7.2
CompoundInterest
Gaston is a talented young musician who earns
a living by playing at coffee houses, university
campuses, and special events such as weddings. Before he buys a new guitar,
Gaston wants to compare various borrowing options that are available to him.
compoundinterest
• interest that is
calculated at regular
compounding periods
• added to the principal
for the following
compounding period
In Section 7.1, you learned that when simple interest is earned, the amount
grows linearly as a function of time. However, most investments and loans are
based on compoundinterest. This means that at the end of each compounding
period, the interest is added to the principal for the next compounding period
so that interest is then paid on this new total amount.
compounding
period
• time interval after
which compound
interest is calculated
Investigate
Tools
• graphing calculator
or
• computer with graphing
software
Howcanyoucalculatecompoundinterest?
Recall from the Investigate in Section 7.1 that Alexis receives $1000 for
her grade 8 graduation. Under the simple interest scenario, the interest
earned at 5% per year for 5 years is $250.
Suppose that Alexis deposits the money into an account that pays the
same annual interest rate of 5%, but compounded annually. How does
this affect the amount of interest she will earn?
1. Copy the table. Calculate the interest using the formula I 5 Prt and
complete the table. Leave room for two more columns to the right.
Year
Balance at
Start of Year ($)
Interest Earned
During the Year ($)
Balance at
End of Year ($)
1
1000
1000(0.05)(1) = 50
1000 + 50 = 1050
2
1050
1050(0.05)(1) = 52.50
1050 + 52.50 = 1102.50
3
4
5
2. a) Look at the values in the Interest Earned column. Describe what
you notice.
b) Reflect What is an advantage of compound interest over simple
interest?
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3. a) Add a column to the right of your table from step 1. Label the
column “First Differences.” Calculate the first differences by
subtracting successive balances at the end of the year. Record the
values in the table.
b) What do the first differences represent?
c) Is the balance at the end of the year versus time a linear
relationship? Explain.
4. a) Add a column to the right of your table from step 3. Label the
column “Common Ratios.” Calculate the common ratios by
dividing successive balances at the end of the year. For example,
calculate 1102.50 ÷ 1050. Record these values in the table.
b) What do you notice about the pattern of common ratios?
c) What type of function does this represent? Explain how you can
tell.
5. a) Graph the Balance at the End of Year versus Year, from step 1,
using graphing technology.
b) Does the shape of the graph support your answer to step 4c)?
Explain.
c) Graph the amount in the simple interest investment given by the
equation A 5 1000  50t on the same set of axes as your graph in
part a). How are these graphs alike? different?
d) What happens to these graphs after a long period of time? Why
does this happen?
6. Reflect
a) Is the interest earned per year the same for every year under
simple interest conditions? under compound interest conditions?
Explain.
b) Suppose that Alexis decides to use her investment to help her
buy a car when she graduates from university. How much more
interest will she have earned in the compound interest account
than in the simple interest account, assuming that the money is
invested for 8 years?
The amount of a compound interest investment or loan can be found
recursively as follows.
Let i represent the annual interest rate per compounding period. The
amount after one compounding period can be determined by adding the
principal to the interest for that compounding period.
A
5PI
5 P  Pit
5 P  Pi(1)
5 P  Pi
5 P(1  i)
Connections
You studied recursive
relationships in
Chapter 6 Discrete
Functions.
After one compounding period, t = 1.
Factor the expression.
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Connections
It is conventional to
use r to represent the
annual interest rate
in situations involving
simple interest and i to
represent the interest
rate per compounding
period in situations
involving compound
interest. Both r and i
refer to the interest rate
expressed as a decimal.
Therefore, the amount after the first compounding period is the product
of the principal at the beginning of the period and the factor (1  i). This
amount becomes the new principal for the second compounding period.
This process can be continued, as shown in the table.
Compounding
Period
Principal for
the Period
Amount at End
of Period
1
P
2
P(1 + i )
A = P(1 + i )(1 + i ) = P(1 + i )
3
P(1 + i )2
A = P(1 + i )2(1 + i ) = P(1 + i )3
P(1 + i )3
4
P(1 + i )
A = P(1 + i ) (1 + i ) = P(1 + i )
P(1 + i )4


Amount Calculation
A = P(1 + i )
P(1 + i )
2
3
3
4

P(1 + i )2

Examine the pattern of terms in the last column.
