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RUDIMENTS
A? T
ONT
NHMEROUS EXERCISES
SLATE AND BTACKBOARD,
FOR BI1--
BY OAMES B. iilOjViSON, A.M.,
AUTHOR OF >*' , EXERCISKS IN ARITn!ft<CTICAIi ANfALYS
PRACTICAL ARITTIMKTIC ; UIGHKR ARITHMETIC; EDITOR OF DAV's
SCHOOL AL^KBK/t, '.KG^NDRE's GEOMSTRV> ETC,
oo
CM
O
,W YOKE:
N , c \ T EY, 48 & 50 WALKER ST.
AGO : S. C. G RIGOS & CO., 89 & 41 LAK T
:vs & co. ST. LOUIS : KEITH & WOODS.
,. x . -23 & CO. DETROIT I EAYMOIO & SKLLiiOK.
-.i.riMS. BTJFFALO: PH1NNKY A CO.
.vaUSfr: T. S. Q[JAOKENBU8H.
.
LIBRARY
OF THE
UNIVERSITY OF CALIFOF
GIFT OF
Received ....
Accession No. 6 J ^ ^ 3~ - Class No.
an& ffifjomsan's Series.
RUDIMENTS
OF
AEITHMETIC;
CONTAINING
NUMEROUS EXERCISES
SI-ATE AND S&ACKBOARD,
FOB BEGINNERS
BY JAMES IB. THOMSON, A.M.,
JSFf P* fllf Tjfl B ^x^^ 1 ^^
AUTHOR OK MENTAL ARiTHMETicr; EXERCISES IN ARIT&MKTICAL ANALYST**
PRACTICAL ARITHMETIC; HtaH^fa ARITHMETIC; E|<ITOR OP DA\ s
SCHOOL ALGEBRA; LEGENT>RK y SGEOMETR.V, ETC.
NEW TOEK:
IVISON & PHINNEY, 48 & 50 WALKER ST.
CHICAGO: S. C. GEIGGS & CO., 39 & 41 LAKE ST.
CINCINNATI : MOORE, WIL8TACH, KEYS & CO. ST. LOUIS : KEITH & WOODS.
PHILADELPHIA; SOWER, BARNES & oo. BUFFALO; PHINNEY fc CO.
NEWBUEQ: T. s. QTTAOKENBUSH. t
1859.
o
Entered according to Act of Congress, in the year 1833, Ifjr
JAMES B. THOMSON,
fa the Clerk's Office for the Southern District of New Yoft
TKREOTTPXD BY THOMAS B. 81CITK,
216 W1LLIAH 8TRKKT, N. V.
PREFACE.
EDUCATION, in its comprehensive sense, is the business of
life. The exercises of the school-room lay the foundation ;
the superstructure is the work of after years. If these exer-
cises are rightly conducted, the pupil oh tains the rudiments
of science, and what is more important, he learns how to
study , how to think and reason, and is thus enabled to appro-
priate the means of knowledge to his future advancement.
Any system of instruction, therefore, which does not embrace
these objects, which treats a child as a mere passive recipient,
is palpably defective. It is destitute of some of the most
essential means of mental development, and is calculated to
produce pigmies, instead of giant intellects.
The question is often asked, " What is the best method of
proceeding with pupils commencing the study of Arithmetic,
or entering upon a new rule ?"
The old method. Some teachers allow every pupil to cipher
u on his own hook;" to go as fast, or as slow as he pleases,
without reciting a single example or rule, or stopping to in-
quire the " why and the wherefore" of a single operation.
This mode of teaching is a relic of by-gone days, and is prima
fade evidence, that those who practice it, are behind the
spirit of the times.
Another method. Others who admit the necessity of teach-
ing arithmetic in classes, send their pupils to their seats, and
tell them to u study the rule." After idling away an hour
or more, up goes one little hand after another with the de-
spairing question : " Please to show me how to do this sum,
sir ?" The teacher replies, " Study the rule ; that will tell
you." At length, to silence their increasing importunity, he
takes the slate, solves the question, and, without a word of
5V PREFACE,
explanation, returns it to its owner. He thus goes through
the class. When the hour of recitation comes, the class is
not prepared with the lesson. They are sent to their seats
to make another trial, which results in no "better success.
And what is the consequence? They are discouraged and
disgusted with the study.
A more excellent way. Other teachers pursue a more ex-
cellent way, especially for young pupils. It is this : The
teacher reads over with the class the preliminary explanations,
and after satisfying himself that they understand the mean-
ing of the terms, he calls upon one to read and analyze the
first example, and set it down upon the blackboard, while
the rest write it upon their slates. The one at the bourd
then performs the operation audibly, and those with their
slates follow step by step.
Another is now called to the board and requested to set
down the second example, while the rest write the same
upon their slates, and solve it in a similar manner. He then
directs them to take the third example, and lets them try
their own skill, giving each such aid as he may require. In
this way they soon get hold of the principle, and if now sent
to their seats, will master the lesson with positive delight.
As to assistance, no specific directions can he given which
will meet every case. The best rule is, to afford the
learner just that kind and amount, which will secure the
greatest degree of exertion on his part. Less than this dis-
courages; more, enervates.
In conclusion, we would add, that this elementary work
was undertaken at the particular request of several eminent
practical teachers, and is designed to fill a niche in primary
schools. It presents, in a cheap form, a series of progressive
exercises in the simple and compound rules, which are
adapted to the capacities of beginners, and are calculated to
form habits of study, awaken the attention, and strengthen
the intellect.
J. B. THOMSON.
KBW YORK, January, 1858.
CONTENTS
SECTION I.
ARITHMETIC defined, ....--7
Notation, 7
Roman Notation, ......--7
Arabic Notation. ...... -9
Numeration, -12
SECTION II.
ADDITION defined, 16
When the sum of a column does not exceed 9, - - -18
When the sura of a column exceeds 9,- - - - -19
General Rule for Addition, - - - - - - -20
SECTION III.
SUBTRACTION defined, 27
When a figure in the lower No. is smaller than that above it, - 28
When a figure in the lower No. is larger than that above it, 29
Borrowing 10, .... 30
General Rule for Subtraction, - - - .. -81
SECTION IV.
MULTIPLICATION defined, 36
When the multiplier contains but one figure, 89
When the multiplier contains more than one figure, - - 41
General Rule for Multiplication, .... 43
To multiply by 10, 100, 1000, <fcc., 45
When there are ciphers on the right of the multiplier, - 46
When there are ciphers on the right of the multiplicand, - 47
When there are ciphers on the right of both, 48
SECTION V.
DIVISION defined, ....... . . 49
Short Division, - 52
VI CONTENTS.
Rule for Short Di nsion, 54
Long Division, 56
Difference between Short and Long Division, - - - 67
Rule for Long Division, 58
To dhide by 10, 100, 1000, <fec., - 61
When there are ciphers on the right of the divisor, - - 62
SECTION VI.
FRACTIONS, - 68
To find what part one given number is of another, - - 66
A part of a number being given, to find the whole, - - 66
To multiply a whole number by a fraction, - - - 67
To multiply a whole by a mixed number, - - - - 69
To divide a whole number by a fraction, 70
To divide a whole by a mixed number, - - - -71
SECTION VII.
TABLES in Compound Numbers, - - - 74
Paper and Books, - - .. . . . -85
Tables of aliquot parts, ------- 87
SECTION VIII.
ADDITION of Federal Money, 90
Subtraction of Federal Money, 92
Multiplication of Federal Money, 93
Division of Federal Money, 94
SECTION IX.
REDUCTION, 96
Rule for Reduction Descending, 97
Rule for Reduction Ascending, 100
Compound Addition, 106
Compound Subtraction, 108
Compound Multiplication, 110
Compound Division, - - - - - - - -11.1
Miscellaneous Exercises, - * - - - 113
Answers to Examples, - - - 119
ARITHMETIC.
SECTION I.
ART, ! ARITHMETIC is the science of numbers.
Any single thing, as a peach, a rose, a book, is called a
unit, or one ; if another single thing is put with it, the
collection is called two ; if another still, it is called three ;
if another, four ; if another, five, &c.
The terms, one, tioo, three, four, <kc., are tke names of
numbers. Hence,
2. NUMBER signifies a unit, or a collection of units.
Numbers are expressed by words, by letters, and bj
figures.
3* NOTATION is the art of expressing numbers by letters
or figures. There are two methods of notation in use, the
Roman and the Arabic.
I. ROMAN NOTATION.
4r The Roman Notation is the method of expressing
numbers by letters ; and is so called because it was invented
by the ancient Romans. It employs seven capital letters,
viz : I, V, X, L, C, D, M.
When standing alone, the letter I, denotes one ; V, fiv-e ;
X, ten ; L, fifty ; C, one hundred ; D, five hundred ; M,
one thousand.
QUEST. 1. What is Arithmetic ? What is a single thing called? If an-
other is put with it, what is the collection called? If another, what ? What
are the terras one, two, three, &c. ? 2. What then is number ? How are
numbers expressed ? 3. What is Notation ? How many methods of notation
are in use? 4. What is the Roman notation? Why so called? How many
etters does it employ? What does the letter I, denote? V? X? L? C? D? M?
NOTATION. !SECT. i.
5. To express the intervening numbers from to one a
thousand, or any number larger than a thousand, we re*
sort to repetitions and various combinations of these let-
ters, as may be seen from the following
TABLE.
I denotes
one.
XXXI
denotes thirty-one.
II "
two.
XL
" forty.
111
three.
XLI
" forty-one.
IV
four.
L
" fifty.
V "
five.
LI
" fifty-one.
VI
six.
LX
" sixty.
VII
seven.
LXI
" sixty-one.
VIII
eight.
LXX
" seventy.
IX "
nine.
LXXX
" eighty.
X "
ten.
xc
" ninety.
XI
eleven.
XCI
" ninety-one.
XII "
twelve.
C
" one hundred.
XIII
thirteen.
CI
" one hund. and one
XIV "
fourteen.
CIV
" one hund. and fout
XV
fifteen.
ex
" one hund. and ten.
XVI
sixteen.
cc
" two hundred.
XVII
seventeen.
ccv
" two hund. and five.
XVIII "
eighteen.
ccc
" three hundred.
XIX
nineteen.
cccc
" four hundred.
XX
twenty.
D
" five hundred.
XXI
twenty-one.
DC
" six hundred.
XXII
twenty-two.
DCC
" seven hundred.
XXIII
twenty-three.
DCCC
" eight hundred.
XXIV "
twenty- four.
DCCCC
" nine hundred.
XXV "
twenty-five.
M
" one thousand.
XXVI "
twenty-six.
MC
" one thousand and
XXVII "
twenty-seven.
one hundred.
XXVIII "
twenty-eight.
MM
" two thousand.
XXIX "
twenty-nine.
MDCCCL " one thousand eight
XXX
thirty.
hundred and fifty.
QUKST. 5. What do the letters IV, denote? VI? VIII? IX? XI? XI V
XVI? XVTII? XIX? XXIV? XL? LXXX? XC? CIV? Express
seven by Id icrs on the slate or black-board. How express eleven ? Thirteen
Tweuty-flvof Nineteen? Forty-lour? Eighty-seven? Ninety-nine?
ARTS. 5 7.] NOTATION. 9
OBS. 1. Every time a letter is repeated, its value is repeated.
Thus, the letter I, standing alone, denotes one ; II, two ones or two,
<fcc. So X denotes ten ; XX, twenty, <fcc.
2. When two letters of different value are joined together, if the
less is placed before the greater, the value of the greater is dimin-
ished as many units as the less denotes ; if placed after the greater,
the value of the greater is increased as many units as the less de-
notes. Thus, V denotes five ; but IV denotes only four ; and VI,
six. So X denotes ten ; IX, nine ; XI, eleven.
Note. The questions on the observations may be omitted, by
beginners, till review, if deemed advisable by the teacher.
II. AKABIC NOTATION.
6. The Arabic Notation is the method of expressing
numbers by figures ; and is so called because it is supposed
to have been invented by the Arabs. It employs the fol-
lowing ten characters or figures, viz :
1234567890
one, two, three, four, five, six, seven, eight, nine, naught.
OBS. 1. The first nine are called significant figures, because each
one always expresses a value, or denotes some number. They are
also called digits, from the Latin word digitus, signifying a finger,
because the ancients used to count on their fingers.
2. The last one is called naught, because when standing alone,
it expresses nothing, or the absence of number. It is also called
cipher or zero.
7 All numbers larger than 9, are expressed by different
combinations of these ten figures. For example, to express
ten, we use the 1 and 0, thus 10 ; to express eleven, we
use two Is, thus 11 ; to express twelve, we use the 1 and
2, thus 12, <fec.
QuKST.Oi*. What is the effect of repeating a letter ? If a letter of lost
value is placed before another of greater value, what is the effect? If placed
after, what ? 6. What is the Arabic notation ? Why so called ? How many
figures does it employ? What are their names? Obs. What are the first nino
called ? Why ? What else are they sometimes called? What is the last one
called? Why? 7. How are numbers larger than nine expressed ? Express
ten by figures. Eleven. Twelve. Fifteen.
10 NOTATION. [SECT, i
Hie method of expressing numbers by figures from
one to a thousand, may be seen from the following
TABLE.
1, one.
36, thirty-six.
71, seventy-one.
2, two.
37, thirty-seven.
72, seventy-two.
3, three.
38, thirty-eight.
73, seventy-three.
4, four.
39, thirty-nine.
74, seventy- four.
5, five.
40, forty.
75, seventy-five.
6, six.
41, forty-one.
76, seventy-six.
7, seven.
42, forty-two.
77, seventy-seven.
8, eight.
43, forty-three.
78, seventy-eight.
9, nine.
44, forty- four.
79, seventy-nine.
10, ten.
45, forty-five.
80, eighty.
11, eleven.
46, forty-six.
81, eighty-one.
12, twelve.
47, forty-seven.
82, eighty-two.
13, thirteen.
48, forty-eight.
83, eighty-three.
14, fourteen.
49, forty-nine.
84, eighty- four.
15, fifteen.
50, fifty.
85, eighty-five.
16, sixteen.
51, fifty-one.
86, eighty-six.
17, seventeen.
52, fifty-two.
87, eighty-seven
18, eighteen.
53, fifty-three.
88, eighty-eight.
19, nineteen.
54, fifty-four.
89, eighty-nine.
20, twenty.
55, fifty-five.
90, ninety.
21, twenty-one.
56, fifty-six.
91, ninety-one.
22, twen'y-two.
57, fifty-seven.
92, ninety-two.
23, twenty-three.
58, fifty-eight.
93, ninety-three.
24, twenty- four.
59, fifty-nine.
94, ninety-four.
25, twenty-five.
60, sixty.
95, ninety-five. /
26, twenty-six.
61, sixty-one.
96, ninety-six.
27, twenty-seven.
62, sixty-two.
97, ninety-seven
28, twenty-eight.
63, sixty-three.
98, ninety-eight.
29, twenty-nine.
64, sixty- four.
99, ninety-nine.
30, thirty.
65, sixty-five.
100, one hundred.
31, thirty-one.
66, sixty-six.
200, two hundred.
32, thirty-two.
67, sixty-seven.
300, three hundred
33, thirty-three.
68, sixty-eight.
400, four hundred.
34, thirty-four.
69, sixty-nine.
900, nine hundred.
35. thirty-five.
70, seventy.
1000, one thousand.
QUEST. How express fifteen? Twenty-five? Forty-seven? Thirty-six
Seventy-three One hundred and one ? One hundred and ten ? One hundrev
and twdnty T Two hundred and fifteen ?
ARTS. 8 11.] NOTATION. 11
8. It will be peiceived from the foregoing table, that
the same figures, standing in different places, have differ-
ent values.
When they stand alone or in the right hand place, they
express units or ones, which are called units of the first
order.
When they stand in the second place, they express tens,
which are called units of the second order.
When they stand in the third place, they express hun-
dreds, which are called units of the third order.
When they stand in the fourth place, they express
thousands, which are called units of the fourth order, <fec.
For example, the figures 2, 3, 4, and 5, when arranged
thus, 2345, denote 2 thousands, 3 hundreds, 4 tens, and 5
units ; when arranged thus, 5432, they denote 5 thousands,
4 hundreds, 3 tens, and 2 units.
9 Ten units make one ten, ten tens make one hundred,
and ten hundreds make one thousand, &c. ; that is, ten of
any lower order, are equal to one in the next higher order
Hence, universally,
O. Numbers increase from right to left in a tenfold
ratio ; that is, each removal of a figure one place towards
the left, increases its value ten times.
11. The different values which the same figures have,
are called simple and local values.
The simple value of a figure is the value which it ex-
presses when it stands alone, or in the right hand place.
QUEST. 8. Do the same figures always have the same value ? When stand-
ing alone or in the right hand place, what do they express? What do they
express when standing in the second place? In the third place? In the
fourth ? 9. How many units make one ten ? How many tens make a hun-
dred ? How many hundreds make a thousand ? Generally, how many of any
lower order are required to make one of the next higher order ? 10. What is
the general law by which numbers increase ? What is the effect upon the value
of a figure to remove it one place towards the left? 11. What are the differ*
ent values of the same figure called ? What is the simple value of a figure ?
What the local value ?
12 NUMERATION. [SECT. 1.
The simple value of a figure, therefore, is the numbel
which its name denotes.
The local value of a figure is the increased value which
: t expresses by having other figures placed on its right.
Hence, the local value of a figure depends on its locality,
or the place which it occupies in relation to other num-
bers with which it is connected. (Art. 8.)
OBS. This system of notation is also called the decimal system,
because numbers increase in a tenfold ratio. The term decimal if
derived from the Latin word decem, which signifies ten.
NUMERATION.
12* The art of reading numbers when expressed by
figures, is called Numeration.
NUMERATION TABLE.
123 861 518 924 263
Period V. Period IV. Period III. Period II. Period I.
Trillions. Billions. Millions. Thousands. Units.
13. The different orders of numbers are divided into
periods of three figures each, 'beginning at the right hand.
QUEST. Upon what does the local value of a figure depend ? Obs. What ia
this system of notation sometimes called ? Why ? 12. What is Numeration ?
Repeat the numeration table, beginning at the right hand. What is the first
place on the right called? The second place? The third? Fourth? Fifth
Sixth? Seventh? Eighth? Ninth? Tenth, &c.? 13. How are the orders of
numbers divided ?
ARTS. 12 14.] NUMERATION. 13
The first, or right hand period is occupied by units, tens,
hundreds, and is called units' period ; the second is oc-
cupied by thousands, tens of thousands, hundreds of
thousands, and is called thousands' period, &c.
The figures in the table are read thus : One hundred
and twenty-three trillions, eight hundred and sixty-one
billions, five hundred and eighteen millions, nine hundred
and twenty-four thousand, two hundred and sixty-three.
1 4 To read numbers which are expressed by figures.
Point them off into periods of three figures each ; then,
beginning at the left hand, read the figures of each period
as though it stood alone, and to the last figure of each, add
the name of the period.
OBS. 1. The learner must be careful, in pointing 0$~ figures, always
to begin at the right hand ; and in reading them, to be^in at the
left hand.
2. Since the figures in the first or right hand period alw lys de-
note units, the name of the period is not pronounced. Hevxce, in
reading figures, when no period is mentioned, it is always u,ider-
stood to be the right hand, or units' period.
EXERCISES IN NUMERATION.
Note. At first the pupil should be required to apply to each fig-
ure the name of the place which it occupies. Thus, beginning at
the right hand, he should say, " Units, tens, hundreds," &c., and
point at the same time to the figure standing in the place which he
mentions. It will be a profitable exercise for young scholars to
write the examples upon their slates or paper, then point them off
into periods, and read them.
QUEST. What is the first period called ? By what is it occupied ? What is
the second period called? By what occupied? What is the third period
called ? By what occupied ? Wnat is the fourth called ? By what occupied f
What is the fifth called? By what occupied? 14. How do you read nurabere
expressed by figures ? Obs. Where begin to point them off? Where to read
them ? Do you pronounce the name of the right hand periol ? When no
period is named, what is understood ?
14
NUMERATION.
Read the following numbers :
[SECT. I
Ex. 1.
97
16.
12642
Si. 7620
2.
110
17.
20871
32. 8040
3.
256
18.
17046
33. 9638
4.
307
19.
43201
34. 11000
5.
510
20.
80600
35. 12100
6.
465
21.
4203
36. 1402C
7.
1248
22.
65026
37. 10001
8.
2381
23.
78007
38. 5 020
9.
4026
24.
90210
39. 18022
10.
6420
25.
5025
40. 30401
11.
8600
26.
69008
41. 2506
12.
7040
27.
100 000
42. 402 321
13.
8000
28.
125 236
43. 65 007
14.
9007
29.
6005
44. 750 026
15.
10000
30.
462 400
45. 804 420
40.
2325672
50. 7289405287
47.
4502360
51. 185205370000
48.
62840285
52. 6423691450896
49.
425026951
53. 75894128247625
EXERCISES IN NOTATION.
15 To express numbers by figures.
Begin at the left hand of the highest period, and write
the figures of each period as though it stood alone.
If any intervening order, or period is omitted in the
given number, write ciphers in its place.
Write the following numbers in figures upon the slate
or black-board.
1. Sixteen, seventeen, eighteen, nineteen, twenty.
2. Twenty-three, twenty-five, thirty, thirty-three.
3. Forty-nine, fifty-one, sixty, seventy-four.
4. Eighty-six, ninety-three, ninety- seven, a hundred.
QUEST. 15. How are numbers expressed by figures ? If any intervening
order is omitted in the example, how is jts place supplied?
ART. 15.J NUMERATION. 19
5. One hundred and ten.
6. Two hundred and thirty-five.
7. Three hundred and sixty.
8. Two hundred and seven.
9. Four hundred and eighty-one.
10. Six hundred and ninety-seven.
11. One thousand, two hundred and sixty-three.
12. Four thousand, seven hundred and ninety-nine.
13. Sixty-five thousand and three hundred.
14. One hundred and twelve thousand, six hundred
and seventy-three.
15. Three hundred and forty thousand, four hundred
and eighty-five.
16. Two millions, five hundred and sixty thousand.
17. Eight millions, two hundred and five thousand,
three hundred and forty-five.
18. Ten millions, five hundred thousand, six hundred
and ninety-five.
19. Seventeen millions, six hundred and forty-five
thousand, two hundred and six.
20. Forty-one millions, six hundred and twenty thou-
sand, one hundred and twenty-six.
21. Twenty- two millions, six hundred thousand, one
hundred and forty-seven.
22. Three hundred and sixty millions, nine hundred
and fifty thousand, two hundred and seventy.
23. Five billions, six hundred and twenty-one millions,
seven hundred and forty-seven thousand, nine hundred
and fifty-four.
24. Thirty-seven trillions, four hundred and sixty-three
billions, two hundred and ninety-four thousand, fire hun-
dred and seventy-two.
1$ ADDITION. [SECT. IL
SECTION II.
ADDITION.
ART. 16. Ex. 1. Henry paid 4 shillings for a pair of
gloves, 7 shillings for a cap, and 2 shillings for a knife :
how many shillings did he pay for all ?
Solution. 4 shillings and 7 shillings are 11 shillings,
and 2 shillings are 13 shillings. He therefore paid 13
shillings for all.
OBS. The preceding operation consists in finding a single num-
ber which is equal to the several given numbers united together,
and is called Addition. Hence,
17 ADDITION is the process of uniting two or more
numbers in one sum.
The answer, or number obtained by addition, is called
the sum or amount.
OBS. When the numbers to be added are all of the same kind, or
denomination, the operation is called Simple Addition.
18 Sign of Addition (+). The sign cf addition is
a perpendicular cross ( + ), called plus, and shows that
the numbers between which it is placed, are to be added
together. Thus, the expression 6 + 8, signifies that 6 is
to be added to 8. It is read, " 6 plus 8," or <: 6 added to 8."
Note. The term plus, is a Latin word, orig'nally signifying
'* rcore." In Arithmetic, it means " added to."
QUEST. 17. What is addition? What is the answer called ? Obs. When
the numbers to bo added are all o? the same denomination, what is the ope-
ration called? 18. What is the si^n of addition? Who* does it show ? JVWe.
What is the moaning of the word plus ?
ARTS. 16 19.] ADDITION. 17
19. Sign of Equality ( ). The sign of equality is
two horizontal lines ( ), and shows that the numbers be-
tween which it is placed, are equal to each other. Thus,
the expression 4 + 3 = 7, denotes that 4 added to 3 are
equal to 7. It is read, " 4 plus 3 equal 7," or " the sum
cf 4 plus 3 is equal to 7." 18 + 5 = 7
ADDITION TABLE.
2 and
3 and
4 and
5 and
1 are 3
1 are 4
1 are 5
1 are 6
2 " 4
2 " 5
2 " 6
2 " 7
3 " 5
3 " 6
3 " 7
3 " 8
4 " 6
4 " 7
4 " 8
4 " 9
5 " 7
5 " 3
5 " 9,
5 " 10
6 " 8
6 " 9
6 " 10
6 " 11
7 " 9
7 " 10
7 " 11
7 " 12
8 " 10
8 " 11
8 " 12
8 " 13
9 " 11
9 " 12
9 " 13
9 " 14
10 " 12
10 " 13
10 " 14
10 " 15
6 and
7 and
8 and
9 and
1 are 7
1 are 8
1 are 9
1 are 10
2 " 8
2 " 9
2 " 10
2 " 11
3 " 9
3 " 10
3 " 11
3 " 12
4 " 10
4 " 11
4 " 12
4 " 13
5 " 11
5 " 12
5 " 13
5 " 14
6 " 12
6 " 13
6 " 14
6 " 15
7 " 13
7 " 14
7 " 15
7 " 16
8 " 14
8 " 15
8 " 16
8 " 17
9 " 15
9 " 16
9 " 17
9 " 18
10 " 16
10 " 17
10 " 18
10 " 19
Note. It is an interesting and profitable exercise for young pupils
to recite tables in concert. But it will not do to depend upon this
method alone. It is indispensable for every scholar who desires to
De accurate either in arithmetic or business, to have the c ommon
QUKST. 19. What is the sign of equality? What does it show?
2
18 ADDITION. [SECT. II.
' tab es distinctly and indelibly fixed in his mind. Hence,
after a taole has been repeated by the class in concert, or individ-
ually, the teacher should ask many promiscuous questions, to prevent
its being recited mechanically, from a knowledge of the regular in-
crease of numbers.
EXAMPLES.
!3O When the sum of a column does not exceed 9.
Ex. 1. George gave 37 cents for his Arithmetic, and
42 cents for his Reader : how many cents did he give for
both?
