Download 0135108178_ppt05 - Dr. Gehan Dhameeth

CHAPTER 5 – Equations
Instructor:
Dr.Gehan Shanmuganathan
5-1
Learning Outcomes
 Solve equations using
multiplication or division.
 Solve equations using
addition or subtraction.
 Solve equations using
more than one operation.
 Solve equations containing multiple
unknown terms.
 Solve equations containing parentheses.
 Solve equations that are proportions.
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
Solve equations using
multiplication or division
5-1-1
Section 5-1
Equations
 An equation is a mathematical statement in
which two quantities are equal.
 Solving an equation means finding the value
of an unknown.
Example:
8x = 24
To solve this equation, the
value of x must be discovered.
Division is used to solve this equation.
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
Solve equations using
multiplication or division
Section 5-1
Equations
Letters, such as (x,y,z) represent
unknown amounts and are called
unknowns or variables.
4x = 16
The numbers are called
known or given amounts.
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
Solve equations using
multiplication or division
Section 5-1
Equations
 Any operation performed on one side of the
equation must be performed on the other
side of the equation as well.
– If you “multiply by 2” on one side, you must
“multiply by 2” on the other side.
– If you “divide by 3” on one side, you must also
“divide by 3” on the other side.
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
Solve equations using
multiplication or division
HOW TO:
Section 5-1
Equations
8x = 24
STEP 1
Isolate the unknown value and determine
if multiplication or division is needed.
STEP 2
Use division to divide both sides by 8.
STEP 3
Simplify: x = 3 3 x 8 = 24
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
Find the value of an unknown
using multiplication
HOW TO:
Section 5-1
Equations
Find the value of a:
a
=6
3
Multiply both sides by 3 to isolate a.
The left side becomes 1a or a.
The right side becomes the
product of 6 x 3, or 18.
a = 18
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
An Example…
Section 5-1
Equations
2b = 40
STEP 1
Determine which operation is needed.
Division
STEP 2
Perform the same operation to both sides.
Divide both sides by 2.
STEP 3
Isolate the variable and solve.
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
40
b=
= 20
2
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
Solve equations using
addition or subtraction
5-1-2
Section 5-1
Equations
 Adding or subtracting any number from one side
must be carried out on the other side as well.
– Subtract “the given amount” from both sides.
Would solving 4 + x = 16 require addition
or subtraction of “4” from each side?
Subtraction
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
Solve equations using
addition or subtraction
HOW TO:
Section 5-1
Equations
4 + x = 10
STEP 1
Isolate the unknown value and determine
if addition or subtraction is needed.
STEP 2
Use subtraction to isolate x.
STEP 3
Simplify: x = 6
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
An Example…
Section 5-1
Equations
b - 12 = 8
STEP 1
Determine which operation is needed.
Addition
STEP 2
Perform the same operation to both sides.
Add 12 to both sides
STEP 3
Isolate the variable and solve.
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
b = 8 + 12 = 20
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
Solve equations using
more than one operation
5-1-3
Section 5-1
Equations
 Isolate the unknown value.
– Add or subtract as necessary first.
– Multiply or divide as necessary second.
 Identify the solution.
– The number on the side opposite the unknown.
 Check the solution by “plugging in” the
number using the original equation.
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
Order of operations
Section 5-1
Equations
 When two or more calculations are written
symbolically, the operations are performed
according to a specified order of operations.
– First — perform multiplication and division as they
appear from left to right.
– Second — perform addition and subtraction as they
appear from left to right.
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
Order of operations
Section 5-1
Equations
 To solve an equation, undo the operations,
working in reverse order
– First — undo the addition or subtraction.
– Second — undo multiplication or division.
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
An Example…
Section 5-1
Equations
7x + 4 = 39
STEP 1
Undo the addition by subtracting 4 from each side.
7x = 35
STEP 2
Divide each side by 7.
35
x=
=5
7
STEP 3
Verify by plugging in 5 in place of x .
