Percentage Change

Understanding Percentage Change

Robert started a business with an initial investment of Rs. 30,000 and in one year, it grew to Rs. 70,000.

Michael started a business at the same time as Robert with an initial investment of Rs. 25,000 and in one year, it grew to Rs. 65,000

Whose business grew rapidly?

Business Initial Value Final Value

Growth
(Final Value - Initial Value)

Robert

Rs. 30,000 Rs. 70,000 Rs. 40,000

Michael

Rs. 25,000 Rs. 65,000 Rs. 40,000

Can we say both businesses grew at the same rate because the growth in each business is Rs. 40,000?

No, we can't do so.

The growth rate should always be calculated with respect to the initial value and only then the rates can be compared.

The percentage change gives the change in a quantity with respect to its initial value.

It gives the growth/decay rate.

This helps us in comparing the quantities.


Definition of Percentage Change

The percent change (or) the percentage change of a quantity is the ratio of the change in the quantity to its initial value multiplied by 100

(OR)

The percentage change of a quantity is the percentage of its initial value either increased or decreased to get its final value.

As the name suggests, percent change is the change in the percentage of the quantity.

Generally, to convert a fraction into a percentage, we multiply it by 100

For example:

Definition of Percentage Change: A fraction 3 over 4 is multiplied by 100 to get 75 percent

In the same way, we multiply the fraction (ratio) of change in quantity to its initial value by 100 to get the percent change.

The percent change gives the change in quantity out of 100


Formula of Percentage Change

The formula of percentage change is:

\({\ \left(\dfrac{ \text{Final Value - Initial Value}}{\text{Initial Value}} \times 100 \right)}\% \)

When Final Value > Initial Value, the formula leads to a positive value which indicates a "percentage increase"

When Final Value < Initial Value, the formula leads to a negative value which indicates a "percentage decrease"


How to calculate Percentage Change Step by Step?

There are two methods to calculate the percentage change.

One is using the above formula and the other is using proportions.

Let us solve the following problem using both the methods.

If 64 is increased to 120, find the percentage change.

To perform these methods, we need to identify the initial and final values.

\[ \begin{aligned} \text{Initial Value}&=64\\[0.2cm] \text{Final Value}&=120 \end{aligned} \]

Method 1

a) Subtract the initial value from the final value

\[\begin{align} \text{Final Value - Initial Value} &= 120-64 \\ &=56\end{align} \]

b) Divide this by the initial value

\[\begin{align}\dfrac{ \text{Final Value - Initial Value}}{\text{Initial Value}} &= \dfrac{56}{64} \\ &=0.875\end{align}\]

c) Multiply the result by 100

\[ \begin{align}&\dfrac{ \text{Final Value - Initial Value}}{\text{Initial Value}} \times 100\\[0.2cm]& = 0.875\times 100\\[0.2cm]& = 87.5\% \end{align}\]

Since the answer is positive, it is a percentage increase.

Percentage Increase = 87.5%

We can understand this process easily using the following flowchart.

The process of finding percentage change is shown using a flowchart.

Method 2

In this method, we assume that

  • the initial value corresponds to 100%
  • the final value corresponds to \(x\)%

Then we can write the following proportion

\[ \dfrac{64}{100}= \dfrac{120}{x}\] Solve for \(x\) by cross multiplication. \[ \begin{aligned} 64x &= 12000\\[0.2cm] x &=187.5 \end{aligned} \]

This is the percentage of the final value when compared to the initial value

To find the percent change, we just subtract 100% from this.

\[ \begin{align} \text{Percentage change}&= 187.5\% -100\% \\&= 87.5\%\end{align}\]

Since it is positive, it is a percentage increase.

Percentage Increase = 87.5%

We can understand this process easily using the following flowchart.

The process of finding percentage change is shown using a flowchart.

We usually use Method 1 to calculate the percentage change.

 
important notes to remember
Important Notes
  1. The formula for percentage change is:
    \({\dfrac{ \text{Final Value - Initial Value}}{\text{Initial Value}} \!\times\! 100} \)
  2. If a quantity is increased by \(\mathbf{r\%}\), then its final value is obtained by multiplying its initial value by \(\mathbf{1+ \dfrac{r}{100}}\)
  3. If a quantity is decreased by \(\mathbf{r\%}\), then its final value is obtained by multiplying its initial value by \(\mathbf{1- \dfrac{r}{100}}\)

Percentage Change Calculator

We can enter the initial value and the final value in the following calculator and it will calculate the percentage change step by step.


