Document 190488

```NUMBERS,
o
AND
TO
HOW
BY
A
J^ATURAL
THE
METHOD.
PRACTICAL
COMPLETE,
SELF-HELP
AND
FOR
ARITHMETIC,
SCHOOLS
FOR
THAN
THEM:
USE
OTHER
PRIMARY.
THE
BY
JOHN
BROWN.
F.
""oj"";oo-
BosftV)^
PUBLISHED
BY
MAILED.
MASS.:
JOHN
POSTPAID.
F.
FOR
BROWN,
\$1.00.
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Ttpogbapht
JOHN
bt
J.
F.
8.
X\$92,
BROWK
Cushino
"
Co.,
Boston.
INTRODUCTION.
"o"
The
Natural
the
presenting
and
now
Method,
Science
formerly
First
and
I
as
chosen
to
differs
from
Arithmetic,
of
in
employed
four
numbers
foremost
have
important
treated
are
it, of
term
methods
respects
abstract.
always
as
:
J
There
is
number
nothing
in
is
mathematics
of
capable
The
used
reference
to
themselves
in
whether
A
phrase
relation
to
ter."
mat-
and
by Wentworth
:
is
employed
particular unit,
abstract
is
mathematics
pure
without
Arithmetic,"
phrase
any
which
quantity
again,
says
as
designate 'numbers
But
8, 10, 21.
all
of thing numbered
kind
the
to
bers
num-
is
is
or
mentioned.*'
not
"
are
This
"
And
abstractly,
Practical
A
"
of
species
figures."
explanations
number.
without
are
of
following
"Abstract
abstract
quantity
or
vocabulary
the
by
Webster
conception of number.
this
that
"
being expressed
"congjiders magnitude
Hill, has
in
novel
Concrete
number.
"
but
concrete,
all
Second^
fundamental
number
the
of
numbers
upon
numbered
Things
are
deduced
the
from
counting.
definitions
formal
Third
meaning.
is abstract."
operations
process
without
and
rules
omitted.
altogether
are
J
Fourth
geometrical
principles
reserved
are
the
for
science
J
of
geometry,
where
they properly
That
geometry
has
no
without
saying.
Yet
the
filled
with
Arithmetics
and
of
are
ideas, which,
For
in
the
instance,
in
place
The
problems.
has
always
geometrical
result
is
a
the
been
of
cases,
perplexity
and
is
that
to
go
otherwise.
tions,
defini-
demonstrations,
vagueness
majority
of
ought
arithmetic
an
practice
great
much
belong.
confusion
grown.
out-
never
in
exists
"
m
"
"
IV
INTRODUCTION.
-
regard
apple
between
the
be cut
into
may
but, the
foundation
employed
as
at fifteen
solved
be
have
in
in
place
no
understands
is
utterly incomprehensible.
In
and
the
find
within
the
of
fact, if
lems
prob-
problems
thoroughly
one
general principles
diflicultywith
little
came
for years,
Such
by algebra.
dent
Presi-
he
fightingover
arithmetic.
will
when
two-thirds
been
hours
numbers
he
for^, simple
indignant as
that
few
an
so
found
and
properly comes
of
anything
of arithmetic.
the scope
simplifyingarithmetic, geometry and algebra may
taught much earlier than they are at present, and students
thus
By
be
well
in the
of
use
book
This
treats
mathematics,
far
as
people
is to treat
all be
present
common
after
of arithmetic
far
so
in
general
requiring
would
come
fundamental
means
powers
b,
of
and
for
its
be
rr, y,
to
general
the
kinds
to
this
of
units
arithmetic
plan,
lars,
(dolbeing
and
formulae.
and
algebraic; then,
of geometry, beginning
principles
almost
it,
comprehension,
According
various
ing
teach-
larger than
no
principlesof
arithmetical
of
public schools,
one
difficult of
algebraic symbols
roots,
the
age.
anything
volume,
mastery.
etc.),the
in
under
more
applied
magnitude
early
an
know
to
connection
arithmetics, no
a,
taught
expected
in
at
and
the ideal method
But
moderate-sized
time
fractions, number
generalized by
be
may
one
more
no
it is
formulae,
only.
commonly
branches
school
bushels, pounds,
would
as
put into
extension
principlesof
equations and
several
the
It may
and
in the
grounded
and
so
consecu-
mere
relegated to algebra.
are
never
a
pure
applying them,
that
was
arithmetic, which
in
could
he
says
Harvard
being
tSemeGf^m,^m^
^
to
equal parts;
subjects reomring
reason,
isfactocv treatment
Eliot
line into
number
unit
numerical
analogous
an
a
An
piece of the apple and parts of the line are
symbols to illustrate \ and ^ ; and numerical
comprehended.
having been taught, are never
a
fractions,never
For
the arithmetical.
equal pieces,or
a
this failure to discriminat
to
geometrical and
principleof
tiveness, a piece of
Yet
directlytraceable
is
fractions
to
this
After
immediately
the
with
INTRODUCTION.
development
the
and
interest, arithmetical
and
topics,would
and
great condensation
If the
that
the
of
reception,
times
Mathematics
and
reasoning."
a
therefore
Of
has
rules"
the
and
lesser
after
not
a
; the
one
fault
If the
degree.
definition should
be
as
the landmarks
of
knowledge
then
should
we
To
illustrate
safer to say
This
or
avoid
having
:
This
that
is
into
do
to
make
in the
number
to
ideas
we
unite
"
to
not
Here
text-books
2
"
and
we
take
in the abstract
and
than
to
What
3 to make
conjure up
the
place
a
of
all
The
"
Or
that
how
of
process
do
unnatural
number
strict
as
a
and
that^ or
or
schools
public
like
more
or
counting
bers
num-
these definitions ?
count
we
until
uniting two
when
objects,
unintelligibleas
of
in the
objectswithout,"
to
is this
conveyed by
false and
by referringit
objects at
applies,these things are
5 ?
of
process
idea is
plete
com-
largely used
is the
and
possible.
as
definitions
two
now
sify
clas-
cannot
thorough
a
serve
consideration, and
it is to say
are
is to
"
We
simple example
a
equivalent number."
Unless
5 ?
mind
"
:
one
together is
How
first given
scientific
of
Mill,
is
the
reason
purpose
much
as
in
he
is because
Stuart
under
definitions
two
Massachusetts
numbers
things
respectivelyfrom
of
the
an
definition
good
a
reason
first have
we
but
formerly,though
of scientific classification."
things correctlyunless
a
him.
for
been
has
main
definitions,in the language of John
frame
multiplicityof
a
for this very
The
thought
to
framed
be
frame
and
tic,
arithme-
for
expected
as
explained,the
deferred.
basis
reform
exists
yet sufficiently
as
outlined,
later.
avoid
to
pupil cannot
been
subjecthas
**
above
work
a
be
not
Still the
same
is
it must
been
definitions.
indifferent
a
tendency
very-
indicate
to
as
Use, comprising
definition
"
allowing
such
appear
may
child should
A
"
be
General
and
good definition," and
late
should
geometry,
that
was
other
treatment.
comprehensive
more
Schools
solid
scheme
general
the present volume
and
idea
the
course,
the
age
Percent-
geometrical progressions,and
simplicityof
Common
for
old
due
ripe for
are
algebra,plane
The
in
follow
and
solids.
lines,surfaces,and
of
measurement
V
2 and
3
theory
thus
in
together
of
"
ideas
dealing with
realityit is only
mathematical
literal
term
expressions.
INTRODUCTION.
VI
I
as
metaphorically,
Taken
these
defined
merely
Definitions
false ideas
create
phrases
than
indulged
in when
should
be
think, that
favorable
divert
to
occupied
intended
to
apply only
consideration
is
are
And
how
of
to
do
to
value
problem,
and
rule is deduced, and
rule
the
directions
occasion
to
manner,
not
to make
even
He
The
the
in
is
repeat
be
will
presented;
which
with
the
rules
it
remarks
subject under
are
so
figuringneed
to do
the
own,
that
he
the
in
but
rule
pupil
by
to
absurd
be
work
under
is trained
If he
to
make
analysis
expected.
has
rules, nor
them
the
of
cation.
appli-
problems
certainlyconduce
the
to
different
a
without
for
the
ing
follow-
and
rule.
look
is unable
vogue,
that
not
to
other
particularproblem,
the
to
In
it.
problems coming
trained
his
long
where
a
is first shown
general principlesin
same
of
how
rules, he works
been
having
recitations; but
to
These
pupil
applied to
given. Thus,
recites
apply
been
that
where
is told
afterwards
there
the chances
has
are
wrought by
easier
background.
cases
then
Rules, definitions, and
which
former
metaphysical questions
performed by referringthem
are
prehensive
com-
are
memorizing
from
rules ?
are
words, general principlesare
a
not.
or
abstruse, like that of number.
what
a
The
opportunity
and
the
in
kept
much
mere
mind
abstruse
to
be
ought
that
mischief
so
an
the
to
which
likely to
requirements
not.
are
The
rule.
second, it is
;
so
that
nition
real defi-
a
first,being usually inexact, they tend
:
it is to
third, they tend
the
latter
is threefold
definitions
those
know
to
It is
complete?
it fulfils the
nition
defi-
true
we
be
to
else
of the term
are
general classes,those
two
exact, and
exception,the
how
is intended
less
or
is, a
the connotation
And
definition
a
whether
sense
of
are
and
to
like
something
to
trouble
the
But
means.
description,more
a
in the fornler
be taken
the term
thing
one
It illustrates
this.
which
well.
very
tells what
is,it
that
or
than
more
that
;
whether
the
do
qiiasi-definitions
may
is much
likenings of
mere
desideratum,
to
glib
ciency
profi-
INTRODUCTION.
After
becoming
definitions hy
of
And
methods
of
the
applying
student
be
may
science, the
the
in
a
framing
exercise.
good
very
understood, and
principlesare
them
be
perhaps
may
grounded
fundamental
after
so,
well
VU
general,specialmodes
in
in
the
of
ating
oper-
the
form
of rules.
That
no
all
but,
principle,
new
before
never
numbers
operations upon
been
far
so
carried
out
(deducingall operations/rom
of
but
obstacles
simple and
so
because
numbers
the
as
may
say
tainted
with
the concrete
of division
In
teacher, what
there
w^ere
have
been
question,
known
instructor
I must
"
high
the
the
to
as
say
school
writes
sorrow
enter
with
solution
of
Another
a
have
129
"
I
grammar
number
that
less and
I
crete
con-
the
are
their treatment
by
your
a
the
of
school
high
lower
the
experience as
use
after
course,
"
well-
a
"
from
less
the
year
abilityto perform
unable
am
to
year
to
get
them
to
pupils
the
use
of
our
ordinary
culations
calin
reason
any
problem."
teacher
repliesin
"
:
say
in arithmetic.
the
with
proficiencyin
the
in
course
with
From
"
"
do you
of
insuperable
whatever
evidenced
fallacy,as
figures of pupils commencing
completion
And
starting-point.
ciated,
appre-
fractions.
the
to
the
and
thought
not
contrary, all -previousarithmetics
writers
to
been
has
there
ability
desir-
the
because
not
scheme
indeed-
is
one
logicalcompleteness by
It is
one.
to
principlehas
the
aware,
its
to
been
as
"'
am
logicala
it has
the way,
in
I
as
reducible
are
the
to
pupils
schools
in
my
Class
There
entering this
On
a
different
IV, nearly
they
school
they
have
been
reviewing percentage
examined
the
weekly
marks
of
the
state,
in arithmetic
with
for
which
arithmetic
a
for four
left
of whom
studied
and
work
all
arithmetic, partly review
at
part
question:
same
June.
last
of years.
note, in
of
weekly
years
my
for
lesson
more.
eight weeks
our
a
in
This
when
I
assistants sup-
Vlii
ply
INTRODUCTION.
I found
me.
70 per
below
that
The
cent.
same
four
eightweeks
for
average
an
taken
37 of them
of 129
out
English,with
and
Some
64
lessons
taken
The
Latin.
up
fell below
70 per
Algebra, 40
English, 20
as
is
studies.
new
proportion of
cent
those
given below
whose
:
of 129
out
29
History,
each,
on
eight weeks
tory,
algebra,general his-
up
week
a
for the
average
""
"
"
"
129
129
*
4
Latin,
Here
larger than
is much
newtstudies
in
The
students.
young
upon
37
"
proportion of failures in arithmetic, an
find that the
we
"
which
the
of
age
average
usually
are
old
study,
counted
class
is
severer
fifteen
near
years."
It
safelysaid
be
can
any
time
spent upon
in the
could
years
which
count
get him
to
failure,as
This
teach
him
a
to
very
count
experiment.
is
and
young
The
different
and
And
idea
;
but
from
old, the
The
to
fain
child to count
Let
shouldn't
why
other
is
objects. Else why
him
the
it
was
who
was
a
but
doubts
be
this
We
the
expression?
other
no
four
two
has
is
hard
are
we
The
all
are
to
easy
to teach
this make
a
ever
revelation.
though
It
tried
and
We
so?
say
could
complete
a
one
husks.
numbers,
of
day
one
nature,
who
thing
some-
years
children.
compound.
compounded
me
me
eat
any
Some
half
a
any
of
In
for.
result
other
children
would
objects.
elementary, the
objects.
and
no
wander
to
prone
fingers.
experiment
was
alike, young
and
to
or
thinking. Noticing
to
me
can
child
his
count
similar
a
accounted
twenty, I called him
up
be
plaint
com-
was
teaching.
of
child of two
a
The
school
one
any
the
for
return
satisfied there
methods
strikingexperience set
a
ago
to
been
results
meagre
with
facility
the
to
the
public schools.
our
I have
years
no
confined
It is not
locality. For
way
is
arithffieticin
general.
is
one
there
quarter, that
from
fear of successful
without
one
the
act
number
ideas, number
of
INTRODUCTION.
the
In
originalpreface
from
on
numbers
Abstract
from
examples
first derived
This
Pestalozzi
Arithmetic
Teaching
Master
objects.
things with
such
no
them
upon
thing
abstract, unless
are
which
be
must
learned
deriving practical
as
abstract
the
have
been
practical."
are
Colburn.
and
is
followed
is
reasoning
sensible
idea of number."
operations
there
;
which
those
from
the
which
those
:
all
to
common
sons,"
Les-
:
and
practicalexamples
is
First
"
acquired by observing
abstract
an
Colburn's
is this passage
quality
obtain
little further
a
"
is first
this
acquainted,we
are
And
that
observed
Having
we
number
of
idea
The
"
Warren
to
published in 1821,
first
IX
the
by
The
from
following is
in
Primary
of
the
disciplesof
numerous
Methods
**
Schools," by Larkin
Boston
Dunton,
School.
Normal
of
Boston,
^1888:
When
"
they
children
distinguisha
can
words.
it
But
for numbers
what
number
for
a
be
give
to
which
the
to
the
are
of
word
not
a
idea
definite
objectsaround
*'
him, and
of
.
.
him
teach
mind
without
abilityto
which
knowing
the words
of words
In
count.
number
direct
his
which
express
take
must
teacher
the
of
should
we
words
knowledge
meaning
him,
to
ideas in the
Later
the
appropriate
signifies. The
mechanical
for the
or
the
know
sounds,
another
this
numbers,
of
of
little ; that is,
a
things by
children
the
or
mistake
to
numbers."
that
succession
a
things one
knowledge
child
as
usually count
can
of similar
numbers
merely
careful
real
few
frequently happens
stand
should
school, they
enter
the
.
order
words
attention
their
place of objects
without."
is the
This
**
good
to
"
schools.
the
the
he
method
And
what
which
has
it
is
of Warren
his work,
he corrected, and
the
the
must
Colburn, the
many
gross
still further
in
followed
now
In
accomplished ?
we
question intelligently
genius
performed
which
object
all
order
take
into
eration
consid-
skill
with
which
in
errors
fact
that
teaching
it is
only
INTI^ODUCTION.
X
owing
last ten
the
during
the
to
present
influence
but
dous
We
as
Method
in
educational
that
of
succession
a
counting
the
perceives
used
those
words
in
unconnected
can
words, such
with
their
relation.
tell.
can
His
Just
I would
and
that
would
doesn't
the
from
other
be
and
know
what
is
to
zero
it.
count
;
a
then
by
by
he
to
natural
After
tens
ones,
he
and
to
would
the
that
be
child
three, etc.,
perceives
sense
of
matter
a
child
takes
from
not
with
numbers.
place.
will
He
unless
doing
he
one,
can
count
zero,
with
of
is
But
instinct.
great-grandparents
taught
twenty
altogetherout
the
perceives, nobody
child
the
some
sense.
two,
one,
some
largely
in
counted
to
eating or walking.
objects,but
process
know
to
fact
in
the
it is
of
of,dog, etc.,
very
will
child
he
of
act
him, though
destitute
he
much
him
first have
backward
any
takes
A
child
a
jingles,like
to
words,
the
repeat
how
the
nonsense
to, the, cat,
is
In
?
merely
"
that
relation.
Thus,
utterly
grandparents,
he
him
of
greater part of their lives, and
the
or
to
stupen-
saying
related, provided
are
Unquestionably
parents,
as
idea
possible evidence
best
is the
the
in
manner
difficulty. The
great
so
that
as
This
counting.
not
in
be
most
three, etc.,
two,
one,
relation.
may
this
in
history.
themselves
learned
that
words
sense
words
is included
there
is the
really serious
one
seventy,
not
Judged
fifteen.
or
its
attained
back,
Arithmetic
in
sounds,** is
these
repeats
ever
ten
repeating
is any
But
granted.
to
our
told
are
only
object method,
schools, has
therefore, go
Object
failure
the
that
normal
must,
sixty,or fiftyyears,
the
the
of
We
sway.
manner,
fifteen years
or
will
by
count
fives
to
you
a
twos, threes,
twenty
colored
papers
Any
explanation
never
imagine
tell him
know
ones
hundred,
etc.
to
one
all
to
so,
he
needs
twenty,
and
and
he
as
to
teach
backward
"
INTRODUCTION.
Nursery jingles,such
XI
as
One, two, buckle
Three, four,
shoe
;
the door
;
my
shut
etc.,
conveniently and
be
may
to the
aids
Those
do
he is
before
who
things.
Number
express
our
rock.
regard
that
to
these
whether
we
tens, or
man
we
two
or
a
always
two
anything
general
;
two
theory is developed
"
The
is then
and
of
method
sooner
by
or
the
which
for the
two
is
mon
com-
is
name
same
thing,
feet,two
units,
to man,
common
;
rock,
;
is
That
theory of
for it
science,
"
the
be
here
Its
is
a
etc., we
name
The
numbers.
can
be
nothing
various
has,
simplicitymust
still further
arithmetical
consistentlyexplained.
:
or
simply
two, three, etc., is number.
method
there
But
later
which
logicallyand
prefacesays
pure
of
tree, tree
groups.
men,
That
fully demonstrated.
must
or
use.
applied to objectsfor practical
naturalness
trust, been
a
as
spoken
name
;
object
shall represent
rock, rock, rock
dealing with objectscalls
certain
precisely the
for one,
name
classifyand
represent that which
means
else.
tree, tree, tree
man
similar
or
with
to
a
the
page.
man,
to
name
sim^plytwo,
The
printed
deal
which
hear
we
the
we
see
objects
origin and
which
name
of
have
we
things. We
have
say
;
to
multiplication
means
the
by
means
all other
and
by
between
objecta
may
have
must
ThuSf
two.
else,
numbers
whenever
the written
We
three.
in
give
tree, rock,
man,
of the
the
counting.
practicalaffairs
one
call it to mind
man,
Our
is
We
it upon
two
In
ideas
in nature.
it and
teach
to
attempt
of number.
nature
but
taught anything
of
part
a
discriminate
sufficiently
not
see
division,and
as
Thus, he will learn
memory.
and
multiplication,
table
profitablyemployed, primarily
reason
also
why
It is the
Colburn
I
be
it
only
operations can
Warren
as
be
in his
INTRODUCTION.
XU
It ifiremarkable
"
of
that
which
examples
involve
operations when
these
all reduce
they
to
principle. And
performing them that they
the
same
In this
in the
impossible
According
findinghow
of
is not
of
Since
of the
one
four
one-fourthof 20
As
this
a
it is
is the
clearly,let
have
is this
follows
that
(5J + 5|
be
5f
+
cents.
either
3
equals
of 20
As
standpoint
20
the
"
a
tial
partioned.
men-
in
of
another,
ion
Thus, divisand
sometimes
equal parts
Fish
explained by
W.
Fish, 1875)
cost
of 1
cents,
(5 +
in
process
of the
as
his
of
is here
as
is
one
is the
cents
correctness
of another
Since
23
5f
cents
is
5f
we
of the
=
:
pencil?
it follows
that
5
20), and
5+5+5=
of the
been
we
of the
as
show
cite
the
of
4 times
four
say
result
cents
we
say
obtained
explanation
an
words, for
taken
23), and
have
4
one
cents
but
is to
To
be
cents
+
method
any
their
way
Daniel
juggling with
mere
the cost
5f
nor
thing
method
circumstance.
5f
he
number."
a
find
they
cents, what
of the
reverse
:
of
one
perfectlysound,
is
a
of
is 5 cents."
cents
accidental
an
do
4 times
equal parts
reasoning
thing
20
that
applying
own
is contained
number
is the
demonstration
division
is
taken
5 cents
such
any
sometimes
how
pencilscost
of
his
equal parts
Here
If 4
Ex.
"
condemns
the
Complete Arithmetic," by
C* The
is
And
number?
a
"
else.
we
artificial method
an
all, division
one
thing,but
one
something
them
to
of the
findingone
use
Neither
in
times
many
minds,
analyze
we
shall find
different ways
we
Colburn
incomplete manner,
and
or
when
if
explaining division, except
of
one,
ety
vari-
a
different form."
a
attempted
ever
perform
to
Indeed,
our
only
strongest possiblemanner.
have
disciples
an
take
in
are
only
Warren
paragraph
them
They
it is
but
perform
we
is able
and
operation
division, recognizes no
he
child, although
a
to
what
say
that
may
be
Now
into
equals
23
of 23
one-fourth
of
1
is
more
we
cents, it
equal parts
which
but
what
23
findingequal parts, the f
equal parts
one
fallacystill
20.
of
cents
cents
must
divided,
or,
INTRODUCTION.
of the 4
1
logically,
more
either
In
and
so
it is
case
we
are
itself.
which
But
number
expression of
an
attempting
to
if
with
is
start
we
be
can
or
almost
one
illustrative
our
23
in
4
is
be
abstract
The
cents.
4
into the
of cents
stration,
demonof the
one
pencils. Right
that
is not
tained
con-
division
irresistible,
and
demonstration
division
that
is
be
not
pencilsare
enter
conclusion
means
complete
and
difficulty
may
to
are
ure
proced-
carefullyselect
we
times
many
cents
explained by
a
the
unscientific
"
to
abstract, this
left to try except to find
nothing
of 23
here, then, is
always
how
cents, and, if
equal parts
cannot
find
cannot
there
else than
this most
that
ion
divis-
by
conception according
a
so
division
explain
and
3 is divided.
performed division,
a
anything
"
examples
We
suggested.
into which
equal parts
result is inevitable, unless
is the
Xlll
numbers
essentially
a
are
reversed
multiplication.
If
5
how
men
a
2
many
the
But
equal parts
but
2
or
their
have
start
we
be
can
with
is not
of
we
of
be
that
else
that
to
in
than
a
This
algebra
the
which
was
change
the
name
in
piece
2
correct
is
in
which
number
is inevitable.
result
an
a
not
mistake
no
to
of
are
arithmetical
of pieces^and
a
second
abstract.
formerly regarded
the
^
would
men
name
given.
does
The
not
one
as
'
^*
result,
There
and
that
always
are
the
^.
c^v^
?
fraction, is 1 of the
a
abstract, this
collection
days
days,
is wrong.
obtaining a
demonstration
a
numbers
call
now
have
we
in 2
obtain
we
is divided.
man
for it would
piece or
a
arithmetic, and
nature
of
in 4
conception according
a
anything
demonstration
What
a
succeeded
not
Here, then, is
fraction
books,
definitions
their
work,
no
in the
work
standpoint this
which
into
amount
do the
to
So, notwithstanding
figuring,we
if
it take
of work
capable of performing work,
men
enough.
