APPLETONS' MATHEMATICAL  SERIES.
A N
ELEMENTARY ARITHMETIC.
BY
G. P. QUACKENBOS, A. M.,
AUTHOR OF
"AN ENGLISH GRAMMAR;" F"FIRST LESSONS IN COMPOSITION;" "ADVANCED
COURSE OF COMPOSITION AND RHlETORIC;   " A NATURAL PHILOSOPIIY;'" ILLUSTRATED SCHOOL HISTORY OF THE UNITED
STATES;" "PRIMARY HISTORY OF THE UNITED
STATES; " ETC.
UPON  THE BASIS OF THE  WORKS OF
GEO. R. PERKINS, LL.D.
NEW YORK:
D. APPLETON AND COMPANY,
443 & 445 BROADWAY.
1867.




Entered, according to Act of Congress, in the year 1868, by
D. APPLETON & CO.,
In the Clerk's Office of the District Court of the United States for the
Southern District of New York.




PRE FA CE.
THIS volume is intended to follow our Primary Arithmetic, or
that of any other series, or may be used as a first book with beginners
that are not too young. It goes over the ground covered by the
Primary, but in a style suited to minds somewhat more mature, enlarging on the subjects there treated, and introducing the pupil to
many new ones. Besides the four fundamental operations, it gives
a comprehensive view of Fractions, Federal Money, Reduction, and
the Compound Rules, presenting under each a large collection of
sums, in every variety, not too difficult, but so constructed as to
require the pupil to think, and thus make the performance intelligent
and not mechanical.
Convinced that too much theory and rule embarrass the young
pupil, the author has in this respect sought to strike a happy mean,presenting necessary explanations, but in few words; giving example
sometimes the precedence over precept, and making rules intelligible
by means of preliminary illustrations.  Definitions are made brief
and simple.  Technical terms unnecessary at this stage of progress
are avoided. The difficulties of beginners being appreciated, it is
believed that they are here so met as to save the teacher the annoyance of constant demands for explanation.
In arrangement we trust some gain will be apparent; particularly
in Compound Numbers, where, in stead of presenting the Tables in
a body, to be confounded together in the pupil's mind, we immediately apply each Table, as soon as learned, in appropriate exercises,
either mental or written. Attention is also invited to the inductive
method used in developing the several subjects.
The teacher is requested to see that every principle is mastered
as the pupil advances. A single defective link makes a whole chain
worthless. If this suggestion is attended to, it is believed that the
present work will make the young student thoroughly acquainted
with the subjects it embraces, and properly prepare him for the next
number of the series, THE PRACTICAL ARITHMETIC.
NEW YORK, August 6, 1863.




CONTENTS.
PAGE
WHAT ARITHMETIC IS,....                      5
NOTATION,.....                          6
NUMERATION,.....16
ADDITION,.....   18
SUBTRACTION,.......        28
MULTIPLICATION,....     37
DIVISION,..........50
Short Division,........     53
Long Division,...  57
FRACTIONS,....                                       66
Reduction of Fractions,.....        72
Addition of Fractions,......      78
Subtraction of Fractions,......81
Multiplication of Fractions,...                      83
Division of Fractions,....91
FEDERAL MONEY,......     95
Addition of Federal Money,.     98
Subtraction of Federal Money,.....      99
Multiplication of Federal Money,.                      101
Division of Federal Money,.....    102
REDUCTION,.. 104
Reduction Descending,....    104
Reduction Ascending,...106
COMPOUND NUMBERS,...    108
COMPOUND ADDITION,.....129
COMPOUND SUBTRACTION,.....    133
COMPOUND MULTIPLICATION,....... 136
COMPOUND DIVISION,.                                         139
MISCELLANEOUS SUMS,.                    142




ELEMENTARY ARITHMAETIC.
WHAT ARITHMETIC Is.
1. WE commence with ONE.  We have one head,
one mouth, one body.
One, a single thing, is called a Unit.
2. A  unit joined to  another unit, makes TWO.
We have two eyes, two hands, two feet.
Another unit joined to two, makes THREE.  Each
of our fingers has three joints.
Another unit joined to three, makes FOIR.
So we may go on.  Adding a unit each time, we
get FIVE, SIX, SEVEN, EIGHT, NINE.
3. One, two, three, four, five, six, &c., are called
Numbers.
A  Number is, therefore, one unit or more.
4. Arithmetic treats of numbers.
5. Repeating the numbers in  order-one, two,
three,fbour,five, six, &c., is called Counting.
QUESTIONS.-1. With what do we commence? What is one, a single
thing, called?-2. Of what is two made up? Of what is three made up? If
we go on, adding a unit each time what do we get?-3. What are one, two,
three, four, &c., called? What is a Number?-4. Of what does Arithmetic
treat?-5. What is Counting? Count nine. Count nine backwards-nine,
eight, seven, &c.




6                       NOTATION.
NOTATION.
6. Every number has a name; as, on0e, two, three.
In stead of writing out the name, however, we may
represent it by a character; as, 1, 2, 3.
Notation is the  art of expressing  numbers by
characters.
7. There are two systems of Notation, the Ar'abic
and the lRoman.
The  Anra'Ae    No ntioln.
8. The Arabic Notation is so called because it
was used by the Arabs.  It employs these ten characters, called Figures:( NAUGHT          2  TWO              6  SIX
0  CIPHER           3  THREE            7  SEVEN
( ZERO           4  FOUR              8  EIGHT
1  ONE              5  FIVE             9  NINE
The first of these figures, 0, implies the absence
of number. 0 cents means not a single cent.
9. The greatest number that can be expressed
with  one figure is nine.  All the numbers above
nine  are  expressed  by  combining  two  or more
figures.
First, 1 is combined with each of the ten figures;
then 2, forming the twenties; then 3, forming the
thirties; then 4, forming theforties, &c.
6. Iow may numbers be represented? What is Notation?-7. How
many systems of notation are there? Name them.-8. Why is the Arabic
Notation so called? How many characters does it use? What are they
called? Learn how to make the ten figures, and their names What does 0
imply?- 9 What is the greatest number that can be expressed with one
figure? How are all numbers above nine expressed?




THE  ARABIC NOTATION.                         7
10, The numbers formed  of two figures are
10  ten              40  forty              70  seventy
11  eleven           41  forty-one         71  seventy-one
12  twelve           42  forty-two'2  seventy-two
13  thirteen         43  forty-three       73  seventy-three
14  fourteen         44  forty-four        74  seventy-four
15  fifteen          45  forty-five        75  seventy-five
16  sixteen          46  forty-six         76  seventy-six
17  seventeen        47  forty-seven       77  seventy-seven
18  eighteen         48  forty-eight       78  seventy-eight
19  nineteen         49  forty-nine        79  seventy-nine
20  twenty           50  fifty             80  eighty
21  twenty-one       51  fifty-one         81  eighty-one
22  twenty-two       52  fifty-two         82  eighty-two
23  twenty-three     53  fifty-three       83  eighty-three
24  twenty-four      54  fifty-four        84  eighty-four
25  twenty-five       55  fifty-five       85  eighty-five
26  twenty-six       56  fifty-six         86  eighty-six
27  twenty-seven      57  fifty-sevens     87  eighty-seven
28  twenty-eight    58  fifty-eight        88  eighty-eight
29  twenty-nine      59  fifty-nine        89  eighty-nine
30  thirty           60  sixty             90  ninety
31  thirty-one       61  sixty-one         91  ninety-one
32  thirty-two       62  sixty-two         92  ninety-two
33  thirty-three     63  sixty-three       93  ninety-three
34  thirty-four      64  sixty-four        94  ninety-four
35  thirty-five      65  sixty-five        95  ninety-five
36  thirty-six       66  sixty-six         96  ninety-six
37  thirty-seven     67  sixty-seven       97  ninety-seven
38  thilty-eight     68  sixty-eight       98  ninety-eight
39  thirty-nine      69  sixty-nine        99  ninety-nine
EXE PR 0ISE.
Count from  1 to 99.  Count from  99 to 1, backwards.
With what figure do the thirties all begin?  The sixties?
Write  the  following  numbers  in  figures: —thirty-seven;
eleven; ninety-eight; eighty-nine; twelve; twenty; five;
fifteen; fifty.  What system  of notation have you just used?




8                         NOTATION.
UNITS, TENS, HUNDREDS.
11. Ten, we  have  seen, is  expressed  thus-10.
Then 1 in the second place denotes one ten.  So, 2
in the second place (20) is two tens, &c.
Any figure standing  in  the  second place represents so many tens. Hence it denotes ten times as
much as if it stood in the first place.
12. The first place is called the units' place.  The
second is the tens' place.
13. Numbers greater than  99 must be expressed
with more than two figures. A third place is thus
required, which is called the hundreds' place.
A figure in the third place denotes ten times as
much as if it stood in the second place, and a hundred times as much as if it stood in the first place.
14. To express the even hundreds, place the several figures in the third place, with  naughts after
them. Thus:100  one hundred                 300  three hundred
200  two hundred                400  four hundred, &c.
15. Observe how  the numbers between  the even
hundreds are expressed:101  one hundred and one          120 one hundred and twenty
102 one hundred and two, &c.    121  one hundred and twenty-one
110 one hundred and ten           201  two hundred and one
111  one hundred and eleven, &c. 202 two hundred and two, &c.
11. How is tent expressed? What does 1 in the second piace denote?
2 in the second place? Any figure in the second place?-12. What is the
first place called? What is the second place called?-13. How must numbers greater than 99 be expressed?'What is the third place called? What
value is denoted by a figure in the third place, compared with the second and
the first?-14. How are the even hundreds expressed — 15. Learn how to express numbers between the even hundreds.




THE ARABIC NOTATION.                      9
E X E R O I S E.
Count from  300 to 400.  Count backwards from  900 to
800.  Writo on your slate the numbers from  500 to 600.
Write 3 units, 6 tens, 9 hundreds (963); 9 units, 6 tens,
8 hundreds; 7 tens, 5 units; 8 hundreds; 3 hundreds, 6
units; five hundreds, one ten; two tens.
Express in figures three hundred and ninety-six.  Two
hundred and twelve.  Eighty-one.  Four hundred and two.
Eight hundred and thirty. Six hundred. Seventeen.
THOUSANDS.
16. The greatest number that can be expressed with
three figures, is 999. Next comes one thousand.
One thousand is expressed thus, 1000-by putting 1 in the fourth place, which is called the
thousands' place.
17. The number of thousands is shown by the
figure in the fourth place.  Thus:
2000  two thousand          6000  six thousand
3000  three thousand        7000  seven thousand
4000  four thousand         8000  eight thousand
5000  five thousand          9000  nine thousand
18. Ten thousand requires five figures to express
it-10000.  The  fifth  place  is  called  that of ten
thousands.
19. A  hundred thousand requires six figures to
express it —100000.  The sixth  place is called that
of hundred thousands.
16. What is the greatest number that can be expressed with three
figures? What comes next to 999? How is one thousand expressed? What
is the fourth place called?-17. By what is the number of thousands shown?
Give examples.-18. How many figures are required to express ten thousand?
What is the fifth place called — 19 How many figures are required to ex.
press a hundred thousand? What is the sixth place called?




10                       NOTATION.
20. We have now had six places named:-units,
tens, hundreds, thousands, ten thousands, hundred
thousands.
These six places are divided into two. Periods, of
three figuries each.  The first Period is that of units;
the second, that of thousands.
6   5   4          3   2   1
Thousands             Units
21. The second period is that of thousands.  To
express a given  number of thousands, write  the
number in  the  second  period.  If there  are no
figures for the units' period, supply naughts.
EXAMPLES.-Write four hundred and twenty-three thousand. To do this, write four hundred and twenty-three, as
already shown-423-for the second period.  Supply naughts
for the units' period, and we have the required number423,000.
So we write seventeen thousand,             17,000.
Five hundred and one thousand,             501,000.
Six hundred and twenty thousand,           620,000.
If there are numbers corresponding to the places of the
units' period, set them there in stead of naughts.
Forty-three thousand, two hundred and ninety,  43,290.
Seven thousand, one hundred and five,             7,105.
One hundred thousand, and sixty-seven.  (As
there are no hundreds in the units' period, supply 0.) 100,067.
20 Name the first six places in order. How are these six places divided?
VWhat is the first period called? What is the second period called? Namo
the places of the first period. Name those of the second period.-21. How
are we to express a given number of thousands? What must be done, if
there are no figures for the units' period? Learn how to torte the examples
given.




THE ARABIC NOTATION.                      11
EXERCISE  IN  NOTATION.
Write the following numbers in figures:1. Five hundred and nine thousand.
2. Sixty-three thousand, two hundred and seven.
3. Eleven thousand, one hundred and eleven.
4. Seven hundred thousand and seventy.
5. Six thousand.  Six hundred thousand.
6. Forty-three thousand and thirty-four.
7. Five hundred and twelve thousand, seven hundred.
8. Eighty thousand, eight hundred and eight.
9. Nine hundred and ninety-nine thousand.
10. Write the greatest number that can be  expressed
with three figures; with four figures; with five figures;
with six figures.
MILLIONS) BILLIONS.
22. The third period is that of millions.
It consists of three places,-millions, ten millions, hundred millions.
EXAMPLES.-Two hundred million,                  200,000,000.
Four hundred and one million,                   401,000,000.
Seventy million, five hundred thousand,'0,500,000.
Six million, seventeen thousand, and seven,       6,017,007.
23. The fourth period is that of billions.
It consists of three places,-billions, ten billions,
hundred billions.
EXAMPLES.-One hundred and two billion,    102,000,000,000.
Eleven billion, eleven thousand, and two,    11,000,011,002.
Four billion, twenty million, and six,        4,020,000,006.
22. WThat is the third period? Of how many places does it consist?
Name them. —23. What is the fourth period? Of how many places does it
consist?  Name them.




12                      NOTATION.
SUMMING UP.
24. Name the periods in order, beginning at the
right.
~      a
h Pr           d Per 2d Per.                s Per.
10 units make 1 ten.
diinishes its value ten  times; removing it one
4th Per.      3d Per,       2d Per.        1st Per.
BILLIONS      MfILLIONS   THOUSANDS        UNITS
Nam te theplaces beginning at the right.
10 units make I ten.
10 tens makhe I hundred.
24hence, removing a figure one place to the right,
dtiminishe    s value ten  times; removing  it one
place to the left, increases it ten times.
25. RULE.-  Write billions in  the fourth period,
millions in the third, thousands in the second, units
in the first, filling the vacant places with naughts,
so as to have three places in each period.
EXERCISE IN  NOTATION.
Write the following numbers in figures, placing units
under units, tens under tens, &c.:
1. Four hundred and seventy-one billion, six thousand.
2. Ninety billion, three million, two thousand and four.
3. Eight hundred million, sixty thousand, one hundred.
24. What is the effect of removing a figure one place to the right?
What is the effect of removing it one place to the left?-25. Give the rule
for notation,




THE ARABIC NOTATION.                13
4. Six hundred and forty thousand, one hundred and one.
5. Nine million, fifty-seven thousand, and eight.
6. Eleven billion, forty-one million, two hundred and ten,
7. Thirty-six million, one hundred thousand, and twelve.
8. Ten billion, ten million, ten thousand, and three.
9. Six hundred and one million, two hundred thousand.
10. Eighty-nine thousand, three hundred and nineteen.
11. Twelve thousand, five hundred and eighty-seven.
12. Four hundred billion, four million, forty thousand.
13. Five million, eight hundred thousand, and seventy-five.
14. One billion, ten million, two hundred thousand, and six.
15. Fifty-seven million, three hundred and twenty-four.
16. Four million, two hundreds and seventeen thousand,
and fifty-eight.
17. Six hundred and nine billion, four hundred and sixtysix million, ninety-two thousand, three hundred and twentyeight.
The Roman  Notation.
26. The Roman Notation is so called because it
was used by the ancient Romans.
It employs seven letters.  I. denotes one; V.,
five; X., ten; L., fifty; C., one hundred; I)., five
hundred; M., one thousand.
27. These letters are combined to express numbers, according to the following principles:1. If a letter is repeated, its value is repeated.
XX. is twenty; III. is three.
2. A letter of less value, placed after one of
greater, unites its value to that of the latter. VI.
is six.
26. Why is the Roman Notation so called? What does it use, to
express numbers — 27. State the principles of the Roman Notation.
2




14                      NOTATION.
3. A letter of less value, placed before one of
greater, takes its value from that of the latter. IV.
is four.
4. A letter of less value, placed between two of
greater, takes its value from  that of the other two
united. LIV. is fifty-four.
5. A  bar over a letter increases its value a thousand times.  V. is five thousand.
TAB L E.
I. is One.                  L. is Fifty.
II. " Two.                 LX. " Sixty.
III. " Three.              LXX. " Seventy,
IV. " Four.              LXXX. " Eighty.
V. " Five.                 XC. " Ninety.
VI. " Six.                    C. " One hundred.
VII. " Seven.                 CI. " One hund. and one.
VIII. " Eight.               CIV. " One hund. and four.
IX. " Nine.                 OX. " One hund. and ten.
X. " Ten.                   CC. " Two hundred.
XI. " Eleven.              CCC. " Three hundred.
XII. " Twelve.            CCCO. " Four hundred.
XIII. " Thirteen.              D. " Five hundred.
XIV.' Fourteen.              DC. " Six hundred.
XV. " Fifteen.              DCC. " Seven hundred.
XVI. " Sixteen.            DCCO. " Eight hundred.
XVII. " Seventeen.        DCCCC. " Nine hundred.
XVI[I. " Eighteen.               M. " One thousand.
XIX. " Nineteen.              MM. " Two thousand.
XX. " Twenty.              MMM. " Three thousand.
XXI. " Twenty-one.        MMMVI. " Four thousand.
XXX. " Thirty.                  V. " Five thousand.
XL. " Forty.                  X. " Ten thousand.
What is the effect of placing a bar over a letter? How is five thousand
denoted? Learn the Tabie.




EXERCISE IN NOTATION.                  15
28. We  may, then, express numbers in  three
ways:
1. With words, as is usual in printed books.
2. With figures, by the Arabic Notation, as in
accounts and calculations.
3. With letters, by the Roman Notation, as in the
headings of chapters.
EXERCISE IN  NOTATION.
Write the following numbers first by the Arabic, and then
by the Roman, Notation:1. Twelve.               6. One thousand.
2. Fifty-seven.          7. Ninety-nine.
3. Nine hundred.         8. Seven hundred.
4. Eighty-six.           9. Sixty-two.
5. Nineteen.            10. Four thousand.
11. Five thousand six hundred and seventy-three.
12. Three hundred and seventy-two.
13. Two thousand eight hundred and forty-one.
14. Nine thousand and twenty-seven.
15. Fifteen hundred and thirty-five.
Express the following numbers according to the Roman
Notation: 12; 1,000; 749; 18; 203; 96; 660; 438; 29;
2,040; 85; 555; 10,801; 79: 5,002; 37; 394; 999; 2,062;
3,186; 119.
Express the following numbers according to the Arabic
Notation: XII. LI. VIII. XLIII. XVI. LXXXIX. XCVIII.
CCI. DXX. XXXIV. MD. IX. MCCXV. DCCCVII. XIV.
MDCLXVI. V.
x. vii. lv. cciv. xxxiii. xix. xlviii. xc. cxxi. xv. lxii.
28. How many ways are there of expressing numbers? What are they,
and where is each used?




16                    TNUMERATION.
N UMERATION.
29. Numeration is the art of reading numbers expressed by figures.
30. In reading numbers, the following principles
apply:1. We read by periods.  Hence, if there are more
than three figures, point off the number into periods
of three figures each, beginning at the right.
2. Always begin to read at the left.
3. The  right-hand  figure  and  the  right-hand
pecod  are never named  as units, the word units
being understood.  We read 7 as seven, not seven
units; 400 is readfour hundred  notfour hundred
6units.
4. Places containing 0 must be passed over in
reading.  We read 1062 one thousand and sixtytwo, not one thousand, no hundred, and sixty-two.
31. RULE.-Beginning at the right, point of  the
number into periods of three igures each.
Beginning at the left, read  the figures in  each
period as if they stood alone, adding the name of
the period in, every case except the last.
EXAMPLES.
10,'09 Ten thousand, seven hundred and nine.
401,840  Four hundred and one thousand, eight hundred and forty.
6,023,070  Six million, twenty-three thousand, and seventy.
29. What is Numeration?-30. Iow do we read numbers? If there are
more than three figures, what do we do? At which side do we begin to
read? What is said of the right-hand figure and the right-hand period?
What must be done in the case of places containing 0?-31. Give the rule
for Numeration.




EXERCISE IN  NUMERATION.                    17
42,110,000  Forty-two million, one hundred and ten thousand.
870,025,002  Eight hundred and seventy million, twenty-five
thousand, and two.
1,,00,000,011  One billion, one million, and eleven.
19,056,007,000  Nineteen billion, fifty-six million, seven thousand.
123,400,789,000  One hundred and twenty-three billion, four hundred
million, seven hundred and eighty-nine thousand.
1100 is read one thousand one hundred, or eleven hundred.
1200 "  " one thousand two hundred, or twelve hundred, &c.
EXERCISE IN  NUMERATION.
Read the following numbers:1.      903   15.       87123645   29. MDCCCLXIII.
2.     8600   16. 476674983429   30.  VCCCXCIX.
3.       21   17.  82000000117   31.  DCLXXXV.
4.   100075   18.           1413   32       XXXVIII.
5.    93282   19.    103600028   33.          MDXIX.
6.  3300000   20.  50500005050   34.    DCCXVII.
7.   463925   21. 442671376000   35.            MCXI.
8.     1650   22.       15000027   36.            LIX.
9.  1040400   23. 998899998998   37.          DXCVI.
10.    26308   24. 203013310031   38.             LXIV.
11.  8005042   25.             410   39.   XCCXXVI.
12.   741607   26.       53413760  40.             LXX.
13.  3821060   27.          83227  41.   MDCXVIII.
14.   600007    28.      14603000   42.   CCCCXLIV.
REVIEW QUESTIONS.-What is a Unit? What is a Number? Of
what does Arithmetic treat? What is Counting? What is Notation?
Name the two systems of notation. What characters are used in the
Arabic Notation? Name the periods in order, beginning at the right.
Name the places. Give the rule for expressing numbers in figures.
What characters are used in the Roman Notation?  State the princi.
ples on which they are combined. What are used to express numbers, in making calculations? What, in accounts?  What, in headings of chapters? What is Numeration? Give the rule for reading
numbers.




18                          ADDITION.
ADDITION.
32. Two men are riding and three are walking;
how many men are there in all?
Here we are required to find one number containing as many
units as 2 and 3 together.  This process is called Addition.
33. Addition  is  the  process  of uniting  two  or
more numbers in one.
The one number thus obtained is called the Sum.
2 and 3 are 5; 5 is the sum.
ADDITION TABLE.
0 and 1 are 1; 0 and 2 are 2; 0 and any number make that number.
1 and           an2 and    3 and         4 and         5 and
I are  2    1 are  3    i are  4    1 are  5    1 are  6
2 are  3    2 are  4    2 are  5    2 are  6    2 are  7
3 are  4    3 are  5    3 are  6    3 are  7    3 are  8
4 are  5    4 are  6    4 are  7    4 are  8    4 are  9
5 are  6    5 are  7    5 are  8    5 are  9    5 are 10
6 are                8 6 are  8    6 are  9    6 are 10    6 are 11
I are  8    7 are  9    7 are 10    7 are 11    7 are 12
8 are  9    8 are 10    8 are 11    8 are 12    8 are 13
9 are 10    9 are 11    9 are 12    9 are 13    9 are 14
10 are 11   10 are 12   10 are 13   10 are 14   10 are 15
6 arnd        7 and a        and         9 and        10 and
1 are  7    1 are  8    1 are  9    1 are 10    1 are 11
2 are  8    2 are  9    2 are 10    2 are 11    2 are 12
3 are  9    3 are 10    3 are 11    3 are 12    3 are 13
4 are 10    4 are 11    4 are 12    4 are 13    4 are 14
5 are 11    5 are 12    5 are 13    5 are 14    5 are 15
6 are 12    6 are 13    6 are 14    6 are 15    6 are 16
7 are 13    7 are 14    7 are 15    7 are 16    7 are 17
8 are 14    8 are 15    8 are 16    8 are 17    8 are 18
9 are 15    9 are 16    9 are 17    9 are 18,  9 are 19
10 are 16   10 are 17   10 are 18   10 are 19   10 are 20




ADDITION.                          19
It is necessary to know the Tables perfectly, so as to say them
backwards or forwards, out of order as well as in order. They must
be mastered before going on.
34. Addition  is denoted  by  an  erect cross -,
called  Plus, placed  between  the  numbers to  be
added.   6 +  5 is read six plus jfve, and means that
six and five are to be added.
35. Two short horizontal lines =, placed between
two quantities or sets of quantities, denote that they
are equal.  6 +  5 =  11 is read six plts five equals
eleven, and means that the sum  of six and five is
eleven.
36. Observe that if
3+2=5           4+5=9            3+7=10          5+8=13
then            then             then            then
13+2=15         34+5=39          53+7=60         25+8=33
23+2=25         44+5=49          63+7=- 70       45+8-53
33+2=35, &c. 54+5=59, &c. 73+'7=80,&c. 85+8=93, &c.
37. Observe that 4 +  5     9 and  5 + 4              9.
Hence, when numbers are to be added, it makes
no difference which is taken first.
EXERCISE  ON  THE  ADDITION  TABLE.
How many are 5 and 4? 4 and 5? 24 and 5? 25 and 4?
4 and 55?  94 and 5      15  and 4?  3 and 2?  2 and 73?
How many are 7 and 1? 1 and 7? 67 and?  7 and 81
11and7?  3and6?  6?  4an           6and3?  46 and3?  6 23?
32. In the example given, what are we required to find? What is this
process called —33. What is Addition? What is the result of addition
called? Recite the Table.-34 By what is addition denoted? What does
6+5 mean?-35. Describe the sign that denotes equality. What does 6+5
=11 mean?-36 If 3+ 2 = 5, then what follows? How much is 5 +8? What
follows?-37. How much is 4+5? I-ow much is 5+4? What principle is
laid down respecting numbers to be added?




20                     ADDITION.
Howmany are 6and2? 66 and2? 2and86? 82 and6?
2and 6? 12 and6? 32 and6? 6and52? 16 and2?
How many are 8 and 2? 2 and 8? 7 and 3? 3 and 7?
18and2? 97 and 3? 32 and 8? 78 and 2? 7and53?
How many are 3 and5? 55 and3? 3and 95? 23 and?  13 and 5? 4and2?  64 and2? 62 and 4? 4and72?
How many are 5and5? 2and7? 45 and5? 52 and7?
12 and7? 37and2? 2and5? 42 and5? 4and6?
How many are 4and 3? 34 and 3? 93 and4? 14 and 3?
4and4? 84 and4? 4and54? 9 and8? 4and9?
Find thesumof 1, 2and3. 8+2+3. 4+1+5. 6+3+1.
3+4+2. 6+3+3. 4+6+7. 3+7+10. 20+3+4.
Count by twos, commencing 2, 4, 6, 8, &c., up to 100.
Count by threes, commencing 3, 6, 9, 12, &c., up to 99.
Count by fours, commencing 4, 8, 12, 16, &c., to 100.
Count by fives, commencing 5, 10, 15, 20, &c., to 100.
MENTAL EXERCISES.
1. Seven metals were known to the ancients; 43 have been
discovered since. How many metals are now known?
Ans. 7+43 metals, or 50 metals.
2. How much will a boy earn in two weeks, if he earns
5 dollars the first week and 2 the second?
3. John Adams was president four years. He was 61
when he entered on the duties of the office; how old was he
when he left it?
4. A gardener set out 9 trees one day, and 8 the next;
how many did he set out both days?
5. If a house has 8 windows on one side, 6 on another, and
4 on a third, how many windows has it in all?
6. Napoleon had 4 brothers and 3 sisters, besides 5 that
died in infancy; how many brothers and sisters had he in all?




