Question

If ratio of AM and GM of two numbers is `5/4`, what are the numbers and what is their HM?

Answer 1

Numbers are `2` and `8`. Their H.M. is `16/5`.

Explanation:

Let the two numbers be `a` and `b`.

Their A.M. is `(a+b)/2` and as G.M. is `sqrt(ab)`

and as ratio between A.M. and G.M. is `5/4`

`((a+b)/2)/(sqrt(ab))=5/4`

or `2a+2b=5sqrt(ab)`

or `4a^2+4b^2+8ab=25ab`

i.e. `4a^2+4b^2-17ab=0`

or `(4a-b)(a-4b)=0`

i.e. either `a/b=4` or `b/a=4`

Hence numbers are of the form `x` and `4x`

and their G.M. is `2x` and H.M.@ will be `(2*x*4x)/(x+4x)` or `(8x^2)/5x` or `(8x/5)`
Further, as difference of G.M. and H.M. is `4/5`

`2x-8/5x=4/5`

or `2/5x=4/5` and `x=2`

Hence numbers are `2` and `8`.

Observe their A.M. is `5`, G.M. is `4` and H.M. is `16/5`.

@ If `h` is H.M. between `a` and `b`,

`1/a,1/h,1/b` are in arithmetic sequence and hence

`2(1/h)=1/a+1/b=(a+b)/(ab)`

and `1/h=(a+b)/(2ab)` i.e. `h=(2ab)/(a+b)`