By selling an article for \[Rs.{\text{ }}240\], a man makes a profit of \[20\% \]. What is his \[C.P.\]? What would his profit percent be if he sold the article for \[Rs.{\text{ 275}}\]?

Hint: We have to find the cost price of an article which a man sells at \[Rs.{\text{ }}240\] by making a profit of \[20\% \]. Here, we will consider the cost price of the article to be \[x\] and profit as \[P\]. As we know, \[\Pr ofit\% = \dfrac{{Profit}}{{C.P.}} \times 100\].Using this we will calculate \[P\] in terms of \[x\]. Then using the formula \[\Pr ofit = S.P. - C.P.\] We will calculate the cost price of the article and using this we will calculate the profit percent.

Complete step by step solution:
Here, given a man who makes a profit of \[20\% \] by selling an article for \[Rs.{\text{ }}240\]. We have to find the cost price of the article.
For this, let's assume the cost price of the article to be \[Rs.{\text{ }}x\] and profit as \[Rs.{\text{ }}P\].
As we know that \[\Pr ofit\% = \dfrac{{Profit}}{{C.P.}} \times 100\].
Putting the values given in the question we get
\[ \Rightarrow 20 = \dfrac{P}{x} \times 100\]
On cross multiplication and rearranging we get
\[ \Rightarrow P = \dfrac{{20}}{{100}} \times x\]
On simplification we get
\[ \Rightarrow P = \dfrac{x}{5} - - - (1)\]
Now, we also know that \[\Pr ofit = S.P. - C.P.\]
Using this we can write,
\[ \Rightarrow P = 240 - x - - - (2)\]
From \[(1)\] and \[(2)\], we can write
\[ \Rightarrow \dfrac{x}{5} = 240 - x\]
Multiplying \[5\] on both the sides, we get
\[ \Rightarrow x = 5\left( {240 - x} \right)\]
On simplification we get
\[ \Rightarrow x = 1200 - 5x\]
Adding \[5x\] on both the sides, we get
\[ \Rightarrow 5x + x = 1200\]
On simplifying
\[ \Rightarrow 6x = 1200\]
Dividing both the sides by \[6\], we get
\[ \Rightarrow x = 200\]
Therefore, \[C.P. = 200\].
Now, we have a new selling price as \[Rs.{\text{ }}275\] with cost price as \[Rs.{\text{ }}200\] and we have to find the \[\Pr ofit\% \].
For this, we will use the formula \[\Pr ofit\% = \dfrac{{Profit}}{{C.P.}} \times 100\] by substituting \[\Pr ofit = S.P. - C.P.\]
Therefore, on substitution we get \[\Pr ofit\% = \dfrac{{S.P. - C.P.}}{{C.P.}} \times 100\].
Putting \[S.P. = Rs.{\text{ }}275\] and \[C.P. = Rs.{\text{ }}200\], we get
\[ \Rightarrow \Pr ofit\% = \dfrac{{275 - 200}}{{200}} \times 100\]
On solving we get
\[ \Rightarrow \Pr ofit\% = \dfrac{{75}}{{200}} \times 100\]
On simplification we get
\[ \Rightarrow \Pr ofit\% = 37.5\]
Therefore, the \[C.P.\] of an article that a man sells for \[Rs.{\text{ }}240\] making a profit of \[20\% \] is \[200\] and the profit percent is \[37.5\% \] if he sold the article for \[Rs.{\text{ 275}}\].

Note: Here, the man was having profit that’s why we have used \[\Pr ofit = S.P. - C.P.\] As in the case of profit, the selling price is more than the cost price. But, in cases where there will be loss then we will use \[Loss = C.P. - S.P.\] because in case of loss we have cost more than the selling price.