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: Define variables.  
Let:

\( s = \text{number of Science students}, \quad a = \text{number of Arts students}, \quad c = \text{number of Commerce students}.

Total students:

\( s + a + c = 1500 \). (1)

Step 2: Use the fee information. 
Fees per student:

Science = 1100, Arts = 1000, Commerce = 800.

Total fee collected:

\( 1100s + 1000a + 800c = 1550000 \).

Divide the whole equation by 100 to simplify:

\( 11s + 10a + 8c = 15500 \). (2)

Step 3: Eliminate \( c \). 
From (1):

\( c = 1500 - (s + a) \).

Substitute into (2):

\( 11s + 10a + 8(1500 - s - a) = 15500 \) 
\( 11s + 10a + 12000 - 8s - 8a = 15500 \) 
\( (11s - 8s) + (10a - 8a) + 12000 = 15500 \) 
\( 3s + 2a = 3500 \). (3)

Step 4: Express \( a \) in terms of \( s \). From (3):

\( 2a = 3500 - 3s \Rightarrow a = \frac{3500 - 3s}{2} = 1750 - 1.5s \).

Step 5: Apply the condition \( s \le a \). 
Given that the number of science students is not more than the number of arts students:

\( s \le a \).

Substitute \( a = 1750 - 1.5s \):

\( s \le 1750 - 1.5s \) 
\( s + 1.5s \le 1750 \) 
\( 2.5s \le 1750 \) 
\( s \le \frac{1750}{2.5} = 700 \).

So, the maximum possible value of \( s \) is \( 700 \), if it is feasible. Step 6: Check feasibility for \( s = 700 \).

\( a = 1750 - 1.5 \cdot 700 = 1750 - 1050 = 700 \), 
\( c = 1500 - (s + a) = 1500 - (700 + 700) = 100 \).

All are non-negative integers, and \( s \le a \) holds (700 = 700). So this distribution is valid. Hence, the maximum possible number of science students is \( \boxed{700} \).