Tan S.T. Finite Mathematics for the Managerial, Life, and Social Sciences
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17. Maximize P 2x y z
subject to x 2y 3z 28
2x 3y z 6
x 2y z 4
x 0, y 0, z 0
18. Minimize C 2x y 3z
subject to 2 x y z 2
x 3y z 6
2x y 2z 12
x 0, y 0, z 0
19. Maximize P x 2y 3
z
subject to x 2y z 20
3x y 30
2x y z 10
x 0, y 0, z 0
20. Minimize C 3x 2y z
subject to x 2y z 20
3x y 30
2x y z 10
x 0, y 0, z 0
21. A
GRICULTURE
—C
ROP
P
LANNING
A farmer has 150 acres of
land suitable for cultivating crops A and B. The cost of cul-
tivating crop A is $40/acre and that of crop B is $60/acre.
The farmer has a maximum of $7400 available for land
cultivation. Each acre of crop A requires 20 labor-hours,
and each acre of crop B requires 25 labor-hours. The
farmer has a maximum of 3300 labor-hours available. He
has also decided that he will cultivate at least 80 acres of
crop A. If he expects to make a profit of $150/acre on crop
A and $200/acre on crop B, how many acres of each crop
should he plant in order to maximize his profit?
22. V
ETERINARY
S
CIENCE
A veterinarian has been asked to pre-
pare a diet for a group of dogs to be used in a nutrition
study at the School of Animal Science. It has been stipu-
lated that each serving should be no larger than 8 oz and
must contain at least 29 units of nutrient I and 20 units of
nutrient II. The vet has decided that the diet may be pre-
pared from two brands of dog food: brand A and brand B.
Each ounce of brand A contains 3 units of nutrient I and
4 units of nutrient II. Each ounce of brand B contains
5 units of nutrient I and 2 units of nutrient II. Brand A costs
3 cents/ounce and brand B costs 4 cents/ounce. Determine
how many ounces of each brand of dog food should be
used per serving to meet the given requirements at mini-
mum cost.
23. F
INANCE
—A
LLOCATION OF
F
UNDS
The First Street branch of
Capitol Bank has a sum of $60 million earmarked for home
and commercial-development loans. The bank expects to
realize an 8% annual rate of return on the home loans and
a 6% annual rate of return on the commercial-development
loans. Management has decided that the total amount of
home loans is to be greater than or equal to 3 times the total
amount of commercial-development loans. Owing to prior
commitments, at least $10 million of the funds has been
designated for commercial-development loans. Determine
the amount of each type of loan the bank should extend in
order to maximize its returns.
24. F
INANCE
—I
NVESTMENTS
Natsano has at most $50,000 to
invest in the common stocks of two companies. He esti-
mates that an investment in company A will yield a return
of 10%, whereas an investment in company B, which he
feels is a riskier investment, will yield a return of 20%. If
he decides that his investment in the stock of company A
is to exceed his investment in the stock of company B by
at least $20,000, determine how much he should invest in
the stock of each company in order to maximize the return
on his investment.
25. M
ANUFACTURING
—P
RODUCTION
S
CHEDULING
A company
manufactures products A, B, and C. Each product is
processed in three departments: I, II, and III. The total
available labor-hours per week for departments I, II, and III
are 900, 1080, and 840, respectively. The time require-
ments (in hours/unit) and the profit per unit for each prod-
uct are as follows:
Product Product Product
ABC
Dept. I $2 $1 $2
Dept. II $3 $1 $2
Dept. III $2 $2 $1
Profit $18 $12 $15
If management decides that the number of units of product
B manufactured must equal or exceed the number of units
of products A and C manufactured, how many units of
each product should the company produce in order to max-
imize its profit?
