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. 40%
• E. 50%

Hint:

Write SP two ways — once from markup and discount, once from the stated profit \(x/2\)% — and equate them to solve for x.

Answer 1

Answer: (E)

Step 1: Set up cost price, marked price and selling price.
Let the cost price (CP) \(=100\).
Marked price (MP) \(=100\left(1+\dfrac{x}{100}\right)\).
A discount of \(\dfrac{x}{4}\%\) on MP gives the selling price (SP):
\(SP = 100\left(1+\dfrac{x}{100}\right)\left(1-\dfrac{x}{400}\right)\).

Step 2: Use the given profit condition.
The profit is \(\dfrac{x}{2}\%\), so \(SP = 100\left(1+\dfrac{x}{200}\right)\).

Step 3: Equate the two expressions for SP.
\(\left(1+\dfrac{x}{100}\right)\left(1-\dfrac{x}{400}\right)=1+\dfrac{x}{200}\)
Expanding the left side:
\(1-\dfrac{x}{400}+\dfrac{x}{100}-\dfrac{x^2}{40000}=1+\dfrac{3x}{400}-\dfrac{x^2}{40000}\)

Step 4: Solve for x.
\(1+\dfrac{3x}{400}-\dfrac{x^2}{40000}=1+\dfrac{2x}{400}\)
\(\dfrac{3x}{400}-\dfrac{2x}{400}=\dfrac{x^2}{40000}\)
\(\dfrac{x}{400}=\dfrac{x^2}{40000}\)
\(100x=x^2 \Rightarrow x(x-100)=0\)
Since \(x\ne0\) (there must be an actual markup), \(x=100\).

Step 5: Find the profit percentage.
Profit\(\% = \dfrac{x}{2} = \dfrac{100}{2}=50\%\).