A PRACTICAL
ARITHMETIC
STEVENS
M
BUTLER
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DATE DUE
MAY 20
1968
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GAYLORD
PRINTEDINU.S.A.
Cornell University Library
QA 106.S84
A practical arithmetic,
3 1924 000 431 621
A PRACTICAL ARITHMETIC
The original of tliis book is in
tine Cornell University Library.
There are no known copyright restrictions in
the United States on the use of the text.
http://www.archive.org/details/cu31924000431621
PEACTICAL ARITHMETIC
BY
F. L. STEVENS
PROFESSOR IN THE NORTH CAROLINA COLLEGE OF AGRICULTURE
AND MECHANIC ARTS, AUTHOR OP " AGRICULTURE
FOR beginners"
TAIT BUTLER
president AMERICAN ASSOCIATION OF FARMERS'
INSTITUTE WORKERS
MRS. F. L. STEVENS
FORMERLY TRAINING TEACHER, COLUMBUS, OHIO
NORMAL SCHOOL ,
r} f
NEW YORK
CHARLES SCRIBNER'S SONS
1910
\
s^ •■
' u ^
''*/,,
meTH
COPYRIGHT, 1909, BY
CHARLES SCRIBNER's SONS
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^^I^\\\\\^,/
li'
PEEFACE
The primary object of arithmetic is to enable tlie
student to acquire skill in computation. In addition to
the attainment of this essential end, great benefit is de
rived from the exercise of the reasoning powers and
their consequent development. While the first of these
must ever remain the fundamental reason for the study
of arithmetic, and the second will always be held in
high esteem, there is a third major object which the
teaching of arithmetic may accomplish, one which is usu
ally almost entirely ignored in the preparation of an
arithmetic ; namely, the incidental teaching of valuable
facts by basing the problems of the book upon the prob
lems of real life.
In the preparation of this book, it has been the aim
of the authors to secure the maximum results in these
three functions of arithmetic teaching.
It is chiefly in the careful consideration which has been
given to the subjectmatter of the problems, and to the
inferences that will unconsciously and unavoidably remain
in the mind of the pupil, that this book differs from other
arithmetics.
Skill in computation comes from learning a few methods,
followed by extensive drill or practice. Methods have
been carefully and clearly presented in this book, and
an abundance of drill problems provided.
The development of the reasoning powers comes from
work with problems requiring careful analysis before pro
ceeding to the more mechanical solution. A large number
VI PREFACE
of carefully graded thought problems, necessitating accu
rate analysis, serves this end.
The special value of this book, however, depends upon
the fact that a large proportion of its problems bring out
clearly in their statement or in their solution important
facts bearing upon the practical activities of life. Since
agriculture is the one fundamental industry of America,
especial attention has been given to this subject, and a
large proportion of the thought problems are based upon
agriculture, without, however, in any way leading to
neglect of other industries.
The problems relating to agriculture are based upon
wholly reliable information, uf)on the most recent find
ings of the State Experiment Stations and of the National
Department of Agriculture. The facts used in these
problems and the legitimate inferences which may be
djiawn from them are trustworthy. In solving these
problems, the pupil will unconsciously absorb and retain
many valuable facts and principles relating to agricul
tural practice, such, for example, as the value of seed
selection, purity and vitality, judicious use of fertilizers,
balancing of animal rations, crop rotation, prevention or
treatment for plant diseases, conservation of soil moisture,
preservation of soil fertility, prevention of insect injury,
economy in methods of harvesting, proper dairy methods,
the improvement of the herd by selection, poultry culture,
value of good roads, etc.
A feature of value is the outline problems to be com
pleted by the pupils with data from their homes.
Teachers, parents and pupils are invited to write to the
authors of this book for information upon any agricultural
points involved.
THE AUTHORS.
Raleigh, N.C, November, 1908.
CONTENTS
FAQS
Notation and Numeration 1
The Arabic System 1
The Roman System 9
United States Money 11
Addition 13
Addition of United States Money 19
Subtraction . 29
Subtraction of United States Money 36
Multiplication 41
Multiplication of United States Money .... 52
Division . . r ^ 57
Short Division 61
Long Division 64
Division of United States Money 68
Cancellation 72
Review Problems 74
Divisors and Multiples 86
Tests of Divisibility, Greatest Common Divisor, Least Com
mon Multiple.
Decimal Fractions 92
Notation and Numeration 92
Addition 95
Subtraction 97
Multiplication 99
Division 103
Review Problems 106
Common Fractions 113
Addition 120
vii
viii CONTENTS
PAGE
Subtraction 122
Multiplication 125
Division 129
Review Problems 138
Accounts and Bills 147
Denominate Numbers 150
Units of Length, Reduction, Metric Units of Measure,
Representation of Magnitudes, Surface Measure (English
and Metric), Surveyor's Measures, Measures of Volume
(English and Metric), Measures of Weight (English and
Metric), Measures of Time, Angle Measure, Counting,
Addition and Subtraction, Multiplication, Division.
