NCERT Class 7 Maths Chapter 8 Rational Numbers Exercise 8.1, 8.2 Solutions
In this page we have provided solutions of the Exercises 8.1, 8.2 of NCERT Class 7 Maths Chapter 8 Rational Numbers. These solutions are made by our team of expert teachers. Practice these solutions carefully for a better understanding of the topic which will help in scoring good marks in the examination.
Publishing Organisation |
NCERT |
Class |
7 |
Subject |
Mathematics |
Chapter |
8: Rational Numbers |
Exercise |
8.1, 8.2 |
Chapter – 8
Rational Number
Exercise – 8.1
(1) List five rational numbers between:
(i) -1 and 0
Ans:- -1/10, -1/20, -1/30, -1/40, -1/50
(ii) -2 and -1
Ans:- -2 = -12/6 and – 1 = -6/6
Five rational numbers are
-11/6, -10/6, -9/6, -8/6, -7/6
(iii) -4/5 and -2/3
Ans:- -4/5 = -4 × 9/5 × 9 = -36/45 and -2/3 = -2 × 15/3 × 15 = -30/45
Five rational numbers are
-35/45, -34/45, -33/45, -32/45, -31/45
(iv) -1/2 and 2/3
Ans:- 1/2 = 1 × 18/2 × 18 = 18/36 and 2/3 = 2 × 12/3 × 12 = 24/36
Five rational numbers are
19/36, 20/36, 21/36, 22/36, 23/36
(2) Write four more rational numbers in each of the following patterns:
(i) -3/5, -6/10, -9/15, -12/20
Ans:- -3/5 = -3 × 1/5 × 1, -6/10 = -3 × 2/5 × 2
-9/15 = -3 × 3/5 × 3, -12/20 = 3 × 4/5× 4
Thus, we observe a pattern in in these numbers. For more, rational numbers would be
-3 × 5/5 × 5 = -15/25, -3 × 6/5 × 6 = -18/30
-3 × 7/5 × 7 = -21/35 and -3 × 8/5 × 8 = -24/40
(ii) -1/4, -2/8, -3/12
Ans:- We have,
-1/4 =-1 × 1/4 × 1, -2/8 = -1 × 2/4 × 2, -3/12 = -1 × 3/4 × 3
Thus, we observe a pattern in these numbers.
For more rational numbers would be
-1 × 4/4 × 4 = -4/16, -1 × 5/4 × 5 = -5/20
-1 × 6/ 4 × 6 = -6/24
and -1 × 7/ 4 × 7 =-7/28 or – 1/4
(iii) -1/6, 2/-12, 3/-18, 4/-24
Ans:- We have,
2/-12 = -1 × (-2) 6 × (-2), 3/-18 = -1 × (-3)/6 × (-3), 4/-24 = -1 × (-4)/6 × (-4)
Thus, we observe a pattern in these numbers.
For more rational numbers would be
-1 × (-5)/6 × (-5) = 5/-30, -1 × (-6)/6 × (-6) = 6/-36, -1 × (7)/6 × (-7) = 7/-42 and -1 × (-8)/6 × (-8) = 8/-48
(iv) -2/3, 2/-3, 4/-6, 6/-9
Ans:- We have,
2/-3 = -2 × (-1)/3 × (-1), 4/-6 = -2 × (-2)/3 × (-2), 6/-9 = -2 × (-3)/3 × (-3)
Thus, we observe a pattern in these numbers,
For more rational numbers would be
-2 × (-4)/3 × (-4) = 8/-12, -2 × (-5)/3 × (-5) = 10/-15, -2 × (-6)/3 × (-6) = 12/-18 and -2 × (-7)/3 × (-7) = 14/-21
(3) Give four rational numbers equivalent to:
(i) -2/7
Ans:- Four rational numbers are
-2 × 2/7 × 2, -2 × 3/7 × 3, -2 × 4/7 × 4, -2 × 5/7 × 5,
-4/14, -6/21, -8/28, -10/35
(ii) 5/-3
Ans:- Four rational numbers are
5 × 2/-3 × 2, 5 × 3/-3 × 3, 5 × 4/-3 × 4, 5 × 5/-3 × 5
10/-6, 15/-9, 20/-12, 25/-15
(iii) 4/9
Ans:- Four rational number are
4 × 2/9 × 2, 4 × 3/9 × 3, 4 × 4/9 × 4, 4 × 5/9 × 5
8/18, 12/17, 16/36, 20/45
(4) Draw the number line and represent the following rational numbers on it:
(i) 3/4
(ii) -5/8
(iii) -7/4
(iv) 7/8
Ans:-
(5) 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.
