A merchant bought 80 articles for Rs. 40 each. He sold n articles at a profit of n% and the remaining at (100-n)%. To find the minimum profit, the problem was modeled as an expression of n2 terms and solved using the technique of completing the square. The minimum profit was found to occur when n = 45, giving a total minimum profit of Rs. 1580. Therefore, the answer is c.

1. Profit & Loss

2. Profit & Loss
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?
(a) Rs. 2160 (b) Rs. 1420
(c) Rs. 1580 (d) Rs. 2210

3. Profit & Loss
CP = 80 × 40
Profit from the n objects = n% × 40 × n.
Profit from the remaining objects = (100 – n)% × 40 × (80 – n).
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?

4. Profit & Loss
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
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?

5. Profit & Loss
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.
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?

6. Profit & Loss
When n = 45, the minimum profit made
 45% × 40 × 45 + 55% × 40 × 35
 18 × 45 + 22 × 35 = 810 + 770
 Rs. 1580
Answer choice (c)
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?

7. For lots more CAT Level Questions, visit
online.2iim.com