Applied Mathematics
Level 3
Worldwide Interactive Network, Inc.
1000 Waterford Place, Kingston, TN 37763 • 888.717.9461
©2008 Worldwide Interactive Network, Inc. All rights reserved.
Copyright © 1998 by Worldwide Interactive Network, Inc. ALL RIGHTS RESERVED.
Printed in the U.S.A. No part of this publication may be reproduced, stored in a retrieval
system, or transmitted in any form or by any means, electronic, photocopying, recording
or otherwise without the prior written permission of Worldwide Interactive Network,
Inc.
ACT™ and WorkKeys® are trademarks of ACT, Inc. Worldwide Interactive Network,
Inc. is not owned or controlled by ACT, Inc.; however, ACT, Inc. has reviewed these
training materials and has determined that they meet ACT, Inc.’s standards for WorkKeys
Training curriculum. The WorkKeys employment system is a product of ACT, Inc.
The use of materials in this manual does not imply any specific results when WIN
materials are used with the ACT WorkKeys system.
Requests for permission to reproduce or make other use of this material should be
addressed to:
Worldwide Interactive Network, Inc.
1000 Waterford Place
Kingston, Tennessee 37763 USA
Tel: (865) 717-3333
Fax: (865) 717-9461
[email protected]
www.w-win.com
2 • Applied Mathematics
INTRODUCTION
Hi, my name is EdWIN. I will be your guide
through Applied Mathematics Level 3. Together we will
proceed through this course at your speed. Look for
me to pop up throughout your lessons to give you
helpful tips, suggestions, and maybe even a pop quiz
question or two. Don’t worry, you can find the answers
to pop quiz questions at the end of the course.
Hi, I’m EdWIN
Now, don’t get nervous. I know how many of you
feel about mathematics, especially when the word
“fraction” is mentioned. We will cover one topic at a
time and I will be there to give you examples to help
you along.
If the content of the lesson is something that you
understand, you should be able to work through it at a
faster pace. On the other hand, if the material is
difficult, read the text several times and then try to
work the exercises one at a time. After you try one
problem, look at the solution. You can learn by
reviewing each step that is provided in the solution
and by concentrating on the process being illustrated.
Now let’s think positive; no negative attitudes allowed!!
Applied Mathematics • 3
INTRODUCTION
Applied Mathematics is a course designed to help
you solve problems that arise in the workplace with
appropriate mathematical techniques. It is important
that you not only have basic mathematical skills, but
that you are able to apply them to problems that arise
on your job. The intention of this level of Applied
Mathematics is for you to be able to solve simple,
straightforward problems using one type of
mathematical operation and possibly one unit
conversion involving either money or time. Addition,
subtraction, multiplication, and division of whole
numbers and/or monetary units are reviewed in this
level. Addition and subtraction involving both positive
and negative values are discussed. Also, this level briefly
covers conversions between fractions, decimals, and
percents.
4 • Applied Mathematics
OUTLINE
LESSON 1
Review of Basic Mathematical Operations
LESSON 2
Introduction to Problem Solving
LESSON 3
Addition and Subtraction of Monetary Units
LESSON 4
Multiplication of Monetary Units
LESSON 5
Division of Monetary Units
LESSON 6
Practice Session with Practical Problems
LESSON 7
Addition and Subtraction of Signed Numbers
LESSON 8
Conversions Involving Whole Numbers, Fractions,
Decimals, and Percents
LESSON 9
Posttest
REFERENCES
Workplace Problem Solving Glossary
Test-Taking Tips
Formula Sheet
Applied Mathematics • 5
LESSON 1
REVIEW OF BASIC MATHEMATICAL
OPERATIONS
Let’s begin by taking a pretest on the skills that
you should already know. You should know how to
add, subtract, multiply, and divide using your calculator
as needed. It is assumed that you understand the
difference between the notation for dollars and cents
as well as how to make basic conversions of time, for
example, converting days to weeks and hours to
minutes.
See if you are ready for this level by completing
the pretest. The answers will be provided on the pages
following the test. You should be able to complete all
of the problems. If you cannot, please review these skills
before you begin this course. There will be review
exercises provided after the pretest. Good luck!
I like thinking about
dollars and cents!
6 • Applied Mathematics
LESSON 1
EXERCISE - PRETEST
Instructions: Solve these problems involving addition, subtraction, multiplication, and
division.
1.
4 + 5 =________
2.
15 ÷ 3 =________
3.
8 - 3 =________
4.
7 × 3 =________
Instructions: Fill in the blank.
5.
7 days = ________ week(s)
6.
1 hour = ________ minute(s)
7.
1 year = ________ day(s)
8.
1 minute = ________ second(s)
Instructions: Answer the following questions.
9.
What is the purpose of the “Clear” key on the calculator? (There
may be a key with “CE” on it and a key with “C” on it.)
10.
How many cents are in one dollar?
Applied Mathematics • 7
LESSON 1
11.
How do you represent 4 cents in dollars?
Instructions: Calculate answers for the following problems on your calculator.
12.
4.52 + 0.08 =________
13.
7.3 × 0.2 =________
14.
1.8 ÷ 0.06 =________
15.
0.124 - 0.008 =________
8 • Applied Mathematics
LESSON 1
ANSWERS TO PRETEST
1.
4 + 5 =________
2.
Answer: 9
3.
8 - 3 =________
Answer:
4.
Answer: 5
5.
15 ÷ 3 =________
5
7 × 3 =________
Answer:
21
7 days = ________ week(s)
Answer: 1
6.
1 hour = ________ minute(s)
Answer: 60
7.
1 year = ________ day(s)
Answer: 365
8.
1 minute = ________ second(s)
Answer: 60
Applied Mathematics • 9
LESSON 1
9.
What is the purpose of the “clear” key on the calculator? (There is
usually a key with “CE” on it and a key with “C” on it.)
Answer: The “CE” clears your last entry. The “C” clears the whole
problem. Some calculators have “AC” which clears all
of the problem and “C” clears the last entry. You should
learn your calculator functions.
10.
How many cents are in one dollar?
Answer: 100
11. How do you represent 4 cents in dollars?
Answer: $.04
12.
4.52 + 0.08 =________
Answer: 4.6
14.
1.8 ÷ 0.06 =________
Answer: 30
13. 7.3 × 0.2 =________
Answer:
1.46
15. 0.124 - 0.008 =________
Answer:
.116
Note: If you solved all of the problems on the pretest correctly, you should begin Lesson
2. If, however, you had any wrong answers, you should spend time practicing basic
operations using a calculator and converting time and money measurements. Review
Exercises are optional.
10 • Applied Mathematics
LESSON 1
REVIEW EXERCISES
These problems are intended to provide practice in conversions of time and money.
1.
52 weeks =________ year(s)
2.
60 seconds =________ minute(s)
3.
14 days =________ week(s)
4.
1 hour =________ minute(s)
5.
1 minute =________ second(s)
6.
1 hour =________ second(s)
7.
1 day =________ hour(s)
8.
23 cents =________ of a dollar
9.
52 cents = $________
10.
$4.63 =________ cents
Applied Mathematics • 11
LESSON 1
ANSWERS TO REVIEW EXERCISE
1. 52 weeks =________ year(s)
Answer: 1
2. 60 seconds =________ minute(s)
Answer: 1
3. 14 days =________ week(s)
Answer: 2
4. 1 hour =________minute(s)
Answer: 60
5. 1 minute =________second(s)
8. 23 cents =________of a dollar
Answer: 60
Answer: 0.23
6. 1 hour =________second(s)
9. 52 cents = $________
Answer: 3,600
Answer: $0.52
7. 1 day =________hour(s)
10.
Answer: 24
Answer: 463
12 • Applied Mathematics
$4.63 = ________cents
LESSON 2
INTRODUCTION TO PROBLEM SOLVING
How did you do on the pretest? I hope you are
ready to move on and tackle problem solving. A strategy
often used in problem solving is the use of estimation
as a tool to predict answers and to check results.
Estimation is the practice of judging an approximate
value, size, or cost.
We will tackle problem
solving together
By using estimation, you can determine if an
answer is reasonable compared to what you already
know. If your answer to a problem indicates an airplane
flew at 5 miles an hour or a car is able to get 200 miles
per gallon, you should recognize that there is an error
and rethink your process to solve the problem.
The practice of rounding numbers, which we will
be discussing in detail later in this course, is often used
in estimation. The following example shows how
estimation could be used to speed up the calculation
process and to check your answers.
A word problem indicates you are to find the total
number of hours Joe worked if his time cards showed
48 hours for week one and 41 hours for week two. You
might estimate the number of hours by adding the
rounded 50 plus 40 to approximate 90 hours. Joe
actually worked 48 plus 41 which equals 89 hours. The
estimate of 90 hours was close to the actual 89 hours.
If your answer was not close to your estimate, you
should check your work. It is easy to touch the wrong
key on a calculator, so always think about your answers
to make sure they make sense.
Applied Mathematics • 13
LESSON 2
Estimation may also be helpful in determining
which operation to use. For instance, read the following
word problem:
Five crates weigh 200 lb Each crate weighs the same amount.
How many lb does each crate weigh?
If you had no idea which mathematical operation to use, you
could estimate an answer by asking yourself if the answer should
be larger or smaller than the facts given. By adding, 5 plus 200,
you get 205 lb which is more than the total 5 crates weigh; by
multiplying 5 times 200, you get 1,000 lb which is also more
than the total 5 crates weigh. Your estimates should let you know
these are the wrong operations. The correct operation is division
indicated by the key words how many does each.
200 divided by 5 equals 40 lb
This is a reasonable answer for each crate to weigh.
14 • Applied Mathematics
LESSON 2
Now, let’s review four steps that make problem solving much easier to do. Read and
become familiar with these four steps before we actually begin working a problem.
Problem solving is generally divided into four parts:
• define the problem
• decide on a plan to solve the problem
• carry out the plan
• examine the outcome to see if it is reasonable
1) Define the Problem
• What am I being asked to do or find?
• What information have I been given?
• Is there other information that I need to know or need to find?
• Will a sketch help?
• Can I restate the problem in my own words?
• Are there any key words?
2) Decide on a Plan
• What operations do I need to perform and in what order?
• On which numbers do I perform these operations?
3) Carry Out the Plan
4) Examine the Outcome
• Is this a reasonable outcome?
• Does the outcome make sense in the original problem?
• If I estimated the answer, would it be close to the result?
• Does the outcome fall outside any limits in the problem? Is it too large or too
small?
Applied Mathematics • 15
LESSON 2
Before we begin working problems, I have a list of key words that indicate the operation
that will be needed. There is also a Workplace Problem Solving Glossary located at the end
of the course.
