8.2. GEOMETRIC SERIES
467
(h) Now we consider larger loans, e.g. automobile loans or mortgages, in which we
borrow a specified amount of money over a specified period of time. In this
situation, we need to determine the amount of the monthly payment we need to
make to pay off the loan in the specified amount of time. In this situation, we
need to find the monthly payment M that will take our outstanding principal
to 0 in the specified amount of time. To do so, we want to know the value of
M that makes P(t) = 0 in formula (8.9). If we set P(t) = 0 and solve for M, it
follows that
M =
r P
1 +
r
12
12t
12
1 +
r
12
12t − 1
.
(i) Suppose we want to borrow $15,000 to buy a car. We take out a 5 year
loan at 6.25%. What will our monthly payments be? How much in total
will we have paid for this $15,000 car? How much total interest will we pay
on this loan?
(ii) Suppose you charge your books for winter semester on your credit card.
The total charge comes to $525. If your credit card has an interest rate of
18% and you pay $20 per month on the card, how long will it take before
you pay off this debt? How much total interest will you pay?
(iii) Say you need to borrow $100,000 to buy a house. You have several options
on the loan:
– 30 years at 6.5%
– 25 years at 7.5%
– 15 years at 8.25%.
(a) What are the monthly payments for each loan?
(b) Which mortgage is ultimately the best deal (assuming you can afford
the monthly payments)? In other words, for which loan do you pay the
least amount of total interest?
467
(h) Now we consider larger loans, e.g. automobile loans or mortgages, in which we
borrow a specified amount of money over a specified period of time. In this
situation, we need to determine the amount of the monthly payment we need to
make to pay off the loan in the specified amount of time. In this situation, we
need to find the monthly payment M that will take our outstanding principal
to 0 in the specified amount of time. To do so, we want to know the value of
M that makes P(t) = 0 in formula (8.9). If we set P(t) = 0 and solve for M, it
follows that
M =
r P
1 +
r
12
12t
12
1 +
r
12
12t − 1
.
(i) Suppose we want to borrow $15,000 to buy a car. We take out a 5 year
loan at 6.25%. What will our monthly payments be? How much in total
will we have paid for this $15,000 car? How much total interest will we pay
on this loan?
(ii) Suppose you charge your books for winter semester on your credit card.
The total charge comes to $525. If your credit card has an interest rate of
18% and you pay $20 per month on the card, how long will it take before
you pay off this debt? How much total interest will you pay?
(iii) Say you need to borrow $100,000 to buy a house. You have several options
on the loan:
– 30 years at 6.5%
– 25 years at 7.5%
– 15 years at 8.25%.
(a) What are the monthly payments for each loan?
(b) Which mortgage is ultimately the best deal (assuming you can afford
the monthly payments)? In other words, for which loan do you pay the
least amount of total interest?