What is the formula to calculate Profit Percentage?

Question

Let cost price of an item be $C$, selling price be $S$. Assume the seller makes a profit.

Then profit would be: $P = S - C$.

Now, what is the formula for calculating Profit Percentage?

1. $P \% = \dfrac{P}{C} \times 100$
2. $P\% = \dfrac{P}{S} \times 100$

Which one is right and why?

Answer 1

The percentage profit $X$ is defined by

$$X = \left(\frac{\textrm{Amount of money you have at the end}}{\textrm{Amount of money you had at the start}} - 1\right) \times 100$$

Since you have $S$ at the end and $C$ at the start (because that's the money you needed to buy the item) then

$$X = \left( \frac{S}{C} - 1\right)\times 100 = \left(\frac{S-C}{C}\right)\times 100 = \frac{P}{C} \times 100$$

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To see why your second decision has to be wrong, consider the case where you buy something for \$1 and sell it for \$1,001, so that $P$=1000. With your first definition,

$$X = 100\times \frac{1000}{1} = 100,000\%$$

which makes sense - you clearly made a huge profit, so you expect your percentage profit to be huge. With your second definition,

$$X = 100\times \frac{1000}{1001} = 99.9\%$$

which is nowhere near big enough.

Answer 2

$C : S = 100 : (100+x) \Rightarrow 100\cdot S=100 \cdot C +C \cdot x \Rightarrow x=\frac {100(S-C)}{C}\Rightarrow x= \frac{100\cdot P}{C}$