Logical Reasoning Questions for CLAT | QB Set 101

During a workshop on time-measuring devices, a mathematics teacher explained how the hands of an analogue clock move. The minute hand completes a full rotation of 360° in 60 minutes, so it moves at 6° per minute. The hour hand completes 360° in 12 hours, meaning that it moves at 30° per hour or 0.5° per minute.

The teacher reminded the students that the hour hand does not remain fixed at a number while the minute hand moves. For example, at 4:20, the minute hand is at 120° from 12, while the hour hand has moved 10° beyond the number 4. Therefore, its position is 130° from 12.

The smaller angle between the hands may be calculated by finding the absolute difference between their positions. If the difference is greater than 180°, it must be subtracted from 360° to obtain the smaller angle. The hands overlap when the angle between them is 0°, form a right angle when the smaller angle is 90°, and point in opposite directions when the angle is 180°. Using these principles, the students were asked to solve several clock-based problems.

Questions

1. What is the smaller angle between the hands of a clock at 4:20?

A. 20°

B. 15°

C. 10°

D. 5°

2. What is the smaller angle between the hands of a clock at 7:20?

A. 90°

B. 95°

C. 105°

D. 100°

3. At what time between 2:00 and 3:00 will the hour and minute hands overlap?

A. Approximately 2:10:55

B. Approximately 2:12:30

C. Approximately 2:15:00

D. Approximately 2:18:20

4. What is the smaller angle between the hands of a clock at 9:30?

A. 105°

B. 90°

C. 75°

D. 120°

5. At what time between 5:00 and 6:00 will the hands of the clock be exactly opposite to each other for the first time?

A. Approximately 5:05:27

B. Approximately 5:08:11

C. Approximately 5:10:55

D. Approximately 5:16:22

Answers and Detailed Explanations

1. Correct Answer: C. 10°

At 4:20:

Position of the minute hand = 20 × 6° = 120°

Position of the hour hand = (4 × 30°) + (20 × 0.5°) = 120° + 10° = 130°

Therefore, the angle between the hands is 130° − 120° = 10°.

Hence, the correct answer is Option C.

2. Correct Answer: D. 100°

At 7:20:

Position of the minute hand = 20 × 6° = 120°

Position of the hour hand = (7 × 30°) + (20 × 0.5°) = 210° + 10° = 220°

The difference between the two positions is 220° − 120° = 100°. Since 100° is less than 180°, it is already the smaller angle.

Hence, the correct answer is Option D.

3. Correct Answer: A. Approximately 2:10:55

At 2:00, the hour hand is at 2 × 30° = 60°.

Let the hands overlap after m minutes. The minute hand moves at 6° per minute, while the hour hand moves at 0.5° per minute. Therefore, the minute hand gains on the hour hand at 6° − 0.5° = 5.5° per minute.

To cover the initial gap of 60°, m = 60 ÷ 5.5 = 10.909 minutes.

The decimal portion, 0.909 minute, equals approximately 55 seconds. Thus, the hands overlap at approximately 2:10:55.

Hence, the correct answer is Option A.

4. Correct Answer: A. 105°

At 9:30:

Position of the minute hand = 30 × 6° = 180°

Position of the hour hand = (9 × 30°) + (30 × 0.5°) = 270° + 15° = 285°

The difference between the hands is 285° − 180° = 105°. Since 105° is less than 180°, it is the smaller angle.

Hence, the correct answer is Option A.

5. Correct Answer: C. Approximately 5:10:55

At 5:00, the hour hand is at 5 × 30° = 150°, while the minute hand is at 0°.

Let the hands become opposite after m minutes. Their positions will be 150° + 0.5m and 6m respectively. For the first opposite position, the minute hand must be 180° away from the hour hand.

Using 150 + 0.5m − 6m = 90, we get 150 − 5.5m = 90. Therefore, 5.5m = 60 and m = 60 ÷ 5.5 = 10.909 minutes.

This is approximately 10 minutes and 55 seconds. Therefore, the hands are opposite at approximately 5:10:55.

Hence, the correct answer is Option C.