Rearranging formulae, change the subject
Learn how to rearrange formulae like unwrapping a present. Worked examples, exam-style questions with answers.
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Download the PDF (643 KB)If v = u + at gives the speed, how do you turn it round to find the time?
Rearranging formulae feels hard because the letter you want is hidden. Think of it like unwrapping a present. You undo steps in reverse order to find the letter you need.
The picture to keep in your head: Unwrapping a present
A formula wraps the letter you want in layers: in y = 3x + 5, x was first multiplied by 3, then had 5 added. To get x out you unwrap in reverse order, like taking off your shoes before your socks: undo the + 5 first (subtract 5), then undo the × 3 (divide by 3). Whatever you do to one side, you do to the other, so the formula stays true.
The method, every time
- Undo in reverse order: Start with the last step done to the letter you want.
- Do the same to both sides: Whatever you do to one side, do to the other.
- Clear the fraction first: If there's a fraction, undo it before other steps.
- Use ± for square roots: When you undo a square, remember to include both positive and negative answers.
- Collect and factorise: If the letter appears twice, group and simplify first.
Words you'll see
- subject
- the letter on its own on one side, e.g. t in t = (v - u)/a
- inverse operation
- the step that undoes another: + undoes -, × undoes ÷, square root undoes square
- reverse order
- undo the last thing done to the letter first
- factorise
- take the letter out as a common factor, e.g. ax - bx = x(a - b)
Undo in reverse order
Think of a formula like a wrapped present. The letter you want is hidden under layers. To get it out, you undo each layer in reverse order, like taking off your shoes before your socks. Whatever you do to one side of the formula, you do to the other. This keeps the formula true.
Watch out
- Dividing only part of one side. Divide the whole side, not just part of it
- Undoing in the wrong order. Undo addition/subtraction first, then multiplication/division
Clear the fraction first
Unwrapping a present is like rearranging a formula. If the formula has a fraction, it’s like the present is inside a box. To get x out, you start by clearing the fraction. Multiply both sides by the denominator to remove the fraction. Whatever you do to one side, you do to the other.
Watch out
- Multiplying only the top by 5. Multiply the whole right-hand side by 5
- Forgetting brackets when writing the answer. Keep brackets around the numerator to show it’s all divided by 2
Square to undo a root, root to undo a square
Squaring and square-rooting undo each other. In A = πr2, r was squared first, then multiplied by π. Unwrap in reverse order: divide by π, then square root. If the letter is under a root instead, square both sides to take the root off. Whatever you do, do it to the whole of both sides.
Watch out
- Square-rooting only part of a side. Always square-root the whole side, not just part of it.
- Writing √(A)/π. Divide inside the square root first, then take the root.
The letter appears twice: collect and factorise
When the same letter appears twice in a formula, it's like having two parts of the present wrapped together. You need to collect all the letter terms on one side and move everything else to the other. Then, you factorise the letter out and divide both sides by what's in the bracket. This way, you unwrap the letter properly.
Watch out
- Trying to divide before collecting the x terms. Always collect like terms first
- Sign errors when moving terms. Change the sign when moving terms across the equals sign
20 exam-style rearranging formulae questions
Show your working: most marks are for the method. Marks are a guide to how much working an exam would expect. Questions written by Bloom Education in the style of GCSE (AQA, Edexcel, OCR) and Cambridge IGCSE papers.
A · One and two steps
- 1Make s the subject of P = 4s.[2 marks]
Show the answer
s = P/4
Divide both sides by 4. - 2Make x the subject of y = mx + c.[2 marks]
Show the answer
x = (y - c)/m
Subtract c, then divide by m. - 3Make w the subject of A = lw.[2 marks]
Show the answer
w = A/l
Divide both sides by l. - 4Make r the subject of C = 2πr.[2 marks]
Show the answer
r = C/(2π)
Divide both sides by 2π. - 5Make u the subject of v = u + at.[2 marks]
Show the answer
u = v - at
Subtract at from both sides.
B · Fractions and brackets
- 6Make v the subject of s = (u + v)t/2.[3 marks]
Show the answer
v = 2s/t - u
Multiply by 2, divide by t, subtract u. - 7Make C the subject of F = 9C/5 + 32.[3 marks]
Show the answer
C = 5(F - 32)/9
Subtract 32, multiply by 5, divide by 9. - 8Make x the subject of y = 3(x - 4).[3 marks]
Show the answer
x = y/3 + 4
Divide by 3, then add 4. - 9Make h the subject of A = (a + b)h/2.[3 marks]
Show the answer
h = 2A/(a + b)
Multiply by 2, then divide by the whole bracket (a + b).
C · Powers and roots (Higher)
- 10Make c the subject of E = mc².[3 marks]
Show the answer
c = √(E/m)
Divide by m, then square root. - 11Make r the subject of V = πr²h.[3 marks]
Show the answer
r = √(V/(πh))
Divide by πh, then square root. - 12Make l the subject of T = 2π√(l/g).[3 marks]
Show the answer
l = gT²/(4π²)
Divide by 2π, square both sides, multiply by g. - 13Make x the subject of y = √(x + 4).[3 marks]
Show the answer
x = y² - 4
Square both sides, then subtract 4.
D · Letter twice (Higher)
- 14Make x the subject of 3x + a = bx.[4 marks]
Show the answer
x = a/(b - 3)
Collect x terms: a = bx - 3x = x(b - 3). - 15Make x the subject of y = (x + 2)/(x - 1).[4 marks]
Show the answer
x = (y + 2)/(y - 1)
y(x - 1) = x + 2, so xy - x = y + 2 and x(y - 1) = y + 2. - 16Make x the subject of p(x + q) = r(x - q).[4 marks]
Show the answer
x = q(p + r)/(r - p)
px + pq = rx - rq, so pq + rq = rx - px = x(r - p). - 17Make t the subject of 5t - k = 2(t + k).[4 marks]
Show the answer
t = k
5t - k = 2t + 2k, so 3t = 3k.
E · Use it
- 18The formula for temperature is F = 9C/5 + 32. Find C when F = 77.[3 marks]
Show the answer
25
Start with F = 9C/5 + 32. Subtract 32: 77 - 32 = 45. Multiply by 5: 45 × 5 = 225. Divide by 9: 225 ÷ 9 = 25. So C = 25. - 19The area of a circle is 50 cm². Rearrange A = πr² to find r, to 1 decimal place.[3 marks]
Show the answer
r = 4.0 cm
r = √(A/π) = √(50/π) = 3.989… - 20The formula for velocity is v = u + at. Find u when v = 30, a = 4, t = 5.[3 marks]
Show the answer
10
Start with v = u + at. Subtract at: 30 - (4 × 5) = 30 - 20 = 10. So u = 10.
Rearranging formulae exam checklist
- Undo in reverse order: Start with the last step done to the letter you want.
- Do the same to both sides: Keep the equation balanced as you work.
- Clear the fraction first: Undo fractions before other operations to avoid mistakes.
- Use ± for square roots: Remember both positive and negative answers when you square root.
- Collect and factorise: If the letter appears twice, group and simplify first.
- Check by substituting: Plug in numbers to make sure your rearranged formula works.
Got stuck on a question?
Find the part of this guide it belongs to and go through the worked example again, line by line. Then put your answer back into the question: if it doesn't work, look for a sign or a step you skipped.
All of this page as a 10-page guide, with every answer at the back. No sign-up.
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