NCERT Solutions Maths Class 10 Chapter 1

# NCERT Solutions Maths Class 10 Chapter 1

Go To:      Exercise 1.1

NCERT Solutions Maths Class 10 Exercise 1.4

Q1. 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) 23/23 × 52         (vii) 129/22 × 57 × 75
(viii) 6/15                         (ix) 35/50                (x) 77/210

Solution:  (i) 13/3125
Factorizing the denominator of the given rational number, we get
3125 =5 × 5 × 5 × 5 × 5 = 55
Since the denominator is in the form of 5m, therefore it is terminating.

(ii) 17/8
Factorizing the denominator of the given rational number, we get
8 = 2 × 2 × 2 = 23
Since the denominator is in the form of 2m, therefore it is terminating.

(iii) 64/455
Factorizing the denominator of the given rational number, we get
455 = 5 × 7 × 13
Since there are 7 and 13 also in the denominator, so the denominator is not in the form of 2m × 5n.

Hence, the given rational number is non-terminating.

(iv) 15/1600
Factorizing the denominator of the given rational number, we get
1600 = 2 × 2 × 2 ×2 × 2 × 2 × 5 × 5 = 26 × 52
Here, the denominator is in the form of 2m × 5n.
Hence, the given rational number is terminating.

(v) 29/343
Factorizing the denominator of the given rational number, we get
343 = 7 × 7 × 7 = 73
Since 7 is also there in the denominator, so the denominator is not in the form of 2m × 5n
Hence, it is non-terminating.

(vi) 23/(23 × 52)
In this rational number, the denominator is in the form of 2m × 5n.
Hence, it is terminating.

(vii) 129/(22 × 57 × 75 )
Since 7 is also there in the denominator, so the denominator is not in the form of 2m × 5n
Hence, it is non-terminating.

(viii) 6/15

First convert the given rational number in its standard form.

Dividing the numerator and the denominator both by 3, we get 2/5.
Since the denominator is in the form of 5m, therefore it is terminating.

(ix) 35/50

First convert the given rational number in its standard form.
Dividing the numerator and the denominator both by 5, we get 7/10
Factorizing the denominator of the given rational number, we get
10 = 2 × 5
Since the denominator is in the form of 2m × 5n, therefore it is terminating.

(x) 77/210
First convert the given rational number in its standard form.
Dividing the numerator and the denominator both by 7, we get 11/30.
Factorizing the denominator of the given rational number, we get
30 = 2 × 3 × 5
There is a 3 also in the denominator, so the denominator is not in form of 2m × 5n.
Hence, it is non-terminating.

Q2. Write down the decimal expansions of those rational numbers in Question 1 above which have terminating decimal expansions.

Solution: In question 1, (i) 13/3125, (ii) 17/8, (iv) 15/1600, (vi) 23/2352, (viii) 6/15 and (ix) 35/50 are terminating decimals. There decimal expansions are as follow:

(i) 13/3125 = 13/55 = 13 × 25/55 × 25 = 416/105 = 0.00416

(ii) 17/8 = 17/23 = 17 × 53/23 × 53 = 17 × 53/103 = 2125/103 = 2.125

(iv) 15/1600 = 15/24 × 102 = 15 × 54/24 × 54 × 102 = 9375/106 = 0.009375

(vi) 23/23 × 52 = 23 × 53 × 22/23 × 53 × 52 × 22 = 11500/105 = 0.115

(viii) 6/15 = 2/5 = 2 × 2/5 × 2 = 4/10 = 0.4

(ix) 35/50 = 7/10 = 0.7

Q3. 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) Solution: (i) 43.123456789

Since this number has a terminating decimal expansion, it is a rational number of the form p/q, and q is of the form 2m × 5n.

(ii) 0.120120012000120000…

The decimal expansion is neither terminating nor recurring. Therefore, the given number is an irrational number.

(iii) Since the decimal expansion is non-terminating recurring, the given number is a rational number of the form p/q, and q is not of the form 2m × 5n.