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Question

Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion:

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Solution

(i)

The denominator is of the form 5m.

Hence, the decimal expansion ofis terminating.

(ii)

The denominator is of the form 2m.

Hence, the decimal expansion of is terminating.

(iii)

455 = 5 × 7 × 13

Since the denominator is not in the form 2m × 5n, and it also contains 7 and 13 as its factors, its decimal expansion will be non-terminating repeating.

(iv)

1600 = 26 × 52

The denominator is of the form 2m × 5n.

Hence, the decimal expansion of is terminating.

(v)

Since the denominator is not in the form 2m × 5n, and it has 7 as its factor, the decimal expansion of is non-terminating repeating.

(vi)

The denominator is of the form 2m × 5n.

Hence, the decimal expansion of is terminating.

(vii)

Since the denominator is not of the form 2m × 5n, and it also has 7 as its factor, the decimal expansion of is non-terminating repeating.

(viii)

The denominator is of the form 5n.

Hence, the decimal expansion of is terminating.

(ix)

The denominator is of the form 2m × 5n.

Hence, the decimal expansion of is terminating.

(x)

Since the denominator is not of the form 2m × 5n, and it also has 3 as its factors, the decimal expansion of is non-terminating repeating.


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