chapter-9: rational numbers

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Chapter-9: Rational Numbers

Exercise 9.1 (Page 182)

Q1. List five rational numbers between:

(i) –1 and 0 (ii) –2 and –1 (iii) 4 2

and5 3

− − 1 2(iv) and

2 3−

Difficulty Level- Low

What is given /known?

Two integers.

What is the unknown?

Five rational numbers between the given two integers.

Reasoning:

These questions can be solved easily with the concept of like fractions, First make the

fractions like by making their denominator equal. You can make denominator equal either

by taking L.C.M of denominator or by multiplying both numerator and denominator by

same integer. By applying these methods, you can get the like fractions and can easily

find out the rational numbers between the given numbers.

Solution:

(i) –1 and 0

Multiplying both numerator and denominator by 6, we get

1 6 6

1 6 6

0 6 0

1 6 6

,− −

=

=

Five rational numbers between –1 and 0 are,

6 5 4 3 2 1 0

6 6 6 6 6 6 6

5 4 3 2 11 0

6 6 6 6 6

− − − − − −

− − − − −−

Thus, the five rational numbers between –1 and 0 are

5 2 1 1 1

6 3 2 3 6, , , ,

− − − − −



(ii) –2 and –1

Multiplying both numerator and denominator by 6, we get

2x6 12

1x6 6

1x6 6

1x6 6

,− −

=

− −=

Five rational numbers between –2 and –1 are,

12 11 10 9 8 7 6

6 6 6 6 6 6 6

11 10 9 8 72 1

6 6 6 6 6

− − − − − − −

− − − − −− −

Thus, the five rational numbers between –2 and –1 are

11 5 3 4 7

6 3 2 3 6, , , ,

− − − − −

4 2(iii) and

5 3

− −

Converting 4

5

− and

2

3

− into like fractions, we get

4 4x9 36

5 5x9 45

2 2x15 30

3 3 15 45

− − −= =

− − −= =

Five rational numbers between 4

5

− and

2

3

− are,

36 35 34 33 32 31 30

45 45 45 45 45 45 45

4 35 34 33 32 31 2

5 45 45 45 45 45 3

− − − − − − −

− − − − − − −

Therefore, the five rational numbers between 4

5

− and

2

3

− are,

7 34 11 32 31

9 45 15 45 45, , , ,

− − − − −



1 2(iv) and

2 3−

Converting 1

2− and

2

3 into like fractions, we get

1 1 3 3

2 2x3 6

2 2 2 4

3 3 2 6

− − −= =

= =

Five rational numbers between 1

2

− and

2

3 are,

3 2 1 2 3 40

6 6 6 6 6 6

− − −

Therefore, the five rational numbers between 1

2− and

2

3 are,

1 1 1 10

3 6 3 2, , , ,

− −

Q2. Write four more rational numbers in each of the following patterns: 3 6 9 12 1 2 3

(i) (ii)5 5 15 5 4 8 12

1 2 3 4 2 2 4 6(iii) (iv)

6 12 18 24 3 3 6 9

, , , , ,

, , , , , ,

− − − − − − −

− −

− − − − − −



Patterns of the numbers.


Four more rational numbers in each of the given patterns.

Reasoning:

This question is based on a definite pattern, while solving such type of questions observe

the numerator and denominator carefully. Here both numerator and denominator are

following a definite pattern, observe this pattern and follow the same pattern, you can

easily find out the next four rational numbers.



Solution:

3 6 9 12i)

5 10 15 20

3 1 3 2 3 3 3 4

5 1 5 2 5 3 5 4

, , ,

, , ,

− − − −

− − − −

Next four rational numbers in the same pattern,

3 5 3 6 3 7 3 8

5 5 5 6 5 7 5 8, , ,

− − − −

Therefore, the numbers are 15 18 21 24

25 30 35 40, , ,

− − − −

1 2 3(ii)

4 8 12

1 1 1 2 1 3

4 1 4 2 4 3

, ,

, ,

− − −

− − −


1 4 1 5 1 6 1 7

4 4 4 5 4 6 4 7

×, , ,

×

− − − −


16 20 24 28, , , .

