Question -
Answer -
(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) 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.
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) 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.
Solutions:
(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) 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?
Solutions:
(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) 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?
Solutions:
(iv) Let x be the number of correct numbers and y be the number of incorrect answers.
According to the question given,
3x – y = 40 …………………………………(i)
4x – 2y = 50
And 2x-y = 25………………………………(ii)
Subtracting equation (ii) from (i), we get
x = 15………………………………………(iii)
Using this in equation (i), we get,
3(15) – 40 = y
y = 5
Therefore, the number of correct answers = 15
And the number of incorrect answers = 5
The total number of questions = 20
(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:
(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