Arithmetic self taught

LITTLE BLUE BOOK NO. QC/I
Edited by E. Haldeman-Julius OjO

Arithmetic
Self Taught

Part I
Lloyd E. Smith

HALDEMAN-JULIUS COMPANY
GIRARD, KANSAS

Copyright, 1925,
Haldeman-Julius Company.

PRINTED IN THE UNITED STATES OF AMERIO

TABLE OF CONTENTS

Part I

Page

Foreword 4

I. Numeration and Notation 5

II. Addition 11

III. Multiplication 17

IV. Subtraction 23

V. Division 26

VI. Factoring and Cancelation 34

VII. Fractions 37

Reduction of Fractions, 39 ; Addition of
Fractions, 41 ; Subtraction of Fractions,
4 4 ; Multiplication of Fractions, 46 ; Di-
vision of Fractions, 47.

VIII. Decimals 51

IX. Percentage 55

X. Averages 60

XI. Ratio and Proportion 61

Answers to Exercises 64

A complete Index to Parts I and II combined
is given on page 61 of Little Blue Book No. 857.

FOREWORD

Arithmetic is the beginning of all mathe-
matics. The child who counts his fingers and
toes is on the way to differential calculus — may
the benevolent deities of digits have mercy on
him! Practical arithmetic confronts all of us
every day of our lives. We cannot buy the
necessities of life or earn a means of sustenance
without using arithmetic.

The subject, in its elementals, is com-
paratively simple. It belongs with the three
"R's" — Reading, 'Riting, and 'Rithmetic. Part
I, the present booklet, deals with all the ele-
mental stages of the subject. Part II, natu-
rally, goes on from where Part I stops, but it
lays particular emphasis on the practical side
of arithmetic, by far the greater portion of the
book being devoted to problems and their so-
lution. In offering the elements of arithmetic,
the extreme and empty simplicity of the texts
used in the lower grades of elementary schools
has been strenuously avoided. It is assumed
that the "setf-teachers" of these Little Blue
Textbooks are students with adult and alert
minds — but, at the same time, those who are
more perspicacious than this text assumes
should be a little patient if the explanation is
in many cases obvious and seemingly unneces-
sary.

The student of arithmetic will find A Hand-
book of Useful Tables (Little Blue Book No.
835), of great value to him in many of his prac-
v^ical calculations.

ARITHMETIC SELF TAUGHT

I. NUMERATION AND NOTATION

Before any subject can be undertaken, its
material and terminology (the names it uses)
must be understood. Arithmetic, according to
the New Standard Dictionary, is the "science
of numbers and the art of reaching results by
their use." It may be divided into abstract
arithmetic, which deals with pure number or
quantity (just 4, and not 4 apples or 4 cents
or 4 "anything"), and practical arithmetic,
which applies the science of pure number to
the problems of everyday life. It is almost
entirely with practical arithmetic that this
book is to deal.

Numeration is the reading or naming of
notation, which is merely to say that notation
is the expression of figures by numbers or let-
ters, and numeration is the naming of those
numbers or letters. Thus:

Notation: 1234567890
Numeration: One, two, three, four, five, six,

seven, eight, nine, zero f(cipherr naught, or

"nothing"). ^ — -*/

The zero (0) may be placed first or last,
since it has no value. It is used in notation
to show the absence of a number or quantity.
This system of notation is commonly called the
Arabic system, but its true origin is claimed
to be Hindu. The Greeks and Romans had no

6 ARITHMETIC SELF TAUGHT. PARTI

zero, and consequently did not progress very
far in arithmetic. The value of the zero is
clearly seen when it is understood that in the
Arabic system numbers take their value by
their position in relation to their component
digits (each of the nine figures, or numerals,
above, is a digit; the digits of the number 485
are 4, 8, and 5). Thus, if 7 stands alone, it
signifies seven units (or "ones"; seven single
sticks, for example); but if a zero is added,
making 70, its value is multiplied by ten, and
it stands for seventy units. The other num-
erals increase the value of their associated
digits in the same way, thus: In the number
765, the nearest digit to the right indicates the
units (here five units), the next to the left
indicates the tens (here six tens or sixty), and
the next to the left the hundreds (here seven
hundreds), so that the number is read "seven
hundred and sixty-five."

Proceeding in numeration from the nine, last
in the notation above, the next number is ten
(10), and the next two are eleven (11) and
(12). So from one to twelve the names are
distinct. But going up from twelve the rest
of the names are compounds. Thus, three and
ten make thirteen (13), the number following
twelve — the name thirteen being a compound of
three (contracted to thir-) and ten (expanded
to -teen). Beginning with this first compound,
thirteen, the next seven numbers end in -teen
(so that girls are said to be "in their teens."
when between the ages of twelve and twenty)
—thirteen (13), fourteen (14), fifteen (15),
sixteen (16), seventeen (17), eighteen (18),

ARITHMETIC SELF TAUGHT. PART I 7

nineteen (19). The manner in which each
units numeral is combined (in name) with the
tens is clearly shown.

Going up from nineteen, the next number is
twice ten, or twenty (20). The series started
with 1, a single digit, a unit, one. By adding
a zero (10) this became ten times as much,
because in the Arabic system, proceeding from
right to left in a number, each digit to the
left is multiplied by ten once for each digit to
its right (thus, in 387, 8 is multiplied by 10
once, for it has one digit, 7, to its right; but
3 is multiplied by 10 twice (10 times 3 is thirty,
and 10 times 30 is 300) because it has two
digits, 8 and 7, to its right). And now twenty
is twice ten (the name being compounded of
twen- for two, and -ty for ten). If one is added
(21), the name one is added to the name also,
making tioenty-one, and, going up, twenty-two
(22), twenty-three (23), twenty-four (24), and
so on. (Notice that in these compound num-
bers there is always a hyphen.)

Numeration is now simply^ a_matter of com-
bining' the fundamental forms. Counting by
tens, the names are ten (10), twenty (20),
thirty (30), forty (40), fifty (50), sixty (60),
seventy (70), eighty (80), ninety (90), and
(ten times ten, for 1 is followed by two digits)
one hundred (100). The intermediate numbers
are simple compounds: sixty-four (64), eighty-
nine (89), fifty-five (55), one hundred and
three (103), two hundred and twenty-four
(224), etc. Ten hundreds (the units digit is
multiplied by ten three ttmes) make one thous-
and (1000) ; ten thousands make ten thousand

8 ARITHMETIC SELF TAUGHT. PART I

(10,000); ten ten-thousands make one hundred
thousand (100,000); ten hundred-thousands
make one million (1,000,000), which is also a
thousand thousands; a thousand millions make
one billion (1,000,000,000) in America, but in
England this is only a thousand millions, an
English billion being a million millions (12
ciphers) ; and a thousand billions make a tril-
lion (12 ciphers), or the English billion, and
so on. In large numbers (four digits or more)
it is customary to divide the numerals into
groups of three by commas, counting from right
to left, as is shown in the preceding numbers
— which makes them easier to read at a glance.

All of the numbers thus far considered have
been iclioje numbers, or integers. They rep-
resent values increasing from 1 up to as high
as one may wish to go. But it is possible to
have only half of a whole — for half an apple
is certainly not a whole apple. This may be
expressed in arithmetic by what is called a -frac-
tion, in this case one-half, written y2 — one over
two, with a. horizontal line separating the top
number from the bottom number. Fractions
vill be- explained in detail a little later.

Or the part of a whole number may be ex-
pressed in arithmetic by what is called a deci-
mal, a name which means specifically a tenth
part. The Arabic system, as has been seen
from the preceding explanation, has the num-
ber ten as its basis — the values of tHe digits
increasing by ten times according to their po-
sition. Just as the numerals increased by ten
times when counting from right to left, ac-
cording to the number of digits on their right,

ARITHMETIC SELF TAUGHT. PARTI 9

so the numerals may be decreased by ten times
when counting from left to right, according to
the number of digits on their left. The starting
point for the counting is a period, or decimal
point — which was not written for the previous
numbers because they were all whole numbers.
When the numbers being used are all integers,
the decimal point is customarily omitted. But
it would be perfectly correct to write it, thus:
758. represents a whole number, seven hun-
dred and fifty-eight, counting its value from
right to left from the decimal point. If the
period were omitted it would be assumed that
the figure was a whole number. But suppose
the position of the period is changed, and the
number is .758 instead of 758. The counting
must now proceed (always from the decimal
point) from left to right, and the 7, instead of
being a unit-digit, is a tenth-digit, and, if it
stood alone (.7) it would be read seven tenths.
Similarly, the 5 is in, not the tens place, but
the hundredths place, and if it stood alone
(.05 — the zero being necessary to distinguish
it from tenths) it would be read five hun-
dredths. The 8 is in the thousandths place, and
.008 would be read eight thousandths. The num-
ber .758 is read seven hundred and fifty-eight
thousandths.

The range of the Arabic system is thus in-
dicated by columns from the right and left of
the decimal point:

5 4321.123 45

ten thou- hun- tens units deci- ten- hun- thou- ten hun*
thou- sands dreds mal ths dred- san- thou- dred

sands point ths dths san- thou-

dths san-
dths

10 ARITHMETIC SELF TAUGHT. PART I

Notice that ciphers standing between the deci-
mal point and a digit on the right or left are
very important, and change the value of the
nuniDer (400. vs. .400 >. But ciphers preceding
a whole number (without themselves being pre
ceded by any other digit), or following a deci-
mal (without themselves being followed by any
other digit) have no significance and are not
usually written (00400. is still four hundred,
and .00400 is still four thousandth.-;).