Review Problems 179
Measurements 186
Practical Measurements 198
Plastering, Painting, Paving, Carpeting, Papering, Masonry
and Brickwork, Wood Measure, Board Measure, Round
Logs, Temperature, Longitude and Time, Standard Time.
Review Problems 216
Percentage 227
Profit and Loss 256
Commission 260
Commercial Discount 263
Insurance .......... 265
Taxes 268
Interest 275
Stocks and Bonds ......... 285
Bank Discount 293
Partial Payments 296
Ratio 302
The Nutritive Ratio, Specific Gravity.
Proportion 308
Levers, Compound Proportion.
CONTENTS ix
PAGE
Powers 318
Roots 321
Miscellaneous Review Problems 326
Appendix 367
Surfaces of Solids, Volumes of Solids, Extraction of Cube
Root, Proof of Fundamental Processes by casting out
Nines, Arithmetical Progression, Geometrical Progression,
Tables of Measures, Weights of Produce, Interest Tables,
Cattlemen's Notation, Lumbermen's Notation.
PKACTICAL ARITHMETIC
oiOic^
NOTATION AND NUMERATION
EXERCISE 1. — ORAL
1. How many ones in 2, 8, 9?
2. How many tens in 20, 30, 50?
3. How many tens and ones in 18, 36, 45, 47, 98 ?
4. How many one hundreds in 200, 400, 600, 900?
5. How many one hundreds, tens, and ones in :
876 425 743 437 982
123 896 456 847 225
378 549 874 953 629
6. How many one hundreds in this number, 1000?
7. What name is given to this number?
8. How many thousands, hundreds, tens, and ones in :
6387 4702 6512 5068 6728
7080 1400 8150 6740 4963
8824 1814 3096 2263 9184
9. How many thousands in this number, 10000?
10. How many tenthousands, thousands, hundreds,
tens, and ones in :
50207 34291 23845 10205 23814
35842 78354 91846 35841 87961
26459 52796 87964 88249 18462
1
PRACTICAL ARITHMETIC
11. How many thousands in this number, 100000?
12. How many hundredthousands, tenthousands, thou
sands, hundreds, tens, and ones in:
259132
271186
350006
495271
468796
660878
504001
116006
203841
398178
802136
275360
275253
134410
884192
1. Hundreds, tens, and ones written together form a
group or period, called Units' Period.
2. Hundredthousands, tenthousands, and thousands
written together form a period, called Thousands' Period.
3. The next period higher than thousands' period is
called Millions' Period ; the next higher, Billions' Period,
and the next Trillions' Period ; but rarely is there use for
these larger numbers.
4. The following diagram will aid in reading large
numbers. Read the numbers given :
Names of
Trill
Bill
Mill
Thou
Units
Periods :
ions
ions
ions
sands
Periods :
1
11
■* Pi
^1
o
Orders :
hundreds
tens
units
hundreds
tens
units
hundreds
tens
units
liundreds
tens
units
hundreds
tens
units
000
12 9
654
00
00
004
20 1
250
000
00
020
045
600
020
006
302
4 63
001
204 875 001 609 451
NOTATION AND NUMERATION 3
5. Numbers of more than four figures are usually
written with a comma between the periods, thus:
1,642,001 63,105,005 78,121
6. To read a number. Begin at the right and point
off into periods of three figures each ; then begin at the
left and read each period as if it stood alone, adding the
name of the period.
7. The place value of a figure. What effect does it
have upon the value of a figure to move it one place to
the left in its period? To move it one place to the right?
jNIoving a figure one place to the left increases its
value tenfold. iNIoving a figure one place to the right
decreases its value tenfold.
exercise 2. — oral
The Meaning of Numbers
There are 5 people in my neighbor's home : father,
mother, daughter, and 2 sons. In 20 such homes there
would be 100 people. In a small village of 100 homes
there are about 500 inhabitants. Ten times as many peo
ple as this in one community would be 5000, and in 100
such towns together there would be 500,000 people.
The following numbers show the population of some
capital cities in 1900. Read the numbers and try to
realize their meaning :
1. Albany,
94,151
3. Richmond,
85,050
2. Harrisburg,
50,167
4. Trenton,
73,307
PRACTICAL ARITHMETIC
5.
Dover,
3,329
15.
Jefferson Cit
y, 9,664
6.
Baltimore,
508,957
16.
Madison,
19,164
7.
Augusta,
39,441
17.
Lansing,
16,485
8.
Boston,
560,892
18.
Columbus,
125,560
9.
Concord,
19,632
19.
Springfield,
34,157
10.
Providence,
175,597
20.
Raleigh,
13,643
11.
Bismarck,
3,319
21.
Jackson,
7,816
12.
Pierre,
2,305
22.
Tallahassee,
2,981
13.
Lincoln,
40,167
23.
Phoenix,
5,544
14.
