Question

If the selling price of an article is five times the discount offered and if the percentage of discount is equal to the percentage profit, what is the ratio of the discount offered to the cost price?

• A. \(11:30\)
• B. \(4:15\)
• C. \(7:30\)
• D. \(1:6\)

Hint:

Let profit percent equal discount percent and express selling price, marked price and discount all in terms of cost price.

Answer 1

The Correct Option is C

Step 1: Relate selling price and discount.
Let the marked price be \(MP\), the selling price \(SP\), the discount \(D=MP-SP\), and the cost price \(CP\).
Given \(SP=5D\), substitute \(D=MP-SP\) to get \(SP=5(MP-SP)\), so \(6SP=5MP\), giving \(SP=\dfrac{5}{6}MP\).

Step 2: Express the discount as a fraction of \(MP\).
\(D=MP-SP=MP-\dfrac{5}{6}MP=\dfrac{1}{6}MP\), so the discount percent is \(\dfrac{1}{6}\times100=\dfrac{50}{3}\%\).

Step 3: Use the condition that discount percent equals profit percent.
Profit percent is also \(\dfrac{50}{3}\%\), so \(SP=CP\left(1+\dfrac{50}{300}\right)=CP\times\dfrac{7}{6}\), which gives \(CP=\dfrac{6}{7}SP\).

Step 4: Write the discount also in terms of \(SP\).
Since \(SP=\dfrac{5}{6}MP\), \(MP=\dfrac{6}{5}SP\), and \(D=\dfrac{1}{6}MP=\dfrac{1}{6}\times\dfrac{6}{5}SP=\dfrac{1}{5}SP\).

Step 5: Form the ratio \(D:CP\).
\(D:CP=\dfrac{1}{5}SP:\dfrac{6}{7}SP=\dfrac{1}{5}:\dfrac{6}{7}=\dfrac{7}{30}:1=7:30\).

Final Answer:
The discount to cost price ratio is 7 is to 30. \[ \boxed{7:30} \]