Division Property of Equality - Easiest Property To Solve Equations

Division is a method of distributing evenly and fairly. Similarly, in an equation you need to operate on both sides of the equation fairly. Whatever you operate on the left side of an equation, you must do it on the right side of the equation as well. You cannot divide something from one side without performing anything on the other side. 

Suppose, you have two cakes of equal size. And you have 8 guests at the party. You have to divide both the cakes evenly throughout. How will you be able to do this?

To divide both the cakes evenly, you can cut each cake into 4 pieces. If you divide each cake into 4 pieces, you will be left with a total of 8 pieces of cakes. Now you can easily distribute an equal piece of cake among your 8 guests. 

In short, cutting both the cakes evenly to get the balance is the division property of equality.

How to Divide an Equation Equally?

The division property of equality states that if you divide both sides of an equation by the same non zero number, the two sides remain equal.

In other words, if a, b, and c are real numbers, such that a = b , and c ≠ 0, then \[\frac{a}{c} = \frac{b}{c} \]

Division Property of Equality

Let us understand with an example:

Consider the equation, 20 = 20

Divide both the sides by 2.

\[\frac{20}{2} = \frac{20}{2} \]

10 = 10

LHS = RHS

The equation remains balanced.

Division Property of Equality Definition

The division property of equality states that dividing both sides of an equation by the same number does not affect the equation.

In other words, it states that if you are given an equation and you divide each side of an equation equally, the equation will always remain balanced and equal. This is termed as division property of equality.

Division Property of Equality Formula

The division property of equality formula states that:

Division Property of Equality Proof

To prove the division property of the equality statement, we will consider the following linear equation.

Consider the equation 5x = 30,

Dividing both sides of the equation by 5:

5x = 30

\[\frac{5x}{5} = \frac{30}{5} \]

x = 6

Therefore, the value of x is 6.

Hence, we applied the division property of equality here to solve for x.

You can verify, if x = 6 is the solution of the given equation by substituting x = 6 in the given equation.

LHS = 5x

= 5 x 6

= 30

RHS = 30

Therefore, LHS = RHS

Hence, we have proved the division property of equality

Division Property of Equality Examples

1. Solve 5x = 40

Solution:

Given Equation: 5x = 40

On dividing 8 from both the side, we get

\[\frac{5x}{8} = \frac{40}{8} \]

\[\frac{5x}{8} \] = 5

5x = 40

x = \[\frac{40}{5} \]

x = 8

To check, we can substitute the value of x in the original equation.

5 x 8 = 40

40 = 40

LHS = RHS

Hence proved

2. Tina brought 100 toffees for ₹ 50. What is the cost of each toffee?

Solution:

Total number of toffees brought by Tina = 100

Cost of 100 toffees = ₹ 50

Let the cost of each toffee be ‘a’. Hence, 100 times ‘a’ is the total cost ₹ 50.

Accordingly, the equation will be

100 a = 50

By dividing both side of the above equation by 10, we get

\[\frac{100}{10} = \frac{50}{10} \]

100 a x 10 = 50 x 10 (Cross multiplication)

1000 a = 500

a = \[\frac{1000}{500} \]

a = 2

To check, we can substitute the value of ‘a’ in the original equation.

500 a = 1000

500 x 2 = 1000

1000 = 1000

LHS = RHS

Hence proved.

In short, the division property of equality states that if we divide both sides of the equation by the same number, the equation remains balanced and equal. Hope you have now understood the division property of equality, now you can easily try questions based on this given on Vedantu’s official website.