− − − −

1 2 3 4(iii)

6 12 18 24

1 1 1 2 1 3 1 4

6 1 6 2 6 3 6 4

, , ,

, , ,

−

− − −

− − − −


1 5 1 6 1 7 1 8

6 5 6 6 6 7 6 8, , ,

− − − −

Therefore, the numbers are5 6 7 8

30 36 42 48, , , .

− − − −



2 2 4 6(iv)

3 3 6 9

2 1 2 1 2 2 2 3

3 1 3 1 3 2 3 3

, , ,

, , ,

−

− − −

−

− − −


2 4 2 5 2 6 2 7

3 4 3 5 3 6 3 7, , ,

− − − −


12 15 18 21, , ,

− − − −

Q3. Give four rational numbers equivalent to:

2 5(i) (ii)

4(ii

7 3i)

9

−

−



Three rational numbers.


Four rational numbers equivalent to each of the given rational number.

Reasoning:

To find out the equivalent fraction of any rational number, multiply the numerator and the

denominator of the given number by the same numbers. Remember here it is asked for

four equivalent rational numbers that means you have to multiply four different numbers,

one by one in both numerator and denominator of the given number.

Solution:

2(i)

7

−

Multiplying both numerator and denominator with the same number, we get

2 2 4 2 3 6 2 4 8 2 5 10

7 2 14 7 3 21 7 4 28 7 5 35, , ,

− − − − − − − −= = = =

Therefore, the equivalent fractions to the number 2

7

− are,

4 6 8 10

14 21 28 35, , ,

− − − −



5(ii)

3−Multiplying both numerator and denominator with the same number, we get

5 2 10 5 3 15 5 4 20 5 5 25

3 2 6 3 3 9 3 4 12 3 5 15, , ,

= = = =

− − − − − − − −

Therefore, the equivalent fractions to the number5

3− are,

10 15 20 25

6 9 12 15, , ,

− − − −4

(iii)9

Multiplying both numerator and denominator with the same number, we get

4 2 8 4 3 12 4 4 16 4 5 20

9 2 18 9 3 27 9 4 36 9 5 45, , ,

= = = =

Therefore, the equivalent fractions to the number 4

9 are,

8 12 16 20and

18 27 36 45, ,

Q4. Draw the number line and represent the following rational numbers on it:

( ) ( ) ( ) ( )3 5 7 7

ii iii iv4 8 4 8

i− −

Solution:



Q5. The points P, Q, R, S, T, U, A and B on the number line are such that,

TR = RS = SU and AP = PQ = QB. Name the rational numbers represented

by P, Q, R and S.

Solution:

Distance between U and T = 1 unit. It is divided into three equal parts.

1TR RS SU

3

1 4R 1

3 3

2 5S 1

3 3

= = =

−= − − =

−= − − =

Similarly, AB = 1unit

It is divided into three equal parts.

1AP PQ QB

3

1 6 1 7P 2

3 3 3 3

2 6 2 8Q 2

3 3 3 3

= = =

= + = + =

= + = + =

Thus, the rational number P, Q, R and S are 7 8 4 5

and3 3 3 3

, ,− −

.

Q 6. Which of the following pairs represent the same rational number?

7 3 16 20 2 2 12(i) and (ii) and (iii) and (iv) and

21 9 20 25 3 3 5 20

8 24 1 1 5 5(v) and (vi) and (vii) and

5 15 3 9

3

9 9

− − − −

− −

− − −

−

−

− −

Difficulty Level - Medium


Two pair of rational numbers.




Which pair represent the same rational number.

Reasoning:

In such type of questions, reduce the rational numbers to the lowest or simplest form.

By reducing them to the simplest form you can easily get out the same rational numbers.

Solution: 7 3

(i) and21 9

−

On reducing them to the simplest form, we get

7 1 3 1and

21 3 9 3

− −= =

1 1Since

3 3,−

Therefore,7

21

− and

3

9 does not represent the pair of same rational numbers.

16 20(ii) and

20 25

−

−


16 4 20 4and

20 5 25 5

− −= =

− −

4 4since,

5 5

−=−

Therefore, 16

20

− and

20

25− represents the pair of same rational numbers.

