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If a shopkeeper sells a TV at $15\%$ profit and a DVD at $12\%$ loss then he earns Rs. 540 as total profit but if he sells the TV at a loss at $12\%$ loss and the DVD at $15\%$ profit then there is no profit or loss. Find the cost price of the TV and the DVD?

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Hint: In order to solve this question suppose that CP of the TV and the DVD be $x$ and $y$ rupees and then formulate two equations using the statements given in the question and applying profit and loss percent formula which is \[profit\%=\dfrac{profit}{CP}\times 100\%\] and \[loss\%=\dfrac{loss}{CP}\times 100\%\]. Solve those two equations and get the values of $x$ and $y$.

Complete step-by-step solution:
Let us suppose that the cost price of TV and DVD be $x$ and $y$ rupees respectively.
Now, according to the first statement, it is given that if the shopkeeper sells a TV at $15\%$ profit and a DVD at $12\%$ loss then he earns Rs. 540 as total profit.
As we know that \[profit\%=\dfrac{profit}{CP}\times 100\%\Rightarrow profit=\dfrac{profit\%\times CP}{100}\]
Now, $15\%$profit on TV means $profit=\dfrac{15\times x}{100}=0.15x$
Similarly, we know that \[loss\%=\dfrac{loss}{CP}\times 100\%\Rightarrow loss=\dfrac{loss\%\times CP}{100}\]
Now, $12\%$ loss on DVD means $loss=\dfrac{12\times y}{100}=0.12y$
$\therefore $ Total profit = profit on TV – loss on DVD
$\Rightarrow 540=0.15x-0.12y...............(1)$
According to the second statement, it is given that if he sells the TV at a loss at $12\%$ loss and the DVD at $15\%$ profit then there is no profit or loss.
Now, $15\%$profit on DVD means $profit=\dfrac{15\times y}{100}=0.15y$
And, $12\%$ loss on TV means $loss=\dfrac{12\times x}{100}=0.12x$
$\therefore $ Net profit = profit on DVD – loss on TV
$\Rightarrow 0=0.15y-0.12x$
$\Rightarrow y=\dfrac{0.12}{0.15}x=\dfrac{4}{5}x................(2)$
Putting the value of $y$ in equation (1), we get
$\Rightarrow 540=0.15x-0.12\times \dfrac{4}{5}x$
$\Rightarrow 540=0.15x-0.096x$
$\Rightarrow 0.054x=540$
$\Rightarrow x=\dfrac{540}{0.054}=10000$
Now, putting the value of $x$ in equation (2), we get
$\Rightarrow y=\dfrac{4}{5}\times 10000=8000$
Hence, the value of $x$ and $y$ is 10000 and 8000 respectively.
So, the cost price of a TV is 10000 rupees and the DVD is 8000 rupees.


Note: This is a simple question of profit and loss where we have to carefully read the statements and formulate the equations based on them and then solve them to get the desired value. Students often get confused with profit and loss like which formula to use and in which case, it is simple to understand that profit means we are getting something positive which is only possible if the Selling price is greater than the cost price and hence profit = SP – CP. Similarly, in case of loss it is reverse and hence loss = CP – SP. So after understanding this question will be easier to solve.