At what time between 3 o'clock and 4 o'clock are the hands of a clock exactly…
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At what time between 3 o'clock and 4 o'clock are the hands of a clock exactly opposite to each other?
- A.
50(6/11) past 3
- B.
49(9/11) past 3
- C.
50(7/11) past 3
- D.
49(1/11) past 3
Attempted by 2 students.
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Correct answer: D
The angle between the hour hand and the minute hand at H hours and M minutes is given by the formula θ = |30H − 5.5M| degrees.
The two hands point in exactly opposite directions when this angle equals 180°, so any ‘hands opposite’ question reduces to solving 30H − 5.5M = 180 or 30H − 5.5M = −180 for M and keeping only the value that lies between 0 and 59.
Take H = 3, since the question asks for the time between 3 o'clock and 4 o'clock.
Set up both cases of the formula: 30(3) − 5.5M = 180, and 30(3) − 5.5M = −180.
Case 1: 90 − 5.5M = 180 ⇒ −5.5M = 90 ⇒ M = −180/11 = −16(4/11). This is rejected because a number of minutes cannot be negative.
Case 2: 90 − 5.5M = −180 ⇒ −5.5M = −270 ⇒ M = 540/11 = 49(1/11).
So the hands of the clock are opposite to each other at 49(1/11) minutes past 3.
Cross-check: substituting M = 540/11 back into the formula gives |30(3) − 5.5 × (540/11)| = |90 − 270| = 180°, confirming that the hands are indeed 180° apart at this time.