Algebra

Algebraic notation, vocabulary and substitution

30 past-paper questions, easiest first. Try each one, then reveal the answer.

Foundation & HigherAll 30 from Foundation papers

Revision guide by The Full Marks

Algebraic notation, vocabulary and substitution: what you need to know

Algebra uses letters to stand for numbers, with its own shorthand for writing things down. These are some of the easiest marks on the paper: simplifying terms, substituting values into formulae and knowing the right word for each kind of statement.

Key facts

$ab$ means $a \times b$ , and $3y$ means $y + y + y$ or $3 \times y$ .
$x$ means $1x$ . Like terms have exactly the same letters and powers; collect them by adding their numbers, keeping the sign in front of each term.
$a^2 = a \times a$ , $a^3 = a \times a \times a$ and $a^2b = a \times a \times b$ .
$\frac{a}{b}$ means $a \div b$ .
An expression has no equals sign ( $3x + 2$ ). An equation can be solved ( $3x + 2 = 11$ ). A formula links different quantities ( $A = lw$ ). An identity is true for every value, written with $\equiv$ . An inequality uses $<$ , $>$ , $\leqslant$ or $\geqslant$ ( $x > 3$ ).
Terms are the parts added or subtracted; factors are the things multiplied together.
Write coefficients as fractions, not decimals: $\frac{1}{2}x$ or $\frac{x}{2}$ , not $0.5x$ .

Simplifying products

Multiply the numbers together.
Write the letters next to each other in alphabetical order.
Use powers for repeated letters: $a \times a = a^2$ .

Substituting into a formula

Write the formula out, replacing each letter with its value in brackets.
Follow BIDMAS: brackets, then powers, then multiply and divide, then add and subtract.
Take extra care with negative numbers, and give units if the question has them.

Worked examples

Example 1 Foundation · grade 2

Simplify (a) $y + y + y + y$ (b) $3a \times 5b$ (c) $m \times m \times m$ (d) $5e + 3f - e - 4f$

Show the solution

(a) Four lots of $y$ is $4y$ (not $y^4$ ).
(b) Multiply the numbers: $3 \times 5 = 15$ . Put the letters together: $ab$ . Answer $15ab$ .
(c) Three $m$ s multiplied is $m^3$ .
(d) $e$ terms: $5e - e = 4e$ . $f$ terms: $3f - 4f = -f$ . Answer $4e - f$ .

Example 2 Foundation · grade 4

The formula $s = ut + \frac{1}{2}at^2$ gives the displacement $s$ metres of an object from its starting point. Work out $s$ when $u = 3$ , $a = -4$ and $t = 5$ .

Show the solution

Substitute with brackets: $s = (3)(5) + \frac{1}{2}(-4)(5)^2$
Power first: $5^2 = 25$
$\frac{1}{2} \times -4 \times 25 = -50$
$s = 15 + (-50) = -35$
$s = -35$ metres

Common mistakes

Writing $y + y + y + y$ as $y^4$ . Adding gives $4y$ ; only multiplying gives a power.
Adding numbers instead of multiplying: $3a \times 5b$ is $15ab$ , not $8ab$ .
Combining unlike terms: $3x + 4y$ cannot be simplified, and is not $7xy$ .
Reading $2g$ as a two-digit number. If $g = 9$ , $2g = 18$ , not $29$ .
Squaring a negative wrongly: $(-4)^2 = 16$ , but $-4^2$ on a calculator gives $-16$ . Always use brackets.

Exam tips

When asked to write a formula, include the subject and the equals sign, e.g. $T = 4n + 3$ . An expression on its own will not get full marks.
Show the substituted numbers before the answer so you can earn a method mark even if the arithmetic slips.

Questions people ask

What is the difference between an expression, an equation and a formula?

An expression is a collection of terms with no equals sign. An equation has an equals sign and can be solved to find a value. A formula shows how one quantity is worked out from others, such as $P = 2l + 2w$ .

What does substitute mean in maths?

It means replacing the letters with given numbers and working out the value. Put each number in brackets so signs and powers are handled correctly.

Exam questions: Algebraic notation, vocabulary and substitution (30, easiest first)

Algebraic notation, vocabulary and substitution1 markGrade 1CalculatorFoundation

June 2024 · Paper 2 Foundation · Q5

Fill in the missing number so that the number times m equals 2m.

