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NCERT Solutions for Class 10 Maths Chapter 1 - Exercise 1.4

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NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers (Ex 1.4) Exercise 1.4

Free PDF download of NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.4 (Ex 1.4) and all chapter exercises at one place prepared by expert teacher as per NCERT (CBSE) books guidelines. Class 10 Maths Chapter 1 Real Numbers Exercise 1.4 Questions with Solutions to help you to revise complete Syllabus and Score More marks. Register and get all NCERT  Solutions in your emails.


Class:

NCERT Solutions for Class 10

Subject:

Class 10 Maths

Chapter Name:

Chapter 1 - Real Numbers

Exercise:

Exercise - 1.4

Content-Type:

Text, Videos, Images and PDF Format

Academic Year:

2024-25

Medium:

English and Hindi

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You can also Download Maths NCERT Solutions Class 10 to help you to revise complete Syllabus and score more marks in your examinations. Vedantu not only provides Solutions for maths but also other subjects as well like you can download Class 10 Science Solutions for free.

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NCERT Solutions for Class 10 Maths Chapter 1

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) $\mathbf{\frac{13}{3125}}$

Ans: Given a rational number $\frac{13}{3125}$.

If the denominator of a rational number has prime factors of the form 2n  5m, where, m and n are positive integers. Then the rational number has a terminating decimal  expansion. If the denominator has factors other than 2and 5, then it has non terminating decimal expansion. 

The denominator of the given number is 3125. 

Then, factors of 3125 are 

⇒312= 5$\times$ 5$\times$ 5$\times$ 5$\times$ 5$\times$ 5 

⇒ 3125 =  55

Here, the factors of  denominator are of the form 5m

Therefore, $\frac{13}{3125}$  has terminating decimal expansion. 

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

Ans: Given a rational number $\frac{13}{3125}$. 

If the denominator of a rational number has prime factors of the form 2n 5m, where, m and n are positive integers. Then the rational number has terminating decimal  expansion. If the denominator has factors other than 2 and 5, then it has non terminating decimal expansion. 

The denominator of the given number is 8 . 

Then, factors of 8 are

⇒ 8= 28 $\times $ 28 $\times $ 2 

 ⇒ 8 =23 

Here, the factors of denominator are of the form 2 n

Therefore,$\frac{13}{3125}$ has terminating decimal expansion. 

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

Ans: Given a rational number $\frac{64}{455}$.

If the denominator of a rational number has prime factors of the form 2n 5m , where, m and n are positive integers. Then the rational number has a terminating decimal  expansion. If the denominator has factors other than 2and 5, then it has non terminating decimal expansion. 

The denominator of the given number is 455. 

Then, factors of 455 are 

⇒ 455 =  5 $\times$ 7 $\times$ 13 

Here, the factors of denominator are not in the form  2n 5m. The denominator has factors  other than 2and 5. 

Therefore, $\frac{64}{455}$ has non-terminating repeating decimal expansion. 

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

Ans: Given a rational number $\frac{15}{1600}$

If the denominator of a rational number has prime factors of the form 2n 5m, where, m and n are positive integers. Then the rational number has terminating decimal  expansion. If the denominator has factors other than 2and 5, then it has non terminating decimal expansion. 

The denominator of the given number is 1600. 

Then, factors of 1600 are 

⇒  1600 =  2$\times$ 2$\times$ 2$\times$ 2$\times$ 2$\times$ 2$\times$ 5$\times$ 5 

 ⇒ 1600 =  26 $\times$ 55

Here, the factors of denominator are of the form 2n 5m .

Therefore,$\frac{15}{1600}$ has terminating decimal expansion. 

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

Ans: Given a rational number $\frac{29}{343}$ 

If the denominator of a rational number has prime factors of the form 2n  5m, where, m and n are positive integers. Then the rational number has a terminating decimal  expansion. If the denominator has factors other than 2and 5, then it has non terminating decimal expansion. 

The denominator of the given number is 343. 

Then, factors of 343 are 

⇒343=  7$\times$ 7$\times $4 7 

⇒343 =  73 

Here, the factors of denominator are not in the form 2n 5m. The denominator has factors  other than 2and 5. 

Therefore, $\frac{29}{343}$  has non-terminating repeating decimal expansion. 

(vi) $\mathbf{\frac{23}{2^3 5^2}}$ 

Ans: Given a rational number $\frac{23}{2^3 5^2}$

If the denominator of a rational number has prime factors of the form 2n 5m, where, m and n are positive integers. Then the rational number has a terminating decimal  expansion. If the denominator has factors other than 2 and 5, then it has non terminating decimal expansion. 

The denominator of the given number is 2352

Here, the denominator is of the form 2n 5m

Therefore,$\frac{23}{2^3 5^2}$ has terminating decimal expansion. 