7 (5) + 4 = 39
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
35 + 4 = 39
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
Solve equations containing
multiple unknown terms
5-1-4
Section 5-1
Equations
 In some equations, the unknown value may occur
more than once.
 The simplest instance is when the unknown value
occurs in two addends, such as 3a + 2a = 25
– Add the numbers in each addend (2+3).
– Multiply the sum by the unknown (5a = 25).
– Solve for a (a = 5).
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
An Example…
Section 5-1
Equations
Find a if: a + 4a – 5 = 39
STEP 1
Combine the unknown value addends.
a + 4a = 5a
5a – 5 = 30
STEP 2
Undo the subtraction.
5a = 35
STEP 3
Undo the multiplication.
a=7
STEP 4
Check by replacing a with 7.
7 + 4(7) = 35
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
Correct!
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
Solve equations containing parentheses
5-1-5
Section 5-1
Equations
 Eliminate the parentheses.
– Multiply the number just outside the parentheses
by each addend inside the parentheses.
– Show the resulting products as addition or
subtraction, as indicated
 Solve the resulting equation.
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
An Example…
Section 5-1
Equations
Solve: 6A + 2 = 24
STEP 1
Multiply 6 by each addend.
6 multiplied by A + 6 multiplied by 2
STEP 2
Show the resulting products.
6A + 12 = 24
STEP 3
Check by replacing a with 7.
7 + 4(7) = 35
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
An Example…
Section 5-1
Equations
5 (x - 2) = 45
TIP: Remove the parentheses first.
5x -10 = 45
5x = 55
x = 11
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
Solve equations that are proportions
5-1-6
Section 5-1
Equations
 A proportion is based on two pairs of related
quantities.
 The most common way to write proportions is
to use fraction notation—also called a ratio.
– When two ratios are equal, they form a proportion.
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
Solve equations that are proportions
5-1-6
Section 5-1
Equations
 A cross product is the product of the numerator
of one fraction, times the denominator of
another fraction.
– An important property of proportions is that the
cross products are equal.
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
Verify that two fractions form a proportion
HOW TO:
Section 5-1
Equations
6
4
Do
and
form a proportion?
12
18
STEP 1
Multiply the numerator from the first fraction by the
denominator of the second fraction.
4 x 18 = 72
STEP 2
Multiply the denominator of the first fraction by the
numerator of the second fraction.
6 x 12 = 72
Are they equal? Yes, they form a proportion.
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
5-2
Learning Outcome
 Use the problem-solving
approach to analyze and
solve word problems.
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
Use the problem-solving approach
to analyze and solve word problems.
5-2-1
Section 5-2
Using Equations to Solve Problems
 Five step problem solving approach:
– What you know.
• Known or given facts.
– What you are looking for.
• Unknown or missing amounts.
– Solution Plan.
• Equation or relationship among known/unknown facts.
– Solution.
• Solve the equation.
– Conclusion.
• Solution interpreted within context of problem.
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
Use the problem-solving approach
to analyze and solve word problems.
5-2-1
Section 5-2
Using Equations to Solve Problems
Key words in Table 5-1
will guide you in using the
problem-solving approach.
See page
533
These words help you interpret the information and
begin to set up the equation to solve the problem.
“of” often implies multiplication.
Example:
“¼ of her salary” means “multiply her salary by ¼”
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
Use the solution plan
HOW TO:
Section 5-2
Using Equations to Solve Problems
Full time employees work more
hours than part-time employees.
If the difference is four per day,
and part-time employees work
six hours per day, how many
hours per day do full-timers work?
What are we looking for?
Number of hours that FT employees work.
What do we know?
PT employees work 6 hours, and the
difference between FT and PT is 4 hours.
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
Use the solution plan
HOW TO:
Section 5-2
Using Equations to Solve Problems
Full time employees work more
hours than part-time employees.
If the difference is four per day,
and part-time employees work
six hours per day, how many
hours per day do full-timers work?
We also know that
Set up a solution plan.
