Understanding the Examination Problem

This question involves calculating the total number of students who appeared for an examination based on the failure rates in two subjects, Mathematics and English, and the number of students who passed both.

We are given the following information:

• Percentage of students who failed in Mathematics: $20\%$
• Percentage of students who failed in English: $15\%$
• Percentage of students who failed in both Mathematics and English: $10\%$
• Number of students who passed in both subjects: $300$

Our goal is to find the total number of students who took the examination.

Calculating the Percentage of Students Failing in at Least One Subject

To find the percentage of students who failed in at least one subject (Mathematics or English or both), we use the principle of inclusion-exclusion for sets. Let M be the set of students who failed in Mathematics and E be the set of students who failed in English.

The formula is:

Percentage(Fail M ∪ Fail E) = Percentage(Fail M) + Percentage(Fail E) - Percentage(Fail M ∩ Fail E)

Substituting the given values:

Percentage(Fail M ∪ Fail E) = $20\% + 15\% - 10\%$

Percentage(Fail M ∪ Fail E) = $35\% - 10\%$

Percentage(Fail M ∪ Fail E) = $25\%$

This means $25\%$ of the students failed in at least one of the subjects.

Determining the Percentage of Students Passing in Both Subjects

The students who did not fail in any subject must have passed in both. Therefore, the percentage of students who passed in both subjects can be calculated by subtracting the percentage of students who failed in at least one subject from the total percentage ($100\%$).

Percentage(Pass Both) = $100\% - \text{Percentage(Fail M ∪ Fail E)}$

Percentage(Pass Both) = $100\% - 25\%$

Percentage(Pass Both) = $75\%$

So, $75\%$ of the total students appeared for the examination passed in both Mathematics and English.

Calculating the Total Number of Students

We know that $75\%$ of the total students is equal to $300$ students (those who passed in both).

Let $T$ be the total number of students who appeared in the examination.

We can set up the equation:

$75\%$ of $T = 300$

Converting the percentage to a decimal:

$0.75 \times T = 300$

Now, we solve for $T$:

$T = \frac{300}{0.75}$

To simplify the division, we can write $0.75$ as a fraction $\frac{3}{4}$:

$T = \frac{300}{3/4}$

$T = 300 \times \frac{4}{3}$

$T = \frac{1200}{3}$

$T = 400$

Conclusion

Therefore, the total number of students who appeared in the examination was $400$. This aligns with option 2.