In every 60 minutes, by how many minutes on the dial does the minute hand gain over the hour hand?

Introduction / Context:
This question is phrased in terms of minutes on the dial rather than degrees. It asks how many dial minutes the minute hand gains over the hour hand in one hour, that is, in 60 minutes. This is another way of talking about relative speed, but instead of using degrees per minute, we use positions on the clock face measured in minute divisions.

Given Data / Assumptions:

• We consider a time interval of 60 minutes.
• On the clock dial, there are 60 minute divisions around the circle.
• The minute hand moves 60 minute divisions in 60 minutes, one full revolution.
• The hour hand moves from one hour mark to the next in 60 minutes.
• We must find how many minute divisions separate the two hands after 60 minutes, that is, the gain of the minute hand over the hour hand.

Concept / Approach:
On a 12 hour clock face, the hour hand moves from one number to the next in 60 minutes. Each hour mark corresponds to 5 minute divisions (because 12 * 5 = 60). Therefore, in 60 minutes, the hour hand moves 5 minute divisions, while the minute hand moves 60 minute divisions. The gain of the minute hand is the difference between these two movements.

Step-by-Step Solution:
Step 1: In 60 minutes, the minute hand completes one full revolution, moving through all 60 minute divisions on the clock face. Step 2: The hour hand moves from one hour mark to the next in 60 minutes. Each hour mark is separated by 5 minute divisions, so the hour hand moves 5 minute divisions in the same 60 minutes. Step 3: The gain of the minute hand over the hour hand is: gain = movement of minute hand minus movement of hour hand. Step 4: Substitute the values: gain = 60 - 5 = 55 minute divisions. Step 5: Therefore, in every 60 minutes, the minute hand gains 55 minutes on the dial over the hour hand.

Verification / Alternative check:
We can also express this in degrees to cross check. One minute division corresponds to 6 degrees (since 360 degrees / 60 divisions = 6 degrees per division). A gain of 55 minute divisions corresponds to 55 * 6 = 330 degrees. If we compute the relative angular speed earlier as 5.5 degrees per minute, then in 60 minutes the gain in degrees is 5.5 * 60 = 330 degrees, which again matches 55 minute divisions. This confirms that the gain is 55 minutes on the dial.

Why Other Options Are Wrong:
53: This would correspond to 318 degrees of gain, not matching the 330 degree result from relative angular speed.

54: This is 324 degrees and still less than the true gain.

56: This would represent 336 degrees, which is larger than the possible relative gain in 60 minutes.

50: This is a round number but does not align with the actual calculations in either minutes or degrees.

Common Pitfalls:
Some learners mistakenly assume that the hour hand does not move during the hour and conclude that the gain is all 60 minutes. Others misinterpret the phrase gain in minutes and confuse it with minutes of time instead of dial divisions. Carefully distinguishing between dial divisions and minutes of real time is essential for solving such questions correctly.

Final Answer:
In every 60 minutes, the minute hand gains 55 minutes on the hour hand.