MATHEMATICS IN EVERYDAY LIFE–8
[Pages:16]MATHEMATICS IN EVERYDAY LIFE?8
Chapter 8 : Linear Equations in One Variable
ANSWER KEYS
EXERCISE 8.1
1. x ? 3 = 8
x =8 + 3
(By transposing ? 3)
x = 11
2. x + 7 = 15
x = 15 ? 7
(By transposing 7)
x=8
3. 5x = 20 Dividing both sides by 5, we get
5x 5
=
20 5
x=4
4. 2x + 6 = 10
2x = 10 ? 6
2x = 4
(By transposing 6)
2x 4
2 =2
(On dividing both sides by 2)
x=2
5. 7x ? 5 = 16
7x = 16 + 5
7x = 21
(By transposing ? 5)
7x 21 7 = 7 (On dividing both sides by 7)
x=3
3x 6. 2 = 18
Multiplying both sides by 2, we get
3x 2 ? 2 = 18 ? 2
3x = 36
3x 36
3 = 3 (On dividing both sides by 3)
x = 12
Mathematics In Everyday Life-8
x 7. 5 = 7
x ?5=7?5 5 (On multiplying both sides by 5)
x = 35
8.
4 7
+ x =
20 7
x =
20 7
?
4 7
(By transposing
4 7)
20 4
x= 7
x = 16
7
9. 7 = 2x ? 5
or 2x ? 5 = 7
2x = 7 + 5
2x = 12
(By transposing ? 5)
2x = 12 (On dividing both sides by 2)
22
x=6
10. 1.6x + 2 = 5.2
1.6x = 5.2 ? 2
1.6x = 3.2
(By transposing 2)
1.6x 3.2 1.6 = 1.6 (On dividing both sides by 1.6)
x=2
11. 13 + 2a = 39
2a = 39 ? 13
2a = 26
(By transposing 13)
2a = 26
(On dividing both sides by 2)
22
a = 13
1
12. 16a ? 4 = 28
16a = 28 + 4
16a = 32
(By transposing ? 4)
16a 32 16 = 16 (On dividing both sides by 16)
a=2
EXERCISE 8.2
1. 2x ? 5 = 3x + 5
Transposing variables on L.H.S. and constant terms on R.H.S., we get
2x ? 3x = 5 + 5
? x = 10
x = 10 (On multiplying both sides by ? 1)
2. 9x + 15 = 20 ? x
Transposing variables on L.H.S. and constant terms on R.H.S., we get
9x + x = 20 ? 15
10x = 5
10x 10
=
5 10
(On dividing both sides by 10)
x= 1
2
3
4
3. 2m ? 5 = m + 5
Transposing variables on L.H.S. and constant terms on R.H.S., we get
2m ? m =
4 5
+
3 5
m=7
5
4. 3m + 4 = 8m + 24
Transposing variables on L.H.S. and constant terms on R.H.S., we get
3m ? 8m = 24 ? 4
? 5m = 20
5m 20 5 =5
(On dividing both sides by 5)
? m =4
m = 4 (On multiplying both sides by ? 1)
m 5. 2 + 6 = m ? 8
Transposing variables on L.H.S. and constant terms on R.H.S., we get
2
m 2 ?m= ?8?6
m = ? 14
2
(?
2)
m 2
=
(? 14) ? (?
2)
(On multiplying both sides by ? 2)
m = 28
6. 7 ? x = 4 ? x 3
Transposing variables on L.H.S. and constant terms on R.H.S., we get
? x +x=4?7 3
x 3x = ? 3 3
2x 3
=? 3
2x 3 ? 3 = ? 3 ? 3 (On multiplying both sides by 3)
2x = ? 9
2x = 9 (On dividing both sides by 2)
22
x= 9
2
72 7. 3t + 5 = 5 + 5t
or 2 + 5t = 3t + 7
5
5
Transposing variables on L.H.S. and constant terms on R.H.S., we get
72 5t ? 3t = 5 ? 5
2t
=
5 5
2t = 1
2t 2
=
1 2
(On dividing both sides by 2)
t= 1
2
8. 3.5x ? 9 = 2.4x + 13
Transposing variables on L.H.S. and constant terms on R.H.S., we get
Answer Keys
3.5x ? 2.4x = 13 + 9
1.1x = 22
1.1x 22
1.1 = 1.1 (On dividing both sides by 1.1)
x = 20
4
5
9. 11x ? 3 = 2x + 3
Transposing variables on L.H.S. and constant terms on R.H.S., we get
54 11x ? 2x = 3 + 3
9
9x = 3
9x = 3
9x 9
=
3 9
(On dividing both sides by 9)
x= 1
3
10.
