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: 13 17 (1) 64 455 (in) (1) (iv) 15 1600 3125 8 6 29 343 (vi) (V) 23 2's? 129 2²57 (viii) (vii) 15 35 77
( ix) (x) 50 210
Answers
Answer:
Without actually performing the long division, the rational numbers 13/3125, 17/8, 15/1600, 23/2352, 6/15, and 35/50 have a terminating decimal expansion whereas, 64/455, 29/343, 129/225775, and 77/210 have a non-terminating repeating decimal expansion.
Answer:
Let x = p/q be a rational number, such that the prime factorization of q is of the form 2n × 5m, where n, m are non-negative integers. Then x has a terminating decimal expansion.
(i) 13/3125
The denominator is of form 20 × 55.
Hence, the decimal expansion of 13/3125 is terminating.
(ii) 17/8
The denominator is of form 23 × 50.
Hence, the decimal expansion of 17/8 is terminating.
(iii) 64/455
455 = 5 × 7 × 13
Since the denominator is not in form of 2m × 5n, and it also contains 7 and 13 as its factors, its decimal expansion will be non-terminating repeating.
(iv) 15/1600
1600 = 26 × 52
The denominator is of form 2m × 5n.
Hence, the decimal expansion of 15/1600 is terminating.
(v) 29/343
343 = 73
Since the denominator is not in form of 2m × 5n, and it has 7 as its factor, the decimal expansion of 29/343 is non-terminating repeating.
(vi) 23/2352
The denominator is of form 2m × 5n.
Hence, the decimal expansion of 23/2352 is terminating.
(vii) 129/225775
Since the denominator is not of the form 2m × 5n, and it also has 7 as its factor, the decimal expansion of 129/225775 is non-terminating repeating.
(viii) 6/15
6/15 = (2 × 3)/(3 × 5) = 2/5
The denominator is of the form 5n.
Hence, the decimal expansion of 6/15 is terminating.
(ix) 35/50
35/50 = (7 × 5)/(10 × 5) = 7/10
10 = 2 × 5
The denominator is of the form 2m × 5n.
Hence, the decimal expansion of 35/50 is terminating.
(x) 77/210
77/210 = (7 × 11)/(30 × 7) = 11/30
30 = 2 × 3 × 5
Since the denominator is not of form 2m × 5n and it also has 3 as its factor, the decimal expansion of 77/210 is non-terminating repeating.
Video Solution:
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
NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.4 Question 1 - Chapter 1 Exercise 1.4 Question 1:
Summary:
Without actually performing the long division, the rational numbers 13/3125, 17/8, 15/1600, 23/2352, 6/15, and 35/50 have a terminating decimal expansion whereas, 64/455, 29/343, 129/225775, and 77/210 have a non-terminating repeating decimal expansion.
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