St. Paul,
163,065
24.
Atlanta,
89,872
Read the following numbers, which express the corn
and wheat produced and the number of milk cows of cer
tain states in 1906 :
COEN, Bushels
Wheat, Bushels
Milk Cows
25.
North Carolina,
41,796,846
6,297,028
282,600
26.
New York,
22,685,000
9,350,180
1,826,211
27.
Georgia,
52,066,596
3,161,070
305,469
28.
Ohio,
141,645,000
43,202,100
919,100
29.
[Mississippi,
40,789,207
17,610
329,664
30.
Iowa,
373,275,000
9,212,218
1,555,300
31.
Texas,
155,804,782
14,126,186
993,122
32.
Kansas,
195,075,000
81,830,611
729,274
8. Any one thing is called a Unit.
9. A unit or collection of units is called a Number.
10. Numbers representing whole units are called Whole
Numbers, Integral Numbers, or Integers.
NOTATION AND NUMERATION 5
11. Figures or Digits are symbols used to express
numbers.
12. The process of reading numbers is called Numeration.
13. Numbers may be expressed by Figures, Letters, or
Words.
EXERCISE 3. — WRITTEN
Write these numbers in figures, using the comma to
separate periods :
1. Six hundred seventyfive.
2. Two hundred thirteen.
3. Four hundred ninetysix.
4. Two hundred twentynine.
5. Fouj hundred eight.
6. One thousand, three hundred fiftytwo.
7. Six thousand, forty.
8. Eighty thousand, eighty.
9. Seven thousand, three hundred.
10. Thirteen thousand, four hundred fifty.
11. Ninetynine thousand, nine.
12. Fortyfour thousand, sixteen.
13. Four hundred six thousand, one hundred fifty.
14. Three thousand, fourteen.
15. Nine thousand, seventyseven.
16. Fifty thousand, sixtyeight.
17. Eleven thousand, nine hundred seventy three.
18. Seven hundred eightyfive thousand, two,
6 PRACTICAL ARITHMETIC
19. Ninetytwo thousand, one hundred six.
20. One million, three hundred ninetyseven thousand.
21. Eight thousand, four hundred eightytwo.
22. Nineteen million, one hundred fiftysix thousand.
23. Eight million, six.
24. Five thousand, one hundred thirtyone.
25. Sixtythree million, sixtyeight thousand, seven.
14. The art of writing numbers is called Notation.
BXBECISE 4.— WRITTEN
Write in figures the following numbers, which express
the wool production and the number of hogs in several
states, arranging in columns and using the comma between
the periods, as in Exercise 2.
POUNDS OF WOOL, PRODUCED IN 1906
1. North Carolina, eight hundred seventyone thou
sand, two hundred fifty.
2. Alabama, five hundred sixtyeight thousand, seven
hundred fifty.
3. Montana, thirtyfive million, eight hundred fifteen
thousand.
4. Florida, three hundred sixteen thousand, six hun
dred two.
5. Wyoming, thirtytwo million, eight hundred forty
nine thousand, seven hundred fifty.
6. Texas, nine million, three hundred sixty thousand.
7. Missouri, four million, six hundred seven thousand,
three.
NOTATION AND NUMERATION 7
NUMBER OF HOGS IN 1907
8. New York, six hundred seventyfive thousand, five
hundred fortyfive.
9. Iowa, eight million, five hundred eightyfour thou
sand, five hundred.
10. Pennsylvania, nine hundred eightynine thousand,
six hundred eightyfive. ,
11. Kentucky, one million, two hundred thirteen thou
sand, three hundred eighty.
12. Illinois, four million, four hundred fortynine thou
sand, seven hundred five.
13. Texas, two million, eight hundred sixty thousand,
eight hundred seventynine.
14. Ohio, two million, four hundred thirtysix thousand,
seven hundred ninetyseven.
15. Alabama, one million, two hundred fiftyone thou
sand, two hundred fiftyone.
16. Nebraska, four million, eighty thousand.
The following are the distances between several impor
tant cities. Write the numbers in figures and try to real
ize what they mean.
17. By rail from Albany, N.Y., to Troy, N.Y., six
miles; from Utica, N.Y., to Rome, N.Y., fifteen miles ;
from Syracuse, N.Y., to Rochester, N.Y., eightyone
miles.
18. From St. Paul, Minn., to Portland, Ore., two
thousand fiftythree miles; from Cleveland, O., to Cin
cinnati, O., two hundred sixtythree miles.
8
PRACTICAL ARITHMETIC
19. From Chattanooga, Teiin., to New Orleans, La.,
xour hundred ninetytwo miles ; from Nashville, Tenn., to
New Orleans, La.,
six hundred twenty
four miles ; from
New Orleans, La.,
to Atlanta, Ga., four
hundred ninetysix
miles.
20. By water from
New York to Liver
pool, three thousand
fiftyeight miles ;
from San Francisco
to Yokohama, four
Street Scene in Atlanta, Ga.
