One very important exponential
equation is the compound-interest formula:

...where "A"
is the ending amount, "P"
is the beginning amount (or "principal"), "r"
is the interest rate (expressed as a decimal), "n"
is the number of compoundings a year, and "t"
is the total number of years.

Regarding the variables,
n
refers to the number of compoundings in any one year, not to the total
number of compoundings over the life of the investment. If interest is
compounded yearly, then n
= 1; if semi-annually,
then n
= 2; quarterly, then
n
= 4; monthly, then
n
= 12; weekly, then
n
= 52; daily, then
n = 365; and
so forth, regardless of the number of years involved. Also, "t"
must be expressed in years, because interest rates are expressed that
way. If an exercise states that the principal was invested for six months,
you would need to convert this to^{
6}/_{12}
= 0.5 years; if it
was invested for 15
months, then t
= ^{15}/_{12} = 1.25
years; if it was invested for 90 days, then t
= ^{90}/_{365} of
a year; and so on.

Note that, for any given
interest rate, the above formula simplifies to the simple exponential
form that we're accustomed to. For instance, let the interest rate r
be 3%,
compounded monthly, and let the initial investment amount be $1250.
Then the compound-interest equation, for an investment period of t
years, becomes:

...where the base is 1.0025
and the exponent is the linear expression 12t.

To do compound-interest
word problems, generally the only hard part is figuring out which values
go where in the compound-interest formula. Once you have all the values
plugged in properly, you can solve for whichever variable is left.

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Suppose that
you plan to need $10,000
in thirty-six months' time when your child starts attending university.
You want to invest in an instrument yielding
3.5% interest, compounded monthly.
How much should you invest?

To solve this, I have
to figure out which values go with which variables. In this case, I
want to end up with $10,000,
so A
= 10,000. The interest
rate is 3.5%,
so, expressed as a decimal, r
= 0.035. The time-frame
is thirty-six months, so t
= ^{36}/_{12} = 3.
And the interest is compounded monthly, so n
= 12. The only remaining
variable is P,
which stands for how much I started with. Since I am trying to figure
out how much to invest in the first place, then solving for P
makes sense. I will plug in all the known values, and then I'll solve
for the remaining variable:

The temptation at this
point is to simplify on the right-hand side, and then divide off to solve
for P.
Don't do that; it tends toward round-off error, and can get you in trouble
later on. Instead, stay exact, and do the dividing off symbolically (and
exactly) first:

Now I'll do the whole
simplification in my calculator, working from the inside out, so everything
is carried in memory and I get as exact an answer as possible:

You should memorize the
compound-interest formula, but you should also memorize the meaning of
each of the variables in the formula. While you might be given the formula
on the test, it is unlikely that you will be given the meanings of the
variables, and, without the meanings, you will not be able to complete
the exercises.