2 2(iii) and

3 3

−

−


2 2 2 2and

3 3 3 3

−= =

−

2 2since,

3 3

−=

−

Therefore, 2

3

−

− and

2

3 represents the pair of same rational numbers.



3 12(iv) and

5 20

− −


3 3 12 3and

5 5 20 5

− − − −= =

3 3since,

5 5

− −=

Therefore,3

5

− and

12

20

− represent the pair of same rational numbers.

8 24(v) and

5 15

−

−


8 8 24 8and

5 5 15 5

− −= =

− −

8 8Since,

5 5

−=

−

Therefore,8 24

and5 15

−

−represent the pair of same rational numbers.

1 1(vi) and

3 9

−


1 1 1 1and

3 3 9 9

− −= =

1 1Since,

3 9

−

Therefore, 1

3 and

1

9

− do not represent the pair of same rational numbers.

5 5(vii) and

9 9

−

− −


5 5 5 5and

9 9 9 9

− −= =

− −

5 5Since,

9 9

−

Therefore, 5

9

−

− and

5

9− do not represent the pair of same rational numbers.



Q7. Rewrite the following rational numbers in the simplest form: 8 25 44 8

(i) (ii) (iii) (iv)6 45 72 10

− − −



Rational numbers.


Simplest form of the given rational numbers.

Reasoning:

While solving such type of questions, find the H.C.F of numerator and denominator and

then divide both numerator and denominator by the H.C.F. After dividing it you will get

the simplest form of the rational number.

Solution:

8(i)

6

−

H.C.F. of 8 and 6 is two. Dividing the numerator and denominator by H.C.F., we get

8 2 4

6 2 3

− −=

25(ii)

45

H.C.F of 25 and 45 is 5. Dividing the numerator and denominator by H.C.F., we get,

25 5 5

45 5 9

=

44(iii)

72

−


44 4 11

72 4 18

− −=

8(iv)

10

−


8 2 4

10 2 5

− −=



Q8. Fill in the boxes with the correct symbol out of >, <, and =.

(i)5

7

− 2

3(ii)

4

5

− 5

7

− 7

8

−(iii)

14

16−

(iv)8

5

− 7

4

−(v)

1

3−

1

4

−(vi)

5

11−

5

11

−

7

6

−(vi)0

Difficulty Level- Medium


Two rational numbers.


Comparison of the two rational numbers i.e., which one is smaller and which one is

greater or are they both equal.

Reasoning:

In this type of questions, first find the LCM of the denominators of both the rational

numbers. Then make denominator of each rational number equal to LCM by multiplying

numerator and denominator with the same number (convert them into like fractions).

Then comparison of the two numbers can be easily made.

Solution:

5 2(i)

7 3

−

( )

5 3 2 7

7 3 3 7

15 14

21 21

15 14Positive number is greater than the negative number

21 21

−

−

−

Therefore, 5 2

7 3

−



4 5(ii)

5 7

− −

4 7 5 5

5 7 7 5

28 25

35 35

28 25

35 35

− −

− −

− −

Therefore, 4 5

5 7

− −

7 14(iii)

8 16

−

−

7 2 14 1

8 2 16 1

14 14

16 16

14 14

16 16

−

−

−

−

− −=

Therefore, 7 14

8 16

−=−

8( v)

7i

5 4

− −

8 4 7 5

5 4 4 5

32 35

20 20

8 7

5 4

− −

− −

− −

Therefore, 8 7

5 4

− −



1 1(v)

3 4

−

−

4

1 4 1 3

3 4 4 3

4 3

12 12

1 1

3

− −

− −

− −

−

−

Therefore, 1 1

3 4−

−

( )5 5

11 1v

1i

−

−

5 5

11 11

5 5

11 11

− −

− −=

Therefore, 5

11 11

5−

−=

( )vii 07

6

−

( )

7

6

70 0 is always greater than negative int

0

eger6

−

−

Q9. Which is greater in each of the following:

2 5 5 4 3 2 (i) (ii) (iii)

3 2 6 3 4 3

1 2 4(iv) (v) 3 3

7 5

1

4 4

, , ,

, ,

− − −

−

−− −

Difficulty Level - Medium


Two rational numbers.




