In an examination 70% of the candidates passed in English, 65% in Mathematics, 27 % failed in both the subjects and 248 candidates passed in both subjects. Find the total number of candidates.

Options

A. a) 200
B. b) 400
C. c) 300
D. d) 100

Answer

To solve the problem step by step, we will use the information provided about the candidates who passed and failed in the subjects.

### Step 1: Define Variables
Let the total number of candidates be \( x \).

### Step 2: Calculate the Percentage of Candidates Passing and Failing
- Candidates passing in English = 70% of \( x \) = \( 0.7x \)
- Candidates passing in Mathematics = 65% of \( x \) = \( 0.65x \)
- Candidates failing in both subjects = 27% of \( x \) = \( 0.27x \)

### Step 3: Calculate the Candidates Passing in At Least One Subject
Using the principle of inclusion-exclusion, we can find the percentage of candidates passing in at least one subject:
- Let \( P(E) \) be the percentage of candidates passing in English, and \( P(M) \) be the percentage of candidates passing in Mathematics.
- The formula for candidates passing in at least one subject is:
\[ P(E \cup M) = P(E) + P(M) - P(E \cap M) \]
Where \( P(E \cap M) \) is the percentage of candidates passing both subjects.

### Step 4: Set Up the Equation
From the problem, we know:
- \( P(E) = 70\% = 0.7 \)
- \( P(M) = 65\% = 0.65 \)
- Candidates passing both subjects = 248

Thus, we can express the candidates passing both subjects as:
\[ P(E \cap M) = \frac{248}{x} \]

Substituting into the inclusion-exclusion formula:
\[ P(E \cup M) = 0.7 + 0.65 - \frac{248}{x} \]

### Step 5: Calculate the Percentage Failing in At Least One Subject
The percentage of candidates failing in at least one subject is given by:
\[ P(\text{Fail}) = 1 - P(E \cup M) \]
Given that 27% of candidates failed in both subjects, we can express this as:
\[ P(\text{Fail}) = \frac{0.27x}{x} = 0.27 \]

### Step 6: Set Up the Equation for Failing Candidates
Setting the two expressions for failing candidates equal:
\[ 1 - \left(0.7 + 0.65 - \frac{248}{x}\right) = 0.27 \]
This simplifies to:
\[ 1 - 1.35 + \frac{248}{x} = 0.27 \]
\[ \frac{248}{x} - 0.35 = 0.27 \]
\[ \frac{248}{x} = 0.27 + 0.35 \]
\[ \frac{248}{x} = 0.62 \]

### Step 7: Solve for \( x \)
Now, we can solve for \( x \):
\[ 248 = 0.62x \]
\[ x = \frac{248}{0.62} \]
Calculating this gives:
\[ x = 400 \]

### Conclusion
The total number of candidates is **400**.