IGCSE Mathematics | Practice Test — Answers & Worked Solutions
Section A — Recall
Questions 1–10
1.
What does the symbol $\neq$ mean? Give one example using numbers.
Answer
$\neq$ means is not equal to. The two values on either side are different.
Example: $3 \neq 7$ (any two different numbers are correct)
Revision: The symbol $\neq$ is the opposite of $=$. Use it when two quantities have different values.
2.
Write the correct symbol ($>$ or $<$) between $-3$ and $2$. $-3$ ___ $2$
Answer
On a number line, $-3$ is to the left of $2$, so $-3$ is smaller.
$-3 < 2$
Revision: All negative numbers are less than all positive numbers. The open end of the symbol always faces the larger number.
3.
Which is larger: $-1$ or $-5$?
Answer
On a number line, $-1$ is to the right of $-5$, so $-1$ is larger.
$-1$ is larger ($-1 > -5$)
Revision: With negative numbers, the one closer to zero is always larger. So $-1 > -5$ even though 1 is smaller than 5 as positive numbers.
4.
What is the result when you multiply a negative number by a negative number? Circle your answer: Positive Negative
Answer
Positive
Revision: Same signs give a positive result. Different signs give a negative result. This applies to both multiplication and division.
5.
Calculate $(-5) + 9$.
Answer
- Start at $-5$ on the number line and move $9$ places to the right: $$(-5) + 9 = 4$$
$(-5) + 9 = 4$
Revision: Adding a positive number moves you to the right on the number line. Since 9 is larger than 5, the result crosses zero and becomes positive.
6.
Calculate $2 – 7$.
Answer
- Start at $2$ and subtract $7$. Since $7 > 2$, the result goes below zero: $$2 – 7 = -5$$
$2 – 7 = -5$
Revision: When you subtract a larger number from a smaller one, the answer is negative. Think of it as: you owe 5 more than you have.
7.
Write out what each letter in BIDMAS stands for.
Answer
B — Brackets
I — Indices
D — Division
M — Multiplication
A — Addition
S — Subtraction
Revision: BIDMAS tells you the order to work through a calculation. D and M have equal priority (work left to right), and so do A and S.
8.
Calculate $5 + 2 \times 4$. Show your working.
Answer
- Multiplication comes before addition (BIDMAS): $2 \times 4 = 8$
- Then add: $5 + 8 = 13$
$5 + 2 \times 4 = 13$
Revision: A common mistake is to add first and get $7 \times 4 = 28$. Always multiply before adding unless brackets tell you otherwise.
9.
Write down the reciprocal of $\dfrac{2}{7}$.
Answer
To find the reciprocal, flip the fraction upside down.
Reciprocal of $\dfrac{2}{7}$ is $\dfrac{7}{2}$
Revision: The reciprocal of $\dfrac{a}{b}$ is $\dfrac{b}{a}$. You use the reciprocal when dividing by a fraction — flip the second fraction, then multiply.
10.
Complete the statement using $\geqslant$ or $\leqslant$. “$x$ is greater than or equal to 3” means $x$ ___ $3$
Answer
“Greater than or equal to” uses the symbol $\geqslant$.
$x \geqslant 3$
Revision: $\geqslant$ means the value can equal 3 or be any number above 3. $\leqslant$ is the opposite — the value can equal 3 or be any number below 3.
Section B — Application
Questions 11–20
11.
Order these numbers from smallest to largest. $-4$ $1$ $-6$ $0$ $3$
Answer
- Place the numbers on a number line. Negative numbers go to the left of zero; more negative = smaller.
- Order from left to right: $-6, -4, 0, 1, 3$
$-6 < -4 < 0 < 1 < 3$
Revision: When ordering integers, remember that the more negative a number is, the smaller it is. $-6$ is less than $-4$ because it is further to the left on the number line.
12.
Write the correct symbol between $-0.5$ and $-0.8$. $-0.5$ ___ $-0.8$
Answer
- Both numbers are negative. Compare their distance from zero: $0.5 < 0.8$, so $-0.5$ is closer to zero.
- The number closer to zero is always larger, so $-0.5 > -0.8$.
$-0.5 > -0.8$
Revision: With negative decimals, be careful — $-0.8$ looks like a larger number but it is actually smaller because it is further from zero on the number line.
13.
Calculate $(-4) \times (-7)$.
Answer
- Multiply the values: $4 \times 7 = 28$
- Check the signs: negative $\times$ negative = positive
$(-4) \times (-7) = 28$
Revision: Same signs always give a positive result. Two negatives multiplied together cancel each other out, giving a positive answer.
