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# University of Toronto Mathematics Network

Answers and Explanations

## Simple and Compound Interest

If a bank account produces "simple interest", the interest is paid
directly and does not become part of the account's interest-bearing
balance. That balance remains constant, and it's easy to figure out
how much money you earn. For instance, if a $100 account earns 10%
simple interest a year, the balance remains at $100 and each year's
interest payment is $10.
The situation is the same for population growth if the offspring
produced are not capable of further reproduction: the fertile
population, and hence the rate of growth, remain constant.

But what if the offspring *can* reproduce further? Or, to return
to the bank example, what if the interest becomes part of the
interest-bearing balance as soon as it is earned, so that you earn
interst on the interest, and interest on the interest on the interest,
etc. (this is called "compound interest")? How much money will you
end up with? How different is it from what you would end up with under
simple interest alone?

The answer depends on how large the simple interest is compared to the
original amount. If it's only a small fraction, then including it with
the original amount won't change that amount by much, so the compound
interest (interest on the original amount plus previously earned
interest) won't be much different from the simple interest on the
original amount alone. The greater the simple interest is compared to
the original amount, the more of an effect compounding will have.

Let's ask the question in the case when the ratio is 1 (simple
interest = 100% of original amount):

**Question**: *If you would earn 100% interest (i.e., your
money would double) under simple interest, how much money would you
end up with under compound interest*?

**Answer**: *You would have e times your original amount*.

This gives us the following physical meaning for the number e:

The number e is the factor by which a bank account earning continually
compounding interest (or a reproducing population whose offspring are
themselves capable of reproduction, or any similar quantity that grows
at a rate proportional to its current value) will increase, if,
without the compounding (or without the offspring being capable of
further reproduction) it would have doubled (increased by 100%).
[Go on]

This page last updated: September 1, 1997

Original Web Site Creator / Mathematical Content Developer:
Philip Spencer

Current Network Coordinator and Contact Person:
Any Wilk - mathnet@math.toronto.edu

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