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 (m – n).
(m + n)(m – n) = 2A(m – n),
m2 – n2 = 2Am – 2An.
m2 – 2Am = n2 – 2An,
Now, add A2 to both sides.
m2 – 2Am + A2 = n2 – 2An + A2.
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?
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., m – n < 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.)