**Multiplicative inverses :**

For every rational number a/b, a ≠ 0, there exists a rational number c/d such that (a/b) x (c/d) = 1. Then c/d is called the multiplicative inverse of (a/b).

That is,

If (a/b) is a rational number, then (b/a) is the multiplicative inverse or reciprocal of it.

- The reciprocal of 2 is 1/2
- The multiplicative inverse of (-3/5) is (-5/3)
- 0 has no reciprocal.
- 1 and – 1 are the only rational numbers which are their own reciprocals.

Let us see some example problems based on the above concept.

**Example 1 :**

What is the multiplicative inverse of 9/11

**Solution :**

Since the given number is a fraction,

Let us consider the given fraction as (a/b)

Multiplicative inverse of (a/b) is (b/a)

Hence multiplicative inverse of 9/11 is 11/9.

**Example 2 :**

What is the multiplicative inverse of -5

**Solution :**

Since the given number is integer,

Let us consider the given number as "x"

Multiplicative inverse of x is (1/x)

Hence multiplicative inverse of -5 is -1/5.

**Example 3 :**

What is the multiplicative inverse of the mixed number 00 2 5/7

**Solution :**

Since the given number is mixed fraction, first we have to convert it as improper fraction.

2 5/7 = [(2x7)+5]/7 = (14+5)/7 = 19/7

From this we need to find the multiplicative inverse of the fraction 19/7.

Multiplicative inverse of the fraction in the form (a/b) is (b/a)

Hence multiplicative inverse of 19/7 is 7/19.

**Example 4 :**

What is the multiplicative inverse of the sum of (2/5) and (7/10)

**Solution :**

From the given question, we come to know that first we have to add the above fractions and find the multiplicative inverse for the answer.

(2/5) + (7/10) = (2/5) + (7/10)

L.C.M of 5 and 10 is 10.

= (2/5) x (2/2) + (7/10)

= (4/10) + (7/10)

= (4 + 7)/10

= 11/10

By combining the above fractions, we get 11/10. Since the answer is in the form (a/b), (b/a) will be its reciprocal.

Hence the multiplicative inverse of 11/10 is 10/11.

**Example 5 :**

What is the multiplicative inverse of the product of (5/21) x (7/20)

**Solution :**

From the given question, we come to know that first we have to simplify the above fractions and find the multiplicative inverse for the answer.

= (5/21) x (7/20)

= (1/3) x (1/4)

= (1/12)

By combining the above fractions, we get 1/12.Since 1/12 is in the form a/b, b/a will be its multiplicative inverse.

Hence multiplicative inverse of 1/12 is 12/1. That is 12.

After having gone through the stuff given above, we hope that the students would have understood "Multiplicative inverses".

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