Ans:- Distance between U and T = 1 unit
It is divided into 3 equal parts.
TR = Rs = SU = 1/3
R = -1 – 1/3 = – 3/3 – 1/3 = -4/3
S = – 1 – 2/3 = – 3/3 – 2/3 = – 5/3
Similarly
AB = 1 Unit
It is divided into 3 equal parts.
P = 2 + 1/3 = 6/3 + 1/3 = 7/3
Q = 2 + 2/3 = 6/3 + 2/3= 8/3
(6) Which of the following pairs represent the same rational number?
(i) -7/21 and 3/9
Ans:- -7/21 is a negative rational number and 3/9 is a positive rational number. So, the given pair does not represent the same rational number.
(ii) -16/20 and 20/-25
Ans:- -16/20 = -16 × (-1)/20 × (-1) = 16/20
= 16 ÷ 4/-20 ÷ 4 = 4/-5
20/-25 = 20 ÷ 5/-25 ÷ 5 = 4/-5
So, the given pair represents the same rational number.
(iii) -2/-3 and 2/3
Ans:- -2/-3 = -2 × (-1)/-3 × (-1) = 2/3
So, the given pair represents the same rational number.
(iv) -3/5 and -12/20
Ans:- -3/5 = -3/ × 4/5 × 4 = -12/20
So, the given pair represents the same rational number.
(v) 8/-5 and -24/15
Ans:- 8/-5 = 8 × (-3)/-5 × (-3) = -24/15
So, the given pair represents the same rational number.
(vi) 1/3 and -1/9
Ans:- 1/3 is a Positive rational number and -1/9 is a negative rational number. So, the given pair does not present the same rational number.
(vii) -5/-9 and 5/-9
Ans:- -5/-9 = -5 × (-1)/-9 × (-1) = 5/9
Thus -5/-9 is a Positive rational number and 5/-9 is a negative rational number. So, the given pair does not represent the same rational number.
(7) Rewrite the following rational numbers in the simplest form:
(i) -8/6
Ans:- -4 × 2/3 × 2 = -4/3
(ii) 25/45
Ans:- 5 × 5/9 × 5 = 5/9
(iii) -44/72
Ans:- -11 × 4/18 × 4 = -11/18
(iv) -8/10
Ans:- -4 × 2/5 × 2 = -4/5
(8) Fill in the boxes with the correct symbol out of >,<, and =.
(i) -5/7 () 2/3
Ans:- -5/7 < 2/3
∵ -5/7 is a negative rational number
Whereas 2/3 is a Positive rational number.
(ii) -4/5 () -5/7
Ans:- -4/5 < -5/7
-4/5 = -4 × 7/5 × 7 = -28/35
-5/7 = -5 × 5/7 × 5 = -25/35
28/35 > 25/35
⇒ -28/35 < -25/35
⇒ -4/5 < -5/7
(iii) -7/8 () 14/-16
Ans:- -7/8 = 14/-16
-7/8 = -7 × (-2)/ 8 × (-2) = 14/-16
(iv) -8/5 ()-7/4
Ans:- -8/5 > -7/4
-8/5 = -8 × 4/5 × 4 = -32/20
-7/4 = -7 × 5/4 × 5 = -35/20
35/20 > 32/20 ⇒ -35/20 < -32/20
⇒ -7/4 <-8/5
⇒ -8/5 > -7/4
(v) 1/-3 () -1/4
Ans:- 1/-3 < -1/4
1/-3 = 1 × (-4)/-3 × (-4) = -4/12
-1/4 = -1 × 3/4 × 3 = -3/12
4/12 > 3/12
⇒ -4/12 < -3/12
⇒ -1/3 < -1/4
(vi) 5/-11 () -5/11
Ans:- 5/11 = -5/11
5/-11 = 5 × (-1)/-11 × (-1) =-5/11
(vii) 0 () -7/6
Ans:- 0 > -7/6
∵ 0 is greater than every negative number.
