NCERT Solutions Class 10th Maths Chapter – 1 Real Numbers Exercise – 1.4

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:

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

Solution: 13/3125
Factorizing the denominator, we get,
3125 = 5 × 5 × 5 × 5 × 5 = 5⁵
Since, the denominator has only 5 as its factor, 13/3125 has a terminating decimal expansion.

(ii) 17/8

Solution: 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

Solution: 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

Solution: 15/ 1600
Factorizing the denominator, we get,
1600 = 2⁶×5²
Since, the denominator is in the form of 2^m ×5ⁿ thus 15/1600 has a terminating decimal expansion.

(v) 29/343

Solution: 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

Solution: 23/2³5²
Clearly, the denominator is in the form of 2^m ×5ⁿ
Hence, 23/2³5² has a terminating decimal expansion.

(vii) 129/2²5⁷7⁵

Solution: 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

Solution: 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

Solution: 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

Solution: 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.

(i) 13/3125

Solution:

13/3125 = 0.00416

(ii) 17/8

Solution:

17/8 = 2.125

(iii) 64/455

Solution: Non terminating decimal expansion

(iv)15/ 1600

Solution:

15/1600 = 0.009375

(v) 29/ 343

Solution: Non terminating decimal expansion

(vi)23/ (2³5²)

Solution: 23/ (2³5²) = 23/(8×25) = 23/200

23/ (2³5²) = 0.115

(vii) 129/ (2²5⁷7⁵)

Solution: Non terminating decimal expansion

(viii) 6/15 

Solution: 6/15 = 2/5

(ix) 35/50 

Solution: 35/50 = 7/10

35/50 = 0.7

(x) 77/210 

Solution: Non-terminating decimal expansion.

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

Solution: 43.123456789
Since it has a terminating decimal expansion, it is a rational number in the form of p/q and q has factors of 2 and 5 only.

(ii) 0.120120012000120000. . .

Solution: 0.120120012000120000. . .
Since, it has non-terminating and non- repeating decimal expansion, it is an irrational number.

Solution: Since it has non-terminating but repeating decimal expansion, it is a rational number in the form of p/q and q has factors other than 2 and 5.