But
from
do
could
a
directions
according to
man,
be
is
would
men
men.
man
perform
can
Following
2^
^
certain
part
alter the
science
INTRODUCTION.
XIV
merely
is
this
be
by
and,
that
he
If
would
this
shall
have
do
from
4
the
we
other
but
out,
existing
methods.
left,
4
and
of
for
which
unscientific.
them
take
we
there
find
how
away,
many
4,
5
are
of
part
a
left
thus
9,
know
we
;
away,
left."
teach
subtraction
to
come
in
and
algebra,
difficulties
has
enough
For
that
"
its
we
and,
operations
illogical
give
other,
what
arithmetic,
have
subtract
to
?
pointed
to
is
have
would
apples
way
when
Numerous
favor
5
arithmetical
should
otherwise,
or
the
?
he
have
is
we
left
many
counting
he
of
and
of
extension
algebra
apples,
9
how
find
To
into
has
would
apples
"
carried
Charles
If
"
and
presentation
any
reason,
cannot
9
outgrowth
an
it
the
be
and
been
said
to
show
method
Natural
judged
might
be
fallacies
of
inconsistencies
fairly
and
the
I
but
have
impartially
one
upon
merits.
J.
May,
1892.
F.
B.
CONTENTS.
PAKT
I.
NUMBEKS.
"
PAOB
The
Expression
Numbers
op
The
Arabic
System,
The
French
Method,
1
the
System
Roman
the
1,
Method
English
4
3
....
6
Subtraction
7
Multiplication
8
10
Division
between
Relation
Division
and
Multiplication
11,
.
The
two
of
factors
Assimilation
of
a
11
product
and
factors
multiplying
together
12
...
Remainder
Divisibility
13,
19
Factors
'Common
Greatest
Factor
Common
Least
20
Multiple
22
25
Fractions
Fractions
What
reduce
and
29
are
and
Multiplication
To
30
15
Numbers
of
30
.
Division
31
Subtraction
improper
an
32
fraction
to
a
whole
or
mixed
number
32
.
To
To
reduce
divide
whole
a
by
a
mixed
or
number
to
an
improper
fraction
fraction
33
Cancellation
Reducing
33
.
33
a
fraction
to
its lowest
34
terms
XV
cosTEyrs.
XVI
PAOB
Fractiom
Decimal
35
reduce
To
fraction
common
a
to
a
35-37
decimal
38
....
PowEBS
A5D
Boots
Sqcabe
Root
39
40
Root
Cube
Roots
45
other
thav
Square
the
ahd
the
Cube
50
....
Tables
Practice
52-55
PART
II.
APPLIED
"
NUMBERS.
Principles
General
Isterpretatioh
Fractiohal
of
States
Uhited
57
Results
58
Mohey
57
Mokey
Ekolish
Weights
aitd
59
Measures
60
Pbrcehtagb
68
Ihterest
74
method,
Accnrate
75
Iitterest
CoMPOUHD
78
Interest
Ahuual
Notes,
method
common
Drafts,
82
Checks
and
83
Payments
Partial
*
85
.
By
United
By
Merchants*
Equation
of
States
Court
Method
87
Method
89
Payments
90
Discount
True
Discount
Bank
Discount
Discount
Cash
Note.
Problems.
and
and
Present
Worth
96
Proceeds
97
98
Discount
Other
applicationsare
98
found
under
the
of
Miscellaneous
1
NUMBEES.
I.
PAET
"
":""o"-
EXPRESSION
THE
Numbers
called
characters
the
are
numerals.
viz.
Arabic,
either
expressed
commonly
are
NUMBERS.
OF
numerals
The
words
by
usually
by
or
employed
:
1234567890
characters
These
used
to
signify
called
are
number
the
figures
for
digits. Digit
or
which
of
one
these
is
also
character
stands.
With
the
only,
the
the
of
of
two,
than
three
the
The
counting
gives
of
name
name
Note.
exception
idea
each
of
zero
;
one,
etc.
one
;
cipher,
of
number
backward.
three,
of
which
the
than
two
;
is
two,
one,
Two
zero.
than
one.
one.
two.
0
nothing,
10
one
ten,
20
two
tens,
or
zero,
nought,
ten.
or
or
consequence
forward
Counting
,
2
as
same
stands.
necessary
one.
less
one
zero,
a
character
is the
character
is almost
backward,
the
to
characters
starting-point
counting
applies
ten
number
a
The
less
these
for
as
which
twenty.
1
is
one
less
AND
NUMBERS,
3 tens
are
HOW
5 tens
thirty
2"xQJiJty^
^
4 tens
QXQ
forty
6 tens
8 tens
are
9 tens
are
,
7 tens
2A
two
04
no
and
tens
The
four, or
twenty-four,
four, ox four.
ten
200
two
hundred,
204
two
hundred
and
264
two
hundred
and
1000
one
thousand,
1004
one
thousand
and
1024
one
thousand
and
2364
two
thousand
three
or
nothing to
the value.
hundred,
100
tens,
ninety.
seventy^
are
and
tens
eighty
^
sixty
are
^
or
THEM.
USE
TO
one
four,
sixty-four,
four,
twenty-four,
and
hundred
and
twerUy-threehundred
sixty-four^
sixty-four.
00
00
CO
0
d
d
o
o
'6
O
00
"'"
00
d
"
o
I
S
'^
"
O
2
'
"
00
rrt
OQ
fl
o
-2 T,
*-"Sd
"H_rj"
"-
rtCj^
rtflO
Sd"^
WehH
WehD
onaSonatgg'rJ
""
0
Eh
;
pq
WhS
.00
6, 3 7 4, 9 6 5. 0 7 2, 1 3 8
^
BilUons'
Gronp.
The
names
of the
groups
Millions' Thousands'
Qroupi
Gronp.
beyond
s.
Units'
Group.
billions
are
:
trillions.
nonillions,
quindecillions,
decillions,
sexdecillions,
quintillions,
undecillions,
septendecillions,
sextillions,
duodecillions,
octodecillions,
septillions,
tredecillions,
novemdecillions,
octillions,
etc.
quatuordecillions, vigintillions,
THE
EXPRESSION
expressionin
The
hundred
three
and
the
diagram
This
is the
French
method
one
third
of
other
every
each.
a
sixtythirty-eight.
is
and
A
States.
is the
different
usuallywritten
each
but
long number,
a
and
and
comma
trillion,
three
half-grouprather than a whole one,
clearlyrepresentedif
constitute
method
United
There,
figureof
Six
naming numbers,
the
in
:
hundred
hundred
one
of
plan prevailsin England.
after every
follows
as
nine
billion^
seventy-four
fivemillion^ seventy-two thousand,
3
NUMBERS.
OP
and
a
and
comma,
the groups
make
the
omit
we
of six
consist
ures
fig-
figures
Thus,
6,374965,072138
V
V
'
/
-^
y
Units'
MUlions'
^onp.
Groapi
and
hundred
In
seventy-fourthousand
and
million, seventy-two
sixty-five
and
thirty-eight.
the
French
trillion,a
method
thousand
million
millions;
beyond
millions
billion is
a
In
billions.
trillion,a
a
are
the
the
universallyfollowed, numbers
billions are quite commonly read as
The
hundred
English method
being, as
the
names
dred
hun-
millions
billion is
English, a
As
billions.
; a
a
numbers
is very
methods.
two
million
French
running
whole
Seventeen
nine
one
there
used, practically
States, although the
the United
thousand
thousand
a
million
often
not
between
little difference
In
hundred
hillion,three
Six
in
method
into
over
England.
is
as
a
the
Thua,
for 1,700,000,000.
billion,trillion,
etc.,
logical,
the
tri-million,etc.)signify,
(bi-million,
is the
second, third, etc.,powers
of
more
a
million.
10
units
make
1 ten.
10
tens
make
1 hundred.
10
hundreds
make
1 thousand,
etc.
4
NUMBERS,
AND
USE
TO
THEM.
The
base of the system
The
system is called the deounal system,
Another
of
way
ia 10.
expressingnumbers
The
In
I
HOW
Roman
is
System.
this system
represents
II represents two.
one,
V
represents^ve.
X
represents ten.
L
representsj^jfy.
C
represents
hundred.
VII
D
represents ^t;6 hundred.
VIII
represents eiffht.
XIII
represents thirteen.
M
represents
CLX
one
Ill
yi represents six.
represents
IV
Vim
seven.
sixty.
I, which
the
on
ished
dimin-
precedes it.
dials
of
than
IIII.
rather
is used
being
cloqks
represents nine.
IX
represents/or^y.
XC
represents ninety.
CX
represents
is used
hundred
one
and
rather
than
Villi.
ten.
represents /owr hundred.
DCC
representsfour
represents
CM
DCCCC
VI
M
the
watches, IV
XL
CD
The
Except
represents nine.
CCCC
MDCCXLII
and
the lesser value
represents four.
and
IX
hundred
one
represents
represents /our,the greater value V
by
nil
represents three hundred.
CCC
thousand.
one
represents three.
seve7i
hundred.
^ ,^
/y
-
hundred.
/:^^
^^
^
^'
2N
^
^^
^
represents nine
hundred.
^'^~
represents nine
hundred.
^^*^-V6"6f^a^
^^'^^
represents six thousand.
represents
one
use
quantitiesover
hundred
of the vinculum
which
\X? ^^7)
''
million.
represents seventeen
general
y^
it is
(
placed are
and
sand.
forty-two thou-
) is
to
to be
indicate
taken
that
together
THE
as
The
quantity.
one
used
EXPRESSION
in
similar
a
that
means
but
the
there
expressionof
The
whole
To
is
Roman
used
numerals
mark
to
in the
;
follows
used
100, millions
:
number
;
point
of
Eight
ings,
writ-
in the
use
and
clocks
They
are
in
entire
;
go
000,
number
billion
ffty-seven
Express
and
Words
one
^
:
14002537.
4
205200.
Six hundred
we
thirty-four.
857100000034.
3.
as
units
are
The
into
mentally
group
below, 34
and
hundred
off
each
857, billions.
17642587.
1.
and
its
dials
upon
first
we
Express
7.
and
chapters,sections, etc.
second
million^ and
2.
This
thousands.
for
occasion
any
figures each, naming
three
thousands
hundred
for
are
vinculum
in old books
found
are
long number,
along. Thus,
as
stands
[ ]
computation.
a
of
groups
longer
no
also
for
is sometimes
the
Romans,
over
quantity
brackets
and
numbers.
watches, and
not
Used
5
NUMBERS.
parenthesis ( )
manner.
of the vinculum
use
01^
in
5.
64029530203.
6.
68507332842630.
Figures
:
seventy-threethousand,
five hundred
twenty-one.
8.
and
Seventy-fivemillion, two
hundred
and
eight thousand,
forty.
9.
10.
and
four thousand,
Seventy-sixtrillion,
Eight hundred
four
11.
twenty-eight.
and seventy million, sixty-five
thousand,
hundred.
Five
hundred
forty-sixmillion, two
seven.
and
and
seventy billion,three
hundred
and
hundred
eighty-four thousand,
and
and
6
AND
NUMBERS,
Express
12.
in
HOW
THEM.
numbers
the
Romans,
USE
TO
from
one
to
further,
we
one
hundred.
If
6.
We
say
3 and
If
3 is 6.
and
8
do
can
2
are
we
13.
are
3, and
with
start
we
this
6,
by
we
count
step,
one
the
mean
eight
3 and
2 and
is
numbers
two
count
3 and
"
second
numbers
8
2
number
beyond
When
we
than
greater
5,
The
13.
are
5.
are
get
get 13.
we
5
is called
process
The
sign of
pliLStwo
Sum
equalsfive,
Sum
5 is called
and
324
6 units
465
6 tens
789
4 hundreds
867
6 units
386
4 units
and
2 tens
and
and
of 3 and
sum
5.
=
2.
8 tens.
are
3 hundreds
13
are
7 hundreds.
are
units
=
1 ten
1 ten
and
8
and
tens
and
3 hundreds
and
6
tens
15
are
test
and
the
and
result, begin
at
tens
=
5 tens.
8 hundreds
are
12
hundreds.
carried.
The
2 is said
to
be
The
1 is said
to
be carried.
1961
To
Three
9 units.
are
7 units
hundred
Sum
the
2
3 +
cross.
3
units.
1253
1 hundred
upright
an
the top and
6, 13, 21
8, 11, 16
3, 12, 15, 19
1961
down.
1
8
AND
NUMBERS,
hundreds,
of the
take
we
tens,
the other
TO
1 and
call It 10 tens.
the other
take
we
HOW
the 3 units, making
13
Now
THEM.
have
we
which
is
changed form,
mentally.
process
term,
the
subtracting;
reduced
1 ;
by
come
to subtract
O's.
What
this
As
:
we
0,
we
to
take
have
we
10
from
term
to
which
shall
we
have
to
7 from
3 from
from
in the
which
the
1 is of
above
higher
are
we
course
when
minuend,
9's in the
have
a
from
take
we
the
from
one
this
in
place
we
of
the
do, then, in the problem justgiven, is
take
take
always
numbers
through
the term
and, if O's intervene
of 4
the
When
along,without
go
to write
go
cannot
we
of 3 from
and
we
1166
to
have
always
we
to
4837
is it necessary
nor
9
599is
subtract
we
is unnecessary
practice,it
units, and
1, call it 10
Leaving
In
Leaving
units.
what
From
USE
take
3, we
6, 4 from
7 from
of 8 from
All
6.
this
actuallyperforming
we
0, 8 from
can
the reduction
see
9 ;
as
even
mentally.
MULTIPLICATION.
Sum
This
The
3 X
3 is
; 3
is
6
6, 6, and
6
Three
6
Or, 3 times 6
are
18.
sixes
are
18.
are
18.
18
Mnltiplioation.
is
sign of multiplication
6
6
18.
Three
an
oblique cross.
tivies six
equals eighteen.
called the midtiplier
; 6, the midtiplioand
; 18,
and
6, factors of the product.
=
the
uct
prod-
MULTIPLICATION.
3
Multiplicand
3
Multiplier
437
'
3
3x6
6x3.
=
6
Product
2622
3
3
18 is
6 and
of 3.
6x7=
42
6 X
3
=
18, + 4
=
22
18
6 X
4
=
24, + 2
=
26
Multiplicand
Multiplier
Product
of
multipleof
_3
Sum
ten
a
number
For, 10
10
Or
as
great
number
Multiplicand
Multiplier
thus
246
=
If
each
times
ten
:
we
before.
was
246
X
10,
annex
246
or
tens.
times
ten
a
if the
And
as
several
parts
great,the wliole
great.
as
100x246
For,
100
Product
=
246x100
=
246
=
246
X
=
246
thousands.
hundreds.
24600
Multiplicand
Multiplier
Product
it
as
are
246
X
cipher to a
number, each digitis removed
one
place
to the left,and consequentlyits value is
2460
times
the
246
246
For, 1000
246
X
1000
1000
246000
Multiplicand
7421
Multiplier
6x
40
246
44526
4x
29684
and
14842
Product
X
10 X
200
X
7421
44526.
7421
10
7421
29684;
29684
7421
X
tens.
14842
hundreds.
4672
3100
4672
14016
Product
Multiply 4672
by 31,
and
14483200
annex
two
7421.
29684
1825566
Multiplicand
Multiplier
4 X
ciphers.
10
JSUyiBEBSj
HOW
A!n"
USE
TO
MoldpUcand
436000
Multiplier
2300
THEM.
1306
872
Product
Haltiplj
by 23,
436
1002800000
and
five
annex
ciphers.
472163
Multiplicand
51002
Multiplier
944326
472163
2360815
Product
Disregard the
24081257326
intermediate
the
ciphersof
multiplier.
DIVISION.
If
start
we
with
7, we
subtract
can
2 three
remainder
subtract
2^
Remainder
3
the
5
2) 7 (3
with
_2
6
find
3
1
tract, and
2^
Remainder
1 ; or,
three
7
Remainder
of
we
can
7
2
X
by
of
times
also the
subtract
no
tracting
subcan
operation,
one
result.
many
is called
1
separate 2's, we
same
how
times, leaving a
That
we
is,we
can
remainder
longer. The
subafter
cess
pro-
Diyision.
the
dividend; 2, the
horizontal
line, like that of
is called
divisor; 3, the quotient; 1, the remainder.
The
sign of
division
subtraction,with
be omitted
place of
or
a
is
dot above
the dots may
the upper
short
a
be
and
a
dot below.
The
omitted, the dividend
dot, the divisor that of the lower.
line may
taking the
*
11
DIVISION.
Thus, 7 divided
by
expressed:
be
2 may
7^2,
7:2,
|.
12
0
Multiplicationbegins
of the
back
run
number.
same
this number
subtract
to
as
The
it is taken.
process
division
factors of
three sixes.
taken
the
even,
is the
process
some
product
a
the
Given
number
are
times,
and
means
of times
6, the number
find how
we
3x6
3 the number
taken, and
product 18,
of
similar.
not
which
times
many
it is
by reversingthe process of multiplication
; that is,by a
is equivalent to subtracting6 from 18, 6 from
which
number
Now,
are
as
long
of 18
and
on
so
as
we
do it,noting the
can
of subtractions.
suppose,
required
equals 18.
may
out
comes
6 is the number
the remainder, and
and
is division.
multiplication.
two
is taken
successive
0.
a
of
reverse
proceeds by
If, reversingthe process, we
times as we
it,we
many
Multiplicationreversed
Whenever
0, and
with
appear,
How
there
to
find
shall
is
we
no
the
do
6, we
number
this ?
direct
method
have
given
which
taken
Simple
by
the
as
which
and
18
3
3,
times
problem
it
can
be
performed.
have
We
seen
that
division
reduces
to
and
multiplication
to
counting. Every operationin
arithmetic
and
tion
subtrac-
is reditcible
12
to the
and
simple process
6
X
prodoeee the
6 is the factor
how
we
are
times
for the
of
purpose
equal parts of
the
the number
But,
6 X
as
18
as
find there
are
bdchvardj
or
before
as
3 ; and
up
six 3's in
so
for
of
3's
18, so
seeking.
involves
always
the
findingthe
to
number
which
taken
a
number
of
metaphor,
common
finding
in another, it may
certain other
a
of
process
is contained
certain
; or,
ploying
em-
of
findingone
number.
of
equal parts
of the three
; one
THEM.
problem.
consider
may
and
a
of the two
One
USE
product
same
amounts
convenient
a
the
number
one
of times
number
we
division
many
TO
By division,we
of 6*8.
So, while
solve
not
present purpose
our
HOW
of counting^either forward
will
coQntiDg
noticed, 3
he
AND
NUMBERS,
number
a
is called
equal parts, one-third
one-half of
of the
; one
equal parts, one-fourth ; two of the three equal parts, twothirds ; three of the four equal parts, three-fonrths ; etc.
four
In
consider
practice,we
rather
of neither
think
we
3 X
together,and
this
6'8
is
6
three
as
3's,but
nor
six 3's,or
6's, or
6
of
and
3 multiplied
for purposes
exact
suflficiently
computation,though in the final analysisthere
multipliedtogether."
thing as
be
can
of
such
no
'*
It is
enabled
otherwise
being
by
virtue
to
have
of this assimilation
several
factors
impossible. Thus,
be
all considered
as
factors
of
2
of
of factors
we
product, which
a
x
that
7
3 X
42,
at
42
=
;
would
2, 3, and
and
one
are
the
7
same
time.
Divide
2379
by
8.
8
8) 2379
and
297
3 remainder,
in
23
times, with
7 hundreds
77 tens.
tens
=
8 in 77 tens, 9 tens
50, + 9
=
59,
8 in
hundred
hundreds,
2
7 hundreds
remainder.
=
times, with
70
tens, +
5 tens
59, 7 times, with
7 tens
remainder.
3 remainder.
=
5
The
remainder
operation
it with
it with
the
the
has
the
three
dividend
been
not
which
upon
and
performed,
the
so
divisor, using the sign of division, and
quotient as part
For
part of the
a
division
of
express
Note.
is
13
DIVISION.
LONG
of the
present
and
until
divided
by eight,and
general result.
otherwise
we
put
Thus, 297f
.
explained,the expression }
similar
expressions in
a
similar
manner.
When,
as
in
recorded, the
above
the
LONG
1786929
Divide
by
DIVISION.
436.
436
436)1786929(409811^
1744
178.
will
not
By
partialmental
a
can
3924
in 1786, 4 times.
3689
4 X
3488
1744
201
42
436
in
429
hundreds
436
in
that
seen
1786
from
thousands
=
1, in 17,
division
436
=
429
=
4290
4292, 9 times.
tens, +
9 X
436
2 tens
=
will
in
it
go
420
42.
hundreds, +
hundreds.
429, 0 times.
=
or
1744.
=
hundreds
in
go
4292
436
are
Short Diyision.
is called
process
final results
the
example, only
=
3924.
4292
tens.
9
14
NUMBERS,
AND
HOW
TO
USE
THEM.
1786929
PROBLEMS
IN
AND
The
to
followingproblems
the
use
attained
only by
the tables furnish
If
can
one
things,neither
been
and
1; ten, 2;
;
the
row
back
;
dividend,
a
the four
so
next
practice
the purpose.
divisor,or
numbers
Test the result
cardboard,
well
shuffled
ten
in
were
hat,
taken
for the
first digit of the firsthorizontal
drawn,
for the
second
each
re-shuffled.
Table
to
were
digitof
drawing, the card
factor,is found
a
of
squares
squares
Practice
columns
digits,
having
100
the whole
a
of the
arrangement
small
100
After
on.
hat, and
From
13.
The
first drawn
and
Of
:
etc.
number
into the
purely chance
a
follows
as
up
the number
row
be
he do the others.
can
tables present
marked
is
of
way
perform such exercises readily,everything else in
follows easilyand
simply. If one cannot do these
arithmetic
The
practice. The kind
is required for
what
constant
which
DIVISION.
given as illustrations of the
Proficiencyin figuringcan
are
Tables.
Practice
PLICATION,
MULTI-
SUBTRACTION,
A, lines
just below it).
In
1
using
reverse
the
was
put
tables,if
9 for 0.
(Page 52.)
1.
to
zontal
first hori-
drawn
to be 0, substitute
No.
the
a
6
45459686,
(i.e.
direction.
and
a
16
therefore,divisible by 10 and
number
A
number
In
other
by
2, both
2.
Note.
words,
It
2
ifits
is
by 2,
mathematically
by
exact
and
of tens
up
0
by
2.
parts,
two
ible
divis-
necessitybe
units, be
divisible
the whole
hence
of tens
up
is divisible
by
number
8 is divisible
divisible
number
with
ending
is
3 units.
of
10.
It
of 10.
is,
Every
divisible
are
5.
by
8.
+
is not
00
for
0.
factor
is divisible
2468
by 5, so
3
Hence,
units
2460
=
tens,
=
by 5, a
ifits
b
0
multiple
a
its units.
2468
by 2, so
is
also
are
number,
any
number
any
0
+
by
that
say
There
-s-
with
ending
2468
Any
into
of
must
to
units.
number.
any
8 is not
Every
2.
number
number
divisible
are
number
and
of 10.
Hence,
units
the
separate
therefore,divisible by 10 and
A
factor
its units.
divisible
are
by
0 is divisible
number
by 2, a
part, the tens,
One
Any
also
+
by
we
THEM.
TO
If,then, the other part, the
parts
is divisible
of tens
is divisible
units.
and
tens
up
USE
HOW
AND
.NUMBERS,
by
2.
divisible
multiple
a
by
6.
It is,
100.
of
therefore, divisible by 100, also by 4, a factor of 100.
Every
units.
Hence,
number
A
as
is divisible
number
number
one
of hundreds
up
are
divisible
j
tens
and
8200
+
28.
