If


of a class is absent and

of those present leave the room, what fraction of the original class remains in the room?

1 answer

To solve the problem, we need to denote the total number of students in the class with a variable. Let's use \( N \) to represent the total number of students in the class.

  1. Calculate the number of students who are absent: If \( x \) is the fraction of students who are absent, then the number of students who are absent is given by: \[ \text{Absent students} = x \cdot N \]

  2. Calculate the number of students present: The number of students present can be calculated by subtracting the number of absent students from the total: \[ \text{Present students} = N - \text{Absent students} = N - xN = (1 - x)N \]

  3. Determine how many present students leave the room: If \( y \) is the fraction of those present who leave the room, then the number of students who leave is calculated as: \[ \text{Students leaving} = y \cdot \text{Present students} = y \cdot (1 - x)N \]

  4. Calculate the number of students remaining in the room: The number of students who remain in the room after some of them leave is: \[ \text{Students remaining} = \text{Present students} - \text{Students leaving} \] Substituting the previous results, we get: \[ \text{Students remaining} = (1 - x)N - y(1 - x)N \] Factor out \( (1 - x)N \): \[ \text{Students remaining} = (1 - x)N(1 - y) \]

  5. Finding the fraction of the original class that remains: To find the fraction of the original class \( N \) that remains in the room, we divide the number of students remaining by the total number of students: \[ \text{Fraction remaining} = \frac{\text{Students remaining}}{N} = \frac{(1 - x)N(1 - y)}{N} = (1 - x)(1 - y) \]

Thus, the fraction of the original class that remains in the room is: \[ \boxed{(1 - x)(1 - y)} \]

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