Substitution into Formulae

Substitution into formulae is one of the most reliable mark-earners in IGCSE Maths: replace each letter with its value, follow the order of operations, and calculate carefully. On this page you will learn a four-step method that works for every formula, see how to handle squares, fractions and negative numbers, and practise the harder skill of substituting known values and then solving for the letter you need. Work through the examples, then test yourself in the auto-marked practice rooms below.

Prior Knowledge This page requires confidence with Order of Operations (BIDMAS) and Solving Equations. Some answers are rounded, so Significant Figures will also help.
⇩ Jump to Practice Questions ⇩

What Is a Formula?

A formula describes a relationship between quantities using algebra. For example, \(v = u + at\) connects the final speed \(v\) of an object to its starting speed \(u\), its acceleration \(a\) and the time \(t\). Substitution means replacing each letter with its known value so the formula becomes a calculation you can complete.

The Four-Step Method

  1. Facts: write down every value you know, with its units, and identify the value you want.
  2. Formula: write the formula out in full before touching any numbers.
  3. Substitute: replace each letter with its value. Put negative numbers in brackets.
  4. Work: calculate one step per line, following BIDMAS, and only round the final answer.

The same four steps work when the value you want is not the subject of the formula. Substitute everything you know first, then solve the equation that is left (Examples 5 and 6 below).

Replace, Then Calculate
Swap every letter for its value before doing any arithmetic. Writing the substitution line in full prevents almost all slips.
BIDMAS Still Rules
Brackets and indices come first: in \(\pi r^2\), square the radius before multiplying by \(\pi\).
Not the Subject? Solve It
If the letter you want is buried inside the formula, substitute the known values and solve the equation that remains.

Formulae You Will Meet

These standard formulae appear throughout the course and in the practice rooms below.

Formula What it finds Letters
\(v = u + at\) Final speed \(u\) start speed, \(a\) acceleration, \(t\) time
\(A = \frac{1}{2}bh\) Area of a triangle \(b\) base, \(h\) perpendicular height
\(A = \frac{1}{2}(a + b)h\) Area of a trapezium \(a\), \(b\) parallel sides, \(h\) distance between them
\(a^2 = b^2 + c^2\) Pythagoras' theorem \(a\) hypotenuse, \(b\), \(c\) shorter sides
\(C = 2\pi r\) and \(A = \pi r^2\) Circumference and area of a circle \(r\) radius
\(s = ut + \frac{1}{2}at^2\) Distance travelled \(u\) start speed, \(a\) acceleration, \(t\) time

Worked Examples

💡 Example 1: Direct Substitution

If \(v = u + at\), find \(v\) when \(u = 4\), \(a = 3\) and \(t = 5\).

\[\begin{aligned} v &= u + at \\ v &= 4 + 3 \times 5 \\ v &= 4 + 15 \\ v &= 19 \end{aligned}\]
What's happening?

Write the formula, then replace each letter with its value. BIDMAS says multiply before adding, so \(3 \times 5\) comes first.

💡 Example 2: Squaring First

A circle has radius \(6\) cm. Find its area to 3 significant figures.

\[\begin{aligned} A &= \pi r^2 \\ A &= \pi \times 6^2 \\ A &= \pi \times 36 \\ A &= 113.097\ldots \\ A &= 113 \text{ cm}^2 \text{ (3 s.f.)} \end{aligned}\]
What's happening?

Indices before multiplication: square the \(6\) first, then multiply by \(\pi\). Keep the full calculator value and only round at the very end.

💡 Example 3: Negative Values

If \(y = 3x^2 - 2x\), find \(y\) when \(x = -4\).

\[\begin{aligned} y &= 3x^2 - 2x \\ y &= 3 \times (-4)^2 - 2 \times (-4) \\ y &= 3 \times 16 + 8 \\ y &= 48 + 8 \\ y &= 56 \end{aligned}\]
What's happening?

Always bracket negative numbers when substituting. \((-4)^2 = 16\), and subtracting \(-8\) becomes \(+8\).

💡 Example 4: A Formula with a Fraction

A trapezium has parallel sides \(5\) cm and \(9\) cm, which are \(6\) cm apart. Find its area.

\[\begin{aligned} A &= \tfrac{1}{2}(a + b)h \\ A &= \tfrac{1}{2}(5 + 9) \times 6 \\ A &= \tfrac{1}{2} \times 14 \times 6 \\ A &= 42 \text{ cm}^2 \end{aligned}\]
What's happening?

Brackets first: add the parallel sides before halving and multiplying by the height.

💡 Example 5: The Unknown Is Not the Subject

A triangle has area \(54\) cm\(^2\) and base \(12\) cm. Find its height.

\[\begin{aligned} A &= \tfrac{1}{2}bh \\ 54 &= \tfrac{1}{2} \times 12 \times h \\ 54 &= 6h \\ h &= 9 \text{ cm} \end{aligned}\]
What's happening?

Substitute everything you know, even though \(h\) is not the subject. The formula becomes an ordinary equation, so solve it.

💡 Example 6: Solving with a Square

A right-angled triangle has hypotenuse \(13\) cm and one shorter side of \(5\) cm. Find the other side.

\[\begin{aligned} a^2 &= b^2 + c^2 \\ 13^2 &= 5^2 + c^2 \\ 169 &= 25 + c^2 \\ c^2 &= 144 \\ c &= 12 \text{ cm} \end{aligned}\]
What's happening?

Substitute the known sides, then solve. Subtract \(25\) from both sides, and square root last. A length must be positive, so take the positive root.

🔑 Key Points

  • Write the formula in full before substituting; it is worth a method mark on its own.
  • Show the substitution line with every letter replaced, then work one step per line.
  • Bracket every negative value: \((-3)^2 = 9\).
  • Only round the final answer, and state the units.
  • If the letter you want is not the subject, substitute first, then solve.

⚠️ Pitfalls

  • Multiplying before squaring in \(\pi r^2\): the square applies to \(r\) only, not to \(\pi r\).
  • Typing \(-3^2\) into a calculator, which gives \(-9\), when the formula needs \((-3)^2 = 9\).
  • Forgetting the brackets in \(2(l + w)\) and working out \(2l + w\) instead.
  • Rounding part-way through a calculation, which loses accuracy marks.
  • Dropping units, or mixing units such as cm and m in the same formula.
⇩ Jump to Practice Questions ⇩

Confident with substituting values? The next step is to rearrange formulae so any letter can become the subject.

Next: Changing the Subject →

Substitution into Formulae: Practice Room

Practise substituting values into formulae across four rooms: Starter, Builder, Challenger and Master. Every reload generates fresh values, and answers are marked when you click out of the box. Enter numbers only (no units). In the Master room circle questions, give your answer to 3 significant figures.

✓ Correct 0
✗ Re-attempts 0
🔥 Streak 0
🏆 Best 0

Input tips: integers like 42, decimals like 12.6, and fractions like 9/2 are all accepted. Use a leading minus sign for negatives.