a fraction and it’s reciprocal is always 1.

Also, read:

Division of Fractions

Division involving a fraction follows certain rules. To perform any division involving fraction just multiply the first number with the reciprocal of the second number. Steps are as follows:

Step 1: First change the division sign (÷) to multiplication sign (×)

Step 2: If we change the sign of division to multiplication, at the same time we have to write the reciprocal of the second term or fraction.

Step 3: Now, multiply the numbers and simplify the result.

These rules are common for:

  1. Division of the whole number by a fraction.
  2. Division of a fraction by a whole number
  3. Division of a fraction by another fraction.

Note: It is to be noted that division of fractions is basically the multiplication of fraction obtained by reciprocal of the denominator (i.e. divisor).

Examples of Divisions of Fractions

Examples for each condition are explained below.

Division of the Whole Number by a Fraction

Example 1: 16 ÷ 4/3

Solution: 16 ÷ 4/3 = 16/1 × 3/4

3/4 is the reciprocal of 4/3

Hence, (16 × 3)/(1×4)

× 3 = 12

Therefore,

16 ÷ 4/3 = 12

Division of a Fraction by a Whole Number

Example 2: Divide 8/3 by 3

Solution: We need to simplify, 8/3 ÷ 3

The reciprocal of 3 is 1/3.

Now write the given expression into multiplication form,

8/3 × 1/3 = 8 /9

Therefore,

8/3 ÷ 3 = 8/9

Division of a Fraction by another Fraction

Example 3: 8/3 ÷ 4/3

Solution: 8/3 ÷ 4/3

Reciprocal of second term 4/3 is 3/4.

Now multiplying first term with the reciprocal of the second term we get;

8/3 × 3/4 = 8/4 = 2

Hence,

8/3 ÷ 4/3 = 2

To perform division involving mixed fraction, convert the mixed fraction into an improper fraction and follow the above steps.

Study more on the related topics such as representing fractions on a number line, visit our Byju’s page.’

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