Question

At what time between 9 and 10 O'Clock are the hands of an ordinary clock 23 minute spaces apart?

• A. At 9 : 28
• B. At 9 : 26
• C. At 9 : 23
• D. At 9 : 37
• E. At 9 : 24

Hint:

Relative speed of minute hand = 5.5° per minute or $\frac{11}{12}$ minute spaces per minute.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
In a clock, the minute hand moves at a speed of 6 degrees per minute, and the hour hand moves at 0.5 degrees per minute. A "minute space" corresponds to 6 degrees. Therefore, 23 minute spaces correspond to \(23 \times 6 = 138^\circ\).
Step 2: Key Formula or Approach:
The angle \(\theta\) between the hands is given by: \[ \theta = |30H - 5.5M| \] where \(H\) is the hour and \(M\) is the minutes. Here, \(H = 9\) and \(\theta = 138^\circ\).
Step 3: Detailed Explanation:
Substitute the values into the formula: \[ 138 = |30(9) - 5.5M| \] \[ 138 = |270 - 5.5M| \] This gives two possible cases: Case 1: \(270 - 5.5M = 138\) \[ 5.5M = 270 - 138 \] \[ 5.5M = 132 \implies M = \frac{132}{5.5} = 24 \] Case 2: \(270 - 5.5M = -138\) \[ 5.5M = 408 \implies M = \frac{408}{5.5} \approx 74.18 \] The only valid solution is \(M = 24\).
Step 4: Final Answer:
The time is At 9 : 24.