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DISTRIBUTION STATEMENT A: Approved for public release; distribution is unlimited.
NONRESIDENTTRAINING
COURSEJune 1985
Mathematics, Basic
Math and Algebra
NAVEDTRA 14139
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DISTRIBUTION STATEMENT A: Approved for public release; distribution is unlimited.
Although the words “he,” “him,” and“his” are used sparingly in this course toenhance communication, they are notintended to be gender driven or to affront ordiscriminate against anyone.
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COMMANDING OFFICER
NETPDTC
6490 SAUFLEY FIELD ROAD
PENSACOLA FL 32509-5237
7 Aug 2002
ERRATA #1
Specific Instructions and Errata for
Nonresident Training Course
MATHEMATICS, BASIC MATH AND ALGEGRA
NAVEDTRA 14139
1. No attempt has been made to issue corrections for errors in
typing, punctuation, etc., which do not affect your ability to
answer problems/questions.
2. In the assignment portion of the course, a question with no
choices is a True/False question. For True/False questions,
answer 1 for True and 2 for False.
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i
PREFACE
By enrolling in this self-study course, you have demonstrated a desire to improve yourself and the Navy.Remember, however, this self-study course is only one part of the total Navy training program. Practical
experience, schools, selected reading, and your desire to succeed are also necessary to successfully round
out a fully meaningful training program.
COURSE OVERVIEW: This course provides a review of basic arithmetic and continues through some ofthe early stages of algebra. Emphasis is placed on decimals, percentages and measurements, exponents,
radicals and logarithms. Exercises are provided in factoring polynomials, linear equations, ratio, proportionand variation, complex numbers, and quadratic equations. The final assignment affords the student anopportunity to demonstrate what he or she has learned concerning plane figures, geometric construction andsolid figures, and slightly touches on numerical trigonometry.
THE COURSE: This self-study course is organized into subject matter areas, each containing learningobjectives to help you determine what you should learn along with text and illustrations to help youunderstand the information. The subject matter reflects day-to-day requirements and experiences of
personnel in the rating or skill area. It also reflects guidance provided by Enlisted Community Managers(ECMs) and other senior personnel, technical references, instructions, etc., and either the occupational or
naval standards, which are listed in the Manual of Navy Enlisted Manpower Personnel Classificationsand Occupational Standards, NAVPERS 18068.
THE QUESTIONS: The questions that appear in this course are designed to help you understand thematerial in the text.
VALUE: In completing this course, you will improve your military and professional knowledge.
Importantly, it can also help you study for the Navy-wide advancement in rate examination. If you arestudying and discover a reference in the text to another publication for further information, look it up.
1980 Edition Reprinted 1985
Published by NAVAL EDUCATION AND TRAINING
PROFESSIONAL DEVELOPMENTAND TECHNOLOGY CENTER
NAVSUP Logistics Tracking Number
0504-LP-026-7940
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ii
Sailor’s Creed
“I am a United States Sailor.
I will support and defend the
Constitution of the United States of America and I will obey the ordersof those appointed over me.
I represent the fighting spirit of theNavy and those who have gonebefore me to defend freedom anddemocracy around the world.
I proudly serve my country’s Navycombat team with honor, courageand commitment.
I am committed to excellence andthe fair treatment of all.”
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INSTRUCTIONS FOR TAKING THE COURSE
ASSIGNMENTS
The text pages that you are to study are listed atthe beginning of each assignment. Study these pages carefully before attempting to answer thequestions. Pay close attention to tables and
illustrations and read the learning objectives.The learning objectives state what you should beable to do after studying the material. Answeringthe questions correctly helps you accomplish theobjectives.
SELECTING YOUR ANSWERS
Read each question carefully, then select theBEST answer. You may refer freely to the text.
The answers must be the result of your ownwork and decisions. You are prohibited fromreferring to or copying the answers of others andfrom giving answers to anyone else taking thecourse.
