Fractions, decimals and percentages

The language of fractions

In the fraction $\frac{3}{8}$ the bottom number, the denominator, says how many equal parts the whole is split into, and the top number, the numerator, says how many of those parts you have.

• A proper fraction is less than 1: the numerator is smaller than the denominator, as in $\frac{3}{8}$.
• An improper fraction is 1 or more: the numerator is at least as big as the denominator, as in $\frac{11}{4}$.
• A mixed number has a whole-number part and a proper fraction part, as in $2\frac{3}{4}$.

Improper to mixed. Divide the numerator by the denominator. The answer is the whole number and the remainder goes over the denominator. $11÷4=2$ remainder 3, so $\frac{11}{4}=2\frac{3}{4}$.

Mixed to improper. Multiply the whole number by the denominator and add the numerator. $2\frac{3}{4}=\frac{2\times 4+3}{4}=\frac{11}{4}$. This works because each whole one is $\frac{4}{4}$, so two wholes are $\frac{8}{4}$, and then three more quarters make $\frac{11}{4}$.

Equivalent fractions and simplest form

Multiplying or dividing the numerator and the denominator by the same number does not change the value of a fraction: $\frac{3}{4}=\frac{6}{8}=\frac{30}{40}$. These are equivalent fractions. It works because you are multiplying by $\frac{2}{2}$ or $\frac{10}{10}$, which is 1.

A fraction is in its simplest form when the numerator and the denominator have no common factor other than 1. To simplify, divide both by their highest common factor. The HCF of 36 and 84 is 12, so $\frac{36}{84}=\frac{3}{7}$. You can also divide in stages, by 2, then 2 again, then 3, as long as you keep going until no common factor is left.

Final answers that are fractions should always be given in their simplest form.

Fractions and decimals

Decimal to fraction. Use place value: the last decimal place tells you the denominator. Tenths, hundredths, thousandths.

• $0.45=\frac{45}{100}=\frac{9}{20}$
• $0.375=\frac{375}{1000}=\frac{3}{8}$
• $2.6=2\frac{6}{10}=2\frac{3}{5}$

Fraction to decimal. A fraction is a division: $\frac{3}{8}=3÷8$. Without a calculator, either use short division, writing 3 as $3.000$, or make the denominator a power of 10: $\frac{7}{20}=\frac{35}{100}=0.35$ and $\frac{3}{25}=\frac{12}{100}=0.12$.

A fraction in its simplest form gives a decimal that stops (a terminating decimal) exactly when its denominator has no prime factors other than 2 and 5, because only then can it be made into a power of 10. Other fractions give decimals that go on for ever, such as $\frac{1}{3}=0.333\dots$ On the Core papers you are not asked to use recurring decimal notation; a decimal like this is rounded, for example $\frac{2}{3}\approx 0.667$ to 3 decimal places.

Percentages

Per cent means per hundred, so a percentage is a fraction with denominator 100: $35\%=\frac{35}{100}$.

• Percentage to decimal: divide by 100. $35\%=0.35$, $7\%=0.07$, $120\%=1.2$.
• Decimal to percentage: multiply by 100. $0.6=60\%$ and $0.045=4.5\%$.
• Percentage to fraction: write it over 100 and simplify. $35\%=\frac{35}{100}=\frac{7}{20}$, and $12.5\%=\frac{12.5}{100}=\frac{125}{1000}=\frac{1}{8}$.
• Fraction to percentage: make the denominator 100, or change to a decimal and multiply by 100. $\frac{9}{25}=\frac{36}{100}=36\%$, and $\frac{3}{8}=0.375=37.5\%$.

A percentage greater than 100 is more than the whole, so it matches an improper fraction or mixed number: $150\%=1.5=\frac{3}{2}=1\frac{1}{2}$.

To compare amounts given in different forms, change them all into the same form. Percentages are usually the clearest, because they all have the same denominator.

Common equivalents

Learn these and you can build others from them: $\frac{3}{8}=3\times 0.125=0.375=37.5\%$, and $\frac{3}{20}=3\times 0.05=0.15=15\%$.