These NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers Ex 1.4 Questions and Answers are prepared by our highly skilled subject experts.
NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers
Exercise 1.4
Question 1.
Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or non-terminating repeating decimal expansion:
(i) \(\frac { 13 }{ 3125 }\)
(ii) \(\frac { 17 }{ 8 }\)
(iii) \(\frac { 64 }{ 455 }\)
(iv) \(\frac{15}{1600}\)
(v) \(\frac { 29 }{ 343 }\)
(vi) \(\frac{23}{2^{3} 5^{2}}\)
(vii) \(\frac{129}{2^{2} 5^{7} 7^{5}}\)
(viii) \(\frac { 6 }{ 15 }\)
(ix) \(\frac { 35 }{ 50 }\)
(x) \(\frac { 77 }{ 210 }\)
Solution:
(i) \(\frac { 13 }{ 3125 }\) = \(\frac{17}{2 \times 2 \times 2}\) = \(\frac{17}{2^{3}}\)
Because the denominator can be in the form 2n 5n, hence it will have terminating decimal expansion.
(ii) \(\frac { 17 }{ 8 }\) = \(\frac { 17 }{ 8 }\)
It will have terminating decimal expansion.
(iii) \(\frac { 64 }{ 455 }\) = \(\frac{64}{5 \times 7 \times 13}\)
Non terminating repeating decimal expansion.
(iv) \(\frac{15}{1600}\) = \(\frac{15}{2^{2} \times 5^{2}}\)
It will have terminating decimal expansion.
(v) \(\frac { 29 }{ 343 }\) = \(\frac{29}{7^{3}}\)
Non terminating repeating decimal expansion.
(vi) \(\frac{23}{2^{3} 5^{2}}\)
It will have terminating decimal expansion.
(vii) \(\frac{129}{2^{2} 5^{7} 7^{5}}\)
Non terminating repeating decimal expansion.
(viii) \(\frac { 6 }{ 15 }\) = \(\frac{6}{3 \times 5}\)
It will have terminating decimal expansion.
(ix) \(\frac { 35 }{ 50 }\) = \(\frac{35}{2 \times 5^{2}}\)
It will have terminating decimal expansion.
(x) \(\frac { 77 }{ 210 }\) = \(\frac{17}{2 \times 3 \times 5 \times 7}\)
Non terminating repeating decimal expansion.
Question 2.
Write down the decimal expansion of those rational numbers in Question 1 above which terminating decimal expansions.
Solution:
(i) \(\frac { 13 }{ 3125 }\) = 0.00146
(ii) \(\frac { 17 }{ 8 }\) = 2.125
(iii) \(\frac { 15 }{ 1600 }\) = 0.009375
(iv) \(\frac{23}{2^{3} 5^{2}}\) = \(\frac { 23 }{ 200 }\) = 0.115
(v) \(\frac { 6 }{ 15 }\) = 0.4
(vi) \(\frac { 35 }{ 50 }\) = 0.7
Question 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 \(\frac { p }{ q }\), what can you say about the prime factors of q?
(i) 43.123456789
(ii) 0.120120012000120000
(iii) \(43 . \overline{123456789}\)
Solution:
(i) 43.123456789
\(=\frac{43123456789}{1000000000}=\frac{43123456789}{2^{9} 5^{9}}\)
it is a rational, number The prime factors of q are 2959
(ii) 0.120120012000120000 …….
\(\begin{aligned}
&=\frac{120120012000012}{100000000000000 \ldots \ldots . .} \\
&=\frac{120120012000012}{\left(2^{1} \times 2^{2} \times 2^{3} \ldots . .\right) \times\left(5^{1} \times 5^{2} \times 5^{3} . \ldots .\right)}
\end{aligned}\)
it is a rational number The prime factors of q are (21 x 2² x 2³ ….) x (51 x 5² x 5³ ….)
(iii) \(43 . \overline{123456789}\)
It is not-terminating decimal expansion.
But it is a rational number, whose prime factors of q will also have a factor other than 2 or 5.