Between 4 o'clock and 5 o'clock, at what exact time (in minutes past 4) will the hour and minute hands of a clock first be at a right angle of 90 degrees to each other?

Introduction / Context:
This problem is a classic example of relative speed and angular movement of clock hands. You are asked to determine the first time between 4 o'clock and 5 o'clock when the hour hand and the minute hand are at right angles (90 degrees). Such questions typically require you to use the standard formula for the angle between the hands and solve a simple linear equation in minutes past a given hour.

Given Data / Assumptions:

• We are working between 4:00 and 5:00 on a clock.
• Let M be the number of minutes past 4 o'clock.
• The hour hand moves 0.5 degrees per minute.
• The minute hand moves 6 degrees per minute.
• The angle between the hands must be 90 degrees for a right angle.

Concept / Approach:
The standard expression for the angle between the hour hand and the minute hand at a given time is based on their positions from 12 o'clock. At H hours and M minutes, the hour hand is at 30H + 0.5M degrees, and the minute hand is at 6M degrees. The angle between them is the absolute difference of these two positions. For a right angle, we set this difference equal to 90 degrees and solve for M.

Step-by-Step Solution:
Step 1: Here H = 4, and M is the unknown minutes past 4 o'clock. Step 2: Hour hand angle = 30 * 4 + 0.5 * M = 120 + 0.5M degrees. Step 3: Minute hand angle = 6M degrees. Step 4: Angle between hands = |(120 + 0.5M) - 6M| = |120 - 5.5M| degrees. Step 5: For the first time they are at right angles after 4:00, we set 120 - 5.5M = 90 (because initially the hour hand is ahead). Step 6: 120 - 5.5M = 90 implies 5.5M = 30. Step 7: M = 30 / 5.5 = (30 * 2) / 11 = 60/11 minutes. Step 8: So the first time they are at right angles between 4 and 5 o'clock is 60/11 minutes past 4, which is approximately 5 minutes and 27 seconds after 4:00.

Verification / Alternative check:
As an approximate check, 60/11 is about 5.4545 minutes. At roughly 4:05, the hour hand has moved slightly from its 4 o'clock position, and the minute hand has moved a bit beyond the 1-minute mark region. The calculated angle using M ≈ 5.45 minutes will still be very close to 90 degrees. Plugging M = 60/11 back into the expression |120 - 5.5M| confirms the angle is exactly 90 degrees, verifying our solution.

Why Other Options Are Wrong:
• 58/11 minutes: This value does not satisfy the equation |120 - 5.5M| = 90 and leads to a slightly different angle.
• 422/11 minutes and 420/11 minutes: These are far beyond 60 minutes and thus cannot represent a time between 4:00 and 5:00. They are clearly invalid in the given interval.

Common Pitfalls:
Learners sometimes forget that the hour hand moves during the hour and treat it as fixed at 120 degrees (4 * 30), leading to incorrect equations. Another frequent error is solving for both 120 - 5.5M = 90 and 5.5M - 120 = 90 without selecting the solution that lies in the correct interval or represents the first occurrence. Always ensure that the computed M is between 0 and 60 and corresponds to the required first time in that hour.

Final Answer:
The two hands of the clock are at right angles for the first time between 4 and 5 o'clock at 60/11 minutes past 4 o'clock.