Declining Balance Interest

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.

Date: 03/23/99 at 11:06:53
From: Doctor Mitteldorf
Subject: Re: Declining balance interest

Start with a borrowed amount P, for principal. Every month, the 
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/