Solving the Shopkeeper's Profit Puzzle: A Detailed Mathematical Analysis

The problem presented involves a shopkeeper who buys two tables for Rs 800 in total. The tables are sold with different profit margins, leading to a different profit recognition based on the context. We will provide a detailed solution to determine the cost prices of the two tables by following a series of algebraic equations and calculations.

Problem Setup

Let's denote the cost price (CP) of the first table as x and the cost price of the second table as y. The following information is provided:

The total cost for the two tables is Rs 800. The first table is sold at a 10% profit. The second table is sold at a 20% profit. When the profit is set to 20% for the first table and 15% for the second table, the total profit amounts to Rs 6.50.

Step-by-Step Solution

We can set up the following equations based on the given information:

Equation 1: Total Cost of the Tables

x y 800

Equation 2: Original Total Selling Price

The selling price for the first table at a 10% profit is:

SP1 x * 1.10

The selling price for the second table at a 20% profit is:

SP2 y * 1.20

The total selling price with the original profits is:

Total SPOriginal SP1 SP2 1.1x 1.2y

Equation 3: Adjusted Total Selling Price

The adjusted selling price for the first table at a 20% profit is:

SP1 x * 1.20

The adjusted selling price for the second table at a 15% profit is:

SP2 y * 1.15

The total selling price with the adjusted profits is:

Total SPAdjusted SP1 SP2 1.2x 1.15y

The difference between the total selling prices with the original and adjusted profits is Rs 6.50:

Total SPAdjusted - Total SPOriginal 6.50

1.2x 1.15y - (1.1x 1.2y) 6.50

1.2x 1.15y - 1.1x - 1.2y 6.50

0.1x - 0.05y 6.50

Which simplifies to:

1 - 5y -650

Equation 4: Simplified Equation

From Equation 1, we express y in terms of x as:

y 800 - x

Substituting y into the simplified equation:

1 - 5(800 - x) -650

1 - 4000 5x -650

5x - 3999 -650

5x 3349

x 669.80

Substituting x back into the equation for y from Equation 1:

y 800 - 669.80 130.20

This gives us the cost prices of the two tables:

First table cost price (CP): Rs 669.80 Second table cost price (CP): Rs 130.20

Conclusion

The detailed algebraic analysis confirms the cost prices of the two tables:

Cost Price of the first table: Rs 669.80 Cost Price of the second table: Rs 130.20

By using algebraic techniques to solve the equations, we can accurately determine the cost prices of the tables, ensuring consistency with the given profit conditions.