EXERCISES IN ADDITION.                  21
7. How many pounds will a pair of chickens weigh, if
each weighs three pounds?
8. My house is at the north end of a lake; Mr. A's is 3
miles south of the south end. If the lake is 5 miles long, how
far is it from my house to Mr. A's?
9. A farmer who has 7 cows, buys 6 more; how many has
he then?
10. In ajar that weighs 6 pounds, I put 11 pounds of butter; how much will the whole weigh?
11. A boy who bought a quire of paper for 20 cents, sold
it so as to gain 5 cents; how much did he sell it for?
12. Washington was born in 1732.  George III. was born
6 years later; what was the date of his birth?
13. A steamboat starts with 72 passengers.  Three miles
down the river, it receives 7 more passengers. How many
has it then?
14. The Earth has 1 moon; Jupiter has 4 moons; Saturn,
8; Uranus, 6; Neptune, 1; how many moons does that make
altogether?
38. Prinnaciples of Addition.
1. We must add things of the same kind.  Therefore, in setting down numbers to be added, place
units under units, tens under tens, &c.
2. The sum of units is units; of tens, tens; &c.
3. Always begin to add at the right.
4. Find the sum  of each column; and, if it is expressed by one figure, set it down under the column
added.
38. How must we set down numbers to be added? Why so? What is
the sum of units? Of tens? Of hundreds? Where must we begin in adding? If the sum of each column is expressed by one figure, where must we
set it?




22                      ADDITION.
EXAMPLE.-Add four million; three hundred and twentysix thousand, two hundred and forty-seven; fifty-three thousand, four hundred and ten; and two hundred and twentyone.
Operation. —Write down the numbers, units under units,
tens under tens, &c.
Begin to add at the right.
1st column. 1 and 7 are 8. Set down 8.
2d. 2 and 1 are 3, and 4 is 7. Set down 7.      4000000
3d. 2 and 4 are 6, and 2 is 8. Set down 8.       326247
4th. 3 and 6 are 9. Set down 9.                   53410
5th. 5 and 2 are 7. Set down 7.                     221
6th. Bring down 3. 7th col. Bring down 4. Ans. 4379878
Proof of Addition.
39. Proving an example is finding whether the
work is correct.
40. Addition is proved by adding the columns
from  the top downward.  If the sum  is the same as
when they are added from  the bottom  upward, we
infer that the sum is right.
This Proof is based on the fact that, when numbers are
to be added, it makes no difference in what order they are
taken.  The sum will be the same. If an error has been
made in adding up, it is not likely to be repeated in adding
down, and will thus be detected.
EXAMPLE.-Prove the above example. Add    4000000
each column from the top downward. 7 and 1          326247
are 8. 4 and 1 are 5, and 2 is 7. 2 and 4 are 6,     53410
and 2 is 8. 6 and 3 are 9.  2 and 5 are 7.             221
Bring down 3. Bring down4. Answer, 4379878 
-the same as before.  Hence the work is right.    43988
Apply these principles in the example given. —9. What is meant by
Proving an example?-40. How is addition proved? On what is this proof
based? Prove the example just given.




EXERCISE IN ADDITION.                 23
EXAMPLES FOR THE SLATE.
41. Read the numbers added. Prove each example.
1. Add 600123, 154235, 34400, and 221.    Ans. 788979.
2. Add 85026371, 41005, and 1810000.   Ans. 86877376.
3. Find the value of 1234+455111+20234+12120.
4. Add 297661031851, 1135204022, 3115, and 1520010.
5. What is the sum of 201+100+2283+364114+322201?
6. Find the sum of thirty-one; one hundred and eleven;
twenty thousand, four hundred and forty-two; seventeen
thousand, one hundred and eleven; and sixty thousand, one
hundred and three.                           Ans. 97798.
7. Add together seventeen million; one hundred and fifty
thousand, one hundred; eight hundred and nine thousand,
two hundred and seventy-two; and forty thousand, three
hundred and sixteen.                      Ans. 17999688.
8. What is the sum of four million, eight hundred and
twelve thousand, one hundred and two; thirty-one thousand,
six hundred and twenty; one million, one thousand, and
forty-five; and three thousand and thirty?   Ans. 5847797.
9. Add together three hundred and twenty; one hundred
and eleven million, two hundred and twelve; forty thousand,
one hundred and thirty-two; and two million, one hundred
and thirty-seven thousand.               Ans. 113177664.
10. Find the sum of twenty billion, one thousand, and one;
four thousand and eleven; four hundred and seven million,
twenty thousand, six hundred and forty-two; and sixty-three
thousand, one hundred and three.       Ans. 20407088757.
11. A merchant sells $10000* worth of goods one day;
$5123 worth, the next; and $2436 worth, the next. How
much does he sell in all?
* This mark ($) denotes dollars. It is always placed before the number. $1000 is read a thouscand dollars.




24                     ADDITION.
12. Three score are sixty. How many are three score and
ten?
13. If I travel 1246 miles by steamer, 732 by railroad, and
21 by stage, how far do I travel altogether?
14. An army contains 23022 infantry privates; 710 infantry officers; 4000 cavalry, including officers; and 164 musicians. How many men in all in the army?
15. A. sells a vessel for $15420, which is $1355 less than
it cost. What did the vessel cost?
16. The estate of a deceased man was divided as follows:
his wife received two hundred and ten thousand dollars; his
daughter, forty thousand two hundred and fifty dollars; his
elder son, twenty-one thousand five hundred and six dollars;
his younger son, twenty thousand one hundred and forty-two
dollars. What was the value of the estate?   Ans. $291898.
17. The planet Mars is 145,205,000 miles from the sun.
Jupiter is 350,610,500 miles farther. How far is Jupiter?
Caarryiing.
42. The sum of a column may make more than
one figure.
EXAMPLE.- Add 487 and 975.                       487
Begin at the right.  5 and 7 are 12-2    975
units and 1 ten.  Set down 2 in the units'  1462
place, and add the 1 ten to the other tens.
1 and 7 are 8, and 8 is 16.  16 tens are 6 tens
and 1 hundred. Set down 6 in the tens' place, and
add the 1 hundred to the other hundreds.
I and 9 are 10, and 4 is 14-14 hundreds, or 4
hundreds and 1 thousand.  Answer, 1462.
42. With the given example, show what is meant by carrying. Give
the rule for carrying.




CARRYING IN ADDITION.                25
43. This adding of the left-hand figure is called
Carrying.
44. RULE FOR CARRYING.- When the sum  of a
column is over 9, set down tle rqight-hand figure,
and carry the left-handfigure or figures to the next
column.
EXAMPLES FOR THE SLATE.
45. Read and add the following numbers. Prove each
example.
(1)             (2)                 (3)
24897         43345678        123423434559
64         1123355          23785432977
234567              7893          9876543696
2357911             54689            751002789
3456T890            734321             10200859
4. Add 123405, 54210, 1794322, and 6541. Ans. 1978478.
5. Add 4275602, 45706, 5567801, and 365. Ans. 9889474.
6. Add 23, 6794, 896423, 597, and 16019.   Ans. 919856.
7. What is the value of 965482190006+4063 +8127299837
+ 102009 + 9238675 + 67.             Ans. 973618834657.
8. Find the number of days in a year, there being 31 days
in January, 28 in February, 31 in March, 30 in April, 31 in
May, 30 in June, 31 in July, 31 in August, 80 in September,
31 in October, 30 in November, and 31 in December.
9. The amount of treasurie exported from California in
1861 was $40639089. This was $1664256 less than in 1860.
What was it in 1860?
10. The two largest cities in Europe are London and Paris.
The population of London in 1851 was 2362236; that of Paris,
1053262. What was the population of both?
11. Benjamin Franklin was born in 1706, and died at the
age of 84. In what year did he die?
3




26                     ADDITION.
12. Rhode Island, the smallest state in the Union, contains
1306 square miles. Texas, the largest state, contains 236198
square miles more than Rhode Island. How many square
miles in Texas?
13. In 1850, South Carolina produced 300901 bales of cotton; Alabama, 564429. How much did both produce?
14. The United States is made up of the Atlantic Slope,
which contains 967576 square miles; the Mississippi Valley,
which contains 1237311 square miles; and the Pacific Slope,
which contains 778266 square miles.  How  many square
miles in the whole United States?
15. In 1850, 4203064 copies of papers and periodicals were
printed in Maine; 3067552, in New Hampshire; 2567662, in
Vermont; 64820564, in Massachusetts; 2756950, in Rhode
Island; 4267932, in Connecticut. How many were printed
in all the New England states?             Ans. 81683724.
46. Rule for Addition.
1. Set units under units, tens under tens, &c.
2. Beginning at the right, find the sum  of each
column.
3. If the sum  is expressed by one jigure, write
it under the column added; if not, set down the
right-hand figure, and carry the left-hand figure or
figures to the next column.
EXERCISE.
47. The following examples are to be practised until they
can be added at sight up and down, naming the results only.
Thus in Example 1:-five, eleven, thirteen, fifteen, eighteen,
twenty-one, thirty, thirty-seven-set down 7, and carry 3.
Three, seven, fourteen, &c.




EXERCISES IN ADDITION.           27
(1)          (2)           (3)          (4)
899697       891985        342687        185232
125429       263882        165431        965368
489843       885487        267994        905596
974583       298688        319416         78748
583162       656599        948668        588933
236972       677494        275773        879370
119876       681395        139915        899477
612345       684923        289076        276431
(5)         (6)            (7)          (8)
958576       758318        669786        895939
328492       272638        359628        765838
223967       364773        694279        678930
225523       525822        946335        514831
221679       294987        834569        455922
120739       162856        179145        379828
653865       175902        579757        263955
862781       943655        954852        675869
426893       682354        864351        416970
4022515      4181305       6082702       5048082
(9)               (10)               (11)
838725889        6158874991          652824777
189404376        8254889782          863928996
397846748         965449873          896956866
578757467        4833437997          885999586
949549259        2511486766          992845696
499699451        1896479693          894896986
887415762          65769958          885699577
739353343        4747786854          864712998
568239774        4598748765          893888579
659128135        2152759921          878994886
487816296        5439765653          893989466
18906147        8901729836          784985886
6814842647       50527180089        10389724299




28                        SUBTRACTION.
SUBTRACTION.
48. Three boys are on the lawn. Two go into
the house; how many are left?
Here we are required to take 2 from  3, or to find the difference between 2 and 3. This process is called Subtraction.
49. Subtraction is the process of taking  one number from  another.
50. The smaller number must always be taken
from the greater. We can take 2 from 3, but not
3 from 2.
SUBTRACTION TABLE.
0 from 1 Altves 1; 0 from 2 leaves 2; 0 from any number leaves
that number.
1 from       2 from        3 from        4 from        5 fiom
1 leaves 0   2 leaves 0   3 leaves 0   4 leaves 0   5 leaves 0
2 leaves 1   3 leaves 1   4 leaves 1    5 leaves   6 leaves 1
3 leaves 2   4 leaves 2   5 leaves 2    6 leaves 2 j 7 leaves 2
4 leaves 3   5 leaves 3   6 leaves 3   7 leaves 3   8 leaves 3
5 leaves 4   6 leaves 4    7 leaves 4    8 leaves 4   9 leaves 4
6 leaves 5   7 leaves 5   8 leaves 5    9 leaves 5  10 leaves 5
7 leaves 6   8 leaves 6   9 leaves 6  10 leaves 6 I 11 leaves 6
8 leaves 7   9 leaves 7  10 leaves 7   11 leaves 7  12 leaves 7
9 leaves 8  10 leaves 8  11 leaves 8  12 leaves 8  13 leaves 8
10 leaves 9  11 leaves 9  12 leaves 9  13 leaves 9  14 leaves 9
6 from       7 from        8 from        9 from        10 from
6 leaves 0   7 leaves 0   8 leaves 0   9 leaves 0  10 leaves 0
7 leaves 1   8 leaves 1   9 leaves 1  10 leaves 1  11 leaves 1
8 leaves 2   9 leaves 2  10 leaves 2  11 leaves 2  12 leaves 2
9 leaves 3  10 leaves 3  11 leaves 3  12 leaves 3  13 leaves 3
10 leaves 4  11 leaves 4  12 leaves 4  13 leaves 4  14 leaves 4
11 leaves 5  12 leaves 5  13 leaves 5  14 leaves 5  15 leaves 5
12 leaves 6  13 leaves 6  14 leaves 6  15 leaves 6  16 leaves 6
13 leaves 7  14 leaves 7  15 leaves 7  16 leaves 7  17 leaves 7
14 leaves 8  15 leaves 8  16 leaves 8  17 leaves 8  18.leaves 8
15 leaves 9  16 leaves 9  17 leaves 9  18 leaves 9  19 leaves 9




SUBTRACTION.                         29
51. The number to  be subtracted, is called the
Subtrahend.  That from  which the subtrahend is to
be taken, is called the Minuend.  The result, or what
is left, is called the Remainder.
2 from  3 leaves 1; 2 is the  subtrahend, 3 the
minuend, 1 the remainder.
52. Subtraction  is denoted by a short horizontal
line -, called Minus, placed before the subtrahend.
3-2 is read three minus two, and means that 2
is to be subtracted from  3.
53. Observe that if
3-2=1          — 3-4           7-3=4            11-10=1
then            then            then             then
13-2=11         47-3=44         27-23=4          31-10=21
23-2-21         57-3=54         77-73=4          51-10=41
33-2=31, &c. 67-3=64, &c. 87-83=4, &c. 91-10=81, &c.
EXERCISE  ON  THE  SUBTRAOTION  TABLE.
How many does 4 from 6 leave?  4 from 16?  4 from 36?
4 from 56?  14 from 16?  74 from 76?  24 from 26?
How much is 9-3?  29-3?  49-3?   69-3?   89-3?
9-6?  39-6?  59-6?  79-6?  99-6?  6-1?  11-1?
Take 2 from 4.  2 from 54.  2 fiom  94.  2 from  24.  3
from 4.  3 from 64.  3 from 34. 4 from 4.  4 from 94.
How much is 8-5? 78-5'? 18-5?  48-5? 8-3? 28
-3?  88-3?  68-3?  68-5?  5-5?  25-5?  35-5?
Subtract 3 from 5.  3 from 55.  33 from 35.  43 from 45.
2 from 5. 2 from 15.  12 from 15.  22 from 25.  2 from 85.
48. In the example given, what are we required to do? What is this
process called?-49. What is Subtraction?-50. Which is the number to be
subtracted? Recite the Table. What does 0 from 1 leave? 0 from 2?
0 from any number?-51. What is the number to be subtracted called?
What is the number from which the subtrahend is to be taken called?
What is the result called — 52. How is subtraction denoted?-53. HIow much
is 3-2? What follows? How much is 11 -10? What follows?




30                    SUBTRACTION.
How many does 5 from 7 leave? 5 from 57?  5 from 87?
2 from7? 2from17? 2 from 67? 4 from7? 4from 27?
How much is 8-4?  18-4?  38-34?  9-2?  69-2?
39-32? 49-42? 9-7? 19-17?  9-4? 59-4?
Take 9 from 10. 9 from 40. 9 from 60. 8 from 10. 8
from 16. 8 from 17. 7 from 11. 6 from 12. 5 from 10.
Count backward by twos from 100. Thus: 100, 98, &c.
Count backward by fives from 100. Thus: 100, 95, &c.
Count backward by tens from 100. Thus: 100, 90, &c.
Count backward by twos from 99. Thus: 99, 97, &c.
MENTAL EXERCISES.
1. Fifty metals are now known. Seven were known to
the ancients. How many have been discovered since?
Ans. 50-7 metals, or 43 metals.
2. A boy buys a paper for 5 cents. He gives the newsman 10 cents. How much change will he get?
3. Ellen is 14 years old, and Jane is 6 years younger.
How old is Jane?
4. A farmer who has 15 sheep, sells 7 of them. How
many has he left?
5. A man is on his way home fiom a town twelve miles
distant. When he has walked nine miles, how much farther
has he to go?
6. A flower-girl who starts with 24 nosegays, comes back
with 4. How many has she sold?
7. Five out of nine eggs turned out bad; how many were
good?
8. If I buy a cow for $29, and sell her for $23, how much
do I lose?
9. Leaving home with $15, I spend $6 and give $3 away.
How much have I left?




PRINCIPLES AND PROOF.                   31
54. Principles of' Subtraction.
1. The smaller number is the one to  be subtracted.  Set it under the greater.
2. As we munst subtract things of the same kind
place units under units, tens under tens, &c.
3. The difference between units and units is units;
between tens and tens, tens; &c.
4. Begin to subtract at the right.
5. Take each figure of the subtrahend from  the
one above it, and set the remainder under the figure
subtracted.
EXAMPLE.-From  eight million six hundred and forty
thousand nine hundred and fifty-seven, subtract two hundred
and ten thousand four hundred and thirty-six.
Operation.-,Write the smaller number under the greater,
units under units, tens under tens, &c. Begin at the right.
6 from 7 leaves I; set it down under the 6.    8640957
3 from 5 leaves 2. 4 fiom 9 leaves 5. 0 from       210, leaves 0. 1 from 4 leaves 3. 2 from 6 leaves    ____
4. Bring down 8. Answer, 8430521.                 8430521
Praoof of ~tkbltrateion.
55. Add the remainder and subtrahend.  If their
sum  is equal to the minuend, the work is tight.
EXAMPLE. —Prove the  cxample just              8430521
given.  Add the remainder to the subsucb.    210436
trahend.  Their sum  is 8640957, which    u        21
is equal to the minuend. Therefore the   sume  8640957.work is right.
54. How are the numbers to be set down in subtraction? Of what lenomination is the difference between units and units? Between tens and
tens   Where must we begin to subtract  How are we to proceed? Solve
the example givene —55. How is subtraction proved? Prove the example just
given.




32                    SUBTRACTION.
EXAMPLES FOR THE SLATE.
56. Read the numbers given and the remainders. Prove
each example.
1. From 1908647 take 2321.    6. 4678759-2678657.
2. From 897628 take 34527.    T. 579583099-20052064.
3. Take 1431 from 11463845.   8. 78199087-60152085.
4. From 673485 take 20432.    9. 3976189653-1042053.
5. Take 38143 from 1159463.  10. 12186947285-103014.
11. From six hundred and fifty-seven thousand eight hundred and forty-nine, subtract five hundred and twenty-one
thousand six hundred and sixteen.            Ans. 136233.
12. Take four million seven thousand one hundred and
thirty-one, from forty-six million eighty-nine thousand five
hundred and forty-two.                     Ans. 42082411.
13. The subtrahend is three million four hundred and
forty-three thousand.  The minuend is five million eight
hundred and forty-nine thousand and six. What is the remainder?                                    Ans. 2406006.
14. From eight hundred and forty-eight billion six hundred and ninety-seven million and ten, take thirty-one billion
one hundred and forty-two million.     Ans. 817555000010.
15. How much more is twelve billion eight hundred and
seventy-nine million three hundred and sixty-four, than one
billion two hundred and thirty-five million and twenty-two?
Ans. 11644000342.
Carrying.
57. The lower figure may be greater than the
one above it.
EXAMPLE.- From  738 take 419.
Begin at the right.  We can not take 9    73
from  8, because 9 is greater than 8.                  ~




CARRYING                       33
Hence from  the 3 tens we take 1 away,              2
leaving 2 tens.  The 1 ten taken away is    738
equal to 10 units, which we add to the 8 
319
units, making 18.
Now  we can subtract 9.  9 from  18 leaves 9;
set it down.  I from  2 (not 3) leaves 1; 4 from  7
leaves 3.  Answer, 319.
To balance the 10 units added to the 8, we took
away 1 of the tens from  the upper line.  But in
stead of diminishing  the upper figure 1, we may
add 1 to the figure below it.  This gives the same
result, and, being more convenient, is the mode
generally pursued.                                    738
Thus: 9 from 18, 9. 1 and 1 are 2; 2 from 3   419
leaves 1. 4 from 7, 3. Answer, 319.                   319
58. This adding of 1 to the lower figure is called
Carrying.
59. We may have to carry several times in succession.
EXAMPLE.-From 10000 take 9999. 9 from 10
leaves 1; set it down and carry 1. 1 and 9 are      9
10; 10 from 10 leaves nothing. Carry 1; I and 9
are 10; 10 from 10 leaves nothing. Carry 1; 1 and 9
9 are 10; 10 from 10 leaves nothing. Answer, 1.
60. RULE FOR CARRYING.-  When  the lower figure is greater than the one above it, add 10 to the
upper figure, subtract, and carry 1 to the next lower
figure.
57. Show, with the example given, how we proceed when the lower figure is greater than the one above it.-58. What is this adding of 1 to the
lower figure called?-59. Show how we may have to carry several times in
succession.-60. Give the rule for carrying.




34                  SUBTRACTION.
EXAMPLES FOR THE SLATE.
61. Read the numbers. Subtract. Prove each example.
(1)             (2)                 (3)
6310475       129060418801          600173240
3501768        68431160424           25823186
(4)              (5)                (6)
19372843       801009647258          415673285
9445276         7771819436            6906356
7. Take four thousand and thirteen, from one million and
eleven.                                   Ans. 995998.
8. From sixty-five thousand and seven, subtract nine hundred and ninety-nine.                      Ans. 64008.
9. From seven hundred and thirty-tour thousand five
hundred and twenty-one, subtract eighty-four thousand two
hundred and eighty-three.                 Ans. 650238.
10. Subtract seven hundred and sixty-two million nine
hundred thousand and seventy, from one billion fourteen
thousand and nine.                     Ans. 237113939.
11. From two hundred and fifty-five thousand, take seven
hundred and two.                          Ans. 254298.
62. Ra ole ftor SriMtraction.
1. Set the smaller number bunder the greater, units
under units, tens under tens, &c.
2. Beginning at the right, take each figuere of the
subtrahend from  the oze above it, and set the remailnder under thefigure subtracted.
3. If the lower Xfgure is greater than the one
above it, add 10 to the upper figure, subtract, and
carry 1 to the next lower figure.




EXAMPLES FOR PRACTICE.                  35
When a whole is given and one' of its parts, what is the
rule for finding the other part?
Subtract the given part from  the whole.
When a whole is given and all its parts but one, what is
the rule for finding that one?
Add the given parts, and subtract their sum from
the whole.
When the year of a person's birth and that of his death are
given, what is the rule for finding his age?
Subtract the earlier date from  the later.
EXAMPLES FOR THE SLATE.
1. Howard, the philanthropist, was born in the year 1726.
He died in 1790. To what. age did he live?
2. Hudson explored the river called by his name in 1609.
How long was this after the discovery of America, which took
place in 1492?
3. How many years have elapsed from the discovery of
America to the present time?
4. A man worth $47650, leaves $29855 to his wife and
the rest to his son. What is the son's portion?
5. At an election 12572 votes were cast, of which the successful candidate received 7698. How many votes did the
other candidate receive?
6. In the above election, what was the majority of the
successful candidate-or, how many votes did he receive
more than the other?                       Ans. 2824 votes.
7. Washington died in 1799, at the age of 67. In what
year was he born?
8. In a state containing 2311786 inhabitants, there are
1143683 females; how many males are there?
Ans. 1168103 males.




36                    SLTBTRACTIOl.
9. A lady buys a house for $3000. She spends $169 on
it for repairs, and then sells it for $3450. Does she gain or
lose, and how much?                       Ans. Gains $281.
10. In 1854, 3038955 bushels of wheat were received in
Chicago. In 1857, the receipts were 10554761 bushels. What
was the increase in the 3 years?      Ans. 7515806 bushels.
11. A house and lot are sold for $10575. If the lot cost
$3250, and the house $8195, does the owner gain or lose, and
how much?                                Ans. Loses $870,
12. The population of the city of New York in 1860 was
814287, which was 184477 more than it was in 1855. What
was the population in 1855?                  Ans. 629810.
13. Two persons are 375 miles apart.  They travel towards each other, one going 93 miles and the other 57. How
far are they then apart?                    Ans. 225 miles.
14. A man owns a house valued at $6250, and stock to
the amount of $4500. If he is in debt $3999, what is he
worth in all?                                  Ans. $6751.
15. If a merchant who has 19000 bushels of corn, sells
one customer 4550 bushels, and another 398, how many
bushels has he left?                   Ans. 14052 bushels,
16. A sells B 60 bushels of wheat for $74, a horse for
$150, a wagon for $95, and $37 worth of butter,  B pays
$125 cash.  How much does he still owe A?        Ans. $231.
17. A  farmer lays up 237 pounds of butter from  his
own dairy, and buys 349 pounds more of a neighbor. After
giving away 50 pounds and selling 488, how much has he
left?                                      Ans. 48 pounds.
18. In a library of 1594 volumes, there are 727 French
books, 586 German, and 138 Spanish. The rest are Italian;
how many of the latter are there?        Ans. 143 volumes.
19. From ten thousand subtract seventy-five. Ans. 9925.




MULTIPLICATION.                            37
M ULTIPLICATION.
63. What will 4 pies cost, at 6 cents eachi 
If 1 pie costs 6 cents, 4 pies will cost 4 times 6 cents, or
24 cents.  Here we are required to take 6 four times.  This
process is called Multiplication.
64. Multiplication is the process of taking a number a certain number of times.
MUILTIPLICATION TABLE.
0 taken any number of times is 0. Once 0 is 0; twice 0 is 0; &c.
O times any number is 0. 0 times 1 is 0; 0 times 2 is 0; &c.
Once any number is that number.  Once 1 is 1; once 2 is 2; &c.
Twice    3 times   4 times   5 times    6 times    7 times
I is  2    1 is  3    1 is  4    1 is  5    1 is  6    1 is  7
2 is  4    2 is  6    2 is  8    2 is 10    2 is 12    2 is 14
3 is  6    3 is  9    3 is 12    3 is 15    3 is 18    3 is 21
4 is  8   4 i12    4i 2      4          20      is 24    4 is 28.S is 10      is 10    5 is    20    5 is 2   5 is 30 5      is 35
6 is 12    6 is 18    6 is.24    6 is 30    6 is 36    6 is 42
7 is 14    7 is 21    7 is 28    7 is 35    7 is 42    7 is 49
8 is 16   8 is 24    8 is 32    8 is 40    8 is 48    8 is 56
9 is 18    9 is 27    9 is 36    9 is 45    9 is 54    9 is 63
10 is 2    1is0 10  is 40   10 is 50   10 is 60   10 is 70
11 is 22  11 is 33   11 is 44   11 is 55  11 is 66  11 is 77
12 is 24   12 is 36   12 is 48   12 is 60   12 is 72   12 is 84
8 times       9.times     10 times      11 times      12 times
I          is   8    I  is   9    1  is  10    I  is  11    I  is  12
2 is  16    2  is  18    2  is  20    2  is  22    2  is  24
3  is 24    3  is  27    3  is  30    3  is  33    3  is  36
4  is 32    4  is  36    4  is  40    4  is: 44    4  is  48
5  is 40    5  is  45    5  is  50    5  is  55    5  is  60
6  is 48    6  is  54    6  is  60    6  is  66    6  is  72
7  is  56    7  is  63    7  is  70    7  is  77    7  is  84
8  is  64    8  is  72    8  is  80    8  is  88    8  is  96
9  is  72    9  is  81    9  is  90    9  is  99    9  is 108
10  is  80   10  is  90   10  is 100   10  is 110   10  is-120
11  is  88   11  is  99   11  is 110   11  is 121   11  is 132
12  is  96   12  is 108   12  is 120   12  is 132   12  is 144
4




38                     MULTIPLICATION.
65. The number to,be multiplied, is called the
MIultiplicand.  That by  which  we are to multiply,
is called the Multiplier.   The result, or number obtained by multiplication, is called the Product.
3 times 2 is 6.  2 is the multiplicand, 3 the multiplier, 6 the product.
66. The multiplicand  and  multiplier are  called
Factors of the product.   2 and 3 are factors of 6.
67. Multiplication  is denoted by a slanting cross
x, placed between the  factors.   2 x 3 is read, and
denotes, two mun tivplied by three.
68. The multiplier shows how  many  times the;multiplicand is to be taken.   Multiplying 2 by 3 is
taking 2 three times: 2+2+2 -6.   2 x 3    6.
Multiplying is therefore a short way of adding a
number to itself.
How  many trees are there in    w'    A1 J            h  A
three rows of four trees each? 
We may do this sum  in  two  it  
ways.  We may say, If one row                  t
contains 4 trees, 3 rows will con-..,.
tain 3 times 4 trees, or 12 trees.           t   
This is doing it by multiplication.
Or we may say, As there are 4 trees in each row, in the
three rows there are 4+4+4 trees, dr 12 trees.  This is
doing it by addition.  The result is the same.
63. Repeat the example given. What are we here required to do?
What is this process called?-64. What is Multiplication? Ilow much is 0,
talen any number of times? How much is 0 times any number? How
much is once any number? Recite the Table.-65. What is the number to
be multiplied called? What is that by which we are to multiply called?
What is the result obtained by multiplication called? —66. What are the
multiplicand and multiplier called? —67. How is multiplication denoted?68.'VWhat does the multiplier show? MIultiplying is a short way of doing
what? Illustrate this.




