Between 3 o clock and 4 o clock, at what exact time (in clock form) will the hour hand and the minute hand coincide?

Introduction / Context:
This question is another version of the classic coincidence problem, but this time the answer is requested in full clock time format instead of just minutes past the hour. We must determine the precise time between 3 o clock and 4 o clock when the hour hand and minute hand are exactly together. Understanding how to convert the computed minutes into standard time notation is an essential skill for such problems.

Given Data / Assumptions:

• Time interval of interest: between 3 o clock and 4 o clock.
• At 3 o clock, the hour hand points at 3 and the minute hand points at 12.
• Minute hand speed: 6 degrees per minute.
• Hour hand speed: 0.5 degrees per minute.
• We must express the final answer as a clock time, such as 3:16 4/11.

Concept / Approach:
As in an earlier question, the approach is to apply relative speed. At the start, the hour hand is ahead in angle. Because the minute hand moves faster, it eventually catches up and coincides with the hour hand. We equate the angular positions of both hands after t minutes and solve for t. Once t is obtained, we add it to 3 o clock and express the result in the required format H:MM fraction.

Step-by-Step Solution:
Step 1: At 3 o clock, the hour hand is at 3 * 30 = 90 degrees. Step 2: The minute hand at 3 o clock is at 0 degrees. Step 3: After t minutes, the minute hand is at 6t degrees. Step 4: After t minutes, the hour hand is at 90 + 0.5t degrees. Step 5: For coincidence, set the angles equal: 6t = 90 + 0.5t. Step 6: Rearranging gives 6t - 0.5t = 90, so 5.5t = 90. Step 7: Solving, t = 90 / 5.5 = 180 / 11 minutes. Step 8: Express 180 / 11 as a mixed number. 11 goes into 180 sixteen times (16 * 11 = 176) with a remainder of 4, so t = 16 4/11 minutes past 3. Step 9: Therefore, the exact clock time is 3:16 4/11.

Verification / Alternative check:
The shortcut formula for the time in minutes after H o clock when the hands coincide is t = (60 * H) / 11. With H = 3, t = (60 * 3) / 11 = 180 / 11 = 16 4/11 minutes. This matches our algebraic solution. Converting this value into clock notation gives 3:16 4/11, confirming the result.

Why Other Options Are Wrong:
3:16 7/11: This corresponds to a slightly larger value of t than the correct 16 4/11 and would result in the minute hand overshooting the hour hand.

3:16 11/4: This expression is not even a valid minute fraction less than one minute and clearly does not correspond to the required small fractional part.

3:30: At 3:30, the minute hand is at 180 degrees, while the hour hand is halfway between 3 and 4 at 105 degrees, so they are not coincident.

3:18 4/11: This is another time slightly later than 3:16 4/11 and does not satisfy the exact coincidence condition.

Common Pitfalls:
A major source of error is mismanaging fractional minutes and incorrectly converting an improper fraction into a mixed number. Some learners also forget that the hour hand moves during the time t and mistakenly set 6t = 90. Using the shortcut formula as a check can prevent such mistakes, but only if the formula is correctly remembered and applied.

Final Answer:
The hands coincide at 3:16 4/11.