In an examination 70% of the candidates passed in English, 80% passed in Mathematics. 10% failed in both the subjects if 144 candidates passed in both, the total number of candidates was :

एक परीक्षा में 70% प्रतियोगी अंग्रेजी में, 80% प्रतियोगी गणित में सफल होते है तथा 10% प्रतियोगी दोनों विषयों में असफल होते हैं। यदि 144 प्रतियोगी दोनों विषयों में सफल होते हैं, तो प्रतियोगियों की कुल संख्या क्या थी ?

Options

A. 125
B. 200
C. 240
D. 375

Explanation

To solve the problem, we need to find the total number of candidates based on the given percentages and the number of candidates who passed in both subjects.

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

### Step 2: Calculate the percentages of candidates passing and failing
- Candidates passing in English = 70% of \( N \)
- Candidates passing in Mathematics = 80% of \( N \)
- Candidates failing in both subjects = 10% of \( N \)

### Step 3: Calculate the percentage of candidates passing in at least one subject
If 10% failed in both subjects, then 90% passed in at least one subject.

### Step 4: Use the inclusion-exclusion principle
Let:
- \( E \) = number of candidates passing in English = \( 0.7N \)
- \( M \) = number of candidates passing in Mathematics = \( 0.8N \)
- \( B \) = number of candidates passing in both subjects = 144

According to the inclusion-exclusion principle:
\[ E + M - B = \text{Candidates passing in at least one subject} \]
Substituting the known values:
\[ 0.7N + 0.8N - 144 = 0.9N \]

### Step 5: Simplify the equation
Combine like terms:
\[ 1.5N - 144 = 0.9N \]

### Step 6: Rearrange the equation
Move \( 0.9N \) to the left side:
\[ 1.5N - 0.9N = 144 \]
\[ 0.6N = 144 \]

### Step 7: Solve for \( N \)
Divide both sides by 0.6:
\[ N = \frac{144}{0.6} \]
\[ N = 240 \]

### Conclusion
The total number of candidates is \( \boxed{240} \).

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