3/9/12 Section 3.6/3.2
Obj: SWBAT solve real world applications using exponential
growth and decay functions.
Bell Ringer:
Get Compound Interest Packet Part 2
HW Requests: pg 296 # 1-16 (3’s) 15, 32, 33
pg 342 #1-9 odds
Homework: pg 342 11-26 (3’s), 47, 50, 53, 55
Announcements: Tuesday -Quiz Log and Exponential
Equations, Exponential Growth and Decay
Function for compound interest,
compounding continuously:
A(t) = Pert
P = the principal amount
r = the interest rate
t = the number of years
Problem
One thousand dollars is invested at 5% interest
compounded continuously.
a. Give the formula for A(t), the compounded
amount after t years.
b. How much will be in the account after 6 years?
c. How long is required to double the initial
investment?
Problem
One thousand dollars is invested at 5% interest compounded
continuously.
a. Give the formula for A(t), the compounded amount after t years.
A(t) = P𝑒 0.05𝑡
b. How much will be in the account after 6 years?
c. How long is required to double the initial investment?
Problem
One thousand dollars is invested at 5% interest compounded
continuously.
a. Give the formula for A(t), the compounded amount after t years.
A(t) = P𝑒 0.05𝑡
b. How much will be in the account after 6 years?
A(t) = 1000𝑒
0.05 (6)
= $1349.86
a. How long is required to double the initial investment?
Problem
One thousand dollars is invested at 5% interest compounded
continuously.
a. Give the formula for A(t), the compounded amount after t years.
A(t) = P𝑒 0.05𝑡
b. How much will be in the account after 6 years?
A(t) = 1000𝑒
0.05 (6)
= $1349.86
a. How long is required to double the initial investment?
2000 = 1000 𝑒 0.05 (𝑡)
2 = 𝑒 0.05 (𝑡)
ln(2) = ln(𝑒 0.05 (𝑡) )
ln(2) = 0.05t ln(e)
ln(2) = 0.05t
ln(2) = t
.05
t = 13.86 years
(0.5)(12)
(0.5)(12)
(0.5)(12)
An annuity is a sequence of equal periodic
payments. We will be working with ordinary
annuities. The deposits are made at the end
of each period and at the same time the interest
is posted or added to the account.
Calculator http://iris.nyit.edu/appack/pdf/FINC201_4.pdf
We can calculate the total of our periodic payments and the interest
accrued. This value is called the Future Value of the annuity.
The net amount of money returned from the annuity is its future value.
We make payments today and get the money some time in the future.
(IRA’s, 401 k, mutual funds)
Future Value Equation:
(1 + 𝑖)𝑛 −1
𝐹𝑉 = 𝑅
𝑖
where
r = the annual interest rate
k = number of times interest is compounded per year
n = number of equal periodic payments
(number of years * compounding period)
R = payment amount
𝑟
i = interest rate ( )
𝑘
I
C/Y
P/Y
PMT
Future Value Equation:
(1 + 𝑖)𝑛 −1
𝐹𝑉 = 𝑅
𝑖
where
r = the annual interest rate
k = number of times interest is compounded per year
n = number of equal periodic payments
(number of years * compounding periods (k))
R = payment amount
𝑟
i = interest rate (𝑘)
Geno contributes $50 per month into Morgan Park Mutual Fund that
earns 7.26 APR. What is the value of Geno’s investment after he
retires in 25 years?
R = $50, k = 12, r = 0.0726, n = 25*12, i = 0.0726/12
(1 + 𝑖)𝑛 −1
𝐹𝑉 = 𝑅
𝑖
Geno contributes $50 per month into Morgan Park Mutual Fund that
earns 7.26 annual interest rate. What is the value of Geno’s investment
after he retires in 25 years?
R = $50, k = 12, r = 0.0726, n = 25*12, I = 0.0726/12
𝐹𝑉 =
(1+.00605)300 −1
50
0.00605
FV= $42211.46
How much money will he need to invest to make .5 million dollars?
FV = .5 million, R = ?
(1 + 𝑖)𝑛 −1
𝐹𝑉 = 𝑅
𝑖
Geno contributes $50 per month into Morgan Park Mutual Fund that
earns 7.26 annual interest rate. What is the value of Geno’s investment
after he retires in 25 years?
