Converting Decimals to Fractions

Change a decimal into a fraction by counting the places after the point, then simplify, including thousandths and decimals bigger than 1.

In the exam for Edexcel IGCSE Maths 4MA1, Cambridge IGCSE Maths 0580, Cambridge O Level Maths 4024 · 26 real past-paper questions below

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What a Decimal Is as a Fraction

Each place after the point has a name: tenths, hundredths, thousandths. The last digit tells you the bottom of the fraction.

Formula

decimal = \frac{the digits after the point}{1 0 ^{n}}

$n$ is the number of places after the point. $1 0^{n}$ is 1 with $n$ zeros after it: $1 0^{1} = 10$ , $1 0^{2} = 100$ , $1 0^{3} = 1000$ .

Decimal	Fraction
0.7 (one place)	$\frac{7}{10}$
0.53 (two places)	$\frac{53}{100}$
0.125 (three places)	$\frac{125}{1000}$

Rule

One zero on the bottom for each place after the point.

Key idea

This is the reverse of the lesson Converting Fractions to Decimals. There you divide; here you read the places.

Read It Off

The most common 1-mark question: write 0.53 as a fraction. Count the places after the point, and put the digits over 10, 100 or 1000.

Worked example

1
$0. 7 = \frac{7}{10}$ — There is one place after the point, so the 7 is tenths. One place means 10 on the bottom.
2
$0. 53 = \frac{53}{100}$ — There are two places, so the last digit is in the hundredths place. Put all the digits, 53, over 100.
3
$0. 09 = \frac{9}{100}$ — Still two places, so still over 100. The 0 only holds the tenths place, so 09 on top is just 9.

Common mistake

Writing 0.09 as $\frac{9}{10}$ . The 9 is in the second place, so it is 9 hundredths: $\frac{9}{100}$ .

Simplest Form

The 2-mark question: write a decimal as a fraction in its simplest form. Read it off, then divide the top and the bottom by the same number until you cannot go further.

Worked example

1
$0.84 = \frac{84}{100}$ — Two places after the point, so the bottom is 100. Write this line down: it can earn a mark on its own.
2
$= \frac{42}{50}$ — 84 and 100 are both even, so halve the top and the bottom. The fraction stays the same size.
3
$= \frac{21}{25}$ — 42 and 50 are still both even, so halve again.
4
$21 = 3 \times 7, 25 = 5 \times 5$ — No number except 1 goes into both 21 and 25, so you can stop. $\frac{21}{25}$ is the simplest form.

Key idea

10, 100 and 1000 are made only of 2s and 5s. So only two checks: is the top even? Halve. Does it end in 5 or 0? Divide by 5. Faster: 4 goes into 84 and 100, so divide by 4 in one go.

Key idea

Show the $\frac{84}{100}$ line. In a 2-mark question, one mark is for a correct fraction like $\frac{84}{100}$ or $\frac{42}{50}$ , even if you do not finish simplifying.

Common mistake

Stopping at $\frac{42}{50}$ . It is right but not simplest, so you lose a mark. Keep dividing until nothing goes into both.

Thousandths and Numbers Bigger Than 1

Three places after the point means over 1000. Then simplify: numbers ending in 5 or 0 share a 5.

Worked example

1
$0.125 = \frac{125}{1000}$ — There are three places, so the last digit is thousandths. The bottom is 1000.
2
$= \frac{25}{200}$ — 125 and 1000 both end in 5 or 0, so divide the top and the bottom by 5.
3
$= \frac{5}{40}$ — 25 and 200 still end in 5 or 0, so divide both by 5 again.
4
$= \frac{1}{8}$ — 5 and 40 share a 5 once more: 5 ÷ 5 = 1 and 40 ÷ 5 = 8. Nothing else goes into 1 and 8, so stop.

If the decimal is bigger than 1, keep the whole number and change only the part after the point.

Worked example

1
$2.4 = 2 + 0.4$ — Split the number at the point. The whole number 2 stays as it is.
2
$= 2 + \frac{4}{10}$ — 0.4 has one place, so it is 4 tenths.
3
$= 2 + \frac{2}{5}$ — 4 and 10 are both even, so halve both to simplify.
4
$= 2 \frac{2}{5}$ — Write it as a mixed number: a whole number and a fraction together.
5
$= \frac{12}{5}$ — If you need a top-heavy (improper) fraction: 2 wholes are 10 fifths, and 10 + 2 = 12 fifths.

Common mistake

Writing 0.125 as $\frac{125}{100}$ . That is bigger than 1. Three places needs 1000.

Zeros Straight After the Point

Exams like decimals such as 0.004. The zeros straight after the point are easy to miss, so count every one.

Worked example

1
$0. 006$ — Count the places: 0, 0, 6. That is three places, so the bottom is 1000.
2
$= \frac{6}{1000}$ — The zeros in front only hold places, so just the 6 goes on top.
3
$= \frac{3}{500}$ — 6 and 1000 are both even, so halve the top and the bottom.
4
$3 and 500$ — 3 does not go into 500 (5 + 0 + 0 = 5, not a multiple of 3), so $\frac{3}{500}$ is the simplest form.

Worked example

1
$0. 05 = \frac{5}{100}$ — Two places, so over 100. Only the 5 goes on top.
2
$= \frac{1}{20}$ — 5 and 100 both end in 5 or 0, so divide both by 5: 5 ÷ 5 = 1 and 100 ÷ 5 = 20.

Common mistake

Writing 0.006 as $\frac{6}{100}$ . That is 0.06. Three places after the point needs 1000 on the bottom.

Which Method? Read the Question

Count the places after the point first.

You see	You do
one or two places	over 10 or over 100
three places	over 1000, then simplify
in its simplest form	divide top and bottom until nothing goes into both
a number bigger than 1	keep the whole number, change the rest
a dot over a digit, or show that using algebra	Recurring Decimals
write a fraction as a decimal	Converting Fractions to Decimals
write it as a percentage	Introduction to Percentages

Common mistake

Leaving a decimal inside the fraction, like 7.3 over 10 for 0.73. A fraction needs whole numbers on top and bottom, so write $\frac{73}{100}$ . The decimal version scores 0.

Key idea

Check by going back: $\frac{21}{25} = \frac{84}{100}$ , which is 0.84 again.

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