GCSE Maths Number - How to Solve: Reverse Percentages – Complete Guide

How to Solve: Reverse Percentages – Complete Guide

(GCSE / A-Level Physics, Chemistry, Biology – Exam-Ready!)

Introduction

"Mastering reverse percentages lets you crack exam questions worth 4–6 marks—like calculating original drug concentrations after dilution, or finding the pre-discount price of lab equipment. Miss this, and you’re leaving easy marks on the table."

WHAT YOU NEED TO KNOW FIRST

1. Percentage increase/decrease: How to calculate a percentage change (e.g., 20% increase = ×1.20).
2. Algebra basics: Rearranging equations (e.g., if A = B × C, then B = A ÷ C).
3. Multiplier method: Converting percentages to decimals (e.g., 15% = 0.15).

KEY TERMS & FORMULAS

Key Terms

• Original value (O): The starting amount before any percentage change.
• New value (N): The amount after the percentage change.
• Multiplier (M): The decimal equivalent of the percentage change (e.g., 30% increase = 1.30).

Formulas

1. For percentage increase:
   N = O × (1 + M)
2. N = New value
3. O = Original value
4. M = Percentage increase as a decimal (e.g., 25% = 0.25)

5. MEMORISE THIS: Rearrange to O = N ÷ (1 + M) for reverse calculations.

6. For percentage decrease:
   N = O × (1 – M)

7. Rearrange to O = N ÷ (1 – M) for reverse calculations.

8. MEMORISE THIS.

9. Multiplier shortcut:

10. If a value increases by x%, multiply by (1 + x/100).
11. If it decreases by x%, multiply by (1 – x/100).

STEP-BY-STEP METHOD

Step 1: Identify the type of change

• Is the question about an increase or decrease?
• Example: "A price increased by 15%" → increase.
• "A solution was diluted by 20%" → decrease.

Step 2: Convert the percentage to a multiplier

• Increase: 1 + (percentage ÷ 100)
• 15% increase → 1 + 0.15 = 1.15
• Decrease: 1 – (percentage ÷ 100)
• 20% decrease → 1 – 0.20 = 0.80

Step 3: Write the equation

• Use N = O × M (where M is the multiplier).
• Rearrange to solve for O:
• O = N ÷ M

Step 4: Plug in the numbers and calculate

• Substitute the given values into O = N ÷ M.
• Use a calculator if needed (exams allow this!).

Step 5: Check units and context

• Does the answer make sense?
• If the original price was higher than the new price, it should be a decrease.
• If the original concentration was lower, it should be an increase.

WORKED EXAMPLES

Example 1 – Basic (Increase)

Question: A lab coat’s price increased by 25%. It now costs £60. What was the original price?

Step 1: Type of change → Increase (25%). Step 2: Multiplier → 1 + 0.25 = 1.25. Step 3: Equation → O = N ÷ M = 60 ÷ 1.25. Step 4: Calculate → 60 ÷ 1.25 = £48. Step 5: Check → £48 + 25% = £60 ✔️.

What we did and why: We reversed a 25% increase by dividing the new price by the multiplier (1.25). This gives the original price before the increase.

Example 2 – Medium (Decrease)

Question: A 500 cm³ solution is diluted by 40%. What was the original volume?

Step 1: Type of change → Decrease (40%). Step 2: Multiplier → 1 – 0.40 = 0.60. Step 3: Equation → O = N ÷ M = 500 ÷ 0.60. Step 4: Calculate → 500 ÷ 0.60 ≈ 833.33 cm³. Step 5: Check → 833.33 cm³ – 40% ≈ 500 cm³ ✔️.

What we did and why: We reversed a 40% decrease by dividing the new volume by the multiplier (0.60). This gives the original volume before dilution.

Example 3 – Exam-Style (Disguised)

Question: A drug’s concentration is reduced to 60% of its original strength after processing. The new concentration is 12 mg/L. What was the original concentration?

Step 1: Type of change → Decrease (to 60% = 40% decrease). Step 2: Multiplier → 0.60 (since it’s now 60% of original). Step 3: Equation → O = N ÷ M = 12 ÷ 0.60. Step 4: Calculate → 12 ÷ 0.60 = 20 mg/L. Step 5: Check → 20 mg/L × 0.60 = 12 mg/L ✔️.

What we did and why: The question says "reduced to 60%," which means the multiplier is 0.60 (not 0.40!). We divided the new concentration by 0.60 to find the original.

COMMON MISTAKES

1. Mistake: Using the wrong multiplier (e.g., 0.25 for a 25% increase instead of 1.25).
   Why it happens: Confusing percentage change with the multiplier.
   Correct approach: Always add/subtract the percentage from 1 first.

2. Mistake: Forgetting to rearrange the equation (e.g., multiplying instead of dividing).
   Why it happens: Not recognising it’s a reverse percentage.
   Correct approach: Write N = O × M first, then rearrange to O = N ÷ M.

3. Mistake: Misinterpreting "reduced to 60%" as a 60% decrease.
   Why it happens: Confusing "reduced by" with "reduced to."
   Correct approach: "Reduced to 60%" = multiplier of 0.60. "Reduced by 60%" = multiplier of 0.40.

4. Mistake: Not checking if the answer makes sense (e.g., original price higher than new price for an increase).
   Why it happens: Skipping the final check.
   Correct approach: Always verify with a quick calculation.

5. Mistake: Using the percentage as a decimal without converting (e.g., 20% = 20 instead of 0.20).
   Why it happens: Forgetting to divide by 100.
   Correct approach: Always convert percentages to decimals first.

EXAM TRAPS

1. Trap: "Increased by 10% then decreased by 10%" – the answer isn’t the original value!
   How to spot it: Two percentage changes in sequence.
   How to avoid it: Calculate each step separately (e.g., O × 1.10 × 0.90).

2. Trap: Questions with "VAT included" or "discount applied" – these are reverse percentages in disguise.
   How to spot it: Words like "inclusive," "after tax," or "final price."
   How to avoid it: Treat it as a percentage increase/decrease and reverse it.

3. Trap: Using the wrong base (e.g., calculating a percentage of the new value instead of the original).
   How to spot it: The question asks for the original amount, but you’re given the new amount.
   How to avoid it: Always start with N = O × M and rearrange.

1-MINUTE RECAP

"Here’s the night-before cheat sheet for reverse percentages: 1. Spot the change: Is it an increase or decrease? ‘Increased by 20%’ = ×1.20. ‘Reduced to 80%’ = ×0.80. 2. Write the equation: New = Original × Multiplier. Rearrange to Original = New ÷ Multiplier. 3. Plug and solve: Divide the new value by the multiplier. Done. 4. Check: Does the answer make sense? If not, you probably used the wrong multiplier. 5. Watch for traps: ‘Reduced to’ ≠ ‘reduced by.’ Two changes? Do them one at a time. You’ve got this—go smash those marks!"