Question

If the angle between the clock hands is \( 60^\circ \), which of the following times is possible?

• A. 2:00
• B. 3:00
• C. 4:00
• D. 6:00

Hint:

For any time that is exactly on the hour (e.g., 2:00, 3:00, 4:00), the angle is simply the hour value multiplied by \( 30^\circ \).

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We need to find out at which of the given times the angle between the hour hand and the minute hand of a standard clock is exactly \( 60^\circ \).


Step 2: Key Formula or Approach:
The angle \( \theta \) between the hands of a clock at \( H \) hours and \( M \) minutes is given by the formula:
\[ \theta = \left| 30H - 5.5M \right| \] Alternatively, for whole hours (where \( M = 0 \)), the hour hand is pointing exactly at the hour mark, and the minute hand is at 12. Since each hour mark represents \( 30^\circ \) (\( 360^\circ / 12 = 30^\circ \)), the angle is simply \( 30^\circ \times H \).


Step 3: Detailed Explanation:
Let us systematically check the angle for each given option:
For 2:00 (\( H = 2, M = 0 \)):
\[ \theta = 30^\circ \times 2 = 60^\circ \] For 3:00 (\( H = 3, M = 0 \)):
\[ \theta = 30^\circ \times 3 = 90^\circ \] For 4:00 (\( H = 4, M = 0 \)):
\[ \theta = 30^\circ \times 4 = 120^\circ \] For 6:00 (\( H = 6, M = 0 \)):
\[ \theta = 30^\circ \times 6 = 180^\circ \] Only at 2:00 is the angle between the clock hands exactly \( 60^\circ \).


Step 4: Final Answer:
The correct time is 2:00.