R = $50, k = 12, r = 0.0726, n = 25*12, I = 0.0726/12
𝐹𝑉 =
(1+.00605)300 −1
50
0.00605
FV= $42211.46 Geno will have $42, 211,46 for his retirement.
How much money will he need to invest to make .5 million dollars?
FV = .5 million, R = ?
(1+.00605)300 −1
𝑅
0.00605
500,000 =
R = $592.26 Geno needs to pay $592.26 monthly to have .5 million
dollars for retirement.
Loans and Mortgages
We want to get a loan or mortgage. We want to get the money today
(present) and pay it back over time. The net amount of money put into
an annuity is its present value. This value is called the Present Value
of the Annuity. (Annual Percentage Rate – APR- the annual interest
Rate charged on consumer loans.)
We can calculate our monthly payments needed to pay back the loan
using the Present Value Equation:
P𝑉 =
where
1−(1+𝑖)−𝑛
𝑅
𝑖
r = the annual interest rate
k = number of times interest is compounded per year
n = number of equal periodic payments
(number of years * compounding period)
R = payment amount
𝑟
i = interest rate ( )
𝑘
Future Value Equation:
1− 1+𝑖
𝑃𝑉 = 𝑅
𝑖
−𝑛
where
r = the annual interest rate
k = number of times interest is compounded per year
n = number of equal periodic payments
(number of years * compounding periods (k))
R = payment amount
𝑟
i = interest rate (𝑘)
Genae buys a black C Class Mercedes Benz Coupe for $40,000.
She gets financing and puts $5000 down for the car. The APR is 6%.
What will be her monthly payments if she finances the car
for 60 months?
Future Value Equation:
1− 1+𝑖
𝑃𝑉 = 𝑅
𝑖
−𝑛
where
r = the annual interest rate
k = number of times interest is compounded per year
n = number of equal periodic payments
(number of years * compounding periods (k))
R = payment amount
𝑟
i = interest rate (𝑘)
Genae buys a C Class Mercedes Benz for $40,000. She gets financing.
and puts $5000 down for the car. The APR is 6%. What will be her
monthly payments if she finances the car for 60 months?
PV = 35000, k = 12, r = 0.06, n =5*12 =60, i = 0.06/12
P𝑉 =
1−(1+𝑖)−𝑛
𝑅
𝑖
Genae buys a C Class Mercedes Benz for $40,000. She gets financing
and puts $5000 down for the car. The APR is 6%. What will be her
monthly payments if she finances the car for 60 months?
PV = 35000, k = 12, r = 0.06, n =5*12 =60, i = 0.06/12
35000=
1−(1+.005)−60
𝑅
0.005
R= $676.65
Genae’s monthly bill will be $676.65 for 5 years to pay for her car.
How much interest will Genae pay in total?
How much interest will Genae pay?
First how much is she paying total? $676.65 * 60 = $40,598.88
Interest Paid: $40,598.88- $35000 = $5598.88
Simple Interest
Simple interest is calculated on the original principal only. Accumulated interest
from prior periods is not used in calculations for the following periods. Simple interest
is normally used for a single period of less than a year, such as 30 or 60 days.
where:
Simple Interest I = P*r*n
p = principal (original amount borrowed or loaned)
i = interest rate for one period
n = number of periods
Compound Interest
Compound interest is calculated each period on the original principal
and all interest accumulated during past periods. Although the
interest may be stated as a yearly rate, the compounding periods can be
yearly, semiannually, quarterly, or even continuously.
Compound Interest
Compound interest is calculated each period on the original principal
and all interest accumulated during past periods. Although the
interest may be stated as a yearly rate, the compounding periods can be
yearly, semiannually, quarterly, or even continuously.
The power of compounding can have an astonishing effect on the
accumulation of wealth. This table shows the results of making a one-time
investment of $10,000 for 30 years using 12% simple interest, and 12%
interest compounded yearly and quarterly.
Type of Interest
Simple
Principal Plus Interest Earned
46,000.00
Compounded Yearly
299,599.22
Compounded Quarterly
347,109.87
10) The Fresh and Green Company has a savings plan for employees.
