Exercise 1.1 ML Aggarwal Class 9 Solutions Rational and Irrational Numbers Best

Exercise 1.1 Exercise 1.2 Exercise 1.3 Exercise 1.4 Exercise 1.5 Chapter Test-1

Exercise 1.1 ML Aggarwal Class 9 Solutions of Mathematics ICSE Textbook contains a total of 9 questions with excellent answers. The questions are based on rational numbers and finding rational numbers between two rational numbers.

Exercise 1.1 ML Aggarwal Class 9 Solutions

Q.1 Q.2 Q.3 Q.4 Q.5 Q.6 Q.7 Q.8 Q.9

Question 1 Exercise 1.1 ML Aggarwal Class 9 Solutions has been given below.

1. Insert a rational number between $\frac{2}{9}$ and $\frac{3}{8}$ , and arrange in descending order.

Answer

The given rational numbers are
$\frac{2}{9}$ and $\frac{3}{8}$

The L.C.M. of 9 and 8 is 72.

$\frac{2}{9} = \frac{2 \times 8}{9 \times 8} = \frac{16}{72}$ , $\frac{3}{8} = \frac{3 \times 9}{8 \times 9} = \frac{27}{72}$

$∵ 16 < 27, \frac{2}{9} < \frac{3}{8}$

A rational number between $\frac{2}{9}$ and $\frac{3}{8}$

$= \frac{\frac{2}{9} + \frac{3}{8}}{2}$

$= \frac{\frac{16 + 27}{72}}{2}$

$= \frac{\frac{43}{72}}{2}$

$= \frac{43}{2 \times 72}$

$= \frac{43}{144}$

The numbers in descending orders are
$\frac{3}{8}, \frac{43}{144}, \frac{2}{9} .$

Question 2 Exercise 1.1 ML Aggarwal Class 9 Solutions has been given below.

2. Insert two rational number between $\frac{1}{3}$ and $\frac{1}{4}$ , and arrange in ascending order.

Answer

The given rational numbers are
$\frac{1}{3}$ and $\frac{1}{4}$ .

The L.C.M. of 3 and 4 is 12.

$\frac{1}{3} = \frac{1 \times 4}{3 \times 4} = \frac{4}{12}$ , $\frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}$ .

$∵ 3 < 4, \frac{1}{4} < \frac{1}{3}$

A rational number between $\frac{1}{4}$ and $\frac{1}{3}$

$= \frac{\frac{1}{4} + \frac{1}{3}}{2}$ $= \frac{\frac{3 + 4}{12}}{2}$ $= \frac{\frac{7}{12}}{2}$ $= \frac{7}{2 \times 12}$ $= \frac{7}{24}$

A rational number between $\frac{1}{4}$ and $\frac{7}{24}$

$= \frac{\frac{1}{4} + \frac{7}{24}}{2}$ $= \frac{\frac{6 + 7}{24}}{2}$ $= \frac{\frac{13}{24}}{2}$ $= \frac{7}{2 \times 24}$ $= \frac{7}{48}$

The numbers in descending orders are
$\frac{1}{4}, \frac{13}{48}, \frac{7}{24}, \frac{1}{3}$ .

Question 3 Exercise 1.1 ML Aggarwal Class 9 Solutions has been given below.

3. Insert two rational number between $- \frac{1}{3}$ and $- \frac{1}{2}$ , and arrange in ascending order.

Answer

The given rational numbers are
$- \frac{1}{3}$ and $- \frac{1}{2}$ .

The L.C.M. of 3 and 2 is 6.

$\frac{- 1}{3} = \frac{- 1 \times 2}{3 \times 2} = \frac{- 2}{6}$ , $\frac{- 1}{2} = \frac{- 1 \times 3}{2 \times 3} = \frac{- 3}{6}$ .

$∵ - 3 < - 2,$ $- \frac{1}{2} < - \frac{1}{3}$

A rational number between $- \frac{1}{2}$ and $- \frac{1}{3}$

$= \frac{\frac{- 1}{2} + \frac{- 1}{3}}{2}$ $= \frac{\frac{- 3 - 2}{6}}{2}$ $= \frac{- 5}{2 \times 6}$ $= \frac{- 5}{12}$

A rational number between $\frac{- 1}{2}$ and $\frac{- 5}{12}$

$= \frac{\frac{- 1}{2} + \frac{- 5}{12}}{2}$ $= \frac{\frac{- 6 - 5}{12}}{2}$ $= \frac{- 11}{2 \times 12}$ $= \frac{- 11}{24}$

The numbers in ascending order are
$- \frac{1}{2}, - \frac{11}{24}, - \frac{5}{12}, - \frac{1}{3}$ .

