Compound interest is an application of solving exponential equations.
There are two formulas that must be memorized
in order to solve these applications.
Plus you will need your calculator in order to evaluate the final answers.
So keep it handy. Let's look at the first example.
It reads: Find the amount that results from 100 dollars
invested at 10 percent compounded monthly, and then for the b part,
we'll look at componded continuously for 2 and 1/4 years.
Okay, so let's look at the a part.
The problem: We have to decide which of the two formulas to use.
We look at the way the money is being compounded.
It's monthly, so we want to use this formula.
Okay it reads, A equals P times in parentheses 1 plus r over n to the nt power.
It's very important besides knowing the formula
to know what the different variables stand for. So let's go over that.
A is your final amount. P is the principal, which in this case,
we know is 100 dollars, or the initial amount that you're investing.
r of course is the rate, which in this problem is 10 percent.
We'll look at how to put that into the formula.
n, both of these spots, is the number of times that the money is compounded per year,
okay, and t of course is time. So looking at the problem,
we want to find the amount. Okay, so that is the unknown.
We've already talked about that the principal is 100 dollars.
Okay, r is the rate that the money is being compounded
and must be written as a decimal.
Percent means per hundred, so it's going to be 0.10.
n is 12, and we'll put the 12 here again and 2 and 1/4.
Now, the next step is just to enter all of these numbers in the proper order in our calculator.
If you would like, it may be easier in the calculator
to enter 2 and 1/4 as 2.25. So we're going to do that,
and we're going to round to the nearest penny.
Turn your calculator on. I'm going to enter 100
parentheses 1 plus 0.1 divided by 12. Close the parentheses.
Now at this point, depending on your calculator,
you can use your carat button or the y to the x button
parentheses to multiply 12 times 2.25. Close the parentheses, equals.
Now, the answer should be more than 100 dollars.
Okay so when look at your answer if you get something that's less than 100 dollars,
you know that you've entered something incorrectly.
To the nearest penny, we have 125 dollars and 12 cents.
Now that finishes the a part, so let's go to the b part.
Okay this time the money is being compounded continuously,
so we use a different formula. You want to make sure that
if the money is compounded yearly, semiannually, quarterly, monthly, daily, or whatever,
you use this formula. If it's continuously, you're going to use this formula.
Okay, some of the same variables: final amount equals
the principal times e, and we know what e is, to the rt power, the rate times the time.
Again, we're looking for the amount. P is still 100,
r is still 0.10, and the time is 2 and 1/4 years.
I'll write it as 2.25 this time so that you can see it.
Okay, let's put an equal here, and then because we're rounding,
we're going to get an approximate answer.
In the calculator, 100, and recall how you enter e:
Second natural log, 0.10 times 2.25.
Now, my calculator put the parentheses around this quantity.
If yours does not, you need to add those.
Okay equals and because we're compounding the interest more times than monthly,
the answer we get here should be greater than what we got in the a part of the problem.
It is. It's 125 dollars and 23 cents.
That should solve this type of problem when we have all of the quantities,
P, r, n, or t, and we're looking for the final amount.
In our next example, we want to look at what happens
when we're given the final amount, and we have to find the time or the rate.
In this example, we will see how many years will it take an investment
an investment of 25,000 dollars to grow to 80,000 dollars.
Assume an interest rate of 7 percent compounded quarterly and then continuously.
We will look at the a part first.
From the first example, remember the key is
to begin the problem, you must decide which formula to use.
What determines that is the way that the money is being compounded.
It's quarterly, so we're going to use our first formula.
A equals P times 1 plus r over n to the nt power.
Notice that this time, we're looking for how many years,
so we want to solve for t. t is in the exponent,
so that creates a little more difficult problem.
A, final amount, is going to be 80,000 dollars.
P, the amount that we start with is 25,000.
Interest rate in this problem is 7 percent. Now 7 percent as a demical is 0.07.
A common mistake that a lot of algebra students make
is they write this as 0.7. That would be 70 percent,
and I don't think any bank is going to give us a 70 percent rate of interest
compounded quarterly or continuously, so be very careful there.
Also, the second mistake that a lot of students make is n.
Quarterly, you do think of it as 1/4 of a year, but n has to be a whole number.
Since this is the number of times the money is compounded per year
every 3 months, so n is 4, and of course t is our unknown.
Now because we're solving for the exponent,
we must use the procedures that we used in solving exponential equations.
The first step of which is to isolate the base to the exponent.
I'm going to divide both sides by 25,000.
On this side when we reduce we will get 16/5, which you can leave,
or you can change it to the decimal 3.2.
Now if the fraction was something like 16/3 or 17/6
where you would not get a terminating decimal,
you definitely would need to leave the fraction.
Also right here, we can enter, because we're dividing by 4,
we're going to get a terminating decimal, so if you want to divide that,
you come up with 1.0175. It may make entering your answer
in the calculator a little simpler.
Now at this point, we need to bring down that exponent.
The way we do that is we take the log or the natural log on both sides.
Let's use log base 10, but natural log would work just as well.
Now this will allow us to bring down the exponent.
Finally we can solve for t by dividing by 4
and by dividing by the log of 1.0175.
Let's come up here. Move the problem up here.
We'll have the log of 3.2 divided by 4 log 1.0175.
Okay finally, we need to enter this in the calculator.
The one thing you want to be careful of is that we need to put parentheses
around the denominator. Let's try that.
Log 3.2 divided by parentheses 4 log 1.0175.
Okay, be careful. I punched in the wrong number here, so be careful.
Press equals, and let's see what our answer is.
We're going to round to two decimal places.
t is going to be approximately 16.76.
Now in our money problems, t is always in years,
so the units here are years. That is our final answer.
In the b part of example two, we're going to see how many years it will take
for the money, the 25,000, to grow to 80,000 at the given rate of 7 percent,
but this time compounded continuously.
Remember we need to decide which formula we're going to use.
For compunded continuously, we're going to use this formula,
which should be very easy to memorize.
Our final amount equals our principal times e to the rate times the time power.
Again, 80,000 is our final amount. P, the principal, is 25,000.
r, we'll just remember that it's 0.07, and t is our unknown.
t is in years. Again, the procedure is exactly the same.
We want to divide both sides by 25,000.
I think I'll leave this as 16/5 this time, so you can see a fraction in action.
Of course, that will give me e to the 0.07 t power.
Now in the previous a part when we were compounding quarterly,
I took log on base 10 on both sides. I mentioned that I could've used the natural log.
Either one would work. Here I do not want to use log on base 10
because I have base e. To undo e, I want to use the natural log of 16/5
equals the natural log of the right side
because that allows me when I bring down the exponent.
I multiply by the natural log of e.
We all know, or we should know, that quantity is 1.
Of course 0.07 t times 1 is 0.07 t.
Last step before putting it in the calculator: we want to divide by 0.07,
and we enter this in the calculator.
Now my calculator puts parentheses here, but if yours does not,
you need to put those in because you want to take the natural log
of the fraction 16/5, not the natural log of 16 divided by 5.
Here we go. Divide, close the parentheses, divide by 0.07. Enter.
Again we're going to round to two decimal places,
and the final answer will be in years. Because we're compounding continuously,
it should take us less time than it did in the a part. We get 16.62 years.
If you become proficient at working these problems,
I think you'll have no trouble with this at all.