An accurate clock shows exactly 7 o clock in the morning. Through how many degrees will the hour hand rotate by the time the clock shows 3 o clock in the afternoon?

Introduction / Context:
This question tests your understanding of how far the hour hand of a clock rotates over a period of time. Instead of asking for the angle between the hour and minute hands, it asks for the total angular rotation of the hour hand from one time to another. Problems like this appear frequently in aptitude exams under the topic of clocks and require only a simple relationship between time and angle.

Given Data / Assumptions:

• Initial time: 7 o clock in the morning.
• Final time: 3 o clock in the afternoon.
• The clock is accurate and moves uniformly.
• We must find the total angle through which the hour hand rotates between these two times.

Concept / Approach:
On a standard clock, the hour hand completes one full revolution of 360 degrees in 12 hours. Therefore, in one hour the hour hand rotates 360 / 12 = 30 degrees. To find the total rotation over a given number of hours, we simply multiply the number of hours by 30 degrees per hour. The key step is to count the number of hours between the two times correctly, noting the change from morning to afternoon.

Step-by-Step Solution:
Step 1: Calculate the time interval from 7 o clock in the morning to 3 o clock in the afternoon. Step 2: From 7 a.m. to 12 noon is 5 hours. From 12 noon to 3 p.m. is another 3 hours. Total time = 5 + 3 = 8 hours. Step 3: The hour hand rotates 30 degrees in one hour. Step 4: Therefore, in 8 hours, the rotation of the hour hand is 8 * 30 degrees = 240 degrees. Step 5: Hence, the hour hand sweeps through an angle of 240 degrees between 7 a.m. and 3 p.m.

Verification / Alternative check:
You can also think in terms of hour marks on the dial. At 7 o clock, the hour hand points at 7. At 3 o clock, it points at 3. Moving forward from 7 to 3 goes through the positions 8, 9, 10, 11, 12, 1, 2, and finally 3. That is 8 hour steps. Each step represents 30 degrees, so 8 * 30 = 240 degrees. This count of hour intervals provides the same result and confirms the calculation.

Why Other Options Are Wrong:
144 degrees: This would correspond to 144 / 30 = 4.8 hours, not 8 hours, so it does not match the actual interval.

168 degrees: This equals 5.6 hours of rotation, again not compatible with the 8 hour time difference.

180 degrees: This would correspond to 6 hours, the rotation from 7 to 1, not all the way to 3 p.m.

150 degrees: This equals 5 hours of rotation, matching only the interval from 7 a.m. to 12 noon.

Common Pitfalls:
A common mistake is miscounting the number of hours between the two times, especially when crossing from morning to afternoon. Another error is confusing the hour hand with the minute hand and using 6 degrees per minute instead of 30 degrees per hour. Always identify which hand is being discussed and check whether the time span crosses noon or midnight.

Final Answer:
The hour hand rotates through 240 degrees between 7 a.m. and 3 p.m.