At 3:40 on an analog clock, what is the angle in degrees between the hour hand and the minute hand (taking the smaller angle between them)?

Introduction / Context:
This question checks your understanding of how to calculate the angle between the hour hand and the minute hand of a clock at a given time. At 3:40, both hands have moved from their original positions, and you must carefully account for each hand's movement to find the smaller angle between them. Such problems are very common in aptitude tests and help build precision in handling relative motion on a circular dial.

Given Data / Assumptions:

• Time considered: 3:40 (3 hours and 40 minutes).
• The minute hand moves 6 degrees per minute (360 degrees / 60 minutes).
• The hour hand moves 30 degrees per hour (360 degrees / 12 hours).
• The hour hand also moves 0.5 degrees per minute (30 degrees / 60 minutes).
• We need the smaller of the two possible angles between the hands.

Concept / Approach:
To compute the angle between the two hands at a given time, we find their absolute positions in degrees from 12 o'clock and then take the absolute difference. The standard formulas are:
Minute hand angle = 6 * minutes. Hour hand angle = 30 * hours + 0.5 * minutes. The required angle is the smaller of the two possible angles between the hands, usually obtained by taking the absolute difference between these angles and, if necessary, subtracting from 360 degrees to ensure it is the acute or smaller angle.

Step-by-Step Solution:
Step 1: At 3:40, the minute hand is at 40 minutes. Step 2: Minute hand angle = 6 * 40 = 240 degrees from 12 o'clock. Step 3: The hour hand at 3:00 is at 30 * 3 = 90 degrees. Step 4: In 40 minutes, the hour hand moves further by 0.5 * 40 = 20 degrees. Step 5: Total hour hand angle at 3:40 = 90 + 20 = 110 degrees. Step 6: Difference between the hands = |240 - 110| = 130 degrees. Step 7: 130 degrees is already less than 180 degrees, so it is the smaller angle between the two hands.

Verification / Alternative check:
We can quickly check the reasonableness: at 3:30, the angle between the hands is 75 degrees. By 3:40, the minute hand has moved 10 more minutes (60 degrees), and the hour hand has moved 5 more degrees. So the angle should increase by about 55 degrees. Starting from 75 degrees, 75 + 55 = 130 degrees, which matches our exact calculation. This confirms that 130 degrees is consistent and correct.

Why Other Options Are Wrong:
• 110 degrees: This would be the hour hand angle alone, not the difference between the two hands.
• 120 degrees: This value does not match the calculated difference and ignores precise movement of both hands.
• 140 degrees: This overestimates the separation and does not fit the accurate calculations using standard formulas.

Common Pitfalls:
A frequent mistake is to ignore the movement of the hour hand within the hour and assume it stays fixed at the hour mark, which would give an incorrect angle. Some students also forget to take the absolute difference or mistakenly compute 360 minus the difference when the direct difference is already the smaller angle. Carefully using the formulas for each hand and then taking the absolute difference helps avoid these errors.

Final Answer:
The smaller angle between the hour and minute hands of the clock at 3:40 is 130 degrees.