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

On the Extramarks website and mobile app, students can avail the NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.4. These NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.4 have been prepared by the subject specialists at Extramarks. They periodically evaluate these NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.4 and make adjustments to make them simpler for the users to understand.

These answers are simple to practise and aid in quick revision for the students. In NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.4, which is the fourth exercise, rational numbers and its decimal expansion are further explained. The NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.4 are provided in a step-by-step fashion to make them simple to understand. Additionally, when preparing, Extramarks professionals made a point of adhering to all NCERT criteria.

NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.4: the fourth exercise in NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.4 is based on Real Numbers. Real Numbers were first introduced in Class 9 and then examined in greater detail in Class 10. The rational numbers and their decimal expansions are explained in this class. The exercise investigates the precise conditions under which a rational integer decimal expansion terminates and when it does not.Rational Numbers and their Decimal Expansions: A Review – There are 3 questions total, with question 1 having 10 parts and question 3 having only 3 parts.

When it comes to exam preparation, choosing the NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.4 is thought to be the best choice for CBSE students. There are numerous exercises in this chapter. The NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.4 are available on Extramarks’ website in PDF format. Students can study these NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.4 straight from the Extramarks website or mobile app, or they can download it as convenient for them.

The problems and questions from the exercise have been meticulously addressed by the internal Extramarks subject matter experts while adhering to all CBSE norms. Any student  in Class 10 who is comprehensive with all the ideas from the Mathematics textbook and quite knowledgeable with all the activities provided in it can easily earn the best scores on the final test. Students may quickly comprehend the types of problems that may be given in the examination from this chapter and learn the chapter’s weight in terms of overall grade points with the aid of these NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.4 so that they can adequately study for the final exam.

There are numerous exercises in this chapter that have numerous questions in addition to these NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.4. As previously noted, Extramarks’ internal topic experts have already resolved or responded to all of these inquiries. Because of this, they are all guaranteed to be of the highest quality, and anyone can use them to study for exams. It is crucial to comprehend all the concepts in the NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.4 and work through the exercises that are provided in order to receive the greatest grades possible in the class.

For better test preparation, students are encouraged to download the NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.4 from the Extramarks website or mobile application. Users can obtain the NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.4 using the application if they already have the Extramarks mobile app installed on their device. The best feature of these NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.4 is that they can be used offline and online.

NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.4 are available on the Extramarks website and mobile app for students to review rational numbers and their decimal expansion. Real Numbers is a remarkably useful tool. Three questions make up NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.4, two of which require lengthy responses while one is simple to answer.

Students are urged to keep in mind a few of the key theorems listed below that were mentioned in this section before moving on to solving the practise questions in the NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.4:

• If x is a rational number for which the decimal expansion comes to an end, then x can be expressed as p/q, where p and q are coprimes, q’s prime factorization is 2n5m, and n and m are non-negative integers.
• The decimal expansion of “x” comes to an end if x = p/q is a rational number and “q” has a prime factorization of the type 2n5m, where “n” and “m” are non-negative integers.
• The decimal expansion of a rational integer, x = p/q, where ‘p’ and ‘q’ are coprimes, is said to be non-terminating repeating if the prime factorization of ‘q’ cannot be represented as 2n5m (recurring).

The decimal expansion of every rational number is either terminating or non-terminating and recurring, as the students will see.

The student will learn as they go through problems that any real number with a terminating decimal expansion can be written as a rational number with a power of 10 as the denominator.

Through the use of pattern recognition, students can use this activity to answer real-world problems.

Through NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.4 students can examine rational numbers’ decimal expansion with the aid of real numbers. Essential terms like terminating and non-terminating decimals will be introduced to students. Certain rational numbers have a portion of the decimal that repeatedly repeats itself when we convert them to decimal form. These are referred to as non-terminating decimals, and examples include 0.333 and others. On the other hand, the term is said to be terminating if no component of the decimal is repeated, for instance, pi= 3.1415926.

The three theorems about the characteristics of rational numbers described above serve as the foundation for the NCERT Solutions for Class 10 Chapter 1 Exercise 1.4 in Class 10 Mathematics. The concepts of terminating and non-terminating decimals are used in all of the questions. Students can learn about these theorems in steps by using the examples that are presented in this activity.

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

NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.4 The Extramarks academic team has created the NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.4 based on Real Numbers. All of the exercises in this chapter have solutions crafted by Extramarks professionals. Below are details on the NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.4. The provided chapter’s formula has been taught to the students. For Class 10 Mathematics, Extramarks experts have created detailed notes and extra questions with concise summaries of all the formulas. Before tackling the NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1. 4 by Extramarks, students are advised to read the theory section as well.

These NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.4 can be availed by students on the Extramarks website and mobile application.

NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.4: Based on Real Numbers, NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.4 is a very useful tool for students to review rational numbers and decimal expansion. Three questions make up the NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.4. It is part of the Class 10 NCERT Solutions Real Numbers Exercise 1.4. As a result, students can resolve these issues rapidly.

Extramarks’ subject matter specialists developed the NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.4 in accordance with the most recent CBSE standards in order to aid students in developing a thorough knowledge of Exercise 1.4 Chapter 1 Real Numbers. The detailed NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.4 are available to students on the Extramarks website and mobile application. To receive the NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.4 in PDF format, keep reading the article.

## NCERT Solutions for Class 10 Maths Chapter 1

Class 10 mathematics is a crucial subject for many competitive admission tests in addition to the Class 10 examinations. Students can use these 10th Grade NCERT Solutions to succeed in exams like JEE Main, JEE Advanced, NEET, and others. Class 10 Mathematics is pretty difficult. Users can improve their exam marks and become an expert in Class 10 Mathematics by using these NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.4.

It is crucial and time-saving to use the NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.4. Students will learn numerous shortcuts for efficiently completing Mathematics problems from the NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.4. These Mathematics solutions cover all of the topics covered in CBSE Class 10 Mathematics, including polynomials, quadratic equations, coordinate geometry, and arithmetic progression.

Students can access the links provided on the Extramarks website and mobile application to acquire the NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.4 of the Real Numbers Chapter. The chapters covered in the CBSE Class 10 Maths book are listed below. All of the Class 10 Mathematics book can be resolved using these answers:

• Chapter 1 – Real Numbers
• Chapter 2 – Polynomials
• Chapter 3 – Pair of Linear Equations in Two Variables
• Chapter 4 – Quadratic Equations
• Chapter 5 – Arithmetic Progressions
• Chapter 6 – Triangles
• Chapter 7 – Coordinate Geometry
• Chapter 8 – Introduction to Trigonometry
• Chapter 9 – Some Applications of Trigonometry
• Chapter 10 – Circles
• Chapter 11 – Constructions
• Chapter 12 – Areas Related to Circles
• Chapter 13 – Surface Areas and Volumes
• Chapter 14 – Statistics
• Chapter 15 – Probability

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

Let us first know in detail about Real Numbers.

In Mathematics, a Real Number is a quantity that may be represented by an endless number of decimal expansions. In contrast to the natural numbers 1, 2, 3,… that result from counting, real numbers are used in measurements of continuously varying quantities such as size and time. They are distinguished from imaginary numbers, which use the symbol I or the square root of 1, by the word “real.” A complex number has a real (1) and an imaginary I component, like 1 + i.

The positive and negative integers, as well as the fractions created from them (also known as rational numbers), as well as the irrational numbers, are all real numbers. Contrary to rational numbers, whose decimal expansions always contain a digit or group of digits that repeats itself, such as 1/6 = 0.16666… or 2/7 = 0.285714285714, irrational numbers have decimal expansions that do not repeat themselves. Since there is no regularly repeating group in the decimal produced as 0.42442444244442, it is irrational.

Thus, to know such complex concepts better and to make sure that students retain them until the very end, it is important that they refer to the NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.4 by Extramarks.

### Decimal Expansions of Rational Numbers

The decimal expansion of a rational number is obtained by dividing the numerator by the denominator of the rational number. Decimal numbers can be of two different types:

• Terminating Decimals
• Non-terminating Decimals
1. Terminating Decimals:

The terminating decimal numbers are those decimal numbers with a limited number of digits following the decimal point. They have a limited number of decimal places. Exact decimal numbers are the name given to these decimal numbers.

1. Non-terminating Decimals:

Non-terminating decimal numbers are those that have an endless number of digits following the decimal point.

1.  Non-Recurring Decimals:

Non-recurring decimal numbers are those with an infinite number of digits after the decimal point and digits that do not repeat after the decimal point at regular intervals.

• Terminating Decimal Expansion of Rational Numbers:

The decimal number terminates after a specific number of digits following the decimal point, according to the terminating decimal expansion.

• Non-terminating Decimal Expansion of Rational Numbers:

Some rational numbers never have a remainder that equals zero after decimal expansion. A non-terminating recurring decimal number is produced by a rational number whose denominator has prime factors other than 2 and 5. In the quotient section, we thus have a block of repeated numerals. The expansion can be described as non-terminating and recurrent.