SUBMITTING YOUR ASSIGNMENTS
To have your assignments graded, you must beenrolled in the course with the Nonresident
Training Course Administration Branch at the Naval Education and Training Professional
Development and Technology Center(NETPDTC). Following enrollment, there aretwo ways of having your assignments graded:
(1) use the Internet to submit your assignmentsas you complete them, or (2) send all the
assignments at one time by mail to NETPDTC.
Grading on the Internet: Advantages to
Internet grading are:
• you may submit your answers as soon as
you complete an assignment, and
• you get your results faster; usually by thenext working day (approximately 24 hours).
In addition to receiving grade results for eachassignment, you will receive course completion
confirmation once you have completed all the
assignments. To submit your assignment
answers via the Internet, go to:
http://courses.cnet.navy.mil
Grading by Mail: When you submit answer
sheets by mail, send all of your assignments atone time. Do NOT submit individual answersheets for grading. Mail all of your assignmentsin an envelope, which you either provideyourself or obtain from your nearest Educational
Services Officer (ESO). Submit answer sheetsto:
COMMANDING OFFICER NETPDTC N331
6490 SAUFLEY FIELD ROADPENSACOLA FL 32559-5000
Answer Sheets: All courses include one“scannable” answer sheet for each assignment.
These answer sheets are preprinted with yourSSN, name, assignment number, and course
number. Explanations for completing the answersheets are on the answer sheet.
Do not use answer sheet reproductions: Useonly the original answer sheets that we
provide—reproductions will not work with ourscanning equipment and cannot be processed.
Follow the instructions for marking youranswers on the answer sheet. Be sure that blocks
1, 2, and 3 are filled in correctly. Thisinformation is necessary for your course to be
properly processed and for you to receive credit
for your work.
COMPLETION TIME
Courses must be completed within 12 months
from the date of enrollment. This includes timerequired to resubmit failed assignments.
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PASS/FAIL ASSIGNMENT PROCEDURES
If your overall course score is 3.2 or higher, youwill pass the course and will not be required to
resubmit assignments. Once your assignmentshave been graded you will receive course
completion confirmation.
If you receive less than a 3.2 on any assignment
and your overall course score is below 3.2, youwill be given the opportunity to resubmit failedassignments. You may resubmit failedassignments only once. Internet students willreceive notification when they have failed an
assignment--they may then resubmit failedassignments on the web site. Internet students
may view and print results for failedassignments from the web site. Students who
submit by mail will receive a failing result letterand a new answer sheet for resubmission of eachfailed assignment.
COMPLETION CONFIRMATION
After successfully completing this course, youwill receive a letter of completion.
ERRATA
Errata are used to correct minor errors or delete
obsolete information in a course. Errata mayalso be used to provide instructions to thestudent. If a course has an errata, it will beincluded as the first page(s) after the front cover.
Errata for all courses can be accessed andviewed/downloaded at:
http://www.advancement.cnet.navy.mil
STUDENT FEEDBACK QUESTIONS
We value your suggestions, questions, and
criticisms on our courses. If you would like tocommunicate with us regarding this course, weencourage you, if possible, to use e-mail. If youwrite or fax, please use a copy of the StudentComment form that follows this page.
For subject matter questions:
E-mail: n3222.products@cnet.navy.milPhone: Comm: (850) 452-1001, Ext. 1520
or 1518DSN: 922-1001, Ext. 1520 or 1518
FAX: (850) 452-1694(Do not fax answer sheets.)
