Carry out the four operations with fractions; convert between fractions, decimals and percentages (including recurring decimals to fractions at Higher tier); and find a percentage of an amount and one quantity as a percentage of another.

What this dot point is asking

This Eduqas number statement pulls together three ways of writing the same proportion: fractions, decimals and percentages. You must add, subtract, multiply and divide fractions; convert freely between the three forms; and carry out percentage calculations, finding a percentage of an amount and writing one quantity as a percentage of another. At Higher tier you also convert a recurring decimal to an exact fraction. These skills appear on both components and feed directly into ratio, proportion and the percentage-change topics, so fluency here pays off widely.

The four operations with fractions

Each operation has its own rule, and mixed numbers must be converted first.

For addition and subtraction, rewrite both fractions over a common denominator, then add or subtract the numerators: $\tfrac{2}{3} + \tfrac{1}{4} = \tfrac{8}{12} + \tfrac{3}{12} = \tfrac{11}{12}$. For multiplication, multiply numerators and denominators, cancelling first where possible: $\tfrac{3}{4} \times \tfrac{2}{9} = \tfrac{6}{36} = \tfrac{1}{6}$. For division, multiply by the reciprocal of the second fraction: $\tfrac{5}{6} \div \tfrac{2}{3} = \tfrac{5}{6} \times \tfrac{3}{2} = \tfrac{15}{12} = \tfrac{5}{4}$.

Converting between the three forms

Fractions, decimals and percentages are three notations for one idea.

So $\tfrac{3}{8} = 3 \div 8 = 0.375 = 37.5\%$, and $45\% = \tfrac{45}{100} = \tfrac{9}{20}$. Knowing the common equivalents by heart ($\tfrac{1}{2} = 0.5 = 50\%$, $\tfrac{1}{4} = 0.25 = 25\%$, $\tfrac{1}{3} = 0.\dot{3} = 33.\dot{3}\%$, $\tfrac{1}{5} = 0.2 = 20\%$) saves time on the non-calculator component, where these conversions appear without a calculator to lean on.

Recurring decimals to fractions (Higher)

A recurring decimal is rational, so it equals an exact fraction.

The power of ten you multiply by must match the length of the repeating block (one digit means $\times 10$, two digits $\times 100$, and so on), so that subtracting cancels the infinite tail exactly.

Percentage calculations

Two standard percentage tasks recur throughout the paper.

To find a percentage of an amount, convert the percentage to a decimal and multiply: $18\%$ of $250$ is $0.18 \times 250 = 45$. To write one quantity as a percentage of another, form the fraction and multiply by $100$: $\frac{18}{40}$ as a percentage is $\frac{18}{40} \times 100 = 45\%$. On the non-calculator component, build up from easy percentages: $10\%$ is dividing by $10$, $5\%$ is half of that, and $1\%$ is dividing by $100$, so any percentage can be assembled from these.

Why this matters

Fractions, decimals and percentages are one idea in three costumes, and converting between them is constant: a probability is a fraction, a discount is a percentage, a measurement is a decimal. Because Eduqas's Component 1 is non-calculator, the conversions and the build-up percentage methods are tested without calculator support, and the Higher recurring-decimal proof is a pure AO2 reasoning task. Securing these underpins the ratio, proportion and percentage-change pages that follow.