# MULTIPLICATIVE INVERSE

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.

Multiplicative inverse of 2 is 1/2 and that of -3/5 is -5/3.

1 and – 1 are the only rational numbers which are multiplicative inverses to themselves.

That is,

multiplicative inverse of 1 = 1

multiplicative inverse of -1 = -1

0 has no multiplicative inverse or multiplicative inverse of 0 is undefined.

Example 1 :

Find the multiplicative inverse of 9/11.

Solution :

Multiplicative inverse of 9/11 = 11/9

Example 2 :

Find the multiplicative inverse of -5.

Solution :

Multiplicative inverse of -5 = -1/5

Example 3 :

Find the multiplicative inverse of 2.

Solution :

Convert the given mixed number to an improper fraction.

2 = 13/5

Reciprocal of 13/5 is 5/13.

Multiplicative inverse of 2 = 5/13

Example 4 :

What is the multiplicative inverse of the sum of 3/4 and 5/6?

Solution :

Two fractions 3/4 and 5/6 have different denominators.

Least common multiple of (4, 6) = 12.

Make the denominator of each fraction as 12 using multiplication.

3/4 = (3 x 3)/(4 x 3) = 9/12

5/6 = (5 x 2)/(6 x 5) = 10/12

3/4 + 5/6 :

= 9/12 + 10/12

= (9 + 10)/12

= 19/12

Reciprocal of 19/12 is 12/19.

Multiplicative inverse of (3/4 + 5/6) = 12/19

Example 5 :

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

Solution :

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

Simplify.

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

= 1/12

Reciprocal of 1/12 is 12.

Multiplicative inverse of (5/21) x (7/20) = 12

Example 6 :

What is the multiplicative inverse of (24/7 ÷ 32/35)?

Solution :

Simplify.

= 3/1 x 5/4

= 15/4

Reciprocal of 15/4 is 4/15.

Multiplicative inverse of (24/7 ÷ 32/35) = 4/15

Example 7 :

What is the multiplicative inverse of 0.32?

Solution :

0.32 = 32/100

= 8/25

Reciprocal of 8/25 is 25/8.

25/8 = 3.125

Multiplicative inverse of 0.32 = 3.125

Example 8 :

What is the multiplicative inverse of 1.25?

Solution :

1.25 = 125/100

= 5/4

Reciprocal of 5/4 is 4/5.

4/5 = 0.8

Multiplicative inverse of 1.25 = 0.8

Example 9 :

If y/3 and 4/5 are multiplicative inverse to each other, then find the value of y.

Solution :

Since y/3 and 4/5 are multiplicative inverse to each other, their product is equal to 1.

y/3 x 4/5 = 1

4y/15 = 1

Multiply each side by 15/4.

y = 15/4

Example 10 :

If 2/5 and 7/z are multiplicative inverse to each other, then find the value of z.

Solution :

Since 2/5 and 7/z are multiplicative inverse to each other, their product is equal to 1.

2/5 x 7/z = 1

14/5z = 1

Take reciprocal on each side.

5z/14 = 1

Multiply each side by 14/5.

z = 14/5

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