Directions. Write the numbers Operation.
under each other, so that units ^ &
may stand under units, tens under jf g
tens, and draw a line beneath them. 3 7 price of Arith.
Then, beginning at the right hand 4 2 " of Read.
or units, add each column sepa- -
rately in the following manner : 7 9 Ans.
2 units and 7 units are 9 units. Write the 9 in units
place under the column added. 4 tens and 3 tens are
Y tens. Write the 7 in tens' place. The amount is 79
cents.
Write the following examples upon the slate or black-
board, and find the sum of each in a similar manner :
(2.) (3.) (4.) (5.)
26 231 623 5734
42 358 145 4253
(6.) (7.) (8.) (9.)
425 3021 5120 3521
132 1604 2403 1043
321 2142 1375 4215
10. What is the sum of 4321 and 2475 ?
11. What is the sum of 32562 and 56214?
12. What is the sum of 521063 and 465725 ?
ARTS. 20 22. J ADDITION. 19
21. When the sum of a column exceeds 9.
13. A merchant sold a quantity of flour for 458 dollars,
a quantity of tea for 887 dollars, and sugar for 689 dol-
lars : how much did he receive for all ?
Having written the numbers as Operation.
Defore, we proceed thus: 9 units 458 price of flour,
and 7 units are 16 units, and 8 887 " of tea.
are 24 units, or we may simply 689 " of sugar,
say 9 and 7 are 16, and 8 are 24. 2034 dollars. Ans.
Now 24 is equal to 2 tens and
4 units. We therefore set the 4 units or right hand figure
in units' place, because they are units ; and reserving the
2 tens or left hand figure in the mind, add it to the column
of tens because it is tens. Thus, 2 (which was reserved)
and 8 are 10, and 8 are 18, and 5 are 23. Set the 3 or
right hand figure under the column added, and reserving
the 2 or left hand figure in the mind, add it to the column
of hundreds, because it is hundreds. Thus, 2 (which was
reserved) and 6 are 8, and 8 are 16, and 4 are 20. Set
the or right hand figure under the column added ; and
since there is no other column to be added, write the 2
in thousands' place, because it is thousands.
N. B. The pupil must remember, in all cases, to set down the
whole sum of the last or l$ft hand column.
22. The process of reserving the tens or left hand fig-
ure, when the sum of a column exceeds 9, and adding it
mentally to the next column, is called carrying tens.
Find the sum of each of the following examples in a
similar manner :
(14.) (15.) (16.) (17.)
856 364 6502 8245
764 488 497 4678
1620 Ans. 602 8301 362
20 ADDITION. [SECT. IL
23. From the preceding illustrations and principles
we derive the following
GENERAL RULE FOR ADDITION".
I. Write the numbers to be added under each other, so
that units may stand under units, tens under tens, &c.
II. Beginning at the right hand, add each column sepa-
rately, and if the sum of a column does not exceed 9, write
it under the column added. But if the sum of a column
exceeds 9, write the units' figure under the column and
carry the tens to the next column.
III. Proceed in this manner through all the orders, and
finally set down the whole sum of the last or left hand
column.
24. PROOF. Beginning at the top, add each column
downward, and if the second result is the same as the
first, the work is supposed to be right t
EXAMPLES FOR PRACTICE.
(1.) (2.) (3.) (4.)
Pounds. Feet. Dollars Yards.
25 113 342 4608
46 84 720 635
_84 2_16 898 43
(5.) (6.) (7.) (8.)
684 336 6387 8261
948 859 593 387
569 698 3045 13
203 872 15 7^
9. What is the sum of 46 inches and 38 inches?
QUEST. 23. How do you write numbers for addition? When tho mim of a
column does not exceed 9, how proceed ? When it exceeds 9, how proceed ?
22. What is meant by carrying the tens ? What do you do with the sum of
the last column ? 24. How is addition proved ?
< 23, 24.] ADDITION. 21
10. What is the sum of 51 feet and 63 feet ?
11. What is the sum of 75 dollars and 93 dollars?
%
12. Add together 45 rods, 63 rods, and 84 rods.
13. Add together 125 pounds, 231 pounds, 426 pounds.
14. Add together 267 yards, 488 yards, and 6 25 yards.
15. Henry traveled 256 miles by steamboat and 320
miles by Railroad : how many miles did he travel ?
16. George met two droves of sheep ; one contained
461, and the other 375 : how many sheep were there in
both droves ?
17. If I pay 230 dollars for a horse, and 385 dollars for
a chaise, how much shall I pay for both ?
18. A farmer paid 85 dollars for a yoke of oxen, 27
dollars for a cow, and 69 dollars for a horse : how much
did he pay for all ?
19. Find the sum of 425, 346, and 681.
20. Find the sum of 135, 342, and 778.
21. Find the sum of 460, 845, and 576.
22. Find the sum of 2345, 4088, and 260.
23. Find the sum of 8990, 5632, and 5863.
24. Find the sum of 2842, 6361, and 523.
25. Find the sum of 602, 173, 586, and 408.
26. Find the sum of 424, 375, 626, and 75.
27. Find the sum of 24367, 61545, and 20372.
28. Find the sum of 43200, 72134, and 56324.
29. A young man paid 5 dollars for a hat ; 6 dollars
for a pair of boots, 27 dollars for a suit of clothes, and 19
dollars for a cloak : how much did he pay for all ?
30. A man paid 14 dollars for wood, 16 dollars for a
stove, and 28 dollars for coal : how many dollars did he
pay for all ?
31. A farmer bought a plough for 13 dollars, a cart
for 46 dollars, and a wagon for 61 dollars : what was the
price of all ?
22 ADDITION. [SECT. II
32. What is the sum of 261+31+256 + 17 ?
33. What is the sum of 163+478+82 + 19 ?
34. What is the sum of 428 + 632 + 76+394 ?
35. W3iat is the sum of 320 + 856 + 100+503?
36. What is the sum of 641+108 + 138+710 ?
37. What is the sum of 700 + 66 + 970+21 ?
38. What is the sum of 304+971+608+496 ?
39. What is the sum of 848 + 683+420+668 ?
40. What is the sum of 868+45 + 17+25+27+38?
41. What is the sum of 641 + 85+580+42+7+63 ?
42. What is the sum of 29+281+7+43+785+46?
43. A farmer sold 25 bushels of apples to one man, IT
bushels to another, 45 bushels to another, and 63 bushels
to another : how many bushels did he sell ?
44. A merchant bought one piece of cloth containing
25 yards, another 28 yards, another 34 yards, and an-
other 46 yards : how many yards did he buy ?
45. A man bought 3 farms ; one contained 120 acres,
another 246 acres, and the other 365 acres : how many
acres did they all contain ?
46. A traveler met four droves of cattle ; the first con-
tained 260, the second 175, the third 342, and the fourth
420 : hOw many cattle did the four droves contain ?
47. A carpenter built one house for 2365 dollars, an-
other for 1648 dollars, another for 3281 dollars, and an-
other for 5260 dollars : how much did he receive for all 9
48. Find the sum of six hundred and fifty- four, eighty-
nine, four hundred and sixty-three, and seventy- six.
49. Find the sum of two thousand and forty-seven,
three hundred and forty-five, thirty-six, and one hundred.
50. In January there are 31 days, February 28, March
81, April 30, May 31, June 30, July 31, August 31, .Sep-
tember 30, October 31, November 30, and December 31 :
how many days are there in a year ?
ART. 24.a.] ADDITION. 23
24ra. Accuracy and rapidity in adding can be ac-
quired only by practice. The following exercises are de-
signed to secure this important object.
OBS. 1. In solving the following examples, it is recommended
to the pupil simply to pronounce the result, as he adds each suc-
cessive figure. Thus, in Ex. 1, instead of saying 2 and 2 are 4,
and 2 are 6, &/c., proceed in the following manner : " two, four, six ;
eight, ten, twelve, fourteen, sixteen, eighteen, twenty." Set down
naught and carry two. " Two, (to carry) three, six, nine," &c.
2. When two or three figures taken together make 10, as 8 and 2,
7 and 3, or 2, 3, and 5, k accelerates the process to add their sum
at once. Thus, in Ex. 4, the pupil should say: " ten (1+9), six-
teen (6), twenty-six (5+5), thirty-six (2+8)," &c.
(1.) (2.) (3.) (4.)
32 654 987 463
32 654 987 647
32 654 987 455
32 654 987 258
32 654 987 572
32 654 987 595
32 654 987 615
32 654 987 346
32 654 987 729
12 114 117 181
(5.) (6.) (7.) (8.)
614 2140 8675 9244
452 8963 2433 1432
528 1232 6182 7234
539 7855 2921 2523
420 2123 2209 8440
385 3333 4863 4346
355 7674 6558 6704
134 4521 5434 1852
976 6589 5276 9258
468 2637 8789 8106
24
ADDITION.
[SECT. IL
(9.)
(10.)
. (11.)
(12.)
4360
9201
42671
62125
7046
7283
68439
31684
5724
4627
32074
22435
5385
6436
47616
16725
8275
9874
30045
94381
9342
8400
26765
25036
6768
6645
10850
85474
5020
4365
25232
10325
9384
8640
43679
42312
(13.)
(14.)
(15.)
(16.)
2720
5764
27856
47639
4382
5346
32534
23421
2640
3042
20631
34323
3047
5268
34327
71036
2163
3161
53102
62342
6741
2560
92763
57654
1360
7304
51834
32103
7056
2723
23452
53728
8554
8459
62327
61342
4275
6715
50632
23201
(17.)
(18.)
(19.)
(20.)
4521
6845
75360
89537
3432
3151
27838
23264
4327
2327
42627
41728
6238
4235
34872
74263
5494
2835
63538
21031
3217
5473
54321
53426
2382
9864
63054
91342
4723
3103
29872
23465
3604
7382
63541
38754
2352
5461
53279
94642
ADDITION.
(21.)
(22.)
(23.)
(24.)
8564
56,430
84,703
341,725
4736
31,932
19,384
227,265
3405
29,754
21,705
311,265
037
46,536
43,641
200,378
6571
86,075
27,469
421,850
7439
30,235
52,267
370,432
4525
41,623
61,383
174,370
3137
45,810
75,604
831,031
2743
56,239
43,876
580,456
(25.)
(26.)
(27.)
(28.)
7243
31,625
68,901
460,732
2034
51,482
50,345
804,045
3710
49,061
75,005
346,325
5634
80,604
29,450
450,673
1730
24,540
80,063
859,721
5613
67,239
91,700
236,548
3005
24,307
43,621
632,462
7206
58,392
47,834
753,324
4354
70,300
83,276
970,300
7821
56,749
25,327
267,436
(29.)
(30.)
(31.)
(32.)
6458
75,340
64,268
346,768
2435
6,731
405
21,380
4678
748
1,708
4,075
4962
68,451
43,671
126,849
5143
396
72,049
257
8437
7,503
492
L305
7643
46,075
1,760
24,350
6850
1,290
25,357
439,871
7063
25,738
1,434
40,306
8324
46,803
84,162
601,734
26
ADDITION.
[SECT. II,
(S3.)
(34.)
(35.)
(36.)
423,674
632,153
317,232
412,783
307,316
420,432
203,671
631,432
730,248
323,680
334,263
572,316
506,213
507,325
210,600
231,254
110,897
383,734
356,237
673,323
206,341
634,156
264,871
217,067
324,563
450,071
531,634
306,421
185,174
803,463
342,106
764,315
364,230
160,705
768,342
207,254
150,176
300,430
407,821
843,552
843,204
461,007
311,289
321,634
370,679
297,313
564,735
502,543
445,168
813,792
470,334
617,405
370,432
200,406
436,216
506,032
5,338,315^4?
is. 6,388,667
Ans. 621,353
762,573
(37.)
(38.)
(39.)
(40.)
674,326
783,457
863,725
958,439
453,403
675,306
755,387
843,670
561,734
858,642
964,845
784,561
789,867
246,468
836,450
976,435
645,275
587,649
645,265
833,406
576,182
523,731
783,842
797,624
934,922
445,372
532,653
845,358
423,641
832,148
647,412
978,262
561,232
465,363
481,735
784,643
143,671
642,742
824,364
865,343
238,406
830,423
537,572
976,736
453,762
256,372
463,489
853,974
984,651
662,456
827,343
467,852
845,359
572,834
642,536
948,685
967,423
864,213
725,342
896,872
A.RTS. 25 27.] SUBTRACTION. 2T
SECTION III.
SUBTRACTION.
ART. 25. Ex. 1. Charles having 15 cents, gave 6 cents
for an orange : how many cents did he have left ?
Solution. 6 cents taken from 15 cents leave 9 cents.
Therefore he had 9 cents left.
OBS. The preceding operation consists in taking a less number
from a greater, and is called Subtraction. Hence,
26. SUBTRACTION is the process of finding the differ-
ence between two numbers.
The answer, or number obtained by subtraction, is called
the difference or remainder.
OBS. 1. The number to be subtracted is often called the subtra-
hend, and the number from which it is subtracted, the minuend.
These terms, however, are calculated to embarrass, rather than
assist the learner, and are properly falling into disuse.
2. When the given numbers are all of the same kind, or denomi-
nation, the operation is called Simple Subtraction.
27. Sign of Subtraction ( ). The sign of subtrac-
tion is a horizontal line (), called minus, and shows
that the number after it is to be subtracted from the one
before it. Thus the expression 7 3, signifies that 3 is to be
subtracted from 7 ; and is read, " 7 minus 3," or " 7 less 3."
Bead the following: 187 = 20 9. 23 10=16 3
35 8 = 31 4.
Note. The term minus is a Latin word signifying less.
QUEST. C 26. What is subtraction? What is the answer called? Obs.
What is the number to be subtracted sometimes called ? That from which it
Is subtracted ? When tho given numbers are of the same denomination, what
is the operation called ? 27. What is the sign of subtraction ? What does it
ihow 1 Note. What is t<he moaning of the term minus?
SUBTRACTION.
[SECT, ill
SUBTRACTION TABLE.
2 from
3 from
4 from
5 from
2 leaves
3 leaves
4 leaves
5 leaves
3 " 1
4 " 1
5 " 1
6 " 1
4 " 2
5 " 2
6 " 2
7 " 2
5 " 3
6 " 3
7 " 3
8 " 3
6 " 4
7 " 4
8 4
9 " 4
7 " 5
8 " 5
9 " 5
10 " 5
8 6
9 " 6
10 " 6
11 " 6
9 " 7
10 " 7
11 " 7
12 " 7
10 " 8
11 " 8
12 " 8
13 " 8
11 " 9
12 " 9
13 " 9
14 9
12 " 10
13 " 10
14 " 10
15 " 10
6 from
7 from
8 from
9 from
6 leaves
7 leaves
8 leaves
9 leaves
7 " 1
8 " 1
9 " 1
10 " 1
8 " 2
9 " 2
10 " 2
11 " 2
9 " 3
10 " 3
11 " 3
12 " 3
10 " 4
11 " 4
12 " 4
13 " 4
11 " 5
12 " 5
13 " 5
14 " 5
12 " 6
13 " 6
14 " 6
15 " 6
13 " 7
14 " 7
15 " 7
16 " 7
14 " 8
15 " 8
16 " 8
17 ' 8
15 " 9
16 " 9
17 " 9
18 " 9 .
16 " 10
17 " 10
18 " 10
19 " 10 1
OBS. This Table is the reverse of Addition Table. Hence, if the
pupil has thoroughly learned that, it will cost him but little time or
trouble to learn this. (See observations under Addition Table.)
EXAMPLES.
28. When each figure in the lower number is smaller
than the figure above it.
1. A farmer raised 257 bushels of apples, and 123
bushels of pears : how many more apples did he rais
than pears ?
ARTS. 28, 29.] SUBTRACTION. 29
Directions. Write the less Operation.
number under the greater, so
that units may stand under units, *g ^ jg
tens mder tos, &c., and draw a Jj _
line beneath them. Beginning 257 apples,
with the units or right hand fig- 123 pears,
ure, subtract each figure in the JRem. 134 bush.
ower numDor from the figure
above it, in the following manner : 3 units from 7 units
leave 4 units. Write the 4 in units' place under the
figure subtracted. 2 tens from 5 tens leave 3 tens ; set
3 in tens' place. 1 hundred from 2 hundred leaves 1 hun-
dred ; write the 1 hundred in hundreds' place.
Solve the following examples in a similar manner :
(2.) (3.) (4.) (5.)
From 45 68 276 698
Take JU 123 453
(6.) (7.) (8.) (9.)
From 54 delis. 76 pounds. 257 yds. 325 shil.
Take Cd dolls. 64 pounds. 142 yds. 103 shil.
10. Samuel having 436 marbles, lost 214 of them:
how many had he left ?
29 When a figure in the lower number is larger than
the figure above it.
11. A man bought 63 bushels of wheat, and after-
wards sold 37: how many bushels had he left?
It is bv> jus tb'.tt we cannot take 7 Ifirst Method.
units fi < t 3 UIP',S, for 7 is larger than 63
3 ; we tl erefove add 10 to the 3 units, 37
and it w U raake 13 units ; then 7 from Rem. 26 bu.
13 leave i 6 ; write the 6 in units' place
under the figure subtracted. To compensate for the 10
30 SUBTRACTION. (SECT [11
units we added to the upper figure, we add 1 ten to the
3 tens or next figure in the lower number, and it makes
4 tens ; and 4 tens from 6 tens leave 2 tens : write the 2
in tens' place Ans. 26 bushels.
We may also illustrate the process of borrowing in tfec
following manner :
63 is composed of 6 tens and 3 Second Method.
units. Taking 1 ten from 6 tens, 63 = 50 + 13*
and adding it to the 3 units, we 37=30+7
have63 = 50+13. Separating the JRem. = 2Q + 6, or 26
lower number into tens and units,
we have 37 = 30 + 7. Now, substracting as before, 7
from 13 leaves 6. Then as we took 1 ten from the 6 tens,
we have but 5 tens left ; and 3 tens from 5 tens leave 2
tens. The remainder is 26, the same as before.
3O The process of taking one from a higher order in
the upper number, and adding it to the figure from which
the subtraction is to be made, is called borrowing ten, and
is the reverse of carrying ten. (Art. 22.)
OBS. When we borrow ten we must always remember to pay it
This may be done, as we have just seen, either by adding 1 to tte
next figure in the lower number, or by considering the nextjigur*
in the upper number 1 less than it is .
12. From 240 subtract 134, and prove the operation.
Since 4 cannot be taken from 0, we Operation.
borrow 10; then 4 from 10 leaves 6. 1 240
added to 3 (to compensate for the 10 we 134
borrowed) makes 4, and 4 from 4 leaves 0. 106 Ans.
1 from 2 leaves 1.
PROOF. We add the remainder Proof.
to the smaller number, and since the 134 less No.
sum is equal to the larger number, 106 remainder,
the work is right. 240 greater No.
ARTS. 30 32. j SUBTRACTION. SI
Solve the following examples, and prove the operation.
(13.) (14.) (15.) (16.)
From 375 5273 6474 8650
Take_238 2657 3204 5447
17. From 8461875, take 3096208.
31 From the preceding illustrations and principles
we derive the following
GENERAL RULE FOR SUBTRACTION.
I. Write the less number under the greater, so that units
may stand under units, tens under tens, &c.
II. Beginning at the right hand, subtract each figure in
the lower number from the figure above it, and set the re-
mainder under the figure subtracted.
III. When a figure in the lower number is larger than
that above it, add 10 to the upper figure ; then subtract as
before, and add 1 to the next figure in the lower number.
32* PROOF. Add the remainder to the smaller num-
ber ; and if the sum is equal to the larger number, the work
is right.
OBS. This method of proof depends upon the obvious principle,
that if the difference between two numbers be added to the less, the
sum must be equal to the greater.
EXAMPLES FOR. PRACTICE.
(1.)
From 325
(2.)
431
(3.)
562
(4.)
600
Take 108
249
320
231
(5.)
From 2230
(6.)
3042
(*)
6500
(8.)
8435
Take 1201
2034
3211
5001
QUEST. 31. How do you write numbers for subtraction ? Where do yon
begin to subtract ? When a figure in the lower number is larger than the one
ibovo it, how do you proceed ? 32, How is subtraction pro\*xl ?
SUBTRACTION. [SEOT. Ill
(11.)
From 45100 826340 1000000
Take 10000 513683 999999
12. From 132 dollars subtract 109 dollars.
13. From 142 bushels subtract 85 bushels.
14. From 375 pounds subtract 100 pounds.
15. From 698 yards subtract 85 yards.
16. From 485 rods subtract 175 rods.
17. Take 230 gallons from 460 gallons.
18. Take 168 hogsheads from 671 hogsheads.
19. Take 192 bushels from 268 bushels.
20. From 275 dollars take 148 dollars.
21. From 468 pounds take 219 pounds.
22. From 3246 rods take 2164 rods.
23. From 45216 take 32200.
24. From 871410 take 560642.
25. From 926500 take 462126.
26. From 6284678 take 1040640.
27. 468423. 37. 1726513167.
28. 675367. 38. 2148020372.
29. 800560. 39. 3067126140.
30. 701643. 40. 4572331203,
31. 963421. 41. 8164757025.
32. 32631242. 42. 265328140300.
33. 41652340. 43. 170643106340.
34. 56003000. 44. 465746241680.
35. 72464161. 45. 694270 590S95.
36. 86707364. 46. 920486500000.
47. A man having 235 sheep, lost 163 of them: ho*
many had he left ?
48. A farmer having 500 bushels of wheat, sold 278
bushels : how much wheat had he left ?
49. A man paid 625 dollars for a carriage and 430
ART. 32.] SUBTRACTION. 33
dollars for a span of horses : how much more did he pay
for his carriage than for his horses ?
50. A man gave 1263 dollars for a lot, and 2385 dol-
lars for building a house : how much more did his house
cost than his lot ?
51. If a person has 3290 dollars in real estate, and
owes 1631 dollars, how much is he worth?
52. A man gave his son 8263 dollars, and his daughter
5240 dollars : how much more did he give his son tha
his daughter?
53. A man bought a farm for 9467 dollars, and sold
it for 11230 dollars : how much did he make by his bar-
gain ?
54. If a man's income is 10000 dollars a year, and his
expenses 6253 dollars, how much will he lay up ?
55. The captain of a ship having a cargo of goods
worth 29230 dollars, threw overboard in a storm 13216
dollars' worth : what was the value of the goods left ?
56. A merchant bought a quantity of goods for 12645
dollars, and afterwards sold them for 13960 dollars:
how much did he gain by his bargain ?
57. A man paid 23645 dollars fora ship and after-
wards sold it for 18260 dollars : how much did he lose
by his bargain ?
58. The salary of the President of the United States is
25000 dollars a year ; now if his expenses are 19265 dol-
lars, how much will he lay up ?
59. A general before commencing a battle, had 35260
soldiers in his army ; after the battle he had only 21316:
how many soldiers did he lose ?
60. The distance of the sun from the earth is 95000000
miles ; the distance of the moon from the earth is 240000
miles : how much farther from the earth is the sun than
the moon ?
2
34 SUBTRACTION. [SECT. Ill
EXAMPLES INVOLVING ADDITION AND SUBTRACTION.
61. Henry bought 63 oranges of one grocer, and 26
of another; he afterwards sold 72: how many oranges
did he have left ?
62. Charles had 47 marbles, and his father gave him
36 more; he afterwards lost 50: how many marbles did
he then have ?
63. A farmer having 158 sheep, lost 30 of them by
sickness and sold 52 : how many sheep did he have left?
64. Sarah's father gave her 60 cents, and her mother
gave her 54 cents ; if she spends 62 cents for a pair of
gloves, how many cents will she have left ?
65. A merchant purchased a piece of silk containing
78 yards; he then sold 18 yards to one lady, and 17 to
another : how many yards had he left ?
66. If a man has property in his possession worth
215 dollars, and owes 39 dollars to one person, and
54 dollars to another, how much money will he have left,
when he pays his debts ?
67. If a man's income is 185 dollars per month, and
he pays 35 dollars for house rent, and 63 dollars for pro-
visions per month, how many dollars will he have left for
other expenses ?
68. George having 74 pears, gave away 43 of them ;
if he should buy 35 more, how many would he then
have?
69. If you add 115 to 78, and from the sum take 134,
what will the remainder be ?
70. If you subtract 93 from 147, and add 110 to the
remainder, what will the sum be ?
71. A merchant purchased 125 pounds of butter of
one dairy-man, and 187 pounds of another ; he afterwards
sold 163 pounds: how many pounds did he have left?
ART. 32.] SUBTRACTION. 35
72. A miller bought 200 bushels of wheat of one
farmer, and 153 bushels of another; he afterwards sold
180 bushels : how many bushels did he have left?
73. A man traveled 538 miles in 3 days ; the first day
he traveled 149 miles, the second day, 126 miles : how
far did he travel the third day ?
74. A grocer bought a cask of oil containing 256 gal-
lons ; after selling 93 gallons, he perceived the cask was
leaky, and on measuring what was left, found he had 38
gallons : how many gallons had leaked out ?
75. A manufacturer bought 248 pounds of wool of one
customer, and 361 pounds of another ; he then worked
up 430 pounds : how many pounds had he left ?
76. A man paid 375 dollars for a span of horses, and
450 dollars for a carriage ; he afterwards sold his horses
and carriage for 1000 dollars; how much did he make
by his bargain ?
77. A grocer bought 285 pounds of lard of one farmer,
and 327 pounds of another; he afterwards sold 110
pounds to one customer, and 163 pounds to another : how
much lard did he have left ?
78. A flour dealer having 500 barrels of flour on hand,
sold 263 barrels to one customer and 65 barrels to an-
other : how many barrels had he left ?
79. Harriet wished to read a book through which con-
tained 726 pages, in three weeks ; the first week she read
165 pages, and the second week she read 264 pages :
how many pages were left for her to read the third week ?
80. A man bought a house for 1200 dollars, and hav-
ing laid out 210 dollars for repairs, sold it for 1300 dol-
lars : how much did he lose by the bargain ?
81. A young man having 2000 dollars, spent 765 the
first year and 843 the second year : how much had he
left?'
36 MULTIPLICATION. [SECT. IV,
SECTION IV.
MULTIPLICATION.
ART. 33. Ex. 1. What will three lemons cost, at 2
cents apiece ?
Analysis. Since 1 lemon costs 2 cents, 3 lemons will
cost 3 times 2 cents ; and 3 times 2 cents are 6 cents.
Therefore, 3 lemons, at 2 cents apiece, will cost 6 cents.