(9) Which is greater in each of the following:
(i) 2/3, 5/2
Ans:- By Converting these into like fractions,
2/3 = 2 × 2/3 × 2 = 4/6
5/2 = 5 × 3/2 × 3 = 15/6
As 15 > 4, therefore, 5/2 is greater.
(ii) -5/6, -4/3
Ans:- -4/3 = -4 × 2/3 × 2 = -8/6
As -5 > -8, therefore, -5/6 is greater.
(iii) -3/4, 2/-3
Ans:- By Converting into like fractions,
-3/4 = -3 × 3/ 4 × 3 = -9/12
-2/3 = -2 × 4/3 × 4 = -8/12
As -8 > – 9, therefore, -2/3 is greater
(iv) -1/4,1/4
Ans:- 1/4 > -1/4
(v) -3 (2/7), -3 (4/5)
Ans:- -23/7, -19/5
By Converting these into like fractions
-23/7 = -23 × 5/7 × 5 = -115/35
-19/5 = -19 × 7/5 × 7 = -133/35
As -115 > -133, therefore -3 (2/7) is greater.
(10) Write the following rational numbers in ascending order:
(i) -3/5, -2/5, -1/5
Ans:- The given rational numbers in ascending order are
-3/5, -2/5, -1/5
∴ -3 < (-2) < (-1)
⇒ -3/5 <-2/5 <-1/5
(ii) -1/3, -2/9, -4/3
Ans:- -1/3 = -1 × 3/3 × 3 = -3/9
-2/9 = -2/9
LCM (3,9,3) = 9
-4/3 = -4 × 3/3 × 3 = -12/9
∵ -12 < (-3) <2
∴ -12/9 < -3/9 < -2/9
∴ -4/3 <-1/3 <-2/9
(iii) -3/7, -3/2, -3/4
Ans:- -3/7 = -3 ×4/7 × 4 = -12/28
1 LCM (7, 2, 4) = 28
-3/2 = -3 × 14/2 × 14 = -42/28
-3/4 = -3 × 7/4 × 7 = -21/28
1 LCM (7, 2, 4) = 28
∵ -42 <(-21) < (-12)
∴ -42/28 < (-21/28) < (-12/28)
∴ -3/2 < (-3/4) < (-3/7)
Exercise – 8.2
(1) Find the sum:
(i) 5/4 + (-11/4)
Ans:- 5/4 + (-11)/4 = 5 + (-11)/4 = -6/4
= -6 ÷ 2/4 ÷ 2 = -3/2
1 HCM (6, 4) = 2
(ii) 5/3 + 3/5
Ans:- LCM of 3 and 5 is 15
So, 5/3 = 5 × 5/3 × 5 = 25/15 and 3/5 = 3 × 3/5 × 3 = 9/15
Thus, 5/3 + 3/5 = 25/15 + 9/15
= 25 + 9/15 = 34/15
(iii) -9/10 + 22/15
Ans:- LCM of 10 and 15 is 30.
So, -9/10 = -9 × 3/10 × 3 = -27/30
and 22/15 = 22 × 2/15 × 2 = 44/30
Thus, -9/10 + 22/15 = -27/30 + 44/30
= -27 + 44/30 = 17/30
(iv) -3/-11 + 5/9
Ans:- LCM of 11 and 9 is 99
So, -3/-11 = -3 × (-9)/-11 × (-9) = 27/99
5/9 = 5 × 11/9 × 11 = 55/99
Thus, -3/-11 + 5/9 = 27/99 + 55/99
= 27 + 55/99 = 82/99
(v) -8/19 + (-2)/57
Ans:- LCM of 19 and 57 is 57.
