MATHEMATICS ESSENTIALS
FOR THE WAR EFFORT
Bulletin No. 40
September, 1942
Prepared at
FLORIDA CURRICULUM LABORATORY
University of Florida
WILLIAM A. GAGER, Director
M. L. STONE, Consultant
DIVISION OF INSTRUCTION
M. W. CAROTHERS, Director
V VIciJ STATE DEPARTMENT OF EDUCATION
 Tallahassee, Florida
COLIN ENGLISH, Superintendent
FOREWORD
On May 7, 8, and 9, 1942, the State Department of
Education and the American Council on Education jointly
sponsored a conference to consider ways and means of re
organizing the secondary school mathematics program to
the end that better training would be provided in this area
for boys and girls entering industry or the armed forces.
Through the cooperation of the Commission on Teacher
Education of the American Council on Education several
outstanding arm and navy officials were secured to serve
as consultants on mathematical needs of the armed forces.
Approximately fifty representatives of colleges and sec
ondary schools attended the conference which was held at
Camp O'Leno.
Since only one unit in mathematics is required for
graduation in most states, it seemed logical that deficiencies
S in mathematics, as revealed by army and navy tests, are
due to the large number of students who have had only one
year of mathematics. Therefore, the conference concluded
that high school boys and girls with special mathematical
aptitude should be guided into the regular fouryear se
quence and that a oneyear course should be planned for
Those who have discontinued mathematics. MATHEMATICS
ESSENTIALS FOR THE WAR EFFORT is presented to the teach
ers and pupils of Florida high schools with the feeling that
it contains much which is needed by the Army, Navy, Air
Corps, and industry. MATHEMATICS ESSENTIALS FOR THE
WAR EFFORT is composed of approximately 35 per cent
arithmetic, 25 per cent algebra, 20 per cent plane geometry,
and 20 per cent trigonometry.
The preparation of this textbook was made possible
through the financial cooperation of the University of Flor
Sida and the State Department of Education. Recognition
and appreciation are extended to the many agencies and
individuals who have contributed time and effort in making
149294
this undertaking a success. The membership of the com
mittee was as follows: Novie Jane Benton, Panama City;
Robert G. Blake, Brooksville; Dr. William A. Gager, St.
Petersburg Junior College, St. Petersburg; Clare Goertz,
Tallahassee; Dr. W. L. Hutchings, Rollins College, Winter
Park; Margaret Murphy, Leesburg; Lula D. Peeples, Fort
Meade; and Alice K. Smith, Tallahassee.
The State Department of Education appreciates the
efforts of the following people: Dr. Charles E. Prall and
Dr. L. L. Jarvie of the American Council on Education;
Commander Burton Davis, Commander James P. Farn
ham, Ensign Hollis M. Leverett, of the Navy; and Mr. C. N.
Smith, Educational Adviser of the Bureau of Naval Per
sonnel. Moreover, for reading and criticizing the manu
script, the State Department of Education is particularly
indebted to Lieutenant Comman'der Norman P. Anderson
of the Jacksonville Naval Air Station; Major Joe M. Robert
son of the U. S. Army; Mr. Robert A. Thompson, Engineer
ing Department, University of Florida; Dr. T. M. Simpson,
Dean of the Graduate School, University of Florida; Dr.
F. W. Kokomoor, Mathematics Department, University of
Florida; and Dr. C. E. Phipps, Mathematics Department,
University of Florida.
Much praise is due the members of the committee who
prepared the material for this book; to Miss Mildred Par
rish, and Mrs. Julia Robbins for the typing; and to Miss
Sue Ella Cason for her assistance with the compilation of
the answers.
^dc zL9
Forew
Prefac
Chapte
I.
II.
III.
IV.
V.
VI.
VII.
VIII.
IX.
X.
XI.
XII.
XIII.
XIV.
'age
3
6
CONTENTS
I
ord ........................
e  .................. ....
*r
Operations with Whole Numbers ..................
Operations with Common Fractions and
Equations ....  
Decimal Measurements ...............
Intuitive Geometry ..........
Literal Numbers . ...................
Statistical Graphs ........ ....... .......... ..... ..
Functions and Equations ........................
Logarithms .................. 
Slide Rule and Conversion Scales ....
Similar Figures ............................
Indirect Measurement ...................................
Oblique Triangles .................... .
Areas and Volumes ..................................
The Earth and Spherical Triangles ...........
Miscellaneous Problems ..............................
Appendix Tables A, B, C, D, E, F ..................
Answers .. 
Index....... ...... ........ ..........
PREFACE
This book has been written primarily for seniors in high
school who have not been majoring in mathematics.
The basic fundamentals of all mathematics, from the
elementary to the most advanced stages, are addition, sub
traction, multiplication, and division of numbers. It is with
these operations that this particular course begins. In the
first three chapters the fundamental operations are applied
to whole numbers, common and decimal fractions, percent
age, and signed numbers. As the course develops emphasis
is given to the equation, to ratio and proportion, to vectors
as used in airplane flight and other practical applications
of geometry, to reading statistical graphs and plotting sta
tistical data, to the application of logarithms and to the
operation of the slide rule, to finding areas and volumes, to
the solving of plane and spherical triangles, and to a large
number of problem solving situations.
A strictly practical viewpoint has been pursued in de
veloping this book. Skill with understanding, and in so far
as possible without proofs, has been the main objective in
selecting and arranging the materials. It is hoped that this
approach will give immediate aid to students who need fur
ther study of mathematics essentials to enable them to serve
efficiently in our war effort.
Outstanding features of this book are: an explanatory
sample preceding each exercise; many review, mastery, and
hurdle tests; and an abundance of practical problems. The
abbreviations used are in accordance with recommended
practice.
It would be unreasonable to assume that a piece of work
of this character would be produced by a committee with
out gathering some ideas from other books. All aids ob
tained from such sources are gratefully acknowledged.
Otherwise, it is the belief of the director and the consultant
that the work in this book is original.
WILLIAM A. GAGER, Director
M. L. STONE, Consultant
CHAPTER I
OPERATIONS WITH WHOLE NUMBERS
Today the armed forces and the industrial world are
looking for young people with the ability to perform the
basic mathematical operations accurately and quickly. This
means that students should be willing to devote more time
to practicing those operations necessary to the perfection
of skills. Students also should do much work with problem
situations in order to develop an analytic technique of prob
lem solving, and to develop rapid, dependable thinking.
Students must, therefore, be able to perform addition, sub
traction, multiplication, and division with whole numbers,
common fractions, and decimal fractions. Skill in using
these processes, along with others to be introduced later,
is absolutely necessary if the individual is to become a
competent participant in our present mechanized society.
READING NUMBERS
Before attempting to operate with numbers it is im
portant to know how to read numbers. In the number
system there are ten digits: namely, 1, 2, 3, 4, 5, 6, 7, 8,
9, 0. Every number is made up of one or more digits. The
position of each digit in the number determines its value;
this system of position is known as the HinduArabic Num
ber System, sometimes called the Place Value System.
8 MATHEMATICS ESSENTIALS FOR THE WAR EFFORT
Table 1. The Place Value System.
(Billions) (Millions) (Thousands) (Units)
CQ
3 'O CQ"
o no a l
a a o a
ffi^ fil
000
000
000
11
a2
~'0 0d9.
EF3
0
'a "8 a
M 43
A r'iZi 
ll4 i l
Sao ja
*000
*00
00r
00
00
00
It is important to keep in mind that all numbers to
the left of the decimal point are whole numbers, while
numbers to the right of this same decimal point have values
less than one and are known as decimal fractions. For
convenience in reading numbers it is customary in the
United States to group the digits by threes, using the
comma. To read a number begin at the left and read each
group as if it were a complete number and supply the name
of the group according to position in the series.
ni~
0c
000
000
000
000
000
000
000
000
000
" 0" Cr)
0 CDC
T C 1
00
00
00
ii
00
00
~0.
1I
OPERATIONS WITH WHOLE NUMBERS 9
Sample la. Read 2,039.
m This number is read "two thousand
g thirtynine." Notice that the two is
followed by the word thousand be
"0 Z cause it is in the thousands group.
FH E The number, 2,039 is not read. "two
2, 0 3 9 thousand and thirtynine." In read
ing numbers and is used only to indi
cate the demical point.
Sample lb. Read 5,237,309.
Irn
SThis number is read "five
S^ million two hundred
Sthirtyseven thousand
Sm three hundred nine."
d 0 d I Note that five is followed
a H a by the word million and
E, E E seven is followed by the
word thousand.
5, 2 3 7, 3 0 9 word thousand
When numbers are given over the telephone or radio,
read for checking, or called out by a surveyor, the "digit
method" of reading is preferred over the magnitude method.
By the digit method 87,042 is read: eight seven zero
four two. The digits should be pronounced clearly and
distinctly. Note particularly that 0 is read zero and never
ought or naught. Later on numbers containing the decimal
point will appear. When the decimal point is reached read
it "point" and read on to the right. For example, to read
3,405.307 say: three four zero five point three zero seven.
10 MATHEMATICS ESSENTIALS FOR THE WAR EFFORT
Sample Ic. Read 6,704 by calling digits.
6, 7 0 4 Read: six seven zero four.
Exercise 1. Write the following numbers in words. Read
in digits.
1. 604 6. 300,507 11. 8,076,256
2. 22,759 7. 3,297,139 12. 13,800,758
3. 457,129 8. 10,007,130 13. 9,307
4. 150,061 9. 756,872,000 14. 7,872,297
5. 20,000,036 10. 82,861,375,279
To write numbers from words refer to Table 1. If the
given words omit positions in the Place Value Syptem, fill
in the blanks with a zero; otherwise, the value of the num
ber will be changed.
Sample 2. Write twenty thousand sixtyseven in digits.
Notice that there are no unit
S r m thousands nor hundreds given.
0 g
0 e Therefore, "zeros" are put in
Sg the Place Value System at
9 Z these places.
2 0, 0 6 7
Exercise 2. Write the following numbers in digits and mark
with commas.
1. Six hundred fiftyeight
2. Eighteen thousand five hundred thirtyseven
3. Three thousand ninetyone
4. Two million seven hundred twenty thousand one hundred
eight
OPERATIONS WITH WHOLE NUMBERS
5. Five hundred eleven thousand two hundred fiftythree
6. Fifteen million four hundred eightyfour
7. Thirtyseven thousand
8. Thirteen billion three hundred forty million two hundred
eighteen thousand ninety
WHOLE NUMBERS
Addition of Whole Numbers
Addition is the process of finding the sum of two or
more numbers. In most practical applications certain units
are attached to numbers. Units may be pounds, gallons,
inches, hours, etc. Obviously one cannot add ten battleships.
and five pounds of sugar and get fifteen battleships or fif
teen pounds of sugar. The point is, therefore, that numbers
to be added must be expressed in the same unit.
Sample 3. There are 327 tires in rack No. 1, 43 in No. 2,
804 in No. 3, 4,951 in No. 4, and 83,791 in No. 5. Find
the total number of tires.
327 tires Note that this addition is possible be
43 cause all numbers are expressed in the
804 same unit. Note also that all unit digits
4,951 are placed in the units' column of the
83,791 Place Value System. Tens are added to
89,916 tires tens, hundreds to hundreds, etc. To add
begin at the bottom of the units' column
and add up. Think 1, 2, 6, 9, 16. Sixteen is 1 ten and
6 units. Write the six in the units' column. In tens'
column think 10, 15, 19, 21. Twentyone tens are 2
hundreds and 1 ten. Write 1 in tens' column. Finish
addition in similar manner. Check by adding down.
12 MATHEMATICS ESSENTIALS FOR THE WAR EFFORT
Practice the following exercises until you feel that you
have increased your accuracy and speed. If you need
further practice refer to material recommended by your
teacher.
Never leave a problem until you have checked it and
are satisfied that it is correct. One good method of check
ing an addition problem is to add the columns in reverse
order.
Exercise 3. Add and check.
1. 49
87
63
45
72
87
94
56
35
75
7. 805
602
56
15
127
2. 98
21
10
79
32
44
67
53
85
16
3. 398
108
97
64
134
227
365
171
299
506
8. 10,765
3,264
17,899
10,666
34,761
4. 456
974
302
556
187
345
962
936
542
660
9. 47,211
394,705
5. 785
9,878
149
1,931
217
9,468
5,110
6. 8,650
7,864
689
5,381
10. 497,561
549,632
487,956
Sample 4. Add horizontally: 4, 7, 3, 16, 27, and 112.
4 + 7 + 3 + 16 + 27 + 112 = 169
Think 2, 9, 15, 18, 25, 29. Write 9 in units' place.
Think 2, 3, 5, 6. Write 6 in tens' place. Write one
in hundreds' place.
OPERATIONS WITH WHOLE NUMBERS
Exercise 4. Add as given.
1.3+5+1+9+6+7+4
2. 9+31+6+14+2+17+8+15+11
3. 38 + 17 + 22 + 56 + 13 + 49
4. 450 + 217 + 793
5. 6,752 + 348 + 23
6. 365 + 942 + 875 + 626
7. 25,965 + 87,392 + 27,645
Subtraction of Whole Numbers
Subtraction is the process of finding the difference be
tween any two numbers. Ordinarily, the procedure is to
place the subtrahend under the minuendunits under units,
tens under tens.
Sample 5. Subtract 6,723 from 9,546.
9,546 Minuend
6,723 Subtrahend
2,823 Remainder
9,546 The Check
To check: Add re
mainder and subtra
hend to get the min
uend.
Be sure units' digit is under
units' digit, tens under ten, etc.
Like things must be subtracted
from like things. Think 6, take
away 3, and 3 is left. Think
4 2 = 2. Seven cannot be
taken from 5. Therefore, bor
row 1 unit from the 9 in the
thousands' column and add it to
the hundreds. This borrowed
unit has a value of 10 when
moved one position to its right. Therefore, think 10
+ 5 7 = 8. Finally, think 8 6 = 2. The re
mainder is 2,823. This is subtraction by the Take
Away Method. If you have learned a different method
which suits you better, use it.
14 MATHEMATICS ESSENTIALS FOR THE WAR EFFORT
Exercise 5. Subtract and check.
1. 3,061
629
6. 2,006
1,875
11. 850,000
352,609
2. 27,036
18,042
7. 8,754
5,714
3. 92,685
886
8. 111,605
5,274
4. 87,591
29,784
9. 30,050
22,708
5. 34,006
2,947
10. 906,781
71,375
12. 8,746,213
2,759,321
13. 5,529 3,667
15. 479,323 37,605
Subtract mentally.
14. 50,075 31,473
16. 955,023 9,833
17. 124
86
18. 212
136
21. 11,241 11,137
23. 23,431 22,986
19. 400 20. 156
87 127
22. 96,425 96,037
24. 5,625 4,836
25. 31,218 31,197
Subtract without rewriting.
29. 166,375
175,616
30. 142,129
142,884
Multiplication of Whole Numbers
Multiplication is a "short cut" for adding equal num
bers. Instead of adding 17 + 17 + 17 + 17 it is shorter
to multiply 17 by 4. The symbols used to indicate multi
plication are X, (), and the dot written above the line.