MENTAL EXERCISES.                     39
69. When two numbers are to be multiplied together, it makes no difference in the result which is
taken as the multiplicand, and which as the multiplier. 4x3=12. 3x4=12.
We have 12 trees in the above engraving, whether we
take them crosswise as forming 3 rows of 4 each, or lengthwise as forming 4 rows of 3 each.
70. The product is of the same kind as the multiplicand.   3 times 2  men  is 6 men; 3 times 2
apples is 6 agpples; &c.
MENTAL  EXERCISES.
1. If there are seven days in a week, how many days are
there in nine weeks?
MODEL.-If there are 7 days in 1 week, in 9 weeks there
are 9 times 7 days, or 63 days. Answer, 63 days.
2. If a man's wages are three dollars a day, how much
will he earn in five days?  How much in ten days?
3. If a stage makes 6 trips a day, of 3 miles each, how
many trips will it make in a week, excluding Sunday?
4. What is the cost of 11 tons of coal, at $6 a ton?
5. How long will it take one boy to do a job which it
takes five boys eight days to do?
6. At the rate of 4 for a cent, how many crackers can be
bought for 7 cents? How many for 12 cents?
7. Four quarts make a gallon.  IHow many quarts will a
five-gallon jug hold?
8. What will 12 tables cost, at $11 apiece?
9. If an omnibus carries 12 passengers each trip, how
many does it carry in 6 trips?  In 8 trips?
69. In multiplying two numbers, what is found to make no difference?
Illustrate this with the engraving.-70. Of what kind is the product?




40                  MULTIPLICATION.
10. If a box of tea lasts 8 persons 11 weeks, how long
will it last 1 person at the same rate?
11. There are 2 pints in a quart. How many pints in 3
quarts? In 5 quarts? In 7 quarts?  In 9 quarts?
12. If two boys do 4 sums apiece every day, how many
will both do in 3 days?
13. Three fields contain 3 trees each. Under each tree
3 cows are lying. How many cows are in the three fields?
14. If 2 apples can be bought for 1 cent, how many can
be bought for 10 cents?
15. Four girls have 2 hens. each, and each hen has G
chickens. How many chickens have they altogether?
EXAMPLES FOR THE SLATE.
71. Multiply 60123 by 3.
Here the multiplier is a single figure.  Set it
under the units' figure of the multiplicand.
Beginning at the right, multiply each fig-   60123
ure of the multiplicand by the multiplier,              3
setting each product in a column with the  180369
figure multiplied.
Three times 3 is 9; 3 times 2 is 6; 3 times 1 is 3; 3 times
0 is 0; 3 times 6 is 18. The last product consists of two figures. Set it down with its right-hand figure under the figure
multiplied. Answer, 180369.
(1)        (2)        (3)          (4)
Multiply   9432       71232       52122       81010
By             2           3          4           5
5. At 7 cents a pound, what will 1010 pounds of cheese
cost?
6. If a man lays up $510 a year, how many dollars will
he be worth in 9 years?                      Ans. $4590.




CARRYING.                        41
Carryinmg.
72. When  a figure  is multiplied, the product
may  consist of two  figures.   As we can  not set
them  both down under the figure multiplied, we
place the right-hand figure there, and add the other
figure to the next product.
This adding of the left-hand figure to the next
product is called Carrying.
EXAMPLE.-Multiply 7608 by 7.                        7608
Begin at the right.  7 times 8 is 56,-5 tens            7
and 6 units.  Set the 6 units in the units' place,    53256
and carry the 5 tens to the next product.
7 times 0 tens are 0 tens, and 5 makes 5 tens. Set it down.
7 times 6 hundreds are 42 hundreds,-or, 4 thousands and
2 hundreds.  Set the 2 hundreds in the hundreds' place, and
carry the 4 thousands to the next product.
7 times 7 thousands are 49 thousands, and 4 makes 53.
This being the last product, set down both figures. Answer,
53256.
73. RULE FOR CARRYING. —          hen c&nyproduct exceeds 9, set down the right-hand figure in the same
column with the figure multizplied, and add the remaining figure or figures to the next product.
EXAMPLES FOR THE SLATE.
1. 23618,x6.    Ans. 141708.    5. 28946735 x9.
2. 41726x7.    Ans. 292082.    6. 9156738x8.
3. 683918547 x 4.                7. 4507956 x 6.
4. 925465893 x 5.                 8. 51708949326 x 8.
72. When, on multiplying a figure, we get a product of two figures, what
must we do? What is this process called? Illustrate the mode of carrying,
with the example given.-73. Recite the rule for carrying in multiplication.




42                   MULTIPLICATIO.o
9. The multiplicand is ninety-three thousand and fortysix; the multiplier is 3; what is the product?  Ans. 279138,
10. How much is nine times seven hundred and fifty-eight
thousand and twenty-nine?                     Ans. 6822261.
11. There are 365 days in a year. How many days are
there in three years?
12. The moon is 240000 miles from the earth. If there
were a comet 6 times that distance from the earth, how far
would it be?                             Ans. 1440000 miles.
13. Nine million eight hundred thousand two hundred
and fifty-seven is one factor; seven is the other; what is the
product?                                     Ans. 68601799.
Mlultplying by two or more figures.
74. Multiply by 10, 11, and 12, in one line.
EXAMPLE.-Multiply 541 by 12.                          41
12 times 1 is 12; set down 2 and carry 1. 12          1
times 4 is 48, and 1 makes 49; set down 9 and carry
4.  12 times 5 is 60, and 4 makes 64. Set it down.   6492
Answer, 6492.
75. When the multiplier is over 12, multiply by
its figures separately.
ExAMPLE.-Multiply 287 by 156.
We can not multiply by 156 at once. H- ence we
first multiply by the 6 units; then by the 5 tens,
or 50; then by the 1 hundred. Thus we get three
Partial Products, as they  are called; and, adding
these, we get the whole product.
74. How are we to multiply by 10, 11, and 12? Give an example.-75.
When the multiplier is over 12, how are we to proceed? What are the results obtained by multiplying by each figure called?




RULE.-PROOF.                       43
Multiplicand   287
Multiplier      156
Partial    1722  =287 x 
t1435   =287 x 50
Products  287     = 287 x 100
Product      44772  = 287 x 156
76. Gevneral Rule.
1. Set the multiplier under the multiplicand.
units under units, tens under tens, &c.
2. Beginning at the right, multiply by each figure
of the multiplier in turn, setting the frst figure of
each partial oproduct in  a columnn with the figure
used in multiplying.
3. Add the partial products, and their sum  will
be the whole product.
Proof.
77. Multiply the multiplier by the multiplicand.
If this product is the same as the former               156
one, the work is right.                                 287
EXAMPLE. —Prove the example given above.           1092
Multiply the multiplier 156, by the multiplicand    1248
287. The product is 44772, which is the same as    312
the former product.  Hence the work is right.        44772
78. When two numbers are to be multiplied together, it is usual to take the one with the fewer
figures for the multiplier.
Multiply 287 by 156.-76. Recite the general rule for multiplication.77. How is multiplication proved? Prove the example last given. —8. Which
of two numbers is it usual to take for the multiplier?




44                   MULTIPLICATION.
EXAMPLES FOR THE  SLATE.
Find the value of the following.  Prove each example.
1. 356x11.                    5. 1678x948.
2. 289 x 12.                  6. 3548x 751.
3. 165 x 234.                 7. 2674 x 863.
4. 329x576.                   8. 9463 x 594.
9. How much is 42 times 6257?              Ans. 262794.
10. How much is 53 times 8164?              Ans. 432692.
11. Multiply 4567031 by 147.             Ans. 671353557.
12. Multiply 4905604 by 263.            Ans. 1290173852.
13. Multiply 65728913 by 5674.       Ans. 372945852362.
14. Multiply 8549412 by 32895.       Ans. 281232907740.
15. Multiply 5076398 by 9168.         Ans. 46540416864.
16. Multiply 765987 by 7896.           Ans. 6048233352.
17. Multiply 542896 by 6892.           Ans. 3741639232.
18. Multiply 99887 by 2673.              Ans. 266997951.
19. Find the product of six thousand four hundred and
twelve, and seventy-five thousand eight hundred and thirtynine.                                       Ans. 486279668.
20. Multiply eighty thousand four hundred and sixtyseven, by nine hundred and seventy-six.      Ans. 78535792.
21. In a certain orchard, there are 15 rows of trees, and
19 trees in each row.  Iow many apples will the whole
orchard produce, if the average yield of each tree is 948
apples?                                  Ans. 270180 apples.
0 in  tae  munltiplier.
79. When  0 occurs in  the multiqplier, bring it
down, and go on multiplying by the next figure, all
in the same line.
79. When 0 occurs in the multiplier, how are we to proceed?




NAUG GHT IN THE MULTIPLIER.               45
EXAMPLE.-Multiply 24T3 by 5008.
2473
Beginning at the right, multiply by 8. Then       5008
bring down the first 0 of the multiplier in the
tens' place, and the second 0 in the hundreds'      19784:
place. Then multiply by 5, placing the product  1236500
in the same line. Finally, add the partial prod-  12384784
ucts. Ans., 12384784.
EXAMPLES FOR  THE  SLATE.
1. Multiply 46893 by 40308.           Ans. 1890163044.
2. Multiply 962734 by 700906.       Ans. 674786037004.
3. Multiply eighty thousand and twenty-nine by five thousand and seven.                            Ans. 400705203.
4. The Morris and Essex Canal, which is 101 miles in
length, cost on an average $30693 a mile. What was the
whole cost?                                 Ans. $3099993.
5. If 19008 pounds of hay are required for the horses of a
cavalry regiment one day, how many pounds will be needed
for 206 days?
6. What would be the cost of constructing 309 miles of
plank road, at $3975 a mile?
7. How many apples will an orchard containing 208 trees
produce, if the average yield is 1269 apples for each tree?
8. In 3 editions of 750 books each, how many pages, if
each book contains 407 pages?           Ans. 915750 pages.
First find how many books there are, then how many pages.
Naughts at the right.
80. When there are naughts at the right of either
factor or both, multiply the, other figures, and annex
to their product as many naughts as are at the right
of both factors.
Solve the example given above.-SO. When there are naughts at the right
of either factor or both, how are we to proceed?




46                   MULTIPLICATION.
EXAMPLE. —MIultiply 450000 by 370              45.0000
Cutting off the naughts, multiply 45 by        3710
37. Then, as there are four naughts at the       315
right of the multiplicand and one at the    135
right of the multiplier, annex five naughts
to the product.  Answer, 166500000.            166500000
81. Multiplying a number by 1 does not affect its
value.  Hence, to multiply by 10, 100, 1000, &c.,
sim2ply annex as many naughts as are in the multiplier.
37 x 10=370    37 x 100=3700    37 x 1000=37000
EXAMPLES FOE THE  SLATE.
Find the value of the following:1. 6706x10.                 6. 8000 x 700.
2. 89271x100.               7. 290000x 98.
3. 50860 x 120.             8. 568 x 11000.
4. 7800 x4300.              9. 74600000 x 56.
5. 867 x10000.             10. 6060x 700.
11. If there are 100 cents in one dollar, how many cents
are there in $60400?                    Ans. 6040000 cents.
12. How many men were there in 11 Roman legions, if
there were 4500 men in one legion?
13. 100 pounds make a hundred-weight.  How many
pounds in a ton, which contains 20 hundred-weight?
14. The Roman soldiers on a march carried 60 pounds'
weight apiece. How many pounds did 3000 soldiers carry?
15. What will five hundred acres of land cost, at thirty
dollars an acre?                               Ans. $15000.
16. What will 1000 acres cost, at $20 an acre?
Multiply 450000 by 370.-81. What is the effect of multiplying a number
by 1  To multiply by 10, 100, 1000, &c., what are we to do?




MULTIPLYING BY FACTORS.                 47
tlIltiplying by factor's.
82. When the multiplier is itself a product, we
may multiply either by the whole, or by its factors
in turn.  The result will be the same.
EXAMrPLE.-Multiply 84 by 36.
36=6x6           or, 9x4         or, 12x 3.
84           84            84             84
36             6             9            12
504          504           756          1008
252              6             4              3
3024         3024          3024           3024
Whether we multiply by 36 at once or by the different
sets of factors that produce it, we get the same product,3q24. Multiplying by factors, therefore, proves whether the
product first obtained is right.
EXAMPLES FOR THE SLATE.
In these examples, first multiply by the multiplier at once.
Then prove the result by multiplying by its factors.
1. 875 x 81 (9 x 9).               4. 9000 x 27.
2. 5761 x 44 (11 x 4).             5. 18274 x 24.
3. 436 x 72 (6 x 12 or 8 x 9).     6. 45000 x 20.
7. How many ounces in 253 pounds of sugar, if there are
16 ounces in one pound?
8. HIow many bricks are there in 18 loads, if each load
contains 1250 bricks?
9. What will 28 horses cost, at $75 apiece?
10. How many pounds in 32 firkins of butter, allowing
56 pounds to the firkin?
82. When the multiplier is itself a product, what two modes of proceeding are there? Illustrate these, with the given example.




48                   MULTIPLICATION.
MISCELLANEOUS EXAMPLES.
1. The sun is 95000000 miles from the earth; the moon
is 240000 miles. How much farther from us is the sun than
the moon?                              Ans. 94760000 miles.
2. A man worth $10000 gains $1000 by one sale, and loses
$500 by another.  What is he then worth?    Ans. $10500.
3. If the railroad from Albany to Buffalo, 326 miles in
length, cost $25649 a mile, what was the whole cost?
Ans, $8361574.
4. How far will a locomotive travel in 7 days of 24 hours
each, if it goes 30 miles an hour?         Ans. 5040 miles.
5. A man left each of his 3 daughters 510 acres of land,
and his wife 100 acres more than all his daughters together.
What was the wife's portion?               Ans. 1630 acres.
6. A farmer raised 1570 bushels of potatoes, and bought
730 bushels more. After selling 4 lots of 500 bushels each,
how many bushels had he on hand?         Ans. 300 bushels.
7. How many hours in 365 days of 24 hours? Ans. 8760.
8. The earth moves in its orbit 68000 miles an hour; how
far will it move in 365 days?         Ans. 595680000 miles.
9. The President's salary is $25000. If he spends $19500
a year, how much will he save during his 4 years' term?
How much will he save in 1 year? How much, then, in 4 years?
Ans. $22000.
10. A has 1060 sheep; B has 849; C has 1276. How
many must C buy, to have as many as A and B together?
Find how many A and B have. Then find the difference between this
number and C's.
Ans. 633 sheep.
11. -Daniel Webster was born in 1782 and died in 1852.
How old was he?
12. At $3 a yard, what will 101 pieces of cloth cost, if
each piece contains 40 yards?                  Ans. $12120.




MISCELLANEOUS EXAMPLES.                  49
13. If a ship sails east from port 350 miles a day for 4
days, and is then driven west 289 miles, how far is she from
port?                                     Ans. 1111 miles.
14. John has 31 marbles; James has 6 times as many;
Henry has as many as John and James together. How many
marbles have they all?                  Ans. 434 marbles.
15. What cost 100 wagons; at $75 apiece?
16. Find the product of 8089 and 9007. Subtract this
product increased by 377, from 100000000. Ans. 271420000.
17. Trinity Church, N. Y., is 284 feet high. St. Peter's at
Rome is 166 feet higher; what is its height?
18. A man who has $1201 in the bank, draws out enough
to pay for 5 lots of land at $195 a lot. How much is left in
the bank?                                      Ans. $226.
19. North America contains 8000000 square miles; South
America, 7000000; Asia, 1000000 more than both Americas
together. How many square miles in Asia? Ans. 16000000.
20. An army of 12100 men lost 631 in killed and wounded,
and twice that number taken prisoners.  How many were
left?                                    Ans. 10207 men.
21. In a school of 127 boys, four times as many study
Arithmetic as study Latin.  The Latin class numbers 28-;
how many study Arithmetic?                  Ans. 112 boys.
22. What is the cost of a carriage and span of horses, if
each horse cost $235, and the carriage $679?    Ans. $1149.
23. A boat makes 208 trips in a season, and carries on an
average 106 passengers each trip; how many does she carry
in all?                              Ans. 22048 passengers.
24. Find the cost of 53 mules, at $100 apiece.
25. A grocer who has 15 firkins of butter, containing 56
pounds each, sells four of his customers 240 pounds. How
much has he left?                         Ans. 600 pounds.
5




50                          DIVISION.
DIVISION.
83. If 2  apples can  be bought for 1  cent, how
"many cents will 6 apples cost'
2 apples cost 1 cent; hence 6 apples will cost as
many cents as 2 is contained times in 6.   Here we
are required to find how  many times 2 is contained
in 6.  This process is called Division.
84. Division is the process of finding how many
times one number is contained in  another.
DIVISION TABLE.
Any number is contained in 0, 0 times.
1 in 1, once.         2 in  2, once.         3 in  3, once.
1 in 2, twice..       2 in' 4, twice.        3 in  6, twice.
I in 3, 3 times.    2 in  6, 3 times.    3 in  9, 3 times.
1 in 4, 4  times.    2 in  8, 4 times.    3 in 12, 4 times.
I in 5, 5 times.    2 in 10, 5 times.    3 in 15, 5 times.
1 in 6, 6 times.    2 in 12, 6 times.    3 in 18, 6 times.
1 in 7, 7 times.    2 in 14, 7 times.    3 in 21, 7 times.
1 in 8, 8 times.    2 in 16, 8 times.    3 in 24, 8 times.
I in 9, 9 times.    2 in 18, 9 times.    3 in 27, 9 times.
4 in  4, once.        5 in  5, once.         6 in  6, once.
4 in  8, twice.       5 in 10, twice.        6 in 12, twice.
4 in 12, 3 times.    5 in 15, 3 times.    6 in 18, 3 times.
4 in 16, 4 times.    5 in 20, 4 times.    6 in 24, 4 times.
4 in 20, 5 times.    5 in 25, 5 times.    6 in 30, 5 times.
4 in 24, 6 times.    5 in 30, 6 times.    6 in 36, 6 times.
4 in 28, 7 times.    5 in 35, 7 times.    6 in 42, 7 times.
4 in 32, 8 times.    5 in 40, 8 times.    6 in 48, 8 times.
4 in 36, 9 times.    5 in 45, 9 times.    6 in 54, 9 times.
83. Repeat the example. What are we here required to do? What is'this process called?-84. What is Division? How many times is any number contained in 0? Recite the Table.




DIVISION.                             51
7 in  7, once.         8 in  8, once.         9 in  9, once.
7 in 14, twice.        8 in 16, twice.        9 in 18, twice.
7 in 21, 3 times.    8 in 24, 3. times.    9 in 27, 3 times.
7 in 28, 4 times.    8 in 32, 4 times.    9 in 36, 4 times.
7 in 35, 5 times.    8 in 40, 5 times.    9 in 45, 5 times.
7 in 42, 6 times.    8 in 48, 6 times.    9 in 54, 6 times.
7 in 49, 7 times.    8 in 56, 7 times.    9 in 63, 7 times.
7 in 56, 8 times.    8 in 64, 8 times.    9 in 72, 8 times.
7 in 63, 9 times.    8 in 72, 9 times.    9 in 81, 9 times.
10 in 10, once.        11 in 11, once.        12 in  12, once.
10 in 20, twice.       11 in 22, twice.       12 in  24, twice.
10 in 30, 3 times.   11 in 33, 3 times.  12 in  36, 3 times.
10 in 40, 4 times.   11 in 44, 4 times.  12 in  48, 4 times.
10 in 50, 5 times.   11 in 55, 5 times.  12 in  60, 5 times.
10 in 60, 6 times.   11 in 66, 6 times.  12 in  72, 6 times.
10 in 70, 7 times.   11 in 77, 7 times.  12 in  84, 7 times.
10 in 80, 8 times.   11 in 88, 8 times.  12 in  96, 8 times.
10 in 90, 9 times.   11 in 99, 9 times.  12 in 108, 9 times.
85. The number to be divided, is called the Dividend; that by which we are to divide, the Divisor.
The  result, or number obtained  by  dividing, is
called the Quotient. It shows how many times the
divisor is contained in the dividend.
2 is contained in 6, 3 times; 2 is the divisor, 6 the dividend, 3 the quotient.
83, The  divisor is not always contained an exact
number of times in the dividend. It may go a certain number of times, and some over. What is left
over is called the Remainder.
5 is contained in 15 exactly 3 times. In 16 it goes 3 times,
and 1 over; 3 is the quotient, 1 the remainder.
85. What is the number to be divided called? What is that by which
we are to divide called? What is the result obtained by dividing called?
What does the quotient show? Illustrate these definitions.-S6. What is
meant by the Remainder? Give an example.




52                       DIVISION.
87. Division is denoted by a short horizontal line
between two dots -.  This sign indicates that the
number before it is to' be divided by the one after
it.  6-2 is read, and denotes, six divided by two.
MENTAL EXERCISES.
How many times is 3 contained in 9? 10 in 40?'T in 56?
11 in 99? 4 in 32?  9 in 81?  6 in42? 12 in 36? 2 in 18
11 in 55   5 in 30?  12 in 96?  8in 72? 7inO?
How many times is 4 contained in 23? (Ans. 5 times, and
3 over.)  2 in 5?  6 in 27?  9 in 20?  8in39? 3 in10? 4
in 29? 5 in 38?  10 in 44?  6 in 59? 12 in 75?
What is the quotient, and what the remainder, in the following?  34-4.  31-6. 22-8.8.   108-12.  37-5. 13-3.
65-.9. 0-4.  89-10. 8-5.  43-8. 72-9. 47-11. 72
8. 24-3. 24-8. 64-12. 28-10. 81- 11.
1. If 64 cherries are divided into 8 equal piles, how many
will there be in each pile?
MODEL.-As many cherries as 8 is contained times in 64,
or 8. Answer, 8 cherries.
2. Twenty-one cents are distributed equally among 7 beggars. What is the share of each?
3. How many coats, at $5 apiece, can I buy for $20?
4. John's father gave him 18 pears, to be divided equally
among his two sisters and himself. How many did each get?
5. How many bushels are there in 16 pecks, there being
4 pecks in 1 bushel?
6. Twelve make a dozen.  How many dozen in 24?
7. Allowing 8 yards of calico for a dress, how many
dresses can be made out of 40 yards?
87. How is division denoted? What does this sign indicate 




MENTAL EXERCISES.                    53
8. If a dozen pineapples are sold for 96 cents, how much
is that apiece?
9. How many stage-coaches, carrying 9 persons each, will
be needed to carry 45 passengers?
10. How many ten-ounce bottles will it take to hold 40
ounces of alcohol?
11. There are 49 days in 7 weeks. How many days are
there in 1 week?
12. If a family use 42 quarts of milk in a week, how much
do they use in a day?
13. If 11 sheep yield 33 pounds of wool, what is the average yield for each sheep?
14. How many caps, at $2 each, can you buy for $12?
15. If I go 50 miles in 10 hours, what is my rate per hour?
16. How many pair will 16 chickens make?
17. If 48 trees are planted in 6 equal rows, how many
trees will there be in a row?
18. Eighty-four eggs make how many dozen?
19. Twelve yards come in a piece of ribbon. How many
pieces will 60 yards of ribbon make?
EXAMIPLES FOR THE SLATE.
88. Divide 27036 by 3.
Here the divisor is a single figure.  Set it at the
left of the dividend, with a curved line           )   0
3) 27036
between.
In dividing, always begin at the left.  See how
many times the divisor is contained in each figure
of the dividend, and  set the quotient under the
figure divided.
88. When the divisor is a single figure, where is it set? Where must we
always begin in dividing? How do we find the quotient?




54                       DIVISION.
3 is not contained in 2.  See, then, how    3)27036
often it will go into 27, the first two figures.       9012
3 in 27, 9 times. Set down 9 under 7, the righthand figure of the two divided.
3 in 0, 0 times.  Set it down, and remember that 0 must
never be omitted in the quotient, unless it is the first figure.
3 in 3, once; set down 1. 3 in 6, twice; set down 2.
Answer, 9012.
89. In the following examples, divide as above: —
(1)            (2)             (3)           (4)
2) 148264      3)219063        4) 28084      5) 40500
5. Divide thirty-six hundred and six, by six.
6. Divide fifty-six million by 7: by 8.
Carry ilig.
90. When, after having divided  all the figures
of the dividend, there is a remainder, set it down
as such.   But if before this a remainder occurs,
prefix it (in the mind) to the next figure of the
dividend, and continue the division.
This prefixing is called Carrying.
EXAMPLE.-Divide 265231 by 6.
21 3
6) 265231
44205 and 1 remainder.
6 is not contained in 2.  6 in 26, 4 times and 2. over; set
down 4 under the 6, and carry 2. 6 in 25, 4 times and 1 over;
set down 4 and carry 1. 6 in 12, twice; set down 2. 6 in 3,
0 times and 3 over; set down 0 and carry 3. 6 in 31, 5 times
and 1 over; set down 5 in the quotient, and 1 as remainder.
Answer, 44205 and 1 remainder.
Go through these steps in the given example.-90. What is meant by
Carrying in division? When is the remainder set down? Show how we
carry, in the given example.




CARRYING.-PROOF.                        55
91. RULE  FOR  CARRYING.-If, in  dividing  any
figure of the dividend except the last, a remainder
occurs, prefix it (in the mind) to the nextfgure to be
divided, and continue the division.
92. When the divisor is greater than the figure
to be divided, set 0 in the quotient, and carry the
latter figure.
Proof off BDBiNIOan.
93. Multiply the quotient by the divisor, and
add in the remainder, if there is one.  If the result
equals the dividend, the work is right.
EXAMPLE.-Prove the example given on the             44205
last page.                                                   6
Multiply the quotient 44205, by the divisor        265230
6.  Add in the remainder, 1.  The result is 
265231, which is the same as the dividend.
Hence the work is right.                              265231
EXAMPLES FOR THE  SLATE.
Find the quotient and remainder.  Prove each example.
1. 50736-.2.  Ans. 25368.    6. 117360194-9. Rem. 5.
2. 271216-.8. Ans. 33902.   7. 219007637-.4.  Rem. 1.
3. 124555806-3.                 8. 470169628-8.  Remn. 4.
4. 2501024085 -5.               9. 401013116-6.  Rem. 2.
5. 5068595742- 7.             10. 303913211-5.  Rern. 1.
11. Divide one million four hundred thousand six hundred
and twelve, by 9.                         Ans. 155623, 5 rem.
12. The dividend is fifteen million; the divisor is seven;
find the quotient.                                     Rem. 1.
91. Give the rule for carrying in division.-92. What must be done when
the divisor is greater than the figure to be divided?-93. How is division
proved? Prove the example just given.




56                        DIVISION.
13. I-ow often is 3 times 2 contained in one hundred and
forty million and four?                 Ans. 23333334 times.
14. If it costs $17068 to build four miles of plank road,
what will one mile cost?
15. A man leaves $317520 to be divided equally between
his two children.  What is the share of each?
16. John gathers 761 nuts, and Jacob 843. If they share
them equally, how many will each have?          Ans. 802 nuts.
17. How many six-acre fields can be laid out in a plantation of 1488 acres?
94. The mode of dividing shown above is called
Short Division.  In  Short Division, the carrying is
done in the mind, and the quotient is written under
the dividend.
]ividimng  by two  or 1sore  iiAres.
95. Divide by 10, 11, and 12, by short division.
EXAMPLE.-Divide 1084608 by 12.
12 is not contained in 1, or in 10. Hence  12) 846
we take the first three figures.  12 in 108,
9 times; set down 9 under 8, the right-hand            90384
figure of those divided.
12 in 4, 0 times; set down 0.  12 in 46, 3 times and 10
over; set down 3 and carry 10.  12 in 100, 8 times and 4
over; set down 8 and carry 4.  12 in 48, 4 times; set down
4. Answer, 90384.
Find the value of the following:1. 8314607-10. Remn..   4. 1109964-12. Ans. 92497.
2. 9923166- 11.  Rem. 0.  5. 1018193 -11. Ans. 92563.
3. 1073285-10.  Rem. 5.  6. 1198872.12. Ans. 99906.
94. What is this mode of dividing called? In Short Division, how is the
carrying done? Where is the quotient written? 95. How are we to divide
by 10, 11, 12? Give an example.




