In this lesson, we'll be comparing how the number of compounds per year
affects the balance of an account for compounded interest.
So, if you deposit two thousand five hundred dollars into a bank that pays four percent
annual interest, the three questions are: If the interest is compounded annually,
what is the account balance after five years? If the interest is compounded quarterly,
what is the account balance after five years? And finally,
if the interest is compounded monthly, what is the account balance after five years?
So here's the formula used for compounded interest, where "A" is the amount after time "T,"
"P" is the principal or starting amount, "R" is the annual rate expressed as a decimal,
"N" is the number of compounds per year, and "T" is time in years. The only thing
that's changing for these three questions is "N," the number of compounds per year,
which means for all three of these questions, we can use the formula:
"A" equals the principal, or starting amount, which is 2500,
times the quantity of one plus the annual interest rate expressed as a decimal. Well,
four percent as a decimal is 0.04. We'll divide this by "N," the number of compounds per year,
and then raise it to the power of "N" times "T," where again, "N" is the number of compounds,
which will change, but "T" is the time in years, and in each case,
"T" is equal to five because it's five years. So, our exponent is going to be five times "N."
Now, looking at the first question, we're told the interest is compounded annually,
which means it's only compounded once a year, so "N" is equal to one. In the second question,
the interest is compounded quarterly, and since there are four quarters per year, "N" is equal to
four. Then, for the last question, the interest is compounded monthly,
and since there are 12 months per year, "N" is equal to 12.
So, when "N" is equal to one, we would have:
"A" equals 2,500 times the quantity of one plus 0.04, raised to the power of five.
Let's set up each of these, and then we'll evaluate them on the calculator at the same time.
For the second question, because the interest is compounded quarterly,
"N" is four, so we would have:
"A" equals 2,500 times the quantity of one plus 0.04 divided by "N," which is now four,
raised to the power of five times "N," or five times four.
And then, for the last question, when the interest is compounded monthly, "N" is 12, so we have:
"A" equals 2,500 times one plus 0.04 divided by 12, raised to the power of five times 12.
Notice, in this example, when we divide the interest rate by four, we're getting a quarterly
interest rate, and then five times four represents the number of quarters in five years. So,
we're really just getting common units by dividing the rate by four and multiplying the number of
years by four. The same thing applies to this equation—dividing the interest rate by 12 gives
us a monthly interest rate, and then five times 12 gives us the number of months in five years.
So, now we'll grab the calculator and determine the account balance
for each type of compounded interest.
First, we'll have 2,500 times the quantity of one plus 0.04, and raise this to the fifth power. If
the interest is compounded annually, the account balance will be 3,041 dollars and some cents.
If the interest is compounded quarterly, or when "N" is equal to four, we would have 2,500
times the quantity of one plus 0.04 divided by four,
raised to the power of five times four, which would be to the 20th power.
Now, the account balance is 3,050 dollars and forty-eight cents.
Finally, when the interest is compounded monthly, we'd have 2,500 times the quantity of one
plus 0.04 divided by 12, and all this is raised to the power of five times 12,
which would be to the 60th power. Now, notice the balance is 3,052 dollars and forty-nine cents.
As we can see from this example, the more compounds per year,
the higher the balance should be.
Okay, I hope this was helpful.