CAT 2025 Slot 3 QA Question 4

First, let's understand the problem. There are 1500 students in total, each choosing one of three streams: science, arts, or commerce. The fees for each stream are Rs 1100, Rs 1000, and Rs 800 respectively. The total fee collected from all students is Rs 15,50,000. Also, it's given that the number of science students isn't more than the number of arts students. We find the maximum possible number of science students.

Okay, so let's denote the number of students in each stream. Let me assign variables:

From the problem, we know that the total number of students is 1500. So, the first equation I can write is:

S + A + C = 1500

That's straightforward. Now, the total fee collected is Rs 15,50,000. Since each stream has a different fee, I can write another equation based on the fees:

1100S + 1000A + 800*C = 15,50,000

So now I have two equations:

I need a third equation or some relationship to solve for S, A, and C. But the problem only gives me two equations. However, there's another condition: the number of science students is not more than the number of arts students. So, S ≤ A.

Since I need to maximize S, I should try to make S as large as possible while satisfying all the given conditions.

Hmm, maybe I can express C from the first equation in terms of S and A. Let's try that.

From equation 1:

C = 1500 - S - A

Now, substitute this into equation 2:

1100S + 1000A + 800*(1500 - S - A) = 15,50,000

Expanding this:

1100S + 1000A + 800*1500 - 800S - 800A = 15,50,000

Calculating 8001500: 8001500 = 1,200,000

So, substituting back:

1100S + 1000A + 1,200,000 - 800S - 800A = 15,50,000

Now, combine like terms:

(1100S - 800S) + (1000A - 800A) + 1,200,000 = 15,50,000

Calculating each:

1100S - 800S = 300S

1000A - 800A = 200A

So, the equation becomes:

300S + 200A + 1,200,000 = 15,50,000

Now, subtract 1,200,000 from both sides:

300S + 200A = 15,50,000 - 1,200,000

15,50,000 - 1,200,000 = 14,30,000

So:

300S + 200A = 14,30,000

Hmm, I can simplify this equation by dividing all terms by 100 to make the numbers smaller:

3S + 2A = 143,000

Wait, that doesn't seem right.

Wait, 15,50,000 minus 1,200,000 is 14,30,000? Let me verify:

15,50,000 - 1,200,000 = 14,30,000. Yes, that's correct.

But 300S + 200A = 14,30,000. Dividing by 100 gives:

3S + 2A = 143,000

Wait, that seems too large because S and A are only up to 1500 each.

Wait, 1100S + 1000A + 800C = 15,50,000

And C = 1500 - S - A

So, substituting:

1100S + 1000A + 800*(1500 - S - A) = 15,50,000

Calculating 800*1500: 1,200,000

So, 1100S + 1000A + 1,200,000 - 800S - 800A = 15,50,000

Now, 1100S - 800S = 300S

1000A - 800A = 200A

So, 300S + 200A + 1,200,000 = 15,50,000

Subtracting 1,200,000:

300S + 200A = 14,30,000

Divide by 100:

3S + 2A = 143,000

Wait, that can't be right because S and A are only 1500 each.

Wait, 15,50,000 is the total fee. If all 1500 students paid Rs 1000, the total would be 1,500,000. But the