OBS. The preceding operation is a short method of finding how
much 2 cents will amount to, when repeated or taken 3 times, and
is called Multiplication. Thus, 2 cents -J- 2 cents -f- 2 cents are
6 cents. Hence,
34r MUTIPLICATION is the process of finding the amount
of a number repeated or added to itself, a given number of
times.
The number to be repeated or multiplied, is called the
multiplicand.
The number by which we multiply, is called the mul-
tiplier, and shows how many times the multiplicand is to
be repeated or taken.
The answer, or number produced by multiplication, is
called the product.
Thus, when we say 5 times 7 are 35, 7 is the multipli-
cand, 5 the multiplier, and 35 the product.
OBS. When the multiplicand denotes things of one kind, or de-
nomination only, the operation is called Simple Multiplication.
QUEST. 34. What is multiplication 1 What, is the numbr r to be repeated
or multiplied called 1 What the number by which we multijyly ? What does
the multiplier show ? What is the answer called ? When wo say 5 times 7
are 35, which is the multiplicand? Which the multiplier? Which the
product^ Obs. When the multiplicand denotes things of one denomination
only, what is the operation called 1
ARTS. 36 39.] MULTIPLICATION. 39
38 The product cf any two numbers will be the same,
whichever factor is taken for the multiplier. Thus,
If a garden contains 3 rows of trees as
represented by the number of horizontal * * * * *
f JL : ^ ' 1 -L * * * * *
rows of stars m the margin, and each row .
has 5 trees as represented by the number of
stars in a row, it is evident, that the whole
number of trees in the garden is equal either to the number
of stars in a horizontal row, taken three times, or to the
number of stars in a perpendicular row taken five times;
that is, equal to 5 X 3, or 3X5.
EXAMPLES.
39. When the multiplier contains but ONE figure.
Ex. 1. What will 3 horses cost, at 123 dollars apiece?
Analysis. Since 1 horse costs 123 dollars, 3 horses
will cost 3 times 123 dollars.
Directions. Write the multi- Operation,
plicr under the multiplicand; 123 multiplicand,
then, beginning at the right 3 multiplier.
hand, multiply each figure of the
u . ,. \\ ., ' 1f . r Dolls. 369 product.
multiplicand by the multiplier.
Thus, 3 times 3 units are 9 units, or we may simply say
3 times 3 a^e 9 ; set the 9 in units' place under the figure
multiplied. 3 times 2 are 6 ; set the 6 in tens' place.
3 times 1 are 3 ; set the 3 in hundreds' place.
Note. The pupil should be required to analyze every example,
and to give the reasoning in full ; otherwise the operation is liable
to become mer* guess-icork, and a habit is formed, which is alike
destructive to mental discipline and all substantial improvement.
Solve the following examples in a similar manner :
(2.) (3.) (4.) (5.)
Multiplicand 34 312 2021 1110
Multiplier 2345
40 MULTIPLICATION. [SECT. IV
(6.) (7.) (8.) (9.)
Multiplicand, 4022 6102 7110 8101
Multiplier, _J3 _4 5 7
10. What will 6 cows cost at 23 dollars apiece.
Suggestion. In this example the product of the differ-
ent figures of the multiplicand into the multiplier, exceeds
9 ; we must therefore write the unit*' figure under the
figure multiplied, and carry the tens to the next product
on the left, as in addition. Thus, begin-
ning at the right hand as before, 6 times Operation.
3 units are 18 units, or we may simply 23 dolls,
say 6 times 3 are 18. Now it requires 6
two figures to express 18; we there- Ans. 138 dollars*
fore set the 8 under the figure multi-
plied, and reserving the 1, carry it to the product of the
next figure, as in addition. (Art. 23.) Next, 6 times 2
are 12, and 1 (to carry) makes 13. Since there are no
more figures to be multiplied, we set down the 13 in full.
The product is 138 dollars. Hence,
4O When the multiplier contains but one figure.
Write the multiplier under the multiplicand, units un-
der units, and draw a line beneath them.
Begin with the units, and multiply each figure of the
multiplicand by the multiplier, setting down the result and
carrying as in addition. (Art. 23.)
Multiply the following numbers together.
11. 78X4. 18. 524X6.
12. 96X5. 19. 360X7.
13. 83X3. 20. 475X4.
14. 120X7. 21. 792X5.
15. 138X6. 22. 820X.8.
16. 163X5. 23. 804x7.
17. 281X8. 24. 968X9.
ARTS. 40, 41.] MULTIPLICATION. 41
25. What will 11 5 barrels of flour cost, at 6 dollars
per barrel ?
26. A man bought 460 pair of boots, at 5 dollars a pair :
,10 w much did he pay for the whole ?
27. What cost 196 acres of land, at 7 dollars per acre?
28. What cost 310 ploughs, at 8 dollars apiece?
29. What cost 691 hats, at 7 dollars apiece?
30. What cost 865 heifers, at 9 dollars per head?
31. What cost 968 cheeses, at 8 dollars apiece?
32. What cost 1260 sheep, at 7 dollars per head?
33. What cost 9 farms, at 2365 dollars apiece?
4: 1 When the multiplier contains more than ONE figure.
34. A man sold 23 sleighs, at 54 dollars apiece : how
much did he receive for them all ?
Suggestion. Eeasoning as before, if 1 sleigh costs
54 dollars, 23 sleighs will cost 23 times as much.
Directions. As it is not Operation.
convenient to multiply by 23 54 Multiplicand,
at once, we first multiply by 23 Multiplier,
the 3 units, then by the 2 162 cost of 3 s.
tens, and add-the two results 108 " " 20 s.
together. Thus, 3 times 4 Dolls. 1242 " "23 s.
are 12, set the 2 under the
figure 3, by which we are multiplying, and carry the 1
as above. 3 times 5 are 15, and 1 (to carry) makes 16.
Next, we multiply by the 2 tens thus : 20 times 4 units
are 80 units or 8 tens ; or we may simply say 2 times 4
are 8. Set the 8 under the figure 2 by which we are
multiplying, that is, in tens' place, because it is tens.
2 times 5 are 10. Finally, adding these two products
together as they stand, units to units, tens to tens, &c.,
we have 1242 dollars, which is the whole product re-
quired.
42
MULTIPLICATION.
[SECT. IV
Note. When She multiplier contains more than one figure, the
several products of the multiplicand into the separate figures of the
multiplier, are called partial products.
35. Multiply 45 by 36, and prove the operation.
Operation.
Beginning at the right hand, we
proceed thus : 6 times 5 are 30 ;
set the under the figure by which
we ar multiplying ; 6 times 4 are
24 and 3 (to carry) are 27, &c.
45 Multiplicand
36 Multiplier.
"270
135
PROOF. We multiply the mul-
tiplier by the multiplicand, and
since the result thus obtained is
the same as the product above,
the work is rio-ht.
1620 Prod.
Proof.
36
45
180
144
1620 Prod.
36. What is the product of 234 multiplied by 165 ?
Operation.
Suggestion. Proceed in the same man- 234
ner as when the multiplier contains but 165
two figures, remembering to place the 1170
right luand figure of each partial product 1404
directly under the figure by which you 234
multiply. 38610 Ant
37. What is the product of 326 multiplied by 205 ?
Suggestion. Since multiplying by a Operation.
cipher produces nothing, in the operation 326
we omit the in the multiplier. Thus, 205
having multiplied by the 5 units, we next ] 630
multiply by the 2 hundreds, and place the 652
first figure of this partial product under 66830 Ans,
the figure by which we are multiplying.
ARTS. 42, 43.] MULTIPLICATION. 43
42 From the preceding illustrations and principles
we derive the following
GENERAL RULE FOR MULTIPLICATION
I. Write the multiplier under the multiplicand, units
under units, tens under tens, &c.
II. When the multiplier contains but ONE figure, begin
with the units, and multiply each figure of the multipli-
cand by the multiplier, setting down the result and carry-
ing as in addition. (Art. 23.)
III. If the multiplier contains MORE than on,e figure,
multiply each figure of the multiplicand by each figure
of the multiplier separately, and write the first figure of
each partial product under the figure by which you are
multiplying.
Finally, add the several partial products together, and
the sum will be the whole product, or answer required.
43 PROOF. Multiply the multiplier by the multipli-
cand, and if the second result is the same as the first, the
work is right.
OBS. 1. It is immaterial as to the result which of the factors is
taken for the multiplier. (Art. 88.) But it is more convenient and
therefore customary to place the larger number for the multipli-
cand and the smaller for the multiplier. Thus, it is easier to mul-
tiply 254672381 by 7, than it is to multiply 7 by 254672381, but
the product will be the same.
2. Multiplication may also be proved by division, and by casting
out the nines; but neither of these methods can be explained here
without anticipating principles belonging to division, with which
the learner is supposed as yet to be unacquainted.
QUEST. 42. How do you write numbers for multiplication? When the
multiplier contains but one figure, how do you proceed ? When the multi-
plier contains more than one figure, how proceed? 41. JVbt. What \& meant
by partial products? What is to be done with the partial products? 43,
How is multiplication proved ?
44 MULTIPLICATION. [^ECT. IV
EXAMPLES FOR PRACTICE.
1. Multiply 63 by 4. 10. Multiply 46 by 10.
2. Multiply 78 by 5. 11. Multiply 52 by 11.
3. Multiply 81 by 7. 12. Multiply 68 by 12.
4. Multiply 97 by 6. 13. Multiply 84 by 13.
5. Multiply 120 by 7. 14. Multiply 78 by 15.
6. Multiply 231 by 5. 15. Multiply 95 by 23.
7. Multiply 446 by 8. 16. Multiply 129 by 35.
8. Multiply 307 by 9. 17. Multiply 293 by 42.
9. Multiply 560 by 7. 18. Multiply 461 by 55.
19. If 1 barrel of flour costs 9 dollars, how much
will 38 barrels cost ?
20. If 1 apple-tree bears 14 bushels of apples, how
many bushels will 24 trees bear ?
21. In 1 foot there are 12 inches : how many inches
are there in 28 feet?
22. In 1 pound there are 20 shillings : how many shil-
lings are there in 31 pounds ?
23. What will 17 cows cost, at 23 dollars apiece ?
24. What will 25 tons of hay cost, at 1 9 dollars per ton ?
25. What will 37 sleighs cost, at 43 dollars apiece ?
26. What will a drove of 150 sheep come to, at 13
shillings per head ?
27. What cost 105 acres of land, at 15 dollars per acre?
28. How much will 135 yards of cloth come to, at 18
shillings per yard ?
29. In 1 pound there are 16 ounces : how many ounces
are there in 246 pounds ?
30. A drover sold 283 oxen, at 38 dollars per head:
how much did he receive for them ?
31. If you walk 22 miles per day, how far will you
walk in 305 days ?
32. In one day there are 24 hours : how many hours
are there in 365 days ?
ARTS. 44, 45.] MULTIPLICATION. 45
33. In 1 year there are 52 weeks : how many weeks
are there in 175 years?
34. In 1 hour there are 60 minutes : how many min-
utes are there in 396 hours?
35. In 1 hogshead there are 63 gallons : how many
gallons are there in 450 hogsheads ?
36. What will 475 horses cost, at 73 dollars apiece?
37. In 1 square foot there are 144 square inches : how
many square inches are there in 235 feet ?
38. How far will a ship sail in 158 days, if she sails
165 miles per day ?
44. It is a fundamental principle of notation, that
each removal of a figure one place towards the left, in-
creases its value ten times; (Art. 10;) consequently an-
nexing a cipher to a number, increases its value ten times,
or multiplies it by 10; annexing two ciphers, increases
its value a hundred times, or multiplies it by 100 ; an-
nexing three ciphers, increases it a thousand times, or mul-
tiplies it by 1000, <fec. ; for each cipher annexed, removes
each figure in the number one place towards the left.
Thus, 12 with a cipher annexed becomes 120, and i?
the same as 12X10; 12 with two ciphers annexed, be-
comes 1200, and is the same as 12X100; 12 with three
ciphers annexed, becomes 12000, and is the same as
12X1000, &c. Hence,
45. To multiply by 10, 100, 1000, &c.
Annex as many ciphers to the multiplicand as there are
ciphers in the multiplier, and the number thus formed will
t>e the product required.
Note. To annex means to place after, or at the right hand.
QUEST. 44. What effect does it have to remove a figure />ne place towards
the left hand ? Two piaces ? 45. How do you proceed when the multiplier
Is 10, 100, 1000, &c ? Note. What is the meaning of the term annex?
46 MULTIPLICATION. [SECT. IT
40. What will 10 dresses cost, at 18 dollars apiece?
Solution. If 1 dress costs 18 dollars, 10 dresses will
cost 10 times 18 dollars. But annexing a cipher to a
number multiplies it by 10. We therefore annex a cipher
to the multiplicand, (18 dollars,) and it becomes 180 dol-
ars. The answer therefore is 180 dollars.
Multiply the following numbers in a similar manner :
41. 26X10. 46. 469X10000.
42. 37X100. 47. 523X100000.
43. 51X1000. 48. 681X1000000.
44. 226X1000. 49. 85612X10000.
45. 341X1000. 50. 960305X100000.
51. What will 20 wagons cott, at 67 dollars apiece?
Suggestion. Since multiplying by Operation,
ciphers produces ciphers, we omit mul- 67
tiplying by the 0, and placing the sig- 20
nificant figure 2 under the right hand Ans. 1340 dollars,
figure of the multiplicand, multiply
by it in the usual way, and annex a cipher to the product.
The answer is 1340 dollars. Hence,
46. When there are ciphers on the right hand of the
multiplier.
Multiply the multiplicand by the significant figures of
the multiplier, and to this product annex as many ciphers,
as are found on the right hand of the multiplier.
(52.) (53.) (54.) (55.)
85 97 123 234
200 3000 40000 50000
(56.)
261
(57.)
329
(58.)
462
(59.)
571
130
2400
3501/0
460000
Q.UET. 46. When there are ciphers on the right of the multiplier, how da
you proceed ?
ARTS. 46, 47.] MULTIPLICATION. 47
60. In one hour there are 60 minutes : how many min-
utes are there in 125 hours?
61. What will 300 barrels of flour cost af 8 dollars per
barrel ?
62. What cost 400 yds. of cloth, at 17 shills. per yd. ?
63. If the expenses of 1 man are 135 dollars per month,
how much will be the expenses of 200 men ?
64. If 1500 men can build a fort in 235 days, how long
will it take one man to build it ?
4:7. When there are ciphers on the right of the mul-
tiplicand.
Multiply the significant figures of the multiplicand by
tJie multiplier, and to tlie product annex as many ciphers,
as are found on the right of the multiplicand.
65. What will 43 building lots cost, at 3500 dollars a
lot?
Having placed the multiplier under Operation.
the significant figures of the multipli- 3500
cand, multiply by it as usual, and to 43
the product thus produced, annex two 105
ciphers, because there are two ciphers 140
on the right of the multiplicand. Ans. 150500 dolls
(66.) (67.) (68.) (69.)
1300 2400 21000 25000
15 17 24 32
70. What is the product of 132000 multiplied by 25 ?
71. What is the product of 430000 multiplied by 34 ?
72. What is the product of 1520000 multiplied by 43 ?
73. What is the product of 2010000 multiplied by 52 ?
74. What is the product of 3004000 multiplied by 61 ?
Q.CBST. 47. When there are ciphers on the right of the multiplinawl) how
do you proceed ?
48
MULTIPLICATION.
[SECT.
48. When the multiplier and multiplicand both have
ciphers on the right.
Multiply the significant figures of the multiplicand by
the significant figures of the multiplier, and to this pro-
duct annex as many ciphers, as are found on the right of
both factors.
75. Multiply 16000 by 3200.
Having placed the significant figures
of the multiplier under those of the mul-
tiplicand, we multiply by them as usual,
and to the product thus obtained, annex
five ciphers, because there are five ci-
phers on the right of both factors.
Solve the following examples :
76. 2100X200.
78. 12000X210.
80. 38000X19000.
82. 2800000X26000.
84. 1000 milesX!40.
86. 120 dollars X 4200.
88. 867 poundsX424.
00. 6726 rodsX627.
92. 25268 penceX4005.
Operation.
16000
3200
32
48
Ans. 51200000
94. 376245X3164.
96. 600400X7034.
98. 432467X30005.
100. 680539X80406.
102. Multiply seventy -three
77. 3400X130.
79. 25000X2600.
81. 500000X42000.
83. 140 yards XI 6000.
85. 20 dollars X 35000,
87. 75000 dolls. X 365.
89. 6830 feetX562.
91. 7207 galls. X 807.
93. 36074 tons X 4060.
95. 703268X5346.
97. 864325X6728.
99. 4567832X27324.
101. 7563057X62043.
thousand and seven by
twenty thousand and seven hundred.
103. Multiply six hundred thousand, two hundred and
three by seventy thousand and seventeen.
QUEST. 48. When there are ciphers on tho right of both tl e inultipliel and
multiplicand, how proceed ?
ABTS. 48 50.] DIVISIDN. 49
SECTIt N V.
DIVISION.
ART. 49o Ex. 1. How many lead pencils, at 2 cents
apiece, can I buy for 1 cents ?
Solution. Since 2 cents will buy 1 pencil, 10 cents
will buy as many pencils, as 2 cents are contained times in
10 cents ; and 2 cents are contained in 10 cents, 5 times.
I can therefore buy 5 pencils.
2. A father bought 12 pears, which he divided equally
among his 3 children : how many pears did each re-
ceive ?
Solution. Reasoning in a similar manner as above, it
is plain that each child will receive 1 pear, as often as 3
is contained in 12 ; that is, each must receive as many
pears, as 3 is contained times in 12. Now 3 is contained in
12, 4 times. Each child therefore received 4 pears.
OBS. The object of the first example is to find how many times
one given number is contained in another. The object of the second
is to divide a given number into several equal parts, and to ascertain
the value of these parts. The operation by which they are solved
is precisely the same, and is called Division. Hence,
5O. DIVISION is the process of finding how many times
one given number is contained in another.
The number to be divided, is called the dividend.
The number by which we divide, is called the divisor.
The ansiuer, or number obtained by division, is called
the quotient, and shows how many times the divisor is
contained in the dividend.
QUEST. 50. What is division ? What is the number to be divided, called ?
The number by which we divide ? What is the answer called? What does
.ne quotient show ?
4
50 DIVISIO.V. [SECT. V
Note. The term quotient is derived from the Latin word qttotiea
which signifies how )ften, or how many times.
51. The number which is sometimes left after division,
in called the remainder. Thus, when we say 4 is con-
tamed in 21, 5 times and 1 over, 4 is the divisor, 21 the
dividend, 5 the quotient, and 1 the remainder.
OBS. 1. The remainder is always less than the divisor; for if it
were equal to, or greater than the divisor, the divisor could be con-
tained once more in the dividend.
2. The remainder is also of the same denomination as the divi-
dend; for it is a part of it.
52. Sign of Division (-r). The sign of Division is
a horizontal line between two dots (-7-), and shows that
the number before it, is to be divided by the number
after it. Thus, the expression 246, signifies that 24 is
to be divided by 6.
Division is also expressed by placing the divisor under
the dividend with a short line between them. Thus the
expression A 7 *, shows that 35 is to be divided by 7, and is
equivalent to 35-7-7.
53* It will be perceived that division is similar in prin-
ciple to subtraction, and may be performed by it. For
instance, to find how many times 3 is contained in 12,
subtract 3 (the divisor) continually from 12 (the dividend)
until the latter is exhausted ; then counting these repeated
subtractions, we shall have the true quotient. Thus, 3
from 12 leaves 9 ; 3 from 9 leaves 6 ; 3 from 6 leaves 3 ;
3 from 8 leaves 0. Now, by counting, we find that 3 has
QUEST. 51. What is the number called which is sometimes left after divi-
sion? When we say 4 is in 21, 5 times and 1 over, what is the 4 cabled? The
21 ? The 5 ? The 1 V Obs. Is the remainder greater or less than the divisor?
Why? Of what denomination is it? Why? 52. What is the sign of divi-
aion ? What does it show ? In what other way is division expressed ?
ARTS. 51 53.] DIVISION.
51
been taken from 12, 4 times; consequently 3 is contained
t~ ?2, 4 times. Hence,
Division is sometimes defined to "be a short way of per-
forming repeated subtractions of the same number.
OBS. 1. It will also be observed that division is the reverse of
multiplication. Multiplication is the repeated addition of the same
number ; division is the repeated subtraction of the same number.
The product of the one answers to the dividend of the other : but
the latter is always given, while the former is required.
2. When the dividend denotes things of one kind, or denominar
tion only, the operation is called Simple Division.
DIVISION TABLE.
1 is in
2 is in
3 is in
4 is in
5 is in
1, once.
2, once.
3, once.
4, once.
5, once.
2, 2
4, 2
6, 2
8, 2
10, 2
3, 3
6, 3
9, 3
12, 3
15, 3
4, 4
8, 4
12, 4
16, 4
20, 4
5, 5
10, 5
15, 5
20, 5
25, 5
6, 6
12, 6
18, 6
24, 6
30, 6
7, 7
14, 7
21, 7
28, 7
35, 7
8, 8
16, 8
24, 8
32, 8
40, 8
9, 9
18, 9
27, 9
36, 9
45, 9
10, 10
20, 10
30, 10
40, 10
50, 10
6 is in
7 is in
8 is in
9 is in
10 is in
6, once.
7, once.
8, once.
9, once.
10, once.
12, 2
14, .2
16, 2
18, 2
20, 2
18, 3
21, 3
24, 3
27, 3
30, 3
24, 4
28, 4
32, 4
36, 4
40, 4
30, 5
35, 5
40, 5
45, 5
50, 5
36, 6
42, 6
48, 6
54, 6
60, 6
42, 7
49, 7
56, 7
63, 7
70, 7
48, 8
56, 8
64, 8
72, 8
80, 8
54, 9
63, 9
72, 9
81, 9
90, 9
60, 10
70, 10
80, 10
90, 10
100, 10
QUEST. -Obs. When the dividend denotes things cf g>ne denomination only,
Irhat iR the operation called ?
52 DIVISION. [SECT. V
SHORT DIVISION.
ART. 54. Ex. 1. How many yards of cloth, at 2 dol
lars per yard, can I buy for 246 dollars ?
Analysis. Since 2 dollars will buy 1 yard, 246 dol-
lars will buy as many yards, as 2 dollars are contained
times in 246 dollars.
Directions. Write the divisor on Operation.
the left of the dividend with a curve w *>'- v ^ n<L
line between them; then, beginning '
at the left hand, proceed thus: 2 is uot ' l
contained in 2, once. As the 2 in the dividend denotes
hundreds, the 1 must be a hundred ; we therefore write
it in hundreds' place under the figure divided. 2 is con-
tained in 4, 2 times ; and since the 4 denotes tens, the 2
must also be tens, and must be written in tens' place. 2 is
in 6, 3 times. The 6 is units ; hence the 3 must be units,
and we write it in units' place. The answer is 123 yards.
Solve the following examples in a similar manner :
2. Divide 42 by 2. 6. Divide 684 by 2.
3. Divide 69 by 3. Y. Divide 4488 by 4.
4. Divide 488 by 4. 8. Divide 3963 by 3.
5. Divide 555 by 5. 9. Divide 6666 by 6.
55 When the divisor is not contained in the first
figure of the dividend, we must find how many times it
is contained hi the first two figures.
10. At 2 dollars a bushel, how much wheat can be
bought for 124 dollars?
Since the divisor 2, is not contained in Operation.
the first figure of the dividend, we find 2)124
how many times it is contained in the first Ans. 62 bu,
two figures. Thus 2 is in 12, 6 times ; set
the 6 under the 2. Next, 2 is in 4, 2 times. The an-
Ewer is 62 bushels.
ARTS 54 57.] DIVISION. 53
11. Divide 142 by 2. 13. Divide 1648 by 4.
12. Divide 129 by 3. 14. Divide 2877 by 7.
56 After dividing any figure of the dividend, if there
is a remainder, prefix it mentally to the next figure of the
dividend, and then divide this number as before.
Note. To prefix means to place before, or at the left hand.
15. A man bought 42 peaches, which he divided
equally among his 3 children : how many did he give to
each?
When we divide 4 by 3, there is 1 re- Operation,
mainder. This we prefix mentally to the 3)42
next figure of the dividend. We then say, 14 Ans.
3 is in 12, 4 times.
16. Divide 56 by 4. 18. Divide 456 by 6.
17. Divide 125 by 5. 19. Divide 3648 by 8.
57. Having obtained the first quotient figure, if the
divisor is not contained in any figure of the dividend, place
a cipher in the quotient, and prefix this figure to the next
one of the dividend, as if it were a remainder.
20. If hats are 2 dollars apiece, how many can be
bought for 216 dollars ?
As the divisor is not contained in 1, Operation.
the second figure of the dividend, we 2)216
put a in the quotient, and prefix the Ans. 108 hats.
1 to the 6 as directed above. Now 2
is in 16, 8 times.
21. Divide 2545 by 5. 23. Divide 6402 by 6.
22. Divide 3604 by 4. 24. Divide 4024 by 8.
25. A man divided 17 loaves of bread equally between
2 poor persons : how many did he give to each ?
Suggestion. Reasoning as before, he gave each as
many loaves as 2 is contained times in 17 :
54 DIVISION. [S2CT. V
Thus, 2 is contained in 17, 8 Opemtion-.
times and 1 over; that is, after 2)17
giving them 8 loaves apiece, there Quot. 8-1 remainder,
is one loaf left which is not divid- Ans. 8-J- loaves.
ed. Now 2 is not contained in 1 ;
hence the division must be represented by writing the 2
under the 1, thus , (Art. 52,) which jnust be annexed to
the 8. The true quotient, is 8-J. He therefore gave eight
and a half loaves to each. Hence,
58 When there is a remainder after dividing the last
figure of the dividend, it should always be written over the
divisor and annexed to the quotient.
Note. To annex means to place after, or at the riglit hand.
59* When the process of dividing is carried on in the
mind, and the quotient only is set down, the operation it
called SHORT DIVISION.
6O From the preceding illustrations and principles, we
derive the following
RULE FOR SHORT DIVISION.
I. Write the divisor on the left of the dividend, with a
curve line between them.
Beginning at the left hand, divide each figure of the
dividend by the divisor, and place each quotient figure
under the figure divided.
