SOLUTION: The arithmetic mean of two numbers is 5 and the geometric means of the same numbers is 3.find the numbers

Algebra ->  Sequences-and-series -> SOLUTION: The arithmetic mean of two numbers is 5 and the geometric means of the same numbers is 3.find the numbers      Log On


   

Question 799680: The arithmetic mean of two numbers is 5 and the geometric means of the same numbers is 3.find the numbers
Found 2 solutions by josgarithmetic, thejackal:
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
The two numbers are x and y.
%28x%2By%29%2F2=5 and sqrt%28xy%29=3.

x%2By=2%2A5=10
y=10-x
Substitute.
sqrt%28x%28y%29%29=sqrt%28x%2810-x%29%29=3
sqrt%2810x-x%5E2%29=3
Square both sides.
10x-x%5E2=9
10x-x%5E2-9=0
x%5E2-10x%2B9=0
%28x-9%29%28x-1%29=0

x=9 or x=1
y=10-9 or y=10-1
y=1 or y=9

ANSWER: Either way, the two numbers are 9 and 1.

Answer by thejackal(72) About Me  (Show Source):
You can put this solution on YOUR website!
Geometric mean: is the root of the products of the number
thus:
root(a.b) = 3
thus (a.b) = 3^2 = 9
Arithmetic mean: is the sum of the numbers divide by the number of number
thus:
(a+b)/2 = 5 or a+b = 10
b can be return as a function of a thus b = 10 - a
take this new function and use it in the first such that
(a.(10-a)) = 9
now you have a quadratic equation
a^2 - 10a + 9 = 0
solve it and you have a = 1 or 9 the same for b