**Question 1: **

**What is the rational numbers? Give 10 examples of rational numbers?**

**Solution:**

The number that can be written in the \(\frac{p}{q}\) form, where p and q are integers and q \(\neq\) 0 are known as rational numbers.

Examples of rational numbers:

(1) \(\frac{1}{2}\) (2) \(\frac{1}{5}\) (3) \(\frac{5}{4}\) (4) \(\frac{4}{1}\) = 1 (5) \(\frac{5}{2}\) (6) \(\frac{1}{7}\)

(7) \(\frac{0}{1}\) = 0

Solution 2:

**Question 2: **

**Represent each of the following rational numbers on the number line:**

**I – 5**

**II — 3**

**III — \(\frac{5}{7}\)**

**IV– \(\frac{8}{3}\)**

**V – 1.3**

**VI — -24**

**VII — \(\frac{23}{6}\)**

**SOLUTION : **

(i) Let

X = \(\frac{1}{4}\) and y = \(\frac{1}{3}\)

Rational number lying between x and y:

\(\frac{1}{2}\) ( x + y) = \(\frac{1}{2}(\frac{1}{4}+\frac{1}{3})\)

= \(\frac{7}{24}\)

(ii) Let:

X = \(\frac{3}{8}\) and y = \(\frac{2}{5}\)

Rational number lying between x and y:

\(\frac{1}{2}\) ( x + y) = \(\frac{1}{2}(\frac{3}{8}+\frac{2}{5})\)

= \(\frac{1}{2}(\frac{15+16}{40})\) = \(\frac{31}{80}\)

(iii) Let:

X = 1.3 and y = 1.4

Rational number lying between x and y:

\(\frac{1}{2}\) (x + y) = \(\frac{1}{2}\) (1.3 + 1.4)

= \(\frac{1}{2}\) (2.7) = 1.35

(iv) Let:

X = 0.75 and y = 1.2

Rational number lying between x and y;

\(\frac{1}{2}\) ( x + y ) = \(\frac{1}{2}\) ( 0.75 + 1.2)

= \(\frac{1}{2}\) (1.95) = 0.975

(v) Let:

X= -1 and y = \(\frac{1}{2}\)

Rational number lying between x and y:

\(\frac{1}{2}\) ( x + y ) = \(\frac{1}{2}\) \((-1 + \frac{1}{2})\)

= \(-\frac{1}{4}\)

(vi) Let:

X = \(-\frac{3}{4}\) and y = \(-\frac{2}{5}\)

Rational number lying between x and y:

\(\frac{1}{2}\) ( x + y ) = \(\frac{1}{2}\) \((-\frac{3}{4}-\frac{2}{5})\)

= \(\frac{1}{2}(\frac{-15-8}{20})= -\frac{23}{40}\)

**Question 3: **

**Find three rational numbers lying between \(\frac{1}{5}\) and \(\frac{1}{4}\)**

**Solution: **

Let:

X = \(\frac{1}{5}\) , Y = \(\frac{1}{4}\) and n = 3

We know:

\(d= \frac{y-x}{n+1}=\frac{\frac{1}{4}-\frac{1}{5}}{3+1} = \frac{\frac{1}{20}}{4}=\frac{1}{80}\)So, three rational numbers lying between x and y are:

(x + d), (x + 2d) and ( x + 3d)

= \((\frac{1}{5}+\frac{1}{80})\) , \((\frac{1}{5}+\frac{2}{80})\) and \((\frac{1}{5}+\frac{3}{80})\)

= \(\frac{17}{80}\) , \(\frac{18}{80}\) and \(\frac{19}{80}\)

**Question 4:**

**Find five rational numbers lying between \(\frac{2}{5}\) and \(\frac{3}{4}\)**

**Solution: **

Let:

X= 25, y = 34 and n= 5

We know:

\(d=\frac{y-x}{n+1} = \frac{\frac{3}{4}-\frac{2}{5}}{5+1} = \frac{\frac{7}{20}}{6}=\frac{7}{120}\)So, five rational numbers between x and y are:

(x + d) ,(x + 2d), (x + 3d), (x + 4d) and (x + 5d)

= \((\frac{2}{5}+\frac{7}{120})(\frac{2}{5}+\frac{14}{120})(\frac{2}{5}+\frac{21}{120})(\frac{2}{5}+\frac{28}{120}) and (\frac{2}{5}+\frac{35}{120})\)

