IGCSE Mathematics | Practice Test — Answers & Worked Solutions
Section A — Recall
Questions 1–10
1.
Write down the definition of a proper fraction and give one example.
Answer
A proper fraction has a numerator that is smaller than the denominator. Its value is always less than 1.
Example: $\dfrac{3}{4}$ (any fraction where the top number is smaller than the bottom number)
Revision: In a proper fraction, the top number (numerator) is always less than the bottom number (denominator). This means the fraction is always less than 1 whole.
2.
Write down the definition of an improper fraction and give one example.
Answer
An improper fraction has a numerator that is equal to or greater than the denominator. Its value is 1 or more.
Example: $\dfrac{7}{4}$ (any fraction where the top number is equal to or greater than the bottom number)
Revision: An improper fraction is “top-heavy” — the numerator is as big as or bigger than the denominator. It can always be written as a mixed number.
3.
Write down the definition of a mixed number and give one example.
Answer
A mixed number has a whole number part and a proper fraction part written together. Its value is always greater than 1.
Example: $2\dfrac{3}{4}$ (a whole number followed by a proper fraction)
Revision: A mixed number is another way to write an improper fraction. For example, $\dfrac{11}{4}$ and $2\dfrac{3}{4}$ represent the same value.
4.
Write $\dfrac{6}{8}$ in its simplest form.
Answer
- Find the HCF of 6 and 8. Factors of 6: 1, 2, 3, 6. Factors of 8: 1, 2, 4, 8. HCF = 2
- Divide both the numerator and denominator by 2: $$\frac{6 \div 2}{8 \div 2} = \frac{3}{4}$$
- Check: HCF(3, 4) = 1, so the fraction is fully simplified.
$\dfrac{6}{8} = \dfrac{3}{4}$
Revision: To simplify a fraction, divide both numbers by their HCF (Highest Common Factor). The fraction’s value does not change.
5.
Convert $\dfrac{1}{4}$ to a decimal.
Answer
- Divide the numerator by the denominator: $$1 \div 4 = 0.25$$
$\dfrac{1}{4} = 0.25$
Revision: To convert any fraction to a decimal, divide the top number by the bottom number.
6.
Convert $\dfrac{1}{2}$ to a percentage.
Answer
- Multiply the fraction by 100: $$\frac{1}{2} \times 100 = 50$$
$\dfrac{1}{2} = 50\%$
Revision: To convert a fraction to a percentage, multiply by 100. This is because percentage means “out of 100”.
7.
Convert $0.6$ to a percentage.
Answer
- Multiply the decimal by 100: $$0.6 \times 100 = 60$$
$0.6 = 60\%$
Revision: To convert a decimal to a percentage, multiply by 100. The decimal point moves two places to the right.
8.
Convert $75\%$ to a decimal.
Answer
- Divide the percentage by 100: $$75 \div 100 = 0.75$$
$75\% = 0.75$
Revision: To convert a percentage to a decimal, divide by 100. The decimal point moves two places to the left.
9.
Convert $25\%$ to a fraction. Write your answer in its simplest form.
Answer
- Write the percentage over 100: $$25\% = \frac{25}{100}$$
- Find HCF(25, 100) = 25. Divide both by 25: $$\frac{25 \div 25}{100 \div 25} = \frac{1}{4}$$
$25\% = \dfrac{1}{4}$
Revision: To convert a percentage to a fraction, write it over 100 and then simplify. Always check you have divided by the HCF.
10.
Convert $\dfrac{9}{4}$ to a mixed number.
Answer
- Divide the numerator by the denominator: $9 \div 4 = 2$ remainder $1$
- The whole number is 2. The remainder (1) becomes the new numerator, and the denominator stays as 4: $$\frac{9}{4} = 2\frac{1}{4}$$
$\dfrac{9}{4} = 2\dfrac{1}{4}$
Revision: To convert an improper fraction to a mixed number: divide the top by the bottom. The answer gives the whole number, and the remainder goes on top of the original denominator.
Section B — Application
Questions 11–20
11.
Simplify $\dfrac{24}{36}$.
Answer
- Find the HCF of 24 and 36.
