Ratio, proportion and rates of change

Sharing in a ratio

14 past-paper questions, easiest first. Try each one, then reveal the answer.

Foundation & Higher13 of 14 from Foundation papers

Revision guide by The Full Marks

Sharing in a ratio: what you need to know

Sharing in a ratio means splitting an amount so the parts are in a given ratio, like sharing money or mixing paint. It is a reliable source of marks if you use the 'value of one part' method every time.

Key facts

Total parts $=$ add the numbers in the ratio.
One part $=$ amount $\div$ total parts.
Each share $=$ one part $\times$ its number in the ratio.
Ratio to fraction: in $a : b$ , the first share is $\frac{a}{a+b}$ of the total. So red : blue $= 1 : 2$ means $\frac{1}{3}$ are red, not $\frac{1}{2}$ .
If you're told a difference (e.g. 'Sam gets £20 more'), divide it by the difference in parts.

Sharing a total

Add the parts of the ratio.
Divide the total by that to find one part.
Multiply one part by each number in the ratio.
Check the shares add up to the total.

When you know one share or a difference

Match the known amount to its number of parts.
Divide to find one part.
Multiply to find whatever the question asks for.

Worked examples

Example 1 Foundation · grade 3

Amir and Beth share £84 in the ratio $3 : 4$ . How much does each get?

Show the solution

Total parts $= 3 + 4 = 7$
One part $= 84 \div 7 = £12$
Amir: $3 \times 12 = £36$ , Beth: $4 \times 12 = £48$
Check: $36 + 48 = 84$

Example 2 Foundation · grade 5

Blue and white paint are mixed in the ratio $2 : 5$ . There are $1.5$ litres more white paint than blue. How much paint is there altogether?

Show the solution

Difference $= 5 - 2 = 3$ parts, and that is $1.5$ litres.
One part $= 1.5 \div 3 = 0.5$ litres
Total $= 7$ parts $= 7 \times 0.5 = 3.5$ litres

Common mistakes

Dividing by one number in the ratio instead of the total parts, e.g. $84 \div 3$ .
Treating a difference as a total. Read whether you're given the total, one share or a difference.
Forgetting part of a three-part ratio when adding the parts.
Turning a ratio straight into a fraction, e.g. thinking $1 : 2$ means a half. Use the total parts on the bottom.

Exam tips

Finding the value of one part usually earns the first method mark, so write it down clearly.
Add your shares to check they make the total; it takes seconds and catches most errors.
In 'is there enough?' or 'is she right?' questions, give a clear yes or no backed up with your figures.

Questions people ask

How do you share an amount in a ratio?

Add the parts of the ratio, divide the amount by that total to get one part, then multiply by each number in the ratio.

How do you share in a ratio with three parts?

Exactly the same way: add all three numbers, find one part, then multiply by each of the three.

Exam questions: Sharing in a ratio (14, easiest first)

Sharing in a ratio1 markGrade 3CalculatorOn both tiers

June 2023 · Paper 2 Foundation · Q22

Decide whether a claim about how many red counters there are, given a red-to-blue ratio, is correct.

Show answer

Answer	No: red is $\frac{1}{3}$ of the counters, so there are 16	1 mark

Worked solution Checked against the mark scheme

The ratio $1 : 2$ has $1 + 2 = 3$ parts.

$48 \div 3 = 16$ , so there are 16 red and 32 blue.

Helen is wrong: 24 red would be half, which is a $1 : 1$ ratio.

Watch out: Agreeing with Helen because 1 is half of 2.

Sharing in a ratio1 markGrade 3CalculatorOn both tiers

November 2022 · Paper 3 Foundation · Q22

Explain the mistake in a worked attempt to share an amount in a ratio.

Show answer

Answer	He should add the parts (3 + 5 = 8) and divide 120 by 8, not by 3 and by 5	1 mark

Worked solution Checked against the mark scheme

The ratio 3 : 5 has $3 + 5 = 8$ parts.

One part $= 120 \div 8 = 15$ , giving $45$ and $75$ .

Rob divided by 3 and by 5 separately instead of by 8.

Watch out: Just giving the correct answer (45 : 75) without saying what went wrong.

Sharing in a ratio3 marksGrade 3No calculatorFoundation

November 2020 · Paper 1 Foundation · Q15

Use a mixing ratio to find the total drink made and how many litre bottles it fills completely.

Show answer

Answer	7	3 marks

Worked solution Checked against the mark scheme

Water: $750 \times 9 = 6750$ ml.

Total drink: $750 + 6750 = 7500$ ml $= 7.5$ litres.

Only whole bottles count, so he can fill 7 bottles.

Watch out: Rounding 7.5 up to 8, or forgetting to add the squash itself.

Sharing in a ratio3 marksGrade 4No calculatorFoundation

November 2024 · Paper 1 Foundation · Q17

Decide whether the paint available is enough to make a mix in a three-part ratio.

