How do you turn a fraction into a decimal with long division?
Put the numerator inside the division bracket and the denominator outside, add a decimal point and zeros, and divide. Each remainder, times 10, becomes the next number you divide.
Here is how to convert a fraction to a decimal with 3/8, the same steps the calculator lays out:
- 3 ÷ 8: 8 doesn't go into 3, so write 0. and bring down a zero to make 30.
- 30 ÷ 8 = 3, remainder 6. Bring down a zero to make 60.
- 60 ÷ 8 = 7, remainder 4. Bring down a zero to make 40.
- 40 ÷ 8 = 5, remainder 0. The division stops.
- Read the digits: 3/8 = 0.375.
Why do some decimals end and others repeat?
It depends on the denominator once the fraction is in lowest terms. If the denominator's only prime factors are 2 and 5, the decimal ends. If any other prime appears, such as 3 or 7, the decimal repeats.
You don't have to divide to know which kind you'll get. The reason is that our number system is based on 10, and 10 = 2 × 5. A denominator built from 2s and 5s can always be scaled up to a power of 10. For example, 3/8 = 375/1000. No power of 10 is a multiple of 3 or 7, so fractions like 1/3 and 1/7 never come out even.
| Fraction | Denominator factors | Decimal |
|---|---|---|
| 3/8 | 2 × 2 × 2 | 0.375 (ends) |
| 7/20 | 2 × 2 × 5 | 0.35 (ends) |
| 11/40 | 2 × 2 × 2 × 5 | 0.275 (ends) |
| 1/3 | 3 | 0.333… (repeats) |
| 5/12 | 2 × 2 × 3 | 0.41666… (repeats) |
| 1/7 | 7 | 0.142857… (repeats) |
The prime factorization calculator will break any denominator into primes so you can tell in advance.
Should you simplify the fraction first?
Yes, if you want to predict whether the decimal ends. The 2s-and-5s rule only works on a fraction in lowest terms.
Take 6/15. The denominator 15 has a factor of 3, so you might expect a repeating decimal. But 6/15 reduces to 2/5, and 5 is fine, so the decimal is 0.4 and it ends. The division itself gives the same answer either way. Simplifying only matters for the prediction.
How does the calculator show a repeating decimal?
It runs the long division until a remainder shows up a second time. From that point the digits must cycle, so the calculator marks that stretch as the repeating block.
A repeat is guaranteed because there are only so many possible remainders. When you divide by 7, every remainder is somewhere from 1 to 6, so within six steps one of them has to come back.
For 1/7, the remainders run through 1, 3, 2, 6, 4, 5 and then 1 again. The result is 0.142857142857… with 142857 repeating. For 5/12, the digits start 0.41 and then 6 repeats, giving 0.41666…. In textbook notation you can write a bar over the repeating digits.
How do you convert a mixed number to a decimal?
Keep the whole number in front and convert only the fraction part. For 2 1/3, the 1/3 becomes 0.333…, so the answer is 2.333….
An improper fraction works the same way with plain division. 7/4 is 7 ÷ 4 = 1.75. Negative fractions keep their sign, so −5/8 = −0.625.
If a mixed number comes from a fraction problem, you can do the whole thing in one place. The fraction calculator shows the decimal form of every answer, and this page gives you the long division behind it.
Is there a shortcut to convert a fraction to a decimal?
Yes, when the denominator can be scaled to 10, 100, or 1000. Multiply the top and bottom by the same number until the bottom is a power of 10, then read off the decimal.
A few of these are worth memorizing. Know that 1/2 = 0.5, 1/4 = 0.25, 3/4 = 0.75, 1/5 = 0.2, and 1/8 = 0.125. Each one has a denominator made only of 2s and 5s, and you'll see them in prices, measurements, and test questions all the time.
It skips the long division entirely when the numbers cooperate. For 7/20, multiply both by 5 to get 35/100, which is 0.35. For 3/16, multiply both by 625 to get 1875/10000, which is 0.1875. When this gets awkward, long division is faster.
How do you round a repeating decimal?
Write out one or two more digits than you need, then round as usual. For 5/6 = 0.8333…, two decimal places gives 0.83.
Read the question to see how many places it wants. Rounding is fine for a final answer in a word problem, like money or measurements. If you still have more steps to do, keep the fraction instead. Rounded decimals add small errors that can grow over several steps, while the fraction stays exact.
What mistakes should you watch for?
The most common mistake is dividing the wrong way. 3/8 means 3 ÷ 8, not 8 ÷ 3, so the answer has to be less than 1. Any fraction with a smaller top than bottom gives a decimal between 0 and 1.
Another is stopping a repeating decimal too early and treating it as exact. Write the dots, draw the bar, or keep the fraction. 1/3 is not 0.33; that is a rounded value. Also watch for a zero right after the decimal point. 1/12 is 0.08333…, and dropping the 0 changes the value.