Address: COMMANDING OFFICER
NETPDTC N32226490 SAUFLEY FIELD ROADPENSACOLA FL 32509-5237
For enrollment, shipping, grading, or
completion letter questions
E-mail: fleetservices@cnet.navy.milPhone: Toll Free: 877-264-8583
Comm: (850) 452-1511/1181/1859DSN: 922-1511/1181/1859FAX: (850) 452-1370
(Do not fax answer sheets.)Address: COMMANDING OFFICER
NETPDTC N331
6490 SAUFLEY FIELD ROADPENSACOLA FL 32559-5000
NAVAL RESERVE RETIREMENT CREDIT
If you are a member of the Naval Reserve, you
may earn retirement points for successfullycompleting this course, if authorized undercurrent directives governing retirement of NavalReserve personnel. For Naval Reserve retire-ment, this course is evaluated at 22 points,which will be credited in units as shown below:
Unit 1 – 12 points upon satisfactorycompletion of Assignments 1 through 6
Unit 2 – 10 points upon satisfactorycompletion of Assignments 7 through 11
(Refer to Administrative Procedures for Naval Reservists on Inactive Duty, BUPERSINST
1001.39, for more information about retirement points.)
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vii
Student Comments
Course Title: Mathematics, Basic Math and Algebra
NAVEDTRA: 14139 Date:
We need some information about you:
Rate/Rank and Name: SSN: Command/Unit
Street Address: City: State/FPO: Zip
Your comments, suggestions, etc.:
Privacy Act Statement: Under authority of Title 5, USC 301, information regarding your military status is
requested in processing your comments and in preparing a reply. This information will not be divulged without
written authorization to anyone other than those within DOD for official use in determining performance.
NETPDTC 1550/41 (Rev 4-00
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MATHEMATICS, VOLUME 1
Figure 5-1.–Place values including decimals.
Figure 5-2.–Conversionof a decimal fractionto shortened form.
are zeros in the denominator of the fractionalform.
Figure 5-3 shows the fraction 24358 and100000
what is meant when it is changed to the short-ened form. This figure is presented to showfurther that each digit of a decimal fractionholds a certain position in the digit sequenceand has a particular value.
By the fundamental rule of fractions, it
should be clear that 5 50 500= = . Writing
10 100 1000
the same values in the shortened way, we have0.5 = 0.50 = 0.500. In other words, the value of a decimal is not changed by annexing zeros atthe right-hand end of the number. This is not
46
Figure 5-3.–Steps in the conversion of adecimal fraction to shortened form.
true of whole numbers. Thus, 0.3, 0.30, and0.300 are equal but 3, 30, and 300 are not equal.
Also notice that zeros directly after the deci-mal point do change values. Thus 0.3 is notequal to either 0.03 or 0.003.
Decimals such as 0.125 are frequently seen. Although the 0 on the left of the decimal pointis not required, it is often helpful. This is par-
ticularlytrue in an expression such as 32 ÷ 0.1.In this expression, the lower dot of the divisionsymbol must not be crowded against the decimalpoint; the 0 serves as an effective spacer. If any doubt exists concerning the clarity of anexpression such as .125, it should be written as0.125.
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MATHEMATICS. VOLUME 1
0.2000.0250.002
0.227
Figure 6-1.–(A) Parts of a micrometer;(B) micrometer scales.
of an inch. Multiplying times 0.025 inch, we
find that each marking on the thimble repre-sents 0.001 inch.
0.025 inch since1
is equal to 0.025. The40
sleeve has 40 markings to the inch. Thus eachspace between the markings on the sleeve isalso 0.025 inch. Since 4 such spaces are 0.1inch (that is, 4 x 0.025), every fourth mark islabeled in tenths of an inch for convenience inreading. Thus, 4 marks equal 0.1 inch, 8 marksequal 0.2 inch, 12 marks equal 0.3 inch, etc.