Show answer

Answer	No: she needs 12 l blue, 9 l yellow, 3 l white but has only 7 l yellow	3 marks

Worked solution Checked against the mark scheme

Total parts $= 4 + 3 + 1 = 8$ , so one part $= 24 \div 8 = 3$ litres.

Blue $= 12$ , yellow $= 9$ , white $= 3$ .

She has only 7 litres of yellow, so no, she does not have enough.

Watch out: Dividing 24 by 3 (the number of colours) instead of 8.

Sharing in a ratio3 marksGrade 4CalculatorOn both tiers

June 2023 · Paper 2 Foundation · Q25

Find a percentage of an amount and then share it in a given ratio to find one part.

Show answer

Answer	168 kg	3 marks

Worked solution Checked against the mark scheme

Paper and glass: $0.57 \times 800 = 456$ kg

Ratio $12 : 7$ has 19 parts: $456 \div 19 = 24$

Glass: $7 \times 24 = 168$ kg

Watch out: Sharing all 800 kg in the ratio instead of the 57%.

Sharing in a ratio3 marksGrade 4No calculatorOn both tiers

November 2021 · Paper 1 Foundation · Q24

Use a difference between two shares of a three-part ratio to find the difference between two other shares.

Show answer

Answer	$33$	3 marks

Worked solution Checked against the mark scheme

Ibrahim minus Matilda: $15 - 7 = 8$ parts $= 24$ , so 1 part $= 3$ .

Ibrahim minus Rosie: $15 - 4 = 11$ parts.

$11 \times 3 = 33$

Watch out: Dividing 24 by 26, the total number of parts.

Sharing in a ratio4 marksGrade 4CalculatorOn both tiers

November 2021 · Paper 3 Foundation · Q22

Work out the cost of cheese needed for a recipe split in a three-part ratio, given a price per pack weight.

Show answer

Answer	£12.86 (or £13.50 if whole packs are bought)	4 marks

Worked solution Checked against the mark scheme

Total parts: $9 + 2 + 1 = 12$

Cheese: $6000 \div 12 \times 2 = 1000$ g

$1000 \div 175 = 5.714...$ lots of 175 g

$5.714... \times 2.25 = £12.857...$ , so about £12.86

Watch out: Using 2 out of 3 parts instead of 2 out of 12.

Sharing in a ratio4 marksGrade 5CalculatorOn both tiers

November 2023 · Paper 2 Foundation · Q26

Use a share in a three-way ratio to find the total, then compare with a new ratio to see how much one person passed to another.

Show answer

Answer	£21	4 marks

Worked solution Checked against the mark scheme

Debbie has 4 parts $= 120$ , so 1 part $= 30$ and the total is $9 \times 30 = 270$ .

At first Errol has $2 \times 30 = 60$ .

New ratio has $2 + 5 + 3 = 10$ parts, so 1 part $= 27$ and Errol now has $3 \times 27 = 81$ .

Chris gave Errol $81 - 60 = £21$

Watch out: Assuming Debbie still has £120 after the change.

Sharing in a ratio5 marksGrade 5No calculatorFoundation

November 2022 · Paper 1 Foundation · Q18

$Share a total in a three-part ratio, remove fractions of two groups, and give one group as a percentage of what remains.$

Show answer

Answer	$60\%$	5 marks

Worked solution Checked against the mark scheme

One part: $240 \div 10 = 24$

Cola 120, lemonade 72, orange 48

Removed: $\frac{1}{2} \times 72 = 36$ and $\frac{1}{12} \times 48 = 4$

Left: $240 - 40 = 200$

$\frac{120}{200} \times 100 = 60\%$

Watch out: Dividing by the original 240 instead of the 200 left.

Sharing in a ratio4 marksGrade 5No calculatorOn both tiers

June 2019 · Paper 1 Foundation · Q27

Use a ratio of packs sold and pens per pack to find how many green pens were sold.

Show answer

Answer	96	4 marks

Worked solution Checked against the mark scheme

Pens in the ratio $7 \times 2 : 3 \times 5 : 4 \times 6 = 14 : 15 : 24$

Total parts $= 53$ , and $212 \div 53 = 4$

Green pens $= 24 \times 4 = 96$

Watch out: Sharing 212 in the ratio 7 : 3 : 4 without allowing for pack sizes.

Sharing in a ratio4 marksGrade 5CalculatorOn both tiers

November 2019 · Paper 2 Foundation · Q23

Use two ratios with the same total to find how many items changed hands.

Show answer

Answer	18	4 marks

Worked solution Checked against the mark scheme

Before: $240 \div 10 = 24$ , so Tom has $3 \times 24 = 72$

After: $240 \div 8 = 30$ , so Tom has $3 \times 30 = 90$

Tom buys $90 - 72 = 18$ stamps

Watch out: Dividing 240 by 3 or 7 instead of by the total number of parts.

