Question

In an MBA entrance exam, 55% failed in QA, & 45% failed in VA, and 25% pass in both QA & VA. Find the % of students who failed in both the subjects.

• A. 15%
• B. 20%
• C. 25%
• D. 30%

Hint:

Drawing a Venn diagram can make set theory problems much clearer. The universe is 100%. The area outside the "failed" circles represents those who passed both.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We are given the percentage of students who failed in two subjects individually (QA and VA) and the percentage who passed in both. We need to find the percentage who failed in both subjects. This is a set theory problem.
Step 2: Key Formula or Approach:
Let F(QA) be the percentage of students who failed in QA.
Let F(VA) be the percentage of students who failed in VA.
The formula for the union of two sets is: \[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \] Here, 'A \(\cup\) B' represents students who failed in at least one subject, and 'A \(\cap\) B' represents students who failed in both.
Step 3: Detailed Explanation:
We are given:
F(QA) = 55%
F(VA) = 45%
Percentage who passed in both subjects = 25%
The opposite of "passing in both" is "failing in at least one".
So, the percentage of students who failed in at least one subject (F(QA \(\cup\) VA)) is:
\[ F(QA \cup VA) = 100% - (\text{Percentage who passed in both}) \] \[ F(QA \cup VA) = 100% - 25% = 75% \] Now, we use the set theory formula:
\[ F(QA \cup VA) = F(QA) + F(VA) - F(QA \cap VA) \] We need to find F(QA \(\cap\) VA), which is the percentage of students who failed in both.
Substitute the known values:
\[ 75% = 55% + 45% - F(QA \cap VA) \] \[ 75% = 100% - F(QA \cap VA) \] Rearranging the equation to solve for F(QA \(\cap\) VA):
\[ F(QA \cap VA) = 100% - 75% = 25% \] Step 4: Final Answer:
The percentage of students who failed in both subjects is 25%.