Section 5.1:
Simple and Compound Interest
Simple Interest
Simple Interest: Used to calculate interest on
loans…often of one year or less.
Formula:
I = Prt
 I : interest earned (or owed)
 P : principal invested (or borrowed)
 r : annual interest rate
 t : time in years
Example 1
To buy furniture for a new apartment,
Jennifer Wall borrowed $5,000 at
6.5% simple interest for 11 months.
a. How much interest will she pay?
Simple interest: I = Prt
I=?
Example 1
To buy furniture for a new apartment,
Jennifer Wall borrowed $5,000 at
6.5% simple interest for 11 months.
a. How much interest will she pay?
Simple interest: I = Prt
I=?
P = $5,000 r = .065
t =11/12
Example 1
To buy furniture for a new apartment,
Jennifer Wall borrowed $5,000 at
6.5% simple interest for 11 months.
a. How much interest will she pay?
Simple interest: I = Prt
I=?
P = $5,000 r = .065
I = Prt = (5000)(0.065)(11/12) =
t =11/12
a. How much interest will she pay?
Simple interest: I = Prt
I=?
P = $5,000 r = .065
t =11/12
I = Prt = (5000)(0.065)(11/12) = $______
Example 1
To buy furniture for a new apartment,
Jennifer Wall borrowed $5,000 at
6.5% simple interest for 11 months.
a. How much interest will she pay?
Simple interest: I = Prt
I=?
P = $5,000 r = .065
t =11/12
I = Prt = (5000)(0.065)(11/12) = $297.92
Example 1
To buy furniture for a new apartment,
Jennifer Wall borrowed $5,000 at
6.5% simple interest for 11 months.
b. What is the total amount to be repaid?
Example 1
To buy furniture for a new apartment,
Jennifer Wall borrowed $5,000 at 6.5%
simple interest for 11 months.
b. What is the total amount to be repaid?
Amount to Repay = Principal + Interest
Example 1
To buy furniture for a new apartment,
Jennifer Wall borrowed $5,000 at 6.5%
simple interest for 11 months.
b. What is the total amount to be repaid?
Amount to Repay = Principal + Interest
= 5000 + 297.92
Example 1
To buy furniture for a new apartment,
Jennifer Wall borrowed $5,000 at 6.5%
simple interest for 11 months.
b. What is the total amount to be repaid?
Amount to Repay = Principal + Interest
= 5000 + 297.92 = $ 5,297.92
Example 1
To buy furniture for a new apartment,
Jennifer Wall borrowed $5,000 at 6.5%
simple interest for 11 months.
b. What is the total amount to be repaid?
Amount to Repay = Principal + Interest
= 5000 + 297.92 = $ 5,297.92
Notice here that we really have:
A = P + I … or A = P + Prt = P(1 + rt)
Example 1
To buy furniture for a new apartment,
Jennifer Wall borrowed $5,000 at 6.5%
simple interest for 11 months.
b. What is the total amount to be repaid?
Amount to Repay = Principal + Interest
= 5000 + 297.92 = $ 5,297.92
Notice here that we really have:
A = P + I … or A = P + Prt = P(1 + rt)
So, if you want a direct formula for A with simple interest, use
A = P(1 + rt)
Example 1
To buy furniture for a new apartment,
Jennifer Wall borrowed $5,000 at 6.5%
simple interest for 11 months.
b. What is the total amount to be repaid?
Amount to Repay = Principal + Interest
= 5000 + 297.92 = $ 5,297.92
Notice here that we really have:
A = P + I … or A = P + Prt = P(1 + rt)
So, if you want a direct formula for A with simple interest, use
A = P(1 + rt)
and, of course if you only want I, then use
I = Prt
Alabama will beat Michigan Saturday in Dallas.
1.
2.
Yes
No
Find simple interest
$10,502 at 4.2%
for 10 months
A.
B.
C.
D.
$370.66
$367.57
$404.33
$330.81
Compound Interest
Compound Interest: more commonly used than simple interest.
With compound interest, the interest itself earns interest.
Formula:
r

