Ratio Given the Difference –

Ratio questions where you are given the difference between two shares — rather than the total — are a common exam trap in GCSE Maths. Many students automatically add the parts and divide the total by this sum, but that only works when you know the total. When the question gives you the difference, you need a different approach: find the difference in parts first, then work out the value of one part. This guide explains the method clearly and provides worked examples at Foundation and Higher level.

What Is a Ratio Given the Difference?

In a standard ratio question, you are told the total amount and asked to share it. In a ratio given the difference question, you are instead told how much more one person (or quantity) gets than another. You must use this difference to find the value of one part.

Key Formulas

Once you know the value of one part, multiply each ratio part by this value to find each share. You can also find the total by multiplying the total parts by the value of one part.

Step-by-Step Method

1. Write down the ratio and identify the two quantities being compared.
2. Find the difference in parts by subtracting the smaller ratio number from the larger.
3. Divide the actual difference (given in the question) by the difference in parts. This gives the value of one part.
4. Multiply each ratio number by the value of one part to find each share.
5. Check that the difference between your answers matches the difference given in the question.

Worked Example 1 — Foundation Level

Question: Ali and Ben share some money in the ratio 5 : 3. Ali receives £40 more than Ben. How much does each person receive?

Working:

Step 1: Difference in parts = 5 - 3 = 2 parts

Step 2: 2 parts = £40, so 1 part = £40 / 2 = £20

Step 3: Ali = 5 x £20 = £100. Ben = 3 x £20 = £60.

Step 4: Check: £100 - £60 = £40 ✓

Answer: Ali receives £100 and Ben receives £60.

Worked Example 2 — Higher Level

Question: The ratio of blue to green to red beads is 7 : 4 : 3. There are 12 more blue beads than red beads. How many beads are there in total?

Working:

Step 1: Difference between blue and red parts = 7 - 3 = 4 parts

Step 2: 4 parts = 12 beads, so 1 part = 12 / 4 = 3 beads

Step 3: Total parts = 7 + 4 + 3 = 14

Step 4: Total beads = 14 x 3 = 42 beads

Check: Blue = 21, Green = 12, Red = 9. Blue - Red = 21 - 9 = 12 ✓

Answer: There are 42 beads in total.

Worked Example 3 — Exam Style

Question: The ages of a father and son are in the ratio 7 : 2. The father is 35 years older than the son. Find both of their ages.

Working:

Difference in parts = 7 - 2 = 5 parts

5 parts = 35 years, so 1 part = 35 / 5 = 7 years

Father = 7 x 7 = 49 years

Son = 2 x 7 = 14 years

Check: 49 - 14 = 35 ✓

Answer: The father is 49 and the son is 14.

Common Mistakes

• Dividing the difference by the total parts instead of the difference in parts. If the ratio is 5 : 3 and the difference is £40, divide by 2 (the difference in parts), not by 8 (the total parts).
• Confusing "difference" with "total." Read the question carefully. "Ali gets £40 more than Ben" gives a difference. "They share £400" gives a total. The method is different for each.
• Getting the subtraction order wrong in three-part ratios. Make sure you subtract the correct pair. If the question says "12 more blue than red," subtract the red part from the blue part.

Exam Tips

• Underline or highlight the word "more," "less" or "difference" in the question so you know which method to use.
• Always write "difference in parts = ..." as your first line of working. This earns a method mark.
• After finding each share, verify that the difference matches the value given in the question. This catches arithmetic errors and is good exam practice.
• Three-part ratio questions may tell you the difference between any two parts (not necessarily the largest and smallest). Read carefully to identify which two are being compared.

Practice Questions

Q1 (Foundation): Two numbers are in the ratio 4 : 9. The larger number is 30 more than the smaller. Find both numbers.

Q2 (Foundation): Mia and Jake share sweets in the ratio 3 : 5. Jake gets 14 more sweets than Mia. How many sweets are there in total?

Q3 (Higher): The ratio of sand to cement to water in a mix is 5 : 2 : 1. There are 18 kg more sand than water. Find the total mass of the mix.

Practise ratio problems with differences and get instant feedback free on GCSEMathsAI.

Related Topics

• Ratio Basics and Sharing
• Ratio and Fractions
• Ratio Problem Solving in Context

Summary

• When given the difference (not the total), find the difference in parts first.
• Value of one part = actual difference / difference in parts.
• Multiply each ratio number by the value of one part to find each share.
• Always check that the difference between your calculated shares matches the given difference.
• Do not confuse difference questions with total questions — the method is different.
• This method extends naturally to three-part ratios, where the difference may be between any two parts.