P(1  i), P(1  i)2, P(1  i)3, P(1  i)4, …
This is a geometric sequence with first term a 5 P(1  i) and common
ratio r 5 1  i. Substitute these into the formula for the nth term of a
geometric sequence.
tn
5
5
5
5
ar n  1
[P(1  i)](1  i)n  1
P(1  i)1  (n  1)
P(1  i)n
Apply the product rule of exponents.
Note that the nth term corresponds to the amount, A, at the end of the
nth compounding period. So, replace tn with A.
A 5 P(1  i)n
This result is the compound interest formula.
The amount, A, of a compound interest investment or loan can be
determined using the formula A 5 P(1  i)n.
P represents the principal.
i represents the interest rate per compounding period, expressed as a
decimal.
n represents the number of compounding periods.
Example 1
Calculate Compound Interest
To buy a new guitar, Gaston borrows $650, which he plans to repay in
5 years. The bank charges 12% per annum, compounded annually.
a) Determine the amount that Gaston must repay.
b) How much interest will Gaston have to pay?
c) Compare this to the amount of interest he would have to pay if the
bank charged simple interest.
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Solution
a) List the given information. Then, apply the compound interest
formula.
P 5 650
i 5 0.12
n 5 5
A 5 P(1  i)n
5 650(1  0.12)5
5 650(1.12)5
 1145.52
Gaston must repay $1145.52 after 5 years.
650

1.12
y x
5

b) To calculate the interest paid, subtract the principal from the amount.
I 5 A  P
5 1145.52  650
5 495.52
Gaston will pay $495.52 in interest charges.
c) If the bank charges 12% simple interest, then the interest can be
found by applying the simple interest formula.
Connections
I 5 Prt
5 650(0.12)(5)
5 390
Gaston would pay only $390 in simple interest, compared to $495.52
in compound interest.
Securing a simple
interest loan for a
significant period of
time is rare in today’s
financial institutions.
Example 2
Vary the Compounding Period
Financial institutions often use other compounding periods rather than
annual. For example, semi-annual compounding means every
6 months, while quarterly compounding means every 3 months, and so
on. Refer to Example 1. What will be the impact on the interest Gaston
pays if the interest is compounded
a) semi-annually?
b) monthly?
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Solution
a) When interest is charged semi-annually (twice a year), the number
of compounding periods is doubled and the interest rate per
compounding period is halved.
P 5 650
0.12
_
​ i 5 ​ 2
5 0.06
n 5 5  2
5 10
Divide by the number of compounding periods in a year, 2.
Multiply by the number of compounding periods in a year, 2.
Substitute the known values into the compound interest formula.
A 5 P(1  i)n
5 650(1  0.06)10
5 650(1.06)10
 1164.05
Calculate the interest paid.
I 5 A  P
5 1164.05  650
5 514.05
Gaston will pay $514.05 in interest charges if interest is compounded
semi-annually.
b) When interest is charged monthly, the number of compounding
periods and the interest rate per compounding period can be
calculated as follows.
P 5 650
0.12 ​ _
i 5 ​ 12
Divide by the number of compounding periods in a year, 12.
5 0.01
n 5 5  12 Multiply by the number of compounding periods in a year, 12.
5 60
A 5 P(1  i)n
5 650(1  0.01)60
5 650(1.01)60
 1180.85
Calculate the interest paid.
I 5 A  P
5 1180.85  650
5 530.85
Gaston will pay $530.85 in interest charges if interest is compounded
monthly.
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The table summarizes the effects of the various compounding
periods.
Amount, A
($)
Interest, I
($)
annual
1145.52
495.52
semi-annual
1164.05
514.05
monthly
1180.85
530.85
Compounding
Period
Note that as the compounding period becomes shorter, the interest
charges increase.
Example 3
DeterminetheInterestRate
Nina is starting a small business. She applies for an $8000 loan, which
she plans to repay in 4 years. She is told by the loan officer that the
amount payable when the loan is due is $11 501.24. What rate of interest,
compounded annually, is Nina being charged?
Solution
List the given information. Then, apply the compound interest formula.
P 5 8000
A 5 11 501.24
n54
A5 P(1  i)n
11 501.24 5 8000(1  i)4
11 501.24 5 (1  i)4
__
8000
1.437 655 5 (1  i)4
Method1:ApplyGraphicalAnalysis
To determine when (1  i)4 5 1.437 655, graph each side of this equation
as a separate function using a graphing calculator. Then, use the
Intersect operation.
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Te c h n o l o g y Tip
To use the
IntersectionPoint(s)
operation on a
TI-Nspire™ CAS
graphing calculator:
• Press b. Select
6:Points&Lines.