Sharing in a ratio4 marksGrade 5CalculatorOn both tiers

June 2019 · Paper 2 Foundation · Q26

Share £360 between four people using a ratio and a multiplier, then find one person's share.

Show answer

Answer	£168	4 marks

Worked solution Checked against the mark scheme

Abby : Ben = 2 : 7. Chloe and Denesh each get $1.5 \times 2 = 3$ parts.

Total parts $= 2 + 7 + 3 + 3 = 15$

One part $= 360 \div 15 = £24$

Ben gets $7 \times 24 = £168$

Watch out: Giving Chloe and Denesh 1.5 parts each instead of 3.

Sharing in a ratio4 marksGrade 5No calculatorHigher

June 2018 · Paper 1 Higher · Q8

Share a triangle's perimeter in the ratio 3 : 4 : 5 and find its area.

Show answer

Answer	$216\text{ cm}^2$	4 marks

Worked solution Checked against the mark scheme

$3 + 4 + 5 = 12$ parts, so 1 part $= 72 \div 12 = 6$ cm.

Sides: $18$ , $24$ and $30$ cm. The right angle is between 18 and 24.

Area $= \frac{1}{2} \times 18 \times 24 = 216\text{ cm}^2$

Watch out: Using the hypotenuse (30 cm) as the base or height.

Sharing in a ratio4 marksGrade 5CalculatorOn both tiers

June 2024 · Paper 2 Foundation · Q25

$Share sweets in a ratio, apply a fraction and a percentage transfer, and check if everyone ends up equal.$

Show answer

Answer	Yes: each person ends with the same amount (e.g. 7 : 7 : 7 per 21 parts, or 147 each out of 441)	4 marks

Worked solution Checked against the mark scheme

Take the shares as $1 : 6 : 14$ parts (21 parts).

Tina gives Andy $\frac{3}{7} \times 14 = 6$ . Now Andy $= 7$ , Tina $= 8$ .

Tina gives Luke $12.5\%$ of $8 = 1$ . Now Luke $= 7$ , Tina $= 7$ .

Final: Andy 7, Luke 7, Tina 7 parts, so Tina is correct.

Watch out: Taking 12.5% of Tina's original share instead of what remains.

Also tested in these questions

Their main topic is different, but part of each one tests this topic.

November 2024 · Paper 3 Foundation · Q21: Share a total in a ratio and use buying and selling prices to find the percentage profit.
June 2024 · Paper 2 Foundation · Q23: Use a probability to find the total number of discs, then share the rest in a ratio to find the red ones.
November 2023 · Paper 1 Foundation · Q23: Use an isosceles triangle, a ratio of angles and angles on a straight line to find an exterior angle.
November 2022 · Paper 1 Foundation · Q17: Find a 'not' probability, use a ratio to find a count, and decide whether the remaining counters can be split equally.
November 2022 · Paper 3 Foundation · Q12: Write a three-part ratio from statements about how many sweets each person has compared with another.
November 2022 · Paper 3 Higher · Q19: Estimate how often a green counter is picked in repeated trials using a probability and a ratio.
November 2022 · Paper 3 Foundation · Q24: Find the volume of four cylindrical tanks and check whether there is enough fertiliser for a 1 : 100 mix.
June 2022 · Paper 3 Higher · Q18: Find the angle between a line in a prism and its base, using a ratio, trapezium area, Pythagoras and trigonometry.
November 2021 · Paper 3 Higher · Q14: Combine two ratios linked by the fact that one pair has half as many sweets as the other, and write the four-part ratio in simplest form.
November 2019 · Paper 1 Foundation · Q21: Work out how many men, women and children are in a group from fraction and multiple facts, then write children to men as n : 1.
November 2019 · Paper 1 Foundation · Q24: Split an angle of a triangle in a given ratio, then use angle facts to find an angle at a point on a side.
November 2019 · Paper 3 Higher · Q13: Find the mass of a cylinder filled with two liquids mixed in a ratio, using their densities.
November 2019 · Paper 3 Foundation · Q16: Fill in two missing probabilities that are in a given ratio.
June 2019 · Paper 3 Higher · Q14: Use the sine area formula for two congruent triangles and a side ratio to find a rectangle's length.
November 2018 · Paper 1 Foundation · Q13: Write three scores as a ratio from relationships between them.
November 2018 · Paper 3 Foundation · Q25: Use an angle ratio in a right-angled triangle to find the angles, then use trigonometry to find the hypotenuse.
June 2018 · Paper 1 Higher · Q2: Combine two linked ratios for houses, flats and bungalows and use the number of bungalows to find the number of houses.
Specimen · Paper 3 Foundation · Q28: Use a ratio of densities and a known density to find the density of a mixture of two liquids.