Question

At what time between 5.00 and 7.00, the hands of clock are straight and point in opposite direction?

• A. 30 min
• B. 40 min
• C. 55 min
• D. 60 min

Hint:

Remember the special positions of clock hands:

\textbf{Together (0°):} Approximately every 65 minutes.
\textbf{Opposite (180°):} Once every hour, except between 5 and 7 where it happens only at 6:00.
\textbf{Right angles (90°):} Twice every hour.
Knowing that the 180° position only happens once at 6:00 between 5 and 7 is a quick way to solve this problem.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
The hands of a clock are straight and in opposite directions when they are 180 degrees apart. This corresponds to a time difference of 30 minutes on the clock face. We need to find when this happens between 5:00 and 7:00. This will happen once between 5:00 and 6:00, and once between 6:00 and 7:00.

Step 2: Key Formula or Approach
The number of minutes past H o'clock when the hands are 180 degrees apart is given by the formula:
\[ M = \frac{60}{11} \times (H \pm 6) \] We use \( (H-6) \) if \(H>6\) and \( (H+6) \) if \(H<6\).
Alternatively, we can use the concept of relative speed. The minute hand gains 5.5 degrees on the hour hand every minute. The hands are 180 degrees apart.

Step 3: Detailed Explanation
Case 1: Between 5:00 and 6:00
Here H = 5. Since H<6, we use (H+6).
\[ M = \frac{60}{11} \times (5 + 6) = \frac{60}{11} \times 11 = 60 \text{ minutes} \] This means the hands are opposite at 60 minutes past 5, which is exactly 6:00.
Case 2: Between 6:00 and 7:00
At 6:00, the hour hand is at 6 and the minute hand is at 12. They are exactly 180 degrees apart. So, 6:00 is one of the times. This corresponds to 60 minutes past 5 o'clock.
Let's check for the time after 6:00. Here H = 6.
\[ M = \frac{60}{11} \times (6 - 6) = 0 \text{ minutes} \] This confirms the time is 0 minutes past 6, i.e., 6:00.
The question asks for a time between 5:00 and 7:00. The only time this happens is exactly at 6:00. The options are given in minutes, likely minutes past some hour. Looking at the options: (A) 30 min: at 5:30, the hands are not opposite. (B) 40 min: at 5:40, the hands are not opposite. (C) 55 min: at 5:55, the hands are not opposite. (D) 60 min: This corresponds to 60 minutes past 5 o'clock, which is 6:00. At 6:00, the hands are exactly opposite. This fits the condition.
Let's re-read the provided answer key. It states D is correct. So 60 min is the answer. This confirms our finding that at 6:00 (i.e. 60 minutes past 5), the hands are opposite. The question is slightly ambiguous about what the minutes refer to, but 60 minutes past 5 is the only logical answer.

Step 4: Final Answer
The hands of a clock are in a straight line and opposite to each other at exactly 6:00. This time is 60 minutes past 5:00. Therefore, the option 60 min is the correct answer.