To summarize notation and numeration, the
notation of a number would be: 48,562,791,006;
and the numeration of the game number is:
forty-eight billion five hundred sixty-two mil-
lion seven hundred ninety-one thousand and
six. The two ciphers indicate that there are
no hundreds and no tens.

The Roman System: The Roman system of
notation uses letters instead of figures, but is
used nowadays only on clock dials and in "dig-
nified" places generally. A complete table of
Roman numerals is given in A Handbook of
Useful Tables (Little Blue Book No. 835, Table
I). To indicate the general nature of the sys-
tem, some of the numbers are: I (one). II
(two), III (three), IIII or IV (four), V (five),
VI (six), VII (seven), VIII (eight), IX (nine),
X (ten), XI (eleven), XII (twelve), XX
(twenty), XXX (thirty), XL (fortv), L (fifty),
LX (sixty), LXX (seventy), LXXX (eighty),
XC (ninety), C (one hundred), D (five hun-
dred), and M (one thousand),

ARITHMETIC SELF TAUGHT. PART I il

II. ADDITION

The fundamental processes of arithmetic are
four: addition, multiplication, subtraction, and
division. Addition and subtraction are habitu-
ally associated, but addition logically belongs
with multiplication, and subtraction with divi-
sion. On these four processes all arithmetical,
even all mathematical, processes are based.

Addition, defined, is the process of finding
the sum or total of two or more numbers. The
sign of addition is + , and is read plus. If you
have two books and put with these two books
three other books, you then have how many?
Five. This is addition, and put in abstract
form the process is 2-J-3 (two plus three) make
5* Or, to put the problem in strict arithmeti-
cal form, 2+3 equals 5, or

2 + 3 = 5

The sign of equality (=) may also be read
is equal to.

Abstract numbers (numbers which stand
alone without referring to quantities of any
substance or thing) may be added indiscrim-
inately. But when dealing with concrete num-
bers (numbers used to name the quantity of
something, as 9 miles, 8 apples, 3 books), only
like concrete numbers can be added. Thus, 2
books may be added to 3 books, but 2 books

12 ARITHMETIC SELF TAUGHT. PARTI

cannot be added to 3 cows or 3 buckets of water.
Not in arithmetic!

The child is taught addition by example, by
placing two balls with one ball, and making
three balls; by placing four sticks with five
sticks and making nine sticks; by adding the
five fingers on one hand to the five fingers' on
the other hand, and making ten fingers; and so
on. This process is clearly indicated in arith-
metic by 2+1=3; 4+5=9; 5+5=10; and so
on.

Familiarity with numbers soon enables any-
one to add such simple sums as these "in
one's head," that is, without the aid of sticks,
or fingers, or figures written on paper. Any-
one can add any two, or any three, simple
digits together without trouble. But as the
numbers become larger the problem grows
complicated, until at last, with many large
numbers, only human wizards, or "lightning
calculators," can perform such sums "in their
heads."

Only simple sums are written with plus and
equals signs. Longer sums are written in
columns, each number being placed directly
beneath the one above it, care being taken
by the writer to place each units figure in
the units column, each tens figure in the tens
column, and so on. Thus, when written
properly, a sum in addition has all the figures
at the extreme right directly under one an-
other in a straight vertical column, and the
next column just as straight, and so on. A
simple example (say 15+25 + 34) would thus
be written:

ARITHMETIC SELF TAUGHT. PART I 13

15

25

+ 34

74

The plus sign is usually placed, as shown, to
indicate the arithmetical process being per-
formed. A line is drawn at the foot of the
column, and the sum is written beneath it.
To add such a column one begins at the right-
hand single column of digits, and adds upward
(preferably), combining in one's head the
units digits (4 and 5 make 9, and 5 make 14;
or simply, when used to figures, 4, 9. 14). The
units digits of the sum (which is here 4. for
the sum is 14), is then written down in the
units column of the total. The other digits
(other than the units digit, here only one, 1)
are "carried over" to the next column, and
serve as a beginning (1 and 3 make 4. and 2
make 6. and 1 make 7), and the tens digit is
placed in the answer.

A longer problem may be made up of "un-
even" figures, so that the column, while even
on the right, will be "ragged" on the left.
Thus, to add 45, 672, S, 59, 7S2. and 5.282:

45

672

8

59

6S48

14 ARITHMETIC SELF TAUGHT. PARTI

The mental process for this example is, in
detail: 2 and 2 make 4, and 9 make 13, and 8
make 21, and 2 make 23, and 5 make 28: put-
ting 8 down, and 2 "to carry"; 2 and 8 make
10, and 8 make 18, and 5 make 23, and 7 make
30, and 4 make 34, putting 4 down, and 3 to
carry; 3 and 2 make 5, and 7 make 12, and 6
make 18, putting 8 down, and 1 to carry; 1
and 5 make 6.

The figures "to carry" from one column to
another may be written down by beginners, in
small numerals, at the tops of each column (as
shown in the written example, Page 16, Plate I,
No. 1), and added to the others when they are
reached instead of being added at the bottom,
so that the example in the illustration is added:
8 and 4 make 12, and 0 make 12, and 9 make 21,
and 6 make 27, and 8 make 35, putting 5 down,
with 3 to carry, and writing the 3 at the top
of the second column; 7 and 3 make 10, and
0 make 10, and 2 make 12, and 5 make 17, and
7 make 24, and 3 make 27.

The addition of decimals is just the same
as the addition of whole numbers. The
decimal points are placed carefully one below
another, and the columns of figures corre
spondingly (as in Plate I, No. 2). The decimal
point of the answer falls directly beneath the
decimal points above. The addition begins at
the extreme right, as always, the only differ-
ence being that in a sum of decimals the
extreme right column is not necessarily com-
plete or "even."

To add rapidly the student roust practise.

ARITHMETIC SELF TAUGHT. PARTI 15

Facility in addition, or in any of the subse-
quent arithmetical processes, comes only
through practise. The following exercises will
serve to aid the student, the correct answers
being given at the end of this book. When he
has done these the student may make up his
own exercises. He can check or verify his
answers by adding his columns first up, and
then dotcn (covering the answer during the
second process so that he won't be unwarily
led into the same answer) — if he gets the same
result both times, without trouble, his answer
is likely to be correct. (Zeros may be written
in, if desired, to fill out decimal columns.)

(Al) (A2) (A3) (A4) (A5)

4285

2766

22,764

22.756

.00015

2

421

4,877

.82

2.6123

464

80

7

3081.

4567.89

10

9

51,006

52.366

3.1416

55b

8988

91

481.2

282.

6000

11

53

5271.008

3.771

987

908

711

32.66

491.876

16 ARITHMETIC SELF TAUGHT. PARTI

PLATE I

7&1

100

34

.4(o
32.541

+ -2.S

f 2o7% 3/4.-]oi

(3) x '42s (4) * %i

3%SAY 211301-

ffl fpj (6) 453.00

3t4£> 325.12

ARITHMETIC SELF TAUGHT. PARTI IT

III. MULTIPLICATION

Multiplication is an expanded and somewhat
special form of addition — it deals with adding
a given number to itself a certain number of
times. Thus, if 1 is added to 1, we have 2,
or: 1+1=2; and, similarly, if we add 1 and
1 and 1, we have 3, or: 1+1 +1=3. This is
addition, certainly. But 1+1 may also be
expressed as 2 times 1, or, in arithmetical
symbols: 2x1=2; and similarly with 1 and
1 and 1, which may be expressed as 3 times 1,
or: 3X1=3. The times symbol (X) there-
fore shows that a number is to be taken and
added to itself a certain number of times. If
two and two make four, it is equally true that
two times two makes four also. And 2X5=10;
3X4=12; 4X8=32; 7x5=35; etc. Expressed
as addition, these appear:

5 4 8 7

+ 5487

+4 8 7

10 +8 7

12 +7

32

35

Since the numbers from 1 to 12, inclusive,
are often multiplied by each other* in the
various arithmetical processes and problems,
it is very important to know at once, by
memory, the product of any two of these low
numbers. The product is the answer obtained
when one number is multiplied by another

18 ARITHMETIC SELF TAUGHT. PARTI

(or, explained in terms of addition, when one
number is added to itself a certain number of
times, the number of times in each particular
case being specified by another number). The
complete list of the products of all the combi-
nations of these numbers (1 to 12) is called
the multiplication tables. Any number multi-
plied by one, as 1X2=2, is not changed in
value. So the tables begin with the 2's:

1X2= 2 1X3= 3 1X4= 4

2X2= 4 2X3= 6 2x4= 8

3X2= 6 3X3= 9 3X4=12

4X2= 8 4X3 = 12 4X4 = 16

5X2 = 10 5X3 = 15 5X4 = 20

6X2 = 12 6X3 = 18 6x4 = 24

7X2 = 14 7X3 = 21 7X4 = 28

8X2 = 16 8X3 = 24 8X4 = 32

9X2 = 18 9X3 = 27 9x4 = 36

10X2 = 20 10X3 = 30 10x4 = 40

11X2 = 22 11X3 = 33 11X4 = 44

12X2 = 24 12X3 = 36 12x4 = 48

Notice that either 9x4 or 4X9 are the same
thing. In any expressed multiplication it
makes no difference which number is put first,
for the answer will always be the same. It
is customary with larger numbers (each two
digits or more), however, to place the smaller
number first, as 42x526.

