When the minute hand of a clock gains on the hour hand, how much time does it gain in 60 minutes?

Steps to Solve

1. Understanding the speeds of the clock hands

The minute hand of a clock completes one full revolution (360 degrees) in 60 minutes. Hence, its speed is:

$$ \text{Speed of minute hand} = \frac{360 \text{ degrees}}{60 \text{ minutes}} = 6 \text{ degrees per minute} $$

The hour hand completes one full revolution in 12 hours (720 minutes). Therefore, its speed is:

$$ \text{Speed of hour hand} = \frac{360 \text{ degrees}}{720 \text{ minutes}} = 0.5 \text{ degrees per minute} $$

2. Calculating the relative speed

Now, we find the relative speed of the minute hand with respect to the hour hand by subtracting the speed of the hour hand from the speed of the minute hand:

$$ \text{Relative speed} = 6 \text{ degrees/min} - 0.5 \text{ degrees/min} = 5.5 \text{ degrees/min} $$

3. Calculating the gain over 60 minutes

To find out how much time the minute hand gains on the hour hand in 60 minutes, we multiply the relative speed by the time duration:

$$ \text{Gain} = \text{Relative speed} \times \text{Time} = 5.5 \text{ degrees/min} \times 60 \text{ min} = 330 \text{ degrees} $$

4. Determining how many hours this gain represents

Next, we convert the gain in degrees into time. Knowing that the hour hand moves 30 degrees per hour (since there are 360 degrees in a full circle and 12 hours in a clock):

$$ \text{Time gain in hours} = \frac{330 \text{ degrees}}{30 \text{ degrees/hour}} = 11 \text{ hours} $$

Since we are looking for how much the minute hand gains in a typical 1-hour period, we know it gains 11 hours of time over the hour.