SELINA Solution Class 9 Chapter 1 Rational and Irrational Numbers Exercise 1A

Exercise 1A

Question 1

Is zero a rational number ? Can it be written in the form $\dfrac{p}{q}$ ,where p and q are integers and q≠0 ?

Sol :

Yes, zero is a rational number.

As it can be written in the form of , where p and q are integers and q≠0 ?

⇒$0= \dfrac{0}{1}$

Question 2

Are the following statement true or false ? Give reason for your answer.
(i) Every whole number is a natural number.
(ii) Every whole number is a rational number.
(iii)Every integer is a rational number.
(iv) Every rational number is a whole number.

Sol :

(i) False, zero is a whole number but not a natural number.

(ii) True, Every whole can be written in the form of, where p and q are integers and q ≠ 0.

(iii) True, Every integer can be written in the form of, where p and q are integers and q ≠ 0.

(iv) False, Every rational number is may or may not be a whole number.

Let us consider some rational numbers,

$\dfrac{1}{4},\dfrac{2}{4},\dfrac{4}{4}\dots$

If you consider

$\dfrac{4}{4}=1$

It is a whole number.

If you consider

$\dfrac{2}{4}=0.5$

It is not a whole number.

So, every rational number is may or may not be a whole number but every whole number is a rational number.

Question 3

Arrange ${\displaystyle -\frac{5}{7},\frac{7}{12},-\frac{2}{3}\text{and}\frac{11}{18}}$ in ascending order of their magnitudes.
Also, find the difference between the largest and smallest of these rational numbers. Express this difference as a decimal fraction correct to one decimal place.

Sol :

Consider the given number : ${\displaystyle -\frac{5}{9},\frac{7}{12},-\frac{2}{3}\text{and}\frac{11}{18}}$

The L.C.M. of 9,12, and 18 is 36.
Thus the given numbers are :

${\displaystyle -\frac{5}{9},\frac{7}{12},-\frac{2}{3}\text{and}\frac{11}{18}}$

${\displaystyle =-\frac{5\times 4}{9\times 4},\frac{7\times 3}{12\times 3},-\frac{2\times 12}{3\times 12}\text{and}\frac{11\times 2}{18\times 2}}$

= ${\displaystyle -\frac{20}{36},\frac{21}{36},-\frac{24}{36}\text{and}\frac{22}{36}}$

Thus the numbers in ascending order are shown below :

${\displaystyle -\frac{24}{36},-\frac{20}{36},\frac{21}{36},\text{and}\frac{22}{36}}$

Thus the given numbers in ascending order are shown below:

${\displaystyle -\frac{2}{3},-\frac{5}{9},\frac{7}{12}\text{and}\frac{11}{18}}$

We need to find the difference between the largest and smallest of the above numbers.

Thus, difference = ${\displaystyle \frac{11}{18}-(-\frac{2}{3})}$

 = ${\displaystyle \frac{11}{18}+\frac{2}{3}}$

 = ${\displaystyle \frac{11}{18}+\frac{2\times 6}{3\times 6}}$

= ${\displaystyle \frac{11}{18}+\frac{12}{18}}$

= ${\displaystyle \frac{11+12}{18}}$

= ${\displaystyle \frac{23}{18}}$

We need to express this fraction as a decimal,
Correct to one decimal place.
Thus, we have ${\displaystyle \frac{23}{18}=1.2\overline{7}}$ ≈ 1.3.

Question 4

Arrange ${\displaystyle \frac{5}{8},-\frac{3}{16},-\frac{1}{4}\text{and}\frac{17}{32}}$ in descending order of their magnitudes.
Also, find the sum of the lowest and largest of these fractions. Express the result obtained as a decimal fraction correct to two decimal places.

Sol :

Consider the given numbers : ${\displaystyle \frac{5}{8},-\frac{3}{16},-\frac{1}{4}\text{and}\frac{17}{32}}$
The LCM of 8, 16, 4 and 32 is 32.
Thus, the given numbers are given below :
${\displaystyle \frac{5}{8},-\frac{3}{16},-\frac{1}{4}\text{and}\frac{17}{32}}$

= ${\displaystyle \frac{5\times 4}{8\times 4},-\frac{3\times 2}{16\times 2},-\frac{1\times 8}{4\times 8}\text{and}\frac{17\times 1}{32\times 1}}$

= ${\displaystyle \frac{20}{32},-\frac{6}{32},-\frac{8}{32},\text{and}\frac{17}{32}}$

Thus, the numbers in descending order are shown below :

${\displaystyle \frac{20}{32},\frac{17}{32},-\frac{6}{32}\text{and}-\frac{8}{32}}$

Thus, the given numbers in descending order are listed below :

${\displaystyle \frac{5}{8},\frac{17}{32},-\frac{3}{16}\text{and}-\frac{1}{4}}$.