16100
+
42.
ifits
^
by
its tens
+
and
unitSy taken
together
4.
by
"
8228=
16142
28
is divisible
42
is not
Any
by 4,
divisible
number
so
=
by 4,
ending
is divisible
8228
so
with
therefore,divisible by 1000
is not
16142
000
and
is
a
4.
by
divisible
multiple of
also
by 8,
a
by
4.
1000.
factor of
It
is,
1000.
and
number
Every
is divisible
number
taken
of thousands
up
togetheras
number,
one
37000
=
+
184.
+
188.
184
is divisible
188
is
divisible
not
6823
=
5000
5000
=
555
20=
X
9 +
5.
X
9 +
8.
2x9
+
2.
3=
by
88 +
+
2
5000
500
=
500
tens
50
=
50=
3
of
up
them), and
of
2 +
8 +
8.
500
=
tens
=
5 tens
=
50
nines
+
nines
+
5 nines
+
500.
50.
5.
the
(in this
9's
its
of
sum
there
case
digits(in this
18). Hence,
=
is divisible
number
A
For
by
3.
number
any
5 +
case,
+
divisible
is not
371*88
so
8.
by
3.
Thus,
555
20
+
is divisible
37184
so
by 8,
800
+
88
=
by 8,
units,
8.
by
divisible
are
37188=37000
are
hundreds, tens,
if its hundreds, tens^ and
S
by
37184
800
its
+
Hence,
units.
A
17
NUMBEKS.
OF
DIVISIBILITY
^
by
ifthe
sum
of its digitsis
divisible
9,
5823
is divisible
divisible
3746
number
by
is not
not
divisible
So, from
A
byS.
the
number
of its
digitsbeing 18, a
ber
num-
by
by
3.
of
sum
its
digitsbeing 20,
a
9.
of the
sum
will
number
9 is divisible
by 9, the
by
If the
remainder
dividing the
sum
9.
divisible
Corollary.
by 9, the
by 9, the
be
the
digitsof
same
a
as
number
the
be divided
remainder
after
9.
number
Any
of
9*8
are
divisible
by
3.
previous demonstration,
is divisible
by
d
if the
su7n
of its digitsis
divisible
18
is divisible
28146
number
divisible
HOW
by 3,
for
number
2 and
Since
A
divisible
not
number
Since
3
4
Take
of the
sums
its
of
sum
digitsbeing 10,
a
by
6
ifit is
by
12
divisible
2 and
by
by
3.
12,
=
such
alternate
0 is
(Note that
a
the factors of 6,
are
number,
any
digitsis 21,
of its
sum
3.
by
is divisible
number
A
by 3, the
is divisible
3 X
the
THEM.
3.
by
divisible
is not
1342
USE
TO
AND
NUMBEKS,
that
digitsis
by
multiple of
11.
0x11=
3 and
by
between
difference
the
some
multiple of
a
divisible
ifit is
4.
the
11.
0.)
Take, for instance, 8263817.
7 +
5 +
8 +
825381
7
8
1 +
28.
=
7 + 10+800+
=
3 +
2
28-6
6.
=
=
22
3,000+50,000+200,000
2x11.
=
+8,000,000.
7
7=
10
10=
800=
8
^x99+
Wx99+
3,000=
"
30
"
'
50,000=
500.
mx99'\%"my99+
200.000=
2,000.
8,000,000-^0,^X^^+80,000.
the
Strikingout
5x99,
we
etc.),
and
being
8
such
and
bdng
the
?0xR^+
20
(8
5 +
of alternate
20
5
^0Px^jr+8OO.
=
11
^x99+
be
7 +
a
8 +
3+2+10
+ 10 + 30 + 20
multiple 11,
5 +
8 +
+ 30+20
is
also
10 +
99, 30
X
=
30 +
other
the
so
1 +
30 +
by
We
set.
20
11.
by
X
99,
20; 7,8,5,
the
took
a
tens'
ber
num-
of the alternate
sums
3 +
8
^x^jr+
99, 500
digitsof 8253817,
be divisible
divisible
800
X
10 +
8 +
that the diflference between
diflfersfrom
if 1 +
80,000
left 7 + 8 +
set
one
digits should
and
2,000=
multiplesof
have
digitsof 10, 30,
500=
2 +
a
10 +
30 +
20
multiple of 11,
by 11, 7+8+5
But
1 +
3 +
+ 8
2 +
10 +
30
7 +
20
+
8 +
5 +
=
11 +
33 +
22,
10 +
30
20
8 +
parts into which
by 11,
Put
A
sums
in
is
alternate
of
sum
by 11,
in connection
If the
If
be
the
of
sum
same
after
subtracted
remainder
the
so
two
ible
divis-
each
11.
after
after
difference
11, this latter
the
number
Practice
numbers
between
Which
of the
the
two
taken
be
these
be
tens,
two
the
the
by
sums
number
units
the
will
by
11.
less than
and
sums
the
same
by
each
if the
by 11,
as
the
11.
Table
a-g
these
the
with
remainder
by
present, and
of
dividing
digits beginning
dividing the
11.
digitsbeginning with
digits beginning with
alternate
of the
the
by
2)
5 is
5.
=
digits beginning with
alternate
remainder
6
"
divisible
dividing the difference
the
dividing
11
3 +
(6 +
151.
271, page
the
22346
is 6.
for
alternate
the
11.
is not
are
following: 2, 3, 4, 5, 6, 8, 9, 10, 11,
9, and
The
are
by
^y
omitted
the
of
sum
after
as
from
Which
24.
11.
by
digitsof
22346
be
of
sum
From
23.
by 11,
if the differencebetween
(4 + 2)
set
Problem
the alternate
of the
remainder,
divisible
8253817
divisible
therefore
the
remainder,
sum
be
other
with
be greater than
11, will
divisible
11
by
following corollary may
tens, the
is
of alternate
set
one
divisible
units
quantity
is divisible
digitsis
of the
Corollary.
the
divisible
sum
later
up
a
general terms,
of its
The
,
separated
8253817
therefore
is
11, the
not
have
we
number
The
+
19
NUMBERS.
OF
DIVISIBILITY
first five numbers
No.
1.
divisible
of the
12?
a-m
are
divisible
by 3,
11, respectively?
FACTORS.
2x6
3
X
No
and
4
=
1 2.
2and6
-=
12.
3 and
other
1.
two
4
numbers
are
factors
of 12.
are
factors
of 12.
are
factors
of
12, except
12
itself
20
AND
NUMBERS,
2x3
2 and
6.
=
2x2
3
There
2
are
the numbers
12
3
is
the
are
3
10
12.
;
the
same
3.
2
30
5.
3
15
10.
=
2 X
=
2x
Or
the factors
are
Common
The
prime
factors
of 98
are
2, 7, and
One
2 and
one
Find
These
and
the
7
are
of 56
is the
another
12.
cannot
factor
method
be
so
may
7.
7.
and
greatest common
greatest common
numbers
3, except
both.
to
common
common/actors
14
or
Factor.
2, 2, 2, and
14.
before.
required.
are
=
2
60
6 X
of 56
7
of
2
=
factors
2 X
as
prime factorsof
prime
are
12,
=
result
factors
are
The
7
4
=
factors of 60 ?
Greatest
2 and
3
numbers.
the
are
prime
6
5
12.
2 X
compositenumber.
a
60
2, 2, 3, and
X
1.
prime
are
2, 2, and
What
and
themselves
2 and
=
2
which
numbers
no
12, 2
=
2 for 4 in 3 X
x
factors of 12
are
6
X
2 X
3 X
3, 2, and
THEM.
USE
of 12.
Substituting2
4.
=
factors
are
TO
factors of 6.
are
x' 3 for 6 in 2
Substituting2
2, 2, and
3
HOW
of 611
98.
factor of 56 and
and
be
98.
1363.
into
tors,
their fac-
21
FACTORS.
The
141)611(4
564
of 564,
564, it is
their
difference,141.
a
of
141.
;
the
Hence,
47
In
above
numbers
141
common
611
same
is the
by
of
remainder,
the
required is
611
47, it is
of
it is
47.
tor
fac-
of
47 is
factor of
either 47
564,
a
a
or
multiple
of their
factor
factor of 1222,
a
and
of 1222
of
factor
a
and
that 47,
of
it is
611
two
numbers.
tiple
mul-
a
factor
a
after the
are
factor of these
two
numbers.
affectingthe
result.
we
required to
omit
may
that 47 itself is
first division, we
of 611.
not
47, is the greatest
a
com-
factorrequired.
and
611,
factor of
1363, and
611*
and
a
and
example,
and
or
greatest common
factor of 141, but
of 141
of
factor
proved
now
factorof
a
factor
1363.
factor of
the
factor
being a
;
is
factor of 564
being a.
have
common
3 is
a
by the last remainder,
required i^ a common
factor of 141
a
being a
of their sum,
mon
given numbers
number
Therefore, the
of 611
We
number
The
being
611;
sum,
of
of 47.
47
141
factor
difference,47.
the last divisor,141,
141.
factor
But
two
of their
factor
of
and
of
a
a
;
and
1222,
requiredbeing a factor of 141 is also a
multiple of 141 ; being a factor of 611 and
a
a
Divide
47
being
122^, it is
of
number
The
factor
611
factor
a
1363
of the
the smaller
of
fac-
requiredbeing a
is also
multiple
141
Divide
number
of 611
tor
47)141 (3
a
by
the less.
1222
141.
divide the greater number
Mrst
611) 1863 (2
141
47)611(13
47_
141
Not
can
being a
common
the
greatest
the factor 3 from
Thus,
141
We
find the
have
that
tor
fac-
141 without
22
AND
ITUMBERS,
Find
HOW
TO
THEM.
USE
the
greatest common
factor of 1547, 1729, and
91 is the
greatest common
factor of 1547
13 is the
greatestcommon
factor
13 is the
greatest common
factor of
From
25.
What
are
prime
a-(7
?
and
1729.
1677.
1547, 1729, and
Table
Practice
the
of 91
and
1677.
No.
factors of each
1677.
1.
of the first thirteen
a-b?
26.
Of
27.
What
28.
Of t-w
29.
Of a-Cf
14-25,
is the
and
ac-z
LEAST
2 X
2 X
280
=
ji!x^x2x
each
COMMON
7 X
=
X
5
2 X
multiple
of
Multiple.
=
3640
13.
5x7.
364
of the first ten
lines?
is
a
S'lso contains
280
not
multiple
all the
of 364
prime
found
in
364
therefore,a multiple of
3640.
both
yz?
MULTIPLE.
of
364
and
wx
?
d-g^ and h-l^ in
364
factor of
greatest common
and
the
smallest
280.
It
is
number
their
factors
; it
280
that
Least
; it
is,
and
is
Oommon
a
24
AND
NUMBERS,
Find
91
the
least
is the
=
some
1547
=
17
X
17
TO
USE
THEM.
multiple of 1547, 1729,
common
factor of 1547
greatest common
1729
1729
HOW
and
1677.
and
1729.
91.
multiple of
91.
X
29393
=
=
least
common
multiple
of
1547
and
multiple
of
1677
and
1729.
Similarly,3791697
29393, and
hence
of
From
30.
lines
31.
What
16-25
Of
=
is the
least
common
1677.
1547, 1729, and
Practice
least
common
Table
c-e,
and/-A,
1.
multiple of
?
a-b,
No.
in lines 6-15?
ah
and
cd
in
25
FRACTIONS.
FRACTIONS.
.1
one
tenth.
.01
one
hundredth,
.001
one
thousandth.
.0001
one
ten-thousandth.
.0024
twenty-four
.0637
six
.4208
is Tea,d
"10
hundred
and
tenth,
one
thirty-seven, ten-thousandths.
hundred
forty-two
.
ten- thousandths.
and
hundredths.
ten
or
The
eight, ten-thousandths.
cipher
nothing
value.
the
to
00
OQ
OQ
00
a
OQ
a
^
o
o
"
"
2
a
-^
CD
a
?::
"
r4
00
a
"="
7
O
ITS
^
no
'^
a
oc
a
"""
fl
2
Billions'
Millions'
(}ronpi
The
whole
part
number
The
;
dot
Thonsands'
CJTonp.
of
the
the
2,
part
is called
at
the
""
a
Q
.2
a
O
13
:72
(3
0
""
"V
n
^
P4
138.03620759
Units'
(}ronp.
number
TS
'C
hWehhWJ^
W"hD
5, 07
"is
-^
rP
"D
WhpqWhSWhh
^
^"
g
2
'='
"
o
^
00
4, 96
a
0
o
.2
-2^
6, 37
"!-"
0
O
o
"
OQ
00
00
a
o
4Q
" r5
^
"4^
TS
"
00
CS
(}ronpi
at
the
the
left
right,
decimal
a
of
the
decimal
point.
dot
is called
fraotioni
or
a
mal.
deci-
26
Al^fD
NUMBERS,
THEM.
USE
TO
IN
EXERCISES
HOW
WRITING
AND
DECIMALS.
Express
To
last
long decimal,
a
to it the
give
of
each
the
hundredths,
The
first find
decimal
digitis
succession
of the
found
from
:
last
whole
a
digit.
the
by naming
the
decimal
second
in the
billion^six hundred
twenty-nine thousand,
and
three
one
hundred
column
number,
The
tion
denomina-
denomination
of
etc.
below
7nilUon, five hundred
and
and
point. Thus, tenths,
thousandths, ten-thousandths,
first decimal
of the
denomination
the
though
as
denomination
last
digitin
the
Words
in
:
teen
nineand
trillionihs.
sixty-tivo,
27
DECIMALS.
EXPBESS
WOBDS:
IN
34.
35.
719601529362.
.019601629362
71960152936.2
.09601529362
7196015293.62
.000601529362
719601529.362
.0061529362
71960152.9362
.01529362
7196015.29362
.000529362
719601.529362
.029362
71960.1529362
.9362
7196.01529362
.00000362
719.601529362
.00062
71.9601529362
.0000002
7.19601529362
.719601529362
Express
Seventy-one billion, nine
36.
hundred
one
six and
two
decimal
one
hundred
One
38.
three
and
:
hundred
fifty-twothousand,
nine
and
sixty million,
and
hundred
thirty-
tenths.
Seventy-one thousand,
37.
three
Figures
in
nine
million, five hundred
and
and
and
sixty
and
twenty-nine thousand,
sixty-two,ten-millionths.
and
hundred
thousand,
hundred
hundred
nine
and
twenty-two
and
decimal
seventy-one,
twentyhundred-
thousandths.
Six hundred
39.
nine
Seven
three
one,
and
hundred
hundred
decimal
and
one
hundred
thousand, three
40.
and
and
and
million, five hundred
and
twenty-
sixty-two, trillionths.
nineteen
five hundred
and
thousand,
and
sixty-two,millionths.
six
hundred
twenty-nine thousand,
28
AND
NUMBERS,
hundred
Five
41.
and
TO
HOW
THEM.
USE
three hundred
twenty-nine thousand,
sixty-two,billionths.
and
42.
Nine
thousand, three hundred
43.
Nine
billion,six hundred
twenty-nine thousand,
and
and
and
three
sixty-two,millionths.
million, five hundred
one
hundred
sixty-two, hun-
and
dred-billionths.
Twenty-nine thousand,
44.
three
hundred
and
sixty-two,
millionths.
hundred
Sixty-one million, ^ve
45.
three
hundred
46.
Decimal
47.
One
forty thousand
and
sand,
twenty-nine thou-
sixty-two,ten-billionths.
seventy-sixmillionths.
and
million, five hundred
hundred
three
and
and
and
twenty-nine thousand,
sixty-two,hundred-millionths.
ten-millionths.
48.
Two
49.
Three,
four
decimal
and
hundred
and
two,
ten-
millionths.
thousand, five hundred
and
three
Sixty-two million,
50.
and
and
hundred
eleven, and
twenty-eight
thousand
four
decimal
six, ten-thousandths.
61.
Sixty-two hundred-thousandths.
62.
Decimal
five
and
hundred
and
billion,six hundred
nineteen
twenty-nine thousand,
three
one
m^lion,
hundred
and
sixty-two, trillionths.
63.
Twenty-four
64.
Three
66.
One
hundred
hundred
ten-thousandths.
and
and
sixty4wo, hundred-millionths.
twenty-eight thousand,
and
decimal
eight hundredths.
66.
hundred
Twenty
and
thousand
fifteen
hundred-millionths.
and
forty-eight,and
thousand, three
hundred
decimal
and
eight
sixty-four,
29
FRACTIONS.
See
As
tenths
10
divided
by 10,
divided
is 248
.248
248
1000,
:
fraction.
neither
10
the
a
nor
number
which
the very
If any
will
decimal
have
be
form
76
=
numerical
numerator
.76, for 3
number.
300
=
hundredths,
or
of
fraction.
a
number
by
and
the
a
1000
3,
.003
X
3.
=
its denominator,
whole
number.
decimal
a
uct
prod-
Hence, the
its
fraction is
the
product would
only
may
a
particular
be
expressed
fraction in the
common
decimal.
hundredths,
.76.
Hence,
and
any
300
hundredths
fraction
has
a
value.
numerical
generalnam,efor
fraction is
Note.
=
form, and
a
more
being
numerator
value, for otherwise
into
being
value.
general. Any
changed
fractionis a
The
as
fractional
a
as
scale, it follows that
decimal
the decimal
fraction in
=
the
multiplied by
But
; the
1000,
or
;
is
; the
numerator
^.
or
nature
fraction
1
or
fraction.
.03
X
10,
of two
product
fraction,it
numerical
a
numerical
a
be
Thus, I
must
the
common
may
of
100
be
value.
of the
the
of the
3,
=
the
:
7 is divided
decimal
a
is the denominator
nature
decimal
has
no
case
.3
is the
is called
general
same
decimal
a
10, 1
-h
by which
is not
as
fraction has
10 X
a
which
terms
are
the decimal
A
number
it is divided
From
A
the number
divided
denominator
-^-4
But
1
by 10,
expressed 248-^
be
may
the
is of
general nature
same
expressed
and
by 1000,
multipliedtogether,J
The
in
be
may
-j^^^. i
or
decimal
lO's
and
divided
tenths
.1 is 10
equal 1,
13.
Page
Note,
It is
a
be established
numerical
is
quantities
number.
number.
only by
Being
qv/intity.
an
a
process
extension
and
of evolution
of
primary
developed before
that
a
fraction
has
become
ideas, underlying principles
this is
possible.
30
NUMBERS,
of
idea
The
division
may
14
number
TO
considered
3
-5-
out
3 X
THEM.
been
now
even,
the
as
4f
-
USE
having
to come
decimal
first three
The
a
as
be
now
may
HOW
hereafter
division
and
fraction
a
AND
and
multiplication
so
of each
reverse
4|
developed,any
other.
14.
=
placesbeing tenths,hundredths, and
tenths; .03, three
three
hundredths;
.003, three thousandths.
.3,.03, and
But
.003
^^ ^^^^
Y(ftnF-S^" A
and
the
are
Y^^3 three thousandths.
three tenths
three
thirds; f, three halves,
2 ; or,
denominator
24
.
^
three
fifths;|,
also be
may
fourths; f,
by
and
half; denominatarf
one
three
3,
numerator
divided
half divided by three
one
and
two
numerator^
or
f
"^, y^,
as
three hundredths;
j^-^,
;
as
and
two
three fractions
same
two
three
2.
thirds,
and
two
thirds.
^
^
fraotions.
oomplez
^
are
^
3
f
^
is
simplefraction.
a
is
2^
f
3|
of
mixed
a
4- is
nmnber.
not
it has
fraction,as
a
not
a
and
numerator
a
denominator.
-
"
-
IS
fraction.
a
6x7
Note.
The
this
evident.
this
idea, the
their size
numbers
a
Fraction
means
compared
the
Applied
is of
denominator
as
fragment
fraction
pieces.
or
quantitiesthat
is
just as
much
a
fragment
denominates
with
Whole
the
means
or
the
entire
a
used
make
piece. Following
numerator
thing,as
out
pieces to indicate
the
names
whole;
an
or
terms
numerates
or
from
distinguished
piece.
to
nitudes, and
geometrical origin. The
the
numbers,
can
be cut
piecesare
an
entire
these
into
have
terms
piecesare
no
or
geometricalquantities,
geometricalfractions.
quantity
as
a
significance.The
whole
A
number
numerical
is.
only
m^g-
fraction
32
AND
NUMBiyElS,
iH
The
least
multiple of
m
5
7
MH-
=
m
4_3x4_12
Fractions.
of
5x7
if H
m
-
and
364
uii
=
+
Ml*
'
is 3640.
280
mi
-
Utt-
=
a^v
=
35
t 4x^, "x4=^-^,
For
THEM,
^^.
common
3
USE
TO
Subtraction
and
and
HOW
and
=
7
7
seventh
one
of
^^
5
5
5
_3x4
"5x7*
A
fraction
whose
is termed
less than
To
an
improper fraction
its denominator
Reduce
This
the division
is the
is
a
Improper
an
OR
Perform
equals or
numerator
ordinary
Mixed
;
case
proper
; one
Number.
=
whose
2^.
of division.
to
nator
denomi-
numerator
fraction.
Fraction
thus, -1
the
exceeds
a
Whole
is
33
FRACTIONS,
Keduce
To
Whole
a
Mixed
or
Improper
Reduce
is the
This
the
2J-to
divisor
The
the
2)^2
=
expressed,f
Reduce
The
2
to
divisor
dividend
Or, since
1
+
is 2
l
2
X
22
=
to
the
4.
=
both
5
having
quotient 2,
and
divisor
2
2 for its denominator.
with
no
remainder,
Ans.
f
=
|.
.
fraction.
^=l
Multiplying
given
1.
'
2xf
=
simple
a
Dividend
fraction
improper
an
have
the remainder
5.
=
We
case.
.
being 2, and
rl
Reduce
previous
2, the quotient 2, and
dividend
thus
are
fraction.
of the
reverse
an
to
Fraction.
improper
an
Number
2j_j.,
of the
terms
fraction
^- by 3,
have
we
5x3
2
5 X
__
11
So,
to divide
denominator
and
by
a
15
3
_
2x11
22
invet't the
fraction^we
fraction^taking the
the numerator
for a numerator,
for a denominator,
multiply.
Cancellation.
is the
What
we
will
the
of
product
of
performing
merely
write
the
V-, ""
the
27, |i, and
several
various
operationsafterwards
;
^
operationsas
numbers
thus,
as
?
we
go
along,
factors,and
form
per-
34
AND
NUMBERS,
;g
gg
X
USE
TO
THEM.
3
3
^
HOW
X
S!jr
X
11 X
1287
13
.
7x^x;^X^^
dividing
Since
does
number
same
of the
in 16
goes
by drawing
4
a
the 16, and
writing 4
27, 3 times
; in
cancels
is
Reducing
the
factor
16.
a
convenient
a
Fraction
4.
321
We
13
16.
4
factor
4287
the
cancel
the
36
3, 3,
are
denominator.
Lowest
its
a
remain
; 9 in
of
method
and
that
the
through
is left of
7 in the
to
line
which
and
way
cancel
we
39, 3 times
factors
4 is
so
in
The
the
4 of the denominator
the 4, another
The
by inspectionthat
see
cancel
the
by
fraction,we
times,
4
through
36, 4 times.
the
to both.
numerator
above
denominator
of
common
in the numerator,
13
Cancellation
We
are
line
the 4 left of the
11, and
and
alter the value
that
^'
7
numerator
not
all factors
strike out
factor
both
Qog
_
_
Terms.
both
of
both
are
factor
terms,
so
divisible
3.
107
we
3.
by
be
cannot
"
107
divided
/Ju
11
X
by 2, 3, 4, 5, 6, 7, 8, 9, 10,
11
121,
=
number
a
greater than
107
"jmf^
A^^
If 107
=
factor
a
-lAnQ
corresponding factor
than
1429
11.
has
107
no
greater than
would
factor
If the
be
all the
factors
common
of the
found
not
and
numerator
and
the
11, its
less
less than
11.
and
of 1429.
factor
being a
the
cannot
greatest
denominator, and
have
we
the denominator
by inspection,find
divisor of the numerator
common
divide
both
this number.
by
To
the
divide
second
one
number
is contained
tte comparative value
is
factors,107
107.
to be
have
Hence, 107 is a prime number,
cancelled
11.
or
f
of 5.
by
another
in
of the
the
is to find how
first.
numbers.
