The grandfather clock at the top of the stairs says it is now 3:20 AM. In how many minutes from now will the hour and minute hand of the clock first meet each other?

To determine when the hour and minute hands of a clock first meet after 3:20 AM, we can follow these steps:

### Step 1: Calculate the positions of the hour and minute hands at 3:20 AM.
- **Hour hand position:** The hour hand is at 3 o'clock, but it's also moved forward because 20 minutes have passed. Every hour, the hour hand moves \(30^\circ\) (since there are 360° in a circle and 12 hours). In 20 minutes, the hour hand moves:
  \[
  \text{Movement of hour hand} = 20 \times \frac{1}{2} = 10^\circ
  \]
  Therefore, the hour hand is at:
  \[
  \text{Hour hand position} = 90^\circ + 10^\circ = 100^\circ
  \]

- **Minute hand position:** The minute hand is at 20 minutes, and since each minute represents 6°, the minute hand is at:
  \[
  \text{Minute hand position} = 20 \times 6^\circ = 120^\circ
  \]

### Step 2: Set up the equation to find the time when they meet.
Let \( t \) be the time in minutes after 3:20 AM when the hands meet. The minute hand moves at \(6^\circ\) per minute, while the hour hand moves at \( \frac{1}{2}^\circ \) per minute.

- **Minute hand's position after \( t \) minutes:**
  \[
  120^\circ + 6t
  \]

- **Hour hand's position after \( t \) minutes:**
  \[
  100^\circ + \frac{1}{2}t
  \]

### Step 3: Set the two positions equal and solve for \( t \).
\[
120 + 6t = 100 + \frac{1}{2}t
\]
Subtract \( \frac{1}{2}t \) from both sides:
\[
120 + 5.5t = 100
\]
Subtract 120 from both sides:
\[
5.5t = -20
\]
Divide both sides by 5.5:
\[
t = \frac{-20}{5.5} \approx -3.64 \text{ minutes}
\]
This negative value indicates that the hands last overlapped about 3.64 minutes before 3:20 AM, which means we should look for the next overlap.

### Step 4: Find the next meeting point.
Since the hands overlap every 65.45 minutes, the next overlap after 3:20 AM will occur:
\[
t = 65.45 - 3.64 \approx 61.81 \text{ minutes after 3:20 AM}
\]

Thus, the hour and minute hands will meet for the first time approximately **61.81 minutes** after 3:20 AM. Therefore, they will meet at approximately **4:21.81 AM**.

Would you like more details or have any questions?

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**Related Questions:**
1. What is the general formula to determine when clock hands overlap?
2. How often do the hands of a clock overlap in a 12-hour period?
3. How do you calculate the angle between the hour and minute hands at a given time?
4. If the time were 4:20 AM, how would the calculation change?
5. What is the next time after 4:21.81 AM when the hands would meet again?

**Tip:** The hands of a clock overlap approximately every 65 minutes, not exactly every hour.

Solve the clock problem to determine when the hour and minute hands will first meet after 3:20 AM. This detailed solution covers key geometric concepts and relative speed calculations, making it suitable for high school students in Grades 9-12.

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