 An important question to ask, what is the outstanding balance on a loan after some of the payments have been made? There are several approaches. One approach is to imagine resetting time after each payment so that each payment corresponds to t equals zero. For example, a five-year loan has payments of t equals one, two, and so on up to sixty if we're measuring time in months. After the t equals twelve payment, we reset the clock so the payments will be at one, two, and so on up to forty-eight. And this approach allows us to use a angle n to compute the balance. For example, you borrow one hundred thirty-five thousand at three point five percent annual interest convertible monthly and make payments of six oh six twenty-two for two years. What's the remaining balance on the loan? So we note that the payments for t equals one, two, three, and so on up to twenty-four were made will reset the clock. So the payment of t equals twenty-five becomes a payment at t equals one, which is twenty-four less. This would make the payment at t equals three sixty, a payment at t equals three thirty-six, which is twenty-four less. Balance of the balance would be six oh six twenty-two a angle three three six at three and a half twelve percent interest and we compute. Now notice that you've paid twenty-four times six oh six twenty-two or fourteen thousand five to forty-nine dollars and twenty-eight cents, but the outstanding balance has only been reduced by about five thousand dollars. And that's because most of your payment when the balance is this large goes to paying off the interest. We can adjust this approach if the borrower misses one or more payments. If there are only a few missed payments, it's easiest to compute the interest separately and add it to the present value. For example, because you splurged on food and heat, you missed the t equals twelve and t equals fifteen payments. Find the outstanding balance after two years. We've already found that if you made the regular payments, the balance would be one twenty-nine seven twenty-eight twenty-eight. The best payment at t equals twelve accrued interest for twenty-four minus twelve twelve months, so its value at two years would be, while the best payment at t equals fourteen accrued interest for ten months, so its value would be, where we round these values up so the bank doesn't lose money. And so our outstanding balance would be increased in addition to whatever fees the bank charges for late or missed payments. In an effort to help customers borrow more money, banks and credit cards might offer variable rate loans. For example, a credit card company might offer an introductory rate of one percent annually for a year and well there's some fine print but nobody ever reads that. You should probably read the fine print. For example, suppose you put one thousand dollars on a credit card with an introductory rate of one percent annually for a year but the rate will go to twenty-four percent annually afterwards. Suppose you want to pay the balance off within two years. During the first year you'll pay fifty dollars per month. What will your monthly payments be during the second year? We'll assume monthly compounding of the interest. So the first twelve payments will have value of fifty, a angle twelve, at one-twelfth percent. Suppose the balance after the first twelve payments is b. This balance would have a present value of v to the twelfth b using our usual v equals the reciprocal of one plus the interest rate. But b should be the value of the remaining twelve payments, p, a angle twelve, at two percent. So the one thousand we receive now should be the same present value as those twelve payments at fifty plus the remaining payments, which are discounted for the twelve months. And b can compute these values, fifty, a angle twelve, at one-twelfth percent will be, while p, a angle twelve, at two percent will be. This gives us our equation, which we solve, and since it's important for the credit card company to make money, we'll round up.