Question

The product of two positive numbers is 100. Find the two numbers so that the sum of the numbers is as small as possible.

Answer 1

The two positive numbers whose product is
`100` and whose sum is the smallest possible are
`10` and
`10`, making the minimum sum
`20`.

Solution:

We're looking for two positive numbers that multiply to 100, but we also want their sum to be as small as possible. It's a bit like a puzzle!

First, let's name these two numbers
`x` and
`y`. We know that when we multiply them, we get 100. That's our first clue:

`xy=100`

Now, we're also trying to make the sum of these numbers really small. So, we're looking at
`x+y` and trying to make this as tiny as we can.

Handling two variables at once can get tricky, can't it? Let's simplify things by expressing one variable in terms of the other. From
`xy=100`, we can say:

`y=(100)/(x)`

This allows us to express the sum of
`x` and
`y` using only
`x`:

`x + (100)/(x)`

We want to find the smallest possible value for this sum. We find the minimum of a function by taking its derivative and setting it to zero.

So, we take the derivative of our sum function with respect to
`x`:

`(d)/(dx)(x+(100)/(x))=1-(100)/(x^2)`

And then set this equal to zero to find where the sum is at its minimum:

`1-(100)/(x^2)=0`

Rearrange this, and we get:

`(100)/(x^2) =1`

`100=x^2`

Solve for
`x`, we find that
`x` is
`10` (since we're only looking for positive numbers). And since
`y=(100)/(x)`,
`y` is also
`10`.

So both numbers are
`10`. They multiply to
`100`, and their sum
`20` is the smallest it can be.