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 130.20Conclusion
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 130.20By using algebraic techniques to solve the equations, we can accurately determine the cost prices of the tables, ensuring consistency with the given profit conditions.