At what time between 7 and 8 o'clock will the hands of a clock be in the same straight line but, not together?

### Concept & Opposite Directions "Straight line but not together" means the hands are pointing in exactly opposite directions, creating an angle of $180^\circ$. $$ \theta = |30H - 5.5M| $$ ### Step-by-Step Solution * Given: $H = 7$ and $\theta = 180^\circ$. * Plug in the values: $180 = |30(7) - 5.5M|$ * Evaluate the absolute value: $180 = |210 - 5.5M|$ * Since $210$ is greater than $180$, we use: $210 - 5.5M = 180$ * Rearrange to solve for $M$: $5.5M = 30$ * Convert decimal to fraction: $\frac{11}{2}M = 30$ * Solve for $M$: $M = \frac{60}{11} = 5 \frac{5}{11}$ minutes. ### Exam Strategy & Shortcut To find when hands are $180^\circ$ apart, identify the opposite hour on the clock face. The opposite of 7 is 1 (which represents 5 minutes). Multiply this minute equivalent by $\frac{12}{11}$. Shortcut: $5 \times \frac{12}{11} = \frac{60}{11} = 5 \frac{5}{11}$ min. ### Common Pitfall Mistakenly selecting "5 min. past 7" by just looking at the opposite number on the dial (1) and forgetting that the hour hand advances slightly past 7 during those 5 minutes. ### Final Answer Therefore, the correct answer is **$5 \frac{5}{11}$ min. past 7**.