Understanding Clock Directions and Time Reasoning

This problem combines time calculation with spatial reasoning based on clock hands pointing in specific directions. We are given the current time, the direction of the hour hand, and asked to find the direction of the minute hand after a certain period.

Analyzing the Initial Information: Time and Hour Hand Direction

The question states the time is quarter past 9 by Rekha's watch. This means the time is $\text{9:15}$.

At $\text{9:15}$:

• The minute hand points directly at the number $\text{3}$ on the clock face.
• The hour hand has moved past $\text{9}$ and is one-fourth of the way between $\text{9}$ and $\text{10}$.

We are told that the hour hand at $\text{9:15}$ is pointing towards the West.

Mapping Clock Positions to Directions

In a standard clock, the numbers correspond to directions if the clock face is aligned with cardinal directions (12 is North, 3 is East, 6 is South, 9 is West). Let's verify if the given information aligns with this standard mapping.

• At $\text{9:15}$, the hour hand is just past $\text{9}$.
• The question states the hour hand points West.
• Since $\text{9}$ is typically associated with West on a standard clock face, the given information confirms the standard mapping:

Determining the Current Minute Hand Direction

At $\text{9:15}$, the minute hand points directly at the number $\text{3}$.

According to our direction mapping derived from the hour hand information, the number $\text{3}$ corresponds to East.

So, at $\text{9:15}$, the minute hand is pointing East.

Calculating the Time After 75 Minutes

We need to find the time $\text{75}$ minutes AFTER $\text{9:15}$.

$\text{75}$ minutes can be broken down as $\text{60}$ minutes + $\text{15}$ minutes, which is $\text{1}$ hour and $\text{15}$ minutes.

Starting time: $\text{9:15}$

Adding $\text{1}$ hour: $\text{9:15} + \text{1 hour} = \text{10:15}$

Adding $\text{15}$ minutes: $\text{10:15} + \text{15 minutes} = \text{10:30}$

The time after $\text{75}$ minutes will be $\text{10:30}$.

Finding the Minute Hand Direction at 10:30

At $\text{10:30}$, the minute hand points directly at the number $\text{6}$ on the clock face.

Using the same direction mapping (from the table above), the number $\text{6}$ corresponds to South.

Therefore, at $\text{10:30}$ (which is $\text{75}$ minutes after $\text{9:15}$), the minute hand will point towards the South.

Summary of Steps:

• Initial time: $\text{9:15}$. Minute hand at $\text{3}$, hour hand slightly past $\text{9}$.
• Hour hand at $\text{9:15}$ points West, confirming standard clock-to-direction mapping (12=N, 3=E, 6=S, 9=W).
• At $\text{9:15}$, minute hand is at $\text{3}$, which is East.
• Time after $\text{75}$ minutes: $\text{9:15} + \text{75 minutes} = \text{10:30}$.
• At $\text{10:30}$, minute hand is at $\text{6}$.
• The direction corresponding to $\text{6}$ is South.

The minute hand will point to the South after $\text{75}$ minutes.

Revision Table: Clock Hand Directions

Additional Information: Clock Reasoning Concepts

Clock reasoning questions often test your understanding of how clock hands move and relate to angles and directions. Key concepts include:

• Hand Speeds: The minute hand moves $\text{360}$ degrees in $\text{60}$ minutes (or $\text{6}$ degrees per minute). The hour hand moves $\text{360}$ degrees in $\text{12}$ hours, which is $\text{30}$ degrees per hour or $\text{0.5}$ degrees per minute.
• Relative Positions: At any given time, the hour hand and minute hand are at specific angles relative to the $\text{12}$ o'clock mark.
• Direction Mapping: Problems often map clock positions (numbers or angles) to cardinal or intercardinal directions. It's crucial to determine this mapping correctly based on the information provided.
• Time Calculation: Accurately adding or subtracting time is essential before determining the new hand positions.

In this problem, the direction mapping was standard, simplified by the hour hand's position. The core task was calculating the new time and identifying the minute hand's position and corresponding direction at that time.