Rearranging Formulae
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How to make a different letter the subject of a formula by undoing each step with its inverse, in reverse order, including formulae with fractions and formulae with several letters.
What the subject of a formula is
A formula is a rule that links letters. For example, P=4s gives the perimeter P of a square with sides of length s. The letter on its own on one side of the equals sign is the subject of the formula. Here the subject is P
Rearranging a formula, or changing its subject, means getting a different letter on its own. Making s the subject of P=4s gives s=\frac{P}{4}, so you can work out the side straight from the perimeter.
In the exam you are usually asked to "Make x the subject of the formula", for 2 marks, in the second half of the paper. Some formulae have a fraction in them. The hardest Foundation questions have several letters, sometimes with a different letter squared. You may also be asked to put numbers into the formula, before or after rearranging it, or to find the mistake in someone's working. Examiners say many students find this hard, so learn the method one step at a time.
Your answer must start with the new subject, such as x=, with nothing else on that side.
Undo with inverse operations
Rearranging works just like solving an equation. Whatever you do to one side, do exactly the same to the other side. The only difference is that your answer has letters in it, not just a number.
An inverse operation undoes another one: - Subtracting undoes adding, so y=x+9 gives x=y-9 - Adding undoes subtracting, so y=x-2 gives x=y+2 - Dividing undoes multiplying, so y=5x gives x=\frac{y}{5} - Multiplying undoes dividing, so y=\frac{x}{4} gives x=4y
Write a division as a fraction, such as \frac{y}{5}. It is clearer than y\div5, especially when more than one term is being divided.
Worked example 1: One-step formulae
(a) Make u the subject of the formula v=u+12 (b) Make s the subject of the formula P=5s
- (a) u has 12 added to it. Subtract 12 from both sides: v-12=u
- Write it with the subject on the left: u=v-12
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Your turn. (b) s is multiplied by 5. Make s the subject: write s= and the rest.
Divide both sides by 5: s=\frac{P}{5}
- Answer: (a) u=v-12 (b) s=\frac{P}{5}
Two steps: undo in reverse order
In y=3x+5, two things happen to x. It is multiplied by 3, then 5 is added, and that makes y
To get x on its own, undo the steps in reverse order, like taking off your shoes before your socks. Undo the +5 first by subtracting 5 from both sides: y-5=3x. Then undo the \times3 by dividing both sides by 3: x=\frac{y-5}{3}
The fraction line goes under all of y-5, because the whole side is divided by 3. Writing y-5\div3 would divide only the 5
Worked example 2: A two-step formula
Make d the subject of the formula L=4d+15
- d is multiplied by 4, then 15 is added. Undo the +15 first, by subtracting 15 from both sides: L-15=4d
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Your turn. Now divide both sides by 4. Write d= and the rest.
d=\frac{L-15}{4}, with all of L-15 on top
- Check with a number. If d=5, then L=4\times5+15=35, and \frac{35-15}{4}=\frac{20}{4}=5, which is the d you started with.
- Answer: d=\frac{L-15}{4}
When the formula has a fraction
If the letter you want is divided by a number, undo the dividing by multiplying. In r=\frac{n}{4}+3, first subtract 3 from both sides: r-3=\frac{n}{4}. Then multiply all of the left side by 4: n=4(r-3)
The bracket matters. 4(r-3) means 4 lots of all of r-3, which is 4r-12. Writing 4r-3 multiplies only the r by 4, which is wrong.
If one whole side is a fraction, clear the fraction first. In k=\frac{x+6}{5}, multiply both sides by 5 to get 5k=x+6. Then subtract 6 from both sides: x=5k-6
Worked example 3: A letter divided by a number
Make k the subject of the formula w=\frac{k}{6}+2
- k is divided by 6, then 2 is added. Undo the +2 first, by subtracting 2 from both sides: w-2=\frac{k}{6}
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Your turn. Now multiply all of the left side by 6. Write k= and the rest.
k=6(w-2), with a bracket so that all of w-2 is multiplied by 6
- You can multiply out the bracket if you like: k=6w-12. Both answers are correct.
