Why Grandma budgeted with money in jars?

As promised, I’m back with the second installment of my thoughts on exponential behaviour.

My Grandmother had money in jars for everything…. Rent, food, housekeeping, buttons, even pin money…  everything!

You may recall my last blog outlining the relationship between exponential numbers, science and finance.  (This has always been a fascination of mine – I mean, interest rates and satellite orbits?? Why?)

Well “e” (or 2.71828…) is a very unique number….

We’ll get to that in a minute.  Let’s examine natural phenomena –  Bacterial growth rates.

Binary Fission
Bacteria reproduce by “binary fission”

Let’s assume a single bacterium is put in a jar at 11pm.   The bacterium reproduces itself every minute, i.e. the number of bacteria in the jar doubles every minute.  The number of bacteria increases in the sequence 1,2,4,8, etc.   After 1 hour the jar is completely full….

I ask you at what minute is the jar half full?

That’s right –  at 11:59 pm

And at 11:58 pm it is a quarter full.

And at the 11:57 pm it is an eighth full.

And so on, right back to the original bacterium.

Why is this important?

Well, interest rates work that way too!  We all know that your money is unlikely to double every minute, but it will double.  For example, at 10% your money will double every seven years.  And this is easy to calculate…. Just divide 70 by your interest rate:

70/10 = 7 years

The math is based on the natural logarithm of 2 (ie doubling), and 100Ln 2 = 69. Whatever, use your calculator…  but trust me its close enough to 70.

Another question…

At 11:58 pm, how many bacteria would have realized that they were running out of room?

And if you retire at age 65 and your money is invested at 10%, how old are you at 11:58pm?

Yep, that’s right….  You are 51 years old.

Have you got enough money in your jar?