26. 5,625
5,776
27. 34,225
34,569
28. 36,864
37,249
OPERATIONS WITH WHOLE NUMBERS 15
Sample 6a. Multiply and check 304 X 942.
304 Factor
942 Factor To multiply place the numbers one below
608 the other, units under units, tens under tens,
1216 etc. Then multiply 304 by 2, putting units
2736 in units' column. Next multiply 304 by 4.
Because the digit 4 is already in the tens'
286368 Product column there will be no unit value, and the
Check: 942 first value is recorded in tens' column.
304 Likewise, 9 is in the hundreds' column, and
9 X 4 has its first value recorded in the
3768 hundreds' column. Check by multiplying
28260 the factors in the opposite order.
286368
One of the most valuable habits a student can form is
estimating the result in a problem situation. This does
not mean to obtain results by careless methods nor does
it mean to accept inaccurate results. The idea is to make
numbers easy to manipulate and by calculation approximate
the result in order to avoid those errors that often occur
in actual computation.
Sample 6b. Estimate the result of 419 X 96.
Computation:
Estimate Actual 400 X 100 = 40,000 which is
an approximate solution of the
400 is about 419
100 is about 96 example. This definitely shows
s aout9 that the result cannot be 4,000
40,000 2514 nor 400,000. The exact result is
3771 40,224.
40224
16 MATHEMATICS ESSENTIALS FOR THE WAR EFFORT
Exercise 6. Estimate the products, find exact results, and
check.
3. 3,201 4.
106
.6 9. 3,465
[5 32,761
428
732
10.
5. 68,794 6. 2,408
549 6,050
67,104 11. 3,076
27 1,005
Division of Whole Numbers
The process of finding how many times one number,
known as the divisor, is contained in another number, known
as the dividend, is called division. Division might be con
sidered as a continued subtraction of the same number in
the sense that multiplication is the continued addition of
the same number. The result obtained by dividing a divi
dend by a divisor is called a quotient. The signs of division
are , a horizontal bar, an oblique bar, and the form used
in Sample 7.
Sample 7. Divide 29,082 by 37.
Divisor 786 Quotient.
37/29082 Dividend
259
318
296
222
222
0 Remainder
Check:
786
37
5502
2358
29082
Quotient
Divisor
Dividend
To solve the problem try the division
of 37 into 29. 37 is greater than 29.
Try the division of 37 into 290. 37
times what number gives a number
equal to or less than 290 ? Seven is
the number. Place 7 above the 0 in
290. Multiply 37 by 7, place the re
sult, 259, under 290, and subtract.
Annex the next digit in the dividend,
making the number 318. Repeat this
process until all the digits in the divi
dend have been used. If the last
subtraction is not zero the final num
ber is called a remainder. A remain
der divided by the divisor is part of
the quotient. Check: Multiply the
quotient by the divisor and add the
remainder to get the dividend.
7,956
56
8. 79,01
'8,24
1. 694 2.
39
7. 87,005
6,835
12. 56,314
4,586
OPERATIONS WITH WHOLE NUMBERS 17
Exercise 7. Estimate quotients, divide, and check.
1. 63/27,027 2. 6,942 78 3. 4,368 8
4. 729/100,602 5. 201/175,473 6. 185,142 354
7. 46/17,664 8. 7/461,909 9. 99/588,060
10. 303,365 83 11. 97,628/9,372,288
12. 3,793/25,610,336
SIGNED NUMBERS
In the main, numbers express quantity or size; they also
express other relationships. For example, an address, 2314
Main Street, may mean 2314 West Main Street, or 2314
East Main Street. In Florida a temperature reading of
500 would be easily understood, but farther north the read
ing 100 is not sufficient. It is necessary to know whether
it is above or below zero degrees. What other examples
can you give?
This idea of "aboveorbelowzero" occurs so frequently
in mathematics that mathematicians have found a very con
venient way to write it. Measurement in one direction is
called positive, and is shown by writing a plus sign (+)
before the number; whereas, measurement in the opposite
direction is shown by writing a minus sign () before the
number. Thus 250 above zero would be shown by +25;
100 below zero would be shown by 100. These positive
and negative numbers which express more than the ordi
nary numbers in arithmetic are called directed or signed
numbers. Observe that the signs, + and , may be used
not only as signs of operation indicating addition and sub
traction, but also as signs indicating direction.
18 MATHEMATICS ESSENTIALS FOR THE WAR EFFORT
Addition of Signed Numbers
The process of adding signed numbers can be made clear
by a study of the number scale.
Negative Numbers
Positive Numbers
10 9 8 7 6 5 4 3 2 1 0 I +2 +3 +++5 +6 +7 8 +9 +10
Fig. 1
Sample 8. Add +3 and +6.
Symbolically, this can be written in two ways:
The second of the expressions
+3 is usually written without the
+6 parentheses and the middle
+9 sign of addition in the form
or +3 + 6 = +9. The +3
(+3) + (+6) = +9 means to start at 0 on the
number scale and go 3 points
to the right. The +6 means to move six more points
to the right of +3 on the number scale. The final
point will be +9. Thus +3 + 6 = +9.
To add three or more directed numbers add the first
two; to this result add the third number; then add the
fourth number; and so on until all numbers are used.
Exercise 8. Find the sums.
2. +4
+7
4. +13 + 5
3. +7
+8
5. +6+ 7
8. +3 5+2
6. 0 + 5
9. +4 + 9
10. +5 +2 + 12
1. +6
3
7. +9
I l l l l l l l ] ] Tl'l l l ]
' "
OPERATIONS WITH WHOLE NUMBERS
Two or more negative numbers are added in the same
manner as positive numbers; on the number scale the nega
tive numbers are found to the left of zero.
Sample 9. Add (3) + (6).
Symbolically the indicated operation is written either:
Although parentheses are not
3 necessary in adding positive
6 numbers they are essential in
9 order to avoid confusion in
or adding negative numbers. The
(3) + (6) = 9 3 means to start at zero on
the number scale and go three
points to the left. The 6 means to start at the 3
and go six more points to the left. The point reached
on the scale is 9. Thus 3 6 = 9.
Exercise 9. Add.
2. 4
2
3. 12
6
7. (14) + (3)
4. 12
0
5. 0
5
8. (12) + (5) + (6)
10. (6) + (0)
Working Rules
1. To add numbers having like signs find the sum of the
numbers and prefix the common sign.
1. 5
3
6. 14
2
5
9. 4
3
2
20 MATHEMATICS ESSENTIALS FOR THE WAR EFFORT
2. To add two numbers having unlike signs find the dif
ference in the two numbers and prefix the sign of the
numerically larger number.
Sample 10. Add +3 and 7.
This can be written The +3 means to start at 0 on
the number scale and go 3 points
+3 to the right. The 7 means to
7 start at +3 on the number scale
4 and go 7 points to the left. Thus
or you go to the left from +3 to
+3 + (7) = 4 4. Therefore, +3 + (7) =
4, applying Rule 2 above.
Exercise 10. Find the values.
1. +5 2. +3 3. 7
2 5 +4
4. 3 5. (6) + 10
6. 5 7. +11 + (3) + (5)
+13
8. +14 9.
5
+2
10. (6) + (+7) + (2)
Do you feel that you understand how to add two or
more positive numbers, two or more negative numbers,
and a mixture of positive and negative numbers?
OPERATIONS WITH WHOLE NUMBERS 21
+ o You have visualized the above problems on
+ 9 a horizontal scale (see Fig. 1). Keep in mind
+ that you can also visualize positive and nega
+ 7 tive numbers on a vertical scale, where the
+ 6 numbers above 0 are considered positive and
S+ 5 the numbers below are negative. Thermometer
Z readings make use of this type of scale. Later
.8 on you will put this vertical scale and the hori
+
Szontal scale together so that the two scales in
f + tersect at right angles forming what is known
+ 1 as a coordinate system. A coordinate system
0 provides the basis for plotting statistical data
_ and various other types of relationships.
2 Subtration of Signed Numbers
3
Subtraction is the inverse of addition. To
subtract two signed numbers find the number
5
Z which added to the subtrahend gives the minu
66
end.
41 7
)Z Working Rule
S To subtract two signed numbers change the
10 sign of the subtrahend and add to the minuend.
Fig. 2
22 MATHEMATICS ESSENTIALS FOR THE WAR EFFORT
Sample 11. Subtract.
(a) + 7 Think: 10 is the number which can be
3 added to 3 to get +7. Refer to Fig. 1
i10 and you will see the reasonableness of the
rule. Seven is 7 points to the right of 0,
and 3 is 3 points to the left of 0. There
fore, the two points are 10 points apart.
(b) 12 Think: 7 is the number which can be
5 added to 5 to get 12. On Fig. 1, 12
S7 is 12 units to the left of 0; 5 is 5 units
to the left of 0. These two points are sep
arated by 7 points on the negative side
of the number scale. Therefore, the num
ber is 7.
Exercise 11. Subtract.
1. +8
3
6. 52
16
2. 11
+44
7. +75
+30
3. 7
8
4. +15
+3
5. 25
+6
8. 25 9. 28 (5)
+38
10. +36 11. +52 12. 273 13. +15 14. 87
48 +16 + 31 50 12
15. Subtract the sum of (2 + 5 3) from the sum of
(8 + 25 3).
Multiplication of Signed Numbers
As has been pointed out before, multiplication is re
peated addition. Thus 4 X 5 gives the same result as
adding 5 + 5 + 5 + 5, which is +20. The result of 4 X
(5) is the same as adding (5) + (5) + (5) +
(5), which is 20.
OPERATIONS WITH WHOLE NUMBERS 23
Working Rules
1. To multiply two numbers with like signs prefix the (+)
sign to the product.
2. To multiply two numbers with unlike signs prefix the
() sign to the product.
Sample 12. Multiply.
According to the working rules are the products
correct?
(a) +5 (b) 5 (c) +5 (d) 5
+3 3 3 +3
+15 +15 15 15
Other ways of writing the above samples:
(a) 5X3= (c) (+5) (3)=
(5) (3) = 5(3)=
(+5) (+3) =
(b) (5) (3) = (d) (5) (+3) =
(5) (3)=
If more than two signed numbers are to be multiplied,
multiply the first two; then multiply their product by the
third number; and so on until all the given numbers are
used.
Exercise 12. Find the product.
1. +5 2. 5 3. 8 4. +7 5. 12
+7 7 5 10 +18
6. (12) (+3) 7. (15) (4) 8. (8) (4)
9. (14) (14) 10. (25) (25) 11. (1) (+2) (3)
12. (5) (+6) (11) 13. (7) (2) (+3)
14. 3 (8) (4) 15. (2) (4) (3) (1)
16. (8) (+2) (3) (+2) 17. (+8) (+5) (+3)
24 MATHEMATICS ESSENTIALS FOR THE WAR EFFORT
Division of Signed Numbers
Working Rule
If the signs of the dividend and the divisor are alike
the (+) sign is placed before the quotient; if the signs of
the dividend and the divisor are unlike a () sign is placed
before thequotient.
This rule is identical to the one given for multiplication.
This happens because division is the inverse of multiplica
tion; dividend divisor = quotient, and inversely, the
quotient X division = dividend.
Sample 13. Divide.
+12
(a) =+4
+ 3
12
(b)  +4
3
12
(c) = 4
+ 3
+12
(d) =4
3
Other ways of writing the above examples:
4 Quotient
(a) Divisor 3 /12 Dividenc
(c) 12 +3 = 4
+4
1 (b) 3 / 12
(d) +12 3 = 4
Checks: Division can be checked by multiplying the
quotient by the divisor to obtain the dividend.
(a) (+3) (+4) = +12
(c) (4) (+3) = 12
(b) (+4) (3) = 12
(d) (4) (3) +12
Exercise 13. Find the quotient and check.
1. 27
9
2. 18
6
3. 15
3
4. 20
5
5. 24
6
OPERATIONS WITH WHOLE NUMBERS 25
6. (15) 5
S 9. 32 8
12. 5/125
7. 25 5
10. 12/+144
13. 16/1024
16. (100) (25)
1. Add
(a)
(b)
(c)
8. (45) (15)
11. 6/54
14. 60
+15
15. 64
4
REVIEW EXERCISE I
3,605 + 75 + 385 + 1,156 + 54 + 287
38,250 + 1,387 + 47,296 + 389
3,082 (d) 75 (e) 176
137 36 28
2,915 84 389
37 91 14
309 28 208
60 59 934
20 19
38 994
2. Subtract
(a) 942 97
(c) 12,000
709
3. Multiply
(a) 101 X 87
(b) 740 X 320
(b) 10,001 3,976
(d) 497,863 (e) 1,541
275,364 697
(c) 384 X 68
(d) 17 X 13 X 12
(e) 8,731 X 509
(f) 34 X 127
4. Divide
(a) 20,306 142
(b) 8,816 29
(c) 593,952 807
(d) 68,322 6
(e) 67,14411
(f) 23/6,072
26 MATHEMATICS ESSENTIALS FOR THE WAR EFFORT
5. Add
(a)
(b)
(c)
18 + 16 15 + 13 + 25
3+7 12+3+6 5
8 (d) 115
+4 + 86
3 + 40
7 57
11 10
+20
6. Subtract
(a) 118
50
(c) 28
(b) 330
118
(70)
(e) 15
8
6
+ 3
28
30
5
+22
+7
(d) 18 (40)
7. Multiply
(a) (8) (3) (2)
S(b) (4) (3) (2)
(c) (25)(6)
8. Divide
(a) 112
4
(b) 308
+7
(38)(7)
(5) (4) (2)
(8) (5) (3) (2)
(c) 16/2,256
(d) 96 12 (e) 125  5 (f) 3,876 12
9. Perform the indicated operations
(a) 6(5) + 8 10 30
130
(b) 8 + 7(6)
5
OPERATIONS WITH WHOLE NUMBERS. 27
10. Write the following in figures
(a) Eight thousand four hundred
(b) Fifteen thousand eightyfive
(c) Six million thirtyone thousand one hundred forty
(d) Seven hundred twentyseven thousand five hun
dred nine
(e) Fiftynine million five hundred sixtytwo thousand
seventeen
ROUNDING OFF NUMBERS
Because it is important to estimate results in problem
situations before carrying out the computation for the cor
rect solutions, some attention should be given to rounding
off numbers.
It is much easier to remember to use 25,000 miles
as the circumference of the earth than it is to use 24,875
miles. Some reasons for rounding off the number 24,875
miles are that other data in the problem may not be ac
curate enough to justify such an exact figure for the cir
cumference; the 24,875 miles may not be an exact value;
the 25,000 miles is sufficiently accurate for an estimate
of the result. The number of significant figures expresses
the accuracy of a number. Since a result is never more
accurate than the least accurate measurement used in its
computation, the final result is in its most adaptable form
when properly rounded off.
Sample 14. Round off 24,875.
The actual rounding off would take place as follows:
To nearest ten change 24,875 to 24,880; to nearest
hundred change 24,880 to 24,900; to nearest thousand
change 24,900 to 25,000; and to nearest ten thousands
change 25,000 to 20,000.
28 MATHEMATICS ESSENTIALS FOR THE WAR EFFORT
Working Rules
1. When the digit dropped is a 5 and the digit to its left
is even, do not change it; when the digit to the left is
odd add 1 to it.