Question 4 Exercise 1.1 ML Aggarwal Class 9 Solutions has been given below.

4. Insert three rational numbers between $\frac{1}{3}$ and $\frac{4}{5}$ , and arrange in descending order.

Answer

The given rational numbers are
$\frac{1}{3}$ and $\frac{4}{5}$ .

The L.C.M. of 3 and 5 is 15.

$\frac{1}{3} = \frac{1 \times 5}{3 \times 5} = \frac{5}{15}$ , $\frac{4}{5} = \frac{4 \times 3}{5 \times 3} = \frac{12}{15}$ .

$∵ 5 < 12, \frac{1}{3} < \frac{4}{5}$

Now, the denominators of both the numbers are equal i.e. 15.

Since we want to find three rational numbers between the given numbers, multiplying the numerator and denominator of each of the above numbers by 3+1= 4, we get

$\frac{5}{15} = \frac{5 \times 4}{15 \times 4} = \frac{20}{60}$ , $\frac{12}{15} = \frac{12 \times 4}{15 \times 4} = \frac{48}{60}$ .

As $\frac{1}{3} = \frac{20}{60} < \frac{21}{60} < \frac{22}{60} < \frac{23}{60} < \frac{24}{60} < \frac{25}{60} < \frac{26}{60}$
$< \frac{27}{60} < \frac{28}{60} < \frac{29}{60} < \frac{30}{60} < \frac{31}{60} < \frac{32}{60} < \frac{33}{60} < \frac{34}{60}$
$< \frac{35}{60} < \frac{36}{60} < \frac{37}{60} < \frac{38}{60} < \frac{39}{60} < \frac{40}{60} < \frac{41}{60} < \frac{42}{60}$
$< \frac{43}{60} < \frac{44}{60} < \frac{45}{60} < \frac{46}{60} < \frac{47}{60} < \frac{48}{60} = \frac{4}{5}$

We can choose any three rational numbers from amongst the rational numbers $\frac{1}{3}$ and $\frac{4}{5}$ .

Hence, the three rational numbers between $\frac{1}{3}$ and $\frac{4}{5}$ are $\frac{27}{60}, \frac{34}{60}, \frac{41}{60}$ or $\frac{27}{60}, \frac{17}{30}, \frac{41}{60}$ .

The descending order of the numbers is $\frac{4}{5}, \frac{41}{60}, \frac{17}{30}, \frac{27}{60}, \frac{1}{3}$ .

Question 5 Exercise 1.1 ML Aggarwal Class 9 Solutions has been given below.

5. Insert three rational numbers between 4 and 4.5.

Answer

The given rational numbers are 4 and 4.5.

A rational number between 4 and 4.5

$= \frac{4 + 4.5}{2} = \frac{8.5}{2} = 4.25$

As 4 < 4.25 < 4.5, a rational number between 4 and 4.25

$= \frac{4 + 4.25}{2} = \frac{8.25}{2} = 4.125$

As 4 < 4.125 < 4.25 < 4.5, a rational number between 4.25 and 4.5

$= \frac{4.25 + 4.5}{2} = \frac{8.75}{2} = 4.375$

We note that 4 < 4.125 < 4.25 < 4.375 < 4.5, therefore, the three rational numbers between 4 and 4.5 are 4.125, 4.25, 4.375.

Question 6 Exercise 1.1 ML Aggarwal Class 9 Solutions has been given below.

6. Find six rational numbers between 3 and 4.

Answer

The given rational numbers are 3 and 4.

These numbers can be written as
$3 = \frac{3}{1}, 4 = \frac{4}{1}$

Since we want to find six rational numbers between the given numbers, multiplying the numerator and denominator of each of the above numbers by 6+1 i.e. by 7, we get
$\frac{3}{1} = \frac{3 \times 7}{1 \times 7} = \frac{21}{7}$ , $\frac{4}{1} = \frac{4 \times 7}{1 \times 7} = \frac{28}{7}$ .

