Substituting into and Interpreting Formulae Revision Notes | GCSE AQA Higher Maths

1. Why Formulae Are the Engine of Mathematics

Every time an engineer calculates the stress on a bridge, or a nurse works out a drug dosage, they are using a formula. A formula is a shortcut — a rule that has been proved to work, packaged so you can use it again and again without re-deriving it from scratch. A formula always has an equals sign and contains at least one variable.

At GCSE, you will meet formulae across every topic: area, volume, speed, density, pressure, sequences, and more. The skill is not memorising them — it is knowing how to use them confidently, whether the question asks you to substitute straight in or to rearrange first.

Think of a formula like a recipe. The ingredients are the variables, the method is the algebra, and the dish is your answer. Change one ingredient and the dish changes — but the recipe stays the same. That consistency is exactly what makes formulae so powerful.

This lesson covers the full process: reading a formula, identifying its variables, substituting values, and rearranging when the unknown is not already the subject. These skills appear on every GCSE Higher maths paper — both with and without a calculator.

2. Reading a Formula: Variables, Symbols, and Structure

Before you can use a formula, you need to read it correctly. Every symbol in a formula stands for a specific quantity. The moment you see a formula, your first job is to build a variable key — a list of what each letter means.

Take the formula for speed: $s = \frac{d}{t}$. Here $s$ is speed, $d$ is distance, and $t$ is time. The subject is $s$ — it is already isolated. If a question gives you distance and time and asks for speed, you can substitute straight away.

But what if the question gives you speed and time and asks for distance? Now $d$ is the unknown, and $d$ is not the subject. You need to rearrange first. This two-step process — rearrange, then substitute — is where most marks are lost, and where this lesson will save you the most time.

A useful memory aid for speed, distance, and time is the formula triangle. Cover the quantity you want to find, and the triangle shows you the operation to use. However, the triangle only works when you already know which formula to use — the algebra of rearranging is the skill that transfers to every other formula you meet.

Here is a quick check on reading formulae before we move to substitution. This is the kind of question that appears in the early marks of an Higher-tier paper.

3. Substituting Values Into a Formula

Substitution is the process of swapping letters for numbers. The key discipline is to write the formula first, then replace each variable with its value on the next line. This two-line habit earns method marks even if you make an arithmetic slip later.

Watch out for one critical trap: the order of operations still applies inside a formula. If the formula contains brackets, powers, or fractions, you must handle those before adding or subtracting. Rushing this step is the single most common source of errors in substitution questions.

Worked example: Substituting into the Area of a Trapezium

A trapezium has parallel sides of length 5 cm and 9 cm, and a perpendicular height of 4 cm. Calculate its area. [3 marks]

1. $A = \frac{1}{2}(a + b)h$ — Always start by writing the formula. This earns the first method mark [M1] and keeps your working structured.
2. $a = 5$ cm, $b = 9$ cm, $h = 4$ cm — Listing values separately before substituting reduces errors. Check the question carefully — parallel sides are $a$ and $b$, height is $h$.
3. $A = \frac{1}{2}(5 + 9) \times 4$ — Replace each letter with its corresponding number. Keep the brackets as they are part of the formula structure. [M1]
4. $A = \frac{1}{2} \times 14 \times 4$ — Calculate the sum inside the brackets: $5 + 9 = 14$. Now the expression becomes $\frac{1}{2} \times 14 \times 4$.
5. $A = \frac{1}{2} \times 14 \times 4 = 28$ — The final answer for the area is $28$ cm².
6. $A = 28 \text{ cm}^2$ — Area must be expressed in square units. Including units is essential as missing units can cost marks [A1].

Answer: $A = 28 \text{ cm}^2$

Notice how each step was written on its own line. This is not just neatness — it is exam strategy. If you make an arithmetic error in step 5 but your substitution in step 3 is correct, the examiner awards the method mark. That is a mark you keep even when you slip up.

Now try a substitution involving a squared term. Powers are where arithmetic errors are most common — especially In questions without a calculator where you cannot use a calculator. The rule is: always expand the power as a multiplication before computing it.

Worked example: Substituting into a Formula with a Squared Term

The kinetic energy of an object is given by $E = \frac{1}{2}mv^2$. Find the kinetic energy when $m = 3$ kg and $v = 6$ m/s. [3 marks]

1. $E = \frac{1}{2}mv^2$ — Write the formula first. [M1] Note: The formula $E = \frac{1}{2}mv^2$ is a Physics formula used here to practise substitution skills, not a formula students need to memorise for GCSE Higher maths.
2. $E = \frac{1}{2} \times 3 \times 6^2$ — Replace $m$ with 3 and $v$ with 6. Do not square the 6 yet — write it as $6^2$ first. [M1]
3. $6^2 = 6 \times 6 = 36$ — Always write the multiplication out longhand. A very common error is writing $6^2 = 12$ (multiplying by 2 instead of squaring). Verify: $6 \times 6 = 36$.
4. $E = \frac{1}{2} \times 3 \times 36$ — Now the expression contains only multiplication.
5. $E = \frac{1}{2} \times 108 = 54$ — $3 \times 36 = 108$, then $\frac{1}{2} \times 108 = 54$.
6. $E = 54 \text{ J}$ — Energy is measured in joules (J). [A1]

Answer: $E = 54 \text{ J}$

This is where students often slip up. Can you spot the trap in the next question?

