Question

You are going to purchase a new car, but being a responsible consumer means doing a little bit of research first.
First, you find the vehicle you are purchasing and its price.

Price: $39,145

Current interest rate: 3%

Using the function A(t)=P(1+
r
n
)nt, create the function that represents your new car loan that is compounded monthly. The principle will be the price of the vehicle you selected, not how much you are putting down.
Being a smart financial planner, you want to figure out how many months it will be until your principal is paid down to $10,000.00. Solve for t

Answer 1

Nice of them to give you the wrong formula.

Try this:

i = 0.03 -- The annual interest rate
j = i/12 -- The monthly interest rate
v = 1/(1+j) -- The monthly discount factor.
P = 39145 -- This is the given purchase price

To find the Payment: \(Pmt\;=\;\dfrac{Price\cdot j}{1-v^{nt}}\)

Your new challenge is to find when this payment level will create a remaining balance less than 10000. Stay with me on this...

Payment look like this...starting from the very last payment.

Pmt -- This is the value of the last payment at the moment you pay it.
Pmt*v -- This is the value of the last payment just after you make the 2nd to last payment.
Pmt + Pmt*v -- This is the value of the last two payments, right before you make the 2nd to last payment.

If we keep on this way, we can see that the value of the payment stream, right after any payment, with exactly "Q" payments left, is this:

Q = 1, then Pmt*v
Q = 2, then Pmt*v + Pmt*v^2
...
Q = 15, then Pmt*v + Pmt*v^2 + ... + Pmt*v^15
and so on.

Since we don't know Q, we write a general expression.

Pmt*v + Pmt*v^2 + ... + Pmt*v^Q = Pmt(v + v^2 + ... + v^Q) = \(Pmt\cdot \dfrac{v-v^{Q+1}}{1-v}\)

OR, \(Pmt\cdot\dfrac{1-v^{Q}}{j} = 10000\)

Putting it all together, substituting for "Pmt": \(\dfrac{Price\cdot j}{1-v^{nt}}\cdot \dfrac{1-v^{Q}}{j} = 10000\).

That can be simplified a little.

This will give a non-integer value of Q. That's no good. Use the next integer value LESS THAN the Q in the expression.

Some FUN mathematics!!

Answer 2

I am going to assume that the formula you meant to write (Kayla) was
\[A(t) = P(1 + rn)^{nt}\]
In this case, P = initial amount, r = rate, n = number of times compounded, t = time passed. If you are still confused, let me know. A quick hint: The initial amount would be the original price of the car.

Answer 3

@wotseit, you had a typo I think you meant\[A(t)=P(1+\frac{ r }{ n })^{nt}\]

Answer 4

You are right, thank you for the correction :)

Answer 5

It doesn't really matter how you correct the wrong formula. With that definition of A(t), you accumulate a single amount for a single period. Nothing is paid off. It's just an accumulation. If you want the balance to go from $39,000 to $10,000, you have to pay it off, not accumulate it.

Answer 6

Fine, but isn't the point of this poorly constructed question is to have algebra 2 students solve a "real world" exponential equation? Most elementary/intermediate algebra students have difficulty learning to solve equations containing variables in the exponent in their early stages of development. Your very interesting treatise on the correct formula and solution might have made this student disengage when she might have profited more from help applying the rules of exponents to set-up and solve the given formula, followed by a brief FYI regarding the use of the correct formula.

Answer 7

@gryphon The one who wrote the problem should be engaged, in order to have the problem statement repaired. The problem statement is WRONG and the intent UNCLEAR. You CANNOT solve an annuity problem in any rational way with a single payment accumulation formula.

I would rather have the student disengaged from this question, rather than have the student frustrated in the belief that the right tools have been presented when the right tools have not been presented.

My demonstration used nothing above Algebra II.

Answer 8

That's not the point. The point is that we are supposed to help her solve the question with the information provided. The teacher wouldn't appreciate your answer, however accurate, on a test.

Either way, let's not start an argument. The asker can decide which method to use.

Answer 9

No, it IS the point. You cannot give me a wheelbarrow and tell me to go move a mountain. You must provide the right tools and they are NOT presented in this problem statement. I would be delighted to have this discussion with the author of the question. You would find that the next version of the question would present the right tools to solve the problem. This one simply doesn't. The asker cannot decide which method to use, because only one correct method has been presented! The original problem statement does not provide a correct method.

@wotseit In compliance with your reasonable definition of the task at hand, to help the student proceed with the given information, I offer this:

@KaylaHake The problem statement includes an incorrect formula. It is essentially impossible to solve this problem with the given information and instruction. Please go have a chat with the author of the question and point out that a single payment accumulation formula has been presented where a multi-payment present value annuity formula would have been more appropriate. The ONLY reasonable way the single payment accumulation formula might be appropriate is if this is supposed to be practice writing Sigma Notation. As there is no such indication, and it is unlikely in Algebra II, I have to conclude that the problem statement is simply wrong. You should not waste any time trying to solve it as presented. I hope you have not been frustrated in your efforts, so far. If you have had some frustration, it's the problem statement's fault, not yours.