A shopkeeper has a certain number of eggs of which 5% are found to be broken. He sells 93% of the remainder and still has 266 eggs left. How many eggs did he originally have?

### Concept & Logic This is a successive percentage problem. When a portion is removed and an operation is performed on the "remainder," you multiply the remaining fractions sequentially to equate them to the final value. $$ \text{Final Value} = \text{Initial Value} \times (1 - \text{Loss}_1) \times (1 - \text{Loss}_2) $$ ### Step-by-Step Solution * Let the original number of eggs be $x$. * Since $5\%$ are broken, the remaining good eggs are $(100\% - 5\%) = 95\%$ of $x$. * The shopkeeper sells $93\%$ of this remainder. This means he is left with $(100\% - 93\%) = 7\%$ of the remainder. * We can set up the equation based on the eggs left: $$0.07 \times 0.95 \times x = 266$$ * Convert decimals to fractions for easier calculation: $$\frac{7}{100} \times \frac{95}{100} \times x = 266$$ * Simplify and solve for $x$: $$x = \frac{266 \times 10000}{7 \times 95}$$ * $$x = \frac{38 \times 10000}{95}$$ * $$x = \frac{380000}{95} = 4000$$ ### Exam Strategy & Shortcut **Fraction Chain:** Think purely in terms of fractions left over. $5\%$ broken means $\frac{19}{20}$ left. $93\%$ sold means $7\%$ or $\frac{7}{100}$ left. $\text{Total remaining} = \frac{19}{20} \times \frac{7}{100} = \frac{133}{2000}$ of the original amount. Equate this to $266$: $$\frac{133}{2000} \times x = 266$$ Notice that $133 \times 2 = 266$. So, $x = 2000 \times 2 = 4000$. ### Common Pitfall A very common mistake is adding the percentages directly (e.g., $5\% + 93\% = 98\%$) and assuming only $2\%$ of the eggs are left. This ignores the crucial phrase "of the remainder." Always apply sequential percentages multiplicatively. ### Final Answer **Therefore, the correct answer is 4000.**