Fractions to Decimals: Why Some Terminate and Others Repeat
October 21, 2026
To convert a fraction to a decimal, divide the numerator by the denominator. The result either stops (a terminating decimal like 1/8 = 0.125) or repeats forever (like 1/3 = 0.333...). A reduced fraction terminates exactly when the prime factors of its denominator are only 2 and 5, because those are the factors of 10, our number base.
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Open the Fraction Calculator →Terminating Decimals
1/2 = 0.5, 1/4 = 0.25, 1/5 = 0.2, 1/8 = 0.125, and 3/16 = 0.1875. Their denominators are 2, 4 (2²), 5, 8 (2³), and 16 (2⁴). Each can be multiplied up to a power of 10: 1/8 = 125/1000, and 3/16 = 1875/10000.
A denominator like 20 (2² × 5) also terminates: 1/20 = 0.05. The number of decimal places equals the larger exponent of 2 or 5 in the denominator.
Repeating Decimals
If the denominator contains any prime factor other than 2 or 5, the decimal repeats. 1/3 = 0.333..., 1/6 = 0.1666..., 1/7 = 0.142857142857..., 1/9 = 0.111..., and 1/11 = 0.0909...
The repeating block has length at most denominator − 1. For 1/7, the cycle is 6 digits. This follows from the fact that the long-division remainders must eventually repeat.
| Fraction | Decimal | Type |
|---|---|---|
| 1/8 | 0.125 | Terminating |
| 3/16 | 0.1875 | Terminating |
| 1/3 | 0.3 (repeating) | Repeating, length 1 |
| 5/6 | 0.83 (the 3 repeats) | Repeating |
| 1/7 | 0.142857 (repeating) | Repeating, length 6 |
Converting a Repeating Decimal Back
To convert 0.777... to a fraction, let x = 0.777.... Then 10x = 7.777..., so 10x − x = 7, which gives 9x = 7 and x = 7/9. For a two-digit repeat like 0.2727..., multiply by 100: 99x = 27, so x = 27/99 = 3/11.
This subtraction trick works for any repeating pattern and shows that every repeating decimal is a rational number.
Why It Matters in Practice
Rounding repeating decimals loses precision, so when exactness matters, keep the fraction until the final step. 1/3 + 1/3 + 1/3 is exactly 1, while 0.33 + 0.33 + 0.33 is 0.99.
Calculators and spreadsheets store decimals in binary, so some values that terminate in base 10, such as 0.1, repeat in binary and carry tiny rounding errors.
Frequently Asked Questions
How do I know if a fraction will terminate?
Reduce the fraction, then factor the denominator. If it contains only 2s and 5s, the decimal terminates.
Why does 1/3 repeat but 1/4 doesn't?
Because 4 = 2², which divides a power of 10 exactly, while 3 never divides any power of 10.
How long can a repeating block be?
At most one less than the denominator. For 1/7 the block is 6 digits long.
Is 0.999... equal to 1?
Yes. Letting x = 0.999..., then 10x − x = 9, so x = 1. The two notations represent the same number.