Reciprocal and Division of Fractions

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 eaten of the whole pizza, such as one-half (½), three-quarters (¾)  etc.

Parts of Fraction-

  1. Numerator
  2. Denominator.

Fraction

Types of Fraction-

(i) Proper Fraction-

If both the numerator and denominator are positive, and numerator is less than the denominator, then such fractions are called proper fraction.

Ex: \(\small \frac{2}{5}\).

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

Ex: \(\small \frac{8}{3}\).

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

Ex: \(\small \frac{8}{3}\).

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Reciprocal-

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

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For example, a reciprocal of 5 is \(\small \frac{1}{5}\). , a reciprocal of \(\small \frac{8}{3}\). is \(\small \frac{3}{8}\).

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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: \(\small \frac{7}{3}\) and \(\small \frac{3}{7}\), where \(\small \frac{3}{7}\) is called reciprocal of \(\small \frac{7}{3}\) or \(\small 2\frac{1}{3}\).

Note: The product of a fraction and its reciprocal is always 1.

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 ×

Step 2: Find the reciprocal of the second term (number after sign) in the question.

Step 3: 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 for each condition are explained below.

Division of Whole Number by a Fraction

Evaluate: \(\small 16\div \frac{4}{3}\)

Solution: \(\small 16\div \frac{4}{3} \Rightarrow \frac{16}{1}\times \frac{4}{3}\)

Here second term is \(\small \frac{4}{3}\) ; take the reciprocal of \(\small \frac{4}{3}\)

\(\small \Rightarrow 16 \div \frac{16}{1}\times \frac{3}{4}=\frac{16\times 3}{1\times 4}=\frac{48}{4}=12\)

Alternatively, here \(\small \frac{16}{1}\times \frac{3}{4}\) ; 16 and 4 can be simplified to get \(\small \frac{4}{1}\times \frac{3}{1}=\frac{4\times 3}{1\times 1} = 12\)

Division of a Fraction by a Whole Number

Evaluate: \(\small \frac{8}{3}\div 3\)

Solution: \(\small \frac{8}{3}\div3 \Rightarrow \frac{8}{3}\times 3\)

Here the second term is \(\small 3 = \frac{3}{1}\) ; take the reciprocal of \(\small \frac{3}{1}\)

\(\small \Rightarrow \frac{8}{3}\times \frac{1}{3}=\frac{8\times 1}{3\times 3}=\frac{8}{9}\)

Division of a Fraction by another Fraction

Evaluate: \(\small \frac{8}{3} \div \frac{4}{3}\)

Solution: Here the second term is \(\small \frac{4}{3}\) , take the reciprocal of \(\small \frac{4}{3}\)

Hence,

\(\small \Rightarrow \frac{8}{3}\times \frac{3}{4} = \frac{8\times 3}{3\times 4}=\frac{8}{4}=\frac{2}{1}=2\)<

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.’


Practise This Question

Simplify the following expression and choose the correct answer:

12×(0×(696142×989789×456989×142696))+56×9×0(989456+498981+598251+198256)×989