At what exact time between 9 o clock and 10 o clock will the hour hand and the minute hand of a watch be together?

Introduction / Context:
Finding the exact time when the hour and minute hands coincide between two specified hours is a central type of clock problem. This question focuses on the interval between 9 o clock and 10 o clock and asks when, during this interval, the two hands are exactly together. It requires understanding of how fast each hand moves and how to set up the catch up equation.

Given Data / Assumptions:

• We consider the period between 9 o clock and 10 o clock.
• At 9 o clock, the hour hand is at the 9 mark, and the minute hand is at the 12 mark.
• The minute hand moves at 6 degrees per minute.
• The hour hand moves at 0.5 degrees per minute.
• We must find the number of minutes t after 9 o clock when both hands coincide.

Concept / Approach:
At 9 o clock, the hour hand is ahead of the minute hand by a certain fixed angle. Because the minute hand rotates faster, it eventually catches up to the hour hand somewhere between 9 and 10. We can calculate the angular position of each hand after t minutes, set them equal for coincidence, and solve for t. A shortcut formula t = (60 * H) / 11, where H is the hour, can also be used after the underlying reasoning is understood.

Step-by-Step Solution:
Step 1: At 9 o clock, the hour hand is at 9 * 30 = 270 degrees from the 12 o clock position. Step 2: At the same moment, the minute hand is at 0 degrees (pointing to 12). Step 3: After t minutes, the angle of the minute hand is 6t degrees. Step 4: After t minutes, the angle of the hour hand is 270 + 0.5t degrees. Step 5: For coincidence, set the angles equal: 6t = 270 + 0.5t. Step 6: Rearranging gives 6t - 0.5t = 270, so 5.5t = 270. Step 7: Therefore, t = 270 / 5.5 = 270 * 2 / 11 = 540 / 11 minutes. Step 8: Convert 540 / 11 to a mixed fraction. 11 goes into 540 forty nine times (49 * 11 = 539) with a remainder of 1. So t = 49 1/11 minutes. Step 9: Hence, the hands are together at 9:49 1/11.

Verification / Alternative check:
Use the shortcut formula for coincidence after H o clock: t = (60 * H) / 11 minutes past H. Here H = 9, so t = (60 * 9) / 11 = 540 / 11 = 49 1/11 minutes, which matches the algebraic result. This confirms that the time of coincidence between 9 and 10 is 9:49 1/11.

Why Other Options Are Wrong:
49 3/11 minutes past 9: This is slightly later than the correct time and would cause the minute hand to have already passed the hour hand.

45 1/11 minutes past 9: This is too early; the minute hand has not yet fully closed the initial angular gap at this time.

45 7/11 minutes past 9: Also too early for coincidence, because the minute hand still lags behind the hour hand.

47 1/11 minutes past 9: This is between the earlier incorrect values and still does not satisfy the equality of angles.

Common Pitfalls:
Learners often forget to include the 0.5t term for the hour hand or mistakenly assume that the hour hand stays fixed at 270 degrees. Another frequent error is to use an incorrect version of the shortcut formula or to round the fraction too early. To avoid these mistakes, always write the full angular expressions for both hands and solve the equation carefully before converting the fraction into mixed form.

Final Answer:
The hands of the watch are together at 49 1/11 minutes past 9.