From a photog:raph.
Copyright, 1907, by Underwood & Dnderwood.
thousand, seven hundred ninetyone miles.
21. From New York to Manila, sixteen thousand, five
hundred miles; from New York to Havana, one thou
sand, four hundred twenty miles.
22. From New York to Strait of Magellan, six thousand
eight hundred ninety miles ; from Strait of Magellan to
San Francisco, six thousand one hundred ninety nine miles.
23. By rail from New York to Omaha, one thousand
three hundred eightyfive miles ; to San Francisco, three
thousand two hundred fifty miles.
24. The railroads of the United States aggregate one
hundred ninetythree thousand miles, bearing thirtyeight
thousand locomotives, fourteen thousand coaches, carry
ing yearly six hundred million passengers, and one bill
ion tons of freight. They cost about five billion dollars.
ROMAN NOTATION
15. The system of notation and numeration already
explained is commonly called the Arabic System. There
is still another system known as the Roman System.
16. In the Roman system of notation seven capital
letters of the alphabet and combinations of these letters
are used to express numbers. The letters and their
values are as follovsrs :,
I V X L C D M
1 5 10 50 100 500 1000
17. A bar placed over a letter increases its value a
thousand fold, e.g., V denotes 5000; X denotes 10,000.
18. When these symbols are used in combination their
values are governed by the following laws :
I. Each repetition of a letter repeats its value, e.g.,
XX denotes 20, XXX denotes 30, CC denotes 200, MMM
denotes 3000.
II. When a letter is placed after one of greater value,
its value is to be added to that of the preceding letter,
e.g., XI represents 10 and 1, or 11 ; VII represents 5 and
2, or 7 ; XVI represents 10 and 5 and 1, or 16 ; CXXI
represents 100 and 10 and 10 and 1, or 121.
III. When a letter is placed before another of greater
value, its value is taken from that of the letter of greater
value, e.g., IX represents 10 less 1, or 9; XL represents
50 less 10, or 40 ; XC represents 100 less 10, or 90.
IV. When a letter is placed between two letters of
9
10 PRACTICAL. ARITHMETIC
greater value, its value is taken from that of the letter
which follows it, e.^., XIX represents 10 and 9, or 19;
CXC represents 100 and 90, or 190.
BXEECISE 5. — WRITTEN
Express in Arabic notation :
1. XI 11. XXVII 21. MDLX
2. XX 12. XCV 22. MDXLVI
3. XIV 13. XLIV 23. MDCCXLIV
4. XXX 14. LXXIV 24. MMDCCXCIII
5. XL 15. CCLIV 25. VCCCLXXVI
6. XVI 16. CDLVI 26. XDCCXCIX
7. LV 17. DCIX 27. DXLIV
8. LIX 18. MCXL 28. MDCCLXXXIII
9. LXXVIII 19. MCXLV 29. MMCCCCLXIV
10. XLIX 20. MDLIV 30. MMMDCCXIX
EXERCISE 6.— WRITTEN
Express in Roman notation :
1.
11
11.
98
21.
421
2.
17
12.
73
22.
943
3.
19
13.
116
23.
719
4.
42
14.
240
24.
1425
5.
33
15.
375
25.
1764
6.
12
16.
480
26.
5861
7.
26
17.
510
27.
24,854
8.
54
18.
450
28.
256,845
9.
88
19.
375
29.
1,450,819
10.
75
20.
741
30.
3,840,006
UNITED STATES MONEY
19. In the money of the United States the unit is the
Dollar. In writing it is expressed by the sign , e.g.,
!? 25 is read twenty five dollars.
20. Our monej' system is based upon the same system of
tens and groups of tens which we studied in Arabic nota
tion. That is, ten units of one order make a unit of the
next higher order.
21. Ten tencent pieces equal one dollar. Ten one
cent pieces equal a tencent piece. A still smaller divi
sion of our money which we do not commonly use is
called Mills. Ten mills equal one cent.
Ten cents is onetenth of a dollar. One cent is one
onehundredth of a dollar. One mill is one onethou
sandth of a dollar.
22. The period used to separate dollars and cents is
called the Decimal Point.
23. The following diagrams serve to show the arrange
ment of dollars, cents, and iiaills as they are written:
F3 CO
a
a>
m — . 7.
1245 . 245
.a +> 3 T3 +3 J5 +J
•t 2 4 5 . 2 4 5
11
12
PRACTICAL ARITHMETIC
EXERCISE 7. — ORAL
Read as dollars and cents :
1. I 8.24 5. I 25.06 9. 1349.99 13. $ 542.89
2. $ 8.72 6. I 91.07 10. ^698.42 14. I 560.90
3. I 9.87 7. f 92.09 11. 1100.10 15. $1845.24
4. $10.25 8. $900.09 12. I 99.99 16. $6291.98
Read as dollars, cents, and mills :
17. $5,842 20. $ 17.001 23. $981,701 26. $699,764
18. $3,205 21. $ 70.070 24. $909,701 27. $263,809
19. $4,998 22. $191,672 25. $340,034 28. $ 89.617
EXERCISE 8. — WRITTEN
Express the following in figures, using the dollar sign
and decimal point :
1. Twenty dollars and fifty cents; thirtyfour dollars
and five cents.