II. When there is a remainder after dividing any fig-
ure, prefix it to the next figure of the dividend and divide
this number as before. If the divisor is not contained in
QUEST. 59. What is Short Division ? GO. How do you write numbers
for short division? Where begin to divide ? Where place each quotient fig-
ure? When there is a remainder after dividing a figure of the dividend,
what must be done with it 1 If the divisor is not contained in a fl ore of th*>
dividend, how proceed? When there is a remainder, after dividing the luat
fl#me of the dividend, what must be done with it ?
ARTS. 58 61.] DIVISION. 55
any figure of the dividend, place a cipher in the quotient,
and prefix this figure to the next one of the dividend, as if
it were a remainder. (Arts. 56, 57.)
III. When there is a ramainder after dividing the last
figure y write it over the divisor and annex it to the quotient,
61 PROOF. Multiply the divisor by the quotient, to
the product add the remainder, and if the sum is equal to
the dividend, the work is right.
OBS. Division may also be proved by subtracting the remainder,
if any, from the dividend, then dividing the result by the quotient.
EXAMPLES FOR PRACTICE.
1. Divide 426 by 3. 10. Divide 3640 by 5.
2. Divide 506 by 5. 11. Divide 6210 by 4.
3. Divide 304 by 4. 12. Divide 7031 by 7.
4. Divide 450 by 6. 13. Divide 2403 by 6.
5. Divide 720 by 7. 14. Divide 8131 by 9.
6. Divide 510 by 9. 15. Divide 7384 by 8.
7. Divide 604 by 5. 16. Divide 8560 by 7.
8. Divide 760 by 8. 17. Divide 7000 by 8.
9. Divide 813 by 7. 18. Divide 9100 by 9.
19. How many pair of shoes, at 2 dollars a pair, cau
you buy for 126 dollars ?
20. How many hats, at 4 dollars apiece, can be bought
for 168 dollars?
21. A man bought 144 marbles which he divided equally
among his 6 children : how many did each receive ?
22. A man distributed 360 cents to a company of poor
children, giving 8 cents to each : how many children were
there in the company ?
23. How many yards of silk, at 6 shillings per yard,
can I buy for 450 shillings ?
QUEST. 61. How is division proved? Obs* What other wav of proving
division is mentioned?
56 DIVISION. [SECT. V
24. A man having 600 dollars, wished to lay it out
in flour, at 7 dollars a barrel: how many whole barrels
could he buy, and how many dollars would he have left ?
25. If you read 9 pages each day, how long will it
take you to read a book through which has 828 pages?
26. If I pay 8 dollars a yard for broadcloth, how many
yards can I buy for 1265 dollars?
27. If a stage coach goes at the rate of 8 miles per
hour, how long will it be in going 1560 miles ?
28. If a ship sails 9 miles an hour, how long will it
be in sailing to Liverpool, a distance of 3000 miles ?
LONG DIVISION.
ART. 62. Ex. 1. A man having 156 dollars laid it
out in sheep at 2 dollars apiece : how many did he buy ?
Analysis. Reasoning as before, since 2 dollars will
buy 1 sheep, 156 dollars will buy as many shee$ as 2
dollars are contained times in 156 dollars.
Directions. Having written the di- Operation.
visor on the left of the dividend as in Di - Divid - ^ uot -
short division, proceed in the follow- 34 .
ing manner :
First. Find how many times the -^
divisor (2) is contained in (15) the
first two figures of the dividend, and place the quotient
figure (7) on the right of the dividend with a curve line
between them. Second. Multiply the divisor by the
quotient figure, (2 times 7 are 14,) and write the product
(14) under the figures divided. Third. Subtract the
product from the figures divided. (The remainder is 1.)
Fourth. Bringing down the next figure of the dividend,
and placing it on the right of the remainder we have 16.
Now 2 is contained in 16, 8 times; place the 8 on the
right hand of the last quotient figure, and multiplying
ARTS. 62, G 3.] DIVISION. 57
the divisor by it, (8 times 2 are 16,) set the product undei
the figures divided, and subtract as before. Therefore 156
dollars will buy 78 sheep, at 2 dollars apiece.
63. When the result of each step in the operation is
set down, the process of dividing is called LONG DIVISION.
It is the same in principle as Short Division. The
only difference between them is, that in Long Division
the result of each step in the operation is written down,
while in Short Division we carry on the whole process
in the mind, simply writing down the quotient.
Note. To prevent mistakes, it is advisable to put a dot under
each figure of the dividend, when it is brought down.
Solve the following examples by Long Division :
2. Divide 195 by 3. Ans. 65.
3. Divide 256 by 2. 6, Divide 2665 by 5.
4. Divide 1456 by 4. V. Divide 4392 by 6.
5. Divide 5477 by 3. 8. Divide 6517 by Y.
OBS. When the divisor is not contained in the first two figures of
the dividend, find how many times it is contained in the first threc t
or the fewest figures which will contain it, and proceed as before.
9. How many times is 13 contained in 10519?
Thus, 13 is contained in 105, Operation.
8 times; set the 8 in the quo- 13)l0519(809- t 2 sr Ans.
tient then multiplying and sub- 104
tracting, the remainder is 1. 119
Bringing down the next figure 117
we have 11 to be divided by 13. 2 rem.
But 13 is not contained in 11 ;
therefore we put a cipher in the quotient, and bring down
the next figure. (Art. 57.) Then 13 is sontained in 119,
CUTEST. 63. What is long division ? Wiiat is the diflerence between long
ed short division ?
58 DIVISION. [SECT "V.
9 times. Set the 9 in the quotient, multiply and sub-
tract, and the remainder is 2. Write the 2 over the di-
visor, and annex it to tho quotient. (Art. 58.)
O4. After the first quotient figure is obtained, for
each figure of the dividend which is brought down, either
a significant figure or a cipher must be put in the quotient.
Solve the following examples in a similar manner :
10. Divide 15425 by 11. Ans. 1402-ft-.
11. Divide 31237 by 15. Ans. 2082-ft.
65. From the preceding illustrations and principles
we derive the following
RULE FOR LONG DIVISION.
I. Beginning on the left of the dividend, find liow many
times the divisor is contained in the fewest figures that will
contain it, and place the quotient figure on the right of
the dividend with a curve line between them.
II. Multiply the divisor by this figure and subtract
the product from the figures divided ; to the right of
the remainder bring down the next figure of the dividend.
and divide this number as before. Proceed in this man-
ner till all the figures of the dividend are divided,
III. When there is a remainder after dividing the last
figure, write it over the divisor, and annex it to the quo-
tient, as in short division.
OBS. 1. Long Division is proved in the same manner as Short
Division.
2. When the divisor contains but one figure, the operation by
Short Division is the most expeditious, and si ould therefore be
practiced; but when the divisor contains two or *r* \re figures, it will
generally be the most convenient to divide by Long Division.
QUEST. 65. How do you divide in long division? Where place the quo-
tient ? Aftei obtaining the first quotient figure, how proceed ? When there is
a remainder after dividing the L'ist figure of the dividend what must be done
with it? Ols. How is long division proved? When should short division
be used ? Wheii long division ?
ARTS. 64, 65.] DIVISION. 59
EXAMPLES FOR PRACTICE.
1. Divide 369 by 8. 10. Divide 675 by 25.
2. Divide 435 by 9. 11. Divide 742 by 31.
3. Divide 564 by 7. 12. Divide 798 by 37.
4. Divide 403 by 10. 13. Divide 834 by 42.
5. Divide 641 by 11. 14. Divide 960 by 48.
6. Divide 576 by 12. 15. Divide 1142 by 53.
7. Divide 274 by 13. 16. Divide 2187 by 67.
8. Divide 449 by 14. 17. Divide 3400 by 75.
9. Divide 617 by 15. 18. Divide 4826 by 84.
19. How many caps, at 7 shillings apiece, can I buy
for 168 shillings?
20. How many pair of boots, at 5 dollars a pair, can
be bought for 175 dollars ?
<}1. A man laid out 252 dollars in beef, at 9 dollars a
barrel : how many barrels did he buy ?
22. In 12 pence there is 1 shilling : how many shillings
are there in 198 pence?
23. In 20 shillings there is 1 pound : how many pounds
are there in 2 1 5 shillings ?
24. In 16 ounces there is 1 pound: how many pounds
are there in 268 ounces ?
25. How many trunks, at 15 shillings apiece, can be
bought for 255 shillings ?
26. If 27 pounds of flour will last a family a week,
how long will 810 pounds last them?
27. How many yards of broadcloth, at 23 shillings per
yard, can be bought for 756 shillings?
28. If it takes 18 yards of silk to make a dress, how
many dresses can be made from 1350 yards?
29. How many sheep, at 19 shillings per head, can be
bought for 1539 shillings?
30. A farmer having 1840 dollars, laid it out in land,
at 25 dollars per acre : how many acres did he buy?
60 DIVISION. [SECT. V
31. A man wishes to invest 2562 dollars in Railroad
stock : how many shares can he buy, at 42 dollars per
share ?
32. In 1 year there are 52 weeks: how many years
are there in 1640 weeks ?
33. In one hogshead there are 63 gallons: how many
hogsheads are there in 3065 gallons ?
34. If a man can earn 75 dollars in a month, he wmany
months will it take him to earn 3280 dollars ?
35. If a young man's expenses are 83 dollars a month,
how long will 4265 dollars support him?
36. A man bought a drove of 95 horses for 4750 dol-
o
lars : how much did he give apiece ?
37. If a man should spend 16 dollars a month, how
long will it take him to spend 172 dollars?
38. A garrison having 2790 pounds of meat, wished to
have it last them 3 1 days : how many pounds could they
eat per day ?
39. How many times is 54 contained in 3241, and how
many over ?
40. How many times is 68 contained in 7230, and how
many over ?
41. How many times is 39 contained in 1042, and how
many over?
42. How many times is 47 contained in 2002, and how
many over?
43. What is the quotient of 1704 divided by 56 ?
44. What is the quotient of 2040 divided by 60 ?
45. What is the quotient of 2600 divided by 49 ?
46. What is the quotient of 2847 divided by 81 ?
47. Divide 1926 by 75. 51. Divide 9423 by 105.
48. Divide 2230 by 85. 52. Divide 13263 by 112,
49. Divide 6243 by 96. 53. Divide 26850 by 123,
50. Divide 8461 by 99. 54. Divide 48451 by 224.
ARTS. 66, 67.] DIVISION. 6!
6G. It has been shown that annexing a cipher to a
number, increases its value ten times, or multiplies it by
10, (Art. 44.) Reversing this process, that is, removing
a cipher from the right hand of a number, will evidently
diminish its value ten times, or divide it by 10; for, each
Sgure in the number is thus restored to its original place,
and consequently to its original value. Thus, annexing a
cipher to 12, it becomes 120, which is the same as 12 X 10.
On the other hand, removing the cipher from 120, it be-
comes 12, which is the same as 12010.
In the same manner it may be shown, that removing
two ciphers from the right of a number, divides it by 100;
removing three, divides it by 1000; removing four, di-
vides It by 10000, &c. Hence,
67. To divide by 10, 100, 1000, &c.
Cut off as many figures from the right hand of the divi-
dend as there are ciphers in the divisor. The remaining
figures of the dividend will be the quotient, and those cut
off the remainder.
55. Divide 2456 by 100.
Since there are 2 ciphers on Operation.
the right of the divisor, we cut 1JOO)24|56
off 2 figures on the right of the Quot. 24 and 56 rera
dividend. The quotient is 24
and 56 remainder, or 24- 1 V<r-
50. Divide 1325 by 10. Ans. 132 and 5 rem.
57. Divide 4620 by 100.
58. Divide 5633 by 1000.
59. Divide 8465 by 1000.
60. Divide 26244 by 1000.
61. Divide 136056 by 10000.
QuEST.~6<5. What is the effect of annexing a cipher to a number? What i*
the effect of removing a cipher from the right of a nr"aber ? 67, How proceed
wben the divisor is 10, 100, 1000, &e.?
DIVISION*
[SECT. V-
62. Divide 2443667 by 100000.
63. Divide 23454631 by 1000000.
68 When there are ciphers on the right hand of tho
divisor.
Cut off the ciphers from the divisor ; also cut of as
many figures from the right of the dividend. Then divide
the remaining figures of the dividend by the remaining fig-
ures of the divisor, and the result will be the quotient.
Finally, annex the figures cut off from the dividend to
the remainder, and the number thus formed will be the true
remainder.
64. At 200 dollars apiece, how many carriages can be
bought for 4765 dollars ?
Having cut off the two ciphers on
the right of the divisor, and two fig-
ures on the right of the dividend, we
divide the 47 by 2 in the usual way.
65. Divide 2658 by 20.
Ans. 132 and 18 rem
Operation.
2|OQ)47|65
Ans. 23 165 rera.
66. 3642 by 30.
68. 76235 by 1400.
70. 93600 by 2000.
72. 23148 by 1200.
74. 50382 by 1800.
76. 894000 by 2500.
78. 7450000 by 420000.
80. 348676 235.
82. 762005 401.
84. 6075071623.
86. 4367238-7-2367.
88. 8230732-7-3478.
90. 93670S58-f-67213.
67. 6493 by 200
69. 82634 by 1600.
71. 14245 by 3000.
73. 42061 by 1500.
75. 88317 by 2100.
77. 9203010 by 3100.
79. 9000000 by 300000.
81. 467342 341.
83. 506725 603.
85. 736241 2764.
87. 6203451-7-3827.
89. 823o762-f-42316.
91. 98765421-7-84327.
QUEST. 68. When there are ciphers on the right of the divisor, how pro
ceed ? \Vhat is to be done with figures cut off fro in the dividend ?
ARTS. 68 73.] FRACTIONS. 63
SECTION VI.
FRACTIONS.
7 1 When a number or thing, as an apple or a pear,
is divided into two equal parts, one of these parts is called
one half. If divided into three equal parts, one of the
parts is called one third ; if divided into four equal parts,
one of the parts is called one fourth, or one quarter / if
into ten, tenths ; if into a hundred, hundredths, &Q.
When a number or thing- is divided into equal parts, as
halves, thirds, fourths, fifths, &c., these parts are called
Fractions. Hence,
72. A FRACTION denotes a part or parts of a number
or thing.
An Integer is a whole number.
Note. The term fraction, is derived from the Latin fractio,
which signifies the act of breaking, a broken part or piece. Hence,
fractions are sometimes called broken numbers.
73. Fractions are commonly expressed by two num-
bers, one placed over the other, with a line between them.
Thus, one half is written, -J-; one third, -J- ; one fourth, J;
three fourths, f- ; two fifths, f ; nine tenths, -f^-, &c.
The number below the line is called the denominator,
and shows into how many parts the number or thing is
divided.
QUEST. 71. What is meant by one half? What is meant by one third 7
What is meant by a fourth ? What are fourths sometimes called ? What is
meant by fifths ? By sixths ? Eighths ? How many sevenths mako a whole
one ? How many tenths ? What is meant by twentieths ? By hundrodths?
72. What is a Fraction? What is an Integer? 73. How are fractions com-
monly expressed 1 What is the number below the line callud ? What does it
show ?
64 FRACTIONS. [SECT. VL
The number above the line is called the numerator, and
shows how many parts are expressed by the fraction.
Thus in the fraction -, the denominator .3, shows that the
number is divided into three equal parts ; the numerator 2,
shows that two of those parts are expressed by the fraction.
The numerator and denominator taken together, are
called the terms of the fraction.
7 4. A proper fraction is a fraction whose numerator is
less than its denominator ; as, -|-, f , -f-.
An improper fraction is one whose numerator is equal
to, or greater than its denominator, as f, f-.
A simple fraction is a fraction which has but one nu-
merator and one denominator, and may be proper, or im-
proper ; as, f,
A compound fraction is a fraction of a fraction ; as, J- of
i of I.
A complex fraction is one which has a fraction in its
2i 4 24- -3-
numerator, or denominator, or in both ; as , -, f, ~.
5 5 8f f
A mixed number is a whole number and a fraction writ-
ten together ; as, 4f , 25i.
7G The vato of a fraction is the quotient of the nu-
merator divided by the denominator. Thus, the value of
f is two ; of % is one ; of -J is one third ; &c.
Read the following fractions, and name the kind of each :
1. f; f; f ; f; if; if; Y; W; W-
2. -foff; f offof-V; * of of of 75.
QJ. 07 AI 3.
3. 2i; 14f ; 86;
QUEST. What is the number above the line called? What does it show?
What are the denominator and numerator, taken together, culled ? 74. What
is a proper fraction ? An improper traction V A simple fraction? A com-
pound fraction ? A complex fraction ? A mixed number ? 70. What is II*
value of a fraction?
ARTS. 74 78.] FRACTIONS. x 65
To find a fractional part of a given number.
Ex. 1. If a loaf of bread costs 4 cents, what will half a
loaf cost ?
Analysis. If a whole loaf costs 4 cents, 1 half a loaf
will cost 1 half of 4 cents ; and 1 half of 4 cents is 2 cents
Half a loaf of bread will therefore cost 2 cents.
2. If a pound of sugar costs 12 cents, what will 1 third
of a pound cost ?
Analysis. Reasoning as before, if a whole pound costs
12 cents, 1 third of a pound will cost 1 third of 12 cents;
and 1 third of 12 cents is 4 cents. Ans. 4 cents.
77. From these examples the learner will perceive that
A half of a number is equal to as many units, as 2 is
contained times in that number.
A third of a number is equal to as many units as 3 is
contained times in that number.
A fourth of a numbei is equal to as many units, as 4 is
contained times in that number, &c. Hence,
78 To find a HALF of a number, divide it by 2.
To find a THIRD of a number, divide it by 3.
To find a FOURTH of a number, divide it by 4, &c.
Note. For mental exercises in Fractions, see Mental Arith-
metic, Section VII.
3. What is half of 257 ?
Dividing 257 by 2, the quotient is 128 Operation.
and 1 over. Placing the 1 over the 2 2)257
and annexing it to the quotient, we ha\e 128i -Ans*
128-J-, which is the answer required.
4. What is a third of 21 ? 33 ? 48 ? 78 ! 151 !
5. What is a fourth of 45 ? 68 ? 72 ? 81 ? 130 I
6. What is a fifth of 7o ? 95 ? 135 ? 163 ?
7. What is an eighth of 73 ? 98 ? 104 ? 128 ?
QCKBT. 78. How do you find a half of a number ? A third ? A fonit t
3
66 FRACTIONS. [SECT. VI.
8. What is a seventh of 88 ? 133 ? 175 ? 250 ?
9. What is a ninth of 126 ? 163 ? 270 ? 316 ?
7 9 To find what part one given number is of another.
Make the number called the part, the numerator, and
the other given number the denominator. The fraction
thus formed will be the answer required.
1. What part of 3 is 2 ? Ans. f.
2. What part of 4 is 1? Is 2 ? Is 3 ? Is5?
3. What part of 7 is 2 ? Is 4 ? Is 5 ? Is 6 ?
4. What part of 9 is 1 ? Is 2 ? Is 4 ? Is 5?
5. 5 is what part of 11 ? Of 12 ? Of 13 ?
6. 8 is what part of 17 ? Of 19 ? Of 45 ?
7. 15 is what part of 38 ? Of 57 ? Of 85 ?
8O A part of a number being given to find the whole.
Multiply the given part by the number of parts into
which the whole is divided, and the product will be the
answer required.
1. 27 is 1 ninth of what number?
Suggestion. Since 27 is 1 ninth, 9 ninths, or the whole,
must be 9 times 27; and 27x9 243. Ans.
The given part is 27, and the Operation.
number of parts into which the 27 = 1 ninth,
whole is divided, is 9 ninths; 9= no. parts,
we therefore multiply 27 by 9. Ans. 2 43= the whole.
2. 18 is 1 fifth of what number?
3. 23 is 1 fourth of what number ?
4. 34 is 1 seventh of what number ?
5. 45 is 1 fifteenth of what number ?
6. 58 is 1 twelfth of what number?
7. 63 is 1 sixteenth of what number ?
QUEST. /9. How do you find what part one number is of another 1
80 When a part of a number is given, how do you find the whole ?
ARTS. 79 83.] FRACTIONS. 67
Multiplying a whole number by a fraction.
81. We have seen that multiplying by a whole num-
ber is taking the multiplicand as many times as there are
units in the multiplier. (Art. 36.) On the other hand,
If the multiplier is only a part of a unit, it is plain we
must take only a part of the multiplicand. Hence,
82 Multiplying by a fraction is talcing a certain
PORTION of the multiplicand as many times as there are
like portions of a unit in the multiplier. That is,
Multiplying by -J-, is taking 1 half of the multiplicand
once. Thus, 6Xi 3.
Multiplying by --, is taking 1 third of the multiplicand
once. Thus, 6xi=2.
Multiplying by f, is taking 1 third of the multiplicand
twice. Thus, 6X1=4.
Obs. If the multiplier is a unit or 1, the product is equal to the
multiplicand ; if the multiplier is greater than a unit, the product
is greater than the multiplicand ; (Art. 36 j) and if the multiplier is
less than a unit, the product is less than the multiplicand. Hence,
83. To multiply a whole number by a fraction.
Divide the given number by tJie denominator, and mul-
tiply the quotient by the numerator.
Obs. 1. The result will be the same if vre first multiply the given
number by the numerator, then divide this product by the denomi-
nator.
2. When the numerator is 1, it is unnecessary to multiply by it;
for, multiplying by 1 does not alter the value of a number. (Art.
82. Obs.)
QUEST. 81. What is meant by multiplying by a whole number? 82. By a
fraction? What is meant by multiplying by 1? By 4? By ? By |?
Obs. If the multiplier is a unit or 1, what is the product equal to ? When the
multiplier is greater than 1, how is the product compared with the multipli-
cand ? When less, how ? 83. How do you multfply a whole number by a frac-
tion ? Obs. What other method is mentioned ? When flie muneiator ie 1, is
tt aece&sas y to multiply by it ? Why not ?
68 FRACTIONS. [SECT. VI,
Ex. 1. If a ton of coal costs 9 dollars, what -will a
ton cost ?
Suggestion. Since a whole ton costs Ojperc&i&ri.
9 dollars, 1 half a ton will cost 1 half of 2)9
9 dollars. Now 2 is contained in 9, 4 Ans. 4 dolls,
times and 1 over. Place the 1 over the
divisor 2, and annex it to the quotient. (Art. 58.)
2. What will f of a yard of cloth cost, at 36 shilling*
per yard ?
Suggestion. First find what 1 third First Operation.
of a yard will cost, then 2 thirds. 3)36
That is, divide the given number by 12
the denominator 3, then multiply the 2
quotient by the numerator 2. Ans. 24 shil.
Or, we may first multiply the given Second Operation
number by the numerator, then di- 36
vide the product by the denomina- 2
tor. The answer is the same as be- 3)72
fore. Ans. 24 shil.
3. If an acre of land costs 30 dollars, what will of
an acre cost ?
4. What will -J- of a barrel of flour cost, at 40 shillings
per barrel ?
5. What will of a hogshead of molasses cost, at 37
dollars per hogshead ?
6. What will -f- of a barrel of apples cost, at 28 shil-
fings per barrel ?
7.. Multiply 48 by f . 12. Multiply 56 by f.
8. Multiply 35 by f. 13. Multiply 72 by f.
9. Multiply 54 by i. 14. Multiply 120 by f.
10. Multiply 49 by f. 15. Multiply 168 by f.
11. Multiply 64 by -f. 16, Multiply 243 by \.
ABT 84.] FRACTIONS. 69
Multiplying a whole number by a mixed number.
17. What will 5 yards of cloth cost, at 18 shillings
per yard ?
Suggestion. Since 1 yard costs 2)18 cost of 1 yd.
18 shillings, 5- yards will cost 5- Si-
times as much. We first multiply 90 cost of 5 yds.
18 shillings by 5, then by , and 9 "of -J- yd.
add the products together. Hence, Ans. 99s. " of 5| yds.
84. To multiply a whole number by a mixed number
Multiply first by the whole number, then by the fraction,
and add the products together. (Art. 83.)
18. Multiply 26 by 2. Ans. 65.
19. Multiply 30 by 2-J-. 25. Multiply 75 by 2-f.
20. Multiply 36 by 3-J-. 26. Multiply 63 by 4-f.
21. Multiply 45 by 4. 27. Multiply 100 by 5f.
22. Multiply 42 by 5K 28. Multiply 165 by 7f.
23. Multiply 36 by 3*- 29. Multiply 180 by 8f.
24. Multiply 56 by H. 30. Multiply 192 by 9$.
31. Multiply 41 rods by 5-J-. Ans. 225 rods.
32. Multiply 68 rods by 16-J-.
33. What cost 21- acres of land, at 35 dollars pei
acre ?
34. What cost 34-J- hundred weight of indigo, at 47
dollars per hundred ?
35. What cost 63f tons of iron, at 96 dollars per ton?
Dividing a whole number by a fraction.
Ex. 1. How many apples at -J- a cent apiece, can you
buy for 5 cents ?
Analysis. If % a cent will buy 1 apple, 5 cents will
buy as many apples, as -J- a cent is contained times in 5
cents ; that is, as many as there are halves in 5 whole ones,
QUEST. 84. How do you multiply a whole number by a mixed number ?
70 FRACTIONS. [SECT. VI
Now in 1 cent there are 2 halves, therefore in 5 cents
there are 5 times 2, which are 10 halves ; and 1 half is
containad in 10 halves, 10 times. Ans. 10 apples.
2. How many plums, at -f of a cent apiece, can you
buy for 8 cents ?
Analysis. Reasoning as before, you can buy as many
plums as | of a cent are contained times in 8 cents. Now
in 1 cent there are 3 thirds, therefore in 8 cents there are
8 times 3, which are 24 thirds, and 2 thirds are contained
in 24 thirds, 12 times. Ans. 12 plums. Hence,
8 5 To divide a whole number by a fraction.
Multiply the given number by the denominator, and
divide the result by the numerator.
OBS. When the numerator is 1, it is unnecessary to divide bj
it ; for it is plain that dividing by 1 does not alter the value of e
number.
3. Divide 17 by i.
We multiply the 17 by the denominator Operation.
2; and since dividing by 1 does not alter 17
the value of a number, we do not divide 2
by it. 34 Ans
4. Divide 19 by -f.
Operation.
Multiply the 19 by 3, and divide the 19
product by 2. Place the remainder 1 3
over the divisor, and annex it to the 28. 2)57
"
6. Divide 25 by -J-. 10. Divide 89 by f.
6. Divide 38 by i. 11. Divide 123 by
7. Divide 47 by \. 12. Divide 156 by
8. Divide 63 by - 2 V 13. Divide 190 by -
9. Divide 72 by f. 14. Divide 256 by -
QUEST. 85. II ow do you divide a whole number by a fraction ? (Ift*.
the numerator is 1, is it necessary to divide by it? Why not?
