A crate of eggs contains one rotten egg out of every 25 eggs, and of all the rotten eggs, 5 out of 8 are unusable. If there are exactly 10 unusable eggs in the crate, how many eggs are there in total in the crate?

Introduction / Context:
This quantitative aptitude question combines ratios and fractions in a real life context. It models a crate of eggs with a certain proportion rotten and, among those, a proportion unusable. Understanding how to move from nested fractions to total quantities is very useful in percentage and ratio based word problems.

Given Data / Assumptions:

• In the crate, 1 out of every 25 eggs is rotten.
• Of all rotten eggs, 5 out of 8 are unusable.
• There are 10 unusable eggs in total.
• We must find the total number of eggs in the crate.

Concept / Approach:
Let the total number of eggs be N. Then the number of rotten eggs is N/25. Of those rotten eggs, 5/8 are unusable, so the number of unusable eggs is (5/8) * (N/25). This quantity is given to be 10. We set up an equation and solve for N. The method is essentially proportion and fraction manipulation.

Step-by-Step Solution:
Let the total number of eggs be N. Rotten eggs are 1 out of every 25, so the number of rotten eggs is N / 25. Of these rotten eggs, 5 out of 8 are unusable, so unusable eggs = (5/8) * (N / 25). We are told there are exactly 10 unusable eggs. Therefore, set up the equation (5/8) * (N / 25) = 10. Simplify the left side: (5N) / (8 * 25) = (5N) / 200. So (5N) / 200 = 10. Multiply both sides by 200 to clear the denominator: 5N = 10 * 200 = 2000. Divide by 5: N = 2000 / 5 = 400.

Verification / Alternative check:
Check the result by reconstructing the situation. With 400 eggs, rotten eggs are 400 / 25 = 16. Among these, unusable eggs are (5/8) of 16, which is 16 * 5 / 8 = 10. This matches the given information exactly, so N = 400 is confirmed as correct.

Why Other Options Are Wrong:
If N = 380, rotten eggs would be 380 / 25 = 15.2, not an integer, which is inconsistent with the scenario. Similarly, N = 420 or 440 would not produce 10 unusable eggs when applying the same fractions. The value 360 gives 360 / 25 = 14.4 rotten eggs, again not a whole number. Only N = 400 results in exactly 16 rotten eggs and precisely 10 unusable eggs.

Common Pitfalls:
Learners sometimes misinterpret the phrase one rotten egg out of every 25 eggs as 1/25 of the total being unusable directly, ignoring the second filter of 5 out of 8. Others invert the fractions or forget to apply both ratios sequentially. Writing the process symbolically as a product of fractions applied to N helps avoid such confusions.

Final Answer:
The crate contains a total of 400 eggs.