If an employee makes an initial deposit of $1000, the company
pays 8% interest compounded quarterly. If an employee withdraws
the money after five years, how much is in the account?
11) Using the information in the question above, find the interest earned
if the money is withdrawn after 35 years.
12) Determine the amount of interest earned if $500 is invested at
an interest rate of 4.25% compounded quarterly for 12 years.
10) The Fresh and Green Company has a savings plan for employees.
If an employee makes an initial deposit of $1000, the company
pays 8% interest compounded quarterly. If an employee withdraws
the money after five years, how much is in the account?
A = $1485.95
11) Using the information in the question above, find the interest
earned.
Simple Interest
Compound Interest
12) Determine the amount of interest earned if $500 is invested at an
interest rate of 4.25% compounded quarterly for 12 years.
10) The Fresh and Green Company has a savings plan for employees.
If an employee makes an initial deposit of $1000, the company
pays 8% interest compounded quarterly. If an employee withdraws
the money after five years, how much is in the account?
A = $1485.95
11) Using the information in the question above, find the interest
earned.
Simple Interest I = prt I = 1000*.08* 5 = $400
Compound Interest = $1485.95 - $1000 = $485.95
12) Determine the amount of interest earned if $500 is invested at an
interest rate of 4.25% compounded quarterly for 12 years.
Simple Interest
Compound Interest
10) The Fresh and Green Company has a savings plan for employees.
If an employee makes an initial deposit of $1000, the company
pays 8% interest compounded quarterly. If an employee withdraws
the money after five years, how much is in the account?
A = $1485.95
11) Using the information in the question above, find the interest
earned.
Simple Interest I = prt I = 1000*.08* 5 = $400
Compound Interest = $1485.95 - $1000 = $485.95
12) Determine the amount of interest earned if $500 is invested at an
interest rate of 4.25% compounded quarterly for 12 years.
Simple Interest $255
Compound Interest $830.41
Future Value Equation: http://www.bugatti.com/en/veyron-16.4.html
(1 + 𝑖)𝑛 −1
𝐹𝑉 = 𝑅
𝑖
where
r = the annual interest rate
k = number of times interest is compounded per year
n = number of equal periodic payments
(number of years * compounding periods (k))
R = payment amount
𝑟
i = interest rate (𝑘)
Genae and Lucky want a Bugatti Veyron sports car. This car costs
$1,700,000. Genae and Lucky are going to use the money Genae was
paying for the Mercedes to save up for the Bugatti. This means Genae
contributes $680 per month @ 7.26 interest rate into the Genae/Lucky
Bugatti fund. How long before Genae and Lucky will get their car?
Future Value Equation: http://www.bugatti.com/en/veyron-16.4.html
(1 + 𝑖)𝑛 −1
𝐹𝑉 = 𝑅
𝑖
where
r = the annual interest rate
k = number of times interest is compounded per year
n = number of equal periodic payments
(number of years * compounding periods (k))
R = payment amount
𝑟
i = interest rate (𝑘)
Genae and Lucky want a Buggatti Veyron sports car. This car costs
$1,700,000. Genae and Lucky are going to use the money Genae was
paying for the Mercedes to save up for the Bugatti. This means Genae
contributes $680 per month @ 7.26 interest rate into the Genae/Lucky
Bugatti fund. How long before Genae and Lucky will get their car?
R = $680, k = 12, r = 0.0726, n = ?, i = 0.0726/12
Future Value Equation: http://www.bugatti.com/en/veyron-16.4.html
(1 + 𝑖)𝑛 −1
𝐹𝑉 = 𝑅
𝑖
where
r = the annual interest rate
k = number of times interest is compounded per year
n = number of equal periodic payments
(number of years * compounding periods (k))
R = payment amount
𝑟
i = interest rate (𝑘)
Genae and Lucky want a Buggatti Veyron sports car. This car costs
$1,700,000. Genae and Lucky are going to use the money Genae was
paying for the Mercedes to save up for the Bugatti. This means Genae
contributes $680 per month @ 7.26 interest rate into the Genae/Lucky
Bugatti fund. How long before Genae and Lucky will get their car?
R = $680, k = 12, r = 0.0726, n = ?, i = 0.0726/12