• Select
3:Intersection
Point(s). Move the
cursor to the first
graph and press ·.
Move the cursor to
the second graph and
press ·.
Similarly, you can use the Intersection
Point(s) operation on a TI-Nspire™ CAS
graphing calculator.
The solution is approximately (0.095, 1.437 655), which indicates that
the interest rate is 9.5%, compounded annually.
Method2:ApplyAlgebraicReasoning
To solve the equation 1.437 655 5 (1  i)4 for i, apply the opposite
operations to isolate the variable.
__________
_______
1.437 655 5 (1  i)4
4
The coordinates of the
intersection point will
be displayed.
4
4
__________
1.437 655 5 1  i
4
__________
1.437 655  1 5 i
Take the fourth root of both sides.
Apply the properties of powers and roots.
__
x

1.437655 2nd 
y 4
1 5
[
]
0.095  i
The interest rate is approximately 9.5%, compounded annually.
KeyConcepts
Compound interest investments or loans add the interest from one compounding period to
the previous principal and use the sum as the principal for the next compounding period.
The compounding effect causes an amount to grow exponentially over time. The amounts
after each compounding period produce a geometric sequence.
The compound interest formula A 5 P(1  i)ncan be used to calculate the amount,
A, if the principal, P; the interest rate per compounding period, i; and the number of
compounding periods, n, are known.
The table shows common methods of compounding.
Frequency of
Compounding
Number of Times Interest Is
Added During a Year
annual
1 (every year)
semi-annual
2 (every 6 months)
quarterly
4 (every 3 months)
monthly
12 (every month)
bi-weekly
26 (every 2 weeks)
daily
365 (every day)
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Communicate Your Understanding
C1 Explain the advantage of compound interest over simple interest, from an investor’s point of
view. Use an example to illustrate your answer.
C2 The table shows the amount in an annually compounded interest
account over time.
a) Look at the values in the Amount column. Is this an arithmetic
b) What is the annual rate of compound interest? How do you
or a geometric sequence? Explain.
know?
Time (years)
Amount ($)
0
400.00
1
416.00
2
432.64
3
449.95
4
467.94
C3 An account with an initial value of $800 earns 6% interest per
year, compounded annually.
a) Describe the shape of the graph of amount versus time.
b) What is the vertical intercept of the graph? What does it represent?
c) Describe what happens to the slope of the graph.
A Practise
For help with questions 1 and 2, refer to
Example 1.
1. Darlene invests $500 for 6 years at
4. For each compounding condition,
determine the number of compounding
periods.
4% interest per year, compounded annually.
a)quarterly compounding for 3 years
a) Determine the amount in the account
b) semi-annual compounding for 4 years
after 6 years.
b) How much interest will Darlene earn?
2. Wes borrows $850 at a rate of 9.5% interest
per year, compounded annually, for 4 years.
a) Determine the amount to be repaid after
4 years.
b) How much interest will Wes have to pay?
For help with questions 3 to 7, refer to
Example 2.
3. For each compounding condition,
_3 c)monthly compounding for ​ ​ of a year
4
d) daily compounding for 2 weeks
e) annual compounding for 6 years
5. Determine the total number, n, of
compounding periods and the interest rate,
i, as a decimal, per compounding period
for each scenario.
a) 8.75% per year, compounded annually,
for 6 years
b) 6% per annum, compounded quarterly,
for 3 years
determine the interest rate per
compounding period, expressed as a
decimal.
c) 2.4% per year, compounded monthly,
a) 9% per year, compounded monthly
d) 4.5% per annum, compounded,
b)8% per annum, compounded quarterly
for 2 years
semi-annually for 7.5 years
c)6% annual interest, compounded
semi-annually
d) 13% per year, compounded bi-weekly
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6. Manjinder invests $1400 in a GIC that
10. Pat borrows $350 for 4 years at an interest
earns 5.75% interest per year, compounded
quarterly, for 3 years.
rate that is compounded quarterly. At the
end of the 4 years, she repays $480.47.
What annual interest rate, compounded
quarterly, was Pat charged?
a) Determine the amount in the account
after 3 years.
b) How much interest was earned?
11. Suppose you
c) Compare this to the amount of interest
that would have been earned if simple
interest had been earned at the same
rate.
7. Susan’s chequing account earns 1.25%
interest per year, compounded daily. How
much interest will she earn if she has
$2000 in the account for 30 days?