The
3
many
quotient
-^
5
=
f,
times
indicates
and
so
3
2 is what
What
To
number
what
which
3 times
We
to
part of 7 ?
is
part off
find
is the
given
find the other
Ans,
^?
part
Ans.
number
one
factor.
What
the
and
This
by
is
The
of what
^
number
=
required
division.
12h-3
=
4.
?
product being 16,
^
?
are
Ans,
16
the
part.
number
is the
^.
|f.
=
divide
we
product,and
do
we
|xt
=
it is the
of which
one
is 12.
factor
one
|-f-f
of another^
is
part by the
number
certain
a
have
35
FRACTIONS.
DECIMAL
and
factor
one
16xj
^, the
other
factor is
ip-22|.
=
T
^
of
number
of
^
-J-fof
a
certain
number
is
What
18^.
is the
?
11
4
11
j;gX
18^
X
17
8228
;g_
X
_
_
A-XttXII"
3x3x^x;^
3Q. ^'
og
27
3
DECIMAL
Decimals
numbers,
be
may
FRACTIONS.
and
subtracted
the
merely observing the positionof
16.437
Minuend
29.081
Subtrahend
^^^"^
same
as
the decimal
whole
point.
840375.3007
64068.470361
Remainder
776306.830339
.03914
391.65714
Sum
In
whole
a
similar
number,
manner,
or
divide
we
a
may
decimal
multiply
by
a
a
whole
decimal
number.
and
a
36
24
X
12362
thousandths
12362
HOW
AND
NUMBERS,
s
thousandths
=
125
thousandths.
296688
=
THEM.
USE
TO
thousandths.
515j^
24
We
may
continue
wish, thus
we
the division
to
as
many
decimal
places a^
:
256)5172.14(20.20367
512
521
512
940
768
1720
1536
1840
1792
48
Divide
42307)
12.03
12.0300
by
42307.
We
(.0002
84614
until
35686
to
contain
obtain
the divisor, point off
the
dividend, and
as
continue
get
we
many
the
the
ciphersto
annex
the
a
the
number
large enough
divisor
first
dividend
at
least once,
significantfigure
decimal
places as
there
division
as
far
desired.
as
are
of
in
Multiply
32.142
1.23.
by
32.142
1.28
32.142-^%^.
1.23
32142
96426
and
for the
32142
123
them
multiply
64284
37
FRACTIONS.
DECIMAL
the
being
numerators,
of the
x\U-
we
3953466
together, obtaining
numerator
^^^
-f^.
product.
WMu'-
39.53466.
=
39.53466
Thus, in multiplication,
we
as
both
of the
Divide
"
36.4728
number
the
right.
dividend
being
by
of
We
We
number
same
there
as
the
a
decimal,
the
point
must
of
also
decimal
are
places in
number
necessary
the
it into
point
number.
same
the
into
point
1223.)
3647.28(2.98
^^^^^.J^JyJ^,.^y^^^^.u^
has
As
affect the
the
then
1201^
and
the
places
dividend,
we
has
as
as
and
used, and
"
Divide
There
there
and
whole
are
then
.6
by
decimal
four
in the
divisor,we
numbers.
to
;i
desired,
as
or,
been
done
in
the
less those
this
the last
until
been
has
in
brought
"
point off
then
"
quotient as
in the
as
there
divisor.
.03275.
being
divide
places
the dividend
places in
"
;i
is
divide
may
274
decimal
to
division
"
^
digit of the dividend
many
come
we
9784
down
the
shifting the point in divisor
of
example,
under
quotient.
continue
a
decimal
mauy
shown
TVhen
^
and
divisor
quotient
-
^
been
places
dividend
the
of
whole
a
of
places,thus multiplying both
not
put
change
may
move
fractions, this does
may
we
the
it, and
are
point off
to
12.23.
by
by moving
the
product
have
always
factors.
divisor
The
to
the
places in
many
shall
as
though
places less
annex
four
dividend
"
in the
ciphersto
and
dividend
the
divisor
than
dividend,
were
both
38
AND
NUMBERS,
HOW
USE
TO
THEM.
(18.
.03275).60000
3275
27250
26200
1050
To
Reduce
Reduce
Common
a
^^
to
Fraction
to
Decimal.
a
three-placedecimal.
a
167) 42.0 (.251+
334
We
860
and
835
perform
250
Jhe
167
mainder.
the
ciphers to
annex
the
division.
indicates
+
numerator,
that
there
is
a
re-
83
PROBLEMS
1-13
;
^,
Reduce
when
1-13;
a-c",m
ef
terms, and
01
r-
and
lowest
to
terms
and
to
the
the fraction
is
an
Reduce
improper
io
lowest
the
fraction
.
p-7'
mixed
number
14-25, find the
to
an
and
sum
improper
also
fraction.
difference
the
a
In
lines
a
one.
-"
0
60.
then
lines
In
59.
1.
No.
n-p
number
58.
Table
.
ij
mixed
FRACTIONS.
Practice
From
57.
IN
14-25, find the
"
sum
of
^, \
gh
kl
and
^.
n
40
AND
NUMBERS,
2x2x2
third root,
8 is the
8.
=
cube
or
.2 X
.2 X
fourth
.2 is the
.2*
8.
=
.2
The
4^
;
The
a
If
of the
show
to
a
root
=
2*.
="-
is the
; 2
2.
fourth
is 4he
.0016^
of
power
.2 ;
.2.
=
fifth power
root
is to' be
=
I
a
(^)i
or
of
-I;lis
=
|.
expressed V4*,
"\/4^or 4^.
of 4 is
little above
a
both, is called
or
fractional
the
exponent.
another
index
The
or
to the
left of
the
=
sign -y/,
2^
(2^)^
=^
2=' X
root.
2"
=
2".
V2^-2'\
2"
process
,of findingpowers
The
process
of
findingroots
to
nent.
expo-
taken, is the index of the
The
or
taken, is called the radical sign.
and
2"x2'=2x2x2x2x2
-
.2, or
rightand
is to be
above
written
what
is the
fifth power,
root,
a
or
fraction,it is
number
or
power
at the
written
indicatingthat
The
=
A^
fourth
of the
root
power
a
.0016
"^/^
^^-
fifth root
indicate
cube, of 2
^.
=
number
8*
2,
=
:^A-
=
the fourth
-^/S
A/J0016
|X|x|Xfx|
(F
or
THEM.
of .0016.
root
the fifth root of
USE
third power,
.0016.
--=
.0016.
=
TO
root, of 8.
2"
.2 X
HOW
is called
is called
Involution.
Evolution.
59049
998001
sign,
SQUARE
Each
above
of the
considerations
To
the
we
digits in
100.
than
number
the
four
annex
number
less than
one
will
good, as
either twice
contains
powers
twice
as
This
many.
the following
from
appear
as
:
smallest
100,
square
or
holds
principle always
Take
second
its root,
digitsas
many
41
ROOT^
number
digitsin
of
and
O's ;
1000,
square
so
of digitsthan
we
To
less
10000,
contain
can
of
O's, one
square
sqioare
number
number
the
three
annex
No
the
twice
less than
1000.
on.
digits,viz., 100.
three
has
O's, one
two
annex
To
that
a
we
less
root, less
in its square
one.
Now
take
999.
999
1000
greatest number
999
is
has
999
X
10000
X
the
number
every
number
of digitsthan
Hence,
ever^
its squxire
or
less than
separate
figures each, by putting
same
the
as
the
with
a
of
twice
that
as
999
And
contain
as
But
999.
X
in its square
a
has.
so
for
greater
root.
digitsas
many
many.
over
periods of
into
every
number
digits in
digits,viz.,
has.
can
number
dot
units, the
number
twice
any
1000
number
square
either
contains
one
If, therefore, we
beginning
the
the number
twice
square
root,
No
three
that 9999
number
of 9's.
up
has
than
twice
only six digits,
has 'twice the
9999
X
number
smaller
a
that
periods will
of
the
alternate
square
root
two
figure
be
thp
of
the
number.
If
we
multiply 42 by 42, settingdown
we
have
We
40
X
40, 40
2, 40
X
represent it thus
may
X
2, and
each
partialproduct,
2x2.
:
42
42
40*
40
X
2
1
"
40
^
X
2
the
=
square
.
twice
"
the
J
2*
the
-
square
of the
tens.
of
product
^
of the
tens
units.
by^ units.
42
AND
NUMBERS,
If
there
all but
consider
The
2
+
than
more
are
the
of
square
last
the
The
the
83
period,49,
the
of the
root
i-equired. The
the
tens
contain
of
for twice
the
than
in 249
goes
be
the units
the
We
If 3
a
be
product of
will
by
twice
the
tens
and
so
number
of
the
second
the
in
Also,
in
18
of tens, and
so
root.
is the
18
is
be
must
greatest square
in 18
units
tens
16, and
the
twice
for the
than
the
Hence,
the
of
the
root
its root
tens
40,
or
4 is
of the
the
nothing
80, for
trial
249,
If 249
square.
and
We
3 to the
3x3
1849, leaves
units
times.
units
by units, and
+
by units,
three
root, and
correct
tens
the
hundreds, from
tens
take
little over
of the
first
greater square
tens
by
of
product
units.
80
83.
of tens
that, dividing 249
give the
16
square,
product
the
twice
were
in
required.
root
Subtracting the
the
some
greatest square
digits.
part of this square.
no
no
the
of
hundreds,
be
of two
is contained
of
root
greatest square
of the
tens'
of
up
consist
square
be
can
must
can
square
square
of
square
hundreds
18
must
root
for the
number
16^
249
the
of
tens
some
3
is therefore
periods,its root
1849(43
249
may
of 1849.
period,18,
80
we
tens.
as
root
two
of
square
the number,
units*.
square
being of
1849
digitsin
two
one
THEM.
USE
TO
number
any
(tens X units)+
Extract
HOW
more
would
trial divisor,
a
will
3 to
assume
divisor,making
root, 3
square
X
80
is twice
of the
units.
If 3
is
3
be
83
X
3 to
3 X
is
83
Extract
this
If
right.
reduce
equal
digits.
root
consists
of 3 to 80, and
first leave
out
4
tens, and
86,
of
is in this
remainder
plus
2 tens
the
first
of the
36589
of
6
as
will
we
hundreds,
8
illustration,
last
8 of the
the
684
tens,
period,
next
of 68486.
root
product
before,
Con-
have
we
of tens
by
units
find the units'
we
root
square
of 68486
is
part of the
root
As
units.
units'
is twice
of
plus
figure,
261, with
next
tens
find
we
find
of the
is
the
the
tens,
by
tens
There
divisor
So
trial divisor.
This
is twice
(twice 20
or
complete
trial divisor.
root
before,
of
to
units
units'
der.
remain-
no
complete.
26
first
product
36589.
=
proceed
the square
out
the
complete divisor, 46,
of the
third
taken
for twice
the
and
period,89,
have
units +
X
to the
tens
We
is the
trial divisor
we
the last
square
2617
The
which
of 365.
leaving
figure.
The
1.
case
6848689.
the
the
68486, 67600, the square
the
As
units.
bring down
2610',
if
the
the
26
as
of
out
for twice
leaving 886
of
the
in
as
find
sidering26
square
root
but
4 units.
will
we
taken
We
present
Now, bringing down
36589
a
all
leftover.
5227)36589
which
four
get 26 for the root, with
521
the
the
Proceeding
tens,
of
of consideration
for
regard
521)886
26
2.
its root
periods, 684,
two
6848689(2617
we
must
we
multiply 82 by
periods,and
four
276
249,
required.
root
of
284
that
know
shall
we
of 6848689.
first
46)
249,
greater than
be
is the
the square
will
than
product
43
We
less
or
just 249.
number
This
to
43
ROOT.
SQUARE
is
20 +
+
The
6.
twice
6)
46, its units
also, 521
simpler
than
+
1
tens
6,
=
we
the
next
;
so
get
tens
doubling the
4:4
If the power
same
AND
NUMBERS,
contains
HOW
decimal,
a
USE
TO
proceed
we
the
of
merely noting the position,
way,
THEM.
ni
exactly the
point ; thus,
372.14623(19.291+
1
29)272
261
The
382)1114
764
decimal
many
Notice
3849)35062
but
34641
that
30.
may
places as
the
There
are
period except
36581)42130
We
complete.
not
that the root
indicates
+
it to
period is
two
figuresin
not
one.
3549
the square
root
of 95986
to
decimal
two
95986(309.81+
9
609) 5986
5481
6188)
50500
49504
61961)99600
61961
37639
2
2
^
3
Extract
the
2 X
3
square
V4
V4
/4
/4
4
2
_
_
3x3
root
V
of
_
121
-x/Q
V9
-^j.
V49
49
7
_
Vl2l
2
_
\9"
^^9
9
11
as
3*
3,
every
38581
Extract
is
wish.
we
last
first
the
carry
found
places.
*
CUBE
Extract
the
of
root
square
45
ROOT.
-Jf
.
43) 16.0 (.37209302
.37209302
129
(.6099+
36
310
1209)12093
301
10881
.
90
12189)121202
86
109701
400
11501
387
"
The
130
be
129
8
evidently
9.
or
We
YQQ^
figure would
next
call the root
may
.6100-.
86
14
From
Extract
68.
places.
Prove
Of
69.
the
Table
Practice
by squaring the
14-19,
^
of
root
square
No.
1-13,
and
root
to five decimal
1.
a-d
to
the
decimal
two
remainder.
places.
"
"
cd
Of 20-25, -5L
70.
to
eight decimal
places.
.
;
"
i
b"d
ROOT.
CUBE
Let
To
100
the
times
us
take
the
by itself,we
multiply 100
again,we
cube
as
many
two
annex
of 100.
as
number
smallest
The
the
cube
root.
of three
two
annex
ciphersmore,
contains
Take
the
two
viz., 100.
digits,
O's ; to
multiply by
making 1,000,000 for
digitsless than three
smallest
number
of four
46
NUMBERS,
AND
The
digits,viz., 1000.
cube
has
of
cube
USE
1000
is
three
less than
two
digits,
ten
TO
HOW
THEM.
'1,000,000,000.
times
as
The
the
as
many
root.
cube
No
take
Now
999
X
The
latter
number
times
X
This
contain
can
cube
rootf
or
in its cube
one
The
+
2
digitsin
square
or
45'
=
45
=
40 +
of
nine
smaller
a
product
three
digits,
is
times
of
we
(40'x5)
=
tens' +
3 X
=
tens' +
(3 X
3(40'
as
periods
third
of the
digitsas
its
many.
If,
of
three
ures
fig-
figure beginning
be
the
same
as
the
number.
as
multiply
we
+
5"
+
5
(40' X 5) + (40
40" +
three times
into
root
many
the square
of
45
get
40
=
as
periods will
If
units'.
three
of digitsthan
times
every
number,
2(40x5)
2
three
over
the third
any
5,
40' +
40" +
45"
is
latter
twelve
number
separate any
(tensX units)+
by 45,
has
largestnumber
9999
X
999.
X
root.
less than
two
or
units, the number
of
number
number
The
9999.
has
either
each, by putting a dot
the
9999
1000
X
the
greater numba^
a
contains
therefore, we
with
Take
x
number
1000
999
9999.
times the number
cube
10000
three times
digits,viz.,
This
999.
as
10000
as
Every
than
999,000,000.
9999
cube
No
three
of
number
many
X
of digitsthan
two.
digits,viz., 9999.
than
many
is
as
999,900,000,000.
as
smaller
a
product
three
digits,
of four
is
999
root, less
largest number
the
X
less number
a
in its cube
the number
999
contain
can
X
+
5) +
tens' X
tens' +
5')
X
+
5"
(40 X 5') +
5"
2(40x5')
3
units
3 X
+
tens
3
X
X
tens
X
units +
units' +
units'
units')X units.
"18
AND
NUMBERS,
Extract
cube
the
of
root
TO
HOW
THEM.
USE
55480717144.
The
cube
with
remainder
a
We
55480717.
taken
tens*
(3
+
units*)
X
3
--
we
1140
1
+
3
the
out
4=
3
cube
tens
of the
units.
which
4*
=
3
16.
divisor.
root
required.
X
root
tens,
units
X
43548300,
the
=
X
cube
+
we
tens
43548300
43594036
X
3810*,
leaving
units*)
for
use
units
3
=
45720
+
X
4
=
X
+
divisor,
16
174376144.
=
=
433200
the
plete
com-
cube
root
with
We
X
1
X
a
mainder
re-
174376.
tens*
X
4
=
have
for
divisor,
3810
3 X
380
X
381,
=
3
trial
a
433200,
units.
174376144
units.
X
units
X
^
The
55480717144.
of
tens
608717,
434341,
=-
55480717
of
the
27,872,-
trial
a
the
=
divisor.
find
+
Unit8*=l*=l.
1140.
Now,
the
380*
X
for
1
units
X
of
tens
X
for
use
obtaining
of
units.
X
tens*
which
3
+
of
root
remainder,
the
=
+
the
27,000,000
=
000),
tens
608.
cube
the
38,
is
Having
out
3 X
55480
of
find
now
(380*
of
root
=
taken
(3
3
X
X
obtaining
3814
3810*
4
for
Units*
45720.
43594036,
tens*
the
is the
plete
com-
cube
Extract
the cube
of 6918
root
49
KOOT.
CUBE
to two
decimal
6918(19.05+
4707375
Extract
the cube
of
root
3^^.
216*
216 \i
/216Y_
\^133iy 1331*
Extract
the cube
of
root
|f|f
6
11
21-f||^.
.242301-f.
=
21.24230i(2.76+
1200
8
420
49
13242
1669
11683
218700
4860
36
223596
1559301
1341576
217725
places.
'
50
and
numbers,
From
Extract
how
Table
Practice
the cube
root
to
71.
Of
1-13, a-g
72.
Of
14-19,
73.
Of 20-25,
decimal
one
use
to
them.
1.
No.
place.
hi
decimal
to three
-"
places.
decimal
to four
places.
7n"t
Roots
Vl6
than
other
V4
4.
=
^J^l25
VT5625=125.
V6561
V81
""/5l2
^8
8.
=
the fifth root
We
first
must
similar
to
Representing
tens' X
+
3 X
+
u', and
Note.
tens
When
tens
t' +
letters
are
Considering
used
("+ w)"
no
understood.
number
any
=
"" +
as
by
units'
2 tw +
gether
It will be alto-
fifth power.
u, the
formula
units',becomes
+
cubes, tens'
for
^ +
3 ^u
to
represent numbers,
if two
So,
of
sign
letters,or
sign of
a
operation between
means
2 x
of tens
up
the
t X
letter
3""M"4-
u.
-f- units,
5"*u
-
^ 4-
{5"* 4-
+
lOt'w*
10""u
"V
tt*
3"V
+
3"u*
+
4- lOtV
+
5"u*
4- u^
e"u"+
"5 +
3 iw'
4- lO^v?
4- 5 iu' 4-
plication
multiand
a
plication
them, multi-
3 "'u 4- 3 "u" 4- v?
=
+
w*.
2 tu
eB + 3t*u+
(e + u)*
a
cubes.
units
x
together,with
always
and
omitted.
commonly
come
is
t, and
by
for
/ormu^a
for squares
units + 3 X
for squares,
is
number,
those
^6561.
=
of 5030919566507.
the
produce
3
3.
=
2=^512.
2.
Extract
V'9
9.
=
=
Cube.
the
5="/l5625.
5.
=
81.
=
and
2=\/l6.
2.
=
Square
the
w*)u.
BOOTS
OTHEH
THAN
ThU
SQUARE
OR
61
CUBE.
5030919566507(347
5th
=
root
required.
_(5^*+
Extract
(t+ uy
the seventh
=
1''+
7^u
=
i^ +
(7^
Thus, whatever
+
+
^u"
21 ^i^ +
to
deal
with
any
+
3ot*u^
35 iV
be the root
principleapplies.
roots
In
10^"w"
+
of 24928547056768.
root
21
10^w
to
35"3w*
+
+
35 ^^3
be
+
+
21 "^u^ +
21 "V
+
extracted, the
practical problems
except the square
and
we
7iu"
7tu^
same
+
+
u'
w*)u.
eral
gen-
rarelyhave
the cube.
52
NUMBERS,
AND
Practice
a
b
cdefghijh
1
4546968663625382717
2
0839988693515916262825
3
59948740583160670179140320
4
436392145137
6
2212254303198160680681072
6
12855886264315323923815302
7
67581903586705777467838781
8
76918628642497807490474739
9
46978743989590524278074627
10
84364458030376770984321954
11
04890223684505101402206478
12
9285900800771967220196643
18
19110769407458521478884803
14
88284560361341478032610794
16
05978151908701279270398788
16
2278347499530974
17
32592101337785572228748558
18
82805120942314062544411878
19
78931290301228894683716725
20
03501925068680097704213363
21
48514670584843619553826028
22
7846356781231472902444576
23
86884246898972648751274910
24
67376004732305941727677807
26
53453347556728796015788643
TO
HOW
Table
T
tyi
THEM.
USE
No.
1.
nopqrstuvwxy
z
648427
1
746
1101033815365
1
0
7
7
'4 52684861
2
1
PRACTICE
Practice
abcdefghijk
1
96121582183995511822622838
2
51163756015082783412609635
3
61331054800986883735679874
4
66298706480301418337028432
6
02090827188646294387519878
63
TABLES.
Table
I
vi
No.
n
o
2.
r
p
q
9
7785
s
t
u
v
iv
0
44698
x
y
6-35877712601471699552139644
7
123683296045764
8
05443994748766404154772024
9
23849820629876446007177304
10
69460752955933142342204
2
66
11
88366228134448646905416
2
33
12
01012411521700252869576384
18
77521422777944332998543392
14
63937835099468872278549310
15
97641566345292657319833774
16
68741577973349285417266697
17
6
18
29264390047844461683642738
19
13583593990741807180578333
20
9218744797246
21
88491503546174080173771742
22
60335489026115614276018856
23
40626672341408701128835937
24
45566925732848866433608098
25
27557798794796916166482142
1236692108778216755984692
8
423650012983
z
54
NUMBERS,
AND
Practice
ahcdefghijk
1
52968823828287473979264754
2
40794194185114171000960639
8
46810532633584507677718434
4
64426872344769601043723827
6
76807441642338679801467886
6
19936091860897753600766911
7
27977987926170461188611015
8
48917697329370139471870239
9
36249408632686262296313003
10
42370149220246213231724685
11
35889664471351336868944095
12
37780388872833287676325857
18
34008376619651176920339685
14
32652756561030796150970207
15
3016814154617
16
328041438030
17
14852816360503871656818158
18
18275847880490863384819364
19
87690777456899725032657162
20
5418
21
70436376089213778781704006
22
40452642326326403225317749
28
40328411898259262174981420
24
81762463452230799205876354
25
44137486524829585761525309
HOW
USE
TO
Table
No.
THEM.
3.
Imnopqrs
3
4^4
4963604929063582733588
716574259533
963810181617
t
u
v
w
x
y
z
PART
APPLIED
IL
NUMBERS.
"
States
United
The
followed
is
dollar.
the
is
applied
\$2347.863
and
forty-seven,
States
United
to
number
by
the
money,
and
which
unit
it is
of
which
hundred
Twenty-three
means,
hundred
eight
the
that
\$ indicates
character
Money.
and
thousandths
sixty-three
dollars.
hundredth
One
a
mill.
tenths
The
worth
If
the
one
We
pencil
is
three
worth
one
a
is
cent
and
three
mills.
is called
cents
ten
of
eighty-six
and
eagle ;
an
dime,
a
A
coin
dollars
twenty
to
If 1
20
1 cent.