LONG DIVISION.                     57
98. When the divisor is over 12, proceed by what
is called Long Division.
97. In Short Division, we set the quotient under
the dividend.  In Long Division, we place it at the
right, with.a curved line between.
In Short Division, we find what is to be carried,
and prefix it to the next figure, in the mind.  In
Long Division, we multiply, subtract, and prefix as
before; but we write down all the figures used.
EXAMPLE.-Divide 9594 by 39.
39 \) 9594(26        The quotient is now to be set at the
39x2=78            right of the dividend.  Beginning at the
-__-     left of the dividend, take as many figures
179          as are required to contain the divisor-in
39x4=156           this case, two. We find on trial that 39
234         is contained in 95 twice.  Set 2 in the
39x 6=  23:4        quotient as the first figure.
~ —~ Multiply the divisor by this 2. Twice
39 is 78; set the product under 95, and subtract. The remainder is 17, which (as in short division) we prefix, by
bringing down 9, the next figure of the dividend.
Now repeat the same steps. Find how often 39 is contained in 179. It goes 4 times. Set 4 in the quotient; multiply the divisor by it; set the product under 179, and subtract. The remainder is 23, to which bring down 4, the next
figure of the dividend.
Find how often 39 is contained in 234. It goes 6 times.
Set 6 in the quotient; multiply the divisor by it; set the
product under 234, and subtract. There is no remainder.
As we have now brought down all the figures of the dividend,
the sum is finished. Answer, 246.
95, 179, and 234, are called Partial Dividends.
96. What process do we use when the divisor is over 12?-97. Show how
Long Division differs from Short Division. Divide 9594 by 39, explaining the
several steps.
3*




58                        DIVISION.
98. We may not always, on the first trial, get
the right quotient figure.
If, when we multiply the divisor by any quotient
figure, the product comes greater than the partial
dividend, the quotient figure is too great, and must
be diminished.
If, on  the  other hand, we  have  a  remainder
greater than the divisor, the quotient figure is too
small, and must be increased.
39)9594(3           EXAMPLES.-In the last example, if we
17T          say 39 is contained 3 times in 95, we get a
product greater than the partial dividend,
and can not subtract.  We must therefore diminish the quotient figure.
If we say it is contained once, on multi-   39) 9594(1
plying and subtracting we get a remainder            39
greater than the divisor. We must therefore          56
increase the quotient figure.
99. For every  figure  of the  dividend  brought
down, a figure. must be placed in the quotient.  To
prevent mistakes, it is well to place a dot under
each figure as it is brought down.
100. When  the  divisor is not contained in the
partial dividend, set 0 in the quotient, and bring
down the next figure of the dividend.
If several figures are brought down before the
divisor will go into the partial dividend, set a 0 in
the quotient for each.
What are the numbers formed each time after a figure is brought down
called?-98. In what are we liable to make mistakes? When may we know
that the quotient figure is too great? When, that it is too small? Give
examples.-99. To prevent mistakes in bringing down the figures, what is
recommended?-100. When the divisor is not contained in the partial divi.
dend, what must be done?




EXAMPLES FOR PRACTICE.                   59
EXAMPLE.-Divide 172602 by 86.
86) 12602 (2007            Here our first remainder is 0.
172                Bring down 6. 86 in 6, 0 times; set
602            0 in the quotient, and bring down
602            the next figure.  86 in 60, 0 times;
set another 0 in the quotient, and
bring down 2, the next figure.  86 in 602, 7 times.  Set 7
in the quotient; multiply and subtract. Answer, 2007.
EXAMPLES FOR THE SLATE.
Prove each example.  In the first nine, if your work is
right, the remainder will come equal to the quotient.
1. Divide 6765 by 122.      10. 259828-34.  Ans. 7642.
2. Divide 80157 by 346.    11. 70338-57.   Ans. 1234.
3. Divide 34075 by 234.    12. 539902  23.    BRem. 0.
4. Divide 159750 by 425.   13. 470112 —354. Ans. 1328.
5. Divide 269459 by 576.   14. 16199569 -1848. Remn. 1.
6. Divide 39298 by 801.    15. 14003794-6974. Remr. 2.
7. Divide 466281 by 926.   16. 13373585- 2643. Rem. 5.
8. Divide 854080 by 1087.  17. 1238025  1536.  Rem. 9.
9. Divide 3054436 by 2953. 18. 789783-.19263..   Rem. 0.
19. Divide 24280295 by 9684.                   Rem. 2507.
20. Divide 1521808704 by 234.               Ans. 6503456.
21. Divide 5763447 by 678509.                Rem. 335375.
22. Divide 24280830966 by 604.                  Rem. 162.
23. Divide 48423768038 by 807009.                  Rem. 2.
24. Divide 22687013 by 310781.                    Ans. 73.
25. Divide 78204820 by 8679548.               Rem. 88888.
26. Divide 36132610 by 8603.                     Remn. 10.
27. Divide 1651877824 by 6503456.                Ans. 2-54.
28. Divide 7807085 by 945.                       2ern. 440.
29. Divide 4848708 by 808.                      Remn. 708.
30. Divide 1917183 by 9682.                     Ren. 147.




60                    DIVISION.
General Rule for Bivision.
1. Set the divisor at the left of the dividend, with
a line between.
2. Take as many figures at the left of the dividend as will contain  the divisor, and find how
many times it will go into them.
3. If the divisor is 12 or less, set this first quotient figure under the figure divided, or under the
right-hand figure of those divided, if more than one
are taken.  Divide each figure of the dividend in
turn, carrying what is over, and setting each quotient figure under the figure divided.
4. Jf the divisor is'over 12, set the first quotient
figure at the right of the dividend.  ]Multiply the
divisor by it, and subtract the prfrduct from the partial dividend.
5. Bring down the next figure of the dividend.
Find the next quotient figure, multiply, and subtract, as before.  Go on thus, till all the figures of
the dividend are brought clown.
6. If eany partial dividend is too small to contain
the divisor, set 0 in the  quotient, bring down the
next figure, and go on as before.
EXAMPLES FOR THE SLATE.
1. The greatest height reached by man in a balloon is
23027 feet. How many miles is this, allowing 5280 feet to
the mile?                       Ans. 4 miles, 1907 feet.
2. The earth's circumference is about 25000 miles. How
many days would it take a person to traverse it, going at the
rate of 125 miles a day?




EXAMPLES FOR PRACTICE.                     6
3. How many barrels of apples, at $3 a barrel, can be
bought for $2568?  If one tree produces 8 barrels, how many
trees will it take to yield this quantity?     Ans. 107 trees.
4. If a ship bound from New York to Canton sails 9225
miles, and is 75 days making the voyage, how many miles a
day does she average?
5. The earth's distance from  the sun is about 95000000
miles.  It takes a sunbeam  about 8 minutes to reach the
earth; how many miles does light travel in a minute?
Ans. 11875000 miles.
6. A flour barrel holds 196 pounds of flour.  How many
barrels will it take to hold 406700 pounds?   Ans. 2075 bbl.
7. If 47 miles of railroad cost $1815375, what is the cost
per mile?                                        Ans. $38625.
8. The product of two factors is 392328687.  One of the
factors is 46391; what is the other? *             Ans. 8457.
9. Twenty-six times a certain number is 1580540.  What
is the number?                                    Ans. 60790.
10. In 1850 there were 2526 newspapers and periodicals in the United States, of which 426409978 copies were
printed yearly.  What was their average yearly circulation?                               Ans. Over 168808 copies.
11. If 218669 bushels of rye are stored in equal lots in 11
warehouses, how much is there in each?         Ans. 19879 bu.
12. I-low long will 15000 pounds of flour last a garrison
of 250 men, allowing them 750 pounds a day?  Ans. 20 days.
13. How many impressions can a printing press make in
one hour, if it makes 633240 impressions in one day of 24
hours?                                Ans. 26385 impressions.
* NOTE.-When a product and one of its factors are given, to find the
other, divide the product by the given factor.
3x 927    Then 27  3   9    Or, 27 - 9= 3
6




62                        DIVISIONo
Wsauaghts at the  righlt of the divisor.
101. TWhen, there are naughts at the rigAt of the
divisor, cut them  off, and  also as many figures at
the right of the dividend.
Divicle the remaininygfigures of the dividend by
those of the divisor.  If there is a remainder, annex
to it the figures cut off from   the dividend for the
true renmainder; if not, the figures cut of  are the
true remainder.
Divisor Dividend( Quo.    Divisor Dividend Quo.    Divisor Dividend
260 )539 (20        4790 )237/ ( 5       3,00 ) 18900fI
52                   235
~-~~~~-   ~-~                 ~63 Quo.
19                    2
Ans. 20, 190 rem.    Ains. 5, 216 rem.    Ans. 63, 1 rem.
102. Dividing a number by 1 does not change its
value.  Ience, to divide a number by 10, 100, 1000,
&c., simply cut off as many figures at the right of
the dividend  as there are naughts in  the divisor.
The remaining figures are the quotient; those cut
of, the remainder.
3500-10=350    3500.100=35    3500.1000=3, 500 rem.
EXAMPLES FOR  THE  SLATE.
1. Divide 186740 by 10.        6. 2294003-3700.  Rem. 3.
2. Divide 729500 by 100.       7. 854009-14000.  Rem. 9.
3. Divide 729500 by 1000.    8. 1782000.9900.  Rem. 0.
4. Divide 729500 by 10000.   9. 405349-.490.  Rem. 119.
5. Divide 901109 by 1000.   10. 253579-510.  Rem. 109.
101. How m-1nst we proceed when there are naughts at the right of the
divisor?-102. What is the effect of dividing a number by 1? Give the rulo
for dividing by 10, 100, 1000, &c.




EXAMPLES FOR PRACTICE.                  63
11. About three million tons of iron are produced yearly
in Great Britain, which is twenty times as much as is produced annually in Sweden.  THow  much is produced in
Sweden?                                  Ans. 150000 tons.
12. A ship bound fiom Boston to Australia, a voyage of
13000 miles, sails at the rate of 100 miles a day. How long
is she on the passage?
13. The product of two factors is 238700. One of the
factors is 770; what is the other?               Ans. 310.
14. A cotton crop of 26320 pounds was put up in bales
averaging 560 pounds each. How many bales did it make?
15. Sound moves 1120 feet in a second. How long before
we hear a cannon, fired at a distance of 12320 feet?
MISCELLANEOUS EXAMPLES FOR THE SLATE.
To find a sum, add.
To find a difference, subtract.
To find a product, multiply.
To find a quotient, divide.
1. Find the sum, then the difference, then the product,
then the quotient, of 125 and 875.
2. A person who had 200 acres in one state, 1173 in another, and 127 in a third, divided his land equally among his
wife and four sons. How much did each receive?
Find how much land he had in all. Among how many did he divide it?
3. If a clock ticks 3600 times in 1 hour, how often will it
tick in two days of 24 hours each?      Arns. 172800 times.
4. How many times is 20 x 50 contained in 50 x 20?
5. A barrel of flour contains 196 pounds. If a family of
5 persons use 4 pounds of flour a day, how long will a barrel
last them?                                  Ans. 49 days.
6. Three men put in $650 each, and invest the whole in
salt, at $2 a sack. How many sacks do they buy? Ans. 975.,




64       EXAMPLES IN ADDITION, SUBTRACTION,
7. A merchant worth $43000, after losing $1750 in trade,
invested the rest in land, at $10 an acre.  How many acres
did he buy?                                 Ans. 4125 acres.
8. What is the difference between 16 x 49 and 49 x 8 x 2?
9. A  lady who had eight children, lost two of them.
Among the survivors she divided her property; which consisted of $1200 cash, $21550 in stock, and $7250 in bonds.
What was the share of each?                      Ans. $5000.
10. The sum of three numbers is 27846.  One of the numbers is 11587; the second is 596; what is the third?
Find the sum of the two given numbers. The third will be the difference between their sum and the whole sum.
Ans. 15663.
11. A man leaves his three sons $36000. To the first he
leaves $14655; to the second, $9875; how much does he
leave the third?                                Ans. $11470.
12. A  soapboiler makes 5977 pounds of soap, of which
657 pounds are soft soap, while the rest is in bars.  He wishes
to pack his bar soap in boxes holding 70 pounds each. How
many boxes must he procure?                   Ans. 76 boxes.
13. How much must I add to 40 times 60, in order to
make 3000?
How much is 40 x 60? What is the difference between this and 3000?
14. A farmer wishes to buy some land for $720. He lays
up $9 a week for one year, or 52 weeks. How much does he
still need?                                       Ans. $252.
15. A grocer, having 3 boxes of tea holding 80 pounds
each, repacks it in six-pound boxes.  How many boxes will
it fill?                                      Ans. 40 boxes.
16. Two men go from  the same point in opposite directions, one 18 miles a day, the other 23. When they are 369
miles apart, how many days have they travelled?
How far apart are they at the end of one day? How many tinees is this
number contained in 369?




MULTIPLICATION, AND DIVISION.               65
17. A company building a railroad, pay $377235 for labor,
and $147690 for other expenses.  The road is 15 miles long;
what is the cost per mile?                    Ans. $34995.
18. A farmer has 917 sheep in one field; 189 in another;
276 in a third; and 379 in a fourth. If he divides them into
three equal lots, how many will there be in each?  Ans. 587.
19. The Vice-President's salary is $8000 a year. If he
spends $15 a day, how much will he lay up during his four
years' term, allowing 365 days to the year?    Ans. $10100.
20. Two men go from the same point in the same direction, one at the rate of 41 miles a day, and the other 24.
When they are 493 miles apart, how many days have they
travelled?                 A/ns. 29 days.
21. I buy 29 horses for $1885, and sell them at a profit of
$290. How much do I get apiece?                   Ans. $75.
22. A planter raises 25963 pounds of cotton, and buys up
35409 pounds more. If he packs the whole in bales of 458
pounds, how many bales will he have?        Ans. 134 bales.
23. Roger Bacon, the inventor of spectacles, lived to the
age of 78, and died 350 years before Sir Isaac Newton was
born. If Newton was born in 1642, what was the year of
Roger Bacon's birth?                            Ans. 1214.
24. How many hogsheads holding 1295 pounds each will
it take to contain 76405 pounds of sugar? Ans. 59 hogsheads.
25. Five partners make $1500 by a speculation. One of
them  divides his share between his two daughters. What
does each daughter get?                         Ans. $150.
26. Find the value of 637+8465367-487. Ans. 8465517.
27. How much is 986743 + 9767 + 98437?  Ans. 1094947.
28. Find the value of 64372159-8524384. Ans. 55847775.
29. Find the product of 79, 86, and 54.    Ans. 366876.
30. Divide 1824962163 by 26837548.  Ans. 68, 8899 rem.




66                        FRACTIONS.
FRACTIONS.
103. When  a whole  is divided  into
I    two  equal parts, each  of these parts is
called one Half
When a whole is divided into three
equal parts, one of these parts is called 
one Third; two are two Thirds; &c.
A    When a whole is divided into four' 1,    equal parts, one of these parts is called
one Fourth (or Quarter); two are called
two Fourths; three, three Fourths; &c.
In the same way we get Fifths, Sixths, Sevenths,
&c., by dividing  a whole into five, six, seven, &c.,
equal parts.  The narme is taken from  the number
of equal parts into which the whole is divided.
104. The value of the part varies according  to
the number of parts into which the whole is divided.
The more parts it is divided into, the smaller they
lmust be.
Half                       Half
Third,      Third       I      Third
Fourth        Frt           Fourth  Fourt Fourth
One half of a thing is greater than one third; one third is
greater than one fourth.
103. When a whole is divided into equal parts, if there are two, what is
each called? If there are three, what is each called? What are two called?
If there are four equal parts, what is each called? What are three such
parts called? How do we get Fifths, Sixths, Sevenths, &c.? From what is
the name taken?-~104. According to what does the value of the part vary?
Wrhich is greater, one half or one third? One fourth or one third?




HOW  WRrTTEN.                        67
105,  These  ecqual parts into  which  a whole is
divided, are called Fractions.
WrJ  ir'Tj f  Y rPactfoxB
106. Learn how fractions are written.
One half!  Five thirteenths 
One third!  Three twenty-seconds        232
One fourth (quarter)         T Twenty sixty-firsts        I
One tenth               A-   Nine three-hundredths  3 - o
One two-hundredth i -h   Three thousandths            1 o3 owr'One thousandth      -   o  Six twelve-hundredths T1 2107. A  fraction, therefore, when written, consists
of two numbers, one below  the other, with  a line
between.
The number below the line is called the Denominator.   It shows into  how  many  equal parts the
whole is divided, and therefore gives name to these
parts.
The number above the line is called the laumerator.  It shows how many of the equal parts denoted
by the Denominator are taken.
The Numerator and the Denominator, taken together, are called the Terms of the fraction.
- is a fraction.  5 and 6 are its Terms.  6 is the Denominator, and shows that the whole is divided into six equal
105. What are such equal parts of a whole called?-106. Write one half;
one third; one quarter; &c. —107. Of what does a written fraction consist?
What is the number below the line called? What does the Denominator
show? AWhat is the number above the line called? What does the Ntmerator show? What are Numerator and Denominator, taken together, called?
Name the terms of tha fraction five sixths. Name its numerator; its denominator.




68                      FRACTIONS.
parts, making each part one sixth.  5 is the Numerator, and
shows thatfive of these equal parts are taken.
In reading, name the numerator first, —five sixths. Always
pronounce th distinctly, in naming the part denoted by the
denominator,-sizTr.
E XE P CI S E.
Read these fractions. Then name the numerator and the
denominator, and tell what each shows.
4    _   8.  5. 2_7_.    1 9. 5 0_   1. 0_19
9   1 2 )  9    1I T  0    20?  02    010       1000 "
Write the following fractions in figures:1. Seven sixteenths.         8. Five seventy-seconds.
2. Nine tenths.              9. Twenty-one ninetieths.
3. Two five-hundredths.    10. Sixty ten-thousandths.
4. Eleven billionths.       11. Sixteen twelve-hundred5. Eighty hundredths.            and-ninety-firsts.
6. Three millionths.        12. Five hundred four-hunT. One thousandth.              dred-thousandths.
De ltfinlftEola.
108. A  Fraction is one or more of the equal parts
into which a whole is divided; as, 1, 
109. A  Proper Fraction is one whose numerator
is less than its denominator; as, 2,   9.
110. An Improper Fraction is one whose numerator is equal to  or greater than  its denominator;
3 24
as,, 92  
111. A   ~iixed lNumber is one that consists of a
whole number and a fraction; as,'    (seven  and
a hayf).
In reading, -which of the terms must be named first?-108. What is a
Fraction?-109. What is a Proper Fraction — 110. What is, an Improper
F action?-lll. What is a Miixed Number?




TAKING'R&ACTIONAL PARTS.                     69
Frae tional Parts of Whole Numbers.
112. A  fraction indicates division:  The fractional
line is the line used in the sign of division -.   The
dividend  is written  in  place  of the  dot above the
line;  the  divisor, in  place  of the  dot below   it.
Hence,
To find ~, divide by 2.  To find, divide by 4.
To find, divide by 3.  To find ~, divide by 5.
And generally, To find  one of  the eualt parts denoted by a denominator, divide by the denominator.
MENTAL  EXERCISES.
1. How  much is - of 14?  Of 8?  Of 18?  Of 2?
2. How  much is I of 20?  Of 45?  Of 10?  Of 50?
3. Find - of 21.  Of 56.  Of 63.  Of 14.  Of 35.
4. Find v of 81.  Of 18.  Of 63.  Of 27.  Of 54.
5.  - of 80?  Of 20?  Of 100?  Of 70?  Of 30?
6. -'-  of T2?  Of 48?  Of 24?  Of 12?  Of 84?
7. - of 16?  Of 80?  Of 48?  Of 24?  Of 72?
S. i — of 22?  Of 66?  Of 99?  Of 33?  Of 55?
9.   of 15?       of 54?   - of 28?  -L  of 36?  6 of 36?
10. I of 24?    of 48?  3 of 27?  3 of 18?  -' of 11?
11. I of 30?    of 9?     of 16?     of 24?     of 12?
12. l-' of 50?  lr  of 500?  — L'  of 4000?  — l  of 640.
13.    of 70?, of 6?  1 of 32?  } of 15?        of 3?
14. fof 42?' of 20?  -  of 77?        of 56?   L of 9?
112. What does a fraction indicate? With what does the fractional line
correspond? Where is the dividend written, and where the divisor? RHow
can we find one half? One third? Give the rule for finding one of the
equal parts denoted by a denominator




0                       FRACTIONS.
113. To find  more than  one of the equal parts
into which a whole is divided, first find one in the
way just shown; then multiply this by the number
of parts required.
EXAMPLES. —. How much is - of 40 
One fifth of 40 is 8; and three fifths are 3 times 8, or 24,
Answer, 24.
2. A boy having 27 cents earned - as much more. How
many cents did he earn?
He earned 4 of 27 cents. One ninth of 27 cents is 3 cents;
and four ninths are 4 times 3 cents, or 12 cents. Answer,
12 cents.
MENTAL EXERCISES.
I. How much is ~ of 18?  3 of 21?     of 108?  -5 of
108s?  -  of 108?   -o2 f           42? X o f 4?      of 36?    of 20 
2. How much is —. of 22?  A- of 44?  ~ of 72?  ~ of
40?  3 of 48?  - of 30? a    of 66? a    of 50?  % of 64?
3. How much is -3y of 200?   of 27?            4 of 24?    of
15?    of 18?  -2 of 81?  L of 60?       of 55?  - of 72?
4. A man having 48 cows sold 3 of them. How many did
he sell?
5. At 81 cents a pound, what will g of a pound of tea cost?
6. A boy having 20 cents gave away one tenth of them and
spent two tenths. How many cents had he left?  Ans. 14c.
EXAMPLES FOR THE SLATE.
1. Find -- of 280.  Ans. 60. 5. Find  f! of 8410. Ans. 348.
2. Find 1  of 441.  Ans. 99. 6. Find -l  of 8832. Ans. 128.
3. Find   of 1610. Ans. 1242. 7. Find L — of 7400. Ans. 275.
4. Find 4 of 4200. Ans. 3010.  8. Find 7W6  of 6100000.
113. How can we find more than one of the equal parts into which a
whole is divided?




REDUCING TO HALVES, ETC.                 71
114. A  fraction  indicates division.  Hence, in
division, when there is a remainder, in  stead  of
writing it as such beside the quotient, it is usual to
place it over the divisor in the form  of a fraction.
EXAMPLE.-Find - of 45.         7)45
Answer, 63.          6, and 3 rem.
115. Redlucnga whole miunm ers to halves, &c.
2 halves make I whole.   5 fifths   make 1 whole.
3 thirds make 1 whole.   6 sixths   make 1 whole.
4 fourths make 1 whole.   7 sevenths make 1 whole.
Hence,  To reduce to halves,  multply by 2.
To reduce to thirds,  nmulti2py by 3.
To reduce to fourths, multiply by 4.
To reduce to fifths,   multiply by 5, &c.
EXAMPLES FOR THE SLATE.
1. How many thirds in 2901? Ans. 8703 thirds; or, -8_.
2. If I cut 45 pies into halves, how many halves have I?
3. How many quarters of beef will 96 oxen make?
4. How many tenths of an inch in 72 inches?
5. Reduce 497 to fiftieths. Reduce 876 to twelfths.
6. A cent is one hundredth of a dollar. How many cents
are there in $75?                        Ans. 7500 cents.
7. A mill is l-oI- of a dollar. How many mills in $34?
8. Find one eleventh of 25304618.      Ans. 2300419 o.
9. Find one thousandth of 19001.          Ans. 19o —O.
10. Find one sixty-eighth of 5375.           Ans. 796.
114. What does a fraction indicate? Hence, in division, how is it usual
to write a remainder?-115. How do we reduce a whole number to halves 
To thirds? To fifths? To twentieths?




T2                       FRACTiONS..
Reductfion  of lFractions.
116. Reducing  a fraction  is  changing  its form
without changing its value.
If we divide a pie into 2 equal parts,
each part is called ~. If we divide each
half into 2 equal
parts, we get four
parts in all, and each is therefore I of
the whole.
Now two of these fourths are made from  one half, and
are therefore equal to one half.  2, therefore, may be written, —or, as it is generally expressed, reduced to ~.
Rledlaetton  to  Lowest Termns.
117. We have just seen that 2 = -. Now what operation
performed on 2 gives'?  Dividing both nu-          2    __    1
merator and denominator by 2.                     4   2 -
We have just seen that l = a. Now what operation performed on I gives?  Multiplying both nu-          x 2 _    2
merator and denominator by 2.                     2 X 2      4
The cvalue of a fraction, therefore, is not changed
by  dividing  or  multi pying  both  numerator  and
denominator by the same number.
118. A  fraction  is said to  be  in its lowest terms
when its numerator and denominator are as small
as they can be made without changing the value of
the fraction.  - is in its lowest terms; I is not.
116. What is meant by Reducing a fraction? To what may - be reduced?
How do you know that a and I are equal?-117. What operation performed
on I gives? WVhat operation performed on a gives a  By what, then, is the
value of a fraction not changed?-11,8. When is a fraction in its lowest terms?




REDUCING  TO LOWEST TERAS.                    73
EXAMPLE. —Reduce Uo to its lowest terms.
Divide both terms by 10, which will make  
them  lower, while it will not change the   10 90  -   9
value of the fraction.
Looking at the resulting fraction  w, we see that its terms
can be divided by 3, which will make them                    2
lower without changing its value.  Dividing,   31 9 
we get ].
As we can find no number greater than 1 that will exactly
divide the terms of this fraction, we know  that it is in its
lowest terms.  Answer, 3.
One division may reduce a fraction to its lowest terms, or
several may be needed.
119. RULE.-To reduce a fraction  to  its lowest
terms, divide its numerator and denominator by any
number greater than 1 that is exactly contained in
both.  Divide the result in the same way, repeating
the process till no number greater than  1 is exactly
contained in both.
EXAMPLES  FOR  THE  SLATE.
Reduce the following fractions to their lowest terms:
1. 3   8    13    9    15    8    1         6    18   21
1. 6.  O    2T 0   12*T  T7   T2   12-    5'  S 35'
2. Reduce  4.       Ans.m 2.   9. Reduce  3         Ans. 16
3. Reduce.       Ans. ~.  10. Reduce -6o.        Ans. 3o
4. Reduce ~5.       Ans..   11. Reduce 3j22.       Ans. -9-.5. Reduce.-%4.    Ans. -.  12. Reduce i:.    Ans.,
6. Reduce,-i'q.    Ans. 1. 13. Reduce -8.    Ans. I.
7. Reduce   n. Ans. 3.  14. Reduce..              Ans. a.
8. Reduce   A. Ans. 7   15. Reduce   ~.    Ans..
Reduce -9 to its lowest terms. How many divisions are needed, to reduce a fraction to its lowest terms?-119. Give the rule for reducing a biaction to its lowest terms.
7




14:                      FRACTIONS.
Reducing ifmproper F'ractioas.
120. An Improper Fraction is one whose numerator is equal to or greater than its denominator.
121. An improper fraction may be reduced to a
whole or mixed number.
EXAMPLES. —1. Reduce -Ui to a whole number.
9 ninths make 1 whole; hence in 18 ninths there are as
many wholes as 9 ninths are contained times in 18 ninths, or
2. Answer, 2.
2. Reduce 2- to a mixed number.
9 ninths make 1 whole; hence in 20 ninths there are as
many wholes as 9 ninths are contained times in 20 ninths, or
22. Answer, 23.
In both these examples, it will be seen, the numerator is
divided by the denominator.  Hence the rule.
122. RULE.-TO  reduce an  improper fraction to
a wlole or mixed number, divide the numerator by
the denomninator.
When the numerator is equal to the denominator, the
value of the fraction is 1.  - =- 1.
When the numerator is greater than the denominator, the
value of the fraction is greater than 1.  - = 1~.
When, on dividing, a remainder occurs, it must be placed
over the denominator in the form of a fiaction.  The answer
is then a mixed number, as in Example 2.
The fraction thus formed, if not already in its lowest
terms, must be reduced.
120. What is an Improper Fraction?-121. To what may an improper
fraction be reduced? Give examples.-122. Give the rule for reducing an
improper fraction to a whole or mixed number. When is the value of the
fraction 1? When is it greater than 1? When, on dividing, a remainder occurs, what must be done? If this fraction is not already in its lowest terms,
what must be done?




