GCSE Maths Practice: percentages

Understanding 75% and Its Fraction Form

In GCSE Maths, percentages and fractions are closely linked. The percentage 75% means 75 parts out of 100, which can be simplified to the fraction \( \dfrac{3}{4} \). Recognising this relationship makes mental maths much easier, especially when working without a calculator. Whenever you see 75%, you can think of it as 'three quarters of' the number.

Concept Explained

To calculate a percentage of a number, multiply the number by the percentage and divide by 100. For example:

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

So to find 75% of any number, use:

\[ 75\% \text{ of } n = \dfrac{75}{100} \times n = 0.75n = \dfrac{3}{4}n \]

Step-by-Step Method

1. Write the percentage as a fraction: \( 75\% = \dfrac{3}{4} \).
2. Multiply the number by \( \dfrac{3}{4} \).
3. Divide by 4 first, then multiply by 3 for mental calculation.
4. Check that your answer is slightly less than the full number (since 75% is less than 100%).

Worked Examples

• Example 1: Find 75% of 200.
  \( \dfrac{3}{4} \times 200 = 150. \)
• Example 2: Find 75% of 60.
  \( \dfrac{3}{4} \times 60 = 45. \)
• Example 3: Find 75% of 240.
  \( \dfrac{3}{4} \times 240 = 180. \)

These examples show that converting percentages into fractions helps you calculate quickly without relying on a calculator.

Common Mistakes to Avoid

• Forgetting to divide by 100 when multiplying by the percentage.
• Using the wrong fraction (e.g., \( \dfrac{4}{3} \) instead of \( \dfrac{3}{4} \)).
• Mixing up percentage of and percentage change — this question is asking for 'of', not increase or decrease.

Real-Life Applications

Understanding 75% has many uses in everyday life. It represents three-quarters of something, a value that appears in discounts, exam results, and statistics. For example:

• If a store offers a 25% discount, you are paying 75% of the original price. For a £200 item, you pay \( 0.75 \times 200 = 150 \).
• If you score 75% on a test with 40 questions, you answered \( 0.75 \times 40 = 30 \) questions correctly.
• If a car travels 75% of a 120 km journey, it covers \( 0.75 \times 120 = 90 \) km.

These examples show that percentage and fraction calculations are practical skills for understanding proportions in daily life.

Quick Mental Maths Tips

• To find 50% of a number, halve it.
• To find 25%, divide by 4.
• To find 75%, combine the two: first divide by 4 (25%), then multiply that by 3.

For example, 75% of 160 = (160 ÷ 4) × 3 = 40 × 3 = 120.

Frequently Asked Questions

Q1: How can I check my answer makes sense?
Estimate roughly. 75% is close to the full amount, so the answer should be a little less than the original number.

Q2: Is 75% the same as 0.75?
Yes. Writing percentages as decimals helps in calculator-based questions. \( 75\% = 0.75 \).

Q3: What is the relationship between 75%, 25%, and 50%?
They are all fractions of a whole: 25% = \( \dfrac{1}{4} \), 50% = \( \dfrac{1}{2} \), and 75% = \( \dfrac{3}{4} \).

Summary

Recognising that \( 75\% = \dfrac{3}{4} \) makes percentage questions faster and more intuitive. To find a percentage of a number, multiply by the percentage and divide by 100, or use its fraction equivalent. This skill is essential in GCSE Maths and real life, from budgeting to interpreting data. Always estimate first — 75% should be roughly three-quarters of the number — to ensure your answer is reasonable.