Question 4 of 24
John’s clock is broken. The minute hand rotates around the clock correctly, but the hour hand is stuck in the three o’clock position. Suppose John first looks at the clock when the hands are aligned and it shows 3:15. He looks at the clock again and sees that the hour and minute hands of the clock form the arms of an angle measuring –135°. How many degrees could the minute hand have rotated to reach its current position?
A. –45°
B. –315°
C. –405°
D. –495°
Since the hour hand is stuck at the 3 o'clock position, it does not move and is always pointing in the same direction. We can imagine it as a fixed reference point.
At 3:15, the minute hand is 1/4 of the way between the 3 and 4 o'clock positions, which is 7.5 minutes away from the 3 o'clock position. Each minute, the minute hand moves 6 degrees (360 degrees/60 minutes), so in 7.5 minutes, it moves 45 degrees. Therefore, the minute hand has moved 45 degrees from the 3:15 position to its current position.
Now we need to figure out how many degrees the hands have moved between the two times that John looked at the clock, and then subtract 45 degrees from that to get the answer.
From 3:15 to the next hour (which the clock would show if the hour hand were working), the minute hand would travel 45 minutes (60 minutes - 15 minutes). Each minute, it moves 6 degrees, so in 45 minutes, it moves 270 degrees. However, the hour hand is stuck, so it does not move at all during this time.
Therefore, the angle between the hands at the second time John looked at the clock is 270 degrees. However, the problem specifies that this angle is measured as -135 degrees. This is because there are two possible angles between the hands: one acute angle and one obtuse angle. The acute angle is always less than 180 degrees, while the obtuse angle is always greater than 180 degrees. To get from the acute angle to the obtuse angle, we need to add 180 degrees. So the acute angle between the hands is 270 degrees - 180 degrees = 90 degrees, and the obtuse angle is 90 degrees + 180 degrees = 270 degrees.
Subtracting the 45 degrees that the minute hand has moved since 3:15, we get that the minute hand could have rotated -225 degrees or -495 degrees (270 degrees - 45 degrees - 180 degrees = -225 degrees; 270 degrees - 45 degrees - 360 degrees - 180 degrees = -495 degrees).
Of these two options, the answer choice that is less than 360 degrees (a full rotation of the clock) is -225 degrees, which is answer choice A.
To solve this problem, we need to find out how much the minute hand has rotated from the 3:15 position.
At 3:15, the minute hand is on the 3 o'clock mark, and the hour hand is stuck at the same position. Since the minute hand moves 360 degrees in 60 minutes (1 hour), at 3:15, the minute hand would have rotated 15 minutes (1/4 hour) * 360 degrees/hour = 90 degrees.
Now, let's consider the angle formed by the hour and minute hand when John looks at the clock again. We are given that this angle measures -135 degrees.
Since the hour hand is stuck in the 3 o'clock position, we can calculate the angle of the minute hand relative to the 12 o'clock position. At 3:15, the hour hand has moved 1/4 hour * 360 degrees/hour = 90 degrees.
Thus, the angle of the minute hand relative to the 12 o'clock position is -135 degrees - 90 degrees = -225 degrees.
However, the minute hand rotates 360 degrees in 60 minutes (1 hour). So, we need to find an equivalent angle between -225 degrees and 135 degrees (since the minute hand could rotate in either clockwise or counterclockwise direction).
To find the equivalent angle, we can add 360 degrees repeatedly until we get to an angle within the desired range.
-225 + 360 = +135 degrees
Therefore, the minute hand could have rotated +135 degrees clockwise or -225 + 360 = +135 degrees counterclockwise to reach its current position.
Since the question asks for the degrees in negative form, the answer is -135 degrees.
Therefore, the correct option is:
A. -45°