Question

find two numbers who sum is 64 and whose product is the greatest number possible

Answer 1

Hello

Answer 2

There is something called "Inequality of Arithmetic and Geometric Means".
\[\frac{ a+b }{ 2 }\ge \sqrt{ab}\]

Answer 3

It states that the arithmetic mean is always greater than the geometric mean.
(If a=b, then they're equal)

Answer 4

The information given is that a+b=64

Answer 5

If you solve the inequality, you get\[1024\ge ab \]
So the greatest possible product is 1024.

Answer 6

Generally speaking, if a sum of 2 numbers are given, and you're asked to find the greatest possible product, the 2 numbers are the same.

ex) The sum is 100. What is the greatest possible product of the 2 numbers?

a+b=100
a=b
a=50
b=50

So, the greatest possible product=ab=2500