A man sells a book and a table at 13% and 9% profit respectively and earns Rs 1060 as profit. But if he sells the book at 16 `2/3`% P and table at 11 `1/9` % loss than bears no profit no loss Find their CP?

Options

A. Rs 7000 Rs 9000
B. Rs 8000 Rs 3000
C. Rs 4000 Rs 6000
D. Rs 3000 Rs 6000

Answer

To solve the problem step by step, we will first define the variables and then set up equations based on the information provided in the question.

### Step 1: Define Variables
Let:
- CP of the book = \( x \)
- CP of the table = \( y \)

### Step 2: Set Up Equations for Profit
From the problem, we know:
- The man sells the book at a profit of 13%, hence the selling price (SP) of the book is:
\[ SP_{\text{book}} = x + 0.13x = 1.13x \]
- The man sells the table at a profit of 9%, hence the selling price (SP) of the table is:
\[ SP_{\text{table}} = y + 0.09y = 1.09y \]

### Step 3: Total Profit Equation
The total profit earned from both sales is Rs. 1060. Therefore, we can write:
\[ (1.13x - x) + (1.09y - y) = 1060 \]
This simplifies to:
\[ 0.13x + 0.09y = 1060 \]

### Step 4: Set Up Equations for No Profit No Loss
When the book is sold at \( 16 \frac{2}{3} \% \) profit (which is \( \frac{50}{3} \% \)) and the table at \( 11 \frac{1}{9} \% \) loss (which is \( -\frac{100}{9} \% \)), the selling prices become:
- Selling price of the book:
\[ SP_{\text{book}} = x + \frac{50}{3} \cdot \frac{x}{100} = x \left(1 + \frac{50}{300}\right) = x \left(1 + \frac{1}{6}\right) = \frac{7}{6}x \]
- Selling price of the table:
\[ SP_{\text{table}} = y - \frac{100}{9} \cdot \frac{y}{100} = y \left(1 - \frac{100}{900}\right) = y \left(1 - \frac{1}{9}\right) = \frac{8}{9}y \]

Since he bears no profit or loss:
\[ \frac{7}{6}x + \frac{8}{9}y = x + y \]

### Step 5: Rearranging the No Profit No Loss Equation
Rearranging gives:
\[ \frac{7}{6}x + \frac{8}{9}y - x - y = 0 \]
This simplifies to:
\[ \left(\frac{7}{6} - 1\right)x + \left(\frac{8}{9} - 1\right)y = 0 \]
\[ \left(\frac{1}{6}\right)x + \left(-\frac{1}{9}\right)y = 0 \]
Multiplying through by 18 to eliminate the fractions:
\[ 3x - 2y = 0 \quad \Rightarrow \quad 3x = 2y \quad \Rightarrow \quad y = \frac{3}{2}x \]

### Step 6: Substitute \( y \) in Profit Equation
Now substitute \( y = \frac{3}{2}x \) into the profit equation:
\[ 0.13x + 0.09\left(\frac{3}{2}x\right) = 1060 \]
This simplifies to:
\[ 0.13x + 0.135x = 1060 \]
\[ 0.265x = 1060 \]
\[ x = \frac{1060}{0.265} = 4000 \]

### Step 7: Find \( y \)
Now substitute \( x \) back to find \( y \):
\[ y = \frac{3}{2} \cdot 4000 = 6000 \]

### Final Answer
Thus, the cost price of the book is Rs. 4000 and the cost price of the table is Rs. 6000.

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