Rearranging Formulae
A formula is written for one job: v = u + at gives you v. Rearranging rewrites it for a different job, so it gives you t instead. The moves are exactly the ones you use to solve an equation; the only difference is that the "numbers" are letters.
National curriculum KS3: rearrange formulae to change the subject; GCSE extends to subjects appearing twice, powers and roots1 · The subject of a formula
The subject is the letter sitting alone on one side. In A = lw the subject is A. If you know l and w, A pops out. But if you know A and l and want w, the formula is pointing the wrong way. You could substitute and solve every time, or you could rearrange once to get w = A/l and use that forever.
Here are the two routes side by side. Same numbers, same answer, but the rearranged formula does the thinking once rather than every time.
2 · It's just an equation with letters
Compare the two columns. On the left, an equation with numbers. On the right, the same shape with letters. Every step is identical: the only difference is that you can't simplify c − b into a single number the way you can simplify 11 − 3 into 8. So you leave it written as c − b.
3 · Peel the onion
A useful way to plan the moves. Look at the formula from the point of view of the letter you want. What's been done to it, in what order? Each operation is a layer wrapped around it. To free the letter, peel the layers off from the outside in, which means undoing the operations in reverse order.
In v = u + at, the letter t was multiplied by a (inner layer), then u was added (outer layer). So first subtract u from both sides, then divide by a. Click "peel" to remove layers one at a time.
Powers and roots are layers too. If the subject is squared, the last layer to peel is the square, undone with a square root. A = πr²: divide by π, then square-root. Remember both sides get rooted: r = √(A/π), with the whole fraction under the root. And when you take a square root in a rearrangement, write ± unless the context (a length, a speed) rules out the negative.
4 · Do it yourself: the balance
Same balance as the equations page, but the weights have names. Choose the letter to make the subject, then click the operation to apply to both sides. The page checks each move is legal; you decide which move is useful.
5 · Checking a rearrangement
You can't "solve and check" because there's no single answer, but you can still check: pick easy numbers, put them into the original formula to get a consistent set, then put the same numbers into your rearranged version and see if it gives the right letter back.
6 · The common slips
| Formula → subject | Wrong | Right | What went wrong |
|---|---|---|---|
| v = u + at → t | t = v − u/a | t = (v − u)/a | The whole of v − u is divided by a. Brackets or a long fraction line. |
| y = mx + c → x | x = y − c − m | x = (y − c)/m | m was multiplying, so the inverse is dividing, not subtracting. |
| A = πr² → r | r = √A/π | r = √(A/π) | Divide by π first, then root the whole thing. |
| s = d/t → t | t = s/d | t = d/s | Multiply both sides by t first (st = d), then divide by s. |
| P = 2(l + w) → l | l = P − 2w / 2 | l = P/2 − w | Divide the whole of P by 2 (or expand first: P = 2l + 2w). |
| T = 3 − 2x → x | x = (3 − T)/−2 | x = (3 − T)/2 | Both are actually equal! But the tidy answer avoids dividing by a negative: 3 − T = 2x. |
7 · Where rearranging is needed
Science practicals
You measure voltage and current and want resistance: V = IR → R = V/I. Density ρ = m/V → V = m/ρ to find the volume of an irregular object from its mass.
Programming
Code runs one way. If you have fahrenheit = celsius * 9/5 + 32 and the user types Fahrenheit, you must rearrange on paper and write celsius = (fahrenheit - 32) * 5/9.
Finance
Simple interest I = Prt. A bank quotes the interest and the rate; how long until you earn £500? t = I/(Pr). Rearrange once, reuse for every customer.
Geometry
Circumference C = 2πr, but you measured the circumference of a tree and want the radius: r = C/(2π). Area of a trapezium rearranged gives the height from the area.
Speed, distance, time
s = d/t is the one everyone memorises as a triangle. The triangle is the three rearrangements: d = st and t = d/s. Rearranging lets you throw the triangle away.
Spreadsheets
Goal Seek in Excel is a brute-force rearrangement: "what input gives this output?" Doing the algebra gives an exact formula in a cell that updates instantly instead.
Worked examples
Make x the subject of y = 3x − 7
Make t the subject of s = d/t
Make r the subject of V = (4/3)πr³
Exam-style (GCSE): make x the subject of y = (2x + 1)/(x − 3)
Quick check
What you've learned
Rearranging a formula is solving an equation whose numbers are letters: do the same to both sides until the letter you want is alone. Plan by looking at what's been done to that letter and undoing it in reverse order, outermost layer first. A letter in a denominator comes up by multiplying through; a squared letter is freed last by square-rooting both sides, keeping everything under the root. Brackets around a whole numerator are not optional. And a quick substitution with easy numbers checks the result.