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 and 25 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
1
1
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
7
2
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 23 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 4 0 6 6 6 6 6 6 − − − Therefore, the five rational numbers between
1
2
−
and 2 3 are, 1 1 1 1 0 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 , , , , , , , , , , , − − − − − − − − − − − − − − −
Difficulty Level- Low What is given /known? Patterns of the numbers. What is the unknown?
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
12
i)
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
,
,
,
,
−
−
−
−
−
−
Next four rational numbers in the same pattern,
1 4 1 5 1 6 1 7 4 4 4 5 4 6 4 7 × , , , × − − − −
Therefore, the numbers are 4 5 6 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
,
,
,
,
,
,
−
−
−
−
−
−
−
−
Next four rational numbers in the same pattern,
1 5 1 6 1 7 1 8
6 5, 6 6, 6 7, 6 8
− − − −
Therefore, the numbers are 5 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
,
,
,
,
,
,
−
−
−
−
−
−
−
−
Next four rational numbers in the same pattern,
2 4 2 5 2 6 2 7
3 4, 3 5, 3 6, 3 7
− − − −
Therefore, the numbers are 8 10 12 14 12, 15, 18, 21
− − − −
Q3. Give four rational numbers equivalent to:
2 5
(i) (ii) (ii 4
7 3 i) 9
−
− Difficulty Level- Low What is given /known? Three rational numbers. What is the unknown?
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 number 5 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 20
and 18, 27, 36 45
Q4. Draw the number line and represent the following rational numbers on it:
( )
3( )
5( )
7( )
7ii iii iv
4 8 4 8
i − −
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.
1
TR
RS SU
3
1
4
R
1
3
3
2
5
S
1
3
3
=
=
=
−
= − − =
−
= − − =
Similarly, AB = 1unitIt is divided into three equal parts.
1
AP
PQ
QB
3
1
6
1
7
P
2
3
3
3
3
2
6
2
8
Q
2
3
3
3
3
=
=
=
= + = + =
= + = + =
Thus, the rational number P, Q, R and S are 7 8 4 and 5 3, 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 What is given /known? Two pair of rational numbers.
What is the unknown?
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) and 21 9 −
On reducing them to the simplest form, we get
7 1 3 1 and 21 3 9 3 − =− = 1 1 Since 3 3 , − Therefore,
7
21
−
and 39 does not represent the pair of same rational numbers.
16 20 (ii) and
20 25
−
−
On reducing them to the simplest form, we get
16 4 20 4 and 20 5 25 5 − =− = − − 4 4 since, 5 5 − = − Therefore, 16 20 − and 20 25
− represents the pair of same rational numbers.
2 2
(iii) and
3 3
− −
On reducing them to the simplest form, we get
2 2 2 2 and 3 3 3 3 − = = − 2 2 since, 3 3 − = − Therefore, 2 3 − − and 2
3 12 (iv) and
5 20
− −
On reducing them to the simplest form, we get
3 3 12 3 and 5 5 20 5 − = − − = − 3 3 since, 5 5 − =− Therefore, 3 5 − and 12 20 −
represent the pair of same rational numbers.
8 24
(v) and
5 15
− −
On reducing them to the simplest form, we get
8 8 24 8 and 5 5 15 5 − − = = − − 8 8 Since, 5 5 − = − Therefore, 8 and 24 5 15 −
− represent the pair of same rational numbers.
1 1
(vi) and
3 9
−
On reducing them to the simplest form, we get
1 1 1 1 and 3 3 9 9 − − = = 1 1 Since, 3 9 − Therefore, 1 3 and 1 9 −
do not represent the pair of same rational numbers.
5 5
(vii) and
9 9
−
− −
On reducing them to the simplest form, we get
5 5 5 5 and 9 9 9 9 − − = = − − 5 5 Since, 9 9 − Therefore, 5 9 − − and 5 9
Q7. Rewrite the following rational numbers in the simplest form:
8 25 44 8
(i) (ii) (iii) (iv)
6 45 72 10
− − −
Difficulty Level- Low What is given /known? Rational numbers. What is the unknown?
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 −
H.C.F of 44 and 72 is 4. Dividing the numerator and denominator by H.C.F., we get,
44 4 11 72 4 18 − = − 8 (iv) 10 −
H.C.F of 8 and 10 is 2. Dividing the numerator and denominator by H.C.F., we get,
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)
0Difficulty Level- Medium What is given /known? Two rational numbers. What is the unknown?
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)i 7 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 1 v 1 i − − 5 5 11 11 5 5 11 11 − − − − = Therefore, 5 11 11 5 − − =( )
vii 0 7 6 −(
)
7 6 70 0 is always greater than negative int
0 eger 6 − −
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 What is given /known? Two rational numbers.
