IGCSE Mathematics | Core Topic
1. Types of Numbers #
IMAGE NEEDED: Diagram of a fraction showing the numerator (top number), fraction bar, and denominator (bottom number) clearly labelled
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Proper Fractions #
Examples: $\dfrac{1}{2}$, $\dfrac{3}{4}$, $\dfrac{5}{8}$
Improper Fractions #
Examples: $\dfrac{5}{4}$, $\dfrac{7}{3}$, $\dfrac{9}{9}$
Mixed Numbers #
Examples: $1\dfrac{1}{4}$, $2\dfrac{3}{5}$, $3\dfrac{7}{8}$
Decimals #
Examples: $0.5$, $1.75$, $0.125$
Percentages #
Examples: $50\%$, $75\%$, $12.5\%$
2. Simplifying Fractions #
To simplify a fraction, divide both the numerator and denominator by their Highest Common Factor (HCF). The value of the fraction stays the same.
- Find the HCF of 12 and 18. HCF = 6
- Divide both the numerator and denominator by 6: $$\frac{12 \div 6}{18 \div 6} = \frac{2}{3}$$
- Check: 2 and 3 share no common factor other than 1, so $\dfrac{2}{3}$ is fully simplified.
3. Converting Between Forms #
Improper Fraction → Mixed Number #
- Divide the numerator by the denominator: $11 \div 4 = 2$ remainder $3$
- The whole number is 2. The remainder becomes the new numerator: $$\frac{11}{4} = 2\frac{3}{4}$$
Mixed Number → Improper Fraction #
- Multiply the whole number by the denominator: $3 \times 5 = 15$
- Add the numerator: $15 + 2 = 17$
- Place over the original denominator: $$3\frac{2}{5} = \frac{17}{5}$$
Fraction → Decimal #
Divide the numerator by the denominator.
Decimal → Fraction #
Write the decimal over a power of 10 based on the number of decimal places, then simplify.
- Two decimal places → denominator is 100: $$0.45 = \frac{45}{100}$$
- Simplify (HCF of 45 and 100 is 5): $$\frac{45 \div 5}{100 \div 5} = \frac{9}{20}$$
Fraction → Percentage #
Multiply the fraction by 100.
Percentage → Fraction #
Write the percentage over 100, then simplify.
- Write over 100: $\dfrac{35}{100}$
- Simplify (HCF = 5): $$\frac{35 \div 5}{100 \div 5} = \frac{7}{20}$$
Decimal → Percentage #
Multiply by 100.
Percentage → Decimal #
Divide by 100.
4. Common Equivalences to Know #
Memorise these — they are very useful in exams.
| Fraction | Decimal | Percentage |
|---|---|---|
| $\dfrac{1}{2}$ | $0.5$ | $50\%$ |
| $\dfrac{1}{4}$ | $0.25$ | $25\%$ |
| $\dfrac{3}{4}$ | $0.75$ | $75\%$ |
| $\dfrac{1}{5}$ | $0.2$ | $20\%$ |
| $\dfrac{2}{5}$ | $0.4$ | $40\%$ |
| $\dfrac{4}{5}$ | $0.8$ | $80\%$ |
| $\dfrac{1}{10}$ | $0.1$ | $10\%$ |
| $\dfrac{1}{8}$ | $0.125$ | $12.5\%$ |
