To solve this problem, let's define the following:

• Let the total number of candidates be 100% of the population.
• Let $P(E)$ represent the percentage of candidates who passed English, which is 60%.
• Let $P(M)$ represent the percentage of candidates who passed Mathematics, which is 65%.
• Let $P(E \cap M)$ represent the percentage of candidates who passed both English and Mathematics, which is 50%.

We are asked to find the percentage of candidates who failed in both subjects.

Step 1: Apply the Principle of Inclusion and Exclusion

To find the total percentage of candidates who passed at least one subject, we use the inclusion-exclusion principle:

$P(E \cup M) = P(E) + P(M) - P(E \cap M)$

Substituting the given values:

$P(E \cup M) = 60\% + 65\% - 50\% = 75\%$

This means 75% of candidates passed at least one subject (either English or Mathematics or both).

Step 2: Find the Percentage Who Failed in Both

The total percentage of candidates is 100%. So, the percentage of candidates who failed in both subjects is:

$\text{Failed in both} = 100\% - P(E \cup M) = 100\% - 75\% = 25\%$

Thus, 25% of candidates failed in both English and Mathematics.

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Would you like further details or explanations? Here are some related questions for further exploration:

1. How can the inclusion-exclusion principle be applied in other contexts (like set theory)?
2. How would this problem change if the percentage of candidates passing both subjects was higher or lower?
3. What does the overlap of passing candidates tell us about the relationship between the two subjects?
4. Can this problem be solved using a Venn diagram? If so, how would you draw it?
5. How can you calculate the percentage of candidates who passed only English or only Mathematics?

Tip: When dealing with percentages in overlapping sets, always remember to subtract the intersection to avoid double-counting those who passed both subjects.