Go To: Exercise 1.1 Exercise 1.2 Exercise 1.3
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:
(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.