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
HCF = 12 - Divide both by 12: $$\frac{24 \div 12}{36 \div 12} = \frac{2}{3}$$
- Check: HCF(2, 3) = 1. Fully simplified. ✓
$\dfrac{24}{36} = \dfrac{2}{3}$
Revision: If you cannot immediately spot the HCF, you can simplify in steps — for example, divide by 2 to get $\dfrac{12}{18}$, then divide by 6 to get $\dfrac{2}{3}$. Both methods give the same final answer.
12.
Convert $2\dfrac{3}{4}$ to an improper fraction.
Answer
- Multiply the whole number by the denominator: $2 \times 4 = 8$
- Add the numerator: $8 + 3 = 11$
- Place over the original denominator: $$2\frac{3}{4} = \frac{11}{4}$$
$2\dfrac{3}{4} = \dfrac{11}{4}$
Revision: Think of it as: “how many quarters are in $2\frac{3}{4}$?” — 2 wholes = 8 quarters, plus 3 more quarters = 11 quarters total, which is $\frac{11}{4}$.
13.
Convert $\dfrac{7}{8}$ to a decimal.
Answer
- Divide the numerator by the denominator: $$7 \div 8 = 0.875$$
$\dfrac{7}{8} = 0.875$
Revision: $\frac{7}{8}$ is a common fraction worth memorising: it equals 0.875. This is also equal to $87.5\%$.
14.
Convert $0.35$ to a fraction. Write your answer in its simplest form.
Answer
- Two decimal places → denominator is 100: $$0.35 = \frac{35}{100}$$
- Find HCF(35, 100).
Factors of 35: 1, 5, 7, 35. Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100.
HCF = 5 - Divide both by 5: $$\frac{35 \div 5}{100 \div 5} = \frac{7}{20}$$
$0.35 = \dfrac{7}{20}$
Revision: Count the decimal places to find the denominator: 1 decimal place → over 10; 2 decimal places → over 100; 3 decimal places → over 1000. Then simplify.
15.
Convert $\dfrac{3}{8}$ to a percentage.
Answer
- Multiply the fraction by 100: $$\frac{3}{8} \times 100 = \frac{300}{8} = 37.5$$
$\dfrac{3}{8} = 37.5\%$
Revision: You can also convert to a decimal first ($\frac{3}{8} = 0.375$) and then multiply by 100 to get $37.5\%$. Both methods work.
16.
Convert $65\%$ to a fraction. Write your answer in its simplest form.
Answer
- Write over 100: $$65\% = \frac{65}{100}$$
- Find HCF(65, 100).
Factors of 65: 1, 5, 13, 65. Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100.
HCF = 5 - Divide both by 5: $$\frac{65 \div 5}{100 \div 5} = \frac{13}{20}$$
$65\% = \dfrac{13}{20}$
Revision: Many percentages that end in 5 or 0 will simplify by dividing by 5. Always check if the fraction can be simplified further.
17.
Which is greater: $\dfrac{5}{8}$ or $60\%$? Show your working.
Answer
- Convert $\dfrac{5}{8}$ to a percentage: $$\frac{5}{8} \times 100 = 62.5\%$$
- Compare: $62.5\% > 60\%$
$\dfrac{5}{8}$ is greater than $60\%$
Revision: To compare values in different forms, convert them all to the same form first (all decimals or all percentages), then compare.
18.
Write $3\dfrac{1}{5}$ as a decimal.
Answer
- Convert the fraction part to a decimal: $\dfrac{1}{5} = 1 \div 5 = 0.2$
- Add the whole number part: $3 + 0.2 = 3.2$
$3\dfrac{1}{5} = 3.2$
Revision: For a mixed number, convert only the fraction part to a decimal. Then add it to the whole number.
19.
Convert $\dfrac{13}{5}$ to a mixed number.
Answer
- Divide the numerator by the denominator: $13 \div 5 = 2$ remainder $3$
- Whole number = 2, remainder = 3, denominator stays as 5: $$\frac{13}{5} = 2\frac{3}{5}$$
$\dfrac{13}{5} = 2\dfrac{3}{5}$
Revision: Check your answer by converting back: $2 \times 5 + 3 = 13$, so $\frac{13}{5}$ ✓
20.
Write $0.125$ as a fraction in its simplest form.
Answer
- Three decimal places → denominator is 1000: $$0.125 = \frac{125}{1000}$$
- Find HCF(125, 1000).
