Math 1304-04 Name: 5.7 Worksheet (Quiz 8)

Math 1304-04 5.7 Worksheet (Quiz 8)

Name:

Compound Interest Formula In this formula:

( r )nt A=P ? 1+

n

P is the amount of money that is invested. (It is also sometimes referred to as the "Principle" or "Present Value.")

r is the interest rate.

n is the number of compounding periods per year.

t is the number of years for which the money is invested.

A is the amount you have at the end of those years. (It is also sometimes called the "Future Value" of the investment.)

Compounding Periods

n in the formula is the number of times the interest is compounded each year. Below is a table with different phrases you might see in problems and what they mean.

Compounded annually

means n = 1

Compounded semiannually means n = 2

Compounded quarterly

means n = 4

Compounded monthly

means n = 12

Compounded daily

means n = 365

Example: (Problem #8 from the book) How much will you have after three years if you invest $50 at 6% compounded monthly?

Solution. We are investing $50, so P = 50. The interest rate is 6%, so r = .06. It's being

compounded monthly, so n = 12. We wish to know how much we will have after three years, so

t = 3, and we want to know how much we will have at the end of those three years, so we're

looking for A. Putting this all in the formula, we get:

(

.06 )12?3

A = 50 ? 1 +

12

and we can simply plug this into a calculator to get A = 59.834. Since we know we're dealing with money, it makes sense that we would always round to the nearest cent, so our answer is: After three years, we will have $59.83.

Now work the following problem: (Problem #9 from the book) How much will you have after two and a half years if you invest $500 at 8% compounded quarterly?

In the two problems you've seen, we've compounded every quarter, and we've compounded every month. If you wanted to get more out of your investment, you would want it to be compounded as often as possible. It could be compounded every day, or every hour, or every second, but the best you could possible do would be for the interest to be compounded "every instant." We call this being compounded continuously. Continuous Compounding

A = P ert In this formula, A, P, r, and t are the same as our first formula. The only difference is there is no n because rather than compounding the interest a specific number of times each year, it's being compounded constantly. Work the following problem: (Problem #13 from the book) How much will you have after 2 years if you invest $1000 at 11% compounded continuously?

It can be difficult to compare different interest rates to see which is better. For instance, which is better, 5% compounded daily or 6% compounded quarterly? One has a higher rate, but the other is compounded more frequently. A way to compare them is by calculating something called the effective rate of interest which is represented by the symbol re.

Effective Rate of Interest Compounding n times per year:

( r )n

re =

1+ n

-1

Compounding continuously:

re = er - 1

Example:(Problem #24 from the book) What is the effective rate of interest for 6% compounded monthly?

Solution. To calculate effective rates, you only need r and n, so in this case r = .06, and n = 12,

so

( .06 )12

re =

1+ 12

- 1 .06168

Thus, the effective rate of interest is about 6.168%.

Now work the following problem: Nathan has some extra money he would like to put into a savings account. He's trying to pick a bank in which to open the account, and he has narrowed it down to three. Bank A offers 3% interest compounded monthly; Bank B offers 3.03% interest compounded annually; and Bank C offers 2.95% interest compounded continuously. Which bank should Nathan choose?

Two more problems for you to work: (Problem #36 in the book) How long does it take for an investment to triple in value if it is invested at 6% compounded monthly?

(Problem #45 in the book) Jerome plans to buy a car for $15,000 in 3 years. How much money should he ask his parents for now so that, if he invests it at 5% compounded continuously, he will have enough to buy the car?

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