A question about The math Formula for "Balance after t Minimum Payments"

Question

So to start of, the formula is

Balance after t Minimum Payments= Initial Balance x (( 1 + r)(1 - m))^t

my teacher is asking

In the balance after t minimum payments formula we usually divide one of the percentage rates by 12 and not the other. Which one gets divided? Why doesn't the other get divided?

Im really stuck I have no idea what he means! I only know how to use the formula, but I dont know what he means.

Answer 1

The form of the question implies that there are two percentage rates (since we divide "one ... and not the other"), so the first thing to do is identify those two rates, then figure out which is which.

If we first stipulate that 't' represents the number of monthly minimum payments, 'm' represents the percentage of the balance that is the monthly minimum, and 'r' represents the nominal annual percentage rate, then the question becomes a bit easier. The initial balance is obviously fixed, and the balance after t payments is what we're calculating, so that's not a rate, either. The number of payments isn't a rate, so that leaves 'r' and 'm' as our two percentage rates.

Think about the simple case of making one monthly payment. If we put in the annual percentage rate for the term that represents the increase due to interest and the monthly payment rate for the term that represents the decrease due to making a payment, we've obviously got a mismatch. Which makes more sense for us to do to make them match?

1. Divide the annual simple interest rate by 12 to make it monthly, or
2. Divide the monthly minimum payment rate by 12 to make it annual.

I hope it's clear from that setup that we should divide the annual percentage rate by 12 to make it a monthly percentage rate. So the answer is that we usually divide 'r' by 12.