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.
Solution:
Let fixed monthly charge = Rs x
Let charge of food for one day = Rs y
According to given conditions, we have
(1), and
(2)
Subtracting equation (1) from equation (2), we get
Rs 30
Putting value of y in (1), we get
Rs 400
Therefore, fixed monthly charges = Rs 400
And, charges of food for one day = Rs 30
(ii) A fraction becomes
when 1 is subtracted from the numerator and it becomes
when 8 is added to its denominator. Find the fraction.
Solution:
Let numerator
Let denominator
According to given condition ,we have
(1)
(2)
(1)
(2)
(1)
(2)
Subtracting equation (1) from (2), we get
Putting value of x in (1), we get
Therefore, numerator = 5
And, denominator = 12
It means 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?
Solution:
Let number of correct answers
Let number of wrong answers
According to given conditions, we have
(1)
And,
(2)
From equation (1), we can say
Putting this in (2), we get
Putting value of x in (1), we get
Therefore, number of correct answers
Number of wrong answers
Total questions
(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?
Solution:
Let speed of car which starts from part A = x km/hr
Let speed of car which starts from part B = y km/hr
According to given conditions, we have
(Assuming x>y)
(1)
And,
(2)
Adding (1) and (2), we get
km/hr
Putting value of x in (1), we get
km/hr
Therefore, speed of car starting from point A = 60 km/hr
And, Speed of car starting from point B = 40 km/hr
(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.
Solution:
Let length of rectangle = x units
Let breadth of rectangle = y units
Area
According to given conditions, we have
(1)
And,
(2)
From equation (1), we can say that
Putting this in (2), we get
units
Putting value of y in (2), we get
units
Therefore, length= 17 units
And, Breadth = 9 units
sonus says
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abin reji says
hi