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I got this question from here.
A man sold $250$ chairs and had a gain equal to selling price of $50$ chairs. His profit percent is?
The answer says it will be $25\%$ of profit. But, how? It should be $20\%$.
My solution is $\dfrac{50}{250}\times 100 = 20\%$.
Can't understand the solution from the website. Please help.
Total Money: Price of $250$ chairs
Gain: Price of $250 - 200 = 50$ chairs
Profit Percent: $\frac{\text{gain}}{\text{spent}} = \frac{250 - 200}{250 - (250 - 200)} = \frac{50}{200} = 25\%$
The total amount of money he brought in was the selling price of $250$ chairs. But $50$ of those chairs were pure gain, so when he bought the chairs, he only spent the equivalent of the selling price of $200$ chairs.
The profit percent is the amount of gain divided by the amount he spent. This is different from what you have calculated (gain divided by total money brought in). So that is $$\frac{\text{selling price of } 50 \text{ chairs gained}}{\text{selling price of } 200 \text{ chairs spent}}\times 100 = 25\%\frac{\text{gained}}{\text{spent}}.$$
The profit is $\pi(p,q)=pq-c(q)$ where $p$ is the selling price, $q$ is the quantity selled and $c(q)$ the cost to produce the quantity $q$. So you have $\pi(p,250)=250p-c(250)$ and you know that $\pi(p,250)=50p$; then you have $c(250)=200p$ and finally $$ \frac{\pi(p,250)}{c(250)}=\frac{250p-200p}{200p}=\frac{50}{200}=25\% $$