### Terminating and Non-terminating Decimals

A fraction is a number that, in Mathematics, designates the portion of a whole. The fraction is therefore a ratio of two numbers. In contrast, a decimal is a number that has a decimal point separating the whole number portion from the fractional portion. Decimal numbers can be categorised into a variety of groups, including repeating and non-repeating decimals and terminating and non-terminating decimals. Since we can readily simplify fractional numbers, converting a decimal to a fractional value is preferred when addressing many mathematical issues.

Terminating and Non-terminating Decimals:

A decimal with an end digit is referred to as a terminating decimal. It is a decimal, which means that its digits are limited .

For instance, 0.15, 0.86, etc.

The decimals that lack an end term are known as non-terminating decimals. There are indefinitely many terms in it.

For instance, 0.5444444, 0.1111111, etc.

### How to Check whether a given Rational Number is Terminating or not?

Any rational number can be expressed as either a terminating decimal or a repeating decimal, which is a fraction in lowest terms. Simply divide the numerator by the denominator, as instructed to the students. They have a terminal decimal if they have a residual of 0. Otherwise, the remainders eventually start to repeat, resulting in a repeating decimal.

If and only if a number is divisible by 10, a decimal expansion of that number is “terminating” (i.e., there are only 0s after a specific number of digits; let’s ignore the ambiguity of substituting 0s with an indefinite period of 9).

• rational (i.e., a fraction of two integers) (i.e., a fraction of two integers)
• a product of a power of 2 and a power of 5 makes up the (reduced) denominator.
• This is the ultimate exception and a small portion of the entire realm of numbers.

### NCERT Solutions for Class 10 Maths Chapter 1 Exercises

The fundamental theorem of arithmetic, prime numbers, composite numbers, Euclid’s division lemma,  HCF and LCM via the prime factorisation method, and irrational numbers are all covered in NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.4: Real Numbers. Rational and irrational numbers are combined to form real numbers.

Studentsalso discover significant information, such as how to prove the absurdity of numerous numbers using the Fundamental Theorem of Arithmetic and how to determine when the decimal expansion of a rational number terminates and when it does not. Understanding the fundamentals is essential since numbers are the basis of Mathematics.

This chapter also covers the method of successive magnification, which is used to depict a decimal expansion on a number line, and how to represent real numbers on a number line. To help the students gain a comprehensive understanding of real numbers and how they are used, the session mixes some ideas that they have already learned in earlier lessons with newer, more difficult concepts.

Users can download the NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.4 and more exercises in pdf format from the link provided on the Extramarks website and mobile application. Some of these are also included in the exercises that are provided below:

Class 10 Maths Chapter 1 Exercise 1.1 of the NCERT Solutions

Class 10 Maths Chapter 1 Exercise 1.2 of the NCERT Solutions

Class 10 Maths Chapter 1 Exercise 1.3 of the NCERT Solutions

NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.4

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Q.1

$\begin{array}{l}\text{Without actually performing the long division},\\ \text{state whether the following rational\hspace{0.17em}\hspace{0.17em}numbers}\\ \text{will have a terminating decimal expansion or}\\ \text{a non}-\text{terminating repeating decimal\hspace{0.17em}\hspace{0.17em}expansion}:\\ \left(\text{i}\right)\frac{\text{13}}{\text{3125}}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\left(\text{ii}\right)\frac{\text{17}}{\text{8}}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\left(\text{iii}\right)\frac{\text{64}}{\text{455}}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\left(\text{iv}\right)\frac{\text{15}}{\text{16}00}\\ \left(\text{v}\right)\frac{\text{29}}{\text{343}}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\left(\text{vi}\right)\frac{\text{23}}{{2}^{3}{5}^{2}}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\left(\text{vii}\right)\frac{\text{129}}{{\text{2}}^{\text{2}}{\text{5}}^{7}{7}^{5}}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\left(\text{viii}\right)\frac{\text{6}}{\text{15}}\\ \left(\text{ix}\right)\frac{\text{35}}{\text{5}0}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\left(\text{x}\right)\frac{\text{77}}{\text{21}0}\end{array}$

Ans.