ARTS. 85, 8G.] FRACTIONS. Yl
Dividing a whole number by a mixed number.
Ex. 1. How many lemons, at 5-J- cents apiece, can you
buy for 22 cents?
Analysis. Since 5j cents will buy 1 lemon, 22 cents
will buy as many lemons, as 5-J- cents are contained times
in 22 cents. Now in 5|- cents there are 11 halves, and in
22 cents there are 44 halves ; but 11 halves are contained
in 44 halves, 4 times. Ans. 4 lemons.
Suggestion. We change the divi- Operation,.
sor to halves by multiplying the whole 51^22
number by the denominator 2, and 2 2
adding the numerator, we have 11 ji ^44
halves ; then reducing the dividend to ^ ns ^ j emong
halves by multiplying it by 2, we have
44 halves. Now 11 is contained in 44, 4 times. Hence,
86. To divide a whole number by a mixed number.
Multiply the whole number in the divisor by the denomi-
nator, and to the product add the numerator. Then mul-
tiply the dividend by the same denominator, and divide as
in ivhole numbers.
Note. For further illustrations of the principles of fractions see
Practical Arithmetic, Section VI. It is incompatible with the
design of the present work to treat of fractions more exten
sively than is necessary to enable the pupil to understand the
operations in Reduction.
EXAMPLES.
2. How many times is 4f- contained in 15 ?
Suggestion. Multiplying the 4 and 15 Q perat i ont
by 3, reduces them to thirds. Now it is
.
plain we can divide thirds by thirds as 33
well as we can divide one whole number
by another ; for the divisor is of the same _. "^ ~Q~T
name or denomination as the dividend.
QUEST. 86. How "lo you divide a whole number by a mixed number 7
72 FRACTIONS. [SECT. VI
3. Divide 18 by 1J. 7. Divide 46 by 7f.
4. Divide 20 by 3-J. 8. Divide 60 by 5i.
5. Divide 25 by 5f. 9. Divide 75 by 8f .
6. Divide 37 by 6-J-. 10. Divide 100 by lOf.
EXAMPLES FOB PRACTICE.
1. How many apples can you buy for 4 cents, if you
pay a cent apiece ?
2. How many peaches can you buy for 6 cents, if you
pay -J of a cent apiece ?
3. How many yards of tape can Sarah buy for 8 cents,
if she pays -J- of a cent a yard ?
4. How many yards of ribbon can Harriet buy for 9
shillings, at -f of a shilling per yard ?
5. How many pounds of tea, at f of a dollar a pound,
can be bought for 6 dollars ?
6. How many yards of calico, at -J of a dollar per yaid,
can you buy for 3 dollars ?
7. At of a penny apiece, how many marbles can
George buy for 14 pence ?
8. At -f of a dollar a bushel, how many bushels of corn
can a man buy for 6 dollars ?
9. At -f of a dollar a yard, how many yards of silk can
a lady buy for 1 5 dollars ?
10. At -fc of a dollar apiece, how many lambs can a
drover buy for 27 dollars?
11. In 1 rod there are 5 yards : how many rods are
there in 88 yards ?
12. In 1 rod there are !<> feet: how many rods are
therein 132 feet?
13. How many yards of cloth, at 5f dollars per yard,
can be bought for 100 dollars?
14. How many cows, at 12 dollars apiece, can be
bought for 125 dollars ?
ART. 86. J FRACTIONS. 73
15. How many acres of land, at 20-f- dollars per acre,
can a man buy for 540 dollars ?
16. A grocer bought a quantity of flour for 239 dol-
1 irs, which was 8- dollars per barrel : how many barrels
did he buy ?
17. A merchant bought a quantity of broadcloth, at
7f dollars per yard, and paid 372 dollars for it: how
many yards did he buy ?
18. A man hired a horse and chaise to take a ride, and
paid 275 cents for the use of it, which was 12-J- cents per
mile : how many miles did he ride ?
19. If a man hires a horse and carriage to go 1 Si-
miles, and pays 315 cents for it, how many cents does he
pay per mile ?
20. A young man hired himself out for 16f dollars per
month, and at the end of his time he received 201 dollars :
how many months did he work ?
21. A farmer having 261 dollars, wished to lay it out
in young cattle which were worth 10-f dollars per head :
how many could he buy ?
22. A man having 100 acres of land, wishes to find
how many building lots it will make, allowing -fa of an
acre to a lot : how many lots will it make ?
23. How many barrels of beef, at 9-J- dollars per barrel,
can be bought for 156 dollars ?
24. How many hours will it take a man to travel 250
miles, if he goes 12-J- miles per hour ?
25. In 1 barrel there are 31 gallons: how many bar-
rels are there in 315 gallons ?
26. A farmer paid 843 dollars for some colts, which
was 35^ dollars apiece : how many did he buy?
27. A wagon maker sold a lot of wagons for 1452 dol-
lars, which was 45f dollars apiece : how many did he
sell?
74 COMPOUND [SECT. VIL
. SECTION VII.
COMPOUND NUMBERS.
ART. 87 SIMPLE Numbers are those which express
units of the same kind or denomination ; as, one, two,
three ; 4 pears, 5 feet, &c.
COMPOUND Numbers are those which express units
of different kinds or denominations ; as the divisions of
money, weight, and measure. Thus, 6 shillings and 7
pence ; 3 feet and 7 inches, &c., are compound numbers.
Note. Compound Numbers are sometimes called Denominate
Numbers.
FEDERAL MONEY.
88. Federal Money is the currency of the United
States. Its denominations are, Eagles, dollars, dimes,
cents, and mills.
10 mills (m.) make 1 cent, marked ct.
10 cents " 1 dime, " d.
10 dimes " 1 dollar, " doll or $.
10 dollars " 1 eagle, " E.
89 The national coins of the United States are of
three kinds, viz : gold, silver, and copper.
1. The gold coins are the eagle, half eagle, and quarter
eagle, the double eagle* and gold dollar.*
2. The silver coins are the dollar, half dollar, quarter
dollar, the dime, half dime, and three-cent-piece.
QUEST. 87. What are simple numbers ? What are compound numbers'?
88. What is Federal Money ? Recite the Table. 89. Of how many kinds are
the coins of the United States ? What are the gold coins ? What are the
silver coins ?
* Added by Act of Congress, Feb. 20th, 1849.
ARTS. 87 91.] NUMBERS. 75
3. The copper coins are the cent and half cent,
Mills are not coined.
Obs. Federal money was established by Congress, August 8th,
1786. Previous to this, English or Sterling money was the princi-
pal currency of the country.
STERLING MONEY.
90, English or Sterling Money is the national cur-
rency of Great Britain.
4 farthings (qr. or far.) make 1 penny, marked d.
12 pence " 1 shilling, " s.
20 shillings " 1 pound or sovereign, .
21 shillings " 1 guinea.
OBS. The Pound Sterling is represented by a gold coin, called
a Sovereign. Its legal value, according to Act of Congress, 1842, is
$4.84; its intrinsic value, according to assays at the U. S. mint, is
$4.861. The legal value of an English shilling is 24-1 cents.
TROY WEIGHT.
91. Troy Weight is used in weighing gold, silver,
jewels, liquors, &c., and is generally adopted in philo-
sophical experiments.
24 grains (gr.) make 1 pennyweight, marked pwt.
20 pennyweights " 1 ounce, " oz.
12 ounces " 1 pound, " Ib.
Note. Most children have very erroneous or indistinct ideas of
the weights and measures in common use. It is, therefore, strongly
recommended for teachers to illustrate them practically, by referring
to some visible object of equal magnitude, or by 'exhibiting the ounce,
the pound ; the linear inch, foot, yard, and rod ; also a square and
cubic inch, foot, &c.
QUEST. What are the copper coins ? Obs. When and by whom was Federal
Money established ? 90. What is Sterling Money ? Repeat the Table. Obs. By
what h the Pound Sterling represented ? What is its legal value in dollars and
tents ? What is the value of an English shilling ? 91. in what is Troy Weight
deed ? Recite the Ttible,
*o COMPOUND [SECT. VII,
AVOIRDUPOIS WEIGHT.
92. Avoirdupois Weiylit is used in weighing groceries
and all coarse articles ; as sugar, tea, coffee, butter, cheese,
flour, hay, &c., and all metals except gold and silver.
16 drams (dr.) make 1 ounce, marked oz.
16 ounces " 1 pound, " Ib.
25 pounds " 1 quarter, " qr.
4 quarters, or 100 Ibs. " 1 hundred weight, ctvt.
20 hund., or 2000 Ibs. " 1 ton, marked T.
OBS. 1. Gross weight is the weight of goods with the boxes, or bags
which contain them, allows 112 Ibs. for a hundred weight.
Net weight is the weight of the goods only.
2. Formerly it was the custom to allow 112 pounds fora hundred
weight, and 28 pounds for a quarter : but this practice has become
nearly or quite obsolete. The laws of most of the states, as well as
general usage, call 100 Ibs. a hundred weight, and 25 Ibs. a quarter.
In estimating duties, and weighing a few coarse articles, as iron,
dye-woods, and coal at the mines, 112 Ibs. are still allowed for a
hundred weight. Coal, however, is sold in cities, at 100 Ibs. for a
hundred weight.
APOTHECARIES' WEIGHT.
93. Apothecaries' Weight is used by apothecaries and
physicians in mixing medicines.
20 grains (yr.) make 1 scruple, marked sc. or S.
3 scruples " 1 dram, " dr. or 3.
8 drams " 1 ounce, " oz. or g.
12 ounces " 1 pound, " Ib.
OBS. 1. The pound and ounce in this weight are the same as the
Troy pound and ounce; the subdivisions of the ounce are different.
2. Drugs and medicines are bought and sold by avoirdupois
weight.
QUEST. 92. In what is Avoirdupois Weight used ? Recite the Table. Obs
What is gross weight? What is net weight? How many pounds were for-
merly allowed for a quarter ? How many for a hundred weight ? 93. In what
is Apothecaries Weight used? Repeat the Table. Obs. To what are the Apo-
thecaries' porn<l and o*ce equal? How are drugs and medicines bought
and sold ?
ARTS. 92 95.] NUMBERS. 77
LONG MEASURE.
O4r Long Measure is used in measuring length or
distances only, without regard to breadth or depth-
12 inches (in.) make 1 foot, marked ft.
3 feet " 1 yard, " yd.
5i yards, or 16 feet " 1 rod, perch, or pole, r. orp
40 rods " 1 furlong, marked fur.
8 furlongs, or 320 rods " 1 mile, " m.
3 miles " 1 league, " L
60 geographical miles, or )
691 statute .miles \" l d ^ ree - " **'"*
360 deg. make a great circle, or the circum. of the eart li,
OBS. 1. 4 inches make a hand; 9 inches, 1 span; 18 inches, 1
cubit ; 6 feet, 1 fathom ; 4 rods, 1 chain ; 26 links, 1 rod.
2. Long measure is frequently called linear or lineal measure.
"Formerly the inch was divided into 3 barleycorns ; but the barley-
corn, as a measure, has become obsolete. The inch is commonly
divided either into eighths, or tenths ; sometimes it is divided into
twelfths, which are called lines.
CLOTH MEASURE.
95 Cloth Measure is used in measuring cloth, lace, and
all kinds of goods, which are bought or sold by the yard.
2 J inches (in.) make 1 nail, marked na.
4 nails, or 9 in. " 1 quarter of a yard, " qr.
4 quarters " 1 yard, " yd.
3 quarters " 1 Flemish ell, " Fl. e.
5 quarters " 1 English ell, " E. e.
6 quarters " 1 French ell, " F. e.
QUEST. 94. In what is Long Measure used 1 Repeat the Table. Draw a
line an inch long upon your slate or black-board. Draw one two inches long.
Draw another a foot long. Draw one a yard long. How long is your teacher 7 !
desk 1 How long is the school-room ? How wide ? Obs. What is Long Meas-
ure frequently called 7 How is the inch commonly divided at the present
dav ? 95. In what is Cloth Measure used 1 Repeat the Table.
COMPOUND
[SECT. VIL
OBS. Cloth mear ire is a species of long measure. The yard is
the eame in both. Cloths, laces, <fec., are bought and sold by the
linear yard, without regard to their width.
SQUARE MEASURE.
96* Square Measure is used in measuring surfaces,
or things whose length and breadth are considered with-
out regard to height or depth ; as land, flooring, plaster-
ing, &c.
144 square in. (sq. in.) make 1 square foot, marked sq.ft.
1 square yard,
sq. yd.
1 sq. rod, perch, ((
or pole,
sq.
1 rood, "
R.
1 acre,
A.
1 square mile, "
M.
9 square feet
30-J- square yards, or ) (( (
2 7 2i square feet ) (
40 square rods
4 roods, or 160 sq.rds. "
640 acres "
OBS. 1. A square is a figure, which has four equal sides, and all
its angles right angles, as seen in the adjoining diagram. Hence,
2. A Square Inch is a square, whose sides 9 sq.ft. I sq. yd.
are each a linear inch in length.
A Square Foot is a square, wiiose sides
are each a linear foot in length.
A Square Yard is a square, whose sides
are each a linear yard or three linear feet in
length, and contains 9 square feet, as re-
presented in the adjacent figure.
3. In measuring land, surveyors use a
chain which is 4 rods long, and is divided
into 100 links. Hence, 25 links make 1 rod, and 7-j^j- inched
make 1 link.
This chain is commonly called Counter's Chain, fiom the name
of its inventor.
4. Square Measure is sometimes called Land Measure, because
?t is used in measuring land.
QUEST. Obs. Of what is Cloth Measure a species? 96. In what is Squaro
Measure used 1 Repeat the Table. Obs. What is a square ? What is a square
inch? What is a square foot? A square jard? Can you draw a
inch ? Can you draw a square foot ? A square yard ?
ARTS. 96, 97.]
NUMBERS.
79
1 cubic yard, "
cu. yd.
1 ton,
T.
1 ton of shipping, "
1 cord foot, or a M
foot of wood,
T.
c.ft.
1 cord,
0.
CUBIC MEASURE.
97. Cubic Measure is used in measuring solid bodies,
or things which have length, breadth, and thickness;
such as timber, stone, boxes of goods, the capacity of
rooms, &c.
1 72 8 cubic inches (cu. in.) make 1 cubic foot, marked cu.ft.
27 cubic feet
40 feet of round, or
50 ft. of hewn timber,
42 cubic feet
16 cubic feet
8 cord feet, or '.
128 cubic feet
OBS. 1. A pile of wood 8 feet long, 4 feet wide, and 4 feet high,
contains 1 cord. For 8 into 4 into 4=128. 27 cu. ft.=l cu. yd.
2. A Cube is a solid body bounded by
six equal squares. It is often called a hex-
acdron. Hence,
A _Cubic Inch is a cube, each of whose
sides is a square inch, as represented by
the adjoining figure.
A Cubic Foot is a cube, each of whose
sides is a square foot.
3. The Cubic Ton is chiefly used for estimating the cartage and
transportation of timber. By a ton of round timber is meant, such
a quantity of timber in its rough or natural state, as when hewn,
will make 40 cubic feet, and is supposed to be equal in weight to
50 feet of hewn timber.
4. The cubic ton or load, is by no means an accurate or uniform
standard of estimating weight ; for, different kinds of timber, are of
very different degrees of density. But it is perhaps sufficiently ac-
curate for the purposes to which it is applied.
QUEST. 97. In what is Cubic Measure used? Recite the Table. How
long, wide, and high, must a pile of wood be to make a cord ? What is a
cube ? What is a cubic inch ? What is a cubic foot ? Can you draw a cubio
Inch on your slate ?
80 COMPOUND [SECT. VIL
WINE MEASURE.
98. Wine Measure is used in measuring wine, alco-
hol, molasses, oil, and all other liquids except beer, ale,
and milk.
4 gills (gi.) make 1 pint, marked pt.
2 pints " 1 quart, " qt.
4 quarts " 1 gallon, " gal.
3 1 gallons " 1 barrel, " bar.orbbl.
42 gallons " 1 tierce, " tier.
63 gallons, or 2 bbls. " 1 hogshead, " hkd.
2 hogsheads " 1 pipe or butt, " pi.
2 pipes " 1 tun, " tun,
OBS The wine gallon contains 231 cubic inches.
BEER MEASURE.
99. Beer Measure is used in measuring beer, ale, and
milk.
2 pints (pt.) make 1 quart, marked qt.
4 quarts " 1 gallon, " gal.
36 gallons " 1 barrel, " bar. or bbl.
54 gals, or 1^ bbls. " 1 hogshead, " hhd.
OBS. The beer gallon contains 282 cubic inches. In many place*
milk is measured by wine measure.
DRY MEASURE.
1 GO. Dry Measure is used in measuring grain, fruit
salt, &c.
2 pints (pts.) make 1 quart, marked qt.
8 quarts " 1 peck, " pJc.
4 pecks, or 32 qts. " 1 bushel, " bu.
8 bushels " 1 quarter, " qr.
32 bushels " 1 chaldron, " cA.
e.-t-ln England, 36 bushels of coal make a chaldron.
QUEST. 98. In what is Wine Measure used? Recite the Table. Obs. Ho*
many cubic inches in a wine gallon? 99. In what is Beer Measure uced
Repeat the Table. Obs. How many cubic inches in a boer gallon ?
ARTS. 98 102.] NUMBERS. 81
TIME.
1O1 Time is naturally divided into days and years ;
the former are caused by the revolution of the Earth on its
axis, the latter by its revolution round the Sun.
60 seconds (sec.) make 1 minute, marked min.
60 minutes " 1 hour, " hr.
24 hours " 1 day, " d.
7 days " 1 week, " wk.
4 weeks " 1 lunar month, " mo.
12 calendar months, or > .. . .,
' x > " 1 civil year, " yr.
365 clays, 6 hrs., (nearly,) $
13 lunar mo., or 52 weeks, " 1 year, " yr.
100 years " 1 century, " cen.
OBS. 1. Time is measured by clocks, watches, chronometers, dials,
hour-glasses, &c.
2. A civil year is a legal or common year ; a period of time es-
tablished by government for civil or common purposes.
3. A solar year is the time in which the earth revolves round
the sun, and contains 365 days, 5 hours, 48 min., and 48 sec.
4. A leap year, sometimes called bissextile, contains 866 days,
and occurs once in four years.
It is caused by the excess of 6 hours, which the civil year con-
tains above 365 days, and is so called because it leaps or rims over
one day more than a common year. The odd day is added to Feb-
ruary, because it is the shortest month. Every leap year, there-
fore, February has 29 days.
1Q2. The names of the days are derived from the
names of certain Saxon deities, or objects of worship. Thus,
Sunday is named from the sun, because this day was dedicated
to its worship.
Monday is named from the moon, to which it was dedicated.
QUEST. 100. In what is Dry Measure used ? Recite the table. 101. How
is Time naturally divided ? How are the former caused ? How the latter t
Repeat the Table. Obs. How is Time measured 1 What is a civil year ? A
solar year? A leap year? How is Leap Year caused ? To which month is
the odd day added 1 From what are the namua of the days derived 1
6
82 COMPOUND [SECT. VII,
Tuesday is derived from Tuisco, the Saxon god of war.
Wednesday is derived from Woden, a deity of northern Europe.
TJiursday is from Thor, the Danish god of thunder, storma, <fec.
Friday is from Frig a, the Saxon goddess of beauty.
Saturday is from the planet Saturn, to which it was dedicated
1O3. The following are the names of the 12 calendar
months, with the number of days in each :
January,
February,
March,
April,
May,
June,
July,
August,
September,
October,
November,
December,
(Jan.) the first month, has 31 days.
(Feb.) " second " " 28 "
(Mar.) " third " "31 "
(Apr.) "fourth " " 30 "
(May) " fifth " "31 "
(June) " sixth " " 30 "
(July) " seventh " " 31 "
(Aug.) " eighth " "31 "
(Sept.) " ninth " " 30 "
(Oct.) " tenth " "31 "
(Nov.) " eleventh " " 30 "
(Dec.) " twelfth " "31 '
OBS. 1. The number of days in each month may be easily re-
membered from the following lines :
" Thirty days hath September,
April, June, and November ;
February twenty-eight alone,
All the rest have thirty-one ;
Except in Leap Year, then is the time,
When February has twenty-nine."
2. The names of the calendar months were borrowed from the
Romans, and most of them had a fanciful origin. Thus,
January was named after Janus, a Roman deity, who WAS sup-
posed to preside over the year, and the commencement of all
undertakings.
February was derived fromfebrno, a Latin word which signifies
to purify by sacrifice, and was so called because this month was
devoted to the purification of the people.
QUEST. 103. What is the origin of the narr.es of the month*?
ARTS. 103 105.] NUMBERS. 88
March was named after Mars, the Roman god of war ; and was
originally the first month of the Roman year.
April, from the Latin aperio, to open, was so called from the
opening of buds, blossoms, cfec., at this season.
May was named after the goddess Maia, the mother of Mercury,
to whom the ancients used to offer sacrifices on ike first day of
this month.
June was named after the goddess Juno, the wife of Jupiter.
July was so called in honor of Julius Ccesar, who was born in
this month.
August was so called in honor of Augustus Ccesar, a Roman
Emperor, who entered upon his first consulate in this month.
September, from the Latin numeral septem, seven, was so called,
because it was originally the seventh month of the Roman year.
It is the ninth month in our year.
October, from the Latin octo, eight, was so called because it was
the eighth month of the Roman year.
November, from the Latin novem, nine, was so called because it
was the ninth month of the Roman year.
December, from the Latin decem, ten, was so called because it
was the tenth month of the Roman year.
104. The year is also divided mtofour seasons of
three months each, viz: Spring, Summer, Autumn or
Fall, and Winter.
Spring comprises March, April, and May ; Summer,
June, July, and August ; Autumn or Fall, September,
October, and November ; Winter, December, Jan. and Feb.
CIRCULAR MEASyRE.
105. Circular Measure is applied to the divisions of
the circle, and is used in reckoning latitude and longitude,
and the motion of the heavenly bodies.
60 seconds (") make 1 minute, marked '
60 minutes " 1 degree,
30 degrees " 1 sign, " s.
12 signs, or 360 " 1 circle, " c.
QUKST. 104. Name the seasons. 105. To what is Circular Measure applied 1
84
COMPOUND
[SECT. VII.
OBS. J. Circular Measure is
c ten tailed Angular Measure,
%nd is chiefly used by astrono-
mers, navigators, and surveyors.
2. The circumference of every
circle is divided, or supposed to
be divided, into 360 equal parts,
called degrees, as in the sub-
icined figure.
3. Since a degree is yfcr part
of the circumference of a circle,
it is obvious that its length
must depend on the size of the circle.
270o
MISCELLANEOUS TABLE.
1O6. The following denominations not included in
the preceding Tables, are frequently used.
12 units
12 dozen, or 144
12 gross, or 1728
20 units
56 pounds
100 pounds
30 gallons
200 Ibs. of shad or salmon
196 pounds
200 pounds
14 pounds of iron, or lead
21 stone
8 pigs
OBS. Formerly 112 pounds were allowed for a quintal.
QUEST. Obs. What is Circular Measure sometimes called ? By whom ia i|
chiefly used ? Into what is the circumference of every circle divided ? On
what does the length of a degree depend ? 10G. How many units make a
dozen ? How many dozen a gross 1 A great gross ? How many units maka
a score ? Pounds a flrjdn ?
make 1 dozen, (doz.)
" I gross.
" 1 great gross.
" 1 score.
" 1 firkin of butter.
" 1 quintal of fish.
" 1 bar. of fish in Mass.
1 bar. in N. Y. and Ct
1 bar. of flour.
1 bar. of pork.
1 stone.
1 P^
1 fother.
ARTS. 106 108.] NUMBERS. 85
PAPER AND BOOKS.
1OT. The terms, folio, quarto, octavo, &c., applied to
books, denote the number of leaves into which a sheet
0f paper is folded.
24 sheets of paper make 1 quire.
20 quires " 1 ream.
2 reams " 1 bundle.
5 bundles " 1 bale.
A sheet folded in two leaves, is called & folio.
A sheet folded in four leaves, is called a quarto, or 4te.
A sheet folded in eight leaves, is called an octavo, or 8vo.
A sheet folded in twelve leaves, is called a duodecimo.
A sheet folded in sixteen leaves, is called a 16 wo.
A sheet folded in eighteen leaves, is called an 18 wo.
A sheet folded in thirty-two leaves, is called a 39mo.
A sheet folded in thirty-six leaves, is called a 3 6 wo.
A sheet folded in forty-eight leaves, is called a 4 8 wo.
1O8 Previous to the adoption of Federal money in
1786, accounts hi the United States were kept in pounds,
shillings, pence, and farthings.
In New England currency, Virginia, Ken- i
tucky, Tennessee, Indiana, Illinois, Mis- >6 shil. make $1.
Bouri, and Mississippi, j
In New York currency, North Carolina, )
8 shil makc & 1 -
, )
]
Ohio, and Michigan,
In Pennsylvania currency, New Jersey, ) _,
Delaware, and Maryland, \ 7s ' 6d ' make $1
In Georgia currency, and South Carolina, 4s. 8d. make $1.
In Canada currency, and Nova Scotia, 5 shil. make $1.
QUEST. 107. When a sheet of paper is folded in two leaves, what is it
called ? When in four leaves, what ? When in eight ? In twelve ? In
sixteen 1 In eighteen 1 In thirty-six ? 108. Previous to the adoption of Fed-
eral Money, in what were accounts kept in the U. S. ? How many shillings
make a dollar in N. E. c\irrency 1 In N. V. currency ^ !\n Penn. currency 1
In Georgia currency 1 In Canada currency 7
86 COMPOUND [SECT. VIL
OBS. At the time Federal money was adopted, the colonial cur*
rency or bills of credit issued by the colonies, had more or less de-
preciated in value : that is, a colonial pound was worth less than a
pound Sterling; a colonial shilling, than a shilling Sterling, &e.
This depreciation being greater in some colonies than in others,
gave rise to the different values of the State currencies.
ALIQUOT PARTS OF $1 IN FEDERAL MONEY.
50 cents =
33i cents =
25 cents =
20 cents =
16| cents =
12 J cents ~
10 cents =*
8| cents =
6| cents =
5 cents =
PARTS OF $1 IN NEW YORK CURRENCY.