Show answer

Answer	$2$	1 mark

Worked solution Checked against the mark scheme

$2m$ means $2 \times m$ .

So the missing number is $2$ .

Watch out: Writing $m$ or 1 in the gap.

Algebraic notation, vocabulary and substitution1 markGrade 1No calculatorFoundation

November 2023 · Paper 1 Foundation · Q3

Simplify a product of two numbers and a letter.

Show answer

Answer	$12a$	1 mark

Worked solution Checked against the mark scheme

Multiply the numbers: $3 \times 4 = 12$ .

Keep the letter: $12a$ .

Watch out: Writing $7a$ by adding the numbers.

Algebraic notation, vocabulary and substitution1 markGrade 1CalculatorFoundation

June 2023 · Paper 2 Foundation · Q4

Simplify a product of a number and an algebraic term.

Show answer

Answer	$12t$	1 mark

Worked solution Checked against the mark scheme

Multiply the numbers: $3 \times 4 = 12$ , and keep the letter.

$3 \times 4t = 12t$

Watch out: Writing $7t$ by adding the numbers.

Algebraic notation, vocabulary and substitution1 markGrade 1CalculatorFoundation

June 2023 · Paper 3 Foundation · Q3

Collect a repeated letter into a single algebraic term.

Show answer

Answer	$4m$	1 mark

Worked solution Checked against the mark scheme

Four lots of $m$ added together is $4 \times m$ , written $4m$ .

Watch out: Writing $m^4$ .

Algebraic notation, vocabulary and substitution1 markGrade 1No calculatorFoundation

November 2022 · Paper 1 Foundation · Q5

Simplify a product of two numbers and two letters.

Show answer

Answer	$15tw$	1 mark

Worked solution Checked against the mark scheme

Multiply the numbers: $3 \times 5 = 15$ .

Write the letters next to it: $15tw$ .

Watch out: Adding the numbers to get $8tw$ .

Algebraic notation, vocabulary and substitution1 markGrade 1No calculatorFoundation

June 2022 · Paper 1 Foundation · Q2

Collect four identical letter terms into a single term.

Show answer

Answer	$4e$	1 mark

Worked solution Checked against the mark scheme

Adding the same term four times gives four lots of it: $e+e+e+e = 4e$

Watch out: Writing $e^4$ , which means multiplying, not adding.

Algebraic notation, vocabulary and substitution2 marksGrade 1CalculatorFoundation

November 2022 · Paper 2 Foundation · Q8

Use a word formula to find the cost of hiring a van for a week.

Show answer

Answer	£315	2 marks

Worked solution Checked against the mark scheme

Cost $= 45 \times 7 = £315$

Watch out: Adding 45 and 7.

Algebraic notation, vocabulary and substitution1 markGrade 1CalculatorFoundation

June 2022 · Paper 3 Foundation · Q4

Simplify a product of a term in m and a number.

Show answer

Answer	$6m$	1 mark

Worked solution Checked against the mark scheme

Multiply the numbers: $2 \times 3 = 6$ , and keep the letter.

$2m \times 3 = 6m$

Watch out: Writing $5m$ by adding instead of multiplying.

Algebraic notation, vocabulary and substitution2 marksGrade 1CalculatorFoundation

November 2021 · Paper 2 Foundation · Q4

Simplify a repeated sum of a letter and divide an algebraic term by a number.

Show answer

(a)	$4m$	1 mark
(b)	$3p$	1 mark

Worked solution Checked against the mark scheme

(a) Four lots of $m$ make $4m$ .

(b) $12p \div 4 = 3p$ because $12 \div 4 = 3$ .

Watch out: Writing $m^4$ in part (a).

Algebraic notation, vocabulary and substitution1 markGrade 1CalculatorFoundation

November 2020 · Paper 2 Foundation · Q3

Collect like terms in a simple expression in e.

Show answer

Answer	$6e$	1 mark

Worked solution Checked against the mark scheme

Combine the numbers in front of $e$ : $3 - 1 + 4 = 6$ .

So the expression simplifies to $6e$ .

Watch out: Treating $e$ as $0e$ and getting $7e$ .

