GCSE Maths Practice: percentages

Question 1 of 10

This question practises reverse-percentage calculations — working out the original number when a percentage and result are known, a key GCSE Maths skill.

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\( \textbf{60 is } 40\% \textbf{ of what number?} \)

Choose one option:

To find the original number, multiply by 100 and divide by the percentage. Check your result by taking 40% of it to see if you get 60.

Understanding Reverse Percentages

In GCSE Maths, reverse-percentage problems ask you to find the original number when you already know a percentage of it. For example, if 60 is 40% of some number, we need to work backwards to discover what that whole number was before the percentage was applied. This technique is essential for topics such as discounts, price increases, and tax problems.

Concept Explained

The general relationship between a number and its percentage is:

\[ \text{Percentage of a number} = \dfrac{\text{Percentage}}{100} \times \text{Original number}. \]

To reverse this process, divide the known value by the percentage fraction:

\[ \text{Original number} = \dfrac{\text{Known value} \times 100}{\text{Percentage}}. \]

This allows you to find the full amount from a partial percentage, without guessing or trial and error.

Step-by-Step Method

  1. Write down what the question tells you. For example, 60 is 40% of a number.
  2. Represent the unknown number as \( x \).
  3. Form the equation: \( \dfrac{40}{100} \times x = 60. \)
  4. Rearrange to make \( x \) the subject: \( x = \dfrac{60 \times 100}{40}. \)
  5. Calculate: \( 60 \times 100 = 6000;\ 6000 \div 40 = 150. \)

The original number is 150.

Worked Examples

  • Example 1: 25 is 20% of what number?
    \( x = \dfrac{25 \times 100}{20} = 125. \)
  • Example 2: 45 is 15% of what number?
    \( x = \dfrac{45 \times 100}{15} = 300. \)
  • Example 3: 72 is 60% of what number?
    \( x = \dfrac{72 \times 100}{60} = 120. \)

Real-Life Applications

  • Shops and Sales: If a jumper costs £60 after a 40% discount, the original price can be found by dividing by 0.6 (the remaining 60%).
  • Finance: If £60 is 40% of a monthly target, the full target is £150.
  • Science: In experiments, if 60 g represents 40% of a solution, the total solution weighed 150 g originally.

Reverse percentages help you undo changes — just like working backwards in equations.

Common Mistakes to Avoid

  • Multiplying by the percentage instead of dividing — remember, you’re finding the whole.
  • Using the wrong remaining percentage (for discounts, always consider what remains after reduction).
  • Forgetting to convert the percentage to a decimal or fraction before using it.

Quick Mental-Maths Tips

When the numbers are easy, you can estimate. For instance, 40% is close to two-fifths. If 60 represents two-fifths, one-fifth is roughly 30, and five-fifths (the whole) is 150 — a good check for your calculation.

Frequently Asked Questions

Q1: How do I find the original number when a discount is given?
If a sale price is £60 after 40% off, you are paying 60% of the original. So divide by 0.6: £60 ÷ 0.6 = £100.

Q2: Why divide by the percentage fraction?
Because percentage ‘of’ means multiplication. To undo multiplication, we use division — the reverse operation.

Q3: How can I check my answer?
Take your result and find 40% of it. If it gives you the known value (60), your calculation is correct.

Summary

Reverse-percentage problems require you to think backwards. Instead of finding a portion of a number, you are reconstructing the original amount from a known part. Multiply the known value by 100 and divide by the percentage. This method is reliable and quick once understood, and it plays a vital role in GCSE topics such as tax, discounts, profit, and data interpretation.