A  P 1  
 m
mt
Compound Interest
Compound Interest: more commonly used than simple interest.
With compound interest, the interest itself earns interest.
Formula:
r

A  P 1  
 m
mt
Where
 A is the compound amount (includes principal and interest)
Compound Interest
Compound Interest: more commonly used than simple interest.
With compound interest, the interest itself earns interest.
Formula:
r

A  P 1  
 m
mt
Where
 A is the compound amount (includes principal and interest)
 P is the initial investment
Compound Interest
Compound Interest: more commonly used than simple interest.
With compound interest, the interest itself earns interest.
Formula:
r

A  P 1  
 m
mt
Where
 A is the compound amount (includes principal and interest)
 P is the initial investment
 r is the annual percentage rate
Compound Interest
Compound Interest: more commonly used than simple interest.
With compound interest, the interest itself earns interest.
Formula:
r

A  P 1  
 m
mt
Where
 A is the compound amount (includes principal and interest)
 P is the initial investment
 r is the annual percentage rate
 m is the number of compounding periods per year:
Compound Interest
Compound Interest: more commonly used than simple interest.
With compound interest, the interest itself earns interest.
Formula:
r

A  P 1  
 m
mt
Where
 A is the compound amount (includes principal and interest)
 P is the initial investment
 r is the annual percentage rate
 m is the number of compounding periods per year:
 Compounded annually, m = 1
 Compounded semiannually, m = 2
 Compounded quarterly, m = 4, etc.
Compound Interest
Compound Interest: more commonly used than simple interest.
With compound interest, the interest itself earns interest.
Formula:
r

A  P 1  
 m
mt
Where
 A is the compound amount (includes principal and interest)
 P is the initial investment
 r is the annual percentage rate
 m is the number of compounding periods per year:
 Compounded annually, m = 1
 Compounded semiannually, m = 2
 Compounded quarterly, m = 4, etc.
 t is the number of years
Compound Interest
Compound Interest: more commonly used than simple interest.
With compound interest, the interest itself earns interest.
r

A  P 1  
 m
Formula:
mt
 P1  i 
n
Where
 A is the compound amount (includes principal and interest)
 P is the initial investment
 r is the annual percentage rate
 m is the number of compounding periods per year:
 Compounded annually, m = 1
 Compounded semiannually, m = 2
 Compounded quarterly, m = 4, etc.
 t is the number of years
 n = mt is the total # of compounding periods over all t years
 i = r/m is the interest rate per compounding period
Example 2
Suppose that $22,000 is invested at 5.5% interest. Find
the amount of money in the account after 5 years if the
interest is compounded annually.
Example 2
Suppose that $22,000 is invested at 5.5% interest. Find
the amount of money in the account after 5 years if the
interest is compounded annually.
mt
r

A  P 1  
m

 P 1  i 
n
Example 2
Suppose that $22,000 is invested at 5.5% interest. Find
the amount of money in the account after 5 years if the
interest is compounded annually.
mt
r

A  P 1  
m

 P 1  i 
A = ?; P = 22,000; r = 0.055; m = 1; t = 5
n
Example 2
Suppose that $22,000 is invested at 5.5% interest. Find
the amount of money in the account after 5 years if the
interest is compounded annually.
mt
r

A  P 1  
m

 P 1  i 
A = ?; P = 22,000; r = 0.055; m = 1; t = 5
 0.055 
A  220001 

1 

(1)( 5 )
n
Example 2
Suppose that $22,000 is invested at 5.5% interest. Find
the amount of money in the account after 5 years if the
interest is compounded annually.
mt
r

A  P 1  
m

 P 1  i 
A = ?; P = 22,000; r = 0.055; m = 1; t = 5
 0.055 
A  220001 

1 

(1)( 5 )
 $28,753.12
n
Example 2
Suppose that $22,000 is invested at 5.5% interest. Find
the amount of money in the account after 5 years if the
interest is compounded annually.
mt
r

