Question

In a school with 1500 students, each student chooses any one of the streams out of science, arts, and commerce, by paying a fee of Rs 1100, Rs 1000, and Rs 800, respectively. The total fee paid by all the students is Rs 15,50,000. If the number of science students is not more than the number of arts students, then the maximum possible number of science students in the school is

Hint:

In word problems with headcount and revenue constraints, first set up two equations: one for the total number of people and another for total money. Then eliminate one variable to get a simple linear relation and apply given inequalities to find extrema.

Answer 1

Correct Answer: 700

Step 1: Let science, arts, commerce counts be \( s,a,c \). Then \( s+a+c=1500 \) and \( 1100s+1000a+800c=1550000 \).
Step 2: Eliminating \( c \): \( 300s+200a=350000 \Rightarrow 3s+2a=3500 \).
Step 3: Since \( s\le a \), substituting \( a=\frac{3500-3s}{2}\ge s \Rightarrow s\le700 \); parity requires \( s \) even, so the maximum is \( s=700 \) (then \( a=700, c=100 \)).
\[ \boxed{s_{max}=700} \]