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

1 Answer
Nov 29, 2016

Numbers are 44 and 1616,

Explanation:

Let the one number be aa and as the geometric mean is 88, product of two numbers is 8^2=6482=64.

Hence, other number is 64/a64a

Now as harmonic mean of aa and 64/a64a is 6.46.4,

it arithmetic mean of 1/a1a and a/64a64 is 1/6.4=10/64=5/3216.4=1064=532

hence, 1/a+a/64=2xx5/32=5/161a+a64=2×532=516

and multiplying each term by 64a64a we get

64+a^2=20a64+a2=20a

or a^2-20a+64=0a220a+64=0

or a^2-16a-4a+64=0a216a4a+64=0

or a(a-16)-4(a-16)=0a(a16)4(a16)=0

i.e. (a-4)(a-16)=0(a4)(a16)=0

Hence aa is 44 or 1616.

If a=4a=4, other number is 64/4=16644=16 and if a=16a=16, other number is 64/16=46416=4

Hence numbers are 44 and 1616,