Percent Increase and Decrease

Percent of change measures how much a value has increased or decreased, expressed as a percentage of the original value. It answers the question: “By what percentage did this go up or down?”

The Percent of Change Formula

$\text{Percent of Change} = \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100$

• If the result is positive, it is a percent increase
• If the result is negative, it is a percent decrease

In practice, many people find it easier to compute the absolute amount of change and then state whether it is an increase or decrease:

$\text{Amount of Change} = |\text{New Value} - \text{Original Value}|$

$\text{Percent of Change} = \frac{\text{Amount of Change}}{\text{Original Value}} \times 100$

Percent Increase

A percent increase occurs when the new value is larger than the original.

Example 1: Price Increase

A product’s price goes from $40 to $52. What is the percent increase?

$\text{Amount of Change} = 52 - 40 = 12$

$\text{Percent Increase} = \frac{12}{40} \times 100 = 30\%$

Answer: 30% increase

Example 2: Population Growth

A town’s population grows from 8,000 to 9,200. What is the percent increase?

$\text{Amount of Change} = 9{,}200 - 8{,}000 = 1{,}200$

$\text{Percent Increase} = \frac{1{,}200}{8{,}000} \times 100 = 15\%$

Answer: 15% increase

Percent Decrease

A percent decrease occurs when the new value is smaller than the original.

Example 3: Sale Price

A jacket originally costs $80 and is on sale for $60. What is the percent decrease?

$\text{Amount of Change} = 80 - 60 = 20$

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

Answer: 25% decrease

Example 4: Weight Loss

A person’s weight goes from 200 pounds to 185 pounds. What is the percent decrease?

$\text{Amount of Change} = 200 - 185 = 15$

$\text{Percent Decrease} = \frac{15}{200} \times 100 = 7.5\%$

Answer: 7.5% decrease

Finding the New Value from a Percent Change

Sometimes you know the original value and the percent change and need to find the new value.

For an increase:

$\text{New Value} = \text{Original} \times (1 + \text{percent as decimal})$

For a decrease:

$\text{New Value} = \text{Original} \times (1 - \text{percent as decimal})$

Example 5: Applying an Increase

A salary of $50,000 gets a 6% raise. What is the new salary?

$\text{New Salary} = 50{,}000 \times (1 + 0.06) = 50{,}000 \times 1.06 = 53{,}000$

Answer: $53,000

Example 6: Applying a Decrease

A car worth $24,000 depreciates by 15%. What is the new value?

$\text{New Value} = 24{,}000 \times (1 - 0.15) = 24{,}000 \times 0.85 = 20{,}400$

Answer: $20,400

Finding the Original Value

If you know the new value and the percent change, work backward:

After an increase:

$\text{Original} = \frac{\text{New Value}}{1 + \text{percent as decimal}}$

After a decrease:

$\text{Original} = \frac{\text{New Value}}{1 - \text{percent as decimal}}$

Example 7: Working Backward from a Sale Price

A shirt is on sale for $45 after a 25% discount. What was the original price?

$\text{Original} = \frac{45}{1 - 0.25} = \frac{45}{0.75} = 60$

Answer: $60

Common Mistake: Using the Wrong Base

The most frequent error is dividing by the new value instead of the original value:

Wrong: $\frac{52 - 40}{52} \times 100 = 23.1\%$

Correct: $\frac{52 - 40}{40} \times 100 = 30\%$

The percent of change is always relative to the original (starting) value — the value you started from.

Successive Percent Changes

When percent changes happen one after another, you cannot simply add the percentages. Each change applies to the new value, not the original.

Example 8: A 20% increase followed by a 20% decrease

Start with $100.

After 20% increase: $100 \times 1.20 = 120$

After 20% decrease: $120 \times 0.80 = 96$

Result: $96, not $100. A 20% increase followed by a 20% decrease results in a net 4% decrease, because the decrease was calculated on the larger amount ($120).

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• Percent of change = (change ÷ original) × 100
• Always divide by the original value, not the new value
• Positive result = increase; negative result = decrease
• To apply a percent change: multiply by $(1 + r)$ for increase or $(1 - r)$ for decrease
• Successive changes do not simply add — each applies to the current value

Return to Arithmetic for more foundational math topics.