Only three of the multiplication tables have
been given (preceding). The student can dis-
cover the others for himself from the follow-
ing device — the product of 5 and 8 is secured
ly locating 5 in the extreme left vertical

ARITHMETIC SEL.F TAUGHT. PART I 1ft

column, and running across on that line of
numbers until the column headed by 8 is
reached. The answer is thus seen to be 40.
Or, going down the second column, the 2's
table is there complete, for, at 4, the top num-
ber, 2, is multiplied by the left number, 2;
and at 6, 2 is multiplied by 3; and at 8, 2 is
multiplied by 4; and so on. These tables must
be memorized.

12345678

2 4 6 8 10 12 14 16

3 6 9 12 15 18 21 24

4 8 12 16 20 24 28 32

5 10 15 20 25 30 35 40

6 12 18 24 30 36 42 48

7 14 21 28 35 42 49 56

8 16 24 32 40 48 56 64

9 18 27 36 45 54 63 72

10 20 30 40 50 60 70 80

11 22 33 44 55 66 77 88

12 24 36 48 60 72 84 96 108 120 132 144

When a product is expressed in arithmetic,
each portion of it has a name:

37 X 548 = 20,276

multi- times multl- equals product

plier sign pli- sign (answer)
cand

The multiplier tells the number of times that
the multiplicand is to be added to itself. This
could be expressed in addition by placing 548,
the multiplicand in the preceding example, in
a column 37 times, and then adding the
column. But, in multiplication, this is simpli-
fied:

9

10

11

12

18

20

22

24

27

30

33

36

36

40

44

48

45

50

55

60

54

60

66

72

63

70

77

84

72

80

88

96

81

90

99

108

90

100

110

120

99

110

121

132

20 AP.ITHMETIC SELF TAUGHT. PAKT I

548 548 3,836 548
X 7 X 30 +16,440 X 37

3,836 16,440 20,276 3836

1644

20,276

Since the multiplier tells us to take 548 and
add it to itself 37 times, suppose we do it first
7 times. Expressing it as in the first example
above, we now use our knowledge of the
multiplication tables (which must be learned
by heart by every student of arithmetic), and
perform the operation something like this:
7 times 8 is 56, putting down 6, with 5 to
carry; 7 times 4 is 28, with 5 carried over
from the previous multiplying, making 33,
putting down 3 and having 3 to carry; 7 times
5 is 35, with 3 to carry, making 38, and, since
this is the last operation, the whole 38 is put
down. The product of 7 times 548 is thus
3836. But we must take 548, not 7, but 37
times, so we now multiply it by 30 (having
already done 7 of the 37).

In multiplying by 30, we first multiply 548
by 0, and since the product of any number
and 0 is always 0 (if you have nothing, and
add it to itself you still have nothing, and
if you multiply it any number of times, you
still and always have nothing!), we write
the 0 down at once, directly under the 0
multiplier. Multiplying 548 by 3 is a simple
matter (3 times 8 is 24, 4 down, 2 to carry;
3 times 4 is 12, adding the 2 carried, is 14, 4
down, 1 to carry; 3 times 5 is 15, adding the
1 carried is 16, and 16 is put down since it is
the last amount), and the result of 548 times

ARITHMETIC SELF TAUGHT. PARTI 21

30 is obtained as 16440. To find the product
of 548 and 37 we now have only to add these
two smaller products. Doing so, we find the
answer is 20,276.

This is simple multiplication. As the
student grows more familiar with his material,
however, he will do his multiplying and adding
all at once. This is shown in the operation of
548 times 37 directly — 548 is first multiplied
by 7, as before, the first figure of the answer
being placed directly under the multiplier (7)
in the units column, and the second figure in
the tens column, and so on; 548 is then multi-
plied by the next figure of the multiplier (3),
the first figure of the answer being placed
directly under the multiplier (3). (It is thus
unnecessary to multiply by 0 here, for the
same thing is accomplished by the indentation
secured when the second product is written
directly under its multiplier.)

Longer problems are performed in the same
way. Write down the multiplicand, and,
directly beneath it, as though for addition, the
multiplier. Multiply the multiplicand by the
first (righthand) figure of the multiplier,
placing the first (righthand or units) figure
of the product directly under the first multi-
plier. Then multiply the multiplicand by the
second (from right to left) figure of the
multiplier, writing the first (righthand)
figure of the product directly under the
second multiplier. And so on. The final
product is obtained by adding the partial
products. A problem in multiplication, when
correctly performed by hand, will appear as
No. 3, in Plate I, Page 16.

22 ARITHMETIC SELF TAUGHT. PARTI

When decimals are multiplied, the process
is exactly the same. The position of the deci-
mal point in the answer is determined by
adding together the "number of places" (count-
ing from left to right from the decimal point)
in the multiplicand and multiplier, and count-
ing off this total (from risht to left) in the
final product. See No. 4. Plate I, Page 16.

Examples for practise (correct answers at
end of book) :

(A6)
135
X2

X34

(All)

(A12)
(A13)

(A14)
(A15)

(A9)

5712

X49 X307

(A10)

20037
X5074

< 66
'01X800

X.32
7 X3.6
$54.76X89

ARITHMETIC SELF TAUGHT. PART I 23

IV. SUBTRACTION

Subtraction is the reverse of addition. If
you have five cents, and lose two of them, you
have how many left? Three. Tw^o is thus
taken from five, leaving three. This is sub-
traction. Or, expressed arithmetically, the
problem is:

5—2=3

The sign of subtraction ( — ) is read minus.
All of the possible subtractions between the
numbers from 1 to 12 inclusive are probably
familiar to the student. If not, he should
practise until he knows them. For the sums
(addition), products (multiplication), and dif-
ferences (subtraction) of or between any two
of these numbers must be known as well as
the student knows his own name.

In subtraction there can never be more than
two numbers. They are arranged as though
they were to be added:

5478 minuend
— 4329 subtrahend

1149 difference (remainder)

The greater of the two numbers is the
minuend, and is always placed "on top." The
lesser of the two numbers is the subtrahend.
The answer is the difference, or remainder.
A minus sign is usually placed at the left, as
shown, to indicate the operation being per-
formed.

SJ ARITHMETIC SELF TAUGHT. PART I

>errorming the operation, we begin always
at the righthand or units column. The first
thing to do is to take 9 from 8, but to do this
we must "borrow one" from the tens column
to make the S into IS, for 9 is smaller than
8, and. we cannot subtract unless the minuend
is greater than the subtrahend. Taking 9,
then, from IS (and remembering we have bor-
rowed one from 7. making the 7 virtually a
6), we write down the difference, 9. Proceed-
ing, we now take 2 from 6 (one was borrowed
from the 7), and write down the difference,
4. Then 3 from 4 is 1, and 4 from 5 is 1. The
complete difference is then 1149.

The way a problem in subtraction should look
is shown by No. 5. Plate I, Page 16. No". 6, Plate
I. shows the subtraction of decimals. The
point keej in addition. But it

should be noted that the subtrahend (as in
the example shown on the plate) may some-
times have more decimal digits than the
minuend. When this happens the minuend is
filled out to the required number of places
with zeros (as shown). The subtraction is
then performed as usual. 1 being borrowed
from the lefthand digit for each cipher,
making the operation proceed: 8 from 10
leaves 2, and 8 from 9 leaves 1. and 7 from
12 leaves 5, and 2 from 4 leaves 2, and 1 from
4 leaves 3.

An example in subtraction may be proved
(that is, the answer may be tested for correct-
ness) by adding the subtrahend and the dif-
ference. If this gives the minuend, the ex-
ample has been done correctly. Thus, in No.

ARITHMETIC SELF TAUGHT. PART I 25

5, Plate I, 3646 plus 1832 is 5478, so the
answer 3646 is correct.

Exercises (correct answers at end of book) :

(A16) (A17) (A18) Take 32 from 566.

5411 245.006 (A19) Take 54.83 fom 465.

— 391 — 1.999 (A20) Take 2,800 from 3,009.

26 ARITHMETIC SELF TAUGHT. PARTI

V. DIVISION

Division is the process of finding how many-
times one number is contained in another.
Thus, if you have six apples and wish to divide
them evenly among three boys, what do you
do? You find out how many times 3 is con-
tained in 6 by "dealing out" the apples to the
boys — each receiving 2, so 3 is contained 2
times in 6. Therefore, 6 divided by 3 equals
2, or 6-f-3=2. The sign of division (-j-) is
read divided by.

The number (here 6) that contains the other
is called the dividend; the number (here 3)
contained in the dividend is the divisor*; and
the number of times (here 2) the divisor is
contained in the dividend is the answer or
quotient.

Division may also be expressed in arithmetic
in the form of a fraction:

6

- = 2 OR 6 f 3 = 2
3

The horizontal line then has the same mean-
ing and force as the division sign.

Division is seen to be a form of subtrac-
tion, for it consists in seeing how many times
one number can be subtracted from another,
with or without a remainder. Thus, in di-
viding 6 by 3 it can be said that 3 is taken
from 6 once, leaving 3, so it can be taken away
again, leaving nothing — or 3 can be subtracted

ARITHMETIC SELF TAUGHT. PARTI 27

from 6 twice, leaving no remainder, so 3 is
contained in 6 two times. Division is also the
reverse of multiplication, for if 6-r-3=2, it is
also true, reversing the expression, that
2X3=6. So in performing the division the
student must memorize the reverse of the
multiplication tables, and not only know that
4X5=20 and 6X7=42, but also that 20^-5=4
and 42--7=6.