We need to find the sum of the largest and smallest of the above numbers.

Thus, sum = ${\displaystyle \frac{5}{8}+(-\frac{1}{4})}$

= ${\displaystyle \frac{5}{8}-\frac{1}{4}}$

= ${\displaystyle \frac{5}{8}-\frac{1\times 2}{4\times 2}}$

= ${\displaystyle \frac{5}{8}-\frac{2}{8}}$

= ${\displaystyle \frac{3}{8}}$

We need to express this fraction as a decimal, correct to two decimal places.
Thus, we have ${\displaystyle \frac{3}{8}}$ = 0.375 ≈ 0.38.

Question 5.1

Without doing any actual division, find which of the following rational numbers have terminating decimal representation : ${\displaystyle \frac{7}{16}}$

Sol :

Given number is ${\displaystyle \frac{7}{16}}$

Since 16 = 2 x 2 x 2 x 2 = 2⁴ = 2⁴ x 5⁰
I.e. 16 can be expressed as 2^m x 5ⁿ

∴ ${\displaystyle \frac{7}{16}}$ is convertible into the terminating decimal.

Question 5.2

Without doing any actual division, find which of the following rational numbers have terminating decimal representation : ${\displaystyle \frac{23}{125}}$

Sol :

Given number is ${\displaystyle \frac{23}{125}}$

Since 125 = 5 x 5 x 5 = 5^{3 =} 2⁰ x 5³
I.e. 125 can be expressed as 2^m x 5ⁿ

∴ ${\displaystyle \frac{23}{125}}$ is convertible into the terminating decimal.

Question 5.3

Without doing any actual division, find which of the following rational numbers have terminating decimal representation : ${\displaystyle \frac{9}{14}}$

Sol :

Given number is ${\displaystyle \frac{9}{14}}$

Since 14 = 2 x 7 = 2¹ x 7¹
I.e. 14 cannot be expressed as 2^m x 5ⁿ

∴ ${\displaystyle \frac{9}{14}}$ is not convertible into the terminating decimal.

Question 5.4

Without doing any actual division, find which of the following rational numbers have terminating decimal representation : ${\displaystyle \frac{32}{45}}$

Sol :

Given number is ${\displaystyle \frac{32}{45}}$

Since, 45 = 3 x 3 x 5 = 3² x 5¹
I.e. 45 cannot be expressed as 2^m x 5ⁿ

∴ ${\displaystyle \frac{32}{45}}$ is not convertible into the terminating decimal.

Question 5.5

Without doing any actual division, find which of the following rational numbers have terminating decimal representation : ${\displaystyle \frac{43}{50}}$

Sol :

Given number is ${\displaystyle \frac{46}{50}}$

Since 50 = 2 x 5 x 5 = 2¹ x 5²
I.e. 50 can be expressed as 2^m x 5ⁿ

∴ ${\displaystyle \frac{43}{50}}$ is convertible into the terminating decimal.

Question 5.6

Without doing any actual division, find which of the following rational numbers have terminating decimal representation : ${\displaystyle \frac{17}{40}}$

Sol :

Given number is ${\displaystyle \frac{17}{40}}$

Since 40 = 2 x 2 x 2 x 5 = 2³ x 5¹
I.e. 40 can be expressed as 2^m x 5ⁿ

∴ ${\displaystyle \frac{17}{40}}$ is convertible into the terminating decimal.

Question 5.7

Without doing any actual division, find which of the following rational numbers have terminating decimal representation : ${\displaystyle \frac{61}{75}}$

Sol :

Given number is ${\displaystyle \frac{61}{75}}$

Since 75 = 3 x 5 x 5 = 3¹ x 5²
i.e. 75 cannot be expressed as 2^m x 5ⁿ

∴ ${\displaystyle \frac{61}{75}}$ is not convertible into the terminating decimal.

Question 5.8

Without doing any actual division, find which of the following rational numbers have terminating decimal representation : ${\displaystyle \frac{123}{250}}$

Sol :

Given number is ${\displaystyle \frac{123}{250}}$

Since 250 = 2 x 5 x 5 x 5 = 2¹ x 5³
i.e. 250 can be expressed as 2^m x 5ⁿ

∴ ${\displaystyle \frac{123}{250}}$ is convertible into the terminating decimal.