Fractions to decimals, and decimals back to fractions

The fractions to decimals worth knowing by heart

Some fraction and decimal pairs come up so often that it's worth knowing them without working them out. Government guidance for teachers expects children in Year 5 to recall these, starting with the unit fractions and then their multiples up to 1:

Fraction	Decimal
1⁄2	0.5
1⁄4	0.25
3⁄4	0.75
1⁄5	0.2
2⁄5	0.4
1⁄10	0.1
3⁄10	0.3
1⁄100	0.01
1⁄8	0.125

Eighths are a Year 6 example. The others follow on: ³⁄₅ is three lots of 0.2, so ³⁄₅ = 0.6. In the same way, ⁷⁄₁₀ = 0.7.

A shaded hundred square shows why. Shade a quarter of a 10 by 10 grid and you've shaded 25 of the 100 squares. That's 25 hundredths, which is written 0.25.

How to change a fraction to a decimal

There are three ways to change a fraction to a decimal, and children use the easiest one for the numbers in front of them.

1. Tenths and hundredths: read them off

If the denominator is 10 or 100, the fraction already tells you the decimal. Seven tenths is 0.7, and twenty-three hundredths is 0.23. A place value chart shows where the digits go:

0.23 in a place value chart

Ones	Tenths	Hundredths
0	2	3

Watch the hundredths with only one digit: ⁷⁄₁₀₀ is 0.07, with a zero in the tenths place. It isn't 0.7.

2. Make the denominator 10 or 100

If the denominator goes into 10 or 100, use equivalent fractions to turn it into tenths or hundredths, multiplying the top and bottom by the same number. Here are two examples:

• ³⁄₅ = ⁶⁄₁₀, so ³⁄₅ = 0.6.
• ⁷⁄₂₅ = ²⁸⁄₁₀₀, so ⁷⁄₂₅ = 0.28.

3. Divide the numerator by the denominator

This works for any fraction. In Year 6, children learn that a fraction is a division: ³⁄₈ means 3 ÷ 8. They work it out with short division (the bus stop method), writing zeros after the decimal point so there's something to divide.

Worked example: changing 3/8 to a decimal

The national curriculum uses this example, 3 ÷ 8. Write 3 as 3.000 under the bus stop, with the decimal point in the answer lined up above the one in 3.000. Then divide one digit at a time, as usual:

1. 8 doesn't go into 3, so write 0 above it, then the decimal point. Carry the 3 ones as 30 tenths.
2. 30 ÷ 8 = 3 remainder 6. Write 3 in the tenths place and carry the 6 as 60 hundredths.
3. 60 ÷ 8 = 7 remainder 4. Write 7 in the hundredths place and carry the 4 as 40 thousandths.
4. 40 ÷ 8 = 5, with nothing left over. Write 5 in the thousandths place.

So 3 ÷ 8 = 0.375, and ³⁄₈ = 0.375. Our bus stop diagram only draws whole numbers, so this one is in words.

A quick check: ³⁄₈ is a bit less than a half, and 0.375 is a bit less than 0.5.

Worked example: fractions bigger than 1

An improper fraction gives a decimal bigger than 1. For ⁵⁄₄, divide 5 by 4: 5 ÷ 4 = 1.25. Or think of it as a mixed number: ⁵⁄₄ = 1 ¹⁄₄, which is 1 and 0.25, so ⁵⁄₄ = 1.25.

Mixed numbers work the same way: the whole number stays in front of the decimal point. 2 ³⁄₅ = 2.6, and 4 ¹⁄₂ = 4.5.

How to change a decimal to a fraction

To express a decimal as a fraction, read it using place value, write it as tenths, hundredths or thousandths, then simplify. The place of the last digit gives the denominator:

• 0.6 is 6 tenths, so 0.6 = ⁶⁄₁₀. Simplified, ⁶⁄₁₀ = ³⁄₅.
• 0.71 is 71 hundredths, so 0.71 = ⁷¹⁄₁₀₀. This is the national curriculum's own example, and it doesn't simplify.
• 0.35 is 35 hundredths, so 0.35 = ³⁵⁄₁₀₀. Divide the top and bottom by 5: ³⁵⁄₁₀₀ = ⁷⁄₂₀.
• 0.125 is 125 thousandths, so 0.125 = ¹²⁵⁄₁₀₀₀. That simplifies all the way down: 0.125 = ¹⁄₈.

