Fraction to Decimal Calculator
This fraction to decimal calculator converts a fraction or mixed number to a decimal and tells you whether it stops or repeats, marking any repeating block in brackets, so 1/7 comes back as 0.(142857). You also get the value rounded to four decimal places and the percentage.
3/8, or 2 3/8 for a mixed number.How to use the fraction to decimal calculator
- Type the fraction as
3/8, with a slash between the numerator and the denominator. - For a mixed number, put a space before the fraction part:
2 3/8. - Read the decimal at the top. If it recurs, the repeating digits appear inside brackets.
- Below that you get the fraction in lowest terms, the value rounded to 4 decimal places, the percentage, whether it terminates or recurs, and the repeating block on its own.
Results appear as you type. A negative numerator is fine, and a negative denominator is tidied up automatically, so 3/-4 is treated as −3/4. A denominator of zero gets a message, because dividing by zero has no answer.
The working shows the denominator broken into its prime factors, because that is what decides which kind of decimal you are about to get. That reasoning is the next section.
A fraction is a division
There is no conversion to learn. The line in a fraction has always meant divide, so turning a fraction into a decimal is just carrying out the division that was written there all along. Top divided by bottom, in that order.
That order matters and is the most common thing to get backwards. 3/8 is 3 ÷ 8 = 0.375, a number less than 1, which is right for a fraction below 1. Doing 8 ÷ 3 gives 2.666…, the decimal for 8/3 instead.
Example: 3/8
Divide 3 by 8. Since 8 does not go into 3, the answer starts 0. and you work into the decimal places by adding zeros.
The remainder hits zero, so it ends: 3/8 = 0.375, which is 37.5%.
Example: 5/16 and 3/40
Both stop. Note 3/40 — the leading zero after the point is a real digit, and 0.75 would be ten times too big.
Because it is only a division, the same job can be done by hand with the long division calculator, which prints the paper layout digit by digit.
Which fractions stop and which repeat
You can tell before dividing anything. Cancel the fraction to its lowest terms, then break the denominator into prime factors.
The reason is that a decimal is a fraction over a power of ten, and 10 = 2 × 5. So a denominator built only from 2s and 5s can always be scaled up into a power of ten, and once it is, you can read the decimal straight off:
Now try it with 1/6. The 6 contains a 3, and no whole number multiplies 6 into 10, 100, 1,000 or any other power of ten, because a 3 can never appear in a number made only of 2s and 5s. The division never closes, so 1/6 = 0.1666…. Same for 1/7: seven divides nothing of the form 10, 100, 1,000, so 1/7 = 0.142857142857…
Cancel first, always. 9/24 looks like it must recur because 24 contains a 3, but the fraction cancels to 3/8 and terminates at 0.375. The rule applies to the simplified denominator, never the one you happened to write down.
| Fraction | Lowest terms | Denominator factors | Decimal |
|---|---|---|---|
| 3/8 | 3/8 | 2 × 2 × 2 | 0.375, terminates |
| 5/16 | 5/16 | 2 × 2 × 2 × 2 | 0.3125, terminates |
| 9/24 | 3/8 | 2 × 2 × 2 | 0.375, terminates |
| 3/40 | 3/40 | 2 × 2 × 2 × 5 | 0.075, terminates |
| 1/6 | 1/6 | 2 × 3 | 0.1(6), recurs |
| 7/12 | 7/12 | 2 × 2 × 3 | 0.58(3), recurs |
| 4/11 | 4/11 | 11 | 0.(36), recurs |
| 1/7 | 1/7 | 7 | 0.(142857), recurs |
Writing a recurring decimal
There are three standard ways to mark the digits that repeat, and they all mean the same thing.
- Brackets. Put the repeating block in parentheses: 0.(142857). This is what the calculator prints, because it survives copying and pasting.
- A bar. Draw a line over the block: 0.142857 with a bar across all six digits. Common in US textbooks.
- Dots. Put a dot over the first and last digit of the block, so 0.142857 gets a dot on the 1 and a dot on the 7. A block of one digit gets a single dot.
Where the block starts matters as much as what it contains. In 1/6 = 0.1666… the 1 does not repeat and the 6 does, so it is written 0.1(6), never 0.(16). Get that wrong and you have written 0.161616…, which is 16/99, a different number.
How long can the block be? At most one digit shorter than the denominator. Dividing by 7, the only possible remainders are 1 to 6, so within six steps a remainder has to repeat and the block closes. That upper limit is reached by 7, whose block is exactly six digits. Others fall short: 11 could in principle run to ten digits but only uses two, and 3 and 9 use one.
| Fraction | Decimal | Block | Fraction | Decimal | Block |
|---|---|---|---|---|---|
| 1/9 | 0.(1) | 1 | 1/7 | 0.(142857) | 142857 |
| 2/9 | 0.(2) | 2 | 2/7 | 0.(285714) | 285714 |
| 3/9 | 0.(3) | 3 | 3/7 | 0.(428571) | 428571 |
| 4/9 | 0.(4) | 4 | 4/7 | 0.(571428) | 571428 |
| 5/9 | 0.(5) | 5 | 5/7 | 0.(714285) | 714285 |
| 8/9 | 0.(8) | 8 | 6/7 | 0.(857142) | 857142 |
The ninths simply repeat their numerator, which is the same 9s pattern that runs in reverse when you go from a decimal back to a fraction. The sevenths are more interesting: every one of them uses the digits 142857 in the same cyclic order, starting at a different point. Note also that 3/9 cancels to 1/3 and 0.(3) is the familiar 0.333…
Mixed numbers
A mixed number is a whole number and a fraction added together, so 2 3/8 means 2 + 3/8. There are two routes to its decimal, and they agree.
