The Mathematics of Money

26 Chapter 1 Simple Interest

If we multiply both sides of this equation by 12, we get:

12(years) =

12(months)

___________ 12

and so:

12(years) = months

This also agrees with common sense. There are 12(1)  12 months in 1 year, 12(2) 

24 months in 2 years, 12(3)  36 months in 3 years, etc. So to convert our answer to

months we multiply by 12 to get:

T = 12(0.4884004884) = 5.860805861 months.

This answer is also unacceptable, though, since we don’t normally talk about messy deci-

mal numbers of months either. We could round this to the nearest month, in which case

we’d conclude that the term is six months. However, checking this answer we’d get:

I = PRT

I = ($9,750)(.0525)(6/12)

I = $255.94

which is close to what it should be, but not quite right. The rounding is the reason for the

discrepancy. The difference between our rounded answer of 6 months and the actual value

of 5.860805861 months is approximately 0.14 months, which works out to be about 4 days.

That isn’t a huge amount of time, but it is enough to cause trouble. We don’t have to accept

this, since we have the option of converting to days to get a more precise measurement of

the term.

Using the same logic for days as we did for months, we multiply by 365 to get

T = 365(0.4884004884) = 178.2661783 days

which we would round to 178 days, following usual rounding rules.

This is still a rounded answer, and in fact since we threw out 0.2661783 in the rounding,

it may actually appear as though the rounding is at least as serious an issue as it was with

the months. However, checking our answer shows that

I = PRT

I = ($9750)(.0525)(178/365)

I = $249.63

which is still not exactly $250, but it is quite a bit closer than before. The rounding is less

of a problem here since 0.2661783 days is considerably less time than 0.14 months.

Sadly, though, rounding is once again keeping our answer from being exact. We could

deal with this by moving to an even more precise measure of time, such as hours, minutes,

or even seconds. For obvious reasons, though, this is almost never done in practice. We

simply have to accept the fact that, using any reasonable units of time, we can get close to

the result we want, but we will not be able to get the interest to come out to exactly $250.

It’s pretty unlikely that Maria is going to care too much about being 37 cents short of her

goal when we are talking about sums of money on the order of $10,000. We conclude that

the best answer to this question is 178 days.^5

Before moving on, let’s sum up our conclusion about how to convert the units of time

for these sorts of problems. Assuming that the interest rate is a rate per year, the solved-for

value of T will be in years. To convert this to months, we need to multiply by 12. To convert

this to days, we multiply by 365 (or 360 if bankers’ rule is being used.) Unless we have a

(^5) You might object that even though Maria is only 37 cents short, she’s still short. The phrasing of the problem
could be taken to mean that she wants at least $10,000, which does not happen until day 179. This is technically
correct, but in most situations following the usual rounding convention is considered close enough. If the situation
did demand that she must have nothing short of $10,000, we would indeed need to use 179. It is in general
understood that insignifi cantly small differences between desired and actual results must necessarily occur and
aren’t worth losing sleep over.