# CAT Quantitative Aptitude Questions | CAT Profit and Loss Questions - Progressions

###### CAT Questions | Profit and Loss Problems | Profit and Loss - Minimum CAT Questions

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 2022. 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 – n)%. What is the minimum profit the merchant could have made on this trade?

1. Rs. 2160
2. Rs. 1420
3. Rs. 1580
4. Rs. 2210

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##### Method of solving this CAT Question from Profit and loss: Form the equation and you are through.

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.

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