Algebraic notation, vocabulary and substitution2 marksGrade 1CalculatorFoundation

November 2019 · Paper 3 Foundation · Q7

Substitute a value into a linear formula.

Show answer

Answer	35	2 marks

Worked solution Checked against the mark scheme

Replace $u$ with 8:

$w = 4 \times 8 + 3 = 32 + 3 = 35$

Watch out: Treating 4u as 48 and getting 51.

Algebraic notation, vocabulary and substitution2 marksGrade 1No calculatorFoundation

November 2018 · Paper 1 Foundation · Q9

Substitute two given values into an algebraic expression.

Show answer

Answer	30	2 marks

Worked solution Checked against the mark scheme

$2g = 2 \times 9 = 18$

$3h = 3 \times 4 = 12$

$18 + 12 = 30$

Watch out: Writing 29 + 34 by treating 2g as 29.

Algebraic notation, vocabulary and substitution3 marksGrade 2CalculatorFoundation

November 2024 · Paper 2 Foundation · Q11

Simplify a product of two terms and collect like terms in an expression.

Show answer

(a)	$6xy$	1 mark
(b)	$5d - 3e$	2 marks

Worked solution Checked against the mark scheme

(a) $2 \times 3 = 6$ and $x \times y = xy$ , giving $6xy$ .

(b) $3d + 2d = 5d$ and $-4e + e = -3e$ , so $5d - 3e$ .

Watch out: In (b), writing $-5e$ by treating $e$ as $-e$ .

Algebraic notation, vocabulary and substitution3 marksGrade 2CalculatorFoundation

June 2024 · Paper 3 Foundation · Q16

Simplify a repeated product using index notation and form an expression for total points.

Show answer

(a)	$m^4$	1 mark
(b)	$5x + 2y$	2 marks

Worked solution Checked against the mark scheme

(a) $m$ multiplied by itself 4 times is $m^4$ .

(b) Wins score $5 \times x = 5x$ , draws score $2 \times y = 2y$ , losses score 0.

Total: $5x + 2y$

Watch out: Writing $4m$ in (a), or $7xy$ in (b).

Algebraic notation, vocabulary and substitution2 marksGrade 2No calculatorFoundation

June 2022 · Paper 1 Foundation · Q14

Substitute a value into a linear formula to find y.

Show answer

Answer	19	2 marks

Worked solution Checked against the mark scheme

$y = 6 \times 4 - 5$

$y = 24 - 5 = 19$

Watch out: Reading 6x as 64 when x = 4.

Algebraic notation, vocabulary and substitution3 marksGrade 2CalculatorFoundation

November 2021 · Paper 3 Foundation · Q6

Simplify a product of letters and a number, then collect like terms in an expression.

Show answer

(a)	$4ab$	1 mark
(b)	$3x + 8$	2 marks

Worked solution Checked against the mark scheme

(a) Write the number first and drop the multiplication signs: $4ab$ .

(b) Collect the $x$ terms: $4x - x = 3x$ .

Collect the numbers: $3 + 5 = 8$ .

So the answer is $3x + 8$ .

Watch out: In (b), treating $-x$ as $-1$ or getting $5x$ by adding instead of subtracting.

Algebraic notation, vocabulary and substitution1 markGrade 2CalculatorFoundation

June 2019 · Paper 2 Foundation · Q7

Write an expression for the number of people in boats with a fixed number per boat.

Show answer

Answer	$7y$	1 mark

Worked solution Checked against the mark scheme

Each boat holds 7 people, so $y$ boats hold $7 \times y$ people.

In algebra we write this as $7y$ .

Watch out: Writing $y + 7$ .

Algebraic notation, vocabulary and substitution2 marksGrade 2CalculatorFoundation

June 2019 · Paper 2 Foundation · Q11

Substitute a positive and a negative value into a formula.

Show answer

Answer	23	2 marks

Worked solution Checked against the mark scheme

$P = 7 \times 5 + 3 \times (-4)$

$= 35 + (-12)$

$= 23$

Watch out: Treating $3 \times (-4)$ as $+12$ , giving 47.

Algebraic notation, vocabulary and substitution2 marksGrade 2CalculatorFoundation

June 2019 · Paper 3 Foundation · Q9

Simplify an expression by collecting like terms.

