Calculating with fractions (the four operations, including mixed numbers), converting between fractions, decimals and percentages including recurring decimals, and working with percentages of amounts.

What this dot point is asking

Fractions, decimals and percentages are three ways of writing the same idea: a part of a whole. Edexcel expects you to calculate fluently with fractions (including mixed numbers and all four operations), to convert freely between the three forms, to turn recurring decimals into fractions at Higher tier, and to find percentages of amounts. Much of this is non-calculator, so secure written methods matter.

Calculating with fractions

The four operations each have a reliable method. The single most important habit is to convert mixed numbers to improper fractions before doing anything else.

For example, $\dfrac{2}{3} + \dfrac{1}{4}$ uses the common denominator $12$: $\dfrac{8}{12} + \dfrac{3}{12} = \dfrac{11}{12}$. To work out $\dfrac{3}{5} \div \dfrac{2}{7}$, flip the second fraction and multiply: $\dfrac{3}{5} \times \dfrac{7}{2} = \dfrac{21}{10} = 2\tfrac{1}{10}$. Always simplify the final answer by dividing top and bottom by their highest common factor.

Converting between the three forms

Conversions are the glue that lets you choose the easiest form for a problem.

• Fraction to decimal: divide the numerator by the denominator. $\dfrac{3}{8} = 3 \div 8 = 0.375$.
• Decimal to percentage: multiply by $100$. $0.375 \times 100 = 37.5\%$.
• Percentage to fraction: write over $100$ and simplify. $40\% = \dfrac{40}{100} = \dfrac{2}{5}$.
• Decimal to fraction: use place value. $0.6 = \dfrac{6}{10} = \dfrac{3}{5}$.

Knowing the common equivalents by heart ($\tfrac{1}{2} = 0.5 = 50\%$, $\tfrac{1}{4} = 0.25 = 25\%$, $\tfrac{1}{3} = 0.\dot{3}$, $\tfrac{1}{5} = 0.2 = 20\%$) saves time in the exam.

Recurring decimals to fractions (Higher)

A recurring decimal has a digit or block of digits that repeats forever, shown with dots over the first and last repeating digits. The trick is to multiply by the right power of ten so the repeating part lines up, then subtract.

Percentages of amounts

To find a percentage of an amount without a calculator, build it from easy "building blocks": $10\%$ is dividing by $10$, $1\%$ is dividing by $100$, $50\%$ is halving, $25\%$ is quartering, and $5\%$ is half of $10\%$. To find $35\%$ of $240$: $10\%$ is $24$, so $30\%$ is $72$, and $5\%$ is $12$, giving $35\% = 72 + 12 = 84$. With a calculator, multiply by the decimal: $35\%$ of $240 = 0.35 \times 240 = 84$.

The same building blocks let you express one quantity as a percentage of another: divide and multiply by $100$. If $18$ out of $40$ students walk to school, the percentage is $\dfrac{18}{40} \times 100 = 45\%$. Knowing which form is easiest is a skill in itself: to find $\tfrac{1}{4}$ of a number, quartering is faster than multiplying by $0.25$, but to compare several proportions, converting them all to percentages makes the comparison clear.

Why three forms

Fractions, decimals and percentages each suit a different job. Fractions keep values exact, which matters in algebra and probability where $\tfrac{1}{3}$ is precise but $0.333$ is not. Decimals are best for calculator work and measurement. Percentages are the natural language of change, interest and comparison, because they put everything "out of $100$". Being able to switch fluently means you can always pick the form that makes a problem easiest, and Edexcel deliberately mixes the three within single questions to test that flexibility.