2. When the digit dropped is less than 5, do not change
the digit to its left.
3. When the digit dropped is more than 5, add 1 to the
digit to the left.
Exercise 14. Round off.
To nearest ten: 1. 23 2. 87 3. 75 4. 94 5. 45
To nearest hundred: 6. 671 7. 823 8. 455 9. 250
10. 550
To nearest thousand: 11. 9,271 12. 4,762 13. 8,755
14. 6,253 15. 2,500
To nearest ten thousand: 16. 23,642 17. 57,703
18. 28,555 19. 95,556 20. 14,279
APPLICATIONS OF WHOLE NUMBERS
TO PROBLEMS
It is not enough to be proficient in addition, subtraction,
multiplication, and division. It is equally important to
determine when and where to use these skills in the solu
tions of various problem situations. Therefore, in solving
any problem certain procedures are essential. These pro
cedures if followed will ultimately lead to understanding
and a feeling of security in problem solving.
Step One. Read and continue to read the problem until
you understand the situation from which the problem was
taken.
OPERATIONS WITH WHOLE NUMBERS
Step Two. Carefully weigh the data given. Ask the
questions, "Are all the data present essential to the solution
of the problem, or are some of the facts just information?"
"Do you have sufficient data?"
Step Three. Make a mental analysis of the data and
their relationships. What is wanted in the problem? If
the problem is complicated it is often necessary to continue
asking what other relations are needed to produce the
relations just established. In this manner the process of
thinking goes on until all the facts are brought into proper
relationship with each other.
Step Four. Before doing any computation ask yourself
if your visualization of relationships is reasonable. A rough
drawing or sketch is helpful. Arrange data in proper order
and estimate the result.
Step Five. This step is concerned with computation and
should involve two things:
1. Computation for the correct result.
2. Checking the result.
Keep in mind that any problem worth solving is worth
checking. These problem solving directions should be re
ferred to constantly.
30 MATHEMATICS ESSENTIALS FOR THE WAR EFFORT
Sample 15. A Boy Scout troop has obtained a plot 85 ft
by 210 ft. One long side is fenced. The boys plan to
cut their own posts, but they just buy fencing for the
three sides. If the fencing costs $1.65 a rod, how much
will the fence cost?
Step One. Keep reading the problem until you under
stand the situationuntil you can visualize a lot with
a fence along one side.
Step Two. The fact that the boys are cutting their
own posts is not essential in solving the problem.
Step Three. Visualize the plot as a rectangle with a
fence on one long side and three sides to be fenced.
Observe that the plot is measured in feet and the fenc
ing is sold in rods. Think: I want number of feet
to be fenced times cost of fence per foot, or number
of rods to be fenced times cost of fence per rod.
Step Four. Make a sketch.
210
85 185
Think: The lot is about 200 ft long and about 80 ft
on each side so that the estimated distance to be fenced
is 360 ft. The fencing at $1.65 per rod is $0.10 per ft.
Therefore, the estimated cost of the fence is $36.00.
Step Five. Fencing required:
210 + 85 + 85 = 380 ft
Cost of fencing: = $.10 per ft
Total cost of fencing: $38.00
Sample 15 has been enlarged to indicate the details of
approaching and thinking through a problem before doing
the computation. You need not write as extensively as is
OPERATIONS WITH WHOLE NUMBERS 31
done in this sample, but you can profit by attacking prob
lems in this manner.
Exercise 15. Solve. If you find problems in the exercise
that require formulas which you do not know, use the
Index or Appendix to find the desired formula.
1. At Camp Blanding where there are 40,000 draftees the
cost per day to feed them is $14,000. It costs $2,000
a day to house these men. What is the cost per man?
With the allowance of $10 per month for clothing and
a pay check of $50 per month, what does each man cost
the Government per month?
2. Recruits from Florida for the armed services during
the week beginning June 14 were: Monday 402, Tues
day 625, Wednesday 358, Thursday 286, Friday 925,
Saturday 811. How many men were recruited during
the week?
3. In 1940 the manufacturers of rubber products paid
$133,715,235 in wages. In 1941 they paid $241,937,840.
What was the increase in wages in 1941?
4. On an aerial map a scale of one inch for each 8 miles
is used. If the distance between two airports as meas
ured on a map is 18 in., what is the air distance be
tween the two airports?
weight of a body
5. Specific gravity =
loss of weight in water
If a piece of copper weighs 356 grams in air and 316
grams in water what is its specific gravity?
6. The Diesel motors in an average sized submarine can
generate enough power in 2 hours on the surface to
allow the sub to operate under water 4 hours. If the
sub had to remain submerged for 16 hours, how many
32 MATHEMATICS ESSENTIALS FOR THE WAR EFFORT
hours must it run its Diesels to recharge its batter(
Subs usually come to the surface during the night.
it is dark at eight o'clock and dawn comes at six o'clock,
the sub can generate enough power for how many hours
of under water travel?
7. The cost of preparing a special type aircraft is as f
lows: experimentation $20,000; raw materials $14,0(
labor $39,000; power $10,000; dies and tools $35,0(
overhead $26,000; accessories $3,500; instruments
$3,200; experimental flying $300; engineering $29,000.
Find the total cost of the first machine.
8. A testing machine shows the tensile strength of sti
to be 81,255 lb per sq in. while that of aluminum
32,134 lb per sq in. Which is stronger? How mu
stronger?
9. This rectangular solid has the following
AB = 28 in., BC = 32 in., and AH = 19
(a) What is the area of
ABCD? D
(b) What is the area of /
ABGH?
(c) What is the area of
BCFG? 
(d) Find the total area.
(e) Find the perimeter of /
the base. n
(f) What is the volume?
dimensions:
in.
10. Find the volume of a cylinder of radius 80 in. and
height 3 ft in terms of cubic feet. Hint: Always give
careful consideration to the units; you cannot add, c
subtract numbers expressed in different units.
11. A bomber crew in flight consists of a pilot, copilol
bombardier, navigator, radioman, and engineer. In
squadrons of 15 planes each, how many bombardiers
OPERATIONS WITH WHOLE NUMBERS
J are there? What is the total number of men? If the
J average weight of each man is 150 lb, what is the crew
weight of each bomber? This particular type bomber
can carry six 500lb bombs. What is the total weight
of crew and bombs carried by the bomber if a full load
Sof bombs is aboard?
A foundry used 6,562 lb of scrap iron, 4,239 lb of
I new pig iron, and 530 lb of nickel in one heat. What
is the total weight of mixture used in this heat?
After machining a 926 lb casting, it weighed 738 lb.
How many pounds of material were machined off?
SHow many 8 in. lengths can be cut frorh a 2 in. pipe
1 62 ft long?
The monthly payroll of an industrial plant is $398,670.
It has 2910 employees. What is the average monthly
income of each employee?
It is possible to make 2 gallons of airplane gasoline
from each 50 gallons of crude oil. Four training schools
used 500 gallons of gasoline in one day. How many
100gallon drums of crude oil must be refined to supply
the daily need of these four training schools? If the
average flying time per gallon was 30 minutes, how
long did the 500 gallons of gasoline last?
A wiring job calls for the following quantities of No. 14
rubber coated wire: kitchen 57 ft, dining room 127
ft, living room, 476 ft, bath 66 ft, bedroom 156 ft,
Sand hall 128 ft. How many feet of wire will the job
require? If you can buy No. 14 rubber coated wire
at $0.02 per foot, what will be the cost of the wire?
T. he dials on an electric meter registered 7,623 kwh on
June 15 and 7,781 kwh on July 15. How many kilowatt
hours of current were consumed during this month?
If the first 100 kwh cost $0.05 per kwh, and all above
34 MATHEMATICS ESSENTIALS FOR THE WAR EFFORT
that cost $0.03 per kwh, what was the cost of current
for the month?
19. A mechanic earns $9 a day. Assuming that he works
every day except Sundays, Thanksgiving Day, and
Christmas, find his total earnings for 1941.
20. A 50,000 gallon oil tanker carrying a full load was sunk
off the eastern seaboard. How many 250 gallon auto
tankers would be needed to replace the ship? What
would be the cost of transporting by ship tanker at the
rate of $0.007 per gallon? If it cost $0.014 per gallon
more to transport the oil by truck, what would be the
additional cost by truck?
MASTERY TEST I
1. Add 2. Subti
9,864 (a)
327
63,976 (c)
1,288
63,932
11,969 3. Multi
67 (a)
358 7,200
93,615 97
4. Divide and check
(a) 36/36,900
(c) 907/758,252
tact and checl
3,718
2,649
2,869,004
L,751,999
ply
and check
(b)
53,614
4,673
(b) 11,067
961
(d) 9,164,317
9,237,811
(c)
79,017
8,245
(b) 379/2,905,035
(d) 72/569,304
5. Write in digits and mark with commas
(a) Four hundred twelve thousand three hundred
seventyone
(b) Thirteen million seventynine
(c) Twelve billion sixtythree thousand one
(d) One hundred million twelve thousand thirtytwo
(d)
86,792
9,704
OPERATIONS WITH COMMON FRACTIONS 35
6. Add
(a) +11
4
6
(b) 29
+15
+17
7. Multiply
(a) (+8) (5) (3) (2)
(c) (4) (9) (11) (5)
8. Divide
(a) 16 (b) 34 '17
4
9. Round off
To nearest ten: (a) 36
(c) 8 + 3 + 1
(d) 120 60 + 32
(b) (6) (+7) (+19)
(d) (13) (8) (2)
(c) 6/54 (d) 156
+ 12
(b) 81 (c) 95 (d) 47
To nearest hundred: (a) 475 (b) 921 (c) 568
(d) 350
To nearest thousand: (a) 8,729 (b) 5,621
(c) 6,354 (d) 2,500
To nearest ten thousand: (a) 26,479 (b) 17,962
(c) 99,721 (d) 35,841
10. The length of a rectangular kitchen is 171 in. Its width
is 156 in. How many feet in the perimeter of the
kitchen ?
11. A rectangular solid has the following dimensions:
length 38 in., width 22 in., height 17 in.
(a) What is the area of one
end?
(b) What is the area of one
side ?
(c) What is the area of the
bottom? 
(d) Find the total area.
(e) Find the perimeter of ,'
the base.
(f) Compute the volume.
36 MATHEMATICS ESSENTIALS FOR THE WAR EFFORT
12. After machining a 1,292 lb casting, it weighed 1,157
Ib. How many pounds of material were cut away?
13. How many 9 in. lengths can be cut from a 3 in. pipe
69 ft long, if the waste is 1/16 in. for each cutting?
14. If a bricklayer earns $12 per day, how much will he
earn in 1942 if he works every day except Sundays and
Christmas ?
15. If 312 man hours are allotted for the performance of
a certain job, and four men work 52 hr each upon the
job and finish it, how many hours were required to
complete the job? If the job was completed in less
hours than the standard allotted time, compute the
number of man hours saved.
16. An excavation for a building is 44 ft x 36 ft x 9 ft.
How many cubic yards of dirt did the contractor have
to move? If he paid $0.80 per cu yd to the day
laborers who did the excavating, how much did he pay
to have the dirt moved? If the contractor received
$500 for the job, how much did he keep for himself?
17. An athletic field is 535 yd long. If its area is 180,830
sq yd, how wide is it?
18. A schoolroom is 25 ft wide, 30 ft long, and 14 ft high.
Will it provide 200 cu ft of air space for each one of
the 50 children in it?
19. A basket ball is 10 in. in diameter. Making no allow
ance for waste, find how many square inches of leather
were necessary to make the ball.
20. A rectangular piece of sheet lead is 36 cm long and
24 cm wide. Find the area in square centimeters.
CHAPTER II
OPERATIONS WITH COMMON FRACTIONS
AND EQUATIONS
PROPER FRACTIONS
Fractions are used when it becomes necessary to deal
with numbers less than one. A fraction expresses one or
more of the equal parts into which a unit is divided and
is written in the form of an indicated division. For ex
ample, 2/3 of an inch means that the inch is divided into
3 equal parts and 2 of these parts are considered. To ex
press the fact that the fraction 2/3 indicates division, it
can be written 2 3.
In the fraction 2/3, the number of equal parts taken,
2, is called the numerator. The number of equal parts into
which the unit is divided, 3, is called the denominator.
2 numerator
3 denominator
The fractions 2/3, 5/7, 11/12, etc., are called proper
fractions because the numerators are less than the de
nominators.
Addition and Subtraction of Fractions
(a) When several fractions having the same denomi
nator are to be added, the addition is performed by addihg
the numerators and placing the sum over the denominator.
38 MATHEMATICS ESSENTIALS FOR THE WAR EFFORT
3241
3 2 4 1
Sample 16a. Add , , , 
5 5 5 5
5555
Solution:
3 2 4 1 10
+++=2
5 5 5 5 5
Since the denominators have the same numerical value,
add the numerators as you would if asked how many
fifths are 3 fifths + 2 fifths + 4 fifths + 1 fifth. The
result would be 10 fifths or 10/5. The numerator is
10; the denominator is 5. The fraction 10/5 is re
duced to the number 2.
(b) When one fraction is to be subtracted from another
and both have the same denominator, subtract the numera
tors and place the difference over the denominator.
6 8
Sample 16b. Subtract from .
15 15
Solution: Since the denominators have the
same numerical value, subtract the
S _6 numerators as you would if asked
15 15 15 how many fifteenths are 8 fifteenths
minus 6 fifteenths. The result would
be 2 fifteenths or 2/15.
(c) When fractions not having the same denominators
are to be added or subtracted, the fractions must first be
changed to equivalent fractions having common denomina
tors. A common denominator is a number which can be
exactly divided by each of the given denominators. There
is more than one number that can be used as a common
OPERATIONS WITH COMMON FRACTIONS 39
denominator, but it is best to use the lowest number that
contains each of the denominators exactly.
Fraction Rule
Both numerator and denominator of any fraction can
be divided or multiplied by any number (except zero) with
out changing the value of the fraction.
4 3 2
Sample 16c. Add , , 
4 4 12 48
 X 
5 5 12 60 By inspection it is found that 60
3 '3 15 45 is the lowest common denomina
X tor. Has the Fraction Rule been
4 4 15 60 properly applied?
2 2 20 40
X 
3 3 20 60
4 3 2 48 45 40 48 + 45 + 40
+ + = _+ _= _
5 4 3 60 60 60 60
133 13
= =2
60 60
One way to find a common denominator is to multiply
all the denominators together. This will frequently give
a common denominator which is larger than necessary. The
lowest common denominator involves the least work and
can usually be obtained by inspection.
40 MATHEMATICS ESSENTIALS FOR THE WAR EFFORT
4 13
Sample 16d. Subtract from .
11 25
Exercise 16. Find.
1. 3/5 2/5
2. 7/9 1/3
3. 11/14 3/5
6. 1/6 + 2/3 3/4 + 4/9
7. 3/4 2/3 + 5/8
8. 5/8 3/16 + 3/4 1/2
4. 1/3 + 2/5 + 13/15 + 3/4 9. 1/2 2/10 + 3/5 + 1/4
5. 5/6 + 7/8 1/3
10. 9/16 + 1/2 3/4 1/8
Multiplication of Fractions
Two or more fractions are multiplied by multiplying
the numerators together to form the new numerator and
by multiplying the denominators together to form the new
denominator. The process of multiplication or division can
often be made less complicated by the use of the same work
ing rule as in Sample 16c: Both numerator and 'denominator
of a fraction can be divided by the same number (except
zero) without changing the value of the fraction. This
Fraction Rule is a vitally important rule and is repeated
here for emphasis.
