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 obtain6y = 180y = 30Substituting this value of y in equation (1), we obtain:x+ 20 x 30 = 1000x = 1000 - 600x = 400Thus, the fixed charge of the food and the charge per day are Rs 400 and Rs 30 respectively. (ii) Let the fraction beAccording to the question,, Subtracting equation (1) from equation (2), we obtain:x = 5Putting the value of x in equation (1), we obtain:15 - y = 3y = 12 Hence the fraction is (iii) Let the number of right answers and wrong answers be x and yrespectively. Subtracting equation (2) from equation (1), we obtain:x = 15Substituting the value of x in equation (2), we obtain:30 - y = 25y = 5Thus, 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/hSpeed of both cars while they are travelling in opposite directions i.e., when they are travelling towards each other = (u + v) km/hDistance travelled = Speed x Time adding the two equations, we get, Substituting the value of u in equation (2), we obtain:v = 40Hence, 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. read less