# How to calculate percentage change

**Why do you need to know how to calculate percentage change?**

The ratio of the difference in the amount to its starting value multiplied by $$100$$ is the percentage change or the percent change of a quantity. When the percentage of a quantity's initial value is either increased or decreased to get its final value, there is always a percentage change or percent change. Percentage change is the difference between the old and the new values after subtracting the old from the latest, dividing by the old and multiplying the final result by $$100$$ to show it as a percentage.

For example, $$\frac{1}{2}\times 100=50\%$$.

To calculate the percentage change, multiply the fraction (ratio) of the difference of the final value and initial value to its initial value by $$100$$. The difference in amount is shown as a percentage change out of $$100$$.

The percentage change formula shows the difference between the final and initial values to the initial value. The percentage change formula is stated mathematically as:

$$ \frac{\text {final value}-\text{ initial value}}{initial value}\times 100$$.

**E1.12: Calculate percentage increase or decrease**

**Percentage increase**

While comparing the increase in quantity through time, you need to subtract the increased value from the original value. The relative increase against the original value is then calculated using this difference and expressed as a percentage. The percentage increase formula is as follows:

**Percentage decrease**

While comparing the decrease in quantity through time, you need to subtract the decreased value from the original value. The relative decrease against the original value is then calculated using this difference and expressed as a percentage. The formula for percentage decrease is as follows:

**Worked examples**

**Example 1:**

Calculate the following:

The annual salary of $$A$$ is increased by $$50,000\text{ dollars}$$ to $$60,000\text{ dollars}$$. Find the percentage increase.

The annual salary of $$B$$ is decreased by $$70,000\text{ dollars}$$ to $$60,000\text{ dollars}$$. Find the percentage decrees.

*Question 1*

**Step 1: Write the given information.**

$$\text{Original salary} = 50,000\text{ dollars}$$

$$\text{Increased salary} = 60,000\text{ dollars}$$

The increase in salary is $$60,000-50,000 = 10,000\text{ dollars}$$

**Step 2: Write the formula.**

$$\text{Percentage increase} = \frac{\text {Increase in salary}}{\text {Original salary}} \times 100$$

**Step 3: Substitute the given value in the formula.**

$$\text{Percentage increase} = \frac{10,000}{50,000} \times 100$$

$$\text {Percentage increase} = \frac{1}{5} \times 100$$

$$\text {Percentage increase} = 20\%$$

Hence, the percentage increase is $$20\%$$.

*Question 2*

**Step 1: Write the given information.**

$$\text {Original salary} = 70,000\text{ dollars}$$

$$\text {Decreased salary} = 60,000\text{ dollars}$$

The decrease in salary is $$70,000-60,000 = 10,000\text{ dollars}$$.

**Step 2: Write the formula.**

$$\text {Percentage decrease} = \frac{\text {Decrease in salary}}{\text {Original salary}} \times 100$$

**Step 3: Substitute the given value in the formula.**

$$\text{Percentage decrease} = \frac{10,000}{70,000} \times 100$$

$$\text{Percentage increase} = \frac{1}{7} \times 100$$

$$\text{Percentage increase} = 14.28\%$$

Hence, the percentage decrease is $$14.28\%$$.

**Example 2**: Due to the outbreak of an epidemic, the population of a small town decreased from $$15,000$$ to $$12,300$$. What is the population decrease as a percentage?

**Step 1: Write the given information.**

$$\text{Original population} = 15,000$$

$$\text{Decreased population} = 12,300$$

The decrease in population is $$15,000-12,300 = 2,700$$

**Step 2: Write the formula.**

$$\text {Percentage decrease} = \frac{\text{Decrease in population}}{\text{Original population}} \times 100$$

**Step 3: Substitute the given value in the formula.**

$$\text{Percentage decrease}= \frac{2,700}{15,000} \times 100$$

$$\text{Percentage increase} = \frac{27}{15} \times 10$$

$$\text{Percentage increase} = 18\%$$

Hence, the population decrease is $$18\%$$

**Example 3:** The number $$72$$ is misread as $$32$$. Find the percentage decrease using the percentage decrease formula.

**Step 1: Write the given information.**

$$\text{Original value} = 72$$

$$\text{New value} = 32$$

There is a decrease in the value as $$72-32=40$$.

**Step 2: Write the formula.**

$$\text{Percentage decrease} = \frac{\text {Decrease in salary}}{\text {Original salary}} \times 100$$

**Step 3: Substitute the given value in the formula.**

$$\text{Percentage decrease}=\frac{40}{72}\times 100$$

$$\text {Percentage decrease} =55.5\%$$

Therefore, the required percentage decrease is $$55.5%$$.

**Example 4: **Solve the following problems

If the marks of $$A$$ is increased from $$50$$ to $$80$$, the percentage change.

If the marks of $$B$$ is decreased from $$60$$ to $$40$$, find the percentage change.

*Question 1*

**Step 1: Write the given information.**

$$\text{Original mark} = 50$$

$$\text{Increased mark} = 80$$

The increase in marks is $$80 - 50 =30$$.

**Step 2: Write the formula for increased percentage.**

$$\text {Percentage increase} = \frac{\text{Increase in mark}}{\text{Original mark}} \times 100$$

**Step 3: Substitute the given value in the formula.**

$$\text{Percentage increase} = \frac{30}{50} \times 100$$

$$\text{Percentage increase} = \frac{3}{5} \times 100$$

$$\text{Percentage increase} = 60\%$$

Hence, the percentage increase is $$60\%$$.

*Question 2*

**Step 1: Write the given information.**

$$\text{Original mark} = 60$$

$$\text{Decreased salary} = 40$$

The decrease in salary is $$60-40 = 20$$.

**Step 2: Write the formula for decreased percentage.**

$$\text{Percentage decrease} = \frac{\text{Decrease in mark}}{\text{Original mark}} \times 100$$

**Step 3: Substitute the given value in the formula.**

$$\text{Percentage decrease} = \frac{20}{60} \times 100$$

$$\text{Percentage decrease} = \frac{2}{6} \times 100$$

$$\text{Percentage decrease} = 33.33\%$$

Hence, the percentage decrease is $$33.33%$$.