What Is Percentage Change?
The percentage change (or) the percentage change of a quantity is the ratio of the difference in the quantity to its initial value multiplied by 100. There is always a change in percentage change (or) the percent change of a quantity when the percent of its initial value is either increased or decreased to obtain its final value.
Definition of Percentage Change
Percentage Change is the difference coming after subtracting the old value from the new value and then divide by the old value and the final answer will be multiplied by 100 to show it as a percentage. Generally, to convert a fraction into a percent, we multiply it by 100.
For example, 3/4 × 100 = 75%
In the same way, we multiply the fraction (ratio) of the difference in quantity to its initial value by 100 to get the percentage change. The percentage change gives the difference in quantity out of 100.
Understanding Percentage Change with help of an example.
Robert started a business with an initial investment of Rs. 30,000 and in one year, it grew to Rs. 70,000. The growth of Robert is 40000 (70000 - 30000). 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. The growth of Robert is 40000 (65000 - 25000). Whose business grew rapidly?
We can say that the growth value of both businesses is 40000, but the growth rate is not the same.
The growth rate should always be calculated with respect to the initial value and only then the rates can be compared. The percent change gives the difference in a quantity with respect to its initial value. It gives the growth/decay rate. This helps us in comparing the quantities.
Example
Percentage Change Calculation
Percentage change is a way to express the relative change between two numbers as a percentage. It helps determine the increase or decrease in a value over time, making it easier to compare changes in quantities.
The general approach to calculating percentage change includes:
- Identifying the original and new values.
- Subtracting the original value from the new value to find the change.
- Dividing the change by the original value.
- Multiplying the result by 100 to convert it into a percentage.
Calculating Percentage Increase
Percentage increase is used when the new value is greater than the original value. The formula for percentage increase is:
\[ \text{Percentage Increase} = \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100 \]Example:
If the original value is $50 and the new value is $75, the percentage increase is calculated as:
- Step 1: Subtract the original value from the new value: \( 75 - 50 = 25 \).
- Step 2: Divide the change by the original value: \( \frac{25}{50} = 0.5 \).
- Step 3: Multiply by 100 to find the percentage increase: \( 0.5 \times 100 = 50\% \).
Calculating Percentage Decrease
Percentage decrease is used when the new value is less than the original value. The formula for percentage decrease is:
\[ \text{Percentage Decrease} = \frac{\text{Original Value} - \text{New Value}}{\text{Original Value}} \times 100 \]Example:
If the original value is $120 and the new value is $90, the percentage decrease is calculated as:
- Step 1: Subtract the new value from the original value: \( 120 - 90 = 30 \).
- Step 2: Divide the change by the original value: \( \frac{30}{120} = 0.25 \).
- Step 3: Multiply by 100 to find the percentage decrease: \( 0.25 \times 100 = 25\% \).
Calculating the New Value after Percentage Change
After finding the percentage increase or decrease, you can calculate the new value by using the following formulas:
For Percentage Increase:
\[ \text{New Value} = \text{Original Value} \times (1 + \frac{\text{Percentage Increase}}{100}) \]For Percentage Decrease:
\[ \text{New Value} = \text{Original Value} \times (1 - \frac{\text{Percentage Decrease}}{100}) \]Example:
If the original value is $200 and you want to increase it by 15%, the new value is calculated as:
- Step 1: Multiply the original value by the percentage increase (in decimal form): \( 200 \times (1 + 0.15) = 200 \times 1.15 \).
- Step 2: The new value is \( 230 \).
Real-life Applications of Percentage Change
Percentage change calculations are used in many real-life situations, such as:
- Determining how much more or less a product costs after a price increase or discount.
- Calculating salary increases or bonuses based on percentage growth.
- Assessing changes in stock prices or financial markets over time.
Common Operations with Percentage Change
Percentage Increase: Use when the new value is greater than the original value.
Percentage Decrease: Use when the new value is less than the original value.
Modifying Values: Adjust original values after a percentage change to determine new quantities, whether for sales, prices, or other applications.
| Problem Type | Description | Steps to Solve | Example |
|---|---|---|---|
| Finding the Percentage Increase | Calculating the percentage increase from an original value to a new value. |
|
For an original value of $50 and a new value of $75, the percentage increase is calculated as: \( \frac{75 - 50}{50} \times 100 = 50\% \). |
| Finding the Percentage Decrease | Calculating the percentage decrease from an original value to a new value. |
|
For an original value of $120 and a new value of $90, the percentage decrease is calculated as: \( \frac{120 - 90}{120} \times 100 = 25\% \). |
| Finding the New Value after Percentage Increase | Calculating the new value after applying a percentage increase. |
|
For an original value of $200 and a 15% increase, the new value is calculated as: \( 200 + (200 \times 0.15) = 230 \). |
| Finding the New Value after Percentage Decrease | Calculating the new value after applying a percentage decrease. |
|
For an original value of $150 and a 20% decrease, the new value is calculated as: \( 150 - (150 \times 0.20) = 120 \). |