Key Words for Word Problems
ADDITION
added to
additional
all together
combined
gain of
how many all together
how many in all
how much all together
in all
increase of
increased by
more than
plus
sum
total
SUBTRACTION
change
decrease
decreased by
difference
dropped
have left
how many more
how many less
how many left
how many fewer
how many remain
how much more
how much less
less
less than
loss of
minus
remaining
save
take away
16 • Applied Mathematics
MULTIPLICATION
double
how many in all (with equal numbers)
how much (with equal amounts)
of (with fractions and percents)
product
times
total (of equal numbers)
triple
twice
twice as much
DIVISION
divided by
divided equally
divided into
evenly
how many in each
how many per
goes into
quotient
what’s half
OPERATION SYMBOLS
+
ADDITION
SUBTRACTION
×
MULTIPLICATION
÷
DIVISION
(Multiplication may be indicated in several ways
i.e., •, ×, ( ). In this course we will use ×.)
SYMBOLS
$
¢
%
#
@
º
DOLLAR
CENT
PERCENT
NUMBER
AT
DEGREE
LESSON 2
Examples:
Addition and subtraction
In an average year, the Smith Co. sells 123,000 washers and
95,000 dryers. What is the total number of appliances sold in
an average year?
Define the Problem
I have 123,000 washers and 95,000 dryers. The problem asks for
the total which is a key word for addition. (You do not have to
write a definition of the problem. Most people complete this
step mentally.)
Decide on a Plan
I will add 123,000 to 95,000. (Again, this is often the thinking
process, though some people like to jot numbers down, make
sketches, etc.)
Carry Out the Plan
123,000 + 95,000 = 218,000
Examine the Outcome
Ask:
Does the answer make sense?
218,000 is a reasonable answer. Let’s suppose for a minute that I
had subtracted:
123,000 - 95,000 = 28,000
28,000 would be less than either the number of washers or dryers.
This would not make sense. Asking this question is one of the
most important steps in solving problems and is often omitted.
We all make mistakes and this gives you an opportunity to catch
them. So, slow down, take your time, and most importantly...
think!
Applied Mathematics • 17
LESSON 2
Last week, Carrie worked 8 hours more than her usual 35 hours.
What was the total number of hours she worked?
Define the Problem
Again, I am asked for a total. I want to know how many hours
Carrie worked.
Decide on a Plan
“Total” and “more than” indicate addition is the appropriate
operation.
Carry Out the Plan
35 hours + 8 more hours = 43 total hours
Examine the Outcome
This answer makes sense because it is more than her usual hours
worked, and it is a reasonable number of hours a person could
work.
18 • Applied Mathematics
LESSON 2
When a corrugated box company begins production at 8:00
a.m., the temperature in the plant is 58° F. At critical locations,
the temperature must be brought up to at least 70° F by using
space heaters. This represents a change of how many degrees.
Define the Problem and Decide on a Plan
I read the problem and locate “change” as a key word. Change is
a key word for subtraction. The problem asks for the change in
temperature from 58° F to 70° F.
Carry Out Plan
70° - 58° = 12° F
Examine the Outcome
This is a change of 12° which is a reasonable difference for the
space heaters to accomplish.
You work at a bookstore. A shipment of 82 boxes of paperbacks
has just arrived. Each box holds 40 books. How many books
were in the shipment?
Define the Problem
I read through the problem and have 82 boxes with 40 books in
each one. I see “how many” as key words. I get confused because
I see the words “how many” under addition and multiplication
on my key word list.
Applied Mathematics • 19
LESSON 2
Decide on a Plan
I draw a sketch to help me.
I have 82 boxes. Each box has 40 books in them. I can either add
40+40+40+40... +40, 82 times but since “how many” with equal
numbers (each box has 40 books) indicates multiplication, I could
multiply 40 × 82. This seems easier.
Carry out the Plan
82 × 40 = 3,280
Examine the Outcome
I have 3,280 books in all. My answer has to be more than 40
because there are 40 in each box and more than 82 if there were
at least 1 book in each box. It seems reasonable to me that there
are 3,280 books.
Sometimes drawing a
picture of the problem
helps me know how to
solve it.
20 • Applied Mathematics
LESSON 2
A woman earns $135 a week. What are her total earnings for
14 weeks?
Define the Problem
I want to know how much money she earns in 14 weeks.
Decide on a Plan
The key word “total” implies addition, but addition of equal
amounts (the same $135 every week) indicates multiplication.
You can add $135 + 135 + 135... 14 times or multiply.
Carry Out the Plan
$135 × 14 = $540
Examine the Outcome
$540 is not much money for 14 weeks! I better try that again.
$135 × 14 = $1,890
$1,890. Now, that’s better! I must have missed a key when I
entered this problem the first time. I told you it is smart to think
about your answers.
Applied Mathematics • 21
LESSON 2
Now that we have looked at some examples, it is
time for you to try some problems. I will work the
answers out following the problems. But, don’t peek.
Solving word problems takes time and effort. The only
way you will learn is to practice. Go back and review
the examples if you are having difficulty. Practicing will
help you score higher on the ACT™ WorkKeys®
Applied Mathematics assessment. So, hang in there!
Don’t peek at the
answers!
22 • Applied Mathematics
LESSON 2
EXERCISE - BASIC OPERATIONS IN PROBLEM SOLVING
Instructions: Use your calculator and the steps for problem solving to answer the following
questions. Remember, look for key words and make sure your answers make
sense.
Bill is employed by the Flower and Shrub Landscaping
Company. His employer sends him to the seed store to buy
grass seed which cost $3 per pound (lb) after tax.
1.
If Bill buys 12 pounds of seed, how much will it cost?
2.
If Bill’s employer sends $50, how much change will he expect when
Bill returns?
Applied Mathematics • 23
LESSON 2
3.
The company is presently landscaping an area of 7,200 sq ft. If a
pound of seed will cover 900 sq ft, how many pounds of seed will
be needed to cover this area?
4.
How many pounds of seed will Bill have left after a 5,400 sq ft area
has been sown?
5.
How many more sq ft would this left over seed cover?
24 • Applied Mathematics
LESSON 2
6.
You need 458 brake linings to fill a customer’s order. You have
already boxed 229. How many more do you need to box?
7.
An employee worked 8 hours on Monday, 5 hours on Tuesday, 10
hours on Wednesday, 9 hours on Thursday, and 7 hours on Friday.
How many hours did he work during this week?
8.
A new machine sorts 28 parts per minute. How many minutes will
it take the machine to sort 952 parts?
Applied Mathematics • 25
LESSON 2
9.
Gary has 4 shipping crates in which to package 240 boxes of cereal.
If each crate should contain the same number of boxes, how many
boxes of cereal should he put in each?
10.
You work for a suit manufacturer. You have an order for 3,345 suits
and you already have sent a partial shipment of 2,390. How many
suits remain to be shipped?
11.
A sewing machine operator makes 125 articles per day. How many
articles does she make in a five-day work week?
26 • Applied Mathematics
LESSON 2
12.
At your workplace, there are 103 people on the day shift and only
43 people on the night shift. How many people are employed all
together?
13.
At the grocery store, your purchases total thirteen dollars after tax
is added. If you hand the cashier a twenty dollar bill, how much
change should you receive?
14.
You are paid $6.00 per hour. How much will you earn in a 42-hour
work week?
Applied Mathematics • 27
LESSON 2
ANSWERS TO EXERCISE
Bill is employed by the Flower and Shrub Landscaping
Company. His employer sends him to the seed store to buy
grass seed which cost $3 per pound after tax.
1.
If Bill buys 12 pounds of seed, how much will it cost?
Answer: Key words – how much (of equal amounts, each lb costs
$3)
$3 (per lb)
× 12 (lb)
$36 total cost
This is a reasonable answer, but $1.50 would not have
been since 1 lb costs $3. $2,000 would also be
unreasonable because no one would pay $2,000 for 12
lb of grass seed.
2.
If Bill’s employer sends $50, how much change will he expect when
Bill returns?
Answer: Key words – how much change
$50 (money sent)
- $36 (cost for seed)
$14 change received
The answer must be less than $50 since Bill has a total of
$50. $14 is reasonable since he spent $36 of the $50.
28 • Applied Mathematics
LESSON 2
3.
The company is presently landscaping an area of 7,200 sq ft. If a
pound of seed will cover 900 sq ft, how many pounds of seed will
be needed to cover this area?
Answer: Key words – how many pounds will cover area. This
may be confusing, since “how many” is a key word for
addition and multiplication, but it does not say “how
many in all.” So, a sketch may be helpful to determine
what operation is needed.
Sketch of process... total area is 7,200 sq ft
• 1 lb covers 900 sq ft or one small area...
• 2 lb covers 1,800 sq ft or two small areas...
This process is dividing up the total 7,200 sq ft. The key
words “how many in each” are implied.
7200 Square Feet
900
900
900
900
900
900
900
900
7,200 ÷ 900 = 8 lb will be needed and this is a
reasonable answer.
Applied Mathematics • 29
LESSON 2
4.
How many pounds of seed will Bill have left if a 5,400 sq ft area has
been sown?
Answer: Bill had 12 pounds of seed (from initial information).
Use the same thought process as in problem 3.
Key word – left
5,400 ÷ 900 = 6 lb used
12 (total lb in original problem)
- 6 (lb used for 5,400 sq ft)
6 lb left
The answer is reasonable. If more than 12 lb were left,
that would have been impossible and you would rework
the problem.
5.
How many more sq ft would this left over seed cover?
Answer: First, let’s define the problem. We want to know how
many sq ft the 6 pounds of left over seed would cover.
Now, decide on a plan.
If 1 pound covers 900 square feet, then 2 pounds would
double that coverage. This sounds like we need to
multiply. If we carry out this plan, 6 × 900 = 5,400 sq ft.
So, 5,400 more sq ft could be covered.
6.
You need 458 brake linings to fill a customer’s order. You have
already boxed 229. How many more do you need to box?
Answer: Key words – how many more
458 (total)
-229 (already boxed)
229 left to box
30 • Applied Mathematics
LESSON 2
7.
An employee worked 8 hours on Monday, 5 hours on Tuesday, 10
hours on Wednesday, 9 hours on Thursday, and 7 hours on Friday.
How many hours did he work during this week?
Answer: Key words – how many
Since the hours per day are not equal, the operation is
addition.
8 + 5 + 10 + 9 + 7 = 39 hours worked
39 hours is a reasonable work week
8.
A new machine sorts 28 parts per minute. How many minutes will
it take the machine to sort 952 parts?
Answer: Key words – per minute
Since you are given a total amount (952 parts) and the
key words - imply division or multiplication...
952 ÷ 28 = 34 (minutes)
It would take 34 minutes to sort 952 parts.
You can check your answer:
28 (parts per minute)
× 34 (minutes)
952 parts
Applied Mathematics • 31
LESSON 2
9.
Gary has 4 shipping crates in which to package 240 boxes of cereal.
If each crate should contain the same number of boxes, how many
boxes of cereal should he put in each crate?
Answer: Key words – how many in each
A sketch of the problem also helps to determine the
operation.
240 ÷ 4 = 60 boxes in each crate
You can visualize the answer to see that 60 boxes per
crate would evenly divide 240 boxes of cereal.
10.
You work for a suit manufacturer. You have an order for 3,345 suits
and you already have sent a partial shipment of 2,390. How many
suits remain to be shipped?
Answer: Key word – remain
3,345 (ordered)
- 2,390 (sent)
955 suits remain to be shipped
32 • Applied Mathematics
LESSON 2
11.
