The Mathematics of Money

6 Chapter 1 Simple Interest

Back to Percents

So now let’s return to the problem proposed a while back of determining 11.35% interest

on an $1,835.49 loan. We must convert 11.35% to a decimal, which gives us 0.1135, and

then multiply by the amount borrowed. So we get:

Interest  (Principal)(Interest Rate as a decimal)

Interest  ($1,835.49)(0.1135)

Interest  $208.33

Actually, multiplying these two numbers yields $208.32811. Since money is measured

in dollars and cents, though, it’s pretty clear that we should round the final answer to

two decimal places. We will follow the usual rounding rules, standard practice in both

mathematics and in business. To round to two decimal places, we look at the third. If

the number there is 5 or higher, we “round up,” moving the value up to the next higher

penny. This is what we did above. Since the number in the third decimal place is an 8, we

rounded our final answer up to the next penny. If the number in the third decimal place is

4 or lower, though, we “round down,” leaving the pennies as is and throwing out the extra

decimal places.

Example 1.1.8 Suppose Bruce loans Jamal $5,314.57 for 1 year. Jamal agrees to

pay 8.72% interest for the year. How much will he pay Bruce when the year is up?

First we need to convert 8.72% into a decimal. So we rewrite 8.72% as 0.0872. Then:

Interest  (Principal)(Interest Rate as a decimal)

Interest  ($5,314.57)(0.0872)

Interest  $463.43

Actually, the result of multiplying was 463.4305, but since the number in the third decimal

place was not fi ve or higher, we threw out the extra decimal places to get $463.43.

We are not done yet. The question asked how much Jamal will pay Bruce in the end, and

so we need to add the interest to the principal. So Tom will pay $5,314.57 
 $463.43 

$5,778.00.

Mixed Number and Fractional Percents

It is not unusual for interest rates to be expressed as mixed numbers or fractions, such as

53 ⁄ 4 % or 8^3 ⁄ 8 %. Decimal percents like those in 5.75% and 8.375% might be preferable, and

they are becoming the norm, but for historical and cultural reasons, mixed number percents

are still quite common. In particular, rates are often expressed in terms of halves, quarters,

eighths, or sixteenths of a percent.^1

Some of these are quite easy to deal with. For example, a rate of 4^1 ⁄ 2 % is easily rewritten

as 4.5%, and then changed to a decimal by moving the decimal two places to the right to

get 0.045.

However, fractions whose decimal conversions are not such common knowledge require

a bit more effort. A simple way to deal with these is to convert the fractional part to a deci-

mal by dividing with a calculator. For example, to convert 9^5 ⁄ 8 % to a decimal, first divide

5/8 to get 0.625. Then replace the fraction in the mixed number with its decimal equivalent

to get 9.625%, and move the decimal two places to get 0.09625.

Example 1.1.9 Rewrite 7 13/16% as a decimal.

(^13) ⁄ 16  0.8125, and so 7 (^13) ⁄ 16 %  7.8125%  0.078125.
(^1) The use of these fractions is supposed to have originated from the Spanish “pieces of eight” gold coin, which
could be broken into eight pieces. Even though those coins haven’t been used for hundreds of years, tradition is
a powerful thing, and the tradition of using these fractions in the fi nancial world has only recently started to fade.
Until only a few years ago, for example, prices of stocks in the United States were set using these fractions, though
stock prices are now quoted in dollars and cents. It is likely that the use of fractions will continue to decline in the
future, but for the time being, mixed number rates are still in common use.