25 X 11 = 275 Common denominator.
13 143 4 100
 ;  Why?
25 275 11 275
13 4 143 100 143 100 43
25 11 275 275 275 275
OPERATIONS WITH COMMON FRACTIONS 41
7 13 8 5 2
Sample 17. Multiply: (a) X ; (b) X X 
9 12 15 12 3
7 13 7 X 13 When possible reduce by divid
X ing the same factor into both a
9 12 9 X 12 numerator and a denominator.
91
108
2 1 Divide 4 into both 8 and 12.
8 5 2 4 Divide 5 into both 5 and 15.
X X Multiply the final numerators,
15 12 3 27
3 3 2 X 1 X 2 4. Multiply the
final denominators, 3 X 3 X 3
= 27.
Exercise 17. Multiply.
1. 2/3 X 7 6. 18/9 X 15/27
2. 4/5 X 6/7 7. 2/3 X 7/8 X 5/14
3. 8 X 3/7 8. 9 X 2/3 X 5/8
4. 3/8 X 16 9. 3/4 X 4/5 X 2/6
5. 8/9 X 6/4 10. 5/8 X 3/15 X 6/11 X 22/33
Division of Fractions
To divide one fraction by another, invert the divisor
and multiply the resulting fractions. This shortcut method
is the one commonly used in dividing fractions.
42 MATHEMATICS ESSENTIALS FOR THE WAR EFFORT
4 2
Sample 18. Divide by .
5 3
4
5
2
3
12
15
10
15
12
10
1
= 1
5
By making the fifths and thirds into a common frac
tional unit, fifteenths, the problem becomes 12 fif
teenths 10 fifteenths = 12/10. This becomes 6/5
when numerator and denominator are each divided
by 2.
By shortcut method.
4 2
Or, 
5 3
2
4
5
2 4
3 5
3 12 6
2 10 5
2 10 5
3 6
2 5
1
Exercise 18. Divide.
1. 2/3 5
2. 1/5 2/7
3. 4/7 5/7
4. 7 2/3
5. 2/13 5/39
6. 15/16 5/8
7. 6 3/4
8. 15 1/2
9. 7/8 14/16
10. 9/12 1/2
OPERATIONS WITH COMMON FRACTIONS 43
IMPROPER FRACTIONSMIXED NUMBERS
An improper fraction is a fraction which has a numera
tor larger than the denominator, as 7/3 and 11/4. An im
proper fraction may be changed to a whole number or mixed
number by dividing the denominator into the numerator,
as 12/3 = 4, and 7/3 = 2 1/3.
Sample 19. Change 17/3 to mixed number.
17/3= 5 2/3 Divide the numerator by the denominator.
Exercise 19. Change the following improper fractions to
whole numbers or mixed numbers.
1. 12/2 5. 75/10 9. 2350/12
2. 19/4 6. 107/3 10. 9001/30
3. 15/7 7. 230/9 11. 60864/16
4. 112/8 8. 331/11 12. 1371/38
A mixed number may be changed to an improper frac
tion by multiplying the whole number by the denominator
of the fraction, adding the numerator of the fraction, and
placing the result over the given denominator.
Sample 20. Change 4 7/8 to an improper fraction.
4 X 8 + 7 32 + 7 39
4 7/8=  
8 8 8
Here a whole number and a frac
tion are being added.
The
7 4 7 32 7 39 The
4 + + = improper
8 1 8 8 8 8 fraction
44 MATHEMATICS ESSENTIALS FOR THE WAR EFFORT
Exercise 20. Change the following mixed numbers to im
proper fractions.
1. 9 1/3
2. 7 3/4
3. 1 7/8
4. 11 1/4
5. 17 1/2
6. 6 8/11
7. 9 5/16
8. 25 3/5
9. 84 1/3
10. 112 1/2
Addition of Mixed Numbers
When it is necessary to add mixed numbers, the frac
tional parts are changed to equivalent fractions with a
common denominator, and then the indicated operations
are performed.
Sample 21. Add 3 1/3 and 7 3/4.
1 4
3 The lowest common denominator is 12.
3 3
3 12 Each fraction is changed to an equiva
lent fraction having 12 for a denomina
3 9
7 7 tor. Add both whole numbers and frac
4 12 tions. Reduce the improper fraction,
1 13/12, to a mixed number, 1 1/12. Then
10 1 1/12 is added to the sum of the whole
12 numbers, making the complete sum
or 11 1/12 111/12.
Exercise 21. Add.
1. 9 3/5+ 7
2. 12 + 3/13
5. 11 1/3 6. 24 4/7
2 3/4 3 1/4
5/6 7 1/2
9. 1 1/8 + 5 1/2 + 1 3/8
3. 31/2 + 4 3/4
4. 4 3/8 + 12 1/4 + 5/6
7. 25 5/16
6 7/8
1 1/2
+ 5/4
8. 11 7/8
4 7/16
15
10. 5 63/64 + 62 7/16 + 23 7/32
OPERATIONS WITH COMMON FRACTIONS 45
Subtraction of Mixed Numbers
To subtract mixed numbers change the fractions to
equivalent fractions having a common denominator. Sub
tract the fractions and the whole numbers.
Sample 22. Subtract 6 3/4 from 8 1/3.
8 1/3 = 8 4/12 = 7 16/12 Change the fractions to
6 3/4 = 6 9/12 = 6 9/12 12ths. 9 cannot be sub
1 7/12 tracted from 4; one unit
(12/12) is borrowed
from the 8 units and added to 4/12. 12/12 + 4/12
= 16/12. Subtracting 9/12 from 16/12 gives 7/12.
The whole numbers 7 6 = 1. The remainder is
1 7/12.
Exercise 22. Perform the indicated operations.
1. From 8 7/16 subtract 4 1/4
2. Subtract 9 7/8 from 12 31/32
3. How much more is 37 1/4 than 23 1/8?
4. 225 1/2 7 3/8 5. 27 9/16 7 3/8
6. How much greater is 17 1/4 than 8 3/8?
7. 54 1/2 39 43/64 8. 103 34 1/3
9. 16 3/4 + 27 1/2 8 5/16 10. 57 1/3 5/8
11. 60 1/6 12. 24 7/8 13. 109 4/5
7 1/2 21 5/16 17 1/3
14. 10 15. 305 1/8
4 1/3 192 5/32
Multiplication of Mixed Numbers
When multiplying mixed numbers change them to im
proper fractions. Reduce the expression by dividing num
erator and denominator by the same number whenever
46 MATHEMATICS ESSENTIALS FOR THE WAR EFFORT
possible; multiply the resulting numerators together and
place the result over the product of the denominators.
Sample 23a. Multiply 2 3/8 by 14 1/4.
1083
2 3/8 X 14 1/4 = 19/8 X 57/4 = = 33 27/32
32
2 3/8 changed to an improper fraction is 19/8; 14 1/4
is 57/4. The product of the numerators, 19 and 57,
is 1083;. the product of the denominators, 8 and 4, is
32. Then the fraction 1083/32 is reduced to the mixed
number 33 27/32.
Sample 23b. Multiply 8 2/5 X 2 1/2.
21 1
42 5 21
8 2/5 X 2 1/2 = X =21
5 2 1
1 1
8 2/5 = 42/5
2 1/2 = 5/2
Divide the denominator 2 and the numerator 42 by 2,
and the denominator 5 and the numerator 5 by 5. Mul
tiply the resulting numerators, 21 and 1, and the de
nominators 1 and 1. The resulting fraction is 21/1
which reduces to 21.
Exercise 23. Perform the indicated operations.
1. 10 1/2 X 1/2
2. Find the product of
28 1/4 and 12.
3. What is 3/4 of 26 1/2?
4. Multiply 15 1/2 by 7/8
5. 9 3/4 X 12/3
7/8 X 4/13 X 17/12
3 1/7 X 55 2/3
8 X 3/4 X 1 1/2
4/9 X 6 2/3 X 1/2
5/8 of 13 3/5
OPERATIONS WITH COMMON FRACTIONS 47
Division of Mixed Numbers
To divide mixed numbers change them to improper frac
tions, invert the divisor, and multiply.
Sample 24. Divide 1 7/8 by 12.
12/1 is the divisor. It has
5 already been explained in
7 15 1 5 the study of common
1 8 12 3 fractions why the divisor
4 is inverted and the two
fractions multiplied.
Exercise 24. Divide.
1. 3 1/5 3
2. 5 1/2 4
3. 3 3/4  7
4. 25  2 7/8
6. 4 5/68 2/3 11. 115 1/2 3 1/7
7. 14 4/53 3/4 12. 2 3/16 4
8. 21/25 1/3 13. 7/15 5 1/2
9. 112 16 2/3 14. 2035 16 3/4
5. 4/9 15/6 10. 11 5/6 9
15. 3000 115 1/3
The reciprocal of any number is one over the given
number.
Sample 25. Write the reciprocals of 5/6, 4, and 7 2/3.
1 5 6
Reciprocal of 5/6 1 1 X = 6/5
5/6 6 5
Reciprocal of 4 = 1/4
Reciprocal of 7 2/3 = reciprocal of
23
23/3 = 1 = 3/23
3
48 MATHEMATICS ESSENTIALS FOR THE WAR EFFORT
Exercise 25. Write the reciprocals.
1. 8 4. 10 1/4 7. 1/3 9. 5 2/3
2. 3/16 5. 9/10 8. 18/5 10. 1
3. 1 1/2 6. 12
EQUATIONS
The equation is the foundation of mathematics. One
of the steps in the solution of many problems is to express
the data in the form of an equation. All formulas are
equations. Therefore, in order to use formulas, equations
must be understood.
An equation is a statement that two expressions are
equal, such as X + 8 = 15. Here X + 8 is called the left
member and 15 the right member. An equation resembles
a pair of scales in that the two members of the equation
balance just as the two pans of the scales balance.
Fig. 3
The load on the scale in Fig. 3 is X + 8 ounces on one
side and 15 ounces of weight on the other. The scales with
these loads represent the equation X + 8 = 15. If an
8ounce load is removed from each side, the new load will
represent the equation X = 7, if the scales balance. The
unknown quantity in this equation is represented by the
literal number X, and the process of finding the value of
this literal number is called solving the equation.
OPERATIONS WITH COMMON FRACTIONS 49
To solve equations it is necessary to make use of certain
statements accepted as being true. These statements are
known as axioms and are recorded below for reference
purposes.
Axiom 1. The same number may be added to each member
of an equation without destroying the equality.
(Addition Axiom)
Axiom 2. The same number may be subtracted from each
member of an equation without destroying the
equality. (Subtraction Axiom)
Axiom 3. Each member of an equation may be divided by
the same number (except zero) without destroy
ing the equality. (Division Axiom)
Axiom 4. Each member of an equation may be multiplied
by the same number without destroying the
equality. (Multiplication Axiom)
Sample 26. Solve the equation X 5 = 11.
X 5 = 11 What number (X) less 5 equals 11?
+ 5 = +5 Add. Axiom
X = 16 16 is readily seen to be the number.
Exercise 26. Solve.
1. X7=12 6. N8=4
2. Z 3 =,15 7. Y 12 =1
3. K 50= 62 8. T 4 = 6
4. S 25 = 75 9. W 2= 8
5. Z11 =13
10. V 1 = 19
50 MATHEMATICS ESSENTIALS FOR THE WAR EFFORT
Sample 27. Solve the equation X + 5 = 11.
X + 5 = 11 What number (X) plus 5 equals 11?
5 = 5 Sub. Axiom
X = 6 The number is 6.
Exercise 27. Solve.
1. X + 5 =8 6. T + 11= 29
2. X + 7= 15 7. Y + 18 = 50
3. Z + 1 = 13 8. N + 10 = 18
4. K + 100 =175 9. W + 6 =13
5. M + 63 = 112 10. R + 15 = 60
Sample 28. Solve the equation 3X + 7 = 22.
3X+ 7 = 22 3 times what number
7 = 7 Sub. Axiom (X) plus 7 22?
3X = 15 Div. Axiom
X = 5 3 times 5 + 7 = 22
Exercise 28. Solve.
1. 4X + 5 = 13 6. 9a 2 =16
2. 2y 1 = 9 7. 2s + 25 = 75
3. 4z 2 = 14 8. 3m 3 = 15
4. 9a + 2 = 20 9. 3t + 10 = 43
5. 6y + 9 = 51 10. 6q 4 = 20
To check an equation substitute in the original equation
the value found for the literal number; if both sides of the
equation reduce to the same number the value found for the
literal number is correct.
OPERATIONS WITH COMMON FRACTIONS 51
Sample 29a. Solve and check 3X + 5 = X + 27.
3X + 5 = X + 27
2X + 5 = 27 Subtracting X
2X = 22 Subtracting 5
X = 11 Dividing by 2
Check: 3(11) + 5 = 1(11) + 27 Substituting 11
33 + 5 = 11 + 27 for X
38 = 38
Therefore, X = 11 is the result.
Terms which contain the same letters, are called similar
terms, such as 5b and 2b, lids and 15 ds. Similar terms
can be combined as: 2y2 and 9y2 = lly2, or 7X 3X = 4X.
Sample 29b. Solve and check 2m 3 + 5m + 12
15m 12 m.
2m 3 + 5m + 12 =
7m + 9 =
9=
21=
3=
15m 12 m
14m 12 Combining similar
terms
7m 12 Sub. Axiom
7m Add. Axiom
m Div. Axiom
Note that literal terms may be collected on either side
of the equal sign.
Check: 2(3) 3 + 5(3) + 12 =
6 3 + 15 + 12 =
30 =
15(3) 121(3)
45 12 3
30
52 MATHEMATICS ESSENTIALS FOR THE WAR EFFORT
Exercise 29. Solve and check.
1. 2X5=X+3
2. 5X+ 6 = 2X+ 21
3. 3k + 7 k = 35 5k
4. 35 3m m 1
5. 25 + 3w 2 = 8w 12
6. 4z 7+ 11=6z8+z
7. 56 9X 6 =2X 20
8. 8X 10 3X=28 + X2
9. 34 2b+ 30 =2b+ 16
10. 5h + 22 h 16 =7h 18
1 1 1
Sample 30. Solve and check X + X 11 = 15 X
2 3 4
1 1 1
X + X 11 = 15 X
2. 3 4
An equation containing fractions is solved by multiply
ing every term in the equation by the lowest common
denominator.
1 1 1
12(X) + 12(X) 12(11) = 12(15) 12(X)
2 3 4
Multiplying by 12
6X + 4X 132 = 180 3X
10X 132 = 180 3X Combining like terms
13X 132 = 180 Adding 3X
13X = 312 Adding 132
X = 24 Dividing by 13
Check: 1/2(24) +, 1/3(24) 11 = 15 1/4(24)
12+ 8 11 = 15 6
9= 9
OPERATIONS WITH COMMON FRACTIONS
Exercise 30. Solve and check.