A sewing machine operator makes 125 articles per day. How many
articles does she make in a five-day work week?
Answer: Key words – how many (equal amounts)
125 × 5 = 625 articles per week
12.
At your workplace, there are 103 people on the day shift and only
43 people on the night shift. How many people are employed all
together?
Answer: Key words – How many (and) all together
103 (day) + 43 (night) = 146 people all together
This number is reasonable. It is more than the number
of people working on either shift, yet it is not double the
number of day shift workers (since you know less people
work at night).
13.
At the grocery store, your purchases total thirteen dollars after tax
is added. If you hand the cashier a twenty dollar bill, how much
change should you receive?
Answer: Key word – change
$20 - $13 = $7
14.
You are paid $6 per hour. How much will you earn in a 42-hour
work week?
Answer: Key words – how much (equal amounts)
$6 × 42 = $252
Applied Mathematics • 33
LESSON 3
ADDITION AND SUBTRACTION OF
MONETARY UNITS
Congratulations!
Congratulations!! You have now made it through
Lesson 2!
Lesson 3 will begin with some addition and
subtraction of monetary units. Money, of course, has
decimals. Since the American monetary system has 100
cents in one dollar, the decimals in money are based
on hundredths or two decimal places. Five dollars and
five cents is written $5.05. When we add or subtract
with decimals, we should always line up the decimals.
For example, $5.05 + $42.50 would be written:
$5.05
+ $42.50
Addition or subtraction would then be carried out
normally, carefully lining up the decimal in the answer.
$5.05
+ $42.50
$47.55
34 • Applied Mathematics
LESSON 3
A whole number has a decimal at the end. $42 is
really $42.00. So, adding $42 to $5.05 would actually
look like:
$5.05
+ $42.00
$47.05
But, your calculator will keep the decimals lined
up so you don’t have to concentrate on it. You will
need to know how to enter the data you have to solve
problems appropriately.
Cents are written with 2 decimal places. 3¢ is
written as $.03. (Don’t let the period at the end of the
sentence confuse you.)
Applied Mathematics • 35
LESSON 3
EXERCISE – ADDING AND SUBTRACTING MONETARY UNITS
Instructions: Use your calculator to add or subtract the following problems. Write your
answers in monetary units ($0.00)
1.
$37.52 + $0.04 =________
2.
$27.89 - $25 =________
3.
$0.04 + 2¢ =________
4.
$25 - 3¢ =________
5.
$10 - 42¢ =________
6.
$142.80 + $31 =________
7.
21¢ - 7¢ =________
8.
$78.21 - $78.20 =________
9.
$1 - 23¢ =________
10.
$1,422.85 + $784.13 =________
36 • Applied Mathematics
LESSON 3
Pop Quiz: Name the 4 steps that are suggested for problem solving. List at
least one question you should ask yourself to complete each step.
Step 1
Question to help complete Step 1
Step 2
Question to help complete Step 2
Step 3
Question to help complete Step 3
Step 4
Question to help complete Step 4
Applied Mathematics • 37
LESSON 3
ANSWERS TO EXERCISE
1.
$37.52 + $0.04 =________
2.
Answer: $37.56
3.
$0.04 + 2¢ =________
Answer:
4.
Answer: $0.06
Be sure to enter 2¢
as .02
5.
$10 - 42¢ =________
6.
21¢ - 7¢ =________
Answer: 14¢ or $0.14
9.
$1 - 23¢ =________
Answer: 77¢ or $0.77
38 • Applied Mathematics
$24.97
$142.80 + $31 =________
Answer:
8.
$2.89
$25 - 3¢ =________
Answer:
Answer: $9.58
7.
$27.89 - $25 =________
$173.80
$78.21 - $78.20 =________
Answer:
1¢ or $0.01
10. $1,422.85 + $784.13 =________
Answer:
$2,206.98
LESSON 4
MULTIPLICATION OF MONETARY UNITS
Now, multiplication of decimals is a little different
than addition and subtraction if you are not using a
calculator. When we multiply using money, we typically
use decimals.
If you did not have a calculator to multiply
$45.36 × 12, we would place the numbers up and
down, not being concerned where any decimals might
be. When we multiply $45.36 × 12, we must carefully
line up our numbers in the calculation as illustrated:
$45.36
× 12
9072
4536
54432
Now, count the number of decimal places in the
original problem. There are two numbers... 3 and 6
...to the right of the decimal. Count back two places
from the right in the last line or result of the
multiplication problem. The decimal goes between the
3 and 4.
See what happens when
you don’t keep track of
decimals!
$544.32 will be your answer.
Applied Mathematics • 39
LESSON 4
In this course, you will not be required to manually
do the mathematical operations. You need to know how
to use your calculator and since this course focuses on
your problem solving skills, it is OK to use that
calculator! When multiplying it does not matter what
number you enter first on your calculator:
$2.32 × 5 = $11.60
5 × $2.32 still equals $11.60
One important fact you should remember is that
money only has two numbers to the right of the
decimal. As stated before, there are 100 cents in one
dollar, so cents are represented by two decimal places
(showing hundredths of dollars).
Examples:
52¢ = $0.52
4¢ = $.04
(Sometimes a zero is placed in the one dollar place
and sometimes it is omitted. Do not let this variation
confuse you. $0.04 and $.04 represent 4 cents.)
40 • Applied Mathematics
LESSON 4
If our answer was $363.180, in calculating
monetary units, we would need to round to the nearest
cent. (Two places to the right of the decimal point.)
Two places past the decimal would be the 8. Now, look
at the next number, the number to the right of the 8.
If that number is a 5 or larger, round the eight up to a
9. If it is less than 5, the number 8 stays the same. In
this case, 0 is less than 5 so the answer is $363.18.
Let’s multiply this monetary problem and round
the answer appropriately.
$48 × .002 =
Answer:
4
8
×
.
0
0
2
=
The display should indicate .096.
.096 rounds to $.10
Note that some calculators may use * for
multiplication.
Since we have 3 decimals places in our original
problem, a zero was placed in front of the nine resulting
in 3 decimal places in our answer. But, we know money
is represented by 2 decimal places. Therefore, we must
round the answer to 2 decimal places. The first two
decimal places are .09 but we must look at the third
decimal place, .096. If this number is 5 or larger (which
6 is), we round the 9 up to 10 which makes our answer
$0.10.
Applied Mathematics • 41
LESSON 4
Now, let’s work some practice problems. The
answers will be on the following page. You should use
your calculator, but be sure you round answers when
appropriate.
Pop Quiz: What
mathematical operation is
indicated by the key
words “have left”?
42 • Applied Mathematics
LESSON 4
EXERCISE - MULTIPLYING MONETARY UNITS
Instructions: Use your calculator to multiply the following problems, but be sure you round
answers when appropriate.
1.
$.24 × 6 =________
2.
$.97 × 21 =________
3.
$13.65 × 2.03 =________
4.
$3.69 × .740 =________
5.
$456.92 × 6.943 =________ 6.
$7.68 × 8 =________
7.
$3.46 × 3.9 =________
8.
$27.95 × 1.5 =________
9.
$47.82 × .890 =________
10. $125 × .20 =________
Applied Mathematics • 43
LESSON 4
ANSWERS TO EXERCISE
1.
$.24 × 6 =________
2.
Answer: $1.44
3.
5.
7.
$13.65 × 2.03 =________
Answer:
4.
$20.37
$3.69 × .740 =________
Answer: $27.71
Answer:
$456.92 × 6.943 =________ 6.
$7.68 × 8 =________
Answer: $3,172.40
Answer:
$3.46 × 3.9 =________
Answer: $13.49
9.
$.97 × 21 =________
$47.82 × .890 =________
Answer:
$42.56
44 • Applied Mathematics
8.
$2.73
$61.44
$27.95 × 1.5 =________
Answer:
$41.93
10. $125 × .20 =________
Answer:
$25 or $25.00
LESSON 5
DIVISION OF MONETARY UNITS
I hope you did well on the multiplication. Now,
let’s look at division. You should know how to do simple
division like:
9÷3=3
Even problems like:
450 ÷ 90 = 5
should not be difficult using your calculator. But,
what if your answer is not a whole number? I want
to quickly make sure you know about remainders.
When you have a problem like:
429 ÷ 9
you will have a remainder.
The correct answer will be 47 with a remainder of
6
6 (sometimes written 47R6). It might be written 47 .
9
(The remainder 6, is placed as a fraction over the
number that you divided by which was 9.) You should
always reduce fractions which we will discuss later.
Applied Mathematics • 45
LESSON 5
When you use a calculator, your display reads
47.6666667. (Calculators vary: some may have more
or fewer decimal places displayed.) Try entering 429 ÷
9 =. Did you get 47.6666667? Your calculator may or
may not round the decimal. The answer should be
rounded to two decimal places if we are dividing
monetary units such as $429.
47.666
In this case, the 3rd position to the right of the
decimal is greater than 5, so we round the second place
up to 7.
Your answer is $47.67.
To round or not to round,
that is the question!
Let’s discuss the rounding process further. If you
wanted to round 47.666 to the nearest whole number,
you would look at the first place after the decimal.
47.666
Since we are rounding in order to make a whole
number, we want to know if the ones place (the place
we are rounding to), 47.666 rounds up to an 8 or stays
a 7, dependent upon the number right of ones place.
If that number (to the right) is 5 or greater, we round
up and eliminate any numbers to the right. So in this
problem, the 7 is rounded up to an 8 making 47.666
round to the whole number 48. If the number to the
right where you are rounding is less than 5, the 7 stays
the same, numbers to the right are eliminated, and the
answer is $47.
46 • Applied Mathematics
LESSON 5
If you have not understood the rounding process,
this can be very confusing. So, let’s practice some more.
If you have mastered this concept, please skip ahead to
the division problems.
To round 51.293 to the nearest whole number,
look at the 2 (51.293) to see if the 1 (ones place) rounds
up or stays the same. The 2 is less than 5, so 1 remains
the same and the nearest whole number to 51.293 is
51.
To round 51.293 to the nearest hundredths place
(like monetary units), look at the 3 (51.293) to see if
the 9 (hundredths place) rounds up or stays the same.
The 3 is less than 5, so 9 remains the same and the
rounded number is 51.29 or $51.29 if we are referring
to money.
To round 51.297 to the nearest hundredths place
(like monetary units), look at the 7 (51.297) to see if
the 9 (hundredths place) rounds up or stays the same.
The 7 is greater than 5, so 9 rounds up to a 10 and the
rounded number becomes 51.30 or $51.30 if we are
referring to money.
Suppose you want to round $2,585.98 to the
nearest dollar or whole number, look at the 9 (2,585.98)
to see if the 5 (ones place) rounds up or stays the same.
The 9 is greater than 5, so 5 rounds up to a 6 and the
nearest whole number is 2,586. Now, let’s think about
our answer because 98 cents is almost one dollar. If we
are going to drop the change from $2,585.98, we would
be closer to the original amount to round up to $2,586
than to $2,585. Right?
