At what time between 4 and 5 o'clock will the hands of a watch point in opposite directions?

### Concept & Clock Opposites Hands pointing in opposite directions indicate an angle of $180^\circ$. We can find the exact minute using the standard clock angle formula or the relative speed multiplier. $$ \theta = |30H - 5.5M| $$ ### Step-by-Step Solution * Given: $H = 4$ and $\theta = 180^\circ$. * Apply formula: $180 = |30(4) - 5.5M|$ * Simplify: $180 = |120 - 5.5M|$ * Since the minute hand must pass the hour hand to be $180^\circ$ ahead: $5.5M - 120 = 180$ * Solve for $5.5M$: $5.5M = 300$ * Substitute $\frac{11}{2}$ for $5.5$: $\frac{11}{2}M = 300$ * Solve for $M$: $M = \frac{600}{11} = 54 \frac{6}{11}$ minutes. ### Exam Strategy & Shortcut Find the hour directly opposite to 4 on the clock face, which is 10. The 10 mark represents 50 minutes. Multiply this by $\frac{12}{11}$: $50 \times \frac{12}{11} = \frac{600}{11} = 54 \frac{6}{11}$ minutes past 4. ### Common Pitfall Picking a simpler minute mark like 50 minutes (the opposite number 10) without accounting for the relative movement of the hour hand. ### Final Answer Therefore, the correct answer is **$54 \frac{6}{11}$ min. past 4**.