4. George deposits $75 at the end of each quarter for 40 years into an account paying 2.8%
annual interest compounded quarterly.
a) How much is in the account at the end of 40 years?
11
nt
r
n
b) How much did George actually contribute to the account?
c) How much interest did the account earn in those 40 years?
87
5. Mr. and Mrs. Lopez have a new son and decide to start an account (sinking fund) for his
college education. They decide to put $100 into the account each month. The account pays
4% annual interest compounded monthly. They start this account when he is 1 year old.
How much will be in the account when the child is 18 years old?
11
nt
r
n
FV PMT r
⋅
⎛⎞
⎛⎞
⎜⎟
+−
⎜⎟
⎜⎟
⎝⎠
⎝⎠
=⋅
6. Mr. and Mrs. Lopez then have a daughter and start an account for her when she is born.
They decide to put $150 into the account each month. Her account pays 3.8% compounded
monthly. How much will be in the account when she is 18?
11
nt
r
n
FV PMT r
⋅
⎛⎞
⎛⎞
⎜⎟
+−
⎜⎟
⎜⎟
⎝⎠
⎝⎠
=⋅
88
7. John and Sally would like to buy a house in 2 years. In order to have money for a down
payment they decide to save $500 a month in an account (sinking fund) that pays 4.8%
annual interest compounded monthly. How much money will they have in the account at the
end of the four years?
11
nt
r
n
FV PMT r
⋅
⎛⎞
⎛⎞
⎜⎟
+−
⎜⎟
⎜⎟
⎝⎠
⎝⎠
=⋅
In order to have the size house they would like, John and Sally decide they need to save 2
more years. How much will be in the account if they continue to save for two more years?
11
nt
r
n
FV PMT r
⋅
⎛⎞
⎛⎞
⎜⎟
+−
⎜⎟
⎜⎟
⎝⎠
⎝⎠
=⋅
89
8. Mr. Johnson decides when is 40 years old to start saving for retirement. He begins
depositing $200 a month into an account that pays 3.75% compounded monthly. How much
will be in the account if he retires when he is 60? if he retires when he is 65? if he retires
when he is 70?
At 60 years At 65 years
11
nt
r
n
FV PMT r
11
nt
r
n
FV PMT r
At 70 years
11
nt
r
n
FV PMT r
9. In order to have $60,000 in an account for their child’s college education, how much
would a couple need to deposit in a sinking fund each quarter if the account pays 5% annual
interest rate compounded quarterly? The couple begins saving when the child is 5 years old
and will begin college when he is 18.
11
nt
r
n
⋅
⎛⎞
⎛⎞
⎜⎟
+−
⎜⎟
⎜⎟
⎝⎠
⎝⎠
10. If the couple decide they will need $80,000 for their child’s education, how much would
they need to deposit each quarter in the same account described in Problem 9 if they begin
saving when the child is born.
11
nt
r
n
FV PMT r
⋅
⎛⎞
⎛⎞
⎜⎟
+−
⎜⎟
⎜⎟
⎝⎠
⎝⎠
=⋅
College Mathematics: Learning Worksheets Chapter 3
91
Name ________________________________ Date ______________ Class ____________
Goal: To find the present value of annuities and solve problems involving amortization
Section 3-4 PV of an Annuity; Amortization
To calculate the present value of an ordinary annuity:
11
nt
r
n
To calculate the payment needed to amortize a debt:
1
nt
−
−
To calculate the interest paid on an amortized loan:
If part or all of the price of an item is paid off with periodic payments,
then to calculate the cash value of the item:
1. Sam makes a deal to pay $500 a month for 4 years on a car loan at 2.9% annual interest
compounded monthly to pay for a car.
a) What is the present value of the car?
11
nt
r
n
PV PMT r
−
⎛⎞
⎛⎞
⎜⎟
−+
⎜⎟
⎜⎟
⎝⎠
⎝⎠
=⋅
b) How much will Sam make in payments for the car?
c) How much interest will he pay on the car loan?
2. Emma makes a deal to pay $200 a month for 5 years on a car loan at 3.6% annual interest
compounded monthly to pay for the car.
a) What is the present value of the car?
11
nt
r
n
PV PMT r
b) How much will Emma make in payments for the car?
c) How much interest will she pay on the car loan?
94
3. A car dealer, figuring interest at 4.9% compounded monthly, offers to sell you a new car
if you trade in your car that is worth $8000 and agree to pay $400 every month for the next
three years.
a) What is the cash value of the car today?
11
nt
r
n
PV PMT DP
−
⎛⎞
⎛⎞
⎜⎟
−+
⎜⎟
⎜⎟
⎝⎠
⎝⎠
=⋅ +
b) How much total interest would you pay with this deal?