1
1. X =8 X
3
1 1
2. y = 1 + y
6 8
1
3. m +
3
6 = 2m 9
k
4. 5k 18 =
2
1 1 1
5. s + s = s + 14
3 4 2
2t
6. 3t = 33 
3
2 1 3
7. X 15 = X + 
5 10 5
6 5 3
8. X  = 35 X
7 7 14
w 5w 1 7w
9. + 8 
2 6 4 12
2 2' 2
10. k  =2k + 
7 5 35
3 13 4
Sample 31. Solve + = 
4X 16 X
3

4X
13 4
 Lowest common denominator is 16X
16 X
3
16X()
4X
13
+ 16X() =
16
12 + 13X =
13X =
X=
Check: 3/4(4) +
3/16 +
4
16X()
X
Multiplying
by 16X
Sub. Axiom
Div. Axiom
13/16 = 4/1(4)
13/16 = 4/4
16/16 =4/4
54 MATHEMATICS ESSENTIALS FOR THE WAR EFFORT
Exercise 31. Solve and check.
1. 1/X = 3/6
2. 1/m 1/6 = 1/3
3. 2/z 3/5 = 2/5
4. 1/2t + 1/3 = 2/3
5. 1/2b + 1/3 = 3/4 2/b
6. 2/X 5/6 = 1/2X
7. 4/9 = 8/t
8. 7/y + 14/3y = 5/3
9. 15/z 11/2 = 25/2z
10. 5/2m 7/m = 3/2
REVIEW EXERCISE II
Solve and check.
1. 2X 3 = 21 4. m/2 = 2m 5 + 1/2
2. 32 5 +3y = 24 + y + 61 5. X/23 = 3
3. 14 + 6X = 99 X 1 6. 4X6=2X +7+1
7. 3/2w + 5/4w = 11/10 2/5w 3/20w
8. 7/k + 14/3k = 35 70/3k
9. 5X + 7 = 9X 4 5C
10. X/3 + 4/5 = X/2 12. 95 = + 32
9
9
11. 20 = (X 32)
5
13. 2X/5 = 38/45
14. The sum of 1/2, 2/3, and 3/5 of a certain number is
212. Find the number.
15. One side of a triangle is 3/2 times the first side, the
third side is twice the first side. Their perimeter is
108 in. Find the length of each side.
RATIO AND PROPORTION
Ratio
For purposes of comparison it is often necessary to ex
press one quantity in terms of another quantity of the
same kind. This relation of one number to another of the
same kind is known as ratio. For example, if one basket
contains 24 eggs and a second basket contains 96 eggs, it
is evident that the second basket contains 4 times as many
OPERATIONS WITH COMMON FRACTIONS
eggs as the first one, and that the ratio between the eggs
in the two baskets is 1 to 4.
One of the common ways of writing the ratio, 1 to 4,
is 1:4. If a horizontal bar is placed between the two verti
cal dots division is indicated. Thus 1 4 is another form
of writing the common fraction 1/4. This, therefore, makes
it clear just why a ratio is a fraction.
Sample 32. Seven men working on a job finish it in 12
days. Twentyone men working on the job can finish
it in 4 days. What is the ratio of the number of men?
Number of men in first crew 7 1
Number of men in second crew 21 3
Therefore, the ratio of the number of men is 1/3.
Exercise 32. Reduce the following ratios to lowest forms.
1. 25/50 5. 3 X 8 X 21 8. 4 1 22
2. 18/27 4 X 6 X 14 4 32
3. 2 n 6. 36 X 22
51 216 X 32 9. 16 X 15 X 42
4. I 32 7. 6 X 18 24 X 45 X 96
I 42 12 X 72
10. The aspect ratio of a wing assumed to be rectangular
is span/chord. Find A. R. for a wing having span
468 in., chord 93 in.
11. What is the ratio of 6 ft to 24 ft? 10 in. to 50 in.?
22 yd to 11 yd?
12. What is the ratio of 5 ft to 9 yd? 4 ft to 30 in.? 8 ft
to 8 yd?
56 MATHEMATICS ESSENTIALS FOR THE WAR EFFORT
13. What is the gear ratio of two gears, one having 30
teeth, and the other 48 teeth, when they are meshed
together?
14. A circle is inscribed in a square whose side is 14 in.
What is the ratio of the area of the circle to the area
of the square?
15. A room is 14 ft by 16 ft. What is the ratio of its width
to its length?
16. Two pulleys have diameters of 18 in. and 8 in. What
is the ratio of their diameters?
17. An alloy consists of 5 parts copper to 2 parts zinc.
What is the ratio of copper to zinc? What part of
the alloy is zinc?
18. If a man can fly to Washington in 2 hr 20 min and
go by bus in 10 hr, what is the ratio of flying time to
the time required to go by bus?
19. An airplane has a ground speed of 130 mph and a true
air speed of 120 mph. What is the ratio of the true
air speed to the ground speed?
20. In a triangle ABC, AB = 18 in., BC = 10 in., and AC
= 20 in. What is the value of the ratio AB/AC?
Proportion
A proportion is a statement of equality between two
or more ratios. The ratio of 2 bu to 3 bu can be equated
to the ratio of 6 men to 9 men. The proportion may be
written either as
2:3 = 6:9 or 2/3 = 6/9
Sample 33. Consider two bombs, one weighing 500 lb and
the other weighing 200 lb, the volume of the first being
15 cu ft. The volume of the second bomb is unknown.
If the volume of any bomb is proportional to its weight,
find the volume of the second bomb.
OPERATIONS WITH COMMON FRACTIONS 57
Volume of first bomb weight of first bomb
Volume of second bomb weight of second bomb
By letting X represent the unknown quantity, we may
write the equation:
15 500
X 200
200(15) = 500(X)
3000 = 500X
6= X
Therefore, 6 cu ft is the volume of the second bomb.
Exercise 33.
1. An airplane travels 600 mi in 3 hr. Set up a propor
tion and find how far the airplane will travel in 15 hr.
Remember that a ratio is a comparison of two like
quantities and a proportion is the expression of equal
ity of two ratios.
2. If an automobile runs 51 mi on 3 gal of gasoline, how
many miles can it run on 18 gal of gasoline under the
same conditions?
3. A certain machinist can finish 9 gears in 3 hr. Work
ing at the same speed, how many gears can he finish
in an 8hr day?
4. If a can can cut 5 bolts in 12 min, how long will it take
him to cut 170 bolts working at the same speed?
5. When 1 in. on a map represents 125 mi, how many
inches will represent 640 mi?
6. The weight carried by beams of the same material and
the same length and width is proportional to the square
of the depth.
First weight carried (wl) Square of first depth (dl)
Square of second depth (d2)
Second weight carried (W2)
58 MATHEMATICS ESSENTIALS FOR THE WAR EFFORT
When 5000 lb is considered a safe load for a 5 in. beam,
what is the safe load for a 6 in. beam?
7. Water pressure varies directly as the depth of the
water. If the pressure on a certain area is 75 lb/sq in.
when submerged15 ft, what will be the pressure in
pounds per square inch when this same area is sub
merged 100 ft?
8. When a 5 ft post casts a shadow 4 ft long, find the
height of a building that casts a shadow 100 ft long.
9. A city block valued at $150,000 is taxed $465. At the
same rate, what is the tax on a block valued at $98,000?
10. If a field 50 ft long and 40 ft wide is to be enlarged
proportionally so that the width is 120 ft, how long
will the enlarged field be?
Sample 34a. A train running at the rate of 60 mph covers
a certain distance in 4 hr. How long would it take to
cover the same distance when traveling at the rate of
75 mph?
time2 mphl
time1 mph2
t2 60
Common denominator
3 75
25t2 = 60 Mult. Axiom
t2 = 2 2/5 hrs. Div. Axiom
Note: Increase in rate gives decrease in time. This
is indirect or inverse proportion.
2 2/5 60
Check:
3 75
180 = 180
OPERATIONS WITH COMMON FRACTIONS 59
Exercise 34.
1. A boat travels at a rate of 429 mi in 39 hr. How long
would it take to cover the same distance when travel
ing 52 mph?
2. Two gears when meshed together have their revolu
tions per minute inversely proportional to the num
ber of teeth.
rpmi teeth2
rpm2 teeth1
Find the revolutions per minute of a 36 tooth gear that
is driven by a 24 tooth gear running at 80 rpm.
3. A gear of 32 teeth running at 150 rpm is to drive a
shaft at 200 rpm. What size gear (number of teeth)
should be used on the shaft?
4. An airplane flying 150 mph covers a certain distance
in 4 hr. Disregarding drift and wind velocity, at what
rate would it have to fly to cover the same distance
in 3 hr?
5. The volume of a quantity of gas is inversely propor
tional to the pressure upon it. If a quantity of gas
measured 250 cu ft at 15 lb pressure per square inch,
how many cubic feet will it measure at 20 lb pressure
per square inch?
6. If a quantity of gas measures 200 cu ft at 5 lb pres
sure per square inch what is the pressure if the gas is
compressed to 100 cu ft?
Pulley speeds and diameters are inversely proportional
to their revolutions per minute.
60 MATHEMATICS ESSENTIALS FOR THE WAR EFFORT
Sample 34b. A 12 in. diameter pulley making 100 rpm is
driving a 6 in. pulley. Find the revolutions per minute
of the 6 in. driven pulley.
D r D = diam. of driving pulley
d = diam. of driven pulley
d R R = rpm of driving pulley
2 r = rpm of driven pulley
12 r
6 100
1
200 = r r
Therefore
r = 200
rpm.
7. An 8 in. diameter pulley running 250 rpm is driving
a 5 in. diameter pulley. Get the revolutions per minute
of the 5 in. driven pulley. Draw a figure to illustrate.
8. A 15 in. diameter pulley making 75 rpm is driving a
second pulley which turns 500 rpm. Find the diameter
of the second pulley. Draw figure.
9. An engine runs at 92 rpm and has a pulley of 4 ft in
diameter. It drives a 30 in. pulley on a jackshaft at
what revolutions per minute?
10. If 8 gal of gasoline will drive a car 144 mi, at the same
rate of using gasoline how many gallons will it take
to drive the same car 1128 mi?
11. At a certain airdrome there are 168 aircraft consisting
of bombers and interceptor aircraft. The ratio of
bombers to interceptors is 2 to 5. Find the number
of each kind of aircraft.
OPERATIONS WITH COMMON FRACTIONS 61
12. Two gears in mesh have a speed ratio of 3 to 5. If
the smaller gear makes 150 rpm, find the revolutions
per minute of the larger.
13. If a post 12 ft high casts a shadow 15 ft long, how
high is an oil derrick that (at the same time) casts a
shadow 60 ft long?
To5un To Sun
I z' Po5t
Derrick
6o'0" Shadow
14. A shop operating.12 lathes finishing 90 drill stems a
day must increase production to 300 stems a day. How
many lathes must it operate daily with the same num
ber of working hours?
15. A company bought 24 lathes for $1080. At the same
rate what will 18 lathes cost?
16. If a train runs 560 mi in 12 hr, how long will it require
to run 6000 mi at the same rate?
17. In a certain plant a mechanic assembles 14 machines
in 5 days. If he works at the same average rate, how
many days will he require to assemble 50 machines?
18. A line shaft runs 196 rpm. It is driven by a motor
running 1200 rpm. The motor pulley is 4 in. in diam
62 MATHEMATICS ESSENTIALS FOR THE WAR EFFORT
eter. What is the diameter of the line shaft driven
pulley ?
19. What force 24 in. from the fulcrum will balance a
weight of 1200 lb that is 8 in. from the fulcrum?
20. A certain nibbler has a 12 in. pulley that runs 450 rpm.
The motor is turning at 1800 rpm. What diameter
of driver pulley is needed on the motor?
21. Two boys on a balanced seesaw weigh 100 lb and 80 lb
respectively. They are 9 ft apart. How far from the
fulcrum is the 80lb boy?
22. A quantity of gas measures 100 cu ft at a pressure of
5 lb per sq in. What will be the pressure in pounds per
square inch if the gas is compressed to 80 cu ft?
23. An airplane flying at 100 mph covers a certain distance
in 8 hr. At what rate would it have to fly to cover
the same distance in 6 hr if drift and wind velocity
are ignored?
24. If a boat drifts downstream 80 mi in 25 hr, how far
will it drift in 16 hr under the same conditions?
REVIEW EXERCISE III
Perform the indicated operations:
3/4 + 5/8 + 1/6
11/12 + 5/16 + 3/8
21/64+2/32+3/4+1/8
9/25 + 1/4
7/100 + 37/50
3/5 11/12
19/20 7/8
39/64 5/12
1/2 + 3/4 2/3
5/9 + 2/5 1/3
1/7 X 2/4 X 5/6
3/8 X 4/9 X 2/3
9/10 X 7/12
3/16 X 1/2 X 4/5
1/2 X 1/2 X 1/4
3/8 7/16
5/9 1/3
6/7 4/11
3/8 X 1/2 5/12
5/6 X 2/5 2/3
3 1/2 X 17 1/2
8 3/4 X 8 3/4 X 8 3/4
17 1/8 5 3/4
4 1/2 + 3 1/4 + 7/8
OPERATIONS WITH COMMON FRACTIONS
337 1/2 48 7/8
28 5/8 6
116 1/2 26 5/6
47 2/3 + 16 1/2 4 3/4
7/9 X 26 1/3
3/2 X 15/16 3
287 22/7
29 3/4 + 7 1/2 4 3/8
(8 7/12)(1 1/2)
58 2/3 16
572 1/5 378 1/2
36. 41/2 X4 1/2 X 3 1/7
37. 34 + 28 1/4 + 9 1/6
N 15
38. Solve for N.
60 42
X 25
39. Solve for X:
5 90
X 2
40. Solve for X: 
11 9
ENGLISH SYSTEM OF MEASUREMENT
The English system of weights and measures is the one
in common use in the United States. The familiar inch,
foot, yard are known to everyone. As the name foot indi
cates, this unit of length has been derived from the length
of the human foot. Similarly, many other units of measure
were derived from parts of the human body: digit, the
width of a finger; hand, the width of the hand; span, the
distance from the end of the thumb to that of the little
finger when they are extended; cubit, the length from the
elbow to the end of the middle finger; and a rod measured
by taking the length of a line composed of the left feet
of the first 16 men who came from church on a Sunday
morning.
Sample 35a. Change 3 1/2 cu ft to cu in.
7 864
3 1/2 X 1728 X 1728 = 6048 cu in.
2
1
There are 1728 cu in. in a cubic foot. Hence, multiply
3 1/2 by 1728.
64 MATHEMATICS ESSENTIALS FOR THE WAR EFFORT
Sample 35b. Change 3,240 lb to tons.
1
2000/3250 There are 2000 lb in a ton. Divide
2000 3250 by 2000.
1250
2000 5/8
1 5/8 tons
Exercise 35. Change each of the following.
6 3/4 ft to inches
19 1/2 gross to units
4 1/2 tons to pounds
36,724 cu ft to cubic
yards
85 in. to yards
2/3 cu yd to cubic inches
4.2 mi to feet.
