NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.4 || CBSE class 10 maths chapter 1 Ex 1.4 Math Solution
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NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.4 || CBSE
class 10 maths chapter 1 Ex 1.4 Math Solution
Exercise-1.4
NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.4 || CBSE class 10 maths chapter 1 Ex 1.4 Math Solution
1.
(i) 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:
\( \frac{13}{3125} \)
\( \frac{13}{3125} \)
Answer
\( \frac{13}{3125} \)Factorize the denominator we get,
\(
3125=5 \times 5 \times 5 \times 5 \times 5=5^{5}
\)
The denominator is of the form 5 m
Hence, the decimal expansion of \( \frac{13}{3125} \) is terminating.
(ii) 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:
\( \frac{17}{8} \)
\( \frac{17}{8} \)
Answer
\( \frac{17}{8} \)Factorize the denominator we get,
\( 8=2 \times 2 \times 2=2^{3} \)
The denominator is of the form \( 2^{\mathrm{m}} \)
Hence, the decimal expansion of \( \frac{17}{8} \) is terminating.
(iii) 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:
\( \frac{64}{455} \)
\( \frac{64}{455} \)
Answer
\( \frac{64}{455} \)Factorize the denominator we get,
\(
455=5 \times 7 \times 13
\)
Since, the denominator is not in the form of \( 2^{\mathrm{m}} \times 5^{\mathrm{n}} \), and it also contains 7 and 13 as its factors,
Its decimal expansion will be non-terminating repeating.
NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.4 || CBSE class 10 maths chapter 1 Ex 1.4 Math Solution
(iv) 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:
\( \frac{15}{1600} \)
\( \frac{15}{1600} \)
Answer
\( \frac{15}{1600} \)Factorize the denominator we get,
\( 1600=2^{6} \times 5^{2} \)
The denominator is in the form \( 2^{\mathrm{m}} \times 5^{\mathrm{n}} \)
Hence, the decimal expansion of \( \frac{15}{1600} \) is terminating.
(v) 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:
\( \frac{29}{343} \)
\( \frac{29}{343} \)
Answer
\( \frac{29}{343} \)Factorize the denominator we get,
\( 343=7^{3} \)
Since the denominator is not in the form of \( 2^{\mathrm{m}} \times 5^{\mathrm{n}} \), it has 7 as its factors.
So, the decimal expansion of non-terminating repeating.
(vi) 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:
\( \frac{23}{2^{3} 5^{2}} \)
\( \frac{23}{2^{3} 5^{2}} \)
Answer
\( \frac{23}{2^{3} 5^{2}} \)The denominator is in the form \( 2^{\mathrm{m}} \times 5^{\mathrm{n}} \)
Hence, the decimal expansion of \( \frac{23}{2^{3} 5^{2}} \) is terminating.
(vii) 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:
\( \frac{129}{2^{2} 5^{7} 7^{3}} \)
\( \frac{129}{2^{2} 5^{7} 7^{3}} \)
Answer
\( \frac{129}{2^{2} 5^{7} 7^{5}} \)Since, the denominator is not in the form of \( 2^{\mathrm{m}} \times 5^{\mathrm{n}} \), as it has 7 in denominator.
So, the decimal expansion of \( \frac{129}{2^{2} 5^{7} 7^{5}} \) is non-terminating repeating.
(viii) 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:
\( \frac{6}{15} \)
\( \frac{6}{15} \)
Answer
\( \frac{6}{15}=\frac{2 \times 3}{3 \times 5}=\frac{2}{5}
\)The denominator is in the form \( 5^{\text {n }} \)
Hence, the decimal expansion of \( \frac{6}{15} \) is terminating.
NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.4 || CBSE class 10 maths chapter 1 Ex 1.4 Math Solution
(ix) 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:
\( \frac{35}{50} \)
\( \frac{35}{50} \)
Answer
\( \frac{35}{50}=\frac{7 \times 5}{10 \times 5}=\frac{7}{10}
\)Factorize the denominator we get,
\( 10=2 \times 5 \)
The denominator is in the form \( 2^{\mathrm{m}} \times 5^{\mathrm{n}} \)
Hence, the decimal expansion of \( \frac{35}{50} \) is terminating.