- Answer: k=6(w-2)
Formulae with more letters
Many formulae have several letters, such as y=mx+c. Treat every letter except the one you want just like a number, and use the same inverse operations in the same order.
For example, the volume of a cuboid is V=lwh. To make h the subject, divide both sides by lw: h=\frac{V}{lw}
If the letter you want is being subtracted, as in k=30-2t, add 2t to both sides first so that it is positive: k+2t=30. Then subtract k from both sides to get 2t=30-k, and divide by 2: t=\frac{30-k}{2}
Take care with the order of a subtraction: 30-k and k-30 are different.
Worked example 4: Rearranging, then substituting
A stack of n identical boxes stands on a pallet. The height of the stack, H cm, is given by the formula H=nb+p, where b cm is the height of one box and p cm is the height of the pallet. (a) Make b the subject of the formula. (b) Work out b when H=135, n=4 and p=15
- (a) b is multiplied by n, then p is added. Subtract p from both sides: H-p=nb
- Divide both sides by n: b=\frac{H-p}{n}
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Your turn. (b) Put H=135, n=4 and p=15 into your formula. What is b?
b=\frac{135-15}{4}=\frac{120}{4}=30
- Check in the original formula: 4\times30+15=135, which is H
- Answer: (a) b=\frac{H-p}{n} (b) b=30, so each box is 30 cm tall.
Dividing by a letter, and squared letters
If the letter you want is divided by another letter, undo the dividing by multiplying by that letter. In y=\frac{x}{t}+4, subtract 4 from both sides: y-4=\frac{x}{t}. Then multiply all of the left side by t: x=t(y-4)
Some formulae have a squared letter, such as y=c^2+3m. If the letter you want is not the squared one, treat c^2 just like any other term and move all of it with an inverse operation. To make m the subject, subtract c^2 from both sides: y-c^2=3m. Then divide both sides by 3: m=\frac{y-c^2}{3}
The square stays where it is. You do not need to take a square root, and c^2 must not turn into c or 2c.
Worked example 5: Dividing by a letter
Make x the subject of the formula g=\frac{x}{w}-6
- x is divided by w, then 6 is subtracted. Undo the -6 first, by adding 6 to both sides: g+6=\frac{x}{w}
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Your turn. Now multiply all of the left side by w. Write x= and the rest.
x=w(g+6), with a bracket so that all of g+6 is multiplied by w
- You can multiply out the bracket if you like: x=wg+6w. Both answers are correct.
- Check with numbers. If x=40 and w=4, then g=\frac{40}{4}-6=4, and 4\times(4+6)=40, which is the x you started with.
- Answer: x=w(g+6)
Worked example 6: A formula with a squared letter
Make h the subject of the formula P=r^2+4mh
- h is multiplied by 4m, then r^2 is added. You want h, not r, so treat r^2 like any other term.
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Your turn. Undo the +r^2 first, by subtracting r^2 from both sides. What is 4mh equal to now?
Subtract r^2 from both sides: P-r^2=4mh
- Now divide both sides by 4m: h=\frac{P-r^2}{4m}, with all of P-r^2 on top and all of 4m underneath
- Check with numbers. If r=3, m=2 and h=5, then P=9+40=49, and \frac{49-9}{8}=\frac{40}{8}=5, which is the h you started with.
- Answer: h=\frac{P-r^2}{4m}
Common mistakes
- Swapping the two letters round. y=3x+4 does not become x=3y+4: undo each operation with its inverse instead.
- Doing something to one side only. Whatever you do to one side, do exactly the same to the whole of the other side.
- Undoing in the wrong order. Undo the last thing that happened to the letter first, so in y=2x+7 subtract 7 before you divide by 2
- Dividing or multiplying only part of a side. Put a fraction line under the whole side, as in x=\frac{y-7}{2}, or use a bracket, as in 4(r-3)
- Getting a subtraction the wrong way round. y-7 and 7-y are different, so keep the order that your working gives.
- Stopping before the letter is on its own. 2x=y-7 is not finished: your answer must start x= with nothing else on that side.
More on this topic: Rearranging Formulae worksheet with full solutions