Applied Mathematics • 47
LESSON 5
In negotiating with a customer, you need to round
$2,585.98 to the nearest hundreds place (not
hundredths place). Look at the 8 right of hundreds
place (2,585.98) to see if the 5 (hundreds place) rounds
up or stays the same. The 8 is greater than 5, so 5 rounds
up and the number nearest hundreds becomes $2,600.
Again, $85.98 is close to one hundred dollars making
585.98 closer to $600 than $500.
That said, let’s look at some problems involving
decimals.
Example:
45.2 ÷ 3.2
When dividing, always enter into your calculator first, the number
you are dividing into (the first number listed). This is called the
dividend. Then enter the number you are dividing by (the second
number listed). This is called the divisor. When we begin word
problems you will have to decide which number is the divisor
(the number that must be entered second). In this case, 3.2 is the
divisor.
45.2 ÷ 3.2 = 14.125
48 • Applied Mathematics
LESSON 5
Let’s do a couple more problems together, and then
you can practice on your own.
79.8 ÷ .24 =
Answer:
332.5
$86.75 ÷ .5 =
Answer:
$173.50
Notice the added zero to 173.5 to make 50 cents
since we are working with money.
EdWIN
Now work the following problems on your own.
The page following the problems will have the answers.
If you get stuck, refer to the solutions. But, before you
practice... it is time for a pop quiz.
Applied Mathematics • 49
LESSON 5
Pop Quiz: List as many “key words” for mathematical operations as you can
remember without referring to your list.
Addition
Multiplication
Subtraction
Division
50 • Applied Mathematics
LESSON 5
EXERCISE – DIVISION OF MONETARY UNITS
Instructions: Use your calculator to solve the following problems. Round answers if necessary
to the nearest hundredth.
1.
$24 ÷ 8 = _________
2.
$12, 096 ÷ 3 =_________
3.
$848 ÷ 16 =_________
4.
$93 ÷ 7 =_________
5.
$78,906 ÷ 46 =_________
6.
$2,817 ÷ 6 =_________
7.
$850.86 ÷ 58 =_________
8.
$3,840,214.72 ÷ 732 =_________
9.
$12.16 ÷ .04 =_________
10. $1,893.72 ÷ .17 =_________
Applied Mathematics • 51
LESSON 5
ANSWERS TO EXERCISE
1.
$24 ÷ 8 = _________
2.
Answer: $3 or $3.00
3.
$848 ÷ 16 =_________
Answer:
4.
Answer: $53 or $53.00
5.
7.
$78,906 ÷ 46 =_________
$93 ÷ 7 =_________
Answer:
6.
$4,032
or $4,032.00
$13.29
13.2857 rounds up
$2,817 ÷ 6 =_________
Answer: $1,715.35
1,715.3478 rounds up
Answer:
$850.86 ÷ 58 =_________
$3,840,214.72 ÷ 732 =_________
Answer: $14.67
9.
$12, 096 ÷ 3 =_________
$12.16 ÷ .04 =_________
Answer: 304
52 • Applied Mathematics
8.
Answer:
$469.50
$5,246.19
5,246.19497
stays the same
10. $1,893.72 ÷ .17 =_________
Answer:
11,139.53
11,139.529
rounds up
LESSON 6
PRACTICE SESSION WITH PRACTICAL
PROBLEMS
Before we move on, we need to practice using what
we have learned. The following pages contain practical
problems and solutions using skills we have discussed.
Refer to Lesson 2 if you have forgotten the 4 steps for
problem solving. If you are having difficulty with a
problem, look for word clues that indicate an operation.
Again, you may need to review key words listed in
Lesson 2. In cases where word clues are not obvious,
restate the question in your own words trying to use
key words such as total, in all, difference, for each, etc.,
to determine which words best fit the meaning of the
question.
Another way to determine which operation you
need to solve word problems is to use the given information such as:
This is key information
for problem solving
• If the given information includes a total value, the
operation is most likely subtraction or division.
• If the problem asks for a total, the operation is always
addition or multiplication. Multiplication is a
shortcut for addition and should be used when the
numbers being totaled are the same.
Remember, you can always refer to the answers if
you really get stuck. Good luck!
Applied Mathematics • 53
LESSON 6
EXERCISE - PRACTICAL APPLICATIONS
Instructions: Perform the indicated operations using your calculator. Remember the four
steps to problem solving. All monetary answers should be rounded appropriately.
Don’t forget to examine your answers to make sure they make sense.
1.
Your plant runs two assembly lines. Line A produces 427 units per
hour and line B produces 519 units per hour. How many more units
per hour does Line B produce than Line A?
2.
Fly Away travel agency is advertising an eight day and seven night
stay in Cancun for $749. If this is a savings of $120, what was the
previous price?
54 • Applied Mathematics
LESSON 6
3.
There are 156 cases of bolts in inventory. If 75 cases are shipped
out, how many cases are left?
4.
You work at the local recycling center. In the last three weeks, 67 lb,
42 lb, and 74 lb of aluminum cans were brought in. How many lb of
aluminum were recycled during this three week period?
5.
Your department is responsible for roadside litter pickup. This year
you have been alloted $57,000 for this purpose. The first month
you spent $5,600 removing trash from county roads. How much
money do you have left for the rest of the year?
Applied Mathematics • 55
LESSON 6
6.
In a given month, your pay checks vary each payday. How much
did you earn all together if your checks were $115, $126, $125, and
$124?
7.
Your department uses 120 file folders per week. You are told to buy
supplies and to get enough for 2 months. How many file folders
should you buy? (Assume 4 weeks equals 1 month.)
8.
You take four prospective buyers out to lunch and everyone orders
the sirloin steak for $8.99. What is the cost of lunch? (Assume tax
is included.)
56 • Applied Mathematics
LESSON 6
9.
A woman earns $135 a week. What are her total earnings for 14
weeks?
10.
A car dealership is advertising a 1994 Dodge Spirit for no money
down and $225 per month for 5 years. What is the cost of this car
based upon this information?
11.
You work at an electronics store. One day you sold 6 VCRs costing
$249 each. What was the total amount of your sales?
Applied Mathematics • 57
LESSON 6
12.
Last year your recycling center took in 12,700 lb of glass. The glass
was sold to a local bottle manufacturer for $.15 per lb. How much
money did the recycling center receive for the glass?
13.
A shipping clerk mailed 15 cartons to each of 740 customers. What
was the total number of cartons mailed?
14.
A textile worker is paid $7.50 per hour overtime. One week he puts
in 8 hours overtime. How much overtime pay does he earn?
58 • Applied Mathematics
LESSON 6
15.
A tree nursery received a contract with the city for planting trees
in the three city parks. In all, 720 trees are to be planted, with each
park receiving an equal number of trees. How many trees will be
planted in each park?
16.
A box contains 6 rolls of tape and sells for $1.86. What is the cost
of one roll of tape?
17.
You have been told to order 500 legal pads for your office. They are
sold in packs of 15. How many packs do you need to order?
Applied Mathematics • 59
LESSON 6
18.
A man purchases a piece of lumber that is 192 inches long. How
many 16 inch long pieces can be cut from it? (Assume there is no
waste from cutting.)
19.
A vial contains 150 cc of penicillin. How many 5 cc injections can
be administered from the vial?
20.
You are volunteering at a local Habitat for Humanity house. The
paint can you are to use indicates a gallon of paint covers 300 sq ft.
How many gallons of paint will you need to cover 1,100 sq ft?
60 • Applied Mathematics
LESSON 6
21.
You work at a shoe manufacturing plant. You have an order for 78
pairs of shoes that need to be boxed for shipment. If each box
holds 4 pairs of shoes, how many boxes will you need to fill the
order?
22.
There are 24 Mars bars in a case. How many cases would you need
to hold 4,032 Mars bars?
23.
A 12-foot board of lumber costs $1.92. What is the cost of one
foot?
Applied Mathematics • 61
LESSON 6
24.
A new copier can produce 600 copies of a document in 5 minutes.
How many copies does it make per minute?
25.
Your office uses 200 pencils per month. You are told to requisition
enough pencils for the next month. If pencils come in boxes of 25,
how many boxes do you need to requisition?
26.
While selling Girl Scout Cookies, each of the 17 Girl Scouts sold
51 cases of cookies. How many cases of Girl Scout Cookies did
the troop sell?
62 • Applied Mathematics
LESSON 6
27.
A sofa normally sells for $225. A customer can save $43 by paying
for the sofa in cash. What is the cash price of the sofa?
28.
During a fund raiser, 37 employees donated $7.25 each. How much
money was raised from the employees?
29.
Your company purchased 7 laser printers for $8,721.86. If each
printer costs the same amount, how much did each printer cost?
Applied Mathematics • 63
LESSON 6
30.
A drill usually costs $29.95. This week it is on sale for $21.86. What
is the difference in price?
What a bargain!
64 • Applied Mathematics
LESSON 6
ANSWERS TO EXERCISE
1.
Your plant runs two assembly lines. Line A produces 427 units per
hour and line B produces 519 units per hour. How many more units
per hour does Line B produce than Line A?
Answer: Key words – how many more
519 - 427 = 92 units
It makes sense that one line produces 92 more than
the other line. It would not, however, make sense if we
incorrectly added and found one line made 946 more
than the other; this is more than either group produced.
Always examine your outcome.
2.
Fly Away travel agency is advertising an eight day and seven night
stay in Cancun for $749. If this is a savings of $120, what was the
previous price?
Answer: No key word is obvious. Ask yourself if the given
information provides a total amount. Remember a given
total amount indicates division or subtraction. If you are
looking for a total amount, multiply or add.
Well, $749 is a total price of the trip, but the question
asked for a previous total price, before the savings.
Eliminate 8 days and 7 nights from your processing
because the question does not address time, only price.
$749 + $120 (savings) = $869 (previous price)
Applied Mathematics • 65
LESSON 6
3.
There are 156 cases of bolts in inventory. If 75 cases are shipped
out, how many cases are left?
Answer: Key words – how many left
156 - 75 (shipped) = 81 cases left
4.
You work at the local recycling center. In the last three weeks, 67
lb, 42 lb, and 74 lb of aluminum cans were brought in. How many lb
of aluminum were recycled during this three week period?
Answer: Key words – how many (amounts not equal)
67 + 42 + 74 = 183 lb of aluminum
5.
Your department is responsible for roadside litter pickup. This year
you have been alloted $57,000 for this purpose. The first month
you spent $5,600 removing trash from county roads. How much
money do you have left for the rest of the year?
Answer: Key words – how much (money do you have) left
$57,000 - $5,600 (used first month) = $51,400 left
6.
In a given month, your pay checks vary each payday. How much
did you earn all together if your checks were $115, $126, $125, and
$124?
Answer: Key words – all together
115 + 126 + 125 + 124 = $490
66 • Applied Mathematics
LESSON 6
7.
Your department uses 120 file folders per week. You are told to buy
supplies and to get enough for 2 months. How many file folders
should you buy? (Assume 4 weeks equals 1 month.)
Answer: Define your problem – How many weeks are you buying
for? In 2 months there are approximately 8 weeks. So,
the problem is how many file folders do you buy for 8
weeks?