Since multiplication and division are oppo-
site processes, one operation can be used to
prove the other. Thus, after obtaining a
product of a multiplier and a multiplicand, the
product can be verified by dividing it by the
multiplier. If this gives the multiplicand,
the answer is correct. Similarly, to prove a
quotient, the quotient may be multiplied by
the divisor. If this gives the dividend, the
answer is correct.

Problems in division between any of the
numbers from 1 to 1«2 inclusive may, after
the tables have been duly learned, be per-
formed in one's head. These are compara-
tively simple, and should be practised until
the student has them all at his tongue's end.

But there is what is known as long division,
which involves numbers beyond 12, ofteD of
many" digits. Short division is simply:

208104 quotient

divisor 4)832416 dividend

The divisor (here 4) is "put into" the first
digit (lefthand) of the dividend (here 8), and
the number ot times it is contained (here 2)
is placed above for the first figure of the

2S ARITHMETIC SELF TAUGHT. PARTI

quotient. Then 4 is put into 3, and goes no
times, so 0 is placed above. The 3 (or what-
ever remainder there may be) is now carried
over to the next digit, making it 32, so that 4
is now put into 32 and goes 8 times. Then
4 into 4 is 1, and 4 into 1 is 0, and 4 into
16 is 4. This problem happens to "come out
even," that is, without an odd remainder. If
the last figure of the dividend were 7, the last
operation would then be 4 into 17, which goes
4 times with a remainder of 1. To express

I divided by 4, the 1 is simply placed over
the 4, making a fraction, %, which is placed
at the end of the quotient, which would then
be 208,10414.

Long division is expressed in a similar way,
but it cannot be performed in the head of an
ordinary mortal, so more figures are needed.
A problem in long division is worked on Plate

II (No. 1), Page 29. Here 272 is the divisor,
4,765,932 is the dividend, and 17,521 55/68 is the
quotient. Since the divisor contains more than
one digit, it must first be put into a portion
of the dividend which will contain it. So 272
is first put into 476, the first three figures
of the dividend. A little examination shows
that it won't go twice, so it must go once
(since 476 is greater than 272, although not
twice as great). The first figure of the quo-
tient is therefore 1, which is placed above the
dividend, directly over the last figure of the
portion of the dividend into which the divisor
was put to obtain the 1 (here 1 is placed over
the 6). The divisor (272) is now multiplied
by this first figure of the quotient (1), and
the product (272) is placed directly beneath the

ARITHMETIC SELF TAUGHT. PARTI 29

PLATE n
, LZ£2df§

'272. 5%3i}t<n 1^20.000

fit) S<?3 ,~\ S23 1

S¥¥ t*' /i/ 2,2,0

fS)

(4)

77* if x %*&&
jk3ox t£ x f

>7xyx/ox^x?x^_^
A3 S3

30 ARITHMETIC SELF TAUGHT. PARTI

476, units under units, tens under tens, as in
addition or subtraction. Now 272 is sub-
tracted from 476, the remainder (204) being
placed directly beneath as shown. — The first
step is now completed, and we have the first
figure of the quotient. To proceed it is nec-
essary to bring down the next figure of the
dividend (here 5), and place it with the 204,
making it 2045. The divisor (272) is now put
into this figure (2045), and until the student
is experienced in division he may have to try
one or two possibilities before he gets the right
number of times that it is contained. (He
may try 8, but if 272 is multiplied by 8 he
will find that the product is greater than 2045.
He may try 6, but if 272 is multiplied by 6,
he will find that when the product is sub-
tracted from 2045 the remainder is greater
than 272, so it will go more than 6 times. It
must then go 7 times.) It is found that 272
goes 7 times into 2045, so 7 is the next figure
of the quotient. The 7 is placed directly above
the figure (5) which was brought down, and
the divisor (272) is then multiplied by 7, the
product (1904) being placed under 2045. The
1904 is then subtracted from 2045, the differ-
ence (141) being written below. — The second
step is now completed. The next figure (9)
is brought down to the 141, making it 1419,
and 272 is put into 1419. Finding that it goes
at least 5 times, 5 is placed in the quotient,
272 is multiplied by 5 as before, and the work
goes on until all the figures of the dividend
have been brought down in turn. After the
last figure of the dividend is brought down,
the division performed, and the subtraction

ARITHMETIC SELF TAUGHT. PART I 31

completed, there may be, as in this case, a
remainder This remainder is then placed
over the divisor (220 over 272) in the form of
a fraction, as shown. It is possible to "reduce"
this big fraction by dividing both top an*1
bottom parts of it by 4 — a process which will
be explained more at length under Fractions.
The remainder is thus 55/68 and is placed in
the quotient.

To simplify the process above, the student,
when dividing 272 into 476, may consider the
operation step by step. The first figure of
the divisor (2) goes into the first figure of
the dividend (4) twice. So 272 may go into
476 twice. Trying it, however, shows that this
is not so, for the other figures (72) won't go
into 76 twice. But this preliminary test serves
to show approximately how many times the ,
divisor may go into a particular portion of
the quotient.

It might be that a three-figure divisor (as
272) wouldn't go *uto the th^ first three figures
of the quotient even once (suppose 476 were
here 176). Since 272 won't go into 176, then, it
must be put into the first four figures, or into
1765. The first figure of the quotint is then
written over the 5, and the first figure to be
brought down will be 9, and the process goes
on as before. Similarly, in one of the remain-
ders, say the second one (204 with the 5 brought
down, 2045), if the figures are smaller than 272
(suppose the remainder was only 4, which with
the 5 brought down became 45), the divisor
won't go at all. The next figure of the quotient
is then 0 (placed in this case over the 5), and
another figure of the dividend must be brought

32 ARITHMETIC SELF TAUGHT. PAilT I

down (here 9, making 459), when the process
goes on as before.

To divide decimals, the divisor must be
treated as a whole number. If the divisor coir -
tains a decimal, the point must be moved to the
right of the last digit, making the divisor in
effect a whole number. But this must be com-
pensated for in the dividend by moving the deci-
mal point of the dividend the same number of
places to the right as the decimal point of the
divisor was moved. Thus, in example No. 2, Plate
II, Page 29, there were two decimal places in the
divisor (52.38), so the point is moved to the
right of the 8 (5238.), or two places to the right.
The decimal point in the dividend (8971.2)
must therefore be moved two places to the right
also (897120.), a zero being added to make the
required second digit. If there are no decimal
digits expressed in the dividend, the point is
moved just the same, beginning at the end of
the whole number, zeros being added for each
placed marked off. The process of division then
goes on as usual, the decimal point of the quo-
tient being placed directly above the decimal
point of the dividend.

Fractional remainders do not occur in the
division of decimals, for the quotient can al-
ways be "carried out" to any desired number of
decimal places. In the example shown, zeros
are added after the decimal point in the divi-
dend, and brought down as required, the quo-
tient being carried out to three decimal places
(171.271). There is, finally, a remainder of
2502 which is not accounted for. This is not
usually put in the quotient, but is discarded
for all practical purposes. If the divisor would

ARITHMETIC SELF TAUGHT. PARTI 33

go into this remainder 5 times or more, it
would be propel lu change the last figure of
the decimal quotient to one more than it is,
making the quotient 171.272. To show (when
desired) that the quotient is a little more than
the actual division, a minus sign would be
added (in this case 171.272 — ) ; and, similarly,
to show that it is a little less than the actual
division, a plus sign may be added ( 171.271 + ).
Exercises (correct answers at end of book) :

(A21) Divide 962721 by 3.

(A22) Divide 243612144 by 12.

(A23) Divide 4170 by 1.5 (until it comes out even).

(A24) Divide 172788 bv 308.

(A25) Divide 3 7 1 2 ;> 9 8 by C91.

(A26) 2'M1S -f- 32.

(A27) 241.4 ■'- 28 (to throe places).

(A28) 3098.22 -f- 2.96 (to four places).

(A29) .0065 -j- 2.3 (to five places).

(A30) 1 — 2 (decimally, to one place).

34 Aki'i LIME JGHT. PARTI

VI. FACTORING AND CANCELATION

The factors of a number are those numbers
«hich- if multiplied together, will give the given
number. Thus, 5 and 2 are the factors of 10,
for if 5 and 2 are multiplied together (5X2)
ihey give 10. Similarly, 8 and 7 are the fac-
tors of 56, for 8X7 = 56. Also 2 and 2 and 2 and
7 are factors of 56, for 2x2x2X7 = 56.

A prime factor is a number which cannot be
divided by any other number (except itself and
1). Thus 3 is a prime factor, for it cannot be
divided (without a remainder) by any num-
bers except 3 (itself) and 1. The prime factors
proceed from 1 as follows: 1, 2, 3, 5, 7, 11, 13,
L7, 19, 23, 29, 31, etc. Any number which is not
a prime number or prime factor may be di-
vided into two or more prime factors. Thus, 4
is not a prime factor, for its prime factors are
2 and 2.

To find the prime factors of a given number,
it is only necessary to divide, beginning with
t^e smallest prime factor which will go evenly
into the number. Thus, to find the prime fac-
ers of 13860:

2)13860
2)6S30

3)3465

3)115;-,

5)385

7 m
D

ARITHMETIC SELF TAUGHT. PARTI 35
The prime factors are therefore:

2X2X3X3X5X7X11=13860.

