Fractions, Decimals & PercentagesEdexcel IGCSE Maths: Revision notes

These three forms describe the same value in different ways, and you must be able to switch between them instantly, with or without a calculator.

Fraction to decimal: divide the numerator by the denominator. $\frac{3}{8} = 3 \div 8 = 0.375$

Decimal to fraction: use place value as the denominator, then simplify. $0.35 = \frac{35}{100} = \frac{7}{20}$

Fraction/decimal to percentage: multiply by 100. $0.6 = 60\%, \quad \frac{3}{4} = 0.75 = 75\%$

Percentage to fraction/decimal: divide by 100. $45\% = \frac{45}{100} = \frac{9}{20} = 0.45$

A recurring decimal has one or more digits that repeat forever. IGCSE uses dot notation above the repeating digit(s):

• $0.\dot{3} = 0.3333...$ (single digit repeats)
• $0.4\dot{5}\dot{6} = 0.456456456...$ (a block of digits repeats)
• $0.1\dot{2}3\dot{4} = 0.1234234234...$ (dots go over the first and last digit of the repeating block)

Any recurring decimal can be written as an exact fraction — this is one of the ways IGCSE tests whether you understand that recurring decimals are rational numbers.

A terminating decimal stops after a finite number of digits (e.g. $0.375$). A decimal that neither terminates nor recurs (like $\pi$) is irrational and cannot be written as an exact fraction.

Use algebra with a variable $x$.

Single repeating digit, e.g. $x = 0.\dot{6}$:

1. $x = 0.6666...$
2. Multiply by 10: $10x = 6.6666...$
3. Subtract: $10x - x = 6.6666... - 0.6666...$, so $9x = 6$
4. $x = \frac{6}{9} = \frac{2}{3}$

Repeating block of 2 digits, e.g. $x = 0.\dot{2}\dot{7}$:

1. $x = 0.272727...$
2. Multiply by 100 (two repeating digits): $100x = 27.272727...$
3. Subtract: $99x = 27$
4. $x = \frac{27}{99} = \frac{3}{11}$

Mixed case (non-repeating part first), e.g. $x = 0.1\dot{6}$:

1. $x = 0.1666...$
2. Multiply by 10 to move past the non-repeating digit: $10x = 1.666...$
3. Multiply by 10 again to shift one full repeating cycle: $100x = 16.666...$
4. Subtract: $100x - 10x = 16.666... - 1.666...$, so $90x = 15$
5. $x = \frac{15}{90} = \frac{1}{6}$

Rule of thumb: multiply by $10^n$ where $n$ is the number of repeating digits, then subtract the original equation so the recurring part cancels exactly.

To order a mixed list, convert everything into the same format first — usually decimals, since they are easiest to compare digit by digit.

Worked example: Order $\frac{3}{5}, 58\%, 0.55, \frac{11}{20}$ from smallest to largest.

1. Convert all to decimals: $\frac{3}{5} = 0.6$, $58\% = 0.58$, $0.55 = 0.55$, $\frac{11}{20} = 0.55$
2. Compare: $0.55 = 0.55 < 0.58 < 0.6$
3. Order: $0.55 \left(=\frac{11}{20}\right), 0.55, 58\%, \frac{3}{5}$

Tip for fractions with different denominators: convert to a common denominator instead of decimals if the numbers are awkward, e.g. comparing $\frac{5}{6}$ and $\frac{7}{9}$ using denominator 18: $\frac{15}{18}$ vs $\frac{14}{18}$.

Knowing these instantly saves time in non-calculator questions:

Notice that any fraction with denominator 9 gives a single repeating digit equal to the numerator (e.g. $\frac{4}{9} = 0.\dot{4}$), and denominators of 3, 6, 7, 9, 11 tend to give recurring decimals, while denominators built only from 2s and 5s (like 2, 4, 5, 8, 10, 20, 25) always terminate.