Converting Fractions to Decimals and Back

A fraction is a division

The fraction bar means divide. That single fact does most of the work here.

$$\frac{3}{4} = 3 \div 4 = 0.75$$

Three whole things shared between four people gives each person $0.75$ of a thing. The fraction and the decimal are two spellings of one number.

Method 1 — rewrite the bottom as 10, 100 or 1000

Decimal places are tenths, hundredths and thousandths. So if you can turn the bottom number into $10$, $100$ or $1000$ using equivalent fractions, you can read the decimal straight off.

$$\frac{3}{4} = \frac{3 \times 25}{4 \times 25} = \frac{75}{100} = 0.75$$ $$\frac{2}{5} = \frac{4}{10} = 0.4 \qquad \frac{7}{20} = \frac{35}{100} = 0.35$$

No division needed. This is the quickest route whenever the bottom number is $2$, $4$, $5$, $8$, $10$, $20$, $25$, $50$ or $100$.

Method 2 — divide

When the bottom number will not cooperate, divide the top by the bottom and keep going past the decimal point.

For $\tfrac{3}{8}$: $3 \div 8$. Since $8$ does not go into $3$, write $3.000$ and divide.

$$3 \div 8 = 0.375$$

For $\tfrac{2}{3}$: $2 \div 3$ gives $0.6666\ldots$, and the $6$s never stop. Write it as $0.\overline{6}$, where the bar marks the digits that repeat.

Why some decimals repeat

This is worth understanding rather than accepting.

Our place-value system is built on $10$, and $10 = 2 \times 5$. So every decimal place is made of twos and fives.

A fraction turns into a terminating decimal exactly when its simplified bottom number is built only from twos and fives.

Fraction	Bottom number as primes	Decimal
$\tfrac{3}{8}$	$2 \times 2 \times 2$	$0.375$, stops
$\tfrac{7}{20}$	$2 \times 2 \times 5$	$0.35$, stops
$\tfrac{2}{3}$	$3$	$0.\overline{6}$, repeats
$\tfrac{5}{6}$	$2 \times 3$	$0.8\overline{3}$, repeats

The $3$ in the bottom of $\tfrac{5}{6}$ is what ruins it. No matter how many zeros you add, $3$ will never divide a power of ten evenly.

Going back: decimal to fraction

Read the place value of the last digit, then simplify.

$$0.6 = \frac{6}{10} = \frac{3}{5}$$ $$0.35 = \frac{35}{100} = \frac{7}{20}$$ $$0.125 = \frac{125}{1000} = \frac{1}{8}$$

One decimal place means tenths, two means hundredths, three means thousandths. Count the places, write that many zeros after the $1$, then simplify.

Worth memorising

These come up constantly, and knowing them saves real time in tests.

Fraction	Decimal	Percent
$\tfrac{1}{2}$	$0.5$	$50\%$
$\tfrac{1}{4}$	$0.25$	$25\%$
$\tfrac{3}{4}$	$0.75$	$75\%$
$\tfrac{1}{5}$	$0.2$	$20\%$
$\tfrac{1}{8}$	$0.125$	$12.5\%$
$\tfrac{1}{3}$	$0.\overline{3}$	$33.\overline{3}\%$

Worked examples

Common mistakes

Practice problems