A part of the total expenses of a hostel varies directly as the number of students and the remaining expense remains constant. When there are 40 students in a hostel, the total expenditure is Rs 4500 and for 48 students in a hostel total expenses are Rs 5300. Then find the total expenditure when there are 55 students.

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Solution

$Let the total expenditure be E and the number of students in the hostel be n . Also, let some part of the expenditure be x . According to the question, x α n \Rightarrow x = k_{1} n . . . (k_{1} is constant .) . . . (1) & E - x = k_{2} . . . (k_{2} i s constant .) From (1), w e g e t : E - k_{1} n = k_{2} \Rightarrow E = k_{2} + k_{1} n N o w, 4500 = k_{2} + k_{1} \times 40 (∵ E = Rs 4500 when n = 40) \Rightarrow k_{2} = 4500 - 40 k_{1} . . . (2) Also, 5300 = k_{2} + k_{1} \times 48 (∵ E = Rs 5300 when n = 48) \Rightarrow k_{2} = 5300 - 48 k_{1} . . . (3) From (2) & (3), w e g e t : 4500 - 40 k_{1} = 5300 - 48 k_{1} \Rightarrow 8 k_{1} = 800 ∴ k_{1} = 100 . . . (4) P u t t i n g eq (4) in eq (2), w e g e t : k_{2} = 4500 - 40 \times 100 ∴ k_{2} = 500 E quation of variation : E = 500 + 100 n \Rightarrow E = 500 + 5500 (∵ n = 55) \Rightarrow E = Rs 6000$

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