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Simplifying and Using Ratios

Simplifying and Using Ratios

Ratios are one of the most frequently examined topics in the Edexcel GCSE Mathematics (1MA1) specification. They appear across all three papers — Paper 1 (non-calculator), Paper 2 (calculator) and Paper 3 (calculator). This lesson covers how to write, simplify and use ratios, including dividing quantities in a given ratio and combining ratios.

What Is a Ratio?

A ratio compares two or more quantities and shows their relative sizes. Ratios are written using a colon, for example 3 : 5, and they have no units — they simply tell us how many times bigger or smaller one quantity is compared to another.

Concept Explanation Example
Ratio A comparison of two or more quantities Boys to girls = 3 : 5
Parts Each number in the ratio represents a "part" 3 parts and 5 parts = 8 parts total
Order The order of the numbers matters 3 : 5 is NOT the same as 5 : 3
Units Both quantities must be in the same unit before writing the ratio Convert cm and m to the same unit first

Writing a Ratio

When writing a ratio, you must ensure the quantities are in the same units.

Worked Example 1: Write the ratio 40 cm to 2 m in its simplest form.

  1. Convert to the same units: 2 m = 200 cm
  2. Write as a ratio: 40 : 200
  3. Simplify by dividing both by the HCF (40): 1 : 5

Edexcel Exam Tip: Always check the units before writing a ratio. A very common mistake is to write 40 : 2 instead of converting metres to centimetres first. On Paper 1 (non-calculator) you must do all simplifying by hand, so look for common factors.

Simplifying Ratios

Simplifying a ratio works exactly like simplifying a fraction — you divide every part by the highest common factor (HCF).

Method

  1. Find the HCF of all the numbers in the ratio.
  2. Divide each part by the HCF.
  3. The result is the ratio in its simplest form.

Worked Example 2: Simplify 24 : 36.

  • HCF of 24 and 36 = 12
  • 24 / 12 = 2
  • 36 / 12 = 3
  • Simplified ratio = 2 : 3

Worked Example 3: Simplify 15 : 25 : 45.

  • HCF of 15, 25 and 45 = 5
  • 15 / 5 = 3, 25 / 5 = 5, 45 / 5 = 9
  • Simplified ratio = 3 : 5 : 9

Simplifying Ratios Involving Fractions and Decimals

If a ratio contains fractions, multiply every part by the lowest common denominator (LCD). If it contains decimals, multiply by a power of 10 to eliminate the decimal places, then simplify.

Worked Example 4: Simplify 0.6 : 1.5.

  • Multiply both by 10: 6 : 15
  • Divide both by HCF (3): 2 : 5

Worked Example 5: Simplify 1/3 : 1/2.

  • LCD of 3 and 2 is 6
  • Multiply both by 6: 2 : 3
  • Simplified ratio = 2 : 3

Worked Example 6: Simplify 2 : 3/4.

  • Multiply both by 4: 8 : 3
  • Simplified ratio = 8 : 3

Edexcel Exam Tip: On Paper 1 (non-calculator), ratio-with-fractions questions appear regularly. Multiply through by the LCD to clear all fractions in one step.

Equivalent Ratios

Two ratios are equivalent if one can be obtained by multiplying (or dividing) every part of the other by the same number. This is identical to the concept of equivalent fractions.

Original Ratio Multiply by Equivalent Ratio
2 : 3 x 4 8 : 12
5 : 2 x 3 15 : 6
12 : 8 / 4 3 : 2

Worked Example 7: The ratio of red beads to blue beads is 3 : 7. If there are 21 blue beads, how many red beads are there?

  • 7 parts = 21, so 1 part = 3
  • Red beads = 3 parts = 3 x 3 = 9 red beads

Unit Ratios (1 : n and n : 1)

A unit ratio expresses one part of the ratio as 1. This is useful for comparing and converting scales.

Worked Example 8: Express 8 : 5 in the form 1 : n.

  • Divide both by 8: 1 : 5/8 = 1 : 0.625

Worked Example 9: Express 8 : 5 in the form n : 1.

