How to Solve: Reverse Percentages

How to Solve: Reverse Percentages

(GCSE & A-Level Maths – Ace Your Exam!)

Introduction

Mastering reverse percentages unlocks real-life money problems—like calculating original prices after discounts, VAT, or pay rises—and can earn you 5-10% of your GCSE/A-Level maths exam marks in just one question.

What You Need To Know First

Before starting, you must understand: 1. Basic percentages – Finding 10%, 20%, etc., of a number. 2. Percentage increase/decrease – How to apply a % change to a value. 3. Algebraic equations – Solving for an unknown (e.g., x).

Key Vocabulary

Formulas To Know

1. Reverse Percentage Formula (Using Multiplier)

Formula: Original Value = New Value ÷ Multiplier

• New Value = The amount after the % change.
• Multiplier = 1 + (% increase as a decimal) OR 1 – (% decrease as a decimal).
• MEMORISE THIS – This is the fastest method.

Example: If a price increased by 15% to £115, the multiplier is 1.15.

2. Reverse Percentage Formula (Using Algebra)

Formula: New Value = Original Value × (1 ± % change as a decimal)

• Rearrange to solve for the Original Value.
• Given on exam sheet (but you must know how to rearrange it).

Example: If £60 is 80% of the original price: 60 = x × 0.80 → x = 60 ÷ 0.80 = £75

Step-by-Step Method

Step 1: Identify the % change and new value

• Read the question carefully.
• Note whether it’s an increase or decrease.
• Write down the new value (the amount after the % change).

Step 2: Convert the % change to a multiplier

• Increase? → 1 + (% as a decimal) (e.g., 15% increase → 1 + 0.15 = 1.15)
• Decrease? → 1 – (% as a decimal) (e.g., 20% decrease → 1 – 0.20 = 0.80)

Step 3: Divide the new value by the multiplier

• Original Value = New Value ÷ Multiplier
• This reverses the % change to find the original amount.

Step 4: Check your answer

• Apply the % change to your answer.
• It should match the new value given in the question.

WORKED EXAMPLE (Using Steps)

Question: A laptop is on sale for £480 after a 20% discount. What was its original price?

Step 1: Identify % change and new value

• % change = 20% decrease (discount).
• New value = £480.

Step 2: Convert % to multiplier

• 20% decrease → 1 – 0.20 = 0.80

Step 3: Divide new value by multiplier

• Original Value = £480 ÷ 0.80 = £600

Step 4: Check

• 20% of £600 = £120
• £600 – £120 = £480 ✅

What we did and why: We reversed the 20% discount by dividing by 0.80 (the multiplier) to find the original price before the discount.

Worked Examples

Example 1 – Basic (Increase)

Question: A salary is £27,500 after a 10% pay rise. What was the original salary?

Solution: 1. % change = 10% increase. 2. Multiplier = 1 + 0.10 = 1.10 3. Original Salary = £27,500 ÷ 1.10 = £25,000 4. Check: 10% of £25,000 = £2,500 → £25,000 + £2,500 = £27,500 ✅

What we did and why: We reversed the 10% increase by dividing by 1.10 to find the salary before the pay rise.

Example 2 – Medium (Decrease with Algebra)

Question: A phone costs £360 after a 25% discount. Find the original price.

Solution (Algebra Method): 1. Let x = original price. 2. 25% discount → New price = x × (1 – 0.25) = 0.75x 3. Given new price = £360 → 0.75x = 360 4. x = 360 ÷ 0.75 = £480 5. Check: 25% of £480 = £120 → £480 – £120 = £360 ✅

What we did and why: We set up an equation to represent the discount, then solved for the original price.

Example 3 – Exam-Style (Disguised Question)

Question: A car’s value depreciates by 12% each year. After 1 year, it’s worth £11,000. What was its value 1 year ago?

Solution: 1. % change = 12% decrease (depreciation). 2. Multiplier = 1 – 0.12 = 0.88 3. Original Value = £11,000 ÷ 0.88 = £12,500 4. Check: 12% of £12,500 = £1,500 → £12,500 – £1,500 = £11,000 ✅

What we did and why: We treated depreciation like a percentage decrease and reversed it using the multiplier method.

Common Mistakes

Exam Traps

1-Minute Recap

(Spoken naturally, as if to a student the night before the exam.)

"Reverse percentages are just about undoing a % change. Here’s the fast version:

1. Find the multiplier – If it’s an increase, add the % to 1 (e.g., 15% → 1.15). If it’s a decrease, subtract from 1 (e.g., 20% → 0.80).
2. Divide the new value by the multiplier – That’s your original amount.
3. Check it – Apply the % change to your answer. If it matches the new value, you’re golden.

Example: If £90 is after a 10% decrease, the multiplier is 0.90. So, £90 ÷ 0.90 = £100. Done.

Watch out for: - Adding instead of dividing. - Mixing up increases and decreases. - Forgetting to check your answer.

You’ve got this—just reverse the % and divide!"