What you'll learn
Transposition of formulae — also called "changing the subject" of a formula — is a key algebra skill in the CSEC Mathematics syllabus. The subject of a formula is the single variable that stands alone on one side, such as A in A = πr². Often you are given a formula written for one variable but need it written for another; transposing rearranges it without changing its meaning. In this guide you will learn the inverse-operation method, how to handle formulae with squares, roots, fractions and brackets, and how to deal with a variable that appears more than once. These skills support physics-style formulae, area and volume work, and almost every topic that uses algebra. Transposition appears in Paper 1 and is frequently embedded in Paper 2 problems.
Key terms and definitions
Subject of a formula — the variable expressed in terms of the others; it sits alone, usually on the left.
Transpose / change the subject — rearrange a formula so a different variable becomes the subject.
Inverse operation — the operation that undoes another: + and −, × and ÷, squaring and square-rooting are inverse pairs.
Term — a part of an expression separated by + or − signs.
Factor — a quantity that multiplies the rest; factorising can collect a repeated variable.
Core concepts
The balance principle
A formula is an equation, so whatever you do to one side you must do to the other to keep it balanced. The goal is to isolate the new subject by undoing, in reverse order, everything that has been done to it — using inverse operations.
Simple rearrangement (one step at a time)
Work from the outside in, just as you would unwrap a parcel. To make r the subject of C = 2πr, the r has been multiplied by 2π, so divide both sides by 2π: r = C ÷ (2π). For v = u + at, to make a the subject, first subtract u (v − u = at), then divide by t: a = (v − u) ÷ t.
Formulae with powers and roots
When the new subject is squared, isolate the squared term, then take the square root of both sides. From A = πr², divide by π to get r² = A ÷ π, then r = √(A ÷ π). When the subject is under a root, isolate the root, then square both sides. From T = √(L ÷ g), square to get T² = L ÷ g, then L = gT².
Formulae with fractions
Multiply through by the denominator to clear fractions early. To make x the subject of y = (x + 3) ÷ 5, multiply both sides by 5: 5y = x + 3, then subtract 3: x = 5y − 3.
The subject appears more than once
When the required variable appears in two places, gather all its terms on one side, factorise it out, then divide. To make x the subject of ax = bx + c, move the bx term: ax − bx = c, factorise: x(a − b) = c, then divide: x = c ÷ (a − b).
Brackets
Either expand the brackets first or divide by the bracketed factor as a whole — whichever keeps the work simplest. From A = ½h(a + b), to make a the subject: multiply by 2 (2A = h(a + b)), divide by h (2A ÷ h = a + b), then subtract b (a = 2A ÷ h − b).
Worked examples
Example 1: Two-step rearrangement (Paper 1 style)
Make t the subject of the formula s = ½t + 4.
Subtract 4 from both sides: s − 4 = ½t. Multiply both sides by 2: t = 2(s − 4) = 2s − 8.
Example 2: A formula with a square root (Paper 2 style)
Make L the subject of T = 2π√(L ÷ g).
Divide both sides by 2π: T ÷ (2π) = √(L ÷ g). Square both sides: [T ÷ (2π)]² = L ÷ g. Multiply by g: L = g[T ÷ (2π)]² = gT² ÷ (4π²).
Example 3: Subject appearing twice (Paper 2 style)
Make x the subject of y = (x + 2) ÷ (x − 1).
Multiply both sides by (x − 1): y(x − 1) = x + 2, so yx − y = x + 2. Gather x-terms: yx − x = y + 2, factorise: x(y − 1) = y + 2, then divide: x = (y + 2) ÷ (y − 1).
Common mistakes and how to avoid them
Doing an operation to one term only. Whatever you do must apply to the whole side. When you divide A = πr² by π, both sides are divided, giving r² = A ÷ π.
Square-rooting too early. Isolate the squared term completely before taking the root. From v² = u² + 2as, find v² first, then v = √(u² + 2as).
Forgetting the ± when square-rooting. In a pure equation a square root gives two values; in many formula contexts the positive root is taken, but be aware of both.
Not factorising a repeated variable. When x appears twice, you must collect and factorise it — you cannot isolate it term by term.
Sign slips when moving terms. A term that is added crosses over as a subtraction, and vice versa. Move one term at a time and re-check.
Exam technique for Transposition of Formulae
Identify the target subject and unwrap in reverse. Undo the operations affecting it in the opposite order to BIDMAS.
Clear fractions and roots early. Multiplying out denominators and squaring roots first usually simplifies the rest.
Show each step. CSEC gives method marks; a single line of fully-worked rearrangement protects you against arithmetic slips.
Watch for a repeated variable — the signal to gather terms and factorise.
Check by substituting numbers. Put simple values into both the original and the rearranged formula; they should agree.
Quick revision summary
Transposing a formula means making a different variable the subject while keeping the equation balanced — whatever you do to one side, you do to the other. Isolate the new subject by applying inverse operations in reverse order: undo addition with subtraction, multiplication with division, squaring with a square root, and a root with squaring. Clear fractions by multiplying through by the denominator, and isolate a squared term or a root completely before undoing it. When the required variable appears more than once, gather its terms on one side, factorise it out, then divide. Handle brackets by expanding or by dividing by the whole bracket, whichever is simpler. Show each step to earn method marks, watch signs when moving terms, remember the ± where relevant, and verify your rearrangement by substituting simple numbers into both versions of the formula.