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## The General Situation

We've seen that *e* is the
factor by which a continually-compounding
bank account will increase if under simple interest it would have doubled
(increased by 100%).
What if the simple interest isn't 100%? In other words, suppose your
money wouldn't have doubled under simple interest, but increased by
some other factor?

Suppose for instance the simple interest is 200%. Then we can split
the time period up into two halves, with the simple interest being
100% for each half (for example, if you earn 200% simple
interest per year, you're earning 100% interest for each six-month period).

At the end of the first half and the beginning of the second half,
the balance under compound interest will be *e* times
the original balance.

At the end of the second half, the balance under compound interest
will be *e* times the balance at the beginning of the second
half. In other words, it will be

*e* times (*e* times original amount)

which is times the original amount. So, if the
simple interest earned is 200% (2 times original amount), the final
balance under compound interest is times the original
amount.
This turns out to be true in general. If the simple interest earned is
*R* times the original balance, then the final balance under
compound interest is times the original balance.

This is one of the reasons why exponentials of the form occur
more frequently in practice than do exponentials with other bases such
as , and why one usually uses exponentials and logarithms
base *e* rather than base 10 or any other base. (The other, more
important, reason has to do with the special role that base *e*
exponentials play in calculus).

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This page last updated: September 1, 1997

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Philip Spencer

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