(vii)  $\mathbf{\frac{129}{2^2 5^7  7^5}}$

Ans: Given a rational number $\frac{129}{2^2 5^7  7^5}$

If the denominator of a rational number has prime factors of the form  2n  5m, where, m and n are positive integers. Then the rational number has terminating decimal  expansion. If the denominator has factors other than 2 and 5, then it has non terminating decimal expansion. 

The denominator of the given number is  225775

Here, the denominator is of the form 2n 5m but also has factors other than 2 and 5. 

Therefore, $\frac{129}{2^2 5^7  7^5}$has non-terminating repeating decimal expansion. 

(viii)  $\mathbf{\frac{6}{15}}$

Ans: Given a rational number $\frac{6}{15}$

If the denominator of a rational number has prime factors of the form 2n 5m, where, m and n are positive integers. Then the rational number has a terminating decimal  expansion. If the denominator has factors other than 2and 5, then it has non terminating decimal expansion. 

The denominator of the given number is 15. 

Then, factors of 3125 are 

⇒ 15 =  3 $\times$  5 

But we can write the numerator of the given number as 

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

Here, the factors of denominator are of the form  5m

Therefore, $\frac{6}{15}$ has terminating decimal expansion. 

(ix)  $\mathbf{\frac{35}{50}}$

Ans: Given a rational number $\frac{35}{50}$

If the denominator of a rational number has prime factors of the form 2n 5m, where, m and n are positive integers. Then the rational number has a terminating decimal expansion. If the denominator has factors other than 2and 5, then it has non terminating decimal expansion. 

The denominator of the given number is 50. 

Then, factors of 50 are 

⇒ 50  = 10  \$times$ 5 

But we can write the numerator of the given number as 

 $\frac{35}{50}$ =  $\frac{7 \times5}{5 \times 10}$

⇒ 10 =  2 $\times$ 5 

Here, the factors of denominator are of the form 2n  5m  . 

Therefore, $\frac{35}{50}$ has terminating decimal expansion. 

(x)  $\mathbf{\frac{77}{210}}$

Ans: Given a rational number  $\frac{77}{210}$

If the denominator of a rational number has prime factors of the form 2n  5m , where, m and n are positive integers. Then the rational number has a terminating decimal  expansion. If the denominator has factors other than 2 and 5, then it has non terminating decimal expansion. 

The denominator of the given number is 210 . 

Then, factors of 210 are 

⇒ 210 =  2 $\times$ 3 $\times$5 $\times$7 

Here, the denominator has factors other than 2and 5. 

Therefore,$\frac{77}{210}$  has non-terminating repeating decimal expansion. 


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

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

Ans: To find the decimal expansion of $\frac{13}{3125}$, we will divide the numerator of the  

number by the denominator using a long division method. We get

$\;\;\;\;\;\;\;\;\;\;0.00416\\3125\;\;)\overline{13.00000}\\\;\;\;\;\;\;\;\;\;\; 0 \\\;\;\;\;\;\;\;\;\;\;\overline{13000}\\\;\;\;\;\;\;\;\;\;\;12500\\\;\;\;\;\;\;\;\;\;\;\overline{5000}\\\;\;\;\;\;\;\;\;\;\;3125\\\;\;\;\;\;\;\;\;\;\;\overline{18750}\\\;\;\;\;\;\;\;\;\;\;18750\\\;\;\;\;\;\;\;\;\;\;\overline{0}$

Therefore, the decimal expansion of  $\frac{13}{3125}$ is 0.00416.  

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

Ans: To find the decimal expansion of $\frac{17}{8}$,  we will divide the numerator of the number  by denominator using long division method. We get 

$\;\;\;\;\;\;\;\;\;\;2.125\\8\;\;)\overline{17}\\\;\;\;\;\;\;16\\\;\;\;\;\;\;\;\overline{10}\\\;\;\;\;\;\;\;8\\\;\;\;\;\;\;\;\overline{20}\\\;\;\;\;\;\;\;16\\\;\;\;\;\;\;\;\overline{40}\\\;\;\;\;\;\;\;40\\\;\;\;\;\;\;\;\overline{0}$

Therefore, the decimal expansion of  $\frac{17}{8}$  is 2.125.  