Solved Examples

Example 1

 

 

The price of a toy car is decreased from Rs. 20 to Rs. 15 after a discount. 

Find the percentage of the discount.

Percentage change word problem: A toy car costs Rs 20. After a discount, it costs Rs 15.

Solution:

Initial price of the car = Rs. 20

Final price of the car = Rs. 15

The percentage change (the percentage of discount) is calculated using:

\[ \begin{aligned} &\text{Percentage Change }\\[0.3cm]&= \left( \dfrac{ \text{Final Value - Initial Value}}{ \text{Initial Value}} \times 100 \right)\%\\[0.3cm] &= \left( \dfrac{15-20}{20} \times 100 \right)\%\\[0.3cm] &= \left( \dfrac{-5}{20} \times 100 \right) \%\\[0.3cm] &= -25 \% \end{aligned} \]

The negative percentage change indicates percentage decrease.

Discount always leads to a percentage decrease.

Note: If you cross-check the answer, 25% of the initial price which is Rs. 20 is Rs. 5,

and Rs. 20 - Rs. 5 = Rs. 15, which is the price given after the discount.

\(\therefore\) The discount percentage = 25%
Example 2

 

 

The length of a tower is increased from 30 m to 50 m.

Find the percentage increase.

Percentage change word problems: Two towers of heights 30 m and 50 m.

Solution:

Initial height of the tower = 30 m

Final height of the tower = 50 m

The percentage change is calculated by the formula:

\[ \begin{aligned} &\text{Percentage Change }\\[0.3cm]&=\!\! \left( \dfrac{ \text{Final Height - Initial Height}}{ \text{Initial Height}} \!\times\! 100 \right)\!\!\%\\[0.3cm] &=\!\! \left( \dfrac{50-30}{30} \times 100 \right)\%\\[0.3cm] &=\!\! \left( \dfrac{20}{30} \times 100 \right) \%\\[0.3cm] &\approx 66.67 \% \end{aligned} \]

Positive percentage change indicates percentage increase.

\(\therefore\) The percentage increase  \(\mathbf{\approx}\) 66.67%
Example 3

 

 

A number 35 is misread to be 53.

Find the percentage error.

Solution:

Original number = 35

Misread number = 53

The percentage error (change) is calculated using:

\[ \begin{aligned} &\text{Percentage Change }\\[0.3cm]&=\!\! \left(\! \dfrac{ \text{Misread number - Original number}}{ \text{Original Number}} \!\times\! 100 \!\!\right)\!\!\%\\[0.3cm] &=\!\! \left( \dfrac{53-35}{35} \times 100 \right)\%\\[0.3cm] &=\!\! \left( \dfrac{18}{35} \times 100 \right) \%\\[0.3cm] &\approx 51\% \end{aligned} \]

\(\therefore\) The percentage error \(\mathbf{\approx}\) 51%
 
Thinking out of the box
Think Tank
  1. A number is increased by 20% and then decreased by 20%

    Find the net or percentage increase or decrease of the number.

Practice Questions

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 
 

Maths Olympiad Sample Papers

IMO (International Maths Olympiad) is a competitive exam in Mathematics conducted annually for school students. It encourages children to develop their math solving skills from a competition perspective.

You can download the FREE grade-wise sample papers from below:

To know more about the Maths Olympiad you can click here


Frequently Asked Questions (FAQs)

1. How do you find the percent of change?

The percentage of change is calculated using the formula:

\(\!\!\!\!\!\!\!  \left(\!\dfrac{ \text{Final Value - Initial Value}}{\text{Initial Value}} \!\!\times\!\! 100 \!\!\right)\!\!\% \)

2. Can you calculate percent change of percentages?

Yes, we just calculate it using the formula:

\(\!\!\!\!\!\left(\!\dfrac{ \text{Final Percentage - Initial Percentage}}{\text{Initial Percentage}} \!\!\times\!\! 100 \!\!\right)\!\!\% \)

3. What is the meaning of percent of change?

The percent change (or) the percentage change of a quantity is the ratio of the change in the quantity to its initial value multiplied by 100.

You can learn more about this in the "Understanding Percentage Change" and "Definition of Percentage Change" sections of this page.

4. How do I find the percentage of multiple changes?

We need to apply the percentage change formula multiple times as required.

  
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