# CAT Practice : Percents, Profits

Completion of squares Ahoy!

## Profits - Maximum

Q.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

Choice C. Rs. 1580

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## Detailed Solution

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

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