2. Eighteen dollars and thirtyfive cents; ninety five
dollars and twenty cents; thirtyone dollars and sixty
cents ; one hundred twenty dollars and four cents.
3. One hundred dollars and ten cents ; fiftyfour dol
lars and nineteen cents ; fiftythree dollars and fiftyfive
cents; nineteen dollars and ninety cents.
4. Eighty dollars and one cent, five mills ; fifty dollars and
fifty cents ; three hundred dollars and six cents, one mill ;
four hundred thirtythree dollars and thirtythree cents.
5. Five hundred dollars and threetenths ; two hundred
dollars and thirtythree hundredths ; one hundred dollars
and three hundred thirtythree thousandths.
ADDITION
24. The growth of an apple twig in 1905 was 4 inches
and in 1906 it was 6 inches. How many inches did it
grow in the two years?
Addition is the process of finding the number that
is equal to two or more numbers taken together.
25. The result obtained by adding numbers is called
the Sum.
26. The sign of addition, +, is called Plus. When
placed between numbers it means that they are to be
added.
27. The sign of equality, =, when placed between
numbers shows that they are equal. Thus 7 + 3 = 10 is
read seven plus three equals ten.
28. Find the sums of the following, which include all
the combinations of two numbers from one to nine :
2 + 7 =
2 + 1 =
3 + 1 =
1 + 9
4 + 2 =
8 + 7 =
1 + 6 =
3 + 5
3 + 4 =
1 + 1 =
2 + 4 =
1 + 7
2 + 3 =
7 + 2 =
3 + 7 =
4 + 3
7 + 1 =
3 + 3 =
5 + 4 =
6 + 1
8 + 2 =
4 + 5 =
1 + 4 =
3 + 2
1 + 9 =
6 + 4 =
2 + 2 =
1 + 2
5 + 2 =
1 + 8 =
5 + 5 =
4 + 6
3 + 6 =
2 + 5 =
8 + 1 =
6 + 2
13
14 PRACTICAL ARITHMETIC
6 + 3 =
4 + 4 =
2 + 8 =
1 + 8 =
5 + 3 =
2 + 6 =
3 + 4 =
4 + 1 =
1 + 5 =
Practise adding these numbers daily until the sum of
any of these combinations can be told at a glance.
EXERCISE 9. — OEAL
1. A man fed a colt 2 quarts of oats, a driving horse
4 quarts, and a draught horse 6 quarts. How many quarts
of oats did he use at a feeding?
2. If the morning and the afternoon each becomes 5
minutes longer during the second week in March, how
much longer are the days of the second week than the
days of the first week ?
3. If the cost of hauling Kansas wheat to the railroad
station is 3 cts. a bushel, and the freight to New York is
11 cts. a bushel, what is the total cost of transportation ?
4. A dairyman had 4 cows. One gave 5 quarts, one
7 quarts, one 8 quarts, and another 9 quarts of milk at a
milking. How many quarts did all give at a milking ?
5. A morning from sunrise to noon in September is
6 hours, and from noon to sunset is 6 hours. What is
the total length of the day ?
6. A ration for a cow is : corn and cob meal, 5 lbs. ;
cottonseed meal, 4 lbs. ; hay, 20 lbs. What is the weight
of the entire ration?
7. The number of cloudy days in January was 9, the
rainy days were 4. How many days were rainy and
cloudy ?
ADDITION 15
8. If a fruit cake requires 3 lbs. of currants, 2 lbs. of
raisins, and 1 lb. of citron, how many pounds of fruit are
used in the cake ?
9. If the increase in temperature at Raleigh, N. C, in
1907 was 26 degrees from February to May and 14 degrees
from May to August, what was the total increase in tem
perature during the six months ?
10. If the temperature decreases 8 degrees from July
to September and 33 degrees from September to March,
what is the total decrease in temperature ?
11. The number of senators from each of several sec
tions of our country in 1890 was as follows : from the
New England States, 12; the Middle States, 8; the
Pacific States, 10. How many senators were there from
all these sections ?
12. The number of senators from the two largest sec
tions of our country in 1890 was as follows : the South, 28 ;
the Northwest, 24. How many senators were there from
these sections?
13. A ration for a cow is 15 lbs. of hay, 30 lbs. of
silage, 4 lbs. of cottonseed meal, 3 lbs. of wheat bran,
and 3 lbs. of corn meal. What is the weight of the
entire ration ?
