# 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.

I am glad you enjoyed the puzzle. I hope it kept you busy for a few minutes, trying all those polynomial factoring possibilities that just don’t quite behave like they are supposed to. The title of the post, of course, is meant to misdirect you.

Thank you for trying the puzzle out. It is the simple ones that get you, isn’t it?

Hmm! But you might want to reread the question. You made a slight silly mistake ;-o !

Ha, ha.

Shawn

• err, yes I did! I shall cheat and fix it and no one will ever know 🙂