120 (per week) × 8 (weeks) = 960 file folders
8.
You take four prospective buyers out to lunch and everyone orders the sirloin steak for $8.99. What is the cost of lunch? (Assume tax is included.)
Answer: Define your problem – Four buyers plus yourself means
5 people ordered steak. The problem is how much did it
cost to buy an $8.99 steak for all 5 people.
Decide on a plan – No key words are obvious, so restate
the question. How much did lunch cost?
Implied key words – how much (of equal amounts)
Carry out the plan – $8.99 × 5 = $44.95 cost for lunch
Examine the outcome – It is reasonable to expect to
pay $45 (rounded) to buy a steak lunch for 5 people.
9.
A woman earns $135 a week. What are her total earnings for 14
weeks?
Answer: Key words – total (of equal numbers, $135 each week)
$135 (per week) × 14 (weeks) = $1,890 total earnings
Applied Mathematics • 67
LESSON 6
10.
A car dealership is advertising a 1994 Dodge Spirit for no money
down and $225 per month for 5 years. What is the cost of this car
based upon this information?
Answer: Define your problem – How many months in 5 years?
12 (months in a year) × 5 (years) = 60 months in 5
years
You want to know the total cost of a car with 60 months
of payments at $225.
$225 (per month) × 60 (months) = $13,500 cost of the car
11.
You work at an electronics store. One day you sold 6 VCRs costing
$249 each. What was the total amount of your sales?
Answer: Key words – total amount (equal price for each VCR)
$249 (cost of each VCR) × 6 (VCRs sold)
= $1,494 total amount of sales
12.
Last year your recycling center took in 12,700 lb of glass. The glass
was sold to a local bottle manufacturer for $.15 per lb. How much
money did the recycling center receive for the glass?
Answer: Key words – how much (of equal amounts)
12,700 (lb of glass) × .15 (for each lb)
= $1,905 from recycling
68 • Applied Mathematics
LESSON 6
13.
A shipping clerk mailed 15 cartons to each of 740 customers. What
was the total number of cartons mailed?
Key words – total number (of equal amounts)
740 (customers) × 15 (cartons to each) = 11,100 cartons
mailed
14.
A textile worker is paid $7.50 per hour overtime. One week he puts
in 8 hours overtime. How much overtime pay does he earn?
Answer: Key words – how much (of equal amounts)
$7.50 (per hour pay) × 8 (hours) = $60.00 overtime pay
15.
A tree nursery received a contract with the city for planting trees
in the three city parks. In all, 720 trees are to be planted, with each
park receiving an equal number of trees. How many trees will be
planted in each park?
Answer: Key words – Don’t be fooled by “how many.” Earlier the
problem stated “In all” there are 720 trees, so we do not
have an addition or multiplication problem. Remember
if the total is given, the operation is often subtraction or
division. We want to know how many trees in each,
which implies division.
720 ÷ 3 = 240 trees in each park
If we had subtracted, that would mean we had 717 trees
for each park. This does not make sense when we only
have a total of 720 trees. Our answer of 240 trees makes
more sense.
Applied Mathematics • 69
LESSON 6
16.
A box contains 6 rolls of tape and sells for $1.86. What is the cost
of one roll of tape?
Answer: Implied key words – one roll of tape (similar to each roll
of tape)
$1.86 ÷ 6 = $0.31 per each roll of tape
17.
You have been told to order 500 legal pads for your office. They are
sold in packs of 15. How many packs do you need to order?
Answer: You are given the total number of legal pads, 500. The
packs are divided into groups of 15, so you need to
divide.
500 ÷ 15 = 33.3 pads needed
Since you cannot buy part of a pack of legal pads (.3),
you must round up to the next whole number. In order
to have enough legal pads (500), we have to purchase
34 packs which will actually give us 510 legal pads. If
we bought 33 packs, we would only have 495 legal pads.
34 × 15 = 510
33 × 15 = 495
18.
A man purchases a piece of lumber that is 192 inches long. How
many 16 inch long pieces can be cut from it? (Assume there is no
waste from cutting.)
Answer: Implied key words – equally divided (he is dividing the
board into equal length pieces)
Another clue is the given information of a total (192)
192 ÷ 16 = 12 pieces of wood
70 • Applied Mathematics
LESSON 6
19.
A vial contains 150 cc of penicillin. How many 5 cc injections can
be administered from the vial?
Answers: Implied key words – divided equally (how many equal
injections from the vial)
150 ÷ 5 = 30 injections
20.
You are volunteering at a local Habitat for Humanity house. The
paint can you are to use indicates a gallon of paint covers 300 sq ft.
How many gallons of paint will you need to cover 1,100 sq ft?
Answer: You might draw a sketch to help determine a plan:
You need to divide the area that needs painted by 300
since one can covers 300 sq ft.
1,100 ÷ 300 = 3.7
Again, you cannot buy .7 cans of paint, so you must
round up to the nearest whole number which is
4 gallons of paint.
Applied Mathematics • 71
LESSON 6
21.
You work at a shoe manufacturing plant. You have an order for 78
pairs of shoes that need to be boxed for shipment. If each box
holds 4 pairs of shoes, how many boxes will you need to fill the
order?
Answer: Implied key words – divided equally (how many boxes
if each box holds 4)
Given a total of 78 pairs of shoes.
78 ÷ 4 = 19.5
Round up to 20 boxes needed.
22.
There are 24 Mars bars in a case. How many cases would you need
to hold 4,032 Mars bars?
Answer: Given a total of 4,032 Mars bars.
4,032 (total) ÷ 24 (divided equally per case)
= 168 cases needed
23.
A 12-foot board of lumber costs $1.92. What is the cost of one
foot?
Answer: Implied key words – one foot (per foot which means
total cost must be divided equally into 12 parts)
Given the total cost of the board.
$1.92 ÷ 12 = $.16 or 16¢ per foot
24.
A new copier can produce 600 copies of a document in 5 minutes.
How many copies does it make per minute?
Answer: Key words – how many per (minute)
600 ÷ 5 = 120 copies per minute
72 • Applied Mathematics
LESSON 6
25.
Your office uses 200 pencils per month. You are told to requisition
enough pencils for the next month. If pencils come in boxes of 25,
how many boxes do you need to requisition?
Answer: No key words are obvious, but it is given in the problem
that a total of 200 pencils are used each month. Since
you are given the total, division is implied but so is
subtraction. The problem restated indicates 25 pencils
are in each box. This problem takes some thought.
Drawing a sketch may be helpful:
200 (pencils) ÷ 25 (per box) = 8 boxes
26.
While selling Girl Scout Cookies, each of the 17 Girl Scouts sold
51 cases of cookies. How many cases of Girl Scout Cookies did
the troop sell?
Answer: Key words – how many (cases) in all (did the troop sell)
Each girl sold the same or equal amounts (51)
17 × 51 = 867 cases of cookies
27.
A sofa normally sells for $225. A customer can save $43 by paying
for the sofa in cash. What is the cash price of the sofa?
Answer: Key word – save ( means less or decrease in price)
$225 (price) - $43 (savings for cash purchase)
= $182 price of sofa if a cash purchase
Applied Mathematics • 73
LESSON 6
28.
During a fund raiser, 37 employees donated $7.25 each. How much
money was raised from the employees?
Answer: Key words – how much (equal amounts of $7.25 given)
37 (people) × $7.25 (given by each person)
= $268.25 donated
29.
Your company purchased 7 laser printers for $8,721.86. If each
printer costs the same amount, how much did each printer cost?
Answer: Key words – how much (did) each
Total amount is given $8,721.86
Total amount must be divided equally into 7 parts to
find cost per unit.
$8,721.86 ÷ 7 = $1,245.98 for each printer
30.
A drill usually costs $29.95. This week it is on sale for $21.86. What
is the difference in price?
Answer: Key word – difference
$29.95 (original price) - $21.86 (sale price)
= $8.09 difference or savings
74 • Applied Mathematics
LESSON 7
ADDITION AND SUBTRACTION OF
SIGNED NUMBERS
I hope you did well on the word problems.
Application of math is what this course is all about.
So, if you had trouble go back and work some problems
again. Repetition is one way you will learn to recognize
key words. The ACT™ WorkKeys ® Applied
Mathematics assessment contains problems similar to
the word problems you encounter in this course.
Remember, practice makes perfect.
Improving workplace skills
improves the paycheck!
Applied Mathematics • 75
LESSON 7
Now, we will begin to look at signed numbers.
This is a number line:
The numbers on the right of zero are positive, and
the numbers on the left are negative. Zero (0) does not
have a sign; it is neutral. Notice that the positive
numbers do not have a sign. A positive number can be
written with or without a sign (for example, 5 or +5).
The farther to the right you move, the larger the
number. The farther to the left you move, the smaller
the number.
If you think about it, you already know how to
add positive numbers:
4+5=9
You have just added two positive numbers.
Sometimes, though, the signs get a little confusing. For
example, you might have a problem that looks like this:
+4 + (+5) =
76 • Applied Mathematics
LESSON 7
Now, that’s exactly what we did in the previous
example, but it looks a little strange. When you see a
problem like that, concentrate on finding two signs
that are together:
Once you have located this, check to see if the signs
are the same or different.
If the two signs are the same, change the sign to a
“plus.” If they are different, change the sign to a
“minus.” Now your problems look like this:
Notice that the signs in the middle have all been
changed to reflect the previous rule.
Applied Mathematics • 77
LESSON 7
We still haven’t added or subtracted yet. Place the
numbers “up and down.”
Now, if the signs of both numbers are the same, you
should add the numbers and carry the sign down.
If the signs are different, subtract and keep the sign
of the larger number.
Same signs:
Different signs:
78 • Applied Mathematics
LESSON 7
It sounds a little complicated, but it just takes
practice. These rules are for anyone who does not have
a calculator. I hope you do, then signed numbers will
be much easier to learn. Your calculator will add and
subtract signed numbers, but you must know how to
enter the information.
So, let’s take a look at our calculators. You should
have a button or key that looks like this if your
calculator will handle signed numbers:
(-)
have the
or
+/-
is not the same as the
(-)
or
+/-
+/-
. If you do not
keys, you will have to use the
signed number rules or invest in a calculator that
computes signed numbers.
Run out and buy an
inexpensive calculator if
you need one.
Applied Mathematics • 79
LESSON 7
This is your negative key +/- . You may have to
press it before the number or after the number. It
depends on the brand and model of your calculator.
Play with your calculator for a minute or two to find
out. Try to get -5 on the display. Press 5, (+/-) or
(+/-),5. See which way will display -5 on your screen.
Now, you can key this problem into your calculator:
5 - (-3) =
Press:
-
5
3
=
+/-
=
+/-
or
-
5
3
You should get “8” on your screen.
-5 + -2 =
Press:
+/-
5
+
+/-
2
=
2
+/-
=
or
5
+/-
+
You should get “-7” on your screen.
Try +7 - +2 =
7
-
2
=
You should get “5” on your screen.
80 • Applied Mathematics
LESSON 7
Practice using the negative key on your calculator
in the following exercise.
