Math homework APR

Question

Assume that you have a balance of $4500 on your Discover credit card and that you make no more charges. Assume that Discover charges 27% APR and that each month you make only the minimum payment of 3% of the balance.

What will the balance be after 32 months? (Round your answer to the nearest cent.)

Answer 1

Let A_n be the account balance at the end of the n^th month. Let

A₀ = the original balance, in this case $4500,

i = the monthly interest rate charge on the outstanding debt, in this case, i = 27% / 12,

p = the minimum percent of the current balance to be paid each month, in this case, 3%.

To get to balance at the end of month n+1, we form this equation:

A_n+1 = A_n + i*A_n - p*A_n

This can be simplified to:

A_n+1 = A_n * (1 + i - p)

Looked at another way,

A_n+1 / A_n = 1 + i - p

Remember that n is arbitrary; it represents 1, 2, 3, etc. So all we have to do to get the balance after the first payment is multiply the original balance by 1 + i - p. To get the balance after the second monthly payment, we multiply the first month's balance by the same number, 1 + i - p.

To get the balance after 32 months, we form this product of 32 fractions:

(A₃₂ / A₃₁₎ * (A₃₁ / A₃₀) * (A₃₀ / A₂₉) * • • • * (A₁ / A₀)

We will simplify this product two different ways.

First, we substitute each fraction with (1 + i - p) to get

(1 + i - p) * (1 + i - p) * (1 + i - p) * • • • * (1 + i - p)

Since there 32 such terms, we get the product:

(1 + i - p)³²

The second way is to note that

(A₃₂ / A₃₁) * (A₃₁ / A₃₀) * (A₃₀ / A₂₉) * • • • * (A₁ / A₀)

can be simplified to just

A₃₂ / A₀

These two simplifications must be equal, so we conclude that

A₃₂ / A₀ = (1 + i - p)³², or

A₃₂ = A_{0 *} (1 + i - p)³²

Now we substitute in the given values as follows:

A₃₂ = (4500) _* (1 + 0.27/12 - .03)³² = 4500 * 0.9925³² = 4500 * 0.78591666 = 3536.62