Which of the following has/have terminating decimal expansion(s)?

(a) \(\frac{2139}{3750}\)

(b) \(\frac{39}{9375}\)

(c) \( \frac{64}{455}\)

(d) \( \frac{245}{1344}\)

Formula used:

A fraction has a terminating decimal expansion if its denominator can be expressed in the form 2^m × 5ⁿ, where m and n are non-negative integers.

Calculation:

(a) \(\frac{2139}{3750}\) = \(\frac{713}{1250}\)

⇒ 1250 = 2 × 625 = 2 × 5⁴

Since the denominator has a factor can be expressed in the form 2m × 5n, the decimal expansion is terminating.

(b) \(\frac{39}{9375}\) = \(\frac{13}{3125}\)

⇒ 3125 = 3 × 5⁵

Since the denominator has a factor can be expressed in the form 2m × 5n, the decimal expansion is terminating.

(c) \(\frac{64}{455}\) is in its simplest form.

455 = 5 × 7 × 13

The denominator is not in the form 2^m × 5ⁿ, so the decimal expansion is non-terminating.

(d) \(\frac{245}{1344} = \frac{35}{192}\)

192 = 2⁶ × 3

The denominator is not in the form 2^m × 5ⁿ, so the decimal expansion is non-terminating.

∴ (a) and (b) have terminating decimal expansions.