After you finish, work the practical problems
containing signed numbers. Remember to use your key
words. The answers are provided following the exercise.
Let’s dive into word
problems!
Applied Mathematics • 81
LESSON 7
EXERCISE – SIGNED NUMBERS ADDITION/SUBTRACTION
Instructions: Complete the following problems using signed numbers.
1. 7 - (-3) =_________
2.
+8 + (-2) =_________
3. -3 + (-2) =_________
4.
0 + 4 =_________
5. +2 + (+4) =_________
6.
-5 - 8 =_________
7. 8 - (+2) =_________
8.
-4 - 8 =_________
9. 17 - (+9) =_________
10.
-18 + (+2) =_________
82 • Applied Mathematics
LESSON 7
Pop Quiz: Solve the following problem:
In preparation for the basketball game, your assignment in the concession stand
is to fill drink carriers which hold 40 drinks. If you expect to sell 2,520 drinks
through vendors who sell in the stands, how many carriers will you have to fill?
Applied Mathematics • 83
LESSON 7
ANSWERS TO EXERCISE
1. 7 - (-3) =
2.
Answer:
7
-
3
+/-
+8 + (-2) =
Answer:
=
or
7
-
+/-
3
=
8
2
+/-
=
+
+/-
2
=
8-2=6
3. -3 + (-2) =
4.
3
+
or
7 + 3 = 10
Answer:
8
+/-
+
2
+/-
=
3
+
+/-
2
=
0+4=
Answer:
0
+
4
=
or
+/-
0+4=4
-3 - 2 = -5
5. +2 + (+4) =
Answer:
Don’t hesitate to
use your mind
instead of your
calculator.
6.
2
+
4
=
-5 - 8 =
Answer:
5
+/-
-
8
=
5
-
8
=
or
2+4=6
+/-
-5 - 8 = -13
84 • Applied Mathematics
LESSON 7
7. 8 - (+2) =
Answer:
8.
8
-
Answer:
=
2
-4 - 8 =
4
+/-
-
8
=
4
-
8
=
or
8-2=6
+/-
-4 - 8 = -12
9. 17 - (+9) =
Answer:
10.
17
-
9
=
-18 + (+2) =
Answer:
18
+/-
+
2
=
18
+
2
=
or
17 - 9 = 8
+/-
-18 + 2 = -16
Applied Mathematics • 85
LESSON 7
EXERCISE – APPLICATION OF SIGNED NUMBERS
Instructions: Solve the following problems using your calculator and your skills we have
been developing.
1. One year the highest temperature in Darbyville was 119 degrees while
the lowest was 18 degrees below zero. What is the difference between
those temperatures?
2. In November, your company had a loss of $2,400. Due to an aggressive
sales campaign, your profits were $4,350 for the month of December.
How much more did the company earn in December than in
November?
86 • Applied Mathematics
LESSON 7
3. The floor of Death Valley is 282 feet below sea level and close by
Owens Telescope Peak is 11,045 feet above sea level. How many feet
would you change in altitude if you went from the bottom of Death
Valley to the top of the peak?
4. A quarterback lost 15 yards in one play and then gained 8 yards on
the next play. What is the net result of the two plays?
5. In a company, 3 employees quit in January and 4 more quit in
February. No new employees were hired. This represents what change
in the total number of employees?
Applied Mathematics • 87
LESSON 7
6. One morning the temperature was -15° F. By noon it increased 7°.
What was the temperature at noon?
7. While scuba diving you noticed that you were 30 feet deep. You went
down another 5 feet. How deep were you then?
88 • Applied Mathematics
LESSON 7
ANSWERS TO EXERCISE
1. One year the highest temperature in Darbyville was 119 degrees while
the lowest was 18 degrees below zero. What is the difference between
those temperatures?
Answer:
Key word – difference
119 - (-18) = 137 ° difference in high and low temperatures
2. In November, your company had a loss of $2,400. Due to an aggressive
sales campaign, your profits were $4,350 for the month of December.
How much more did the company earn in December than in
November?
Answer:
Key words – how much more
4,350 - (-2,400) = $6,750
3. The floor of Death Valley is 282 feet below sea level and close by
Owens Telescope Peak is 11,045 feet above sea level. How many feet
would you change in altitude if you went from the bottom of Death
Valley to the top of the peak?
Answer:
Key word – change
11,045 - (-282) =11,327 feet from the valley to the peak
4. A quarterback lost 15 yards in one play and then gained 8 yards on
the next play. What is the net result of the two plays?
Answer:
-15 + 8 = -7 yards or 7 yards lost
Applied Mathematics • 89
LESSON 7
5. In a company, 3 employees quit in January and 4 more quit in
February. No new employees were hired. This represents what change
in the total number of employees?
Answer:
Key words – total number
-3 + -4 = -7
The total number of employees decreased by 7.
6. One morning the temperature was -15° F. By noon it increased 7°.
What was the temperature at noon?
Answer:
Keyword – increased
-15 + 7 = -8 ° F
by noon it was -8° F
90 • Applied Mathematics
LESSON 7
7. While scuba diving you noticed that you were 30 feet deep. You went
down another 5 feet. How deep were you then?
Answer:
Key words – deep and down (indicates negative numbers
as opposed to above and up)
-30 - 5 = -35 (or 35 feet deep)
Applied Mathematics • 91
LESSON 8
CONVERSION INVOLVING WHOLE
NUMBERS, FRACTIONS, DECIMALS, AND
PERCENTS
Lesson 8 will deal with conversions involving whole
numbers, fractions, percents, and decimals. Remember
whole numbers have an implied decimal at the end
(on the right side) of the number.
Pop Quiz: If the given
information in a problem
includes a total amount,
what 2 operations are
your “likely” choices?
92 • Applied Mathematics
LESSON 8
42 and 42.0 represent the same number. If we are
referring to money, we would write: $42.00 or $42
(the decimal is implied).
First, we will convert whole numbers and decimals
to percents. When converting whole numbers and
decimals to percents, always move the decimal two
places to the right.
Examples:
0.45 = 45%
0.003 = .3%
1.2 = 1.20 = 120%
Notice if the decimal includes a whole number (like
the last example 1.2), the percentage will be greater
than 100%.
When converting percents to decimals, always
move the decimal two places to the left.
Examples:
45% = .45
35.2% = .352
Applied Mathematics • 93
LESSON 8
Now how are you going to remember when to
move which way?
Think about your math problem. If you are
changing a decimal to percent or percent to a decimal
write D P, always D first since alphabetically D comes
before P. (This is an association with something you
already know.) If you are given a decimal to change to
a percent, put your pencil on D (for decimal) and from
D to P you must move right (always two places). Are
you given a percent to change to a decimal? If so, put
your pencil on P (for percent) and from P to D you
must move left (always 2 places).
Many calculators have percentage keys which will
make these conversions for you. If you have this
function, use it. If not, remember the D P, D P trick
and let’s practice a few problems together.
94 • Applied Mathematics
LESSON 8
Change 20% to a decimal.
You might think why would I ever want to do that?
Well, you cannot do math operations with
percentages. You must first convert percentages to
decimals. So, if you are told your work hours must
be cut by 20% because of budget cuts, you would
first change 20% to a decimal and then calculate
how many hours you are expected to work.
20% (Hint: the decimal is implied after the zero
(0) and using D P we are starting with a P,
percent, so move left to make a decimal.)
20% = 20% = .20 or .2
Now, if your boss really told you to reduce your
hours by 20%, you would multiply .2 times the
number of hours you regularly work. This answer
is how many hours your regular hours must be cut.
If you work 40 hours a week, then:
20% of 40 =
.2 × 40 =
8 hours
To calculate how many hours you will be working
subtract the 20% from your regular work hours.
40 (regular hours) - 8 (20%) = 32 hours per week
for new schedule.
Applied Mathematics • 95
LESSON 8
I hope you can see how important it is for you to
know how to use percentages and the conversion
process. Many workplace problems involve percents.
Change 52% to a decimal.
52% = .52 (D P)
Change .15 to a percent.
.15 = 15% (D P)
Change 2% to a decimal.
2% = .02 (D P)
Change 1.25 to a percent.
1.25 = 125% (D P)
Not only do we need to convert decimals to
percents and percents to decimals, but problem solving
sometimes requires the conversion of fractions.
96 • Applied Mathematics
LESSON 8
To convert fractions to decimals, divide the bottom
number into the top number.
Examples:
1
= 1 ÷ 8 = .125
8
1
= 1 ÷ 4 = .25
4
2
= 2 ÷ 3 = .666667 = .67 (rounded to
3
hundredths)
If you wanted to convert these fractions to percents,
you would simply move the decimal to the right after
dividing.
Let’s review this process again. First, change the
fraction to a decimal. Second, remember D P moves
the decimal right two places. So, move the decimal and
add the % sign.
Examples:
1
1
= .125 = 12.5%
= .25 = 25%
8
4
2
= .67 = 67%
3
Now, let’s reverse the order and start with a percent.
Sometimes word problems require this process. First, I
change the percent to a decimal. I will drop the percent
sign, move the decimal two places to the left (D P).
Then I will convert to a fraction by placing the decimal
number without the decimal point over the appropriate
place value. All of these examples indicate 2 decimal
places which is hundredths, so we place the decimal
number (without the actual decimal point) over 100.
We then reduce the fraction.
Applied Mathematics • 97
LESSON 8
Examples:
45% = .45 =
45
9
=
100 20
50% = .50 =
50 1
=
100 2
Some calculators even have a key that will reduce
fractions. It looks like:
/
ab
or
ab/c
0
0
Key in:
4
5
ab/c
1
=
and on your screen you will see:
9
or 9
20
98 • Applied Mathematics
20 or 9
20
LESSON 8
If the percentage has a decimal with it, first change
the percent to a decimal. You are starting with a percent
so go to the decimal in the percent and move two places
to the left. 45.2% = .452 (D P moves left). From
there, you must change the decimal to a fraction. You
must count the places past the decimal. In this case,
there are 3 places past the decimal. This tells you how
many 0s (zeros) to put on the bottom of the fraction.
.452 =
3 places
452
1000
3 zeros
Now reduce the fraction.
452
226
113
(if you divide top and bottom by 2) =
(need to divide by 2 again) =
1000
500
250
Don’t forget your calculator might reduce the
fraction.
4
5
2
ab/c
1
0
0
0
=
113
which is 452 ÷ 1000 = 250
Applied Mathematics • 99
LESSON 8
Let’s practice some problems of each kind.
Change 37.5% to a decimal:
D P moves left. You are starting with a percent
(37.5%) so go first to the decimal in the percent
and move left.
37.5% = .375 is the decimal.
Now complete the change of 37.5% to a
fraction (3 decimal places means three zeros):
375
75 3
.375 =
=
=
1000 200 8
Now, look at the table in the following exercise.
We have learned that we can convert percents, decimals,
and fractions from one form to the other. You will need
to become comfortable with these processes because
application or word problems frequently require you
to do so before you can solve the problems.
