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

Solution:

Let x = p/q be a rational number, such that the prime factorization of q is of the form 2ⁿ × 5^m, 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 2^m × 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 2^m × 5ⁿ.

Hence, the decimal expansion of 15/1600 is terminating.

(v) 29/343

343 = 7³

Since the denominator is not in form of 2^m × 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 2^m × 5ⁿ.

Hence, the decimal expansion of 23/2³5² is terminating.

(vii) 129/2²5⁷7⁵

Since the denominator is not of the form 2^m × 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 2^m × 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 2^m × 5ⁿ and it also has 3 as its factor, the decimal expansion of 77/210 is non-terminating repeating.

☛ Check: NCERT Solutions for Class 10 Maths Chapter 1

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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 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, 129/2²5⁷7^5, and 77/210 have a non-terminating repeating decimal expansion.

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☛ Related Questions:

• Prove that 3 + 2√5 is irrational.
• Prove that the following are irrationals: (i) 1/√2 (ii) 7√5 (iii) 6 + √2
• Write down the decimal expansions of those rational numbers in Question 1 above which have terminating decimal expansions. (i) 13/3125 (ii) 17/8 = 2.125 (iii) 64/455 (iv) 15/1600 (v) 29/343 (vi) 23/ (23 x 52) (vii) 129/(22 x 57 x 75) (viii) 6/15 (ix) 35/50 (x) 77/210
• The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form p/q, what can you say about the prime factor of q? (i) 43.123456789 (ii) 0.120120012000120000.... (iii) 43.123456789