= \(\frac{55}{120}\) , \(\frac{62}{120}\) , \(\frac{69}{120}\) , \(\frac{76}{120}\) and \(\frac{83}{120}\)

**Question 5:**

** Insert 6 rational numbers between 3 and 4.**

**Solution:**

Let:

X= 3, y =4 and n=6

We know:

d = \(\frac{y-x}{n+1}\) = \(\frac{4-3}{6+1}\) = \(\frac{1}{7}\)

So, six rational numbers between x and y are:

(x + d) ,(x + 2d), (x + 3d), (x + 4d), (x + 5d) and (x + 6d)

= \((3+\frac{1}{7})\), \((3+\frac{2}{7})\), \((3+\frac{3}{7})\) , \((3+\frac{4}{7})\), \((3+\frac{5}{7})\), \((3+\frac{6}{7})\)

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

**Question 6: Insert 16 rational numbers between 2.1 and 2.2 .**

**Solution:**

Let x = 2.1, y = 2.2 and n= 16

We know,

d = \(\frac{y-x}{n+1}\) =\(\frac{2.2-2.1}{16+1}\)= \(\frac{0.1}{17}\) = \(\frac{1}{170}\) = 0.005 (approx)

So, 16 rational numbers between 2.1 and 2.2 are:

(x + d), (x + 2d), . . . . . (x + 16d)

= [ 2.1 + 0.005], [ 2.1 +2( 0.005)]. . . . . . . . . [2.1 + 16(0.005)]

= 2.105, 2.11, 2.115, 2.12, 2.125 , 2.13, 2.135, 2.14, 2.145, 2.15, 2.155, 2.16, 2.165, 2.17, 2.175 and 2.18

**Question 7: Without actual division find which of the following rational are terminating decimals**

**\(\frac{13}{80}\)****\(\frac{7}{24}\)****\(\frac{5}{12}\)****\(\frac{8}{35}\)****\(\frac{16}{125}\)**

Solution:

(i) Denominator of \(\frac{13}{80}\) is 80.

And,

80 = 2^{4} x 5

Therefore, 80 has no other factors than 2 and 5.

Thus, \(\frac{13}{80}\) is a terminating decimal.

(ii) Denominator of \(\frac{7}{24}\) is 24

And 24 = 2^{3}x 3

So, 24 has a prime factor 3, which is other than 2 and 5.

Thus, \(\frac{7}{24}\) is not a terminating decimal.

(iii) Denominator of \(\frac{5}{12}\) is 12.

And,

12 = 2^{2}x 3

So, 12 has a prime factor 3, which is other than 2 and 5.

Thus, \(\frac{5}{12}\) is not a terminating decimal.

(iv) Denominator of \(\frac{8}{35}\) is 35

And,

35 = 7 x 5

So, 35 has a prime factor 7, which is other than 2 and 5.

Thus, \(\frac{8}{35}\) is not a terminating decimal.

(v) Denominator of \(\frac{16}{125}\) is 125.

And,

125 = 5^{3}

Therefore, 125 has no other factors than 2 and 5.

Thus, \(\frac{16}{125}\) is a terminating decimal.

**Question 8: Convert each of the following into a decimal.**

**\(\frac{5}{8}\)****\(\frac{9}{16}\)****\(\frac{7}{25}\)****\(\frac{11}{24}\)****\(2\frac{5}{12}\)**

Solution:

(i) \(\frac{5}{8}\) = 0.625

By actual division, we have:

(ii) \(\frac{9}{16}\) = 0.5625

By actual division, we have:

(iii) \(\frac{7}{25}\) = 0.28

By actual division, we have:

(iv) \(\frac{11}{24}\) = 0.4583333

By actual division, we have:

(v) \(2\frac{5}{12}\) = \(\frac{29}{12}\) = 2.416666

By actual division, we have:

**Question 9: Express each of the following as a fraction in simplest form.**

**0. \(\overline{3}\)****1. \(\overline{3}\)****0. \(\overline{34}\)****3. \(\overline{14}\)****0. \(\overline{324}\)****0. 1\(\overline{7}\)****0.5\(\overline{4}\)****0.1\(\overline{63}\)**

**Solution: **

(i) 0. \(\overline{3}\)

Let x = 0. \(\overline{3}\)

Therefore, X = 0.3333 – – – – – – – – – – – – – – – (1)

