Fractions ToolkitCambridge IGCSE Maths: Revision notes

• Proper fraction: numerator smaller than denominator, e.g. $\frac{3}{4}$
• Improper fraction: numerator larger than (or equal to) the denominator, e.g. $\frac{7}{4}$
• Mixed number: a whole number combined with a proper fraction, e.g. $1\frac{3}{4}$

All three forms, along with decimals and percentages, describe parts of a whole and can be used interchangeably depending on the context of a question.

Simplifying: divide both the numerator and denominator by their highest common factor until no further division is possible, e.g. $\frac{12}{18} = \frac{2}{3}$ (divide by 6).

Converting an improper fraction to a mixed number: divide the numerator by the denominator; the whole number part is the quotient, and the remainder becomes the new numerator, e.g. $\frac{11}{4} = 2\frac{3}{4}$.

Converting a mixed number to an improper fraction: multiply the whole number by the denominator, add the numerator, and keep the same denominator, e.g. $2\frac{3}{4} = \frac{11}{4}$.

1. Find a common denominator (usually the lowest common multiple of the two denominators)
2. Convert each fraction to an equivalent fraction with that common denominator
3. Add or subtract the numerators, keeping the denominator the same
4. Simplify the result if possible

For mixed numbers, either convert to improper fractions first, or add/subtract the whole numbers and fraction parts separately (careful with 'borrowing' when subtracting a larger fraction part).

Multiplying: multiply the numerators together and the denominators together, simplifying before or after, e.g. $\frac{2}{3} \times \frac{3}{5} = \frac{6}{15} = \frac{2}{5}$.

Dividing: multiply by the reciprocal (flip the second fraction), e.g. $\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}$.

For mixed numbers, always convert to improper fractions first before multiplying or dividing.

A recurring decimal (e.g. $0.\overline{3}$ or $0.1\overline{7}$) can always be written as an exact fraction.

Method:

1. Let $x$ equal the recurring decimal
2. Multiply $x$ by a power of 10 so the recurring part lines up
3. Subtract the original equation from the new one to eliminate the recurring part
4. Solve for $x$ as a fraction and simplify

Example: $x = 0.\overline{3}$. Then $10x = 3.\overline{3}$. Subtracting: $9x = 3$, so $x = \frac{3}{9} = \frac{1}{3}$.