Profit and Loss - Q1 - Linc and Cello Pens

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

A.

Let Selling Price of 1 Linc Pen = $S_L$
Let Cost Price of 1 Linc Pen = $C_L$
Let Selling Price of 1 Cello Pen = $S_C$
Let Cost Price of 1 Cello Pen = $C_C$

From "he can buy 650 Linc pens with the sale of 481 Linc pens",
$$650C_L=481S_L$$
$$\to \frac{S_L}{C_L}=\frac{650}{481}=\frac{13\times 50}{13\times 37}=\frac{50}{37}$$
Hence, Profit Percentage on 1 Linc Pen = $$\%P_L=\frac{S_L-C_L}{C_L}=\frac{50-37}{37}=\frac{13}{37}$$

From "he can buy only 408 Cello pens with the sale of 629 Cello pens",
$$408C_L=629S_L$$
$$\to \frac{S_C}{C_C}=\frac{408}{629}=\frac{17\times 24}{17\times 37}=\frac{24}{37}$$
Hence, Profit Percentage on 1 Cello Pen =
$$\%P_C=\frac{S_C-C_C}{C_C}=\frac{24-37}{37}=-\frac{13}{37}$$

Cost of 3850 Linc Pens = $3850C_L$
Cost of 1848 Cello Pens = $1848C_C$

From "A retailer bought 3850 Linc pens and 1848 cello pens at the same price",
$$3850C_L=1848C_C$$

Overall Profit Percentage is the cost-weighted average of the individual Profit Percentages. Hence,
$$\%P=\frac{\left[3850C_L\times \frac{13}{37}\right]+\left[1848C_C\times \left(-\frac{13}{37}\right)\right]}{3850C_L+1848C_C}$$
Since, $3850C_L=1848C_C$
$$\%P=\frac{\left[1848C_C\times \frac{13}{37}\right]+\left[1848C_C\times \left(-\frac{13}{37}\right)\right]}{1848C_C+1848C_C}=0$$

Hence, overall the retailer makes No Profit, No Loss.