🔄 Reverse Percentages

Cambridge Lower Secondary · Grade 7 · Percentages

The Big Idea
After 20% increase: divide by 1.20
£96 ÷ 1.20 = £80 original
After a Decrease
After 15% decrease: divide by 0.85
£68 ÷ 0.85 = £80 original
The Golden Rule
Original × Multiplier = Final
So: Original = Final ÷ Multiplier

Forward vs Reverse — spot the difference!

FORWARD ➡️
"Find 20% of £80"
£80 × 1.20 = £96
We know the original. Find the new value.
REVERSE 🔄
"£96 is AFTER a 20% increase"
£96 ÷ 1.20 = £80
We know the final value. Find the original.

The SAME multiplier — but we divide instead of multiply!

What you'll learn:

  • The key difference between forward and reverse percentages
  • Finding the original price before a % increase
  • Finding the original price before a % decrease / discount
  • Working out the pre-VAT price from a VAT-inclusive price
  • Spotting the common error (subtracting % from final — WRONG!)
  • Multi-step word problems: salary rises, sale prices, depreciation
Created by Mr Josh

📖 Learn: Reverse Percentages

Part 1: What Is a Reverse Percentage?

A reverse percentage problem gives you the value after a percentage change has already happened, and asks you to find the original value before the change.

Forward: "A coat costs £80. It goes up 20%. What is the new price?" → £80 × 1.20 = £96
Reverse: "A coat costs £96 after a 20% increase. What did it cost originally?" → £96 ÷ 1.20 = £80
💡 In both cases the multiplier is 1.20. Forwards you multiply; in reverse you divide.

Part 2: Building the Multiplier

The multiplier is based on 100% = 1.00 (the "whole"):

Change What it means Multiplier To reverse: divide by
+ 20% 100% + 20% = 120% 1.20 ÷ 1.20
+ 5% 100% + 5% = 105% 1.05 ÷ 1.05
− 15% 100% − 15% = 85% 0.85 ÷ 0.85
− 30% 100% − 30% = 70% 0.70 ÷ 0.70
VAT + 20% Same as +20% 1.20 ÷ 1.20

Part 3: The Method — Step by Step

Step 1: Identify the percentage change and whether it was an increase or decrease.
Step 2: Build the multiplier. Increase → 1 + (% ÷ 100). Decrease → 1 − (% ÷ 100).
Step 3: Divide the final value by the multiplier. Original = Final ÷ Multiplier.
Step 4: Check: multiply your original back by the multiplier — do you get the final value?
📌 Formula: Original = Final ÷ Multiplier

Part 4: The Common Error Trap

A jacket costs £120 after a 20% increase. Find the original price.

❌ WRONG METHOD
Student thinks: "take off 20%"
£120 × 0.20 = £24
£120 − £24 = £96
✗ £96 is WRONG

Check: £96 × 1.20 = £115.20 ≠ £120

✅ CORRECT METHOD
Multiplier = 1.20 (20% increase)
Original = £120 ÷ 1.20
= £100
✓ £100 is CORRECT

Check: £100 × 1.20 = £120 ✓

⚠️ Why is subtracting 20% wrong? Because the 20% was taken of the original (£100), not of the final price (£120). 20% of £100 = £20. 20% of £120 = £24. These are different!

Part 5: VAT and Real-Life Contexts

VAT (Value Added Tax) in the UK is 20%. Shop prices often include VAT. To find the pre-VAT price:

A phone costs £360 including 20% VAT. Find the pre-VAT price.
Multiplier = 1.20 → Pre-VAT = £360 ÷ 1.20 = £300
💡 The VAT amount itself = £360 − £300 = £60. You can verify: £300 × 0.20 = £60 ✓

Salary problems work the same way:

After a 15% pay rise, Maya earns £23,000. What was her original salary?
Multiplier = 1.15 → Original = £23,000 ÷ 1.15 = £20,000
Created by Mr Josh

💡 Worked Examples

Example 1: After a Percentage Increase

A TV costs £552 after a 15% increase. Find the original price.

Step 1: 15% increase → multiplier = 1 + 0.15 = 1.15
Step 2: Original = Final ÷ Multiplier = £552 ÷ 1.15
Step 3: £552 ÷ 1.15 = £480
Check: £480 × 1.15 = £552 ✓
💡 Increase → multiplier > 1. Dividing by something greater than 1 makes the answer smaller — the original should be less than the final.

Example 2: After a Percentage Decrease

A dress is in a sale at £51 after a 32% discount. What was the original price?

Step 1: 32% decrease → multiplier = 1 − 0.32 = 0.68
Step 2: Original = £51 ÷ 0.68
Step 3: £51 ÷ 0.68 = £75
Check: £75 × 0.68 = £51 ✓
💡 Decrease → multiplier < 1. Dividing by something less than 1 makes the answer larger — the original should be more than the final sale price.

Example 3: Reverse VAT

A laptop costs £840 including 20% VAT. Find: (a) the pre-VAT price, (b) the VAT amount.

Step 1: VAT = 20% increase → multiplier = 1.20
Step 2: Pre-VAT price = £840 ÷ 1.20 = £700
Step 3 (b): VAT amount = £840 − £700 = £140
Check: £700 × 0.20 = £140 ✓ and £700 + £140 = £840 ✓

Example 4: Salary Word Problem

After a 15% pay rise, Ahmed's salary is £28,175. What was his salary before the rise?

Step 1: 15% rise → multiplier = 1.15
Step 2: Original = £28,175 ÷ 1.15
Step 3: £28,175 ÷ 1.15 = £24,500
Check: £24,500 × 1.15 = £28,175 ✓
💡 Word problems often say "after a rise/fall of X%". That's your signal to use the reverse method.
Created by Mr Josh

⚙️ Interactive Visualizer

🔄 The Reverse Percentage Machine

Enter the final value and the percentage change. Watch the machine run in reverse to find the original!

🎪 Before/After Revealer

A product's after-sale price is shown. Enter the original price, then lift the curtain!

Curtain closed — enter your answer below!

⚠️ Common Mistake Trap — Spot the Error

Two students attempt the same problem. Can you see who's right and who's wrong?

Student A
Student B

💼 Salary Timeline — Work Backwards!

A worker's salary has changed over 3 years. You know the current salary. Work backwards through each year to find the starting salary.

?
Year 0
Starting Salary
?
Year 1
After Year 1
?
Year 2
After Year 2
Now
Current Salary
Enter the salary after Year 2:
Enter the salary after Year 1:
Enter the starting salary:

Created by Mr Josh

✏️ Exercise 1 — Find Original After Increase

Each value is the result AFTER a percentage increase. Find the original.

Created by Mr Josh

✏️ Exercise 2 — Find Original After Decrease

Each value is the result AFTER a percentage decrease. Find the original.

Created by Mr Josh

✏️ Exercise 3 — Reverse VAT (20%)

Each price includes 20% VAT. Find the pre-VAT price.

Created by Mr Josh

✏️ Exercise 4 — Mixed (Increase & Decrease)

Read carefully — some are increases, some are decreases!

Created by Mr Josh

🌍 Exercise 5 — Word Problems

Identify the percentage change from the context, then solve.

Created by Mr Josh

📝 Practice — 20 Questions

A full mix of reverse percentage types. Round to 2 decimal places where needed.

Created by Mr Josh

🏆 Challenge — 8 Questions

Multi-step problems, reasoning questions, and tricky contexts.

Created by Mr Josh