GCSE Maths Practice: simplifying-ratios

Question 10 of 10

This question focuses on simplifying three-part ratios, an important GCSE Higher Maths skill.

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\( \begin{array}{l}\text{Simplify the three-part ratio } 16:24:32.\end{array} \)

Choose one option:

Always divide every part of the ratio by the same highest common factor.

Simplifying Three-Part Ratios (GCSE Maths – Higher Tier)

At GCSE Higher level, ratio questions often involve more than two quantities. Three-part ratios are used to compare three different values at the same time and are commonly seen in problems involving sharing, scaling, recipes, and proportional reasoning. Although they look more complex than two-part ratios, the method for simplifying them is exactly the same.

What Is a Three-Part Ratio?

A three-part ratio compares three quantities using two colons, for example a:b:c. Each part of the ratio represents the size of one quantity relative to the others. Simplifying a three-part ratio means reducing all three numbers so that they are in their smallest whole-number form while keeping the same relationship.

Key Principle: All Parts Must Be Treated Equally

When simplifying a three-part ratio, it is essential that every term is divided by the same value. Dividing only two of the numbers or using different divisors will change the meaning of the ratio. This rule is the same whether the ratio has two parts or more.

Step-by-Step Method

  1. Write the ratio clearly with all parts included.
  2. Find the highest common factor (HCF) of all the numbers.
  3. Divide every term in the ratio by the HCF.
  4. Check that the final ratio has no further common factor.

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

Worked Example 1

Simplify the ratio 12:18:24.

The highest common factor of 12, 18, and 24 is 6. Dividing all three terms by 6 produces a simplified ratio.

Worked Example 2

Simplify the ratio 9:15:21.

The HCF of 9, 15, and 21 is 3. Dividing each term by 3 reduces the ratio to its lowest terms.

Worked Example 3

Simplify the ratio 20:30:50.

The highest common factor is 10. Dividing all three parts by 10 simplifies the ratio correctly.

Common Mistakes to Avoid

  • Dividing only two of the three terms.
  • Using a factor that is not the highest common factor.
  • Forgetting to simplify all three parts.
  • Assuming the ratio is simplified just because the numbers look smaller.

Why Three-Part Ratios Matter at Higher Tier

Three-part ratios appear frequently in Higher GCSE questions involving sharing in given ratios, comparing multiple quantities, and proportional reasoning problems. Being confident with these ratios allows you to move smoothly into multi-step questions without losing marks on simplification.

Frequently Asked Questions

Do I always need to find the HCF of all three numbers?
Yes. The ratio is only fully simplified when all parts share no common factor greater than 1.

Can three-part ratios include variables?
Yes. At Higher level, ratios may include both numbers and algebraic terms.

Does order matter in three-part ratios?
Yes. Changing the order changes the meaning of the ratio.

Study Tip

When simplifying three-part ratios, always check all three numbers carefully. A small mistake in one term can change the entire ratio.

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