Two Car Loan Problems You Can Try!

Jeannie Wants to Buy A Car!

(A Car Loan Amortization Problem)
Jeannie wishes to purchase a car and she would like to have payments no higher than $350 and for no more than five years. She has saved enough money for a pretty good down. She can go up to $20,000 for a loan through her credit union. She would like to get a good percent loan but the best she maybe able to get is 11%.

The way her loan works is that interest is figured monthly over the life of the loan.

The payments she makes of $350 are deducted each month from the principal after the interest for the monthly time period is taken out of the payment. For example, if the interest for one month on the remaining principal was $100, then $250 would be deducted from the principal, while $100 of the $350 payment would simply pay off that month's interest. This new value of the remaining loan principal would then has interest figured for the next month, and so on.

You may be able to see that over time, the amount going towards interest decreases as the principal part of the loan is reduced, and more and more of the $350 goes towards the principal. You can also see that the higher the interest, the longer it takes to pay off the oloan, because so much of the loan payment is going towards interest and the balance isn't going down very fast. This is why, for example, it takes so long for people to pay off cars and houses.

The Problem To Solve

Anyway, getting back to the problem, find out how much Jeannie would owe after one year and two years, showing the monthly balances and all the interest. Also show the amount of total interest she will pay over two years. Will she have paid for more than half of her original loan of $20,000? Try to estimate by graphing when she will pay off her loan. What will you graph to find this out?

Use: I = P(r/n)

where I is the amount of the interest for any month, r is the rate of interest, n is the number of payments per year, and P is the remaining Principal. Also:

P (new) = P(1 + r/n) - L

The NEW PRINCIPAL may be found by using the above equation where L is the amount of the loan payment, P is the original remaining balance, r is the interest rate and n is the number of payments in a year.

Here is an example, with a different interest, payments and principal:

P(1) = 7500 r = 9% n = 12 payments per year $250 payment per month

I = 7500(.09/12) I = $56.25

P(2) = 7500(1+.09/12) - $250 = (7500 + 56.25) - $250 = $7306.25

This becomes the new balance for the next month and interest is accumulated on this new principal balance.

Make a table something like this: Frist Row of Table

Principal:7500
Interest: 56.25
Payment - Interest: 250-56.25 =193.75
New Balance:7306.25
Interest Paid: 56.25

Second Row Of Table

New Principal: 7306.25

Remember that Jeannie's payments are $350 on an original principal balance of $20000, and her interest rate is 11%.

MORE CREDIT: can you make up a program which will produce a table of these values over several years or give values for any particular month? These are called Amortization Rates/Tables and are used quite a bit in real life situations for housing loans or many other loans over a period of time.

Jeannie's Car Loan Problem Revisited

Remember from the first problem, that Jeannie felt she could spend up to $350 per month on a new car or utility vehicle. But although she thought she could go as high as $20000 in terms of a loan principal balance, in turned out that it would take far too long to pay it off. No bank of finance company will carry a loan over more than 5 years (or possibly six years if a person's credit is excellent). Jeannie would really like to pay off a loan in 4 or 5 years , but still can only afford $350 per month for payments.

What is the maximum amount of money she can request for a loan balance after her down and all charges? She is going to try to negotiate a slightly lower interest rate as well, 10.5% as opposed to 11%, so she asks for a 9% interest rate, but will settle at 10-10.5% with the car sales manager. This time you will also need to figure out about the tax on the sale. This will include the tax on the total sales price, including her down payment which will be $4000. Figure the tax to be 7.5% of the total price. See below for examples.

You will also need to graph it out so that she can see how much she can get for both a 4 and a 5 year loan which pays off the entire amount of the principal balance. You also need to have this for both 9% and 10.5% so she can get some idea of how it would change her payments and time or possible balances. There are several VARIABLES here: time (4 or 5 years) interest (9% or 10.5%) loan principal balances after taxes and down ( more than two possibilities?!). You might try to do just the 5 year, 9% to start.

One suggestion is to use a "guess and check" method of coming up with a possible maximum loan. There may be other ways, including developing a simple (!?) computer program to do the work.

ONE PARTIAL EXAMPLE of how to start:

Suppose Jeannie's loan is $22000. Her down is $4000.

Then: The tax is .075 x 22000 = $1650.

Then: $22000 + 1650 = 23650 (total price) (we are neglecting the registration and other fees, but hey....this is only an estimate of how it really works!)

Then: $23650 - $4000 (her down) = $19650 This is her principal balance.

Then: At 9% Compounded monthly over. The first two months are worked out below:

Then:
P(1) = 19650
r = 9% (.09)
n = 12 paymts per year
Paymt = 350 per month

Then:
I(1) = 19650(.09/12) = $147.38
P(2) = 19650(1+.09/12) - $350 = 19447.38 (new balance)
$350 - 147.38 = 202.62 (Paid Towards Principal)

First Row of Table

Principal: $19650.00
Interest: 147.38
Paymt -Interest: 202.62
New Balance: 19447.38
Interest Paid: 147.38
Towards Principal: 202.62
Second Row of Table

Principal: 19447.38
Interest: 145.86
Paymt -Interest: 204.14
New Balance: 19243.26
Interest Paid: 293.24
Towards Principal: 406.76
Third Row (incomplete)

Principal: 19243.26

The reason I made a 6th column, which is the sum of the Payment-Interest amounts, is so that one might see another way to graph this to see how long it will take to get to the original principal. Hint: try looking at the differences of the growth of the amount towards the principal. Of course, you can also graph the changing (declining) balance amounts.

Partial credit will be given for any reasonable effort in this problem, but at least one or more graphs must be included which show when the loan will be paid off for such credit!

Good luck! After doing this problem, you may have a better idea of what to think about when you are buying your next car! Knowing such math CAN BE a powerful tool!

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