4/5 degrees to minutes
16/100 ft to inches
5 gal to quarts
1 gross 5 doz to units
42 oz to pounds
4,250 cu in. to gallons
2 bu to pints
5 1/2 days to hours
472,381 sq ft to acres
1000 oz to pounds
18. 1 1/4 rd to feet
19. 5/9 min to seconds
20. 640 sq in. to square feet
21. 27 pk to bushels
22. 17 sq yd to square feet
23. 75 gal of water to pounds
of water
24. 29,376 cu in. to cubic feet
25. 3 2/3 tons to pounds
Perform indicated operations
26. Add: 3 yd, 2 ft; 5 yd, 1
ft; 3 yd, 2 ft
27. Add: 700 58' 10", 280 25'
40", 30 46' 20"
28. 1800 1300 22' 15"
29. 9 ft 6 3/4 in. 4 ft
7 1/2 in.
30. 16 lb 5 oz X 5
OPERATIONS WITH COMMON FRACTIONS
APPLICATIONS OF COMMON FRACTIONS
TO PROBLEMS
Sample 36. The radius of the inner surface of a steel pipe
is 6 3/8 in. and the radius of the outer surface is 7 1/4
in. Using Pi as
ference.
3 1/7 find the difference in circum
Computation:
(a) 2 X 3 1/7
(b) 2 X 3 1/7
Read problem until understood. It
is observed that no irrelevant data
are present. It is reasoned that if
the circumference (2 1 r) of the
smaller circle is obtained and sub
tracted from the circumference
of the larger one, the desired
measurement would be obtained.
Roughly, the result would be
2 H (7) 2 n (6) = 2 r = 6 2/7.
11
1 22 29 319
< 7 1/4 = 2 X X 
7 4 7
2
1
K
11
1 22 51 561
6 3/8 = 2 X X 
7 8 14
4
2
319 561 638 561
(c)  =  77/14 = 5 1/2 in.
7 14 14 14
The difference in circumference is 5 1/2 in.
66 MATHEMATICS ESSENTIALS FOR THE WAR EFFORT
Exercise 36.
1. A dealer had 18 gal of oil tb sell. He sold 2 1/2 gal
to one customer, 3 3/4 gal to another, 6 1/4 gal to
another, and the remainder to a fourth. How much
did he sell to the fourth?
2. The distance from outside edge to outside edge between
two holes in a metal plate is'8 2/3 in. If one hole is
2 1/8 in. in diameter and the other is 1 1/6 in. in di
ameter, find the length of metal between the holes.
3. If a motor makes 2100 rpm, how many revolutions
does it make in 2 3/4 hr?
4. In the blue print of a house, 1/4 in. in the print repre
sents one foot in the actual house. Find the dimen
sions of the rooms that measure as follows on the blue
print: 3 1/2 by 3 1/2 in., 4 1/8 by 4 5/8 in., 4 1/16 by
4 5/32 in.
5. A bolt is 1 3/4 in. long. What is. the length of another
bolt 3 1/2 times as long?
6. The circumference of a circle is approximately 3 1/7
times the diameter. What is the circumference of a
wheel when the diameter is 21 in.?
7. How many flooring boards wide is a hall 5 ft 5 in. wide,
if each board is 3 1/8 in. wide?
8. An airplane travels 70 mi in 18 3/5 min. Find its
velocity in miles per hour.
9. Find the number of threads on a bolt threaded 3 1/4
in. if the width of the thread is 1/8 in.
10. How many board feet of lumber will there be in 22
pieces 12 ft long, 3 in. by 4 in.? A board foot of lum
ber is the amount of lumber in a piece 1 ft long, 1 ft
wide, and 1 in. or less thick.
11. Change 28 1/4 gal to cubic inches. (231 cu in. in one
gallon)
12. Express 2 ft in eighths of an inch.
OPERATIONS WITH COMMON FRACTIONS 67
13. How many pieces of steel 2 1/3 ft long are there in
46 1/2 ft?
14. A bar of iron weighed 15 3/4 lb. It was machined
down to 8 7/8 lb. How many pounds were machined
off?
15. What is the perimeter of
this triangle?
16. If a machinist works 38 hr 73 10"
and 45 min at 87 1/2 cents
per hour, 'how much pay
does he receive?  12 B
17. What is the area of the ring
of a 6 in. shaft with a 3 in. hole through the center?
18. How many lengths of streamline tubing each 3 3/4 in.
long can be cut from an 84 in. length? Allow 1/32 in.
for each cut. What is the length of the remnant?
19. If a pilot flies 357 mi in 2 hr 15 min, how far will he
travel at the same speed in 6 hr 45 min under the same
conditions ?
20. What is the total length of the 4 steel bands which
must be buttwelded around an oil drum with a diam
eter of 2 ft 4 in.?
MASTERY TEST II
Perform the indicated operations
1. (a) 7/8 5/8 2. (a) 3/5 X 2/3 X 5/12
(b) 7/8 5/6 (b) 3/5 X 8 X 15/24
(c) 9/13 2/11 1/143 (c) 15/27  5/9
(d) 7/12 1/4 2/3 5/6 (d) 15 3/4
68 MATHEMATICS ESSENTIALS FOR THE WAR EFFORT
Solve and check
3. (a) 2X 11 =23 (b) 5w + 60 = 2w + 200
4. 2y 2/35 + 7 = 2y/7 +2/5 7
17 51
5.
X 21
6. A train travels 900 mi in 5 hr. Set up a proportion
and find how far the train will travel in 13 hr. (No
allowance being made for stops)
7. When one inch on a map represents & mi, how many
inches will represent 70 mi?
8. When 6000 lb is considered a safe load for a 6 in. beam,
what is the safe load for a 7 in. beam of the same
material and length?
9. If the water pressure on a certain area is 100 lb/sq in.
when submerged 20 ft, what will be the pressure in
pounds per square inch when this same area is sub
merged 55 ft?
10. A train traveling at the rate of 80 mph covers a cer
tain distance in 4 hr. How long will it take to cover
the same distance when running at the rate of 90 mph?
11. Find the revolutions per minute of a 42 tooth gear
that is driven by a 36 tooth gear running at 105 rpm.
12. An airplane flying 200 mph covers a certain distance
in 3 1/2 hr. Disregarding drift and wind velocity, at
what rate would it have to fly to cover the same dis
tance in 2 1/3 hr?
13. If a quantity of gas measures 300 cu ft at a pressure
of 25 lb/sq in., how many cubic feet will it measure
at a pressure of 28 lb/sq in.?
14. A 6 in. diameter pulley turning 350 rpm is driving a
4 in. diameter pulley. Find revolutions per minute of
the 4 in. pulley. Draw illustration.
OPERATIONS WITH COMMON FRACTIONS
15. Two gears in mesh have a speed ratio of 5 to 7. If the
smaller gear makes 210 rpm, find the revolutions per
minute of the larger.
16. If a 15 ft vertical pole casts a shadow 28 ft long, how
high is a building that (at the same time) casts a
shadow 196 ft long?
17. A pulley with a 12 in. diameter making 100 rpm is
driving a second pulley which turns 400 rpm. Find
the diameter of the second pulley.
18. What force 12 in. from the fulcrum will balance a
weight of 875 lb which is 7 in. from the fulcrum?
19. Two boys on a balanced teeter board weigh 96 lb and
75 lb respectively. The plank is 10 ft long. How far
is the fulcrum from the 96lb boy?
20. Perform the indicated operations
(a) 12 3/4 + 7 1/3 2 1/2
(b) 7 1/6 2 1/3 + 3 1/7
(c) How much greater is 15 1/5 than 11 1/11?
(d) Find the product of 26 1/4 X 17 1/8.
CHAPTER III
DECIMAL MEASUREMENTS
Decimal fractions are read as is shown in Table II.
m
,.o V
a
Q) F Ce ,Q V4
ct
E, ; ;
~o mo
C 0 "ZC1
0 o
a 'm fl F2 =
Ca 111^
S'g' oo S61 2"'0 O
' ' g g
QuaXEragEFfi
*00000rrcr
00000
*000
*00
*0
r1 r1 i4
000
000
000
000
000
000
*00
*0
Sample 37. Read 0.3, 0.45, 0.892, 15.66.
0.3 is read three tenths or zero point three.
0.45 is read fortyfive hundredths or zero point four
five.
0.892 is read eight hundred ninetytwo thousandths or
zero point eight nine two.
15.66 is read fifteen and sixtysix hundredths or one
five point six six.
Notice that in a mixed number the decimal point is
read "and" or "point".
The value possessed by a digit because of its position
in a number is known as its place value. The position of
DECIMAL MEASUREMENTS
a digit in a series of digits forming a decimal fraction de
termines whether it represents tenths, hundredths, thou
sandths, etc. The name given to a decimal fraction is de
termined by the place value of the last digit in the fraction.
Exercise 37a. State the place value.
1. 0.8 2. 0.08 3. 0.008 4. 0.0008 5. 0.00008
Notice that as the 8 is moved one place to the right its
value is one tenth as great as its former value.
Exercise 37b. Read these numbers both by digit and mag
nitude methods.
1. 0.7 4. 1.0125 7. 0.3004 10. 0.00045
2. 0.80 5. 0.62 8. 0.725 11. 0.92
3. 0.875 6. 0.0002 9. 0.708 12. 0.36894
Sample 38. Express as a decimal fractionthirteen thou
sand six hundred five hundredthousandths.
Thirteen thousand six hundred five hundred thou
sandths must have five decimal places for the name
hundredthousandths appears five places to the right
of the decimal place in Table II. Therefore, it would
be written 0.13605.
Exercise 38. Write these numbers.
1. Thirtyfive hundredths
2. Four and eight tenths
3. Four thousandths
4. One hundred twentyfive tenthousandths
5. Seven and' six hundredths
6. Four thousand six hundred twentyfive tenthousandths
7. Six hundredthousandths
8. Sixtynine and two hundred sixteen thousandths
9. One hundred three and two thousandths
72 MATHEMATICS ESSENTIALS FOR THE WAR EFFORT
10. Thirtyeight thousand four hundred fiftyseven
hundredthousandths
11. Four hundred and three hundredths
12. Nine and one thousand six tenthousandths
For most practical work decimals are used only to the
thousandths place. The type of work being done will de
termine the degree of precision required. Decimal fractions
are rounded off by applying the same rules used in rounding
off whole numbers.
Sample 39. Express to the nearest thousandths 0.3212,
0.4855, 0.4865, 0.1047.
0.3212 to the nearest thousandth would be 0.321 for
the first digit dropped is less than five.
0.4855 to the nearest thousandth would be 0.486 for
the first digit dropped is 5 and the preceding digit
is odd.
0.4865 to the nearest thousandth would be 0.486 for
the first digit dropped is 5 and the preceding digit
is even.
0.1047 to the nearest thousandth would be 0.105 for
the first digit dropped is greater than 5.
Exercise 39. (a) Express each number to the nearest
thousandth. (b) Express each number to the nearest
hundredth.
1. 6.8405 3. 0.1809 5. 0.0835 7. 0.755
2. 3.1416 4. 22.37654 6. 0.6853 8. 0.6329
(c) Express each decimal fraction to the nearest tenth.
9. 0.96 10. 0.55 11. 0.65 12. 0.078 13. 0.146
To change common fractions to decimal equivalents
divide the numerator by the denominator.
DECIMAL MEASUREMENTS
Sample 40a.
Express 7/8 as an equivalent decimal fraction.
.875 Annex three zeros to 7 and divide by 8 as
8/7.000 in division of whole numbers. Place deci
mal point in quotient in a vertical line with
decimal point in dividend.
Sample 40b. Express 7/16 in. to the nearest hundredths
inch.
0.437
16/7.000 Since the division to hundredths gives a
6 4 remainder of 12, it is necessary to carry
60 the division one place beyond the second
48 decimal place and round off. 0.437 in.
120 when rounded off is 0.44 in.
112
Exercise 40.
hundredth.
1. 1/4
2. 5/6
Express as decimal fractions to the. nearest
5. 3/2
6. 7/12
9. 2/3
13. 623/1728
10. 3 5/16 14. 3/125
7. 2 3/4' 11. 6/5
8. 5/9
15. 984/231
12. 13/25
To convert a decimal fraction to a common fraction
write the digits of the decimal as the numerator. For the
denominator use 1 and as many zeros to the right of 1 as
there are decimal places in the original fraction.
3. 5/32
4. 4/7
74 MATHEMATICS ESSENTIALS FOR THE WAR EFFORT
Sample 41. Express 0.0475 as a common fraction.
475 19 Why are the 1 and the 4
0.0475 = = zeros placed in the denomi
10000 400 nator?
Fraction Ruledividing by
25.
Exercise 41.
tions.
1. 0.125
2. 0.0375
3. 0.008
4. 0.06
Express decimal fractions as common frac
5. 0.9
6. 0.095
7. 0.7250
8. 0.0004
9. 0.8665 13. 0.017
10. 0.00005 14. 0.170
11. 0.625 15. 0.000042
12. 0.0075 16. 0.37 1/2
Addition of Decimal Fractions
To add decimal fractions it is necessary to keep in the
same column all digits having the same place value: the
tenths must be placed under tenths, hundredths under
hundredths, thousandths under thousandths. Keep the
decimal points of the numbers to be added in a vertical
line and the digits of a decimal fraction will always be in
their proper place.
Sample 42. Add 4.06, 0.057, 0.6, 1.72, and 3.125.
4.06
0.057
0.6
1.72
3.125
9.562
9.6, rounded off
Place in column form, units under
units, decimal points in a vertical
line, tenths under tenths, etc.
DECIMAL MEASUREMENTS
Exercise 42. Add
1. 12.04 + 7.8 + 0.03 3. 6 + 2.;
2. 4.75 + 2.5 + 3.25 4. 365.68'
5. 18.200 + 12.375 + 0.5 + 0.1225
6. 0.03 7. 4.96 8.
0.08 5.05
0.93 7.00
0.54 3.50
0.36 13.77
0.87 8.07
0.61 0.12
9. 0.007 + 36.00 + 123.75 + 347.286
10. 12 + 0.85 + 0.378 + 6.4 + 27
11. 0.0695 + 0.00842 + 0.068 + 0.4
12. 8.25 + 7.675 + 0.7604 + 98.007
13. 4. 14. 0.75 1
0.5 0.9
4.3 1.0
8 + 0.24
2 385.246
6.500
7.000
0.840
0.008
74.040
.5. 0.45
0.7
0.8
2. 3.
Subtraction of Decimal Fractions
In subtracting decimal fractions in vertical columns keep
the decimals points in a straight line.
Sample 43. Subtract 9.26 from 25.5.
25.50 Place the decimal point in the remainder
9.26 in the vertical line formed by the other
16.24 decimal points. Annex zero to the minu
.2 end and subtract. Does annexing zero to
16.2
25.5 change its value?
76 MATHEMATICS ESSENTIALS FOR THE WAR EFFORT
Exercise 43. Perform the indicated operations.
38.5 19.375
678.08 27.6
5.0075 2.38
200 38.68
0.63 0.4
0.7 0.045
380.98 250
27.376 2.125
2.0167 0.0804
69.72 10.004
73.44 cu in. 8.7365
cu in.