Now, we can list some of the rational numbers between $\frac{21}{7}$ and $\frac{28}{7}$ as follows.

$3 = \frac{21}{7} < \frac{22}{7} < \frac{23}{7} < \frac{24}{7} < \frac{25}{7} < \frac{26}{7} < \frac{27}{7} < \frac{28}{7} = 4$

Clearly, the six rational numbers between 3 and 4 are $\frac{22}{7}, \frac{23}{7}, \frac{24}{7}, \frac{25}{7}, \frac{26}{7}, \frac{27}{7}$ .

Question 7 Exercise 1.1 ML Aggarwal Class 9 Solutions has been given below.

7. Find five rational numbers between $\frac{3}{5}$ and $\frac{4}{5}$ .

Answer

The given rational numbers are $\frac{3}{5}$ and $\frac{4}{5}$ .

Since we want five rational numbers between $\frac{3}{5}$ and $\frac{4}{5}$ , multiplying the numerator and denominator of each of the given numbers by 5+1 i.e. 6, we get
$\frac{3}{5} = \frac{3 \times 6}{5 \times 6} = \frac{18}{30}, \frac{4}{5} = \frac{4 \times 6}{5 \times 6} = \frac{24}{30}$

Now, we can list some of the rational numbers between $\frac{18}{30}$ and $\frac{24}{30}$ as follows.

$\frac{3}{5} = \frac{18}{30} < \frac{19}{30} < \frac{20}{30} < \frac{21}{30} < \frac{22}{30} < \frac{23}{30} < \frac{24}{30} = \frac{4}{5}$

$\Rightarrow \frac{3}{5} = \frac{18}{30} < \frac{19}{30} < \frac{2}{3} < \frac{7}{10} < \frac{11}{15} < \frac{23}{30} < \frac{24}{30} = \frac{4}{5}$

Therefore, five rational numbers between $\frac{3}{5}$ and $\frac{4}{5}$ are:
$\frac{19}{30}, \frac{2}{3}, \frac{7}{10}, \frac{11}{15}, \frac{23}{30}$

Question 8 Exercise 1.1 ML Aggarwal Class 9 Solutions has been given below.

8. Find ten rational numbers between $- \frac{2}{5}$ and $\frac{1}{7}$ .

Answer

The given rational numbers are $- \frac{2}{5}$ and $\frac{1}{7}$ .

L.C.M. of the denominators 5 and 7 is 35.

We can write the given numbers with the same denominator as follows:
$\frac{- 2}{5} = \frac{- 2 \times 7}{5 \times 7} = \frac{- 14}{35}, \frac{1}{7} = \frac{1 \times 5}{7 \times 5} = \frac{5}{35}$

Now, we can see that there is enough gap between the numerators of $\frac{- 14}{35}$ and $\frac{5}{35}$ , so we do not need to multiply by a number further.

As $\frac{- 2}{5} = \frac{- 14}{35} < \frac{- 13}{35} < \frac{- 12}{35} < \frac{- 11}{35} < \frac{- 10}{35} < \frac{- 9}{35}$
$< \frac{- 8}{35} < \frac{- 7}{35} < \frac{- 6}{35} < \frac{- 5}{35} < \frac{- 4}{35} < \frac{- 3}{35} < \frac{- 2}{35} < \frac{- 1}{35}$
$< \frac{0}{35} < \frac{1}{35} < \frac{2}{35} < \frac{3}{35} < \frac{4}{35} < \frac{5}{35} = \frac{1}{7}$ .

Now, we can easily select any ten rational numbers from the above list of rational numbers.

The ten rational numbers between $\frac{- 2}{5}$ and $\frac{1}{7}$ are $\frac{- 13}{35}, \frac{- 12}{35}, \frac{- 11}{35}, \frac{- 10}{35}, \frac{- 9}{35}, \frac{- 8}{35}, \frac{- 7}{35},$ $\frac{0}{35}, \frac{1}{35}, \frac{2}{35}$

After simplification, we get
$\frac{- 13}{35}, \frac{- 12}{35}, \frac{- 11}{35}, \frac{- 2}{7}, \frac{- 9}{35}, \frac{- 8}{35}, \frac{- 1}{5}, 0, \frac{1}{35}, \frac{2}{35}$

Question 9 Exercise 1.1 ML Aggarwal Class 9 Solutions has been given below.