14. Count from 3 to 99 by threes.
15. Count from 4 to 100 by fours.
16. Count from 6 to 96 by sixes.
17. Count from 9 to 99 by nines.
18. At Raleigh, N.C., in 1907 the lowest temperature
for the month of October was 36 degrees, the highest
16 PRACTICAL ARITHMETIC
temperature was 45 degrees higher. What was the high
est temperature ?
19. Two skilled laborers earned 13 and $5 respectively
a day. How much did the two earn together ?
20. Two unskilled laborers earned respectively il.25
and f 1.50 a day. How much did the two earn ?
21. The cultivation of an acre of corn costs $6, the fer
tilizers $ 3, harvesting and other expenses f 3. What is
the cost of the crop per acre ?
22. A ration for a horse vreighing 1000 lbs. when doing
moderately hard work is 6 lbs. of corn, 8 lbs. of oats, and
15 lbs. of hay. What is the weight of the ration?
23. A ration for a fattening beef animal weighing
1000 lbs. is 30 lbs. of corn silage, 12 lbs. of corn stover,
5 lbs. of cottonseed meal, and 4 lbs. of cotton seed.
What is the weight of the ration?
24. If a man vs^ishes to seed an acre for a meadow and
iises 11 lbs. of timothj', 6 lbs. of red top, and 5 lbs. of
clover seed, how many pounds of seed does he sow on the
acre ?
25. If 1000 lbs. of an average mixed stable manure
contain 5 lbs. of nitrogen, 6 lbs. of potash, and 3 lbs. of
phosphoric acid, how many pounds of these plant foods
does it contain ?
26. If the plants in a ton of dry clover hay used 39
lbs. of nitrogen, 37 lbs. of potash, and 11 lbs. of phos
phoric acid in growth, how many pounds of these mate
rials were used by the plants ?
27. If a ton of wheat straw iased in growing 11 lbs.
ADDITION 17
of nitrogen, 23 lbs. of potash, and 4 lbs. of phosphoric
acid, how many pounds of these materials were taken
from the soil ?
28. If a ton of oat grain used in growing 35 lbs. of
nitrogen, 9 lbs. of potash, and 13 lbs. of phosphoric acid,
how many pounds of these materials were used ?
EXERCISE ID.
— WRITTEN
Find the sum
of:
1. 2.
3.
4.
5.
6.
7.
8.
9 4
5
8
7
2
5
6
2 4
2
7
5
3
4
4
3 5
1
2
4
5
2
4
4 8
5
4
2
4
3
y
5 7
7
_5_
1
6_
2
9. 10. 11. 12. 13. 14. 15. 16.
76954862
63887888
95796964
88654979
56873294
4 _8_711i
29. Add 259, 634, and 872.
259 The sum of the units' column is 15, or 5 units and
g34 1 ten ; the 6 representing units is written under the
units' column. The sum of the tens' column + 1 ten
carried from the first column is 16. The figure repre
senting the 6 tens is placed under the tens' column,
872
1765
and the 1 representing hundreds is added with the hundreds'
18 PRACTICAL ARITHMETIC
column. The sum of the hiiudreds' column + 1 from the tens'
column is 17. The last sum is placed under the column
added.
30. Rule for Addition. Write the immbers to be added
so that the figures representing units stand under units,
tens under tens, hundreds under hundreds, etc. Begin
at the riglit and add each column separately, placing the
sum underneath if it is less than ten. If the sum of any
column exceeds nine, set down the righthand figure and
add the other figure or figures to the next column at the
left.
EXERCISE 11. — WRITTEN
Add:
1.
2.
3.
4.
5.
6.
56
78
14
4
126
696
76
43
18
25
205
348
99
21
7
79
811
319
4
7
39
64
79
871
18
51
88
23
236
296
15
75
78
49
815
64
37
64
99
86
429
337
7.
8.
9.
10.
11.
12.
526
237
124
58
321
379
448
213
87
222
118
672
973
716
116
972
961
789
87
89
615
736
69
709
988
825
793
845
296
69
147
298
611
47
348
879
268
794
989
623
799
27
ADDITION OF UNITED STATES MONEY
31. Add 14.25, $16.50, f .45, 1150, 13.455.
S4 '"'5
'^ ■^' Write the numbers to be added so ihat units
of the same order shall stand in the same col
umn, with the decimal points in a vertical line.
Add as in integral numbers. The decimal point
in the sum should stand directly in line with
the decimal points of the numbers added.
:;3
16.50
.45
150.00
3.455
1174.655
EXERCISE 12. — "WRITTEN
Add:
1.
12.50
3.05
.90
.06
7.25
2.
11.006
21.05
3.425
42.14
75.141
3.
1340.006
10.01
62.62
324.05
231.005
4.
$30,006
300.10
106.001
10.50
3.75
1. Add
2. Add
Add
Add
Add
Add
$6798.98.
3.
4.
5.
6.
EXERCISE 13. — WRITTEN
$3.05, $20,006, $45.2.5, $6.25, $3,755.
