Exercise 1.4

NCERT Solutions for Class 10 Maths
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/(2³5²) (vii) 129/(2²5⁷7⁵) (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 ×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 2^m × 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 2^m × 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 2^m × 5ⁿ thus 29/343 has a non-terminating decimal expansion.

(vi)23/(2³5²)

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

Hence, 23/ (2³5²) has a terminating decimal expansion.

(vii) 129/(2²5⁷7⁵)

As you can see, the denominator is not in the form of 2^m × 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 2^m × 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 2^m × 5ⁿ .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) 23/2³ 5²    = 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 / 10⁹

= 43123456789 / 2⁹ x 5⁹

Here, the denominator is of the form 2^m5ⁿ
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^m5ⁿ