GCSE Maths Practice: fractions

Question 4 of 11

This question links fraction skills to real-life ratios. It asks you to reverse a recipe comparison: if water : juice = \(\tfrac{5}{6}\), what is juice : water? You’ll need to apply the concept of reciprocals to interpret the change correctly.

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\( \begin{array}{l}\text{A fruit drink is mixed in the ratio water : juice = }\frac{5}{6}.\\\text{What is the ratio juice : water written as a single fraction?}\end{array} \)

Choose one option:

When a question reverses a comparison (like A : B → B : A), take the reciprocal of the fraction form. Always state clearly which quantity each number represents.

Many GCSE Maths questions present fractions through real-life ratios. Fractions are simply ratios written in a single line: \( \tfrac{5}{6} \) means 5 parts of one quantity for every 6 parts of another. Being able to reverse that relationship is vital when comparing rates, scaling recipes, or converting unit ratios.

Scenario

A fruit drink is mixed in the ratio 5 : 6 (water : juice). This can be written as a fraction of water to juice — \( \tfrac{5}{6} \). But if a question asks for juice : water, you must invert the ratio. The reciprocal of \( \tfrac{5}{6} \) is \( \tfrac{6}{5} \), meaning for every 6 parts of juice there are 5 parts of water.

Reciprocal relationships like this appear whenever a comparison is reversed — speed vs. time, density vs. volume, or price per item vs. items per pound. Each pair is connected by multiplication that equals 1.

Worked Examples

Example 1 – Fuel efficiency:
A car travels 60 miles on 4 gallons. The rate is \( \tfrac{60}{4}=15 \) miles per gallon. To find gallons per mile, take the reciprocal: \( \tfrac{1}{15} \) gallons per mile.

Example 2 – Cooking ratio reversal:
A sauce recipe uses water : oil = \( \tfrac{3}{8} \). To express oil : water, take the reciprocal → \( \tfrac{8}{3} \). This helps when scaling ingredients the other way round.

Example 3 – Speed and time:
If speed = distance/time = \( \tfrac{120}{2}=60 \) km/h, then time per kilometre = reciprocal of speed = \( \tfrac{1}{60} \) h per km (1 minute). Reciprocal thinking quickly converts between “per hour” and “per kilometre”.

Common Mistakes

  • Swapping numerators and denominators without understanding what the ratio represents — always state what each part measures.
  • Forgetting that reciprocals are only valid for non-zero values.
  • Confusing the reciprocal with subtraction or difference.

Real-World Links

Reciprocals appear in unit conversions (e.g., £ per kg ↔ kg per £), mechanical problems (period = 1/frequency), and proportional reasoning in finance (interest rate vs. time). Recognising when a reciprocal is needed allows you to switch viewpoints between quantities effortlessly.

Quick FAQs

  • Q: Why is reversing a ratio the same as taking a reciprocal?
    A: Because a ratio written as \( a:b \) becomes \( \tfrac{a}{b} \), so swapping terms gives \( \tfrac{b}{a} \).
  • Q: Can a ratio of 0 have a reciprocal?
    A: No — division by zero is undefined.
  • Q: Are reciprocals always greater than 1?
    A: No — if the original fraction is less than 1, its reciprocal is greater than 1, and vice versa.

Study Tip

Whenever a question reverses wording — “per hour” → “hours per”, “A : B” → “B : A” — think reciprocal. Writing the fraction both ways prevents direction errors in multi-step GCSE problems.

Try These Yourself

  • The paint ratio red : blue is \( \tfrac{4}{7} \). Write blue : red.
  • A train’s speed is \( \tfrac{5}{9} \) km per minute. Find minutes per kilometre.
  • For every 2 litres of concentrate, 3 litres of water are added. Express water : concentrate as a single fraction.