Navigation Panel: Previous | Up | Forward | Graphical Version | PostScript version | U of T Math Network Home

# University of Toronto Mathematics Network

Answers and Explanations

## 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 e^2 times the original amount (with "^"
standing for "to the power of" on text-only browsers; if your
browser can display graphics, switch to the graphical
version for better notation). So, if the
simple interest earned is 200% (2 times original amount), the final
balance under compound interest is e^2 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 e^R times the original balance.

This is one of the reasons why exponentials of the form e^x occur
more frequently in practice than do exponentials with other bases such
as 10^x, 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).

[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

Navigation Panel:

Go backward to Simple and Compound Interest

Go up to Does the Number e Have Special Meaning?

Go forward to The Number e as a Limit

Switch to graphical version (better pictures & formulas)

Access printed version in PostScript format (requires PostScript printer)

Go to University of Toronto Mathematics Network
Home Page