7 0.035
98.005 7.7
36.05 9.74
4.735 + 0.165 2.08
Multiplication of Decimal Fractions
To multiply decimal fractions, multiply as with whole
numbers. To place the decimal point begin at the right
digit of the product and count off towards the left as many
digits as the sum of the digits in the decimal parts of the
original numbers. Zeros may be prefixed, if necessary, to
fill out the required number of decimal places..
Sample 44a. Multiply 0.092 by 1.2.
0.092 3 decimal places Estimate: 0.092 is about
1.2 1 decimal place .1; 1.2 is about 1. One
184 tenth of 1 is .1.
92
0.1104 4 decimal places should be pointed off
Sample 44b. Multiply 6.38 by 0.0055.
6.38 Estimate 6 X 0.006 = 0.036
0.0055 Since only 5 places were obtained by mul
3190 tiplying and 6 places are required, prefix
3190 a zero to the number and place the deci
0.035090 mal point.
DECIMAL MEASUREMENTS
Exercise 44. Estimate and find the products.
1. 0.85 X 3.4
2. 7.75 X 0.25
3. 3.1414 X 0.36
4. 1.35 X 12
5. 6.875 X 0.75
6. 504 X 0.009 11. 25 X 2.5 X 42.5
7. 6.3 X 7.035 12. 0.82 X 0.68
8. 253.1 X 0.04 13. 0.2269 X 0.039
9. 9024 X 10.08 14. 45.738 X 0.001
10. 0.0074 X 225.7 15. 178 X 0.291
Division of Decimal Fractions
To divide a decimal fraction by a whole number divide
as with whole numbers. Place the decimal point in the
quotient above the decimal point in the dividend.
To divide a decimal fraction by a decimal fraction move
the decimal point to the right of the last digit in the divisor;
then move the decimal point to the right the same number
of places in the dividend, annexing zeros if necessary. Di
vide as usual.
Sample 45a. Divide 46.44 by 0.6.
46.44 464.4
 Estimate: 460/6 = about 77
0.6 6
Move decimal point one place to
77.4
right in both numerator and de
S6/464.4 nominator. This procedure makes
42 it possible to always avoid a deci
44 mal fraction as a divisor.
42
24
24
78 MATHEMATICS ESSENTIALS FOR THE WAR EFFORT
Sample 45b. Divide
36.3 363000
0.0012 12
30250
= 12/363000
36
30
24
60
60
36.3 by 0.0012.
Estimate: 360000/12 = 30000
Decimal point is moved 4 places to
right. The quotient may be car
ried to any number of decimal
places by annexing zeros to the
dividend.
Exercise 45. Place the decimal point. Annex zeros if neces
sary. The digits in the quotient are in the order 7, 4,
and 9.
1. 393/294.357 5. 0.0393/29.4357 9. 3.93/294.357
2. 3.93/29.4357 6. 3930/294.357 10. 39.3/29435.7
3. 39.3/2943.57 7. 3.93/294357
4. 393/2.94357 8. 0.393/294.357
Estimate and divide rounding off the quotient to two deci
mal places.
11. 133.7 0.834 16. 0.483 4.45 21. 0.6084 0.068
12. 194.37 38.6 17. 69.216 309 22. 9801.9 0.9
13. 200.2 0.002 18. 7301 3.4 23. 157/17.3
14. 16.88 0.04 19. 47.61 0.79 24. 265.5/18
15. 1942.2/0.036 20. 165/3.14 25. 15.7/1728
DECIMAL MEASUREMENTS
REVIEW EXERCISE IV
1. Add
(a) 36.7 + 4.25 + 16.375 + 0.9
(b) 437.2 + 6.56 + 0.7 + 16 + 7.1
(c) 0.83 + 0.6 + 0.755 + 0.2 + 0.25
(d) 4 + 2.5 + 7.53 + 18
2. Subtract
(a) 457 3.85 (c) 56.095 27
(b) 27.45 5.7 (d) 0.367 0.176
3. Multiply
(a) 87.48 X 0.012 (c) 25X 7.5 X 3.5
(b) 3.7 X 4.2 X 0.8 (d) 56,304 X 5.62
4. Divide correct to nearest hundredth
(a) 375 0.25 (c) 0.3645 3.7
(b) 254.06 0.8 (d) 4.8928 0.74
5. Write in figures
(a) Fortyeight tenthousandths
(b) Fiftyfour and twentynine hundredths
(c) One hundred four and three hundred fiftyfive
thousandths
(d) Three thousand nine hundred eightyone ten
thousandths
Sample 46. How many 4.5 in. lengths can be cut from a
30 in. piece of piping if 0.06 in. waste is allowed for
each cut?
Estimate: The length of each piece plus the cut
30 5 = 6 ting is 4.56 in. Rounding this off to
5 and dividing into 30 gives 6 as the
6 estimate of the number of pieces. There
4.56/30.00 will be 6 pieces with a remnant 2.64 in.
27 36 long.
264
80 MATHEMATICS ESSENTIALS FOR THE WAR EFFORT
Exercise 46.
1. If a 7/8 in. rivet weighs 0.375 lb, how many rivets are
there in 28 lb?
2. If the fuel consumption of a 110 hp engine is 0.655
lb/hphr, how many pounds of fuel will be used in an
hour?
(b) The weight of the fuel is 6.25 lb/gal. How many
gallons of fuel will be used per hour?
(c) At 23 cents a gallon what will the fuel cost to run
the engine for an hour?
(d) The fuel tank will hold 38.2 gal. How many hours
will the engine run on this amount?
(e) This airplane flies 120 mi/hr. How many miles
can be flown on a tank full of gas, allowing a 30 min
reserve?
3. A pump delivers 0.975 cu ft of water at every stroke.
If it makes 48 strokes per minute, how many cubic feet
of water does it deliver in an hour? If there are 7.48
gal in a cubic foot, how many gallons does it deliver
in an hour?
4. An electrician bought a roll of copper wire weighing
22.1 lb. If this wire weighs 0.2008 lb/ft, how many
feet did he buy? If the roll of wire cost him $4.00,
how much did it cost per foot?
5. The outside diameter of a steel tube is 0.620 in., the
wall thickness is 0.045 in. What is the inside diameter?
6. The Wasp 425 hp engine runs for 4 1/2 hr. Its specific
fuel consumption is 0.48 lb/hphr. How many pounds
will it consume? Hint: First find horsepowerhours,
then multiply by specific consumption.
7. The tip speed of a propeller of an airplane is the num
ber of feet the tip of the propeller travels in one second.
DECIMAL MEASUREMENTS 81
What will be the tip speed of a 7 ft 6 in. propeller run
ning 1600 rpm?
8. A steel plate is 8.75 in. wide. How long will it have
to be to have an area of 1 sq ft?
9. The model of the German bomber, Heinkel (He177),
has a wing span of 17 1/8 in. If the scale for the
model is 1 to 72, what is the wing span of the plane?
10. The wing area of a plane is found by multiplying the
span by the chord. Find the area of a wing, assumed
to be a rectangle, whose span is 26.5 ft and chord
4.25 ft.
Spn hrd
Rectangular wing
11. If each square foot of the wing of a monoplane can
carry an average of 22.6 lb, how many pounds gross
weight can a wing area of 365 sq ft support?
12. Wing loading of an airplane is the number of pounds
of gross weight that each square foot of the wing must
support in flight. It is found by dividing the gross
weight by the wing area. If the Gruman G21A has
a gross weight of 8000 lb and a wing area of 375 sq ft,
what is its wing loading?
13. The inside diameter of a cast iron cylindrical pipe is
3.375 in. The wall thickness is 0.4375 in. What is the
outside diameter of this pipe?
14. At $37.50 per 1000 board feet, what will 14,280 bd ft
of lumber cost?
15. It is necessary to drill 65 holes in a sprinkler pipe 15
ft 4 in. long. If these holes are to be evenly spaced,
and the end holes are 1 in. from the ends of the pipe,
how far apart should the holes be drilled?
82 MATHEMATICS ESSENTIALS FOR THE WAR EFFORT
16. On a circular saw the diameter is 20 in. If there are
50 teeth evenly spaced on the saw, what is the spacing
between the points?
17. Find the weight of a bar of steel 2 in. by 2 in. and
20 in. long if one cubic inch weighs 0.283 lb.
18. A book 0.5 in. thick contains 90 pages. How many
pages of the same thickness are there in a book 1.3 in.
thick?
19. The diameter of a piston is 3.125 in. Find the area
of the piston head.
20. How many square inches does this trapezoid contain?
The area of a trapezoid equals one half of the sum of
the bases times the altitude.
3.25L
55".5
METRIC MEASUREMENTS
The metric system of measurement is used in many
foreign countries and in many types of scientific work in
the United States. Although it was legalized in the United
States as early as 1866 it has not been generally adopted
by American industry. Manufacturing companies, however,
do quote the sizes of reamers, drills, etc., in the metric units.
Some companies who handle large foreign orders have found
it to be good business to carry out their manufacturing pro
cesses using metric measures.
The fundamental unit of length in the metric system
is the meter, which is 39.37 in. long.
DECIMAL MEASUREMENTS 83
The gram is the principal unit of weight. It is the
weight of one cubic centimeter of water at its maximum
density. 1 gram = 0.03527 oz = 0.002205 lb.
The liter is the unit of volume. One liter = 1.057 qt.
Each of these three units of measure uses the same pre
fixes to designate smaller and larger values. The prefixes
which are most commonly used are:
milli which means 0.001 of a unit
centi which means 0.01 of a unit
kilo which means 1000 times a unit.
The complete table of metric measures is given in the
Appendix.
Sample 47. Change 15 yards to meters.
1 ft = 30.4801 cm = 0.304801 meters
1 yd = 3 ft = 0.304801 m X 3 = 0.914403 meters
15 yd = 0.914403 m X 15 = 13.716045 meters
13.716045 should be rounded off to the same number
of digits significant in the given data.
Check:
15 yd = 45 ft
1 m = 3.28083 ft
Exercise 47. Change
1.
2.
3.
4.
5.
6.
7.
5000 m to km
6 kg to g
75 cm to m
5 mi to km
30 in. to cm
8 yd to m
352 kg to lb
Then
45
= 13.716045
3.28083
= 14 meter
to unit indicated.
8. 98 mm to in.
9. 1.6 m to mm
10. 120 cm to m
11. 6 m 15 cm to cm
12. 12 km to m
13. 2 gal to liter
14. 118 km/hr to mi/hr
84 MATHEMATICS ESSENTIALS FOR THE WAR EFFORT
15. 80 m to km
16. 16 m + 62 cm 8 mm to m
17. 8 ft to cm
18. 10 km to mi
19. Radium which cost $25 per mg would cost how much
per ounce?
20. One liter of water weighs 1 kg. Fifty gallons would
weigh how much?
SQUARES AND SQUARE ROOTS OF NUMBERS
If a number is multiplied by itself the product is said
to be the square of that number. The original number,
then, is the square root of the product. Since 3X 3' = 9,
9 is the square of 3, and 3 is the square root of 9. The
square of a number is usually written with a small 2 slightly
above the lines of writing and to the right of the number
to be squared. This 2 indicates that the number is being
used twice as a factor.
Squares
Sample 48. Square the number 7.5.
7.5 The square of 7.5 is written (7.5)2
7.5 (7.5)2 means 7.5 X 7.5 = 56.25
375
525
56.25
Exercise 48. Square.
1. All numbers from 1 to 20
2. 4.8
3. 0.7
4. 0.88
5. 85
6. 3/4
7. 6 1/2
8. 37.2
9. 136
10. 5.65
DECIMAL MEASUREMENTS
Square Roots
The square root of a number is one of the two equal
factors of that number. It is indicated by the sign (V)
called the radical sign.
Sample 49. Find the square root of 36.
V36 = 6 What number squared, that is, used
twice as a factor, is 36?
Check: When the square root of a number is
6 X 6 = 36 multiplied by itself the original number
is obtained.
Exercise 49. Find the square roots.
1. 9
2. 81
4. 1
5. 121
7. 16
8. 25
10. 49
13. 9/16
11. 1/4 14. 0.0049
3. 144 6. 64
9. 100 12. 0.25 15. 1.44
The square root of a number cannot always be accurately
found by inspection, because not all numbers are perfect
squares such as 9, 16, 25, or 64. Numbers such as 18, 27,
1/5, 0.125, are not perfect squares and their square roots
must be determined by other methods.
86 MATHEMATICS ESSENTIALS FOR THE WAR EFFORT
Sample 50a. Find the square root of 18.
4.24
V18. 00 00 00 Estimate: 4 X 4 16 which
16 is less than 18. 5 X 5 = 25
which is more than 18. There
82 / 2 00
1 6 fore, the square root must be
between 4 and 5.
844 / 36 00
33 76
8482 / 2 24
Separate the number into periods of two digits start
ing at the decimal point. If the root is to be correct
to hundredths, place a decimal point after the 18 and
annex three periods of zeros. The largest perfect
square up to 18 is 16, so 4 will be the first number
in the root. Square 4, which is 16, and subtract from
18 and bring down the next two digits, which in this
case are two zeros. To get the trial divisor, multiply
the 4 in the root by 2. Disregard the last digit in the
new dividend and estimate, 8 into 20, about 2 times.
Place the 2 in the root and also after the 8 in the trial
divisor. Multiply the complete divisor 82 by 2 in the
root and subtract from 200. Bring down the next
period, two zeros. Multiply the root thus far obtained
by 2 to get the next trial divisor, 84. Again disregard
the last digit and estimate 84 into 360, about 4. Place
the 4 in the root and also after the 84 in the trial
divisor. Multiply 844 by 4 and write the product under
3600 and subtract. This process can be repeated until
the root is carried to as many decimal places as desired.
To locate the decimal point, begin at the left of the
root and point off as many digits as there are periods
of digits to the left of the decimal point.
DECIMAL MEASUREMENTS
Sample 50b. Find the square root of 43,264.
2 0 8
V4 32 64
4
408 / 32 64
32 64
Estimate: 200 X 200 = 40,000 which
is not enough. 210 X 210 = 44,100
which is too much. Therefore, the
square root is between 200 and 210.
Group in twos beginning at the decimal point. The
largest perfect square contained in 4 is 4. Therefore
place 2 in the root. Subtract 4 from 4 and bring down
32 the next period of digits. Multiply the 2 in the
root by 2 and place the 4 as a trial divisor. Since 4
will not divide into 3, place a zero in the root, and also
after the 4 in the trial divisor. Bring down another
period, 64. Then estimate how many times 40 will
divide into 326. Place the 8 in the root and a!so after
40 in the trial divisor. Multiply the complete divisor,
408 by 8 and place under 3264. Since 43,264 is a per
fect square there is no remainder.
Exercise 50. Estimate and extract the roots.
1. V42.25
4. V125.46 7. V59
2. V2079.36 5. V2.87
3. V390,625 6. V4258
10. V2 2/3
8. V29,910
9. V1/8
A convenient way to find a square root is to use the
table of Square Roots found in the Appendix. This table
gives the square roots of integers from 1 to 100. By proper
placing of the decimal point and by interpolation the table
can be used to find the square roots of any number.
88 MATHEMATICS ESSENTIALS FOR THE WAR EFFORT
Sample 51. Find the square root of 98 by using the table.
