At a university entrance exam, 40 percent of the candidates are actually capable and 60 percent are incapable. Among the capable candidates, 80 percent pass the test, and among the incapable candidates, 30 percent pass the test. What is the ratio of capable students who pass the test to the total number of students who pass?

Introduction / Context:
This problem is a probability and ratio question framed in an examination setting. We are given information about how many candidates are genuinely capable versus incapable and the probabilities of passing for both groups. The objective is to find the fraction of passing candidates who actually belong to the capable group. This is conceptually similar to a Bayes theorem style reasoning, although we can handle it with direct percentage arithmetic and careful counting.

Given Data / Assumptions:
- 40 percent of candidates are capable.
- 60 percent of candidates are incapable.
- 80 percent of capable candidates pass the test.
- 30 percent of incapable candidates pass the test.
- We are interested in the ratio of capable candidates who pass to the total number of passing candidates.

Concept / Approach:
Assume a convenient total number of candidates, such as 100, to convert percentages into actual counts. Then compute how many of these 100 are capable and how many are incapable. Next, find how many pass in each group using the given pass percentages. Adding the passes from both groups gives the total number of students who pass the test. Finally, the ratio asked for is the number of capable students who pass divided by the total number of students who pass, which can be simplified to lowest terms.

Step-by-Step Solution:
Step 1: Assume total candidates = 100 for easy calculation. Step 2: Capable candidates = 40 percent of 100 = 40. Step 3: Incapable candidates = 60 percent of 100 = 60. Step 4: Passing capable candidates = 80 percent of 40 = 0.8 * 40 = 32. Step 5: Passing incapable candidates = 30 percent of 60 = 0.3 * 60 = 18. Step 6: Total passing candidates = 32 + 18 = 50. Step 7: Ratio of capable passing candidates to total passing candidates = 32 / 50. Step 8: Simplify 32 / 50 by dividing numerator and denominator by 2 to get 16 / 25.

Verification / Alternative check:
We can compute the same ratio using decimal probabilities. The proportion of capable passers is 0.4 * 0.8 = 0.32, and the proportion of incapable passers is 0.6 * 0.3 = 0.18. Total pass proportion is 0.32 + 0.18 = 0.50. The required ratio is 0.32 / 0.50 = 0.64, which is identical to 16 / 25 when written as a fraction. This confirms the consistency of our earlier calculation with percentage based reasoning.

Why Other Options Are Wrong:
11/24 is approximately 0.458 which is much smaller than 0.64 and does not match the computed ratio. 1/5 equals 0.2 and is far too small. 6/5 equals 1.2, which is impossible because it implies more capable passers than total passers. Only 16/25 matches the correct fraction 32 divided by 50 obtained from the step by step calculations.

Common Pitfalls:
A common mistake is to average the pass percentages directly or to confuse the proportion of capable candidates with the proportion of passing candidates. Some students mistakenly treat 80 percent and 30 percent as if they were applied to the same base group. Another pitfall is failing to choose a convenient base total, like 100, which makes calculations much easier and clearer.

Final Answer:
The proportion of capable students among all students who pass the test is 16/25.