Between 6 and 7 o'clock, the two hands of a clock will coincide at
2023
Between 6 and 7 o'clock, the two hands of a clock will coincide at
Answer: B. 6 hours \(32\frac8{11}\) minutes — CONCEPTFor a clock-hands coincidence, compare angular positions measured from 12 o’clock. The minute hand advances at 6° per minute and the hour hand at 0.5°…
- A.
6 hours \(32\frac57\) minutes
- B.
6 hours \(32\frac8{11}\) minutes
- C.
6 hours \(32\frac9{11}\) minutes
- D.
More than one of the above
- E.
None of the above
Show answer & explanation
Correct answer: B
CONCEPT
For a clock-hands coincidence, compare angular positions measured from 12 o’clock.
The minute hand advances at 6° per minute and the hour hand at 0.5° per minute, so their relative speed is 5.5° per minute.
APPLICATION
Let t be the number of minutes after 6:00. At 6:00, the hour hand is 180° ahead of the minute hand.
After t minutes, the minute-hand angle is 6t and the hour-hand angle is 180 + 0.5t.
At coincidence, 6t = 180 + 0.5t, so 5.5t = 180.
Therefore \(t = 180/5.5 = 360/11 = 32\frac{8}{11}\) minutes.
CROSS-CHECK
At t = 360/11, both hands are at 2160/11° from 12, so their positions coincide.
Thus, the hands coincide at 6 hours \(32\frac{8}{11}\) minutes.