Find the angle between the minute hand and the hour hand of a clock when the time shown is 8:30.

Introduction / Context:
This is another angle between hands problem, this time at 8:30. It checks the same core concept as earlier angle questions but at a different time. The key idea is to remember that both hands move and that the hour hand moves half a degree per minute in addition to its 30 degrees per hour component.

Given Data / Assumptions:

• The time is 8:30.
• Minute hand moves 6 degrees per minute.
• Hour hand moves 30 degrees per hour plus 0.5 degree per minute.
• We want the smaller angle between the two hands.

Concept / Approach:
Use the standard formulas: minute hand angle = 6 * minutes, hour hand angle = 30 * hours + 0.5 * minutes. Compute both angles and take their absolute difference. Choose the smaller angle if the reflex angle is produced as well.

Step-by-Step Solution:
Step 1: Compute minute hand angle. Minutes = 30. Minute hand angle = 6 * 30 = 180 degrees. Step 2: Compute hour hand angle. Hours = 8, minutes = 30. Hour hand angle = 30 * 8 + 0.5 * 30. Hour hand angle = 240 + 15 = 255 degrees. Step 3: Find the angle between the hands. Difference = |255 - 180| = 75 degrees. Since 75 degrees is less than 180 degrees, this is the required angle.

Verification / Alternative check:
Reflex angle would be: 360 - 75 = 285 degrees, which is larger and therefore not the standard angle between hands unless specifically requested. The 75 degree result is therefore correct for the usual interpretation.

Why Other Options Are Wrong:
Option a (80 degrees) and option d (105 degrees) are far from the computed difference. Option c (60 degrees) might appear from an incorrect assumption that the hour hand is at exact 240 degrees, ignoring the half degree per minute term. Only 75 degrees correctly represents the actual angular separation.

Common Pitfalls:
Students often forget to include the 0.5 degree per minute contribution to the hour hand angle and instead assume it stays at the hour mark for the entire hour. Another issue is mixing up which angle is taken as the difference, but using the absolute difference of the two angles always avoids sign mistakes.

Final Answer:
The angle between the minute hand and hour hand at 8:30 is 75 degrees.