Solving the Profit and Loss Problem for Table and Chair Prices

Let's break down this word problem involving the purchase and sale of a table and a chair at different profit percentages. We are given the total cost price, the individual profit percentages, and the total profit earned. Our goal is to find the original cost price of each item and then calculate the difference between them.

Defining Variables

We can represent the unknown original prices using variables:

• Let $\text{CP}_T$ be the cost price of the table in Rupees.
• Let $\text{CP}_C$ be the cost price of the chair in Rupees.

Setting Up Equations from the Problem Statement

Based on the information given, we can form two linear equations:

1. The total purchase price of the table and chair is Rs. 3,900.
   Equation 1: $\text{CP}_T + \text{CP}_C = 3900$
2. Ramesh sells the table at an 8% profit and the chair at a 16% profit, earning a total profit of Rs. 540.
   • Profit on the table is 8% of $\text{CP}_T$, which is $0.08 \times \text{CP}_T$.
   • Profit on the chair is 16% of $\text{CP}_C$, which is $0.16 \times \text{CP}_C$.
   The total profit is the sum of the profit on the table and the profit on the chair.
   Equation 2: $(0.08 \times \text{CP}_T) + (0.16 \times \text{CP}_C) = 540$

Solving the System of Equations

We now have a system of two linear equations:

1) $\text{CP}_T + \text{CP}_C = 3900$

2) $0.08 \times \text{CP}_T + 0.16 \times \text{CP}_C = 540$

Let's simplify Equation 2 by multiplying the entire equation by 100 to remove the decimals:

$100 \times (0.08 \times \text{CP}_T + 0.16 \times \text{CP}_C) = 100 \times 540$

$8 \times \text{CP}_T + 16 \times \text{CP}_C = 54000$

Now, let's divide this new equation by 8 to make the coefficients smaller:

$\frac{8 \times \text{CP}_T}{8} + \frac{16 \times \text{CP}_C}{8} = \frac{54000}{8}$

$\text{CP}_T + 2 \times \text{CP}_C = 6750$ (Equation 3)

We can use the elimination method to solve the system using Equation 1 and Equation 3:

Equation 1: $\text{CP}_T + \text{CP}_C = 3900$

Equation 3: $\text{CP}_T + 2 \times \text{CP}_C = 6750$

Subtract Equation 1 from Equation 3:

$(\text{CP}_T + 2 \times \text{CP}_C) - (\text{CP}_T + \text{CP}_C) = 6750 - 3900$

$\text{CP}_C = 2850$

Now that we have the cost price of the chair ($\text{CP}_C$), we can substitute this value back into Equation 1 to find the cost price of the table ($\text{CP}_T$):

$\text{CP}_T + \text{CP}_C = 3900$

$\text{CP}_T + 2850 = 3900$

$\text{CP}_T = 3900 - 2850$

$\text{CP}_T = 1050$

Calculating the Difference in Original Price

We have found the original price of the table is Rs. 1,050 and the original price of the chair is Rs. 2,850.

The question asks for the difference between the original price of the table and the chair.

Difference = $|\text{CP}_T - \text{CP}_C|$ or $|\text{CP}_C - \text{CP}_T|$

Difference = $|1050 - 2850| = |-1800| = 1800$

Or

Difference = $|2850 - 1050| = |1800| = 1800$

The difference between the original price of the table and the chair is Rs. 1,800.

Checking the Total Profit

Let's quickly verify if these cost prices yield the total profit of Rs. 540.

• Profit on Table = 8% of 1050 = $0.08 \times 1050 = 84$
• Profit on Chair = 16% of 2850 = $0.16 \times 2850 = 456$

Total Profit = Profit on Table + Profit on Chair = $84 + 456 = 540$

This matches the given total profit, confirming our calculated cost prices are correct.

Revision Table: Profit and Loss Concepts

Additional Information: Solving Simultaneous Equations

Problems like this often require solving two or more equations with multiple variables simultaneously. Common methods include:

• Substitution Method: Solve one equation for one variable and substitute that expression into the other equation.
• Elimination Method: Multiply one or both equations by constants so that the coefficients of one variable are opposites, then add the equations to eliminate that variable.

In this solution, we used a form of the elimination method by subtracting Equation 1 from Equation 3 after simplifying Equation 2.

Setting up the correct equations based on the problem statement is the crucial first step in solving such word problems.