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Hello everyone, This is my first question in math.stackexchange. I want to know how the minute hand of the clock "gains" 55 minutes in 60 minutes. I have read this article-: Wiki Answers

about how an hour hand gains 55 mins in 60 mins. But my understanding of it is that the hour hand of the clock only moves 5mins in 60 mins. So how is that being called a "gain" of 55 mins? I mean obviously the hour hand moves much slower than the minute hand. I mean I am pretty confused here.

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The answer you link to says the minute hand gains $55$ minutes on the hour hand, not the other way around. As you say, in one hour the minute hand moves $360^\circ$, which the answer calls $60$ minutes. In one hour, the hour hand moves $30^\circ$, so the minute hand gains $330^\circ=55$ minutes.

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What the article means is that the hour hand has a sort of an advantage- it just has to move 5 minutes for 60 minutes. So technically they are expressing it as a gain for the hour hand against the minute hand.

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55 min. spaces are gained by minute hand in 60 min period.

to find how many spaces it has actually gained, we need to fix a standard point first..!! with respect to it, we need to see the difference by how much is it actually varying..!!

so let us assume the standard point to be the place where the minutes hand and hours hand has been coincided..!!

i may be 12:00 ,1.06, 2.11,3.17...etc..from there..60 minutes implies the minute hand must come back to the same point where it has started..!!

is it not..??

now, 60 minute passed and so minute hand covers 60 minute spaces. And the hour hand advances by 5 minute spaces..!!

so from the standard point fixed initially (we assumed the standard point is where the minute point and hours hand were coinciding..also..60 minutes will be passed when the minute hand comes back to the same position from where it started)

now, there is an absolute 60 min spaces covered by minute hand in 60 min and then there is 5 min spaces advanced by hour hand in 60 min period..!!

so on total.. total advancement is 60-5 = 55 minute spaces..!!

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I guess we need to look at it this way, when we compare the two hands, let us take instances where the two hands are at the same position.Then as the minute hand moves forward it obviously gains over the hour hand.(just like in a running race!)

So let's say the hands are together at 2:10 pm (starting point), the race starts & the minutes hand moves forward every minute. The minute hand runs a full circular track back to the starting point at 2.

Suppose the hour hand hadn't moved an inch, so clearly in terms of minutes the minute hand has won the race by 60 minutes.But that is not the case, the hour hand also moves and with respect to minutes it moves by 5 minutes i.e,from 2pm to 3pm. So if you know look at it ,the minute hand wins the race or gains by not 60 mins but by 55 mins, 5 minutes less because the hour hand moved.

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****We can see this visually as-****

H = hour hand

M = min hand

This is like a race between the hour and the minute hand as shown below

SP = Start Point

EP = End Point

(SP)I________________________________I(EP)

[H/M]<------------------60 mins---------------------->

After an hour, we see that

(SP)I_______________________________________________I(EP)
 <---5 mins--->[H]
               <--------------55 mins----------->[M]

As we see above, minute hands leads hour hand by 55 mins after 1 hour time

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