100 • Applied Mathematics
LESSON 8
EXERCISE – PERCENT, DECIMAL, AND FRACTION CONVERSION
Instructions: Complete the table by filling in the missing values. Use the given information
in each row to calculate the other 2 forms of that number. Use your calculator
as needed.
Applied Mathematics • 101
LESSON 8
ANSWERS TO EXERCISE
102 • Applied Mathematics
LESSON 8
The next table contains some common conversions. You may want to memorize these
because they are used frequently. Many people assume you know these common
conversions.
You should try to memorize
this information.
Applied Mathematics • 103
LESSON 8
EXERCISE – APPLICATION OF PERCENTS, DECIMALS, AND FRACTIONS
Instructions: Solve each problem using the 4 steps, described in Lesson 2, looking for key
words, and using the conversion process as needed.
1.
Two-fifths of the people in your office have worked for the company
for more than 10 years. What percent is this?
2.
An order for computer disks to be shipped to a customer is 70%
ready. What fraction of the order is ready?
104 • Applied Mathematics
LESSON 8
3.
Last year,
5
8
of the trash picked up beside state roads originated
from fast food restaurants. Express this as a decimal and as a
percent.
4.
An engine manufacturer discovered that .08 of a certain production
run was defective. What fraction of the run does this represent?
5.
The time needed by an employee to do a particular task is .30 of an
hour. What fraction of an hour is needed?
Applied Mathematics • 105
LESSON 8
6.
At Barker Printing Company, 38% of the employees are female.
What fraction of the employees are female?
7.
In a shipment of 40 stoves, 2 are defective. What percent is
defective?
106 • Applied Mathematics
LESSON 8
ANSWERS TO EXERCISE
1.
Two-fifths of the people in your office have worked for the company
for more than 10 years. What percent is this?
Answer:
Define the problem – What percent (10 years is not
relevant to this question)
2
= 2 ÷ 5 = .40 = 40%
5
2.
3.
An order for computer disks to be shipped to a customer is 70%
ready. What fraction of the order is ready?
Answer:
70% = .70 =
Last year,
5
8
70
7
=
100 10
of the trash picked up beside state roads originated
from fast food restaurants. Express this as a decimal and as a
percent.
Answer:
4.
5
= .625 = 62.5% (Remember to convert fractions
8
divide the bottom number into the top number)
An engine manufacturer discovered that .08 of a certain production
run was defective. What fraction of the run does this represent?
Answer:
.08 =
8
2
=
100 25
Applied Mathematics • 107
LESSON 8
5.
The time needed by an employee to do a particular task is .30 of an
hour. What fraction of an hour is needed?
Answer:
6.
30
3
=
100 10
At Barker Printing Company, 38% of the employees are female.
What fraction of the employees are female?
Answer:
7.
.30 =
38% =
38 19
=
100 50
In a shipment of 40 stoves, 2 are defective. What percent is
defective?
Answer:
108 • Applied Mathematics
2
= 2 ÷ 40 = .05 = 5%
40
LESSON 9
Well, you have now completed this level of Applied
Mathematics. Congratulations!! I hope you did not find
it too difficult. Now, if you feel confident enough,
complete the posttest. If you still feel doubtful, go back
and review the information in Level 3. Take the Posttest
until you make a good score. Personally, I think 95%
is pretty good, but why not go for 100%? Good luck...
I know you can do it.
Answers for the Posttest questions are provided at
the end of the workbook... but don’t peek! If you peek,
your score will not be accurate and it will not reflect
whether or not you have learned the information in
this course thoroughly!!
No fair peeking on the
test.
Applied Mathematics • 109
POSTTEST
EXERCISE - POSTTEST
Instructions: Solve the following word problems using your new skills. Remember to
examine your answers to make sure they make sense. Round decimals to the
nearest hundredth.
1.
In 1997, Smith Brothers, Inc., sold 123,690 washers and 95,125
dryers. What was the total number of appliances sold that year?
2.
Last week, you worked 8 hours more than your usual 35 hours.
What was the total number of hours you worked?
110 • Applied Mathematics
POSTTEST
3.
A bank needs part-time tellers. The pay is $7.50 per hour and parttime employees may work a maximum of 20 hours per week. What
is the most a part-time employee can earn per week?
4.
Your department manufactures thermostats. In one week they
average making 2,150, but 25 thermostats are usually found
defective and are eliminated. How many thermostats, on an average,
does your department contribute to inventory per week?
5.
A single mother is entitled to a welfare grant of $633 per month. If
she works, however, part of her earnings must be applied to this
amount. In Sally Lewis’s case, $69 must be deducted. How much
money does Sally receive from the welfare grant?
Applied Mathematics • 111
POSTTEST
6.
To calculate the tax charged on an item, you multiply the original
price by the rate. If the tax rate is 9.2% and a ladder costs $59.95,
how much tax is due?
7.
When Larry’s wife was hospitalized, his co-workers wanted to show
their support by donating some money. John gave $100, Chuck
contributed $225, Ellen gave $170, and Casey put in $55. How much
money did they collect?
112 • Applied Mathematics
POSTTEST
8.
Tony is paid $7.25 an hour and time and a half for overtime hours.
Overtime begins after 40 hours of work in one week. Last week he
worked 42.5 hours. What were his total earnings?
9.
To receive a $125 rebate, Teresa must place an order before
January 1. If she purchases a copier for $1,475 by the deadline,
how much is her net cost?
10.
It takes Howard 5 minutes to press and cut one needle. How many
needles can he produce per hour?
Applied Mathematics • 113
POSTTEST
11.
A checking account contained $6,274.54. After a $385.79 check
was drawn, what was left in the account?
12.
A produce plant processes 5,424 pounds of beans each day. The
plant packages the beans in 4-pound bags. How many bags do
they package each day?
13.
Wecandoit, Inc. reported a loss of $17,225 in March, but in April
showed a profit of $32,500. How much more did Wecandoit, Inc.
make in April than in March?
114 • Applied Mathematics
POSTTEST
14.
Your department employed 35 laborers in January 1996, lost 12
employees in July, and regained 3 in October. How many total
laborers are employed in October?
15.
The temperature on Monday morning was -2°F. By noon it had
warmed up to 5°F. How many degrees did the temperature change?
Applied Mathematics • 115
POSTTEST
16.
Any Company showed a profit of $12,250 dollars for the first quarter,
a loss of $2,575 for the second quarter, another loss in the third
quarter of $5,100, and a slight profit of $875 in the last quarter.
What profit/loss did Any Company have for the total year?
17.
One out of every 25 workers at Dean Manufacturing claimed
Worker’s Compensation this year. What percent of the workers does
this indicate have had accidents?
116 • Applied Mathematics
POSTTEST
18.
A manufacturer of engineered metal structures, claims that a new
system helped them reduce man-hours by 50 hours per unit per
week. If each unit averages 200 man-hours a week, by what percent
did they reduce the number of hours per unit?
19.
A company initially gave assessments to 45 applicants and filled
27 positions. What percentage of these 45 applicants were hired?
20.
A manufacturer of engines and clutches, increased employment
from 125 in 1991 to 750 in 1997. What percentage was this increase?
Applied Mathematics • 117
POSTTEST
ANSWERS TO EXERCISE
1.
In 1997, Smith Brothers, Inc., sold 123,690 washers and 95,125
dryers. What was the total number of appliances sold that year?
Answer:
218,815
Assuming washers and dryers are the only appliances
sold, 218,815 appliances were sold in 1997.
Key word-total 123,690 + 95,125 = 218,815.
2.
Last week, you worked 8 hours more than your usual 35 hours.
What was the total number of hours you worked?
Answer:
43 hours
Key word-total 8 + 35 = 43
3.
A bank needs part-time tellers. The pay is $7.50 per hour and parttime employees may work a maximum of 20 hours per week. What
is the most a part-time employee can earn per week?
Answer:
$150.00
Key words (implied) – how much money can he/she make
(equal amounts per hour)
$7.50 x 20 = $150
118 • Applied Mathematics
POSTTEST
4.
Your department manufactures thermostats. In one week they
average making 2,150, but 25 thermostats are usually found
defective and are eliminated. How many thermostats, on an average,
does your department contribute to inventory per week?
Answer:
2,125 thermostats per week
Key words – how many (implied) left or remain from the
word eliminated
2,150 - 25 = 2,125
5.
A single mother is entitled to a welfare grant of $633 per month. If
she works, however, part of her earnings must be applied to this
amount. In Sally Lewis’s case, $69 must be deducted. How much
money does Sally receive from the welfare grant?
Answer:
$564 a month
Key words – how much money... deducted
$633 - $69 = $564
6.
To calculate the tax charged on an item, you multiply the original
price by the rate. If the tax rate is 9.2% and a ladder costs $59.95,
how much tax is due?
Answer:
$5.52 tax due
Remember you cannot use math operations with %.
Change 9.2% to a decimal (D P decimal moves left)
.092
$59.95 × .092 = 5.5154 rounded to $5.52 (monetary units
must be rounded to 2 decimal places)
Applied Mathematics • 119
POSTTEST
7.
When Larry’s wife was hospitalized, his co-workers wanted to show
their support by donating some money. John gave $100, Chuck
contributed $225, Ellen gave $170, and Casey put in $55. How much
money did they collect?
Answer:
$550
Key words-how much (not equal amounts)
100 + 225 + 170 + 55 = $550
8.
Tony is paid $7.25 an hour and time and a half for overtime hours.
Overtime begins after 40 hours of work in one week. Last week he
worked 42.5 hours. What were his total earnings?
Answer:
$317.20
Key word – total (equal amounts) but at 2 different rates
Base Rate – $7.25
Overtime Rate – $7.25 × 1
1
= $10.88
2
$7.25 × 40 = $290.00
$10.88 × 2.5 = $27.20
$290.00 + 27.20 = $317.20 total earnings
120 • Applied Mathematics
POSTTEST
9.
To receive a $125 rebate, Teresa must place an order before January
1. If she purchases a copier for $1,475 by the deadline, how much
is her net cost?
Answer:
$1,350
Key word – rebate (money back from original price)
$1,475 - $125 = $1,350
10.
It takes Howard 5 minutes to press and cut one needle. How many
needles can he produce per hour?
Answer:
12 needles
Key words – how many... per hour
One hour equals 60 minutes
60 (minutes) ÷ 5 (minutes for each needle) = 12
11.
A checking account contained $6,274.54. After a $385.79 check
was drawn, what was left in the account?
Answer:
$5,888.75
Key words – what was left
$6,274.54 - $385.79 = $5,888.75
Applied Mathematics • 121
POSTTEST
12.
A produce plant produces 5,424 pounds of beans each day. The
plant packages the beans in 4-pound bags. How many bags do
they package each day?
Answer:
1,356 bags
Key words – how many divided equally into 4 pound bags
5,424 ÷ 4 = 1,356
13.
Wecandoit, Inc. reported a loss of $17,225 in March, but in April
showed a profit of $32,500. How much more did Wecandoit, Inc.
make in April than in March?
