### 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.

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?

e” – where Finance meets Science

Many of you would know I am  a trained accountant.  Most would not know that my original training was in science and engineering.  It’s that aspect of my life the I would like to write about now.

All of you would however be familiar with the concept of interest rates.  A nightly topic on the news, interest rates determine how much spare cash we have after the mortgage payment.  And the total mortgage paid is a simple calculation.  To save time, I’m going to skip the math…  but the formula is:

FV = PV(1+r)^t

Where:

FV = Future value

PV = Present Value

r  =  Interest Rate

t  = time

its the formula for calculating the total amount you pay – and, (just like me), its not very sexy!

The Cool Thing though, is if you have a greedy banker!!