Edexcel GCSE Maths Foundation (1MA1), Foundation tier · Ratio, proportion and rates of change
Practise Ratio. 2 exam-style questions plus unlimited generated ones on this subtopic, at up to four difficulty levels, with full mark schemes and a progress tracker. Free, no account needed.
A ratio compares the sizes of two or more quantities, e.g. 3 : 5. Ratio questions are common on both tiers: simplifying and sharing on both and, at Higher tier, multi-step problems where you combine ratios or form an equation when a ratio changes.
Grade by grade
What you need to be able to do, from the first marks up to the top grade.
2
Simplify a ratio using the HCFDivide every part by the highest common factor, e.g. 12 : 18 = 2 : 3.
3
Share an amount in a given ratioFind the value of one part first, e.g. £40 shared in the ratio 3 : 5 gives £15 and £25.
3
Write one quantity as a fraction of anotherUse the same units, then write the first over the second and simplify, e.g. 40 minutes out of an hour is \(\frac{2}{3}\).
4
Convert between ratios and fractionsIn the ratio 2 : 5 the first share is \(\frac{2}{7}\) of the total, not \(\frac{2}{5}\).
4
Use one share or a differenceDivide the known amount by the number of parts it stands for to find one part.
4
Write ratios in the form 1 : nDivide both parts by the first part, e.g. 4 : 10 = 1 : 2.5.
5
Combine two ratios with a shared quantityMake the shared quantity's parts equal, e.g. a : b = 1 : 2 and b : c = 3 : 4 give a : b : c = 3 : 6 : 8.
Notes
Simplifying ratios
A ratio compares quantities in order: if boys : girls = 3 : 4, then girls : boys = 4 : 3.
Simplify by dividing every part by the HCF: 18 : 24 : 30 = 3 : 4 : 5 (divide by 6).
Change to the same units first, then leave the units out: 40 cm : 2 m = 40 : 200 = 1 : 5.
Clear decimals or fractions by multiplying every part: 1.5 : 2 = 3 : 4, and \(\frac{1}{2} : \frac{1}{3}\) = 3 : 2 (multiply by 6).
For the form 1 : n, divide both parts by the first part: 8 : 20 = 1 : 2.5 (n can be a decimal).
Sharing and finding one part
Add the numbers in the ratio to get the total parts, divide the amount by this to find one part, then multiply.
£56 in the ratio 3 : 5 is 8 parts, so one part = £7 and the shares are £21 and £35.
Given one share, divide it by its number of parts. In 2 : 7, if the smaller share is 16, one part is 8 and the larger share is 56.
Given a difference, divide by the difference in parts. In 5 : 2, a difference of 45 is 3 parts, so one part is 15.
Ratios, fractions and proportion
In a : b the first quantity is \(\frac{a}{a+b}\) of the total: red : blue = 3 : 5 means \(\frac{3}{8}\) are red.
'A is \(\frac{2}{3}\) of B' means A : B = 2 : 3.
One quantity as a fraction of another: same units, then first over second. 45 minutes out of 2 hours is \(\frac{45}{120} = \frac{3}{8}\).
Equal ratios are in proportion: 4 : 6 = 10 : 15, as both simplify to 2 : 3.
A ratio gives a straight-line rule: if y : x = 3 : 2, then \(y = \frac{3}{2}x\).
Combining ratios
If a : b = 2 : 3 and b : c = 4 : 5, make the b parts equal. The LCM of 3 and 4 is 12.
a : b = 8 : 12 and b : c = 12 : 15, so a : b : c = 8 : 12 : 15.
Cheatsheet
Total parts = the numbers in the ratio added together
One part = amount ÷ total parts
In a : b, the first share is \(\frac{a}{a+b}\) of the total
Form 1 : n: divide both parts by the first part
a : b = c : d means \(\frac{a}{b} = \frac{c}{d}\) (equal ratios are in proportion)
Combine a : b and b : c by making the b parts equal (use the LCM)
How to answer each type of question
Share an amount in a given ratio
2 marks3
Add the numbers in the ratio to get the total number of parts.
Divide the amount by the total number of parts to find one part.
Multiply one part by each number in the ratio.
Check that the shares add up to the amount you started with.
Example. Priya and Tom share £84 in the ratio 3 : 4. Work out how much money each of them gets.
Show the model answer
3 + 4 = 7 parts, so one part = 84 ÷ 7 = £12 (M1) Priya gets 3 × 12 = £36 and Tom gets 4 × 12 = £48 (A1)
Write a fraction as a ratio in the form 1 : n
2 marks4
Turn the fraction into a ratio: if one share is \(\frac{2}{5}\) of the total, the other share is \(\frac{3}{5}\), so the ratio is 2 : 3.
Divide both parts by the first part.
