**NCERT Solutions for Class 10 Maths**

Chapter 1 Real Numbers Exercise 1.4

Chapter 1 Real Numbers Exercise 1.4

**Page 17**

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/(2 ^{3}5^{2}) (vii) 129/(2^{2}5^{7}7^{5}) (viii) 6/15 (ix) 35/50 (x) 77/210**

**Answer :**

**Note:** If the denominator has only factors of 2 and 5 or in the form of 2^{m} ×5^{n} 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 = 5^{5}

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 = 2 ^{3}**

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 2^{m} × 5^{n}, thus 64/455 has a non-terminating decimal expansion.

**(iv) 15/ 1600**

Factorizing the denominator, we get,

1600 = 2^{6}5^{2}

Since, the denominator is in the form of 2^{m} × 5^{n}, thus 15/1600 has a terminating decimal expansion.

**(v) 29/343**

Factorizing the denominator, we get,

343 = 7×7×7 = 7^{3} Since, the denominator is not in the form of 2^{m} × 5^{n} thus 29/343 has a non-terminating decimal expansion.

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

Clearly, the denominator is in the form of 2^{m} × 5^{n}.

Hence, 23/ (2^{3}5^{2}) has a terminating decimal expansion.

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

As you can see, the denominator is not in the form of 2^{m} × 5^{n}.

Hence, 129/ (2^{2}5^{7}7^{5}) 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 2^{m} × 5^{n} 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 2^{m} × 5^{n} .Hence, 77/210 has a non-terminating decimal expansion.

**2. Write down the decimal expansions of those rational numbers in Question 1 above which have terminating decimal expansions.**

**Answer :**

(i) 13/3125

= 0.00416

0R

13/3125 = 13 x 32 / 3125 x 32 = 416/ 100000 = 0.00416

(ii) 17/8 = 2.125

(iii) 15/1600 = 0.009375

(iv) 23/2^{3} 5^{2 }= 23/200

(v) 6/15 = 0.4

(vi) 35/50 = 0.7

**3. 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 factors of q?**

**(i) 43.123456789
(ii) 0.120120012000120000. . .
(iii) 43.123456789**

**Answer :**

**(i) 43.123456789**

= 43123456789 / 10^{9}

= 43123456789 / 2^{9} x 5^{9}

Here, the denominator is of the form 2^{m}5^{n}

Hence, the number is a rational number, specifically a terminating decimal.^{
}

**(ii) 0.120120012000120000. . .**

Since the given decimal number is nonterminating non-repeating, it is not rational.

**(iii) 43.123456789 **

Since the given decimal number is nonterminating repeating, it is rational, but the denominator is not of the form 2^{m}5^{n}