Question

The geometric mean of two numbers is 8 and their harmonic mean is 6.4. What are the numbers?

Answer 1

Numbers are `4` and `16`,

Explanation:

Let the one number be `a` and as the geometric mean is `8`, product of two numbers is `8^2=64`.

Hence, other number is `64/a`

Now as harmonic mean of `a` and `64/a` is `6.4`,

it arithmetic mean of `1/a` and `a/64` is `1/6.4=10/64=5/32`

hence, `1/a+a/64=2xx5/32=5/16`

and multiplying each term by `64a` we get

`64+a^2=20a`

or `a^2-20a+64=0`

or `a^2-16a-4a+64=0`

or `a(a-16)-4(a-16)=0`

i.e. `(a-4)(a-16)=0`

Hence `a` is `4` or `16`.

If `a=4`, other number is `64/4=16` and if `a=16`, other number is `64/16=4`

Hence numbers are `4` and `16`,