REDUCING IMPROPER FRACTIONS.                  75
EXAMPLE.-Reduce 40 to a whole or mixed number.
Divide the numerator              26) 40 (1       Quotient.
by the denominator.                   26            1l
14
Reduce 14 to its lowest terms.
2) 14  -  7           Answter, 1 
EXAMPLES FOE  THE  SLATE.
Reduce these fractions to whole or mixed numbers:
i6i   - _     15    1 2    7    1 6    1     8
~: *    ~  II*  - 7' *  4T  15    i6'  5'
2. Reduce -.      Ans. 13.   8. Reduce  A5-. Ans. 11-1.
3. Reduce'       Ans. As. 3-.   9. Reduce -ot.'Ans. 25 5
4. Reduce "1.   Ans. 7,.   10. Reduce -9. Ans. 20J9.
5. Reduce L-,4    Ans. 5.  11. Reduce 96O.  Ans. 37 —.
6. Reduce -l-P-.  Ans. 9-P.  12. Reduce  _0.   Ans. 863.
7. Reduce 24 0     Ans. 60.  13. Reduce 9%-.    Ans. 37.
Redauce ig lixedl lBnmibers to Siasproi per
Fractionas.
123. A  mixed  number may  be reduced  to  an
improper fraction.
Reduce 12} to an improper fraction.   12In 1 there are 4 fourths; and in 12, twelve     4
times 4 fourths, or 48 fourths.  48 fourths     - +- 1 -  49
and 1 fourth make 49 fourths.  Ans. 4 —.
Hence the following rule:-                   Ans. -4
124, ItuLE.-iTo  recduce a  mizxed  number to an
imprroper fraction, multiply the whole nnumber by
the  denominator of the fraction, addc iqn  the numerator;' and set the result over the denomz2iator.
123. To what may a mixed number be reduced? Give an example.124. What is the rule for reducing a mixed number to an improper fraction?




1TO8~ FRACTIONS.
125. We have seen, in ~ 115, how to reduce a whole number to an improper fraction.  The process is the same as that
just shown, except that there is no numerator to be added in.
A whole number may also be reduced to a fractional form
by giving it 1 for a denominator.  3 =     7 =  7.
EXAMPLES FOR THE  SLATE.
1. Reduce 81- to an improper fraction.          Ans.?.
2. Reduce 4-.                  6 Reduce 1374.
3. Reduce 20~-.                7. Reduce 29l-.
4. Reduce 39'-.                8. Reduce 47~?.
5. Reduce 251.                 9. Reduce 151.
10. Reduce 20 to a fractional form.              An/s. —.
11. Reduce 15 to ninetieths.                   Ans. 193,0
12. How many quarters in 52 yards?
13. How many twelfths in 43-1?  In 5 -9?
14. How many nineteenths in 43?  In 8 1?
15. Reduce 100-7  to an improper fraction.
16. Reduce 999-l to an improper fraction.
17. Reduce 640-v  to an improper fraction.
Reducieig to a Co1inmmoi ienomia ator.
126. Fractions are said to  have  a comnnon denominatorj when their denominators are the same.
-2 and 4 hayve a common denominator.
127. Fractions whose denominators are different
maybe reduced  to  others having  a common  denominator.
125. How may a whole number be reduced to an improper fraction? In
what other way may a whole number be reduced to a fractional form? To
what whole number is 9 equivalent?-126. When are fractions said to have
a common denominator?-127. What may be done with fractions whose denominators are different?




COIMMON DENOMINATOR.                      77
EXAMPLE.-Reduce 2, 4 and, to fractions that
have a common denominator.
The three denominators are 5, 4, and 7.  Now, the
product of three factors is the same, in whatever order they
are taken.  Hence, if we multiply each denominator by the   5 x 4 x 7 =  140 
other two, we shall get the   4 x 5 x   = 140  Codmon
same number, and this will be   7 x 5 x 4 = 140 
the common denominator.
But the value of the fractions
2 X 4  7  ~ AL6             must not be changed.  We must,
T    x  4  x 7  14  ]
5 x   _ 4 x 7   A,4o    therefore, multiply each numera4   5x 7  47     0          f tor by the same multipliers as its
S X 5 x 4     1 0 0         denominator.  H-ence the follow7 x 5 X 4     1 40)J
ing rule:128. RUtLE.-~To reduce fractions to others having
a common denominator,  multiply both terms of each
fraction by all the denominators except its owgn.
129. Whole numbers must first be reduced to a fractional
form, and mixed numbers to improper fractions.
EXAMPLES  FOR  THE  SLATE.
Reduce the following fractions to equivalent ones having
a common denominator:
1. Reduce - and -9.                           Ans. 2-1, 27
4                                434  42I
2. Reduce 2 3 and 4.                      Ans. 15  50 24
27 37     5~                         3 5  35  3 6
3. Reduce.3, 5, and i.                Ans. -1-   450.4
4. Reduce $,, and l.                  An 6s. 63 67,  4-1.,
5. Reduce 3, 1,5 and -.                Ans.  13  0  4       75
6. Reduce -1, 43 and 6.                 Ans. 2    -3,   30.
With the given example, show how fractions may be reduced to others
having a common denominator.-128. Give the rule.-129. To what must
whole numbers first be reduced? To what must mixed numbers first be
reduced?




78                       FRACTIONS.
7. Reduce 4, -, 4, and -, to fractions having a common
denominator.                         Ans.  2    4        2 0 o 
8. Reduce 4,7 9  1,  3           Arn.. T-%3, S 3~7 -1-  TU33 3   10Ans.                      3 6  5 
9. Reduce, 1-, 1.6 Ans.,,-.5 2
10. Reduce I,  and 1.                        Ans. 3 78, 3.o
l1. Reduce    A  3j 5.                 ns.      3   63  21.
R 7 17 27                      427 427 427 42
12. Reduce  3  2 5, 34            ARs2.     T7S 4,, 43.65
13. Reduce -   54 and.                 Ans. 3ans   30, 3
14. Reduce -5 and  70.                        As. -6  6-27
15. Reduce 31, ^ and 2.                   Ans. 0-b, - 3,
Addition  off ]Fractions.
130. Halves and halves, thirds and thirds, &.,
can  be  added, just as we  add  pears  and  pears,
dollars and dollars.
EXAMPLE.-sWhat is the sum  of 5 halves
and 4 halves?  Answer, 9 halves.'2   ~   - 2
Here the denominators are the same, and we simply add
the numerators, and place the sum  over the common denominator.
1. Add l, -, and  5             5. Add  5,, and -.
2. Add l, 2, and 4.             6. Add 4,,', and.
3. Add, 8, and  6.          7. Add  5,,  and _d
4. Add L  2  1 0             5  17        161
4. Add l, I, and —.            8. Add -, - -,- 7  and.
131. Halves and thirds, halves and fourths, &c.,
can not be thus directly added, any more than we
can  add  pears and  dollars.   They  are things of
different kinds.
130. Can halves and halves, thirds and thirds, &c., be added? In what
way? Give an example.-131. Can halves and thirds, halves and fourths,
&c., be added directly? Why not?




ADDITION OF FRACTIONS.                   79
EXAMPLE.-What is the sum of 5 thirds and 3 halves?
The parts being of different value, we can not put them
together, and say they make 8 halves or 8 thirds. But, if we
reduce them to parts of the same kind or value, we can then
add them.
By reducing to a common de-               5      I 1 0
nominator, we find that 5 thirds are  
equal to 10 sixths, and 3 halves to 9_              _
sixths.  10 sixths and 9 sixths make         6 
19 sixths.  "9- being an improper           o         a
fraction, we reduce it to 3}. Answer, 3-.
132. PRuLE.-1. To aCddfraCtions, when they have
a common denominator,  add their numerators, and
place the sum  over the common, denominator.
When  they have not a common denonmiator, reduce them  to fractions that have, and then proceed
as above.
lf the resulting fraction is not in its lowest terms,
reduce it.  If it is an improper fraction, reduce it
to a whole or mixed number.
2. To add mixed numbers, or fractions and whole
numbers together, find the sum  of the fractions separately, and add it to the sum  of the whole numbers.
EXAMPLE.-Add together ~, 41, 2, and 5.
Add the fractions:         i + ~ +    -  1l3.
Add the whole numbers: 4 + 5 =  9.
123
Add these two sums:'        s
9
Ans. 103
Show how we add 5 and q. —132. Give the rule for adding fractions.
Give the rule for adding mixed numbers, or fractions and whole numbers
Arply this latter rule in the given example.




80                      FRACTIONS.
EXAMPLES FOR  THE SLATE.
1. Find the sum of 4 and  -.                   Ans. 1 -
2. Find the sum  of 5 and                       A Ans. 4-.
9      9.                        4 5'
3. Find the sum  of ~  and 5.                 As. 1-.
4. Add together 2, 1j, and 8.                  Ans. 2.
5. Add together 3  and 2-.                    Ains. 2?.
6. Add together 2, -1- and'.                 A7ns. 5.
7. Add together 4, 10, and 8.               Ans. 1825.
8. Add together 7-, E-, 4, and.             Ans. 12-3.
9. Add together 3, 4,, and 1-.               Ans. 9^.
10. Add together 5,,,, and 2-.               Ans. 41.
11. A hackman earns $21 one day, $31 the next, $4 the
next, and $51 the next.  How much does he earn in all four
days?                                              Ans. $15.
12. How much land is there in 3 fields, containing 14-,
7j, and 231 acres?                            Ans. 45 acres.
13. If I buy $23 worth of paper, and $61 worth of books,
and give the storekeeper a ten-dollar bill, how much change
will I receive?                                     Ans. $1.
14. Three men, buying a meadow, put in respectively
$30-gV, $25i-, and $19g. What does the meadow cost?
15. A five-pound jar contains 32 pounds of bread and 24
pounds of cake; what does the whole weigh?
16. A peddler walks 81 miles one day, 51 the next, 10]
the next, and 12 the next; how far does he walk in all?
17. How many pecks of peaches in four baskets, containing respectively 21, 3}, 23, and 31 pecks?
18. If 5- gallons of brandy are mixed with 12  gallons of
water, how many gallons are there of the mixture?
19. A lady hires a gardener for 15 cents an hour. How
much must she pay him, if he works 6-L5 hours the first day,
75 the second, and 53 the third?             Ans. 300 cents.




SUBTRACTION OF FRACTIONS.                81
~iablraction  of Fractions.
133. The same principle applies in subtracting,
as in  adding,  fractions.   Before subtracting, the
parts must be made of the same kind or value, if
they are not already so; that is, the denominators
must be made the same.
EXAMPLES.-1. From 5 halves take 4 halves.     
Answer, I half.                   2  2
2. From 5 thirds take 3 halves.
Thirds and halves being parts of dif-        5 - _?
ferent value, we must reduce them  to           3    9
parts of the same value. Reducing to a,,,
common denominator, we find that 5 thirds   "    6    
are equal to 10 sixths, and.3 halves to 9     Ans. 6
sixths. 9 sixths from 10 sixths leave 1 sixth. Answer, -.
134. RuLE.-1.  To subtract one fraction from
another, when they have a common  denominator,
take the numerator of the subtrahend from  that of
the minuend, and place the remainder over the common denominator.
When they have not a common denominator, reduce them to fractions that hcve, and then proceed
as above.
Reduce the resulting fraction to its lowest terms,
or to a whole or mixed number, as may  be necessary.
2. Whole and mixed numbers may be reduced to
improper fractions, before subtracting.
133. What principle applies in subtracting fractions? Illustrate this,
with the examples given.-134. Recite the rule. What may be done with
whole and mixed numbers, before subtracting 




82                       FRACTIONS.
EXAMPLES FOR  THE SLAT].
1. From  6 take.             4. From  4I  take
2. From,4 take  3            5. From  i   take -.
3. Take -7- from  T.            6. Take 29 from  4 —.
7. Subtract - from  9-,                          Ans. 78. From  -3- subtract                            Ans. 41.
9. Subtract -5  from  4.                        Ans. 1-.
10. From  6 — subtract -V.                          Ans. 5.
11. Take -  from  2.            14. From  -I take --
12. Take:-3  from  5.          15. From -2- take 5
13. Take -  from  4.            16. From  -9- take 4.
17. From  1 subtract -.                             Ans. 3.
[Reduce 1 to thirds; then proceed as before.]
18. From  1 take -.  From  1 take 4.
19. From  4 subtract —.                           Ans. 33.
[Take 1 of the 4 units, and redu.ce it  4 =  3 A ZMin.
to thirds.  Then subtract the frac-.            Sub.
tion. ~ from  3 leaves 3. Ans. 34.           34 Rem.]
20. From  5 subtract -.  [5 = 41]                  Ans. 44.
21. From  17 subtract  4.                       Ans. 16-.
22. Take ya from  3. Take - from  3.  Take 2 from  2.
23. Subtract 3 from  53.                         Ans. 55.
[Take -4 from 4, and annex the result to 5.]
24. Subtract - from  2'.  From  24.  Take 2 from  64.
25. Subtract 4 from  6-.                          Ans. 61.
26. Subtract 25 from  4-.                          ns. 12.
[- is greater than -.  HIence reduce both mixed numbers to improper fractions.  The sum  then becomes,
Subtract -'1- from  -s. Now proceed as before.]
27. Subtract 1-  from  342.                      Ans. 1-'.
28. Subtract 204 from  24.                        Ans. 31-.
29. Subtract -7  from  12.                     Ans. 11.
80. Subtract I from  1ns. An. 
" 1o   I        




MULTIPLICATION OF FRACTIONS.                83
31. Subtract 14 fron 9-.                       Ans. 7T.
32. A grocer, having mixed 14- pounds of tea with 26pounds of a different kind, sold all the mixture but 11 pounds.
How much did he sell?
33. If a person owning two farms, one of 70] acres, and
the other of 120-3 acres, sells 90-,- acres, how much land has
he remaining?                               Ans. 100 acres.
34. How much paper has a printer left, if he had on hand
27 r n has u  rms r ne j n  reams, and has used  reams for one job, and 6 rea
for another?                               Ans. 13-9 reams.
Ml ltiplicationl  of lFractlons.
FRACTION X WHOLE NUMBER.
135. To multiply any number of equal parts by
2, we may either take twice as many such parts, or
make each part twice as great.  That is, we may
double the number of parts, or double their size.
ExAMPLE.-Multiply 3 by 2.
Double the?number of parts.  Twice 2 is {.
Or, double the size of the parts. A half is twice as great
as a fourth.  Hence, twice - is 1.
2 may be reduced to -. The two answers agree.
Now, in the first case, we multiplied the numerator by 2.
In the second case, we divided the denominator by 2. The
latter mode is better, because it brings the fraction at once in
its lowest terms. Hence the rule.
136. RULE.-To multiply a fraction by a whole
number, divide its denominator by the whole number, if  this can be done without a remainder; if
not, multi)ply its numerator.
135. To multiply any number of equal parts by 2, what may we do?
Illustrate these two methods, in multiplying I by 2. Which method is
better — 136. Give the rule for multiplying a fraction by a whole number.




84                      FRACTIONS.
EXAMPLES.-1. Multiply i-   by 4.
16 can be divided by 4. Divide it. Answer, 4.
2. Multiply - by 4.
17 can not be divided by 4 without a remainder.
Multiply the numerator.  Answer, -9.
137. Find the value of the following: —
1.    x 7.      Ans. 42.       7.,-   X  1 12. - x 2.       Ans. 1         8. -3 x 4.        Ans..
3. -7 x 4.      Ans. 2-.       9. Ay  x 12.       Ans. 1.
4. -3- X 5.      An?. -       10. I-    x 12.    4ns. 143.
5.                           113 X 6. 45 X 8.
6. -  x 6.                   12.  l  x 10.
MIXED NUMBER X WHOLE NUMBER.
138. RULE.-TO  mnultiply a mzixed number by a
whole number, nltiply the fractional part and the
whole part of the nixed number separately, and add
the products.
EXAMPLE.-Multiply 20   by 5.
Multiply the fraction:'   x  5 =         2 1 
Multiply the whole part:  20 x 5         100
Add the products:                        101~ Ans.
1. What will 10 clocks cost, at $5' each?       Ans. $52.
2. There are 51 yards in a rod; how many yards are
there in 40 rods?                            Ans. 220 yards.
3. If 12 acres of land produce, on an average, 35- bushels
of wheat each, what is the yield of the whole?
4. A locomotive is moving at the rate of 25J miles an
hour; how far will it go in 3 hours?         Ans. 77j miles.
5. What will 16 chairs cost, at $23 apiece?
138. Give the rule for multiplying a mixed numoer by a whole number.




MULTIPLICATION OF FRACTIONS.                85
WHOLE INUMBER X FRACTION.
139. Multiplying a number by a fraction is simply taking such a part of it as is denoted by the
fraction.
Multiplying a number by - is taking - of it, or dividing
it by 3, as shown in ~112. Multiplying by 2 is taking -
twice, or dividing by the denominator 3, and multiplying by
the numerator 2, as shown in ~ 113.
In such cases, it is best to multiply first, as there may be
a remainder on dividing. Hence the rule.
140. RULE.-To multiply a whtole?lnumber by a
fraction, multiply it by the numerator of the fraction, and divide by the denominator.
EXAMPLE.-Multiply 140 by 2.                   140
Multiply 140 by the numerator 2, and             2
divide the product by the denominator 3.  3)280
Answer, 93.                                        93} Ans.
1. Multiply 160 by -.         4. 5713 x -.   Ans. 4080.
2. Multiply 199 by -.         5. 8296 x.   Ans. 64529.
3. Multiply 200 by W.         6. 2644 x-. Ans. 2403 -.
7. A farmer owning 467 acres of land, sells 3 of it; how
many acres does he sell?                    Ans. 3501 acres.
8. How many pounds in {  of a ton of coal, a ton being
2000 pounds?                            Ans. 14661 pounds.
9. How many pounds in a of a barrel of flour, there being
196 pounds in a barrel?
10. If a hogshead holds 63 gallons, and Il of its contents
has leaked out, how much remains?           Ans. 57 gallons.
139. To what is multiplying a number by a fraction equivalent? To
what, for instance, is multiplying by 1 equivalent? By i?-140. Give the
rule for multiplying a whole number by a fraction.
8




86                      FRACTIONS.
WHOLE NUMBER X MIXED NUMIBER.
141. RULE.-TO  mdltiply a whole number by a
mixed number, nultiply it first by the fractional
part of the mixed number, then by the whole part,
and add the products.
317
EXAMPLE.-Multiply 317 by 63.                     63
Multiply 317 by 2:            2113
Multiply 317 by 6:           1902
Add the products:            2113S  Ans.
1. Multiply 2163 by 7-.                     Ans. 18472^.
2. Multiply 5698 by 6                        ns. 37606-.
3. Multiply 1275 by 39,.                     Ans. 50065.
4. Multiply 5416 by 85`.                   Ans. 4611721.
5. Multiply 8570 by 73.                    Ans. 627752g.
6. Multiply 9087 by 20-~.                Ans. 19000011
7. Multiply 2639 by 625.                    Ans. 164633.
8. Multiply 4471 by 49-7.                   Ans. 220920.
9. Multiply 6381 by 569.                    Ans. 360172.
FRACTION X FRACTION.
142. [Multiplying by a fraction, we have seen, is
equivalent to taking such a part as is denoted by
the fraction.  Multiplying  I by  I is equivalent to
taking - of ~.
143. A  fraction of a fraction, such as - of —, is
called a Compound Fraction.
The same process is used in multiplying fractions
and in reducing compound fractions to simple ones.
141. Give the rule for multiplying a whole number by a mixed number.
142. To what is multiplying by a fraction equivalent?-143. WThat is a fraclion of a fraction called? In what two operations is the same process used 




MULTIPLICATION  OF FRACTIONS.                   87
144. Multiply  - by  3.
These fractions indicate division.  The numerators are
the dividends.  The denominators are the divisors.  Multiply
the numerators together to find the total dividend, and the
denominators to find the total divisor.
Then  set the former product over the               x  — =   Ans.
latter, in the form of a fraction.
1. Multiply -- by.                                Ans.  5,.'6      81                                   UP
2. Multiply, 1, and   t together.                Ans. 1-.
3. Multiply 2, 4, and  r together.                 Ans. 3.
4. Multiply  -, -,, and - together.               Ans. x —.
5. Reduce 2 of  f of -7 to a simple fiaction.    Ans. 17.
145. CANCELLING.-Whlen  the  same number appears as a numerator and  a denominator, draw  a
line through  it, and omit it in  multiplying.  This
is called Cancelling.
Ex.-Reduce 2 of, of 4 of I to a simple fraction.
2         4          8
2 of   of 4 of    =  8-  Ants.
I    9         9 9 1
Cancel 3 and 3, 5 and 5.  Then multiply the remaining
numerators and denominators.  Observe that 9 and 9 are not
cancelled, because they are both denominators.
1. Multiply - by }.                                  Ans. {.
2. Multiply 3, and 4 together.                     Ans. -.
3. Multiply,  -- -:'-, and 3 together.             Ans. 3.
4. Reduce   of   of -f    to a simple fraction.    Ans. Io
5. Reduce - of 5- of a to a simple fraction.    Ans..
6. Reduce -5 of 2 of Io! to a simple fraction.
7. Reduce 7 of 3- of 7- of - to a simple fraction.
144. In multiplying. by ~, show how we proceed, and why.-145. When
the same number appears as a numerator and a denominator, what do we
do? What is this called? Give an example.




88                      FRACTIONS.
146. Equal factors in a numerator and a denominator may also be cancelled.
Ex.-Reduce 2 of - of 35 to a simple fraction.
Throw the whole number 35 into a fractional form.
2      5
-2   f ~ of     _  20 _   6- Ans.
3
Cancel 3 (that is, divide by 3) in the first denominator
and second numerator.  Cancel 7 (that is, divide by 7) in the
second denominator and third numerator.  Then multiply
the remaining factors.
1. Multiply!, 4, and s together.                Ans. 2-.
2. Multiply 1, 6, and 14 together.                 Ans. 4.
3. Multiply 6,,  j, and % together.              Ans.  {-.
4. Reduce - off  f  o  - of 3.                    Azs..
147. Cancelling  is dividing.                 _   1 
When, therefore, a numerator    A                     1 
or denominator disappears by cancelling, 1 (not 0)
is left in its place.
148. Cancelling is dividing.  Hence, by cancelling before
we multiply, we save the trouble of dividing after we multiply, to reduce the result to its lowest terms.
149. GENERAL RULE.-1. 0To multiZply one fraction by another, or to reduce a compou0nd fraction
first cancel factors common to camy numerator and
denominator;  then  mulntiply  the numerators  together for a new numerator, and  the denominators
for a n ew denominator.
146. In what other way may we cancel? Give an example.-147. When
we cancel, what operation do we perform? When a numerator or denominator disappears by cancelling, what is left?-148. What trouble do we
avoid, by cancelling before we multiply?-149. Give the general rulo for the
multiplication of fractions.




MULTIPLICATION OF FRACTIONS.                 89
2. TWlole numbers occurring in a compound fraiction muzst frst be reduced to a fractionalform, and
mixed numbers to improper jractions.
EXAMPLES rFO   THE  SLATE.
1. Multiply together %, -, 14, and -1-.       Ans. 3r.
2. Multiply together 3, 2, 4 and 15.             Ans. 1.
3. Multiply together 3 -        and.        Ans. -3
4. Find the product of 9, 4s, and 31-.         Ans. 3.
5. Find the product of 4, 14, ^  and.    Ans. 2.
6. Reduce 9of  of  of  of.                   Ans. 1i.
7. Reduce - of I of - of —.                  Ans. r4B.
8. Reduce 2 of I of 3 of 9.                    Ans. 14.
9. Reduce 4- of - of - of 41.                  Ans. I.
10. Peduce - of 7 of -1 of 12.                 Ans. 7.
MIXED NuIMBEE X MIXED NUMBEPR.
150.. RULE. —T     multiply  twoo  or  more  mixed
numbers together, reduce themn to improper factions,
and proceed as in multiplication of fractions.
EXAMPLE.-Multiply 41, 5- 4, and 1  together.
Reduce the mixed numbers to improper fractions. Then,
cancelling, we get  t, or 35'.
X   71  X            71        35s   Ans.
Jo      2      2
Find the value of the following:1.26}x3.       Ans. 931. 4.  33x41xl.  Ans. 17.
2. 7, x1.       Ans. 13'. 5. 11 x    xl-. As. 24-,.
3. l1-x9-.      Ans. 13g.  6. 2-x3-x5-.  Ans. 44t_.
150. Recite the rule for multiplying two or more mixed numbers together.




90                    FRACTIONS.
EXAMPLES FOR  THE SLATE.
1. A merchant owning 3 of a ship, sold — 7 of his share.
What part of the ship did he sell?                   Ains. 2.
2. What -will 151  yards of velvet cost, at 41 dollars a
yard?                                           Als. $67-153.
3. A person having 4231 acres of land, left - of it to his
son. What was the son's share?               Ans. 254 acres.
4. A  farmer has three wheat fields, of 4s acres each.
Their average yield is 333 bushels to the acre. What is the
yield of the whole?                           Ans. 425k bu.
5. General Putnam  lived to be 72 years old. Patrick
Henry attained  of that age; how old was he at the time of
his death?
6. IHow much flour must be laid in for a garrison of 355
men, to allow each man 563 pounds?
7. The British House of Commons contains 654 members.
24 8 of this number are from England and Wales; how many
does that make?                           Ans. 496 members.
8. How many yards are there in a bale of linen, containing 56 pieces, if there are 251 yards in each piece?
9. If a clock ticks sixty times in a minute, how many
times will it tick in 151 hours, there being sixty minutes in
an hour?                                   Anzs. 56160 times.
10. If 680 persons subscribe for a work in three volumes,
costing half a guinea a volume, what is the whole amount
of the subscription?
11. A owns 2 of a factory.  IIe sells half his share to B,
who in turn sells 6 of his share to C. What part of the factory belongs to C?                                 Ans. i.
12. What will be the cost of three boxes of oranges, allowing 96 oranges to the box, at 1- cents apiece?
13. Multiply 51, 5, and -'- together.           Ans. 1.




DIVISION OF FRACTIONS.                  91
Bvliyon  of  Fractions.
FRACTION  -- WHOLE NUMBER.
151. To divide any number of equal parts by 2,
we may  either take half as many  such  parts, or
make each part half as great.
EXAMPLE.-Divide - by 2.
Take half the number of parts. Half of 4 is 2.
Or, make each part half as great. A tenth is half as great
as a fifth. Hence half of 4 is,4.
-^ can be reduced to 2. The two answers agree.
Now, in the first case, we divided the numerator by 2.
In the second case, we multiplied the denominator by 2.
The former mode is better, because it brings the fraction at
once in its lowest terms. Hence the following rule.
152. RuLE. —.   To divide a fraction by a whole
number, divide its numerator by the whole number,
if this can be cone without ca remainder; if not,
multiply its denominator.
2. To divide a mixed number by a whole number,
reduce the mixed number to an inproper fraction;
tAen proceed as above.
ExAMPLES.-1. Divide,-5 by 6.
If the numerator 5 contained 6 exactly, we should divide
it by 6. As it does not, we multiply 
the denominator.                        T6  6        6   ns.
2. Divide 24 by 6.
Reduce 24 to an improper frac-              -   3  A.
tion, —. As the numerator 18 con-    7 
tains 6 exactly, divide it by 6..
151. To divide any number of equal parts by 2, what may we do? Illustrate these two methods, in dividing I by 2. Which method is better?-152.
Give the rule for dividing a fraction by a whole number.




92                       FRACTIONS.
153. Find the value of the following:1. " +  7.      Jn4s. -.       8. -- 9.            Ans. 5.
2. 63    9                     9.  - 11.
3. 2 —    11.                 10. -4  5.
4. 7-       10.  Ans. 5.    11. 1-  13.            Ans.'
5. 2       4.                 12. 2   S3.
6. 31 - 6.                    13. 5   23.
7. 5. 17.                     14. 55  40.
IFACTION  4- FRACTION.
154. Divide  3 by.
That is, find how many times - is contained in 2.  1 is
contained in 1, 7 times.  Hence, in 5 it is contained _ of 7
times, or 21 times.
But q is twice as great as I, and hence is contained only
half as many times.  - of  t- is %1. Answer, 2, or 21.
Now, what have we done to the dividend -, to produce'l}?  We have multiplied its numerator by the denominator
of the divisor -, and multiplied its denominator by the numerator of the divisor.  Or, in other words,
we have inverted the divisor, and then mul-      3 X  7 -  21
tiplied the fractions.  Hence the rule.
155. RULE. —. To divide onefraction by another,
multiply the dividend by the divisor inverted.
2. VWhole and  mixed nutnmbers  nzust first be redutcd to improper fractions.
EXAMPLE. —Divide 4, by 23.
Reduce the mixed nupnbers:   3           ~ 5
Invert the divisor; cancel,2       3    9
- -          1 
equal factors; multiply.             5 X                  As
154. Divide A by 2. What have we done to the dividend, to produce
this result?-155. Give the rule for dividing one fraction by another.




