Percentage Calculations

Percentages are tested extensively throughout the Edexcel GCSE Mathematics (1MA1) specification. You need to be confident with finding percentages of amounts, expressing one quantity as a percentage of another, calculating percentage change, and working with reverse percentages. This lesson covers all of these skills.

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Key Vocabulary

Term	Meaning
Percentage	A number expressed as a fraction of 100 (per cent = per hundred)
Multiplier	The decimal used to calculate a percentage (e.g. 0.35 for 35%)
Percentage increase	The amount has gone up by a given percentage
Percentage decrease	The amount has gone down by a given percentage
Reverse percentage	Working backwards from the final amount to find the original
Percentage change	The change expressed as a percentage of the original

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Finding a Percentage of an Amount

Non-Calculator Method (Paper 1)

Use known percentages and build up from them.

Percentage	How to Find It
50%	Divide by 2
25%	Divide by 4
10%	Divide by 10
5%	Half of 10%
1%	Divide by 100

Worked Example 1

Find 35% of 240 without a calculator.

• 10% of 240 = 24
• 30% of 240 = 24 x 3 = 72
• 5% of 240 = 24 / 2 = 12
• 35% of 240 = 72 + 12 = 84

Calculator Method (Papers 2 and 3)

Convert the percentage to a decimal (the multiplier) and multiply.

Worked Example 2

Find 17.5% of 360.

• Multiplier = 17.5 / 100 = 0.175
• 0.175 x 360 = 63

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Expressing One Quantity as a Percentage of Another

Formula: (Part / Whole) x 100

Worked Example 3

In a class of 30 students, 12 got an A. What percentage got an A?

• (12 / 30) x 100 = 40%

Worked Example 4

A shop buys a coat for 45 pounds and sells it for 63 pounds. What is the profit as a percentage of the cost price?

• Profit = 63 - 45 = 18 pounds
• Percentage profit = (18 / 45) x 100 = 40%

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Percentage Increase and Decrease

Using Multipliers

This is the most efficient method, especially for calculator papers.

Change	Multiplier
Increase by 15%	1.15
Increase by 3.5%	1.035
Decrease by 20%	0.80
Decrease by 8%	0.92
Decrease by 2.5%	0.975

Rule: To increase by r%, multiply by (1 + r/100). To decrease by r%, multiply by (1 - r/100).

Worked Example 5

A laptop costs 580 pounds. It is increased by 12%. What is the new price?

• Multiplier = 1.12
• New price = 580 x 1.12 = 649.60 pounds

Worked Example 6

A sofa normally costs 750 pounds. It is reduced by 35% in a sale. Find the sale price.

• Multiplier = 0.65
• Sale price = 750 x 0.65 = 487.50 pounds

Without a Calculator (Paper 1)

Worked Example 7

Increase 360 by 15%.

• 10% of 360 = 36
• 5% of 360 = 18
• 15% of 360 = 36 + 18 = 54
• New amount = 360 + 54 = 414

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Percentage Change

The formula for percentage change is:

Percentage change = (Change / Original) x 100

This formula works for both increases and decreases.

Worked Example 8

A house was bought for 180,000 pounds and is now worth 207,000 pounds. Find the percentage increase.

• Change = 207,000 - 180,000 = 27,000
• Percentage increase = (27,000 / 180,000) x 100 = 15%

Worked Example 9

A car's value drops from 15,000 pounds to 12,750 pounds. Find the percentage decrease.

• Change = 15,000 - 12,750 = 2,250
• Percentage decrease = (2,250 / 15,000) x 100 = 15%

Edexcel Exam Tip: Always divide by the original amount, not the new amount. This is the most common error students make in percentage change questions.

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Reverse Percentages

Reverse percentage problems give you the final amount after a percentage change, and ask you to find the original amount.

Method

1. Identify the multiplier used (e.g. 1.20 for a 20% increase, or 0.75 for a 25% decrease).
2. Divide the final amount by the multiplier.

Worked Example 10

After a 20% increase, a bill is 54 pounds. What was the original bill?

• Multiplier for 20% increase = 1.20
• Original = 54 / 1.20 = 45 pounds

Worked Example 11

In a sale, prices are reduced by 30%. A jacket costs 42 pounds in the sale. What was the original price?

• Multiplier for 30% decrease = 0.70
• Original = 42 / 0.70 = 60 pounds

Worked Example 12

A car depreciates by 15% in one year. It is now worth 10,200 pounds. What was it worth a year ago?

• Multiplier = 0.85
• Original = 10,200 / 0.85 = 12,000 pounds

Edexcel Exam Tip: Reverse percentage questions are worth 2-3 marks and appear frequently on Higher papers. A common wrong approach is to find 20% of the final amount and add it on — this gives the wrong answer because the percentage should be of the original, not the final amount.

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Expressing as a Percentage — Further Examples

Worked Example 13

A test has 80 marks. Priya scores 62. What is her score as a percentage?

• (62 / 80) x 100 = 77.5%

Worked Example 14

In 2023, a company made 1.2 million pounds profit. In 2024, profit rose to 1.38 million. Find the percentage increase.

• Change = 1.38 - 1.2 = 0.18 million
• Percentage increase = (0.18 / 1.2) x 100 = 15%

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Practice Problems