RECIPROCAL IN MATHEMATICS

Reciprocal of a number is nothing but its multiplicative inverse. 

Let us consider the fraction 3/4. 

If we want to get its reciprocal, we have write the denominator 4 as numerator and numerator 3 as denominator. 

In simple words, we have to flip the fraction. 

So, the reciprocal of 3/4 is 4/3. And also the reciprocal of 4/3 is 3/4. 

Therefore, 3/4 and 4/3 are reciprocal to each other. 

Important note :

If two numbers are reciprocal to each other, then their product is 1. 

That is, 

(3/4) x (4/3)  =  1

Reciprocal of a Whole Number

Let us consider the whole number 8. 

If we want to get reciprocal of the whole number 8, first we have write 8 as a fraction with denominator 1. 

That is 8/1. 

To get reciprocal of 8/1, we have write the denominator 1 as numerator and numerator 8 as denominator. 

That is 1/8. 

Therefore, the reciprocal of the whole number 8 is 1/8.

And also, 

8 x 1/8  =  1

It has been illustrated in the diagram given below. 

Shortcut : 

If we want to get reciprocal of any whole number, we have to write 1 as numerator and the given whole number as denominator. 

Reciprocal of 1

The reciprocal of 1 is 1 itself.

Let us see, how the reciprocal of 1 is 1 itself. 

To get the reciprocal of 1, let us write 1 as a fraction with denominator 1. 

That is 1/1. 

To get reciprocal of 1/1, we have write the denominator 1 as numerator and numerator 1 as denominator. 

That is 1/1. 

Therefore, the reciprocal of the whole number 1 is 1 itself.

Reciprocal of Zero

The reciprocal of zero is undefined. 

Let us see, how the reciprocal of zero is undefined. 

To get the reciprocal of 0, let us write 0 as a fraction with denominator 1. 

That is 0/1. 

To get reciprocal of 0/1, we have to write the denominator 1 as numerator and numerator 0 as denominator. 

Then, we have  1/0.

Here, 1 is divided by 0.

In Math, any non zero number divided by zero is undefined.   

Therefore, the reciprocal of zero is undefined. 

Reciprocal of ∞

The reciprocal of ∞ is zero. 

Let us see, how the reciprocal of infinity is zero. 

In Math infinity (∞) is the symbol that we use for the term "Undefined".

To get the reciprocal of ∞, let us write ∞ as a fraction 1/0.

Because, any non zero number divided by zero is undefined or infinity. 

To get reciprocal of 1/0, we have write the denominator 0 as numerator and numerator 1 as denominator. 

Then, we have  0/1.

Here, 0 is divided by 1.

In Math, zero divided by any non zero number is zero.    

Therefore, the reciprocal of infinity (∞) is zero. 

Reciprocal of Mixed Number

Let us consider the mixed number 2 3/4. 

To get the reciprocal of 2 3/4, let us convert the given mixed number 2 3/4 into improper fraction. 

Then, we have

2 3/4  =  11/4

To get reciprocal of 11/4, we have to write the denominator 4 as numerator and numerator 11 as denominator. 

Then, we have  4/11.

Therefore, the reciprocal of 2 3/4 is 4/11.

Example 1 :

Divide  2/5  by  6/7.

Solution :

Using the method explained above, we have 

 2/5 ÷ 6/7  =  2/5 x 7/6

2/5 ÷ 6/7  =  (2x7) / (5x6)

2/5 ÷ 6/7  =  7/15

Example 2 :

Divide  7/5  by  3/2.

Solution :

ained above, we have 

7/5 ÷ 3/2  =  7/5 x 2/3

7/5 ÷ 3/2  =  (7x2) / (5x3)

7/5 ÷ 3/2  =  14/15

Example 3 :

Divide  5/12  by  20/13.

Solution :

Using the method explained above, we have 

5/12 ÷ 20/13  =  5/12 ÷ 20/13

5/12 ÷ 20/13  =  5/12 x 13/20

5/12 ÷ 20/13  =  (5x13) /  (12x20)

5/12 ÷ 20/13  =  13/48

Example 4 :

Divide 2/19  by  6 1/2.

Solution :

First, let us convert the mixed number 6 1/2 in to improper fraction.

6 1/2  =  13/2

Now,m we have 2/19 ÷ 6 1/2  =  2/19 ÷ 13/2

 Using the method explained above, we have 

2/19 ÷ 13/2 =  2/19 x 13//2

2/19 ÷ 13/2 =  (2x13) / (19x2)

2/19 ÷ 13/2 =  13 / 19

Example 5 :

One pizza can be made in 1/2 hour. How many pizzas can be made in 5/2 hours ?

Solution :

Time taken to make one pizza  =  1/2 hour  

No. of pizzas made in 5/2 hours  =  5/2 ÷ 1/2

No. of pizzas made in 5/2 hours  =  5/2 x 2/1

No. of pizzas made in 5/2 hours  =  (5x2) / (2x1)

No. of pizzas made in 5/2 hours  =  5

Example 1 : 

Simplify : 2/5 ÷  6 

Solution :

Using method 1, we have 

2/5 ÷ 6  =  2/5 ÷ 6/1

 2/5 ÷ 6  =  2/5 x 1/6

2/5 ÷ 6  =  (2x1) / (5x6)

2/5 ÷ 6  =  1/15

Example 2 :

Simplify : 5/12 ÷ 20

Solution :

Using method 2, we have 

5/12 ÷ 20  =  5 / (12x20)

5/12 ÷ 20  =  1/48

Example 3 :

David eats 1/4 of a pizza and divides the remaining in to two equal parts for his two kids. What is the part of the pizza will each kid receive ?

Solution :

Part of the pizza eaten by David  =  1/4

Remaining pizza  =  3/4

Given : Remaining pizza is divided in to equal parts for his two kids. 

So, part of the pizza received by each kid is 

=  3/4 ÷ 2

=  3/(4x2)

=  3/8

Hence, each kid will receive 3/8 part of the pizza.

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