Download Compound Interest Name Suppose you invest $1000 in a bank that

Compound Interest
Name ____________________________
1. Suppose you invest $1000 in a bank that pays 2.4% interest compounded annually. Fill in the
blank cells in the left side of the table (using the specific example already described), then use
what you learned to fill in the right side of the table (using a generic starting amount, 𝑷, and a
generic interest rate, 𝒓).
# of years
0
Amount at
start of
period
1000
Amount of
interest
-------
Amount at
end of
period
-------
1
1000
24
Amount at
start of
period
P (for
“principal”)
P
1024
Amount of
interest
Amount at
end of period
-------
-------
P∙r
P+P∙r
= P(1 + r)
P(1 + r)
2
1024
24.58
1048.58
3
1048.58
…
n
2. Formula:
If a principal P is invested at a fixed annual interest rate r, calculated at the end of each year, then the
value of the investment after n years is
A = _____________________
3. Example:
Suppose Quan Li invests $500 at 7% interest compounded annually. Find the value of her investment
10 years later. Show your work. If you do the work correctly, the solution should be $983.58.
4. Suppose the $1000 described in problem 1 is invested at a bank that pays the same 2.4%, but
compounds the interest quarterly.
a. What would the interest rate be per quarter? __________
b. How many times would the interest be paid over the course of three years? ________
c. How much would be in the bank at the end of three years? ________________
d. How much more money is this than what you found in the table in example 1?__________
e. Why does it make sense that the final amount of money if compounded quarterly is more
than the final amount of money compounded annually?
________________________________________________________.
5. Another example:
Suppose Roberto invests $500 at 4% annual interest compounded monthly. Find the value of
his investment 5 years later. Show your work. If you do the work correctly, the solution
should be $610.50.
6. Another formula: Use the formula you found in #2 and the process you used in #4 and #5 to
find the formula for interest compounded more than annually. That is, suppose a principal 𝑃
is invested at an annual interest rate 𝑟 compounded 𝑘 times per year for 𝑡 years. Then _____
is the interest rate per compounding period, and _____is the number of compounding
periods. So, the amount 𝐴 in the account after 𝑡 years is
𝐴 = ______________________________
7. Another example:
Use the tabling or graphical capabilities of your calculator to solve this problem… Jason has
$500 to invest at 4% annual interest compounded monthly. How many years will it take for
his investment to triple? Show your work. If you do the work correctly, the answer should be
28 years.
8. One more example: Salha also has $500 to invest. What interest rate compounded quarterly
(four times per year) is required to double his money in 12 years? Round your answer to the
nearest hundredth of a percent. Show your work. If you do the work correctly, the answer
should be 5.82%.
Do the following problems from your textbook to practice your new skills:
p. 341-342 #2, 4, 6, 8, 22, 24, 26, 28, 30, 46.
9.
We have worked with some problems involving the number “e”. The number “e” is often called the
“natural exponential,” because it arises naturally in “real-life” math and physical science situations.
One area that it occurs in math is in the area of finance, specifically in the area of compound interest.
Suppose we had $1 to invest in a bank that paid us 100% annual interest for 1 year, compounded over
different periods. This is not one of those real-life situations, but the numbers will help us explore the
value of “𝑒.” Fill in the missing cells in the table below.
Frequency of
compounding
Number of times
interest is compounded
in a year (n)
One Year Accumulation
n
End of Year Balance (A)
1
$2.00
1

A  1  
n

annual
n=1

A  1 

semiannual
n=2
1

A  1  
2

2
$2.25
quarterly
n=4
1

A  1  
4

4
$2.44141
monthly
n = 12
weekly
n=
daily
n=
hourly
n=
every minute
n=
every second
n=
1
1 
As you can see, the more often you compound, the larger and larger the computed value gets. But near the
end of the table, the growth is slowing down quite a bit. As the number of compoundings increases, the
computed value appears to be getting closer and closer to some fixed value that starts out "2.71828". The
number we're approaching is called "𝑒".
n
1

e  lim 1   , as n approaches  e  2.7182818284590 ...
n

e is an irrational number ( a non-repeating, infinite decimal).
10. Formula: Use the formulas you found in problems 2 and 6 and what you’ve learned about the number
“𝑒” to find the formula for interest compounded continuously. Suppose a principal 𝑃 is invested at a
fixed annual interest rate 𝑟. The value of the investment after 𝑡 years is
𝐴 = ________________________ when interest compounds continuously.
11. Example:
Suppose LaTasha invests $100 at 5% annual interest compounded continuously. Fill in a table below
with the value of her investment at the end of each of the years 1, 2, …, 7. Show your equation. If you
do the work correctly, the value of her investment at the end of year 7 should be $141.91.
12. A “non-finance” example:
Certain bacteria, given favorable growth conditions, grow continuously at a rate of 4.6% per day. Find
the bacterial population after thirty-six hours, if the initial population was 250 bacteria. Show your
equation. If you do the work correctly, the solution should be approximately 268 bacteria.
Do the following problems in your book to practice your new skills:
p.341-342 #9, 12, 25, 31-40.