KC Sinha: Exercise 1.4 - Mathematics Solution Class 10 Chapter 1 Real numbers

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Question 1 C 

Without actually performing the long division, state whether the following rational numbers have terminating or non-terminating repeating (recurring) decimal expansion.

$\dfrac{29}{343}$
Sol :
Given rational number is $\frac{29}{343}$
$\frac{p}{q}$ is terminating if
a) p and q are co-prime &
b) q is of the form of 2ⁿ 5^m where n and m are non-negative integers.Firstly, we check co-prime

29 = 29 × 1
343 = 7 × 7 × 7
⇒29 and 343 have no common factors
Therefore, 29 and 343 are co-prime.
Now, we have to check that q is in the form of 2ⁿ5^m
343 = 7³
So, denominator is not of the form 2ⁿ5^m
Thus, $\frac{29}{343}$ is a non-terminating repeating decimal.

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Question 1 D 

Without actually performing the long division, state whether the following rational numbers have terminating or non-terminating repeating (recurring) decimal expansion.

$\dfrac{13}{125}$

Sol :
Given rational number is $\frac{13}{125}$
$\frac{p}{q}$ is terminating if
a) p and q are co-prime &
b) q is of the form of 2ⁿ 5^m where n and m are non-negative integers.Firstly, we check co-prime

13 = 13 × 1
125 = 5 × 5 × 5
⇒13 and 125 have no common factors
Therefore, 13 and 125 are co-prime.
Now, we have to check that q is in the form of 2ⁿ5^m
125 = 5³
= 1 × 2³
= 2⁰ × 5³
So, denominator is of the form 2ⁿ5^m where n = 0 and m = 3
Thus, $\frac{13}{125}$ is a terminating decimal.

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Question 1 E 

Without actually performing the long division, state whether the following rational numbers have terminating or non-terminating repeating (recurring) decimal expansion.

$\dfrac{27}{8}$

Sol :
Given rational number is $\frac{27}{8}$
$\frac{p}{q}$ is terminating if
a) p and q are co-prime &
b) q is of the form of 2ⁿ 5^m where n and m are non-negative integers.Firstly, we check co-prime

27 = 3 × 3 × 3
8 = 2 × 2 × 2
⇒27 and 8 have no common factors
Therefore, 27 and 8 are co-prime.
Now, we have to check that q is in the form of 2ⁿ5^m
8 = 2³
= 1 × 2³
= 5⁰ × 2³
So, denominator is of the form 2ⁿ5^m where n = 3 and m = 0
Thus, $\frac{27}{8}$ is a terminating decimal.

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Question 1 F 

Without actually performing the long division, state whether the following rational numbers have terminating or non-terminating repeating (recurring) decimal expansion.

$\dfrac{7}{80}$
Sol :
Given rational number is $\frac{7}{80}$
$\frac{p}{q}$ is terminating if
a) p and q are co-prime &
b) q is of the form of 2ⁿ 5^m where n and m are non-negative integers.Firstly, we check co-prime

7 = 7 × 1
80 = 2 × 2 × 2 × 2 × 5
⇒7 and 80 have no common factors
Therefore, 7 and 80 are co-prime.
Now, we have to check that q is in the form of 2ⁿ5^m
80 = 2⁴ × 5
So, denominator is of the form 2ⁿ5^m where n = 4 and m = 1
Thus, $\frac{7}{80}$ is a terminating decimal.

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Question 1 G 

Without actually performing the long division, state whether the following rational numbers have terminating or non-terminating repeating (recurring) decimal expansion.

$\dfrac{64}{455}$

Sol :
Given rational number is $\frac{64}{455}$
$\frac{p}{q}$ is terminating if
a) p and q are co-prime &
b) q is of the form of 2ⁿ 5^m where n and m are non-negative integers.Firstly, we check co-prime

64 = 2⁶
455 = 5 × 7 × 13
⇒64 and 455 have no common factors
Therefore, 64 and 455 are co-prime.
Now, we have to check that q is in the form of 2ⁿ5^m
455 = 5 × 7 × 13
So, denominator is not of the form 2ⁿ5^m
Thus, $\frac{64}{455}$ is a non-terminating repeating decimal.

