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### Declining Balance Interest

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Date: 03/22/99 at 23:34:54
From: Maria Santigate
Subject: Declining balance interest

I have been asked by my cooperative teacher to explain declining
balance interest to a 12th grade business class. I have been unable
to find any information in the business math books. Can you provide
me with a simple way to explain this concept to the students.
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Date: 03/23/99 at 11:06:53
From: Doctor Mitteldorf
Subject: Re: Declining balance interest

principal gets multiplied by a factor f because of interest.
(Depending on the students, you may take a whole lesson dispelling
their implicit assumption that every month a constant amount is ADDED
for interest.) For example, if the annual interest rate is quoted as
r, usually f is computed as 1+r/12.

So, at the end of a month, the balance is fP, but then you pay a
constant amount x: the payment is the same every month. So, at the end
of one month, the remaining balance is

fP - x

The next month, the same thing happens to this new principal: it gets
multiplied by f and x is subtracted. You have

f (fP - x) - x

At the end of 3 months, you have

f(f(fP - x) - x) - x

Ask the students to multiply this out and separate the terms.

Now you start to see a pattern. For the balance at the end of n
months, you can separate out the two kinds of terms:

P*f^n - x(1 + f + f^2 + f^3 + ... + f^(n - 1))

Now you come to the summation formula for a geometric series. How to
handle this?  They have seen this formula derived before, but probably
not a single student in the class has a deep understanding of it. You
can do the 'standard American' thing and lecture to them again, doing
the derivation in front of them. A few might follow momentarily, but a
month from now none of them will have any recollection of the formula
or where it comes from.

In my view, the only effective way to teach is to have students work
on problems singly or, better, in twos or threes, and make discoveries
based on their own experience. This is a very un-American way to teach
(did you see the article in the March 1999 issue of Science magazine
comparing math teaching in America to Germany and Japan? We come out a
distant third).

It seems very inefficient to let students proceed on their own,
discovering mathematics rather than having it fed to them, pre-
digested. And in fact, you will find that you cover a whole lot less
material. But the result will be students who can think flexibly and
apply their methodology to a new situation. Most American students,
confronted with a new situation, pore endlessly through their notes
trying to find a question just like it that the teacher solved for
them.

If you choose to follow this route, you can set some up with
calculators adding 1+1/2+1/4+1/8... then trying 1 + 2/3 + 4/9 +
8/27... or 1 + 1/5 + 1/25 + 1/125...  Let them try different examples
and see if they can perceive a pattern. Then let them translate that
pattern into a formula (a hard step) and finally, after several days,
lead them gingerly toward the algebraic trick for deriving that
formula. The infinite case is actually easier than the finite case,
and that is what you should work on first.

If they have studied repeating decimals, you can relate this to the
fact that 1.11111 is the decimal expression for 1/9.  If they have
studied other number bases, you can show them how the binary
expression for 111111 is always just less than 1000000; that means
that 1 + 2 + 4 + 8 + 16 + 32 is the same as 64 - 1. Let them discover
variations in other number bases.

The result, at the end of this exercise, is the familiar formula
1 + f + f^2 + f^3 + ... + f^(n - 1)... out to infinity = 1/(1 - f) and
1 + f + f^2 + f^3 +... + f^(n - 1) = (1 - f^n)/(1 - f) for the finite
sum.

If the loan is for a period of n months, you want the balance to be
zero after n payments. So the above expression is set to zero.

They should all be able to appreciate that P is a simple multiple of
x. If you borrowed twice as much money, you would expect your monthly
payments to be twice as big. Therefore, all this summing just computes
a proportionality factor between P and x. What does that
proportionality factor depend on? How does it behave with r? Plot it
as a function of r. Why does this make sense? How does it behave as a
function of n? Again, plot it and make sense of the plot.

A final exercise would be for them to calculate how long it takes to
repay a loan for a given P and x. In other words, they are solving for
n. Let them struggle with this on their own. Some of them will not get
it until you review logarithms for them. Again you are probably
standing on the shoulders of midgets. They did not carry any
understanding at all out of their unit on logarithms two years ago.
But, in my view, better to teach thoroughly and promote a lasting
understanding at a more basic level than to "cover" a lot of advanced
material of which the students have no mastery.

Try to trick them with an x and a P where there is no solution for
finite x. Why is this? Can it be that you can go on paying a loan
forever and it is still never paid off? Can you see a simple reason
why this should be so? Can you go from that simple reason to a formula
for the minimum monthly payment needed to assure that the loan is paid
off in less than infinite time?

If you have done your job well, they will discover on their own that
this corresponds exactly to the monthly interest. For the principal to
go down the first month, x must be greater than (f-1)P.

You could teach them a lot of mathematics in this lesson, if you are
so inclined. Or you can do what most business math teachers would do:
put the formula on the board, make them memorize it, and tell them on
the test they are responsible for plugging in and grinding out.

Here is an excerpt from the Science article:

Teaching practices were remarkably uniform within each country, but
they differed sharply from nation to nation. In Japan, teachers
usually present their students with a problem and then let them work
on it individually and in small groups so that students have to
struggle with the relevant mathematical concepts. In the United
States and Germany, teachers tend to drill students on concepts they
have just described. The result, says the principal investigator for
the study, James Stigler of the University of California, Los
Angeles, is that "American and German students tend to practice
routine procedures, while Japanese students are doing proofs."

- Doctor Mitteldorf, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
High School Interest
High School Sequences, Series

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