10
cents
=
1 dime.
10
dimes
=
1 dollar.
10
dollars
=
1
=
is
a
Ans.
20
20
=
applied
and
to
not
upon
pencils
This
4x5
means
the
operation
cents.
The
being
factors
cents,
how
many
pencils
can
be
?
Operation:
^
=
?
cents.
cents.
cents,
of 4
cost
formed
perare
4
20.
5
costs
20.
eagle.
is the
what
cents,
cents
20
=
product
pencil
cents
=
numbers
upon
5, the
mills
4x5
say
cents
5
X
10
5
costs
4
:
may
applied
for
of
be
may
tenth
One
oent.
a
eagle.
Operation
and
value
dollars
ten
double
\$.863
is
eighty-six cents
or
of
coin
dollar
a
decimal
cents,
A
of
4.
67
bought
58
NUMBERS,
If 4
Operation: -2^
5.
Note.
problems, and
=
The
last two
In
same
and
the
factor
is what
\$14f
how
men
a
be
not
of 3
true
for 4
to men,
and
cannot
be
2
as
Or, if
1
A
fraction
there
boy,
There
stated, 2^
being
all
the work
numbers.
pure
in 2
in 4
days,
days?
men.
3
Yet
men.
3
would
men
require the
not
labor
entire
imaginary quantity as applied
of the problem
the conditions
an
We
1
working
man
could
boy
a
cost?
of work
with.
working
is,however,
but
do
4
2
days, would
the work
half
days,
would
to
one
that
can
of
also
2
just
a
be
man,
just
problem
the
men.
is also
no
by dividing the
1 ton
amount
take
that
days, and
4
does
to do
2^
is
2^
fullycomplied
and
enough.
Ans.
know
we
the work.
men
factor,
one
__.
certain
a
for it would
days.
so
working
men
do
we
performedupon
are
it would
sense
men
and
product
a
the
1265
it take
2J^.
=
certain
a
under
\$6.
arithmetio
would
men
Operation: f
In
?
following,come
This
factor.
\$4, what
Ans.
perform
can
many
pencil cost
=
costs
6.
=
t
operationBin
If 5
1
115x11
of coal
Operation:
All
does
given
^=_^^
ton
a
other
THEM.
given.
14f
,.
Operatum:
-Jof
the
USE
the two
have
we
TO
part of \$371^?
^
If
each
find
required to
are
product by
do
cents, what
20
pencilscost
HOW
AND
coins
be
numbers
may
6J
cents
each, the
of
computation.
an
imaginary quantity as applied to cents,
of cent
applied.
numerical
4 X
6^
=
to which
denominations
If handkerchiefs
quantity
25, 2
X
6^
6
-:
J
are
is taken
12^.
fractional
the basis
as
If
at
we
4
handkerchiefs,
13
can
The
cents.
first case,
be
we
25
pay
the
however,
applied;
in
cents
is the
price
;
if
we
in
same
result
the
59
NUMBERS.
APPLIED
both
obtained
second
instances.
by
the
In
pay
the
multiplication
it cannot
case,
must
we
be, without
modification.
\$1014.10
27.
\$3871.
1924.67
234.86
Sum
=
\$284.62
\$1275.96
Difference
=
5
\$1946.33
72)\$6722.64 (\$93.37
Product
=\$1423.
10
Quotient.
=
648
242
216
266
216
504
504
Money.
English
farthings(far.)
1 penny
12
pence
=
1
shilling(s.).
20
shillings
=
1
pound ("1
4
=
(cf.).
or
1
L).
A
coin worth
5
is
shillings
a
orown.
A
coin
worth
20
is
shillings
a
Bovereign.
A
coin
worth
21
is
shillings
a
guinea.
guinea is no longer coined, but
meaning 21 shillings.
The
Farthings are
expressedas
fractions
of
the
a
term
penny.
is still used,
Thus, life?.
60
16s.
"42
7
13
Sum
AND
NUMBERS,
"63
=
11
14
6
14
l". +
nd.
USE
THEM.
+ ^d.==21d,
14".+35.
+ "13
"1
16
7
=
TO
+ "7
+
16s.
ls.9d.
=
34s.
=
"114s.
=
-="63.
+ "42
9d.
s.
"42
Difference
ed. +
^d.
3
HOW
3
s.
Product
13
"105
=
llrf.("7 3s. 5i|rf.
s.
Quotient.
=
161
12 inches
(in.)
yards
40
rods
8 fur.
or
or
5280
or
16i feet
320
144
square
9 square
BOi square
square
160
square
square
640
acres
1 rod
=
1
|
(rd.).
furlong.
^ ^.^^
1
league.
=
1 square
foot
=
1 square
yard (sq.yd.).
j
^^^
(sq.in.)
feet
yard (yd.).
Measure.
Surface
yards or 272*
_
^^^^
(sq.ft.).
(^
J
or
43560
)
i
__
feet
=
(ft.).
j
feet
feet
rods
1
=
or
inches
=
rds. )
3 miles
Square
1 foot
=
3 feet
5}
3 c?.
6
bd.
s.
19s.
23)"164
12
"17
11
12
"35
4rf.
s.
^^-g
)
=
1 square
mile.
^^
)
6d,
Cubic
1728
1 cubic
foot
feet
=
1 cubic
yard (cu. yd.).
feet
=
1
inches
27
cubic
16
cubic
feet
cord
(cu. ft.).
=
cubic
8
Measure.
Solid
or
61
MEASURES.
AND
WEIGHTS
or
(cu. in.)
128
(cd. ft.).
foot
cord
)
cu-
i
-i
__
'
bic
feet
)
Weight.
Avoirdupois
16
(oz.)
ounces
pounds
2000
28
4
20
1
=
1 ton
quarters
or
112
hundred-weight
pounds
or
quarter
=
1
hundred-weight
)
i
for
weighing
/rn
i.
^
^^
v
'^
Weight.
gold, silver,
and
precious
grains (gr.)
=
1 carat.
24
grains
=
1
20
pennyweights
=
1
ounce
12
ounces
=
1
pound
4
(cwt.).
J
pounds
Troy
Used
(qr.).
1
_
2240
(T.).
=
"
pounds
(lb.).
Table.
Ton
Long
pound
=
stones.
pennyweight
(oz.)
(lb.).
(dwt.).
62
AND
NUMBERS,
Reduce
7 lbs. 11
oz.
HOW
13 dwt.
TO
18
USE
THEM.
grains.
gr. to
7652
3826
45912
18
7 lbs. 11
45930
Reduce
oz.
13
dwt.
18
gr.
grains to pounds,
=
45930
ounces,
gr.
pennyweights,
and
grains.
24)45930(1913
dwt.
24
20) 1913
219
12}95 oz.
216
"
13
dwt.
7 lbs. 11
"
oz.
33
24
"90
72
18
gr.
Ans,
7 lbs. 11
oz.
13 dwt.
18
gr.
64
NUMBERS,
AND
HOW
USE
TO
THEM.
Time.
A
by 100,
will
is
year
not
a
if its number
if its number
or
be
leap year
is divisible
The
leap year.
a
is divisible
year
by
2000
units
12
dozen
^
-
year
1 dozen.
24
sheets
1 gross.
20
quires or
1
gross
1
great gross.
units
1
score.
480
-.
sheets j
Books.
A
book
formed
of sheets
folded
in
2 leaves
is
in
4 leaves
is
a
8 leaves
is
an
ti
a
folio,
quarto.
octavo
n
12
leaves
is
a
duodecimo
n
16
leaves
is
a
16mo.
n
18
leaves
is
an
n
24
leaves
is
a
24mo.
n
32
leaves
is
a
32mo.
n
64
leaves
is
a
64mo.
18mo.
or
quire.
)
~
20
1900
Paper.
_
12
not
will be.
Numbers.
12
The
400.
4 and
by
8vo.
or
12mo.
ream.
WEIGHTS
The
of
sizes
AND
taken
commonly
most
paper
65
MEASURES.
the
as
basis
are
the
Cap
i4 inches
by
17 inches.
Crown
15
inches
by
19
inches.
16
inches
by
21
inches.
19
inches
by
24
inches.
(de-my')
Demy
.
.
Royal
Thus,
each
we
a
sheet
being
have
may
standard, and
books
being
depending
can
still
printed page.
sizes
York)
7^
Large
8vo
Small
Svo
dimensions
variety of
used
was
forms
and
of
dimensions
following
lisher
pub-
though
the
definite idea of the size
of the
of the
type page
mon
com-
in
7
X
4 J inches.
6f
X
3|
....
.
.
.
"
"
18mo
5
((
the
margins:
type
pages,
for instance,
x3
an
inch
a
bound
or
two
12mo
must
book
be
inches, etc."
blank
still used,
books, the
though
the
always correspond.
terms
names
cap
and
quarto, crown
the
;
sizes,
the
etc.,
a
the
as
standard, printed
5|x3f
the
For
name
12mo
being
5
Similarly,
:
"These
X
other
no
longergive any
.
for
when
paper
the
are
17 in.,
by
taken
formerly
was
endless
an
used, no
of the
Following
royal
Consequently, 8vo, 12mo,
are
(New
the
sizes,is of whatever
terms
"
in.
14
leaves.
16
in
desire.
may
make
be said to be any
the
upon
of sheets
hardly
established
certain
to
as
understood
was
there
now
so
quarto, a demy folio,a royal 8vo.
crown
a
book
a
folded
printed books,
For
but
is
16mo
cap
page
8vo, etc., are
dimensions
do not
66
and
numbers,
how
to
use
them.
Problems.
The
74.
ending
profitsof
net
June
30, 1891,
certain
a
were
follows
as
1887
1888
265,147.08
1889
i243,854.87
1890
265,448.52
268,960.87
.....
At
76.
the
was
35
:
\$269,862.33
1891
What
for the five years
company
profitper year?
average
what
hundred,
cts. per
is the
of
cost
655
lbs. of
ice?
Express
the
7
operationsand
cancel
thus,
;
^^^y^
thus,
Or
131
229i.
=
6.55
-35
\$2.29.
^725.
1965
4
2.29
is the value
76.
What
77.
If coal
78.
oranges
is the
28
cts.
cost
a
long
each
profiton
14
cts. per
is sold
ton
for
dozen
?
\$6.25
per
lbs. ?
2000
is the
dozen, what
a
pint of peanuts
rate
same
Callingthe
of
81.
How
82.
What
per
at
eggs
cost
of
14
doz. and
7
?
at the
80.
the
At
If half
79.
cost
costing \$4.10
ton, what
short
of 1004
a
many
of
to three
of the
a
bushel
pound sterling\$4.8665, what
peanuts in United
inches
square
decimal
would
l^c?.,what
?
value
bushel
cost
of
decimal
a
cubic
States
in 1 square
yard
places.
is 13
cu.
is
money?
mile?
ft. 1124^
cu.
in. ?
1472
3 tons
83.
pounds,
How
86.
1900
1891
year
many
how
many
grains?
be
there
will
leap years
many
followingthe
equivalent of
is the
oz.
drams, scruples,and
ounces,
How
84.
lbs. 13
67
MEASURES.
AND
WEIGHTS
in
the 2000
years
?
seconds
in the nineteenth
seconds
does
century (1801
to
?
inclusive)
How
86.
many
revolutions
How
many
87.
\$2.25
what
How
there
in 27
of each
is the cost
half-pintbottles
many
to make
the earth
100
?
sun
pencilsare
gross,
per
88.
the
it take
doz. ?
11^
gross
At
pencil?
be filled from
can
hogs-
a
containing 68 gals.2^ qts.?
89.
How
90.
What
I
91.
If
92.
in
of
a
13
cord
of wood?
of
acres
land
2^
at
cts. per
16
a
invested
7 cts. per
of
note
what
I
net
Taxes
\$1800.
\$654.92.
to
I sell
12^ cts.,and
ft. at
foot, 1322
profit?
6
weighs
paper
must
for
amount
is my
What
cts.
ream
quarter of land
a
charge
per
of
pages
lbs., and
quire
to
costs
double
18
my
?
money
How
93.
from
at
acre
at
and
money
pound,
per
14
leaves
94.
many
If the
steps in
a
a
12mos
man
what
in
second,-how
164
measuring
to each
paper
pound,
If
cap
of paper
reams
cts. per
95.
value
acre
on
an
the balance
cts.
an
interest
half
one
is the
inches
foot?
square
and
cubic
many
for waste
book
weighs
is the
X
lbs. to
the
cost
of the
paper
takes
will
32
it take
in.
him
?
and
ream,
per
at
can
in.,allowing 8
28
leaves
32
walking
long
17 in.
each
a
to go
book
costs
12
?
2
step, and
14
miles
?
68
wheel
If the
96.
the tire,how
48
HOW
AND
NUMBERS,
of
a
USE
THEM.
make
the wheel
will
in. around
94.248
measures
wagon
revolutions
many
TO
in
going
miles ?
PERCENTAGE.
The
terms
centum,
6 per
per
and
cent
meaning, by
cent, or 6 %,
percentage
from
the Latin per
is the
base, 9 is the
come
the hundred.
.06 of 150
or
9.
=
6 is the rate per cent, .06 is the rate, 150
peroentage.
(1) troods which
is the
What
14%.
\$162.50
cost
(2) Goods
14%.
sold at
which
a
loss of
14%,
\$162.50
cost
is the
What
the
are
loss is
sold
at
\$22.75.
an
sellingprice?
57
325
'
IM^xiM=.
^
\$185,25.
r
Goods
is the
which
\$162.50
cost
are
sold at
a
loss of
14%.
What
sellingprice?
325
43
IM#_Xi^
\$139.75.
=
r
(3)
is the
Goods
which
of
\$22.75.
=
are
an
7
"M2li^
they
at
?
profit
325
If
sold
are
cost
\$162.50
are
?
profit
\$185.25
162.50
\$22.75
sold
for
\$185.25.
What
of
69
PERCENTAGE.
which
Goods
\$162.50
cost
sold
are
\$139.75.
for
What
is
the loss ?
\$162.50
139.75
\$22.75
Goods
(4)
\$185.25.
The
is the
What
profitis ^^
cost, the cost
the
sold at
are
is
sellingpriceis
?
profit
sellingpriceis \^^ of
sellingprice,the profitis ^^
cost, the
of the
of the
^|
The
profitof 14%.
a
the
of
sellingprice.
325
^^"^'^^
X
li
\$22.75.
=
Goods
sold
are
\$139.75. .What
loss is
The
the
cost
at
14%.
loss of
The
sellingprice is
is the loss?
y^^
of the cost, the
of the
-y^
a
sellingprice^^
sellingprice,the loss ^ of
of the cost,
the
selling
price.
7
325
\$Mlixi^
\$22.75.
=
(5)
what
Goods
was
are
sold
for
\$185.25.
The
profitbeing \$22.75,
the cost ?
\$185.25
22.75
"
\$162.50
Goods
was
are
sold
for
\$139.75.
The
the cost ?
\$139.75
22.75
\$162.50
loss
being \$22.75, what
70
AND
NUMBERS,
Goods
(6)
of the
14%
sold
are
at
What
cost.
HOW
a
TO
profit(or loss) of \$22.75,
the
was
325
the
was
sold
is
50
1162.50.
-.
are
which
cost?
mM"im
(7) Goods
THEM.
USE
for
\$185.25.
The
profitis 14%.
What
cost?
325
m?^
50
.
Goods
the
sold
are
\$139.75.
for
The
14%.
loss is
What
was
cost ?
325
50
?lM^"li22
=
(8)
Goods
profitis
that
what
per
\$162.50
cost
cent
of the
are
\$162.50.
sold
7
per
\$162.50
that
cost
cent
of the
\$162.50
139.75
The
2
%%n^m^Y\
162.50
what
\$185,25.
cost?
\$185.25
Goods
for
cost
are
sold
for
\$139.75.
?
7
2
mi2"M.^u
The
loss is
72
AND
NUMBERS,
Goods
(12)
sold at
are
is the
wm
Problems
bbls. of
be sold per barrel
A
98.
much
How
I
99.
sell
How
cost.
I
100.
will he
more
house
a
and
shoeing the horse, \$7.50
whip,
sell the
and
per
week
than
is
raised
5%.
formerly ?
20%
carriagefor \$355.
than
more
\$450.
for
\$2.25 for
I pay
is the per
What
it
?
repairingthe carriage,\$1.50
for
outfit
has his wages
the transaction
by
they
must
?
day
\$4200, which
I make
horse
a
per
for
do
much
\$164.32, for what
cost
earning \$2.80
man
Percentage.
yield a profitof 28%
to
is
\$139.75.
=
in
apples
which
43
mMj"M
\$186.25.
=
If 90
THEM,
325
m
X
USE
sellingprice?
57
325
97.
TO
profit(or loss)of \$22.75,
a
What
of the cost.
14%
HOW
for
cent
a
of
profit?
A
101
of
\$1232.16, which
I
102.
market
What
value
per
103.
for 152
An
of
\$.62
grain
at
I
compelled
am
was
of
lot at
What
cost.
dealer
is
22%
sells
more
a
lot of
than
they
grocer
sells flour at
of the cost.
architect
a
profit
a
the cost?
was
bushel.
per
The
sell at
to
potatoes
cost.
for
\$.57.
60
What
profitof \$1.10
What
charges 5%
is his
cts.
did
he
the house
the commission
including commission
?
a
per barrel.
sellingprice?
commission
plans and superintending the building of
cost
and
bu. ?
profitis 18%
105.
bu.
produce
pay
His
of the
16f %
house
a
do I lose ?
bushel, which
A
sells
declines, and
cent
A
is
1000
per
104.
dealer
real estate
.
house.
for
making
The
being \$6240.21,
total
what
73
PERCENTAGE.
An
106.
He
I send
He
coffee.
A
108.
the
A
109.
at
a
of the par
from
The
in
commission, and
America
?
98
at
30,800 lbs.
J-(per cent
much
J, how
(of
of
profit is
the
value)
par
selling?
stock
of
broker's
The
much
value, how
10%.
to invest
bought
113
at
shares
13J%.
for his
of
goods ?
\$4000
being \%
for
same
sells 28
of
the broker
110.
paper
the
broker
it is sold
for the
coffee.
is
commission
broker's
for
If
commission
a
in South
value).
par
of the money
pound
per
on
America
transportingthe
in
the cost
was
in South
4%
reserves
goods
agent
my
expends \$144
What
What
\$152.47.
remits
107.
sells
auctioneer
is the
for
\$3171, which
is
being ^%
commission
principalentitled
?
from
following is
an
in
daily
a
:
Vases
worth
\$2.00 for
73c.
Vases
worth
\$1.00
for
44c.
Vases
worth
50c.
Flower
Globes
for
worth
19c.
75c. for
29c.
....
Jars
Rose
Bonbon
Trays
worth
49c.
75c. for
32c.
....
and
Finger
Bowls
Water
Pitchers, \$3.98,
Creams
worth
75c. for
42c.
.
worth
50c.
.
.
for
22c.
79c.
now
Pitchers,98c., now
Cream
39c.
Pitchers, 98c., now
24c.
Sugar Bowls, \$1.98, now
49c.
Teapots,\$2.29, now
49c.
is the
entire
\$1.25 for
Sugars
Milk
What
worth
one
per
cent
of
discount
article of each
purchase ?
on
each
kind, what
article?
is the per
If
cent
a
on
chaser
pur-
the
74
AND
NUMBERS,
In
111.
the
same
Pears
in
Choice
is the
paper
Yellow
Choice
TO
HOW
THEM.
following :
Peaches, 12c.
per^doz.
can,
a
glassjars,qts. 35c.,
L.
USE
2
qts.
65c.
.
.
^.
.
Raisins, 9c. lb., 6 lbs. for
M.
60c.
.
Best
Mince
Pure
Meat,
Lustre
25c.
31c.
.
.
.
.
.
Starch, 9c., 3 for
and
Mocha
.
25c.
25c.
.
.
Best
.
box, 3 for
Salt, 5-lb. bag 6c., 5 bags
Table
Electric
9c.
Tartar, ^ lb. 18c., 1 lb.
Cream
Best
Java
.
.
\$1.00.
Coffee, 35c., 3 lbs.
.
What
saved
is
cent
per
\$1.20.
.
each
on
.
article
the
in
larger quantities?
INTEREST.
paid
Money
loaned
money
is the
for each
much
If
;
^uL^
\$575
is termed
money
The
at
interest
is
certain
a
interest.
a
rate
The
percentage
per
of
i.e,
annmn,
year.
for
2|
at
years
6%
interest, \$500
is the
3
5
rate
principal.
\$500 be loaned
principal:
of
use
is reckoned
principal,and
the
so
the
for
is the
^^
=\$75,
"
is
the
interest;
6%
is
the
amount.
Thus,
The
number
The
What
interest
is the
product of
the
principal^ the
raie, and
the
of years.
is the
amount
is the
9, 1891, at
5%
interest
?
of the principal
sum
on
\$642.71
from
and
Dec.
the interest.
13, 1888,
to
Sept.
75
INTEREST.
Method,
Accurate
Dec.
13, 1888, to Dec.
13, 1890
Dec.
31, 1890
13, 1890,
Dec.
to
2yrs.
.
18
31
1891,
Jan.
dys.
.
Feb.
28
Mar.
31
Apr.
30
May
31
June
30
July
31
Aug.
31
9
Sept. 1-9
2 yrs.
2 yrs. 270
dys.
2fH
=
7^.
=
^^W^
dys.
270
yrs.
10
\$642.71xgx;000_^33Q^
73
This
is
observed
a
in
greater degree
computing
the interest
for
month, and
360
days
days
of
interest.
is reckoned
a
accuracy
By
as
than
is
commonly
ordinary methods,
a
the
year.
Oommon
13, 1890, 2 yrs.
Dec.
Dec.
13, 1888,
to
Dec.
13, 1890,
to
Aug.
13
to
Aug.
Aug.
31
to
Sept. 9
Method.
8
Aug. 13, 1891,
mos.
18
31
...
.
9
dys.
"
....
2 yrs.
8
mos.
27
dys.
76
AND
NUMBEES,
Interest
\$1
on
"
"
HOW
for 2 yrs. at
\$1
"
\$1
"27dys.
8
"
TO
6%
"
-=
2 X
=
^
"
mos.
THEM.
USE
of
6 cts.
=\$.12
8 cts.
=
Iof 27
"
=
mills
.04
.004^
=
\$.164^
642.71
.164i
257084
385626
64271
32135
6)\$105.72579
17.62
"
'"
"
seems
27
mos.
but
the time
directlyfrom
in
"
long
is the
when
the dates
the time,
July 18, 1891,
interest
at
8%
the
so
on
1^.
"
"
"
"
"
"
"5%.
written
out
in detail in
this method
given,and
the
we
interest
this
write
on
\$1
operationis reallyquite short.
\$243.84
from
April 22, 1891,
?
Accurate
Method.
8
31
30
18
87
dys.,at 6%.
"
performing a problem by
manner,
What
for 2 yrs. 8
=
process
directlyfrom
\$642.71
on
=
\$88.10
The
Int.
=
days.
1812
to
77
INTEREST.
Method.
Common
243.84
"014i
97536
2
26
mos.
\$.014J
dys.
Int.
=
the time.
=
oaqqa
\$1
on
the
for
6%
at
8128
time.
given
3)3.49504
=
Int. at
6%.
=
Int. at
8%
1.165
\$4.66
What
is the interest
\$10,000
on
for 24
6%
at
?
Method.
Common
Method,
Accurate
days
=
\$10000
20
?00
.004
\$^0000X6X24^^3^^^
;00
"
*
m
X
73
Problems
In
of
a
the
following interest
month,
months
stated,consider
What
the rate
is the interest
problems, consider
12ths
as
Interest.
in
to
be
of
a
6%.
on
\$1 for each
number
of
113.
\$1
for each
number
of months
114.
\$1 for each
number
of years
115.
\$146.24 for
4 yrs. 6
mos.
18
116.
\$64 for
mos.
19
dys.?
117.
\$2245.86 for
mos.
26
dys.?
118.
