GCSE Maths Practice: percentages

Question 3 of 10

An item is reduced by 10% and then by another 5%. The final selling price is £108. Work out the original price before any discounts. Give your answer to two decimal places.

‹ ›

\( \begin{array}{1}\text{An item is reduced by 10% and then by another 5%.}\\ \text{The final selling price is £108.}\\ \text{What was the original price before any discounts?}\\ \text{Give your answer to two decimal places.}\end{array} \)

Choose one option:

Combine multipliers, not percentages. For two decreases of 10% and 5%, use 0.9 × 0.95 = 0.855, then divide £108 by 0.855.

Reverse Multi-Step Percentage Decrease (Higher Tier)

In GCSE Higher Tier problems, you are often asked to reverse more than one percentage change. For example, an item is reduced by 10% and then by another 5%. The final sale price is known, but the task is to find the original price before either discount occurred. This requires working backwards through compound multipliers.

Understanding the Process

Each reduction has its own multiplier:

  • 10% decrease → multiplier \(0.90\)
  • 5% decrease → multiplier \(0.95\)

When two reductions happen successively, multiply the two factors to find the combined multiplier:

\[ 0.90 \times 0.95 = 0.855. \]

This means the final price is 85.5% of the original value.

Setting Up the Equation

If the original price is \(x\), then:

\[ 0.855x = 108. \]

To reverse the change, divide by the combined multiplier:

\[ x = \dfrac{108}{0.855} = 126.32. \]

The original price was approximately \(\pounds126\).

Step-by-Step Method

  1. Convert each percentage decrease into a decimal multiplier.
  2. Multiply the multipliers together to form the total multiplier.
  3. Write an equation relating the final price to the original: final = multiplier × original.
  4. Rearrange to find the original: \( \text{original} = \dfrac{\text{final}}{\text{multiplier}}. \)
  5. Use a calculator and round appropriately at the end.

Why Division Works

Every decrease is a multiplication. To reverse multiplication, we divide. If an 8% growth used \(1.08\), you would divide by \(1.08\) to find the starting amount. The same logic applies for decreases — divide by the total multiplier.

Worked Examples (Different Numbers)

  • Example 1: A laptop is reduced by 20% then 10%, final price £720.
    Combined multiplier: \(0.8 \times 0.9 = 0.72.\)\(x = 720 / 0.72 = 1000.\)
  • Example 2: A watch is discounted by 15% then 5%, final £161.50.
    Combined multiplier: \(0.85 \times 0.95 = 0.8075.\)\(x = 161.5 / 0.8075 = 200.\)

Real-World Context

Shops often apply consecutive reductions during clearance sales. A 10% discount followed by 5% is not a 15% total discount — it’s slightly less (14.5%) because the second reduction applies to an already reduced price. Understanding this distinction prevents errors in both exams and everyday reasoning.

Common Mistakes

  • Adding percentages (10 % + 5 %) instead of multiplying multipliers.
  • Dividing by only one of the reductions instead of the combined factor.
  • Using rounded multipliers like 0.86 instead of 0.855, causing accuracy loss.

How to Check Your Answer

Once you find \(x\), apply both reductions forward to ensure the final value matches £108. \(126.32 \times 0.9 \times 0.95 = 108.00.\) This confirms accuracy.

Advanced Extension (Compound Change)

In algebraic form, if a price changes by \(a\%\) then \(b\%\), the overall multiplier is:

\[ M = (1 - \tfrac{a}{100})(1 - \tfrac{b}{100}). \]

To reverse the change, divide the final value by \(M\). For compound increases, replace the minus signs with plus.

Frequently Asked Questions

Q1: Can two 10% discounts be treated as one 20% discount?
No. Two successive 10% discounts multiply to \(0.9^2 = 0.81\), an overall 19% decrease.

Q2: How do I handle percentage increases and decreases together?
Multiply both factors. For example, +10% then −10% → \(1.1 \times 0.9 = 0.99\), a 1% overall decrease.

Q3: Why are reverse questions harder?
Because they require algebraic rearrangement and understanding of the inverse operation — you are undoing multiple multiplications in one step.

Summary

Multi-step reverse-percentage questions demand strong control of multipliers. Combine all multipliers, form \( \text{final} = Mx, \) and divide by \(M\) to find the starting value. Avoid adding percentages, and check by re-applying the changes. Mastery of this concept prepares you for exam topics like compound interest, VAT reversal, and real-life modelling of discounts or inflation.