NCERT Solutions Class 10 Maths Chapter 1 Real Numbers Ex 1.4

NCERT Solutions Class 10 Maths
Chapter – 1 (Real Numbers)

The NCERT Solutions in English Language for Class 10 Mathematics Chapter – 1 Real Numbers Exercise 1.4 has been provided here to help the students in solving the questions from this exercise.

Chapter : 1 Real Numbers

Exercise – 1.4

1. 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:
NCERT Class 10 Maths Solution
Solutions –
(i) $\mathbf{\frac{13}{3125}}$
Factorizing the denominator, we get,
3125 = 5 × 5 × 5 × 5 × 5 = 5⁵The denominator is of the form 5^m.
Hence, the decimal expansion of $\frac{13}{3125}$ is terminating.

(ii) $\mathbf{\frac{17}{8}}$
Factorizing the denominator, we get,
8 = 2 × 2 × 2 = 2³The denominator is of the form 2^m.
Hence, the decimal expansion of $\frac{17}{8}$ is terminating.

(iii) $\mathbf{\frac{64}{455}}$
Factorizing the denominator, we get,
455 = 5 × 7 × 13
Since, the denominator is not in the form of 2^m × 5ⁿ, thus $\frac{64}{455}$ has a non-terminating decimal expansion.

(iv) $\mathbf{\frac{15}{1600}}$
Factorizing the denominator, we get,
1600 = 2 × 2 × 2 × 2 × 2 × 2× 5 × 5 = 2⁶× 5²Since, the denominator is in the form of 2^m × 5ⁿ, thus $\frac{15}{1600}$ has a terminating decimal expansion.

(v) $\mathbf{\frac{29}{343}}$
Factorizing the denominator, we get,
343 = 7 × 7 × 7 = 7³
Since, the denominator is not in the form of 2^m × 5ⁿ thus $\frac{29}{343}$ has a non-terminating decimal expansion.

(vi) $\mathbf{\frac{23}{2^3 5^2}}$
Clearly, the denominator is in the form of 2^m × 5ⁿ.
Hence, $\frac{23}{2^3 5^2}$ has a terminating decimal expansion.

(vii) $\mathbf{\frac{129}{2^5 5^7 7^5}}$
As you can see, the denominator is not in the form of 2^m × 5ⁿ.
Hence, $\frac{129}{2^5 5^7 7^5}$ has a non-terminating decimal expansion.

(viii) $\mathbf{\frac{6}{15}}$

$\frac{6}{15} = \frac{2}{5}$
Since, the denominator has only 5 as its factor, thus, $\frac{6}{15}$ has a terminating decimal expansion.

(ix) $\mathbf{\frac{35}{50}}$

$\frac{35}{50} = \frac{7}{10}$
Factorising the denominator, we get,
10 = 2 × 5
Since, the denominator is in the form of 2^m × 5ⁿ thus, $\frac{35}{50}$ has a terminating decimal expansion.

(x) $\mathbf{\frac{77}{210}}$

$\frac{77}{210}$ = $\frac{11}{30}$
Factorising the denominator, we get,
30 = 2 × 3 × 5
As you can see, the denominator is not in the form of 2^m × 5ⁿ.
Hence, $\frac{77}{210}$ has a non-terminating decimal expansion.

2. Write down the decimal expansions of those rational numbers in Question 1 above which have terminating decimal expansions.
Solutions –
(i) $\mathbf{\frac{13}{3125}}$

NCERT Class 10 Maths Solution

$\frac{13}{3125}$ = 0.00416

(ii) $\mathbf{\frac{17}{8}}$

NCERT Class 10 Maths Solution
$\frac{17}{8}$ = 2.125

(iii) $\mathbf{\frac{64}{455}}$ has a non terminating decimal expansion

(iv) $\mathbf{\frac{15}{1600}}$

NCERT Class 10 Maths Solution
$\frac{15}{1600}$ = 0.009375

(v) $\mathbf{\frac{29}{343}}$ has a non terminating decimal expansion

(vi) $\mathbf{\frac{23}{2^3 5^2}}$ = $\frac{23}{8\times 25} = \frac{23}{200}$

NCERT Class 10 Maths Solution
$\frac{23}{2^3 5^2}$ = 0.115

(vii) $\mathbf{\frac{129}{2^5 5^7 7^5}}$ has a non terminating decimal expansion

(viii) $\mathbf{\frac{6}{15}}$ = $\frac{2}{5}$

$\frac{2}{5}$ = 0.4

(ix) $\mathbf{\frac{35}{50}}$

NCERT Class 10 Maths Solution
$\frac{35}{50}$ = 0.7

(x) $\mathbf{\frac{77}{210}}$ has a non-terminating decimal expansion.

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, p q what can you say about the prime factors of q?
(i) 43.123456789
(ii) 0.120120012000120000. . .
(iii) 43. $\mathbf{\overline{123456789}}$

Solutions –

(i) 43.123456789
Since it has a terminating decimal expansion, it is a rational number in the form of p/q and q has factors of 2 and 5 only.

(ii) 0.120120012000120000 . . .
Since, it has non-terminating and non- repeating decimal expansion, it is an irrational number.

(iii) 43. $\mathbf{\overline{123456789}}$
Since it has non-terminating but repeating decimal expansion, it is a rational number in the form of p/q and q has factors other than 2 and 5.

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NCERT Solutions Class 10 Maths Chapter 1 Real Numbers Ex 1.4