For a decimal bigger than 1, keep the whole number. 2.4 is 2 and 4 tenths, so 2.4 = 2 ⁴⁄₁₀, which simplifies to 2 ²⁄₅. Simplifying fractions has more on that last step.

Saying the decimal in words helps. Government guidance points out that children usually say 0.35 as "nought point three five", and they need to practise saying it as "thirty-five hundredths" too. Once they say it that way, the fraction is already there.

Recurring decimals: fractions that never stop

Some fractions give a decimal that goes on for ever, called a recurring decimal. ¹⁄₃ is 1 ÷ 3: the short division keeps leaving 1 behind, so the 3s never end, 0.333... and on and on. ²⁄₃ gives 0.666... in the same way.

In Year 6, children learn to round these, usually to three decimal places: ¹⁄₃ is about 0.333, and ²⁄₃ is about 0.667. Changing a recurring decimal back into an exact fraction isn't taught at primary school: it's part of GCSE maths.

Common mistakes to watch for

• Writing the digits side by side. ³⁄₄ isn't 3.4 and ¹⁄₅ isn't 1.5. A fraction less than 1 is a decimal less than 1.
• Dividing the wrong way round. ³⁄₈ is 3 ÷ 8, not 8 ÷ 3. Top divided by bottom.
• Missing the zero in hundredths. ⁷⁄₁₀₀ is 0.07, not 0.7.
• Thinking 0.5 is a fifth. 0.5 is five tenths, which is a half. A fifth is 0.2.
• Thinking more digits means bigger. 0.125 is smaller than 0.5, even though it has more digits.
• Leaving the fraction unsimplified when asked. ²⁵⁄₁₀₀ is right, but the question may want ¹⁄₄.

How to help at home

• Use money. A pound is 100 pence, so 0.25 of a pound is 25p, which is a quarter. 0.1 of a pound is 10p, a tenth. Children often find decimals easiest with money.
• Use measures. Half a litre is 0.5 litres, and a quarter of a metre is 0.25 metres. Look at bottles, rulers and recipes together.
• Say both versions. Read 0.6 aloud as "six tenths" and "nought point six", and ask your child to say both.
• Quiz the common pairs. A few at a time: "What's ³⁄₄ as a decimal?" and "What's 0.2 as a fraction?"
• Keep division facts quick. The division method leans on the times tables, especially the 8 times table for eighths.

When do children learn fractions to decimals?

Children start in Year 4 and finish the primary part in Year 6. In Year 4 maths they write tenths and hundredths as decimals, and learn the decimals for a quarter, a half and three quarters. In Year 5 maths they write decimals as fractions, work with thousandths and recall the common pairs. In Year 6 maths they see a fraction as a division and work out decimals like 0.375 for ³⁄₈, which leads on to fractions, decimals and percentages together.

If fractions still feel shaky, start with what a fraction is and why the parts must be equal.

Fractions to decimals questions to try

1. Write ⁹⁄₁₀ as a decimal.
2. Write ³⁄₁₀₀ as a decimal.
3. Write ⁴⁄₅ as a decimal.
4. Write ⁷⁄₄ as a decimal.
5. Write 0.45 as a fraction in its simplest form.
6. Write 0.625 as a fraction in its simplest form.

Answers

1. 0.9
2. 0.03 (three hundredths, with a zero in the tenths place)
3. 0.8 (⁴⁄₅ = ⁸⁄₁₀)
4. 1.75 (⁷⁄₄ = 1 ³⁄₄, and 1 ³⁄₄ = 1.75)
5. ⁹⁄₂₀ (0.45 = ⁴⁵⁄₁₀₀, then divide the top and bottom by 5)
6. ⁵⁄₈ (0.625 = ⁶²⁵⁄₁₀₀₀, then divide the top and bottom by 125)

For the rest of what children learn in these years, see the KS2 maths guide. Place value explains tenths, hundredths and thousandths in more detail.