The geometric mean of two numbers is found by multiplying the two items, then applying the square root
GM = geometric mean
GM = sqrt(x*y)
4 = sqrt(x*y)
xy = 4^2
xy = 16
y = 16/x
y/16 = 1/x
1/x = y/16
The harmonic mean (HM) is found by first adding the reciprocals of the numbers.
Once you determine the sum, you divide n over it.
This is the same as multiplying n by the reciprocal of the sum calculated.
This is what the harmonic mean looks like for 2 numbers x and y:
x,y = original numbers
1/x,1/y = reciprocals of those said numbers
Add those reciprocals
1/x + 1/y
y/16+1/y ... replace 1/x with y/16 due to the previous equation above
(y^2)/(16y) + 16/(16y)
(y^2+16)/(16y)
The reciprocal of that is 16y/(y^2+16)
Multiplied with n = 2 gets us 32y/(y^2+16)
Set that equal to the stated HM of 3.2 and solve for y
HM = 3.2
32y/(y^2+16) = 3.2
32y/(y^2+16) = 32/10
32y/(y^2+16) = 16/5
5*32y = 16(y^2+16)
160y = 16y^2+256
16y^2-160y+256 = 0
16(y^2-10y+16) = 0
y^2-10y+16 = 0
(y-8)(y-2) = 0
y-8 = 0 or y-2 = 0
y = 8 or y = 2
If y = 8, then,
xy = 16
x = 16/y
x = 16/8
x = 2
Or if y = 2, then,
xy = 16
x = 16/y
x = 16/2
x = 8
We get the same pair of numbers just in a different order.
The order doesn't matter.
The two numbers are 2 and 8
We can verify that sqrt(2*8) = sqrt(16) = 4 is the correct geometric mean
And that n/(1/x + 1/y) = 2/(1/2 + 1/8) = 3.2 is the correct harmonic mean
This fully verifies our answers.
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