RD Sharma Solutions Class 9 Number System Exercise 1.1

RD Sharma Solutions Class 9 Chapter 1 Exercise 1.1

RD Sharma Class 9 Solutions Chapter 1 Ex 1.1 Free Download

Q1. Is 0 a rational number? Can you write it in the form \(\frac{P}{Q}\), where P and Q are integers and Q ≠ 0?

Solution:

Yes, 0 is a rational number and it can be written in \(P\div Q\) form provided that Q ≠ 0

0 is an integer and it can be written various forms, for example

\(0\div 2 , 0\div 100 , 0\div 95\) etc.

Q2. Find five rational numbers between 1 and 2

Solution:

Given that to find out 5 rational numbers between 1 and 2

  • Rational number lying between 1 and 2

= \(\frac{1+2}{2}\)

= \(\frac{3}{2}\)

= \(1<\frac{3}{2}<2\)

  • Rational number lying between 1 and \(\frac{3}{2}\)

= \(\frac{1+\frac{3}{2}}{2}\)

= \(\frac{5}{4}\)

\(1< \frac{5}{4}<\frac{3}{2}\)

  • Rational number lying between 1 and \(\frac{5}{4}\)

= \(\frac{1+\frac{5}{4}}{2}\) Rational number lying between \(\frac{3}{2}\) and 2

= \(\frac{9}{8}\)

= \(1<\frac{9}{8}<\frac{5}{4}\)

  • Rational number lying between \(\frac{3}{2}\) and 2

= \(\frac{\frac{3}{2}+2}{2}\)

= \(\frac{7}{4}\)

= \(\frac{3}{2}<\frac{7}{4}<2\)

  • Rational number lying between \(\frac{7}{4}\) and 2

\(\frac{\frac{7}{4}+2}{2}\)

= \(\frac{15}{8}\)

= \(\frac{7}{4}<\frac{15}{8}<2\)

Therefore, \(1<\frac{9}{8}<\frac{5}{4}<\frac{3}{2}<\frac{7}{4}<\frac{15}{8}<2\)

Q3. Find out 6 rational numbers between 3 and 4

Solution:

Given that to find out 6 rational numbers between 3 and 4

We have,

3 \(\times \frac{7}{7}\) = \(\frac{21}{7}\) and

4 \(\times \frac{6}{6}\) = \(\frac{28}{7}\)

We know 21 < 22 <  23 < 24 < 25 < 26 < 27 < 28

  • \(\frac{21}{7}<\frac{22}{7}<\frac{23}{7}<\frac{24}{7}<\frac{25}{7}<\frac{26}{7}<\frac{27}{7}<\frac{28}{7}\)
  • \(3<\frac{22}{7}<\frac{23}{7}<\frac{24}{7}<\frac{25}{7}<\frac{26}{7}<\frac{27}{7}<4\)

Therefore, 6 rational numbers between 3 and 4 are

\(\frac{22}{7},\frac{23}{7},\frac{24}{7},\frac{25}{7},\frac{26}{7},\frac{27}{7}\)

Similarly to find 5 rational numbers between 3 and 4, multiply 3 and 4 respectively with \(\frac{6}{6}\) and in order to find 8 rational numbers between 3 and 4 multiply 3 and 4 respectively with \(\frac{8}{8}\) and so on.

Q4. Find 5 rational numbers between \(\frac{3}{5}\) and \(\frac{4}{5}\)

Solution : Given to find out the 5 rational numbers between \(\frac{3}{5}\) and \(\frac{4}{5}\)

To find 5 rational numbers between \(\frac{3}{5}\) and \(\frac{4}{5}\), \(\frac{3}{5}\) and \(\frac{4}{5}\) with \(\frac{6}{6}\)

We have,

\(\frac{3}{5}\) \(\times \frac{6}{6}\) = \(\frac{18}{30}\)

\(\frac{4}{5}\) \(\times \frac{6}{6}\) = \(\frac{24}{30}\)

We know  18 < 19 < 20 < 21 < 22 <  23 < 24

  • \(\frac{18}{30}<\frac{19}{30}<\frac{20}{30}<\frac{21}{30}<\frac{22}{30}<\frac{23}{30}<\frac{24}{30}\)
  • \(\frac{3}{5}<\frac{19}{30}<\frac{20}{30}<\frac{21}{30}<\frac{22}{30}<\frac{23}{30}<\frac{4}{5}\)

Therefore, 5 rational numbers between \(\frac{3}{5}\) and \(\frac{4}{5}\) are \(\frac{19}{30},\frac{20}{30},\frac{21}{30},\frac{22}{30},\frac{23}{30}\)

Q5. Answer whether the following statements are true or false? Give reasons in support of your answer.

 (i) Every whole number is a rational number

(ii) Every integer is a rational number

(iii) Every rational number is an integer

(iv) Every natural number is a whole number

(v) Every integer is a whole number

(vi) Every rational number is a whole number

Solution:

(i)  True. As whole numbers include and they can be represented

For example – \(\frac{0}{10},\frac{1}{1},\frac{2}{1},\frac{3}{1}\)…. And so on.

(ii) True. As we know 1, 2, 3, 4 and so on, are integers and they can be represented in the form of \(\frac{1}{1},\frac{2}{1},\frac{3}{1} \frac{4}{1}\).

(iii) False. Numbers such as \(\frac{3}{2},\frac{1}{2},\frac{3}{5},\frac{4}{5}\) are rational numbers but they are not integers.

(iv) True. Whole numbers include all of the natural numbers.

(v) False. As we know whole numbers are a part of integers.

(vi) False. Integers include -1, -2, -3 and so on….. .which is not whole number