“difference” implies
What are we looking for?
subtraction.
FT – PT = 4
Number of hours that FT employees
work.
FT = N [unknown] PT = 6 hours
What do we know?
Conclusion:
PT employees
andtime
the employees
Solution plan:
N = 4 + 6 =work
10 6 hours,Full
difference between FT and PT iswork
4 hours.
10 hours.
N–6=4
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
HOW TO:
Section 5-2
Use the solution plan
Using Equations to Solve Problems
Jill has three times as many trading cards as Matt. If the
total number both have is 200, how many does Jill have?
1. What are you looking for?
The number of cards that Jill has.
2. What do you know?
The relationship in the number of cards is 3:1; total is 200
3. Set up a solution plan.
x(Matt’s) + 3x(Jill’s) = 200
4. Solve it.
x + 3x = 200; 4x = 200; x = 50
5. Draw the conclusion.
Jill has 3x, or 150 cards
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
HOW TO:
Section 5-2
Use the solution plan
Using Equations to Solve Problems
Diane’s Card Shop spent a total of $950 ordering 600 cards
from Wit’s End Co., whose humorous cards cost $1.75 each
and whose nature cards cost $1.50 each. How many of each
style of card did the card shop order?
What are you looking for?
How many humorous cards were ordered and how many nature
cards were ordered—the total of H + N = 600 or N = 600 – H.
If we let H represent the humorous cards, Nature cards will be
600 – H, which will simplify the solution process by using only
one unknown: H.
MORE
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
HOW TO:
Use the solution plan
Section 5-2
Using Equations to Solve Problems
Diane’s Card Shop spent a total of $950 ordering 600 cards
from Wit’s End Co., whose humorous cards cost $1.75 each
and whose nature cards cost $1.50 each. How many of each
style of card did the card shop order?
What do you know?
A total of $950 was spent.
Two types of cards were ordered.
The total number of cards ordered was 600.
Humorous cards cost $1.75 each and nature cards
cost $1.50 each.
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
MORE
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
Use the solution plan
HOW TO:
Section 5-2
Using Equations to Solve Problems
Diane’s Card Shop spent a total of $950 ordering 600 cards
from Wit’s End Co., whose humorous cards cost $1.75 each
and whose nature cards cost $1.50 each. How many of each
style of card did the card shop order?
Solution Plan
Set up the equation by multiplying the unit
price of each by the volume, represented by
the unknowns equaling the total amount spent.
$1.75(H) + $1.50 (600 – H) = $950.00
Unit
prices
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
Volume
unknowns
Total
spent
MORE
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
HOW TO:
Section 5-2
Use the solution plan
Using Equations to Solve Problems
Diane’s Card Shop spent a total of $950 ordering 600 cards
from Wit’s End Co., whose humorous cards cost $1.75 each
and whose nature cards cost $1.50 each. How many of each
style of card did the card shop order?
Solution
$1.75H + $1.50(600 - H) = $950.00
$1.75H + $900.00 - $1.50H = $950.00
$0.25H + $900.00 = $950.00
$0.25H = $50.00
MORE
H = 200
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
HOW TO:
Section 5-2
Use the solution plan
Using Equations to Solve Problems
Diane’s Card Shop spent a total of $950 ordering 600 cards
from Wit’s End Co., whose humorous cards cost $1.75 each
and whose nature cards cost $1.50 each. How many of each
style of card did the card shop order?
Conclusion
The number of humorous cards ordered is 200.
Since nature cards are 600 – H, we can conclude that 400 nature
cards were ordered.
Using “200” and “400” in the original equation proves that the
volume amounts are correct.
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
HOW TO:
Section 5-2
Use the solution plan
Using Equations to Solve Problems
Denise ordered 75 dinners for the awards banquet. Fish
dinners cost $11.75 and chicken dinners cost $9.25 each.
If she spent a total of $756.25, how many of each type of
dinner did she order?
