Janet and Donald agree to meet for lunch between 11:30 and 12:30. They each arrive at a random time in that interval. If Janet has to wait more than 15 minutes for Donald, she gets bored and leaves. Donald is busier so will only wait 5 minutes for Janet. What is the probability that the two will eat together? Express your answer as a fraction.
1. **Define the problem in terms of a coordinate system:**
   - Let $x$ be the number of minutes after 11:30 that Janet arrives.
   - Let $y$ be the number of minutes after 11:30 that Donald arrives.
   - Both $x$ and $y$ range from $0$ to $60$.

2. **Set up the coordinate plane:**
   - The total area of possible arrival times is a $60 \times 60$ square on the coordinate plane, which has an area of $3600$ square units.

3. **Determine the conditions under which they will meet:**
   - Janet will wait for Donald if $|x - y| \leq 15$.
   - Donald will wait for Janet if $|y - x| \leq 5$.
   - Therefore, they will meet if $|x - y| \leq 5$.

4. **Graph the inequalities:**
   - The line $y = x + 5$ represents the upper bound where Donald will wait for Janet.
   - The line $y = x - 5$ represents the lower bound where Donald will wait for Janet.
   - The line $y = x + 15$ represents the upper bound where Janet will wait for Donald.
   - The line $y = x - 15$ represents the lower bound where Janet will wait for Donald.

5. **Identify the region of interest:**
   - The region where they will meet is bounded by $y = x + 5$, $y = x - 5$, $y = x + 15$, and $y = x - 15$.
   - This region is a parallelogram centered on the line $y = x$.

6. **Calculate the area of the region where they will meet:**
   - The region where they will meet is the intersection of the two strips defined by the inequalities.
   - The area of the region where $|x - y| \leq 5$ is a band of width 10 units (from $x - 5$ to $x + 5$) and height 60 units.
   - The area of the region where $|x - y| \leq 15$ is a band of width 30 units (from $x - 15$ to $x + 15$) and height 60 units.
   - The intersection of these two bands is a parallelogram with a base of 10 units and a height of 60 units.

7. **Calculate the area of the intersection:**
   - The area of the intersection is $10 \times 60 = 600$ square units.

8. **Calculate the probability:**
   - The probability that they will meet is the ratio of the area of the intersection to the total area of the square.
   - Therefore, the probability is $\frac{600}{3600} = \frac{1}{6}$.

The final answer is $\boxed{\frac{1}{6}}$.