. A retailer bought 3850 linc pens and 1848 cello pens at the same price. He sells linc pens in such a way that he can buy 650 linc pens with the selling price of 481 linc pens but he can only buy 408 cello pens with the selling price of 629 cello pens. What is his overall profit percentage?

Answer

To solve the problem, we need to determine the overall profit percentage from the transactions involving linc pens and cello pens. Let's break this down step by step.

### Step 1: Understand the Cost Price (CP) and Selling Price (SP) relationship for Linc Pens

The retailer can buy 650 linc pens with the selling price of 481 linc pens. This gives us the following relationship:

- Let the cost price of one linc pen be \( CP_L \).
- The selling price of 481 linc pens is \( SP_L = 481 \times CP_L \).
- The selling price of 650 linc pens is \( SP_L = 650 \times SP_L \).

From this, we can set up the equation:

\[ 481 \times CP_L = 650 \times SP_L \]

### Step 2: Calculate the Profit or Loss for Linc Pens

From the equation derived in Step 1, we can find the ratio of the cost price to the selling price:

\[ \frac{CP_L}{SP_L} = \frac{481}{650} \]

This implies that for every 481 linc pens sold, the cost price is equivalent to 481, and the selling price is equivalent to 650.

### Step 3: Understand the Cost Price (CP) and Selling Price (SP) relationship for Cello Pens

Similarly, for cello pens, the retailer can buy 408 cello pens with the selling price of 629 cello pens. This gives us the following relationship:

- Let the cost price of one cello pen be \( CP_C \).
- The selling price of 629 cello pens is \( SP_C = 629 \times CP_C \).
- The selling price of 408 cello pens is \( SP_C = 408 \times SP_C \).

From this, we can set up the equation:

\[ 629 \times CP_C = 408 \times SP_C \]

### Step 4: Calculate the Profit or Loss for Cello Pens

From the equation derived in Step 3, we can find the ratio of the cost price to the selling price:

\[ \frac{CP_C}{SP_C} = \frac{629}{408} \]

### Step 5: Combine the Information

Since the cost price of both types of pens is the same, we can denote it as \( CP \).

Let’s denote the total cost price for linc pens and cello pens:

- Total CP for linc pens = \( 3850 \times CP_L \)
- Total CP for cello pens = \( 1848 \times CP_C \)

### Step 6: Calculate Total Selling Price

Now, we need to calculate the total selling price for both types of pens.

- Total SP for linc pens = \( 650 \times \text{(number of linc pens sold)} \)
- Total SP for cello pens = \( 408 \times \text{(number of cello pens sold)} \)

### Step 7: Calculate Overall Profit Percentage

To find the overall profit percentage, we will use the formula:

\[ \text{Profit Percentage} = \left( \frac{\text{Total SP} - \text{Total CP}}{\text{Total CP}} \right) \times 100 \]

### Final Calculation

1. Calculate the total cost price and total selling price.
2. Substitute these values into the profit percentage formula.