Question

There were 35 students in a hostel. If the number of students is increased by 7, the expenditure on food increases by 42 per day while the average expenditure of students is reduced by 1. What was the initial expenditure on food per day?
a) 433
b) 442
c) 420
d) 421

Answer 1

Hint: Suppose the original expenditure per day per student is Rs x. Now, find the total expenditure for 35 students and if 7 are included in the hostel as well. Hence, form an equation in a single variable ‘x’ with the help of the given condition. Solve it further to get ‘x’ and the total initial expenditure.

Complete step-by-step answer:
As it is given in the question that there are 35 students in the hostel and if the number of students is increased by 7, the expenditure increased by 42 per day while average expenditure is reduced by 1 and hence we need to determine the initial expenditure of the food per day.
So, let the original expenditure be Rs x per day per student. Hence, the total expenditure at 35 students per day is $35\times x$ Rs i.e. Rs 35x. Now, if 75 students are added to the hostel, so the total number of students is 35 + 7 = 42.
Now, it is given that average expenditure per day will decrease by 1 i.e. current expenditure will reduce by 1 of the initial expenditure food per day. So, we know the original initial expenditure is Rs x per day. Hence, current expenditure will be (x – 1) Rs. Per day. So, total expenditure for 42 students can be calculated as 42 (x – 1) Rs per day. And we are given that expenditure on food increases by 42 per day, if 7 more students are added to the hostel. Hence, we can write an equation as current expenditure = initial expenditure + 42
42 (x – 1) = 35x + 42
42x – 42 = 35x + 42
7x = 42 + 42 = 84
$\text{x}=\dfrac{84}{7}=12\text{Rs}$
Hence, the initial expenditure per student per day is Rs 12. So, total initial expenditure for 35 students per day will be
$35x=35\times 12=420\text{Rs}$
So option (c) is the correct answer.

Note: Writing the mathematical equation from the given problem in words is the key point of the question. Don’t miss the statement that expenditure will reduce by 1 if 7 students are added to the hostel, otherwise one may get the calculation in ‘x’ as 42x = 35x + 42, which will give the wrong value of x. So, be careful with the information given in the problem.
Term average expenditure in the problem is used for the initial expenditure for the 35 students in the hostel.