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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.

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.

Grade:12th pass

1 Answers

Pawan Prajapati
askIITians Faculty 60787 Points
2 years ago
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

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