A man bought a number of clips at 3 for a rupee and an equal number at 2 for a rupee. At what price per dozen should he sell them to make a profit of 20%?

### Concept & Formula When equal quantities of items are bought at two different rates, the average Cost Price (CP) must be calculated based on the total cost divided by the total number of items, not just the simple average of the rates. $$SP = CP \times \left(1 + \frac{\text{Profit } \%}{100}\right)$$ ### Step-by-Step Solution * **Given:** CP1 = 3 for ₹ 1; CP2 = 2 for ₹ 1. Target Profit = 20%. Quantity is equal. * **Calculation:** 1. Let the number of clips bought of each kind be the LCM of 3 and 2, which is 6. 2. CP of first 6 clips = $\frac{1}{3} \times 6 = \text{₹ } 2$ 3. CP of second 6 clips = $\frac{1}{2} \times 6 = \text{₹ } 3$ 4. Total CP for 12 clips (1 dozen) = $2 + 3 = \text{₹ } 5$ 5. To make a 20% profit, SP = $CP \times 1.2$ 6. SP of 1 dozen = $5 \times 1.2 = \text{₹ } 6$ ### Exam Strategy & Shortcut Average CP per unit = $\frac{\frac{1}{3} + \frac{1}{2}}{2} = \frac{5/6}{2} = \frac{5}{12}$. So, CP for 1 dozen (12 units) = $12 \times \frac{5}{12} = 5$. Adding 20% profit to 5 gives $5 + 1 = 6$. ### Common Pitfall Taking a simple average of the rates ($\frac{3+2}{2} = 2.5$) instead of equating the quantities first. The costs must be weighted equally by item count, not by transaction batch. ### Final Answer Therefore, the correct answer is **₹ 6**.