Some fractions end (3/8 = 0.375), others repeat forever (1/7 = 0.142857142857…). The difference depends only on the denominator: in base 10 a fraction terminates exactly when the denominator has no prime factors other than 2 and 5.
How to use the Terminating Decimals Calculator
- Numerator
- Denominator
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
The fraction is reduced, the denominator is factorised and the decimal expansion is computed by exact long division, with the repeating part shown in brackets.
Formula
n/d in lowest terms terminates ⇔ d = 2ᵃ × 5ᵇ
Worked example
Inputs — Numerator: 3; Denominator: 12.
Results — Terminating or repeating? = Terminating decimal; Decimal expansion = 0.25; Why = In lowest terms the denominator is 4, whose only prime factors are 2 and 5, so the decimal ends.; Fraction in lowest terms = 1/4.
Key terms
- Terminating decimal
- A decimal with finitely many digits.
- Repeating decimal
- A decimal with a block that repeats forever.
Frequently asked questions
Does 1/3 terminate?
No – 0.333… repeats because 3 is not 2 or 5.
Does 7/20 terminate?
Yes – 20 = 2² × 5, giving 0.35.
How long can the repeating block be?
At most d − 1 digits for denominator d.
Is a repeating decimal rational?
Yes, always.