$125 = 5 \times 5 \times 5$ and $1000 = 8 \times 125$, so HCF = 125. - Divide both by 125: $$\frac{125 \div 125}{1000 \div 125} = \frac{1}{8}$$
$0.125 = \dfrac{1}{8}$
Revision: $0.125 = \frac{1}{8}$ is a common equivalence worth memorising. You can also simplify in smaller steps: $\frac{125}{1000} \rightarrow \frac{25}{200} \rightarrow \frac{5}{40} \rightarrow \frac{1}{8}$.
Section C — Challenge
Questions 21–25
21.
Write these values in order from smallest to largest.
$\dfrac{7}{12}$ $0.62$ $\dfrac{3}{5}$ $58\%$
Answer
- Convert all values to decimals so they can be compared directly:
$\dfrac{7}{12} = 7 \div 12 = 0.5833…$
$0.62 = 0.62$
$\dfrac{3}{5} = 3 \div 5 = 0.60$
$58\% = 58 \div 100 = 0.58$ - Order the decimals from smallest to largest: $$0.58 \; < \; 0.5833... \; < \; 0.60 \; < \; 0.62$$
- Write back in the original forms: $$58\% \; < \; \frac{7}{12} \; < \; \frac{3}{5} \; < \; 0.62$$
Smallest to largest: $58\%$, $\dfrac{7}{12}$, $\dfrac{3}{5}$, $0.62$
Revision: When ordering mixed forms, always convert everything to the same form first. Decimals are usually the easiest form for comparing values.
22.
Show that $\dfrac{14}{21}$ and $\dfrac{2}{3}$ are equivalent fractions.
Answer
- Simplify $\dfrac{14}{21}$ by finding the HCF of 14 and 21.
Factors of 14: 1, 2, 7, 14. Factors of 21: 1, 3, 7, 21. HCF = 7 - Divide both by 7: $$\frac{14 \div 7}{21 \div 7} = \frac{2}{3}$$
- $\dfrac{14}{21}$ simplifies to $\dfrac{2}{3}$, so the two fractions are equivalent. ✓
$\dfrac{14}{21} = \dfrac{14 \div 7}{21 \div 7} = \dfrac{2}{3}$ Therefore $\dfrac{14}{21} = \dfrac{2}{3}$ ✓
Revision: Two fractions are equivalent if one simplifies to give the other. You can also check by cross-multiplying: $14 \times 3 = 42$ and $21 \times 2 = 42$ — since the products are equal, the fractions are equivalent.
23.
Write $4\dfrac{3}{8}$ as a decimal.
Answer
- Convert the fraction part to a decimal: $\dfrac{3}{8} = 3 \div 8 = 0.375$
- Add the whole number part: $4 + 0.375 = 4.375$
$4\dfrac{3}{8} = 4.375$
Revision: This question requires knowing that $\frac{3}{8} = 0.375$ (or being able to calculate $3 \div 8$). Keep the whole number part separate and only convert the fraction.
24.
Write $0.875$ as a fraction in its simplest form.
Answer
- Three decimal places → denominator is 1000: $$0.875 = \frac{875}{1000}$$
- Find HCF(875, 1000).
$875 = 5 \times 175 = 5 \times 5 \times 35 = 5 \times 5 \times 5 \times 7 = 125 \times 7$
$1000 = 125 \times 8$
HCF = 125 - Divide both by 125: $$\frac{875 \div 125}{1000 \div 125} = \frac{7}{8}$$
$0.875 = \dfrac{7}{8}$
Revision: If finding HCF of large numbers is difficult, simplify in steps: $\frac{875}{1000} \rightarrow \frac{175}{200} \rightarrow \frac{35}{40} \rightarrow \frac{7}{8}$ (dividing by 5 each time). Notice that $0.875 = \frac{7}{8}$ — the reverse of Q13.
25.
Convert $\dfrac{5}{6}$ to a decimal. Give your answer correct to 2 decimal places.
Answer
- Divide the numerator by the denominator: $$5 \div 6 = 0.8333…$$
- Round to 2 decimal places. The third decimal digit is 3 (less than 5), so round down: $$0.8333… \approx 0.83$$
$\dfrac{5}{6} \approx 0.83$ (to 2 decimal places)
Revision: $\frac{5}{6}$ produces a decimal that goes on forever (0.8333…). For IGCSE you do not need to write the recurring notation — just perform the division and round to the number of decimal places asked.