14.
Calculate $(-15) \div 3$.
Answer
- Divide the values: $15 \div 3 = 5$
- Check the signs: negative $\div$ positive = negative
$(-15) \div 3 = -5$
Revision: Different signs give a negative result. This rule is the same for both division and multiplication.
15.
Calculate $\dfrac{1}{2} + \dfrac{3}{5}$. Give your answer as a mixed number.
Answer
- Find the lowest common denominator (LCD) of 2 and 5: LCD = 10
- Convert both fractions: $$\frac{1}{2} = \frac{5}{10} \qquad \frac{3}{5} = \frac{6}{10}$$
- Add the numerators: $$\frac{5}{10} + \frac{6}{10} = \frac{11}{10}$$
- Convert to a mixed number: $11 \div 10 = 1$ remainder $1$ $$\frac{11}{10} = 1\frac{1}{10}$$
$\dfrac{1}{2} + \dfrac{3}{5} = 1\dfrac{1}{10}$
Revision: Always find a common denominator before adding fractions. The LCD is the smallest number that both denominators divide into evenly.
16.
Calculate $\dfrac{5}{6} – \dfrac{1}{4}$.
Answer
- Find the LCD of 6 and 4: LCD = 12
- Convert both fractions: $$\frac{5}{6} = \frac{10}{12} \qquad \frac{1}{4} = \frac{3}{12}$$
- Subtract the numerators: $$\frac{10}{12} – \frac{3}{12} = \frac{7}{12}$$
- Check if it simplifies: HCF(7, 12) = 1, so $\dfrac{7}{12}$ is already in its simplest form.
$\dfrac{5}{6} – \dfrac{1}{4} = \dfrac{7}{12}$
Revision: The LCD of 6 and 4 is 12, not 24. Always use the lowest common denominator to keep numbers manageable.
17.
Calculate $\dfrac{3}{5} \times \dfrac{10}{9}$. Give your answer in its simplest form.
Answer
- Cross-cancel before multiplying:
3 and 9 share factor 3 → 3 becomes 1, 9 becomes 3
10 and 5 share factor 5 → 10 becomes 2, 5 becomes 1 $$\frac{\cancelto{1}{3}}{\cancelto{1}{5}} \times \frac{\cancelto{2}{10}}{\cancelto{3}{9}} = \frac{1 \times 2}{1 \times 3} = \frac{2}{3}$$
$\dfrac{3}{5} \times \dfrac{10}{9} = \dfrac{2}{3}$
Revision: Cross-cancelling before multiplying keeps the numbers small and avoids having to simplify a large fraction at the end.
18.
Calculate $\dfrac{3}{4} \div \dfrac{9}{16}$. Give your answer as a mixed number.
Answer
- KCF — Keep, Change, Flip: $\dfrac{3}{4} \div \dfrac{9}{16}$ becomes $\dfrac{3}{4} \times \dfrac{16}{9}$
- Cross-cancel before multiplying:
3 and 9 share factor 3 → 3 becomes 1, 9 becomes 3
4 and 16 share factor 4 → 4 becomes 1, 16 becomes 4 $$\frac{\cancelto{1}{3}}{\cancelto{1}{4}} \times \frac{\cancelto{4}{16}}{\cancelto{3}{9}} = \frac{4}{3} = 1\frac{1}{3}$$
$\dfrac{3}{4} \div \dfrac{9}{16} = 1\dfrac{1}{3}$
Revision: Always flip the second fraction (the one you are dividing by), not the first. Then cross-cancel diagonally before multiplying.
19.
Calculate $24 \div (2 + 4) + 3 \times 2$. Show all steps.
Answer
- Brackets first: $2 + 4 = 6$ → $24 \div 6 + 3 \times 2$
- D and M, left to right: $24 \div 6 = 4$, then $3 \times 2 = 6$ → $4 + 6$
- Addition: $4 + 6 = 10$
$24 \div (2 + 4) + 3 \times 2 = 10$
Revision: Division and multiplication have equal priority — work through them from left to right. Without the brackets, $24 \div 2 = 12$ would give a completely different (wrong) answer.
20.
A diver is at $-15\,\text{m}$ (15 m below sea level). She swims up 6 m. What is her new depth? Write your answer using a negative number.
Answer
- Swimming up means increasing (adding) the depth value: $$-15 + 6 = -9$$
New depth = $-9\,\text{m}$ (9 m below sea level)
Revision: Negative depth means below sea level. Adding a positive number moves the value towards zero (closer to the surface). Since the answer is still negative, the diver is still below sea level.