95
4. Another car dealer, figuring interest at 4.6% compounded monthly, offers to sell you a
new car if you trade in your car that is worth $8000, pay an additional $1000 down, and
agree to pay $500 for the next two years.
a) What is the cash value of the car today?
11
nt
r
n
PV PMT DP
−
⎛⎞
⎛⎞
⎜⎟
−+
⎜⎟
⎜⎟
⎝⎠
⎝⎠
=⋅ +
b) How much total interest will you pay with this deal?
c) Which is a better deal, the one from Problem 3 or from Problem 4? Write a short
paragraph to give the advantages and disadvantages of each deal.
96
5. Mark and Natalie want to buy a house selling for $120,000. They will put $20,000 down
as a down payment and finance $100,000.
a) If the bank offers a 30-year mortgage at 5.4% annual interest compounded
monthly, what will their monthly payment be?
1
11
nt
rr
PMT PV nn
b) Approximately how much interest will they pay if they make all the loan
payments on time?
6. Jeff and Theresa are buying a $125,000 house and putting 20% down. The bank is
offering a 20-year loan at 5% compounded monthly.
a) What is the loan amount?
b) What will the monthly payment be?
College Mathematics: Learning Worksheets Chapter 3
c) What is the (approximate) total amount of interest paid during the term of the
loan?
7. Wake and Elizabeth are buying a $200,000 house and putting 10% down. The bank is
offering a 30-year loan at 4.5% compounded monthly.
a) What is the loan amount?
b) What will the monthly payment be?
c) What is the (approximate) total amount of interest paid during the
term of the loan?
College Mathematics: Learning Worksheets Chapter 3
98
(For Problems 8–10, round monetary answers to nearest penny.)
8. Margaret buys new stereo equipment for $500. The store agrees to finance the purchase
price for 4 months at 12% annual interest rate compounded monthly, with approximately
equal payments at the end of each month.
Her first 3 monthly payments will be $128.14. The amount of the fourth payment will be
$128.14 or less (depending on the balance after the third payment). Use this information to
complete the amortization schedule below.
Payment
Number
Balance,
start of
month
Amount of
Payment
Interest due,
at end of
month
Principal
due, at end
of month
Balance
after
Payment
1 $500 $128.14 $5 $123.14 $376.86
IPrt
IPrt
IPrt
PPMTI
PPMTI
PPMTI
newBal prevBal P
newBal prevBal P
newBal prevBal P
9. A student buys a $2400 computer. The price will be amortized at 15% annual interest
compounded monthly, and repaid in 6 substantially equal monthly payments.
a) What monthly payment is required?
1
11
nt
rr
PMT PV nn
b) Complete the amortization schedule below.
Payment
Number
Balance,
start of
month
Amount of
Payment
Interest due,
at end of
month
Principal
due, at end
of month
Balance
after
Payment
1 $2400 $417.69 $30 $387.69 $2012.69
2 $2012.69 $417.69 $25.15 $392.54 $1619.77
IPrt
IPrt
IPrt
PPMTI
PPMTI
PPMTI
newBal prevBal P
newBal prevBal P
newBal prevBal P
College Mathematics: Learning Worksheets Chapter 3
100
IPrt
IPrt
IPrt
PPMTI
PPMTI
PPMTI
newBal prevBal P
newBal prevBal P
newBal prevBal P
c) How much did the student actually pay for the computer (starting balance +
sum of interest column)?
10. After using your computer for a year or so, you decide to upgrade to a better system. A
friend buys your old computer for $750, and you use this for a down payment on a new
system that costs $4200. You finance the balance through the computer store in a sequence
of nine substantially equal monthly payments, and the dealer charges 18% annual interest
compounded monthly.
a) What monthly payment is required?
1
11
nt
rr
PMT PV nn
College Mathematics: Learning Worksheets Chapter 3
101
b) Complete the amortization schedule below.
Payment
Number
Balance,
start of
month
Amount of
Payment
Interest
due,
at end of
month
Principal
due, at end
of month
Balance
after
Payment
1 $3450 $412.66 $51.75 $360.91 $3089.09
2 $3089.09 $412.66 $46.34 $366.32 $2722.77
3 $2722.77 $412.66 $40.84 $371.82 $2350.95
IPrt
IPrt
IPrt
PPMTI
PPMTI
PPMTI
newBal prevBal P
newBal prevBal P
newBal prevBal P
IPrt
IPrt
IPrt
PPMTI
PPMTI
PPMTI
newBal prevBal P
newBal prevBal P
newBal prevBal P
College Mathematics: Learning Worksheets Chapter 3
IPrt
IPrt
IPrt
PPMTI
PPMTI
PPMTI
newBal prevBal P
newBal prevBal P
newBal prevBal P
c) How much did you actually pay for the computer (starting balance + sum of
interest column + down payment)?