GCSE Maths Practice: simplifying-ratios

Question 8 of 10

This question focuses on simplifying ratios that include variables, a key GCSE Higher Maths skill.

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\( \begin{array}{l}\text{Simplify the ratio } 18x:30y.\end{array} \)

Choose one option:

Simplify the numerical coefficients fully and keep the variables unchanged.

Simplifying Ratios That Contain Variables (GCSE Maths – Higher Tier)

At GCSE Higher level, ratio questions often include algebraic terms such as variables. These questions test whether you can apply the same simplifying principles used for numerical ratios while also working confidently with algebra. Although the presence of variables may look more challenging, the underlying method remains the same.

What Is an Algebraic Ratio?

An algebraic ratio is a ratio that includes variables, such as x or y, instead of (or as well as) numbers. Each part of the ratio may contain a coefficient (a number) multiplied by a variable. Simplifying an algebraic ratio means reducing the numerical coefficients while keeping the variables unchanged, provided they are different.

Key Principle: Treat Variables Like Units

When simplifying ratios with variables, it is helpful to think of the variables as labels or units. Just as you would not cancel different units when simplifying numerical ratios, you must not cancel different variables. Only common numerical factors can be divided out unless the variables are identical.

General Method for Simplifying Algebraic Ratios

  1. Identify the numerical coefficients in each part of the ratio.
  2. Find the highest common factor (HCF) of the coefficients.
  3. Divide each part of the ratio by the HCF.
  4. Keep the variables exactly as they are.
  5. Check that no further numerical simplification is possible.

This method ensures the ratio is fully simplified while remaining mathematically correct.

Worked Example 1

Simplify the ratio 12a : 20b.

The highest common factor of 12 and 20 is 4. Dividing both parts by 4 reduces the numerical coefficients, while the variables remain unchanged.

Worked Example 2

Simplify the ratio 15x : 25y.

The HCF of 15 and 25 is 5. Dividing both parts by 5 simplifies the ratio correctly.

Worked Example 3

Simplify the ratio 8m : 12n.

The highest common factor is 4. After dividing both parts by 4, the simplified ratio can be written clearly.

Common Mistakes to Avoid

  • Trying to cancel variables that are different.
  • Forgetting to simplify the numerical coefficients.
  • Dividing only one part of the ratio.
  • Leaving the ratio partially simplified.

Why Algebraic Ratios Matter at Higher Tier

Algebraic ratios appear in many Higher GCSE topics, including proportional reasoning, similar shapes, and algebraic problem solving. Being confident with these ratios helps you handle multi-step questions where simplifying is required before substitution or further calculation.

Frequently Asked Questions

Can variables ever be cancelled?
Only if the same variable appears in both parts of the ratio.

Do I always simplify coefficients first?
Yes. Numerical simplification is always required.

Does order matter in algebraic ratios?
Yes. Reversing the order changes the meaning of the ratio.

Study Tip

When simplifying algebraic ratios, focus on the numbers first. Treat the variables as labels and simplify the coefficients just as you would with a numerical ratio.

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