To enable measurement of a partial turn,the beveled edge of the thimble is divided into25 equal parts. Thus each marking on the
thimble is1
of a complete turn, or1
of 1
25 25 401
25
READING THE MICROMETER
It is sometimes convenient when learning toread a micrometer to writedown the component
62
parts of the measurement as read on the scalesand then to add them. For example, in figure6-1 (B) there are two major divisions visible(0.2 inch). One minor division is showingclearly (0.025 inch). The marking on the thimblenearest the horizontal or index line of the sleeve
is the second marking (0.002 inch). Addingthese parts, we have
Thus, the reading is 0.227 inch. As explainedpreviously, this is read verbally as "two hun-dred twenty-seven thousandths." A more skill-ful method of reading the scales is to read alldigits as thousandths directly and to do anyadding mentally. Thus, we read the major divi-sion on the scale as “two hundred thousandths”and the minor division is added on mentally.The mental process for the above setting thenwould be “two hundred twenty-five; two hundredtwenty-seven thousandths.”
Practice problems:
1. Read each of the micrometer settings shownin figure 6-2.
Figure 6-2.–Micrometer settings.
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MATHEMATICS, VOLUME 1
The foregoing example could be followedthrough for any distance between markings.Suppose the 0 mark fell seven tenths of the dis-tance between ruler markings. It would takeseven vernier markings, a loss of one-hundredthof an inch each time, to bring the marks in lineat 7 on the vernier.
The vernier principle may be used to getfine linear readings, angular readings, etc.The principle is always the same. The vernierhas one more marking than the number of mark-ings on an equal space of the conventional scaleof the measuring instrument. For example, thevernier caliper (fig. 6-5) has 25 markings onthe vernier for 24 on the caliper scale. Thecaliper is marked off to read to fortieths (0.025)of an inch, and the vernier extends the accuracyto a thousandth of an inch.
0.30000.07500.00800.0004
0.3834
Figure 6-5.–A vernier caliper.
Vernier Micrometer
By adding a vernier to the micrometer, it is
possible to read accurately to one ten-thousandthof an inch. The vernier markings are on thesleeve of the micrometer and are parallel tothe thimble markings. There are 10 divisionson the vernier that occupy the same space as 9divisions on the thimble. Since a thimble spaceis one thousandth of an inch, a vernier space is1
of 9
inch, or9
inch. It is1
inch10 1000 10000 10000
less than a thimble space. Thus, as in the pre-ceding explanation of verniers, it is possible toread the nearest ten-thousandth of an inch byreading the vernier digit whose marking coin-cides with a thimble marking.
In figure 6-6 (A), the last major divisionshowing fully on the sleeve index is 3. Thethird minor division is the last mark clearly
64
showing (0.075). The thimble division nearestand below the index is the 8 (0.008). The ver-nier marking that matches a thimble markingis the fourth (0.0004). Adding them all together,we have,
The reading is 0.3834 inch. With practice thesereadings can be made directly from the microm-eter, without writing the partial readings.
Figure 6-6.–Vernier micrometer settings.
Practice problems:
1. Read the micrometer settings in figure 6-6.
Answers:
1. (A) See the foregoing example.
(B) 0.1539 (E) 0.4690
(C) 0.2507 (F) 0.0552
(D) 0.2500
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1-64. The product, in simpli fied form, of themultiplication, problem; 4 (2 hours22 minutes 32 seconds) is1. 8 hours 88 minutes 128 seconds
2. 8 hours 90 minutes 8 seconds3. 9 hours 28 minutes 8 seconds4. 9 hours 30 minutes 8 seconds
1-65. The product of 12 miles and 13 miles is
1. 156 miles2. miles3. 156 milessquare4. 156
1-66. The product of 2 feet 8 inches times3 feet 4 inches may be found by1. mu l tip ly ing 2 - fee t t imes 3 fee t then
multiplying 8 inches times 4 inches2. mult iplying 3 feet t imes 2 feet 8
inches then multiplying 4 inches times2 feet 8 inches
3. converting 2 feet 8 inches to 2 feet23
and 3 feet 4 inches to 3then mult ip ly ing
1 fee t and3
4. changing 2 feet 8 inches to 3 feet and3 feet 4 inches to 4 feet and thenmul t ip ly ing
1-67. If a pipe 22 feet 6 inches long is cut into3 equal lengths, how long are the pieces?(Neglect the width of the saw cuts.)1. 7 feet 1 inch2. 7 feet 3 inches3. 7 feet 4 inches4. 7 feet 6 inches
1-68. The result of dividing 23 hours 31 minutes20 seconds by 5 is
1. 43 hours 6 minutes 4 seconds1
52. 4 hours 0 minutes 4224 seconds3. 4 hours 42 minutes 16 seconds4. 4 hours 6 minutes 4 seconds
1-69. In which of the following series of operations is the order in which theoperations are performed important?l . 2 + 3 + 5
2. (3) (9) (7)3. 6 x 8 x 94. 48 ÷ 6 x 3
1-70. The answer to the problem 24 ÷ 4 ÷ 3 ÷ 2i s1. not def ined2. 13. 4
4. 9
1-71. The order of operations is important if division or mult ipl icat ion is involvedwith other operations, Use the rulespertaining to a series of mixed operationsto calculate the value of 6 x 4 + 8 ÷ 2.