DIVISION  OF FRACTIONS.                   93
EXAMPLES  FOR  THE SLATE.
156. Find the value of the following:1.  * 6        Ans. 5-.      7. 4.,        Ans. 7  1
2. A"            Ans. 7.       8. 7  -. 2       Ans. 15.
3. 4    2.      Ans. 6.       9. 9  -  1.    Ans. 6-4.
4. 6  ~.         Ans. 8.      10.    -- 21-.      Ans..
5. 9-.                        11. 5.9
6.                            12..2.   -8  31.
13. How many clocks, at $5   apiece, can be bought for
$21?                                           Ans. 4 clocks.
14. If it takes 61 yards of lace to trim  one dress, how
many dresses will 60 yards trim?               Ans. 9 dresses.
15. A  flock of sheep yield 104} pounds of wool.  How
many sheep are there, if they average 36 pounds of wool
each?                                          Ans. 27 sheep.
16. If a locomotive goes 156- miles in 5- hours, what is
its rate per hour?                           Ans. 27-1 miles.
17. If a cow is allowed   -,of a bushel of turnips a day,
how long will half a bushel last her?
18. Divide 8-3  by 2-.                            Ans. ~3.
MISCELLANEOUS QUESTIONS ON FRACTIONS.-What operation does
a fraction indicate?'What part of the fraction corresponds with the
divisor?  What corresponds with the dividend?  What corresponds
with the quotient?  Ans. The value of the fraction.  Which is
greater, - or -? When we increase the denominator of a fraction,
do we increase or diminish its value? Which is greater, 1 or 2?
When we increase the numerator of a fraction, do we increase or
diminish its value?  What kind of a fraction is -?  What is its
value?  Is the value of a proper fraction greater or less than 1?
Does multiplying a number by - increase or diminish it?  Does
dividing a number by i increase or diminish it? When we take
i of a number, do we multiply or divide by?  To what is cancelling equivalent? When can we cancel?




94                      FRACTIONS.
MIISCELLANEOUS EXAMPLES.
1. Find the sum, then the difference, then the product, of
and,-. Divide I by -,.
2. If a boy's wages are $4} a week, what will they
amount to in 52 weeks?                            Ans. $221.
3. Allowing 240 pins to a paper, how many pins are there
in - of a paper?
4. Three boys agreed to share their earnings for one week
equally.  The first earned $51; the second, $4:; the third,
$3.  What was the share of each?                 Ans. $413o.
5. A planter who has 56 hogsheads of sugar, sells -I of
them to one merchant, and 3 to another.  How many hogsheads has he left?                        Ans. 21 hogsheads.
6. If a man has to make a journey of 15  -3  miles, how
far will he have to go when he has travelled 485 miles?
Ans. 126- miles.
7. How many quarter-dollars are there in $1250?
8. A person owning 200 acres of land, leaves I of it to
his wife, and she divides her portion equally among her 5
sons. What fraction of the whole does each son get, and
how many acres?                            Ans. -, 20 acres.
9. If a merchant sells three dresses, of 10- yards each,
from a piece of calico containing 40 yards, how many yards
will he have left?                            Ans. 7` yards.
10. If a farm of 143- acres is sold for $2575, what is the
price per acre?                                 Ans. $71T}.
11. Three partners buy some silks for $1250', and sell
them for $13251.  What profit.has each?           Ans. $25.
12. A tailor sold a coat for $20-. If the materials cost
him $91, and he paid $8{ for making it, what was his profit?
Ans. $2.4




FEDERAL  IONEY.                         90
FEDERAL MONEY.
157. Different countries have different currencies,
or kinds of money.  The currency  of the United
States is called Federal  Honey.
TABLE OF FEDERAL MONEY.
10 mills (m1.) make  1 cent,... c., ct.
10 cents,                 1 dime,.. c.
10 dimes,                 I  dollar,.. $
10 dollars,               1 eagle,.  E.
158. All the denominations in this Table, except
mills, are represented by  coins.  For convenience,
other coins also have been issued.  The coins of the
United States are as follows:
GOLD.   Double eagle,    worth $20.
Eagle,               "    $10.
Half-eagle,          "    $ 5.
Three-dollar piece, "    $ 3.
Quarter-eagle,       "    $ 2{.
Dollar,              "    $ 1.
SILVER.  Dollar,               "    $ 1
Half-dollar,         "         50 c.
Quarter-dollar,                25 c.
Dime,                 4        10 c.
Half-dime,           "           5 c.
Three-cent piece,   "            3 c.
CoPPin. Two-cent piece,    "               2 c.
Cent,                           1 c.
157. What is said of different countries? What is the currency of the
United States called?  Recite the Table of federal money.-15S. What denomination of this table is not represented bv a coin? Name the gold coins,
and their value. The silver coins. The copper coins.




S96                 FEDERAL MONEY.
MENTAL EXERCISES.
1. How many dollars are 5 double eagles worth?
MODEL.-1 double eagle is worth $20; and 5 are worth
5 times $20, or $100. Answer, $100.
2. How many dollars are 3 half-eagles worth? 4 quartereagles   6 eagles? 2 double eagles?
3. How many cents in 2 half-dollars? In 4 quarter-dollars?   In 7 dimes?  In 3 half-dimes?
4. How many dollars in an eagle and a half-eagle?  In
4 eagles and a half-eagle?
5. How many cents in 11 three-cent pieces? In 12 dimes
and a half-dime?
6. How many eagles in 80 dollars?
MODEL.-$10 make 1 eagle; in $80 there are as many
eagles as 10 is contained times in 80, or 8. Ans., 8 eagles.
7. How many dimes in 50 cents? In 6 half-dimes?
8. How many dollars in 18 half-dollars?  In 24 quarterdollars? In 70 dimes?
9. How many double eagles in $60? In $100?
Wrating and Beadiang lFederal Ilosaey.
159. In writing and reading federal money, the
only  denominations used  are  dollars, cents, and
mills.
160. Dollars are denoted by this sign $, always
placed  before  the number.   They  are  separated
from  cents and mills by a point.  The first two
figures at the right of the point denote cents; the
third figure, mills.
159. In writing and reading federal money, what denominations alone
are used? 160. How are dollars denoted? Where are cents and mills
found?




WRITING  AND READING IT.                  97
161. RULE.-To write federal money, set down
the dollars with apoint at the right.  Set the cents
in the first two places at the right of the point, and
the mills in the third place.
If the cents are expressed by one figure, fill the
vacant 5place with a naught.  If there are mills, but
no cents, fill both vacant places with naughts.
Ex. Six dollars,                       $6.
Six dollars, fifty cents,         $6.50
Six dollars, fifty cents, one mill,  $6.501
Six dollars, five cents, one mill,   $6.051
Six dollars, one mill,            $6.001
162. Write the following: let the points range
in line.
1. Nine dollars, seventy cents.
2. Ninety dollars, five mills.
3. One hundred and forty dollars, seven cents.
4. Five dollars, seventy-five cents, seven mills.
5. Thirteen dollars, three cents, three mills.
6. Forty-one dollars, fourteen cents.
163. RULE.-In federal money, read what is at
the left of the point as dollars, the frst two figures
at the right of the point as cents, and  the third
figrte as mills.
164. Read the following:$400.276           $350.70             $54.
$112.009            $41.06           $6000.606
$907.072          $1011.004           $789.001
161. Recite the rule for writing federal money.-163. Recite the rule for
reading federal money.
9




98                   FEDERAL MONEY.
Addltion  Of Federal MoS ey.
165. Add $72.25, 37V  cents, $9, and $15.625.
$72.25         Here we are required to find the sum  of sev-.375   eral items in Federal Money.  We must add
9.00    things of the same kind.  Therefore set dollars
15.625   under dollars, cents under cents, &c., letting the
q97.250   points all range in line.  Represent the half-cent
as 5 mills.  Then add in the usual way, and set
off dollars in the result by placing a point under the points
in the items added. Answer, $97.25.
166. RULE. —Write  the several items, with their
points ranging in line.  Add, and place apoint in
the result undber the points in the items added.
NOTE.-As there are no mills coined, less than 5 mills in
a result are disregarded in business dealings, and 5 mills or
more are counted as an additional cent.
EXAMIPLES FOR  THE SLATE.
Read the following expressions: find their sum.
(1)             (2)             (3)            (4)
$842.75        $1269.454        $327.00          $27.50
34.03           73.401         562.009           4.516
460.983           1.011          437.09.375
908.625         856.875          591.90          89.008
376.009          28.652         875.63           43.921
5. Bought a box of raisins for $1.32, a bushel of apples
for 88 cents, a cheese for $5.94, and a barrel of sugar for
$27.62. What did the whole amount to?
165. Set down the given example. How must we place dollars, cents, &c.?
How do we represent the half-cent? IIow do we set off dollars in the result?
-166. Give the rule for the addition of federal money.




ADDITION OF FEDERAL MONEY.                 99
6. A farmer receives $15.37 for a cow, $75 for a horse,
$3.13 for some potatoes, and $5.55 for some poultry. How
much does he receive in all?                   Ans. $99.05.
7. Sold some velvet for $3.33, broadcloth for $18.75, silk
for $12.50, muslin for $5.40, carpeting for $30.05, a shawl
for $12.25. What is the amount of the bill?   Ans. $82.28.
8. If a house cost $3487.75; repairs, $53.37; painting,
$119.23; furniture, $1563.39; moving, $9; what was the
whole cost?                                 Ans. $5232.74.
9. A lady gives 25 cents for needles, $17.50 for a dress,
$2.63 for trimmings, $1.50 for a cap, and 12 cents for thread.
How much does she lay out?                       Ans. $22.
10. A man lends $68 to one friend, $443.75 to another,
and $19.05 to a third. How much does he lend in all?
11. Add together sixty dollars; five cents, six mills; sixty
cents; six hundred and fifty dollars, five mills; four mills;
fifty-nine cents.                           Ans. $711.255.
12. There were taken up in a church collection, 16 cents,
3 three-cent pieces, 10 half-dimes, 8 dimes, 4 quarter-dollars,
3 half-dollars, and a two-dollar bill.  What did the whole
amount to?                                      Ans. $6.05.
slubtaction  of Federal loney.
167. If a person owing  $143 pays $27.37, how
much does he still owe?
$143.00         We are here required to find the difference
27.37    between $143 and $27.37. Set the subtrahend
115- 63    under the minuend, filling the vacant places of
*  the latter with naughts. Place dollars under
dollars, cents under cents, &c. Subtract in the usual way,
and set off dollars in the remainder by placing a point under
the other points. Answer, $115.63.




100                 FEDERAL MONEY.
168. RIULE.- VWrite  the  subtrahend  under  the
minuend, with tAeir points ranging in line.  Subtract, andplace a point in the remainder under the
other points.
EXAMPLES FOE  THE  SLATE.
(1)           (2)            (3)           (4)
From  $43.69        $101.467        $50.001       $64.
Take   17.748         88.35           9.099.625
5. From forty-six dollars, two mills, subtract eighteen
cents, nine mills.                             Ans. $45.813.
6. From  one hundred dollars, three cents, take seven
dollars, seven mills.                          Ans. $93.023.
7. A  person, having bought goods to the amount of
$65.76, gave the storekeeper a hundred-dollar bill.  How
much change must he receive?                   Ans. $34.24.
8. If a merchant sells goods that cost him  $151.32 for
$99.99, does he gain or lose, and how much?
9. Bought a cow  for $37.25; paid on account $6.87.
How much remains unpaid?
10. Paid for a lot $947.25; for erecting a house, $2345.47;
for furniture, $1159. I sold the whole for $4500. Did I gain
or lose, and how much?                 Ans. Gained $48.28.
11. A man worth $10000 gave away $956.38, and lost
$1197.82. What was he then worth?             Arns. $7915.80.
12. If a lady gives 12 cents for ink, 63 cents for pens,
$13.30 for books, and $1.87 cents for paper, how  much
change must she get for a twenty-dollar bill?    Ans. $4.08.
13. Bought $75 worth of hay, and $25.25 worth of corn;
paid $49.88. How much is still due?             Ans. $50.37.
14. From fifty dollars, take fifty cents, five mills.
168. Give the rule for the subtraction of federal money.




MULTIPLICATION  OF FEDERAL MONEY.              101
TlaltipticatioEn  of Feederal Money.
169. What do 15 cows cost, at $16.345 each?
$16.345        If 1 cow  costs $16.345, 15 cows cost 15
15  times $16.345. Find the product, and point off
81i25   from the right three figures for cents and mills,
16345    because there are three  figures representing
-2~~     ~cents and mills in the multiplicand. Answer,
$2455.17 5   $245.175.
170. IRULE.-    ulti2ply in the usual way; p2oint
of from,  the right of the product, for cents and
mills, as mcany figures as represent cents and mills
in the multiplicand.
EXAMPLES FOR THE  SLATE.
(1)          (2)         (3)           (4)
Multiply  $64.275       $15.89      $37.59       $293.872
By               9          12          13             56
5. What will 42 calves cost, at $3.75 apiece?
6. At 37} cents, apiece, what will 75 geese cost?
7. What will 890 cords of wood cost, at $3.78 a cord?
8. What will be the cost of 141 yards of black silk, at
$1.20 a yard?                                   Ans. $17.40.
9. If a boy's wages are $4.75 a week, how much will he
earn in a year, or 52 weeks?                     Ans. $247.
10. If a clerk earns $8 a week, and spends $4.75 a week,
how much will he lay up in a year?               As. $169.
11. What will it cost six persons to board for a year, at
the rate of $5 apiece each week?                 Ans. $1560.
169. Go through the several steps in the example.-170. Recite the rule
for the multiplication of federal money.




102                 FEDERAL MONEY.
ivIBsan ~Of Federal 1tVoney.
171. If 4 piano-fortes cost $1501, how  much do
they cost apiece?
4) $1501         If 4 pianos cost $1501, one piano will cost
3751   one fourth of $1501.  Dividing, we find the
4  quotient to be $3751.
If cents and mills are required in the  4)$1501.00
answer, in stead of the fraction of a dollar,       375 25
annex ciphers and continue the division.
Point off in the product, for cents and mills, as many figures
as represent cents and mills in the dividend. Ans., $375.25.
25 cents make a quarter of a dollar.  The answers agree.
172. RULE.-Divide in the usqal way; point off
from  the right of the quotient, for cents ccd mills,
as many figures as represent cents and mills in the
dividend.
EXAMPLES FOR  THE SLATE.
(1)            (2)           (3)           (4)
8) 143.816      9)$47.88      4) $106       7)$82.53
$5.47           $              $           $11.79
5. Divide $12.48 equally among 12 persons.
6. If 36 hats cost $88.92, how much is that apiece?
7. If a farmer sells 240 bushels of oats for $132.48, how
much does he get a bushel?                       Ans. $.552.
8. Find T  of $424.632.                     Ans. $4.083.
9. Four partners bought some land for $1150. They sold
it for $940 cash and $500 worth of grain. How much did
each make by the bargain?                      Ans. $72.50.
171. Divide $1501 by 4 in both the ways shown above.-172. Recite the
rule for the division of federal money.




'IISCELLAN:EOUS EXAMPLES.                 103
MISCELLANEOUS EXAMPLES IN FEDERAL MONEY.
1. If a person spends $410.28 in a year, how much is that
a week, allowing 52 weeks to a year?             Ans. $7.89.
2. A man buys 4 barrels of flour, at $5.95 a barrel; 18
chickens, at 29 cents each; and 56 pounds of butter, at 271
cents a pound.  What does the whole cost?    Ans. $44.42.
Find the cost of each ite; then add the three amounts.
3. Bought 3 pair of gloves, at $.75 a pair; 12 yards of
lace, at $1.38 a yard; 10 yards of sheeting, at 45 cents a yard.
What is the cost of the whole?                  Ans. $23.31.
4. A lady buys 2 turkeys at $1.25 each, and 5 bushels of
potatoes at 94 cents a bushel. How much change will she
receive for a $20 bill?                         Ans. $12.80Q
5. A man pays for some land $400 cash and $192.80 in
produce.  If there were 57 acres, how much does the land
cost him per acre?                              Ans. $10.40.
6. If a boy buys 12 knives for $7.50, and sells them at 75
cents apiece, how much does he make on each?  Ans. $.125.
Find the cost of 1 knife, and subtract it from the selling price.
7. Divide $2117.71 equally among 35 families, and find
the share of each.                             Ans. $60.506.
8. Four persons contribute $8000 for a speculation. The
first puts in $99.05; the second, $2460.30; the third, $986.
What does the fourth put in?                  Ans. $4454.65.
9. If 184 pounds of coffee are sold for $52.44, what is the
rate per pound?                                  Ans. $.285.
10. A father who has $2450 in stock, a house valued at
$4750, and bonds to the amount of $15040, divides the whole
equally among his two sons and three daughters. What is
the share of each?                              Ans. $4448.
11. If 40 acres of meadow land are worth $1260, what
is 2 of the tract worth?                          Ans. $840.




104                     REDUCTION.
REDIUCTIO N.
173. How many cents in five dollars?
In 1 dollar there are 100 cents, and in 5 dollars 5 times
100 cents, or 500 cents. Answer, 500 cents.
We have here changed the denomination from  dollars to
cents, without changing the value.  This process is called
Reduction.  We have reduced dollars to cents.
174. Reduction is the process  of changing  the
denomination  of a numlier without changing  its
value.
175. There are two kinds of Reduction:1. Reduction Descending, in which we change a
higher denomination to a lower, as dollars to cents.
Here we must multiply.
2. Reduction Ascending, in  which  we change  a
lower denomination to a higher, as cents to dollars.
Here we must divide.
RIeduct~iol  Elesceildmga
176. Reduce $27 to mills.
2^7  ~       100 cents male $1; in $27, therefore,
1007           there are 100 times 27 cents, or (annexing
two naughts) 2700 cents.
2T00 e.          There are 10 mills in 1 cent; in 2700
10        cents, therefore, there  are 10 times 2700
27000 m.   mills, or (annexing one naught) 27000 mills.
Answer, 27000 mills.
17T. Solve the example given. What have we done in this example?174. What is Reduction?-175. How many kinds of reduction are there?
What do we do in Reeuction Descending? What operation must we perform?
What do we do in Reduction Ascending? What operation must we perform 
-176. Reduce $27 to mills, explaining the steps.




REDUCTION DESCENDING.                  105
177. Redluce $27.465 to mills.
Reduce $27 to cents:           27 x 100 = 2700 c.
Add in 46 cents:              2700 +46 = 2746 c.
Reduce 2746 cents to mills:  2746 x 10 = 27460 m.
Add in 5 mills:               27460+5 = 27465 m. Ans.
Compare this result with $27.465, the amount to be reduced.  It is the same, with the dollar mark and point
omitted.
178. GENERAL RULE FOR REDUCTION DESCENDING.
-1-Multiply the highest given denominaction by the
number that it ta/kes of the next lower to make one
of this higher, and add in the number belonging to
such lower denomination, if any be given.
Go on thus with each denomination in turn, till
the one required is reached.
179. Federal  M2oney.-In  federal  money, the
above rule is applied as follows.  See the examples
in ~176, 177.
1. To reduce dollars to mills, annex three naughts;
to reduce dollars  o to cents, two; to reduce cents to
mill[s, one.
2. To reduce dollars and cents to cents, or dollars,
cents, and mills, to mills, simply remove the dollar
mark and the point.
180. Reduce the following:1. $624 to cents.            4. $450.63 to mills.
2. $125 to mills.            5. $29.172 to mills.
3. $.485 to mills.           6. $5,0000 to mills.
177. Reduce $27.465 to mills, explaining the steps.  How does the result
compare with the original amount?-178. Recite the general rule for reduction descending —179. Give the rules for the reduction of federal money.
IG




106                    REDUCTION.
7. Reduce 5 eagles to mills.            Ans. 50000 mn
8. How many cents in 28 eagles?          Ans. 28000 c.
9. How many mills in 15 double eagles?
10. How many half-dimes in 73 eagles?
11. Reduce $450.59 to mills.
12. How many dimes are $671 worth?
13. A lady gets a half-eagle changed to dimes.  How
many dimes should she receive?                   Ans. 50 d.
14. If a boy gets a $10 bill changed to half-dimes, how
many should he receive?                Ans. 200 half-dimes,
Reductiona   Ascending.
181. Redice 2T465 mills to dollars.
10 mills make 1 cent; therefore in 27465 mills there are
as many cents as 10 is contained times in 27465. Dividing
~10) 2o465       by 10 (cutting off one figure), we get
10)27465 m.           2746 cents, and 5 mills over.
100) 2746., 5 m.        100 cents make 1 dollar; thereAns. $27.465.          fore in 2746 cents there are as many
dollars as 100 is contained times in
2746. Dividing by 100 (cutting off two figures), we get $27,
and 46 cents over.  Answer, $27, 46 cents, 5 mills; or,
$27.465.
Compare this result with 27465 mills, the amount to be
reduced. We have simply pointed off three figures from the
right, and inserted the dollar mark.
182. In ~ 177 we reduced $27.465, and obtained
27465  mills.  In  ~181 we reduced  27465  mills,
and obtained $27.465. Thus it will be seen that
Reduction  Descending  and Reduction Ascending
prove each other.
181. Reduce 27465 mills to dollars, explaining the steps. How does the
result compare with the amount given to be reduced —182. By comparing
the examples in ~ 177 and 181, what do we find?




REDUCTION ASCENDING.                   107
183. GENERAL RULE FOR IEDUCTION  ASCENDING.
— Divide  the  given  denomnination  by  the number
that it takes of it to make one of the next higher.
Divide  the quotient in  the sacme way, and go on
thzus  till  the'required  denomination  is reached.
The last quotient and thie different remainders form
the answer.
184. Federal  iMloney.-In  federal  money, the
above rule is applied as follows:To reduce mills to dollars, point of threefigures
from  the right; to reduce cents to dollars, two; to
reduce mills to cents, one.
EXAMPLES FOR THE  SLATE.
1. Reduce 4790 mills to' cents.              Ans. 479 c.
2. Reduce 59195 mills to dollars.
3. Reduce 461063 cents to dollars.
4. How many dollars in 70000 mills?
5. How many dollars in 85310 cents?
6. I-low many eagles are 50 gold dollars worth?
7. Reduce 2500 dimes to eagles.
8. How many dimes are equal to 600 mills?
9. Reduce 46000 cents to double-eagles.
10. Reduce 4676 mills to cents.
11. How many half-eagles in 1500 cents?
12. How many dollars in 200 half-dimes?
13. How many double eagles in 1200 half-dimes?
14. Reduce 1623487 cents to dollars.
15. How many mills in 5 three-dollar pieces?
16. How many cents in 4 quarter-eagles?
183. Recite the general rule for reduction ascending.-184. Give the rule
for the reduction of federal money.




108               COMPOUND NUMBERS.
COMPOUND NUMBERS.
185. A  Compound Number is one  consisting  of
different denominations: as, 3  dollars, 14 cents;
5 feet, 10 inches.
186. To show the relations that different denominations bear to each other, Tables are constructed.
They must be learned perfectly.
En glish or Sterliag A5oxaey.
187. The  currency  of Great Britain  is called
English or Sterling Mloney.
TABLE.
4 farthings (far., qr.), 1 penny,.. d.
12 pence,                  1 shilling,. s.
20 shillings,              1 pound,.. ~.
21 shillings,              1 guinea,.. guin.
188. The pound mark always precedes the number; as, ~2.
Farthings are sometimes written as the fraction of a penny;
2d.3 far., or 2cd.; 3d. 2 far., or 38d.
The pound is simply a denomination. A gold coin called
the Sovereign represents it. The Sovereign is worth $4.84.
The English penny is worth about 2 of our cents.
Guineas, originally made of gold brought from Guinea, are
no longer coined.
The Crown is a silver coin, worth 5 shillings..185. What is a Compound Number?-187. What is the currency of Great
Britain called? Recite the Table of Sterling Money.-188. How must the
pound mark stand? How are farthings sometimes written? What represents the pound'  What is the sovereign worth? The penny? What is
said of guineas? What is the crown worth?




STERLING MONEY.                  109
EXAMPLES FOR THE SLATE.
189. Recite the rules for Reduction, ~ 178, 183.
1. In ~7 5s. 1 far. how many farthings?
Multiply the ~7 by 20, to reduce
them  to shillings, because 20 shillings     7   s. 
20
make a pound. Add in the 5 shillings. 
Multiply 145s., thus obtained, by 12,     12
to reduce them  to pence, because 12
pence make a shilling. There are no          4
pence in the given number to add in.
Multiply the 1740d., thus obtained,    6961far.   s.
by 4, to reduce them to farthings, because 4 farthings make
a penny. Add in the 1 farthing. Answer, 6961 far.
2. Reduce 15383 far. to pounds, shillings, &c.
4)15383 far.         Divide by 4, to reduce to pence.
12) 385 3 f     Divide the quotient, 3845d., by 12, to
12)3845 3 far.
12) 385.    reduce it to shillings.
2j0) 32j0 td.       Divide the quotient, 320s., by 20, to
~16           reduce it to pounds.  The quotient and
Ans. ~16 5ad.   remainders form  the answer.  Always
mark the denominations throughout, as in these examples.
3. Reduce ~75 to pence.                 Ans. 18000d.
4. Reduce 19s. 6d. to pence.               Ans. 234d.
5. Reduce 15s. 3d. 2 far. to farthings.  Ans. 734 far.
6. Reduce 8670d. to pounds, &c.      Ans. ~36 2s. 6d.
7. Reduce 16255s. to pounds.          Ans. ~812 15s.
8. Reduce 24681 far. to guineas, &c. Ans. 24 guin., &e.
9. Reduce ~3 14s. 7Ed. to farthings.   Ans. 3582 far.
10. Reduce 1920 far. to pounds, &c.
11. Reduce 3 guin. 10s. 6d. to farthings.
12. Reduce 16s. 104d. to farthings.
13. Reduce 1080 farthings to pounds, &c.
14. Reduce 8628 pence to pounds, &c.
10




110               COMPOUND NUMBERS.
Troy Weight.
190. Troy Weight'is used in weighing gold, silver,
precious stones, and liquors; also in philosophical
experiments.
TABLE.
24 grains (gr.) make 1 pennyweight, pwt.
20 pennyweights,    1 ounce,.... oz.
12 ounces,             1 pound,... lb.
EXAMPLES FOR THE SLATE.
1. Reduce 48494 gr. to pounds, &c. Ans. 8 lb. 5 oz. 14 gr.
Divide by 24; then by 20; then by 12.
2. In 41b. 6 oz. 13 pwt. how many grains?  Ans. 26232.
Multiply 4 lb. by 12; add in 6. Multiply the result by 20; add in
i3. Multiply this result by 24. Mark the denominations throughout.,3. In 100 lb. 1 gr. how many grains?    Ans. 576001 gr.
4. Reduce 8976 pwt. to pounds. Ans. 37 lb. 4 oz. 16 pwt.
5. Reduce 9 oz. 5 pwt. 20 gr. to grains.    Ans. 4460 gr.
6. How many pounds, &c., in 1180 oz.?'7. Reduce 16 lb. 10 oz. to pennyweights.
8. Reduce 5 lb. 5 pwt. to pennyweights.
19. Reduce 12 lb. 1 oz. 9 pwt. to grains.   Ans. 69816 gr.
10. If a miner digs 20 pwt. of gold one day, and 37 the
next, how many ounces will he have?      Ans. 2 oz. 17 pwt.
11. How many pounds in two bars of silver, one of which
weighs 240 pwt., and the other 360 pwt.?
12. How many pennyweights in 4 rubies, weighing 8 gr.,
10 gr., 12 gr., and 24 gr.?              Ans. 2 pwt. 6 gr.
13. A lady has 36 tablespoons weighing 34pwt. each, and
24 teaspoons weighing 19 pwt. each. How many pounds do
all her spoons weigh?                           Ans. 7 lb.




APOTIECARIES' WEIGHT.                111
Apoltecaries' Weight.
191. Apothecaries' Weight is used by apothecaries
in  mixing  medicines.  They  buy  and  sell their
drugs, in quantities, by Avoirdupois Weight, given
on the next page.
TABLE.
20 grains (gr.) make 1 scruple, sc. or 3.
3 scruples,           1 dram,.. dr. or 3.
8 drams,              1 ounce,..  oz. or.
12 ounces,             1 pound,.. lb. or lb.
192. The  ounce  and  pound  of Apothecaries'
Weight are the same as in Troy Weight.
MENTAL EXERCISES.
1. How many ounces in 7t pounds of senna?
2. Reduce 1 ounce to grains.
3. Reduce 9} ounces to drams.
4. Which is greater, 15 5 or 1 S?
5. How many grains in 2 3 of magnesia?
6. How many scruples in 9~ drams?
7. How many doses of 12 grains each in 13 of musk?
8. I-ow many powders of ten grains each can a druggist
make out of 2D of calomel?
9. Divide a dram of jalap into six powders; how many
grains will there be in each?
10. If a druggist sells four customers 6 oz. of blue vitriol
each, out of a. five-pound package, how many pounds has he
left?
11. How many grains in the following mixture: nitrate
of silver, 5 gr.; opium,  53; camphor, 13?




