In an examination `65%` students pass in history and `55%` students pass in hindi. If `5%` student failed in both the subject, then what is the percentage of students who have passed in both the subjects ?

Options

A. `15`
B. `20`
C. `25`
D. `30`

Answer

To find the percentage of students who passed in both subjects, we can follow these steps:

### Step 1: Understand the given percentages
- Let the total number of students be represented as 100% for simplicity.
- According to the problem, 65% of students pass in History and 55% pass in Hindi.
- Additionally, 5% of students failed in both subjects.

### Step 2: Calculate the percentage of students who passed in at least one subject
Since 5% of students failed in both subjects, the percentage of students who passed in at least one subject is:
\[ 100\% - 5\% = 95\% \]

### Step 3: Use the formula for the union of two sets
Let \( x \) be the percentage of students who passed in both subjects. According to the principle of inclusion-exclusion, the formula for the union of two sets is:
\[ \text{Pass in History} + \text{Pass in Hindi} - \text{Pass in both} = \text{Pass in at least one} \]
Substituting the known values:
\[ 65\% + 55\% - x = 95\% \]

### Step 4: Solve for \( x \)
Now, we can rearrange the equation to solve for \( x \):
\[ 120\% - x = 95\% \]
\[ x = 120\% - 95\% \]
\[ x = 25\% \]

### Conclusion
Thus, the percentage of students who passed in both subjects is **25%**.

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