B Connect and Apply
8. Karin borrows
$600 for 5 years at
12% interest per
year.
a) Compare the
Selecting Tools
Representing
Problem Solving
Connecting
Reflecting
Communicating
interest charges
under each
condition. Then, rank these scenarios
from best to worst from Karin’s
perspective.
i) simple interest
ii) annual compounding
iii) quarterly compounding
iv) monthly compounding
b) Explain the effect of the compounding
period on this loan.
For help with questions 9 and 10, refer to
Example 3.
9. Ramone borrows $10 000 as start-up
capital for his new business. He plans
to repay the loan in 4 years, at which
point he will owe $15 180.70. What rate
of interest is Ramone being charged,
assuming that it is compounded annually?
Selecting Tools
Representing
• First Provincial
Problem Solving
Connecting
Reflecting
Communicating
Bank: earns 5% per year, compounded
semi-annually
• Northern Credit Union: earns 4.8% per
year, compounded monthly
Reasoning and Proving
Reasoning and Proving
have $1500 to
invest for 5 years.
Two options are
available:
Which investment would you choose and
why?
12. Chapter Problem Chloe places $2000 in a
chequing account that earns 1.8% interest
per year, compounded monthly, for 4 years.
Chloe can access this money at any time
with no penalty except lost interest.
a) Determine the amount in this account
after 4 years, assuming no other
transactions take place.
b) How much interest was earned?
c) Compare this with the interest earned
by Chloe’s investment in question 10 in
Section 7.1. Why might Chloe open this
chequing account?
13. The Rule of 72 states that the number of
years required for an investment to double
when interest is compounded annually can
be estimated by dividing 72 by the annual
interest rate.
a) Use the Rule of 72 to determine how
many years it will take for an amount
to double for each interest rate,
compounded annually.
i) 8%
ii) 9%
iii)12%
b) Verify your results in part a) using the
compound interest formula. Is the
Rule of 72 exact? Explain.
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14. Pavel invested $720 in an account
that earned 7.25% interest per year,
compounded quarterly. When he closed
the account, it contained $985. For how
long did Pavel invest his money?
Achievement Check
15. Consider these investment options:
online tool.
d) Verify the accuracy of the result using
the compound interest formula.
e) Discuss the advantages and the
disadvantages of the online tool.
18. Math Contest The nominal interest rate
year, compounded annually
is the rate quoted. The effective rate is
the actual rate charged if it was simple
interest. Some credit cards charge
19% per annum, compounded daily. What
is the effective rate, to two decimal places?
a) Write an equation to represent the
A 20.92
B 28.00
amount of each investment as a
function of time.
C 69.35
D 19.00
Option A: $800 invested at 10% per year,
simple interest
c) Perform a sample calculation using the
Option B: $800 invested at 7% interest per
b) Graph both equations on the same set of
axes.
c) Which is the better investment option?
Explain.
d) How does your answer change if the
compounding period in Option B
changes? Provide a detailed
explanation.
C Extend
16. Refer to question 13. Is the Rule of 72
valid for compounding periods other
than annual compounding? Explore this
question and use mathematical reasoning
to justify your response.
Connections
Go to the Functions 11 page on the McGraw-Hill
Ryerson Web site and follow the links to Chapter 7 to
learn more about the Rule of 72, doubling your money,
and more.
17. There are several compound interest
calculators available on the Internet. Go to
the Functions 11 page on the McGraw-Hill
Ryerson Web site and follow the links to
Chapter 7 to see one such calculator. Use
this compound interest calculator or an
alternative one to answer the following.
19. Math Contest Sumeet invested $1200 for
5 years. He put part of the money into a
high-yielding risky investment at
18% per annum, compounded
semi-annually. He put the rest into another
investment at 7% per annum, compounded
annually. At the end of 5 years, he had
earned $869 in interest. In which interval
does the interest, I, earned at 18% per
annum, compounded semi-annually, fall?
A I  530
B 530  I  540
C 540  I  550
D I  550
20. Math Contest The constant
π is used
in many applications in mathematics.
Another mathematical constant is e.
One of the applications for e is
continuous compounding. This means the
compounding period is infinitely small.
The formula for continuous compounding
is A 5 Pe rt, where A represents the
amount; P represents the principal;
e  2.718 281 828 459; r represents the
annual rate of interest, as a decimal; and t
represents the number of years. Determine
the amounts for $1000 invested for 5 years
at 10% for each compounding period.
a) annual
b) monthly
c) daily
d) continuous
a) Why is the online calculator useful?
b) Explain what each field means.
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