## A bit of fun: Shawn Urban’s math puzzle

Before you read the rest, maybe you should have a look at Shawn’s blog… as what’s below may spoil the puzzle.

He writes:

Every math week, I plan to provide a math challenge which could take you seconds to hours to solve, assuming you don’t cheat by using technology to solve it for you.

Today, I offer a time filler I occasionally use while substituting a junior or senior high math class that has “completed” all of its work, usually before I even enter class. I learned this problem from Blaine Dowler of Sylvan Learning. Usually I reward the first person who solves it, and usually it is the “weakest” student who does it.

The challenge is: If the product of two numbers is two and their sum is five, what is the simplest sum of their inverses?

I think I understand why the “weakest” student answers first, because the naive answer is the correct one:

Let our numbers be $m$ and $n$. So:

$mn = 2$ and $n + m = 5$

We are looking to solve $\frac{1}{m} + \frac{1}{n}$

$\frac{1}{m} + \frac{1}{n} = \frac{n + m}{nm} = \frac{5}{2}$

Have to say this did not occur to me immediately at all, though I could hear a faint voice telling me to try it for some time before I actually did.