Show answer

Answer	$11e + 5f$	2 marks

Worked solution Checked against the mark scheme

$4e + 6f + 7e - f$

$e$ terms: $4e + 7e = 11e$

$f$ terms: $6f - f = 5f$

Answer: $11e + 5f$

Watch out: Writing 7f by treating −f as +f.

Algebraic notation, vocabulary and substitution3 marksGrade 2CalculatorFoundation

Specimen · Paper 2 Foundation · Q4

Simplify three short algebraic expressions by collecting like terms and multiplying.

Show answer

(a)	$6f$	1 mark
(b)	$16mn$	1 mark
(c)	$2t^2$	1 mark

Worked solution Checked against the mark scheme

(a) $5f - f + 2f = 4f + 2f = 6f$

(b) Multiply the numbers: $2 \times 8 = 16$ , then write the letters: $16mn$

Watch out: In (c), writing $t^4$ or $2t^4$ .

Algebraic notation, vocabulary and substitution3 marksGrade 3CalculatorFoundation

June 2024 · Paper 2 Foundation · Q13

Simplify a product of two algebraic terms, and substitute a negative number into a formula.

Show answer

(a)	$6cd$	1 mark
(b)	$T = -5$	2 marks

Worked solution Checked against the mark scheme

(a) $2c \times 3d = 2 \times 3 \times c \times d = 6cd$

(b) $T = 3 + 2 \times (-4) = 3 - 8 = -5$

Watch out: In (b), working out $3 + 2 = 5$ first and then multiplying by $-4$ .

Algebraic notation, vocabulary and substitution2 marksGrade 3CalculatorFoundation

November 2024 · Paper 3 Foundation · Q12

Write an algebraic expression for the total number of fruit sold from packs and bags.

Show answer

Answer	$6x + 7y$	2 marks

Worked solution Checked against the mark scheme

Each pack has 6 apples, so $x$ packs give $6x$ apples.

Each bag has 7 oranges, so $y$ bags give $7y$ oranges.

Total: $6x + 7y$

Watch out: Writing $13xy$ by combining unlike terms.

Algebraic notation, vocabulary and substitution6 marksGrade 3No calculatorFoundation

November 2023 · Paper 1 Foundation · Q12

Substitute values into linear formulae, including finding an unknown and using a negative number.

Show answer

(a(i))	26	2 marks
(a(ii))	13	2 marks
(b)	−11	2 marks

Worked solution Checked against the mark scheme

(a)(i) $P = 2 \times 3 + 4 \times 5 = 6 + 20 = 26$

(ii) $38 = 2g + 12$ , so $2g = 26$ and $g = 13$

(b) $V = 3 \times (-3) - 2 = -9 - 2 = -11$

Watch out: In (b), getting −7 by working out −9 + 2.

Algebraic notation, vocabulary and substitution3 marksGrade 3No calculatorFoundation

June 2023 · Paper 1 Foundation · Q18

Write a formula for the total number of apples from bags and crates.

Show answer

Answer	$T = 5b + 28c$	3 marks

Worked solution Checked against the mark scheme

Each bag has 5 apples, so $b$ bags hold $5b$ .

Each crate has 28 apples, so $c$ crates hold $28c$ .

$T = 5b + 28c$

Watch out: Leaving out 'T =' or writing b + c = T.

Algebraic notation, vocabulary and substitution4 marksGrade 3CalculatorFoundation

November 2020 · Paper 2 Foundation · Q15

Substitute into a two-variable formula, including a negative value, then find an unknown by working backwards.

Show answer

(a)	$-13$	2 marks
(b)	$y = 5$	2 marks

Worked solution Checked against the mark scheme

(a) $T = 3(5) + 4(-7) = 15 - 28 = -13$

(b) $38 = 3(6) + 4y = 18 + 4y$

$4y = 20$ , so $y = 5$

Watch out: In (a), getting 43 by ignoring the negative sign.

Algebraic notation, vocabulary and substitution3 marksGrade 3No calculatorFoundation

November 2019 · Paper 1 Foundation · Q17

Solve a simple linear equation, then substitute the result into an expression with a square.

