GCSE Maths Practice: fractions

Question 3 of 11

This question asks you to find a reciprocal in context: a ratio is written as \(\tfrac{2}{3}\). Remember, the reciprocal reverses the relationship so that the two quantities multiply to 1.

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\( \begin{array}{l}\text{A fraction is written as }\frac{2}{3}.\\\text{What is its reciprocal?}\end{array} \)

Choose one option:

To find a reciprocal, swap the numerator and denominator. Always check by multiplying with the original to ensure the result equals 1.

In GCSE Maths, the term reciprocal means the multiplicative inverse — the number that multiplies by the original to make 1. This concept connects directly to division, ratios, and algebraic manipulation. Understanding reciprocals allows you to reverse proportions quickly and solve equations more efficiently.

What Is a Reciprocal?

For any non-zero number or fraction, its reciprocal is found by swapping the numerator and denominator. In symbols, the reciprocal of \(\tfrac{a}{b}\) is \(\tfrac{b}{a}\). If you multiply them, the result is always 1:

\[ \tfrac{a}{b} \times \tfrac{b}{a} = 1 \]

This relationship is used when dividing fractions, because dividing by a fraction is the same as multiplying by its reciprocal.

Worked Examples

Example 1 – Simple fraction:
Find the reciprocal of \( \tfrac{5}{8} \).

  1. Swap numerator and denominator → \( \tfrac{8}{5} \).
  2. Check: \( \tfrac{5}{8} \times \tfrac{8}{5} = 1 \).
  3. Answer: \( \tfrac{8}{5} \).

Example 2 – Mixed number:
Find the reciprocal of \( 1\tfrac{1}{2} \).

  1. Convert to improper fraction → \( \tfrac{3}{2} \).
  2. Swap → \( \tfrac{2}{3} \).
  3. Verify: \( \tfrac{3}{2} \times \tfrac{2}{3} = 1 \).
  4. Answer: \( \tfrac{2}{3} \).

Example 3 – Negative fraction:
Find the reciprocal of \( -\tfrac{7}{3} \).

  1. Keep the sign; swap top and bottom → \( -\tfrac{3}{7} \).
  2. Check: \( -\tfrac{7}{3} \times -\tfrac{3}{7} = 1 \).
  3. Answer: \( -\tfrac{3}{7} \).

Example 4 – Decimal or whole number:
The reciprocal of 4 is \( \tfrac{1}{4} \); the reciprocal of 0.25 is 4. This shows how reciprocals and decimals connect.

Common Mistakes

  • Forgetting that zero has no reciprocal because dividing by 0 is undefined.
  • Swapping incorrectly (e.g., writing \( \tfrac{-a}{b} \) instead of \( -\tfrac{b}{a} \)).
  • Thinking the reciprocal means “negative” — it does not change sign.

Real-Life Links

Reciprocals appear in many practical contexts: for instance, the speed–time relationship (if speed doubles, time halves), electrical resistance in parallel circuits, and unit-rate conversions such as kilometres per hour versus hours per kilometre. In each case, one value is the reciprocal of another in a proportional relationship.

Quick FAQs

  • Q: Does every number have a reciprocal?
    A: All numbers except 0 have reciprocals.
  • Q: What happens if a number equals its own reciprocal?
    A: Only 1 and −1 are their own reciprocals.
  • Q: Why do we use reciprocals in division?
    A: Because dividing by a fraction is equivalent to multiplying by its reciprocal.

Study Tip

Whenever you divide fractions, write a small arrow reminder “flip and multiply.” Recognising reciprocals instantly saves time and reduces careless errors in GCSE fraction questions.

Try These Yourself

  • Find the reciprocal of \( \tfrac{4}{9} \).
  • Find the reciprocal of −\( \tfrac{5}{2} \).
  • Find the reciprocal of 0.4.

After practising, test by multiplying your answer by the original — if the result is 1, you’re correct.