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* = 2*A*

Multiply both sides by (*m* – *n*).

(*m* + *n*)(*m* – *n*) = 2*A*(*m* – *n*),

*m*^{2} – *n*^{2} = 2*Am* – 2*An*.

*m*^{2} – 2*Am* = *n*^{2} – 2*An*,

Now, add *A*^{2} to both sides.

*m*^{2} – 2*Am* + *A*^{2} = *n*^{2} – 2*An* + *A*^{2}.

Thus (*m* – *A*)^{2} = (*n* – *A*)^{2}.

So (*m* – *A*) = (*n* – *A*), 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?

^{2}= 5

^{2}but that doesn’t mean that -5 = 5.

*n*>

*m*, i.e.,

*m*–

*n*< 0.

A = | m + n |
, so (m – A) = m – |
m + n |
= | m |
– | n |
= | m – n |
< 0. |

2 | 2 | 2 | 2 | 2 |

*m*– A) is negative.

But n – A = n – |
m + n |
= | n |
– | m |
= | n – m |
> 0 |

2 | 2 | 2 | 2 |

*n*– A is positive.

*m*– A)

^{2}= (

*n*– A)

^{2}, we can’t say that

*m*– A =

*n*– A.