At what time between 5 oclock and 6 oclock are the hour hand and minute hand of a clock exactly 3 minute divisions apart on the dial?

Introduction / Context:
This problem deals with the relative positions of the hour and minute hands of a clock. The phrase "3 minutes apart" refers to 3 minute divisions on the clock dial. Each minute division corresponds to 6 degrees, so the required angular separation is 18 degrees. The question tests the ability to set up and solve a simple equation using the angular speeds of the two hands.

Given Data / Assumptions:

• We consider the time between 5 oclock and 6 oclock.
• The hands of the clock are to be 3 minute divisions apart, that is 18 degrees apart.
• The minute hand moves 6 degrees per minute.
• The hour hand moves 0.5 degree per minute.

Concept / Approach:
At any time t minutes after 5 oclock: hour hand angle = 30 * 5 + 0.5 * t = 150 + 0.5 t degrees, minute hand angle = 6 * t degrees. The separation between the two hands is the absolute difference of these angles. We set this difference equal to 18 degrees, solve for t, and ensure that the solution lies between 0 and 60 minutes.

Step-by-Step Solution:
Step 1: Write the angles. Hour hand angle H = 150 + 0.5 t. Minute hand angle M = 6 t. Step 2: Separation condition. We need |M - H| = 18 degrees. Between 5 and 5:24 the minute hand is behind but catching up, so initially H is ahead. Hence use H - M = 18 for the relevant interval. Step 3: Form the equation. (150 + 0.5 t) - 6 t = 18. 150 + 0.5 t - 6 t = 18. 150 - 5.5 t = 18. 5.5 t = 150 - 18 = 132. t = 132 / 5.5 = 24 minutes. So the hands are 3 minute divisions apart at 24 minutes past 5.

Verification / Alternative check:
At 5:24: Hour hand angle = 150 + 0.5 * 24 = 150 + 12 = 162 degrees. Minute hand angle = 6 * 24 = 144 degrees. Difference = 162 - 144 = 18 degrees, which is exactly 3 minute divisions (3 * 6 degrees). Therefore, the result is correct.

Why Other Options Are Wrong:
Option b (12 minutes) and option c (13 minutes) would give smaller angles than 18 degrees between the hands when substituted into the formula, whereas option d (14 minutes) also does not satisfy the 18 degree separation condition. Only at 24 minutes past 5 do we get the exact required separation.

Common Pitfalls:
Learners sometimes treat "3 minutes apart" as 3 minutes of time instead of 3 minute markings on the dial. Another common mistake is to forget that the hour hand moves as minutes pass, incorrectly fixing it at 150 degrees. Ignoring the absolute value and choosing the wrong sign in the angle difference equation can also lead to a negative separation or a wrong time.

Final Answer:
The hands of the clock are exactly 3 minute divisions apart at 24 minutes past 5.