1. **Problem statement:** We need to find the time(s) when the angle between the hour hand and the minute hand of a clock is exactly 50°. 2. **Formula and rules:** - The angle between the hour and minute hands at time $h$ hours and $m$ minutes is given by: $$\theta = \left|30h - 5.5m\right|$$ where 30° per hour for the hour hand and 6° per minute for the minute hand are used. 3. **Set up the equation:** We want: $$\left|30h - 5.5m\right| = 50$$ which gives two cases: $$30h - 5.5m = 50 \quad \text{or} \quad 30h - 5.5m = -50$$ 4. **Solve for $m$ in terms of $h$:** - Case 1: $$30h - 5.5m = 50 \implies 5.5m = 30h - 50 \implies m = \frac{30h - 50}{5.5} = \frac{60h - 100}{11}$$ - Case 2: $$30h - 5.5m = -50 \implies 5.5m = 30h + 50 \implies m = \frac{30h + 50}{5.5} = \frac{60h + 100}{11}$$ 5. **Check for valid times:** - $h$ is an integer from 0 to 11 (12-hour clock) - $m$ must be between 0 and 59 6. **Calculate valid $m$ values:** - For $h=1$ to $11$, compute $m$ for both cases and check if $0 \leq m < 60$ and $m$ is a valid minute (can be fractional but usually rounded to nearest second). 7. **Examples:** - For $h=2$: Case 1: $m = \frac{60\times2 - 100}{11} = \frac{120 - 100}{11} = \frac{20}{11} \approx 1.818$ minutes Case 2: $m = \frac{60\times2 + 100}{11} = \frac{120 + 100}{11} = \frac{220}{11} = 20$ minutes Both are valid times: 2:01.82 and 2:20 8. **Final answer:** The times when the angle between the hour and minute hands is 50° are given by $$m = \frac{60h - 100}{11} \quad \text{or} \quad m = \frac{60h + 100}{11}$$ for integer $h$ where $m$ is between 0 and 59. For example, at 2:01.82 and 2:20 the angle is 50°.