Between 3 o clock and 4 o clock, after how many minutes past 3 will the hour hand and minute hand of a clock exactly coincide?

Introduction / Context:
This is another version of the standard clock problem asking when the hour and minute hands coincide between 3 o clock and 4 o clock. The focus is again on computing the precise number of minutes after 3 o clock at which the hands are exactly together. Such repeated practice reinforces the method of forming and solving the coincidence equation.

Given Data / Assumptions:

• We consider the time interval between 3 o clock and 4 o clock.
• At 3 o clock, the hour hand is at the 3 mark and the minute hand 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 a time t minutes after 3 o clock when the two hands are coincident.

Concept / Approach:
As before, we model the movement of each hand as uniform circular motion. At 3 o clock the hour hand leads the minute hand by a fixed angle. Over time, the faster minute hand gains on the hour hand. When the hands coincide, their angular positions measured from 12 o clock are equal. Setting these angular expressions equal leads to a simple linear equation in t, which we solve and then convert to a mixed fraction in minutes.

Step-by-Step Solution:
Step 1: At 3 o clock, the hour hand angle from 12 is 3 * 30 = 90 degrees. Step 2: The minute hand angle at 3 o clock is 0 degrees because it points at 12. Step 3: After t minutes, the minute hand angle is 6t degrees. Step 4: After t minutes, the hour hand angle is 90 + 0.5t degrees. Step 5: For coincidence, set these angles equal: 6t = 90 + 0.5t. Step 6: Rearranging, 6t - 0.5t = 90 gives 5.5t = 90. Step 7: Solve for t: t = 90 / 5.5 = 90 * 2 / 11 = 180 / 11 minutes. Step 8: Convert 180 / 11 to mixed form. Since 11 * 16 = 176, t = 16 4/11 minutes. Step 9: Hence, the hands coincide exactly 16 4/11 minutes past 3.

Verification / Alternative check:
Using the standard formula for the time after H o clock when the hands are together, t = (60 * H) / 11. Substituting H = 3, we get t = (60 * 3) / 11 = 180 / 11 = 16 4/11 minutes. This matches our full derivation and confirms that the computed time is correct. Substituting t back into the angles 6t and 90 + 0.5t also shows that they are numerically equal.

Why Other Options Are Wrong:
11 1/11 minutes: This is too early; the minute hand has not yet closed the full 90 degree gap by that time.

15 4/11 minutes: Slightly less than the correct value and does not satisfy the equation 6t = 90 + 0.5t exactly.

14 4/11 minutes: Even earlier and farther from the true coincidence moment.

12 4/11 minutes: Clearly too early for the faster minute hand to have caught up with the hour hand.

Common Pitfalls:
As with similar questions, errors often arise from forgetting the continuous motion of the hour hand and incorrectly setting 6t equal to 90. Another issue is mishandling the fraction 180 / 11 when converting to mixed form, leading to an incorrect expression of the final answer. Remembering the shortcut formula and verifying that it matches the detailed calculation is a good way to avoid mistakes.

Final Answer:
The hands coincide 16 4/11 minutes after 3 o clock.