It is exactly 10:30 by the clock. If, at this moment, the minute hand is pointing towards the south direction, then in which compass direction will the hour hand be pointing?

Introduction / Context:
Clock and direction problems connect the positions of clock hands with compass directions. At certain times the hour and minute hands form specific angles relative to the 12 o clock position. If one hand is aligned with a known cardinal direction, we can deduce the direction of the other hand. In this question we know it is 10:30 and the minute hand points south, and we must determine the direction of the hour hand.

Given Data / Assumptions:

• The time is exactly 10:30.
• The minute hand is pointing towards the south direction.
• We assume a standard analog clock where 12 is at the top, 3 at the right, 6 at the bottom and 9 at the left.
• We must find the compass direction (such as north west or south east) of the hour hand.

Concept / Approach:
At 10:30, the minute hand is at the 6 on the clock, which is directly opposite the 12. The hour hand is between 10 and 11, slightly closer to 11. On a standard clock that is not rotated, 12 corresponds to north, 3 to east, 6 to south and 9 to west. Because we are told specifically that the minute hand points south at 10:30, the clock's normal orientation is already aligned with compass directions. Hence we identify the geometric position of the hour hand on the dial and match that with the appropriate compass direction.

Step-by-Step Solution:
Step 1: At 10:30, the minute hand points at the 6. On a standard clock face, 6 is directly downward from the centre, opposite 12. Step 2: The problem states that at this time, the minute hand points towards the south. This matches the usual convention where the downward direction corresponds to south, so no rotation is required. Step 3: Now we analyse the hour hand. At exactly 10:00 the hour hand points at the 10, which is between 9 (west) and 11 (north west region). Step 4: Between 10:00 and 11:00, the hour hand slowly moves from the 10 mark towards the 11 mark. At 10:30 it is halfway between the 10 and 11 positions. Step 5: The 9 mark is directly west, the 12 mark is directly north. The 10 and 11 marks lie between these two, with both leaning towards the north west quadrant. Step 6: Therefore, halfway between 10 and 11 is still firmly in the north west region of the clock face. Step 7: Since the clock is aligned such that 12 is north and 9 is west, the intermediate position of the hour hand at 10:30 corresponds to a north west direction.

Verification / Alternative check:
An alternative is to think in terms of angles. The full circle is 360 degrees, and there are 12 hours, so each hour marking is 30 degrees apart. At 10:30, the hour hand is at 10.5 hours from the top. Measured clockwise from 12, the angle is 10.5 multiplied by 30 degrees, which equals 315 degrees. On a compass, 0 degrees corresponds to north and 90 degrees to east. A bearing of 315 degrees lies between 270 degrees (west) and 360 degrees (north), nearer to north and west, which is precisely the north west direction.

Why Other Options Are Wrong:

• Option A, South-West, points downwards and to the left, inconsistent with the upper left position of the hour hand at 10:30.
• Option C, North-East, describes an upwards and rightwards direction, opposite to the hour hand's location on the left side of the clock.
• Option D, South-East, lies in the bottom right quadrant and is incompatible with the hour hand being above the centre between 10 and 11.
• Option E, West, would align with the 9 o clock position, but at 10:30 the hour hand is clearly above and to the right of west.

Common Pitfalls:
Students sometimes forget that the hour hand moves continuously and is not fixed exactly on the hour marks at half past times. Others may incorrectly rotate the clock instead of matching it with the given minute hand direction. Always check whether the description of the minute hand matches the standard orientation. If it does, no extra rotation is needed, simplifying the reasoning considerably.

Final Answer:
At 10:30, with the minute hand pointing south, the hour hand points towards the North-West direction.