At what time, in minutes, between 3 o'clock and 4 o'clock, both the needles will coincide each other?

### Concept & Clock Angles When the hands of a clock coincide, the angle between them is $0^\circ$. The angle $\theta$ between the hour and minute hand is calculated using the formula: $$ \theta = |30H - 5.5M| $$ ### Step-by-Step Solution * Given: The time is between 3 and 4 o'clock, so $H = 3$. The angle $\theta = 0$. * Substitute into the formula: $0 = |30(3) - 5.5M|$ * Simplify: $90 - 5.5M = 0$ * Solve for $M$: $5.5M = 90$ * Convert decimal to fraction: $\frac{11}{2}M = 90$ * Calculate $M$: $M = \frac{180}{11} = 16 \frac{4}{11}$ ### Exam Strategy & Shortcut To find the time when hands overlap between $H$ and $(H+1)$ o'clock, simply multiply the starting hour $H$ by $\frac{60}{11}$. Shortcut: $3 \times \frac{60}{11} = \frac{180}{11} = 16 \frac{4}{11}$ minutes past 3. ### Common Pitfall Misinterpreting the fraction conversion. Sometimes students calculate $90 / 5.5$ incorrectly by forgetting to move the decimal point, leading to arithmetic errors. ### Final Answer Therefore, the correct answer is **$16 \frac{4}{11} '$ past**.