10th Maths 1.4
NCERT Class 10th solution of Exercise 1.1
NCERT Class 10th solution of Exercise 1.2
NCERT Class 10th solution of Exercise 1.3
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) 133125
Sol. :
133125=1355
Since the denominator is in the form 2n.5m, where n and m are non-negative integers.
Answer :
Thus, the decimal expansion is terminating.
ii) 178
Sol. :
178=1723
Since the denominator is in the form 2n.5m, where n and m are non-negative integers.
Answer :
Thus, the decimal expansion is terminating.
iii) 64455
Sol. :
64455=6451×71×131
Since the denominator has 7 and 13 also other than 2 and 5 as a prime factor.
Answer :
Thus, the decimal expansion is non-terminating recurring.
iv) 151600
Sol. :
156000=5×326×52=326×5
Since the denominator is in the form 2n.5m, where n and m are non-negative integers.
Answer :
Thus, the decimal expansion is terminating.
v) 29343
Sol. :
29343=2973
Since the denominator has the power of 7 other than 2 and 5 as a prime fator.
Answer :
Thus, the decimal expansion is non-terminating recurring.
vi) 232³5²
Sol. :
232³5²=232³5²
Since the denominator is in the form 2n.5m, where n and m are non-negative integers.
Answer :
Thus, the decimal expansion is terminating.
vii) 1292²5⁷7⁵
Sol. :
1292²5⁷7⁵
Since the denominator has the power of 7 other than 2nand 5m.
Answer :
Thus, the decimal expansion is non-terminating recurring.
viii) 615
Sol. :
615=25
Since the denominator is in the form 2n.5m, where n and m are non-negative integers.
Answer :
Thus, the decimal expansion is terminating.
ix) 3550
Sol.
3550=710=72×5
Since the denominator is in the form 2n.5m, where n and m are non-negative integers.
Answer :
Thus, the decimal expansion is terminating.
x) 77210
Sol. :
77210=1130=112×3×5
Since the denominator has 3 also other than 2n.5m.
Answer :
Thus, the decimal expansion is non-terminating recurring.
Q2. Write down the decimal expansions of those rational numbers in Question 1 above which have terminating decimal expansions.
Sol.
We solve those rational numbers in Question 1 above which have terminating decimal expansions
i)
133125
=13×2555×25
=416105
=0.00416
Answer :
=0.00416
ii)
178
=1723
=17×5323×53
=2125103
=2.125
Answer :
=2.125
iii)
151600
=3×526×52
=3×5×5426×52×54
=9375106
=0.009375
Answer :
=0.009375
iv)
2323×52
=23×523×52×5
=115103
=0.115
Answer :
=0.115
v)
615
=2×35×3
=2×25×2
=410
=0.4
Answer :
=0.4
vi)
3550
=7×52×52
=72×5
=710
=0.7
Answer :
=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 pq, what can you say about the prime factors of q.
i) 43.123456789 ii) 0.120120012000120000.. iii) 43.¯123456789
Sol. :
43.123456789
=43123456789109
=pq
Since the decimal expansion is terminating.
Answer :
Thus, the given number is a rational number that can be expressed in the form pq and the prime factors of q are 2n.5m.
ii)
0.120120012000120000..
Since the given decimal expansion is non-terminating non-recurring.
Answer :
Thus, the number is an irrational number.
iii)
43.¯123456789
Since the given decimal expansion is non-terminating recurring.
Answer :
Thus, the number is rational and is in the form pq in which q has prime factors other than 2n.5m also.
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