9. Find six rational numbers between $\frac{1}{2}$ and $\frac{2}{3}$ .

Answer

The given rational numbers are $\frac{1}{2}$ and $\frac{2}{3}$ .

L.C.M. of 2 and 3 are 6.

We can form equivalent rational numbers of the given rational numbers with the same denominator as follows:
$\frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6}, \frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6}$

Since the numerators of the equivalent rational numbers $\frac{3}{6}$ and $\frac{4}{6}$ have not enough gap so that we could get six rational numbers, we multiply by 6+1 i.e. 7 in both the numerators and denominators. This is done to increase the gap between the numerators.

$\frac{3}{6} = \frac{3 \times 7}{6 \times 7} = \frac{21}{42}, \frac{4}{6} = \frac{4 \times 7}{6 \times 7} = \frac{28}{42}$

Now, the gap between the numerators of $\frac{21}{42}$ and $\frac{28}{42}$ is good enough to find six rational numbers easily.

As $\frac{1}{2} = \frac{21}{42} < \frac{22}{42} < \frac{23}{42} < \frac{24}{42} < \frac{25}{42} < \frac{26}{42} < \frac{27}{42} < \frac{28}{42} = \frac{2}{3}$

$\Rightarrow \frac{1}{2} = \frac{21}{42} < \frac{11}{21} < \frac{23}{42} < \frac{4}{7} < \frac{25}{42} < \frac{13}{21} < \frac{9}{14} < \frac{28}{42} = \frac{2}{3}$

Therefore, the six rational numbers between $\frac{1}{2}$ and $\frac{2}{3}$ are

$\frac{11}{21}, \frac{23}{42}, \frac{4}{7}, \frac{25}{42}, \frac{13}{21}, \frac{9}{14}$

Exercise 1.1 ML Aggarwal Class 9 Solutions

1. Insert a rational number between $\frac{2}{9}$ and $\frac{3}{8}$ , and arrange in descending order.

2. Insert two rational number between $\frac{1}{3}$ and $\frac{1}{4}$ , and arrange in ascending order.

3. Insert two rational number between $- \frac{1}{3}$ and $- \frac{1}{2}$ , and arrange in ascending order.

4. Insert three rational numbers between $\frac{1}{3}$ and $\frac{4}{5}$ , and arrange in descending order.

5. Insert three rational numbers between 4 and 4.5.

6. Find six rational numbers between 3 and 4.

7. Find five rational numbers between $\frac{3}{5}$ and $\frac{4}{5}$ .

8. Find ten rational numbers between $- \frac{2}{5}$ and $\frac{1}{7}$ .

9. Find six rational numbers between $\frac{1}{2}$ and $\frac{2}{3}$ .

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Exercise 1.1 ML Aggarwal Class 9 Solutions

1. Insert a rational number between 29 and 38, and arrange in descending order.

2. Insert two rational number between 13 and 14, and arrange in ascending order.

3. Insert two rational number between −13 and −12, and arrange in ascending order.

4. Insert three rational numbers between 13 and 45, and arrange in descending order.

5. Insert three rational numbers between 4 and 4.5.

6. Find six rational numbers between 3 and 4.

7. Find five rational numbers between 35 and 45.

8. Find ten rational numbers between −25 and 17.

9. Find six rational numbers between 12 and 23.

1. Insert a rational number between $\frac{2}{9}$ and $\frac{3}{8}$ , and arrange in descending order.

2. Insert two rational number between $\frac{1}{3}$ and $\frac{1}{4}$ , and arrange in ascending order.

3. Insert two rational number between $- \frac{1}{3}$ and $- \frac{1}{2}$ , and arrange in ascending order.

4. Insert three rational numbers between $\frac{1}{3}$ and $\frac{4}{5}$ , and arrange in descending order.

7. Find five rational numbers between $\frac{3}{5}$ and $\frac{4}{5}$ .

8. Find ten rational numbers between $- \frac{2}{5}$ and $\frac{1}{7}$ .

9. Find six rational numbers between $\frac{1}{2}$ and $\frac{2}{3}$ .