Percentages, fractions and decimals

Shops say "20% off". Report cards show 85%. Banks talk about 7% inter­est. A per­cent­age is just a frac­tion with 100 at the bot­tom, and that makes amounts easy to com­pare. In this les­son you will learn what a per­cent­age is, and how to change per­cent­ages into frac­tions and dec­i­mals and back again. The next lessons use these changes all the time.

What a per­cent­age is#

Sup­pose you got 20 marks out of 100. You can write this as a frac­tion:

$$\displaystyle \frac{20}{100} = 20\%$$

The sign % is read "per cent" or "per­cent­age". "Per cent" means "out of a hun­dred". So 20% means 20 out of every 100.

Rule 1. For any num­ber $x$,

$$\displaystyle x\% = \frac{x}{100}$$

Here are some com­mon ones:

$$\displaystyle 5\% = \frac{5}{100} = \frac{1}{20} \qquad 10\% = \frac{10}{100} = \frac{1}{10} \qquad 15\% = \frac{15}{100} = \frac{3}{20}$$

$$\displaystyle 25\% = \frac{25}{100} = \frac{1}{4} \qquad 50\% = \frac{50}{100} = \frac{1}{2} \qquad 75\% = \frac{75}{100} = \frac{3}{4}$$

A whole is 100%#

Rule 2. One whole is 100%, because $\displaystyle 1 = \frac{100}{100}$.

So half of some­thing is 50%, and a quar­ter of it is 25%.

Frac­tions you should know#

Here is a table of frac­tions with 1 on top. Learn them. They make later work much quicker. The sign $\approx$ means "is about".

The dots mean that the dig­its go on for ever.

Short­cuts for the table#

You do not have to work out every row from the start. Some rows come from oth­ers by halv­ing.

• $\displaystyle \frac{1}{8}$ is half of $\displaystyle \frac{1}{4}$. Half of 25% is 12.5%.
• $\displaystyle \frac{1}{6}$ is half of $\displaystyle \frac{1}{3}$. Half of $\displaystyle 33\frac{1}{3}\%$ is $\displaystyle 16\frac{2}{3}\%$.
• $\displaystyle \frac{1}{9}$ is one third of $\displaystyle \frac{1}{3}$. A third of $\displaystyle 33\frac{1}{3}\%$ is $\displaystyle 11\frac{1}{9}\%$.
• $\displaystyle \frac{1}{12}$ is half of $\displaystyle \frac{1}{6}$. Half of $\displaystyle 16\frac{2}{3}\%$ is $\displaystyle 8\frac{1}{3}\%$.
• $\displaystyle \frac{1}{14}$ is half of $\displaystyle \frac{1}{7}$. Half of $\displaystyle 14\frac{2}{7}\%$ is $\displaystyle 7\frac{1}{7}\%$.
• $\displaystyle \frac{1}{16}$ is half of $\displaystyle \frac{1}{8}$. Half of 12.5% is 6.25%.

A pat­tern in sev­enths#

Divide 1 by 7 by long divi­sion and you get $0.142857142857...$ The six dig­its 142857 repeat for ever. Now look at the other sev­enths.

Every one uses the same six dig­its, 1, 4, 2, 8, 5, 7, in the same cir­cu­lar order. Each one just starts at a dif­fer­ent place. Once you know $\displaystyle \frac{1}{7}$, you know them all.

Chang­ing from one form to another#

1. A per­cent­age into a frac­tion. Divide by 100 and can­cel.

$$\displaystyle x\% = \frac{x}{100} \qquad 35\% = \frac{35}{100} = \frac{7}{20}$$

2. A frac­tion into a per­cent­age. Mul­ti­ply by 100 and put the % sign.

$$\displaystyle \frac{a}{b} = \left(\frac{a}{b} \times 100\right)\%$$

For exam­ple, $\displaystyle \frac{1}{11} = \left(\frac{1}{11} \times 100\right)\% = \frac{100}{11}\% = 9\frac{1}{11}\%$, which is about 9.09%. And $\displaystyle \frac{1}{13} = \frac{100}{13}\%$, which is about 7.69%.

3. A ratio into a per­cent­age. Write the ratio as a frac­tion first.

$$\displaystyle a : b = \frac{a}{b} = \left(\frac{a}{b} \times 100\right)\%$$

For exam­ple, $\displaystyle 1 : 2 = \frac{1}{2} = \left(\frac{1}{2} \times 100\right)\% = 50\%$.

4. A dec­i­mal into a per­cent­age. Mul­ti­ply by 100.

$$0.5 = (0.5 \times 100)\% = 50\%$$

Mul­ti­ply­ing by 100 moves the dec­i­mal point two places to the right. So $0.36 = 36\%$ and $1.25 = 125\%$.

5. A per­cent­age into a dec­i­mal. Divide by 100. This moves the point two places to the left. So $45\% = 0.45$ and $8\% = 0.08$.

Exam­ple 1#

Write $\displaystyle \frac{3}{8}$ as a per­cent­age.

$$\displaystyle \frac{3}{8} = \left(\frac{3}{8} \times 100\right)\% = \frac{300}{8}\% = 37.5\%$$

Or: $\displaystyle \frac{1}{8}$ is 12.5%, so $\displaystyle \frac{3}{8}$ is $3 \times 12.5 = 37.5$ per cent.

Exam­ple 2#

Write 12% as a frac­tion in its low­est terms, and as a dec­i­mal.

$$\displaystyle 12\% = \frac{12}{100} = \frac{3}{25}$$

As a dec­i­mal, $12\% = 0.12$.

Exam­ple 3#

Write the ratio $7 : 20$ as a per­cent­age.

$$\displaystyle 7 : 20 = \frac{7}{20} = \left(\frac{7}{20} \times 100\right)\% = 35\%$$

Exam­ple 4#

Write $\displaystyle 2\frac{1}{4}$ as a per­cent­age.

$$\displaystyle 2\frac{1}{4} = \frac{9}{4} = \left(\frac{9}{4} \times 100\right)\% = 225\%$$

A per­cent­age can be more than 100. It just means more than one whole.

Prac­tice#

1. Write 35% as a frac­tion in its low­est terms.
2. Write 4% as a frac­tion in its low­est terms.
3. Write $\displaystyle \frac{3}{5}$ as a per­cent­age.
4. Write $\displaystyle \frac{7}{8}$ as a per­cent­age.
5. Write the ratio $3 : 4$ as a per­cent­age.
6. Write 0.36 as a per­cent­age.
7. Write 1.25 as a per­cent­age.
8. Write $\displaystyle \frac{2}{3}$ as a per­cent­age.
9. Write 45% as a dec­i­mal.
10. Write $\displaystyle \frac{1}{16}$ as a per­cent­age and as a dec­i­mal.

Answers#