GCSE Maths Practice: simplifying-ratios

Question 9 of 10

This question focuses on simplifying ratios that include variables and numerical coefficients.

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\( \begin{array}{l}\text{Simplify the ratio } 2x:6y.\end{array} \)

Choose one option:

Simplify the numerical coefficients fully and remove any coefficient of 1 in the final ratio.

Simplifying Algebraic Ratios with Coefficients (GCSE Maths – Higher Tier)

At GCSE Higher level, you are expected to simplify ratios that include algebraic terms. These questions test your ability to combine knowledge of ratios with algebraic manipulation. Although variables may make the ratio look more complex, the process of simplifying is very similar to simplifying numerical ratios.

Understanding Algebraic Ratios

An algebraic ratio is a ratio that includes variables, such as x or y, alongside numerical coefficients. Each part of the ratio consists of a coefficient multiplied by a variable. Simplifying an algebraic ratio means reducing the numerical coefficients while leaving the variables unchanged, provided the variables are different.

Key Principle: Simplify the Coefficients Only

When working with algebraic ratios, you should focus on simplifying the numerical part first. Variables act like labels and must not be cancelled unless they are exactly the same on both sides of the ratio. If the variables are different, only the numerical coefficients can be simplified.

Step-by-Step Method

  1. Identify the numerical coefficient in each part of the ratio.
  2. Find the highest common factor (HCF) of the coefficients.
  3. Divide both parts of the ratio by the HCF.
  4. Rewrite the ratio without unnecessary coefficients of 1.
  5. Check that the ratio cannot be simplified further.

This method ensures the ratio is fully simplified and written in standard GCSE form.

Worked Example 1

Simplify the ratio 4a : 12b.

The highest common factor of 4 and 12 is 4. Dividing both parts by 4 simplifies the coefficients while keeping the variables unchanged.

Worked Example 2

Simplify the ratio 6x : 9y.

The HCF of 6 and 9 is 3. Dividing both parts by 3 reduces the ratio to its simplest algebraic form.

Worked Example 3

Simplify the ratio 10m : 5n.

The highest common factor is 5. Dividing both parts by 5 simplifies the ratio correctly.

Common Mistakes to Avoid

  • Cancelling variables that are different.
  • Leaving a coefficient of 1 in the final answer.
  • Dividing only one part of the ratio.
  • Stopping before the ratio is fully simplified.

Why Algebraic Ratios Are Important

Algebraic ratios appear frequently in Higher GCSE questions involving proportion, similar shapes, and algebraic expressions. Being able to simplify them confidently allows you to move smoothly into multi-step problems without losing easy marks.

Frequently Asked Questions

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

Why do we remove coefficients of 1?
Because writing x instead of 1x is the standard mathematical convention.

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

Study Tip

When simplifying algebraic ratios, always deal with the numbers first and treat variables as labels. This keeps your working clear and helps avoid common Higher-tier mistakes.

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