GCSE Maths Practice: simplifying-ratios

Question 4 of 10

This question helps you practise simplifying ratios using the highest common factor, a key GCSE Maths skill.

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\( \begin{array}{l}\text{Simplify the ratio } 35:14 \\ \text{to its simplest form.}\end{array} \)

Choose one option:

Always divide both parts of a ratio by the highest common factor and check your final answer carefully.

Simplifying Ratios Using the Highest Common Factor

Simplifying ratios is a key topic in GCSE Maths and forms the foundation for many more advanced ratio and proportion problems. A ratio shows the relationship between two quantities by comparing their sizes. Writing a ratio in its simplest form makes this relationship clearer and easier to use.

What Does It Mean to Simplify a Ratio?

When you simplify a ratio, you reduce it to the smallest whole numbers that represent the same comparison. This process is similar to simplifying fractions. The important rule is that both parts of the ratio must be divided by the same number so that the comparison remains fair and accurate.

The Role of the Highest Common Factor (HCF)

The highest common factor is the largest number that divides exactly into both numbers in a ratio. Using the HCF ensures the ratio is fully simplified. If you divide by a smaller factor, the ratio may still be reducible, which would not be considered the simplest form in GCSE exams.

Step-by-Step Method

  1. Write the ratio clearly using a colon.
  2. Find the highest common factor of both numbers.
  3. Divide each number in the ratio by the HCF.
  4. Check that the resulting numbers have no common factor greater than 1.

This method works for all Foundation-level simplifying ratio questions.

Worked Example 1

Simplify the ratio 21:28.

The highest common factor of 21 and 28 is 7. Dividing both numbers by 7 produces a simplified ratio.

Worked Example 2

Simplify the ratio 18:6.

The highest common factor is 6. Dividing both parts by 6 gives a much simpler comparison.

Worked Example 3

Simplify the ratio 45:30.

The HCF of 45 and 30 is 15. Dividing each number by 15 reduces the ratio to its lowest terms.

Common Mistakes to Avoid

  • Dividing only one part of the ratio.
  • Using a factor that is not the highest common factor.
  • Stopping before the ratio is fully simplified.
  • Mixing up the order of the ratio when rewriting it.

Real-Life Applications of Ratios

Ratios are used widely in everyday life. Recipes rely on ratios to mix ingredients correctly. Sports statistics use ratios to compare wins and losses. In classrooms, ratios can describe numbers of students or resources. Being able to simplify ratios ensures that comparisons are accurate and easy to understand.

Frequently Asked Questions

Do ratios always need to be simplified?
Yes. Unless stated otherwise, GCSE exam questions expect ratios in their simplest form.

Can a ratio be simplified more than once?
Yes. If a ratio is divided by a factor that is not the HCF, it may still be reducible.

What if one number divides exactly into the other?
You should still divide both numbers by the HCF to ensure the simplest form.

Study Tip

Practise finding the highest common factor quickly. Strong HCF skills make simplifying ratios faster and help avoid errors in GCSE Maths exams.

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