26. M
ANUFACTURING
—P
RODUCTION
S
CHEDULING
Wayland Com-
pany manufactures two models of its twin-size futons, stan-
dard and deluxe, in two locations, I and II. The maximum
output at location I is 600/week, whereas the maximum out-
put at location II is 400/week. The profit per futon for stan-
dard and deluxe models manufactured at location I is $30
and $20, respectively; the profit per futon for standard and
deluxe models manufactured at location II is $34 and $18,
respectively. For a certain week, the company has received
an order for 600 standard models and 300 deluxe models. If
prior commitments dictate that the number of deluxe mod-
els manufactured at location II not exceed the number of
standard models manufactured there by more than 50, find
how many of each model should be manufactured at each
location so as to satisfy the order and at the same time max-
imize Wayland’s profit.
27. N
UTRITION
—D
IET
P
LANNING
A nutritionist at the Medical
Center has been asked to prepare a special diet for certain
252
4 LINEAR PROGRAMMING: AN ALGEBRAIC APPROACH
87533_04_ch4_p201-256 1/30/08 9:52 AM Page 252
xyu √ P Constant
01
1
3
1
3
010
10
2
3
1
3
030
00
7
3
1
3
1
冨
90
xyu√ P Constant
1 1100 40
1 2010 10
2 3001
冨
0
xyu√ P Constant
11100 40
1 2010 10
2 30 01 0
xyu√ P Constant
03110 30
1 2010 10
0 7021 20
xyu√ P Constant
01
1
3
1
3
010
1 2010 10
0 7021 20
The last tableau is in final form, so we have obtained the
following solution
x 30 y 10 u 0 √ 0 P 90
2. Let x denote the number of acres of crop A to be cultivated
and y the number of acres of crop B to be cultivated. Since
there is a total of 150 acres of land available for cultivation,
we have x y 150. Next, the restriction on the amount
of money available for land cultivation implies that 40x
60y 7400. Similarly, the restriction on the amount of
time available for labor implies that 20x 25y 3300.
Also, since the farmer will cultivate at least 70 acres of
crop A, x 70. Since the profit on each acre of crop A is
$150 and the profit on each acre of crop B is $200, we see
that the profit realizable by the farmer is P 150x 200y.
Summarizing, we have the following linear programming
problem:
Maximize P 150x 200y
subject to x y 150
40x 60y 7400
20x 25y 3300
x 70
x 0, y 0
To solve this nonstandard problem, we rewrite the fourth
inequality in the form
x 70
patients. He has decided that the meals should contain a
minimum of 400 mg of calcium, 10 mg of iron, and 40 mg
of vitamin C. He has further decided that the meals are to
be prepared from foods A and B. Each ounce of food A
contains 30 mg of calcium, 1 mg of iron, 2 mg of vitamin
C, and 2 mg of cholesterol. Each ounce of food B contains
25 mg of calcium, 0.5 mg of iron, 5 mg of vitamin C, and
5 mg of cholesterol. Use the method of this section to
determine how many ounces of each type of food the nutri-
tionist should use in a meal so that the cholesterol content is
minimized and the minimum requirements of calcium, iron,
and vitamin C are met.
28. M
ANUFACTURING
—S
HIPPING
C
OSTS
Steinwelt Piano manu-
factures uprights and consoles in two plants, plant I and
plant II. The output of plant I is at most 300/month, whereas
the output of plant II is at most 250/month. These pianos are
shipped to three warehouses that serve as distribution centers
for the company. To fill current and projected orders, ware-
house A requires a minimum of 200 pianos/month, warehouse
B requires at least 150 pianos/month, and warehouse C
requires at least 200 pianos/month. The shipping cost of each
piano from plant I to warehouse A, warehouse B, and ware-
house C is $60, $60, and $80, respectively, and the shipping
cost of each piano from plant II to warehouse A, warehouse B,
and warehouse C is $80, $70, and $50, respectively. Use the
method of this section to determine the shipping schedule that
will enable Steinwelt to meet the warehouses’ requirements
while keeping the shipping costs to a minimum.
