In an examination `70%` of the candidates passed in History and `50%` in Geography and `20%` students failed in both the subjects. If 500 students passed in both the subjects, then how many candidates appeared for the exam?

Options

A. 1000
B. 1500
C. 2500
D. none of these

Answer

To solve the problem step by step, we can use the information given about the percentages of students who passed and failed in the subjects.

### Step 1: Define the total number of candidates
Let the total number of candidates who appeared for the exam be \( x \).

### Step 2: Calculate the number of students who passed in each subject
- The number of students who passed in History = \( 70\% \) of \( x \) = \( 0.7x \)
- The number of students who passed in Geography = \( 50\% \) of \( x \) = \( 0.5x \)

### Step 3: Calculate the number of students who failed in both subjects
According to the problem, \( 20\% \) of the students failed in both subjects. Therefore, the number of students who failed in both = \( 20\% \) of \( x \) = \( 0.2x \).

### Step 4: Use the principle of inclusion-exclusion
Let:
- \( A \) = number of students who passed in History
- \( B \) = number of students who passed in Geography
- \( n(A) = 0.7x \)
- \( n(B) = 0.5x \)
- \( n(A \cap B) = 500 \) (students who passed in both subjects)

Using the principle of inclusion-exclusion, we have:
\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \]
Where \( n(A \cup B) \) is the number of students who passed in at least one subject.

### Step 5: Calculate the number of students who passed in at least one subject
The total number of students who failed in both subjects is \( 0.2x \), so the number of students who passed in at least one subject is:
\[ n(A \cup B) = x - n(\text{failed in both}) = x - 0.2x = 0.8x \]

### Step 6: Set up the equation
Now, we can set up the equation:
\[ 0.8x = n(A) + n(B) - n(A \cap B) \]
Substituting the values we have:
\[ 0.8x = 0.7x + 0.5x - 500 \]

### Step 7: Simplify the equation
Combine like terms:
\[ 0.8x = 1.2x - 500 \]
Rearranging gives:
\[ 500 = 1.2x - 0.8x \]
\[ 500 = 0.4x \]

### Step 8: Solve for \( x \)
Now, divide both sides by \( 0.4 \):
\[ x = \frac{500}{0.4} = 1250 \]

### Conclusion
The total number of candidates who appeared for the exam is \( 1250 \).

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