How to Check Whether a Fraction Has a Terminating Decimal

How do you test a fraction without dividing?

Write the rational number as p/q in lowest terms, with q positive. Its decimal terminates exactly when q has no prime factors other than 2 and 5. A denominator of 1 also qualifies: an integer has a terminating decimal. Always cancel common factors before applying this test.

Why do 2 and 5 matter?

A terminating decimal can be written with denominator 10, 100, 1000 or another power of 10. Since 10 = 2 × 5, every power of 10 contains only these two primes. Once a fraction is reduced, a different prime in its denominator cannot disappear when you multiply the numerator and denominator by the same integer.

For example, 7/40 has denominator 2³ × 5. Multiply top and bottom by 25 to obtain 175/1000 = 0.175. In contrast, 7/30 is already reduced and its denominator contains 3, so its decimal is non-terminating recurring.

The cancellation trap

The denominator of 21/84 contains 3 and 7, but testing it immediately gives the wrong conclusion. Cancel 21 first: 21/84 = 1/4 = 0.25. The reduced denominator is 2², so the decimal terminates.

For 13/125, the denominator is 5³. Multiply by 8/8 to get 104/1000 = 0.104. For a reduced denominator 2ᵃ × 5ᵇ, at most max(a, b) decimal places are needed. A negative sign changes the sign of the answer, not whether the decimal terminates.

Practice with answers

• 18/45 reduces to 2/5: terminating, 0.4.
• 11/24 is reduced and contains a factor 3 below: recurring.
• 35/140 reduces to 1/4: terminating, 0.25.

Your checklist is short: reduce, factor, then decide. This test concerns rational numbers; it is not a method for turning an irrational number into a fraction.

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