The Mathematics of Money

1.1 Simple Interest and the Time Value of Money 7

The Impact of Time

Let’s return again to Tom and Larry. Suppose that Larry returns to the original plan of

borrowing $200, but instead of paying it back in 1 year, he offers to pay it back in 2 years.

Could he reasonably expect to still pay the same $50 interest, even though the loan is now

for twice as long?

The answer is obviously no. Of course, Tom should receive more interest for letting

Larry have the use of his money for a longer term. Once again, though, common sense sug-

gests the proper way to deal with this. If the loan is for twice as long, it seems reasonable

that Larry would pay twice as much interest. Thus, if the loan is extended to 2 years, Larry

would pay (2)($50)  $100 in interest.

Example 1.1.10 Suppose that Raeshawn loans Dianne $4,200 at a simple interest

rate of 8½% for 3 years. How much interest will Dianne pay?

We have seen that to fi nd interest we need to multiply the amount borrowed times the inter-

est rate, and also that since this loan is for 3 years we then need to multiply that result by 3.

Combining these into a single step, we get:

Interest  (Amount Borrowed)(Interest Rate as a decimal)(Time)

Interest  ($4,200)(0.085)(3)

Interest  $1,071.00

One question that may come up here is how we know whether that 8^1 ⁄ 2 % interest rate

quoted is the rate per year or the rate for the entire term of the loan. After all, the problem

says the interest rate is 8^1 ⁄ 2 % for 3 years, which could be read to imply that the 8^1 ⁄ 2 % covers

the entire 3-year period (in which case we would not need to multiply by 3).

The answer is that unless it is clearly stated otherwise, interest rates are always assumed

to be rates per year. When someone says that an interest rate is 8^1 ⁄ 2 %, it is understood that

this is the rate per year. Occasionally, you may see the Latin phrase per annum used with

interest rates, meaning per year to emphasize that the rate is per year. You should not be

confused by this, and since we are assuming rates are per year anyway, this phrase can

usually be ignored.

The Simple Interest Formula

It should be apparent that regardless of whether the numbers are big, small, neat, or messy, the

basic idea is the same. To calculate interest, we multiply the amount borrowed times the interest

rate (as a decimal) times the amount of time. We can summarize this by means of a formula:

FORMULA 1.1

The Simple Interest Formula

I 
 PRT

where

I represents the amount of simple INTEREST for a loan

P represents the amount of money borrowed (the PRINCIPAL)

R represents the interest RATE (expressed as a decimal)

and

T represents the TERM of the loan

Since no mathematical operation is written between these letters, we understand this to be
telling us to multiply. The parentheses that we put around numbers for the sake of clarity
are not necessary with the letters.
At this point, it is not at all clear why the word simple is being thrown in. The reason is
that the type of interest we have been discussing in this chapter is not the only type. Later
on, in Chapter 3, we will see that there is more to the interest story, and at that point it will
Copyright © 2008, The McGraw-Hill Companies, Inc.become clear why we are using the term “simple interest” instead of just “interest.” In the