4.3 THE SIMPLEX METHOD: NONSTANDARD PROBLEMS 253
4.3 Solutions to Self-Check Exercises
1. We are given the problem
Maximize P 2x 3y
subject to x y 40
x 2y 10
x 0, y 0
Using the method of this section and introducing the slack
variables u and √, we obtain the following tableaus:
Ratio
10
1
10
40
1
40
Pivot
씮
row
앖
Pivot
column
R
2
⎯→
Ratio
—
30
3
10
R
1
R
2
⎯⎯⎯→
R
3
2R
2
Pivot
row
씮
앖
Pivot
column
R
2
2R
1
⎯⎯⎯→
R
3
7R
1
R
1
⎯→
1
3
87533_04_ch4_p201-256 1/30/08 9:52 AM Page 253
xyu√ wzPConstant
1 110000 150
40 6001000 7,400
20 2500100 3,300
⫺1 000010 ⫺70
⫺150 ⫺20000001 0
xyu√ wzPConstant
1 1100 00 150
40 60 0 1 0 0 0 7,400
20 25001 00 3,300
1 0000⫺10 70
⫺150 ⫺200000 01 0
xyu√ wzPConstant
0 1100 10 80
0 60010 400 4,600
0 25001 200 1,900
1 0000 ⫺10 70
0 ⫺200 0 0 0 ⫺150 1 10,500
xyu√ wzPConstant
0 110 0 10 80
0 60 0 1 0 40 0 4,600
0 100
ᎏ
2
1
5
ᎏ
ᎏ
4
5
ᎏ
076
1 000 0 ⫺10 70
0 ⫺200 0 0 0 ⫺150 1 10,500
Using the method of this section with u, √, w, and z as slack
variables, we have the following tableaus:
This last tableau is in final form, and the solution is
x ⫽ 70 y ⫽ 76 u ⫽ 4 √ ⫽ 40
w ⫽ 0 z ⫽ 0 P ⫽ 25,700
Thus, by cultivating 70 acres of crop A and 76 acres of
crop B, the farmer will attain the maximum possible profit
of $25,700.
254
4 LINEAR PROGRAMMING: AN ALGEBRAIC APPROACH
Pivot
씮
row
앖
Pivot
column
Ratio
⫺70
⫺1
⫽ 70
3300
20
⫽ 165
7400
40
⫽ 185
150
1
⫽ 150
⫺R
4
⎯→
Ratio
—
1900
25
⫽ 76
4600
60
⫽ 76
2
3
80
1
⫽ 80
R
1
⫺ R
4
R
2
⫺ 40R
4
⎯⎯⎯⎯→
R
3
⫺ 20R
4
R
5
⫹ 150R
4
앖
Pivot
column
Pivot
row
씮
R
1
⫺ R
3
⎯⎯⎯⎯→
R
2
⫺ 60R
3
R
5
⫹ 200R
3
R
3
⎯→
1
25
standard maximization problem (202) pivot column (205) standard minimization problem (227)
slack variable (202) pivot row (205) primal problem (228)
basic variable (203) pivot element (205) dual problem (228)
nonbasic variable (203) simplex tableau (206) nonstandard problem (242)
basic solution (204) simplex method (207) mixed constraints (242)
TERMS
Summary of Principal Terms
CHAPTER 4
Concept Review Questions
CHAPTER 4
Fill in the blanks.
1. In a standard maximization problem: the objective function
is to be
_____
; all the variables involved in the problem are
_____
; and each linear constraint may be written so that
the expression involving the variables is
_____ _____
or
_____ _____
a nonnegative constant.
2. In setting up the initial simplex tableau, we first transform
the system of linear inequalities into a system of linear
_____
, using
_____ _____
; the objective function is rewrit-
ten so that it has the form
_____
and then is placed
_____
the system of linear equations obtained earlier. Finally, the
initial simplex tableau is the
_____
matrix associated with
this system of linear equations.
3. In a standard minimization problem: the objective function
is to be
_____
; all the variables involved in the problem are
_____
; and each linear constraint may be written so that
the expression involving the variables is
_____ _____
or
_____ _____
a constant.
