Benchmark Fractions as Decimals and Percentages

1. Fractions refresher: equivalent fractions and simplest form

A fraction describes equal parts of a whole. In $\frac{5}{6}$, the denominator 6 tells you the whole is cut into 6 equal parts, and the numerator 5 tells you how many of those parts you take. If a rectangle is divided into 6 equal squares, shading 5 of them shows $\frac{5}{6}$ of the rectangle.

Equivalent fractions name the same amount. You make them by multiplying (or dividing) the numerator and the denominator by the same number. Cutting every piece into 2 doubles the number of pieces and doubles the number shaded, but the amount shaded does not change. So three fractions equivalent to $\frac{2}{3}$ are $\frac{4}{6}$, $\frac{6}{9}$ and $\frac{8}{12}$.

▶ Notice that the shaded length never changes: the fractions look different but they are equal.

To write a fraction in its simplest form, divide the numerator and denominator by their highest common factor (HCF). For $\frac{6}{10}$ the HCF of 6 and 10 is 2, so $\frac{6}{10}$ = $\frac{6÷2}{10÷2}$ = $\frac{3}{5}$. For $\frac{12}{20}$ the HCF is 4, so $\frac{12}{20}$ = $\frac{3}{5}$ as well.

An improper fraction has a numerator bigger than (or equal to) its denominator, such as $\frac{12}{5}$. Divide to change it to a mixed number: 12 ÷ 5 = 2 remainder 2, so $\frac{12}{5}$ = $2\frac{2}{5}$. To go back, multiply the whole number by the denominator and add the numerator: $2\frac{5}{9}$ = $\frac{2\times 9+5}{9}$ = $\frac{23}{9}$.

2. Changing a fraction to a decimal and a percentage

A percentage is simply a fraction with denominator 100, and a decimal uses tenths and hundredths. So the easiest way to change a fraction is to make an equivalent fraction with denominator 10 or 100.

1. 1Find a number that turns the denominator into 10 or 100. For $\frac{1}{4}$, 4 × 25 = 100.
2. 2Multiply the numerator by the same number: $\frac{1\times 25}{4\times 25}$ = $\frac{25}{100}$.
3. 3Read it as a decimal: 25 hundredths = 0.25.
4. 4Read it as a percentage: 25 out of 100 = 25%.

▶ One quarter of the 100 small squares is 25 squares: $\frac{1}{4}$ = 25% = 0.25.

When the denominator cannot become 10 or 100 (like 3), divide the numerator by the denominator. $\frac{1}{3}$ = 1 ÷ 3 = 0.333… where the 3 repeats forever. To get a percentage, multiply the decimal by 100: 0.333… × 100 = 33.333…% which we write as $33\frac{1}{3}$%.

3. The benchmark table — learn it by heart

Halves, thirds, fourths (quarters), fifths and tenths appear everywhere: discounts, exam scores, sharing of farm produce and sports statistics. These are called benchmark fractions. Learn this table the way you learnt your times tables — it will save you a lot of time in the BECE.

4. Using benchmarks to compare quantities

Benchmarks help you judge the size of any fraction. The three most useful landmarks are 0, **$\frac{1}{2}$ and 1**. A fraction is close to 0 when the numerator is very small compared with the denominator ($\frac{1}{10}$, $\frac{1}{8}$). It is close to $\frac{1}{2}$ when the numerator is about half the denominator ($\frac{2}{5}$, $\frac{5}{9}$). It is close to 1 when the numerator is almost equal to the denominator ($\frac{9}{10}$, $\frac{7}{8}$).

▶ Watch where each fraction lands: its nearest landmark (0, ½ or 1) tells you its size at a glance.

To compare two quantities given in different forms, change them to the same form — usually percentages or decimals, because these are easy to compare. Is $\frac{3}{5}$ bigger or smaller than $\frac{2}{3}$? From the table, $\frac{3}{5}$ = 60% and $\frac{2}{3}$ = $66\frac{2}{3}$%. So $\frac{3}{5}$ < $\frac{2}{3}$.

Benchmarks also work for fractions that are not in the table. Compare $\frac{7}{12}$ and $\frac{8}{10}$: $\frac{7}{12}$ is just over a half (because $\frac{6}{12}$ = $\frac{1}{2}$), while $\frac{8}{10}$ = 0.8 is close to 1. So $\frac{8}{10}$ is greater.

Worked examples

Try each one yourself first, then open the solution and compare every step.