Percentage Change and Reverse Percentages

Percentages are essential throughout GCSE Mathematics and everyday life. This lesson focuses on calculating percentage increase and decrease, expressing one quantity as a percentage of another, and solving reverse percentage problems — where you must work backwards from a final amount to find the original. These skills are tested frequently on AQA GCSE papers at both Foundation and Higher tiers.

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Expressing One Quantity as a Percentage of Another

To express A as a percentage of B:

Formula: (A / B) x 100

Worked Example 1: There are 12 girls in a class of 30. What percentage are girls?

• (12 / 30) x 100 = 40%

Worked Example 2: A student scored 56 out of 80 on a test. What percentage is this?

• (56 / 80) x 100 = 70%

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Calculating a Percentage of an Amount

Method 1: Divide and Multiply

• Find 1% by dividing by 100, then multiply by the required percentage.

Method 2: Use a Decimal Multiplier

Percentage	Decimal Multiplier
10%	0.10
15%	0.15
25%	0.25
50%	0.50
5%	0.05
17.5%	0.175

Worked Example 3: Find 35% of 240.

• 240 x 0.35 = 84

Exam Tip: Using the multiplier method is faster and less error-prone than finding 10% then 5% separately. Learn to convert any percentage to a decimal quickly.

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Percentage Increase and Decrease

Method Using a Multiplier

To increase by a percentage, the multiplier = 1 + (percentage / 100).

To decrease by a percentage, the multiplier = 1 - (percentage / 100).

Change	Percentage	Multiplier
Increase	20%	1.20
Increase	5%	1.05
Increase	12.5%	1.125
Decrease	15%	0.85
Decrease	30%	0.70
Decrease	3%	0.97

Worked Example 4: Increase 350 by 20%.

• 350 x 1.20 = 420

Worked Example 5: Decrease 800 by 15%.

• 800 x 0.85 = 680

graph TD
    A[Percentage Change] --> B[Increase?]
    A --> C[Decrease?]
    B --> D[Multiplier = 1 + percentage/100]
    C --> E[Multiplier = 1 - percentage/100]
    D --> F[Multiply original by multiplier]
    E --> F

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Finding the Percentage Change

To find the percentage change between an original value and a new value:

Formula: Percentage change = (change / original) x 100

Worked Example 6: A house was bought for 200,000 pounds and sold for 230,000 pounds. What is the percentage increase?

• Change = 230,000 - 200,000 = 30,000
• Percentage change = (30,000 / 200,000) x 100 = 15%

Worked Example 7: A car was bought for 18,000 pounds and is now worth 13,500 pounds. What is the percentage decrease?

• Change = 18,000 - 13,500 = 4,500
• Percentage change = (4,500 / 18,000) x 100 = 25%

Exam Tip: The percentage change formula ALWAYS uses the original value as the denominator, not the new value. This is the most common mistake on percentage questions.

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Reverse Percentages

A reverse percentage problem gives you the final amount (after a percentage increase or decrease) and asks you to find the original amount.

The Key Idea

If you know what percentage the final amount represents, you can find 1% and then 100%.

Method

1. Identify what percentage the given amount represents.
2. Divide the given amount by that percentage (as a decimal) to find the original.

Worked Example 8: A coat costs 68 pounds after a 15% reduction. What was the original price?

• After a 15% decrease, the price represents 100% - 15% = 85% of the original.
• 85% = 0.85
• Original price = 68 / 0.85 = 80 pounds

Worked Example 9: A population increased by 12% to 8,960. What was the original population?

• After a 12% increase, the value represents 100% + 12% = 112% of the original.
• 112% = 1.12
• Original population = 8,960 / 1.12 = 8,000

Worked Example 10: A laptop costs 552 pounds including 15% VAT. What is the price before VAT?

• 552 represents 115% of the pre-VAT price.
• Pre-VAT price = 552 / 1.15 = 480 pounds

graph TD
    A[Reverse Percentage Problem] --> B[Identify the percentage the final amount represents]
    B --> C[Convert to a decimal multiplier]
    C --> D[Divide the final amount by the multiplier]
    D --> E[Original amount found]

Exam Tip: In reverse percentage questions, you must NOT calculate the percentage of the given amount and add/subtract it. The given amount is NOT the original — it is the result of the change. Always divide by the multiplier.

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Expressing a Change as a Percentage

Some questions ask you to compare two values and express the difference as a percentage of one of them.

Worked Example 11: Last year, 480 students passed an exam. This year, 540 students passed. Express the increase as a percentage of last year's figure.

• Increase = 540 - 480 = 60
• Percentage increase = (60 / 480) x 100 = 12.5%

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Common Mistakes

Mistake	Correct Approach
Adding/subtracting the percentage of the final amount in reverse problems	Divide the final amount by the multiplier
Using the new value as the denominator for percentage change	Always use the original value
Forgetting that 20% increase means the multiplier is 1.20, not 0.20	The multiplier includes the original 100%

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Summary

• Express A as a percentage of B: (A / B) x 100.
• Use multipliers for percentage increase (1 + p/100) and decrease (1 - p/100).
• Percentage change = (change / original) x 100.
• For reverse percentages, divide the given amount by the percentage it represents as a decimal.
• Never calculate the percentage of the final amount and add/subtract it in reverse problems.