NCERT Solutions for Class 10 Maths Chapter 1 - Real Numbers Exercise 1.4: Detailed Guide for Assam State Board Syllabus 2025-2026 (English Medium)

Free Solutions for Assam State Board Class 10 Maths Chapter 1 Exercise 1.4 in English Medium

 

NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers Exercise 1.4 – Digital Pipal Academy

Digital Pipal Academy brings you well-structured NCERT Solutions for Class 10 Maths Chapter 1, Exercise 1.4, covered under the Assam State Board Syllabus 2025-2026. This exercise focuses on exploring the concepts of irrational numbers and their representation. Our expert faculty provides step-by-step solutions to simplify learning and enhance understanding. Aligned with NCERT guidelines, these solutions help students prepare effectively and perform well in their exams.

Real Number	Solution Link
Exercise 1.1 (Real Number)	Click Here
Exercise 1.2 (Real Number)	Click Here
Exercise 1.3 (Real Number)	Click Here
Exercise 1.4 (Real Number)	Click Here

 

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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:

(i) $\frac{13}{3125}$

$$\frac{13}{3125}$$ $$3125 = 5^5$$

Since the denominator can be represented as a power of 5,

$$\frac{13}{3125}$$

has a terminating decimal expansion.

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

$$\frac{17}{8} = \frac{17}{2^3}$$

Since the denominator has factors of the form $2^m 5^n$,

$$\frac{17}{8}$$

has a terminating decimal expansion.

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(iii) $\frac{64}{455}$

$$\frac{64}{455}$$ $$455 = 5 \times 7 \times 13$$

Since the denominator contains prime factors other than 2 and 5,

$$\frac{64}{455}$$

has a non-terminating but repeating decimal expansion.

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

$$\frac{15}{1600} = \frac{3}{320}$$ $$320 = 2^6 \times 5$$

Since the denominator has factors of the form $2^m 5^n$,

$$\frac{15}{1600}$$

has a terminating decimal expansion.

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(v) $\frac{29}{343}$

$$\frac{29}{343} = \frac{29}{7^3}$$

Since the denominator contains prime factors other than 2 and 5,

$$\frac{29}{343}$$

has a non-terminating but repeating decimal expansion.

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(vi) $\frac{23}{2352}$

$$\frac{23}{2352} = \frac{23}{2^4 \times 3 \times 7}$$

Since the denominator is of the form $2^m 5^n$,

$$\frac{23}{2352}$$

has a terminating decimal expansion.

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(vii) $\frac{129}{225775}$

Since the denominator has prime factors other than 2 and 5,

$$\frac{129}{225775}$$

has a non-terminating but repeating decimal expansion.

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

$$\frac{6}{15} = \frac{2}{5}$$

Since the denominator has factors of the form $2^m 5^n$,

$$\frac{6}{15}$$

has a terminating decimal expansion.

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

$$\frac{35}{50} = \frac{7}{10}$$ $$10 = 2 \times 5$$

Since the denominator has factors of the form $2^m 5^n$,

$$\frac{35}{50}$$

has a terminating decimal expansion.

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

$$\frac{77}{210} = \frac{11}{2 \times 3 \times 5}$$

Since the denominator contains prime factors other than 2 and 5,

$$\frac{77}{210}$$

has a non-terminating but repeating decimal expansion.

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2. Write down the decimal expansions of those rational numbers in Question 1 above which have a terminating decimal expansion.

(i) $\frac{13}{3125}$

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

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

$$\frac{17}{8} = 2.125$$

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

$$\frac{15}{1600} = 0.009375$$

(vi) $\frac{23}{2352}$

$$\frac{23}{200} = 0.115$$

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

$$\frac{6}{15} = 0.4$$

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

$$\frac{35}{50} = 0.7$$

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3. The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not.

(i) $43.123456789$

Since this number has a terminating decimal expansion, it is rational and can be written in the form $\frac{p}{q}$, where the denominator is of the form $2^m 5^n$.

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(ii) $0.120120012000120000...$

This number has a non-terminating, non-repeating decimal expansion, so it is irrational. It cannot be written in the form $\frac{p}{q}$.

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(iii) $43.123456789\overline{123456789}$

This number has a non-terminating but repeating decimal expansion, so it is rational. It can be written in the form $\frac{p}{q}$, where the denominator has factors other than just 2 and 5.

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