The Mathematics of Money

Copyright © 2008, The McGraw-Hill Companies, Inc.

1.1 Simple Interest and the Time Value of Money 5

precisely what you would have gotten by moving that decimal two places to the left. This

is not a coincidence, and in fact we can always convert percents into their decimal form

simply by moving the decimal place.

So, when using percents, we can either go to the trouble of actually dividing by 100, or

instead we can just move the decimal place.

Example 1.1.4 Convert 25% to a decimal.

By dividing: 25%  25/100  0.25

By moving the decimal: 25%  0.25

Why did we place that extra zero to the left of the decimal? The zero placed to the left of

the decimal place is not really necessary. It would be just as good to have written “.25”.

Tacking on this zero does not change the numerical value in any way. It only signifies

that there is nothing to the left of the decimal. There is no mathematical reason to prefer

“0.25” over “.25” or vice versa; they both mean exactly the same thing. However, we

will often choose to tack on the zero because the decimal point is so small and easy to

miss. It is not hard to miss that tiny decimal point on the page and so .25 can be easily

mistaken for 25. This tiny oversight can lead to enormous errors; 0.25 is far less likely

to be misread.

Example 1.1.5 Convert 18.25% to a decimal.

Moving the decimal two places to the left we see that 18.25%  0.1825.

Example 1.1.6 Convert 5.79% to a decimal.

Here, there aren’t two numbers to the left of the decimal. Simply moving the decimal point

two places to the left would leave us with “0._579”. The blank space is obviously a problem.

We deal with it by placing a 0 in that position to “hold the space.” So 5.79%  0.0579.

Let’s put this all together to recalculate the interest on Larry’s $1,000 loan once again.

Example 1.1.7 Rework Example 1.1.3, this time by converting the interest rate

percent to a decimal and using it.

We have seen that 25%  0.25, and that to use it we multiply it by the principal. Thus:

Interest  Principal * Interest Rate as a decimal

Interest  $1,000 * 0.25

Interest  $250

This answer agrees with our previous calculations.

Notation for Multiplication

There are a number of different ways to indicate multiplication. Probably the most familiar

is the  symbol, though the asterisk * that we used above is also widely used, especially

with computers. It is also a standard mathematical convention that, when no symbol is

written between two quantities, multiplication is assumed. From this point forward, we will

be following that convention. To indicate “1,000 times 0.25” we will write:

(1,000)(0.25)

The parentheses are used to make the separation between the numbers clear. If we simply

wrote the two numbers next to each other without them, “1,000 0.25” could be easily

misread as the single number “10,000.25”. However, we don’t really need both sets of

parentheses to avoid this, and so we could equally well put parentheses around only one of

the numbers. So, to indicate “1,000 times 0.25” we may write any of the following:

(1000)(0.25) or (1000)0.25 or 1000(0.25).