A  P 1  
m

 P 1  i 
A = ?; P = 22,000; r = 0.055; m = 1; t = 5
 0.055 
A  220001 

1 

Find the amount of interest earned.
(1)( 5 )
 $28,753.12
n
Example 2
Suppose that $22,000 is invested at 5.5% interest. Find
the amount of money in the account after 5 years if the
interest is compounded annually.
mt
r

A  P 1  
m

 P 1  i 
A = ?; P = 22,000; r = 0.055; m = 1; t = 5
 0.055 
A  220001 

1 

(1)( 5 )
 $28,753.12
Find the amount of interest earned.
Compound Amount (A) = Principal (P) + Interest (I), so
I=A–P
n
Example 2
Suppose that $22,000 is invested at 5.5% interest. Find
the amount of money in the account after 5 years if the
interest is compounded annually.
mt
r

A  P 1  
m

 P 1  i 
A = ?; P = 22,000; r = 0.055; m = 1; t = 5
 0.055 
A  220001 

1 

(1)( 5 )
 $28,753.12
Find the amount of interest earned.
Compound Amount (A) = Principal (P) + Interest (I), so
I=A–P
= 28,753.12 – 22,000 = $ 6,753.12
n
Example 3
If $22,000 is invested at 5.5% interest. Find the amount of
money in the account after 5 years if interest is
compounded monthly. (Round answer to nearest dollar.)
Example 3
If $22,000 is invested at 5.5% interest. Find the amount of
money in the account after 5 years if interest is
compounded monthly. (Round answer to nearest dollar.)
r

A  P 1  
m

mt
 P 1  i 
n
Example 3
If $22,000 is invested at 5.5% interest. Find the amount of
money in the account after 5 years if interest is
compounded monthly. (Round answer to nearest dollar.)
mt
r

n
A  P1    P 1  i 
m

A = ?; P = 22,000; r = 0.055; m = 12; t = 5
Example 3
If $22,000 is invested at 5.5% interest. Find the amount of
money in the account after 5 years if interest is
compounded monthly. (Round answer to nearest dollar.)
mt
r

n
A  P1    P 1  i 
m

A = ?; P = 22,000; r = 0.055; m = 12; t = 5
 0.055 
A  220001 

12 

(12)( 5 )
Example 3
If $22,000 is invested at 5.5% interest. Find the amount of
money in the account after 5 years if interest is
compounded monthly. (Round answer to nearest dollar.)
mt
r

n
A  P1    P 1  i 
m

A = ?; P = 22,000; r = 0.055; m = 12; t = 5
 0.055 
A  220001 

12 

(12)( 5 )
 $28,945
to the nearest DOLLAR
Example 3
If $22,000 is invested at 5.5% interest. Find the amount of
money in the account after 5 years if interest is
compounded monthly. (Round answer to nearest dollar.)
mt
r

n
A  P1    P 1  i 
m

A = ?; P = 22,000; r = 0.055; m = 12; t = 5
 0.055 
A  220001 

12 

Find the amount of interest earned.
(12)( 5 )
 $28,945
to the nearest DOLLAR
Example 3
If $22,000 is invested at 5.5% interest. Find the amount of
money in the account after 5 years if interest is
compounded monthly. (Round answer to nearest dollar.)
mt
r

n
A  P1    P 1  i 
m

A = ?; P = 22,000; r = 0.055; m = 12; t = 5
 0.055 
A  220001 

12 

(12)( 5 )
 $28,945
to the nearest DOLLAR
Find the amount of interest earned.
Compound Amount (A) = Principal (P) + Interest (I), so
I=A–P
Example 3
If $22,000 is invested at 5.5% interest. Find the amount of
money in the account after 5 years if interest is
compounded monthly. (Round answer to nearest dollar.)
mt
r

n
A  P1    P 1  i 
m

A = ?; P = 22,000; r = 0.055; m = 12; t = 5
 0.055 
A  220001 

12 

(12)( 5 )
 $28,945
to the nearest DOLLAR
Find the amount of interest earned.
Compound Amount (A) = Principal (P) + Interest (I), so
I=A–P
= 28,945 – 22,000 = $ 6,945
Find the compound amount
$9000
At 3% compounded
semiannually for 5 years
A.
B.
C.
D.
$10,444.87
$10,433.47
$10,350.00
$9,695.56
Example 4: Effective Rate
The Effective Annual Rate (EAR) is the rate that
would be paid using simple interest in order to be
equivalent to a stated compounded rate.
Example 4: Effective Rate
The Effective Annual Rate (EAR) is the rate that
would be paid using simple interest in order to be
equivalent to a stated compounded rate.
Financial institutions are usually required by law to provide the effective
rate so that consumers can easily compare ‘apples to apples’.
Example 4: Effective Rate
The Effective Annual Rate (EAR) is the rate that
would be paid using simple interest in order to be
equivalent to a stated compounded rate.
Financial institutions are usually required by law to provide the effective
rate so that consumers can easily compare ‘apples to apples’.
Ex. Find the effective annual rate corresponding to
a rate of 8% compounded quarterly.
Example 4: Effective Rate
The Effective Annual Rate (EAR) is the rate that
would be paid using simple interest in order to be
equivalent to a stated compounded rate.
Financial institutions are usually required by law to provide the effective
rate so that consumers can easily compare ‘apples to apples’.
Ex. Find the effective annual rate corresponding to
a rate of 8% compounded quarterly.
This question is easy to answer if we notice a simplifying fact: The interest
rate doesn’t change based on the principal or the amount of time.
Example 4: Effective Rate
The Effective Annual Rate (EAR) is the rate that
would be paid using simple interest in order to be
equivalent to a stated compounded rate.
Financial institutions are usually required by law to provide the effective
rate so that consumers can easily compare ‘apples to apples’.
Ex. Find the effective annual rate corresponding to
a rate of 8% compounded quarterly.
This question is easy to answer if we notice a simplifying fact: The interest
rate doesn’t change based on the principal or the amount of time.
So, in our formulas, we can just calculate using $1 for 1 year.
Example 4: Effective Rate
The Effective Annual Rate (EAR) is the rate that
would be paid using simple interest in order to be
equivalent to a stated compounded rate.
Financial institutions are usually required by law to provide the effective
rate so that consumers can easily compare ‘apples to apples’.
Ex. Find the effective annual rate corresponding to
a rate of 8% compounded quarterly.
This question is easy to answer if we notice a simplifying fact: The interest
rate doesn’t change based on the principal or the amount of time.
So, in our formulas, we can just calculate using $1 for 1 year.
First see how much would be
earned with compounding:
Example 4: Effective Rate
The Effective Annual Rate (EAR) is the rate that
would be paid using simple interest in order to be
equivalent to a stated compounded rate.
Financial institutions are usually required by law to provide the effective
rate so that consumers can easily compare ‘apples to apples’.
Ex. Find the effective annual rate corresponding to
a rate of 8% compounded quarterly.
This question is easy to answer if we notice a simplifying fact: The interest
rate doesn’t change based on the principal or the amount of time.
So, in our formulas, we can just calculate using $1 for 1 year.
First see how much would be
earned with compounding:
A 
 r
P 1 

 m
mt
 .08 
 1 1 

 4 
4 (1)
Example 4: Effective Rate
The Effective Annual Rate (EAR) is the rate that
would be paid using simple interest in order to be
equivalent to a stated compounded rate.
Financial institutions are usually required by law to provide the effective
rate so that consumers can easily compare ‘apples to apples’.
Ex. Find the effective annual rate corresponding to
a rate of 8% compounded quarterly.
This question is easy to answer if we notice a simplifying fact: The interest
rate doesn’t change based on the principal or the amount of time.
So, in our formulas, we can just calculate using $1 for 1 year.
First see how much would be
earned with compounding:
A 
A 
 r
P 1 

 m
1.02 4
mt
 .08 
 1 1 

 4 
 1.0824
4 (1)
Example 4: Effective Rate
The Effective Annual Rate (EAR) is the rate that
would be paid using simple interest in order to be
equivalent to a stated compounded rate.
Financial institutions are usually required by law to provide the effective
rate so that consumers can easily compare ‘apples to apples’.
Ex. Find the effective annual rate corresponding to
a rate of 8% compounded quarterly.
This question is easy to answer if we notice a simplifying fact: The interest
rate doesn’t change based on the principal or the amount of time.
So, in our formulas, we can just calculate using $1 for 1 year.
First see how much would be
earned with compounding:
A 
A 
 r
P 1 

 m
1.02 4
mt
 .08 
 1 1 

 4 
4 (1)
 1.0824
So $1 would turn into $1.0824 in 1 year.
Example 4: Effective Rate
The Effective Annual Rate (EAR) is the rate that
would be paid using simple interest in order to be
equivalent to a stated compounded rate.
Financial institutions are usually required by law to provide the effective
rate so that consumers can easily compare ‘apples to apples’.
Ex. Find the effective annual rate corresponding to
a rate of 8% compounded quarterly.
This question is easy to answer if we notice a simplifying fact: The interest
rate doesn’t change based on the principal or the amount of time.
So, in our formulas, we can just calculate using $1 for 1 year.
First see how much would be
earned with compounding:
A 
A 
 r
P 1 

 m
1.02 4
mt
 .08 
 1 1 

 4 
4 (1)
 1.0824
So $1 would turn into $1.0824 in 1 year.
Now use A = $1.0824 in the
simple interest formula & solve
for r. (This will be the EAR.)
Example 4: Effective Rate
The Effective Annual Rate (EAR) is the rate that
would be paid using simple interest in order to be
equivalent to a stated compounded rate.
Financial institutions are usually required by law to provide the effective
rate so that consumers can easily compare ‘apples to apples’.
Ex. Find the effective annual rate corresponding to
a rate of 8% compounded quarterly.
This question is easy to answer if we notice a simplifying fact: The interest
rate doesn’t change based on the principal or the amount of time.
So, in our formulas, we can just calculate using $1 for 1 year.
First see how much would be
earned with compounding:
A 
A 
 r
P 1 

 m
1.02 4
mt
 .08 
 1 1 

 4 
4 (1)
 1.0824
So $1 would turn into $1.0824 in 1 year.
Now use A = $1.0824 in the
simple interest formula & solve
for r. (This will be the EAR.)
A = P(1+rt)
Example 4: Effective Rate
The Effective Annual Rate (EAR) is the rate that
would be paid using simple interest in order to be
equivalent to a stated compounded rate.
Financial institutions are usually required by law to provide the effective
rate so that consumers can easily compare ‘apples to apples’.
Ex. Find the effective annual rate corresponding to
a rate of 8% compounded quarterly.
This question is easy to answer if we notice a simplifying fact: The interest
rate doesn’t change based on the principal or the amount of time.
So, in our formulas, we can just calculate using $1 for 1 year.
First see how much would be
earned with compounding:
A 
A 
 r
P 1 

 m
1.02 4
mt
 .08 
 1 1 

 4 
4 (1)
 1.0824
So $1 would turn into $1.0824 in 1 year.
Now use A = $1.0824 in the
simple interest formula & solve
for r. (This will be the EAR.)
A = P(1+rt)
1.0824 = 1[1 + r(1)]
Example 4: Effective Rate
The Effective Annual Rate (EAR) is the rate that
would be paid using simple interest in order to be
equivalent to a stated compounded rate.
Financial institutions are usually required by law to provide the effective
rate so that consumers can easily compare ‘apples to apples’.
Ex. Find the effective annual rate corresponding to
a rate of 8% compounded quarterly.
This question is easy to answer if we notice a simplifying fact: The interest
rate doesn’t change based on the principal or the amount of time.
So, in our formulas, we can just calculate using $1 for 1 year.
First see how much would be
earned with compounding:
A 
A 
 r
P 1 