Section C — Challenge
Questions 21–25
21.
Order these values from smallest to largest. Show your working. $-0.6$ $-\dfrac{3}{4}$ $\dfrac{1}{3}$ $-0.5$
Answer
- Convert all values to decimals so they can be compared directly:
$-0.6 = -0.6$
$-\dfrac{3}{4} = -(3 \div 4) = -0.75$
$\dfrac{1}{3} = 1 \div 3 = 0.333…$
$-0.5 = -0.5$ - Order from smallest to largest (most negative first): $$-0.75 < -0.6 < -0.5 < 0.333…$$
- Write back in original forms: $$-\frac{3}{4} < -0.6 < -0.5 < \frac{1}{3}$$
Smallest to largest: $-\dfrac{3}{4}$, $-0.6$, $-0.5$, $\dfrac{1}{3}$
Revision: When ordering a mix of fractions and decimals, convert everything to decimals first. With negative values, the most negative is always the smallest.
22.
Calculate $2\dfrac{2}{3} + 1\dfrac{3}{4}$. Show all steps.
Answer
- Convert both mixed numbers to improper fractions: $$2\frac{2}{3} = \frac{(2 \times 3) + 2}{3} = \frac{8}{3} \qquad 1\frac{3}{4} = \frac{(1 \times 4) + 3}{4} = \frac{7}{4}$$
- Find the LCD of 3 and 4: LCD = 12. Convert: $$\frac{8}{3} = \frac{32}{12} \qquad \frac{7}{4} = \frac{21}{12}$$
- Add: $$\frac{32}{12} + \frac{21}{12} = \frac{53}{12}$$
- Convert back to a mixed number: $53 \div 12 = 4$ remainder $5$ $$\frac{53}{12} = 4\frac{5}{12}$$
$2\dfrac{2}{3} + 1\dfrac{3}{4} = 4\dfrac{5}{12}$
Revision: Always convert mixed numbers to improper fractions before calculating. Convert back to a mixed number at the end if the result is an improper fraction.
23.
Calculate $2\dfrac{1}{4} \div 1\dfrac{1}{2}$. Show all steps.
Answer
- Convert both mixed numbers to improper fractions: $$2\frac{1}{4} = \frac{(2 \times 4) + 1}{4} = \frac{9}{4} \qquad 1\frac{1}{2} = \frac{(1 \times 2) + 1}{2} = \frac{3}{2}$$
- KCF — Keep, Change, Flip: $\dfrac{9}{4} \div \dfrac{3}{2}$ becomes $\dfrac{9}{4} \times \dfrac{2}{3}$
- Cross-cancel before multiplying:
9 and 3 share factor 3 → 9 becomes 3, 3 becomes 1
4 and 2 share factor 2 → 4 becomes 2, 2 becomes 1 $$\frac{\cancelto{3}{9}}{\cancelto{2}{4}} \times \frac{\cancelto{1}{2}}{\cancelto{1}{3}} = \frac{3}{2} = 1\frac{1}{2}$$
$2\dfrac{1}{4} \div 1\dfrac{1}{2} = 1\dfrac{1}{2}$
Revision: For dividing mixed numbers: (1) convert to improper fractions, (2) KCF to flip the second fraction, (3) cross-cancel, (4) multiply. Never divide mixed numbers directly.
24.
Calculate $(-3) \times 4 + 20 \div (-5)$. Show all steps clearly.
Answer
- Multiplication and Division first (left to right):
$(-3) \times 4 = -12$ (different signs → negative)
$20 \div (-5) = -4$ (different signs → negative)
→ $-12 + (-4)$ - Addition: $$-12 + (-4) = -12 – 4 = -16$$
$(-3) \times 4 + 20 \div (-5) = -16$
Revision: Apply BIDMAS even when the numbers are negative. Do all multiplication and division first, then addition and subtraction. Remember: adding a negative is the same as subtracting.
25.
Calculate $2 \times 3^2 – (12 \div 4) + (-5)$. Show all steps clearly.
Answer
- Brackets: $12 \div 4 = 3$ → $2 \times 3^2 – 3 + (-5)$
- Indices: $3^2 = 9$ → $2 \times 9 – 3 + (-5)$
- Multiplication: $2 \times 9 = 18$ → $18 – 3 + (-5)$
- A/S left to right: $18 – 3 = 15$, then $15 + (-5) = 15 – 5 = 10$
$2 \times 3^2 – (12 \div 4) + (-5) = 10$
Revision: Work through BIDMAS one step at a time. Rewrite the expression after each step to keep track. Note that $+(-5)$ is the same as $-5$.
