Understanding the Profit and Loss Problem

This problem involves a series of transactions where a watch is sold from one person to another, with profits or losses at each step. We are given the profit percentages for the first two sales, a loss percentage for the third sale, and a key relationship between the profits of the first two sellers. Our goal is to find the final price paid by the last buyer, D.

• A sells to B at a 20% profit.
• B sells to C at a 30% profit.
• C sells to D at a 10% loss.
• The profit earned by B is Rs. 80 more than the profit earned by A.

Setting up the Equations

Let's denote the cost price for each person. Let \(C_A\) be the cost price for A.

• A's Selling Price (\(S_A\)) is the cost price for B (\(C_B\)).
• B's Selling Price (\(S_B\)) is the cost price for C (\(C_C\)).
• C's Selling Price (\(S_C\)) is the cost price for D (\(C_D\)).

Now, let's express the selling prices and profits/losses based on the percentages:

• A's Profit = 20% of \(C_A\) = \(0.20 \times C_A\).
• \(S_A = C_B = C_A + 0.20 \times C_A = 1.20 \times C_A\).
• B's Profit = 30% of \(C_B\) = \(0.30 \times C_B\).
• \(S_B = C_C = C_B + 0.30 \times C_B = 1.30 \times C_B\).
• C's Loss = 10% of \(C_C\) = \(0.10 \times C_C\).
• \(S_C = C_D = C_C - 0.10 \times C_C = 0.90 \times C_C\).

Using the Profit Difference Information

We are told that B's profit is Rs. 80 more than A's profit.

\(\text{B's Profit} = \text{A's Profit} + 80\)

\(0.30 \times C_B = 0.20 \times C_A + 80\)

We know that \(C_B = 1.20 \times C_A\). Substitute this into the equation:

\(0.30 \times (1.20 \times C_A) = 0.20 \times C_A + 80\)

\(0.36 \times C_A = 0.20 \times C_A + 80\)

Solving for the Initial Cost Price (C_A)

Now, we solve the equation for \(C_A\):

\(0.36 \times C_A - 0.20 \times C_A = 80\)

\(0.16 \times C_A = 80\)

\(C_A = \frac{80}{0.16} = \frac{8000}{16}\)

\(C_A = 500\)

So, the cost price for A was Rs. 500.

Calculating the Prices at Each Step

Now that we know \(C_A\), we can find the price at each transaction stage.

Step 1: A sells to B

• \(C_A = 500\)
• A's Profit = 20% of 500 = \(0.20 \times 500 = 100\)
• \(S_A = C_B = C_A + \text{A's Profit} = 500 + 100 = 600\)

B bought the watch for Rs. 600.

Step 2: B sells to C

• \(C_B = 600\)
• B's Profit = 30% of 600 = \(0.30 \times 600 = 180\)
• \(S_B = C_C = C_B + \text{B's Profit} = 600 + 180 = 780\)

C bought the watch for Rs. 780.

Let's quickly check the profit difference condition: B's Profit (180) - A's Profit (100) = 80. This matches the problem statement.

Step 3: C sells to D

• \(C_C = 780\)
• C's Loss = 10% of 780 = \(0.10 \times 780 = 78\)
• \(S_C = C_D = C_C - \text{C's Loss} = 780 - 78 = 702\)

D bought the watch for Rs. 702.

Summary of Transactions

Final Answer

Based on the calculations, D bought the watch for Rs. 702.

Revision Table: Profit and Loss Concepts

Additional Information: Solving Multi-step Percentage Problems

When dealing with a series of transactions involving percentages, it's crucial to remember that the percentage change at each step is calculated based on the cost price for that specific step (which is the selling price of the previous step). Do not calculate percentages based on the initial cost price (\(C_A\)) for subsequent steps unless explicitly stated.

Here are some tips for solving such problems:

• Clearly define variables for the cost/selling price at each stage.
• Express each transaction's outcome (selling price) in terms of the cost price for that step using percentage formulas.
• Use any given relationships (like the difference in profits in this problem) to form an equation.
• Substitute variables to get an equation with a single unknown, preferably the initial cost price, as it often makes calculations easier.
• Solve the equation to find the unknown value.
• Work through the transactions step-by-step using the calculated values to find the final required price.
• Always check if the values calculated satisfy all the conditions given in the problem.