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 1/3 when 1 is subtracted from the numerator and it becomes 1/4 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 the fixed charge of the food and the charge for food per day be x and y respectively.
According to the question,
x + 20y = 1000     ...(1) 
x + 26y = 1180   ...(2) 

Substituting this value of y in equation (1) from equation (2), we obtain 
6y = 180 
y = 30 

Substituting this value of y in equation (1), we obtain:
x+ 20 x 30 = 1000 
x = 1000 - 600 
x = 400 

Thus, the fixed charge of the food and the charge per day are Rs 400 and Rs 30 respectively.

(ii)  Let the fraction be  According to the question, 

, 

Subtracting equation (1) from equation (2), we obtain: 
x = 5
Putting the value of x in equation (1), 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.

Subtracting equation (2) from equation (1), we obtain:
x = 15
Substituting the value of x in equation (2), we obtain:
30 - y = 25 
y = 5 

Thus, the number of right answers and the number of wrong answers is 15 and 5 respectively.
Therefore, the total number of questions is 20. 

(iv)  Let the speed of first car and second car be u km/h and v km/h respectively.

Speed of both cars while they are travelling in same direction = (u - v) km/h
Speed of both cars while they are travelling in opposite directions i.e., when they are travelling towards each other = (u + v) km/h

Distance travelled = Speed x Time

adding the two equations, we get, 

Substituting the value of u in equation (2), we obtain:
v = 40
Hence, speed of the first car is 60 km/h and speed of the second car is 40 km/h.

(v)  Let length and breadth of rectangle be x unit and y unit respectively. 

Area = xy

By cross multiplication method, 

Thus, the length and breadth of the rectangle are 17 units and 9 units respectively.