A fraction has a terminating decimal when, once reduced, its denominator has only the prime factors 2 and 5. If any other prime remains, the decimal repeats. For example, 6/24 reduces to 1/4 and gives 0.25, while 7/30 has a 3 in the denominator and its decimal repeats.
This guide helps you decide without dividing. If you already have the fraction and want to see the decimal, use the fraction to decimal converter. If you have a repeating decimal and want the fraction, see the repeating decimal to fraction tool.
| Reduced denominator | Decimal | Example |
|---|---|---|
| Only 2s and 5s (or it is 1) | Terminates | 9/20 = 0.45 |
| No 2 or 5, with another prime | Purely repeating | 1/3 = 0.3̅ |
| A 2 or 5, plus another prime | Mixed repeating | 7/30 = 0.23̅ |
The table sums up the guide: all that matters is which primes are left in the denominator after you simplify.
If you had factored 24 without reducing, you would have seen a 3 and concluded that the decimal repeats. That 3 cancels with the 6 in the numerator, which is why the order of the steps matters. To find the GCF or the factors when the numbers are large, use simplify fractions and prime factorization.
Why only 2 and 5 count
A decimal with n digits after the point is a fraction with denominator 10ⁿ, and 10ⁿ = 2ⁿ × 5ⁿ. So a reduced fraction gives a terminating decimal only if its denominator is made of 2s and 5s. Conversely, if the reduced denominator is 2ᵃ × 5ᵇ, you multiply top and bottom by the missing factors until you reach 10ⁿ, with n equal to the larger of a and b.
In 3/40, the denominator is 40 = 2³ × 5. To reach 10³ = 1000 you are missing two factors of 5, which means multiplying by 25. So 3/40 = 75/1000 = 0.075. The decimal has as many digits as the larger exponent, here three.
The test depends on base ten and does not measure which fraction is more precise: 1/3 is as exact a rational number as 1/4, even though its decimal form does not end.
Worked example: 6/24
Does the decimal of 6/24 terminate?
- Check the denominator. 24 is not zero and is positive, so there is nothing to fix.
- Reduce. The GCF of 6 and 24 is 6, and 6/24 = 1/4.
- Factor the reduced denominator. 4 = 2 × 2, so there are only 2s.
- Build up to a power of 10. 1/4 × 25/25 = 25/100.
Result: 6/24 = 0.25, a two-digit decimal.
Check: 4 × 0.25 = 1, and 24 × 0.25 = 6.
When the decimal repeats: purely or mixed
When the reduced denominator has a prime other than 2 and 5, long division never reaches a zero remainder. Sooner or later a remainder you already had comes back, and from there the same digits repeat. In 1/6 the remainder 4 appears again and produces another 6.
If the denominator has neither 2 nor 5, the repeating block starts right after the decimal point and the decimal is purely repeating, like 1/7 = 0.1̅4̅2̅8̅5̅7̅. If it has a 2 or 5 plus another prime, some digits come first that do not repeat, and the decimal is mixed repeating, like 7/30 = 0.23̅. The bar marks the digits that repeat: 0.23̅ and 0.2333… are the same number. By contrast, 0.23 is only an approximation: 7/30 − 23/100 = 1/300, so 7/30 ≈ 0.23 with an error of 1/300.
The next table lists every reduced denominator up to 30. Just find the row where yours falls.
| Type | Denominators |
|---|---|
| Terminates | 2, 4, 5, 8, 10, 16, 20, 25 |
| Purely repeating | 3, 7, 9, 11, 13, 17, 19, 21, 23, 27, 29 |
| Mixed repeating | 6, 12, 14, 15, 18, 22, 24, 26, 28, 30 |
The table works only for fractions in lowest terms. For example, 9/21 reduces to 3/7, so it stays in the purely repeating row, and 15/30 reduces to 1/2 and moves to the terminating row.
More examples
- 9/20 is already reduced and 20 = 2² × 5. You multiply by 5 and get 45/100 = 0.45.
- In 7/30, 30 = 2 × 3 × 5. The 3 stays, so it does not terminate. Long division gives 0.2333…, that is, 0.23̅.
- A whole number written as a fraction. In 12/4 = 3/1, the denominator 1 has no prime factors and the decimal is 3, with no digits after the point.
For practice, classify 8/12, 9/40 and 7/28 without dividing, then check with the fraction to decimal converter. If you are curious about the difference between a decimal fraction (denominator 10, 100, 1000…) and a terminating decimal, see decimal fractions. The notation 0.9̅ = 1 and how to write the bar are covered in terminating and repeating decimals.
Sources
LibreTexts Español, Preálgebra (Arnold), 5.6 (in Spanish, accessed October 16, 2026) states that if the denominator of the simplified fraction factors only into 2s and/or 5s, the decimal representation terminates. OpenStax, Contemporary Mathematics, 3.4 (in English, accessed October 16, 2026) defines a terminating decimal as one with a last decimal digit and a repeating decimal as one that repeats a pattern.