Question

Manoj purchased a video for Rs.12000. He sold it to Rahul at a gain of 10%. If Rahul sells it to Rakesh at a loss of 5%, what did Rakesh pay for it? \[\]

Answer 1

Hint: We find the selling price for Manoj from the formula of percentage gain $ \% \text{Gain}=\dfrac{SP-CP}{CP}\times 100$ and using the given data. The selling price for Manoj is the cost price for Rahul. We find the selling price for Rahul using the formula for percentage loss $ \% \text{Loss}=\dfrac{CP-SP}{SP}\times 100$ and putting the obtained and given values. The selling price for Rahul is the cost price for Rakesh.\[\]

Complete step-by-step solution
We know that the cost price abbreviated as CP of an article is the price at which the article is bought or the price that was spent on making the article from materials plus additional expenditure. The additional expenditure may tax, transportation, etc.\[\]
The selling price abbreviated as SP is the price at which the article is sold. If the selling price is greater than the cost price then we say the sellers earned profit or gain. The gain amount is given by
\[\text{Gain}=\text{SP}-\text{CP}\]
The percentage gain is calculated on cost price as
\[ \% \text{Gain}=\dfrac{\text{Gain}}{\text{CP}}\times 100=\dfrac{SP-CP}{CP}\times 100\]
If the selling price is less than the cost price then we say the sellers earned a loss. The loss amount is given by
\[\text{Loss}=\text{CP}-\text{SP}\]
The percentage gain is calculated on cost price as
\[ \% \text{Loss}=\dfrac{\text{Loss}}{\text{CP}}\times 100=\dfrac{CP-SP}{SP}\times 100\]
Let us denote the cost price of a video for Manoj, Rahul and Rakesh be $C{{P}_{1}}, C{{P}_{2}}, C{{P}_{3}}$, and the selling price of a video for Manoj, Rahul is $S{{P}_{1}}, S{{P}_{2}}$ respectively. We are given the question that Manoj purchased a video for Rs.12000. He sold it to Rahul at a gain of 10%. So we have $C{{P}_{1}}=12000,%\text{Gain}=10$. We put these values in the formula in the percentage gain formula and find $S{{P}_{1}}$. We have,
\[\begin{align}
  & 10=\dfrac{S{{P}_{1}}-12000}{12000}\times 100 \\
 & \Rightarrow 1200=S{{P}_{1}}-12000 \\
 & \Rightarrow S{{P}_{1}}=12000+1200=13200 \\
\end{align}\]
So Rahul buys the video at Rs.13200. So the cost price for Rahul is
\[C{{P}_{2}}=S{{P}_{1}}=13200\]
We are further given in the question that Rahul sells the video to Rakesh at a loss of 5%. So we have$%\text{Loss}=5$. We put $C{{P}_{2}}$ and %Loss in the formula for %Loss to find out $S{{P}_{2}}$. We have
\[\begin{align}
  & \Rightarrow 5=\dfrac{13200-S{{P}_{2}}}{13200}\times 100 \\
 & \Rightarrow 660=13200-S{{P}_{2}} \\
 & \Rightarrow S{{P}_{2}}=13200-600=12540 \\
\end{align}\]
The selling price for Rahul is the cost price for Rakesh. So the cost price for Rakesh is
\[C{{P}_{3}}=S{{P}_{2}}=\text{Rs}.12540\]

Note: We note that the percentage profit and loss are always calculated on cost price not on selling otherwise mentioned. The key to solving this problem is the selling price of one person is the cost price of another. If there is a profit of $x$% and a loss of $y$% then net percentage profit or loss is $\dfrac{x-y-xy}{100}$. We can use this formula for short calculation.