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:
(i) | (ii) | (iii) | (iv) |
(v) | (vi) | (vii) | (viii) |
(ix) | (x) |
(i) 
Factorize the denominator we get,
3125 = 5×5×5×5×5 = 55
The denominator is of the form 5m
Hence, the decimal expansion of 
is terminating.
(ii) 
Factorize the denominator we get,
8 = 2 × 2 ×2 = 23
The denominator is of the form 2m
Hence, the decimal expansion of 
is terminating.
(iii) 
Factorize the denominator we get,
455 = 5×7×13
Since, the denominator is not in the form of 2m × 5n, and it also contains 7 and 13 as its factors,
Its decimal expansion will be non-terminating repeating.
(iv) 
Factorize the denominator we get,
1600 = 26×52
The denominator is in the form 2m × 5n
Hence, the decimal expansion of 
is terminating.
(v) 
Factorize the denominator we get,
343 = 73
Since the denominator is not in the form of 2m × 5n, it has 7 as its factors.
So, the decimal expansion of 
non-terminating repeating.
(vi) 
The denominator is in the form 2m×5n
Hence, the decimal expansion of 
is terminating.
(vii) 
Since, the denominator is not in the form of 2m × 5n, as it has 7 in denominator.
So, the decimal expansion of 
is non-terminating repeating.
(viii) 
The denominator is in the form 5n
Hence, the decimal expansion of 
is terminating.
(ix) 
Factorize the denominator we get,
10 = 2×5
The denominator is in the form 2m×5n
Hence, the decimal expansion of 
is terminating.
(x) 
Factorize the denominator we get,
30 = 2×3×5
Since the denominator is not in the form of 2m × 5n, as it has 3 in denominator.
So, the decimal expansion of 
non-terminating repeating.