112                COMPOUND NUMBERS.
Avoirdupois Weight.
193. Avoirdupois Weight is that in  common use.
By  it are weighed  all articles not named  under
Troy and Apothecaries' Weight; such as groceries,
meat, coal, cotton, and all the metals except gold
and silver.
TABLE.
16 drams (dr.) make 1 ounce,...... oz.
16 ounces,               1 pound,... lb.
25 pounds,               I quarter,....    qr.
4 quarters,             1 hundred-weight, cwt.
20 hundred-weight, 1 ton.....   T.
194. The ounce of Avoirdupois Weight is less
than the Troy ounce, but its pound is greater than
the Troy pound.
195. It was formerly customary to allow  112 pounds to
the hundred-weight, and 28 pounds to the quarter. But this
is now seldom  done, except in the case of coal, iron, and
plaster bought in large quantities, and English goods passing
through the Custom House.
Twenty hundred-weight of 112 pounds make a ton of
2240 pounds, which is distinguished as a Long Ton.
MISCELLANEOUS QUESTIONS.-In what denominations do British
merchants keep their accounts? How much is a half-crown worth?What weight is used in weighing emeralds?  In weighing hay?
Coins?  Cotton? In weighing drugs for a physician's prescription?'
Recite the Table used in philosophical experiments. Recite the Table
used in weighing cheese. Recite the Table. used in mixing drugs.
Which is greater, the Troy pound or Apothecaries' pound? The
Troy ounce or Avoirdupois ounce?  The Apothecaries''pound or
Avoirdupois pound? How many pounds were formerly allowed to
the hundred-weight? In what alone is this now done? What is a
Long Ton?




AVOIRDUPOIS' WEIGHT.                      113
EXAMPLES FOR  THE  SLATE.
1. In 873450 drams, Avoirdupois Weight, how many tons,
&c.?                     Ans. I T. 14 cwt. 11 lb. 14 oz. 10 dr.
Does this example fall under Reduction Descending or Reduction Ascending? Repeat the rule for Reduction Ascending. Name the numbers in order,
by which we have to divide. In all examples in Reduction, be careful to mark
the denominations throughout.
Prove this example by reducing your answer to drams. If the result
agrees with the number of drams given above, your work is right.
2. Reduce 5 cwt. 21 lb. 4 oz. to ounces.    Ans. 8340 oz.
Which kind of Reduction does this fall under? Repeat the rule for
Reduction Descending. Name the numbers in order, by which we have to
multiply. Why do we not first multiply by 20? As there are no quarters to
add in, we may multiply the 5 cwt. by 100 at once, to reduce them to pounds,
in stead of by 4 and 25.
3. Reduce   T. 1 cwt. 1 dr. to drams.   Ans. 537601dr.
4. Reduce 856702 drams to tons, hundred-weight, &c.
Ans. 1T. 13cwt. lqr. 211b. 7oz. 14 dr.
5. How  many tons in 60000 lb. of lead?            Ans.. 30 T.
6. Reduce 3 cwt. 15 oz. to drams.              Ans. 77040 dr.
7. How  many pounds in 401 tons?               Ans. 815001b.
8. How  many ounces in 67  cwt.?
9. Reduce 602000 oz. to tons.        Ans. 18 T. 16 cwt. 1 qr.
10. What will 51 cwt. of poultry cost, at 9c. a pound?
Reduce 5j cwt. to lb.; then multiply by the price of 1 lb.
11. What cost 231 cwt. of beef, at 12c. a lb.?  Ans. $285.
12. How many tons, &c., in 13 bales of cotton, weighing
550 lb. apiece?                          Ans. 3 T. 11 cwt. 2 qr.
13. How  many pounds in 50 tons?  In 50 long tons?
14. If I buy 370 long tons of coal, and sell 370 ordinary
tons, how  many pounds have I left?               Ans. 88800 lb.
15. Bought 7 T. 18 cwt. of iron.  What does it cost, at 4
cents a pound?                                        Ans. $632.
16. How  many ounce balls can bo moulded out of 25
pounds of lead?




114               COMPOUND NUMBERS.
196. In  connection  with  Avoirdupois Weight,
learn the following
MISCELLANEOUS TABLE.
14 pounds,..  1 stone of iron or lead.
56 pounds,.. I firkin of butter.
100 pounds,...  1 quintal of dried fish.
196 pounds,... 1 barrel of flour.
200 pounds,...    bl. of beef, pork, or fish.
EXAMPLES FOR THE SLATE.
1. What is the cost of a firkin of butter, at 20 cents a
pound?                                         Ans. $11.20.
2. How many packages of 7 pounds each can be made
out of a barrel of flour?
3. How many pounds in 421 stone?
4. If a grocer who has 7 barrels of pork, sells half a
barrel, how many pounds has he left?
5. Iow many firkins will 364 lb. of butter fill? Ans. 61.
6. How many pounds in 17' quintals of codfish?
7. Bought 50 quintals of fish, at 6 cents a pound; what
do they cost?                                    Ans. $300.
8. If half a barrel of flour is sold for $2.94, what is the
price per pound?                                    Ans. 3c.
9. How many more pounds are there in 20 barrels of
salted beef than in 20 barrels of flour?
10. If a barrel of pork brings $11.50, how much is that
a pound?                                          Ans. 51c.
I11. How many ounces in 5 stone?
196. Recite the Miscellaneous Table. Is the pound here spoken of, the
Avoirdupois or Troy pound? Is dried salted fish sold by the barrel or
quintal? Is fish in pickle sold by the barrel or quintal 




LONG MEASUmR:.                     1 15:LoTEg  IMeasure,
197. Long Measure is used  in  measuring  length
and distance. It begins with the inch.
1 inch.
TABLE.
12  inches (in.) make 1 foot,... ft.
3  feet,                 1 yard,.. yd.
51 yards,                1 rod,... rd.
40  rods,                  1 furlong,. fur.
8  furlongs,             1 mile,.. mi.
198. It is well to remember that (8x40) 320 rods or
(320 x 5, x 3) 5280 feet make a mile.
199. The Hand, used as a measure of the height of horses,
is 4 inches. The Fathom, used as a measure of depths at sea,
is 6 feet.
MENTAL EXERCISES.
1. How many inches will a man 6 feet high measure?
2. If a horse is 15 hands high, how many feet is that?
3. How many feet in 10 yards?  In 2 rods?
4. Reduce 4 ft. 8 in. to inches.
5. How many inches in 2 fathoms?
6. How many yards in 6 rods?  In 8 rods?
7. Reduce 108 inches to yards.
8. How many furlongs in 10~ miles?
9. How many inches in a quarter of a yard?
197. In what is Long Measure used? Draw a line an inch long. Recite
the Table.-198. How many rods make a mile? How many feet make a
mile?-199. What is the Hand used in measuring? How many inches make
a hand? What is the Fathom used in measuring? How many feet make a
fathom?




116                COMPOUND NUMBERS.
EXAMPLES FOR  THE  SLATE.
1. Reduce 103 yards to rods.
5- yards make a rod; hence we must divide 103 yards by
51, or -. To divide by -'!, multiply
103 yds.                 2                 2 1^^^    ^
2              I'by the divisor inverted, il.  Multi11 )206 half-yd.     plying by the numerator 2, and dird. 18  8 hf.-yd.   viding by the denominator 11, we
get 18 rods, and 8 remainder.
Ans. 18 rd. 4 yd.           This remainder is not 8 yards,
but 8 half-yards, since we reduced the original yards to halfyards when we multiplied by 2. Hence, to get the remainder
in yards, divide it by 2. Answer, 18 rd. 4yd.
In reducing yards to rods, then, if there is a remacinde;r
divide it by 2, to bring it to yards.
2. Reduce 464 yd. to rods.              Ans. 84rd. 2yd.
3. Reduce 1765 yd. to miles, &c.         Ans. 1 mi. 5 yd.
4. Reduce 4355 in. to yards.    Ans. 120 yd. 2 ft. 11 in.
5. Reduce 248 mi. to inches.            Ans. 15713280in.
6. How many miles, furlongs, &c., are there in 1051907
inches?            Ans. 16 mi. 4fur. 32rd. 3 yd. 1 ft. 11 in.
7. How many inches are there in 10mi. I fur. 29 rd.
3 yd. 2 ft. 10 in.?                          Ans. 647404 in.
8. Mt. Everest, a peak of the Himalayas, the highest mountain as yet surveyed, is 29002 feet above sea level.  How
many miles is this?                      Ans. 5 mi.. 2602 ft.
200. In  measuring drygoods, as cloth, muslin,
lace, &c., the yard of long measure is used, divided
into halves, quarters, and eighths.
1. What cost 191 yd. of lace, at 90c. a yard?
2. How many pieces a quarter of a yard long can be cut
from 12 yards of silk?
3. If 2 dresses of 68 yd. and 101 yd. are cut from a piece
of calico containing 40 yd., how much is left?




SQUARE MEASURE.                   11t
Square Measure.
201. Square Mleasure is used in measuring surfaces,
which have length and breadth; such as land, the
sides of rooms, floors, &c.
202. A Square is a figure that has four equal
sides perpendicular one  to  an-          A SQUARE INCH.
other-that is, leaning no more                 lich.
to one side than the other.
A  Square  Inch is a square 
whose sides are each an inch long. 
A  Square Foot is a square whose
sides are each a foot long.                   linch.
TABLE.
144  square inches (sq. in.), 1 square foot, sq. ft.
9  square feet,              1 square yard, sq. yd.
301 square yards,             1 square rod, sq. rd.
40  square rods,               1 rood,... R.
4  roods,                    1 acre,...  A.
640  acres,                     1 square mile, sq. mi.
1. Reduce 40 A. 2 R. to square rods.    Ans. 6480 sq. rd.
2. Reduce 14245 sq. rd. to acres.   Ans. 89 A. 5 sq. rd.
3. Reduce 3 sq. mi. to square rods.   Ans. 307200 sq. rd.
4. How many acres in 59 lots, of 2 roods each?
5. What will it cost to plaster four walls, each containing
270 sq. ft., at 20 cents a square yard?         Ans. $24.
6. How many farms of 32 acres will 1 sq. mi. make?
201. In what is Square Measure used?-202. What is a Square? What
is a Square Inch  A Square Foot? Recite the Table.




118              COMPOUND NUDMBERS.
Cubic'MSeasure.
203. Cubic Measure is used in measuring bodies,
which have length, breadth, and depth or thickness;
as stone, timber, earth.
204. A  Cube is a  body bounded by  six equal
squares.
A  Cubic Inch is a cube, one inch long, one inch
broad, and one inch thick.  Each of its six sides is
a square inch.
205. The engraving repre1 yd. or 3 ft.
sents a Cubic Yard.  It is                     y. o  f.
I yard, or 3 feet, in length,
breadth, and depth.  It will
be seen that each of its six            i       lli; i  t1
sides is I square yard, or 9            i        iBilB
square feet.                              I  ll ji.'
1 yd. or 3 ft.
The top of this cube contains 9 square feet.  Hence,
if it were only 1 foot deep, it would contain 9 cubic
feet.  As it is 3 feet deep, it contains 3 times 9, or
27, cubic feet.
TABLE.
1X28 cubic inches (cu. in.),   1 cubic foot,  cu. ft.
27 cubic feet,                 1 cubic yard, cu. yd.
40 cu. ft. of round or                orlod   T.
50 cu. ft. of hewn timber,    
16 cubic feet,                 1 cord foot,. cd. ft.
8 cord feet,                  1 cord,....  Cd.
203. In what is Cubic Measure used?-204. What is a Cube? What is a
Cubic Inch?-205. What does the engraving represent?  Describe it. How
many cubic feet does it contain? Recite the Table.




CUBIC MEASURE.                     119
206. The ton in this Table is a measured ton; the Avoirdppois ton is a ton of weight. Round timber is wood in its
natural state. A ton of round timber consists of as much as,
when hewn, will make 40 cubic feet.
207. A cord of wood is a pile 8 feet long, 4 feet wide,
and 4 feet high.  Multiplying these dimensions together, we
find 128 cubic feet in the cord.  One foot in length of such a
pile is called a cord foot.
208. Cubic Measure is often used in  estimating the
amount of work in solid masonry, in digging cellars, making
embankments, &o.
EXAMPLES FOE  THE SLATE.
1. Reduce 9 cubic yards, 16 cubic feet, 862 cubic inches
to cubic inches.                          Ans. 448414 cu. in.
2. Reduce 746496 cu. in. to cubic yards. Ans. 16. cu. yd.
3. What will it cost to dig a cellar having a capacity of
4860 cu. ft., at 22 cents a cubic yard?         Ans. $39.60.
Find the number of cubic yards; multiply by the price.
4. What will it cost to make an embankment containing
108270 cu. ft. of earth, at 27c. a cubic yard?
5. Find the price of 1536 cd. ft. of wood, at $5 a cord.
6. How many cubic feet in 638 cords?
7. Reduce 8 cu. yd. 469 cu. in.       Ans. 373717 cu. in.
8. How many feet of hewn timber in 472 tons?
MISCELLANEOUS QUESTIONS.-What measure is used in finding
the amount of surface in a floor? In estimating the amount of work
in digging a well? In ascertaining the contents of a block of marble?
In finding the distance between two places? In measuring one dimension, such as length?  In measuring what has two dimensions,
length and breadth?  In measuring what has three dimensions,
length, breadth, and thickness? How does the ton of Cubic Measure
differ from the Avoirdupois ton? What is round timber?  How
much round timber makes a ton? What is meant by a cord of
wood? A cord foot? In what is Cubic Measure often used?




120              COMPOUND NUMBERS.
Liquid MVeasure.
209. Liquid or Wine 1easure is used in measuring liquids generally; as, liquors (beer sometimes
excepted), water, oil, milk, &c.
TABLP.
4  gills (gi.) make  1 pint,... pt.
2  pints,              1 quart,.. qt.
4  quarts,             1 gallon,.. gal.
311 gallons,            1 barrel,.. bar.
42  gallons,            1 tierce,.. tier.
2  barrels,(63 gal.), 1 hogshead, hhd.
2  hogsheads,          1 pipe,... pi.
2  pipes,              1 tun,.. tun.
210. The wine gallon contains 231 cubic inches.
211. Liquids are put up in casks of different size, distinguished as barrels, tierces, hogsheads, pipes, and tuns; but
these casks do not uniformly contain the number of gallons
assigned to them in the Table, but only about that quantity.
The contents are found by gauging, or actual measurement.
When the barrel is used in connection with the capacity of
cisterns, vats, &c., 81- gallons are meant; in Massachusetts,
32 gallons.
MENTAL EXERCISES.
1. How many gills in 2 qt.? In 1 gallon?
2. Reduce 3 qt. 1 pt. to gills.
3. How often can you fill a quart measure from a fourgallon tub? From a six-gallon tub?
4. What will a six-gallon can of milk cost, at 3c. a qt.?
5. If a milkman sells 10 qt. out of a four-gallon can, how
many quarts remain?
6. How many gallons in 100 pints?




WINE AND  BEER MEASUZE.                      121
Beer Measure.
212. Beer Measure was formerly used in measuring  beer  and  milk.   It is still used  by  some  for
beer, though Wine Measure is fast taking its place.
TABLE.
2   pints (pt.) make 1 quart,.. qt.
4   quarts,                1 gallon,.. gal.
36  gallons,                I barrel,.. bar.
1- barrels (54 gal.), I hogshead, hhd.
213. The beer gallon  contains 282 cubic inches.  The
gallon, quart, and pint of this measure, are therefore greater
than those of Wine Measure.
EXAMPLES  FOR  THE  SLATE.
1. Reduce 10000 gills to gallons.          Ans. 312 gal. 2 qt.
2. In 16 hogsheads of wine, of 63 gallons each, how many
gills?                                              Ans. 32256 gi.
3. How many tuns will it take to hold 201600 gills of
wine, allowing 252 gallons to the tun?             Ans. 25 tuns.
4. How many quarts in 42 tierces of oil, of 42 gal. each 
5. Reduce 11 hhd. of beer to pints.
6. In 100000 pints of beer, how many gallons?
7. If a person buys a barrel of beer, containing 36 gallons,
for $9, what does it cost him a quart?
8. How many pints in 18 gal. 3 qt. 1 pt.?
209. What is Liquid or Wine Measure used in measuring? Recite the
Table.-210. How many cubic inches in a wine gallon?-211. What is said
of the different denominations, barrels, tierces, &c.? How are the contents
of a cask ascertained? How many gallons go to the barrel in Massachusetts?
-212. For what was Beer Measure formerly used? For what is it still used?
Recite the Table.-218. How many cubic inches in a beer gallon? Which is
greater, the beer quart or the wine quart?
11




122                COMPOUND NUMBERS.
DIry Nleassure.
214. Dry Measure  is used  in  measuring  grain,
seeds, vegetables, fruit, salt, coal, &c.
TABLE.
2 pints (pt.) make 1 quart,... qt.
8 quarts,             1 peck,..  pk.
4 pecks,              1 bushel,.. bu.
36 bushels,            1 chaldron,. chal.
215. The quart of Dry Measure is greater than that of
Liquid Measure.-What is called the Small Measure contains
2 qt.
216. Foreign coal is imported by the chaldron.  American
coal is bought and sold, in large quantities, by the ton; in
small quantities, by the bushel.
EXAMPLES FOR THE  SLATE.
1. Reduce 56bu. 2pk. 3 qt. to quarts.       Ans. 1811 qt.
2. Reduce 8256 pt. to bushels.               Ans. 129 bu.
3. How many bushels in 121  chaldrons?  Ans. 4380 bu.
4. Reduce 1597qt. to bushels.    Ans. 49bu. 3pk. 5 qt.
5. How many small measures in a bushel?
6. How many chaldrons in 1843 bushels of coal?
7. If a bushel of apples is bought for 80c., and retailed at
14c. a half-peck, what is the profit on them?      Ans. 32c.
8. If 6 bushels of peaches are sold for $8.64, what do they
bring a quart?
214. In what is Dry Measure used? Recite the Table.-215. How does
the quart of Dry Measure compare with that of Liquid Measure?-216. What
is imported by the chaldron? How is American coal bought and sold?




MEASURE OF TIME.                   123
Time l    easlure.
217. The natural divisions of time are the year
and the day.  The year is the period in which the
Earth makes one revolution  round the Sun; the
day, that in which it makes one revolution on its
axis.
The year is divided into twelve calendar months,
differing in length; the day, into hours, minutes,
and seconds.
TABLE.
60 seconds (sec.) make  1 minute,.. min.
60 minrutes,               1 hour,... h.
24 hours,                  1 day,....  da.
7 days,                   1 week,... wk.
12 calendar months or            year,. yr.
365 days,
366 days,                   1 leap year.
100 years,                  1 century,.. cen.
218. The twelve calendar months (mo.), and the
number of days in each, are as follows:DAYS. I                      DAYS.
1st month, January,      31.'7th month, July,        31.
2d month, February,    28.  8th month, August,    31.
3d month, March,         31.   9th month, September,  30.
4th month, April,        30.  10th month, October,    31.
5th month, May,          31.  11th month, November,  30.
6th month, June,         30.  12th month, December,  31.
217. What are the natural divisions of time? What is the year? What
is the day? How is the year divided? The day? Recite the Table.-218.
Name the twelve calendar months in order, and the number of days in each.




124               COMPOUND NUMBERS.
219. The days in these months, added together,
make 365 days in the year.   Every  fourth  year
(except three in four centuries) is a Leap Year;
then February has 29 days, and the year 366.
220. The leap years are those that can be divided by 4
without a remainder; as, 1864, 1868, 1872, &c. But, of the
even hundreds, only those that can be divided by 400 are leap
years. The year 1900 will not be a leap year, but 2000 will be.
221. When we speak of a month, we mean a calendar month.
The following lines will help the pupil to remember the number of days in each:Thirty days hath September,
April, June, and November;
All the rest have thirty-one,
Except February alone;
Which has but four and twenty-four,
Till Leap Year gives it one day more."
EXAMPLES FOR THE SLATE.
1. In 30 days how many seconds?       Ans. 2592000 sec.
2. Reduce 81920 min. to weeks. Ans. 8 wk. 21 h. 20 min.
3. Find the number of seconds in  13wk. 3 da. 19h.
25min. 39sec.                             Ans. 8191539sec.
4. How many seconds in 4 successive years, three common
years and one leap year?              Ans. 126230400 sec.
5. Reduce 106847 sec. to hours. Ans. 29 h. 40 min. 47 sec.
6. How often will a clock, ticking once a second, tick in
24 hours?
7. When 12~ hours of a day have gone, how many minutes
remain?
219. How often does Leap Year occur? How many days in a leap year?
Which month receives the additional day?-220. Which years are leap yearsl
-221. Repeat the lines giving the number of days in the months.




COMPOUND NUMBERS.                  125
CrctialaLr   lesasilreo
222. Circular Mleasure is used chiefly in measuring angles and parts of circles, in determining latitude and longitude, and estimating the motions and
positions of the heavenly bodies.
223. Every circle may be divided into 360 equal
parts, called Degrees.  The actual length  of the
degree will of course depend on the size of the
circle.  The Sign is a division of the circle used
only in Astronomy.
TABLE.
60 seconds ("),   1 minute,...
60 minutes,         1 degree,..
30 degrees,         1 sign,...  S.
12 signs (3600),   1 circle,...
Paper -Men-ure.
24 sheets make  1 quire.
20 quires,         1 ream.
2 reams,          1 bundle.
5 bundles,        1 bale.
Caolectioas of Uints,
12 units make  I dozen, doz.
12 dozen,         1 gross.
12 gross,         1 great gross.
20 units,         1 score.
222. In what is Circular Measure chiefly used? —223. Into what may
every circle be divided? On what will the actual length of the degree depend? Recite the Tables.




126              COMPOUND NUMBERS.
EXAMPLES FOR THE SLATE.
1. Reduce 6 signs, 9 degrees, to degrees.
2. In 1000 minutes how many degrees?
3. Reduce 3 8. 18~ to seconds.          Ans. 388800".
4. In 10000" how many degrees, &c.? Ans. 2~ 46' 40".
5. Reduce 45~ 45' 35" to seconds.       Ans. 164735".
6. In 1000 bottles of porter how many dozen?
7. How many buttons in 181 dozen?
8. In 80064 tacks how many gross?
9. What will 480 bottles of ink cost, at $1 a dozen?
10. Pens are put up in boxes containing a gross; how many
pens in 5 dozen boxes?                    Ans. 8640 pens.
11. If a box of pens is bought for 72 cents, what is the
price of each pen?                            Ans. 5 mills.
12. If a paper of tacks containing a gross is sold for 6 cents,
how many does that make for 1 cent?
13. How many great gross in 1378941?
14. How many sheets in one ream of paper?
15. How many sheets in 15  quires?
16. How many reams will 22480 sheets make?
17. How many sheets in 27 reams, 3 quires?
18. If half a ream of paper costs $2.40, what is the cost
of each sheet?                                  Ans. 1 ct.
19. If a stationer buys paper at $2.50 a ream, and retails
it at 20c. a quire, what does he make on a ream?
20. How much paper will it take to make 1000 books,
containing 6 sheets each?                Ans. 121 reams.
21. If 2 circulars are printed on a sheet, how many can be
printed on 5 bundles of paper?         Ans. 9600 circulars.
22. How many reams of paper will it take to print 9696
such circulars?                    Ans. 10 reams, 2 quires.




MISCELLANEOUS EXAMPLES.                  127
MISCELLANEOUS EXAMPLES IN COMPOUND NUMBERS.
1. In 11959 grains, Apothecaries' Weight, how  many
pounds, ounces, &c.?                    Ans. 2 lb. 7 3 19 gr.
2. The highest point of the globe ever attained by man is
the top of Mt. Chimborazo, 19699 feet above sea level. What
is this height in miles?                  Ans. 3 mi. 3859 ft.
3. Reduce 1 square mile, 29 acres, 8 square rods, 7 square
yards, to square feet.                  Ans. 29143881 sq. ft.
4. -low many tumblerfuls, of half a pint each, Will it take
to fill a half-gallon pitcher?
5. Reduce 3  weeks to seconds.          Ans. 2116800 sec.
6. In 1000 cd. ft. of wood, how many cords?
7. 1-low many pages will there be in an edition of 2000
books, each book made of 12 sheets, and each sheet containing 24 pages?                             Ans. 576000 pages.
8. In 4 lb. 5 oz. 1 dr. 2sc. 10 gr., how many grains are
there?                                         Ans. 25550 gr.
9. How many tons of hewn timber in 2500 cu. ft.?
10. Reduce 85274 pt. to bushels.   Ans. 1332 bu. 1 pk. 5 qt.
11. How many lb. of silver will it take to make 4 dozen
spoons weighing 18 pwt. each?         Ans. 3 lb. 7 oz. 4 pwt.
12. What will 15 firkins of butter come to, at 231 cents a
pound?                                         Ans. $197.40.
13. How many doses of 6 drams each will 4 pounds of
epsomn salts make?
14. A farmer wishes to put up 367k bushels of potatoes in
barrels holding 3 bu. 2 pk. each.  How many barrels must he
procure?                                    Ans. 105 barrels.
How many pecks are to be put up? How many pecks will each
barrel hold?
15. At 2 cents a dozen, how much will a great gross of
buttons cost?                                    Ans. $2.88.




128       EXAMPLES IN COMPOUND NUMBERS.
16. I-ow  many days are there in the Spring months,
March, April, and May?
17. How many days in. the Summer months, June, July,
and August?
18. How many days in the Autumn or Fall months, September, October, and November?
19. How many days in the Winter months, December,
January, and February, when February falls in a leap year?
20. Reduce 19 cu. ft. to cubic inches.  Ans. 32832 cu. in.
21. Reduce 5740 pwt. to pounds.        Ans. 231b. 11 oz.
22. If from   of an ounce of gold a jeweller takes enough
to make 6 rings weighing 2? pwt. each, how many pennyweights will he have left?
23. The depth of water at a certain spot is found to be 31
fathoms, 3 ft. How many inches is this?       Ans. 2268 in.
24. How many brushes, at 2s. each, can be bought for
~8?
25. How long will 12 bushels of oats last a horse, if he is
fed 8 quarts a day?
26. At the rate of $4 a cord, what is the value of a pile
of wood, 8 feet long, 4 feet wide, and 4 feet high?
27. If sound moves at the rate of 1120 feet in a second,
how many miles off is a cannon that is heard 11 seconds after
it is discharged?                        Ans. 2 mi. 1760 ft.
28. If a family use 28 lb. of'flour in a week, how long
will 2 barrels last them?
29. How many pounds sterling will 6 dozen combs cost,
at 9d. apiece?                               Ans. ~2 14s.
30. If a locomotive goes a mile in 2 minutes, how many
hours will it take to go 150 miles?              Ans. 5 h.
31. Reduce 15 A. 3 R. 20 sq. rods, 15 sq. yards, 2 sq. ft.
to square inches.                     Ans. 99597888 sq. in.




COMPOUND ADDITION.                  129
COMPOUND ADDITION.
224. Compound numbers may be  added, subtracted, multiplied, and divided.
225. When compound numbers are added, the
process is called Compound Addition.  It combines
addition and reduction ascending.
226. A person spends in one store ~6 5s. 7d.; in
another, ~7 Is. 2d. I far.; in a third, ~1 13s.; and
in a fourth, ~4 18s. Id. 3 far.  How much does he
lay out altogether.
We are here required to find the sum of several compound
numbers. We must add things of the same kind; therefore
write numbers of the same denomination in the same column.
Mark the denominations over the top.
Beginning at the right, add the first
column. Its sum  is 4 farthings, which,         s.  d. far.
by dividing by 4, we reduce to Id. Set, 0      5 6     0
in the column of farthings, and carry 1 to     1  2   1
the next column.                              13   0   0
The sum  of the next column is 11.   4 18  1   3
lid. is not reducible to shillings, since    9     
it takes 12d. to make Is. Set down 11,
therefore, under the column of pence.
The sum of the shillings is 37s. = ~1 I7s. Set 17 under
the column of shillings, and carry ~1 to the next column.
The sum of the next column is ~19. As pounds can not
be reduced to any higher denomination, we set 19 at once
under the column added.  Answer, ~19 T1s. lid.  Hence
the following rule:
224. What operations may be performed on compound numbers?-225.
When compound numbers are added, what is the process called? What
operations does Compound Addition combine —226. Explain the several
steps in the'given example.