Greater rational number of the two given rational numbers.

Reasoning:

In such type of questions take the L.C.M of denominator of both the rational numbers or

convert them into like fractions. After converting them into like fractions comparison will

be easy. Solution:

2 5(i)

3 2,

L.C.M of 3 and 2 is 6

2 2 6 5 3 15and

3 2 9 2 3 6

4 15 2 5Since, so

6 6 3 2,

= =

5 4(ii)

6 3,

− −


5 1 5 4 2 8 and

6 1 6 3 2 6

5 8 5 4Since, So,

6 6 6 3

− − − −= =

− − − −

3 2(iii)

4 3,

−

−


3 3 3 9 2 2x4 8and

4 4x3 12 3 3 4 12

9 8 3 2Since, ,

12 1 3o

4S

2

− − − −= = = =

− −

− − −

−

1( v)

4i

1

4,

−

( )1 1

Negative number is always smaller than the positive number4 4

−

2 4(v) 3 3

7 5,− −

2 23 4 193 and 3

7 7 5 5

− −− = =




23 23 5 115

7 7 5 35

− − −= =

19 19 7 133

5 7 355

− − −= =

Since,115 133

35 35

− −

So, 2 4

3 37 5

− −

Q10. Write the following rational numbers in ascending order: 3 2 1 1 2 4 3 3 3

(i) (ii) (iii)5 5 5 3 9 3 7 2 4

, , , , , ,− − − − − − − − −

Difficulty Level - Low


Three rational numbers.


Ascending order of the given rational numbers.

Reasoning:

In such type of questions take the L.C.M of denominator of the rational numbers or

convert them into like fractions. After converting them into like fractions comparison will

be easy.

Solution:

3 2 1(i)

5 5 5, ,

− − −

Since denominator is same in all the rational numbers, these can be easily arranged into

ascending order , 3 2 1− − −

Hence, the required ascending order is,

3 2 1

5 5 5

− − −

1 2 4(ii)

3 9 3, ,

− − −

L.C.M of 3,9 and 3 is 9

1 1 3 3

So,3 3 3 9

− − −= =

2 2 1 2

9 1 99

− − −= =



and4 4 3 12

3 3 3 9

− − −= =

Arranging them into ascending order we get,

12 3 2

9 9 9

4 1 2Or

3 3 9

− − −

− − −

3 3 3(iii)

7 2 4, ,

− − −

L.C.M of 7, 2 and 4 is 28

3 3 4 12

7 7 4 28

− − −= =

3 3 14 42

2 2 14 28

− − −= =

3 3 7 21

4 4 7 28

− − −= =

Arranging them into ascending order we get,

42 21 12

28 28 28

− − −

Therefore, 3 3 3

2 4 7

− − −



Chapter-9: Rational Numbers

Exercise 9.2 (Page 190)

Q1. Find the sum:

5 11 5 3 9 22(i) (ii) (iii)

4 4 3 5 10 15

3 5 8 ( 2) 2(iv) (v) (vi) 0

11 9 19 57 3

1 3(vii) 2 4

3 5

− − + + +

− − − −+ + +

−

− +

Difficulty Level: Low


Two rational numbers


Sum of two rational numbers.

Reasoning:

In such type of questions, take the L.C.M of denominator or convert the given fractions

into like fractions and then find their sum. You can also reduce the fractions to the

lowest form.

