A Little Algebra

Here's a problem that's nearly impossible to solve until you use a (very) little algebra, and then you can generate an infinite number of solutions:

Can you think of two numbers that give you the same result when you add them as when you multiply them? For example, 2 + 2 = 2 x 2, and there's another trivial solution, but are there any solutions to this when the numbers aren't the same? What if you include fractions or decimals?

If you try to guess and check your way to a solution, it seems impossible. But believe it or not, there are an infinite number of pairs that work. All you have to do is use a little algebra.

Translate this sentence into symbols: "Multiplying two numbers, x and y, is the same as adding x and y."

xy = x + y

I agree, this doesn't automatically lead to a list of solutions for me, either, but plug in a number for y and solve for x:

3x = 3 + x

2x = 3

x = 3/2

Sure enough, 3 (3/2) = 3 + 3/2 = 4.5

Plug in 4:

4x = 4 + x

3x = 4

x = 4/3

4 (4/3) = 4 + 4/3 = 5 1/3

See a pattern yet? It's pretty easy to generate tons of pairs of numbers that work now, having taken the most elementary step into algebra land. And that's the only reason to go there, in my opinion: to solve a real problem.