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

Ans: To find the decimal expansion of $\frac{15}{1600}$, we will divide the numerator of the  

number by the denominator using a long division method. We get 

$\;\;\;\;\;\;\;\;\;\;0.005375\\1600\;\;)\overline{15.000000}\\\;\;\;\;\;\;\;\;\;0\\\;\;\;\;\;\;\;\;\;\;\overline{150}\\\;\;\;\;\;\;\;\;\;\;0\\\;\;\;\;\;\;\;\;\;\;\overline{15000}\\\;\;\;\;\;\;\;\;\;\;14400\\\;\;\;\;\;\;\;\;\;\;\overline{6000}\\\;\;\;\;\;\;\;\;\;\;4800\\\;\;\;\;\;\;\;\;\;\;\overline{12000}\\\;\;\;\;\;\;\;\;\;\;11200\\\;\;\;\;\;\;\;\;\;\;\overline{8000}\\\;\;\;\;\;\;\;\;\;\;8000\;\;\;\;\;\;\;\;\;\;\\ \overline{0}$

Therefore, the decimal expansion of $\frac{15}{1600}$  is 0.009375.  

(iv) $\mathbf{\frac{23}{2^3 5^2}}$

Ans: To find the decimal expansion of    $\frac{23}{2^3 5^2}$

we will divide the numerator of the number  

by the denominator using a long division method. We get

$\frac{23}{2^3 5^2}$ = $\frac{23}{200}$

$\;\;00.115\\\;\;\;\;200)  \overline{23.000}\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; 0\\\;\;\;\;\;\;\;\;\;\; \overline{23}\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\\ \;\;\;\;\;\;\;\;\;\;\;\;\;0 \\\;\;\;\;\;\;\;\;\;\;\; \overline {230}  \\\;\;\;\;\;\;\;\;\;\;\; 200\\\;\;\;\;\;\;\;\;\;\;\; \overline{1000}\\ \;\;\;\;\;\;\;\;\;\;\; 1000 \\\;\;\;\;\;\;\;\;\;\;\; \overline{0}$

Therefore, the decimal expansion of  $\frac{23}{200}$ is 00.115.

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

Ans: To find the decimal expansion of $\frac{6}{15}$, we will divide the numerator of the number  by the denominator using a long division method. We get 

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

$\;\;\;\;\;\;\;\;0.4\\\;\;\;\;\;5)\overline{2.0}\\ \;\;\;\;\;\; \;\;\;0 \\ \;\;\;\;\;\;\;\;\overline{20}\\\;\;\;\;\;\;\;\;20\\ \;\;\;\;\;\;\;\; \overline{0}$ 

Therefore, the decimal expansion of $\frac{6}{15}$ is 0.4. 

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

Ans: To find the decimal expansion of $\frac{35}{50}$,  we will divide the numerator of the number  by the denominator using a long division method. We get 

$\;\;\;\;\;\;\;\;0.7\\\;\;\;\;\;50)\overline{35.0}\\ \;\;\;\;\;\; \;\;\;0 \\ \;\;\;\;\;\;\;\;\overline{350}\\\;\;\;\;\;\;\;\;350\\ \;\;\;\;\;\;\;\; \overline{0}$

Therefore, the decimal expansion of $\frac{6}{15}$  is 0.7. 


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 $\mathbf{\frac{p}{q}}$ what can you say about the prime factors of q? 

(i) 43.123456789 

Ans: Given a decimal expansion 43.123456789. 

The given number has terminating expansion, we can write the number as $\frac{43124356789}{1000000000}$, , which is of the form pq. 

Therefore, the number 43.123456789is a rational number. 

Since the number has terminating decimal expansion, the factors of q must be of the  form 2n 5m

(ii) 0.120120012000120000...... 

Ans: Given a decimal expansion 0.120120012000120000.......

When we observe the given expansion we can say that the number has non-terminating  and non-repeating decimal expansion. Hence we cannot express it in the form of $\frac{p}{q}$ Therefore, the number is irrational. 

(iii) 43.123456789 

Ans: Given the decimal expansion 43.123456789. 

The given number has non-terminating but repeating decimal expansion. So the  number will be of the form$\frac{p}{q}$ . 

Therefore, the number 43.123456789is a rational number. 

But the factors of denominator are not of the form 2n 5m. Denominator also has factors  other than 2and 5.


NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers Exercise 1.4

Opting for the NCERT solutions for Ex 1.4 Class 10 Maths is considered as the best option for the CBSE students when it comes to exam preparation. This chapter consists of many exercises. Out of which we have provided the Exercise 1.4 Class 10 Maths NCERT solutions on this page in PDF format. You can download this solution as per your convenience or you can study it directly from our website/ app online.


Vedantu in-house subject matter experts have solved the problems/ questions from the exercise with the utmost care and by following all the guidelines by CBSE. Class 10 students who are thorough with all the concepts from the Maths textbook and quite well-versed with all the problems from the exercises given in it, then any student can easily score the highest possible marks in the final exam. With the help of this Class 10 Maths Chapter 1 Exercise 1.4 solutions, students can easily understand the pattern of questions that can be asked in the exam from this chapter and also learn the marks weightage of the chapter. So that they can prepare themselves accordingly for the final exam.


