What is the smaller angle between the hour hand and the minute hand of a clock when the time shown is 9:15?

Introduction / Context:
Angle between hands questions are classic clock problems in aptitude exams. They test your ability to convert a given time into angular positions for the hour and minute hands, and then find the angle between those positions. Here, the time is 9:15, and you are asked to find the smaller angle between the hour hand and the minute hand at that instant.

Given Data / Assumptions:

• Time shown on the clock is 9:15.
• The clock is accurate and circular with 360 degrees total.
• The minute hand moves 360 degrees in 60 minutes, that is, 6 degrees per minute.
• The hour hand moves 360 degrees in 12 hours, that is, 30 degrees per hour, or 0.5 degrees per minute.
• We must report the smaller angle between the two hands.

Concept / Approach:
The standard method is to compute the separate angular positions of the minute and hour hands measured from the 12 o clock position. The minute hand position is straightforward: minutes multiplied by 6 degrees. The hour hand position depends on both the hour and the extra movement caused by the minutes past the hour. Once both angles are known, we find their absolute difference. If this difference is greater than 180 degrees, we subtract it from 360 degrees to obtain the smaller angle between the two hands.

Step-by-Step Solution:
Step 1: Find the angle of the minute hand at 9:15. The minute hand is at 15 minutes, so its angle from 12 is 15 * 6 = 90 degrees. Step 2: Find the angle of the hour hand at 9:15. At exactly 9:00, the hour hand is at 9 * 30 = 270 degrees. In the next 15 minutes, it moves further by 0.5 degrees per minute, that is, 15 * 0.5 = 7.5 degrees. Step 3: Total angle of the hour hand from 12 o clock at 9:15 is 270 + 7.5 = 277.5 degrees. Step 4: Compute the difference between the two angles: difference = |277.5 - 90| = 187.5 degrees. Step 5: The smaller angle between the hands is 360 - 187.5 = 172.5 degrees. Step 6: Therefore, the required smaller angle is 172.5 degrees.

Verification / Alternative check:
A quick consistency check is to note that at exactly 9:00 the angle between the hands is 90 degrees (hour at 270 degrees and minute at 0 degrees). As time passes to 9:15, the minute hand moves forward and the hour hand also moves forward slightly, increasing the separation. The result being slightly less than 180 degrees is reasonable. Using the general angle formula |30H - 5.5M| with H = 9 and M = 15 gives |30 * 9 - 5.5 * 15| = |270 - 82.5| = 187.5; then the smaller angle is 360 - 187.5 = 172.5 degrees, confirming the answer.

Why Other Options Are Wrong:
165 degrees: This is close but does not match the precise calculation from the standard formula.

112.5 degrees: This would be the difference if the separation were much smaller; it is not supported by the positions of the hands.

125.5 degrees: This is not derived from any correct application of the 30H minus 5.5M formula.

187.5 degrees: This is the larger reflex angle between the hands, not the smaller angle requested in the question.

Common Pitfalls:
A common mistake is to forget that the hour hand moves continuously with the minutes, so taking the hour hand only at 270 degrees for 9 o clock leads to an incorrect difference. Another error is to stop at the raw difference of 187.5 degrees and forget to find the smaller angle by subtracting from 360 degrees. Always apply the formula carefully and remember to choose the smaller of the two possible angles between the hands.

Final Answer:
The smaller angle between the hour hand and minute hand at 9:15 is 172.5 degrees.