-8/19 = -8 × 3/19 × 3 = -24/57;
-2/57 = -2/57
Thus, -8/19 + (-2)/57 = -24/57 + (-2)/57
= (-24) + (-2)/57
= -26/57
(vi) -2/3 + 0
Ans:- -2/3 + 0/3 = -2 + 0/3 = -2/3
(vii) -2 (1/3) + 4 (3/5)
Ans:- LCM of and 5 is 15
-7/3 = -7 × 5/3 × 5 = -35/15
23/5 = 23 × 3/5 × 3 = 69/15
Thus, -7/3 + 23/5 = -35/15 + 69/15
= -35 + 69/15 = 34/15
(2) Find:
(i) 7/24 – 17/36
Ans:- L.C.M of 24 and 36 is 72
7/24 – 17/36 = 7 × 3/24 × 3 – 17 × 2/36 × 2 = 21/72 – 34/72 = 21 – 34/72 = -13/72
(ii) 5/63 – (-6/21)
Ans:- L.C.M of 63 and 7 is 63.
5/63 + 2/7 = 5/63 + 2 × 9/7 × 9 = 5/63 + 18/63 = 5 + 18/63 = 23/63
(iii) -6/13 – (-7/15)
Ans:- L.C.M of 13 and 15 is 195
-6/13 + 7/15 = -6 × 15/13 × 15 + 7 × 13/15 × 13 = -90/195 + 91/195 = -90 +91/195 = 1/195
(iv) -3/8 – 7/11
Ans:- L.C.M of 8 and 11 is 88.
-3/8 – 7/11 = – 3 × 11/8 × 11 – 7 × 8/11 × 8 = – 33/88 – 56/88 = -33 – 56/88 = -89/88
(v) -2 (1/2) – 6
Ans:- L.C.M of 9 and 1 is 9.
- 19/6 – 6/1 = – 19/9 – 6 × 9/1 × 9 = – 19/9 – 54/9 = -19 – 54/9 = -73/9
(3) Find the product:
(i) 9/2 × (-7/4)
Ans:- 9 × -7/2 × 4 = -63/8
(ii) 3/10 × (-9)
Ans:- 3 × (-9)/10 = -27/10
(iii) -6/5 × 9/11
Ans:- -6 × 9/5 × 11 = -54/55
(iv) 3/7 × (-2/5)
Ans:- 3 × (-2)/ 7 × 5 = -6/35
(v) 3/11 × 2/5
Ans:- 3 × 2/11 × 5 = 6/55
(vi) 3/-5 × -5/3
Ans:- 3 × (-5)/-5 × 3 = -15/-15 = 1.
(4) Find the value of:
(i) (-4) ÷ 2/3
Ans:- -4 × 3/2 = -12/2 = -6
(ii) -3/5 ÷ 2
Ans:- -3/5 × 1/2 = -3 × 1/5 × -3/10
(iii) -4/5 ÷ (-3)
Ans:- -4/5 × 1/-3 = (-4) × 1/5 × (-3) = -4/-15 = 4/15
(iv) -1/8 ÷ 3/4
Ans:- 1/8 × 4/3 = -1 × 4/8 × 3 = -4/24 = -1/6
(v) -2/13 ÷ 1/7
Ans:- -2/13 × 7 = -14/13
(vi) -7/12 ÷ (-2/13)
Ans:- -7/12 × 13/-2 = (-7) × 13/12 × (-2) = -91/-24 = 91/24
(vii) 3/13 ÷ (-4/65)
Ans:- 3/13 × 65-/4 = 3 × 65/13 × (-4) = 195/-52 = – 15/4
Chapter 7 Comparing Quantities