Let's begin by understanding the rate at which both Ben and Karl address their envelopes and then determine how many envelopes Karl will have left when Ben finishes addressing his share.
### Step-by-Step Solution:
1. Ben's Progress:
- Ben has 48 envelopes to start with.
- He addresses 12 envelopes per hour.
To find out how many hours it will take Ben to finish addressing all his envelopes, we divide the total number of envelopes by the number of envelopes he addresses per hour:
[tex]\[
\text{Time for Ben to finish} = \frac{48 \text{ envelopes}}{12 \text{ envelopes/hour}} = 4 \text{ hours}
\][/tex]
2. Karl's Progress:
- Karl also starts with 48 envelopes.
- Each hour, Karl addresses half of the envelopes he has left at the beginning of that hour.
We need to determine how many envelopes Karl has left after each hour for the duration it takes Ben to finish (which is 4 hours).
- At the beginning (Hour 0): Karl has 48 envelopes.
- After 1st hour: Karl addresses half of 48, so he addresses [tex]\( \frac{48}{2} = 24 \)[/tex] envelopes.
Thus, envelopes left after the 1st hour [tex]\( = 48 - 24 = 24 \)[/tex] envelopes.
- After 2nd hour: Karl addresses half of the 24 remaining envelopes, so he addresses [tex]\( \frac{24}{2} = 12 \)[/tex] envelopes.
Thus, envelopes left after the 2nd hour [tex]\( = 24 - 12 = 12 \)[/tex] envelopes.
- After 3rd hour: Karl addresses half of the 12 remaining envelopes, so he addresses [tex]\( \frac{12}{2} = 6 \)[/tex] envelopes.
Thus, envelopes left after the 3rd hour [tex]\( = 12 - 6 = 6 \)[/tex] envelopes.
- After 4th hour: Karl addresses half of the 6 remaining envelopes, so he addresses [tex]\( \frac{6}{2} = 3 \)[/tex] envelopes.
Thus, envelopes left after the 4th hour [tex]\( = 6 - 3 = 3 \)[/tex] envelopes.
### Conclusion:
After 4 hours, which is the time it takes for Ben to finish addressing all of his envelopes, Karl has 3 envelopes left that he still needs to address.
[tex]\[
\boxed{3}
\][/tex]