Leave n as a decimal if it does not divide exactly.
Example. Ben and Ali share some money. Ben gets \(\frac{2}{5}\) of the money and Ali gets the rest. Write the ratio of Ben's share to Ali's share in the form 1 : n.
Show the model answer
Ali gets \(\frac{3}{5}\), so Ben : Ali = 2 : 3 (M1) 2 ÷ 2 : 3 ÷ 2 = 1 : 1.5 (A1)
Use one share or a difference to find an amount
3 marks4
Work out how many parts the known amount stands for (one share, or the difference between two shares).
Divide to find the value of one part.
Multiply to find what the question asks for (a share or the total).
Example. A bag contains only red sweets and green sweets. The ratio of red sweets to green sweets is 7 : 3. There are 28 more red sweets than green sweets. How many sweets are in the bag altogether?
Show the model answer
The difference is 7 − 3 = 4 parts, so one part = 28 ÷ 4 = 7 (P1) Total = (7 + 3) × 7 (P1) = 70 sweets (A1)
Combine two ratios
3 to 4 marks5
Find the quantity that appears in both ratios.
Multiply each ratio so that this quantity has the same number of parts in both (use the LCM).
Write one three-part ratio and add the parts.
Find one part, then the amount the question asks for.
Example. An orchestra has only string, wind and brass players. The ratio of string players to wind players is 5 : 2. The ratio of wind players to brass players is 4 : 3. There are 68 players in the orchestra. How many brass players are there?
Show the model answer
string : wind = 10 : 4 and wind : brass = 4 : 3 (P1) string : wind : brass = 10 : 4 : 3, which is 17 parts (P1) One part = 68 ÷ 17 = 4 (P1) Brass = 3 × 4 = 12 players (A1)
Shortcuts and memory tricks
Think 'one part first': almost every ratio question is easy once you know what one part is worth.
Draw a bar model: one box per part, write the known amount across its boxes, then share it out.
Check that your shares add up to the total and simplify back to the ratio you were given.
On the calculator paper, entering 18 ÷ 24 as a fraction simplifies it to \(\frac{3}{4}\), so 18 : 24 = 3 : 4.
Where marks are lost
Dividing the amount by one number in the ratio instead of by the total number of parts.
Writing the ratio in the wrong order: keep the order the question uses.
Simplifying before changing to the same units, e.g. writing 40 cm : 2 m as 20 : 1.
In difference questions, dividing by the total number of parts instead of the difference in parts.
Reading a ratio as a fraction: in 2 : 3 the first share is \(\frac{2}{5}\) of the total, not \(\frac{2}{3}\).
Stopping at x in a changing-ratio question instead of working out the amount asked for.
Exam technique
Write 'one part = ...' clearly: it is usually the first method or process mark.
Label each share with its name and units, e.g. 'Tom: £48'.
'Simplest form' means no whole number other than 1 divides into every part.
In multi-step problems, write a few words next to each calculation (e.g. 'rest of the money') so the examiner can follow your process.
Sample questions
Written for this site in the style of Edexcel exam questions. They are not taken from real past papers.
Question 1Easy3 marks
In a class, the ratio of boys to girls is 3 : 5. Leah says, '\(\frac{3}{5}\) of the class are boys.'
(a) Explain why Leah is wrong.[1]
(b) There are 32 students in the class. How many of the students are girls?[2]
Show the answer and mark scheme
(a)Answer: The ratio compares boys with girls. There are \(3 + 5 = 8\) parts, so \(\frac{3}{8}\) of the class are boys.
C1 for explaining that there are 8 parts altogether, so the fraction of boys is \(\frac{3}{8}\)
Worked solution: 3 : 5 means 3 boys for every 5 girls, so 3 out of every 8 students are boys: \(\frac{3}{8}\), not \(\frac{3}{5}\).
(b)Answer: 20
M1 for \(32 \div 8\) (= 4) or \(\frac{5}{8} \times 32\)
A1 for 20
Worked solution: \(32 \div 8 = 4\), so there are \(5 \times 4 = 20\) girls.
Question 2Medium3 marks
Amir and Bea share some sweets in the ratio 5 : 3. Amir then gives Bea 12 of his sweets. Now they each have the same number of sweets.
How many sweets were there altogether?[3]
Show the answer and mark scheme
Answer: 96
P1 for recognising that the difference of 2 parts is \(2 \times 12 = 24\), or \(5k - 12 = 3k + 12\)
P1 for 1 part = 12 (or \(k = 12\))
A1 for 96
Worked solution: After the swap they are equal, so Amir had 24 more than Bea (he lost 12 and she gained 12). The difference is \(5 - 3 = 2\) parts, so 2 parts = 24 and 1 part = 12. Total \(= 8 \times 12 = 96\) sweets.