Converting Fractions, Decimals and Percentages

A fraction, a decimal and a percentage can show exactly the same amount. For example, \frac{1}{2}, 0.5 and 50\% are all one half. Converting changes how the amount is written, not how big it is.

Per cent means “out of 100”. So 35\% means \frac{35}{100}, which is 35 hundredths. As a decimal that is 0.35: 3 tenths and 5 hundredths.

In the exam, this topic often opens a paper with a 1 mark question: write a decimal as a percentage, a percentage as a decimal or a fraction, or a fraction as a decimal. Questions worth 1 or 2 marks ask you to write a mixture of fractions, decimals and percentages in order of size, or to write a decimal as a fraction in its simplest form. Later questions may ask you to write one amount as a percentage of another.

To change a decimal to a percentage, multiply by 100. To change a percentage to a decimal, divide by 100.

Multiplying by 100 moves every digit two places to the left. So 0.3 = 30\%, not 3\%, and 0.06 = 6\%, not 60\%. If it helps, write 0.3 as 0.30 first: that is 30 hundredths.

One whole is 100\%, so an amount bigger than one whole is more than 100\%. For example, 1.25 = 125\% and 140\% = 1.4

If you can, make an equivalent fraction with denominator 10 or 100. Multiply the top and the bottom by the same number, so the value does not change. For example, \frac{3}{5} = \frac{6}{10} = 0.6 = 60\%

A method that always works is to divide, because the fraction line means “divided by”. So \frac{1}{8} = 1 \div 8 = 0.125. On a calculator paper you can simply type the division in.

These are worth knowing by heart.

• \frac{1}{2} = 0.5 = 50\%
• \frac{1}{4} = 0.25 = 25\% and \frac{3}{4} = 0.75 = 75\%
• \frac{1}{5} = 0.2 = 20\% and \frac{1}{10} = 0.1 = 10\%
• \frac{1}{3} = 0.333\ldots and \frac{2}{3} = 0.666\ldots, where the digit goes on for ever.

Look at the last decimal place. One decimal place means tenths, two mean hundredths and three mean thousandths. So 0.7 = \frac{7}{10}, 0.29 = \frac{29}{100} and 0.051 = \frac{51}{1000}

A percentage always goes over 100, never over 10 or 1000. For example, 41\% = \frac{41}{100}

Then simplify. Divide the top and the bottom by their highest common factor (HCF). The fraction is in its simplest form when no number bigger than 1 divides into both.

To write 7 out of 20 as a percentage, first write it as a fraction: the part over the total, \frac{7}{20}. Then make the denominator 100: \frac{7}{20} = \frac{35}{100} = 35\%

On a calculator paper you can divide instead: 7 \div 20 = 0.35, which is 35\%

To compare fractions, decimals and percentages, change them all into the same form. Decimals are usually easiest. Work to at least 2 decimal places, or 3 if two numbers are close.

Fill gaps with zeros, so every decimal has the same number of places. 0.5 = 0.50, which is bigger than 0.45, even though 5 is smaller than 45.

In your answer, write the numbers as they were given, not your decimals. Check whether the question says start with the smallest or the largest.

Changing to the same form also lets you add amounts and find the rest. If 0.3 of a class walk to school and 45\% come by bus, write 0.3 as 30\%. Together that is 30\% + 45\% = 75\%, so the rest is 100\% - 75\% = 25\%

• Moving the digits only one place. Multiply or divide by 100, not 10: 0.3 = 30\%, not 3\%
• Just removing the % sign. As a decimal, 35\% is 0.35, not 35 or 0.35\%
• Giving the count as the percentage. 27 out of 60 is not 27\%. Write \frac{27}{60}, then change it to hundredths.
• Answering in the wrong form. If the question asks for a decimal, give a decimal: 9\% = 0.09, not \frac{9}{100}
• Comparing the digits, not the values. Fill in zeros first: 0.50 is bigger than 0.45
• Ordering fractions by the bottom number. Change every number into a decimal first, then check whether to start with the smallest.