\$862 for
9
mos.
2
Unless
year.
112.
1 yr. 3
days
19
days
dys.?
from
1 to 30?
from
from
dys.?
1 to 11?
1 to 10?
as
30ths
otherwise
78
at
at
8
\$352.46
120.
\$1226.83 for
121.
\$864.41
for 2 yrs.
122.
\$256.30
for 11 mos.?
123.
\$22.98
124.
\$684.37 for 11
What
at
for 7 yrs.
119.
is the
9
dys.?
7
15
dys.?
for 1 yr. 3
mos.
mos.
?
mos.
23
mos.
dys.?
of
amount
17, 1886,
125.
\$624.84
from
June
126.
\$146.22
from
July 4, 1890,
127.
\$1042
128.
\$6234.16
6^%
Feb.
from
28, 1888,
June
from
130.
\$324.72 from
131.
\$1216.66
?
10%
?
30, 1888,
at
Sept. 19, 1891,
to
Nov.
July 18, 1889.
to
30,
at
Aug. 4, 1888,
to
1890,
Aug.
to
7^^%
?
5%
?
at
29, 1891,
?
Jan.
from
\$623.50
133.
\$2241.62
16, 1891,
Dec.
from
to
Dec.
1887,
29,
14, 1891,
24,
Aug.
to
at
3|%
?
1889,
?
June
\$26.50 from
30, 1868,
interest is considered
When
due
of time, the interest
unpaid, may
principalto
as
compound
be
form
well
interest
a
as
new
on
as
after
regarded
to
Nov.
at
7^%?
as
the
a
payable at specifiedintervals
lapse of
new
loan
principal. Thus,
the
29, 1891,
Interest.
Compound
interest
7%?
at
Sept. 18, 1890,-"it8%
Nov.
to
April 6, 1887,
from
132.
12%
to
Aug. 7, 1889,
?
\$24.98 from Jan. 14, 1876,
4f %
to
11, 1891,
129.
134.
if
USE
?
mos.
4 yrs. 3
THEM.
TO
HOW
AND
NUMBERS,
every
to
such
interval,
interest
originalprincipal.
This
is
to
the
paid
on
is termed
80
AND
NUMBERS,
What
interest
is the
of
amount
compounded
TO
HOW
USE
\$10,000
THEM.
1 yr.
for
6
at
mos,
4|%,
semi-annually?
10000.
1.02|
10237.5
=
amount
of
\$1
=
amount
of
=
amount
=
amount
for 6
4f %.
mos.
at
\$10,000
for 6
mos.
of
\$10,000
for 1 yr. at
of
\$10,000 for
at
4f %.
1.02f
204750
102375
12797
25594
10480.64
4f %.
1.021
2096128
1048064
131008
262016
\$10729.55
Problems
What
135.
mos.
at
4|%.
Interest.
of
is the amount
\$613.72
Compound
in
18
for 3 yrs. 7
mos.
16
dys.,interest compounded
annually?
136.
\$221.32
for 4 yrs. 11
at
mos.
5%,
interest
compounded
annually?
137.
\$950
for
1
yr.
7
at
mos.
8%,
interest
compounded
semi-annually?
138.
What
July 14, 1891,
is the interest
interest
on
\$617.25
compounded
from
July 1, 1889,
semi-annually?
to
81
INTEREST.
different
In
regard
to
simple
interest
Savings
be
can
In
due, the
taken
immediately
back
states, where
some
it is allowable
of
to
interest
by
as
a
and
in
only
unpaid
law.
credited
Being
interest*
to
is, in effect,paid and
loan.
new
be
interest cannot
compound
other
rule is that
due
Interest
at
prevail
usages
general
interest
charge simple
; in
and
action
compound
pay
depositor when
the
The
charged.
be collected
banks
laws
interest.
compound
however,
may,
different
states
interest
on
deferred
each
states, compound
collected,
interest
ment
pay-
may
be
collected.
What
is the interest
6%,
interest
paid
until
allowed
2 yrs. 4
the
^expirationof
the deferred
16
mos.
being payable semi-annually, but
.the
on
\$416.12 for
on
not
dys. at
actually
time, simple interest
being
payments ?
416.12
.03
\$12.48
Ist deferred
paym't
of
=
6
interest.
mos.
\$12.48 bears
int. for 1 yr. 10
mos.
16
dys.
4
mos.
16
dys.
10
mos.
16
dys.
4
mos.
16
dys.
4 yrs. 6
mos.
4
dys.
1 yr.
((
It
('
It
=
\$3,378
interest
\$12.48 for
on
=
interest
on
\$12.48
=
interest
on
principal.
=
interest
on
interest.
=
total interest.
for 4 yrs. 6
416.12
.142f
59.366
3.378
\$62.74
mos.
4
dys.
82
AND
NUMBERS,
interest
Simple
is sometimes
What
called annnal
is the
TO
USE
principaland
the
on
HOW
annual
THEM.
each
on
year's interest
interest.
interest
for
\$10,500
on
3 yrs.
4
mos.
24dy8. at6%?
2 yrs. 4
1
24
mos.
4
24
4
24
4 yrs. 2
dys.
dys.
12
mos.
m
630.
=
interest
\$10,500
on
for 1 yr.
.252
\$158.76
10500.
.204
2142.
158.76
\$2300.76
Problems
What
is the annual
=
interest
on
principal.
=
interest
on
interest.
=
total interest.
Annual
in
interest
139.
\$462.17 for
140.
\$1226.82 for
141.
\$617.25 for
4 yrs. 3
What
is the amount
at
142.
\$1654.26 for
143.
\$891.82
144.
\$1682.36 from
146.
\$642.50
from
from
7
3 yrs.
2 yrs.
mos.
mos.
annual
2 yrs. 3
June
on
mos.
6
Interest.
mos.
17
dys.?
?
12
dys.?
interest of
at
19, 1887,
to
Apr. 30, 1888,
Nov.
16, 1887,
7%
to
?
Jan. 6, 1891, at
Dec.
to Feb.
6, 1890,
28, 1891,
at
at
5J%
?
5%
?
7^%
?
NOTES,
AND
DRAFTS,
NOTES,
Two
months
Jordan,
Marsh
after date
d
One
17, 1891.
I
promise
order
the
to
to pay
of
Co.
hundred
and
-^5^dollars.
thirty-two
An
obligationof
payable
of
a
time
after
days
termed
the
days of
at
this
note
time
or
specified. These
Unless
grace.
until
notes
without
interest
after
a
Notes
that time
The
ment
pay-
until
three
days
notes
specified,
due.
bear
are
are
do not
maturity,
After
interest
the
at
legal
differs in different states.
New
At
Seventh
National
days sightpay
three
Bank,
York, Aug. 25, 1891.
William
current
to the order
of
NY.
Eighty-six
To
note.
three
become
\$86.42.
Value
^-^^
demand.
on
otherwise
they
until
called
j^^^
legallydemanded
be
cannot
interest
rate, which
descriptionis
spegifiedtime
a
bear
The
CHECKS.
Boston, Nov.
\$132.47.
Value
83
CHECKS.
AND
DRAFTS,
dollars.
^^
of exchange
rate
on
York.
New
F. 8. Francis.
Jmies,
Boston, Mass.
This
William
of
is
a
bill of
Jones
\$86.42, and
him.
exchange
being
the
indebted
account
or
draft,as
to
F.
it is
S. Francis
being due,
the
usually termed.
to the
creditor
amount
draws
on
84
above
The
draft is indorsed
National
Seventh
J. D,
Bank
.
Boston,
of
York.
of New
Cashier.
W.
:
drawer, F. S. Francis, leaves the draft with the Seventh
The
of
Bank
National
Bank
National
28.
he
National
debtor
The
presents it
Bank
National
Seventh
The
collection.
for
the
Jones, Aug.
the draft,
York,
New
sends
The
Boston.
and
Bank,
follows
as
order,for collection,account
or
of
the back
upon
National
Pay
THEM.
USE
TO
HOW
AND
NUMBERS,
writes
Bank
liam
to Wil-
face of
the
across
"
then
in
stands
similar
a
the
position to
maker
of
a
note.
Drafts
certain
On
drafts
The
three
in
draft
a
specifiedtime,
sight,or
at
a
sight.
York
it.New
draft
at
days' grace
sight drafts, and
on
grace
allows
the
after
time
time
allow
payable
are
does
are
do not.
some
The
not.
payable always
allowed.
law
of
Some
Massachusetts
the
and
where
state
governs.
question being presented Aug. 28,
days' sightexpire Aug. 31,
states
the draft is
the
three
payable three days
later, Sept. 3.
bank
A
by
one
to pay
check.
check
who
a
has
is
a
a
depositat
specifiedsum
Being
without
speciesof
drawn
grace.
to
draft.
the bank, and
the order
upon
It is
a
of
the
calls upon
the payee
deposit,it
is
draw^n
presumably
named
payable
on
bank
in the
tation
presen-
in
calculating
in
Although
the
determining
not
We
so.
1
If
must
month
the
last
than
days
is the
of
of
Jan.
grace
next
the
grace,
1888,
How
note?
month,
drafts
it
is
Feb.
legal holiday
a
note
or
In
preceding.
day
28.
next
draft
is
(Sunday,
payable
counting
days
following
the
on
other
holiday
taken.
one
demand
has
1
as
and
notes
expires
upon
etc.), the
count
days.
1891,
falls
days
of
actual
31,
PARTIAL
A
30
maturity
the
Christmas,
day
interest,
of
count
day
Thanksgiving,
the
time
from
85
PAYMENTS.
PAHTIAL
note
the
much
for
following
shall
be
PAYMENTS.
\$6240
at
payments
paid
July
6%
interest,
indorsed
14,
dated
the
upon
1890,
to
take
June
back
up
6,
:
the
86
The
natural
most
payments
is the
HOW
AND
NUMBERS,
and
obvious
following
TO
USE
THEM.
in
procedure
of
case
partial
:
6240.
1.0l^
24960
6240
6240
1040
4160
6332.66
for 2
=
amount
of
=
amount
due
Sept. 4,
=
amount
due
Dec.
22, 1888.
=
amount
due
June
7, 1889.
=
amount
due
May
14, 1890.
=
amount
due
July 14,
\$6240
mos.
446.21
5886.35
1888.
1.018
5992.304
800
5192.304
1.027J
5335.958
110.25
5225.708
'
1.0561
5519.218
500
5019.218
1.01
15069.41
1890.
29
dys.
88
AND
NUMBERS,
Problems
A
146.
note
is indorsed
under
United
for
|1642.71 with
follows
as
TO
HOW
USE
THEM.
Method.
Court
States
Feb.
interest from
17, 1888,
:
6, 1888, \$164.42
July
;
Sept.17, 1889, \$55.12;
15, 1890, \$350.
Dec.
is the amount
What
A
147.
Partial
due
\$1500,
for
note
payments
June
?
7, 1891
is dated
6%,
at
Sept. 6,
1890.
:
Dec.
12, 1890, \$45
Dec.
31, 1890, \$65.50
;
;
Apr. 16, 1891, \$655.
What
due
is the amount
A
148.
14.
July
months'
four
Payments
16, 1891
Nov.
for
note
\$540,
?
interest,is dated
5%
at
:
Aug. 17, \$100
;
Aug. 30, \$50.
What
due
is the amount
A
149.
maturity ?
\$10,000,
for
note
at
is dated
6%,
at
:
Apr. 16, 1888. \$750
The
United
by
ourselves
some
with
Dec.
States
of
these
principle,local
31, 1891
Court
the
;
4, 1891, \$5000.
Apr.
is due
;
22, 1888, \$2500
Dec.
much
21, 1886.
'
Indorsements
How
Nov.
state
?
Method
courts,
modifications.
usages
can
has
but
If
been
we
one
be followed
fied
variouslymodi-
need
not
understands
without
concern
the general
diflSculty.
PARTIAL
Paetial
this
By
time
Payments
a
year,
and
on
by
if the
method,
Merchants'
the
does
account
interest is allowed
each
until
the
than
either of the
Method.
for
run
the
the entire
time
favorable
more
than
more
principalfor
from
It is
is settled.
not
the
on
partialpayment
account
89
PAYMENTS.
to the debtor
methods
w
\$1240
is loaned
are
June
follows
as
4, 1890,
28, 1890, \$462.20
Feb.
20, 1891, \$112.
due
in 7
\$462.20
"
\$112.
"
;
Dec.
is the amount
\$240.
Payments
:
Sept. 6, 1890, \$240
What
interest.
6%
at
April 10,
mos.
3
"
1
"
4
1891
;
?
dys. amounts
\$248.56
to
13
"
"
"
21
"
"
"
470.134
112.952
\$831.65
\$1240
in 10
6
mos.
dys. amounts
\$1303.24
to
831.65
Amount
due
If the account
calculated
end
of
1891
for much
\$471.59
=
than
more
the
on
payment
till the
runs
April 10,
principalfor
during the year,
1
the
balance
The
year.
and
year,
from
a
the
year,
on
time
is taken
interest
each
it
partial
was
for
is
a
new
principal.
Problems
150.
A
Sept. 18,
How
much
demand
1890.
is due
note
Method.
Merchants*
under
for
Indorsements
\$600,
at
6%
:
Dec.
12, 1890, \$45
Mar.
16, 1891, \$240.
Aug. 14,
1891
?
;
interest, is dated
90
AND
HOW
\$4000
at
NUMBERS,
A
151.
How
for
note
is due
much
PAYMENTS.
is due
1246.10
1460.
131.84
3, 1889.
May
?
OF
EQUATION
is dated
6%
30, 1891
Mar.
THEM.
USE
TO
in 1 month.
"
"
2 months.
"
"
3 months.
11837.94
When
the
may
loss to either
without
We
will
which
the
the
as
first amount
becomes
date
what
were
lose the interest
date
for
due.
is assumed,
the
at
be
amounts
paid
all at
once,
?
party
first assume,
convenience
payment
these
of
sum
time
It is
any
the
payment,
date
a
matter
mere
will
assumed, the
date
of
If
debtor
on
would
on
\$1460.
131.84
fori
month
for 2 months
=,
at
6%, \$7.30
=,
at
6%,
1.32
\$8.62
The
date
assumed,
assumed
so
be
required
must
that
interest
to the
date
the
enough
on
required,will
later
\$1837.94,
be
\$8.62.
than
from
the
the
one
date
EQUATION
The
interest
ffj
30
X
The
28
payment
If
we
have
would
of
As
amount
have
is the rate
the rate
may
be
may
be done
of interest
in the
of
^
that.
The
per
taking
followingmanner
are
interest
day.
problem
into account.
rate
any
final
tables
interest
mill
we
\$8.62, and
this rate, the
At
a
\$8.62
of
affect the result,the
does not
without
performed
Unless
same.
and
days.
fifth less than
usually taken.
month,
per
28
fifth less than
one
one
the
been
5%,
at
amount
assumed.
in 1 month
is due
=|9.19.
6%
at
date
interest
an
\$9.19, an
is 1 cent
for 1 month
\$1837.94
obtained
used, 12%
\$1
of
91
PAYMENTS.
after the
days
=
calculated
result would
on
\$1837.94
on
=
OP
It
:
246
1460
132
X
1
=
1460
X
2
=
264
By paying
the
at
the debtor
time
sumed,
as-
would
lose
1724
1838
equal
amount
an
30
\$1724
on
1838) 51720(28
\$51,720
3676
est
for 1
for
day
\$1838 for
on
est
to the inter1
month,
or
the inter-
=
28
days.
14960
14704
In
Arts,
Average
the
followingaccount
Mr.
a
Debtor
1 month
28
days.
:
to
B.
Mr.
Credits,
Due
April 17, 1891, \$240.38
"'
July
2,
"
"
Sept. 26,
"
100.46
56.20
\$397.04
360.
\$37.04
May
17, 1891, by cash, \$160.
June
14,
on
2
mos.
"
**
mdse.
credit,
200.
\$360.
92
AND
NUMBERS,
Interest
on
\$100.46
HOW
for 2
56.20
dys. at 12%
9
"
THEM.
USE
15
mos.
5
"
TO
"
\$2.51
=
2.98
""
"
=
\$5.49
Interest
on
\$160 for
200
"
1
3
"
12%
at
mo.
28
"
dys.
"
\$1.60
=
7.87
"
=
\$9.47
5.49
\$3.98
Interest
on
\$37.04
for 1
at
mo.
37)398(10
12% =\$.37.
dys.
23
mos.
370
"28
30
840
74_
100
The
assumption
full amount
of
\$5.49
\$200
in
gain,the
for this
beforethe
\$3.98
is the
for
mos.
on
on
less 10
of the
June
time, to the
of
these
making
loss to the
May
on
gained \$3.98.
In
of the account
must
debtor
17, and
payments
the
on
Striking the
interest.
in
the
which
on
order
to
make
up
be considered
as
assumed.
interest
payment
Suppose
has
a
by paying \$160
gained \$9.47
balance
date
April 17, 1891,
2
But
balance, he
the date
as
due, involves
was
interest.
date, he has
interest
time
April 17, 1891,
\$397.04
Aug. 14,
on
assumed
due
of
of
\$37.04 for
23
mos.
dys.
balance
14, 1891, B
amount
of
10
=
23
mos.
May 25,
dys.
1890=
equated
\$37.04.
of
\$300.
merchandise
The
account
of
now
A
on
stands,
EQUATION
OF
93
PAYMENTS.
Dr.
Or.
April 17, 1891, \$240.38
July
2,
""
Sept. 26,
**
17, 1891, \$160.
May
100.46
300.
"
Aug. 14,
i^
66.20
397.04
\$397.04
\$62.96
Without
is
counting interest,there
Interest
\$100.46
on
66.20=
"(
Interest
\$2.61
=
2.98
A.
\$62.96 due
\$160
on
"
\$1.60
=
300=
"
H.80
\$6.49
\$13.40
6.49
\$7.91
Interest
for
\$62.96
on
"
There
being
balance
of the
the balance
gain
of
be deferred
may
12%
\$.629.
=
17dys. =\$7.91.
"
being
account
at
mo.
12
"
net
a
1
interest
due
the debtor, and
to
the
to
the
of
debtor, payment
long enough
to
make
for this
up
gain.
April 17,
date
for B
If
on
are
if the
the account
of
assumed
balance
There
"
In
balance
is
fact, the
of
Equated
\$62.96.
and
the
and
the account
really nothing
6%
in
is
rate
partialproducts
methods
The
=
balance
of
be
are
the assumed
balance
the
date will
interest
of
earlier
interest
than
the
date.
Note.
rate, the
4, 1892
May
=
oppositesides^the equcUed
on
of 6.
dys.
the balance
A
of
17
mos.
side,the equated date will be later than
same
;
12
+
to pay
the balance
the
date
1891
are
interest,Part
question that
of the
account
to be
altogetherpreferable.
smaller
and
easier
to
the
With
handle.
See
smaller
cut
short-
III.
usually arises is,not
due, but how
much
at
date
what
is due
at
some
is the
par-
94
NUMBERS,
ticular date.
may
be, how
it under
the
HOW
AND
Thus, in the
much
does
B
just
case
THEM.
considered
Oct. 1, 1891
A
owe
of
USE
TO
the
?
question
This
brings
partialpayments.
\$405.15
Or,
Amount
i(
of
"c
\$160
300
for 4
1
"
14
mos.
dys.
17
"
=
\$163.57
302.35
"
=
\$465.92
405.15
Balance
due
A
Oct.
1, 1891
=
\$60.77
TABLE
Showing
the number
of
days
date
from
in any
any
other
date
in
month.
one
month
to the
same
96
AND
NUMBERS,
HOW
THEM.
USE
TO
Or.
June
1, 1891, by balance
Oct.
4,
"
"
cash,
Oct.
16,
"
"
mdse.
account, \$142.76
ol
;
251.76;
150.42.
dys.,
30
Dr.
Jan.
156.
;
44.88;
"
60
"
30
"
321.84;
"
936.90.
May
6,
"
Aug.
4,
"
"
16,
"
**
Nov.
dys.,\$120.60
60
to mdse.
17, 1891,
90
Cr.
Apr. 16, 1891, by
cash
Oct.
"...
14,
.
"
.
\$640.
.
500.
DISCOUNT.
True
debt
A
is
of
required
considered
The
2
mos.
the
6%
18
dys.
=
6%
at
2
at
18
mos.
dys.
the
once,
use
principalwill
amount
true
for the
\$1
between
\$246.10
given
of the
to
158.
\$1000
159.
\$164.76
due
due
\$246.10
in
time.
\$242.94
mos.
account.
present worth
\$1642.80 due
and
for 2
\$246.10
on
157.
being
money
Am.
discount
is the
of
amount
?
""242.94.
difference
What
annum?
per
of
amount
in
debt
the
pay
present worth
What
is due
question is,what
^1Sir
is the
to
worth
\$1,013
The
\$246.10
Discount.
in 4
mos.
in 2 yrs. 5
in 3
mos.
of
3
mos.
17
dys.?
?
dys.?
18
is
\$242.94.
\$3.16.
dys.
This
\$242.94
is
97
DISCOUNT.
160.
\$500.25 due
161.
\$1600 due
162.
\$5647.28
163.
\$223.50 due
in 30
due
10
mos.
dys.?
in 90
in 60
dys. at 10%
dys. at 9%
due
in 30
dys.
} \$700
due
in 60
dys.
90 dys.
\$247.62 due
I
in
in
2
I hold
suppose
15
mos.
writing my
name
to
who
one
any
its
due, provided I
checks
provided they
of
indorsement
Wanting
dys. to
18
mos.
for 2
mos.
avails
What
or
?
of
all be
the
not
the
failure
to
by
th" order
is
blank
a
particularperson
names
no
use,
I take
me
it when
pay
transferred
payable
by
payable
note
makes
and
does
order
my
the note
legal owner,
just mentioned
it
the money
discounted.
2
month
to
pay.
ment,
indorse-
of.
An
ment,
indorseto whom
payment
18
makes
if the maker
are
called because
so
This
may
kind
the
per
I indorse
legallynotified
am
%
\$246.10, payable to
its
payment
Notes, drafts, and
1
at
interest.
become
may
responsiblefor
for
note
a
the back.
on
?
Discount.
without
dys.
?
.
Bank
Now
dys.?
\$250
f
164.
in 2
to
the three
bank
The
run.
the note
days'grace,
deducts
the
the
to
a
note
interest
to be
bank
has
2
mos.
\$246.10 for
on
dys. \$3.20. This is the bank discount on \$246.10
\$3.20
\$242.90, the
\$246.10
=
-
proceedsof
the note.
is the bank
discount
in 30
and
the
proceeds of
165.
\$1560 due
166.
\$1860 due in
167.
\$520.94 due in
168.
\$257.42 due
in 4
mos.
at
7%
?
169.
\$320.40
due
in 2
mos.
at
8%
?
dys.?
60
dys.?
90
=
dys.?
a
note
for
98
AND
NUMBERS,
HOW
TO
USE
THEM.
_
*
the list or
Frequently
40%
sellingprice
being
the list
\$60.
is
If sold
this
is reckoned
and
40
Goods
listed at
\$110, discount
171.
Goods
listed at
\$560.50, discount
172.
5 doz. articles at
173.
16
174.
150
bu. at
175.
325
bbls. at
\$14.40
at
per
commercial
As
may
of
goods
listed
\$100, which
have
been
sold
3%
in
for cash
be
is the cash
176.
16
6%
Note.
in 10
on
30
and
60
5?
6?
?
4
may
a
were
time, with
mos.
dys.
be
on
cash
discount
sold at 40
a
further
this basis, the
On
tion
30
and
also.
10,
discount
dys. price
\$52.38.
What
and
at
50, 10, and
?
commonly
are
the
would
or
transactions
there
Thu9
?
Discount.
discount
a
35%
10%
to
The
50.
10?
discount
5%
\$5.20, discount
Cash
60 and
gross,
\$1.35, discount
counts.
dis-
goods
make
doz., discount
per
count
dis-
a
\$60.
on
170.