Estimate: 9 X 9 = 81 In the number column, find
10 X 10 = 100 98. At the right of it, be
low the column headed
Therefore, the root is be "square root" find 9.899
tween 9 and 10. which is the square root
V98 = 9.899 of 98.
Exercise 51. From the tables find the square roots.
1. 69 4. 8
7, 77 10. 0.44 13. 0.0063
2. 93 5. 48 8. 87 11. 0.26 14. 730
3. 60 6. 11
9. 0.89 12. 0.0047 15. 7300
Interpolation
Interpolation is the process of finding the number which
lies between and is consistent with two known numbers.
This process is necessary to find the square root of a num
ber between two given numbers in a square root table, and
to find the number between two numbers in many other
tables.
DECIMAL MEASUREMENTS
Sample 52. Find the square root of 94.3.
Estimate: 9 X 9= 81
10 X 10 = 100
Therefore the square root is
between 9 and 10.
J (94.0 9.695
13 194.3 ? J 0.052
[ 95.0 9.747 j
0.3 X
1 0.052
X = 0.0156, or 0.016
.. '/94.3 = 9.695 + 0.016
9.711
The given number is
between 94 and 95.
From the table the
square root of 94 is
9.695; the square root
of 95 is 9.747. The
square root of 94.3 is
9.695 plus 0.3 of the
difference between
9.695 and 9.747. This
difference is 0.052. 0.3
X 0.052 = 0.0156 or
0.016 rounded to three
decimal places. There
fore the square root
of 94.3 is 9.695 plus
0.016, or 9.711.
Exercise 52.
polation.
1. 16.8
Estimate and
5. 81.1
2. 37.25 6. 7.63
3. 9.4
4. 6.56
7. 897
find square roots by inter
9. 0.00691
10. 86.42
11. 0.1396
8. 0.0165 12. 847.3
13. 3 1/4
14. 19 1/2
15. 0.6 3/4
16. 1876
90 MATHEMATICS ESSENTIALS FOR THE WAR EFFORT
THE RIGHT TRIANGLE
If two sides of a triangle are perpendicular to each
other, the figure is called a right triangle, and the side
opposite the 90 angle is the hypote
nuse.
SSide a is an altitude of the triangle.
Side b is a base.
5 Side c is the hypotenuse.
Fig. 4 In any right triangle the square of the
hypotenuse is equal to the sum of the
squares of the other two sides. This may be expressed:
c2 : a2 + b2
and is known as the Pythagorean Theorem. By using the
Pythagorean Theorem any side of a right triangle can be
found if the other two
sides are known. In fig
ure 5, squares are con
structed upon the three
sides, of the right tri
angle. The hypotenuse
is 5 units long, the base
3 units, and the altitude
4 units. Substituting in
the equation
c2 a2 + b2
52 = 42 + 32
25 = 16 + 9
25 = 25 Fig. 5
It may be observed from figure 5 that the square on the
hypotenuse contains as many square units as the sum of
the squares on the other two sides.
DECIMAL MEASUREMENTS
Sample 53a. In a right triangle the base is 24 in. and the
altitude is 18 in. How long is the hypotenuse?
 24"'4
C2 = a2 + b2
c2 = 324 + 576
c2 = 900
c = V900
c = 30 in.
Here are two identical factors, c and c, whose product
is c2. A square root of a number is one of the two
equal factors. Therefore the square root of c2 is c;
the square root of 900 is 30; hence, c = 30. Power
Root Axiom.
PowerRoot AxiomLike powers or like roots of equals
are equal.
Sample 53b. Find the altitude of a right triangle whose
hypotenuse is 11 ft and whose base is 5 ft.
c2 = a2 + b2
121 = a2 + 25
96 = a2 Subtraction Ax.
V96 = a PowerRoot Ax.
9.79 = a
.. a = 9.79 ft
V5' i
92 MATHEMATICS ESSENTIALS FOR THE WAR EFFORT
APPLICATIONS OF RIGHT TRIANGLES
Exercise 53.
1. How long will the braces be which extend from corner
to corner of a rectangular frame 4 ft by 6 ft?
2. The length of a lake
was needed. A right
angle was carefully
laid out as shown in
the diagram. The Lake
sides were measured
and found to be 420
ft and 500 ft respec
tively. How long is
the lake?
3. The roof of a house,
as shown in the dia r 500'
gram, has a run of 12 ft
and a rise of 8 ft. How
long will the rafter be if
1 ft extra is allowed for l
overhang ? Sp&a n
4. What is the rise of a rafter 20 ft long if the span is
30 ft? (The run is one half the span.)
5. If the radius of the circle, OA,
is 18 in. and AN is 5 in., how
long is ON?
6. How long is the diagonal of a
cube 42 in. on an edge? Hint:
First find diagonal of the base;
then find the diagonal of the
cube.
7. Find the altitude of an equi
lateral triangle whose side is A
DECIMAL MEASUREMENTS
9 in. Hint: The altitude divides the triangle into two
equal right triangles.
8. A ladder.25 ft long leans against a wall with its base
7 ft from the wall. How high on the wall does the
ladder reach?
9. A gun is located at point A. The //
distance from A to C is 1200 yd,
from C to B 1000 yd. If the angle
C is a right angle, what is the range C\
(distance) from A to B? ?
10. An airplane leave town A, 200 mi 12001
south of town C. Two hours later
the plane is over town B, which is A
60 mi east of C. How far has the
plane traveled? What is its ground speed?
11. An enemy plane is reported at low level over point C
traveling west at 200 mi/hr. An interceptor plane
takes off from point A, 60 mi north of C, with a ground
speed of 250 mi/hr and flies a course to intercept the
enemy craft at B. How many minutes will it be before
the planes meet? How far will the interceptor plane
have traveled? How far is the enemy plane from C
when they meet? Hint: The two planes will have
traveled the length of time, t. Then AB and BC may
be represented as 250t and 200t.
12. A pilot wishes to fly north. His plane has a TAS of
80 mi/hr. The wind is from the east with a speed of
15 mi/hr. What is the speed of the airplane with
respect to the ground? In this case the ground speed
will be represented by one side of a right triangle with
the hypotenuse equal to the TAS and the shorter side
the wind speed. If the plane has enough fuel in addi
tion to its reserve to travel for 3 1/2 hr, how far can it
travel?
94 MATHEMATICS ESSENTIALS FOR THE WAR EFFORT
PERCENTAGE
The words "per cent" come from the Latin words "per
centum" which mean by the hundred. Per cent, therefore,
is another name for hundredths, and can be designated by
the symbol %. Is this definition clear? Does 1% mean
0.01 to you? Does 17% mean 0.17 to you? Does 90%
mean 90 hundredths to you? It is not wise to go forward
until the fact that per cent means hundredths is thoroughly
understood.
Another approach to percentage is to assume that a
given quantity is divided into 100 equal parts. Then each
one of these equal parts is taken as 1% of the original
quantity. Thus from a fractional viewpoint 1% means
1/100, 17% means 17/100, 90% means 90/100. Notice the
idea of ratio appearing as you express the relationship of
the part to the whole to obtain the per cent. Part/whole
= per cent.
Sample 54. Write 8%
8%== 0.08
8% = 0.08 =
2
25
8
100
as a common fraction.
To change from per cent to deci
mal fraction remove the symbol
and move the decimal point two
places to the left.
To change from a decimal frac
tion to a common fraction write
the given number without a dec
imal in the numerator. In the de
nominator place 1 and as many
zeros to the right of one as there
are decimal places in the decimal
fraction.
DECIMAL MEASUREMENTS 95
Exercise 54. Write as common fractions.
5. 37 1/2%
6. 5%
9. 87 1/2% 13. 4.5%
10. 93 3/4% 14. 27.6%
3. 12 1/2% 7. 56 1/4% 11. 112%
8. 60%
15. 106.44%
12. 97%
Sample 55. Write 1/3 as a per cent.
33 1/3 Change the given fraction to a simi
1/3 =  lar fraction with a denominator of
100 100 by multiplying the numerator
= 0.33 1/3 and the denominator by 33 1/3, write
in decimal form, drop decimal point
S33 1/3%symbol for per cent.
and annex symbol for per cent.
Exercise 55a. Write as per cent.
3. 6/25
4. 3/8
5. 3/5
6. 0.06
7. 0.22
8. 0.45
9. 0.62 1/2 13. 11 1/2
10. 0.37 1/2 14. 99 9/10
11. 3.75
12. 2 1/4
15. 0.0435
A further way of perfecting skill in changing per cents
to fractions and fractions to per cents is to determine the
proper results for the blanks in the following columns.
1. 2%
2. 15%
4. 25%
96 MATHEMATICS ESSENTIALS FOR THE WAR EFFORT
Exercise 55b.
Per Cent Decimal Fractions
0.01
0.03
0.05
0.20
0.33 1/3
18 3/4
0.75
16 2/3 (Sample) 0.16 2/3
0.12 1/2
Common Fractions
1/100
2/100
1/4
1/2
2/3
16 2/3
 or 1/6
100
There are three types of problems in percentage, usually
expressed as rate = percentage/base.
1). To find the rate (per cent) when the percentage (part)
and the base (whole) are given, divide the percentage
by the base. r = p/b
2). To find the percentage when the rate and base are
given, multiply the base by the rate. p = br
3). To find the base when the rate and percentage are
given, divide the percentage by the rate. b = p/r
DECIMAL MEASUREMENTS 97
Sample 56. Solve problems for finding (1) rate, (2) per
centage, and (3) base.
1). There were 50 planes which landed at the Pan
American airport. Four of them were Havana Clip
pers. What per cent were Clippers?
Percentage r = 4/50 = .08 = 8%
Rate =
Base
Check: 50 X 0.08 = 4
2). In an air raid 10% of the 480 planes were shot
down. How many were destroyed?
Percentage = Base X Rate
p = 480 X .10 = 48 planes
Check: 48/480 = .10 = 10%
3). A motorist was complaining because he got behind
20 army trucks and could not pass them. He was told
that this was only 5% of the trucks at Camp Blanding.
Find the number of trucks at the camp.
Percentage b = 20/.05 = 2000/5 = 400
Base =
Rate
Check: 400 X .05 = 20
Exercise 56.
1. In an air raid on the French coast there were 21 British
planes and 16 American planes shot down. What per
cent of the planes lost were American? What per cent
were British?
2. Dr. Brown had an income of $5,000 a year. When he
was drafted his income was reduced to $2,700 a year.
98 MATHEMATICS ESSENTIALS FOR THE WAR EFFORT
What per cent of his original salary did the army pay
him?
3. A class of 400 cadets entered basic training at Carl
strom Field. Twentysix were washed out in the
ground school. One week after flight training began,
34 more were washed out. What per cent of the orig
inal class lost out in each division of training?
4. As soon as synthetic rubber is on the market, there
will most likely be a rationing program. If 15 lb of
each 100 lb are allotted for civilian tires, 7 lb for mis
cellaneous civilian use, and the remainder for the army
and navy, what per cent of the rubber can be used for
civilians? What per cent for the armed forces?
5. In many localities the housing situation has become
acute because of the influx of people. One landlord
raised the rent on an apartment from $15 to $25 a
week. What was the rate of increase?
6. If all men between the ages of 18 and 35 were put in
the armies of the principal belligerents in World War
II, there would be 85,203,000 men in the field, ex
cluding China, India, and the Netherlands Indies.
56,643,000 men would serve under Allied flags, and
28,560,000 under Axis flags. What per cent of the
total would serve under Allied flags? What per cent
under Axis flags? What is the ratio between those
under Allied flags and those under Axis flags?
7. The Triborough Bridge in New York City consists of
a 1380 ft suspension span, a 705 ft side span, a 310 ft
vertical lift span, and a 350 ft fixed truss span. What
per cent of the total length is each span?
8. On July 1, 1940, there were 41,000 licensed pilot fliers
in the United States. 22.9% were solo pilots; 54.1%
private pilots; 17.9% commercial pilots; and 3% air
line transport pilots. How many pilots were there in
each classification?
DECIMAL MEASUREMENTS 99
9. In October, 1940, the major navies of the world owned
the following number of battleships: United States,
32; British Empire, 23; Japan, 18; France, 5; Italy, 8;
Germany, 8; Russia, 6. What per cent of the total was
owned by each country?
10. In 1939 the Middle East produced 985,141 tons of rub
ber. This was 96.81% of the world production. How
much rubber was produced that year?
REVIEW EXERCISE V
1. Express as per cent
(a) 1 1/4 (b) 5/8 (c) 9/16 (d) 3
2. Write in decimal form
(a) 38% (e) 0.04% (i) 32%
(b) 0.9% (f) 1/2% (j) 7%
(c) 30.4% (g) 8 1/4%
(d) 6.75% (h) 200%
3. Express as common fractions
(a) 0.375 (b) 0.048 (c) 0.65 (d) 0.008
4. Square the following numbers
(a) (1.2)2 (b) (0.5)2 (c) (3.75)2 (d) (28)2
5. Find the square root by interpolation of square root
tables
(a) 37.62 (b) 28.5 (c) 1672 (d) 1.452
6. Find the square root
(a) V0.15 (b) V5.63 (c) V4.5 (d) V8,954
7. Round off these numbers to the nearest hundredth
(a) 25.478 (b) 5.365 (c) 0.055 (d) 2.0972
8. Round off to the nearest thousandth
(a) 0.3795 (b) 0.0485 (c) 0.0035 (d) 0.0037
9. Find
(a) 2.4% of 62 (b) 350% of 724
(c) 60 is what per cent of 96?
(d) 108 of 30% of what number?
100 MATHEMATICS ESSENTIALS FOR THE WAR EFFORT
10. Find
(a) 1.06% of 725
(b) 4.5% of what number is 22.5?
(c) 1137 is what per cent of 1250?
(d) 62 is what per cent of 56?
11. In a right triangle a = 16, b = 20. Find c.
12. In a right triangle a = 3.4, c = 17.2. Find b.
13. In a right triangle b = 72, c = 180. Find a.
14. In a right triangle a = 112 ft, c = 300 ft. Find b.
15. In a right triangle ABC, a = 3.4 in., b = 7.2 in. Find c.
MASTERY TEST III
1. Express as decimal fractions
(a) Eleven thousand four hundred twentyseven hun
dredthousandths
(b) Fiftyfive and seven hundred fourteen thousandths
(c) Seven and one thousand four tenthousandths
(d) Three hundred thirtyeight thousand four hun
dred fiftythree and seven hundredths.
2. Express to the nearest thousandths
(a) 0.0465 (b) 0.7275 (c) 0.10579 (d) 0.93949
3. Express as equivalent decimal fractions, correct to
nearest hundredth
(a) 1/3 (b) 5/642 (c) 17/25 (d) 894/327
4. Express as common fractions
(a) 0.375 (b) 0.065 (c) 0.0000125 (d) 0.0875
5. Add
(a) 17.253 + 0.71 + 10.042
(b) 0.0065 + 9.0642 + 11 + 1.640
Subtract
(a) 964.005 691.7
(b) 37.062 15.11 9.049
6. Multiply
(a) 89.06 X 0.0425 (c) 394 X 0.0001
(b) 19.7 X 6.3101 (d) 0.647 X 9.63
