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: At 9:00, minute hand at 12 (0 min spaces), hour hand at 9 (45 min spaces ahea(d). Difference = 45 min spaces.
Step 2: Relative speed of minute hand over hour hand = $5.5^\circ$ per minute or 0.5 min spaces per minute? Actually in terms of minute spaces: minute hand speed = 1 min space per minute, hour hand speed = $\frac{1}{12}$ min space per minute. Relative speed = $1 - \frac{1}{12} = \frac{11}{12}$ min spaces per minute.
Step 3: Need the hands to be 23 min spaces apart. Two possibilities: minute hand 23 spaces ahead of hour hand, or 23 spaces behind.
Step 4: Initially at 9:00, hour hand is 45 spaces ahead. For minute hand to be 23 spaces ahead, it needs to gain $45 + 23 = 68$ spaces? Actually if minute hand is ahead, it needs to cover the 45 space gap plus 23 more = 68 spaces. Time = $\frac{68}{11/12} = 68 \times \frac{12}{11} = \frac{816}{11} \approx 74.18$ minutes, which is >60 min, so not between 9-10.
Step 5: For minute hand to be 23 spaces behind hour hand, it needs to gain $45 - 23 = 22$ spaces. Time = $\frac{22}{11/12} = 22 \times \frac{12}{11} = 24$ minutes.
Step 6: So at 9:24, the hands are 23 spaces apart.
Step 7: Final Answer: At 9 : 24.