Show answer

Answer	$18$	3 marks

Worked solution Checked against the mark scheme

$x - 1 = 2$ , so $x = 3$

$2x^2 = 2 \times 3^2 = 2 \times 9 = 18$

Watch out: Working out $(2x)^2 = 36$ instead of $2x^2$ .

Algebraic notation, vocabulary and substitution1 markGrade 3No calculatorFoundation

June 2019 · Paper 1 Foundation · Q13

Write an algebraic expression for a length in millimetres given it in centimetres.

Show answer

Answer	$10x$	1 mark

Worked solution Checked against the mark scheme

There are 10 mm in 1 cm, so $x$ cm is $10x$ mm.

Watch out: Writing $x + 10$ or $x \div 10$ .

Algebraic notation, vocabulary and substitution4 marksGrade 3CalculatorFoundation

November 2019 · Paper 2 Foundation · Q18

Write a formula for the total of three ages in terms of n and pick out the identity from a list.

Show answer

(a)	$T = 4n - 5$	3 marks
(b)	$5m - 3m = 2m$	1 mark

Worked solution Checked against the mark scheme

(a) Ben $= n$ , Chloe $= 2n$ , Dan $= n - 5$

$T = n + 2n + n - 5 = 4n - 5$

(b) An identity is true for every value of the letter. $5m - 3m$ always equals $2m$ , so tick $5m - 3m = 2m$ .

Watch out: In (a), writing $4n - 5$ without $T =$ , so it is not a formula.

Algebraic notation, vocabulary and substitution3 marksGrade 3CalculatorFoundation

Specimen · Paper 3 Foundation · Q10

Complete a table for an expression in n: substitute a value and work backwards from an output.

Show answer

Answer	38 (when $n = 12$ ) and 15 (when $3n + 2 = 47$ )	3 marks

Worked solution Checked against the mark scheme

Top row: $3 \times 12 + 2 = 36 + 2 = 38$ .

Bottom row: $3n + 2 = 47$ , so $3n = 45$ and $n = 15$ .

Watch out: Working out 47 ÷ 3 − 2 instead of undoing the steps in reverse order.

Algebraic notation, vocabulary and substitution6 marksGrade 4No calculatorFoundation

June 2018 · Paper 1 Foundation · Q16

Substitute values into a formula, expand a single bracket and solve a linear equation with brackets.

Show answer

(a)	$14$	2 marks
(b)	$4e^2 + 8e$	2 marks
(c)	$m = 11$	2 marks

Worked solution Checked against the mark scheme

(a) $P = 4 \times 5 + 3 \times (-2) = 20 - 6 = 14$

(b) $4e \times e = 4e^2$ and $4e \times 2 = 8e$ , so $4e^2 + 8e$

Watch out: In (a), treating $3 \times -2$ as $+6$ to get 26.

Also tested in these questions

Their main topic is different, but part of each one tests this topic.

June 2024 · Paper 2 Foundation · Q22: Use a formula for time taken by a number of machines to find the difference in hours between two days.
June 2024 · Paper 3 Foundation · Q4: Simplify an expression by collecting like terms.
June 2024 · Paper 3 Foundation · Q29: Substitute a negative value into a formula and rearrange a formula to change its subject.
November 2023 · Paper 3 Foundation · Q13: Simplify two algebraic expressions by collecting like terms.
June 2023 · Paper 1 Foundation · Q13: Simplify a simple algebraic fraction, collect like terms, and factorise a linear expression.
November 2022 · Paper 2 Foundation · Q14: Simplify an expression by collecting like terms, solve an equation with a bracket, and write an expression for a total.
November 2021 · Paper 3 Foundation · Q11: Convert minutes into hours and minutes, then write an expression for average speed.
November 2021 · Paper 3 Foundation · Q17: Expand a bracket, factorise an expression, solve a linear equation with a bracket, and simplify a product using index laws.
November 2019 · Paper 2 Foundation · Q20: Substitute into a formula for a building's height and show the error is under 5% of the real height.
November 2019 · Paper 2 Foundation · Q28: Use an inverse proportion formula to compare the numbers of workers two companies need.
November 2018 · Paper 2 Foundation · Q7: Collect like terms and simplify a product of numbers and letters.
June 2018 · Paper 1 Foundation · Q6: Simplify a product of a number and an algebraic term, then collect like terms.