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1. 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) 13/3125 (ii) 17/8 (iii) 64/455 (iv) 15/1600 (v) 29/343 (vi) 23/(2352) (vii) 129/(225775) (viii) 6/15 (ix) 35/50 (x) 77/210

1. 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) 13/3125 (ii) 17/8 (iii) 64/455 (iv) 15/1600 (v) 29/343 (vi) 23/(2352) (vii) 129/(225775) (viii) 6/15 (ix) 35/50 (x) 77/210

Grade:12th pass

1 Answers

Pawan Prajapati
askIITians Faculty 60787 Points
2 years ago
Solutions: Note: If the denominator has only factors of 2 and 5 or in the form of 2m ×5n then it has terminating decimal expansion. If the denominator has factors other than 2 and 5 then it has a non-terminating decimal expansion. (i) 13/3125 Factorizing the denominator, we get, 3125 = 5 × 5 × 5 = 55 Since, the denominator has only 5 as its factor, 13/3125 has a terminating decimal expansion. (ii) 17/8 Factorizing the denominator, we get, 8 = 2×2×2 = 23 Since, the denominator has only 2 as its factor, 17/8 has a terminating decimal expansion. (iii) 64/455 Factorizing the denominator, we get, 455 = 5×7×13 Since, the denominator is not in the form of 2m × 5n, thus 64/455 has a non-terminating decimal expansion. (iv) 15/ 1600 Factorizing the denominator, we get, 1600 = 2652 Since, the denominator is in the form of 2m × 5n, thus 15/1600 has a terminating decimal expansion. (v) 29/343 Factorizing the denominator, we get, 343 = 7×7×7 = 73 Since, the denominator is not in the form of 2m × 5n thus 29/343 has a non-terminating decimal expansion. (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 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.

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