Form the pair of linear equations in the following problems and find their solutions (if they exist) by any algebraic method:
(i) A part of monthly hostel charges is fixed and the remaining depends on the number of days one has taken food in the mess. When a student A takes food for 20 days she has to pay Rs 1000 as hostel charges whereas a student B, who takes food for 26 days, pays Rs 1180 as hostel charges. Find the fixed charges and the cost of food per day.
(ii) A fraction becomes 
when 1 is subtracted from the numerator and it becomes 
when 8 is added to its denominator. Find the fraction.
(iii) Yash scored 40 marks in a test, getting 3 marks for each right answer and losing 1 mark for each wrong answer. Had 4 marks been awarded for each correct answer and 2 marks been deducted for each incorrect answer, then Yash would have scored 50 marks. How many questions were there in the test?
(iv) Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds, they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of the two cars?
(v) The area of a rectangle gets reduced by 9 square units, if its length is reduced by 5 units and breadth is increased by 3 units. If we increase the length by 3 units and the breadth by 2 units, the area increases by 67 square units. Find the dimensions of the rectangle.
(i) Let x be the fixed charge of the food and y be the charge for food per day. According to the given information,
X + 20y = 1000 .......... (1)
X + 26y = 1180 ........... (2)
Subtracting equation (1) from equation (2), we obtain
6y = 180
Y = 30
Substituting this value in equation (1), we obtain
X + 20 × 30 = 1000
x = 1000 - 600 = 400
x = 400
Hence, fixed charge = Rs 400 And charge per day = Rs 30
(ii) Let the fraction be 
.
According to the given information,
.......(i)
.......(ii)
Subtracting equation (i) from equation (ii), we obtain
x = 5 ............. (iii)
Putting this value in equation (i), we obtain
15 – y = 3
y = 12
Hence, the fraction is 
.
(iii) Let the number of right answers and wrong answers be x and y respectively.
According to the given information,
3x –y = 40
4x – 2y = 50 .......(i)
2y – y = 25 .........(ii)
Subtracting equation (ii) from equation (i),
we obtain x = 15 (iii)
Substituting this in equation (ii), we obtain
30 – y = 25
y = 5
Therefore,
number of right answers = 15
And number of wrong answers = 5
Total number of questions = 20
(iv) Let the speed of 1st car and 2nd car be u km/h and v km/h.
Respective speed of both cars while they are travelling in same direction = (u - v) km/h
Respective speed of both cars while they are travelling in opposite directions i.e.,
travelling towards each other = (u + v) km/h
According to the given information,
5(u - v) = 100
= u - v = 20......... (i)
1(u + v) = 100....... (ii)
Adding both the equations, we obtain
2u = 120
u = 60 km/h ...... (iii)
Substituting this value in equation (ii), we obtain v = 40 km/h
Hence, speed of one car = 60 km/h and speed of other car = 40 km/h
(v) Let length and breadth of rectangle be x unit and y unit respectively.
Area = xy
According to the question,
(x - 5) (y + 3) = xy - 9
= 3x – 5y – 6 = 0 ..........(i)
(x + 3) (y + 2) = xy + 67
= 2x + 3y – 61 = 0 ........(ii)
By cross-multiplication method, we obtain
x = 17, y =9