A formula is a rule that connects quantities, like A = l × w for the area of a rectangle, or v = u + at in physics. Formulas are everywhere — converting temperatures, working out speed, sizing a circle. Two skills go with them: putting numbers in (substituting), and rearranging to get a different letter on its own (changing the subject).
Why this matters: a single formula can answer many questions if you can rearrange it. C = 2πr finds the circumference from the radius — but rearranged, the very same formula finds the radius from the circumference. Learn to change the subject once and you never need a second formula.
Changing the subject: rearrange to get r on its own
Substituting into a formula
Replace each letter with its value, then work it out — minding BIDMAS so that multiplying happens before adding.
v = u + at, with u = 3, a = 2, t = 5: v = 3 + 2×5 = 3 + 10 = 13. (Do the 2×5 first, not 3 + 2.)
BIDMAS: do the multiply (2×5) before the add.
Changing the subject
The subject is the letter sitting alone on one side, the one the formula tells you how to find. Changing the subject means rearranging so a different letter stands alone. This is exactly like solving an equation — use inverse operations on both sides until the letter you want is by itself.
Analogy: think of unwrapping a parcel. To reach the letter you want, peel off the operations in reverse order — undo the "+", then undo the "×", each time doing the same thing to both sides so the scales stay balanced.
Every operation is undone by its opposite:
To undo…
Do the opposite
+ 4
subtract 4
− 4
add 4
× 3
divide by 3
÷ 3
multiply by 3
squaring (²)
square-root (√)
square root (√)
square (²)
Make x the subject of y = 3x − 4: add 4 to both sides → y + 4 = 3x, then divide both sides by 3 → x = (y + 4) / 3.
Keep it balanced: do the same to both sides every step.
Make r the subject of C = 2πr: r is being multiplied by 2π, so divide both sides by 2π → r = C / (2π).
Make r the subject of A = πr²: divide both sides by π → A/π = r², then square-root both sides → r = √(A/π). Undo one layer at a time.
Peel off layers: undo ×π first, then undo the square.
Common mistake: whatever you do to one side, do to the whole other side — not just part of it. And undo operations in the right order: to make x the subject of y = 3x − 4, deal with the "− 4" before the "× 3". To undo a square, square-root both sides; to undo a square root, square both sides.
In a nutshell: substitute by swapping letters for numbers (minding BIDMAS); change the subject by undoing operations on both sides — outermost first — until your letter is by itself.