Solving the Clock Direction Problem at Quarter Past Eight

This problem asks us to determine the direction of the hour hand on a clock at a specific time, given the direction of the minute hand. The time is quarter past eight, which is 8:15.

Understanding Clock Hand Positions at 8:15

On a standard analog clock:

• The minute hand points directly at the number 3 when it is 15 minutes past the hour.
• The hour hand at 8:15 is past the number 8, moving towards the number 9. It has moved 15 minutes' worth of distance past 8.

Relating Clock Positions to Directions

The problem states that the minute hand points towards the East. On a standard clock face, if 12 is at the top (North) and 6 is at the bottom (South), then 3 is to the right (East) and 9 is to the left (West).

Since the minute hand at 8:15 points to the number 3, and the problem says the minute hand points East, this confirms that the clock is oriented in the standard way:

• 12 is North
• 3 is East
• 6 is South
• 9 is West

The directions between these main points are:

• Between 12 (North) and 3 (East) is North-East.
• Between 3 (East) and 6 (South) is South-East.
• Between 6 (South) and 9 (West) is South-West.
• Between 9 (West) and 12 (North) is North-West.

Determining the Hour Hand Direction at 8:15

At 8:15, the hour hand is between the numbers 8 and 9 on the clock face. It is one quarter of the way from 8 towards 9.

Let's locate the numbers 8 and 9 on the standard clock face relative to the main directions:

• The number 9 is West.
• The number 6 is South.
• Numbers between 6 and 9 are in the South-West quadrant.
• Number 7 is between 6 and 9, closer to 6.
• Number 8 is between 6 and 9, closer to 9.

So, the number 8 on the clock face is in the South-West direction.

At 8:15, the hour hand has moved past the 8 mark towards the 9 mark. Since the 8 mark is in the South-West quadrant and the hour hand is just past it, moving towards the 9 mark (West), the hour hand is still pointing in the South-West general direction.

Detailed Angle Calculation (Optional but helpful)

We can also calculate the angle of the hour hand from the 12 (North) position.

• The hour hand moves \(360^\circ\) in 12 hours, or \(30^\circ\) per hour (\(360/12\)).
• It also moves \(0.5^\circ\) per minute (\(30/60\)).
• At 8:00, the hour hand is at the 8 mark. The angle from 12 is \(8 \times 30^\circ = 240^\circ\) clockwise.
• In 15 minutes, the hour hand moves an additional \(15 \times 0.5^\circ = 7.5^\circ\).
• So, the total angle of the hour hand from 12 is \(240^\circ + 7.5^\circ = 247.5^\circ\) clockwise.

Comparing this angle to the standard directions:

• North: \(0^\circ\) or \(360^\circ\)
• East: \(90^\circ\)
• South: \(180^\circ\)
• West: \(270^\circ\)
• South-West: \(225^\circ\) (midway between South and West)

The angle \(247.5^\circ\) is between \(225^\circ\) (South-West) and \(270^\circ\) (West). This confirms that the hour hand is pointing in a direction between South-West and West, which is considered the South-West general direction in the context of the given options.

Conclusion

At quarter past eight (8:15), the minute hand is at 3, pointing East. The hour hand is between 8 and 9, specifically at an angle of \(247.5^\circ\) from North. This direction falls within the South-West quadrant.

Therefore, the hour hand would point towards the South-west direction.

Revision Table: Clock Direction Concepts

Additional Information on Clock and Direction Reasoning

Clock and direction problems often involve understanding the relationship between the numbers on a clock face and the cardinal and intercardinal directions (North, South, East, West, North-East, South-East, South-West, North-West). The key is to establish the mapping between clock numbers and directions based on the information given in the problem.

Steps to solve such problems generally include:

1. Identify the time given.
2. Determine the exact position of the hour and minute hands on a standard clock face for that time.
3. Use the given information about the direction of one hand to determine the orientation of the clock face relative to the standard directions. For example, if the 12 points South, then 6 points North, 3 points West, and 9 points East.
4. Based on the established orientation, determine the direction that the other hand is pointing towards.
5. Angles can be used for precise calculations, where each minute mark is \(6^\circ\) apart, and each hour mark is \(30^\circ\) apart. The hour hand also moves as minutes pass.