Which of these fractions is not equivalent to a terminating decimal number?

• A. 4/5
• B. 12/14
• C. 5/8
• D. 24/64

Answer: (B)

A decimal terminates exactly when, after reducing a fraction to lowest terms, its denominator has only the prime factors 2 and 5 (that is, it can be written as 2^a·5^b).

• Four-fifths already has denominator 5, so it becomes 0.8, a terminating decimal.

• Twelve fourteenths simplifies to six-sevenths. The denominator 7 has a prime factor other than 2 or 5, so the decimal repeats and does not terminate.

• Five-eighths has denominator 8 = 2^3, so it becomes 0.625, a terminating decimal.

• Twenty-four sixty-four simplifies to three-eighths. Denominator 8 is 2^3, so it terminates as 0.375.

Therefore, the one that does not terminate is the fraction that reduces to a denominator with a prime factor other than 2 or 5. In this set, that’s the simplified form six-sevenths.