Dividing in a Given Ratio

Dividing an amount in a given ratio is one of the most common question types on the AQA GCSE Mathematics papers. You must be able to share amounts between two or more parts, work backwards from a known share, and handle three-part ratios confidently.

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Sharing an Amount in a Given Ratio

To divide (share) an amount in a given ratio, follow this method:

1. Add the parts of the ratio together 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 ratio number by the value of one part.

Worked Example 1

Divide 240 in the ratio 3 : 5.

Step	Calculation	Result
Total parts	3 + 5	8
Value of one part	240 / 8	30
First share	3 x 30	90
Second share	5 x 30	150

Check: 90 + 150 = 240 (correct)

graph TD
    A[Total Amount: 240] --> B[Total parts: 3 + 5 = 8]
    B --> C[One part = 240 / 8 = 30]
    C --> D[First share: 3 x 30 = 90]
    C --> E[Second share: 5 x 30 = 150]

Exam Tip: Always check your two (or three) shares add up to the original amount. This is a quick way to spot calculation errors.

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Worked Example 2: Money Problem

Amy and Ben share 420 pounds in the ratio 2 : 5. How much does each person get?

• Total parts = 2 + 5 = 7
• One part = 420 / 7 = 60
• Amy = 2 x 60 = 120 pounds
• Ben = 5 x 60 = 300 pounds

Check: 120 + 300 = 420 (correct)

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Three-Part Ratios

The method works in exactly the same way when there are three or more parts.

Worked Example 3

Divide 600 in the ratio 1 : 3 : 6.

• Total parts = 1 + 3 + 6 = 10
• One part = 600 / 10 = 60
• First share = 1 x 60 = 60
• Second share = 3 x 60 = 180
• Third share = 6 x 60 = 360

Check: 60 + 180 + 360 = 600 (correct)

Worked Example 4

The angles in a triangle are in the ratio 2 : 3 : 4. Find each angle.

• Total parts = 2 + 3 + 4 = 9
• Total angle sum = 180 degrees
• One part = 180 / 9 = 20
• Angles: 2 x 20 = 40 degrees, 3 x 20 = 60 degrees, 4 x 20 = 80 degrees

Check: 40 + 60 + 80 = 180 (correct)

Exam Tip: Ratio questions involving angles in triangles or quadrilaterals are very popular. Remember that angles in a triangle sum to 180 degrees and angles in a quadrilateral sum to 360 degrees.

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Finding the Original Amount

Sometimes you are given one share and need to work backwards to find the total or the other shares.

Method

1. Work out how many parts the known share represents.
2. Divide the known share by that number of parts to find one part.
3. Multiply up to find the total or other shares.

Worked Example 5

Two numbers are in the ratio 3 : 7. The smaller number is 18. Find the larger number.

• The smaller number corresponds to 3 parts.
• One part = 18 / 3 = 6
• The larger number = 7 x 6 = 42

Worked Example 6

Ali and Beth share some money in the ratio 4 : 9. Beth gets 45 pounds more than Ali. How much does each person get?

• The difference in parts = 9 - 4 = 5 parts
• 5 parts = 45 pounds
• One part = 45 / 5 = 9 pounds
• Ali = 4 x 9 = 36 pounds
• Beth = 9 x 9 = 81 pounds

Check: 81 - 36 = 45 (correct)

graph TD
    A["Difference: Beth - Ali = 45 pounds"] --> B[Difference in parts: 9 - 4 = 5]
    B --> C[One part = 45 / 5 = 9]
    C --> D[Ali = 4 x 9 = 36]
    C --> E[Beth = 9 x 9 = 81]

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Given One Share, Find the Total

Worked Example 7

Claire and Dan share some money in the ratio 2 : 3. Dan receives 75 pounds. What was the total amount shared?

• Dan's share = 3 parts = 75 pounds
• One part = 75 / 3 = 25 pounds
• Total parts = 2 + 3 = 5
• Total amount = 5 x 25 = 125 pounds

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Combining Ratios

Sometimes you need to combine two separate ratios that share a common quantity.

Worked Example 8

The ratio of cats to dogs is 3 : 4. The ratio of dogs to birds is 2 : 5. Write the ratio of cats to dogs to birds.

• Dogs appear in both ratios. Make the dog value the same:
  • Cats : Dogs = 3 : 4 — multiply by 1 gives 3 : 4
  • Dogs : Birds = 2 : 5 — multiply by 2 gives 4 : 10
• Now dogs = 4 in both.
• Combined ratio: Cats : Dogs : Birds = 3 : 4 : 10

Exam Tip: When combining ratios, find the LCM of the shared quantity so that you can link the two ratios together. This is a favourite higher-tier question.

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Common Mistakes