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Question 1 H 

Without actually performing the long division, state whether the following rational numbers have terminating or non-terminating repeating (recurring) decimal expansion.

$\dfrac{6}{15}$
Sol :
Given rational number is $\frac{6}{15}$
$\frac{6}{15}=\frac{2}{5}$
$\frac{p}{q}$ is terminating if
a) p and q are co-prime &
b) q is of the form of 2ⁿ 5^m where n and m are non-negative integers.Firstly, we check co-prime

⇒2 and 5 have no common factor
Therefore, 2 and 5 are co-prime.
Now, we have to check that q is in the form of 2ⁿ5^m
5 = 5¹ × 1
= 5¹ × 2⁰
So, denominator is of the form 2ⁿ5^m where n = 0 and m = 1
Thus, $\frac{6}{15}$ is a terminating decimal.

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Question 1 I 

Without actually performing the long division, state whether the following rational numbers have terminating or non-terminating repeating (recurring) decimal expansion.

$\dfrac{35}{50}$
Sol :
Given rational number is $\frac{35}{50}$
$\frac{35}{50}=\frac{7}{10}$
$\frac{p}{q}$ is terminating if
a) p and q are co-prime &
b) q is of the form of 2ⁿ 5^m where n and m are non-negative integers.Firstly, we check co-prime

7 = 1 × 7
10 = 2 × 5
⇒ 7 and 10 have no common factor
Therefore, 7 and 10 are co-prime.
Now, we have to check that q is in the form of 2ⁿ5^m
10 = 5¹ × 2¹
So, denominator is of the form 2ⁿ5^m where n = 1 and m = 1
Thus, $\frac{35}{50}$ is a terminating decimal.

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Question 1 J 

Without actually performing the long division, state whether the following rational numbers have terminating or non-terminating repeating (recurring) decimal expansion.

$\dfrac{129}{2^{2} \times 5^{7} \times 7^{5}}$
Sol :
Given rational number is $\frac{129}{2^{2} \times 5^{7} \times 7^{5}}$
$\frac{p}{q}$ is terminating if
a) p and q are co-prime &
b) q is of the form of 2ⁿ 5^m where n and m are non-negative integers.Firstly, we check co-prime

129 = 3 × 43
Denominator = 2² ×5⁷ ×7⁵
⇒129 and 2² ×5⁷ ×7⁵ have no common factors
Therefore, 129 and 2² ×5⁷ ×7⁵ are co-prime.
Now, we have to check that q is in the form of 2ⁿ5^m
Denominator = 2² ×5⁷ ×7⁵
So, denominator is not of the form 2ⁿ5^m
Thus, $\frac{129}{2^{2} \times 5^{7} \times 7^{5}}$ is a non-terminating repeating decimal.

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Question 1 K 

Without actually performing the long division, state whether the following rational numbers have terminating or non-terminating repeating (recurring) decimal expansion.

$\dfrac{2^{2} \times 7}{5^{4}}$
Sol :
Given rational number is $\frac{2^{2} \times 7}{5^{4}}$
$\frac{p}{q}$ is terminating if
a) p and q are co-prime &
b) q is of the form of 2ⁿ 5^m where n and m are non-negative integers.Firstly, we check co-prime

28 = 7 × 2²
625 = 5⁴
⇒ 28 and 625 have no common factors
Therefore, 28 and 625 are co-prime.
Now, we have to check that q is in the form of 2ⁿ5^m
625 = 5⁴ × 1
= 5⁴ × 2⁰
So, denominator is of the form 2ⁿ5^m where n = 0 and m = 4
Thus,$\frac{2^{2} \times 7}{5^{4}}$ is a terminating decimal.

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Question 1 L 

Without actually performing the long division, state whether the following rational numbers have terminating or non-terminating repeating (recurring) decimal expansion.

