At 3:40 on an analog clock, what is the angle between the hour hand and the minute hand?

Introduction / Context:
Clock angle questions are a classic part of quantitative aptitude and reasoning tests. They check whether you understand how fast the hour hand and minute hand move and how to translate a given time into angles on the dial. In this question, we are asked to find the angle between the hour hand and the minute hand when the clock shows 3:40, which means we must carefully account for the fact that the hour hand has moved away from the exact 3 o clock position.

Given Data / Assumptions:

- Time shown on the clock is 3:40. - A full circle on the clock is 360 degrees. - The minute hand completes one full revolution in 60 minutes. - The hour hand completes one full revolution in 12 hours. - We are interested in the smaller angle between the two hands.

Concept / Approach:
The minute hand moves 360 degrees in 60 minutes, so it moves 6 degrees per minute. The hour hand moves 360 degrees in 12 hours, which is 360 degrees in 720 minutes, so it moves 0.5 degrees per minute. To find the angle of each hand from the 12 o clock position, we multiply the time in minutes by the appropriate rate and then take the difference between the two positions. The absolute difference gives the angle between the hands, and if needed we choose the smaller angle inside the clock face.

Step-by-Step Solution:
Step 1: At 3:40, the minute hand is at 40 minutes. Step 2: Angle of the minute hand from 12 is 40 * 6 = 240 degrees. Step 3: At 3:40, the hour hand has moved 3 full hours plus 40 minutes. Step 4: In 3 hours, the hour hand moves 3 * 30 = 90 degrees, because each hour mark is 30 degrees apart. Step 5: In 40 minutes, the hour hand moves 40 * 0.5 = 20 degrees further. Step 6: Total angle of the hour hand from 12 is 90 + 20 = 110 degrees. Step 7: The difference between the hands is |240 - 110| = 130 degrees. Step 8: The reflex angle would be 360 - 130 = 230 degrees, but the standard convention is to report the smaller angle, which is 130 degrees.

Verification / Alternative check:
Another way is to remember that at exactly 3:00 the hour hand is at 90 degrees. In 40 minutes it moves an extra 40 * 0.5 = 20 degrees, reaching 110 degrees. The minute hand at 40 minutes is at 240 degrees. The same subtraction 240 minus 110 gives 130 degrees. Since 130 degrees is less than 180 degrees, it is clearly the smaller angle between the hands, confirming our result.

Why Other Options Are Wrong:
120 degrees is too small because it assumes the hour hand did not move enough from the 3 o clock position. 125 degrees and 135 degrees are near the correct result but do not match the precise calculation using 6 degrees per minute and 0.5 degrees per minute. 150 degrees is even larger and ignores the exact motion of the hour hand during the 40 minutes after 3:00.

Common Pitfalls:
Many learners forget that the hour hand moves continuously and not in jumps of one hour. They sometimes keep the hour hand fixed at 90 degrees for 3:00 and only move the minute hand, which leads to an incorrect angle. Another common mistake is to compute the reflex angle instead of the smaller interior angle without checking which one the question is asking. Always compute both the absolute difference and, if needed, 360 minus that difference, and then choose the smaller angle unless the question specifies reflex angle.

Final Answer:
The correct smaller angle between the hour hand and the minute hand at 3:40 is 130 degrees.