Question

Two merchants sell, each an article for Rs. 1000. If merchant A computes his profit on cost price, while merchant B computes his profit on selling price, they end up making profits of 25%. By how much is the profit made by Merchant B greater than that of Merchant A?\[\]
A.66.67\[\]
B.125\[\]
C. 50\[\]
D. 100\[\]

Answer 1

Hint: We first calculate the cost price $C{{P}_{B}}$ for Merchant B by putting given values in the percentage profit loss formula on selling price ${{P}_{S}}=\dfrac{SP-CP}{SP}\times 100$ and then the profit ${{P}_{B}}=S{{P}_{B}}-C{{P}_{B}}$. We use to put the given values to find the cost price $C{{P}_{A}}$ of merchant A in the percentage profit formula on cost price ${{P}_{C}}=\dfrac{SP-CP}{CP}\times 100$ and then the profit ${{P}_{A}}=S{{P}_{A}}-C{{P}_{A}}$. We find ${{P}_{B}}-{{P}_{A}}$.

Complete step-by-step solution:
It is given in the question that the name of two merchants are A and B. Let us denote the cost price, selling price and profit of merchant A as $C{{P}_{A}},S{{P}_{A}},{{P}_{A}}$ respectively. Similarly let us denote Let us denote the cost price, selling price and profit of merchant B as $C{{P}_{B}},S{{P}_{B}},{{P}_{B}}$respectively. \[\]
It is also given in the question that both the merchants sell the article at a price 1000 rupees. So we have,
\[S{{P}_{A}}=S{{P}_{B}}=1000\]
We also know the percentage profit on sell price ${{P}_{S}}$formula,
\[\% {{P}_{S}}=\dfrac{SP-CP}{SP}\times 100....\left( 1 \right)\]
We know the formula for profit on cost price ${{P}_{C}}$ as
\[\% {{P}_{C}}=\dfrac{SP-CP}{CP}\times 100....\left( 2 \right)\]
Merchant B computes his profit at the selling price at a profit 25%. Here the for the merchant B is $ \% {{P}_{S}}= \% {{P}_{B}}=25 \% $, $\text{S}{{\text{P}}_{B}}=1000$. Putting these in formula (1) we find out the cost price of merchant B that is $C{{P}_{B}}$
\[\begin{align}
  & \% {{P}_{B}}=\dfrac{S{{P}_{B}}-C{{P}_{B}}}{S{{P}_{B}}}\times 100 \\
 & \Rightarrow 25=\dfrac{1000-C{{P}_{B}}}{1000}\times 100 \\
 & \Rightarrow 250=1000-C{{P}_{B}} \\
 & \Rightarrow C{{P}_{B}}=850 \\
\end{align}\]
So the profit earned by merchant B is
\[{{P}_{B}}=S{{P}_{B}}-C{{P}_{B}}=1000-850=250\]
It is given in question that the merchant A computes his profit on cost price and he makes the same percentage profit $ \% {{P}_{C}}= \% {{P}_{A}}= 25 \% $ on cost price and with same selling price $S{{P}_{A}}=1000$rupees. Putting these in formula (2) we find out the cost price of merchant A that is $C{{P}_{A}}$
 \[\begin{align}
  & \% {{P}_{A}}=\dfrac{S{{P}_{A}}-C{{P}_{A}}}{C{{P}_{A}}}\times 100 \\
 & \Rightarrow 25=\dfrac{1000-C{{P}_{A}}}{C{{P}_{A}}}\times 100 \\
 & \Rightarrow 25C{{P}_{A}}=1000\times 100-100C{{P}_{A}} \\
 & \Rightarrow 125C{{P}_{A}}=100000 \\
 & \Rightarrow C{{P}_{A}}=\dfrac{100000}{125}=800 \\
\end{align}\]
So the profit earned by merchant A is \[ {{P}_{A}}=S{{P}_{A}}-C{{P}_{A}}=1000-800=200 \]
Now the difference in profit between merchant A and B is ${{B}_{p}}-{{A}_{p}}=250-200=50$ rupees. So the correct option is C.

Note: We can directly find the cost price from the selling price with the same profit using the formula $CP=\dfrac{100\times SP}{100+P}$. If there is loss is instead of profit then the loss $L$ is calculated by $L=CP-SP.$ If the two articles are sold at percentage profit ${{p}_{1}},{{p}_{2}}$ on the same price then net percentage $p$ is given by \[p=\dfrac{100{{p}_{1}}{{p}_{2}}+2{{p}_{1}}{{p}_{2}}}{200+{{p}_{1}}+{{p}_{2}}}\]