12x ? 2 7
13 = 14 ? 5x
Transposing variables on L.H.S. and constant terms on R.H.S., we get
13 2 12x + 5x = 14 + 7
17x = 17
14
17x 17
=
17 14 17
(On dividing both sides by 17)
x= 1
14
EXERCISE 8.3
4x 1 1. 5x 2 = 1
4x ? 1 = 5x + 2
? 1 ? 2 = 5x ? 4x
?3=x
x=3
(By cross multiplication) (By transposing)
Verification: L.H.S. = 4( 3) 1 = 12 1 5( 3) 2 15 2
= 13 = 1 = R.H.S. 13
Mathematics In Everyday Life-8
2.
x 5 3 2x
=
5 3
3(3x + 5) = 5(3 ? 2x)
9x + 15 = 15 ? 10x
9x + 10x = 15 ? 15
19x = 0
0 x = 19 = 0
x=0
(By cross multiplication) (By transposing)
Verification: L.H.S. =
3(0) 5 3 2(0)
=
05 30
= 5 = R.H.S. 3
2x 3 9 3. 3x 2 = 7
7(2x + 3) = 9(3x + 2)
14x + 21 = 27x + 18
21 ? 18 = 27x ? 14x
13x = 3
3 = 13x
(By cross multiplication) (By transposing)
13x 13
=
3 13
(On dividing both sides by 13)
x= 3
13
Verification: L.H.S. =
2
3 13
3
=
6 3 13
3
3 13
2
9 13
2
6 39 45
=
13 9 26
=
13 35
13
13
= 45 = 9 = R.H.S. 35 7
4. (3x 4) 2x = ? 9
(2 5x) 7x
58
3x 4 2x 2 5x 7x
=?
9 58
x4 =? 9
2 12x
58
58(x + 4) = ?9(2 ? 12x)
(By cross multiplication)
58x + 232 = ? 18 + 108x
3
58x ? 108x = ? 18 ? 232
(By transposing)
? 50x = ? 250
50x = 250
50 50
(On dividing both sides by ?50)
x=5
Verification: L.H.S. = (3 5 4) 2 ? 5 (2 5 ? 5) 7 ? 5
= (15 4) 10 = 19 10 (2 25) 35 23 35
= 9 = ? 9 = R.H.S.
58
58
1
1
5. 2 (m ? 1) + 5 = 3 (2m + 1)
Multiplying both sides by 6, the L.C.M. of 2 and 3,
we get
6 ? 1 (m ? 1) + 6 ? 5 = 6 ? 1 (2m + 1)
2
3
3(m ? 1) + 30 = 2(2m + 1)
3m ? 3 + 30 = 4m + 2
? 3 + 30 ? 2 = 4m ? 3m
(By transposing)
25 = m
m = 25
Verification: L.H.S. = 1 (25 ? 1) + 5 = 1 ? 24 + 5
2
2
= 12 + 5 = 17
R.H.S. = 1 (2 ? 25 + 1) = 1 (50 + 1)
3
3
= 1 ? 51 = 17 3
L.H.S. = R.H.S.
6. 5x = ? 5 3x 4
5x = ? 5(3x + 4)
5x = ? 15x ? 20
5x + 15x = ? 20
20x = ? 20
(By cross multiplication) (By transposing)
20x = 20 (On dividing both sides by 20)
20 20
x=1
Verification: L.H.S. = 5 ? ( 1) = 5 3 ? ( 1) 4 3 4
= 5 = ? 5 = R.H.S. 1
4
2y 7. 3y 1 = ? 3
2y = ? 3(3y + 1)
2y = ? 9y ? 3
2y + 9y = ? 3
11y = ? 3
(By cross multiplication) (By transposing)
y = 3 (On dividing both sides by 11)
11
Verification: L.H.S. =
2
3 11
=
6 11
3
3 11
1
9 1 11
6
6
=
11 9 11
=
11 2
11
11
= 6 = ? 3 = R.H.S. 2
8.
1.2x 3 3.2x 6
=
9 8
8(1.2x + 3) = 9(3.2x + 6)
(By cross multiplication)
9.6x + 24 = 28.8x + 54
24 ? 54 = 28.8x ? 9.6x (By transposing)
? 30 = 19.2x
30
x = 19.2
(On dividing both sides by 19.2)
x = 300
192
x = 25
16
Verification: L.H.S. =
1.2
?
25 16
3
3.2
?
25 16
6
=
30 3 16 80 6 16
30 48 18
=
16 80 96
=
16 16
16
16
= 18 = 9 = R.H.S. 16 8
Answer Keys
9.