$\begin{array}{l}\left(\text{i}\right)\\ \frac{\text{13}}{\text{3125}}=\frac{\text{13}}{\text{125}×\text{25}}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}=\frac{\text{13}}{{\text{5}}^{5}}\\ {\text{Here denomiantor is of the form of 2}}^{\mathrm{m}}{5}^{\mathrm{n}},\mathrm{where}\mathrm{m}\mathrm{and}\mathrm{n}\mathrm{are}\mathrm{non}–\mathrm{negative}\mathrm{integers}.\\ \text{So},\text{}\frac{\text{13}}{\text{3125}}\text{has a terminating decimal expansion.}\\ \left(\text{ii}\right)\\ \frac{\text{17}}{8}=\frac{\text{17}}{{\text{2}}^{3}}\\ {\text{Here denomiantor is of the form of 2}}^{\mathrm{m}}{5}^{\mathrm{n}},\mathrm{where}\mathrm{m}\mathrm{and}\mathrm{n}\mathrm{are}\mathrm{non}–\mathrm{negative}\mathrm{integers}.\\ \text{So},\text{}\frac{\text{17}}{8}\text{has a terminating decimal expansion.}\\ \left(\text{iii}\right)\\ \frac{64}{455}=\frac{64}{5×7×13}\\ {\text{Here denomiantor is not of the form of 2}}^{\mathrm{m}}{5}^{\mathrm{n}},\mathrm{where}\mathrm{m}\mathrm{and}\mathrm{n}\mathrm{are}\mathrm{non}–\mathrm{negative}\mathrm{integers}.\\ \text{So},\text{}\frac{64}{455}\text{has a non-terminating repeating decimal expansion.}\\ \left(\text{iv}\right)\\ \frac{15}{1600}=\frac{3×5}{{2}^{6}×{5}^{2}}=\frac{3}{{2}^{6}×{5}^{1}}\\ {\text{Here denomiantor is of the form of 2}}^{\mathrm{m}}{5}^{\mathrm{n}},\mathrm{where}\mathrm{m}\mathrm{and}\mathrm{n}\mathrm{are}\mathrm{non}–\mathrm{negative}\mathrm{integers}.\\ \text{So},\text{decimal expansion of}\frac{15}{1600}\text{is terminating.}\\ \left(\text{v}\right)\\ \frac{29}{343}=\frac{29}{{7}^{3}}\\ {\text{Here denomiantor is not of the form of 2}}^{\mathrm{m}}{5}^{\mathrm{n}},\mathrm{where}\mathrm{m}\mathrm{and}\mathrm{n}\mathrm{are}\mathrm{non}–\mathrm{negative}\mathrm{integers}.\\ \text{So},\text{decimal expansion of}\frac{29}{343}\text{is non-terminating repeating.}\\ \left(\text{vi}\right)\\ \frac{23}{{2}^{3}{5}^{2}}\\ {\text{Here denomiantor is of the form of 2}}^{\mathrm{m}}{5}^{\mathrm{n}},\mathrm{where}\mathrm{m}\mathrm{and}\mathrm{n}\mathrm{are}\mathrm{non}–\mathrm{negative}integers\\ \text{So},\text{decimal expansion of}\frac{23}{{2}^{3}{5}^{2}}\text{is terminating.}\\ \left(\text{vii}\right)\\ \frac{129}{{2}^{2}{5}^{7}{7}^{5}}=\frac{43×3}{{2}^{2}{5}^{7}{7}^{5}}\\ {\text{Here denomiantor is not of the form of 2}}^{\mathrm{m}}{5}^{\mathrm{n}},\mathrm{where}\mathrm{m}\mathrm{and}\mathrm{n}\mathrm{are}\mathrm{non}–\mathrm{negative}\mathrm{integers}.\\ \text{So},\text{decimal expansion of}\frac{129}{{2}^{2}{5}^{7}{7}^{5}}\text{is non-terminating repeating.}\\ \left(\text{viii}\right)\\ \frac{6}{15}=\frac{2×3}{3×5}=\frac{2}{5}\\ {\text{Here denomiantor is of the form of 2}}^{\mathrm{m}}{5}^{\mathrm{n}},\mathrm{where}\mathrm{m}\mathrm{and}\mathrm{n}\mathrm{are}\mathrm{non}–\mathrm{negative}\mathrm{integers}.\\ \text{So},\text{decimal expansion of}\frac{6}{15}\text{is terminating.}\\ \left(\text{ix}\right)\\ \frac{35}{50}=\frac{35}{2×{5}^{2}}\\ {\text{Here denominator is of the form of 2}}^{\mathrm{m}}{5}^{\mathrm{n}},\mathrm{where}\mathrm{m}\mathrm{and}\mathrm{n}\mathrm{are}\mathrm{non}–\mathrm{negative}\mathrm{integers}.\\ \text{So},\text{decimal expansion of}\frac{35}{50}\text{is terminating.}\\ \left(\text{x}\right)\\ \frac{77}{210}=\frac{77}{2×3×5×7}=\frac{11}{2×3×5}\\ {\text{Here denominator is not of the form of 2}}^{\mathrm{m}}{5}^{\mathrm{n}},\mathrm{where}\mathrm{m}\mathrm{and}\mathrm{n}\mathrm{are}\mathrm{non}–\mathrm{negative}\mathrm{integers}.\\ \text{So},\text{decimal expansion of}\frac{77}{210}\text{is non-terminating repeating.}\end{array}$

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

Ans.

