Question

If

of a class is absent and

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

Answer 1

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:
\[
\boxed{(1 - x)(1 - y)}
\]