# Division is inverse operation of multiplication

Let us first understand the meaning of inverse operation.

## What is Inverse Operation?

Any math calculation which have opposite affect in the calculation is called inverse operation.

For Example, Addition and Subtraction have opposite affect in the calculation.

Similarly multiplication and division have opposite effect, hence they are considered as inverse operation in mathematics.

### Division and Multiplication as Inverse operation

Division and multiplication have opposite affect in number calculation.

To understand the concept, let us consider a number 5.

Multiply number 5 with 7, you will get number 35;
⟹ 5 x 7 = 35

Now, if you want to get number 5 back, divide number 35 by 7;
⟹ 35 ÷ 7 = 5

Here we have used division to get back to the original number.
Hence, division works opposite to multiplication.

### How this inverse property helpful for students?

The property is helpful to solve complex algebra problems in fast and easy way.

If the given equation is: A ÷ B = C

Then using the above property, the equation can be written as:
A = B x C

Example:
(a) 6 ÷ 2 = 3
We can also write; 3 x 2 = 6

(b) 45 ÷ 5 = 9
Or, 9 x 5 = 45

(c) 72 ÷ 12 = 6
Or, 12 x 6 = 72

(d) 96 ÷ 8 = 12
Or, 12 x 8 = 96

(e) 75 ÷ 15 = 5
Or, 15 x 5 = 75

### Using inverse property to solve Algebraic Equation

In algebraic equation, moving from one side to another side the division will change into multiplication or vice versa.

Observe the below image.
Note how the division is changed into multiplication in algebraic equation.

Similarly the multiplication can be changed into division on moving to other side of the algebra equation.

Let us see some examples to fully understand the concept.

Example 01
Find the value of x in given equation
\mathtt{\frac{x}{2} \ =\ 7}\\\ \\

\mathtt{The\ equation\ can\ be\ written\ as:}\\\ \\ \mathtt{x\ =\ 7\ \times \ 2}\\\ \\ \mathtt{x\ =\ 14} \\\ \\

Hence x = 14 is the solution.

Example 02
Find the value of y in below equation
\mathtt{\frac{8y}{3} \ =\ 16} \\\ \\

\mathtt{The\ equation\ can\ be\ written\ as:}\\\ \\ \mathtt{8y\ =\ 16\ \times \ 3}\\\ \\ \mathtt{8y\ =\ 48}\\\ \\ \mathtt{Move\ 8\ towards\ right}\\\ \\ \mathtt{y\ =\ \frac{48}{8}}\\\ \\ \mathtt{y\ =\ 6}

### If inverse of multiplication is division then what is the inverse of subtraction?

Addition and Subtraction are inverse to each other.

Suppose we have number 5 and we add 2 in it.
5 + 2 = 7

Now in order to get 5 back, subtract 2 from 7
7 – 2 = 5

Hence, both addition and subtraction works in opposite way.

## Solved Problems – Division as Inverse of Multiplication

(01) Find the value of x from below equation
\mathtt{\frac{2x\ +\ 8}{4} \ =\ 7} \\\ \\

\mathtt{Move\ 4\ to\ the\ right}\ \\\ \\ \mathtt{2x\ +\ 8\ =\ 7\ \times \ 4}\ \\\ \\ \mathtt{2x\ +\ 8\ =\ 28}\ \\\ \\ \mathtt{Move\ 8\ to\ the\ right}\ \\\ \\ \mathtt{2x\ =\ 28\ -\ 8}\ \\\ \\ \mathtt{2x\ =\ 20}\ \\\ \\ \mathtt{Move\ 2\ to\ the\ right}\ \\\ \\ \mathtt{x\ =\ \frac{20}{2}}\ \\\ \\ \mathtt{x\ =\ 10 } \\\ \\

Hence, the value of x is 10.

(02) Find the value of y in given equation
\mathtt{\frac{3y\ }{11} \ =\ 5y\ -\ 1} \\\ \\

\mathtt{Moving\ 11\ to\ the\ right.}\ \\ \\ \mathtt{Division\ will\ turn\ into\ multiplication}\ \\\ \\ \mathtt{3y\ =\ 11\ \times \ ( 5y\ -1)}\\\ \\ \mathtt{3y\ =\ 55y\ -\ 11}\ \\\ \\ \mathtt{Moving\ 11\ to\ the\ left}\ \\ \\ \mathtt{Subtraction\ will\ become\ addition}\ \\\ \\ \mathtt{3y\ +\ 11\ =\ 55y}\ \\\ \\ \mathtt{Moving\ 3y\ towards\ right}\ \\ \\ \mathtt{Addition\ will\ become\ subtraction}\ \\\ \\ \mathtt{11\ =\ 55y\ -\ 3y\ }\ \\\ \\ \mathtt{11\ =\ 52y}\ \\\ \\ \mathtt{y\ =\ \frac{11}{52}} \\\ \\

Hence, the value of y is 11/52

(03) Find the value of y in given equation
\mathtt{y-2\ =\ \frac{3y}{7}} \\\ \\

\mathtt{Move\ 7\ to\ the\ left}\\\ \\ \mathtt{Division\ will\ turn\ into\ multiplication}\ \\\ \\ \mathtt{7\ ( y\ -\ 2) \ =\ 3y}\ \\\ \\ \mathtt{7y\ -\ 14\ =\ 3y}\ \\\ \\ \mathtt{Move\ 3y\ towards\ left}\ \\ \\ \mathtt{Addition\ becomes\ subtraction}\ \\\ \\ \mathtt{7y\ -\ 3y\ -\ 14\ =\ 0}\ \\\ \\ \mathtt{4y\ -\ 14\ =\ 0}\ \\\ \\ \mathtt{Move\ 14\ towards\ right}\ \\ \\ \mathtt{Subtraction\ become\ addition}\ \\\ \\ \mathtt{4y\ =\ 14} \\\ \\ \mathtt{y\ =\ \frac{14}{4}} \\\ \\

Hence, the value of y is 14/4