Turn a ratio into a fraction of the whole, or into an equation like y=kx, and back again. It is on Foundation and Higher, aimed at grade 5. Free questions with answers are on this page. Worked examples show each line before you try the questions.
A ratio 2:5 can be rewritten as a fraction of the whole (72 and 75) or as a linear relationship y=25x if x and y stay in that ratio. Exam questions hop between those languages: 'the ratio of boys to girls is 3:4' then 'write the number of boys as a fraction of the class' then 'an expression for g in terms of b'.
To write one amount as a fraction of another, put the asked amount on top. To write a formula, pick a letter for one quantity and scale the other by the ratio multiplier. Keep the order of the ratio matching the order of the letters.
A ratio a:b is a parts out of a+b of the whole, so the first part is a+ba. Writing q in terms of p from p:q=3:8 is q=38p. Going the other way, 53 red means red:not red =3:2.
Objectives
Evaluate y = mx + c at a given x.
Use a fractional multiplier such as 3/4 of x.
Run a two-step machine, and find x from y.
Key points
Write each part of a ratio as a fraction of the total (add the parts for the denominator).
Turn a fraction of a whole back into a ratio of part : rest.
Write a ratio x:y=a:b as the equation y=abx.
Use that equation to find one quantity when you know the other.
Worked examples
Example 1
y = 2x + 3. When x = 4, find y.
2 marks
Substitute x = 4 into the formula: y = 2(4) + 3.
Multiply before adding: 2 × 4 = 8, then 8 + 3 = 11.
The resulting value of y is 11.
Answer: 11
Common slips
Slip: Add 2, 3 and 4 directly. The gradient 2 must multiply x before adding the intercept 3.
Slip: Amy wrote 2 × 4 + 1 = 9. The constant added is 3, not 1.
Slip: Conclude that y = 9. Adding 3 to 8 gives 11.
Example 2
y = (3/4)x when x = 8, find y.
2 marks
Multiply 8 by 3 and divide by 4.
Compute: (3 × 8) / 4 = 24 / 4 = 6.
The value of y is 6.
Answer: 6
Common slips
Slip: Divide 8 by 4 and forget the numerator 3. The numerator 3 is part of the fractional scale factor.
Slip: Ben wrote 8 / 4 = 2. You must multiply by the numerator 3 as well as dividing by 4.
Slip: Conclude y = 2. Three quarters of 8 is 6.
Example 3
A machine multiplies by 2 and then adds 3. Work out the output when the input is 4.
3 marks
Apply operations in the stated order: multiply by 2 first.
Multiply: 4 × 2 = 8, then add 3: 8 + 3 = 11.
The final output of the machine is 11.
Answer: 11
Common slips
Slip: Add 3 to 4 before multiplying. The machine multiplies first and adds second.
Slip: Chloe added first: (4 + 3) × 2 = 14. Following operations out of order changes the result.
Slip: Conclude output is 14. Multiplying 4 by 2 then adding 3 produces 11.
Questions and answers
GCSE Maths ratio as a fraction or linear function questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
y = 3x + 1. When x = 2, find y.
Show answer
Answer: 7
Substitute x = 2: y = 3(2) + 1.
Calculate: 6 + 1 = 7.
M1 — Substitute x = 2: y = 3(2) + 1.
A1 — Calculate: 6 + 1 = 7.
Question 2 Easy2 marks
y = 4x - 2. When x = 3, find y.
Show answer
Answer: 10
Substitute x = 3: y = 4(3) - 2.
Calculate: 12 - 2 = 10.
M1 — Substitute x = 3: y = 4(3) - 2.
A1 — Calculate: 12 - 2 = 10.
Question 3 Medium2 marks
y = (2/5)x when x = 10, find y.
Show answer
Answer: 4
Divide by denominator: 10 / 5 = 2.
Multiply by numerator: 2 × 2 = 4.
M1 — Divide by denominator: 10 / 5 = 2.
A1 — Multiply by numerator: 2 × 2 = 4.
Question 4 Medium2 marks
y = (3/2)x when x = 6, find y.
Show answer
Answer: 9
Divide by denominator: 6 / 2 = 3.
Multiply by numerator: 3 × 3 = 9.
M1 — Divide by denominator: 6 / 2 = 3.
A1 — Multiply by numerator: 3 × 3 = 9.
Question 5 Hard2 marks
A machine multiplies by 3 and then adds 1. Work out the output when the input is 5.
Show answer
Answer: 16
First operation: 5 × 3 = 15.
Second operation: 15 + 1 = 16.
M1 — First operation: 5 × 3 = 15.
A1 — Second operation: 15 + 1 = 16.
Question 6 Hard2 marks
A machine multiplies by 4 and then adds 2. Work out the output when the input is 3.
Show answer
Answer: 14
First operation: 3 × 4 = 12.
Second operation: 12 + 2 = 14.
M1 — First operation: 3 × 4 = 12.
A1 — Second operation: 12 + 2 = 14.
Question 7 Exam-style2 marks
y = (5/2)x. Work out the value of y when x = 8.
Show answer
Answer: 20
Divide 8 by the denominator 2 to get 4.
Multiply by the numerator 5: 5 × 4 = 20.
M1 — 8 ÷ 2 = 4, or 5 × 8 = 40
A1 — 20
Common mistakes
Most lost marks on this topic are one slip.
Adding the gradient, the intercept and x
Multiply x by the gradient first
Ignoring the numerator of a fractional multiplier
Multiply by the top and divide by the bottom
Adding before multiplying in a two-step machine
Follow the order in the sentence
Spot the slip
Ada reads y = 2x + 3. When x = 4, she adds 2, 3 and 4. What is the value of y?
11
9
24
Show the slip
11 2×4 + 3 = 8 + 3 = 11.
Exam tips
Be ready to do this without a calculator. Paper 1 is the non-calculator paper.
If the question says “show that”, the result is given. The marks are for the steps that reach it.
The method is the same on AQA-style and Edexcel-style papers.
Write the total number of parts down first — it is usually the method mark.
Always test your y=kx formula with the numbers from the ratio; a one-line check catches the upside-down error.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.