The reflex angle between the hands of a clock at 10.25 is

### Concept & Reflex Angle Calculation First, calculate the interior (acute/obtuse) angle $\theta$ between the hands using the formula $\theta = |30H - 5.5M|$. The "reflex angle" is the larger outer angle, defined mathematically as an angle greater than $180^\circ$. It is found by subtracting the interior angle from the full circle ($360^\circ$): $$\text{Reflex Angle} = 360^\circ - \theta$$ ### Step-by-Step Solution 1. **Given:** The time is 10:25. $H = 10, M = 25$. 2. Calculate the interior angle first: $$\theta = \left| 30(10) - 5.5(25) \right|$$ 3. $30 \times 10 = 300^\circ$ 4. $5.5 \times 25 = \frac{11}{2} \times 25 = \frac{275}{2} = 137.5^\circ$ 5. $\theta = |300^\circ - 137.5^\circ| = 162.5^\circ$. This is the interior angle. 6. Now, compute the reflex angle: $$\text{Reflex Angle} = 360^\circ - 162.5^\circ = 197.5^\circ$$ 7. Convert to fraction form: $197\frac{1}{2}^\circ$. ### Exam Strategy & Shortcut If a question asks for the "reflex angle" and you calculate an initial result greater than $180^\circ$ from the absolute difference formula, that *is* the reflex angle. In this case, our initial calculation was $162.5^\circ$ (less than $180^\circ$), so the $360^\circ$ subtraction step was mandatory. ### Common Pitfall A classic error is failing to read the word "reflex" and simply selecting the interior angle if it happens to be present in the options. Always double-check if the question asks for the interior or exterior (reflex) angle. ### Final Answer Therefore, the correct answer is **$197\frac{1}{2}^\circ$**.