# All the Money You Need

If you are tired of feeling broke all the time, here is some algebra that will cheer you up.
Let m be the amount of money that you actually have now,
and let n be the amount of money that you think you need.

A, their average, is given by
A = (m + n) / 2, as usual.

Now, follow this argument and see how much better you feel.
m + n = 2A
Multiply both sides by (mn).
(m + n)(mn) = 2A(mn),
m2n2 = 2Am – 2An.
m2 – 2Am = n2 – 2An,
Now, add A2 to both sides.
m2 – 2Am + A2 = n2 – 2An + A2.
Thus (mA)2 = (nA)2.
So (mA) = (nA), and then m = n.
So you have all the money you need.

Isn’t mathematics wonderful? But how can this be? Can you find a flaw with this reasoning?

The flaw involves that same old thing about square roots:
(-5)2 = 52 but that doesn’t mean that -5 = 5.

In the problem, we’re assuming that n > m, i.e., mn < 0.
 A = m + n , so (m – A) = m – m + n = m – n = m – n < 0. 2 2 2 2 2
Thus, (m – A) is negative.

 But n – A = n – m + n = n – m = n – m > 0 2 2 2 2
Thus, n – A is positive.

So, while we can say that (m – A)2 = (n – A)2, we can’t say that m – A = n – A.

(And, by the way, for those who have more money than they think they need, there’s a similar argument.)