00:00Now, our third average type, friends.
00:03We are in the harmonic mean.
00:04What is the harmonic mean?
00:06The harmonic mean of two numbers, X and Y, is inversely...
00:29The sum of the multiplicative inverses of numbers...
00:34The sum of the multiplicative inverses of numbers...
00:55...is multiplied by one-half.
01:02These two numbers are for friends.
01:05The harmonic mean of the upper number...
01:23...reverse...
01:25...again, they're the opposite, friends.
01:28The multiplicative inverses of numbers...
01:40...according to multiplication, friends...
01:47...their opposites...
01:48...what is it, friends?
01:51It is the sum multiplied by one-third.
02:05So, this is what the harmonic mean looks like, folks.
02:08Let's write this in general terms now.
02:09If X and Y are...
02:11...and two numbers...
02:12X and Y...
02:16...harmonic mean...
02:17...I write it as he.
02:18As an abbreviation, friends.
02:19We write it as one divided by H.
02:22This changes.
02:23How many numbers are there, friends?
02:24Two of them.
02:25Then we call it one-half.
02:27Times...
02:27...what are we doing, guys?
02:29We add them together with their multiplicative inverses.
02:31How does that happen?
02:31What are the multiplicative inverses of the number two?
02:33It is one divided by x.
02:35I'm saying plus one divided by y.
02:38Alright...
02:39...these were X and Y.
02:41X...
02:43...Y...
02:43...and Z's...
02:47...and that's the harmonic mean, folks.
02:49In the same way again...
02:50...we call it one divided by the harmonic mean.
02:53It's different.
02:54One third.
02:54Because there are three numbers.
02:56All three numbers...
02:57...we multiply by the sum of their multiplicative inverses.
03:02That's it.
03:03To summarize generally, friends...
03:06...in the form of a formula...
03:06...x1...
03:09...x2...
03:09...the numbers xn...
03:12...n numbers...
03:15...harmonic mean...
03:17...h o'su...
03:20...if h then...
03:24...what is this, guys?
03:25...one part h...
03:26...that is equal to...
03:28...one section...
03:29...however many numbers there are...
03:30...reverse...
03:31...times...
03:33...one section...
03:34...x1...
03:34...plus...
03:35...one section...
03:36...x2...
03:37...plus...
03:39...one section...
03:40...xn...
03:40...is...
03:42...we say.
03:43Alright...
03:44...now let's move on to the minor features, folks.
03:46...n numbers...
03:53...let's say...
03:55...arithmetic mean...
04:00...I'm saying...
04:03...as a short six-part sentence...
04:04...let it be.
04:05Alright...
04:06...this...
04:06...geometric mean...
04:12...if g...
04:15...let's just say it happens...
04:16...and its harmonic mean is...
04:22...h then...
04:25...what is this all about, folks...?
04:27...in this situation?
04:29...h...
04:31...harmonic mean...
04:32...must be less than or equal to...
04:34...from the geometric mean...
04:35...it must also be less than or equal to...
04:36...is the arithmetic mean...
04:38...we say.
04:40...
04:41...this is important.
04:42...now, friends...
04:43...another feature...
04:44...again...
04:44...n numbers...
04:46...Friends...
04:46...in the same way again...
04:47...arithmetic mean...
04:48...a...
04:48...geometric mean...
04:49...g...
04:49...harmonic mean...
04:50...h then...
04:50...the same applies.
04:52...in this case, friends...
04:55...and here...
04:56...valid for two numbers...
04:57...But...
04:57...let's pay attention to this...
05:01...geometric means...
05:03...square...
05:04...that is equal to...
05:05...a...
05:06...well...
05:06...armitic means...
05:08...times...
05:09...are their harmonic averages.
05:11...now, let me give a small example...
05:13...let's try to pick it up...
05:15...let's say...
05:16...forty...
05:17...And...
05:18...let it be sixty...
05:22...of the numbers...
05:24...what is it, friends...?
05:25...harmonic mean...
05:26...how many, let's say...
05:27...possible...
05:30...how many...
05:31...how were we doing it, guys...?
05:33...we were saying h to each other...
05:34...that is equal to...
05:35...because there are two numbers...
05:37...we used to say one to two...
05:38...times...
05:39...two numbers...
05:40...what was it...
05:41...according to the multiplication...
05:41...we were getting the opposite of them...
05:42...together 40...
05:44...plus...
05:45...together 60...
05:46...when we perform this operation...
05:48...h from here...
05:48...that is, the harmonic mean...
05:49...we find it from here...
05:50...now, friends...
05:51...how are we going to do this...
05:52...let's do the part inside the parentheses first...
05:53...from the priority of operations...
05:55...together 40...
05:55...plus...
05:56...together 60...
05:56...what are the denominators equalized at, friends...?
05:58...one hundred and twenty...
05:59...I'll put 3 here...
06:00...I'm putting 2 here...
06:01...what's going on, guys...?
06:02...two times each...
06:03...three plus two...
06:04...five divided by 120...
06:07...let's make this happen...
06:09...One...
06:09...what is it, friends...?
06:10...24...
06:11...one in 48...
06:13...in this situation...
06:15...our harmonic mean...
06:16...equals...
06:17...if one part H...
06:18...if one divided by 48 equals...
06:19...What is H, friends...?
06:20...that is, the harmonic mean...
06:24...it is 48...
06:25...let's give another example...
06:27...Friends...
06:30...two numbers...
06:31...geometric mean...
06:33...let's say...
06:35...let there be two numbers...
06:37...again in love...
06:38...possible...
06:40...why...
06:42...if it is 18...
06:43...And...
06:45...harmonic mean...
06:46...if it is 12...
06:48...if possible...
06:50...why...
06:52...let them be divisible...
06:53...12...
06:55...whereas...
06:56...the arithmetic mean...
06:58...let's ask, friends...
06:59...how do we do this...
07:00...of course...
07:01...from our formula...
07:02...what is that, friends...?
07:03...the geometric mean...
07:04...square...
07:05...we use the one over there...
07:06...Friends...
07:07...arithmetic mean...
07:08...times...
07:09...harmonic mean...
07:09...was equal to...
07:10...if that's the case...
07:12...geometric mean...
07:13...what was our question...
07:13...18 squared...
07:16...that is equal to...
07:17...a multiplication...
07:18...it is 12...
07:19...isn't it...
07:19...harmonic mean...
07:20...what was it, friends...?
07:21...and 12...
07:22...the arithmetic mean...
07:23...was asking...
07:23...our arithmetic mean...
07:25...what can we conclude from this, friends...?
07:26...18 times...
07:28...18...
07:29...section...
07:29...from 12...
07:30...simplify by 6...
07:32...this place...
07:32...isn't that right, friends...?
07:33...Yes...
07:34...2...
07:36...simplify by 6...
07:37...3...
07:37...simplify by 1...
07:39...9...
07:39...3...
07:41...27...
07:41...our arithmetic mean...
07:43...according to our question...
07:44...27...
07:44...apparently...
07:45...
07:46...
07:46...
07:46...
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