4. The fundamental theorem of duality states that a primal
problem has a solution if and only if the corresponding
_____
problem has a solution. If a solution exists, then the
_____
functions of both the primal and the dual problem
attain the same
_____ _____
.
xyu√ wzPConstant
0010⫺
ᎏ
2
1
5
ᎏ
ᎏ
1
5
ᎏ
04
0001⫺
ᎏ
1
5
2
ᎏ
⫺80 40
0100
ᎏ
2
1
5
ᎏ
ᎏ
4
5
ᎏ
076
1000 0 ⫺10 70
0000 8 101 25,700
87533_04_ch4_p201-256 2/18/08 10:35 AM Page 254
In Exercises 1–8, use the simplex method to solve each
linear programming problem.
1. Maximize P 3x 4y 2. Maximize P 2x 5y
subject to x 3y 15 subject to 2x y 16
4x y 16 2x 3y 24
x 0, y 0 y 6
x 0, y 0
3. Maximize P 3x 2y
subject to x 3y 18
3x 2y 19
3x y 15
x 0, y 0
4. Maximize P 5x 3y
subject to 3x 4y 42
x y 12
2x y 20
x 0, y 0
5. Maximize P 2x 3y 5z
subject to x 2y 3z 12
x 3y 2z 10
x 0, y 0, z 0
6. Maximize P x 2y 3z
subject to 2x y z 14
3x 2y 4z 24
2x 5y 2z
10
x 0, y 0, z 0
7. Minimize C 4x 7y
subject to 3x y 8
x 2y 6
x 0, y 0
8. Minimize C x 3y 2z
subject to 2x y 4z 24
x 2y z 8
2x y 2z 16
x 0, y 0, z 0
9. Suppose that the primal problem for a linear programming
problem is
Minimize C 2x 5y
subject to 3x 2y
8
x 4y 6
x 0, y 0
and the final simplex tableau for the dual problem associ-
ated with the primal problem is
Find the solution to the primal problem and the solution to
the dual problem.
10. Construct the dual problem associated with the following
primal problem:
Minimize C 2x 4y 3z
subject to x 2y z 14
4x y 2z 21
3x 2y 5z 12
x 0, y 0, z 0
Do not solve.
11. Minimize C 3x 2y 12. Minimize C x 2y
subject to 2x 3y 6 subject to 3x y 12
2x
y 4 x 4y 16
x 0, y 0 x 0, y 0
13. Minimize C 24x 18y 24z
subject to 3x 2y z 4
x y 3z 6
x 0, y 0, z 0
14. Minimize C 4x 2y 6z
subject to x 2y z 4
2x y 2z 2
3x 2y z 3
x
0, y 0, z 0
15. Maximize P 3x 4y 16. Minimize C 2x 3y
subject to x y 45 subject to x y 10
x 2y 10 x 2y 12
x 0, y 02x y 12
x 0, y 0
17. Maximize P 2x 3y
subject to 2x 5y 20
x 5y 5
x 0, y 0
18. Minimize C 3x
4y
subject to x y 45
x y 15
x 30
y 25
x 0, y 0
REVIEW EXERCISES 255
u √ xyP Constant
10 0
01 0
00 2 11 9
11
10
3
10
1
5
3
10
1
10
2
5
Review Exercises
CHAPTER 4
87533_04_ch4_p201-256 1/30/08 9:52 AM Page 255
19. M
INING
—P
RODUCTION
Perth Mining Company operates
two mines for the purpose of extracting gold and silver.
The Saddle Mine costs $14,000/day to operate, and it
yields 50 oz of gold and 3000 oz of silver each day. The
Horseshoe Mine costs $16,000/day to operate, and it yields
75 oz of gold and 1000 oz of silver each day. Company
management has set a target of at least 650 oz of gold and
18,000 oz of silver. How many days should each mine be
operated at so that the target can be met at a minimum cost
to the company? What is the minimum cost?
20. I
NVESTMENT
A
NALYSIS
Jorge has decided to invest at most
$100,000 in securities in the form of corporate stocks. He
has classified his options into three groups of stocks: blue-
chip stocks that he assumes will yield a 10% return (divi-
dends and capital appreciation) within a year, growth stocks
that he assumes will yield a 15% return within a year, and
speculative stocks that he assumes will yield a 20% return
(mainly due to capital appreciation) within a year. Because
of the relative risks involved in his investment, Jorge has
further decided that no more than 30% of his investment
should be in growth and speculative stocks and at least 50%
of his investment should be in blue-chip and speculative
stocks. Determine how much Jorge should invest in each
group of stocks in the hope of maximizing the return on his
investments. What is the maximum return?