Besides these NCERT solutions for Class 10 Maths Chapter 1 Exercise 1.4, there are plenty of exercises in this chapter which contain innumerable questions as well. All these questions are solved/answered by our in-house subject experts as mentioned earlier. Hence all of these are bound to be of superior quality and anyone can refer to these during the time of exam preparation. In order to score the best possible marks in the class, it is really important to understand all the concepts of the textbooks and solve the problems from the exercises given next to it. 


Do not delay any more. Download the NCERT solutions for Class 10 Maths Chapter 1 Exercise 1.4 from Vedantu website now for better exam preparation. If you have the Vedantu app in your phone, you can download the same through the app as well. The best part of these solutions is these can be accessed both online and offline as well. 


Decimal Expansions of Rational Numbers

Rational Numbers: Rational numbers are numbers that are written in the form p/q, (p in the numerator and q in the denominator) where p and q are integers and q≠0. We can say all fractions are rational numbers.


Examples of rational numbers: 1/3, 3/10, -7/10,−3


Terminating and nonterminating decimals

1. Terminating decimals: The decimal values whose value is definite and terminate at a certain digit are said to be Terminating decimals.

Example: 0.4, 5.8, and so on.


2. Non-terminating: The decimal values in which the digits after the decimal point do not terminate and the decimal value is not exactly known are called Non-terminating decimals.

Example: 0.333333….., 0.13135235343…


Non-terminating decimals can be classified again as:

  • Non-terminating recurring Decimals: In this type of decimal number, a part of the decimal indefinitely repeats itself. Eg: 0.142869142869142869….(where 142869 is repeating after the decimal point).

  • Non-terminating non-recurring Decimals: In this type of decimal number, no part of the decimal repeats indefinitely. Example: π=3.1415926535…


How to check whether a given rational number is terminating or not?

If a rational number a/b is given, then its decimal expansion of the rational number will terminate if it satisfies the below two conditions:

a) The H.C.F of a (numerator) and b (denominator) of the given rational number should be 1.

b) The denominator of the given rational number(b) can be expressed as a prime factorization of 2 and 5 i.e b=2m×5n where m or n, or both can be 0. If the prime factorization of b contains any number other than 2 or 5, then the decimal expansion of that number will be a non-terminating decimal.


NCERT Solutions for Class 10 Maths Chapter 1 Exercises

Chapter 1 - Real Numbers Exercises in PDF Format

Exercise 1.1

5 Questions & Solutions (4 Long Answers, 1 Short Answer)

Exercise 1.2

7 Questions & Solutions (4 Long Answers, 3 Short Answers)

Exercise 1.3 

3 Questions & Solutions (3 Short Answers)

FAQs on NCERT Solutions for Class 10 Maths Chapter 1 - Exercise 1.4

1. What is a Rational number?

Any real number that takes the form p/q and has q not equal to zero is a rational number. Any fraction with a non-zero denominator is considered rational. Half, one-fifth, three-quarters, and so on are a few examples of rational numbers. Since it may be written as 0/1, 0/2, 0/3, etc., the number "0" is also a rational number. But because they give us endless values, 1/0, 2/0, 3/0, etc., are not reasonable.

2. What do you mean by terminating decimal expansion?

A decimal with an end digit is referred to as a "terminating decimal expansion." It is a decimal, which means that its digits are limited. After a limited number of steps, the decimal expansion comes to an end. Terminating decimals is the name given to these kinds of decimal expansions. It indicates that the numbers reach their maximum value after the decimal point. For instance, 1/2 is a rational number, and its decimal expansion is 0.5.

3. What do you mean by non-terminating repeating decimal expansion?

A decimal with an infinite number of digits after the decimal point is referred to as a non-terminating repeating decimal expansion. A decimal where some of the digits after the decimal point repeat without ending is known as a non-terminating, repeating decimal.

4. Why should one opt for Vedantu for the NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers Exercise 1.4?

When it comes to exam preparation, choosing the NCERT solutions for Ex. 1.4 Class 10 Math is thought to be the best choice for CBSE students. There are numerous exercises in this chapter. On this page, in PDF format, we have the Exercise 1.4 Class 10 Maths NCERT solutions. You can study this solution directly from our website or mobile app, or you can download it as needed.

5. Is NCERT Class 10 Chapter 1 Real Numbers Exercise 1.4 difficult?

No, Real Numbers Exercise 1..4 is not difficult at all. It only includes questions related to the expansion of decimals in real numbers. With a clear conceptual understanding of decimal expansion, one can easily solve all the questions of that particular exercise. Constant practice will enhance your solving capacity and knowledge of real numbers and their decimal expansion.