How Do Fractions and Decimals Convert?

Fractions and decimals are two notations for the same thing. Which one is easier depends entirely on the job, which is why both survive and why converting between them comes up constantly.

Fraction to decimal

Divide the top number by the bottom number. That is the whole method.

• 1/2 is 1 ÷ 2 = 0.5
• 3/8 is 3 ÷ 8 = 0.375
• 5/16 is 5 ÷ 16 = 0.3125
• 2/3 is 2 ÷ 3 = 0.6666 recurring

A fraction is a division waiting to happen. The line between the numbers is the division sign, which is worth saying explicitly because a surprising number of people learn fractions without ever being told it.

Decimal to fraction

Read the decimal by its last place value, write it over that place value, then simplify.

0.375 has three decimal places, so the last digit sits in the thousandths. That makes it 375/1000.

Then simplify by dividing both numbers by their greatest common divisor. Here that is 125:

375 ÷ 125 = 3, and 1000 ÷ 125 = 8, giving 3/8.

For 0.6: one decimal place, so 6/10, which simplifies to 3/5. For 0.25: two places, so 25/100, which simplifies to 1/4.

Simplifying without hunting for the divisor

If the greatest common divisor is not obvious, divide both numbers by any common factor you can see and repeat until nothing divides both.

For 375/1000: both are divisible by 5, giving 75/200. Both again by 5, giving 15/40. Both again by 5, giving 3/8. Three easy steps beat one hard one.

Why some decimals never end

This is the part people find genuinely interesting once they see the answer.

Our number system is base ten, and ten factorises into 2 × 5. A fraction terminates as a decimal only when its denominator, once fully simplified, contains no prime factors other than 2 and 5.

• 1/8: eight is 2 × 2 × 2, so it terminates, at 0.125
• 1/20: twenty is 2 × 2 × 5, so it terminates, at 0.05
• 1/3: three is neither 2 nor 5, so it repeats, as 0.333 recurring
• 1/7: seven is neither, so it repeats, as 0.142857 recurring with a six-digit cycle

Nothing is wrong with a third. It is a perfectly exact quantity with a perfectly exact fractional representation. It simply has no exact representation in base ten, in the same way a base-three system would struggle to write a half.

Recurring decimals back to fractions

There is a neat trick worth knowing.

• One repeating digit goes over 9: 0.777 recurring is 7/9
• Two repeating digits go over 99: 0.1212 recurring is 12/99, which simplifies to 4/33
• Three go over 999, and so on

Which is why 0.999 recurring equals 9/9, which equals exactly 1. That is not an approximation or a rounding convention. The two expressions denote the same number, and the fraction rule is one of the cleaner ways to see why.

The conversions worth knowing by heart

Fraction	Decimal	Percentage
1/8	0.125	12.5%
1/5	0.2	20%
1/4	0.25	25%
1/3	0.333…	33.3%
3/8	0.375	37.5%
1/2	0.5	50%
5/8	0.625	62.5%
2/3	0.666…	66.7%
3/4	0.75	75%
7/8	0.875	87.5%

The eighths are the most useful set, because imperial measurements, spanner sizes, drill bits and timber thicknesses are all built on repeated halving. Knowing that 5/8 is 0.625 saves reaching for anything.

Improper fractions and mixed numbers

A fraction where the top exceeds the bottom, such as 11/4, converts the same way: 11 ÷ 4 = 2.75.

As a mixed number it is 2 and 3/4, found by asking how many whole 4s fit into 11, which is 2 with 3 left over. Going back, multiply the whole number by the denominator and add the numerator: 2 × 4 + 3 = 11, over 4.

Recipes and measurements tend to use mixed numbers; calculations are easier with improper ones.

When each notation is better

Fractions are exact, they handle thirds and sevenths without rounding, and they scale cleanly. Doubling a recipe calling for 3/4 of a cup is instant. They also preserve the relationship in the original: 3/4 tells you something about quarters that 0.75 does not.

Decimals are easier to compare at a glance and are what calculators, spreadsheets and measuring instruments use. Deciding whether 7/16 is larger than 3/7 takes a moment as fractions and no time at all once both are decimals, at 0.4375 and 0.4286.

Where the conversion goes wrong

Rounding too early. Converting 1/3 to 0.33 and then multiplying by 300 gives 99, not 100. The error looks trivial in one step and compounds through a calculation.

Keep fractions as fractions through the working and convert at the end, or carry enough decimal places that the rounding cannot accumulate into something visible. For money, that usually means working to four decimal places and rounding once, at the final figure.

Percentages, the third notation

A percentage is simply a fraction with a denominator of 100, which makes it a decimal shifted two places.

• 0.375 is 37.5 percent
• 3/8 is 37.5 percent
• 37.5 percent is 0.375

To convert a decimal to a percentage, multiply by 100. To go back, divide by 100. To turn a fraction into a percentage, convert to a decimal first and then multiply.

This is why the three notations are genuinely interchangeable and why a discount of a third, 0.333 and 33.3 percent all describe the same reduction. Being fluent between them is most of what everyday numeracy consists of.