4y 8 5y 8
=
5 6
6(4y + 8) = 5(5y + 8)
24y + 48 = 25y + 40
48 ? 40 = 25y ? 24y
8 =y
(By cross multiplication) (By transposing)
y=8
Verification: L.H.S. = 4 ? 8 8 = 32 8 5 ? 8 8 40 8
xb xb 10. a b = a b
= 40 = 5 = R.H.S. 48 6
(x + b) (a + b) = (a ? b) (x ? b)
(By cross multiplication)
ax + bx + ab + b2 = ax ? ab ? bx + b2
ax + bx ? ax + bx = ? ab + b2 ? ab ? b2
(By transposing)
2bx = ? 2ab
2ab
x = 2b
(On dividing both sides by 2b)
x=a
Verification: L.H.S. = a b = (a b) = ? 1
ab
a b
R.H.S. = a b = (a b) = ? 1
ab
a b
11. 7x 4 = ? 4
x 2
3
3(7x + 4) = ? 4(x + 2)
21x + 12 = ? 4x ? 8
21x + 4x = ? 8 ? 12
25x = ? 20
(By cross multiplication) (By transposing)
x = ? 20 (On dividing both sides by 25)
25
x= 4
5
Verification: L.H.S. =
7
4 5
4
42
=
28 4 5 4 2
5
5
28 20 8
=
5 4 10
=
5 6
5
5
= 8 = 4 = R.H.S.
6
3
Mathematics In Everyday Life-8
12.
17(2 x) 5(x 12) 1 7x 7x 1
= 8
34 17x 5x 60 1 7x 7x 1
=8
34 17x 5x 60
1 7x 1 7x
=8
34 17x (5x 60) = 8 (1 7x)
34 17x 5x 60
(1 7x)
=8
22x 26 = 8
1 7x
? 22x ? 26 = 8(1 ? 7x)
(By cross multiplication)
? 22x ? 26 = 8 ? 56x
? 22x + 56x = 8 + 26
(By transposing)
34x = 34
x=1
(On dividing both sides by 34)
Verification: L.H.S. =
17(2 1) 5(1 12) 17?1 7?11
= 17 ? 1 5 13 = 17 65 17 71 6 6
4x 3 0.05 13. 10x 9 = 0.25
= 17 65 = 48 = 8 = R.H.S.
6
6
4x 3 5 1 10x 9 = 25 = 5
5(4x ? 3) = 1(10x ? 9) 20x ? 15 = 10x ? 9 20x ? 10x = ? 9 + 15
(By cross multiplication) (By transposing)
10x = 6
x= 6 =3
10 5
(On dividing both sides by 10)
x= 3
5
5
Verification: L.H.S. =
4
3 5
3
10
3 5
9
=
12 3 5 30 9 5
12 3 12 15 3
=
5 69
=
5 3
= 5 3
= 3 = 1 5( 3) 5
R.H.S. = 0.05 = 5 = 1 0.25 25 5
L.H.S. = R.H.S.
3
1
14. 2x (3 4x) = 3
3 2x 3 4x
=
1 3
3
1
6x 3 = 3
3 ? 3 = 6x ? 3 (By cross multiplication)
9 = 6x ? 3
9 + 3 = 6x
(By transposing)
12 = 6x
6x = 12
x=2
(On dividing both sides by 6)
Verification: L.H.S. =
3 2 ? 2 (3 4 ? 2)
=
3 4 (3 8)
=
3 4 ( 5)
=
3 45
= 3 = 1 = R.H.S. 9 3
5(x 6) 15(2 x)
15.
3x 1
= 10
5x 30 30 15x
3x 1
= 10
20x = 10
3x 1
20x = 10(3x ? 1)
(By cross multiplication)
20x = 30x ? 10
20x ? 30x = ? 10
(By transposing)
? 10x = ? 10
x=1
(On dividing both sides by ? 10)
Verification: L.H.S. = 5(1 6) 15(2 1) 3?11
= 5 ? 7 15 ? 1 = 35 15
31
2
= 20 = 10 = R.H.S. 2
x5 x4 16. x 3 = x 6
(x ? 5)(x ? 6) = (x ? 3)(x ? 4) (By cross multiplication)
x2 ? 11x + 30 = x2 ? 7x + 12
(Cancelling x2 from both sides)
? 11x + 30 = 7x + 12
? 11x + 7x = 12 ? 30
(By transposing)
? 4x = ? 18
18 9
x = 4 = 2
(On dividing both sides by ? 4)
x= 9
2
Verification: L.H.S. =
9 5 2 9 3 2
=
9 10
2 96
2
=
1 2 3
2
= ?
1 3
R.H.S. =
9 4 2 9 6
=
98
2 9 12
=
1
2 3
= 1 3
2
2
2
L.H.S. = R.H.S.
2
7
17. x 9 = 4x 3
2(4x ? 3) = ? 7(x + 9)
(By cross multiplication)
8x ? 6 = ? 7x ? 63
8x + 7x = ? 63 + 6
(By transposing)
15x = ? 57
6
Answer Keys
x =
57 15
=
19 5
(On dividing both sides by 15)
x = 19
5
Verification: L.H.S. =
2 19 9
5
=
2 19 45
5
=
2 26 5
= 10 = 5 26 13
R.H.S. =
7
4
19 5
3
=
7 76 3
5
=
7 76 15
=
7 91
5
5
= 7?5 = 5 91 13
L.H.S. = R.H.S.
