IGCSE · Practice questions · Algebraic notation and substitution

Swap the letter, keep the order.

Six original Cambridge-style questions on algebraic notation: writing expressions and formulas, and substituting into them accurately.

Original questions All questions on this page are original work, written in the Cambridge IGCSE style. They are not from past papers. They test the same concepts and skills the syllabus rewards.

The method

Brackets round every value, then the order of operations.

Notation: $3a = 3 \times a$, $ab = a \times b$, $3a^{2} = 3 \times a^{2}$.
Words to algebra: if something happens first, it may need a bracket: $3(n + 5)$.
Substitute: write each value in brackets, especially negatives.
Order: brackets, powers, multiply/divide, add/subtract.
Check signs: minus a negative is plus; a negative squared is positive.

Recall

[2 marks]

Write an expression for (a) 7 less than $y$ (b) $x$ divided by 4, then add 3.

(a) $y - 7$ ✓
(b) $\dfrac{x}{4} + 3$ ✓

Calculation

[3 marks]

Without a calculator, find the value of each expression when $a = 5$ and $b = -3$.

(a) $a - b$ (b) $ab^{2}$ (c) $\dfrac{a + b}{a - b}$

(a) $5 - (-3) = 8$ ✓
(b) $b^{2} = 9$, so $5 \times 9 = 45$ ✓
(c) $\dfrac{5 + (-3)}{8} = \dfrac{2}{8} = \dfrac{1}{4}$ ✓

Calculation

[2 marks]

The time, $T$ seconds, for one swing of a pendulum of length $l$ metres is $T = 2\pi\sqrt{\dfrac{l}{g}}$.

Find $T$ when $l = 0.8$ and $g = 9.8$.

$\dfrac{0.8}{9.8} = 0.08163...$, and $\sqrt{0.08163...} = 0.2857...$ ✓
$T = 2\pi \times 0.2857... = 1.795...$, so $T = 1.80$ s (3 s.f.) ✓

Analysis

[2 marks]

Sam says that $2x^{2}$ and $(2x)^{2}$ are always equal. Use $x = 3$ to show that Sam is wrong.

$2x^{2} = 2 \times 9 = 18$ ✓
$(2x)^{2} = 6^{2} = 36$. $18 \ne 36$, so Sam is wrong. ✓

Calculation

[2 marks]

Use the formula $F = \dfrac{9C}{5} + 32$ to find $F$ when $C = -15$.

$\dfrac{9 \times (-15)}{5} = -27$ ✓
$F = -27 + 32 = 5$ ✓

Analysis

[4 marks]

A taxi charges $3.50 plus $1.20 for each kilometre travelled.

(a) Write a formula for the cost, $C$ dollars, of a journey of $d$ kilometres.

(b) Find the cost of a 12 km journey.

(a) $C = 3.5 + 1.2d$ ✓
(b) $C = 3.5 + 1.2 \times 12 = 3.5 + 14.4 = 17.9$, so $17.90 ✓
(c) $3.5 + 1.2d = 11.3$, so $1.2d = 7.8$ ✓
$d = 7.8 \div 1.2 = 6.5$ km ✓

Mark this once you have attempted all six and checked your working. It records a Practiced badge on the topic and adds a one-time bonus. Revealing the solutions alone does not count.