It is now 3:16. How much time will pass before the hour and minute hands coincide?

To determine how much time will pass before the hour and minute hands coincide again after 3:16, we need to calculate when the positions of the hour and minute hands are the same.

First, let's convert the given time into a single unit (degrees).

### Step 1: Calculate the positions of the hour and minute hands at 3:16

- **Minute hand:** 
  - The minute hand moves 6 degrees per minute (360 degrees / 60 minutes).
  - At 16 minutes past the hour, the minute hand is at \( 16 \times 6 = 96 \) degrees.

- **Hour hand:** 
  - The hour hand moves 0.5 degrees per minute (30 degrees per hour / 60 minutes per hour).
  - At 3:00, the hour hand is at \( 3 \times 30 = 90 \) degrees.
  - In 16 minutes, the hour hand moves \( 16 \times 0.5 = 8 \) degrees.
  - Therefore, at 3:16, the hour hand is at \( 90 + 8 = 98 \) degrees.

### Step 2: Determine the angle difference between the two hands at 3:16

The difference in their positions is \( 98 - 96 = 2 \) degrees.

### Step 3: Calculate the relative speed of the hour and minute hands

- The minute hand moves at 6 degrees per minute.
- The hour hand moves at 0.5 degrees per minute.
- The relative speed between the two hands is \( 6 - 0.5 = 5.5 \) degrees per minute.

### Step 4: Calculate the time needed for the hands to coincide

We need to find the time \( t \) in minutes for the 2-degree difference to be covered by the relative speed of 5.5 degrees per minute.

\[
t = \frac{\text{angle difference}}{\text{relative speed}} = \frac{2}{5.5}
\]

\[
t = \frac{2}{5.5} \approx 0.3636 \text{ minutes}
\]

### Step 5: Convert this time into minutes and seconds

\[
0.3636 \times 60 \approx 21.82 \text{ seconds}
\]

Thus, the hour and minute hands will coincide approximately \( 0.3636 \) minutes or about 22 seconds after 3:16.

### Conclusion

The hands will coincide at approximately 3:16 and 22 seconds.

Would you like more details or have any questions?

Here are 5 questions you might want to ask:
1. How do you calculate the exact position of clock hands at any given time?
2. Can the same method be used to find when the hands coincide after any other time?
3. What are the common methods to solve clock problems?
4. How often do the clock hands coincide in a 12-hour period?
5. What are other interesting problems involving clock hands?

**Tip:** When dealing with clock problems, it's helpful to remember the relative speeds of the hands: the minute hand moves 12 times faster than the hour hand.

Learn how to calculate the time it takes for the hour and minute hands of a clock to coincide at 3:16. This problem involves using geometry and trigonometry to determine the angular positions of clock hands and their relative speeds. Suitable for students in Grades 7-9 studying clock mathematics.

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