Which of the following numbers has a terminating decimal representation?
- (a)3⁄8
- (b)1⁄6
- (c)2⁄7
- (d)4⁄9
Answer
Why
Correct — A. Rule: a fraction in lowest terms terminates only when its denominator has no prime factor other than 2 and 5.
All four fractions are already in lowest terms, so test the denominators.
3⁄8: 8 = 2³, only 2s → it terminates
Divide to confirm: 3 ÷ 8 = 0.375 → option (a)
Why the others are wrong
- (b)1⁄6 — 6 = 2 × 3. No power of 10 has a factor 3, so 1⁄6 cannot terminate: 1⁄6 = 0.1666… with the 6 recurring.
- (c)2⁄7 — 7 is prime and is neither 2 nor 5, so 2⁄7 recurs: 0.285714285714…, a six-digit repeating block.
- (d)4⁄9 — 9 = 3², so 4⁄9 recurs: 0.444… with the 4 repeating forever.
Concept
A terminating decimal is a whole number over a power of 10, and 10ⁿ = 2ⁿ × 5ⁿ. So a reduced fraction terminates exactly when its denominator divides some power of 10, which means its prime factors are 2s and 5s alone.
The number of decimal places is the larger of the two powers: 3⁄8 = 3⁄2³ needs 3 places, giving 0.375.
Any other prime in the denominator, such as 3 or 7, forces a repeating block.
The same test by multiplication: 3⁄8 = 375⁄1000, because 8 × 125 = 1000. No whole number times 6, 7 or 9 gives a power of 10.
Key facts
- A reduced fraction p⁄q terminates exactly when q = 2ᵐ × 5ⁿ.
- Its number of decimal places is the larger of m and n: 3⁄8 = 3⁄2³ has 3 places.
- A prime factor 3 or 7 in the reduced denominator gives a recurring decimal, as in 1⁄6 = 0.1666… and 2⁄7 = 0.285714…
Study next
Common traps
- Testing the denominator before reducing: 3⁄6 has a 3 in its denominator yet equals 1⁄2 = 0.5, which terminates.
- Looking at the numerator: whether a reduced fraction terminates depends on its denominator alone.
Here four reduced fractions are offered and the task is to spot the terminating one; factorising each denominator settles it without long division.
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