CAT 2025 Slot 3 · Quant · Question 13

CAT 2025 Slot 3 previous year question on Linear Equations, with the answer and a step-by-step solution.

Linear EquationsAlgebraType in the answer (TITA)Hard

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 ₹1100, ₹1000, and ₹800, respectively. The total fee paid by all the students is ₹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

Answer & solution

Answer: 700

Let s, a and c be the numbers taking science, arts and commerce, so s+a+c=1500.

The fee equation is 1100s+1000a+800c=1550000.

Substituting c=1500-s-a into it: 1100s+1000a+800\left( 1500-s-a \right)=1550000.

Expanding, 1100s+1000a+1200000-800s-800a=1550000.

Collecting terms, 300s+200a=350000, that is 3s+2a=3500.

The condition s\le a with a=\frac{3500-3s}{2} gives 2s\le 3500-3s.

So 5s\le 3500 and s\le 700.

At s=700: a=\frac{3500-2100}{2}=700 and c=1500-700-700=100, all non-negative.

Checking, 1100\left( 700 \right)+1000\left( 700 \right)+800\left( 100 \right)=770000+700000+80000=1550000, so the maximum is 700.

Shortcut. Eliminate the cheapest stream by substituting for c — the 1500 and the 800 both disappear into constants, leaving the single clean relation 3s+2a=3500. The constraint then bounds s in one line.

CAT 2025 Slot 3 · Quant · Question 13

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