MULTIPLICATIVE INVERSE

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 2⅗.

Solution :

Convert the given mixed number to an improper fraction.

2⅗ = 13/5

Reciprocal of 13/5 is 5/13.

Multiplicative inverse of 2⅗ = 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 :

= 24/7 ÷ 32/35

Simplify.

= 24/7 x 35/32

= 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

Example 11 :

Which equation illustrates the multiplicative inverse property ?

a) 1 ⋅ x = x      b)  x ⋅ (1/x) = 1   c) 1⋅0 = 0   d)  -1⋅ x = -x

Solution :

Multiplicative inverse :

Multiplying a numerical value with its reciprocal, we will get the result 1.

When x is the quantity, its reciprocal will be 1/x. So,

x ⋅ (1/x) = 1

Option b is correct.

Example 12 :

By what rational number -8/39 should be multiplied to obtain 5/26 ?

Solution :

Let x be the required rational number.

x ⋅ (-8/39) = 5/26

To solve for x, we have to ignore -8/39 in the left side. To do that we have to multiply the reciprocal or multiplicative inverse of -8/39 on both sides.

x ⋅ (-8/39) ⋅ (-39/8) = 5/26 ⋅ (-39/8)

Using 13 times table, simplifying it, 

x = 5/2 ⋅ (-3/8)

x = -15/16

So, the required numbers is -15/16.

Example 13 :

Find the reciprocal of 

(-2/3) ⋅ (-5/7) + (2/9) ÷ (1/3) ⋅ (6/7)

Solution :

(-2/3) ⋅ (-5/7) + (2/9) ÷ (1/3) ⋅ (6/7)

To get the reciprocal of the given, first we have to simplify the expression using order of operation.

= (10/21) + (2/9) ÷ (2/7)

While changing the division sign as multiplication, we have to convert 2/7 as its reciprocal.

= (10/21) + (2/9) ⋅ (7/2)

= (10/21) + (7/9)

LCM(21, 9) = 63

= 30/63 + 49/63

= (30 + 49)/63

= 79/63

Its reciprocal is 63/79

Example 14 :

Multiplicative inverse of 0/1 is

a)  1   b)  -1     c) 0   d) not defined

Solution :

While simplifying 0/1, we get 0

The reciprocal of 0/1 is 1/0 which is undefined. So, option d is correct.

Example 15 :

The product of a non-zero rational number and its reciprocal is ________.

Solution :

Let x be the non zero number, then its reciprocal will be 1/x.

Multiplying a number and its reciprocal, we get

x (1/x) = 1

We get 1 as result.

Example 16 :

state whether the given statements are true or false.

Every rational number has a reciprocal.

Solution :

False

Example 17 :

The product of two rational numbers is –7. If one of the number is –10, find the other.

Solution :

Product of two rational numbes = -7

Let x be the required number.

One number = -10.

-10(x) = -7

To ignore -10 which is multiplied, we get the reciprocal of -10 which is -1/10

x = -7 (-1/10)

x = 7/10

So, the required number be 7/10.

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