130                COMPOUND ADDITION.
227. IPULE. —To add comnpound nu2bers, set tlem
down so that the samC e denominations may stind in
the    e same column.
Beginning at the right, add  the denominations
separately.  Set each sum  under the column  added,
unless it cacn be reduced to a higher denromination.
If so, divide by the number that it takes to make
one of that denomination; set the remainder under
the column added, and carry the quotient.
PROOF.-Prove  the addition, by addin'g in  the
opposite direction.
EXAMPLES FOR  THE  SLATE.
Add the following compound numbers. Always mark the
denominations over the top.
(1)                (2)                    (3)
~   s.  d.       lb. oz. pwt. gr.       b  3   3
55 5             7 3 05                211 6 2
8 1   74        11   2  17 22         10   8   3   1
2 0   1{        40   0   0  20        1410   2   2
13   0  11             6  18  16             6   5   0
6   6   6        2  10  15  17         7   5   4   1
34  14   8        6111  13   8         36   6  6        0
(4)                            (5)
T. cwt. qr. lb. oz. dr.         mi. fur. rd. yd. ft. in.
10  18  2   5  15   1            2  6  37  4  1   9
1  15  0  20  14  15            8  0  30  5  2   2
12   0  1   3   010              14   0  3  2   7
130   7   1  11             7  1   I  0  2  10
2      2  20   7   8            2  025        1  11
27   9  3   7   7  13           21  5  15  5  2   3
227. Recite the rule for Compound Addition. How may the addition be
proved?




EXAMPLES IN OOMPOUND ADDITION.                  131
(6)                     (7)                   (8)
sq. yd. sq. ft. sq. in.  cu. yd. cu. ft. cu. in.   Cd. cd. ft.
100    8      130        4     26   1000            3    7
50     0    100          1     10   1541          10    4
10     5       0              20       80        12    1
8    143        10    17       11          8    6
13    2        8        8    25        59        15    3
175     7      93       26    18       963        50    5
(9)                             (10)
pi. hhd. gal.  qt. pt.    tun  pi. hhd. gal.  qt. pt. gi.
1    1   30   3   1          1   1   1   37   3   1   3
10   25   1   1        10   0   0   50   0   1   2
25    0   2   0        11   0   1   13   1   0   1
7   60   1   I         4   1   0   25   2   0   0
1    1   45   3   1          8   0   1   18   0   1   3
(11)                  (12)                  (13)
bu. pk. qt. pt.      da.  h. min. sec.,,
10  I  1  1          15  18  50  49           13  10  19
2  3  6  0           1  13  59  59            1  40  35
5  2  3  1           4  23  47   2            2  48  39
8  3  1  1               2  10  15               30  40
15  2  4  0          10  11   1   4           10  45  45
14. Find the sum  of 2 hhd. 50gal. 3 qt. 1 pt. (Beer
Measure); 10 hhd. 30 gal. 1 qt.; 11 hhd. 25 gal. 1 pt.; 25
hhd. 1 gal. 1 qt.; and 6 hhd. 52 gal. 3 qt. 1 pt.
Ans. 56hhd. 52 gal. 1 qt. 1pt.
15. Add together ~7 13s. 3d.; ~3 5s. 10d. 2 far.; ~6 18s.
7d.; 2s. 5d. 3 far.; ~4 3d.; and ~17 15s. 4d. 2 far.
Ans. ~39 15s. 9d. 3 far.
16. Add together 1 dr. 18gr.; 2 dr. 1 so. 15gr.; 3 dr. 2sc.
13gr.; 4 dr.; and 6dr.  sc. 7gr.         Ans. 2oz. 2 dr. 13gr.
17. A stable-keeper uses 8 bu. 2 pk. of oats, one day; 7 bu.
3pk. 7 qt., the next; 7 bu. 6 qt., the third; 6 bu. 2pk. 1pt.,




132       EXAMPLES IN COMPOUND ADDITION,
the fourth; 8bu., the fifth; 7bu. 1 pk. 6 qt. 1 pt., the sixth;
and 6 bu. 3 pk. 5 qt., the seventh. How much does he use
during the week?                      Ans. 52 bu. 2 pk. 1 qt.
18. If one piece of cloth contains 40yd.; another, 391
yd.; a third, 381 yd.; a fourth, 391 yd.; and a fifth, 40~ yd.;
how many yards are there in all?               Ans. 1981 yd.
19. What is the weight of four lots of iron, the first
weighing 4 cwt. 3 qr. 20 lb.; the second, 2 T. 5 cwt. 14 b.;
the third, 1T. 2 cwt. 2 qr.; and the fourth, 10 T. 19 cwt.
1 qr. 24 lb.?                        Ans. 14 T. 12 cwt. 8 lb.
20. How much paper will a printer use for three jobs, if
the first job requires 6 bales, 1 bundle; the second, 4 bundles, 1 ream, 15 quires; and the third, 2 reams, 10 quires,
12 sheets?      Ans. 7 bales, 2 bundles, 15 quires, 12 sheets.
21. A person owns five farms. The first contains 100A.
1 R. 30 sq. rd.; the second, 600 A. 2 R. 10 sq. rd.; the third,
40 A. 1 R. 12 sq. rd.; the fourth, 250 A. 3 R. 2 sq. rd.; the
fifth, 144 A. 20 sq. rd. How much land does he own in all?
Ans. 1136 A. 34 sq. rd.
22. A manufacturer makes four lots of pens. The first
consists of 20 great gross, 7 gross, 5 dozen, and 6; the second,
of 9 gross, 10 dozen, and 5; the third, of 15 great gross, 11
dozen; and the fourth, of 17 great gross, 3 gross. What is
the whole amount made?
Ans. 53 great gross, 9 gross, 2 dozen, and 11.
23. Find the sum of 1 wk. 2 days 13 h. 40 min. 30 sec.;
2 wk. 6 days 10 h. 8 min. 3 sec.; 5 days 22 h. 55 min. 45 sec.;
4h. 1 min. 15 sec.; and 1 wk. 2 days 4h. 5 min.
Ans. 6 wk. 3 days 6 h. 50 min. 33 sec.
24. Add together 10 rd. 4 yd. 2 ft. 8 in.; I rd. 3 yd. 5 in.;
8rd. 2yd. Ift, 6in.;   rd. 4in.; and 2yd. 1 ft. 9 in.
Ans. 22 rd. 2 yd. 8in.




COMPOUND SUBTRACTION.                 133
COMPOUND SUBTRACTION.
228. When one compound number is taken from
another, the process is called Compound Subtraction.
It combines subtraction and reduction descending.
229. A  person who had  1 cwt. 8 b. 15 oz. of
cheese, sold 3 qr. 19 lb. 7 oz.  How  much had he
left.
We are here required to find the difference between two
compound numbers. Write the subtrahend under the minuend,
placing numbers of the same denomination in the same column.
Mark the denominations over the top.
Begin to subtract at the right. 7 oz.  cwt. qr.  lb,  oz,
from  15 oz. leave 8 oz.; set down  8    1  0    8  15
beneath, in the same column.  19 lb.          3  19    7
can not be taken from 8 lb. We therefore  As.     14    8
take one of the next higher denomination, 1 qr., reduce it to pounds, and add it to 8 lb. 25 + 8=33;
19 lb. from 33 lb. leave 14 lb. Set down 14.
To balance the quarter added to the minuend, we must
add I qr. to the subtrahend. This we do, by carrying 1 to
the next column.
1 and 3 are 4. 4 qr. can not be taken from 0 qr. We
therefore take 1 cwt., reduce it to quarters, and add it to 0 qr.
4 + 0 - 4 qr. 4 qr. from 4 qr. leave O qr. Carry 1. 1 from
1, 0, Answer, 14 lb. 8 oz. Hence the rule.
230. RULE. —To subtract a compound  number,
set it under the minuend, placing numbers of the
same denomination in the same column.
228. When one compound number is taken from another, what is the
process called? What does Compound Subtraction combine?-229. Go
through the example, explaining the steps.-230. Give the rule for Compound
Subtraction.
12




134              COMPOUND SUBTRACTION.
Beginning at the right, subtract each denomination separamtely  and place the remainder in the same
column with the number slbtracted.
If, in any denomination, the subtrahend is greater
than the minuend, add to the latter as many as make
one of the next higher denomination.  Subtract, and
carary I to the next denomination of the subtrahend.
PRooF. —Add   the  remainder  and  subtrahend.
If  the sum   is equal to the minuend, the work is
rilght.
EXAMPLES FOR  THE  SLATE.
(1)                        (2)
yr. mo. wk. da.       T. cwt. qr. lb.  oz. dr.
From  17  8  3  1            13  18  1  20   0  13
Take   4  1  2  6            10   0  3  21  12   0
Ans.  13  7  0  2             3  17  1  23   4  13
(3)                    (4)                (5)
lb    5  3        gr.      mi. fur. rd.        A. R. sq rd.
24   7  2  1  16            60  0   0          69  3  25
16  10  3  2  17           40  7  39           10  0  38
7 8  6  1  19             19  0   1           59  2  27
(6)                   (7)                    (8)
cu.yd. cu.ft. cu.in.    ch. bu. pk.         tun pi. hhd. gal. qt.
144  12  123           30  10  1           101 1  150  1
89  23  869           10   8  3            1  0  0  60  3
9. A grocer buys 15 cwt. 20 lb. of sugar, and sells 10 cwt.
23 lb.  How much remains unsold?  Ans. 4 cwt. 3 qr. 22 lb.
10. From  a piece of cloth containing 373 yd., have been
cut off 61 yd. for one dress and 106 yd. for another.  How
many yards remain in the piece?                   As. 21 3yd.
What is the proof in Compound Subtraction?




EXAMPLES IN  COMPOUN-D SUBTRACTION.    135
11. A person having ~20 18s., spends ~5 18s. 7Vd.  How
much lias he left?                        Ans. ~1 4  19s. 4-d.
12. A farmer raises 100 bu. 3 pk. 2 qt. of wheat from one
field, and  87 bu. 1 pk. I qt. I pt. from  another.  He sells
53 bu. to one person, and 37 bu. 2 pk. 1 qt. to another. How
much has he left?                 Ans. 97 bu. 2 pk. 2 qt. I pt.
13. From  a pile of wood containing 100 cords, I sold
10 Cd. 100 cu. ft. to one customer, and  18 Cd. 59 cu. ft. to
another.  How many cords remained? Ans. 70 Cd. 97 cu. ft.
14. The subtrahend is 19 mi. 7 fur. 8 rd. 2 ft. 10 in.; the
minuend is 24 mi. 5 fur. 18 rd. 2 yd. 1 ft. 7 in. What is the
remainder?           Ans. 4 mi. 6 fur. 10 rd. I yd. 1 ft. 9 in.
15. A printer who has a bale of paper, uses 1 bundle,
1 ream, 5 quires, and 6 sheets.  How much remains on hand?
16. A tailor uses 5 gross, 7 dozen, of buttons.  Iow many
has he on hand, if he had 1 great gross at first?
17. Poughkeepsie is 75 miles from  New  York.  A  man
who starts to walk- there, goes 22 mi. 3fur. 15 rd. the first
day, and 19 mi. 5 fur. 30 rd. the second. How  much farther has he to go?                   Ans. 32 mi. 6 fur. 35 rd.
18. A jeweller, having a bar of silver weighing 2 lb. 6 oz.,
used 5 oz. 7pwt. 12 gr. for one job, and 1 lb. 18 pwt. 7 gr.
for another.  How much was left?  Ans. 11 oz. 14 pwt. 5 gr.
19. From  a barrel of beer containing 54 gallons, a person drew 12 gal. 3 qt. one day, and 9 gal. 2 qt. 1 pt. another.
How much was left?
20. From 39 sq. rd. 29 sq. yd. 128 sq. in., subtract 17 sq. rd.
16 sq. yd. 5 sq. ft.
21. A  grocer has lcwt. 18 lb. of sugar in one barrel,
3qr. 21 b. in another, and lcwt. 2qr.  llb. in a third.
After selling lcwt. 3qr. 15lb., how  much will he have
left?                                  Ans. 1 cwt. 3qr. 10llb.




136           COMPOUND MULTIPLICATION.
COMPOUND MULTIPLICATION.
231. When a compound number is multiplied,
the process is called  Compound miultiplication.  It
combines multiplication and reduction ascending.
232. Multiply 4 bu. 3pk. 7 qt. 1pt. by 27.
27 being the product of 9 and 3, it is best to multiply by
the factors in turn. Set 9, the first multiplier, under the
lowest denomination of the multiplicand.
Begin to multiply at the right. 9 times   bu. pk. qt. pt.
1 pint' is 9 pt.; which, by dividing by 2,    4   3  7  1
we reduce to 4 qt. 1 pt. Set 1 pt. in the               9
column of pints, and carry 4 qt. to the    44   3  3   i
next product.                                           3
9 times T qt. is 63 qt., and the 4qt.            
carried make 67qt., equal to 8pk. 3 qt.  1
Set 3 in the column of quarts, and carry 8 pk. to the next
product.
9 times 3 pk. is 27 pk., and the 8 pk. carried make 35 pk.,
equal to 8 bu. 3 pk. Set 3 in the column of pecks, and carry
8 bu. to the next product.
9 times 4 bu. is 36 bu., and the 8 bu. carried make 44 bu.
Set down 44.
Now multiply this product by 3, reducing and carrying in
the same way. Answer, 134 bu. 2 pk. 2 qt. 1 pt.
233. Had we multiplied by 27 at once, we should have
proceeded in the same way. As, however, we could not then
have multiplied and divided in the mind, we should have had
to write out the figures elsewhere, and set the results only
under the line.-Multiply by 12 or less at once, in one line.
231. When a compound number is multiplied, what is the process called?
What does Compound Multiplication combine — 232. Go through the example, explaining the steps.-233. Had we multiplied by 27 at once, how
should we have proceeded? How must we multiply by 12 or less?




COMPOUND MULTIPLICATION.                   137
234. RULE.-Set the multiplier under the lowest
denomnination of the nultiplicand.
Ildtiply each denomination in turn, and set the
product under the number multiplied, unless it can
be reduced to a higher denomination.  If so, divide
it by the number that it' takes to make one of that
denomination; set the remncinder under the number multiplied, and carry the quotient to the next
product.
PRooF. —TW en factors have been used in qnultiplying, multiply by the factors in reverse order.
If the results agree, the work is right.
EXAMPLES FOE  THE  SLATE.
(1)              (2)              (3)
yd. ft. in.      ~   s.   d.          oz. pwt. gr.
Multiply  1   0   9        10  10  10        8   7   5   2
By                 4                 3                   7
Ans.      4   3   0       31  12   6       60   21514
(4)                            (5)
T. cwt. qr. lb. oz. dr.        hhd. gal. qt. pt. gi.
8   0   2  4   5             1   2   3   1   2
6                           17
2   8   013   9 14              1749  3   1   2
6. Multiply 5 cubic yards, 21 cubic feet, 643 cubic inches
by 12.                    Ans. 69 cu. yd. 13 cu. ft. 804 cu. in.
7. How much cloth will it take for 7 suits of clothes, if
each suit requires 7 yd. 3 qr.?             Ans. 54 yd. 1 qr.
8. How much wood can a horse draw in 13 loads, if he
draws 1 Cd. 1 cd. ft. each load?
234. Recite the rule for Compound Multiplication. What is the mode of
proof, when factors have been used in multiplying?




138            COMPeOUND MULTIPLICATION.
9. How long will a man be in sawing 6 cords of wood, if
he takes 7 h. 30 min. 45 sec. to saw 1 cord, allowing 10 working hours to each day?       Ans. 4 days 5 h. 4 min. 30 sec.
Multiply in the usual way; then reduce the hours to working days
by dividing by 10.
10. H-ow much sugar is there in 21 hhd., each containing
11 cwt.   qr. 15 lb.?           Ans. 12T. 9cwt. 3qr. 15 lb.
11. Bought 15 yd. of broadcloth, at ~1 3s. 6d. a yard, and
22 yd. of silk, at 7s. 8d. 2 far. a yd.  What was the amount of
the bill?                                    Ans. ~26 2s. Id.
Find the cost of each item; then add.
12. The exact time in a year is 365 days 5h. 48 min.
49-0- sec. What is the exact time in 50 years?
(50=5 x 10) Ans. 18262 days 2 h. 41 min. 25 sec.
13. How much brandy in 84 pi., each containing 128 gal.
2qt. 1 pt. 3gi.?                    Ans. 10812 gal. 1 qt. 1 pt.
14. If a man owning 5 farms, of 120A. 1R. 12 sq. rd.
each, sells 450 A. 3E. 25sq.rd., how  much land has he
left?                              Ans. 150 A. 2 R. 35 sq. rd.
15. Bought 17 boxes of raisins, at 12s. 4d. a box; 5 bar.
of flour, at ~1 10s. 6d. a barrel; and 16 lb. of tea, at 5s. 3Id.
a lb.  Paid on account ~19 10s.; how much remains unpaid?                                        Ans. ~2 16s. 6d.
16. What will be the yield of 32 acres of wheat, at the
rate of 24 bu. 2 pk. 7 qt. per acre?              Ans. 791 bu.
17. If 2 gal. 2 qt. 1 pt. 1 gi. leak out of a water pipe in 1
hour, what will be the waste in 1 day?    Ans. 63 gal. 3 qt.
18. Suppose a person to walk, on an average, 3 mi. 2 fur.
every morning, and 3 mi. 20rd. I yd. every afternoon; how
far will he walk in two weeks?  Ans. 88 mi. 3 fur. 2 rd. 3 yd.
19. If from 2 lb. of silver enough is taken to make a dozen
spoons, weighing 1 oz. 10 pwt. 2 gr. each, how much will be
left?                                       Ans. 5 oz. 19 pwt.




cOMPOTND DIVISION.                   139
COMPOUND DIVISION.
235. When a compo-und number is divided, the
process is called Compound Division.   It combines
division and reduction descending.
236. Divide 148 gal. 3 qt. 1 pt. 3 gi. by 23.
Here we must use Long Division. Remember that a quotient is of the same denomination as the dividend from which
it arises.  Begin to divide at the left.
Divide 148 gal. by 23: quo-                     
ient, 6 gal.; remainder, 10 gaL     23)148  3  1  3 (6 gal
Reduce the remainder to qt.,            138
and add in the 3 qt. in the divi-       -1 g
dend.  10x4=40  40+3=43.                  4
Divide 43 qt. by 23: quo-         23)43 qt. (1 qt.
tient,   qt.; remainder, 20qt.           23
Reduce the remainder to pt.,             20 qt.
and add in the 1 pt. in the               2
dividend. 20x2=40  40+1              23 ) 41 pt. (1 pt.
=41.                                     23
Divide 41 pt. by 23: quo-            18 pt.
tient, 1 pt.; remainder, 18 pt.           4
Reduce the remainder to gi.,         23)75 gi. (3 — g1
69
and add in the 3 gi. in the divi-        6
dend. 18x4=72  72+3=T5.                   6 gi.
Divide 75 gi. by 23: quo-  Ans. 6gal. 1 qt. I pt. 34- gi.
tient, 3 gi.;  remainder, 6 ri.
As there is no lower denomination to reduce this remainder
to, write it over the divisor in the form of a fraction, -.
Collect the several quotients for the answer.
If in any case there is no remainder, bring down the next
denomination of the dividend, and proceed as above.
235. When a compound number is divided, what is the process called?
What does Compound Division combine?-236. Go through the example, explaning the steps.




140                COMPOUND DIVISION,
If the divisor is not contained in any dividend, set 0 in
the quotient for that denomination, and reduce the dividend
to the next.
237. RULE.-Beginning at the left, divide each
deCnominzctlion in turnI. WTIhen there is a remcci2,der;
reduce it to the next lower denominationT, cdd in the
number of that denomination in the given divideqnd
if any, acnd cogntinue the division.
When there is a remnainder after the last division,
place it over the divisor, in the form, of a fraction,
anCd ca ex it to the last quotient.  STe several quotients, each of the same denominz ation as its dividend, form  the entire quotient.
PnooF.-JMultiply  the qutotient by  the  divisor,
If theirproduct is equal to the dividend, the work
is right.
238. When the divisor is less than 12, use Short Division,
EXBAMPLES FOR  THE SLATE.
(1)                            (2)
mi. fur. rd. yd. ft.          cwt. qr. lb.  oz. dr.
7)19  2  35  4  1             9)27  3  17  13  9
2  6   5  0  1                3  0  10   5  1
3. Divide 8 oz. 16 pwt. by 13.     Ans. 13 pwt. 121, gr
4. Divide 22 rd. 1 yd. 1 ft. 10 in. by 11. Ans. 2 rd. 5-a- in
5. Divide ~6 15s. 3d. by 10.        Ans. 13s. 6d. 1l far.
6. Divide 86 bu. I pk. 1 pt. by 14. Ans. 6 bu. 5 qt. -t5 pt.
7. Divide 102A. 1 B. 11 sq. rd. by 51. Ans. 2.. I sq.rd.
8. Divide 4 gal. 2 qt. by 144.                 Ans. 1 gl.
9. Divide 40cu. yd. 10 cu. ft. by 18.
237. Recite the rule for Compound Division. What is the proof?




EXA APLES IN COMPOUND DIVISION.               141
10. If 31. clocks cost ~113 13s. 4d., how  much is that
apiece?                                    Ans. ~3 13s. 4d.
11. If 6 oz. 7 dr. (Apothecaries' Weight) of magnesia are
put up in 60 equal parcels, how much will each weigh?
Ans. 2sc. 15gr.
12. A silversmith makes seven teapots, of equal weight,
out of 9 lb. I oz. 14 pwt 5 gr. of silver.  What is the weight
of each?                      Ans. I lb. 3 oz. 13pwt. 11 gr.
13. If 47 casks, of the same size, hold 1686 gall. 1 pt. of
beer, how much will each contain?   Ans. 35 gal. 3 qt. I pt.
14. If a traveller goes 600 miles in 1 day 7h. 35 min. 20
sec., what is his average time per mile? Ans 3 min. 9 -a sec.
15. Divide 6 bales, 3 bundles, 1 ream, of paper into 8
equal parts.              Ans. 4 bundles, 7 quires, 12 sheets.
16. A farmer puts up 1000 bushels of apples in 350 barrels
of uniform  size. How many bushels, &c., does each barrel
contain?                              Ans. 2 bu. 3 pk. 3 qt.
17. An estate worth ~2570 is to be divided as follows:
the widow is to have one third of the whole, and the rest is
to be divided equally between seven children. What is the
widow's share, and what each child's?
An. (Widow's, ~856 13s. 4d.
Child's,  ~244 15s. 2d. 33 far.
18. What is the weight of 13 crowns, each weighing
18 pwt. 44- gr.; 14 shillings, each weighing 3 pwt. 15- gr.;
and 9 sixpences, each weighing 1 pwt. 19-7 gr.?
Ans. I lb. 3 oz. 3pwt. 15- Lgr.
19. A farmer having 450 bu.  pk. I qt. of corn, after
selling 425 bu. 3 pk. 6 qt., distributed the rest equally among
5 poor families. How much did each receive?
Ans. 4 bu. 3pk. 3qt. 1 pt.
20. Divide 182~ 5' 12" by 12.




142            MISCELLANEOUS EXAMPLES.
MISCELLANEOUS EXAMPLES.
1. A person has $741.35 in one bank, $350 in another,
and $1129.88 in a third; how  much has he in bank altogether?
2. At 40 cents a yard, what will be the cost of 5 miles
of telegraph cable?                             Ans. $3520.
8. The silk-worm's thread is about -oog of an inch thick;
the spider's web is about I as thick.  How thick is it?
4. Reduce 27 to its lowest terms.
5. How many pens in 8-^ gross?
6. Three boys, gathering nuts, agree to divide equally all
they get.  The first gathers 540, the second 960, the third
720. How many does each receive?             Ans. 740 nuts.
7. Cut from a piece of cloth containing 36- yd., enough
cloth to make 10 coats, each requiring 2} yd., and how much
will be left?
8. The product of two factors is 294. One of the factors
is 81; what is the other?                           Ans. 3-.
9. If 1 sovereign is worth $4.84, what are 27 sovereigns
worth?
10. A man who has ^ mile the start of another, gains on
him 30 rods more. How'many feet is he then ahead of the
other?
11. If a boy who has been in the habit of sleeping 9 h.,
rises an hour earlier every day, how many days will he save
in 5 years, allowing for one leap year?      Ans. 76 da. 2h.
12. From  2} subtract 1l, and multiply the remainder
by 3^.                                              Ans. 2.
13. If a man has 3 small farms, of 8 fields each, and each
field contains 2 A. 3 R. 22 sq. rd., how much land has he in
all?                                Ans. 69 A. I R. 8 sq. rd.




MISCELLANEOUS EXAMPLES.                  143
14. From a piece of cloth containing 1211 yd., a tailor
made 18 coats, which took one third of the whole piece.
How many yards did each coat contain?            Ans. 2\ yd.
15. How many pints in 261 gal.?            Ans. 2101 pt.
16. A man worth $10000, at his decease, left $225.75 to
the poor, and the rest to his five sons, in equal shares. How
much did each get?                           Ans. $1954.85.
17. How many five-acre lots are there in a square mile?
18. A trader, making four speculations, in the first gains
$1520, in the second loses $460, in the third loses $720.50,
and in the fourth gains $986.25.  Does he gain or lose on the
whole, and how much?                   Ans. Gains $1325.75.
19. A man owning a mine, sells - of it to one person,
O df it to another, and ~ of it to a third. How much does
lie sell altogether?                               A ns.'-7.
20. How many lots of half a rood each can be laid out
in 6  acres?                                    Ans. 52 lots.
21. How many guineas are equivalent to ~42?
22. A man buys 7 head of cattle for $210.  Two die, and
he sells the rest for $34.50 apiece. How much does he lose?
23. Add f of 90, 2 of 129, and - of 1260.      Ans. 1001.
24. If from I gal. of wine 3 qt. 11 pt. are sold, how much
will remain?                                       Ans. ~ pt.
25. How many sheets of paper in three lots, consisting
of 1  reams, 5 quires, and 1 bundle?      Ans. 1800 sheets.
26. Three persons, going on an excursion, agree to share
the expense equally.  They hire a carriage for $2.25, a boat
for 75c.  Their dinner costs them  50c. each, and their tea
$1.20 for all four.  How much has each to pay?  Ans. $1.90.
27. Which contains the most units, 4  dozen, I of a gross,
or 3e of a great gross?
28. From  4 of 6- subtract I of 21.               Ans. 4.




144           MISCELLANEOUS EXAMPLES.
29. How much is a man worth, over and above his debts,
whose whole property is valued at ~5100, and who owes one
person ~75 10s. and another ~427 19s. 6d. 2 far.?
Ans. ~4596 1Os. 5d. 2 far.
30. When a man who has 3- mi. to walk, has gone halfway, how many miles has he yet to go?
31. How many dozen in 3 of a great gross?
32. How many times is 8- contained in 92?     Ans. 1.
33. A crown is worth 5s. How many crowns are equivalent to ~25 10s.?
34. Two regiments, of 1125 men each, reach a river.
They have 5 boats, capable of carrying 25 men each. How
many trips must the boats make, to get them over 
35, Reduce to a simple fraction 2 of 4 of - of 27 of f
of " of 1.                                        Ans. 5.
36. An officer, pursuing a thief, gains on him 4 of a mile
every hour. If the thief is 7T miles ahead, how many hours
will it be before he is overtaken?             Ans. 9- h.
37. How many more seconds does a leap year contain
than an ordinary year?                     Ans. 86400 sec.
38. A lad who has 288 marbles, loses ~ of them, sells 4,
and gives away D-U of them. How many marbles has he
left?                                     Ans. 48 marbles.
39. How many cows, at $30 apiece, must a farmer give
for 6 acres of land, at $15 an acre?          Ans. 3 cows.
Find the value of the land; how many cows, at $30, will pay for it?
40. A miller buys 22- cords of wood at $4 a cord, and
pays for it in flour at $6 a barrel. How many barrels must
he give?
41. A weekly paper has been in iexistence 10  years. If
each paper contains 8 pages, and each page 6 columns, how
many columns has it contained during that time?