A shopkeeper purchases 510 eggs at Rs 20 per dozen. On the way, 30 eggs are broken and cannot be sold. At what selling price per dozen must he sell the remaining eggs so that he earns an overall profit of 20%?

Introduction / Context:
This is a profit and loss problem involving damaged goods and selling price per dozen. The shopkeeper buys eggs in bulk, some eggs are broken, and he wants to maintain a target profit percentage on his entire purchase by adjusting the selling price of the remaining eggs.

Given Data / Assumptions:
- Number of eggs bought = 510. - Purchase rate = Rs 20 per dozen. - Number of broken eggs = 30, which are unsellable. - Desired overall profit = 20% on total cost. - Remaining eggs are sold at a uniform rate per dozen.

Concept / Approach:
First compute the total cost price based on dozens. Then subtract the broken eggs to find the number of eggs that can actually be sold. Next compute the total selling amount required to achieve a profit of 20%, and finally divide this required total by the number of dozens sold to get the selling price per dozen.

Step-by-Step Solution:
Step 1: Number of dozens bought = 510 / 12 = 42.5 dozen. Step 2: Total cost price C = 42.5 * 20 = Rs 850. Step 3: Number of broken eggs = 30, so sellable eggs = 510 - 30 = 480. Step 4: Sellable dozens = 480 / 12 = 40 dozen. Step 5: Desired profit = 20% of 850 = 0.20 * 850 = Rs 170. Step 6: Required total selling price S = C + profit = 850 + 170 = Rs 1020. Step 7: Selling price per dozen = 1020 / 40 = Rs 25.50.

Verification / Alternative check:
Check the effective cost per dozen sold. Effective cost per dozen = total cost / dozens actually sold = 850 / 40 = Rs 21.25 per dozen. Shopkeeper sells at Rs 25.50 per dozen. Profit per dozen = 25.50 - 21.25 = Rs 4.25. Profit percent = 4.25 / 21.25 * 100 ≈ 20%. This reaffirms that the chosen selling price achieves the desired profit.

Why Other Options Are Wrong:
Option Rs. 22.50 per dozen gives too little profit and would result in a profit percentage much lower than 20%. Options Rs. 26 and Rs. 26.50 give higher profits than required and correspond to profit percentages above 20%. Only Rs. 25.50 per dozen exactly satisfies the 20% target on total cost.

Common Pitfalls:
Learners sometimes compute average cost only on the eggs that remain and ignore the broken ones, which understates true cost and skews the profit percentage. Another mistake is to treat 510 as 51 dozen instead of 42.5 dozen. Always convert eggs to dozens carefully and include the cost of broken items in the overall cost price.

Final Answer:
The shopkeeper must sell the remaining eggs at Rs 25.50 per dozen to earn a 20% profit.