Solution:

5 11 5 11 5 11 6 3(i)

4 4 4 4 4 4 2

− − − − + = − = = =

5 3(ii)

3 5+

Taking L.C.M of 3 and 5, we get 15

5 3 5 5 3 3 25 9 25 9 34

3 5 15 15 15 153 5 5 3

++ = + = +

= =

9 22(iii)

10 15

−+


9 22 9 3 22 2 27 44 17

10 15 10 3 30 3015 2 30

− − −+ = + = + =



3 5(iv)

11 9

−+

−


3 5 3 9 5 11 27 55 82

11 9 11 9 99 99 9911 9

− − + = + = + =

− −

8 ( 2)(v)

19 57

− −+


8 ( 2) 8 3 2 1 24 2 24 2 26

19 57 57 57 57 5719 3 57 1

− − − − − − − − −+ = + = + = =

2(vi) 0

3

−+


2 2 1 0 3 2 0 2 0 20

3 3 1 1 1 3 3 3 3

− − − − + −+ = + = + = =

1 3 7 23(vii) 2 4

3 5 3 5

−− + = +


7 5 23 3 35 69 35 69 34

3 5 5 3 15 15 15 15

− − − ++ = + = =

Q2. Find

7 17 5 6 6 7 (i) (ii) (iii)

24 36 63 21 13 15

3 7 1(iv) (v) 2 6

8 11 9

− − − − − −

−− − −

Difficulty Level - Low




Difference between the given two rational numbers.



Reasoning:

In such type of questions take the L.C.M of denominator or convert them into like

fractions, then find their difference between them. You can also reduce them to the lowest

or simplest form.

Solution:

7 17(i)

24 36−


7 17 7 3 17 2 21 34 21 34 13

24 36 24 3 2 72 72 72 236 7

− −− = − = − = =

5 6(ii)

63 21

− −


5 6 5 1 6 3 5 18 5 18 23

63 21 63 1 3 63 63 63 6321

− + − = + = + = =

6 7(iii)

13 15

− −−


6 7 6 15 7 13 90 91 90 91 1

13 15 13 15 13 195 195 195 19515

− − − − − +− = + = + = =

3 7(iv)

8 11

−−


3 7 3 11 7 8 33 56 33 56 89

8 11 11 8 88 8 88 1 81 8 8 8

− − − − − −− = − = − = =

1(v) 2 6

9− − =

1 6 19 62

9 1 9 1− − = − −


19 6 19 1 6 9 19 54 19 54 73

9 1 9 1 1 9 9 9 9 9

− − − − −− − = − = − = =



Q3. Find the product:

( ) ( ) ( ) ( )

( ) ( ) ( )

9 7 3 6 9i ii 9 iii

2 4 10 5 11

3 2 3 2 3 5iv v vi

7 5 11 5 5 3

− − −

− − −

Difficulty Level: Low




Product of two rational numbers.

Reasoning:

In such type of questions, find the product of numerator and denominator .

Solution:

( )9 7

i2 4

− =

9 7 9 7 63

2 4 2 4 8

− − − = =

( ) ( )3 3 3 9 27

ii 910 10 1 10 10

9 − − − = = =

−

( )6 9 6 9 54

iii5 11 5 11 55

− − − = =

( )3 2 3 2 6

iv7 5 7 5 35

− − − = =

( )3 2 3 2 6

v11 5 11 5 55

= =

( )3 5 3 5 15

vi 15 3 5 3 15

− − − = = =

− − −



Q4. Find the value of:

( ) ( ) ( ) ( )

( ) ( ) ( )

( )

2 3 4(i) 4 ii 2 iii 3

3 5 5

1 3 2 1 7 2iv vi

8 4 13 7 12 13

3 4vii

13 65

v

− −− −

− − − −

−

Difficulty Level: Moderate


Two numbers.


Value of given numbers.

Reasoning:

In such type of questions, convert the denominator into its reciprocal. When you

convert the denominator into its reciprocal, the sign of division will also convert

into multiplication and now you can find out the product of the given numbers.

Solution:

( ) ( )2 3

i 4 4 2 3 63 2

− = − = − = −

( )3 3 1 3

ii 25 5 2 10

− − − = =

( )4 4 1 4

(iii) 35 5 3 15

− − − − = =

( )1 3 1 4 4 1

iv8 4 8 3 24 6

− − − − = = =

( )2 1 2 7 2 7 14

v13 7 13 1 13 13

− − − − = = =

( )7 2 7 13 7 13 91 91

vi12 13 12 2 12 2 24 24

− − − − − = = = = − − −

( )3 4 3 65 3 65 3 5 15

vii13 65 13 4 13 4 4 4

− − = = = = − − −



chapter-9: rational numbers

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