This process of factoring may be used in di-
vision to simplify the divisor and dividend.
When so used it is called cancelation, for it con-
sists in canceling out certain common factors.
When two numbers have among their prime
factors one or more of the same prime factors,
these duplicated factors are common to both
numbers.

Thus, if it is desired to divide the product
of 2 X 12 X 49 X 50 by the product of 3 X 14 X 35 X 40,
the problem may be written:

2X12X49X50 1

3X14X35X40 ~~ 1 ~
•

The process of canceling out the prime factors,
as shown on Plate II, No. 3, Page 29, thus brings
us to the answer, 1, in very short order. This is
certainly easier than multiplying out to find
that the divisor and dividend are equal. For di-
vision is rapid: 3 goes into 12 4 times, and 4
goes into 40 10 times, and 10 goes into 50 5
times, and 5 goes into 35 7 times, and 7 into 49
7 times, and 7 into 14 twice, and 2 into 2 once.
When the factors go once, the 1 is usually not
written. Since all the factors of the dividend
become 1, we have lXlXlXl=:l, and the same
for the divisor. And l-=-l is certainly 1, so
that's the answer.

It is often possible to greatly simplify a prob-
lem in division — even to perform the entire di-
vision— by cancelation;

36 ARITHMETIC SELF TAUGHT. PART I

210, 576
3 631,728
7 4,422,096
2 8.844,192 210,576

2 184,254 4,837
7 92,127

3 13,161

4,387

Thus 210,576 divided by 4,387 is far simpler
than the original problem. It so happens that
cancelation can be carried further if it is seen
that 41 is a prime factor of both numbers — but
to test factors all the way up to 41 is often as
arduous as it is to perform the indicated di-
vision, so, in canceling, factors beyond 11 are
not usually considered (unless, of course, they
are evident for some reason). The factors
used in the above cancelation are indicated by
<ie figures at the left.

Exercises (correct answers at end of book) :

(A31; P^ind the prime factors of (a) 3,080; (b)
735; (c) 1,188; (d) 7,812; (e) 33,096.

(A32) Divide 37X64X210 by 74X16X56X6.
(Simplify first by cancelation.)

(A33) Divide 108 X 1000 X 49 by 24 X 81 X 625 X 56.
(Simplify first by cancelation.)

(A34) Divide 169X42x121x150 by 26X39X77X
33. (Simplify first bv cancelation.)

(A35) Divide 194,040 by 166,320. (Simplify by
cancelation.)

arithmetic self Caught, parti &

VII. FRACTIONS

Fractions, as the term is now applied in
arithmetic, include only what were once dis-
tinguished as common fractions, and are ex-
clusive of decimal fractions, which are now
called simply decimals. That decimals and
fractions are closely related will soon be seen.
As before explained, a whole number or integer
may be subdivided into several equal parts —
these parts being fractions of the whole. Thus,
if a pie is cut in half, each half is % of the
whole pie — and % is a fraction. Similarly, if
the pie is cut in six pieces, each piece is 1/6 of
the whole pie — and 1/6 is another fraction. If
someone takes one piece, five pieces are left,
making up 5/6 of the whole pie — and 5/6 is
another friction. As seen, a fraction (as said,
exclusive of decimals) is written by placing
one number over another, the top number (or
numerator) telling hoAv many times the bottom
number (or denominator) is to be taken Th. s,
when speaking of 5/6 of a pie, *\e numerator
(5) names the number of pieces, sc to Ljeak,
and the denominator (6) tells what part each
piece is as related to the whole — tells us, indeed,
that the whole pie has been divided into six
pieces, and that we are here considering five of
them.

Although we certainly will not always be
dealing with pies, we cannot get along in arith-
metic without using fractions. We must be fa-
miliar enough with them to add, subtract, mul-

38 ARITHMETIC SEL<F TAUGHT. JPART T

tiply, and divide them, just as we can do with
whole numbers.

The more familiar fractions range from %
to 1/16 — V2 being a great deal larger than 1/16,
for the larger the denominator the smaller the
part that the fraction represents. It is clear
that something divided into two parts, so that
each is one-half, has larger parts than some-
thing divided into sixteen parts, so that each
is one-sixteenth. Similarly, y2 is larger than
1/3 and 1/3 is larger than 1/4. Tabulating, the
everyday fractions are:

1/8 (one-eighth)
3/8 (three-eighths)
5/8 (five-eighths)
7/8 (seven-eighths)

1/9 (one-ninth)
2/9 (two-ninths)
4/9 (four-ninths)
5/9 (five-ninths)
7/9 (seven-ninths)
8/9 (eight-ninths)

1/10 (one-tenth)
3/10 (three-tenths)
7/10 (seven-tenths)
9/10 (nine-tenths)

1/16 (one-sixteenth)

3/16 (three-sixteenths)

5/16 (five-sixteenths)

7/16 (seven-sixteenths)

11/16 (eleven-sixteentns)

13/16 (thirteen-sixteenths)

15/16 (fifteen-sixteenths)

The above are all what are known as proper
fractions because the numerators are all smaller
than their denominators. An improper -fraction
is a fraction in which the numerator is larger

1/2 (one-half)

1/3 (one-third)
2/3 (two-thirds)

1/4 (one-fourth)
3/4 (three-fourths)

1/5 (one-fifth)
2/5 (two-fifths)
3/5 (three-fifths)
4/5 (four-fifths)

1/6 (one-sixth)
5/6 (five-sixths)

1/7 (one-seventh)
2/7 (two-sevenths)
3/7 (three-sevenths)
4/7 (four-sevenths)
5/7 (five-sevenths)
6/7 (six-sevenths)

ARITHMETIC SEEK TAUGHT. PART I 39

than the denominator, as 5/4 or 7/3. Every im-
proper fraction is more than a whole number,
just as every proper traction is less than a
whole number. Thus, 7/3 is one whole number
(3/3), and another (3/3), making twro whole
numbers, with 1/3 left over — or 2 1/3. A wrhole
number with a fraction, as this 2 1/3, is called
a mixed number. It is read two and one-third.

A fraction in which the numerator and de-
nominator are equal is equ? to 1. Thus 3/3 — 1,
because both numerator and denominator can
be divided by 3 (cancelation), and wre have
1/1, or 1.

The terms (numerator and denominator) of
a fraction may be large or small. Thus,

numerator 2S91

denominator 3276

is just as much a fraction as 1/2. But we
learned before that this was another way to
express division, and so it is. For a fraction
is nothing more than an expressed division —
1/2 really signifies 1-^-2, for it is one (whole
one) divided into two parts, of which one part
is here taken. So, if we say 2/3, we mean 2-^-3;
one whole one is divided into three parts, of
which two are here taken. To read 2/3 as two-
thirds is really a short wTay of saying two one-
thirds.

REDUCTION OF FRACTIONS. — Fractions
may be changed in form without being changed
"in value. Thus, by cancelation, it is seen that
4/12-1/3. The fraction 4/12 is thus said to be
reduced to 1/3, for its value has not been
changed. The process of cancelation as applied

40 ARITHMETIC SELF TAUGHT. PART I

to fractions is sometimes called simplifying.

By cancelation it is seen that dividing both
numerator and denominator of a fraction by the
same number does not change the value of the
fraction. Similarly, if both numerator and de-
nominator are multiplied by the same number,
the value is not changed either. Turning the
preceding example about, it can be said that
both numerator and denominator of 1/3 are
multiplied by 4, giving 4/12, and, as we have
seen, the value remains the same.

A fraction may be changed from one denom-
inator to another, providing that one denomi-
ator is a factor or multiple of the other. (A
multiple of a number is a product of the num-
ber and some other number. Thus, 25 is a mul-
tiple of 5 — a product of 5 and 5.) Thus, 5/6
may be reduced to thirty-sixths (that is, to a
fraction with the denominator 36), since 36 is
a multiple of 6. Reversing, 30/36 may be re-
duced to sixths, for 6 is a factor of 36. To re-
duce sixths to thirty-sixths, the new denomina-
tor, 36, is first divided by the old (if it is
larger, as here), and the quotient (here 6) is
then used as a multiplier for both numerator
and denominator of the fraction to be reduced:

5X6 30

6X6 36

And 30/36 can be reduced to 5/6 by the already
familiar process of cancelation, or simplifying.
When 30/36 is thus simplified, it is reduced to
lower terms, for 5 and 6, the terms, are both
lower than 30 and 36, the former terms.

A fraction is said to be reduced to its lowest

ARITHMETIC SELF TAUGHT. PARTI 41

terms when both numerator and denominator
are such that they do not possess any common
factors — or when numerator and denominator
cannot both be divided by the same number
without a remainder. Thus 5/8 is in its lowest
terms, for 5 and 8 have no common factors.

A whole number or a mixed number may be
readily changed to an improper fraction of the
same value. Suppose you wish to change 9 to
thirds. How many thirds are there in 9 whole
units? There are 3 thirds in every unit, for
each whole can be divided into thr3e equal
parts, so there must be 9 times as many in 9
units, or 27. Expressing this as a fraction (im-
proper), we have 9 — 27/3. (Turning about,
27/3 may be reduced to 9.)

Similarly, 15 1/4 may be changed to an im-
proper fraction by changing the 15 to fourths,
and adding the extra fourth. If there are 4
fourths in 1, there are 15 times as many in 15,
so, to change a mixed number to an improper
fraction, multiply the whole number by the
denominator of the fraction, add the numera-
tor of the fraction to the product, and place
the final result as the new numerator over the
old denominator:

4 X 15 + 1 61

15y4 = = _

4 4

(Turning about, 61/4 may be reduced to 15 1/4
by dividing 4 into 61, and placing the remainder
1, over the denominator, 4.)

