+466 The positive geometric mean of two numbers is 4 and the arithmetic mean of the two numbers is 5. What are the two numbers?
+128577 Call the two numbers m and n
Arithmetic mean = [ m + n] / 2
Geometric mean = √[ m * n]
So we have that
[ m + n] / 2 = 5
m + n = 10
n = 10 - m
So, subbing this into the geometric mean formula, we have
√ [ m * (10 - m) ] = 4 simplify
√ [10m - m^2 ] = 4 square both sides
-m^2 + 10m = 16
-m^2 + 10m - 16 = 0 multiply through by -1
m^2 - 10m + 16 = 0 factor
(m - 8) ( m - 2) = 0 and setting both factors to 0 m = 8 or m = 2
And when m = 8 n = 2 or vice-versa
So......the two numbers are 8 and 2
