Clock Direction Analysis at 7:45

This problem involves determining the direction of the minute hand on a standard analog clock after a specific time interval, given the initial position of the hour hand.

Clock Basics and Standard Directions

• A clock face represents 360 degrees.
• There are 12 numbers, so each number represents ${360^\circ}/{12} = 30^\circ$.
• Standard directions: 12 = North ($0^\circ$), 3 = East ($90^\circ$), 6 = South ($180^\circ$), 9 = West ($270^\circ$).
• The minute hand moves $360^\circ$ in 60 minutes ($6^\circ$ per minute).
• The hour hand moves $360^\circ$ in 12 hours ($30^\circ$ per hour, or $0.5^\circ$ per minute).

Step 1: Analyze Initial Time (7:30 am)

At 7:30:

• Minute Hand: Points directly at the number 6. The angle from 12 (North) is $6 \times 30^\circ = 180^\circ$. This corresponds to the South direction.
• Hour Hand: Is halfway between the numbers 7 and 8. The angle calculation confirms this position aligns with South-West. Angle = $(7.5 \times 30^\circ) = 225^\circ$. This is exactly South-West ($180^\circ$ South + $45^\circ$ towards West). The given condition is consistent with a standard clock orientation.

Step 2: Determine Time After 15 Minutes

Starting time = 7:30 am.

Time elapsed = 15 minutes.

New time = 7:30 am + 15 minutes = 7:45 am.

Step 3: Find Minute Hand Direction at 7:45 am

At 7:45, the minute hand points directly at the number 9.

The angle for the number 9 from the 12 (North) position is $9 \times 30^\circ = 270^\circ$.

A direction of $270^\circ$ clockwise from North corresponds to West.

Conclusion

Therefore, after 15 minutes, when the time is 7:45 am, the minute hand will point West.