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) 23/(2352)
(ii) 129/(225775)
(iii) 6/15
(iv) 35/50
(v) 77/210
(vi)23/(2352)
Clearly, the denominator is in the form of 2m × 5n.
Hence, 23/ (2352) has a terminating decimal expansion.
(vii) 129/(225775)
As you can see, the denominator is not in the form of 2m × 5n.
Hence, 129/ (225775) has a non-terminating decimal expansion.
(viii) 6/15
6/15 = 2/5
Since, the denominator has only 5 as its factor, thus, 6/15 has a terminating decimal expansion.
(ix) 35/50
35/50 = 7/10
Factorising the denominator, we get,
10 = 2 x 5
Since, the denominator is in the form of 2m × 5n thus, 35/50 has a terminating decimal expansion.
(x) 77/210
77/210 = (7× 11)/ (30 × 7) = 11/30
Factorising the denominator, we get,
30 = 2 × 3 × 5
As you can see, the denominator is not in the form of 2m × 5n. Hence, 77/210 has a non-terminating decimal expansion.