4 shillings ==
2 shil. 8 pence =
2 shillings =
1 shil. 4 pence =
1 shilling =
6 pence =
OBS. 1. In New York currency, it will be seen, (Art. 108,^ that
A six-pence, written 6d. = 6^ cents,
A shilling, " Is. = 12 "
One (shil.) and 6 pence, 1/6. = 18J "
Two shillings, " 2s. = 25 "
PARTS OF $1 IN NEW ENGLAND CURRENCY.
3 shillings = $- 1 shilling = $f
2 shillings = $ -J- 9 pence = $fc
I shil. and 6d. = $i 6 pence = $fV
OBS. 2. In New England currency, it will be seen, that
A four-pence-half-penny, written 4d. = 6 cents.
A six-pence, " Gd. = 8 "
A nine-pence, " 9d. = 12^ {C
A shilling, " Is. = 16 f
One (shil.) and six-pence, " 1/6. = 25 "
Two shillings, " 2s. = 33 J "
QITEST. What are the aliquot parts of $1 in Federal Money 7 In New York
currency 7 In Now England currency 7 What are the aliquot parts of a pou
Sterling 7 Of a shilling 7
ART. 108.]
NUMBERS.
87
ALIQUOT PARTS OF STERLING MONEY.
Parts of 1.
Parte o/ Is.
10 shil. = i
6 pence = -J- shil.
6s. 8d. = i
4 pence = - shil.
5 shil. = i
3 pence = } shiL
4 shil. = i
2 pence = -J- shil.
3s. 4d. ~ |
1 pence = i shil.
2s. 6d. = i
1 penny = iV shil.
2 shil. = 1 1 (r
1 far. = i penny.
Is. 8d. -iV
2 far. = -J- penny.
1 shil. = -2 1 o
3 far. = f penny.
ALIQUOT PARTS OR A TON.
10 hund. lbs.=i ton.
5 hund. lbs.=-J- ton.
4 hund. lbs.=-- ton.
2 hund. 2 qrs.=i ton.
2 hund. Ibs. =-fo ton.
1 hund. Ibs. ^ ton.
ALIQUOT PARTS OF A POUND AVOIRDUPOIS.
8 ounces =% pound.
4 ounces =-J- pound.
2 ounces =i pound.
1 ounce =iV pound.
ALIQUOT PARTS OF TIME.
Parts of 1 year.
Parts of 1 month.
6 months = year.
15 days=i- month.
4 months =% year.
10 day s==^- month.
3 months =-J- year.
6 days= month.
2 months =-J- year.
5 days=-J- month.
1J- month =-J- year.
3 days=-iV month.
1-J- month =| year.
2 days^iV month.
1 month ==-i 1 2 year.
1 day =-sV month.
QUEST. How many shillings in half a pound Ster. ? In a fourth ? A fifth ?
A tenth ? A twentieth V How many pence in half a shilling ? In a third ? A
fourth? A sixth 7 A twelfth? How many hundreds in half a ton? In a
fourth ? A fiftn ? A tenth ? How many ounces in half a pound ? In a fourth ?
An eighth? A sixteenth? How many months in half a year? la a third?
A fourth? A sixth? A twelfth 7
88 FEDERAL MONEY. [SECT. VIIL
SECTION VIIL
FEDERAL MONEY.
1 1 0. Accounts in the United States are kept in dol-
lars, cents, and mills. Eagles are expressed in dollars, and
dimes in cents. Thus, instead of 4 eagles, we say, 40 dol-
lars ; instead of 5 dimes, we say, 50 cents, &c.
Ill* Dollars are separated from cents by placing a
point or separatrix ( . ) between them. Hence,
112. To read Federal Money.
Call all the figures on the left of the point, dollars ; the
first two figures on the right of the point, are cents ; the
third figure denotes mills ; the other places on the right are
parts of a mill. Thus, $5.2523 is read, 5 dollars, 25
cents, 2 mills, and 3 tenths of a mill.
OBS. 1. Since two places are assigned to cents, when no cents
are mentioned in the given number, two ciphers must be placed
before the mills. Thus, 5 dollars and 7 mills are written $ 5.007.
2. If the given number of cents is less than ten, a cipher must
always be written before them. Thus, 8 cents are written .08, <fcc.
1. Read the following expressions: $83.635 ; $75.50.
$126.607; $268.05; $382.005; $2160.
2. Write the following sums : Sixty dollars and fifty
cents. Seventy-five dollars, eight cents, and three mills.
Forty-eight dollars and seven mills. Nine cents. Six
cents and four mills.
QUEST. 88. What is Federal Money? What are its denominations? Re-
cite the Table. 110. How are accounts kept iu the United States? How are
Eagles expressed ? Dimes? 111. How are dollars distinguished from cents
and mills ? 112. How do you read Federal Money ? Obs. How many places
are aseigned to cents ? When the number of cents is less than ten, what must
be done ? When no cents are mentioned, what do you do ?
ARTS. 110 113.] FEDERAL MONEY. 89
REDUCTION OF FEDERAL MONEY.
CASE I.
Ex. 1, How many cents are there in 65 dollars?
Suggestion. Since in 1 dollar there are Operation.
100 cents, in 65 dollars there are 65 times as 65
many. Now, to multiply by 10, 100, &c.,we
annex as many ciphers to the multiplicand, 6500 cents.
as there are ciphers in the multiplier. (Art. 45.) Hence,
113. To reduce dollars to cents, annex TWO ciphers.
To reduce dollars tc mills, annex THREE ciphers.
To reduce cents to mills, annex ONE cipher.
OBS. To reduce dollars and cents to cents., erase the sign of dollars
and the separatrix. Tfefes, $25.36 reduced te cents, become 2536
cents.
2. Reduce $4 to cents. Ans. 400 cents.
3.* Reduce $15 to cents. 7. Reduce $96 to mills.
4. Reduce $27 to cents. 8. Reduce $12.23 to cents.
5. Reduce $85 to cents. 9'. Reduce $86.86 to cents.
6. Reduce $93 to cents. 10. Reduce $9.437 to mills.
CASE II.
1. In 2345 cents, how many dollars ?
Suggestion. Since 100 cents make 1 dol- Operation.
lar, 2345 cents, will make as many dollars 1|00)23|45
as 100 is contained times in 2345. Now to Ans. $23.45
divide by 10, 100, &c., we cut off as many
figures from the right of the dividend as there are ciphera
in the divisor. (Art. 67.) Hence,
QUEST. 113. Ho\v are dollars reduced to cents? Dollars to mills ? Centa
to mills ? Obs. Dollars and cents to cents ?
90 FEDERAL MONET. [SECT. VIIL
114* To reduce cents to dollars.
Point off TWO figures on the right ; the figures remain*
ing on the left express dollars ; the two pointed off, cents.
1 1 5. To reduce mills to dollars.
Point off THREE figures on the right ; the remaining
figures express dollars ; the first two on the right of the
point, cents ; the third one, mills.
116* To reduce mills to cents.
Point off ONE figure on the right, and the remaining
figures express cents ; the one pointed off, mills.
2. Reduce 236 cts. to dolls. 3. Reduce 21 63 cts. to dolls.
4. Reduce 865 mills to dolls. 5. Reduce 906 mills to cts.
6. Reduce 2652 cts. to dolls. 7. Redfce 3068 cts. to dolls.
ADDITION OF FEDERAL MONEY.
Ex. 1. What is the sum of $8.125, $12.67, $3.098, $11 ?
Suggestion. Write the dollars under
dollars, cents under cents, mills tinder 1267
mills, and proceed as in Simple Addition. 3 098
From the right of the amount point off 11.00
three figures for cents and mills. Ans. $34.893
117* Hence, we derive the following general
RULE FOR ADDING FEDERAL MONEY.
Write dollars under dollars, cents under cents, mills
under mills, and add each column, as in simple numbers.
From the right of the amount, point off as many figures
for cents and mills, as there are places of cents and mills
in either of the given numbers.
QUEST.- 114. How are cents reduced to dollars? 115. Mills to dollars? 117
How do you add Federal Money? How point off the amount?
ARTS. 114 11*7.] FEDERAL MONEY. 91
OBS. If either of the given numbers have no cents expressed,
upply their place by ciphers.
(2.)
$375.037
(3.)
$4869.45
$760.275
(5.)
$4607.375
60.20
344.00
897.008
. 896.084
843.462
6048.07
965.054
95.873
(6.)
$782.206
$609.352
(8.)
$2903.76
(9.)
$4668.253
84.60
830.206
453.06
430.064
379.007
408.07
25.89
307.60
498.015
631.107
6842.07
7452.349
10. What is the sum of $63.072, $843.625, and $71:60 !
11. Add $873.035, $386.23, $608.938, $169.176.
12. Add 463 dolls. 7 cts. ; 248 dolls. 15 cts. ; 169 dolls.
9 cts. 7 mills.
13. Add 89 dolls. 8 cts.; 97 dolls. 10 cts. 3 mills; 40
dolls. 6 cts. ; 75 dolls.
14. Add 365 dolls. 20 cts. 2 mills; 68 dolls. 6 cts. 5
mills ; 7 cts. 3 mills ; 286 dolls. ; 80 dolls. 6 mills ; 30 dolls.
15 cts.
15. A man bought a cow for $16.375, a calf for $4.875,
and a ton of hay for $13.50 : how much did he pay foi
the whole ?
16. A lady paid $23 for a cloak, $7.625 for a hat, $25.75
for a muff, and $18 for a tippet : how much did she pay
for all ?
17. A farmer sold a cow for $16.80, a calf for $4.08, a
horse for $78, and a yoke of oxen for $63.18 : how much
did he receive for all ?
QUEST. Obs. When any of the given numbers have no cents expressed,
how ia their place supplied?
92 FEDERAL MONEY. [SECT. VIIL
SUBTRACTION OF FEDERAL MONEY.
Ex. 1. What is the difference between $845.634, and
$86.087 ?
Suggestion. Write the less number Operation.
under the greater, dollars under dollars, $845.634
&c., then subtract, and point off the an- 86.087
swer as in addition of Federal Money. Ans. $759.547.
118* Hence, we derive the following general
EULE FOR SUBTRACTING FEDERAL MONEY.
Write the less number under the greater, with dollars
under dollars, cents under cents, and mills under mills ;
then subtract, and point off the answer as in addition
of Federal Money.
(2.)
From $856.453
Take $387.602
(3.)
$960.78
$463.05
(4.)
$605.607
78.36
(5.)
$6243.760
327.053
(6.)
From 965.005
Take 87.85
840.000
378.457
(8.)
483.853
48.75
(9.)
4265.76
2803.98
10. From $86256.63 take $4275.875 ?
11. From $100250, take $32578.867 1
12. From 1 dollar, subtract 11 cents.
13. From 3 dolls. 6 cts. 7 mills, take 75 cents.
14. From 110 dolls. 8 mills, take 60 dolls, and 8 cents.
15. From 607 dolls. 7 cents, take 250 dolls, and 3 cts.
16. A lad bought a cap for $2.875, and paid a five-
dollar-bill : how much change ought he to receive back ?
17. Henry has $7.68, and William has $9.625 : how
much more has the latter than the former ?
18. From $865275.60, take $340076.875.
QUEST. 106. How do you subtract FederaJ Money 1 How point off tht
nnswer 1
ARTS. 118, 119.] FEDERAL MONEY. 93
MULTIPLICATION OF FEDERAL MONEY.
Ex. 1. What will 3 caps cost, at $1.625 apiece ?
Suggestion. Since 1 cap costs $1.625, Operation.
8 caps will cost 3 times as much. We *^ ^25
therefore multiply the price of 1 cap by 3, 3
the number of caps, and point off three Ans. $4.875.
places for cents and mills. Hence,
111). When the multiplier is a whole number, we have
the following
RULE FOR MULTIPLYING FEDERAL MONEY.
Multiply as in simple numbers, and from the right of
the product, point off as many figures for cents and mills,
as there are places of cents and mills in the multiplicand.
OBS. 1. In Multiplication of Federal Money, as well as in simple
numbers, the multiplier must always be considered an abstract
number.
2. In business operations, when the mills are 5 or over, it is
customary to call them a cent ; when under 5, they are disregarded.
(2.) (3.) (4.) (5.)
Multiply $633.75 $805.625 $879.075 $9071.26
By 8 9 24 37
(6.)
(7.)
(8.)
(9.)
Multiply $4063.36
$5327.007
$6286.69
$8265.68
By 63
86
123
264
10. What cost 8 melons, at 17 cents apiece ?
11. What cost 12 lambs, at 87 cents apiece ?
12. What cost 8 hats, at $3.875 apiece 1
13. At $8.75 a yard, what will 9 yards of silk come to!
14. At $1.125 apiece, what will 11 turkeys cost?
QUEST. 119. How do you multiply Federal Money ? How point off the
product? 065. What must the multiplier always be considered ? When
the mills are 5, or over, what is it customary to call them ? When lees than
5 what may be done with them ?
94 FEDERAL MONEY. [SECT. VIIL
15. At $2.63 apiece, what will 15 chairs come tof
16. What costs 25 Arithmetics, at 37-^ cents apiece!
17. "What cost 38 Readers, at 62|- cents apiece?
18. What cost 46 over-coats, at $25.68 apiece ?
19. What cost 69 oxen, at $48.50 a head ?
20. At $23 per acre, what cost 65 acres of land ?
21. At $75.68 apiece, what will 56 horses come to ?
22. At 7-J cents a mile, what will it cost to ride 100
miles ?
23. A farmer sold 84 bushels of apples, at 87-J- cents per
bushel : what did they come to 1
24. If I pay $5.3 7|- per week for board, how much will
it cost to board 52 weeks ?
DIVISION OF FEDERAL MONEY.
Ex. 1. If you paid $18.876 for 3 barrels of flour, how
much was that a barrel ?
Suggestion. Since 3 barrels cost $18.-
876, 1 barrel will cost 1 third as much,
We therefore divide as in simple division,
and point on three places for cents and
mills, because there are three in the dividend. Hence,
1 2O. When the divisor is a whole number, we have
the following
RULE FOR DIVIDING FEDERAL MONEY.
Divide as in simple numbers, and from the right of the
quotient, point off as many figures for cents and mills, as
there are places of cents and mills in the dividend.
OBS. "When the dividend contains no cents and mills, if there
is a remainder annex three ciphers to it ; then divide a& before,
and point off three figures in the quotient.
QUKST. 320. How do you divide Federal Money ? How point off the
quotient ? Obs> When the dividend contains no cents and mills, how f roceed ?
AAT. 120.] FEDERAL MONEY 95
Note. For a more complete development of multiplication and
division of Federal Money, the learner is referred to the author's
Practical and Higher Arithmetics.
When the multiplier or divisor contain decimals, or cents and
mills, to understand the operation fully, requires a thorough
knowledge of Decimal Fractions, a subject which the limits of this
work will not allow us to introduce,
(2.) (3.) (4.)
6) $856.272. 8) $9567.648. 9) $7254,108.
5. Divide $868.36 by 27. 6. Divide $3674.65 by 38.
7. Divide $486745 by 49. 8. Divide $634.075 by ofi.
9. Divide $6634.25 by 60. 10. Divide $5340.73 by 78
11. Divide $7643.85 by 83. 12. Divide $4389.75 by 89.
13. Divide $836847 by 94. 14. Divide $94321.62 by 97.
15. A man paid $2563.84 for 63 sofas : what was that
apiece ?
16. A miller sold 86 barrels of flour for $526.50 : how
much was that per barrel ?
17. If a man pays $475.56 for 65 barrels of pork, what
is that per barrel ?
18. A man paid $1875.68 for 93 stoves: how much
was that apiece ?
19. If $2682.56 are equally divided among 100 men,
how much will each receive ?
20. A cabinet-maker sold 116 tables for $968.75 : how
much did he get apiece ?
21. A farmer sold 168 sheep for $465 : how much did
he receive apiece for them ?
22. A miller bought 216 bushels of wheat for $375.50 :
how much did he pay per bushel ?
23. If $2368.875 were equally divided among 348 per-
sons, how much would each person receive ?
98 REDUCTION. [SECT. IX.
SECTION IX
REDUCTION.
ART. 121* REDUCTION is the process of changing
Compound Numbers from one denomination into another
without altering their value.
REDUCING HIGHER DENOMINATIONS TO LOWER.
122. Ex. 1. Reduce 2, to farthings.
Suggestion. First reduce the Operation.
given pounds (2) to shillings, by 2
multiplying them by 20, because 20s. in l.
20s. make l. Next reduce the 40 shillings,
shillings (40) to pence, by multi- I2d. in Is.
plying them by 12, because 12d. 480 pence.
make Is. Reduce the pence (480) _1 far - in ld -
to farthings, by multiplying them ^ ns - 192 farthings.
by 4, because 4 far. make Id.
2. Reduce l, 2s. 4d. and 3 far. to farthings.
Suggestion. In this example Operation.
there are shillings, pence, and far- * d. far.
things. Hence, when the pounds * 2 . 4 3
are reduced to shillings, the given
shillings (2) must be added men- itd^iJT
tally to the product. When the
' x , , , 268 pence,
shillings are reduced to pence, the 4 f - ' id
given pence (4) must be added; ^ ^^ ^ '
and when the pence are reduced to
farthings, the given farthings (3) must be added.
Q,UEST. 121. What is reduction? 122. Ex. 1. How reduce pounds to shil-
lings? Why multiply by 20 ? How are shillings reduced to pence? Why ?
How pence to farthings 1 Why ?
ARTS. 121 124.] REDUCTION. 0Y
OBS. lu these examples it is required to reduce higher denomi
nations to lower, as pounds to shillings, shillings to pence, <fcc.
123* The process of reducing higher denominationa
to lower, is usually called Reduction Descending.
It consists in successive multiplications, and may with
propriety be called Reduction by Multiplication.
124* From the preceding illustrations we derive the
following
RULE FOB REDUCTION DESCENDING.
Multiply the highest denomination given by the num-
ber required of the next lower denomination to make ONE
of this higher, and to the product, add the given number of
this lower denomination.
Proceed in this manner with each successive denomina-
tion, till you come to the one required.
EXAMPLES.
3. Reduce 4 miles, 2 fur., 8 rods and 4 feet to feet.
Operation.
Suggestion. Having reduced the m.fur. r. ft.
miles and furlongs to rods, we have
1368 rods. We then multiply by .
10-J-, because 16^- feet make 1 rod. ,Q
(Art. 94.) Now 16 J is a mixed 2 )1368 rods,
number; we therefore first multi- jgi
ply by the whole number (16), 8212
then by the fraction (-), and add 1368
the products together. (Art. 84.) 684
Ans. 22576 feet.
QUEST. 123. What is reducing compound numbers to lower denominations
usually called? Which of the fundamental rules is employed in reduction
descending? 124. What is the rule for Reduction Defending ?
98 REDUCTION. | SECT. IX
4. In 5, 16s. 7d., how many farthings ? Ans. 5596 far
5. In 18 how many pence?
6. In 23, 9s., how many shillings ?
7. In 17s. 2d. 3 far., how many farthings?
8. Reduce 5 Ibs. 6 oz. Troy weight, to grains.
Ans. 31680 grs.
9. Reduce 13 Ibs. Troy, to ounces.
' 10. Reduce 4 Ibs. 3 oz. Troy, to penny weights.
11. Reduce 15 oz. 6 pwts. 4 grs. Troy, to grain*.
12. In 2 cwt. 3 qrs. 7 Ibs. 4 .oz. 3 drams, avoirdupow
weight, how many drams? Ans. 72259 drams.
13. In 13 Ibs. 4 oz. avoirdupois, how many ounces?
14. In 2 qrs. 17 Ibs. avoirdupois, how many pounds f
15. In 6 Ibs. 12 oz. avoirdupois, how many drams?
16. In 12 cwt. 1 qr. 6 Ibs. avoir., how many ounces *
17. In 16 miles, how many rods?
18. In 28 rods and 2 feet, how many inches ?
19. In 19 fur. 4 rods and 2 yds., how many feet ?
20. In 25 leagues and 2 m., how many rods ?
21. Reduce 14 yards cloth measure to quarters.
22. Reduce 21 yards 2 quarters to nails.
23. Reduce 17 yards 3 quarters 2 nails, to nails.
24. How many quarts in 23 gallons, wine measure ?
25. How many gills in 30 gallons and 2 quarts?
26. How many gills in 63 gallons ?
27. How many quarts in 41 hogshead* ?
28. How many pecks in 45 bushels ?
29. How many pints in 3 pecks and 2 quarts ?
30. How many quarts in 52 bu. and 2 peck* ?
31. How many hours in 15 weeks?
32. How many minutes in 25 days ?
83. How many seconds in 265 hours ?
34. How many minutes in 52 weeks ?
35. How many seconds in 68 days ?
ARTS. 125, 126.] REDUCTION. 99
REDUCING LOWER DENOMINATIONS TO HIGHER.
125. Ex. 1. Reduce 1920 farthings to pounds.
Suggestion. First reduce the given far- Operation.
things (1920) to pence, the next higher 4)1920 far,
denomination, by dividing them by 4, be- 12)480d.
cause 4 far. make Id. Next reduce the 20)40s.
pence (480) to shillings, by dividing them 2 Ans.
by 12, because 12d. make Is. Finally, re-
duce the shillings (40) to pounds, by dividing them by 20,
because 20s. make l. The answer is 2. That is, 1920
far. are equal to 2.
2. In 1075 farthings, how many pounds?
Suggestion. In dividing the Operation.
given farthings by 4, there is a 4)1075 far.
remainder of 3 far., which should 12)268d. 3 far. over,
be placed on the right. In di- 20)22s. 4d. over,
viding the pence (268) by 12, 1, 2s. over,
there is a remainder of 4d., which Ans. l, 2s. 4d. 3 far.
should also be placed on the
right. In dividing the shillings (22) by 20, the quotient
is l and 2s. over. The last quotient with the several
remainders is the answer. That is, 1075 far. are equal to
1, 2s. 4d. 3 far.
OBS. In the last two examples, it is required to reduce lower de-
nominations to higher, as farthings to pence, pence to shillings, &c.
The operation is exactly the reverse of that in Reduction Descending.
126* The process of reducing lower denominations to
higher is usually called Reduction Ascending.
It consists in successive divisions, and may with propri-
ety be called Reduction by Division.
QUFST. 125. Ex. 1. How are farihings reduced to pence ? Why divide by 4 1
How reduce pence to shillings 1 WLy? How shillings to pounds? Why?
120. What is reducing compound numbers to higher denominations usually
called ' Which ol the fundamental rules ia employed in Reduction Ascending ?
100 REDUCTION. [SECT. IX.
127. From the preceding illustrations we derive the
following
RULE FOR REDUCTION ASCENDING.
Divide the given denomination ~by that number which it
takes of this denomination to ma fee ONE of the next higher.
Proceed in this manner with each successive denomination)
till you come to the one required. The last quotient, with
tlie several remainders, will be the answer sought.
128* PROOF. Reverse the operation; that is, reduce
back tlie answer to the original denominations, and if the
result correspond with the numbers given, the work is right.
OBS. Each remainder is of the satne denomination as the divi-
dend from which it arose. (Art. 51, Obs. 2.)
EXAMPLES.
3. In 429 feet, how many rods ? Operation,
Suggestion. We first reduce the feet 3 )429 feet.
to yards, then reduce the yards to rods 5i-)143 yds.
by dividing them by 5-J-. (Art. 86.) 2 2
Or, we may divide the given feet by 11 )286
16, the number of feet in a rod, and the Ans. 26 rods,
quotient will be the answer.
Proof.
We first reduce the rods back to yards, 26 rods.
(Art. 84,) then reduce the yards to feet. 5^
The result is 429 feet, which is the same 130
as the given number of feet. 13
Or, we may multiply the 26 by 16i, 143 yds.
and the product will be 429. 3
429 feet.
4. Reduce 256 pence to pounds. Ans. l, Is. 4d.
5. Reduce 324 pence to shillings.
QUEST. 127, What is the rule for reduction ascending ? 328, Hovr is re-
duction proved 1 Obs. Of what denomination is each remainder 7
ARTS. 127 129.] REDUCTION. 101
6. Reduce 960 farthings to shillings.
7. Reduce 1250 farthings to pounds.
8. In 265 ounces Troy weight, how many pounds ?
9. In 728 pwts., how many pounds Troy? %
10. In 54'8 grains, how many ounces Troy?
11. In 638 oz. avoirdupois weight, how many pounds?
12. In 736 Ibs. avoirdupois, how many quarters?
13. In 1675 oz. avoirdupois, how many hundred weight ?
14. In 1000 drams avoirdupois, how many pounds?
15. In 4000 Ibs. avoirdupois, how many *,ons?
16. How many yards in 865 inches ?
17. How many rods in 1000 feet?
18. How many miles in 2560 rods ?
19. How many miles in 3261 yards ?
20. How many leagues in 2365 rods ?
EXAMPLES IN REDUCTION ASCENDING AND DESCENDING.
129* In solving the following examples, the pupil
tnust first consider whether the question requires higher
denominations to be reduced to lower, or lower denomina-
tions to higher. Having settled this point, he "ill find no
difficulty in applying the proper rule.
FEDERAL MONEY. (ART. 88.)
1. In 3 dollars and 16 cents, how many cents ?
2. In 81 cents and 2 mills, how many mills?
3. In 245 cents, how many dollars?
4. In 321 mills, how many dimes?
5. In 95 eagles, how many cents ?
6. In 160 dollars, how many cents ?
7. In 317 dollars, how many dimes?
8. In 4561 mills, how many dollars?
9. In 8250 cents, how many eagles ?
10. In 61 dolls., 12 cts., *md 3 mills, how many milk?
02 REDLCTION. [SECT. IX.
STERL .NG MONEY. (ART. 90.)
11. Keduce 17, 16s. to shillings.
12: Reduce 19s. 6d. 2 far. to farthings.
13. ^Reduce 1200 pence to pounds.
14. Reduce 3626 farthings to shillings.
15. Reduce 19 to farthings.
16. Reduce 2880 farthings to shillings.
17. Reduce 21, 3s. 6d. to pence.
18. Reduce 3721 farthings to pounds.
TROY WEIGHT. (ART. 91.)
.19. In 7 Ibs., how many ounces ?
20. In 9 Ibs. 2 oz., how many pennyweights ?
21. In 165 oz., how many pounds?
22. In 840 grains, how many ounces ?
23. In 3 Ibs. 5 oz. 2 pwts. 7 grs., how many grains?
24. In 6860 grains, how many pounds?
AVOIRDUPOIS WEIGHT. (ART. 92.)
25. In 200 oz., how many pounds ?
26. In 261 Ibs., how many ounces?
27. In 3 tons, 2 cwt., how many pounds?
28. In 1 cwt. 2 qrs., how many ounces ?
29. In 1000 oz., how many pounds?
30. In 4256 Ibs., how many tons ?
APOTHECARIES' WEIGHT. (ART. W.)