\$1.36
at
more
do
not
certain
a
or
and
10
the
termed, the
above-mentioned
sellingprice of
gross
it is
two
are
is the
What
of
sold
are
off,as
there
\$100, the 10
on
40
is not
10, the
40
case
\$100
at
price,or
goods
discount
a
Sometimes
at
In
bring \$54.
40%
there
from
of
catalogue price
Thus, if goods listed
percentage.
of
Discount.
doz.
price of
articles
at
off for
spot cash ?
Cash
a
dys. or
is
30
flexible
dys.
\$.85
term.
per
It
may
doz., discount
mean
payment
40
and
10,
immediately,
99
DISCOUNT.
4
6
177.
cash
for
gross
\$35
at
and
drafts,
as
maturity.
Nominally,
become
and
in
due
two
the
before
bears
debt
any
E.
invoice
species
or
bill
of
Atwood,
:
30
dys. 2%
bear
10; and
2%
after
interest
interest
Boston,
10
dys.
discount
of
goods
after
it
has
are
illustrated
:
Conn.,
Mar.
30, 1891.
Mass.
To
Terms
and
payable.
last-mentioned
following
stated, always
Norwich,
F.
60
?
Notes
The
discount
gross,
per
CHASE
MFG.
CO.,
Dr.
PART
CONTRACTIONS
III.
AND
"
EXPEDIENTS.
-odi^oo-
BY
in
Suppose,
24, and
easily
This
the
next
is much
27.
and
4
tens
by 1,
The
the
the
thus
is
smaller
3
8
51299 1
1
5"47
12
8
7"9
7^'0
5
7"0
9"4
5
7
2"5"4^"3
5
6
3
of
the
of
time
34.
and
then
3, 7,
increase
the
operation.
one
the
form
to
can
we
group
can
digits in
the
grouping,
easy
united
expressed
sums
idea, first disregard the
get the
smaller
;
8"
these
get
expressed by
3, 12, 10, 6,
figures (9, 11, 7, 6;
small
the
latter
the
take
Now
etc.).
to
numbers
the
columns,
of
so
group
and
figure numbers,
numbers
columns
those
the
in
columns.
5"
16
9
2
4
9
6
3
After
Never
one
can
single digits, he
niaterially by
make
we
same
7,
24
to
by practically
digits, and
as
0
6
24
to
3
4,
reached
24, making
to
be
the
at
example
rpo
0
2* 9^*2"1'
2
24, and
10
to
7
beyond
14
an
be
and
3
are
it would
next
to
has
sum
figures :
11
9"3
3
4
figures;our
and
10
being mentally
51051471141156
4
be
following
columns
digits to
than
digit
into
of
7 into
easier
If
the
and
3
column
a
two
combine
7 to
by
GROUPING.
skip
simple
a
digit
can
increase
and
smoothly
his
practising grouping.
combinations
and
return
that
to
it
can
speed
very
At
be
by
first
easily
afterwards,
101
bijt
102
clean
everything
up
the process,
to
into
a
Most
and
the
three
much
grouping.
and
of
three
should
be
put digitsenough
to
little
A
10.
over
sum
used
becoming
After
rapid figuringgive directions
on
columns
do
THEM.
along.
go
the
make
at
practical. Unless
can
you
USE
TO
practice
produce astonishingresults.
will
books
as
HOW
general rule
to
group
day
every
he
AND
NUMBERS,
better
The
^o be
work
columns
be
may
a
method
of
indication
following general
at
not
two
very
lightningcalculator,*'
"
the
by
is
This, however,
once.
happens
one
for
is
once
singlecolumn
the
of
given
two
way
for the
sake
completeness:
161
The
2190
process
42 +
57
=
50
99, +
If while
So, if
carry.
15486
91805
43
45172
46
47114
82878
653103
to
46
38
we
161.
columns
to
necessary
how
out
perform
a
first,and
for any
column,
the
process
rupted,
be inter-
go
back
to
the
there
were
many
get
a
required,and
still further
must
we
unnecessary
61
full amount
653103
work,
of
we
each
as
difierent
a
reason
the
the
second
of the
is
COLUMN
find
Thus:
combinations.
by
=
column
83930
31135
LONG
it becomes
91799
98214
shortened
149, + 12
=
IN
CARRYING
65570
be
may
may
to
a
test
footing
back
for
To avoid
this
express
the
go
column,
partialamounts.
vious
pre-
and
then
\
104
We
may
the remainder
number, the remainder
third
number, and
USE
1 with
the
Also,
on.
THEM.
digitsof the second
with the digitsof the
obtained
thus
so
TO
HOW.
AND
NUMBERS,
may
omit
Drop
the
we
9*8
as
we
go
along; thus,
12, 19
1
(6 + 6=12,+7
drop 4
+ 4
5, 13, 24-6(l
12, 13, 23-5
10, 12,
(6+
6
5
+
19.
=
9 t)f
19, also
9).
=
13,+ ll
=
5, + 8
=
12, etc.).
=
2+4
24.
=
=
6).
15-6
9, 8, 14
5
-
12,19, 9-0(5
+ 7
1 +
12, 15
7
12,+
=
8
19.
=
Drop
the 9 of 19.
9).
=
6
10, 16, 21-3
5
8
9, 6,12-3
number
Any
Let
r
be
may
resolved
9's +
be
X
9's +
9'8 X
up
into 9*s +
r
+
9's X
up of 9's +
9's,each
that
the
9 +
r^ Xr
first one
4
3
=
3
2316656167293
.
=
have
we
of
of 9's.
Product
12
remainder
except the
term
543639
=
number
remainder.
a
a
multiplyingout,
4261387
rV
other
Any
resolved
rV ; for
+ r'
der.
certain remain-
(read r prime) represent this remainder.
The product of the two num-
rV
9's +
a
Let r'
r
9's + r'
=
9's +
represent this remainder.
may
Product
into
3.
The
product
product obtained
To
=
9's +
.
3.
The
.
inference
is correct.
to division,we
apply the principle
that a dividend
quotientX divisor
=
3
12
.
have
+
only to
remainder.
ber
remem-
is
OUT
CASTING
105
NINES.
THE
3075
and
9's is the
the
Casting out
best
possibletest for multiplication
division.
Application
of
the
38"
=
54872
38
=
9's +
Application
op
the
9's
=
Involution.
to
8.
9's +
2.
9's
2^
8.
=
Evolution.
to
176413(420.015+
16
420015
82) 164
=
9's +
164
3
3
8400iyi30000
84001
9-0
399775
=
9's +
4
840025)4599900
4200125
399775
4
176413
=
9'8 +
106
Perform
the
1-13;
179.
14-25
180.
1-25
181.
the
what
Table
Practice
178.
of
USE
TO
THEM.
and
following problems,
From
record
HOW
the
test
results
by
the 9*8.
castingout
No.
a-",f-j,k-o, p-t,
a-A, i-q,
whole
there
r-z.
Table
Practice
of
is to
each
carry
downward
u-z.
n-z,
; a-m,
;
53.)
(Page
2.
of
No.
3.
the
Test
time.
and
upward,
Keep
also
the
result
by casting
the 9's.
out
Multiply Or-jin
182.
No.
2
by
the
same
same
same
186.
in
a-h
187.
in
a-o
eight
first
lines of Table
each
line from
9
to
16
by
i-7n
in
line
each
from
17
to
hj
25
p-um
line.
a-j
in each
the
of
eight lines by
first
h-m
in
line.
Divide
a-h
in
Divide
a-o
each
line
line
What
is the
square
189.
What
is the
cube
190.
What
is the
fourth
191.
What
is the
fifth power
192.
Extract
534533475567.
by
i-^m
in the
from
17 to 25
by
jh-u
in the
places.
188.
the
9 to 16
from
places.
in each
line, to 2 decimal
same
the
of
line.
same
line, to 3 decimal
same
each
line.
Divide
185.
the
in the
Multiply
184.
the
A-m
Multiply
183.
of
AND
NUMBERS,
of
61341?
of 627?
square
of
of
power
root
of
4585?
243
47?
of
4231406?
of
?
of
of
65?
828051;
621
of
?
227834749;
193.
Extract
194.
Extract
407458
; of
23140625
the
of
sixth
and
To
"
denominators
the
WHOSE
1.
A
for
the
numerator
the
denominator
the
for
difference
the
;
product
is the
What
also the
iandi?
iandi?
^^andl?
is the
product
99
"
and
6749
60741
;
sum.
for
of
^ij-and^?
99 ?
674900
6749
denominators
sum
SUBTRACTION.
BY
of 6749
the
of the
difference
^andi?
MULTIPLY
of
denominator.
iandi?
TO
A"-
=
the
between
for the
and
sum
A
+
iandi?
What
of
NUMERATORS
A-
denominators
numerator
196.
A- i
=
subtract, take
the
place,
^.
the
multiply
decimal
1
to
root,
FRACTIONS
1
places, of
75567287960.
TWO
J
decimal
2
root, to
ARE
numbers.
three
same
3259210133.
;
of the
root
fourth
the
Extract
195.
TO
the cube
107
MULTIPLICATION.
AND
668151=
=
=
x
6749
1 X
6749
100
99x6749
60741
668151
By annexing
6
figuresless
two
than
O's and
is
subtracting,we
required by
the
do
the work
ordinary method.
with
108
AND
NUMBERS,
The
O's need
well
just as
not
thus
be
HOW
TO
The
written.
USE
THEM.
subtraction
be
may
done
:
6749
6749
668151
To
multiply by 999, multiply by
multiplicand ;
twice
to
the
1000
subtract
and
multiply by 98, multiply by
and
100
tract
sub-
multiplicand.
Multiply
197.
3216
by
99.
3007
by
999.
32167
5648
by
98.
4067
by
997.
6750
WHEN
THE
MULTIPLIER
EITHER
2467
7345
16
31
14802
OF
We
one
HAS
WHICH
let
can
of the
the
by
TWO
IS
by
998.
9994.
DIGITS,
1.
multiplicand
stand
partialproducts ; thus,
7345
7345
2467
2467
22035
14802
22035
39472
227695
39472
227695
198.
the
Multiply
57
13.
by
287
142
by
16.
2612
307
by
19.
479
1286
540
by
by
41.
71.
by
31.
by
by
140.
107.
1642
by
112.
3756
by
321.
1635
by 13^.
1503
by
610.
329
2287
by
109.
1641
by 107^.
by 301^.
for
THE
THE
67436
In
11
67436
the
^^^"
^^ ^^"
Then
do
this
result
the
multiplicand for
tens
come
thousands, and
well
write
and
of the
tens
product.
as
have
partial products, we
and
and
just
can
these
units
hundreds
67436
LOCK-STITCH.
109
LOCK-STITCH.
on,
so
mentally
without
as
and
to
the
hundreds,
pairs.
in
we
go
along,
the
down
putting
We
partialproducts.
27
X
11
X
there
11
737.
=
we
may
of the
to
put the
of
sum
being 13,
the
digitis
3 ;
6 to 7.
raise
we
intermediate
the
47
by
11.
2284
by
11.
15641
41
by
11.
1365
by
11.
141089
by
11.
97
by
11.
2289
by
11.
365414
by
11.
the
of
each
the
be
must
X
14
carry
and
that
=
the
digit of
the left-hand
605696.
4 X
4
=
16
29
1 +
4x6
+
4
=
2 +
4x2
+
6
=
16
1 +
4x3
+
2
=
16
1 +
4x4
+
3
=
2 +
4=
next
20
6
is 1,
the units
lower
digitof
of the
the tens
multipliedby
11.
first of which
multiplicandby
the
digit of
multiplier,and
to
digits,the
two
by
Principle.
Lock-Stitch
the
multiplierhas
multiply
remembering
43264
merely
Multiply
Extension
When
7
6 +
1 to carry,
being
199.
have
them.
digitsbetween
67
We
297.
=
order,
tiplicand
the mul-
multiplier.
110
When
we
multiply each
the
multiplicand
the
be
must
first of
which
multiplicand by
the
and
to carry,
THEM.
digits,the
double
USE
TO
two
digitof
multiplierand
order, remembering
the
HOW
multiplierhas
the
may
of
AND
NUMBERS,
digitof
the
the
the
multiplied by
2,
units
lower
next
left-hand
that* the
is
digit of
tens
of
the
multiplier.
67431
X
26
1753206.
=
1=
6x
6x3+
2
=
20
2 +
6x4+
6
=
32
3 +
6x7+
8
=
53
5 +
6x6+14
=
55
=
17
5 +
2341
654
by
by
13.
82403759
Total
units
of
the
AND
19
teens
of the
X
annex
units.
a
26.
28.
by
NUMBERS
24.
by
23.
BETWEEN
INCLUSIVE.
(15 + 3) + (3 X 5)
smaller,
by the product
10
by
254037925
17.
TWO
=
by
5671543
16.
ANY
product
multiply
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MULTIPLY
11
18067
19.
by
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To
12
Multiply
200.
TO
6
to
the
cipher,and
larger number
increase
this
the
result
112
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202.
165
by 33^.
168
by
25.
175
by
54.
186
by
55.
218
by
150.
16| by 15^.
76i by
75.
347
by
180.
86207
281
by
45.
674
by 112^.
162
by
225.
164
by 37f
1567
169
by 66| (= i^).
by
by
95.
75.
MIXED
MULTIPLYING
NUMBERS
MIXED
MULTIPLY
TO
is the cost 'of
If
17^
3f
of tens
and
The
fraction
of
l
+
of 17 +
,
products taken
to the
I
of 17
nearest
11
=
17^
?
cents
numbers
consisting
+
be
may
3x17.
omitted, and
other
the
cent.
17
=
86
;
i
of 85
11).
=
1
Sxi
=
3x17
=51
is the cost
@
two
3x^
(5 X
63
What
THE
shall have
fraction
a
would
we
units, we
I of^
cloth
Sf yds. of
multiply as
we
TO
UNIT.
NEAREST
What
113
NUMBEKS.
of 13
cents.
lbs. 11
oz.
10
(3x13
=
39;
3
(4x11
=
44;
of sugar
iof89
@
=
T^of44=
4f
cents
?
10).
3).
52
65
cents.
Multiply 12| by 6^.
12^
11
6
72
89
6 X
5J.
f
product, which
for this
=
But
is
the
fractions
dropped, will
by calling6
X
6.
-J^,
be
both
being large,
considerable.
We
make
their
up
114
AND
NUMBERS,
Multiply
203.
to
TO
HOW
the nearest
MULTIPLICATION
227374
MULTIPLES.
BY
in
14
147
THEM.
unit
16241
113687
USE
the
7, we
twice
product by
7.
7, then
product by
14.
by
multiplierbeing
multiply by
first
this
partial product
2.
2387427
should
Care
make
67.034
come
52.15
mistakes
134068
335170
1005510
3495.82310
product by
2.
product by
5.
product by
15.
the
digitsof
in
to avoid
footingthe
when
Put
figure
product
under
figureof
its
of
the
each
the
to
order
same
line,in order
products.
hand
taken
be
tial
par-
rightpartial
right-hand
multiplier.
CRISS-CROSS
115
MULTIPLICATION.
Multiply
204.
1647
279.
by
953
by
30535.
34681
by
27381.
866.
by
20371
23146
by
6024.
134437
by
43619.
by
48612.
857032
by
360724.
165432
346281
by
56728.
8562371
12406
by
213363.
by
314065817
8643.
CRISS-CROSS
972648.
by
MULTIPLICATION.
Analysis.
21
units,
3x1.
(3x2
43
tens
units
X
units
=
units.
1
units
X
tens
=
tens.
J
tens
X
units
=
tens.
tens
X
tens
=
hundreds.
,
*
63
84
(4
tens
X
1
4 tens
X
2 tens,
multiply 21 by
43
hundreds,
903
If
the
we
partialproducts
total
and
63
Units
product
tens.
These
write
the
at
tens
of
the
little
the written
ordinary method,
tens, which
find
we
we
by unitSy
so
we
write
may
Units
by
the
what
total
tens
may
the
tens
and
is to
units
tens.
the
the
give
mentally, and
Tens
carry,
by
which
by
tens
the
units
figure of
combine
product.
there
unit's
for
have
we
and
obtain
we
only partialmultiplicationof
once.
product, and
only.
tens
partialproducts we
hundreds.
a
84
and
tens
is the
units,
is
product
After
by
units
by
total
units
the
this process,
product. Analyzing
by units,
with
by
by
we
tens
express
give
this
multiplicationis complete.
practice it
is
just
partialproducts,and
as
express
easy
to
do away
the final result
116
KIJMBEBS,
AKD
HOW
USE
TO
THEM.
903
The
foregoing
process,
should
and
not
be
as
given
other
The
43x1
Multiply
the
reason
For
way.
=
67
43;
by
as
indication
an
performed.
for that
some
is
of
by
37.
93
by
48.
62
by 51.
'
of
a
general
particularproblem
multiplicationcould
be
done
instance,
43x2
=
86, +4
48.
Multiply
54
this
how
digitsare
3216
205.
illustration
an
86
by
43.
29
by
26.
87
by
54.
=
90.
Total, 903.
small,
easier
Multiply
121
by
117
MULTIPLICATION.
CRISS-CROSS
243.
Analysis,
121
3x1
243
3x2
tens
363
3x1
hundred
484
4 tens
X
1
242
4 tens
X
2 tens
29403
4 tens
X
1 hundred
2 hundreds
X
1
2 hundreds
x
2 tens
2 hundreds
X
1 hundred
Or, differently
arranged.
units,
3x1
3x2
{
tens
tens
4 tens
hundred
3x1
I
hundreds
2 hundreds
3 xl-3
thousands,
1
X
4 tens
X
2 tens
4 tens
X
1 hundred
1
thousands
ten
1
X
2 hundreds
X
2 tens
2 hundreds
X
1 hundred
21
121
12
X
I
X
43
243
24
3x2
=
6
4x1
=
4
10
3x1
=
3
1
tocarrj
2x1=2
4x2
4x1=4
to
carry
2x2
=
8
2
4
2x1=2
1
4
9
14
29403
Notice
written
that
but
the
the
work
is
final result.
entirely mental.
Nothing
is
118
AND
mJMBERS,
HOW
Modifications
One
who
performed
has
some
employ
modifications
digits,and
are
simple,and
apparent.
to
fall back
THEM.
Cbiss-Ceoss.
the
the
Any
fairlyapt
one
with
the
method
are
special cases
of
shortening of
the
a
after
be
of
USE
previous pages understandingly,and
how
to
multiply criss-cross.
may
TO
a
little
However,
upon.
the
always
has
use
The
criss-cross
process.
practice the
one
in the
figures
following
with
They
short-cut
the
of
will
are
two
very
usually
general principle
120
NUMBERS,
When
AND
Tens
the
aee
HOW
TO
the
same
36
4x3
tens
hundreds,
34
10 X
3 tens
In all
cases
+ 3 tens
like
units
X
for
=
Units
the
of
the
4 tens
multiply
the hundreds
This
3 tens.
X
of the
units
3 tens
and
10
=
product
this,we
higher number
next
tens
and
be the tens
will
1224
the
so
THEM.
10.
TO
6x3
+
TTSE
units
makes
(4x6
=
24)
X
12 hundreds.
3 tens=
the number
of
by
the
Units
by
tens
of the product.
completestheprodiLcL
68
7
62.
X
6
42.
=
Annexing
2 X
8
=
16,
we
have
4216.
4216
To
35
35
1225
210.
number
a
square
oj
tens
number
by
the
48.
65
by
65.
24
by
26.
79
by
71.
15
by
15.
35
by
35.
83
by
87.
92
by
98.
55224
6, inultiplythe
higher number
and
annex
^
Multiply
by
234
next
with
25.
42
236
ending
Considering
annex
4x6.
23
as
tens,
we
multiply
23
by 24,
and
THE
121
CRISS-CROSS.
Multiply
211.
547
by
543.
996
by
995.
872
by
878.
635
by
635.
425
by
425.
659
by
651.
406
by
401
438
by
432.
174
by
176.
745
by
745.
885
by
885.
351
by
359.
723
by
727.
125
by
125.
238
by
232.
375
by
375.
Tbe
whole
*^
principleapplies
same
is alike
number
both
in
numbers
mixed
to
and
factors
the
wben
fractions
tbe
^-
3f
15|
240H
12f
Multiply
212.
14^ by Uf
12i by 12J.
17| by 17^.
62f by 62|.
I^byl8}i.
225iby225i.
36^ by 36f
27^ by 27f
When
Units
the
are
the
TO
same
.
and
Tens
the
10.
"
This
67
47
6 tens
we
digits.
This
givesthe
rest,
63
43
but
a
slight modification
of
the
previous
principle.
3149
-So
is
The
X
7 +
4 tens
the number
gives
of the
hundreds
product
of
cipher occupiesthe
units
lens*
x7
=
units
of
by
the
100x7
to the
7 hundreds.
=
product of the
Units
units
place in
being
the
X
less than
tens*
units
10,
a
122
AND
NUMBERS,
167
47
16
Considering
of
hundreds
the
TO
HOW
USE
tens, the
as
the
THEM.
tens
have
we
4 X
So
20.
to
16
for
7.
twice
+
7849
214.
Multiply
163
by
43.
186
141
by
61.
59
by
159.
102
118.
65
by
145.
87
by
98
When
Digits
the
of
THOSE
44
73
are
OTHER
the
number
the
Make
Factor
one
OF
74
26.
by
whose
the
same,
and
10.
the
digits are
3
X
10
X
4 +
7
X
10
X
4
=
10
=
4 hundreds.
'
7x4
Multiply
of the
127.
the
same
multiplicand.
3212
than
102.
by
by
TO
134.
by
the number
the rmraber
hundreds
of tens
Units
+
4 hundreds
the
in
in
multiplier.
by
units
10
8x4
=
of tens
the
X
X
4
=
gives the
rest.
X
hundreds.
multiplicand by
This
100
one
more
gives hundreds
4
MULTIPLICATION
MULTIPLICATION
DECIMALS
OF
EEVEESE
What
is the
BY
THE
METHOD.
and
of 1246.329
product
123
DECIMALS.
OF
to two
decimal
Reverse
Method.
1.435
places?
1246.329
1246,329
1246.33
1.435
5341
1.435
i
6231645^
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00
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1788,482115
There
multiply
places
decimal
three
decimal
the
usual
manner
in
the
If, in order
thousandths
6
over
9 thousandths
.01.
The
best
we
saves
only
do
can
four
and
in
figures,and
simple
Hundredths
hundredths
in order
one,
by
;
units
units
hundredths.
under
the
So, if
the
of to the
each
the
time
with
we
hundredths
remaining digttsof
what
right,and
;
digitof
the
tenths
it is
a
by tenths,
the
of
in
a
do.
to
units
by
the
that
sandths,
thou-
plier
multi-
multiplicand, and
multiplierreversed,
But
through
go
; tens
the
of
call the
it is.
hundredths
write
of
to
the
error
particularcase
produces hundredths
by hundredths,
is to
way
of
drop
an
multiplieras
this
in
find out
to
of
have
we
be
would
off the
error
an
should
we
sort
the
retain
preliminary calculation,though
very
If
this
be
two
don't
we
strike
would
multiplicand,there
the
multiplicand 1246.33,
this
obtained.
product
of
should
figures,we
save
that
we
off six
get the
to
more
multiplier,there
the
of
the
in
to
four
get
we
point
to
order
is, in
factor, if
each
have
shall
we
want,
in
places
That
product.
figures we
want.
5
being
in
1788.48
1788.48355
the
is,4, 3, 5
to
multiplying,commence
multiplicand directlyover
the
124
of
one
lower
the
each
of the
two
decimal
What
THEM.
using, disregarding the
with
multiplicand,
is to carry
partialproduct
partialproducts
the next
from
be
will
exception
lower
ination,
denom-
be
will
the
and
hundredths,
in
obtained
thus
the
to
correct
places.
is the
decimal
there
USE
TO
then
are
the
of
in what
sum
HOW
multiplier we
denominations
of
AND
NUMBERS,
product
and
of .23047651
35.12043
three
to
places?