$\dfrac{29}{243}$
Sol :
Given rational number is $\frac{29}{243}$
$\frac{p}{q}$ is terminating if
a) p and q are co-prime &
b) q is of the form of 2ⁿ 5^m where n and m are non-negative integers.Firstly, we check co-prime

29 = 29 × 1
243 = 3⁵
⇒29 and 243 have no common factors
Therefore, 29 and 243 are co-prime.
Now, we have to check that q is in the form of 2ⁿ5^m
243 = 3⁵
So, the denominator is not of the form 2ⁿ5^m
Thus, $\frac{29}{243}$ is a non- terminating repeating decimal.

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Question 2 A 

Write down the decimal expansions of the following numbers which have terminating decimal expansions.

$\dfrac{17}{8}$
Sol :
We know, $\frac{17}{8}=\frac{17}{2^{3} \times 5^{0}}$
Multiplying and dividing by 5³
$=\frac{17 \times 5^{3}}{2^{3} \times 5^{0} \times 5^{3}}$
$=\frac{17 \times 125}{2^{3} \times 1 \times 5^{3}}$
$=\frac{2125}{(2 \times 5)^{3}}$
$=\frac{2125}{(10)^{3}}$
$=\frac{2125}{1000}$
= 2.125

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Question 2 B 

Write down the decimal expansions of the following numbers which have terminating decimal expansions.

$\dfrac{3}{8}$
Sol :
We know, $\frac{3}{8}=\frac{3}{2^{3} \times 5^{0}}$
Multiplying and dividing by 5³
$\frac{3 \times 5^{3}}{2^{3} \times 5^{0} \times 5^{3}}$
$=\frac{3 \times 125}{2^{3} \times 1 \times 5^{3}}$
$=\frac{375}{(2 \times 5)^{3}}$
$=\frac{375}{(10)^{3}}$
$=\frac{375}{1000}$
= 0.375

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Question 2 C 

Write down the decimal expansions of the following numbers which have terminating decimal expansions.

$\dfrac{29}{343}$
Sol :
We know, $\frac{29}{343}=\frac{29}{7^{3}}$
Given rational number is $\frac{29}{343}$
$\frac{p}{q}$ is terminating if
a) p and q are co-prime &
b) q is of the form of 2ⁿ 5^m where n and m are non-negative integers.Firstly we check co-prime

29 = 29 × 1
343 = 7 × 7 × 7
⇒29 and 343 have no common factors
Therefore, 29 and 343 are co-prime.
Now, we have to check that q is in the form of 2ⁿ5^m
343 = 7³
So, the denominator is not of the form 2ⁿ5^m
Thus, $\frac{29}{343}$ is a non-terminating repeating decimal.

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Question 2 D 

Write down the decimal expansions of the following numbers which have terminating decimal expansions.

$\dfrac{13}{125}$
Sol :
We know, $\frac{13}{125}=\frac{13}{2^{0} \times 5^{3}}$
Multiplying and dividing by 2³
$=\frac{13 \times 2^{3}}{2^{0} \times 5^{3} \times 2^{3}}$
$=\frac{13 \times 8}{1 \times 2^{3} \times 5^{3}}$
$=\frac{104}{(2 \times 5)^{3}}$
$\frac{104}{(10)^{3}}$
$=\frac{104}{1000}$
= 0.104

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Question 2 E 

Write down the decimal expansions of the following numbers which have terminating decimal expansions.

$\dfrac{27}{8}$

Sol :
We know, $\frac{27}{8}=\frac{3^{3}}{2^{3} \times 5^{0}}$
Multiplying and dividing by 5³
$\frac{27 \times 5^{3}}{2^{0} \times 2^{3} \times 5^{3}}$
$=\frac{27 \times 125}{1 \times 2^{3} \times 5^{3}}$
$=\frac{3375}{(2 \times 5)^{3}}$
$\frac{3375}{(10)^{3}}$
$=\frac{3375}{1000}$
= 3.375

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Question 2 F 

Write down the decimal expansions of the following numbers which have terminating decimal expansions.

$\dfrac{7}{80}$
Sol :
We know, $\frac{7}{80}=\frac{7}{2^{4} \times 5^{1}}$
Multiplying and dividing by 5³
$=\frac{7 \times 5^{3}}{2^{4} \times 5^{1} \times 5^{3}}$
$=\frac{7 \times 125}{2^{4} \times 5^{4}}$
$=\frac{875}{(2 \times 5)^{4}}$
$=\frac{875}{(10)^{4}}$
$=\frac{875}{10000}$
= 0.0875

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Question 2 G 

Write down the decimal expansions of the following numbers which have terminating decimal expansions.