31. Reduce 45 pounds to ounces.
32. Reduce 71 oz. to scruples.
33. Reduce 93 Ibs. 2 oz. to grains.
34. Reduce 165 oz. to pounds.
35. Reduce 962 drams to pounds.
LONG MEASURE. (ART. 94 )
36. In 636 inches, how many yards ?
87. In 763 feet, how many rods?
ART. 129 ] REDUCTION'. 103
38. In 4 miles, how many feet ?
39. In 18 rods 2 feet, how many inches ?
40. In 1760 yards, how many miles?
41. In 3 leagues, 2 miles, how many inches?
CLOTH MEASURE. (ART. 95.)
42. How many yards in 19 quarters?
43. How many quarters in 21 yards and 3 quaiters?
44. How many nails in 35 yards and 2 quarters?
45. How many Flemish ells in 50 yards ?
46. How many English ells in 50 yards ?
47. How many French ells in 50 yards ?
SaUARE MEASURE. (ART. 96.)
48. In 65 sq. yards and 7 feet, how many feet ?
49. In 39 sq. rods and 15 yds., how many yards?
50. In 27 acres, how many square feet?
51. In 345 sq. rods, how many acres ?
52. In 461 square yards, how many rods?
53. In 876 sq. inches, how many sq. feet ?
CUBIC MEASURE. (ART. 97.)
54. In 48 cubic yards, how many feet ?
55. In 54 cubic feet, how many inches ?
56. In 26 cords, how many cubic feet ?
57. In 4230 cubic inches, how many feet?
58. In 3264 cubic feet, how many cords ?
WINE MEASURE. (ART. 98.)
59. Reduce 94 gallons 2 qts. to pints.
60. Reduce 68 gallons 3 qts. to gills.
61. Reduce 10 hhds. 15 gallons to quarts.
62. Reduce 764 gills to gallons.
63. Reduce 948 quarts to hogsheads.
64. Reduce 896 gills to gallons.
J V4 REDUCTION. [SECT. IX.
BEER MEASURE. (ART. 99.)
65. How many quarts in 1 1 hogsheads of beer ?
66. How many pints in 110 gallons 2 qts. of beer ?
67. How many hogsheads in 256 gallons of beer?
68. How many barrels in 320 pints of beer?
69. How many pints in 46 hhds. 10 gallons ?
70. How many hhds. in 2592 quarts ?
DRY MEASURE. (ART. 100.)
71. In 156 pecks, how many bushels ?
72. In 238 quarts, how many bushels ?
73. In 360 pints, how many pecks ?
74. In 58 bushels, 3 pecks, how many pecks ?
75. In 95 pecks, 2 quarts, how many quarts ?
76. In 373 quarts, how many bushels ?
77. In 100 bushels, 2 pecks, how many pints?
TIME. (ART. 101.)
78. How many minutes in 16 hours ?
79. How many seconds in 1 day?
80. How many minutes in 365 days ?
81. How many days in 96 hours ?
82. How many days in 3656 minutes ?
83. How many seconds in 1 week ?
84. How many years in 460 weeks?
CIRCULAR MEASURE. (ART. 105.)
85. Reduce 23 degrees, 30 minutes to minutes.
86. Reduce 41 degrees to seconds.
87. Reduce 840 minutes to degrees.
88. Reduce 964 minutes to signs.
89. Reduce 2 signs to seconds.
90. Reduce 5 signs, 2 degrees to minutes.
91. Reduce 960 seconds to degrees.
92. Reduce 1800 minutes to signs.
ART. 117.] REDUCTION. 105
93. In 45 guineas, how many farthings \
94. In 60 guineas, how many pounds ?
95. In 62564 pence, how many guineas ?
96. In 84, how many guineas ?
97. How many grains Troy, in 46 Ibs. 7 oz. 5 pwts. ?
98. How many pounds Troy, in 825630 grains ?
99. Reduce 62 Ibs. 10 pwts. to grains.
100. In 16 tons, 11 cwt. 9 Ibs., avoir., how many pounds ?
101. Reduce 782568 ounces to tons.
102. In 18 rods, 2 yds. 3 ft. 10 in., how many inches ^
103. How many feet in 3 leagues, 2 miles, 12 rods ?
104. In 2738 inches, how many rods ?
105. In 2 tons, 3 cwt. 2 qrs. 15 Ibs., how many ounces !
106. Reduce 53 Ibs. 11 pwts. 10 grs. Troy, to grains.
107. How many English ells in 45 yards ?
108. How many yards in 45 English ells ?
109. How many Flemish ells in 54 yards 1
110. How many French ells in 60 yards ?
111. In 13 m. 2 fur. 6 ft. 7 in., how many inches ?
112. In 84256 feet, how many leagues ?
113. In 135 bu. 3 pks. 2 qts. 1 pt. how many pints ?
114. In 84650 pints, how many quarters ?
115. How many gills in 48 hhds. 18 gal. wine measure ?
116. How many pipes in 98200 quarts?
117. How many seconds in 15 solar years ?
118. How many weeks in 8029200 seconds?
119. How many square feet in 82 acres, 36 rods, 8 yds. !
120. How many cords of wood in 68600 cubic inches ?
121. How many inches in 10 cords and 6 cubic feet ?
122. In 246 tons of round timber, how many inches I
123. In 65200 square yards, how many acres?
124. In 8 signs, 43 deg. 18 sec., how many seconds I
125. In 75260 minutes, how many signs ?
106 COMPOUND ADDITION. [SECT. VII.
COMPOUND ADDITION.
ART. 129. Compound Addition is the process of
Uniting two or more compound numbers in one sum.
Ex. 1. What is the sum of 2, 3s. 4d. 1 far.; 1, 6s.
9d. 3 far. ; 7, 9s. 7d. 2 far.
Suggestion First write the Operation.
numbers under each other, pounds St $ f art
under pounds, shillings under shil- 2 " 4 " 4 " 1
lings, &c. Then, beginning with 1 " 6 " 9 " 3
H It Q ff H ft o
the lowest denomination, we find * tf ' *
the sum is 6 farthings, which is Ans. 11 '0 '9 '2
equal to 1 penny and 2 far. over. Write the 2 far. under
the column of farthings, and carry the Id. to the column of
pence. The sum of the pence is 21, which is equal to Is.
and 9d. Place the 9d. under the column of pence, and
carry the Is. to the column of shillings. The sum of the
shillings is 20, which is equal to l and nothing over.
Write a cipher under the column of shillings, and carry the
l to the column of pounds The sum of the pounds is
11, which is set down in full.
13O* Hence, we derive the following general
RULE FOB COMPOUND ADDITION.
I. Write the numbers so that the same denominations
shall stand under each other.
II. Beginning at the right hand, add each column sepa-
rately, and divide its sum by the number required to make
ONE of the next higher denomination. Setting the remain-
der under tke column added, carry the quotient to the next
column, and thus proceed as in Simple Addition. (Art. 23.)
PROOF. The. proof is the same as in Single Addition.
QUEST. 121). What ia Compound Addition 1 130. How do you write com-
pound numbers for addition 1 Where do you begin to add, and how pro*
coed 1 How is Compound Addition proved 1
ARTS. 129, 130.] COMPOUND ADDITION.
(2-)
(3.)
W .
s.
d.
far.
Ib.
oz.
pwt. gr.
m.
r.
/*
^fl
1
3
6
2
2
5
7
4
7
15
20
8
3
8
3
2
5
19
6
4
8
7
9
18
9
1
6
8
3
9
6
4
4
14 3 2^W6\11 1 13 2^/15.22 27
(5.)
5. d. /ar.
10 17 1
(6.)
Ib. os. pwt. gr.
17 10 13 5
(*)
r. yd. ft. in.
4426
19 6 5 2
8928
6602
7820
10 4 11 3
6814
8263 21 11 16 6 2 3 2 S
(8.) (9.) (10.)
cwt. qr. Ib. oz. wk. d. hr. mm. yd. qr. na. in.
5345 13 4 19 30 6312
6298 15 13 16 3231
8172 73 5 10 7024
6096 12 14 25 5112
11. Add 4 tons, 5 cwt. 3 qrs. 2 Ibs. 10 oz. 4 drs. ; 6
tons, 4 cwt. 17 Ibs. 15 oz. 9 drs. ; 3 tons, 2 cwt. 1 qr. 15 Ibs.
12. Add 4 hhds. 10 gals. 3 qts. 1 pt. ; 15 hhds. 19 gals.
2 qts. ; 8 hhds. 7 gals. 2 qts. 1 pt. wine measure.
13. Add 1 pipe, 1 hhd. 8 gals. 2 qts. 1 pt. 2 gills ; 1
pipe, 6 gals. 1 qt. ; 3 pipes, 1 hhd. 3 gals. 3 qts. 1 pint.
14. A man sold the following quantities of wheat : 5 bu.
3 pks. 2 qts. ; 10 bu. 1 pk. 4 qts. ; 21 bu. 2 pks. 5 qts. :
how much did he sell in all ?
15. A merchant bought 3 pieces of silk, one of which
contained 21 yds. 2 qrs. 3 nails ; another 19 yds. 3 qrs. 1
nail ; and the other 26 yds. 1 qr. and 2 nails : how many
yards did they all contain ?
108 COMPOUND SUBTRACTION. [SECT. VIIL
COMPOUND SUBTRACTION.
AIIT. 131. Compound Subtraction is the process of
finding the difference between two compound numbers.
Ex. 1. From 11, 8s. 5d. 3 far., subtract 5, 10s. 2d. 1
farthing.
Suggestion. Write the less number Operation.
under the greater, pounds under pounds, s . d.far.
shillings under shillings, &c. Then, be- 11 8 5 3
ginning with the lowest denomination, 5 10 2 1
proceed thus : 1 far. from 3 far. leaves 2 5 18 3 2
far. Set the remainder 2 under the farthings. Next, 2d.
from 5d. leave 3d. Write the 3 under the pence. Since
10 shillings cannot be taken from 8 shillings ; we borrow
as many shillings as it takes to make one of the next
higher denomination, which is pounds ; and 1, or 20s.,
added to the 8s. make 28 shillings. Now 10s. from 28s.
leave 18s., which we write under the shillings. Finally,
carrying 1 to the next number in the lower line, we have
6 ; and 6 from 11 leave 5, which we write under
the pounds. The answer is o, 18s. 3d. 2 far.
132, Hence, we derive the following general
RULE FOR COMPOUND SUBTRACTION.
I. Write the less number under the greater, so that the
same denominations may stand under each other.
II. Beginning at the right hand, subtract each lower
number from the number above it, and set the remainder
under the number subtracted.
III. When a number in the lower line is larger than
that above it, add as many units to the upper number as it
UUKST. 131. What is Compound Subtraction ? 132. How do you write
compound numbers for subtraction 1 Where begin to subtract, and how
proceed? When a number in the lower line is laiger than that above it,
what is to be done 1 *
ARTS. 131, 132.] COMPOUND SUBTRACTION. 109
takes to make ONE c/ the next higher denomination ; then
subtract as before, and adding I to the next number in the
lower line, proceed as in Simple Subtraction.
PROOF. The proof is the same as in Sim. Subtraction.
(2.) (3.)
From 13, 7s. 8d. 3 far. 19 Ibs. 3 oz. 7 pwts. 12 grs.
Take 6, 5s. lid. 1 far. 15 Ibs. 8 oz. 3 pwts. 4 grs.
(4.) (5.)
From 12 T. 7 cwt. 1 qr. 3 Ibs. 15 m. 3 fur. 10 r. 8 ft. 4 in.
Take 7 T. 9 cwt. 3 qrs. 4 Ibs. 9 m. 6 fur. 3 r. 4 ft. 7 in.
6. From 24 yds. 2 qrs. 3 nails, take 16 yds. 3 qrs. 2
nails.
7. A lady having 18, 4s. 7d. in her purse, paid 8, 7s.
3d. for a dress : how much had she left ?
8. If from a hogshead of molasses you draw out 19 gals.
3 qts. 1 pi, how much will there be left in the hogshead ?
9. A person bought 8 tons, 3 cwt. 19 Ibs. of coal, and
having burned 3 tons, 6 cwt. 45 Ibs. sold the rest: how-
much did he sell ?
10. From 17 years, 7 mos. 16 days, take 15 years, and
4 months.
11. From 39 yrs. 3 mos. 7 days, 4 min., take 23 yrs. 5
mos. 3 days, 16 hrs.
12. From 43 A. 2 roods, 15 rods, take 39 acres and 11
rods.
13. From 38 leagues, 2 miles, 5 fur. 17 rods, take 29
leagues, 2 miles, 7 fur. 13 rods.
14. From 125 bushels, 3 pecks, 4 quarts, 2 pints, take
108 bushels, 2 pecks, 7 quarts.
15. From 85 guineas, 13 shillings, 4 pence, 2 far. take
39 guineas, 15 shillings, 8 pence.
QUEST, -How is Compound Subtradfcon proved 7
110 COMPOUND MULTIPLICATION. [SjECT. VIIL
COMPOUND MULTIPLICATION.
ART. 133. Compound Multiplication is the process
of finding the amount of a compound number repeated or
added to itself, a given number of times.
Ex. 1. What will 3 barrels of flour cost, at 1, 7s. 5d. 2
far. per barrel ?
Suggestion. Write the multiplier un- ~ .
der the lowest denomination of the multi- ~ , 1
plicand, and proceed thus : 3 times 2 far. i " V " 5* 2 *
are 6 far. which are equal to Id. and 2 3
far. over. Write the remainder 2 far. 4 2 4 2
under the denomination multiplied, and
carry the Id. to the next product. 3 times 5d. are 15d.,
and 1 to carry makes 16d., equal to Is. and 4d. over.
Write the 4d. under the pence, and carry the Is. to the
next product. 3 times 7s. are 21s. and 1 to carry makes
22s., equal to l, and 2s. Write the 2 under the shillings
and carry the l to the next product. Finally, 3 times
1 are 3, and 1 to carry makes 4. Write the 4 under
the pounds. The answer is 4, 2s. 4d. 2 far.
134;* Hence, we derive the following general
RULE FOR COMPOUND MULTIPLICATION".
Beginning at the right hand, multiply each denomina-
tion of the multiplicand by the multiplier separately, and
divide its product by the number required to make ONE of
the next higher denomination, setting down the remainder
and carrying the quotient as in Compound Addition.
2. Multiply 4, 6s. 2d. 3 far. by 15.
3. Multiply 19 Ibs. 8 oz. 9 pwts. 7 grs. by 12.
4. If a man walks 3 miles, 3 fur. 18 rods in 1 hour, how
far will he walk in 10 hours ?
QUEST. 133. What is Compound Multiplication ? 134. What is the role
for Compound Multiplication ?
ARTS. 133, 134.] COMPOUND DIVISION. Ill
5. Multiply 7 leagues, 1 m. 31 rods, 12 ft. 3 in. by 9.
6. Multiply 18 tons, 3 cwt. 10 Ibs. 7 oz. 3 drs. by 11.
7. A man has 7 pastures, each containing 6 acres, 25
rods, 5 1 square feet : how much do they all contain ?
8. A man bought 9 loads of wood, each containing 1
cord and 21 cu. ft. : how much did they all contain ?
9. Multiply 17 yds. 3 qrs. 2 nails by 35.
10. Multiply 53 days, 19 min. 7 sec. by 41.
11. Multiply 36 years, 3 weeks, 5 days, 12 hours, by 63.
12. Multiply 65 hhds. 23 gals. 3 qts. 1 pt. by 72.
COMPOUND DIVISION.
135* Compound Division is the process of dividing
compound numbers.
Ex. 1. A father divided 10, 5s. 8d. 2 far. equally
among his 3 sons : how much did each receive ?
Suggestion. Write the divisor ~
A , , ., ~ ,. ,. ., , Operation.
on the left of the dividend, and x , -
, . , TV . . ' , s. d. far.
proceed as in Snort Division. Thus, 3^10 " 5 " 8 2
3 is contained in 10, 3 times and A Q // Q ///>// Ql '
n . ^f, . . , ^LnS. O O O O-ar
1 over. We write the 3 under
the pounds, because it denotes pounds ; then reducing the
.remainder l to shillings and adding the given shillings 5,
we have 25s. Again, 3 is in 25s. 8 times and Is. over. We
set the 8 under the shillings, because it denotes shillings ;
then reducing the remainder Is. to pence and adding the
given pence 8, we have 20d. Now 3 is in 20d. 6 times
and 2d. over. We set the 6d. under the pence, because it
denotespence. Finally, reducing the rem. 2d. to farthings
and adding the given far. 2, we have 10 far. ; and 3 is in
10, 3 times and 1 far. over. Write the 3 under the far
QUEST. 135. What is Compound Division 2
112 COMPOUND DIVISION. [SECT. VIIL
136. Hence, we derive the following general
KULE FOR COMPOUND DIVISION.
1. Beginning at the left hand, divide each denomination
of the dividend by the divisor, and write the quotient fig-
ures under the figures divided.
II. If there is a remainder, reduce it to the next lower
denomination, and adding it to the figures of the correspond-
ing denomination of the dividend, divide this number as
before. Thus proceed through all the denominations, and
the several quotients will be the answer required.
OBS. 1. Each quotient figure is of the same denomination as that
part of the dividend from whitfia it arose.
2. When the divisor exceeds 12, and is a composite number, we
may divide first by one factor and that quotient by the other.
2. Divide 14 Ibs. 5 oz. 6 pwts. 9 grs. by 3.
3. Divide 5, 17s. 8cl. 1 far. by 4.
4. Divide 25 Ibs. 3 ounces, 8 pwts. 7 grs. by 5.
5. Divide 15 T. 15 cwt. 3 qrs. 10 Ibs. by 6,
6. Divide 23 yards, 2 qrs. 1 nail, by 7.
7. Divide 35 leagues, 1 rn. 3 fur. 17 rods by 8.
8. Divide 45 hhds. 18 gals. 39 qts. 1 pint by 9.
9. A farmer had 34 bu. 3 pks. 1 qt. of wheat in 9 bags .
how much was in each bag?
10. If you pay 25, 17s. 8^d. for 5 cows, how much
will that be apiece ?
11. Divide 38 tons, 5 cwt. 2 qrs. 15 Ibs. by 17.
12. Divide 41 hhds. 13 gals. 2 qt. wine measure by 23
13. Divide 54 acres, 2 roods, 25 rods, by 34.
14. Divide 29 cords, 19 cu. feet, 18 cu. inches by 41.
15. Divide 78 years, 17 weeks, 24 days, by 63.
QUEST. 136. What is the rule for Compound Division ? Obs. Of what de-
nomination is each quotient figure ?
MISCELLANEOUS EXERCISES. 118
MISCELLANEOUS EXERCISES.
1. From the sum of 463279 + '734658, take 926380.
2. To the difference of 856273 and 4671 9, add 420376.
3. To 476208 add 5207568 4808345.
4 Multiply the sum of 863576 + 435076 by 287.
5. Multiply the difference of 870358 640879 by 365.
6. Divide the sum of 439409 + 87646 by 219.
7. Divide the difference of 607840 23084 by 367
8. Divide the product of 865060X406 by 1428.
9. Divide the quotient of 55296+144 by 89.
10. What is thesum of 4845 + 76 + 1009 + 463+407 ?
11. What is the sum of 836X46, and 784x76?
12. What is the sum of 1728+72, and 2828-+ 96?
13. What is the sum of 85263 45017, and 68086?
14. What is the difference between 38076 + 16325, and
20268 + 45675?
15. What is the difference between 40719 + 6289, and
31670 18273.
16. What is the difference between 378X 96, and 9419 I
17. What is the difference between 7560-7-504, and
7560X504?
18. Froml45X87, take 12702+87.
19. Multiply 83X19 by 75X23.
20. How many times can 34 be subtracted from 578 f
21. How many times can 1512 be taken from 7569 ?
22. How many times can 63 X 24 be taken from 27640 1
23. How many times is 68 + 31 contained in 45600?
24. Divide 832 + 1429 by 45 + 84.
25. Divide 467 + 2480 by 346 187.
26. Divide 6824016226 by 10405 6200.
27 Divide 320X160 by 2125 960.
28. Divide 826340 36585 by 126X84.
29. From 62345 + 19008, take 2134X38.
30. From 2631X216, take 57636.
1 14 MISCELLANEOUS EXERCISES.
33. A young man having 50 dollars, bought a coat frf
15 dollars, a pair of pants for 8 dollars, a vest for 5 dol-
lars, and a hat for 3 dollars : how much money did he
have left ?
34. A farmer sold a cow for 18 dollars, a calf r or 4
dollars, and a lot of sheep for 35 dollars: how much
more did he receive for his sheep than for his cow and
calf?
35. A man having 90 dollars in his pocket, paid 27
dollars for 9 cords of wood, 35 dollars for 7 tons of coal,
and 1 1 dollars for carting both home : how much money
had he left ?
36. A young lady having received a birthday present
of 100 dollars, spent 17 dollars for a silk dress, 26 dol-
lars for a crape shawl, and 8 dollars for a bonnet : how
many dollars did she have left ?
37. A dairy- woman sold 23 pounds of butter to one
customer, 34 pounds to another, and had 29 pounds left:
how many pounds had she in all ?
38. A lad bought a pair of boots for 16 shillings, a
pair of skates for 10 shillings, a cap for 17 shillings, and
had 20 shillings left : how many shillings had he at first ?
39. A grocer having 500 pounds of lard, sold 3 kegs
of it ; the first keg contained 43 pounds, the second 45
pounds, and the third 56 pounds : how many pounds did
he have left ?
40. A man bought a horse for 95 dollars, a harness
for 34 dollars, and a wagon for 68 dollars, and sold them
all for 225 dollars : how much did he make by his bar-
gain?
41. A person being 1000 miles from home, on his re-
turn, traveled 150 miles the first day, 240 miles the sec-
ond day, and 310 miles the tfiird day: how far from
home was he then ?
MISCELLANEOUS EXERCISES. 115
\ 42. George bought a pony for 78 dollars and paid 3
dollars for shoeing him ; he then sold him for 100 dol-
lar : how much did he make by his bargain ?
43. A man bought a carriage for 273 dollars, and paid
27 dollars for repairing it ; he then sold it for 318 dol-
lars : how much did he make by his bargain ?
44. A man bought a lot for 275 dollars, and paid a
carpenter 850 dollars for building a house upon it : he
then sold the house and lot for 1200 dollars : how much
did he make by the operation ?
45. A farmer having 150 sheep, lost 17 and sold 65 ;
he afterwards bought 38 : how many sheep had he then ?
46. A man bought 27 cows, at 31 dollars per head:
how many dollars did they all cost him ?
47. A miller sold 251 barrels of flour, at 8 dollars ^
barrel : how much did it come to ?
48. A merchant sold 218 yards of cloth, at 8 dollars
per yard : how much did it come to ?
49. A merchant sold 18 yards of broadcloth, at 4 dol-
lars a yard, and 21 yards of cassimere, at 2 dollars a yard :
how much did he receive for both ?
50. A farmer sold 12 calves, at 5 dollars apiece, and
35 sheep, at 3 dollars apiece : how much did he receive
for both ?
51. A grocer sold to one person 25 firkins of butter,
at 7 dollars a firkin, and 13 to another, at 8 dollars a fir-
kin : how much did both lots of butter come to ?
52. A shoe dealer sold 100 pair of coarse boots to one
customer, at 4 dollars a pair, and 156 pair of fine boots
to another, at 5 dollars a pair: what did both lots of
boots come to ?
53. A miller bought 165 bushels of corn, at 5 shillings
a bushel, and 286 bushels of wheat, at 9 shilliigs a
bushel : how much did he pay for both ?
. 16 MISCELLANEOUS EXERCISES.
54. A man bought 45 clocks, at 3 dollars apiece, and
sold them, at 5 dollars apiece : how much did he make by
his bargain?
55. A bookseller bought 87 books, at 7 shillings apiece,
and afterwards sold them, at 6 shillings apiece : how much
did he lose by the operation?
56. How many yards of calico, at 18 cents a yard, can
be bought for 240 cents ?
57. A little girl having 326 cents, laid it out in ribbon,
at 25 cents a yard : how many yards did she buy ?
58. If a man has 500 dollars, how many acres of land
can he buy, at 15 dollars per acre?
59. How many cows, at 27 dollars apiece, can be bought
for 540 dollars ?
60. How many barrels of sugar, at 23 dollars per bar-
rel, can a grocer buy for 575 dollars?
61. Henry sold his skates for 87 cents, and agreed to
take his pay in oranges, at 3 cents apiece : how many
oranges did he receive ?
62. William sold 80 lemons, at 4 cents apiece, and took
his pay in chestnuts, at 5 cents a quart : how many chest-
nuts did he get for his lemons ?
63. A milkman sold 110 quarts of milk, at 6 cents a
quart, and agreed to take his pay in maple sugar, at 11
cents a pound : how many pounds did he receive ?
64. A farmer bought 25 yards of cloth, which was
worth 6 dollars per yard, and paid for it in wood, at 2
dollars per cord : how many cords did it take ?
65. A pedlar bought 4 pieces of silk, at 24 dollars
apiece : how much did he pay for the whole ?
66. A farmer sold 8-j- bushels of wheat, at 96 cents
per bushel : how much did he receive for his wheat ?
67. A man sold a lot of land containing 15f acres, at
16 dollars per acre : how much did he receive for it ?
MISCELLANEOUS EXERCISES. 117
\ 68. If a man can walk 45 miles in a day, how far caa
ha wak in 2 Of days?
69. What cost 75 yds. of tape, at f of a cent per yd. ?
70. What will 100 pair of childrens' gloves come to,
at -ft of a dollar a pair ?
71. What will 160 boys' caps cost, at f of a dollar
apiece ?
72. What will 210 pair of shoes cost, at -f- of a dollar
a pair ?
73. How many childrens' dresses can be made from a
piece of lawn which contains 54 yards, if it takes 4 yards
for a dress ?
74. A farmer wishes to pack 100 dozen of eggs in
boxes, and to have each box contain 6-J- dozen : how many
boxes will he need ?
75. A lad having 275 cents, wishes to know how many
miles he can ride in the Railroad cars, at 2 cents per mile :
how many miles can he ride ?
76. How many apples, at a cent apiece, can Horatio
buy for 75 cents ?
77. If Joseph has to pay f of a cent apiece for marbles,
how many can he buy for 84 cents ?
78. At f of a dollar apiece, how many parasols can a
.shopkeeper buy for 168 dollars?