4. Rearranging a Formula to Change the Subject

When the unknown in a question is not already the subject of the formula, you need to rearrange before substituting. The principle is identical to solving an equation: apply inverse operations to both sides until the target variable is isolated.

Think of rearranging like undoing a knot. Each operation that has been applied to your target variable needs to be undone in reverse order. If the variable has been multiplied and then added to something, you undo the addition first, then the multiplication.

Worked example: Rearranging the Density Formula to Find Volume

A block of aluminium has a density of 2.7 g/cm³ and a mass of 81 g. Calculate its volume. [3 marks]

1. $\rho = \frac{m}{V}$ — Start with the standard form. The subject is $\rho$, but we need $V$. [M1]
2. $V = \frac{m}{\rho}$ — Multiply both sides by $V$: $\rho V = m$. Then divide both sides by $\rho$: $V = \frac{m}{\rho}$. [M1]
3. $V = \frac{81}{2.7}$ — Replace $m$ with 81 and $\rho$ with 2.7. Always substitute after rearranging, not before.
4. $V = 81 \div 2.7 = 30$ — To divide by 2.7 without a calculator: $81 \div 2.7 = 810 \div 27 = 30$. Multiply numerator and denominator by 10 to remove the decimal.
5. $V = 30 \text{ cm}^3$ — Volume is in cubic centimetres because mass was in grams and density in g/cm³. Units must be consistent. [A1]

Answer: $V = 30 \text{ cm}^3$

Earlier we saw how substitution follows a formula directly. Now we are seeing the same structured approach applied after rearranging — the steps are the same, just with an extra algebraic move at the start. This pattern repeats across every formula you will meet at GCSE.

Rearranging becomes trickier when the target variable appears in a more complex position — for example, inside a bracket or under a square root. The next worked example shows how to handle a two-step rearrangement.

Worked example: Two-Step Rearrangement: Finding Initial Velocity

The formula $v = u + at$ connects final velocity $v$, initial velocity $u$, acceleration $a$, and time $t$. Find $u$ when $v = 25$ m/s, $a = 3$ m/s², and $t = 5$ s. [3 marks]

1. $v = u + at$ — The subject is $v$. We need to make $u$ the subject. [M1]
2. $v - at = u$ — $at$ is added to $u$, so subtract $at$ from both sides to isolate $u$.
3. $u = v - at$ — Rewriting with the subject on the left is conventional and clearer. [M1]
4. $u = 25 - (3 \times 5)$ — Replace $v$ with 25, $a$ with 3, and $t$ with 5. Use brackets around $at$ to keep the multiplication clear.
5. $u = 25 - 15 = 10$ — $3 \times 5 = 15$, then $25 - 15 = 10$.
6. $u = 10 \text{ m/s}$ — Velocity is in metres per second. [A1]

Answer: $u = 10 \text{ m/s}$

This is a classic two-step rearrangement. Notice that we rearranged the formula in its general form first, then substituted numbers. This order — rearrange, then substitute — is strongly recommended because it keeps the algebra clean and reduces arithmetic errors.

5. A Formula Reference Bank for GCSE

The exam provides a formulae sheet for some topics (such as the quadratic formula and certain volume formulae), but many standard formulae are expected to be memorised. The table below collects the formulae you are most likely to need across the GCSE course.

Use this as a rapid-recall revision tool. For each formula, practise covering the right-hand column and recalling the formula from the name alone. Then practise rearranging each one to make every variable the subject in turn.

Notice that several of these formulae share the same structure: one variable equals a product or quotient of two others. Once you can rearrange $s = \frac{d}{t}$ confidently, you can rearrange $\rho = \frac{m}{V}$ and $P = \frac{F}{A}$ using exactly the same moves. The algebra transfers.

The graph above shows three linear relationships of the form $y = kx$, which is the structure of many formulae (such as $d = st$ when speed is constant). A steeper line means a larger constant $k$ — for example, a higher speed means distance grows faster with time. Recognising this proportional structure helps you sense-check answers: if you double the time, you should double the distance.

6. Deliberate Practice: Apply the Methods

The only way to make substitution and rearrangement automatic is to practise them on fresh problems. Work through each question below before checking the solution. Write the formula first, then substitute — do not skip steps even when the numbers look straightforward.

Before moving to the exam strategy section, check your understanding of the rearrangement process with this ordering question. Arrange the steps in the correct sequence for making $r$ the subject of $A = \pi r^2$.