21. M
AXIMIZING
P
ROFIT
A company manufactures three prod-
ucts, A, B, and C, on two machines, I and II. It has been
determined that the company will realize a profit of
$4/unit of product A, $6/unit of product B, and $8/unit of
product C. Manufacturing a unit of product A requires
9 min on machine I and 6 min on machine II; manufac-
turing a unit of product B requires 12 min on machine I
and 6 min on machine II; manufacturing a unit of product
C requires 18 min on machine I and 10 min on machine
II. There are 6 hr of machine time available on machine I
and 4 hr of machine time available on machine II in each
work shift. How many units of each product should be
produced in each shift in order to maximize the com-
pany’s profit? What is the maximum profit?
22. I
NVESTMENT
A
NALYSIS
Sandra has at most $200,000 to
invest in stocks, bonds, and money market funds. She
expects annual yields of 15%, 10%, and 8%, respectively,
on these investments. If Sandra wants at least $50,000 to be
invested in money market funds and requires that the
amount invested in bonds be greater than or equal to the
sum of her investments in stocks and money market funds,
determine how much she should invest in each vehicle in
order to maximize the return on her investments. What is
the maximum return?
256
4 LINEAR PROGRAMMING: AN ALGEBRAIC APPROACH
Before Moving On . . .
CHAPTER 4
1. Consider the following linear programming problem:
Maximize P x 2y 3z
subject to 2x y z 3
x 2y 3z 1
3x 2y 4z 17
x 0, y 0, z 0
Write the initial simplex tableau for the problem and iden-
tify the pivot element to be used in the first iteration of the
simplex method.
2. The following simplex tableau is in final form. Find the
solution to the linear programming problem associated
with this tableau.
3. Using the simplex method, solve the following linear pro-
gramming problem:
Maximize P 5x 2y
subject to 4x 3y 30
2x 3y 6
x 0, y 0
4. Using the simplex method, solve the following linear pro-
gramming problem:
Minimize C x 2y
subject to x y 3
2x 3y 6
x 0, y 0
5. Using the simplex method, solve the following linear pro-
gramming problem:
Maximize P 2x y
subject to 2x 5y 20
4x 3y 16
x 0, y 0
xyzu √ wP Constant
0
1
2
01
1
2
00 2
0
1
4
10
5
4
1
2
011
1
1
4
00
3
4
1
2
02
0
1
4
3
00
1
4
1
2
128
87533_04_ch4_p201-256 1/30/08 9:52 AM Page 256
How much can the Jacksons afford
to borrow from the bank for the
purchase of a home? They have
determined that after making a
down payment they can afford a
monthly payment of $2000. In
Example 4, page 290, we learn how
to determine the maximum amount
they can afford to borrow if they
secure a 30-year fixed mortgage at
the current rate.
I
nterest that is periodically added to the principal and thereafter
itself earns interest is called compound interest. We begin this
chapter by deriving the compound interest formula, which gives
the amount of money accumulated when an initial amount of
money is invested in an account for a fixed term and earns
compound interest.
An annuity is a sequence of payments made at regular
intervals. We derive formulas giving the future value of an
annuity (what you end up with) and the present value of an
annuity (the lump sum that, when invested now, will yield the
same future value as that of the annuity). Then, using these
formulas, we answer questions involving the amortization of
certain types of installment loans and questions involving sinking
funds (funds that are set up to be used for a specific purpose at
a future date).
MATHEMATICS
OF FINANCE
5
© Thomas Barwick/Getty Images
87533_05_ch5_p257-312 1/30/08 9:55 AM Page 257
258 5 MATHEMATICS OF FINANCE
Simple Interest
A natural application of linear functions to the business world is found in the compu-
tation of simple interest—interest that is computed on the original principal only.
