A merchant buys 80 articles, each at Rs. 40. He sells n of them at a profit of…

CP = 80 × 40

Profit from the n objects = n% × 40 × n.

Profit from the remaining objects = (100 – n)% × 40 × (80 – n).

We need to find the minimum possible value of n% × 40 × n + (100 – n)% × 40 × (80 – n).

Or, we need to find the minimum possible value of n2 + (100 – n) (80 – n).

Minimum of n2 + n2 – 180n + 8000

Minimum of n2 – 90n + 4000

Minimum of n2 – 90n + 2025 – 2025 + 4000

We add and subtract 2025 to this expression in order to crate an expression that can be expressed as a perfect square.

This approach is termed as the “Completion of Squares�? approach. We keep revisiting this in multiple chapters.

Minimum of n2 – 90n + 2025 + 1975 = (n – 45)2 + 1975

This reaches minimum when n = 45.

When n = 45, the minimum profit made

45% × 40 × 45 + 55% × 40 × 35

18 × 45 + 22 × 35 = 810 + 770 = 1580

The question is "What is the minimum profit the merchant could have made on this trade?"

Hence, the answer is Rs. 1580.

Choice C is the correct answer.