$\begin{array}{l}\left(\text{i}\right)\\ \frac{\text{13}}{\text{3125}}=0.00416\\ \\ \text{3125}\begin{array}{c}0.00416\\ \overline{)\text{13000}}\end{array}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\underset{¯}{12500}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\underset{¯}{\begin{array}{l}5000\\ \text{3125}\end{array}}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\underset{¯}{\begin{array}{l}18750\\ 18750\end{array}}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\underset{¯}{\text{0}}\\ \\ \left(\text{ii}\right)\\ \frac{\text{17}}{8}=2.125\\ 8\begin{array}{c}2.125\\ \overline{)17}\end{array}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\underset{¯}{16\text{\hspace{0.17em}}}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\underset{¯}{\begin{array}{l}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}10\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}8\text{\hspace{0.17em}}\end{array}}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\underset{¯}{\begin{array}{l}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}20\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}16\text{\hspace{0.17em}\hspace{0.17em}}\end{array}}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\underset{¯}{\begin{array}{l}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}40\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}40\end{array}}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0}\\ \\ \left(\text{iv}\right)\\ \frac{15}{1600}=0.009375\\ \\ 1600\begin{array}{c}0.009375\\ \overline{)15000}\end{array}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\underset{¯}{14400}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\underset{¯}{\begin{array}{l}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}6000\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}4800\end{array}}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\underset{¯}{\begin{array}{l}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}12000\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}11200\end{array}}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\underset{¯}{\begin{array}{l}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}8000\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}8000\end{array}}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0}\\ \\ \left(\text{vi}\right)\\ \frac{23}{{2}^{3}{5}^{2}}=\frac{23}{200}=0.115\\ \\ 200\begin{array}{c}0.115\\ \overline{)230}\end{array}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\underset{¯}{200}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\underset{¯}{\begin{array}{l}\text{\hspace{0.17em}\hspace{0.17em}}300\\ \text{\hspace{0.17em}\hspace{0.17em}}200\end{array}}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\underset{¯}{\begin{array}{l}1000\\ 1000\end{array}}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0}\\ \\ \left(\text{viii}\right)\\ \frac{6}{15}=\frac{2}{5}=0.4\\ \\ 5\begin{array}{c}0.4\\ \overline{)20}\end{array}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} \hspace{0.17em}}\underset{¯}{20}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} \hspace{0.17em}0}\\ \\ \left(\text{ix}\right)\\ \frac{35}{50}=0.7\\ \\ 50\begin{array}{c}0.7\\ \overline{)350}\end{array}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\underset{¯}{350}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} 0}\end{array}$

Q.3

$\begin{array}{l}\text{The following real numbers have decimal expansions}\\ \text{as given below. In each case decide whether they are}\\ \text{rational or not. If they are rational, and of the form}\frac{\mathrm{p}}{\mathrm{q}},\\ \text{what can you say about the prime factors of}\mathrm{q}?\\ \\ \text{(i) 43.123456789 (ii) 0.120120012000120000}...\\ \text{(iii) 43.}\overline{123456789}\end{array}$

Ans.

$\begin{array}{l}\text{(i) 43.123456789}\\ \text{The given number has a terminating decimal expansion.}\\ \text{So, it is a rational number of the form}\frac{\mathrm{p}}{\mathrm{q}}\text{and}\mathrm{prime}\mathrm{factors}\mathrm{of}\mathrm{q}\mathrm{will}\mathrm{be}\mathrm{either}2\mathrm{or}5\mathrm{or}\mathrm{both}only\text{.}\\ \text{(ii) 0.120120012000120000}..\\ \text{​​​​​​ The given number is neither terminating nor recurring.}\\ \text{So, it is an irrational number.}\\ \text{(iii) 43.}\overline{123456789}\\ \text{​​​​​​ The given number is non-terminating repeating.}\\ \text{So, it is a rational number of the form}\frac{\mathrm{p}}{\mathrm{q}}\text{and}\mathrm{q}\mathrm{will}\mathrm{also}\mathrm{have}\mathrm{a}\mathrm{prime}\mathrm{factor}\mathrm{other}\mathrm{than}2\mathrm{or}5.\end{array}$