$\dfrac{64}{455}$
Sol :
We know, $\frac{64}{455}=\frac{2^{6}}{5 \times 7 \times 13}$
Since the denominator is not of the form 2ⁿ5^m
$\frac{64}{455}$ has a non-terminating repeating decimal expansion.

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Question 2 H 

Write down the decimal expansions of the following numbers which have terminating decimal expansions.

$\dfrac{6}{15}$
Sol :
We know, $\frac{6}{15}=\frac{2}{5}=\frac{2}{2^{0} \times 5}$
Multiplying and dividing by 2¹
$=\frac{2 \times 2}{2^{0} \times 5^{1} \times 2}$
$=\frac{4}{1 \times 10}$
$=\frac{4}{10}$
= 0.4

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Question 2 I 

Write down the decimal expansions of the following numbers which have terminating decimal expansions.

$\dfrac{35}{50}$
Sol :
We know, $\frac{35}{50}$
$=\frac{7}{10}$
$=\frac{7}{2 \times 5}$
$=\frac{7}{10}$
= 0.7

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Question 2 J 

Write down the decimal expansions of the following numbers which have terminating decimal expansions.

$\frac{129}{2^{2} 5^{7} 7^{5}}$

Sol :
Given rational number is $\frac{129}{2^{2} \times 5^{7} \times 7^{5}}$
Since the denominator is not of the form 2ⁿ5^m
$\frac{129}{2^{2} \times 5^{7} \times 7^{5}}$has a non-terminating repeating decimal expansion.

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Question 2 K 

Write down the decimal expansions of the following numbers which have terminating decimal expansions.

$\dfrac{2^{2} \times 7}{5^{4}}$
Sol :
We know, $\frac{2^{2} \times 7}{5^{4}}=\frac{2^{2} \times 7}{2^{0} \times 5^{4}}$
Multiplying and dividing by 2⁶
$\frac{2^{2} \times 7 \times 2^{6}}{2^{0} \times 5^{4} \times 2^{6}}$
$=\frac{2^{2} \times 7 \times 2^{6}}{1 \times 5^{4} \times 2^{6}}$
$=\frac{7 \times 2^{6}}{(2 \times 5)^{4}}$
$=\frac{448}{(10)^{4}}$
$=\frac{448}{10000}$
= 0.0448

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Question 2 L 

Write down the decimal expansions of the following numbers which have terminating decimal expansions.

$\dfrac{29}{243}$
Sol :
Given rational number is $\frac{29}{243}$
$\frac{29}{243}=\frac{29}{3^{5}}$
Since the denominator is not of the form 2ⁿ5^m.
$\frac{29}{243}$ has a non-terminating repeating decimal expansion.

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Question 3 

The following real numbers have decimal expansions as given below. In each case examine whether they are rational or not. If they are a rational number of the form p/q, what can be said about q?

(i) 7.2345
(ii) $5 . \overline{234}$
(iii) 23.245789
(iv) $7 . \overline{3427}$
(v) 0.120120012000120000…
(vi) 23.142857
(vii) 2.313313313331…
(viii) 0.02002000220002…
(ix) 3.300030000300003…
(x) 1.7320508…
(xi) 2.645713
(xii) 2.8284271…
Sol :

(i) 7.2345
Here, 7.2345 has terminating decimal expansion.
So, it represents a rational number.
i.e. 7.2345 $=\frac{7.2345}{10000}=\frac{p}{q}$
Thus, q = 10⁴, those factors are 2³ × 5³

(ii) $5 . \overline{234}$
$5 . \overline{234}$ is non-terminating but repeating.
So, it would be a rational number.
In a non-terminating repeating expansion of $\frac{p}{q}$ ,
q will have factors other than 2 or 5.

(v) 0.120120012000120000…
0.120120012000120000… is non-terminating and non-repeating.
So, it is not a rational number as we see in the chart.