Question

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?

• A. 20%
• B. 25%
• C. 30%
• D. 50%

Hint:

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.

Answer 1

The Correct Option is D

Step 1: Assume a cost price and write the marked price.
Let the cost price (CP) be Rs. 100. Marking up by \(x\%\) gives a marked price \[ MP = 100+x \]

Step 2: Apply the discount to get the selling price.
A discount of \((x/4)\%\) on the marked price gives \[ SP = (100+x)\left(1-\frac{x}{400}\right) \]

Step 3: Write the selling price using the given profit.
Since CP is 100 and the profit is \((x/2)\%\), \[ SP = 100+\frac{x}{2} \]

Step 4: Set the two expressions for SP equal and simplify.
\[ (100+x)\left(1-\frac{x}{400}\right) = 100+\frac{x}{2} \]
Expanding the left side: \[ 100-\frac{x}{4}+x-\frac{x^2}{400} = 100+\frac{x}{2} \]
The 100 cancels from both sides, leaving \[ \frac{3x}{4}-\frac{x^2}{400} = \frac{x}{2} \]

Step 5: Solve for x.
\[ \frac{3x}{4}-\frac{x}{2} = \frac{x^2}{400} \] \[ \frac{x}{4} = \frac{x^2}{400} \] Cross multiplying, \(400x = 4x^2\), so \(x^2 = 100x\), giving \(x=100\) (rejecting \(x=0\), since there is a genuine markup).

Final Answer:
Profit percentage \(=x/2 = 100/2 = 50\%\). \[ \boxed{50\%} \]