Question

If two numbers have arithmetic mean 2700 and harmonic mean 75, find their geometric mean.

Answer 1

does that seem correct @vinhu ?

Answer 2

Not quite ! Since the answer is less than 999

Answer 3

might call in the big guns, hang on @Jhannybean and @oldrin.bataku can you guys help...?

Answer 4

let the two numbers be a,b.. arithmetic mean=a+b/2.. geometric mean=sqrt(ab).. two equations, two variables.. find a and b first..

Answer 5

once you have the two numbers.. (1/a + 1/b)/2 gives you the harmonic mean...

Answer 6

ah, cheers @yrelhan4

Answer 7

:)

Answer 8

and.. i should also mention it here, arith mean*geom mean is not equal to harmonic mean..
but for n=2. harmonic mean=G^2/A

Answer 9

we have n=2 here, so doesnt that mean that sqrt (harmonic x Arith) = geometric...?

Answer 10

geometric^2 on the rhs.

Answer 11

sqrt on the LHS...

Answer 12

ah, missed it. my bad. yes thats right.

Answer 13

so my initial guess was correct, and we dont have to solve for a and b...?

Answer 14

no you dont, if you know this formula. but again, the best and the safest thing is to start from the basics.. these formulas are certainly not for beginners..
and i thought you wrote, arith mean*geom=harmonic mean or something like that. sorry about that.

Answer 15

and i hope @vinhu got it?

Answer 16

O_o

Answer 17

no, good point, how you get it is important
but with this:
"once you have the two numbers.. (1/a + 1/b)/2 gives you the harmonic mean..."
i think harmonic is n/(1/a + 1/b), so it should be 2 / (1/a + 1/b)... yeah?

Answer 18

err wth. i am really messing it up today.
yes. 1/H=(1/a + 1/b)/2.. so H=2/(1/a + 1/b).

Answer 19

cool, thanks man, i learnt something new just then, cheers @yrelhan4

Answer 20

I was wrong twice. thanks to you too. :)

Answer 21

Could anyone summarize the point for me since I haven't learned this kind of mathematics before !

Answer 22

ok look..
arithmetic mean for two numbers= a+b/2
geometric mean= sqrt(ab)
harminic mean= 2/(1/a+1/b)..
you are given the values of arith and geom mean.. solve those two equations and find the values of a and b.. then put those values in the third equation to get your answer.

Answer 23

I'm still in the middle of nowhere. More detail please!!! My a , b is extremely huge. Also they are decimal numbers !

Answer 24

Just round them off to up to one decimal points.. and you have all the details you need to solve this.

Answer 25

a= 37.7 ish, b = 5362.2 ish... yeah? @vinhu

Answer 26

all u need so do is solve them like a series of simultaneous equations
arithmetic mean for two numbers= a+b/2
2700 = a + b / 2
5400 = a + b
so a = 5400 - b

plug that into:
harmonic mean= 2/(1/a+1/b)
75 = 2/(1/a+1/b)

and a = 5400 - b
so
75 = 2/(1/(5400 - b)+1/b)
solve for b
use that to find a

use a and b to find geometirc mean ( sqrt (axb))

Answer 27

just an alternate way....might be easier.
5400 = a+b
75= 2ab/ (a+b) = 2ab/ 5400
so, ab = 75*5400/2 = 202500
and Geometric mean is just \(\sqrt{ab}=\sqrt{202500}=450\)
see, not much solving was involved :)

Answer 28

and definitely no need to find a and b.