Answer:
$49,725
Key words – how much more, loss, profit
$32,500 - $-17,225 = $49,725
<——————|———————0—————————|———>
-17,225
32,500
How much more-indicates a difference which means the
operation needed is subtraction. There are 49,725 units
between -17,225 and 32,500. If you set your problem up
-17,225 - +32,500 and calculated $-49,725, step 4 of
problem solving should help you find your mistake.
Examine your outcome. Did the company’s earnings
increase or decrease in April?
When moving from -17,250 to 32,500 you move in a
positive direction. Think about the problem. The company
increased their profits in April indicating a gain (a positive
answer).
122 • Applied Mathematics
POSTTEST
14.
Your department employed 35 laborers in January 1996, lost 12
employees in July, and regained 3 in October. How many total
laborers are employed in October?
Answer:
26 laborers
Key word – total, lost, regained
July: 35 - 12 = 23
October: 23 + 3 = 26
15.
The temperature on Monday morning was -2°F. By noon it had
warmed up to 5°F. How many degrees did the temperature change?
Answer:
7°
Key word – change
5 - (-2) = 7
16.
Any Company showed a profit of $12,250 dollars for the first quarter,
a loss of $2,575 for the second quarter, another loss in the third
quarter of $5,100, and a slight profit of $875 in the last quarter.
What profit/loss did Any Company have for the total year?
Answer:
$5,450 profit
Key words – total, profit, loss
+12,250 + (-2,575) + (-5,100) + 875 = $5,450
Applied Mathematics • 123
POSTTEST
17.
One out of every 25 workers at Dean Manufacturing claimed
Worker’s Compensation this year. What percent of the workers does
this indicate have had accidents?
Answer:
4% of the workers
one out of 25 =
1
25
Change the fraction to a decimal ...
1 ÷ 25 = .04
... then, change the decimal to a percent.
.04 (D P) 4%
18.
A manufacturer of engineered metal structures, claims that a new
system helped them reduce man-hours by 50 hours per unit per
week. If each unit averages 200 man-hours a week, by what percent
did they reduce the number of hours per unit?
Answer:
25%
50 number of hours reduced
= .25 = 25%
200 total hours per week
124 • Applied Mathematics
POSTTEST
19.
A company initially gave assessments to 45 applicants and filled
27 positions. What percentage of these 45 applicants were hired?
Answer:
60%
27 number of applicants hired
45 total number of applicants
= .6
.6 change decimal to percent 60%
20.
A manufacturer of engines and clutches, increased employment
from 125 in 1991 to 750 in 1997. What percentage was this increase?
Answer:
600%
750 employees in 1997
= 6 = 600%
125 employees in 1991
Applied Mathematics • 125
CALCULATING YOUR SCORE
Calculate your score counting the number of questions you answered correctly. Divide
the number of your correct answers by 20. Change the decimal answer to a percent by
moving the decimal two places to the right.
126 • Applied Mathematics
SUMMARY
Well, how did you do on the Posttest? If you scored
95% or higher, you have a reasonable chance to pass
Level 3 of the ACT WorkKeys® Applied Mathematics
assessment. Remember the basic steps for solving
mathematics problems. Take your time and think about
each question, and you will do fine. But, you may want
to complete Level 4 with me before you take the
assessment. Hope to see you there.
Now don’t be discouraged if you scored below 95%.
There is a lot of information to remember. You can do
it! And, your enhanced work skills will pay off in the
long run.
Take time to review the Workplace Problem Solving
Glossary and Test-Taking Tips provided at the end of
this workbook. Good luck improving your work skills
and attaining your goals!
You should be proud of
your progress.
Applied Mathematics • 127
REFERENCE
WORKPLACE PROBLEM SOLVING GLOSSARY
The following is a partial list of words that has been compiled for you to review before
taking the ACT WorkKeys® Applied Mathematics assessment. The assessment consists of
approximately 30 application (word) problems that focus on realistic workplace situations.
It is important that you are familiar with common workplace vocabulary so that you
may interpret and determine how to solve the problems.
Annual - per year
Asset - anything of value
Budget - estimate of income and expenses
Capital - money, equipment, or property used in a business by a person or corporation
Capital gain (loss) - difference between what a capital asset costs and what it sells for
Commission - an agent’s fee; payment based on a percentage of sales
Contract - a binding agreement
Convert - to change to another form
Deductions - subtractions
Denominate number - numbers with units i.e., 5 feet, 10 seconds, 2 pounds
Depreciation - lessening in value
Difference - answer to a subtraction
Discount - reduction from a regular price
Dividend - money a corporation pays to its stockholders
Expense - cost
128 • Applied Mathematics
REFERENCE
Fare - price of transportation
Fee - a fixed payment based on a particular job
Fiscal year - 12-month period a corporation uses for bookkeeping purposes
Gross pay - amount of money earned
Gross profit - gross pay less immediate cost of production; difference in sales price of
item or service and expenses attributed directly to it
Interest - payment for use of money; fee charged for lending money
Interest rate - rate percent per unit of time i.e., 7% per year
Liquid asset - current cash or items easily converted to cash
Markup - price increase
Measure - a unit specified by a scale, such as an inch
Net pay - take-home pay; amount of money received after deductions
Net profit (income) - actual profit made on a sale, transaction, etc., after deducting all
costs from gross receipts
Overtime - payment for work done in addition to regular hours
Per - for each
Percent off - fraction of the original price that is saved when an item is bought on sale
Product - answer to a multiplication problem
Profit - income after all expenses are paid
Proportion - an equation of 2 ratios that are equal
Applied Mathematics • 129
REFERENCE
Quotient - answer to a division problem
Rate - a ratio or comparison of 2 different kinds of measures
Ratio - a comparison of 2 numbers expressed as a fraction, in colon form, or with the
word “to”
Regular price - price of an item not on sale or not discounted
Return rate - percentage of interest or dividends earned on money that is invested
Revenue - amount of money a company took in ( interest, sales, services, rents, etc.)
Salary - a fixed rate of payment for services on a regular basis
Sale price - price of an item that has been discounted or marked down
Sum - answer to an addition problem
Yield - amount of interest or dividends an investment earns
130 • Applied Mathematics
REFERENCE
EDWIN’S TEST–TAKING TIPS
Preparing for the test . . .
Complete appropriate levels of the WIN Instruction Solution self-study courses. Practice
problems until you begin to feel comfortable working the word problems.
Get a good night’s rest the night before the test and eat a good breakfast on test day.
Your body (specifically your mind) works better when you take good care of it.
You should take the following items with you when you take the assessment: (1)
pencils; pens are not allowed to be used; it is a good idea to have more than one pencil
since the test is timed and you do not want to waste time sharpening a broken pencil
lead; and (2) your calculator; be sure your batteries are strong if you do not have a
solar-powered calculator and that your calculator is working properly.
Allow adequate time to arrive at the test site. Being in a rush or arriving late will likely
upset your concentration when you actually take the test.
About the test . . .
The test is comprised of approximately 33 multiple-choice questions. All test questions
are in the form of word problems which are applicable to the workplace. You will not
be penalized for wrong answers, so it is better to guess than leave blanks. You will
have 45 minutes to complete the test.
The test administrator will provide a Formula Sheet exactly like the one provided in
this workbook. You will not be allowed to use scratch paper, but there is room in your
assessment booklet to work the problems.
Applied Mathematics • 131
REFERENCE
During the test . . .
Listen to instructions carefully and read the test booklet directions. Do not hesitate
to ask the administrator questions if you do not understand what to do.
Pace yourself since this is a timed test. The administrator will let you know when you
have 5 minutes left and again when you have 1 minute remaining. Work as quickly as
possible, but be especially careful as you enter numbers into your calculator.
If a problem seems too difficult when you read it, skip over it (temporarily) and move
on to an easier problem. Be sure to put your answers in the right place. Sometimes
skipping problems can cause you to get on the wrong line, so be careful. You might
want to make a mark in the margin of the test, so that you will remember to go back
to any skipped problems.
Since this is a multiple-choice test, you have an advantage answering problems that
are giving you trouble. Try to eliminate any unreasonable answers and make an
educated guess from the answers you have left.
If the administrator indicates you have one minute remaining and you have some
unanswered questions, be sure to fill in an answer for every problem. Your guess is
better than no answer at all!
If you answer all of the test questions before time is called, use the extra time to check
your answers. It is easy to hit the wrong key on a calculator or place an answer on the
wrong line when you are nervous. Look to see that you have not accidentally omitted
any answers.
132 • Applied Mathematics
REFERENCE
Dealing with math anxiety . . .
Being prepared is one of the best ways to reduce math or test anxiety. Study the list of
key words for solving word problems. If your problem does not include any key
words, see if you can restate the problem using your key words. Feeling like you know
several ways to try to solve problems increases your confidence and reduces anxiety.
Do not think negatively about the test. The story about the “little engine that could”
is true. You must, “think you can, think you can, think you can.” If you prepare
yourself by preparing properly, there is no reason why you cannot be successful.
Do not expect yourself to know how to solve every problem. Do not expect to know
immediately how to work word problems when you read them. Everyone has to read
and reread problems when they are solving word problems. So, don’t get discouraged;
be persistent.
Prior to the test, close your eyes, take several deep breaths, and think of a relaxing
place or a favorite activity. Visualize this setting for a minute or two before the test is
administered.
During the test if you find yourself tense and unable to think, try the following
relaxation technique:
1.
2.
3.
4.
5.
Put feet on floor.
Grab under your chair with your hands. (hope there are no surprises!)
Push down with your feet and up on your chair at the same time - hold for 5
seconds.
Relax 5 seconds (especially try to relax your neck and shoulders).
Repeat a couple of times as needed, but do not spend the entire 45 minutes of
test trying to relax!
Studying with a partner is another way to overcome math anxiety. Encouragement
from each other helps to increase your confidence.
Applied Mathematics • 133
REFERENCE
FORMULA SHEET
(≈ indicates estimate, not equal)
134 • Applied Mathematics
REFERENCE
POP QUIZ QUESTION ANSWER KEY
1. Page 37 – any of the following questions would be correct
(answers do not need to be word for word as long as
the meaning is similar)
Step 1 – Define the Problem
• What am I being asked to do or find?
• What information have I been given?
• Is there other information that I need to know or need to
find?
• Will a sketch help?
• Can I restate the problem in my own words?
• Are there any key words?
Step 2 – Decide on a Plan
• What operations do I need to perform and in what order?
• On which numbers do I perform these operations?
Step 3 – Carry Out the Plan
Step 4 – Examine the Outcome
• Is this a reasonable outcome?
• Does the outcome make sense in the original problem?
• If I estimated the answer would it be close to the result?
• Does the outcome fall outside any limits in the problem? Is it
too large or too small?
2. Page 42 – subtraction
3. Page 50 – refer to page 17 to check your answers.
4. Page 83 – 2,520 ÷ 40 = 63 drink carriers
5. Page 92 – subtraction or division
Applied Mathematics • 135
WIN Career Readiness Courseware - ©2008 Worldwide Interactive Network, Inc. All rights reserved.
Worldwide Interactive Network, Inc.
1000 Waterford Place
Kingston, TN 37763
Toll-free 888.717.9461
Fax 865.717.9461
www.w-win.com