.23047651
2153
6914
1152
23
6
8.094
Write
the units of the
multiplicand,and
To
be
the rest
the
of
sure
be well
to
one
really wanted,
But
if
look
we
out
of the
last
and
then
sharplyfor
216.
the
product, it
further
place
omit
the
what
there
usually be exactlyright without
of
the
multiplierreversed.
figure of
decimal
one
go
thousandths
multiplierunder
than
is to carry,
this
the
last
figure obtained.
last
taking
times
some-
may
shall
we
precaution.
Multiply
12.491
by
.36 to two
.17653
by
122.5
60.31654224
by
340.6582716354
to
decimal
one
521.2
by
decimal
to
four
places.
place.
decimal
3.7012407853612
places.
to three
.1243
by
3.256
to the
nearest
thousandth.
1243.
by
325.6
to the
nearest
thousand.
33546.86
by
5728.6
to
the nearest
million.
decimal
"
places.
CONTRACTED
CONTRACTED
The
DIVISION
DECIMALS.
OF
be
principle may
same
125
DECIMALS.
OF
DIVISION
applied
the
to
division
of
decimals.
Divide
will
not
go
in
quotientwill
the
As
soon
be found
is
be tens, and
bringingdown
what
last
dropped.
may
sometimes
number
two
there
of
each
To
be
be
1764
well
3681
the
delay
of
divisor,we
the last
digitof
the
the
decimal
3681) 1764 (.48
29
figureof
quotient,it
until
contraction
in the divisor.
147
stop
In
quotient remaining
to two
figureof
multiplying,we
of the digit
multiplication
of the last
to
the
the number
by
drop
partialdivision.
sure
the first
in the
number
from
quotient.
five.
figures,
quotientremaining to
the
figuresof
the
is to carry
figuresot
less than
Divide
than
; so
total number
the dividend, and
from
the divisor before
of
the
places.
will be in the
18, but will go in 189
less
one
decimal
figuresthere
many
the number
as
to three
32.467
by
first find how
We
32
1897.5431
places.
to
be
found
the
is
126
AND
NUMBERS,
HOW
TO
USE
THEM.
Divide
217.
by
286
176.241
by
METHODS
places.
decimal
four
3.7542134
by
132674
Reverse
to
decimal
two
to
378.2
by
41
13.47
9321765
places.
six decimal
to
three
to
decimal
CALCULATING
OF
places.
places
INTEREST.
multiplicationis especiallyapplicable to figuring
interest.
is the
What
\$1642.27
of
amount
for
2 yrs. 3
16
mos.
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at6%?
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the
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and
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saved
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easy
order,
reverse
figures are
many
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Note
manner.
denomination
same
this
in
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of
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What
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from
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26
19
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July
26
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23
mos.
1642.27
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|32
of
the
to
see
where
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the first
should
be
figure
placed.
3285
Unless
want
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the
result
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492
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go
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multiplicand; hundredths,
placesbeyond.
128
There
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be
AND
NUMBERS,
are
great majority of
is the amount
\$1246.11 for
4
\$245.81
for 63
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?
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1 yr. 6
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is the interest
\$3,214
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interest for 20
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of
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On
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\$2644.32
What
THEM.
upon.
What
218.
the
cases
USE
TO
specialrules
many
improved
HOW
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int. for 30
dys.
dys.
be
may
convenient
in
a
found
in this way
periods,but the
few
simple cases
by
method
like the
STATES
UNITED
AND
All
specialinterest rules are
formula, Interest
principalX
=
there is
no
shortening the
of
way
shown.
FIND
THE
TO
founded
rcUe
United
as
States
of the
value
;
further than
VALUE
and
has
OF
MONEY.
ENGLISH
The
general
of years
any
STATES
the
upon
number
X
process
UNITED
129
MONEYS.
ENGLISH
be taken
pound sterlingmay
ent
at differ-
\$4.8665, though the proportionvaries somewhat
times.
first reduce
We
of
As
20s.
pound,
a
shillings,
pence,
and
then
It, 28. =.11.
=
and
multiply
So
we
farthingsto
this number
divide' the number
the
4.8665.
by
of
mal
deci-
shillings
if there
pound, callingthe odd shilling,
there
be one, .05.
Pence
reduce
to farthings. As
are
we
960 farthingsin a pound, 1 farthing .001 of a pound, and a
by
2 for tenths
of
a
=
little
pound,
we
exactly equal to .0121 of a
farthings are
36 farthings
to .0371.
So, if there are 12 farthings
sandths.
thou2 extra
extra
thousandth; if 36 farthings,
12
over.
and
one
As
to
a
there
4
are
and
shilling,
pence
farthingsto
a
and
penny,
farthingstogethermake
12
not
farthings.
What
is the United
"2.75
States
="2
49
\$13.62
=
value
15s.
llfrf.
of "2
15s.
llfrf.?
pence
over
47
130
AND
NUMBERS,
What
is
the
value
HOW
\$13.62
of
USE
TO
in
48665)136200(2.799
THEM.
English
?
money
pounds.
9733
.75/.
3887
M9L
3406
=
=
15
shillings.
47
far.
11|";.
=
"481
438
43
219.
What
is
the
"13
175.
6^.
"22
lis.
4cf.
value
Ans.
in
United
States
"47
?
"1127
?
"2
of
money
18s.
Hid.
3s.
llfd
lbs.
?
7c?.
?
,
"241
"56
220.
Os.
19s.
What
is
ll|c?.
S^d.
the
"462
17s.
Sid.
"286
12s.
4d
?
?
value
in
English
money
?
?
of
\$46.87?
\$3342.76?
\$3,7.41?
\$216.34?
\$207.29?
\$286.53?
\$706.51?
\$361.42?
\$541.89?
I
.
PROBLEMS.
MISCELLANEOUS
-^:*?"
221.
Complete
by casting
out
the
following table, proving
tlie
9's
the
divisions
:
MASSACHUSETTS.
131
132
NUMBERS,
SIMPLE
AND
HOW
INTEREST
ONE
DOLLAR.
TO
USE
THEM.
TABLE.
SIMPLE
SIMPLE
IKTEBE3T
INTEREST
ONE
DOLLAR.
TABLES.
TABLE.
133
134
NUMBERS,
AND
SIMPLE
360
Days
to
HOW
TO
INTEREST
Year.
8
Months
USE
THEM.
TABLE.
Days.
4%.
\$9000
136
NUMBEES,
AHD
COMPOUND
HOW
TO
USE
TABLE.
INTEREST
Amoubt
of
THEM.
\$1.00.
INTEREST
COMPOUND
TABLES.
INTEREST
COMPOUND
Amount
of
TABLE.
\$1.00.
137
I'
.
138
Complete
222.
The
AND
NUMBERS,
calculations
HOW
the interest
12) .04
=
\$1
on
1 third
previouspages.
:
for 1 yr. at
interest
=
THEM.
the six
follows
as
interest
and
.003333
USE
tables oh
may
TO
for 1
mo.
1
3333
.006666
2
3333
"
"
"
"
2mos.
=
Jl
.010000
0
"
3
=
etc.
30).003333^
and
.000111
1 ninth
Ill
1
.000222
2
.000333
interest for 1
=
"
"
2dys.
"
"
3
=
'3
dy.
=
"
etc.
The
different
last amount
ones
in
depended
be
may
having
amounts
been
each
upon,
found
by
series is
except
successive
right,the
for the last decimal
figure.
Proof.
.055
=
interest
on
\$1
for 11
mos.
at
6%.
2
3).110
.036667
.005
=
=
interest
interest
on
on
\$1
\$1
for 11
for 30
mos.
at
4%.
dys. at 6%.
3) .01
.003333
=
interest
on
\$1
for 30
vious
pre-
dys. at
4
%
.
INTEREST
If
we
now
last decimal
the
over
go
work
verifythe figuresin
and
be
shall
place,we
139
TABLES.
the results obtained
sure
the
are
absolutely accurate.
3)40.
13.33^
26.66|
interest
=
lll=
"
\$1000
'*
.
26.77
and
7 ninths
11
for 8
\$1000
on
4%.
Idy.
"'
interest
=
at
mos.
for 8
"
,
1
mos.
dy.
1
etc.
8
X
2677^
and
21422
=
88
2 ninths
=
\$8000
int. on
for 8
mos.
1
dy.
8
mos.
2
dys.
8
mos.
3
dys.
8
"
\$8000
"
"
\$8000
"
"
\$8000
"
8
mos.
4
dys.
"
\$8000
"
8
mos.
5
dys.
=
=
=
21777
etc.
Proof.
.045
=
interest
on
\$1
for 9
mos.
at
6%.
1000.
3)
46.
"
"
"
"
=
\$1000
for 9
mos.
at
6%.
9
mos.
at
4%.
15
30.
=
\$1000
"
140
AND
NUMBERS,
Verify
^/y ^J?
73
2
the
figures in
the
I.I30137
=
decimal
last
laOia?
=
THEM.
USE
TO
HOW
place
before.
as
cents
day
=
accurate
int.
on
\$1000
for 1
=
accurate
int.
on
\$1000
for 151
at
4|%
13.0137
151
1965.0687
cente
dys. at 4f %,
13.0137
1978.0824."
dys. at 4|%
"
"
\$1000
"
152
"
"
\$1000
"'
153dy8.at4f%.
=
13.0137
*'
1991.0961
=
etc.
90
^
73
%
1.03
=
of
amount
\$1
for
1 yr.
at
3%
compound
interest.
309
1.0609
"
\$1
"2yrs.
at3%
"
\$1
"3yrs.
at3%
"
\$1
"
"
\$1
"
=
31827
1.092727
=
327818
1.1255088
=
4
yrs.
at
3%
yrs.
at
3%
337653
1.1592741
=
5
4"
(I
INTEREST
141
TABLES.
/
1.036
=
amount
of
\$1
for
1 yr.
=
amount
of
\$1
for
2
yrs.
=
1.035
=
amount
of
\$1
for
3
yrs.
=
amount
of
\$1
for
4
at
1.035
6175
3105
1035
1.071225
5301
3i^%.
at
reversed.
10712250
321368
53561
1.1087179
53
1
11087179
332615
55436
1.1475230
yrs.
etc.
Take
Proof.
2
year,
the
first
etc.,
also
we
make
obtained
similar
3
years,
the
bears
the
from
amount
a
from
manner.
to
the
first
the
the
form
column
first, and
the
second,
below.
series.
the
term
preceding.
next
one
of
amounts
series, each
a
the
The
column.
etc.,
years,
ratio
same
get
1^%
third
second
These
\$1
for
1
which
of
Subtracting
the
from
numbers
The
second
from
the
third,
should
column
second,
is
in
a
142
AND
NUMBERS,
HOW
THEM,
USE
TO
15225
22
Q
15920
ifl
^l
5
16159
^l
4
16402
^l
2
16647
16898
;^V
o^5
2
17151
f.^
4
17408
17669
;^{
i^i
17934
;^^
5
18204
;;"
2
18476
^'^
5
15686
1
6
4
4
18753
5
QQo
19320
i%i
^T
6
19611
o^i
2
19035
3
293
19904
Suppose
we
of the twelfth
year,
mistake
a
We
1.195628.
year
in the
obtained
should
then
fifth decimal
the
as
place
for
amount
that
have
17408
271
17679
245
17924
280
18204
and^ the
A
would
error
slighterror
by
in
have
the
this process.
originalfiguringshould
been
apparent.
last decimal
To
be
make
gone
place might
.of the last
sure
far
as
over
not
as
be discovered
digit,the
necessary
for
the purpose.
The
use
of interest
examples
What
at
4%
be
illustrated
by
the following
:
is the
?
tables may
amount
of
\$642.37 for
2
yrs.
6
mos.
14
dys.,
144
AND
NUMBERS,
What
is the
compounded
9^ years
1^%
at
TO
\$1324.83
of
amount
HOW
USE
for
9^
THEM.
years,
3%,
at
semi-annually?
3%
semi-annually is equivalent to
19
annually.
1.32695
384231
132695
39809
2654
530
106
4
1757.98
Compute
223.
the
What
followingfrom
is the
\$756.42
\$29.60
for 4 yrs.
12
the
completed
for 1 yr. 3
\$190.99
for 8
for 2 yrs. 4
dys.,at 4%
bank
discount
30
days' note
for
\$480.12,
90
days' note
for
\$2500.,
60
days' note
for
\$542.84,
What
for
is the interest
\$856.41
for 8
mos.
dys.,at 7%
11
is the
note
?
dys.,at 10%
mos.
on
a
6%
at
at
7%
at
?
?
10%
at
?
8%
?
on
dys.,at 4%
?
?
dys.,at 8%
\$3486.22,
17
?
?
25
mos.
What
4 months'
dys.,at 4%
16
mos.
29
mos.
tables
on
dys.,at 4%
for 93
\$564.80
\$1560.
225.
interest
for 3 yrs. 4
\$2845.75
224.
est
inter-
?
?
:
years
at
INTEREST
for 8
\$1652.50
I
mos.
3
\$582.40
for 8
mos.
15
for 3 yrs., at
\$266.50
for 18 yrs., at
for 4 yrs. at
for 3 yrs. at
?
annually, wljat
is
the
?
5%
?
7%
?
4^%
?
7 mos.,
\$761.48
for 11
\$876.29
for 5 yrs. 2 mos.,
\$342.17 for
yrs.
2 yrs. 3
at
14
mos.
5%
6%
at
?
?
dys.,at 2^%
.
semi-annually,what
being compounded
Interest
227.
3%
\$762.41
\$92.56
is the
of
amount
for 7 yrs. 6 mos.,
\$2284.39
\$854.17
\$84.40
for 5 yrs., at
\$100.
for 2 yrs.
for 20
(First find,by
take
for 10
to the
the
3
of
the
for
a
3%
?
?
at
5%
13
mos.
3%
yrs., at
at
?
dys.,at 6%
?
table, the
for
amount
principaland
new
?
10
years,
find the amount
more.)
\$322.50
his
use
this amount
years
value
7%
for 3 yrs. 6 mos.,
\$1231.84
form
dys.,at 4%
being compounded
\$1241.62
As
?
of
mount
then
dys.,at 4%
for 8
?
dys.,at 4%
20
mos.
\$342.17
Interest
226.
146
TABLES.
for 35
best form
in the
own
yrs. 8
for interest
solution
tables
As
dys. at 5%
tables, and
of interest
conclusions.
compiling of
19
mos.
an
as
to their
problems, the
exercise
is invaluable.
?
in the
tical
prac-
use
of
can
ures,
fig-
146
AND
NUMBERS,
the
Complete
228.
1885)
HOW
TO
USE
following table
THEM.
(Mass.
Classified
Details
AND
Capital
Capital
op
(2,366 Establishments).
Percent-
Invested.
ages.
Amounts.
\$896,310
Buildingsand
fixtures
3,256,603
Machinery
4,613,370
and
Implements
tools
696,281
capital
20,354,644
Supplied by partners
stockholders
or
.
payable, accounts
long time,
on
1,215,355
.
3,380,858
etc.
Total
100.00
Classified
and
Land, buildings,
Summaey.
fixtures
.
Machinery, implements,
Cash
and
.
.
.
.
.
.
\$4,152,913
tools
5,209,651
capital
Credit
20.354,644
capital
4.596,213
Aggregate
A
229.
payable
bank
each
100.00
for
note
in 30
bearing the
be
there
A
date
same
is dated
second
is
the
20, 1891, and
Dec.
for the
note
payable
of
each
on
in
same
What
month.
1
notes, and
above
amount
when
and
is the
must
paid ?
The
followingis
interest
Rule.
\$12,000
days.
discount
230.
as
of
SHOES.
Land
Bills
Census
:
BOOTS
Cash
State
given
general rule
as
a
as
many
for calculating
:
Multiply
are
often
the
days, and
principalby
then
divide
as
follows
one-hundredths
:
4
5
6
7
8
9
10
12
90
72
60
52
45
40
36
30
Percent,
Divide
by
how
Show
147
PROBLEMS.
MISCELLANEOUS
is deduced
rule
the
the
general
formula
yr.
9
mos.
6
yrs.
7
mos.
27
dys. was
12
dys. was
from
for
interest.
The
231.
\$29.68.
interest
What
The
232.
\$93.15.
interest
What
The
234.
dys.
was
for
\$584
2
the rate ?
interest
What
\$295.89.
1
?
rate
on
was
The
233.
the
was
for
\$240
on
\$340.10
on
the
was
interest
for 8 yrs.
8
mos.
?
rate
on
\$682.41
on
\$1421.95
7%
at
was
\$371.80.
What
\$29.15.
What
the time ?
was
The
235.
interest
the time
was
A
236.
rate
A
3
for 1 yr.
what
239.
more
Method
discount
what
dys.
long
days' note
60
a
for the
\$128.13
The
the amount
What
Why
on
how
the
at
bank
yielded
the note
to run?
\$8.01. The
was
the amount?
was
14
was
discounted
being 6%,
rate
pays
mos.
rate
of the
principalwill
use
of
of interest
a
certain
sum
charged being
loan?
amount
to
in 6
\$684.67
mos.
12
?
Merchants'
is the
favorable
?
\$2250
borrower
dys.,at 10%
240.
for
bank
being 6%,
238.
8%,
The
The
237.
was
?
note
\$2237.62.
6%
at
debtor
the
to
Illustrate
\$1000 bears
interest
is the amount
due
by
the
from
Dec.
Method
1 ?
than
the
of Partial
United
following problem
Aug.
1.
\$500
is
paid
Payments
States
:
A
Oct.
Court
debt
1.
What
of
148
AND
NUMBERS,
If 4
241.
men
of 10 hours
do
men
The
the
work
same
of
number
42
days, 7
of
do it in 4 times
required by
analysis.
the
put
how
men
do the work
can
as
1
in
smaller
men
days
would
7
be
in
7
number
of
in
the
in
many
days.
days, 1
to
require less
in
certain
work
a
go
time
would
of
days
through
this
than'^
the
numerator,
(If4
man
^ the number
isn't necessary
men
number
the
^^as
men
will
it in
do
days, 7
that
do
can
given
a
It
man.
days, the
many
can
many
Notice
each
9 hours
of
in 42
?
If 4
days.
of work
amount
days
many
THEM.
USE
TO
certain
a
each, in how
question being
terms
men
do
can
HOW
so
men,
the
larger in
denominator.)
hours
10
If, working
9 hours
days, working
number
certain
a
it will take
day
a
it takes
day,
a
-^
as
of
days.
many
2
^^i^
?
=
=
26|days.
3
If 9 boxes
242.
at the
each
rate
same
weighing
90
each
weighing
lbs.,cost \$61.20,
80
will be the cost
what
pound,
per
of 14 boxes,
lbs. ?
of flour cost
barrels
If 56
243.
of soap,
\$317.52, what
bbls.
will 126
cost?
If it costs
244.
and
18
high, for
ton,
as
high, with
feet
will it cost
to heat
the
former, and
many
to
If
a
marks
\$.193, and
36
room
will
of
of
40
room
long, 25
time, with
coffee
oz.
\$.238?
ft.
ft. wide,
long, 22
per
ton, how
ft. wide, and
coal
much
17^
costing\$4.75
coal
heat
dimensions
7 lbs. 13
mark
ft.
latter
that the
their
a
costing \$6.50
coal
the
pound
a
to heat
length
that
proportionalto
245.
a
same
assuming
the
\$48
gives out -^
required for the
much
as
two
ft.
per
heat
rooms
is
?
costs
cost,
half, how
a
franc
and
a
franc
being equivalent
a
If 12
246.
in dollars
and
At
247.
When
minute
60
12
be
next
In
60
the
minutes
gains 11
minutes
of
a
being
39.37
watch
are
inches
?
together.
together?
the hour-hand
spaces,
metre
hands
is the cost
francs, what
28
cost
3^ yards, a
o'clock, the
they
Solution,
of
cents
12
will
of cloth
metres
149
PROBLEMS.
MISCELLANEOUS
0
minute-hand
one
How
spaces.
The
space.
12
goes
five-
minute-hand
it take
long will
in
it to
gain
spaces ?
60x12
'"
n.s
65^
=
"
"
"
mm.
1 o'clock
An8,
At
248.
The
249.
numbers
of
sum
ratio
three
other
each
to
hands
the
of
apart will they
far
How
apart.
same
o'clock
6
be
watch
a
7.45
at
is
numbers
that 3, 4, and
30
are
?
161.
minutes
8.15
at
?
bear
They
What
7 do.
min.
5^
the
the
are
?
250.
i + i + i of
251.
On
a
is 165.
number
the
the
number
is the
What
?
of
freezing-point
,
is marked
water
these
68.
What
hang
should
What
should
be
the
252.
What
253.
What
boiling-point100.
marked
points are
thermometers
two
the
0, and
side
be
and
by
side.
the
below
zero
is the square
is
32
the
square
of
heit,
the Fahren-
respectively. The
212
The
On
Fahrenheit
of
the
the Fahrenheit
ters
regis-
when
the Cen?
?
17^^
?
root
of
17-j^ to
three
decimal
places?
254.
Perform
the
operations indicated
expressions,results being required to
vTr,
three
in
the
decimal
following
places:
^/3^ 14*, 5^ (16|)*,.653', .OOOOli
150
NUMBEBS,
AND
266.
Simplify the
266.
What
and
?
-J-
257.
What
268.
How
subtracted
260.
Shopworn
What
sold at
of
A
and
bbls.
150
are
?
f
4|^ be
fractional, may
35%
when
been
an
Goods
If
mos.
.
For
cost, bring
they
\$500 be paid
what
been
an
at
and
once,
paid ?
bbls. at
\$4.80
bbls.-at
he sells 100
per
\$2.90,
ance
he sell the bal-
price must
What
40%.
be
1000
sell at
to
down
if
profitof 8%
average
marked
the balance
flour dealer
marked
below
?
should
\$3.40.
at
35%
at
portion being damaged,
to make
263.
sold
have
in 4
wholesale
A
barrel.
cost
would
in 2 mos.,
262.
and
,
denominators.)
common
multiple of 3^, ^, and
common
goods,
is due
\$2240
\$1000
12|^,^, f
part of 12f is 11| ?
\$164.16.
261.
having
of
?
16f
What
divisor
common
times, whole
many
259.
an
fractions
to
THEM.
^
greatest
is the least
from
USE
TO
fraction
is the
(Change
HOW
?
cent
per
65%
of
of
the
on
this
profitdoes
leave ?
by sellingcloth
264.
If
25%,
what
of
A
265.
6%.
at
266.
4 months'
What
I
owe
are
note
the
\$1.40 per
sellingprice be
for
\$1000
yard
there
in order
is
to
is
a
loss
gain 25%
discounted
at
a
?
bank
proceeds?
\$1000 due
to settle the
pay
the
must
at
in 4
How
mos.
much
account, interest being allowed
should
cash
at
the rate
I
of
6%?
267.
cash
A
4%.
bill of
How
goods
much
amounts
cash
to
\$1000.
Terms
will settle the bill?
:
4
mos.
or
152
NUMBERS,
AND
HOW
TO
USE
THEM.
List
Priee.
Price
Extensions.
Net
Price
Eztensioni.
MISCELLANEOUS
153
PROBLEMS.
\$201.33
Verify
actual
of
cost
price per
net
dozen
;
gross
is
otherwise, the
Example.
at
kind
\$5.40
of
goods, and
per
given, find
cost
and
the
Less
The
first article
in
gross,
60%, 10%,
and
price
price
or
sellingprice per
bill.No. 2538, is listed
the
5%
off.
=
5.40
reversed.
==
cost
of No.
14
\$1.85
.36
Net
list
171
.1=4
of .1
the
and
cost
45
"lo
^
When
sellingprice
.342
.60
Less
also the
the
sellingpriceper singlearticle.
1.00
Less
footings. Ascertain
yield a profitof 40%.
to
necessary
each
the
and
price extensions
the
=
=
18
^dd
.342 of list
pr.
.4=
.74
12)2.59
\$.22
2538
per
=
sellingpr.
"
=
gross.
per
"
gro.
doz.
```