ADDITION OF FRACTIONS.— Fractions, to
be added, must have the same denominators —
that is, they must have common denominators.
It is possible to add 3/4, 1/4, and 5/4 at once:

42 ARITHMETIC SELF TAUGHT. PARTI

3 1 5 9

4 4 4 4

The numerators only are added, the resultant
fraction having the same denominator. For, in
principle, 3/4 signifies three one-fourths, and
5/4 five one-fourths, so if you add three to one
to five, you have nine one-fourths.

But to add 5/9, 7/16, 9/10, and 5/8, a common
denominator must be found. It is reasonable
to perceive at once that the product of all four
denominators will provide a common denomina-
tor, but this may be a great deal larger than
necessary. If the fractions are added by using
a common denominator larger than necessary,
the resultant sum will have to be simplified.
In arithmetic every process is made as simple
and labor-saving as possible, so it is necessary
to find not only the common denominator, but
the least common denominator (often abbrevi-
ated 1. c. d.).

To find the least common denominator of
5/9, 7/16, 9/10 and 5/8, we must examine the
denominators (9, 16, 10, 8) for a multiple that
will be the smallest multiple of each. We can
hit upon this by guessing, but this may be an
arduous method. So we do it arithmetically:
2 ) 9, 16, 10, 8

2 ) 9,

8,

5,

4

2 ) 9,

4,

.".

2

2 ) 9,

2,

5.

1

3 ) 9,

1,

5,

1

3 ) 3,

1,

5,

1

5) 1,

1,

i,

I

) 1,

1,

1,

1

ARITHMETIC SELF TAUGHT. PART I *.

The least common denominator is therefore
2X2X2X2X3X3X5, or 720— not a very small
one, to be sure. {Explanation: To find the
least common denominator of several unlike de-
nominators, place the denominators in a row,
separating one from another by commas. Then
divide by factors that will go into at least one
of the denominators without remainder, usually
proceeding with the prime factors from 2 up,
in each division bringing down unchanged any
denominators into which the factor-divisor will
not go evenly. Continue this until the last row
is entirely l's. The product of the resultant
factors will be the least common denominator.)
Using this least common denominator, we
now reduce each fraction to a fraction with
this denominator:

5 X 80 400

7 X 45 315

9 X 80 ~~ 720

16 X 45 ~ 720

9 X 72 648

5 X 90 450

10 X 72 ~~ 720

8 X 90 ~~ 720

And, adding:

400 315
720 720

+

648

+

720

400

315

648

450

450 1813

720 _ 720

1813
Thus, fractions with large or small denomi-
nators may be added. To add mixed numbers,
you must first change the mixed numbers to im
proper fractions, and then fir'1 the least com-

44 ARITHMETIC SELF TAUGHT. PARTI

mon denominator of those fractions. Or, in
more simple form, find the least common de-
nominator of the fractional portions of the
mixed numbers. Thus, to add 15 5/6 and 18 1/15
and 24 7/30 — find first the least common de-
nominator of 5/6, 1/15, and 7/30. A little ex-
amination shows that this is 30, so these frac-
tions are reduced to fractions each having 30
as a denominator: 25/30, 2/30, 7/30. Aud,
adding:

25
15—

30

2
IS—

30

7
+ 24—

30

34 4 2

57— = 5S— = 58—

30 30 15

Ex2)lanation: Adding the fractions first, the
total is found to be 34/30. This is an improper
fraction which may be simplified to 1 2/15. The
whole unit is therefore added to the sum of the
whole numbers, making the final result 58 2/15.

SUBTRACTION OF FRACTIONS. — Before
fractions with unlike denominators can be sub-
tracted one from another they must be reduced
to fractions with the same denominators. So,
as in addition, the least common denominator
must first be found. The numerators are then
subtracted, and the difference is placed over
the common denominator to form a new frac-
tion— the answer. Thus, 8/15 minus 9/60 is
(I.e. d. = 60) :

ARITHMETIC SELF TAUGHT. PART I 45
32 9 23
60 60 ~~ 60

Mixed numbers are subtracted in a way very
similar to that used in adding them:

9% —5% (I. c. d. = 24)

16

9—

24

15

24

1

4—

24

But the lower fraction, in subtraction, may
sometimes be greater than the upper:

16

9—

24

18

~~ 24

22
5— = 5 11/12
24

In this case, a whole unit, equal to 24/24, is
borrowed from the 9, making the 16/24 into
40/24. 18/24 is then subtracted from 40/24,
leaving 22/24,- or 11/12; and 3 is then subtracted
from 8, for 1 has been borrowed from the 9.
Similarly, if there is no fraction at all in the
minuend, a whole unit is borrowed from the
whole number, and considered as a fraction
with the denominator of the fraction of the sub-
trahend.

40 ARITHMETIC SELF TAUGHT. PART I

MULTIPLICATION OP FRACTIONS.— To
multiply fractions it is not necessary to change
them to fractions with a least common denomi
nator. The numerators are multiplied together,
forming by their product the numerator of the
answer; and the denominators are multiplied
together, and their product is the denominator
of the answer. Thus:

3 7 3X7 21

5 16 ~~ 5 X 16 ~ 80
Or, if a fraction is multiplied by a whole num-
ber, the whole number is regarded as a frac-
tion with 1 for a denominator (of may stand
for the times sign, as "% of 6"=%X6):

5 5 X 6 30 15

-X6=- _=—=_= 3%
8 8X1 8.4

If the product can be simplified, it is always
best to do so, as shown.

If several fractions are multiplied together,
it may be possible to simplify by cancelation:
1

118

15 11 32 1

16 30 44 ~~ 4
2 2 4

1
(In this example the l's are printed because
the numbers canceled are not crossed out.)

Mixed numbers are multiplied by first chang-
ing them to improper fractions:

19
3 5 38 17 323 8

I 6 ~ 5 6 ~~ ~15~ ~~ 15

3

ARITHMETIC SELF TAUGHT. PARTI 47

The example was simplified a little, as shown,
by canceling 2 from 38 and 6.

To multiply a mixed number by a whole num-
ber, the fractional part of the mixed number
is treated separately, and, if it is an improper
fraction, any units derived therefrom are added
to the product of the whole numbers:

7 29
38- 5

8 X 5-
X 4 9

_28

8
152

1551/2

1

161-

9

The principle is the same in both examples.,
but when the fraction is in the multiplier in-
stead of in the multiplicand, the simplification
of the improper fraction is accomplished at once

5 145

in the manner shown. Thus, - X 29 — ,

9 9

and this is simplified by dividing the numera-
tor, 145, by the denominator, 9, without more
ado. This may always be done where preferred.

blVISION OF FRACTIONS.— To divide one
fraction by another it is only necessary to in-
vert the divisor and proceed as in multiplica-
tion. To invert a fraction is to turn It "upside

48 ARITHMETIC SELF TAUGHT. PARTI

down," so that what was the denominator be-
comes the numerator, and what was the num-
erator becomes the denominator. Thus, 3/4
inverted becomes 4/3.

The principle of thus dividing is shown when
it is seen that fractions may be divided by
dividing the numerators or multiplying the
denominators:

8 8 -r- 4 _ 2

9 ~ ~~ - 9 ""9

8 8 8 2

'9' " 9 X 4 " 36 "" 9

Or, what has really been done in the second
ease, the whole number is regarded as a frac-
tion with the denominator 1, and is inverted,
and the resultant fractions are multiplied:

8 8 4 8 18 2

9' ~~ 9 ' 1 "~ 9 4 _ 36 _ 9

Similarly, with fraction divided by fraction:

3 7 _ 3 8 _ 24

5 ~ 8 ~ 5 7 _ 35

It should be remembered that 4 over 9, or
4/9, signifies 4-=-9, the horizontal line taking
the place of the division sign. Thus, a long prob-
lem in division, where fractional factors are in-
volved, may be expressed as in No. 4, Plate II,
Page 29, and be correspondingly simplified by
inversion of the fractions, and canceling. Thus,
considering each fraction separately, it can be

ARITHMETIC SELF TAUGHT. PART I 49

said that 5/11 is to be divided by all below the
long horizontal line. Since this is so, we can
invert and multiply, so we put 5 in the numera-
tor and 11 in the denominator of the "big" frac-
tion— considering all above the long line as the
numerator and all below it as the denominator.
Similarly with 4/7 in the denominator: we
put 7 in the numerator and leave 4 in the
denominator. Which amounts to transferring
only the denominators of the fractions. Notice
that the mixed number, 8 1/3, is first changed
to an improper fraction, 25/3.

Important Note: This cancelation cannot
be dene if a plus or minus sign occurs above
or below the line. Only when the numerator
and denominator are continuous (expressed)
products is cancelation permissible. When can-
celation, for any reason, cannot be used, the
result must be obtained by multiplying out the
numerator and denominator and performing the
expressed division.

Exercises in fractions (correct answers at end
of book):

(A36) Reduce (a) % to fortieths; (b) % to
fifteenths; (c) 25/100 to fourths.

(A37) Reduce the following fractions to their
lowest terms: 54/81; 27/9; 10/200; 55/99.

(A38) Change the following mixed numbers tr
improper fractions: 24%; 10 %; 52%; 19%.

(A39) Add %, %, % and %.

