Percentage Change

Definition of Percentage Change

Percentage change measures how much a quantity has changed compared to its original value, expressed as a percentage. It shows the size of the change relative to where you started. This change can be an increase or a decrease.

To find the percentage change, you first find the difference between the original value and the new value, then express this difference as a percentage of the original value.

Calculating Percentage Change

The formula to calculate percentage change is:

Percentage change = $\frac{\left|\text{Change}\right|}{\text{Original value}} \times 100\%$

Here, Change means the difference between the new value and the original value. Use the absolute value of this difference to calculate the size of the change, then determine if it is an increase or a decrease by comparing the new value to the original.

For example, if a shop reduces the price of a jacket from $\text{\u00A3}80$ to $\text{\u00A3}60$, the change is $\text{\u00A3}60 - \text{\u00A3}80 = -\text{\u00A3}20$. The absolute change is $\text{\u00A3}20$.

The percentage change is:

$$\frac{20}{80} \times 100 = 25\%$$

Since the price decreased, the percentage change is a decrease of $25\%$.

Interpreting Percentage Change

The sign of the percentage change tells you whether the value has increased or decreased:

• A positive percentage change means an increase.
• A negative percentage change means a decrease.

Understanding the context is important. For example, a $10\%$ increase in population means the population grew by $10\%$ compared to the original number. A $15\%$ decrease in sales means sales dropped by $15\%$ from the original amount.

Always check which value is original and which is new before calculating. The original value is the starting point, and the new value is the result after the change.

For instance, if a town's population was 12,000 last year and is now 13,200, the change is $13,200 - 12,000 = 1,200$. The percentage change is:

$$\frac{1,200}{12,000} \times 100 = 10\%$$

This is a $10\%$ increase in population.

Applications of Percentage Change

Percentage change is widely used in everyday life and exams to compare changes over time or between different quantities. Common examples include:

• Changes in prices of goods or services
• Population growth or decline
• Changes in measurements such as distance, weight, or temperature
• Comparing exam results or sports statistics over time

Percentage change helps to understand the significance of a change relative to the original size. For example, a $\text{\u00A3}5$ increase on a $\text{\u00A3}20$ item is a $25\%$ increase, but the same $\text{\u00A3}5$ increase on a $\text{\u00A3}100$ item is only a $5\%$ increase.

When solving problems involving percentage change, always:

• Identify the original and new values clearly
• Calculate the absolute change
• Use the formula to find the percentage change
• Interpret the result in context (increase or decrease)

For example, if a population decreases from 50,000 to 47,500, the change is $-2,500$. The percentage change is:

$$\frac{2,500}{50,000} \times 100 = 5\%$$

This means a $5\%$ decrease in population.

Percentage change can also be used to compare changes in different contexts, such as comparing price changes in two different shops or population changes in two different cities.

For example, if Shop A increases the price of a product from $\text{\u00A3}40$ to $\text{\u00A3}50$, and Shop B increases the price of the same product from $\text{\u00A3}60$ to $\text{\u00A3}66$, which shop has the larger percentage increase?

Calculate Shop A's percentage increase:

$$\frac{50 - 40}{40} \times 100 = \frac{10}{40} \times 100 = 25\%$$

Calculate Shop B's percentage increase:

$$\frac{66 - 60}{60} \times 100 = \frac{6}{60} \times 100 = 10\%$$

Shop A has the larger percentage increase at $25\%$.

This shows how percentage change allows fair comparison even when original values differ.

For instance, if a car's value drops from $\text{\u00A3}12,000$ to $\text{\u00A3}9,000$, the change is $-\text{\u00A3}3,000$. The percentage change is:

$$\frac{3,000}{12,000} \times 100 = 25\%$$

So, the car's value has decreased by $25\%$.