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.

Solutions: (i) Let x be the fixed charge and y be the charge of food per day. According to the question, x + 20y = 1000……………….. (i) x + 26y = 1180………………..(ii) Subtracting (i) from (ii) we get 6y = 180 y = Rs.30 Using this value in equation (ii) we get x = 1180 -26 x 30 x= Rs.400. Therefore, fixed charges is Rs.400 and charge per day is Rs.30. (ii) Let the fraction be x/y. So, as per the question given, (x -1)/y = 1/3 => 3x – y = 3…………………(1) x/(y + 8) = 1/4 => 4x –y =8 ………………..(2) Subtracting equation (1) from (2) , we get x = 5 ………………………………………….(3) Using this value in equation (2), we get, (4×5)– y = 8 y= 12 Therefore, the fraction is 5/12. (iii) Let the number of right answers is x and number of wrong answers be y According to the given question; 3x−y=40……..(1) 4x−2y=50 ⇒2x−y=25…….(2) Subtracting equation (2) from equation (1), we get; x = 15 ….….(3) Putting this in equation (2), we obtain; 30 – y = 25 Or y = 5 Therefore, number of right answers = 15 and number of wrong answers = 5 Hence, total number of questions = 20 (iv) Let x km/h be the speed of car from point A and y km/h be the speed of car from point B. If the car travels in the same direction, 5x – 5y = 100 x – y = 20 …………………………………(i) If the car travels in the opposite direction, x + y = 100………………………………(ii) Solving equation (i) and (ii), we get x = 60 km/h………………………………………(iii) Using this in equation (i), we get, 60 – y = 20 y = 40 km/h Therefore, the speed of car from point A = 60 km/h Speed of car from point B = 40 km/h. (v) Let, The length of rectangle = x unit And breadth of the rectangle = y unit Now, as per the question given, (x – 5) (y + 3) = xy -9 3x – 5y – 6 = 0……………………………(1) (x + 3) (y + 2) = xy + 67 2x + 3y – 61 = 0…………………………..(2) Using cross multiplication method, we get, x/(305 +18) = y/(-12+183) = 1/(9+10) x/323 = y/171 = 1/19 Therefore, x = 17 and y = 9. Hence, the length of rectangle = 17 units And breadth of the rectangle = 9 units