79. If I am charged -f- of a dollar apiece for fans, how
many can I buy for 265 dollars ?
80. How many yards of silk, which is worth ^ of a
dollar a yard, can I buy for 227 dollars ?
81. How many pair of slippers, at -J of a dollar a pair,
can be bought for 448 dollars ?
82. In 45, 13s. 6d., how many pence ?
83. In 63, 7s. 8d. 2 far., how many farthings?
84. How many yards of satin can I buy for 75, 10s.,
If I have to pay 5 shillings per yard ?
118 MISCELLANEOUS EXERCISES.
85. How many six-pences are there in 100 ?
86. A grocer sold 10 hogsheads of molasses, at 3 shit
lings per gallon : how many shillings did it come to ?
87. A milkman sold 125 gallons of milk, at 4 cents pel
quart : how much did he receive for it ?
88. A man made 30 barrels of cider which he wished to
put into pint bottles : how many bottles would it require ?
89. How much would 85 bushels of apples cost, at 12
cents a peck ?
90. What will 97 pounds of snuff cost, at 8 cents per
ounce ?
91. What will 5 tons of maple sugar come to, at 11
cents a pound ?
92. A farmer sold 34 tons of hay, at 65 cents per hun-
dred : how much did he receive for it ?
93. A blacksmith bought 53 tons of iron for 3 dollars
per hundred : how much did he pay for it ?
94. A young man returned from California with 50
pounds of gold dust, which he sold for 16 dollars per
ounce Troy : how much did he receive for it ?
95. A man bought 36 acres of land for 3 dollars per
square rod : how much did his land cost him ?
96. John Jacob As tor sold five building lots in the city
of New York, containing 560 square rods, for 13 dollars
per square foot : how much did he receive for them ?
97. A laboring man engaged to work 5 years for 16
dollars per month : what was the amount of his wages ?
98. What will 17 cords of wood cost, at 6 cents per
cubic foot ?
99. If it takes 35 men 18 months to build a fort, how
many years would it take 1 man to build it ?
] 00. If it takes 1 man 360 days to build a house, how
many weeks would it take 15 men to build it, allowing 6
working days to a week ?
ANSWERS TO EXAMPLES.
119
ANSWERS TO EXAMPLES.
ADDITION.
Sx. Aus.
Ex. Ana.
Ex. Ans.
ART. 20.
4. 5286 yards.
28. 171658.
1. Given.
5. 2404.
29. 57 dollars.
2. 68.
6. 2765.
30. 58 dollars.
3. 589.
7. 10040.
31. 120 dollars.
4. 768.
8. 8668.
32. 565.
5. 9987.
9. 84 inches.
33. 742.
6. 878.
10. 114 feet
34. 1530.
V. 6767.
11. 168 dollars.
35. 1779.
8. 8898.
12. 192 rods.
36. 1597.
9. 8779.
13. 782 pounds.
37. 1757.
10. 6796.
14. 1380 yards.
38. 2379.
11. 88776.
15. 576 miles.
39. 2619.
12. 986788.
16. 836 sheep.
40. 1020.
17. 615 dollars.
41. 1418.
ART. 22.
18. 181 dollars.
42. 1191.
13, 14. Given.
19. 1452.
43. 150 bushels.
15. 1454.
20. 1255.
44. 133 yards.
16. 15300.
21. 1881.
45. 731 acres.
17. 13285.
22. 6693.
46. 1197 cattle.
23. 20485.
47. 12554 dollars.
ART. 24.
24. 9726.
48. 1282.
1. 155 pounds.
25. 1769.
49. 2528.
2. 413 feet.
26. 1500.
50. 365 days.
3. 1960 dollars.
27. 106284.
ART. 2 4. a.
10. 65471.
20. 551452.
30. 279,075.
1. 300.
11. 327371.
21. 46157.
31. 295,306.
2. 6000.
12. 390497.
22. 424634.
32. 1,606,895.
3. 9000.
13. 37938.
23. 430032.
35. 6,140,704.
4. 4861.
14. 50342.
24. 3458772.
36. 7,569,904.
5. 4871.
15. 449458.
25. 48350.
37. 9,253,854.
6. 47067.
16. 466789.
26. 514299.
38. 9,247,176.
7. 53340.
17. 40290.
27. 595522.
39. 10,531,960
8. 59139.
18. 50676.
28. 5781566.
40. 12,811,860.
9. 61304.
19. 508302.
29. 61993.
120
ANSWERS. [PAGES 28 35.
SUBTRACTION.
Ex. A.ns.
Ex. Ans.
Ex. Ana.
ART. 28.
14. 275 pounds.
48. 222 bushels.
1. Given.
15. 613 yards.
49. 195 dollars.
2. 24.
16. 310 rods.
50. 1122 dollars.
3. 12.
17. 230 gallons.
51. 1659 dollars.
4. 153.
18. 503 hhds.
52. 3023 dollars.
5. 24S.
19. 76 bushels.
53. 1763 dollars.
6. 31 dollars.
20. 127 dollars.
54. 3747 dollars.
7. 12 pounds.
21. 249 pounds.
55. 16014 dollars.
8. 115 yards.
22. 1082 rods.
56. 1315 dollars.
9. 222 shillings.
23. 13016.
57. 5385 dollars.
10. 222 marbles.
24. 310768.
58. 5735 dollars.
25. 464374.
59. 13944 soldiers
ART. 3O.
26. 5244038.
60. 94760000 m.
11, 12. Given.
27. 45.
61. 17 oranges.
13. 137.
28. 308.
62. 33 marbles.
14. 2616.
29. 240.
63. 76 sheep.
15. 3270.
30. 58.
64. 52 cents.
16. 3203.
31. 542.
65. 43 yards.
17. 5365667.
32. 2021.
66. 122 dollars.
33. 1825.
67. 87 dollars.
ART. 32.
34. 2600.
68. 66 pears.
1. 217.*
35. 3085.
69. 59.
2. 182.
36. 1306.
70. 164.
3. 242.
37. 4098.
71. 149 pounds.
4. 369.
38. 1108.
72. 164 bushels.
5. 1029.
39. 4531.
73. 263 miles.
6. 1008.
40. 14520.
74. 125 gallons.
7. 3289.
41. 24622.
75. 179 pounds.
8. 3434.
42. 125028.
76. 175 dollars.
9. 35100.
43. 64303.
77. 339 pounds.
10. 312657.
44. 224066.
78. 172 barrels.
11. 1.
45. 103875.
79. 297 pages.
12. 23 dollars.
46. 420486.
80. 110 dollars.
13. 57 bushels.
47. 72 sheep.
81. 392 dollars.
* It is an excellent exercise for the pupil to prove all the examples. This is
one of the beet means to give him confidence in his own powers.
PAGES 39 46 ] ANSWERS.
121
MULTIPLICATION.
Ex. Ans.
Ex. Ans.
Ex. Ana.
ART. 39.
ART. 41.
33. 9100 weeks.
1. Given,
34 37. Given.
34. 23760 min.
2. 68.
35. 28350 gallons.
3. 936.
ART. 43.
36. 34675 "dolls.
4. 8084.
1. 252.
37. 33840 sq. in.
5. 5550.
2. 390.
38. 26070 miles.
6. 12066.
3. 567.
7. 24408.
4. 582.
ART. 45.
8. 35550.
5. 840.
40. Given.
9. 56707.
6. 1155.
41. 260.
10. Given.
7. 3568.
42. 3700.
8. 2763.
43. 51000.
ART. 40.
9. 3920.
44. 226000.
11. 312.
10. 460.
45. 341000.
12. 480.
11. 572.
46. 46900*00.
13. 249.
12. 816.
47. 52300000.
14. 840.
13. 1092.
48. 681000000.
15. 828.
14. 1170.
49. 856120000.
16. 815.
15. 2185.
50. 96030500000
17. 2248.
16. 4515.
51. Given.
18. 3144.
17. 12306.
19. 2520.
18. 25355.
ART. 46.
20. 1900.
19. 342 dollars.
52. 17000.
21. 3960.
20. 336 bushels.
53. 291000.
22. 656C,
21. 336 inches.
54. 4920000.
23. 5628.
22. 620 pounds.
55. 11700000
24. 8712.
23. 391 dollars.
56. 33930.
25. 1050 dollars.
24. 475 dollars.
57. 789600.
26. 2300 dollars.
25. 1591 dollars.
58. 16170000.
27. 1372 dollars.
26 1950 shil.
59. 262660000.
28, 2720 dollars.
27. 1575 dollars.
60. 7oOO minutes.
29. 4837 dollars.
28. 2430 shil.
61. 2400 dollars.
30, 7785 dollars.
29. 3936 ounces.
62. 6800 shillings.
31. 7744 dollars.
30. 10754 dollars.
63 27000 dollars.
S2. 8820 dollars.
31. 6710 miles.
64. 352500 days.
&3. 2 1285 dollars.
32. 8760 hours.
122
ANSWERS.
[PAGES 47 55.
MULTIPLICATION CONTINUED. ARTS. 47, 48.
Ex. Ana.
Ex. Ans. Ex. Ans.
65. Given.
78. 2520000. [ 91. 5816049 galls.
66. 19500.
79. 65000000. 92. 101198340 d.
67. 40800.
80. 722000000. 93. 146460440 T.
68. 504000.
81. 21000000000. 94. 1190439180.
69. 800000.
82. 72800000000. 95. 3759670728.
70. 3300000.
83. 2240000yds. 96. 4223213600.
71. 14620000.
84. 140000 miles. 97. 5815178600.
72. 65360000.
85. 700000 dolls. 98. 12976172335.
73. 104520000.
86. 504000 dolls. 99. 124811441568
74. 183244000.
87. 27375000 d. 100. 54719418834.
75. Given.
88. 367608 Ibs. 101. 469234745451
76. 420000.
89. 3838460ft. 102. 197118900.
77. 442000.
90. 4217202 r. 103. 420152303451,
SHORT DIVISION.
ART. 54.
17. 25.
9. 116*.
1. Given.
18. 76.
10. 728.
2. 21.
19. 456.
11. 1552f.
3. 23.
12. 1004f
4. 122.
ART. 57.
13. 400f.
5. 111.
20. Given.
14. 903*.
6. 342.
21. 509.
15. 923.
7. 1122.
22. 901.
16. 1222f.
8. 1321.
23. 1067.
17. 875.
9. 1111.
24. 503.
18. 1011-|.
25. Given.
19. 63 pair.
ART. 55.
20. 42 hats.
10. Given.
ART. 61.
21. 24 marbles.
11. 71.
1. 142.
22. 45 children.
12. 43.
2. 101-J-.
23. 75 yards.
13. 412.
3. 76.
24. 85 barrels, an C
14. 411.
4. 75.
5 dolls, over.
5. 102f
25. 92 days.
ART. 56.
6. 56|.
26. 158-J- yards.
15. Given.
7. 120f.
27. 195 hours.
16. 14.
8. 95.
28. 333| hours
PAGES 56 62.] ANSWERS.
123
LONG DIVISION.
far
Ex.
Ex.
ART. 62.
1, 2. Given.
3. 128.*
4. 364.
5. 1825f.
6. 533.
7. 732.
8. 931.
911. Given.
ART. 65.
1. 46-i
2. 48-f.
3. 80f.
4. 40^.
5. 58-ft.
6. 48.
7.
8.
9. 41A-
10. 27.
11. 23f.
12. 21-fj,
13. 19ff.
14. 20.
15.
16,
17. 45f|.
18. 57ff.
19. 24 caps.
20. 35 pair.
21. 28 barrels.
82. 1900W.
83. 840ff.
QA g>7 \ 5 5
OT:. O t ^r~i ,;' .>T,
1 22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
85.
86.
87.
88.
16-ft- shillings
10^ pounds.
16|f pounds.
17 trunks.
30 weeks.
32f yards.
75 dresses.
81 sheep.
73-J-f- acres.
61 shares.
3 Iff years.
48ii hhds.
43ff- months.
5 Iff months.
50 dollars.
lOif months.
90 pounds.
60, and 1 over.
106, and 22
over.
26, and 28
over.
42, and 28
over.
30|f.
34.
53^.
25f.
51.
52.
53. 218-ftV
54. 216-iftfr.
ART. 67.
55. 56. Given.
57.
58.
59.
60.
61.
62.
63.
1620-fiif
ART. 68.
64, 65. Given,
66.
67.
68.
69.
70. 46<HHH!.
71.
72.
73.
74. 27rH.
75.
76.
77.
78.
79. 30.
80. 14834fJ-.
89.
90.
91.
124
ANSWERS.
[PAGES 68 -92,
FRACTIONS.
Ex, Ans.
Ex. Ana.
Ex. Ans.
Ex. Ans.
ART. 83.
15. 147.
30. 1896.
11. 307.
3. 10 doll.
16. 135.
31. Given.
12. 273.
4. 10 shill.
19. 70.
32. 1122 r.
13. 304.
5. 7f doll.
20. 117.
33. 752| dol.
14. 329+.
6. 21 shill.
21. 189.
34. 1609fd.
ART. 86.
7. 36.
22. 217.
35. 6120 dol.
3. 12.
8. 28.
23. 112.
ART. 85.
4. 6.
9. 6.
24. 399.
5. 100.
5. 4-fr
10. 14.
25. 200.
6. 190.
6. 5ff.
11. 40.
26. 270.
7. 423.
V. 5.
12. 21.
27. 575.
8. 1260.
8. 11H.
13. 32.
28. 1287.
9. 108.
9. 9/ ff .
14. 100.
29. 1540.
10. 118f.
10. 9.
EXAMPLES FOR PRACTICE.
1. 8 a.
8. 9| bus.
15. 25 W a.
22. 277-f 1.
2. 8 p.
9. I7f yds.
16. 28 W bar.
23. 16 bar.
3. 24 yds.
10. 30 lambs.
17. 48 yds.
24. 20 hrs.
4. 13| yds.
11. 16 rods.
18. 22 miles.
25. 10 bar.
5. lOlbs.
12. 8 rods.
19. 20-H- cts.
26. 24 colts.
6. 24 yds.
13. 1 7-2^ yds.
20. 12 mo.
27. 32 wag.
7. 49 mar.
14. 10 cows.
21. 24 cattle.
ADDITION OF FEDERAL MONEY. ART. 117.
2.^1278.699.
6. $1743.828.
10. $978.297.
14.$829.49d
3.111261.52.
7.12478.735.
11. $2037.379.
15. $34.75.
4. $2622.337.
8.110224.78.
12. $880.317.
16. $74.375.
5.85599.332.
9. $12858.266,
13. $301.243.
17. $162.06.
SUBTRACTION OF FEDERAL MONEY. ART. 118.
2. $468.851.
6. $877.155. 1
0. $81980.755.
14. $49.928.
3. $497.73.
7. $461.543. ]
il. $67671. 133.
15. $357.04.
4. $527.247.
8. $435.103. 1
L2. $0.89.
16. $2.125.
5. 5916.707.
9. $1461.78.1
13. $2.317.
17. $1.945.
PAGES 93 100.] ANSWERS. 125
MULTIPLICATION OF FEDERAL MONEY. ART. 119.
Ex. Ana,
Ex. Aria.
Ex. Ana.
Ex. Ans.
1. Given.
2. $5070.
3. $7250.625.
4, $21097. 80.
5. $335636.62
6. $255991.68
7. $458122.602
8. $773262.87.
9.$2182139.52
10. $1.36.
11. $10.44.
12. $31.
13. $78.75.
14.$12.375.
15. $39.45.
16. $9.375.
17. $23.75.
18.11181.28
19. $3346.50
20. $1495.
21. $4238.08
22. $7.50.
23. $73.50.
24. $279.50.
DIVISION OF FEDERAL MONEY. ART. 120.
1. Given.
7. $9933.57.
13. $8902.627.
19. $26.82
2. $142.712.
8. $11.322.
14. $972.38.
20. $8.35.
3. $1195.956.
9. $110.57.
15. $40.69.
21. $2.767.
4. $806.012.
10. $68.47.
16. $6.12.
22. $1.738.
5. $32.16.
11. $92.09.
17. $7.31.
23. $6.807.
6. $96.70.
12. $49.32.
18. $20.16.
REDUCTION DESCENDING.
ART. 124.
1-4. Given.
5. 4320d.
6. 469s.
11. 7348 gr.
13. 212 oz.
14. 67 Ibs.
15. 1728 dr.
20. 24640 r.
21. 56 qrs.
22. 344 na.
23. 286 na.
28. 180 pk.
29. 52 pts.
30. 1680 qts.
31. 2520 hrs.
7. 827 far.
8. Given.
9. 156 oz.
10. 1020pwt.
16. 19696 oz.
17. 5120 r.
18. 5568 in.
19. 12614 ft.
24. 92 qts.
25. 976 g.
26. 2016 g.
27. 10332 q.
32. 36000 m.
33. 954000s.
34. 524160 m.
35. 5875200s
REDUCTION ASCENDING.
ART. 12T.
14. Given.
5. 27 shillings.
6. 20 shillings.
7. l,6s.0d.2far.
8. 22 Ibs. 1 oz.
9. 3 Ibs. oz. 8
pwts.
10. 1 oz. 2 pwts.
20 grs.
11. 39 Ibs. 14 oz.
12. 29 qrs. 11 Ibs.
13. 1 cwt. 4 Ibs.
11 oz.
14. 3 Ibs. 14 oz. 8
drs.
15. 2 tons.
16. 24 >ds. 1 in.
17. 60 r. 10 ft.
18. 8 miles.
19. 1 m. 6 fur. 32
r. 5 yds.
20. 2 lea. 1 m. 3
fur. 5 r.
126
ANSWERS. L PAaE * 101 1
REDUCTION ASJENDING AND DESCENDING.
Ex. Ans.
Ex. Ans
Ex. Ans.
ART. 129.
31. 540 ounces.
62. 23 gals. 3 qts.
1. 316 cents.
32. 1704 scruples.
Ipt.
2. 812 mills.
33. 536640 grs
63. 3hhds. 48 gls,
3. 2 dolls. 45 cts.
34. 13 Ibs. 9 oz.
64. 28 gals.
4. 3 dimes 2 cts.
35. 10 Ibs. oz.
65. 2376 qts.
1 mill.
2 drs.
66. 884 pints.
5. 95000 cents.
36. 17 yds. 2 ft.
67. 4hhds.40gls.
6. 16000 cents.
37. 46 rods 4 ft.
68. 1 bbl. 4 gals.
7. 3170 dimes.
38. 21120 feet.
69. 19952 pts.
8. 4 dolls. 56 cts.
39. 3588 inches.
70. 12 hhds.
1 mill.
40. 1 mile.
71. 39 bushels.
9. 8E.2dolls.50c.
41. 696960 in.
72. 7 bu. 1 pk
10. 61123 mills.
42. 4 yds. 3 qrs.
6 qts.
11. 356 shillings.
43. 87 qrs.
73. 22 pks. 4 qts.
12. 938 farthings.
44. 568 nails.
74. 235 pecks.
13. 5.
45. 66Fl.e.2qrs.
75. 762 quarts.
14. 75s. 6d. 2 far.
46. 40 E. e.
76. 11 bu. 2 pks
15. 18240 far.
47. 33 F. e. 2 qrs.
5 qts.
16. 60 shillings.
48. 592 sq. ft.
77. 6432 pints.
17. 5082 pence.
49. 1194f sq. yds.
78. 960 minutes.
18. 3, 17s 6d.
50. H76120sq.ft.
79. 86400 sec.
1 far.
51. 2 A. 25 sq.r.
80. 525600 min.
19. 84 ounces.
52. 15 sq. r. 7f
81. 4 days.
20. 2:00 pwts.
sq. yds.
82. 2 days 12 hr?
21. 13 Ibs. 9 oz.
53. 6 sq. ft. 12
56 min.
22. 1 oz. 15 pwts.
sq. in.
83. 604800 sec,
23. 19735 grains.
54. 1296 cu. ft.
84. 8 yrs. 11 mo.
24. 1 Ib. 2 oz. 5
55. 93312 cu. in.
85. 1410'.
pwts. 20 grs.
56. 3328 cu. ft.
86. 147600".
25. 12 Ibs. 8 oz.
57. 2 cu. ft. 774
87. 14.
26. 4176 ounces.
cu. in.
88. Os. 16 4-.
27. 6200 Ibs.
58. 25 cords, 64
89. 216000".
28. 2400 ounces.
cu. ft.
90. 9120'.
29. 62 Ibs. 8 oz.
59. 756 pts.
91, 16'.
30. 2 tons, 2 cwt. 60. 2200 gills.
92. 1 sign..
2 qrs. 6 Ibs. 1 61. 2580 qts.
PAGES 105 111.] ANSWERS. 127
REDUCTION ASCENDING AND DESCENDING.
Ex. Ans.
Ex. Ans.
Ex. Ans.
93. 45360 far.
94. 63.
95. 248G.5s.8d.
96. 80 G.
104. 13r.l3f. 8 i.
105. 69840 oz.
106. 205554 grs.
107. 36 E. ells.
116. 194 p. Ik 43
gals.
117. 473353920s
118. 13 wks. 1 d.
97. 268440 grsr
98. 143 1. 4 o. 1
108. 56 yds. 1 qr.
109. 72. Fl. ells.
22 hrs. 20 min.
119. 3581793s. ft.
p. 6g.
99. 357360 grs.
100. 33109 Ibs.
110. 40 F. ells.
111. 839599 in.
112. 5 1. 306 r. 7 f.
120. 39- ft. 1208 i.
121. 2222208 c.in.
122. 17003520 in.
101. 24 T. 9 cwt.
10 Ibs. 8 oz.
102. 3682 in.
103. 58278 ft.
113. 8693 pts.
114. 165 qrs.2bu.
2 pks. 5 qts.
115. 97344 gills.
123. 13 A. 75 r.
11^ yds.
124. 1018818 sec.
125. 418.24, 20'.
COMPOUND ADDITION.
5. 40, 14s.2d.2f.
9. 35 w. 4h. 21m.
13. 6 pi. 18 gals.
6. 59 1. 2 p. 22 g.
10. 23 yds. 3 na.
3 qts. 2 gi.
7. 22 r. 1 yd. 5 in.
11. 13T.12c.lqr.
14. 37 bu. 3 pks.
8. 26 cwt. 3 qrs. 5
10 1. 9 o. 13 d.
3 qts.
Ibs. 5 oz.
12. 27hhds. 38 g.
15. 67 y. 3 q. 2 na.
COMPOUND SUBTRACTION.
2. 7, Is. 9d. 2 far.
6. 7 yds. 3 qrs. 1 n.
12. 4 A. 2 roods, 4
3. 3 Ibs. 7 oz. 4
7. 9, 17s. 4d.
rods.
pwts. 8 grs.
8. 44 gals. 1 pt.
13. 8 lea. 2 mi.
4. 4 T. 17 cwt. 1
9. 4T. 16 c. 74 Ib.
fur. 4 r.
qr. 24 Ibs.
10. 2y. 3 mo. 16 d.
14. 17 bu. 5 q. 2 p.
5. 5 m. 5 fur. 7 r.
11. 15 y. 10 mo. 3
15. 45 G. 18s. 8d
3 ft. 9 in.
d. 8 h. 4 m.
2 far.
COMPOUND MULTIPLICATION.
1. Given.
5. 661. 285 r. 11 f.
9. 625 y. 2 q. 2 n
2. 64, 13s. 5d. 1
3 i.
10. 2173 d. 13 h. 3
farthing.
6. 199 T. 14 c. 14
m. 47 s.
3. 236 1.5 o. 11 p.
1. 15 o. 1 I.
11. 2272 y. 30 w
12 g.
7. 43A.16r.84f f.
3 d. 12 h.
4, 34 mi. 2 f. 20 r.
8. 10 cords, 61 c. f.
12. 4707 h. 18 g.
128
ANSWERS. [PAGES 113 118.
COMPOUND DIVISION.
Ex. A us.
Ex. Ans.
Ex. Ans.
1. Given.
2. 4 1. 9 oz. 15 p.
11 g.
3, 1, 9s. 5d. f.
4 51. 13 p. 15f g.
5. 2 T. 12 c. 2 q.
14*1
6. 3 y. 1 q. If na.
7. 4 1. 1 in. 2 f.
I7i r.
8. 5h.2g.4q.0-fp.
9. 3 b. 3 p. 3f q.
10. 5, 3s. 6d. 2 f.
11.2 T.5 c. 3-H-l-
12. 1 hhd. 49 gals.
3|f qts.
13. 1 A. 2 rx>ds,
l**r.
14. 91 c. f. OH i-
15. 1 yr. 12 wks,
4 d.
MISCELLANEOUS EXERCISES.
1. 271557.
23. 460|f.
51. $279.
76. 150 ap.
2. 1229930.
24. 17-ftV
52. $1180.
77. 112 mar.
3. 875431.
25. 18-ftV
53. 3399 s.
78. 192 par.
4. 372713-
26. 12HH-
54. $90.
79. 371 fans.
124.
27. 43i+H-
55. 87 s.
80. 25 2| yds.
5. 837598-
28. 74V L ( /V i 8V
56. 13-f-g yds.
81. 512 pair.
35.
29. 261.
57. 13-/g- yds.
82. 10962d.
6. 24064H-
30. 568280.
58. 33-ft- a.
83. 60850 f.
7. 1593i.
33. $19.
59. 20 cows.
84. 302 yds.
8. 245948,
34. $13.
60. 25 bar.
85. 4000.
616 rem.
35. $17.
61. 29 or.
86. 1890 s.
9. 4-f*
36. $49.
62. 64 quarts.
87. $20.
10. 6800.
37. 86 Ibs.
63. 60 Ibs.
88. 7560 bot
11. 98040.
38. 63 s.
64. 75 c.
89. $40.80.
12. 53f.
39. 356 Ibs.
65. $108.
90. $124.16
13. 108332.
40. $28.
66, $7.92.
91. $1100.
14. 11542.
41. 300 m.
67. $252.
92. $442.
15. 33611.
42. $19.
68. 936 m.
93. $3180.
16. 26869.
43. $18.
69. 45 cents.
94. $9600.
17. 3810225.
44. $75.
70. $30.
95- $17280.
18. 12469.
45. 106 sk
71. $140.
96. $1981980.
19. 2720325.
46. $837.
72. $150.
97. $960.
20. 17.
47. $2008.
73. 12 dress's.
98. $130.56.
21. 5 and 9 r.
48. $1744.
74. 16 boxes.
99. 52 y. 6 m.
22. 18 and
49. 8114.
75. 110 miles.
1^0. 4 weeks,
23. 424 over.
50.:$165.
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