A man who went out between 3 and 4 and returned between 8...

A man who went out between 3 and 4 and returned between 8 and 9 , found that the hands of the watch had exactly changed places. He returned at
(a) 14 mins. past 8
(b) $21 \frac{1}{13}$ mins. past 8
(c) $19 \frac{2}{13}$ mins. past 8
(d) $18 \frac{6}{13}$ mins past 8

Key Concepts

Clock Angle Calculation

This concept involves determining the positions of the hour and minute hands in terms of angles measured from the top of the clock. The hour hand moves at 0.5 degrees per minute (30 degrees per hour) while the minute hand moves at 6 degrees per minute. Knowing these rates is fundamental to solving any problem related to the angles between the hands or their positions at any given time.

Relative Motion of Clock Hands

This key concept addresses how the minute and hour hands move relative to each other due to their different speeds. In many clock problems, including cases where the hands swap positions, the relative motion allows one to form relationships between the times shown by each hand. This is essential in setting up equations that capture the dynamics of the hands’ movements.

Formulating Algebraic Equations

Solving clock problems typically involves translating the conditions given into algebraic equations. By assigning variables to the unknown quantities (like the minutes past the hour) and using the relationships provided by the hand speeds and positions, one can establish equations that, when solved, reveal the exact times of interest.

Swapped Positions of Clock Hands

In some clock problems, the scenario is that the positions of the hour and minute hands are interchanged at different times. Understanding and setting up this condition requires knowing the expression for the angle of each hand at any time and equating them appropriately when the hands swap roles. This concept is crucial to analyze and solve such problems.

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