Marked Price Problem Analysis

This question involves calculating the original marked price of an object based on information about how two different salespeople, A and B, sell it and receive commissions.

Both salespeople sell the object at a price less than the marked price and earn a commission based on their respective selling prices. The key piece of information is that they both receive the same amount of commission.

Defining Variables

Let's denote the marked price of the object as \( MP \).

• Salesperson A sells the object at a price Rs. 5 less than the marked price.
• Salesperson B sells the object at a price Rs. 15 less than the marked price.

Based on this, we can write the selling price for each salesperson:

• Selling Price for A (\( SP_A \)) = \( MP - 5 \)
• Selling Price for B (\( SP_B \)) = \( MP - 15 \)

Calculating Commission

The commission for each salesperson is a percentage of their selling price:

• Commission for A (\( C_A \)) = 5% of \( SP_A \)
• Commission for B (\( C_B \)) = 15% of \( SP_B \)

We can write these commissions using the selling price expressions:

• \( C_A = 0.05 \times (MP - 5) \)
• \( C_B = 0.15 \times (MP - 15) \)

Equating Commissions

The problem states that both A and B receive the same amount in commission. Therefore, we can set their commission amounts equal to each other:

\( C_A = C_B \)

\( 0.05 \times (MP - 5) = 0.15 \times (MP - 15) \)

Solving for Marked Price

Now we need to solve this equation for \( MP \).

First, distribute the percentages on both sides:

\( 0.05 \times MP - 0.05 \times 5 = 0.15 \times MP - 0.15 \times 15 \)

\( 0.05 MP - 0.25 = 0.15 MP - 2.25 \)

Next, gather the \( MP \) terms on one side and the constant terms on the other side. Subtract \( 0.05 MP \) from both sides and add \( 2.25 \) to both sides:

\( 2.25 - 0.25 = 0.15 MP - 0.05 MP \)

\( 2.00 = 0.10 MP \)

Finally, divide by \( 0.10 \) to find the value of \( MP \):

\( MP = \frac{2.00}{0.10} \)

\( MP = 20 \)

The marked price of the object is Rs. 20.

Verification

Let's check if the commissions are indeed equal when \( MP = 20 \).

• For A: \( SP_A = 20 - 5 = 15 \). \( C_A = 5\% \text{ of } 15 = 0.05 \times 15 = 0.75 \).
• For B: \( SP_B = 20 - 15 = 5 \). \( C_B = 15\% \text{ of } 5 = 0.15 \times 5 = 0.75 \).

Since both \( C_A \) and \( C_B \) are equal to 0.75, our calculated marked price of Rs. 20 is correct.