$11.75(F) + $9.25(75 - F) = $756.25
$11.75F + $693.75 - $9.25F = $756.25
$2.50F + $693.75 = $756.25
$2.50F = $62.50
F = 25
Conclusion:
25 fish dinners and 50 chicken dinners were ordered.
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
Proportions
Section 5-2
Using Equations to Solve Problems
 The relationship between two factors is often
described in proportions.
– You can use proportions to solve for unknowns.
Example:
The label on a container of weed killer gives directions to mix
three ounces of weed killer with every two gallons of water.
For five gallons of water, how many ounces of weed killer
should you use?
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
Proportions
Section 5-2
Using Equations to Solve Problems
 The relationship between two factors is often
described in proportions.
– You can use proportions to solve for unknowns.
1. What are you looking for?
Example:
Number of ounces of weed killer needed for 5 gallons of water.
The label on a container of weed killer gives directions to mix
2. What
do you
know?killer with every two gallons of water.
three
ounces
of weed
2 gallons
of water,
need
3 oz. ofof
weed
killer.
For For
fiveevery
gallons
of water,
how you
many
ounces
weed
killer
should you use?
2
5
Set up a solution plan.
=
Conclusion.
3
x
Solve it.
You need 7.5 oz of weed
killer for 5 gal of water.
Cross multiply: 2x = 15; x = 7.5
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
Proportions
Section 5-2
Using Equations to Solve Problems
 Many business-related problems that involve pairs of
numbers that are proportional involve direct proportions.
– An increase (or decrease) in one amount causes an increase
(or decrease) in the number that pairs with it.
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
An Example…
Section 5-2
Using Equations to Solve Problems
Your car gets 23 miles to the gallon.
How far can you go on 16 gallons of gas?
1 gallon
16 gallons
=
23 miles
x miles
Cross multiply: 1x = 368 miles
Conclusion:
You can travel 368 miles on 16 gallons of gas.
In this example, an increase in the amount of gas would
directly and proportionately increase the mileage yielded.
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
5-3
Learning Outcomes
 Evaluate a formula.
 Find an equivalent formula
by rearranging the formula.
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
5-3-1
Evaluate the formula
Section 5-3
Formulas
 Write the formula.
 Rewrite the formula substituting known values
for the letters of the formula.
 Solve the equation for the unknown letter or
perform the indicated operations, applying the
order of operations.
 Interpret the solution within the context of the
formula.
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
An Example…
Section 5-3
Formulas
A plasma TV that costs $2,145 is marked up
$854. What is the selling price of the TV?
Use the formula S = C + M where S is the
selling price, C is the cost, and M is Markup.
S = $2,145 + $854
S or Selling Price = $2,999
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
5-3-2
Section 5-3
Find an Equivalent Formula
by Rearranging the Formula
Formulas
 Determine which variable of the formula is to be
isolated (solved for).
 Highlight or mentally locate all instances of the
variable to be isolated.
 Treat all other variables of the formula as you
would treat numbers in an equation, and
perform normal steps for solving an equation.
 If the isolated variable is on the right side of the
equation, interchange the sides so that it
appears on the left side.
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
An Example…
Section 5-3
Formulas
The formula for
Square Footage = Length x Width or S = L x W.
Solve the formula for W or width.
Isolate W by dividing both sides by L.
The new formula is:
S
=W
L
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
EXERCISE SET A
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
EXERCISE SET A
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
EXERCISE SET A
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
EXERCISE SET A
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
EXERCISE SET A
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
EXERCISE SET A
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
PRACTICE TEST
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
PRACTICE TEST
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
PRACTICE TEST
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
PRACTICE TEST
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
PRACTICE TEST
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
PRACTICE TEST
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
PRACTICE TEST
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
PRACTICE TEST
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved
Business Math, Ninth Edition
Cheryl Cleaves, Margie Hobbs & Jeffrey Nobel
© 2012 Pearson Education, Inc.
Upper Saddle River, NJ 07458 All Rights Reserved