  • Divide both by 5: 8/5 : 1 = 1.6 : 1

Edexcel Exam Tip: Map scale questions often require you to express a ratio in the form 1 : n. You may also be asked to find a real-life distance given a map scale — always check the units of your final answer.

Dividing a Quantity in a Given Ratio

This is one of the most common ratio question types in the Edexcel exam.

Method

  1. Add the parts of the ratio to find the total number of parts.
  2. Divide the total amount by the total number of parts to find the value of one part.
  3. Multiply each part of the ratio by the value of one part.

Worked Example 10: Divide 180 in the ratio 2 : 3 : 4.

  1. Total parts = 2 + 3 + 4 = 9
  2. One part = 180 / 9 = 20
  3. The three shares are: 2 x 20 = 40, 3 x 20 = 60, 4 x 20 = 80

Check: 40 + 60 + 80 = 180

Worked Example 11: Alex and Beth share 240 pounds in the ratio 5 : 7. How much does Beth receive?

  1. Total parts = 5 + 7 = 12
  2. One part = 240 / 12 = 20
  3. Beth's share = 7 x 20 = 140 pounds

Worked Example 12: Two numbers are in the ratio 3 : 8. The difference between them is 30. Find both numbers.

  1. Difference in parts = 8 - 3 = 5
  2. One part = 30 / 5 = 6
  3. Smaller number = 3 x 6 = 18; Larger number = 8 x 6 = 48

Check: 48 - 18 = 30

Combining Ratios

Sometimes you need to combine two separate ratios into a single ratio.

Worked Example 13: A : B = 2 : 3 and B : C = 4 : 5. Find A : B : C.

  1. B appears in both ratios but has different values (3 and 4).
  2. Find the LCM of 3 and 4 = 12.
  3. Scale A : B so that B = 12: multiply by 4 to give A : B = 8 : 12.
  4. Scale B : C so that B = 12: multiply by 3 to give B : C = 12 : 15.
  5. Combined ratio: A : B : C = 8 : 12 : 15

Worked Example 14: X : Y = 5 : 3 and Y : Z = 6 : 7. Find X : Y : Z.

  1. LCM of 3 and 6 = 6.
  2. X : Y = 10 : 6 and Y : Z = 6 : 7.
  3. Combined ratio: X : Y : Z = 10 : 6 : 7

Edexcel Exam Tip: Combining ratios questions are typically worth 3-4 marks and appear on both Foundation and Higher papers. Always make the common term equal before combining.

Practice Problems

  1. Simplify the ratio 36 : 48. Answer: 3 : 4
  2. Write 0.75 : 1.25 in its simplest form. Answer: 3 : 5
  3. Simplify 2/5 : 3/4. Answer: 8 : 15
  4. Divide 450 in the ratio 2 : 3 : 4. Answer: 100, 150, 200
  5. P : Q = 3 : 4 and Q : R = 6 : 5. Find P : Q : R. Answer: 9 : 12 : 10
  6. The ratio of cats to dogs at a shelter is 5 : 8. There are 40 dogs. How many cats are there? Answer: 25
  7. Two amounts of money are in the ratio 7 : 3. The larger amount is 28 pounds more than the smaller. What is the total? Answer: 70 pounds
  8. Express the ratio 4 : 9 in the form 1 : n. Answer: 1 : 2.25

Common Mistakes

Mistake How to Avoid It
Forgetting to convert to the same units Always check units before writing a ratio
Dividing by different numbers when simplifying Always use the same HCF for all parts
Getting the order wrong in the ratio Read the question carefully — the order matters
Using the difference instead of the total when dividing in a ratio Re-read whether the question gives a total amount or a difference
Not simplifying fully Check there is no further common factor

Summary

  • A ratio compares quantities using the same units.
  • Simplify by dividing all parts by the HCF.
  • Equivalent ratios are found by multiplying or dividing all parts by the same number.
  • Unit ratios (1 : n or n : 1) are useful for scales and comparisons.
  • Divide in a ratio by finding the value of one part first.
  • Combine ratios by making the common term equal using the LCM.
  • Always show your working and check your answer adds back to the original total.

Edexcel Exam Tip: Ratio questions can appear as part of bigger problems involving fractions, percentages or algebra. Be ready to use ratio skills alongside other topics.

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