(A40) Find the 1. c. d. of 5/6, 7/12, %, %, %,
and %. Then add the fractions together.

(A41) Add 3/15, 9/20, and 11/30; add 11/16.
%, M, and 3/2.

tA4tf) Add 24 5/9 and 56 7/18 and 12%.

50 ARITHMETIC SELF TAUGHT. PARTI

CA43) Subtract: 3/7 from 13/14; 4/5 from 9/10;
49/50 from 199/200; 39% from 42; 14 5/9 from
1/11 from 18 7/11.
CA4.4) Multiply 15/16 by %; 35 bv 2/7; 24 1/6

(A45) Divide 12 by 34 ; 7/15 bv %; 12% by 5/6;
% by 3.

(A46) Slmnlify 22 X 14 5/6 X 11/16 X 72 ovei
33 9/11 X 173 X 11/12.

ARITHMETIC SELJT TAUGHT. PART I 51

Vin. DECIMALS

A decimal, as has been explained, is a frac-
tion expressed in some multiple oi: ten — tenths,
hundredths, thousandths, ten-thousandths, etc.
The nature of decimal fractions is very clearly
shown in the monetary system of the United
States, which is known as a decimal system.
The unit is the dollar, which is equal to 100
cents. Ten cents is then one-tenth of a dollar,
and is expressed decimally by .10, or, with the
dollar sign, $.10. Read decimally, this is ten-
hundredths, but one-hundredtn is one cent, so
this is ten cents. {Cent comes from the Latin
centum, meaning "hundred.") Similarly,
$.25 = 25c or twenty-five cents, and nay be read
decimally as twenty-five hundredths. This is
one-quarter of a dollar, so that the coin of thi'j
value is colloquially called a quarter.

Decimals are equivalents of common frac-
tions, which they largely replace because it is
much easier to perform operations with deci-
mals than with common fractions, especially
when the fractions are large ones. The equiva-
lent relation between decimals and common
fractions has already been seen in the above
example of twenty-five cents — .25=%. Any
decimal may be readily written as a common
fraction by placing the decimal (without the
decimal point) over whatever denominator it
needs to express it. Thus, .25 becomes 25/100
(twenty-five hundredths), which, reduced to
its lowest terms, becomes %,

52 ARITHMETIC SELF TAUGHT. PART I

It is possible also to change any common
fraction to a decimal by dividing (decimally)
the numerator by the denominator, thus:
.25

4 ) 1.00
Since 1 is smaller than 4, a decimal point is
placed immediately after it to separate it from
its decimal portion, which is entirely ciphers.
The division then proceeds normally.

There are a number of common fractions
with decimal equivalents which ought to be
^earned by heart by the student, for their oc-
currence is frequent and the ability to use
readily the fractional or decimal equivalent, as
may be desired, will be found of great assist-
ance in performing rapid calculations. A table
of the more important of these follows:

1 12

- = .5 - = .16-

2 6 3

1 11
-=.25 _-.12-
4 8 2

3 3 1

- = .75 - = .37-

4 8 2

11 5 1

- = .33- - = .62-
3 3 8 2

2 2 7 1

- = .6b- - = .87-

3 3 9 2

1 1

- = .2 — =: y
6 10

ARITHMETIC SELF TAUGHT. PARTI 58

Supposing, knowing the preceding values
very well, the student desires to find the cost
of 28 yards of cloth at 37 V2c a yard. He knows
at once that .37% is equivalent to 3/8 of a
dollar, so he takes 3/8 of 28 (3/8X28), and
finds the cost to be $10.50 — a much simpler
process than multiplying 28 by .375 or .37%.
Similarly, if a wholesale discount is 33 1/3%
(see Percentage), he can find it very readily
by figuring it as "1/3 off."

The use of decimals is entirely a matter of
thinking. The student should be able, for in-
stance, to reduce inches to the decimal part of
a foot. Suppose he is asked to change 8 inches
into a decimal fraction of a foot. He thinks
about ie for a moment, and decides t7iat 8 inches
is 8/12 of a foot, since a foot contains 12 inches.
Reducing 8/12, he finds it to equal 2/3, and
this, he knows, is equivalent to .66 2/3 — so 8
inches is equal to .66 2/3 of a foot. If he did
not know at once the decimal equivalent of the
fraction, he could easily find it by dividing the
numerator by the denominator:

.66%

3 ) 2.00

If it is desired to express a decimal as a
common fraction with a given denominator,
this may readily be done by multiplying the
decimal by a fraction with numerator and de-
nominator the same as the denominator desired
(a whole unit reduced to a fraction with the
given denominator). Thus, to change .2814 to
fifteenths:

i>4 ARITHMETIC SELF TAUGHT. PARTI

15 4.221

.2814 X — =

15 15

The fraction (15/15) is 1 expressed as a frac-
tion with the denominator 15, for when the
numerator and denominator are equal the frac-
tion is always equal to 1. The result may be
considered as approximately 4/15. This is prob-
ably accurate enough for all practical purposes.

Exercises (correct answers at end of book) :

(A47) Change to common fractions: .6; .98;
83% ; .39856; and reduce to lowest terms.

(A48) Change to decimals: 4/5; 9/10; 11/16;
7/12; 583/797.

(A49) Express .5614 in twelfths and .2674 in
sixteenths.

(A50) What decimal part of a yard is 2% feet?

ARITHMETIC SELF TAUGHT. PART I 55

IX. PERCENTAGE

Percentage is the rate per hundred, or the
proportion of something in a hundred parts —
as the percentage of alloy in a gold coin would
be the number of parts of alloy in 100 parts of
gold and alloy together, the total of both equal-
ling 100. That is, in percentage the basis is
100, which is the whole. Every whole is valued
at 100, and 1 percent (the sign of percent is
%, so 1%=1 percent) is therefore one part in
a hundred. (NOTE: Percent is etymologicaJ-
ly an abbreviation of the Latin per centum, "by
the hundred" or "per hundred" — hnt the cus-
tom of writing it with a period — per cent. — is
slowly giving way to the adoption of a single
word, percent.)

Since percentage is based on 100 as a wholo
one, it is very closely related to decimals. So
that since 6% signifies six parts in 100, 6%
may be written and thought of as six-hun-
dredths, or .06 — a decimal. (Notice carefulb
the difference between 6% = .06, and 60% =.60-
the percent sign, %, takes the place of the
decimal pcint, so that .06% would not be 6%,
but 6/100%.)

The United States monetary system may also
be used to illustrate percentage, since its basis
is a dollar divided into 100 equal parts. One
cent is, as we have seen, one-hundredth of a
dollar, and is written decimally as $.01. Since
it is one part in one hundred, one cent is also
1% of a dollar. Twenty-five cents ($.25) is
25% of a dollar.

Something else is thus seen very clearly.
Twenty-five cents is twenty-five hundredths of

66 ARITHMETIC SELF TAUGHT. PARTI

a dollar, twenty-five percent of a dollar, and
one-quarter of a dollar. Therefore 25% = .25 = 14.
And so on through the table given under
Decimals:

50 % = V2 16 % % = 1/6 87% % = %_

To find a certain percent of any numbe. it
is oDly necessary to multiply the number by
the amount of percent desired (expressed as
a decimal). Thus, to find 45% of 780, simply
multiply 780 by .45:

780
X.45

3900
3120

Sil.tt

And 45% of 780 is found to be exactly 351.
Sometimes, when the percent is equivalent to
some very common fraction (as in the table of
equivalents under Decimals), it is easier to find
the fractional part. Thus, it is easier to think
of 50% as y2— and 50% of 244 is obtained at
once as % of 244, or 122.

It is frequently desired to know what per-
cent of a certain number another number is.
This is exactly the same thing as finding the
decimal part that one number is of another.
For if we want to find what percent 43 is of
983, we must first know what decimal part of
983 the number 43 is. To find this we divide as
in decimals:

.043

983)43.000
39 32

ARITHMETIC SELF TAUGHT. PART I 57

Therefore 43 is about 4% of 983, for .04 i?
equal to 4% (the extra figure 3 being dropped,
since two places is enough for all practical
purposes).

If it is known that 43 is 4% of some number,
the number may be found by dividing the
number by the percent:

1075

.04)43. 4)4300.
Accordingly, 1075 is exactly the number of
which 43 is 4%. This can be proved:

1075
.04

43.00
Above, 43 was only approximately 4% of 983.
If the percent is figured as nearer .044, and
43 is divided by .044, the result is about 977 —
which is near enough to show the correctness
of the process. (If the results in such odd cal-
culations are carried out several places, the an-
swers will be more accurate — but two or three
places are usually considered enough when
working with decimals, unless, of course, some
problem specifically requests or requires a cer-
tain number of decimal places.)

The preceding paragraph is only a reverse of
finding the percentage, for 4% of 1075 is found,
as shown, by multiplying 1075 by .04. The prin-
ciple is therefore nothing new, for it was ex-
plained under Multiplication.

In problems it is usually desired to find the
net amount after an increase or decrease of
a certain percent of the whole. Thus, if a
town's population was 3,500 and has increased
18% — the present population may be found by

68 ARITHMETIC SELF TAUGHT. PART I

adding 18% to 3,500. Or it may all be done
in one operation by multiplying 3,500 by 1.18
instead of by .18 only, thus:

3500
X.18

3500
XL 18

280 00
350 0

280 00
350 0
3500

630.00
+ 3500

4130.00

4130
And the present population is 4,130.

Again, supposing that the same town's popu-
lation decreased 18%, the whole operation can
be done at once by finding the net value of
the present population in percent. If it de-
creased 18%, then its present population must
be 1 minus .18, or:

1.00

— .18

.82

Since the present population is 82% of what
it w^s when it was 3,500:

3,500 3,500 3,500

X.82 X.18 —630

70 00 280 00 2,870

2800 0 350 0

2870.00 630.00

The present population is 2,870, the same re-
sult being obtained either way.

