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TOPICS
Exercise - 2.5

Question-1 :-  Solve x/2 - 1/5 = x/3 + 1/4 .

Solution :-
```  x/2 - 1/5 = x/3 + 1/4
x/2 - x/3 = 1/4 + 1/5
(3x - 2x)/6 = (5 + 4)/20
x/6 = 9/20
x = (9 x 6)/20
x = 54/20
x = 27/10
Verification :
L.H.S = x/2 - 1/5 = 27/20 - 1/5 = (27 - 4)/20 = 23/20
R.H.S = x/3 + 1/4 = 27/30 + 1/4 = (108 + 30)/120 = 138/120 = 23/20
L.H.S = R.H.S
```

Question-2 :-  Solve n/2 - 3n/4 + 5n/6 = 21 .

Solution :-
```  n/2 - 3n/4 + 5n/6 = 21
(6n - 9n + 10n)/12 = 21
7n/12 = 21
7n = 21 x 12
n = (21 x 12)/7
n = 3 x 12
n = 36
Verification :
L.H.S =  n/2 - 3n/4 + 5n/6
= 36/2 - (3 x 36)/4 + (5 x 36)/6
= 18 - (3 x 9) + (5 x 6)
= 18 - 27 + 30
= 48 - 27
= 21 = R.H.S
```

Question-3 :- Solve x + 7 - 8x/3 = 17/6 - 5x/2 .

Solution :-
```   x + 7 - 8x/3 = 17/6 - 5x/2
x - 8x/3 + 5x/2 = 17/6 - 7
(6x - 16x + 15x)/6 = (17 - 42)/6
(21x - 16x)/6 = -25/6
5x/6 = -25/6
5x = (-25 x 6)/6
5x = -25
x = -25/5
x = -5
Verification :
L.H.S = x + 7 - 8x/3
= -5 + 7 - (-40)/3
= -5 + 7 + 40/3
= -5 + (21 + 40)/3
= -5 + 61/3
= (-15 + 61)/3
= 46/3
R.H.S = 17/6 - 5x/2
= 17/6 - (-25)/2
= 17/6 + 25/2
= (17 + 75)/6
= 92/6
= 46/3
L.H.S = R.H.S
```

Question-4 :-  Solve (x - 5)/3 = (x - 3)/5 .

Solution :-
```  (x - 5)/3 = (x - 3)/5
5 (x - 5) = 3 (x - 3)
5x - 25 = 3x - 9
5x - 3x = -9 + 25
2x = 16
x = 16/2
x = 8
Verification :
L.H.S = (x - 5)/3 = (8 - 5)/3 = 3/3 = 1
R.H.S = (x - 3)/5 = (8 - 3)/5 = 5/5 = 1
L.H.S = R.H.S
```

Question-5 :-  Solve (3t - 2)/4 - (2t + 3)/3 = 2/3 - t .

Solution :-
```  (3t - 2)/4 - (2t + 3)/3 = 2/3 - t
[3 x (3t - 2) - 4 x (2t + 3)]/12 = (2 - 3t)/3
[9t - 6 - 8t - 12]/12 = (2 - 3t)/3
(t - 18)/12 = (2 - 3t)/3
3 (t - 18) = 12 (2 - 3t)
3t - 54 = 24 - 36t
3t + 36t = 24 + 54
39t = 78
t = 78/39
t = 2
Verification :
L.H.S = (3t - 2)/4 - (2t + 3)/3
= (6 - 2)/4 - (4 + 3)/3
= 4/4 - 7/3
= 1 - 7/3
= (3 - 7)/3
= -4/3
R.H.S = 2/3 - t = 2/3 - 2 = (2 - 6)/3 = -4/3
L.H.S = R.H.S
```

Question-6 :- Solve m - (m - 1)/2 = 1 - (m - 2)/3 .

Solution :-
```   m - (m - 1)/2 = 1 - (m - 2)/3
m - (m - 1)/2 + (m - 2)/3 = 1
[6m - 3 x (m - 1) + 2 x (m - 2)]/6 = 1
[6m - 3m + 3 + 2m - 4] = 6
5m - 1 = 6
5m = 6 + 1
5m = 7
m = 7/5
Verification :
L.H.S = m - (m - 1)/2
= 7/5 - (7/5 - 1)/2
= 7/5 - (7 - 5)/10
= 7/5 - 2/10
= 7/5 - 1/5
= (7 - 1)/5
= 6/5
R.H.S = 1 - (m - 2)/3
= 1 - (7/5 - 2)/3
= 1 - (7 - 10)/15
= 1 - (-3)/15
= 1 + 1/5
= (5 + 1)/5
= 6/5
L.H.S = R.H.S
```

Question-7 :-  Solve and Simplify 3(t – 3) = 5(2t + 1) .

Solution :-
```  3(t – 3) = 5(2t + 1)
3t - 9 = 10t + 5
3t - 10t = 5 + 9
-7t = 14
t = 14/(-7)
t = -2
Verification :
L.H.S = 3(t – 3) = 3(-2 - 3) = 3 x (-5) = -15
R.H.S = 5(2t + 1) = 10t + 5 = 10 x (-2) + 5 = -20 + 5 = -15
L.H.S = R.H.S
```

Question-8 :-  Solve and simplify 15(y – 4) –2(y – 9) + 5(y + 6) = 0 .

Solution :-
```  15(y – 4) – 2(y – 9) + 5(y + 6) = 0
15y - 60 - 2y + 18 + 5y + 30 = 0
20y - 2y - 60 + 48 = 0
18y - 12 = 0
18y = 12
y = 12/18
y = 2/3
Verification :
L.H.S = 15(y – 4) – 2(y – 9) + 5(y + 6)
= 15y - 60 - 2y + 18 + 5y + 30
= 15 x 2/3 - 60 - 2 x 2/3 + 18 + 5 x 2/3 + 30
= 10 - 60 + 18 + 30 - 4/3 + 10/3
= 58 - 60 + (-4 + 10)/3
= -2 + 6/3
= -2 + 2
= 0 = R.H.S
```

Question-9 :- Solve and simplify 3(5z – 7) – 2(9z – 11) = 4(8z – 13) – 17 .

Solution :-
```   3(5z – 7) – 2(9z – 11) = 4(8z – 13) – 17
15z - 21 - 18z + 22 = 32z - 52 - 17
15z - 18z - 32z = -52 - 17 + 21 - 22
15z - 50z = -91 + 21
- 35z = -70
z = 70/35
z = 2
Verification :
L.H.S = 3(5z – 7) – 2(9z – 11)
= 15z - 21 - 18z + 22
= -3z + 1
= -3 x 2 + 1
= -6 + 1
= -5
R.H.S = 4(8z – 13) – 17
= 32z - 52 - 17
= 32 x 2 - 69
= 64 - 69
= -5
L.H.S = R.H.S
```

Question-10 :-  Solve and simplify 0.25(4f – 3) = 0.05(10f – 9) .

Solution :-
```  0.25(4f – 3) = 0.05(10f – 9)
1.00f - 0.75 = 0.50f - 0.45
1.00f - 0.50f = -0.45 + 0.75
0.50f = 0.30
f = 0.30/0.50
f = 30/50
f = 3/5
f = 0.6
Verification :
L.H.S = 0.25(4f – 3)
= 0.25(4 x 0.6 - 3)
= 0.25(2.4 - 3)
= 0.25 x (-0.6)
= -0.150
R.H.S = 0.05(10f – 9)
= 0.05(10 x 0.6 - 9)
= 0.05(6.0 - 9)
= 0.05 x (-3.0)
= -0.150
L.H.S = R.H.S
```
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