Reciprocal and Division of Fractions

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What is Reciprocal and Division of Fractions?

Reciprocal and Division of fractions are different from each other. When two terms, such that the second is the inverse of first, gives a product of 1 upon multiplication, then the fractions are called reciprocal or multiplicative inverse of each other. It can be achieved by interchanging the numerator and denominator of the fraction. If a term is x/y, then its reciprocal will be y/x. A fraction is a numerical quantity that represents a part of the whole number.


Dividing fractions requires inverting the divisor (reciprocal of the divisor) and then follow the steps of multiplication. The term x/y, when divided by a non-zero fraction p/q, then the expression looks like:

x/y ÷ p/q = x/y ÷ q/p

Division of fractions involves multiple steps.


Parts of Fraction

The parts of a fraction are:

  • Numerator: The numerator is the number which is on the top of the line. It shows how many equal parts of the whole or collection is taken.

  • Denominator: The number below the line is the denominator. It shows the total divisible number of equal parts or the total number of equal parts which they are in a collection.

Types of Fraction

There are three different types of fractions:

  • Proper Fraction: A fraction in which the numerator and denominator are positive, and the numerator is smaller than the denominator; those fractions are called proper fractions.

Example: 4/5, 1/6, 7/9, 3/7, etc.

  • Improper Fraction: A fraction in which the numerator is greater than the denominator, those fractions are called improper fractions.

Example: 5/2, 7/3, 8/5, 5,3, etc.

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

Example: 4²/₃, 3¹/₂, etc.

Reciprocal of Fractions

Interchanging or swapping the numerator or denominator with each other gives us the reciprocal of the fraction. Like for example, the reciprocal of 1/2 is 2/1, or that of 4 is 1/4.


In order to obtain a reciprocal from a mixed fraction, it must be converted to an improper fraction, and then the numerator and denominator must be swapped. For example, to find the reciprocal of 4²/₃, it is first converted into an improper fraction.


4²/₃ =14/3[improper fraction]

14/3 = 3/14

Therefore, the reciprocal of 4²/₃ is 3/14.

The product of a fraction and its reciprocal is always 1, as it is nothing but its inverse.


Division of Fractions

Division of fractions involves certain rules and follows multiple steps. To perform Division, we have to multiply the first fraction with the reciprocal of the second. Division involves some steps to be followed:

  • Step 1: Changing the division sign (÷) to the multiplication sign (×).

  • Step 2: If we change the sign to multiplication, we also have to write the reciprocal of the second term or fraction.

  • Step 3: Multiplying the fractions and simplifying the result.

Here, is an example of Division of fractions:

15³/₇ ÷ 1²³/₄₉

First, we have to change the mixed  to an improper fraction.

= 108/ 7 ÷ 72/49.

Step 1: Changing the sign to multiplication from division and writing the reciprocal of the second term [72/49 = 49/72]

 = 108/7 × 49/72


Step 2: Multiplying the first with the reciprocal of the second fraction.

= (108 × 49)/ (7 × 72)

= (3 × 7)/ (1 × 2)

= 21/2


Step 3: Getting the simplified result of the expression.

Division of fractions is the multiplication of fractions by just changing the second fraction to its reciprocal.


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Solved Example:

1. 5/9 ÷ 2/3

[Step I: Turning over the second fraction upside-down (it becomes a reciprocal): 2/3 becomes 3/2.]

= 5/9 × 3/2 

 [Step II: Multiplying the first fraction by the reciprocal of the second: (3 × 5)/(2 × 9)]

= 5/6 

[Step III: It is the simplified expression, hence no further simplifications].


2. 4/9 ÷ 2/3

[Step I: Turning over the second fraction upside-down (it becomes a reciprocal): 2/3 becomes 3/2.]

= 4/9 × 3/2

[Step II: Multiplying the first fraction by the reciprocal of the second]

= (4 × 3)/ (9 × 2)

= (2 × 1)/(3 × 1)

= 2/3

[Step III: It is the simplified expression, hence no further simplifications]

FAQ (Frequently Asked Questions)

Q1. How do you Divide Fractions with the Whole Numbers?

Answer: Suppose, a fraction 2/3 is to be divided by a whole number 8, as in the following case.

2/3 ÷ 8

Firstly, we have to convert the Division into multiplication by writing the reciprocal of the divisor, i.e. a fraction. 


[Step I: Turning over the second fraction upside-down (it becomes a reciprocal): 8 = 8/1 becomes 1/8.]


= 2/3 × 1/8

 

= (2 × 1)/(3 × 8) [Step II: Multiply the first fraction by the reciprocal of the second]


By Solving the expression, we get the simplified fraction in the following step, which cannot be further simplified.


[Step III: Simplify the fraction]


= 1/12

Q2. What are the Different Cases Involved in the Division of Fractions?

Answer: The different types that exist in the Division of a fraction are:


Division of a Fraction by a Whole Number

The whole number is converted to its reciprocal and then the fraction is multiplied to it. An instance of it is given below:

8/3 ÷ 4 = 8/3 × 1/4


Division of a Whole Number by a Fraction

In this case, the fraction is converted to its reciprocal, and then it is multiplied with the whole number. An example of this case is given below:

4 ÷ 8/3 = 4 × 3/8


Division of a Fraction by Another Fraction

Here, the second fraction is converted to its reciprocal and then both the fractions are multiplied. An example of it is given below:

8/3 ÷ 4/5 = 8/3 × 5/4