$63.43, $25,002, $23,025, $300.45, $62,725.
$90.,93, $84,005, $2,005, $16.85, $4.35, $1.98.
$63.05, $400.62, $50.50, $200.20, $3.75, $4.25.
$50,005, $560.35, $428.72, $34.91, $863,100.
$678.78, $67249.16, $9381.62, $862.98,
19
20 PRACTICAL ARITHMETIC
EXBHCISB 14. — WBITTEN
Find the sum of :
1.
2.
3.
4.
5.
6314
6798
8569
7873
2361
7581
8347
7863
2578
5671
6384
6437
9088
5447
1983
2678
8089
8637
9138
3612
7896
1463
4475
7819
9278
6.
7.
8.
9.
$4798.75
19862.79 $
428.69
17963.07
267.98
299.89
1796.99
629.67
495.12
4621.00
289.78
1198.79
490.07
7978.91
964.68
682.87
989.75
7842.29
629.29
298.
862.93
7096.07
729.62
391.09
Find the sum of :
10. 213, 14, 594, 672, 10, 756, 1875, 67.
11. 310, 64, 236, 79, 118, 296, 51, 9.
12. 716, 615, 1857, 9241, 19, 65, 452.
13. 463, 7, 8700, 1207, 1439, 4245, 5402.
14. $7.42, $28.75, $30,015, $500.00, $89,675, $620,908.
15. $62.50, $934.25, $245.63, $72.00, $786.21, $5.04.
16. $650.30, $12,645.32, $100.84, $100,084, $256.25.
17. 7,682,963, 842,981, 428,792, 6879, 84,289, 831.
18. 6,687,328, 64,298, 179,632, 281, 7, 698, 768,342,
19. 1,786,984, 6,827,341, 92,712, 863,298, 347,829.
20. $862.79, $9341.05, $67,294,73, $76,932.06.
21. 1,796,328, 6,297,348, 291, 7,283,409, 8624.
ADDITION
21
EXERCISE 15. — WRITTEN
The exports of cheese for 19011905 were
Country
1901
1902
1903
1904
1905
Pounds
Pounds
Pounds
Pounds
Pan ml a
Bulgaria . . .
6,449,020
5,651,335
7,064,385
6,624,517
7,227,816
Canada . . .
200,946,401
229,099,925
233,980,716
215,733,259
215,834,543
France ....
17,795,274
20,545,803
23,119,970
20,711,480
22,125,152
Germany . . .
3,211,693
3,119,981
2,813,539
2,5117,927
2,650,397
Italy . . . .
24,104,455
28,841,967
33,158,617
30,299,443
37,694,647
Netherlands
104,2i;S),090
104,785,152
109,025,968
103,069,081
98,438,575
New Zealand .
11 ,680,928
8,371,552
8,375,360
9,466,912
9,918,944
Russia . . . .
1,610,414
1,655,230
1,406,557
1,396,951
1,119,497
Switzerland . .
60,075,729
54,491,422
53,642,.S6:!
56,688,989
61,383,731
United States .
27,203,184
18,987,178
23,335,172
10,134,424
16,562,451
Other countries
7,924,000
9,469,000
8,833,000
7,050,000
5,092,000
15. Find the total exports for each year.
616. Find the total exports from each country.
The pupil should add as many of the following problems
as need be to attain speed and accuracy.
U
h
c
d
e
1
86793
84928
76321
42894
78621
2
84697
68296
98988
86796
78629
3
17986
79826
42391
76842
68792
4
28649
78421
97632
49988
68499
5
99763
89792
88767
84921
69837
6
89798
66787
62849
76639
62897
7
56845
86283
32918
74289
88997
8
67632
97987
98999
77877
68298
9
74911
83172
64829
79638
28429
10
64523
68791
48796
42981
78429
11
69354
81786
86429
36827
64287
12
98782
88793
34287
42963
96832
22 PEACTICAL ARITHMETIC
1721. Find the sum of columns a, h, c, d, e.
2233. Find the sum of each line of numbers from 1 to
12 without copying.
3447. Find the sums of the numbers in diagonal rows,
thus : b 1, a 2, also c 1, J 2, a 3, and all other possible rows
having the same direction, as d 1 to a 4, etc.
4861. Similarly, find the sums of the numbers in diag
onal rows dl, e 2, also c 1, d 2, e S, and by all other pos
sible rows having the same direction.
EXERCISE 16. — WRITTEN
1. A dairyman has 6 cows. The first gives 18 lbs. of
milk, the second 11 lbs., the third 23 lbs., the fourth
26 lbs., the fifth 32 lbs., the sixth 40 lbs. How much do
they all give?
2. If a 200lb. sack of fertilizer contains 8 lbs. of
nitrogen, 10 lbs. of potash, and 16 lbs. of phosphoric
acid, how many pounds of these plant foods are there in
a sack ?