2 3y 4 18. 1 5y = 3
3(2 + 3y) = 4(1 + 5y) 6 + 9y = 4 + 20y 9y ? 20y = 4 ? 6 ? 11y = ? 2
(By cross multiplication) (By transposing)
y = 2 (On dividing both sides by ? 11)
11
Verification: L.H.S. =
2
3
2 11
1
5
2 11
=
2
6 11
1 10 11
=
22 6
11 11 10
11
28 = 11 = 28 = 4 = R.H.S.
21 21 3
11
3x 2 3x 1 19. 5x 1 = 5x 2
(3x ? 2)(5x + 2) = (5x ? 1) (3x + 1)
15x2 + 6x ? 10x ? 4 = 15x2 + 5x ? 3x ? 1
15x2 ? 4x ? 4 = 15x2 + 2x ? 1
(Cancelling 15x2 from both sides)
? 4x ? 4 = 2x ? 1
? 4x ? 2x = ? 1 + 4 (By transposing)
Mathematics In Everyday Life-8
? 6x = 3
3 1 x= 6 = 2
(On dividing both sides by ? 6)
x= 1
2
Verification: L.H.S. =
3
1 2
2
5
1 2
1
=
32 2 5 1 2
34 7
=
2 52
=
2 7
= 1
2
2
R.H.S. =
3
1 2
1
5
1 2
2
=
3 1 2 52 2
32 1
=
2 54
=
2 1
= 1
2
2
L.H.S. = R.H.S.
EXERCISE 8.4
1. Let the number to be added be x. Therefore,
3
?
7 4
+
x
=
3 7
21
3
4 +x= 7
x
=
3 7
+ 21 4
(By transposing 21 ) 4
12 147
x = 28
159
x = 28
159 Hence, the required number is 28 .
2. The lengths of sides of a triangle are 2x, 2x + 3 and 2x + 5.
Perimeter of triangle = Sum of the lengths of sides
56 = 2x + 2x + 3 + 2x + 5
6x + 8 = 56
7
6x = 56 ? 8
(By transposing)
6x = 48
x =
48 6
= 8
x =8
Hence, lengths of sides of triangle are 2 ? 8 = 16, 2 ? 8 + 3 = 19 and 2 ? 8 + 5 = 21.
3. Let the required number be x. Then according to the given condition,
8x ? 3x = 35
5x = 35
35
x= 5 =7
x =7
Hence, the required number is 7.
4. Let the two numbers be 4x and 7x.
7x ? 4x = 9
3x = 9
x =3
Hence, the required numbers are 4 ? 3 = 12, 7 ? 3 = 21.
5. Let the required number be x.
Therefore,
according to condition,
xxx 342
= 169
4x 3x 6x
12
= 169
( L.C.M. of 3, 4 and 2 is 12)
13x
12 = 169
13x = 169 ? 12
169 12
x = 13
x = 13 ? 12
x = 156
The required number is 156.
6. Let the number of boys in the class be x.
3x Therefore, number of girls in the class = 5
Total number of students in the class = x +
3x 5
3x
x + 5 = 96
5x 3x
5 = 96
8x
5 = 96
8
8x = 96 ? 5
96 5
x= 8
x = 12 ? 5 = 60
Hence, number of boys = 60
Number of girls = 3 ? 60 = 36. 5
7. Let the three consecutive integers be x, (x + 1) and (x + 2). Therefore, according to condition,
x + (x + 1) + (x + 2) = 48
3x + 3 = 48
3x = 48 ? 3
(By transposing)
3x = 45
x
=
45 3
=
15
x + 1 = 15 + 1 = 16
x + 2 = 15 + 2 = 17
Hence, the required three consecutive integers are 15, 16 and 17.
8. The ratio of ages of A and B = 5 : 7
Let the ages of A and B are 5x and 7x respectively.
After 4 years, age of A = (5x + 4) years
and age of B = (7x + 4) years
According to the condition.
(5x + 4) + (7x + 4) = 56
12x + 8 = 56
12x = 56 ? 8
(By transposing)
12x = 48
48
x = 12 = 4
Hence, present age of A = 5 ? 4 = 20 years
present age of B = 7 ? 4 = 28 years
9. Let the required number be x.
Then, according to the given condition,
7x + 7 = 28
7x = 28 ? 7
(By transposing)
7x = 21
21
x= 7 =3
Hence, the required number is 3.
10. Let the present age of Ashwin be x years.
After 20 years, Ashwin's age = (x + 20) years.
Now, as per condition,
(x + 20) = 3x
3x ? x = 20
(By transposing)
2x = 20
Answer Keys
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