GCSE Maths Practice: percentages

Question 10 of 10

This question practises higher-tier percentage reasoning. You must interpret the percentage as a decimal or fraction and apply it precisely, using estimation to check accuracy.

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\( \textbf{What is } 18\% \textbf{ of } 250? \)

Choose one option:

Check by estimation: 10% of 250 = 25, 20% = 50, so 18% must be slightly below 50. Use both decimal and fraction methods for accuracy.

Exploring Advanced Percentage Techniques

At GCSE Higher Tier, percentage problems often go beyond simple 'find this percentage' tasks. You may need to deal with unusual percentages, mixed numbers, or situations that require you to rearrange formulas or use proportion reasoning. This question, which involves 18% of 250, demonstrates both procedural accuracy and proportional reasoning.

Decimal and Fraction Conversions

The percentage \(18\%\) can be expressed in several equivalent forms:

  • As a decimal: \( 0.18 \)
  • As a fraction: \( \dfrac{18}{100} = \dfrac{9}{50} \)

Higher-tier questions often expect you to switch between these representations confidently. Using fractions is sometimes faster when dealing with terminating decimals or common multiples.

Step-by-Step Calculation

  1. Convert \(18\%\) to a decimal or fraction.
  2. Multiply the given number by this value:

\[ 250 \times 0.18 = 45. \]

Alternatively, using the fraction method:

\[ 250 \times \dfrac{9}{50} = \dfrac{2250}{50} = 45. \]

Both yield the same result, confirming consistency across methods.

Applying Proportion Reasoning

Percentages are simply fractions with a denominator of 100. This means \(18\%\) of 250 is the same as 18 parts out of 100 equal parts of 250. Using a unitary method:

\[ 1\% = \dfrac{250}{100} = 2.5, \quad 18\% = 18 \times 2.5 = 45. \]

This reasoning helps verify results without depending purely on a calculator — a useful strategy in non-calculator GCSE papers.

Common Higher-Tier Variations

Examiners may adapt this style of question in more complex forms, such as:

  • Reverse percentage: 45 is 18% of what number?
  • Compound percentage: A value increases by 18% then decreases by 10%. What is the net change?
  • Percentage comparison: How much greater is 18% of 250 than 15% of 300?

Understanding the base (what the percentage is taken from) is crucial for these harder variations.

Real-World Higher Examples

  • Finance: An investment of £250 gains 18% interest in one year. The interest earned is £45, making the final balance £295.
  • Taxation: A £250 service charge increases by 18% VAT, resulting in a new total of \(250 + 45 = £295.\)
  • Data Analysis: If 18% of survey respondents prefer an option, and there are 250 participants, then \(45\) people chose that option.

These applications show how percentages underpin real calculations in business, science, and everyday life.

Common Higher-Tier Mistakes

  • Using 18 ÷ 250 instead of \(18\%\) of 250 — mixing up the order of operations.
  • Forgetting that 18% = 0.18, not 1.8.
  • Incorrect rounding: when intermediate steps include decimals, avoid rounding until the end.

Check by Estimation

Always check that your answer makes sense. 10% of 250 = 25, and 20% would be 50. The result 45 lies between them — correct and logical.

Advanced Discussion

Percentage problems at higher level may combine algebraic or functional reasoning. For instance, if \( y = 250 \times \dfrac{p}{100} \), then \( p = \dfrac{100y}{250} = 0.4y \). Understanding how to rearrange this formula allows you to solve any forward or reverse percentage scenario algebraically — a key Higher Tier expectation.

Summary

To find \(18\%\) of \(250\): multiply \(250 \times 0.18 = 45.\) Always understand what the percentage represents, check by estimation, and link your working to fraction or algebraic reasoning. This approach ensures precision across calculator and non-calculator GCSE contexts.