Understanding the Vendor's Profit Calculation

This problem involves calculating the number of items (pens) a vendor needs to buy and sell to achieve a specific profit target. We are given the purchase details (cost price) and selling details (selling price) in batches, and the desired total profit.

Calculating Cost Price (CP) and Selling Price (SP) per Pen

First, let's determine the cost price and selling price for a single pen.

• Cost Price Calculation: The vendor buys 20 pens for ₹15. To find the cost price of one pen, we divide the total cost by the number of pens:

  CP per pen = ₹15 / 20

  CP per pen = ₹$ \frac{15}{20} = ₹ \frac{3}{4} $ or ₹0.75

• Selling Price Calculation: The vendor sells 15 pens for ₹20. To find the selling price of one pen, we divide the total selling amount by the number of pens sold:

  SP per pen = ₹20 / 15

  SP per pen = ₹$ \frac{20}{15} = ₹ \frac{4}{3} $

Determining Profit per Pen

Profit is the difference between the selling price and the cost price. Let's calculate the profit earned on each pen sold.

Profit per pen = SP per pen - CP per pen

Profit per pen = ₹$ \frac{4}{3} - ₹ \frac{3}{4} $

To subtract these fractions, we find a common denominator, which is 12:

Profit per pen = ₹$ \left( \frac{4 \times 4}{3 \times 4} \right) - \left( \frac{3 \times 3}{4 \times 3} \right) $

Profit per pen = ₹$ \frac{16}{12} - \frac{9}{12} $

Profit per pen = ₹$ \frac{16 - 9}{12} = ₹ \frac{7}{12} $

So, the vendor earns a profit of ₹$ \frac{7}{12} $ on each pen.

Calculating the Number of Pens for Target Profit

The vendor wants to earn a total profit of ₹245. We know the profit made per pen. To find the total number of pens needed, we divide the total desired profit by the profit per pen.

Number of pens = Total Desired Profit / Profit per pen

Number of pens = ₹245 / ₹$ \frac{7}{12} $

To divide by a fraction, we multiply by its reciprocal:

Number of pens = $ 245 \times \frac{12}{7} $

First, divide 245 by 7:

$ \frac{245}{7} = 35 $

Now, multiply the result by 12:

Number of pens = $ 35 \times 12 $

Number of pens = 420

Conclusion

Therefore, the vendor needs to buy and sell 420 pens to earn a profit of ₹245.