Thus, if I denotes the interest on a principal P (in dollars) at an interest rate of r per
year for t years, then we have
I Prt
The accumulated amount A, the sum of the principal and interest after t years, is
given by
and is a linear function of t (see Exercise 36). In business applications, we are nor-
mally interested only in the case where t is positive, so only that part of the line that
lies in Quadrant I is of interest to us (Figure 1).
EXAMPLE 1
A bank pays simple interest at the rate of 8% per year for certain
deposits. If a customer deposits $1000 and makes no withdrawals for 3 years, what
is the total amount on deposit at the end of 3 years? What is the interest earned in
that period of time?
Solution
Using Equation (1b) with P 1000, r 0.08, and t 3, we see that the
total amount on deposit at the end of 3 years is given by
or $1240.
The interest earned over the 3-year period is given by
I Prt Use (1a).
1000(0.08)(3) 240
or $240.
100031 10.0821324 1240
A P11 rt2
P11 rt2
A P I P Prt
A
t
P
A = P + P
rt
FIGURE 1
The accumulated amount is a linear func-
tion of t .
5.1 Compound Interest
Simple Interest Formulas
Interest: (1a)
Accumulated amount: (1b)
A P11 rt2
I Prt
Exploring with
TECHNOLOGY
Refer to Example 1. Use a graphing utility to plot the graph of the function
A 1000(1 0.08t), using the viewing window [0, 10] [0, 2000].
1. What is the A-intercept of the straight line, and what does it represent?
2. What is the slope of the straight line, and what does it represent? (See Exer-
cise 36.)
87533_05_ch5_p257-312 1/30/08 9:55 AM Page 258
APPLIED EXAMPLE 2
Trust Funds An amount of $2000 is invested
in a 10-year trust fund that pays 6% annual simple interest. What is the
total amount of the trust fund at the end of 10 years?
Solution
The total amount of the trust fund at the end of 10 years is given by
or $3200.
Compound Interest
In contrast to simple interest, compound interest is earned interest that is periodically
added to the principal and thereafter itself earns interest at the same rate. To find a for-
mula for the accumulated amount, let’s consider a numerical example. Suppose $1000
(the principal) is deposited in a bank for a term of 3 years, earning interest at the rate
of 8% per year (called the nominal, or stated, rate) compounded annually. Then,
using Equation (1b) with P 1000, r 0.08, and t 1, we see that the accumulated
amount at the end of the first year is
or $1080.
To find the accumulated amount A
2
at the end of the second year, we use (1b)
once again, this time with P A
1
. (Remember, the principal and interest now earn
interest over the second year.) We obtain
or $1166.40.
Finally, the accumulated amount A
3
at the end of the third year is found using (1b)
with P A
2
, giving
or approximately $1259.71.
If you reexamine our calculations, you will see that the accumulated amounts at
the end of each year have the following form.
First year:
Second year:
Third year:
These observations suggest the following general result: If P dollars is invested over
a term of t years, earning interest at the rate of r per year compounded annually, then
the accumulated amount is
(2)
Formula (2) was derived under the assumption that interest was compounded
annually. In practice, however, interest is usually compounded more than once a year.
The interval of time between successive interest calculations is called the conversion
period.
A P11 r2
t
A
3
100011 0.082
3
, or A
3
P11 r2
3
A
2
100011 0.082
2
, or A
2
P11 r2
2
A
1
100011 0.082, or A
1
P11 r2
100031 0.08 4
3
100011.082
3
⬇ 1259.71
100031 0.081124
2
31 0.081124
A
3
P11 rt2 A
2
11 rt2
100031 0.08 4
2
100011.082
2
1166.40
100031 0.08112431 0.0811 24
A
2
P11 rt2 A
1
11 rt2
100031 10.0821124 100011.08 2 1080
A
1
P11 rt2
200031 10.06211024 3200
A P11 rt2
5.1 COMPOUND INTEREST 259
87533_05_ch5_p257-312 1/30/08 9:55 AM Page 259
If interest at a nominal rate of r per year is compounded m times a year on a prin-
cipal of P dollars, then the simple interest rate per conversion period is
For example, if the nominal interest rate is 8% per year (r 0.08) and interest is com-
pounded quarterly (m 4), then
or 2% per period.
