Question

The price of a jewel, passing through three hands rises on the whole by \[65\% \]. If the first and the second sellers earned \[20\% \]and \[25\% \] profit respectively, find the percentage profit earned by the third seller.
A.\[25\% \]
B.\[20\% \]
C.\[15\% \]
D.\[10\% \]

Answer 1

Hint: We will assume two things here, the price of the jewel and the percentage profit earned by the third seller respectively. The percentage of the profit earned by each seller is on the selling price of the previous seller. This means that the second seller earns a profit on the selling price of the first seller. The selling price of the first seller is the cost price of the second seller.

Complete step-by-step answer:
From looking at the question, we get to know that the price of the jewel gets increased by \[65\% \]. The first seller earns a profit by increasing the price of the jewel by \[20\% \] and the second seller earns a profit by increasing the price of the jewel by \[25\% \] on the price of the first seller.
Let us assume that \[P\]is the cost price of the jewel before the increase.
Let \[x\% \]be the percentage profit earned by the third seller.
So, the final price of the jewel after the increase
\[
   = P + 65\% \times P \\
   = P(1 + 65\% ) \\
   = P\left( {1 + \dfrac{{65}}{{100}}} \right) \\
   = P\left( {\dfrac{{100 + 65}}{{100}}} \right) \\
   = \dfrac{{165}}{{100}}P \\
\]
The selling price of the jewel after the first seller
\[
   = P + 20\% \times P \\
   = P(1 + 20\% ) \\
   = P\left( {1 + \dfrac{{20}}{{100}}} \right) \\
   = P\left( {\dfrac{{100 + 20}}{{100}}} \right) \\
   = \dfrac{{120}}{{100}}P \\
\]
The selling price of the jewel after the second seller
\[
   = \dfrac{{120}}{{100}}P + 25\% \times \dfrac{{120}}{{100}}P \\
   = \dfrac{{120}}{{100}}P(1 + 25\% ) \\
   = \dfrac{{120}}{{100}}P\left( {1 + \dfrac{{25}}{{100}}} \right) \\
   = \dfrac{{120}}{{100}}P\left( {\dfrac{{100 + 25}}{{100}}} \right) \\
   = \dfrac{{120}}{{100}}P \times \dfrac{{125}}{{100}} \\
\]
The selling price of the jewel after the third seller
\[
   = \dfrac{{120}}{{100}}P \times \dfrac{{125}}{{100}} + x\% \times \dfrac{{120}}{{100}}P \times \dfrac{{125}}{{100}} \\
   = \dfrac{{120}}{{100}}P \times \dfrac{{125}}{{100}}(1 + x\% ) \\
   = \dfrac{{120}}{{100}}P \times \dfrac{{125}}{{100}}\left( {1 + \dfrac{x}{{100}}} \right) \\
   = \dfrac{{120}}{{100}}P \times \dfrac{{125}}{{100}}\left( {\dfrac{{100 + x}}{{100}}} \right) \\
\]
Now, according to the question we can write
\[
  \dfrac{{120}}{{100}}P \times \dfrac{{125}}{{100}}\left( {\dfrac{{100 + x}}{{100}}} \right) = \dfrac{{165}}{{100}}P \\
   \Rightarrow \dfrac{{120}}{{100}} \times \dfrac{{125}}{{100}}\left( {\dfrac{{100 + x}}{{100}}} \right) = \dfrac{{165}}{{100}} \\
   \Rightarrow \left( {\dfrac{{100 + x}}{{100}}} \right) = \dfrac{{165}}{{100}} \times \dfrac{{100}}{{125}} \times \dfrac{{100}}{{120}} \\
   \Rightarrow \left( {\dfrac{{100 + x}}{{100}}} \right) = \dfrac{{165 \times 100}}{{125 \times 120}} \\
   \Rightarrow 100 + x = \dfrac{{165 \times 100 \times 100}}{{125 \times 120}} \\
   \Rightarrow 100 + x = 110 \\
   \Rightarrow x = 110 - 100 \\
   \Rightarrow x = 10 \\
\]
Therefore, the profit percentage earned by the third seller is \[10\% \]

Thus, the answer is option D.

Note: We might make the mistake of calculating the price for each seller directly on the starting cost price and not on the selling price of each seller. We need to remember that the profit percentage is to be multiplied with the selling price of the previous seller or the cost price of the current seller as they both are the same.