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A trader marks up a Music system \(x\%\) over the cost price and gives a discount of \((x/4)\%\) to get a profit of \((x/2)\%\). What is his profit percentage?

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Assume CP = 100, write MP and SP in terms of x, then set the resulting selling price equal to a profit of (x/2)% and solve for x.
Updated On: Jul 21, 2026
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The Correct Option is D

Solution and Explanation

Step 1: Call the profit percentage P and work backward.
Let the profit percentage itself be \(P\), so by the question \(P=x/2\), which means \(x=2P\). Rewriting everything in terms of \(P\) lets us solve for the answer directly, without a separate step to divide by 2 at the end.

Step 2: Write MP, discount and SP using P.
With CP = 100: \[ MP = 100+2P,\qquad \text{discount} = \frac{2P}{4}\% = \frac{P}{2}\% \] \[ SP = (100+2P)\left(1-\frac{P}{200}\right) \]

Step 3: Use the fact that profit percentage is P.
Since CP is 100 and the profit percentage is \(P\), \[ SP = 100+P \]

Step 4: Equate and simplify.
\[ (100+2P)\left(1-\frac{P}{200}\right) = 100+P \] \[ 100-\frac{P}{2}+2P-\frac{2P^2}{200} = 100+P \] \[ 100+\frac{3P}{2}-\frac{P^2}{100} = 100+P \] The 100 cancels: \[ \frac{3P}{2}-P = \frac{P^2}{100} \] \[ \frac{P}{2} = \frac{P^2}{100} \]

Step 5: Solve directly for P.
Dividing both sides by \(P\) (P is not zero): \[ \frac{1}{2} = \frac{P}{100} \] \[ P = 50 \]
This directly gives the profit percentage without needing a separate halving step at the end. \[ \boxed{50\%} \]
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