In an examination `52%` of the candidates failed in Science,`42%` in Mathematics and `17%` in both . The no of those who passed in both the subjects, is :

Options

A. `83%`
B. `64%`
C. `23%`
D. `55.55%`

Answer

To solve the problem step by step, we can use the principle of inclusion-exclusion to find the number of candidates who passed in both subjects.

### Step 1: Understand the percentages of failures
- **Percentage of candidates who failed in Science**: 52%
- **Percentage of candidates who failed in Mathematics**: 42%
- **Percentage of candidates who failed in both subjects**: 17%

### Step 2: Calculate the percentages of candidates who passed
- **Percentage of candidates who passed in Science**:
\[ 100\% - 52\% = 48\% \]

- **Percentage of candidates who passed in Mathematics**:
\[ 100\% - 42\% = 58\% \]

### Step 3: Use the principle of inclusion-exclusion
To find the percentage of candidates who passed in both subjects, we can use the formula:
\[ \text{Passed in Science or Mathematics} = \text{Passed in Science} + \text{Passed in Mathematics} - \text{Passed in both} \]

Let \( P(S) \) be the percentage that passed in Science, \( P(M) \) be the percentage that passed in Mathematics, and \( P(B) \) be the percentage that passed in both subjects.

From the previous calculations:
- \( P(S) = 48\% \)
- \( P(M) = 58\% \)
- The percentage of candidates who failed in both subjects is \( 17\% \), so the percentage that passed in both subjects is:
\[ P(B) = 100\% - (P(S) + P(M) - \text{Failed in both}) \]
Substituting the values:
\[ P(B) = 100\% - (48\% + 58\% - 17\%) \]

### Step 4: Calculate the percentage of candidates who passed in both subjects
\[ P(B) = 100\% - (48\% + 58\% - 17\%) = 100\% - 89\% = 11\% \]

### Step 5: Conclusion
Thus, the percentage of candidates who passed in both subjects is \( 11\% \).