Why is "Reading $3n$ as the digits joined, e.g. $3n$ with $n = 4$ giving 34." a common mistake in Substitution and Formulae?

Why it happens: The hidden multiplication sign is forgotten, so the number and letter look stuck together. How to avoid it: 3n means 3 × n. With n = 4 it is 3 × 4 = 12.

Why is "Working left to right after substituting, e.g. $2 \times 5 + 3$ giving $2 \times 8 = 16$." a common mistake in Substitution and Formulae?

Why it happens: Once numbers replace the letters, the order of operations is overlooked. How to avoid it: Follow BIDMAS. Multiply before adding: 2 × 5 + 3 = 10 + 3 = 13.

Why is "Treating a power as a multiplication, e.g. $n^2$ with $n = 4$ giving $4 \times 2 = 8$." a common mistake in Substitution and Formulae?

Why it happens: The small raised 2 is read as 'times 2' rather than a count. How to avoid it: n^2 means n × n. With n = 4 it is 4 × 4 = 16.

Why is "Putting a number in place of the wrong letter when using a formula." a common mistake in Substitution and Formulae?

Why it happens: The numbers are substituted in a rush without matching them to letters. How to avoid it: Check which quantity each number describes, then place it where its letter is.

Why is "Giving a final answer with no units, such as writing 18 instead of \$18." a common mistake in Substitution and Formulae?

Why it happens: The focus is on the number, so the unit gets dropped. How to avoid it: Always include the unit (\$, cm, kg) so the answer means something real.

Substituting n = 3 into n + 5

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Substitution and formulae — Cambridge Lower Secondary Maths, Grade 6

A formula is a recipe written in letters — feed in your numbers and out comes an answer. This guide walks you through substituting values into expressions and formulae, writing your own simple formulae from words, and using them to solve real problems — one short step at a time.

At a glance

Substituting means replacing a letter with a number, then working the answer out.
Put each number in carefully, then follow BIDMAS to calculate.
$3n$ means $3 \times n$ , so if $n = 4$ , then $3n = 12$ — never $34$ .
A formula is a rule connecting quantities, written with letters and an equals sign.
Real formulae include $A = l \times w$ for the area of a rectangle.
To write a formula from words, name each quantity with a letter, then describe the link.
Using a formula is just substitution: put the known numbers in and calculate.
Brackets and powers need extra care when you substitute — work them out in order.

What you’ll learn

Mapped to the Cambridge Lower Secondary Mathematics curriculum framework.

Stage 7 — Substitute positive numbers into expressions and evaluate the result.
Stage 7 — Construct simple formulae expressed in words and in symbols.
Stage 7 — Substitute values into a formula to find the value of an unknown quantity.
Stage 7 — Apply the correct order of operations when substituting.

What is substitution?

Substitution means swapping a letter for a number, then calculating.

Substitution is one of the most useful moves in algebra. It simply means replacing a letter with a number, then working out the answer.

Think of an expression like $n + 5$ as a vending machine slot. The letter $n$ is a gap waiting for a number. If you decide $n = 3$ , you drop a 3 into the gap and the expression becomes $3 + 5 =$ .

Substituting with order of operations

Once the numbers are in, BIDMAS tells you which calculation to do first.

When an expression has several operations, the order of operations (BIDMAS) decides what you do first: Brackets, Indices, Division and Multiplication, then Addition and Subtraction.

Take the expression $2a + b$ with $a = 5$ and $b = 3$ . Substitute first: Now BIDMAS: multiply before you add, so .

What is a formula?

A formula is a rule linking quantities, written with letters.

A formula is a rule that connects two or more quantities, written using letters and an equals sign. It is like a recipe — a reliable instruction you can use again and again.

You already know some formulae without thinking of them that way:

The area of a rectangle: $A = l \times w$ (length times width).
The perimeter of a rectangle: $P = 2(l + w)$ .

Writing a formula from words

Name each quantity with a letter, then write the link between them.

Often a real-life situation is described in words, and your job is to turn it into a formula. Here is a reliable method.

Example: "A taxi charges a $4 fixed fee, plus $2 for every kilometre travelled."

Name the quantities with sensible letters. Let $C$ be the total cost and $k$ be the number of kilometres.
Describe the link in words: total cost = fixed fee + (cost per km × kilometres).
Write it in symbols: $C = 4 + 2k$ .

