The question is from CAT Profit and Loss. CAT exam is known to test on basics rather than high funda ideas. 2IIMs CAT question bank provides you with CAT questions that can help you gear for CAT Exam CAT 2020. This question is from profit and loss and discusses about a merchant who sells a product at two different prices. Completion of squares Ahoy!

Question 13: A merchant buys 80 articles, each at Rs. 40. He sells n of them at a profit of n% and the remaining at a profit of (100 â€“ n)%. What is the minimum profit the merchant could have made on this trade?

- Rs. 2160
- Rs. 1420
- Rs. 1580
- Rs. 2210

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CP = 80 Ã— 40

Profit from the n objects = n% Ã— 40 Ã— n.

Profit from the remaining objects = (100 â€“ n)% Ã— 40 Ã— (80 â€“ n).

We need to find the minimum possible value of n% Ã— 40 Ã— n + (100 â€“ n)% Ã— 40 Ã— (80 â€“ n).

Or, we need to find the minimum possible value of n^{2} + (100 â€“ n) (80 â€“ n).

Minimum of n^{2} + n^{2} â€“ 180n + 8000

Minimum of n^{2} â€“ 90n + 4000

Minimum of n^{2} â€“ 90n + 2025 â€“ 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 â€œCompletion of Squaresâ€? approach. We keep revisiting this in multiple chapters.

Minimum of n^{2} â€“ 90n + 2025 + 1975 = (n â€“ 45)^{2} + 1975

This reaches minimum when n = 45.

When n = 45, the minimum profit made

45% Ã— 40 Ã— 45 + 55% Ã— 40 Ã— 35

18 Ã— 45 + 22 Ã— 35 = 810 + 770 = 1580

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

Choice C is the correct answer.

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