At Lucknow Public School `1/9` students were absent in an exam and only `19/24` of those who appeared for the exam passed it. Now we know that 500 students failed in the exam. Total number of students registered for the exam is :

Options

A. 2000
B. 2400
C. 2700
D. 3000

Answer

To find the total number of students registered for the exam, we can follow these steps:

### Step 1: Define the total number of students
Let the total number of students registered for the exam be \( N \).

### Step 2: Calculate the number of absent students
According to the problem, \( \frac{1}{9} \) of the students were absent. Therefore, the number of students who were absent is:
\[ \text{Absent students} = \frac{1}{9} N \]

### Step 3: Calculate the number of students who appeared for the exam
The number of students who appeared for the exam is:
\[ \text{Appeared students} = N - \text{Absent students} = N - \frac{1}{9} N = \frac{8}{9} N \]

### Step 4: Calculate the number of students who passed
It is given that \( \frac{19}{24} \) of the students who appeared passed the exam. Therefore, the number of students who passed is:
\[ \text{Passed students} = \frac{19}{24} \times \frac{8}{9} N = \frac{19 \times 8}{24 \times 9} N = \frac{152}{216} N \]

### Step 5: Calculate the number of students who failed
The number of students who failed is:
\[ \text{Failed students} = \text{Appeared students} - \text{Passed students} = \frac{8}{9} N - \frac{152}{216} N \]
To simplify this, we need a common denominator. The common denominator of 9 and 216 is 216:
\[ \text{Failed students} = \frac{192}{216} N - \frac{152}{216} N = \frac{40}{216} N \]

### Step 6: Set the number of failed students equal to 500
We know from the problem that 500 students failed:
\[ \frac{40}{216} N = 500 \]

### Step 7: Solve for \( N \)
To find \( N \), we can rearrange the equation:
\[ N = 500 \times \frac{216}{40} \]
Calculating this gives:
\[ N = 500 \times 5.4 = 2700 \]

### Conclusion
The total number of students registered for the exam is:
\[ \boxed{2700} \]