Using a formula

Using a formula is just substitution — put the known numbers in and calculate.

Once you have a formula, using it is exactly the substitution skill from earlier. You put the known numbers in place of the letters and calculate the subject.

Example: A taxi uses the formula $C = 4 + 2k$ . How much does a 7 km trip cost?

Write the formula: $C = 4 + 2k$ .

Where you'll use this next

Substitution and formulae power a huge amount of the maths ahead.

Getting confident with substitution and formulae now helps across many topics:

Equations are solved by checking — and checking is substitution.
Graphs are drawn by substituting $x$ -values into a rule to find $y$ -values.
Geometry and measure rely on formulae for area, perimeter, volume and angles.
Sequences use the nth-term rule, which you apply by substituting the position $n$ .

In everyday life, formulae are everywhere — phone bills, recipes scaled up or down, currency conversion, fuel costs and sports statistics all run on the skills in this guide.

If formula work feels hard later, it is usually a substitution slip or a BIDMAS step underneath. Come back to this guide whenever you need a refresher — there is no prize for rushing.

Checking equations is substitution in action.
Graphs are built by substituting into a rule.

Quick recap

Substitution means replacing a letter with a number, then calculating.
Remember $3n$ means $3 \times n$ — put the times sign back in.
After substituting, follow BIDMAS to work out the answer.
A formula is a rule linking quantities, written with letters and an equals sign.
To write a formula, name each quantity and describe the link in words first.
Using a formula is substitution — put known numbers in and calculate the subject.
Always include units in your final answer so it has real meaning.

Sources: Cambridge Lower Secondary Mathematics curriculum framework (Stage 7). Last reviewed 2026-05-19.

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3 + 5 = 8

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Substitution and Formulae — Cambridge Lower Secondary Mathematics — Stage 7 Revision Notes & Practice | Tutopiya

Substitution and Formulae

Detailed Study Notes

Take this whole topic with you

Substitution and formulae — Cambridge Lower Secondary Maths, Grade 6

At a glance

What you’ll learn

What is substitution?

Substituting with order of operations

What is a formula?

Writing a formula from words

Using a formula

Where you'll use this next

Quick recap

Step-by-step worked examples — Substitution and Formulae

1Substituting into a simple expression

2Substituting with a hidden times sign

3Substituting with order of operations

4Writing a formula from words

5Using a formula with a power

Key Definitions and Keywords — Substitution and Formulae

Substitute

Evaluate

Formula

Subject of a formula

Expression

Variable

Quantity

Order of operations

Fixed amount

Units

Power

Coefficient

Common Mistakes and Misconceptions — Substitution and Formulae

✕Reading $3n$ as the digits joined, e.g. $3n$ with $n = 4$ giving 34.

✕Working left to right after substituting, e.g. $2 \times 5 + 3$ giving $2 \times 8 = 16$ .

✕Treating a power as a multiplication, e.g. $n^2$ with $n = 4$ giving $4 \times 2 = 8$ .

✕Putting a number in place of the wrong letter when using a formula.

✕Giving a final answer with no units, such as writing 18 instead of $18.

Substitution and Formulae

Detailed Study Notes

Take this whole topic with you

At a glance

What you’ll learn

Quick recap

1Substituting into a simple expression

2Substituting with a hidden times sign

3Substituting with order of operations

4Writing a formula from words

5Using a formula with a power

Substitute

Evaluate

Formula

Subject of a formula

Expression

Variable

Quantity

Order of operations

Fixed amount

Units

Power

Coefficient

✕Reading 3n3n3n as the digits joined, e.g. 3n3n3n with n=4n = 4n=4 giving 34.

✕Working left to right after substituting, e.g. 2×5+32 \times 5 + 32×5+3 giving 2×8=162 \times 8 = 162×8=16.

✕Treating a power as a multiplication, e.g. n2n^2n2 with n=4n = 4n=4 giving 4×2=84 \times 2 = 84×2.

✕Putting a number in place of the wrong letter when using a formula.

✕Giving a final answer with no units, such as writing 18 instead of $18.

✕Reading $3n$ as the digits joined, e.g. $3n$ with $n = 4$ giving 34.

✕Working left to right after substituting, e.g. $2 \times 5 + 3$ giving $2 \times 8 = 16$ .

✕Treating a power as a multiplication, e.g. $n^2$ with $n = 4$ giving $4 \times 2 = 8$ .