Keep the whole part aside. Convert only the fraction, then put the whole number back in front. 3/8 is 0.375, so 2 3/8 is 2.375. This is usually the quicker route in your head, because the fraction part is small and familiar.
Or make it improper first. Multiply the whole number by the denominator, add the numerator, and keep the denominator. That is what the calculator does before dividing.
For a negative mixed number, the sign covers the whole thing. −2 3/8 is −(2 + 3/8) = −2.375, not −2 + 0.375. Converting between mixed numbers and top-heavy fractions on its own is the job of the improper fraction calculator.
Rounding a recurring decimal
A recurring decimal has no last digit, so any decimal you write down is a rounded version. The results give the value to four decimal places alongside the exact bracketed form, and the two answer different questions.
Look at 2/3. Rounded to two places it is 0.67, not 0.66. The digits after the cut are 6666…, which is more than half, so the last kept digit goes up. Chopping instead of rounding is a common slip and it always makes the number too small.
When the decimal is the final answer, match the precision to the job: money to 2 places, most school work to 2 or 3, and lab measurements to whatever the instrument justified. To read the same value as a percentage instead, the decimal to percent calculator shifts the point for you.
Common mistakes
- Dividing bottom by top. 3/8 is 3 ÷ 8 = 0.375. Doing 8 ÷ 3 gives 2.666…, which is the decimal for a completely different fraction.
- Applying the terminating rule before cancelling. 9/24 has a 3 in the denominator but reduces to 3/8 and terminates. Simplify first, then look at the factors.
- Marking the wrong repeating block. 1/6 is 0.1(6), not 0.(16). The first digit after the point does not repeat.
- Truncating instead of rounding. 2/3 to two places is 0.67. Writing 0.66 throws away a digit rather than rounding it.
- Treating a rounded decimal as exact. 0.333 is not 1/3, and multiplying it up magnifies the gap.
- Losing a leading zero. 3/40 is 0.075. Writing 0.75 is out by a factor of ten.
- Converting only the fraction part of a mixed number. 2 3/8 is 2.375, not 0.375.
- Assuming a long decimal means an error. 1/7 really does need six digits before it repeats. That is the fraction behaving normally, not a fault in the arithmetic.
Frequently asked questions
What is 3/8 as a decimal?
3/8 = 0.375. Divide 3 by 8: 8 into 30 goes 3 times with 6 left, 8 into 60 goes 7 times with 4 left, 8 into 40 goes 5 times exactly. It terminates because 8 is 2 × 2 × 2, all twos. As a percentage it is 37.5%.
What is 1/7 as a decimal?
1/7 = 0.(142857), meaning 0.142857142857… forever. Seven has no factors of 2 or 5, so the division never closes, and the block runs the full six digits — the longest a denominator of 7 allows. Rounded to four places it is 0.1429, or about 14.29%.
How do you turn a fraction into a decimal?
Divide the numerator by the denominator. The fraction bar is a division sign, so 5/16 means 5 ÷ 16 = 0.3125. If the top is smaller than the bottom the answer starts with 0., and you keep adding zeros after the point and dividing until the remainder reaches zero or starts repeating.
Which fractions give a terminating decimal?
Only those whose denominator, after cancelling, is built from 2s and 5s alone. 1/8, 5/16 and 3/40 all stop. 1/6 and 1/7 do not, because 6 contains a 3 and 7 is a prime other than 2 or 5. The reason is that 10 = 2 × 5, so only those factors scale up into a power of ten.
What do the brackets in 0.(36) mean?
They mark the digits that repeat forever, so 0.(36) is 0.36363636… It is the same thing as a bar drawn over the 36, or dots above the 3 and the 6. Brackets are used here because they copy and paste cleanly. That value is 4/11.
How do you convert a mixed number to a decimal?
Convert the fraction part and put the whole number back in front, or turn the whole thing into an improper fraction first. Both give the same answer: 2 3/8 is 2 + 0.375 = 2.375, and 19/8 = 2.375. For a negative, the sign applies to the whole value, so −2 3/8 = −2.375.
How long can a repeating block be?
At most one digit fewer than the denominator. Dividing by 7 there are only six possible remainders, so the pattern must return within six steps, and 1/7 uses all six. Denominators often use fewer: 1/11 repeats in a block of two, and 1/9 in a block of one.
What is 7/12 as a decimal?
7/12 = 0.58(3), that is 0.58333… The 12 factors as 2 × 2 × 3, and the 3 forces it to recur. The 5 and 8 come out before the pattern settles, so only the final 3 repeats. To four decimal places it is 0.5833.
Why do the sevenths all use the same digits?
1/7 through 6/7 all repeat the block 142857, each starting at a different point in the cycle: 2/7 = 0.(285714), 3/7 = 0.(428571), and so on. Dividing by 7 cycles through the same six remainders in the same order, so the digits it produces are the same six, rotated.
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