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be smaller than the denominator (the bottom). Fractions whose numerators are larger are
called improper but there really is nothing mathematically improper about them at all.
There are cultural reasons why people may prefer to avoid such fractions—a recipe that
called for^3 ⁄ 2 cups flour would seem strange, while a recipe calling for 1^1 ⁄ 2 cups wouldn’t—
but these reasons are a matter of tradition and style, not mathematical necessity. While we
could rewrite^20 ⁄ 12 as 1^8 ⁄ 12 , simplify that to 1^2 ⁄ 3 , and then convert it to a decimal, this would
accomplish nothing except needlessly adding steps. We will freely use “improper” frac-
tions whenever they show up.Loans with Terms in Days—The Exact Method
After we have dealt with loans whose terms are measured in months, it’s not surprising
that our next step is to consider loans with terms in days. The idea is the same, except that
instead of dividing by 12 months, we divide by the number of days in the year.Example 1.2.4 Nick deposited $1,600 in a credit union CD with a term of 90 days
and a simple interest rate of 4.72%. Find the value of his account at the end of its
term.Since there are 365 days in a year, we divided by 365 instead of 12, since 90 days is^90 ⁄^365
of a year.I PRT
I ($1,600)(0.0472)(90/365)
I $18.62And so Nick’s ending account value will be $1,600 + $18.62 $1,618.62.Since we divided by 12 when the term was in months (since there are 12 months in the
year), it only makes sense that we should divide by 365 when the term is in days (since
there are 365 days in the year) as we did in this example.
Unfortunately, this is not quite as clear cut as it might seem. While there are exactly
12 months in each and every year, not every year has exactly 365 days. Leap years, which
occur whenever the year is evenly divisible by four^2 (such as 1996, 2000, 2004, 2008,.. .)
have an extra day, and if the year is a leap year we really should use 366.
This example didn’t state whether or not it occurred in a leap year, so we don’t know
for certain whether to use 365 or 366. And heaven help us if the term of the loan crosses
over two calendar years, one of which is a leap year and the other isn’t! Calculating
interest based on days can clearly become quite complicated. But even that is not the
end of the story; we can carry things even further if we really want to be precise. It actu-
ally takes the earth 365^1 ⁄ 4 days to circle the sun (the extra^1 ⁄ 4 is why leap years occur one
out of every 4 years). In some cases interest may be calculated by dividing by 365.25
regardless of whether or not the year is a leap year. Taking that approach might be a little
bit extreme, and it is unusual but not completely unheard of to see it used in financial
calculations.^3
Some businesses always use the correct calendar number of days in the year (365 in
an ordinary year, 366 in a leap year). Others simply assume that all years have 365 days,
while still others use 365.25. Having this many different approaches can be confusing, but
it is an unfortunate fact of life that any one of them could be used in a given situation. The
good news is that the difference among these methods is very small, as the next example
will illustrate.(^2) Actually, the rule is a bit more complicated: A year is a leap year if it is divisible by 4, except in cases where it is
also divisible by 100. But even this exception has an exception: if the year is also divisible by 400, it is a leap year
after all! Since the last time a year divisible by 4 was not a leap year was 1900, and the next time it will happen is
2100, for all practical purposes we can ignore the exceptions.
(^3) For the truly obsessive, an even more exact value for the time required to circle the sun is 365.256363051 days,
called a sidereal year. The pointlessness of carrying things this far should be obvious.
1.2 The Term of a Loan 15