What is the unknown?
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 15 and 3 2 9 2 3 6 4 15 2 5 Since, so 6 6 , 3 2 = = 5 4 (ii) 6 , 3 − − L.C.M of 6 and 3 is 6 5 1 5 4 2 8 and 6 1 6 3 2 6 5 8 5 4 Since, So, 6 6 6 3 − =− − = − − − − − 3 2 (iii) 4 , 3 − − L.C.M of 4 and 3 is 12 3 3 3 9 2 2x4 8 and 4 4x3 12 3 3 4 12 9 8 3 2 Since, , 12 12 So 4 3 − =− =− = =− − − − − − − 1 ( v) 4 i 1 4 , −
(
)
1 1Negative number is always smaller than the positive number
4 4 − 2 4 (v) 3 3 7, 5 − − 2 23 4 19 3 and 3 7 7 5 5 − − − = =
L.C.M of 7 and 5 is 35 23 23 5 115 7 7 5 35 − = − = − 19 19 7 133 5 7 35 5 − = − = − Since, 115 133 35 35 − − So, 32 34 7 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 What is given /known? Three rational numbers. What is the unknown?
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 So, 1 1 3 3 3 3 3 9 − = − = − 2 2 1 2 9 1 9 9 − = − =−
and 4 4 3 12
3 3 3 9
− − −
= =
Arranging them into ascending order we get,
12 3 2 9 9 9 4 1 2 Or 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 What is given /known? Two rational numbers What is the unknown? 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 3 5 5 3 15 15 15 15 + + = + = + = = 9 22 (iii) 10 15 − +
Taking L.C.M of 10 and 15, we get 30
9 22 9 3 22 2 27 44 17
10 15 10 3 15 2 30 30 30
− + = − + = − + =
3 5 (iv)
11 9 − + −
Taking L.C.M of 11 and 9, we get 99
3 5 3 9 5 11 27 55 82 11 9 11 9 11 9 99 99 99 − + = − + = + = − − 8 ( 2) (v) 19 57 − − +
Taking L.C.M of 19 and 57, we get 57
8 ( 2) 8 3 2 1 24 2 24 2 26 19 57 19 3 57 1 57 57 57 57 − − − − − − − − − + = + = + = = 2 (vi) 0 3 − +
Taking L.C.M of 3 and 1, we get 3
2 2 1 0 3 2 0 2 0 2 0 3 3 1 1 1 3 3 3 3 − + = − + = − + = − + = − 1 3 7 23 (vii) 2 4 3 5 3 5 − − + = +
Taking L.C.M of 3 and 5, we get 15
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 What is given /known? Two rational numbers What is the unknown?
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
Taking L.C.M of 24 and 36, we get 72
7 17 7 3 17 2 21 34 21 34 13 24 36 24 3 36 2 72 72 72 72 − − − = − = − = = 5 6 (ii) 63 21 − −
Taking L.C.M of 63 and 21, we get 63
5 6 5 1 6 3 5 18 5 18 23 63 21 63 1 21 3 63 63 63 63 − + − = + = + = = 6 7 (iii) 13 15 − −−
Taking L.C.M of 13 and 15, we get 195
6 7 6 15 7 13 90 91 90 91 1 13 15 13 15 15 13 195 195 195 195 − − − − − + − = + = + = = 3 7 (iv) 8 11 − −
Taking L.C.M of 8 and 11, we get 88
3 7 3 11 7 8 33 56 33 56 89 8 11 8 11 11 8 88 88 88 88 − − = − − = − − = − − = − 1 (v) 2 6 9 − − = 21 6 19 6 9 1 9 1 − − = − −
Taking L.C.M of 9 and 1, we get 9
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 9 i ii 9 iii 2 4 10 5 11 3 2 3 2 3 5 iv v vi 7 5 11 5 5 3 − − − − − − Difficulty Level: Low What is given /known? Two rational numbers What is the unknown?
Product of two rational numbers. Reasoning:
In such type of questions, find the product of numerator and denominator .
Solution:
( )
9 7 i 2 4 − = 92−47=92 4 −7 = −863 ( )
3( )
3 3 9 27 ii 9 10 10 1 10 10 9 − − − = − = = ( )
6 9 6 9 54 iii 5 11 5 11 55 − = − = − ( )
3 2 3 2 6 iv 7 5 7 5 35 − − − = = ( )
3 2 3 2 6 v 11 5 11 5 55 = = ( )
3 5 3 5 15 vi 1 5 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 2 iv vi 8 4 13 7 12 13 3 4 vii 13 65 v − − − − − − − − − Difficulty Level: Moderate What is given /known? Two numbers.What is the unknown? 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:
