Section 5.1: Simple and Compound Interest

1
Section 5.1 Simple and Compound Interest
2
Simple Interest

• Simple Interest Used to calculate interest on
  loansoften 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

3
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 ?

4
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

5
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)

6
a. How much interest will she pay? Simple
interest I PrtI ? P 5,000 r .065 t
11/12I Prt (5000)(0.065)(11/12) ______







7
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

8
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?

9
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

10
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

11
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

12
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)

13
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)
14
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
15
Alabama will beat Michigan Saturday in Dallas.

1. Yes
2. No

16
Find simple interest
10,502 at 4.2 for 10 months

1. 370.66
2. 367.57
3. 404.33
4. 330.81

17
Compound Interest

• Compound Interest more commonly used than simple
  interest.
• With compound interest, the interest itself earns
  interest.
• Formula
  

18
Compound Interest

• Compound Interest more commonly used than simple
  interest.
• With compound interest, the interest itself earns
  interest.
• Formula
  
• Where
• A is the compound amount (includes principal and
  interest)

19
Compound Interest

• Compound Interest more commonly used than simple
  interest.
• With compound interest, the interest itself earns
  interest.
• Formula
  
• Where
• A is the compound amount (includes principal and
  interest)
• P is the initial investment

20
Compound Interest

• Compound Interest more commonly used than simple
  interest.
• With compound interest, the interest itself earns
  interest.
• Formula
  
• Where
• A is the compound amount (includes principal and
  interest)
• P is the initial investment
• r is the annual percentage rate

21
Compound Interest

• Compound Interest more commonly used than simple
  interest.
• With compound interest, the interest itself earns
  interest.
• Formula
  
• 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

22
Compound Interest

• Compound Interest more commonly used than simple
  interest.
• With compound interest, the interest itself earns
  interest.
• Formula
  
• 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.

23
Compound Interest

• Compound Interest more commonly used than simple
  interest.
• With compound interest, the interest itself earns
  interest.
• Formula
  
• 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

24
Compound Interest

• Compound Interest more commonly used than simple
  interest.
• With compound interest, the interest itself earns
  interest.
• Formula
  
• 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

25
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.
26
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.

27
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.
• A ? P 22,000 r 0.055 m 1 t
  5

28
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.
• A ? P 22,000 r 0.055 m 1 t
  5

29
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.
• A ? P 22,000 r 0.055 m 1 t
  5

30
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.
• A ? P 22,000 r 0.055 m 1 t
  5

Find the amount of interest earned.
31
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.
• A ? P 22,000 r 0.055 m 1 t
  5

Find the amount of interest earned. Compound
Amount (A) Principal (P) Interest (I), so
I A P
32
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.
• A ? P 22,000 r 0.055 m 1 t
  5

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
33
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.)
34
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.)

35
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.)
• A ? P 22,000 r 0.055 m 12 t
  5

36
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.)
• A ? P 22,000 r 0.055 m 12 t
  5

37
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.)
• A ? P 22,000 r 0.055 m 12 t
  5

to the nearest DOLLAR
38
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.)
• A ? P 22,000 r 0.055 m 12 t
  5

to the nearest DOLLAR
Find the amount of interest earned.
39
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.)
• A ? P 22,000 r 0.055 m 12 t
  5

to the nearest DOLLAR
Find the amount of interest earned. Compound
Amount (A) Principal (P) Interest (I), so
I A P
40
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.)
• A ? P 22,000 r 0.055 m 12 t
  5

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
41
Find the compound amount
9000 At 3 compounded semiannually for 5 years

1. 10,444.87
2. 10,433.47
3. 10,350.00
4. 9,695.56

42
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.
43
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.
44
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.

45
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 doesnt
change based on the principal or the amount of
time.
46
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 doesnt
change based on the principal or the amount of
time. So, in our formulas, we can just calculate
using 1 for 1 year.
47
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 doesnt
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
48
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 doesnt
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
49
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 doesnt
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
50
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 doesnt
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
So 1 would turn into 1.0824 in 1 year.
51
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 doesnt
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
Now use A 1.0824 in the simple interest
formula solve for r. (This will be the EAR.)
So 1 would turn into 1.0824 in 1 year.
52
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 doesnt
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
Now use A 1.0824 in the simple interest
formula solve for r. (This will be the EAR.) A
P(1rt)
So 1 would turn into 1.0824 in 1 year.
53
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 doesnt
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
Now use A 1.0824 in the simple interest
formula solve for r. (This will be the EAR.) A
P(1rt) 1.0824 11 r(1)
So 1 would turn into 1.0824 in 1 year.
54
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 doesnt
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
Now use A 1.0824 in the simple interest
formula solve for r. (This will be the EAR.) A
P(1rt) 1.0824 11 r(1) 1.0824 1 r
So 1 would turn into 1.0824 in 1 year.
55
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 doesnt
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
Now use A 1.0824 in the simple interest
formula solve for r. (This will be the EAR.) A
P(1rt) 1.0824 11 r(1) 1.0824 1 r
r .0824
So 1 would turn into 1.0824 in 1 year.
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.
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 doesnt
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
Now use A 1.0824 in the simple interest
formula solve for r. (This will be the EAR.) A
P(1rt) 1.0824 11 r(1) 1.0824 1 r
r .0824 So, the EAR is 8.24
So 1 would turn into 1.0824 in 1 year.
57
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

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!
58
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.

59
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.

60
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.

A 300,000 P ? r 0.123 m 2
t 15
61
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.

A 300,000 P ? r 0.123 m 2
t 15
62
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.

A 300,000 P ? r 0.123 m 2
t 15 P 50,063.51
63
How much of this did you understand well today?

1. All or most
2. A lot of it
3. About half of it
4. Not too much of it
5. None or hardly any of it