At what exact time between 4 oclock and 5 oclock will the hour and minute hands of a clock be in opposite directions and form a straight line?

Introduction / Context:
This question involves relative motion of the hour and minute hands on a clock. We are asked to find the exact time between 4 oclock and 5 oclock when the two hands are in opposite directions, that is when the angle between them is 180 degrees. Such clock problems test a candidate's understanding of angular speed, relative speed, and algebraic setup using simple equations.

Given Data / Assumptions:

• We consider a standard clock face with 12 hours marked equally.
• The minute hand moves 360 degrees in 60 minutes.
• The hour hand moves 360 degrees in 12 hours, that is 30 degrees per hour.
• We are observing the period between 4 oclock and 5 oclock.
• We want the hands to be in exactly opposite directions, so the angle between them is 180 degrees.

Concept / Approach:
The minute hand moves faster than the hour hand. We measure angles from 12 oclock in degrees. At any given time: minute hand angle = 6 * m hour hand angle = 30 * h + 0.5 * m Here h is the hour and m is the minutes past that hour. Between 4 and 5, h = 4. For opposite directions, we need the absolute difference between the angles to be 180 degrees.

Step-by-Step Solution:
Step 1: Set up the angles at time 4:m. Hour hand angle = 30 * 4 + 0.5 * m = 120 + 0.5 m. Minute hand angle = 6 * m. Step 2: Condition for opposite directions. We need |6 m - (120 + 0.5 m)| = 180. Since after 4 oclock the minute hand moves ahead of the hour hand, we can take: 6 m - (120 + 0.5 m) = 180. Step 3: Solve the equation. 6 m - 120 - 0.5 m = 180 5.5 m - 120 = 180 5.5 m = 300 m = 300 / 5.5 = 300 * 2 / 11 = 600 / 11 minutes. m = 54 6/11 minutes. Thus the time is 4 hours 54 6/11 minutes.

Verification / Alternative check:
We can compute the actual angles to confirm. Minute hand angle at this time: 6 * (600 / 11) = 3600 / 11 degrees. Hour hand angle: 120 + 0.5 * 600 / 11 = 120 + 300 / 11 = (1320 + 300) / 11 = 1620 / 11 degrees. Difference: 3600 / 11 - 1620 / 11 = 1980 / 11 = 180 degrees. So the condition for opposite directions is satisfied exactly.

Why Other Options Are Wrong:
Option a (54 minutes) is close but ignores the fractional part and gives a slightly different angle. Options b and c propose other fractional minutes but they do not give an exact 180 degree separation when substituted into the formula. Only 54 6/11 minutes produces exactly 180 degrees between the hands.

Common Pitfalls:
Many learners forget that the hour hand also moves as minutes pass and incorrectly fix the hour hand at 120 degrees for the whole hour. Another common mistake is ignoring the absolute value and choosing the wrong sign for the equation. Rounding off the fractional minute too early also leads to approximate answers which do not match any option exactly.

Final Answer:
The hands of the clock are in opposite directions at 54 6/11 minutes past 4.