1. 162. 28
3. 3 64. 48
1-72. The number 36 is a multiple of 1, 2, 3, 4, 6, 9, 18, 36, and1. 102. 11
3. 124. 13
1-73. An odd number when divided by 2 producesa remainder of 1.
1-74. The number 7 is a factor of
l . 112. 17
3. 24
4. 35
1-75. What is the value of x if x = (3)(0)(4)(6)?l . 02. 18
3. 24
4. 72
11
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Assignment 7
Fundamentals of Algebra; Factoring Polynomials
Textbook Assignment: Chapters 9, 10 (111-117)
7-1. The literal numbers a, x, and p are moregeneral than the numbers 9, 8, and 7.
7-2. The commutative law for addition is il-lustrated by the equation1 . ab = ba2. a + a = b + b3. a + b = b + a4. a(b + c) = ab + ac
7-3. The associat ive law of mult ipl icat ion isi l lustrated by the equation1. abc = acb = cba2. a x (b + c) = c x (b + a)3. a (b + c + d) = ab + ac + ad4. a· (b·c) = a·b·c = (a·b)·c
7-4. If a = 2, b = -3, and c = 4, the algebraicsum of a - b - c equals
1. -52. 13 . 34. 9
7-5. If r = 1, s = 3, t = 12, and x = 15, whatis the value of the expression
2rx?t -
s
1 . - 3 3. 22. - 2 4. 3
7-6. The algebraic expression
is considered to be three numbers.
7-7. What is the value of the algebraic ex-pression
5x2 - 2xy + (3x)
2
when x = 2 and y = -3?1 . 442. 683. 1244. 14 8
7-8. What name is given to the algebraic ex-pression ax - 2bx + cx
2- 4?
1. Monomial2 . B in om ia l3 . T ri n om i al4 . Po lynom ial
7-9. Which statement is true regarding themonomial 17xyz?1. 17 i s t he coe f fi c i en t o f xyz .2. 17x is the coef f ic ient of yz.3. 17xy i s the coe ff i c i en t of z .4. Each of the above is a true statement.
7-10. In the expression xy, what is the co-e f f i c ien t o f xy?1. 12. x
3. y
4 . x y
7-11. What is the coef f ic ient of x2y in theexpression x
2y - ab?
1. 12. 23. x
2
4. y
7- 12 . Two terms of an expression are said to bel ike i f they con ta in1. at least one factor in common2. the same numerical coefficient3. the same l i tera l factors wi th only
their exponents d i f ferent4. the same l i teral factors raised to the
same powers
7-1 3. The like terms in the expression
2 ac2 - 2bc
2 + ac
2 - 2c
2 are
1 . 2ac2 and 2bc
2
2 . 2ac2 and ac
2
3. 2ac2 and 2c
2
4. 2bc2 and 2c2
40
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