To find a general formula for the accumulated amount when a principal of P dol-
lars is deposited in a bank for a term of t years and earns interest at the (nominal) rate
of r per year compounded m times per year, we proceed as before, using (1b) repeat-
edly with the interest rate i . We see that the accumulated amount at the end of
each period is
First period:
Second period:
Third period:
nth period:
There are n mt periods in t years (number of conversion periods per year times the
term in years). Hence the accumulated amount at the end of t years is given by
A P11 i2
n
A
n
A
n 1
11 i2 3P11 i2
n 1
411 i 2 P11 i 2
n
oo
A
3
A
2
11 i2
3P11 i2
2
411 i 2 P11 i 2
3
A
2
A
1
11 i2
3P11 i2411 i 2 P11 i 2
2
A
1
P11 i2
r
m
i
r
m
0.08
4
0.02
Annual interest rate
Periods per year
i
r
m
260 5 MATHEMATICS OF FINANCE
Compound Interest Formula (Accumulated Amount)
(3)
where and
A Accumulated amount at the end of n conversion periods
P Principal
r Nominal interest rate per year
m Number of conversion periods per year
t Term (number of years)
i
r
m
, n mt,
A P11 i2
n
Exploring with
TECHNOLOGY
Let A
1
(t) denote the accumulated amount of $100 earning simple interest at the rate
of 10% per year over t years, and let A
2
(t) denote the accumulated amount of $100
earning interest at the rate of 10% per year compounded monthly over t years.
1. Find expressions for A
1
(t) and A
2
(t).
2. Use a graphing utility to plot the graphs of A
1
and A
2
on the same set of axes,
using the viewing window [0, 20] [0, 800].
3. Comment on the growth of A
1
(t) and A
2
(t) by referring to the graphs of A
1
and A
2
.
87533_05_ch5_p257-312 1/30/08 9:55 AM Page 260
EXAMPLE 3
Find the accumulated amount after 3 years if $1000 is invested at
8% per year compounded (a) annually, (b) semiannually, (c) quarterly, (d) monthly,
and (e) daily (assume a 365-day year).
Solution
a. Here, P 1000, r 0.08, and m 1. Thus, i r 0.08 and n 3, so
Equation (3) gives
or $1259.71.
b. Here, P 1000, r 0.08, and m 2. Thus, and n (3)(2) 6, so
Equation (3) gives
or $1265.32.
c. In this case, P 1000, r 0.08, and m 4. Thus, and n (3)(4) 12,
so Equation (3) gives
or $1268.24
d. Here, P 1000, r 0.08, and m 12. Thus, and n (3)(12) 36, so
Equation (3) gives
or $1270.24.
e. Here, P 1000, r 0.08, m 365, and t 3. Thus, and n
(3)(365) 1095, so Equation (3) gives
or $1271.22. These results are summarized in Table 1.
⬇ 1271.22
A 1000 a1
0.08
365
b
1095
i
0.08
365
⬇ 1270.24
A 1000 a1
0.08
12
b
36
i
0.08
12
⬇ 1268.24
A 1000 a1
0.08
4
b
12
i
0.08
4
⬇ 1265.32
A 1000 a1
0.08
2
b
6
i
0.08
2
⬇ 1259.71
A 100011 0.082
3
5.1 COMPOUND INTEREST 261
TABLE 1
Nominal Conversion Interest Rate/ Initial Accumulated
Rate, r Period Conversion Period Investment Amount
8% Annually (m 1) 8% $1000 $1259.71
8 Semiannually (m 2) 4 1000 1265.32
8 Quarterly (m 4) 2 1000 1268.24
8 Monthly (m 12) 2/3 1000 1270.24
8 Daily (m 365) 8/365 1000 1271.22
87533_05_ch5_p257-312 1/30/08 9:55 AM Page 261