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

Introduction / Context:
This is a standard clock question that asks for the exact time between two hours when the hour hand and the minute hand coincide. Problems of this type are very common in competitive exams because they combine angle speed, relative speed, and simple algebra. Here, we want to know how many minutes after 3 o clock the two hands will be exactly together between 3 and 4.

Given Data / Assumptions:

• We are interested in the interval between 3 o clock and 4 o clock.
• At 3 o clock, the hour hand is exactly at the 3 mark.
• The minute hand is at the 12 mark at 3 o clock.
• The hour hand moves at 0.5 degrees per minute, and the minute hand at 6 degrees per minute.
• We must find the number of minutes t after 3 o clock when both hands coincide.

Concept / Approach:
The idea is to treat the motion of both hands as uniform circular motion and then use relative speed. At 3 o clock, the hour hand is ahead of the minute hand by a certain angle. As time passes, the minute hand moves faster and gradually catches up. When they coincide, both hands are at the same angular position measured from the 12 o clock reference. We set up an equation equating the positions of the two hands after t minutes and solve for t.

Step-by-Step Solution:
Step 1: At 3 o clock, the hour hand is at 3 * 30 = 90 degrees from the 12 o clock position. Step 2: The minute hand at exactly 3 o clock is at 0 degrees because it points to 12. Step 3: After t minutes, the minute hand angle becomes 6t degrees. Step 4: After t minutes, the hour hand angle becomes 90 + 0.5t degrees. Step 5: For coincidence, these two angles must be equal: 6t = 90 + 0.5t. Step 6: Rearranging, 6t - 0.5t = 90 gives 5.5t = 90. Step 7: So t = 90 / 5.5 = 90 * 2 / 11 = 180 / 11 minutes. Step 8: Convert 180 / 11 into a mixed fraction: 11 goes into 180 sixteen times (16 * 11 = 176) with a remainder of 4. So t = 16 4/11 minutes. Step 9: Therefore, the hands coincide 16 4/11 minutes past 3 o clock.

Verification / Alternative check:
There is a well known shortcut formula for the time in minutes after H o clock when the hands coincide: t = (60 * H) / 11. For H = 3, t = (60 * 3) / 11 = 180 / 11 = 16 4/11 minutes. This confirms the result from the detailed angular method. Both methods agree, so the value is reliable.

Why Other Options Are Wrong:
11 4/11 minutes: This is too early for the minute hand to fully catch up to the hour hand, as the angular gap would still be positive.

13 4/11 minutes: Also smaller than 16 4/11 and does not satisfy the equation 6t = 90 + 0.5t.

15 4/11 minutes: Close, but still not enough time for the minute hand to close the initial 90 degree gap at a relative speed of 5.5 degrees per minute.

18 4/11 minutes: This overshoots; by this time the minute hand has already moved ahead of the hour hand, so they no longer coincide.

Common Pitfalls:
A common error is to forget that the hour hand also moves during these minutes and to equate 6t directly to 90, ignoring the extra 0.5t term. Another pitfall is to miscalculate the mixed fraction from 180 / 11 or to round the decimal instead of keeping the exact fractional expression, which is usually required in exam answers. Remembering and correctly using the shortcut formula 60H / 11 can save time once the concept is understood.

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