# 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) 23/2^{3}5^{2 }(vii) 129/2^{2}5^{7}7^{5 }(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 2^{n} × 5^{m}, where n, m are non-negative integers. Then x has a terminating decimal expansion.

(i) 13/3125

The denominator is of form 2^{0} × 5^{5}.

Hence, the decimal expansion of 13/3125 is terminating.

(ii) 17/8

The denominator is of form 2^{3} × 5^{0}.

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} × 5^{n}, and it also contains 7 and 13 as its factors, its decimal expansion will be non-terminating repeating.

(iv) 15/1600

1600 = 2^{6} × 5^{2}

The denominator is of form 2^{m} × 5^{n}.

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

(v) 29/343

343 = 7^{3}

Since the denominator is not in form of 2^{m} × 5^{n}, and it has 7 as its factor, the decimal expansion of 29/343 is non-terminating repeating.

(vi) 23/2^{3}5^{2}

The denominator is of form 2^{m} × 5^{n}.

Hence, the decimal expansion of 23/2^{3}5^{2} is terminating.

(vii) 129/2^{2}5^{7}7^{5}

Since the denominator is not of the form 2^{m} × 5^{n}, and it also has 7 as its factor, the decimal expansion of 129/2^{2}5^{7}7^{5 }is non-terminating repeating.

(viii) 6/15

6/15 = (2 × 3)/(3 × 5) = 2/5

The denominator is of the form 5^{n}.

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} × 5^{n}.

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} × 5^{n} 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

**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, 23/2^{3}5^{2}, 6/15, and 35/50 have a terminating decimal expansion whereas, 64/455, 29/343, 129/2^{2}5^{7}7^{5,} and 77/210 have a non-terminating repeating decimal expansion.

**☛ 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