10x = 3.3333 – – – – – – – – – – – – – – (2)

On subtracting (1) from (2) we get:

9x = 3

= x = \(\frac{1}{3}\)

Therefore, 0. \(\overline{3}\) = \(\frac{1}{3}\)

(ii) 1. \(\overline{3}\)

Let x = 1. \(\overline{3}\)

Therefore, X = 1.33333 – – – – – – – – – – – – – (i)

10x = 13.33333 – – – – – – – – – – – (ii)

On subtracting (i) and (ii) we get:

9x = 12

= x = \(\frac{4}{3}\)

Therefore, 1. \(\overline{3}\) = \(\frac{4}{3}\)

(iii) 0. \(\overline{34}\)

Let x = 0. \(\overline{34}\)

Therefore, X = 0.3434… – – – – – – – – – – – – (1)

100x = 34.343434… – – – – – – – – – – (2)

On Subtracting (1) and (2) we get:

99x = 34

= x = \(\frac{34}{99}\)

Therefore, 0. \(\overline{34}\) = \(\frac{34}{99}\)

(iv) 3. \(\overline{14}\)

Let x = 3. \(\overline{34}\)

Therefore, X = 3.1414.. – – – – – – – – – – – (1)

100x = 314.1414… – – – – – – – – – – (2)

On subtracting (1) from (2), we get:

99x = 311

= x = \(\frac{311}{99}\)

Therefore, 3. \(\overline{34}\) = \(\frac{311}{99}\)

(v) 0. \(\overline{324}\)

Let x = 0. \(\overline{324}\)

Therefore, X = 0.324324… – – – – – – – – – – – – – – – (1)

1000x = 324.324324.. – – – – – – – – – – – – (2)

On subtracting (1) from (2), we get:

999x = 324

= x = \(\frac{324}{999}\) = \(\frac{12}{37}\)

Therefore, 0. \(\overline{324}\) = \(\frac{12}{37}\)

(vi) 0. 1\(\overline{7}\)

Let x = 0.1 \(\overline{7}\)

Therefore, X = 0.1777.. – – – – – – – – – – – – – (1)

10x = 1.7777.. – – – – – – – – – – – – – (2)

On subtracting (1) from (2) we get:

90x = 16

= x = \(\frac{8}{45}\)

Therefore, 0.1\(\overline{7}\) = \(\frac{8}{45}\)

(vii) 0.5\(\overline{4}\)

Let x = 0.5\(\overline{4}\)

Therefore, X = 0.54444…

10x = 5.4444.. – – – – – – – – – – – – – – – (1)

100x = 54.4444… – – – – – – – – – – – – – – (2)

On subtracting (1) from (2) we get:

90x = 49

= x = \(\frac{49}{90}\)

Therefore, 0.5\(\overline{4}\) = \(\frac{49}{90}\)

(viii) 0.1\(\overline{63}\)

Let x = 0.1\(\overline{63}\)

Therefore, X = 0.1636363…

10x = 1.6363.. – – – – – – – – – – – – – – – (1)

1000x = 163.6363… – – – – – – – – – – – – – – (2)

On subtracting (1) from (2) we get:

990x = 162

= x = \(\frac{162}{990}\)= \(\frac{9}{55}\)

Therefore, 0.1\(\overline{63}\) = \(\frac{9}{55}\)

**Question 10: Write, whether the given statement is true or false. Give reason.**

**Every natural number is a whole number.****Every whole number is a natural number.****Every integer is a rational numbers.****Every rational number is a whole number.****Every terminating decimal is a rational number.****Every repeating decimal is a rational number.****0 is a rational number**

**Solution:**

(i) True

Natural numbers start from 1 to infinity and whole numbers start from 0 to infinity; hence, every natural number is a whole number.

(ii) False

0 is a whole number not a natural number, so every whole number is not a natural number.

(iii) True

Every integer can be expressed in the \(\frac{p}{q}\) form.

(iv) False

Because whole numbers consist only of number of the form \(\frac{p}{1}\) , where p is a positive number. On the other hand, rational numbers are the numbers whose denominator can be anything except 0.

(v) True

Every terminating decimal can be easily expressed in the \(\frac{p}{q}\) form.

(vi) True

Every terminating decimal can be easily expressed in the \(\frac{p}{q}\) form.

(vii) True

0 can be expressed in the form \(\frac{p}{q}\), so it is a rational number.

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