## 1730.41 – Need More Money?

Feeling tired and run down? Broke all the time? Need more money? Try a little algebra and enjoy as much money as you need.

Let $m$ be the money you have now and $n$ be the amount that will satisfy your needs. Now the average of the two is, as usual, $A = \frac{m+n}{2}$. We proceed as follows, employing powerful, timeless, well-tested tools of algebra. Watch carefully:

\begin{aligned} m+n &=& 2A \\ (m+n)(m-n) &=& 2A(m-n) \\ m^2 - n^2 &=& 2Am - 2An \\ m^2 - 2Am &=& n^2 - 2An \\ m^2 - 2Am + A^2 &=& n^2 - 2An + A^2 \\ (m-A)^2 &=& (n-A)^2 \\ m - A &=& n - A \\ m &=& n. \end{aligned}

Voila! It turns out that $m = n$, and you have exactly as much money as you need.

Find the flaw in the reasoning.

Solution

The flaw is an old one: just because $x^2 = (-x)^2$, it doesn't mean $x = -x.$ Often $x$ and $-x$ are different. So you can't go from $(m-A)^2 = (n-A)^2$ to $m - A = n - A$ .

(This doesn't mean that you can't trust algebra: you just have to be knowledgeable in how you use it.)

By the way, there is a similar argument for those who think they have more money than than they need.