But supposing the town's present population
is given as 4,130, and we are told that this is
18% more than it was ten years ago. How shall

ARITHMETIC SELF TAUGHT. PART I 59

we find the papulation of ten years ago? By-
reversing our process — adding .18 to 1, and di-
viding cur amount (4,130) by 1.18:

3500.

118)413000. (The decimal point has

354 been moved in divisor

and dividend.)

590
590

The population ten years ago was therefore
3,500. Similarly, if we are given the present
population as 2,870, and are told that this is
18% less than it was ten years ago, we can
find the population of ten years ago by sub-
tracting .18 from 1, and dividing 2,870 by .82
(the difference).
Exercises (correct answers at end of book):

(A51) Find 15% of 500; 32% of 784; 37%% of
240; 90% of 412; .2% of 1000.

(A52) What percent of 1000 is 50; oi 2986 is
57 ; of 394 is 55.16?

(A53) 5 is 20% of what number? 32 is what
percent of 256? 1,247 is what percent of 2,900?

(A54) If the daily output of a department has
been 275 toys a day, and new machinery in-
creases this by 24%, what is the present output?

(A55) Tf the value of a plot of land, of which
the purchase price was $4,980, has decreased by
12%, what is the present value?

(A56) The apple harvest of a large orchard
was 493 barrels five years ago. Due to lack of
care, this is 44% more than the harvest for this
year. How many barrels were picked this y^ar?

(A57) The number of savings banks in the
U. S. reached a high water mark in 1915 vith
2.159 brinks. The lowest number was in 1830, and
the figure for 1915 represents an increase of
about 5,897%. About how many banks were
there in 1830?

60 ARITHMETIC SELrF TAUGHT. PART I

X. AVERAGES *

The computation of averages is an important
fuaction of arithmetic, and a comparatively
simple one. Suppose that over a period of six
days, a man succeeded in winning between five
and ten sets of tennis each day. If his tabu-
lated results were exactly:

Monday 5 sets

Tuesday 8 sets

Wednesday 6 sets

Thursday 10 sets

Friday 8 sets

Saturday 6 sets

43 sets
He won in all 43 sets, during six days. What
was his average winning score per day? This
value called the average is obtained by divid-
ing the total by the number of items, or here,
by dividing 43 by 6. The average number of
sets he won per day is therefore about 7 (ex-
actly 7.16 2/3).

Exercises (correct answers at end of book):

(A5S) If the attendance of a certain grammar
school for four weeks of five school days each
was for each of the twenty school days in suc-
cession: 252. 251, 24S, 255, 256. 250, 253, 252. 252,
254, 256, 257, 258, 256. 253, 252. 249, 2-19. 248, 247.
what was the average attendance per day? If
a perfect attendance (total enrollment) was 258.
what percentage of perfection does this average
represent?

(A59) If a runner ran the 220-yard dash on
three consecutive days in 28 5/6. 27%, and 25 1/5
seconds, respectively, what was his average
time for the dash?

(A60) In New York state there were 144.469
marriages in 1920; 130,110 in 1921: and 138,242 in
1922. What was the average number of mar-
riages per year for the three years?

ARITHMETIC SELF TAUGHT. PARTI 61

XL RATIO AND PROPORTION

The ratio between two numbers is their rela-
tion to each other as to size. Thus, the ratio
of 3 to 6 is 2, for 6 is twice as large as 3.
Ratio may be expressed as a fraction, but it is
customary to use a colon:

3 4

- = - or 3:6::4:8

6 8

The first form is merely an equality between
fractions. The second form is an expression
in ratio, and is read "three is to six as four
is to eight." Ratio is thus a comparison be-
tween two numbers (two abstract numbers, or
two concrete numbers of the same kind). The
fractional form of expression receives great fa-
vor in higher mathematics, particularly in al-
gebra, for it has all the advantages of being in
a form readily used and simplified. It may be
read the same as the form with the colons
(note the double colon instead of an equals
sign, though "— ' may be substituted).

The terms of a ratio are the numbers com-
pared (3 and 6, and 4 and 8, are the terms of
the preceding ratios; the two ratios together
form a proportion). Proportion is merely an
equality of ratios. In a proportion the first
term of the first ratio, and the last term of
the last ratio, are called the extremes] and the
inner terms (second term of the first ratio, and
first term of the second ratio) are called tbe

62 ARITHMETIC SELF TAUGHT. PARTI

means. In every* proportion the product of the
means is always equal to the product of the ex-
tremes. (If expressed fractionally, the de-
nominator of the first and numerator of the
second form the means, the other two the ex-
tremes.) Thus, in the preceding example,
3x8r=6x4. It is thus possible, with any three
terms of a proportion given, to find the miss-
ing term:

3:?::4:8 ?:6::4:8 3:6::?:S 3:6::4:?

3XS 6X4 3X8 6X4
= 6 =3 = 4 = 8

This is what is familiarly termed the rule of
three.

Any ratio (since it is in effect a fraction, or
an expressed division) may be raised to any
power (see Powers and Roots, Part II), or its
terms may be multiplied or divided by the same
number, without being altered in value. Sim-
ilarly, the same root of each of its terms may
be taken without changing its value. If this
holds true for ratios, it must hold true also for
proportions, since proportions are composed of
ratios.

The principle of all calculations in propor-
tion is that, three terms being given, the fourth
is to be found. This "rule of three" is one of
the most important and valuable principles of
arithmetic. Thus, if 3 men earn $15 in one
week, how much will 6 mean earn? Why.
simply (a simple example anyway!) a matter
of proportion:

$1? : Amount 6 men earn : : 3 men : 6 men

ARITHMETIC SELF TAUGHT. PART I e*>

Reading, "$15 is to the number of dollars 6
men will earn as 3 men are to 6 men." And the
answer is obviously $30.

Exercises (correct answers at end of book):

(A61) What are the following ratios: 2:6;
48:144; 100:1000; 100:10?

(A62) Add a simple ratio in equality with
each of the above ratios, making of each a pro-
portion.

(A63) In the following proportions, find the
missing terms: (a) 14:28:: ?: 56; (b) 23; ? ::
3:9; (c) 16:64::22: ? .

(A64) If 7 men can unload a carload of keg*
of nails in two hours, how long will it take ?
men?

(A65) If a post iV2 feet high casts a shadov
3 feet long, at a certain hour of the day, ho\*
high is a flagpole that casts a shadow 38 feet
long at the same time of day?

64

ARITHMETIC SELF TAUGHT. PART I

ANSWERS TO EXERCISES

<A1, 12,303

'v s\ t ; i 3 , 1 8 a

(A3) 79,509

--■ - 8941.?*

rA5) 5351.29105

(A6) 270

(A,) 23.052

(A3) 279,888

(A9) 1,620,3-16

(A10) 101.6(57,738

(All) 1,907,994

(A12) 1.902,400,800

fA13) 75.04192

(A14) 7.625052

(A15) $4,873.64

(A16) 5.020

(A17) 243.007

(A18) 534

(A19) 410.17

(A20) 209

(A21) 320.907

(A22) 20,301,012

(A23) 2.7S0

(A24) 561

(A25) 4.166 7/9

(A26) 731 13/16

(A27) 8.G21 +

(A28) 1046.6959-f-

(A29) .00282

(A30) .5

A 31) (a) 2x2x2x5x7x11

(b) 3x5x7x7

(c) 2x2x3x3x3x11

(d) 2x2x3x3x7x31

(e) 2x2x2x3x7x197
'A32) 35-^-28 = 1%

(A 33) 7 -f 90=. 078 —

(A 34) 50

(A3 5) 7-s- 6=i 1/6

rA36) (a) r x>
(b) 10/15

re j w

.A37> 2/3; 3; 1/20; 5/9
**) 73/3; 21/2; 211/4;
U9/I

(A39) 16/8 = 2

(A40) L.<J.1>.=24; sum=

=-95/24 = 3 23/24
(A41) 61/t»0 = i 1/6*:

45/16 = 2 13/16
(A42) 93 11 18
(A43) % ; 1/10; 3/200;

2% ; 13 1/9; IS 3/11
(A44) 15/32; 6/35; 54%;

795 3/7
(A45) 16; 7/10; 14 7/10;

7/24
(A46) 11X11X3

(A47) 3/5; 49/50; 250-3
2491

6250
(A48) .8; .9; .68% ; .58% ;

.7315
(A49) 6.7368 4.2784

12 16

(A50) .875
(A51) 75; 250. SS: 90;

370 S ■ 2
(A52) 5%; about 2%; 14%
(A53) 25; 12V-%; 43%
(A54) 341 toys
(A55) S4.382.40
(A56) About 342 bbls.
(A57) About 36 banks
(A58) 252.4 per dav = 98%
(A 59) 2 7 8/45 seconds
(A60) 137.607 marriages

per \
(A 61) 3; 3; JO; 1 10
(A62) Possibly 2:6::1:3
4^:144::2:6
100:1000::2:20
100:10::20:2
'A63) 28; 69; 83
(A64) 1 5/9 bours
(A65) 57 feet