 m
1.02 
4
mt
 .08 
 1 1 

 4 
4 (1)
 1.0824
So $1 would turn into $1.0824 in 1 year.
Now use A = $1.0824 in the
simple interest formula & solve
for r. (This will be the EAR.)
A = P(1+rt)
1.0824 = 1[1 + r(1)]
1.0824 = 1 + r
Example 4: Effective Rate
The Effective Annual Rate (EAR) is the rate that
would be paid using simple interest in order to be
equivalent to a stated compounded rate.
Financial institutions are usually required by law to provide the effective
rate so that consumers can easily compare ‘apples to apples’.
Ex. Find the effective annual rate corresponding to
a rate of 8% compounded quarterly.
This question is easy to answer if we notice a simplifying fact: The interest
rate doesn’t change based on the principal or the amount of time.
So, in our formulas, we can just calculate using $1 for 1 year.
First see how much would be
earned with compounding:
A 
A 
 r
P 1 

 m
1.02 
4
mt
 .08 
 1 1 

 4 
4 (1)
 1.0824
So $1 would turn into $1.0824 in 1 year.
Now use A = $1.0824 in the
simple interest formula & solve
for r. (This will be the EAR.)
A = P(1+rt)
1.0824 = 1[1 + r(1)]
1.0824 = 1 + r
r = .0824
Example 4: Effective Rate
The Effective Annual Rate (EAR) is the rate that
would be paid using simple interest in order to be
equivalent to a stated compounded rate.
Financial institutions are usually required by law to provide the effective
rate so that consumers can easily compare ‘apples to apples’.
Ex. Find the effective annual rate corresponding to
a rate of 8% compounded quarterly.
This question is easy to answer if we notice a simplifying fact: The interest
rate doesn’t change based on the principal or the amount of time.
So, in our formulas, we can just calculate using $1 for 1 year.
First see how much would be
earned with compounding:
A 
A 
 r
P 1 

 m
1.02 
4
mt
 .08 
 1 1 

 4 
4 (1)
 1.0824
So $1 would turn into $1.0824 in 1 year.
Now use A = $1.0824 in the
simple interest formula & solve
for r. (This will be the EAR.)
A = P(1+rt)
1.0824 = 1[1 + r(1)]
1.0824 = 1 + r
r = .0824
So, the EAR is 8.24%
Example 4: Effective Rate
If you would rather have a formula for EAR, here it is:
The effective rate corresponding to a
stated rate of interest r compounded m
times per year is
m
r

re  1   1
m

This formula gives the same answer that you would get if you
just ‘figured it out’ as we did earlier. Try it yourself and see!
Example 5
A family plans to retire in 15 years and expects to need
$300,000. Determine how much they must invest
today at 12.3% compounded semiannually to
accomplish their goal.
Example 5
A family plans to retire in 15 years and expects to need
$300,000. Determine how much they must invest
today at 12.3% compounded semiannually to
accomplish their goal.
r 

A  P 1  
m

mt
Example 5
A family plans to retire in 15 years and expects to need
$300,000. Determine how much they must invest
today at 12.3% compounded semiannually to
accomplish their goal.
r 

A  P 1  
m

A = $300,000
mt
P=?
r = 0.123
m=2
t = 15
Example 5
A family plans to retire in 15 years and expects to need
$300,000. Determine how much they must invest
today at 12.3% compounded semiannually to
accomplish their goal.
r 

A  P 1  
m

A = $300,000
mt
P=?
r = 0.123
 0.123 
300000  P1 

2 

m=2
( 2 )(15)
t = 15
Example 5
A family plans to retire in 15 years and expects to need
$300,000. Determine how much they must invest
today at 12.3% compounded semiannually to
accomplish their goal.
r 

A  P 1  
m

A = $300,000
mt
P=?
r = 0.123
 0.123 
300000  P1 

2 

P = $50,063.51
m=2
( 2 )(15)
t = 15
How much of this did you understand well today?
1.
2.
3.
4.
5.
All or most
A lot of it
About half of it
Not too much of it
None or hardly any of it