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

1 Answer
Nov 29, 2016

Answer:

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#,