(x) 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:
\( \frac{77}{210} \)
\( \frac{77}{210} \)
Answer
\( \frac{77}{210}=\frac{11 \times 7}{30 \times 7}=\frac{11}{30}
\)Factorize the denominator we get,
\(
30=2 \times 3 \times 5
\)
Since the denominator is not in the form of \( 2^{\mathrm{m}} \times 5^{\mathrm{n}} \), as it has 3 in denominator.
So, the decimal expansion of \( \frac{77}{210} \) non-terminating repeating.
2.
(i) Write down the decimal expansions of those rational numbers which have
terminating decimal expansions.
\( \frac{13}{3125} \)
\( \frac{13}{3125} \)
Answer
\( \frac{13}{3125}=0.00416 \)
(ii) Write down the decimal expansions of those rational numbers which have
terminating decimal expansions.
\( \frac{17}{8} \)
\( \frac{17}{8} \)
Answer
\( \frac{17}{8}=2.125 \)
(iv) Write down the decimal expansions of those rational numbers which have
terminating decimal expansions.
\( \frac{15}{1600} \)
\( \frac{15}{1600} \)
Answer
\( \frac{15}{1600} 0.009375\)
(vi) Write down the decimal expansions of those rational numbers which have
terminating decimal expansions.
\( \frac{23}{2^{3} 5^{2}} \)
\( \frac{23}{2^{3} 5^{2}} \)
Answer
\( \frac{23}{2^{3} \times 5^{2}}=0.115 \)
NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.4 || CBSE class 10 maths chapter 1 Ex 1.4 Math Solution
(viii) Write down the decimal expansions of those rational numbers which have
terminating decimal expansions.
\( \frac{6}{15} \)
\( \frac{6}{15} \)
Answer
\( \frac{6}{15}=\frac{2 \times 3}{2 \times 5}=\frac{2}{5}=0.4
\)
(ix) Write down the decimal expansions of those rational numbers which have
terminating decimal expansions.
\( \frac{35}{50} \)
\( \frac{35}{50} \)
Answer
\( \frac{35}{50}=0.7 \)
3.
(i) 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 ?
43.123456789
43.123456789
Answer
43.123456789 Since this number has a terminating decimal expansion, it is a rational number \( \frac{p}{q} \) of the form and q is of the form \( 2 \mathrm{m} \times 5 \mathrm{n} \)
That is, the prime factor of \( q \) will be 2 or 5 or both.
(ii) 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 ?
\( 0.120120012000120000 \ldots \)
\( 0.120120012000120000 \ldots \)
Answer
\( 0.120120012000120000 \ldots \)The decimal expansion is neither terminating nor recurring.
Therefore, the given number is an irrational number.
(iii) 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 ?
\( 43 . \overline{123456789} \)
\( 43 . \overline{123456789} \)
Answer
\( 43 . \overline{123456789} \)Since the decimal expansion is non-terminating but recurring, the given number is a rational number of the form \( \frac{p}{q} \) and \( q \) is not of the form \( 2^{\mathrm{m}} \times 5^{\mathrm{n}} \) that is, the prime factors of q will also have a factor other than 2 or 5.
NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.4 || CBSE class 10 maths chapter 1 Ex 1.4 Math Solution
Central Board of Secondary Education Official Site
Class 10 : NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.1
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Class 10 : CBSE Class 10 Maths Chapter 13 Surface Areas and Volumes solutions Ex 13.3
Class 10 : NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.1
Class 10 : NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.2
Class 10 : NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.3
Class 10 : CBSE Class 10 Maths Chapter 2 Polynomials Ex 2.1
Class 10 : CBSE Class 10 Maths Chapter 5 Arithmetic Progressions Ex 5.3
Class 10 : NCERT Solutions for Class 10 Maths Chapter 8 Exercise 8.2
Class 10 : CBSE Class 10 Maths Chapter 10 Circles solutions Ex 10.2
Class 10 : CBSE Class 10 Maths Chapter 13 Surface Areas and Volumes solutions Ex 13.2
Class 10 : CBSE Class 10 Maths Chapter 13 Surface Areas and Volumes solutions Ex 13.3