3. If a reader costs 43 cts., a history 86 cts., an arith
metic 49 cts., a geography $1.08, a tablet 8 cts., pens and
pencils 7 cts., what is the cost of the equipment?
4. In the stomach and crop of a bobwhite there were
found 400 pigweed seeds, in another 500 ragweed seeds,
in another 550 sheepsorrel seeds, in another 640 seeds of
pigeon grass. How many of these weed seeds were con
sumed by these four bobwhites ?
5. When thoroughly dry, the weight of the different
parts of the cotton plants grown on an acre of land was
as follows: lint 300 lbs., seed 507 lbs., bolls 363 lbs..
ADDITION ■ 23
leaves 566 lbs., roots 130 lbs., stems 604 lbs. What was
the total weight of plants produced to yield 300 lbs. of
lint cotton ?
6. If the material grown on an acre in corn weighs as
follows : ears 4325 lbs., stover, which consists of stalks,
2379 lbs., leaves 1190 lbs., and shucks 397 lbs., how
much did the ears and stover together weigh? How
much did the stover weigh ?
7. A farmer sold the following products during the
year: 1 horse 145, 9 beef animals 468, 18 hogs $234,
13 lambs 152, 309 bus. potatoes $12Q, 10 bales cotton
1528, 150 bus. seed oats 75, 100 bus. seed corn 100, 50
chickens $22, 200 doz. eggs $38, 12 turkeys $21, 220
bus. wheat $191. To what did his total sales amount?
8. The different parts of the dressed carcass of a beef
animal that, when alive, weighed 1550 lbs. weigh as fol
lows : neck 40 lbs., chuck 237 lbs., prime ribs 117 lbs.,
porterhouse steak 103 lbs., sirloin steak 87 lbs., rump
36 lbs., round steak 183 lbs., shank 30 lbs., flank, 52 lbs.,
ribs plate 138 lbs. What is the weight of the dressed
carcass ?
9. A New England cottage was built at a cost as
follows: masonry $250, lumber and mill work $700, car
pentering $400, plumbing $170, painting $90, hardware
$20, heater, $200. What did the house cost when com
pleted ?
10. A cottage planned to cost $1000 was built for the
following cost: foundation and brickwork $428.80, lum
ber $370.15, carpentering $264.87, painting and plaster
24 PRACTICAL ARITHMETIC
ing 8253.25, hardware 138.90, tin work, 13.78. What
did the house actually cost ?
11. A farm costing $5250 is equipped as follows:
house 11579.42, farm mules 1850.00, 1 horse 1175.00,
cattle $275.00, hogs 1112.00, implements and tools
1672.00. What is the total value of the farm and its
equipment ?
COST OF FURNISHING A HOUSE
Two young people equip their home with the following
articles. Find the cost of each room separately.
12. Kitchen: Linoleum, 120.00; range, 145.00; table,
11.50; chair, $1.00; utensils, $13.00; laundry outfit,
$8.25; refrigerator, $21.00; lamp, 45 cts.; clock, $1.00;
cutlery, 75 cts.
13. Dining Room : Table, $20.00 ; six chairs, $24.00;
sideboard, $30.00; rug,$12.00; clock,$4.98; silence cloth,
$1.25 ; lamp, $2.97 ; 3 dozen napkins, $9.48 ; 2 dozen nap
kins, $4.30; six tablecloths, $15.00; six plated knives,
$4.50 ; six solid silver forks, $12.00 ; 12 solid silver spoons,
$10.00; three tablespoons, $6.00; dishes, $15; glass
ware, $7.45.
14. Living Room: Lamp, $3.50; table, $15.00; Mor
ris chair, $14.60; chair, $5.45; rattan chair, $9.25;
armchair, $7.50 ; bookcase, $9.45; magazine stand, $6.25;
rug, $34.50.
15. Hall: Lamp, $3.25; hall rack, $16.35; chair,
$3.45; rug, $6.00.
16. Bedroom: Bureau, $22.35; chiffonier, $11.25;
washstand, $4.00; toilet set, $6.25; chair, $2.75; lamp.
ADDITION 25
11.50; bed, $13.00; springs, 15.00 ; mattress, 115.00; 12
sheets, 19.60; 12 pillow cases, 15.00; 2 pairs blankets,
f 10.00; 1 comfortable, 11.00; 2 counterpanes, 13.45;
2 dozen towels, 16.00; 1 dozen yards toweling, $1.80;
rug or matting, $5.80.
17. Find the total amount spent in furnishing the house.
18. If a family of two persons spends for rent $130,
food $210, clothing $80, fuel $30, light $6, insurance
$24, replenishing $ 10, carfare $5, literature $5, charity
$10, and saves $20, what is the income?
19. A second family of five persons spends for house
rent $240, table expense $364, clothing $175, fuel $52,
light $3, hired help $50, renewals $50, dentist $25, boys
spending money $13, spending money of other children
$5. What is the income?
20. What is the cost of raising an acre of corn, estimat
ing rent at $3.03, fertilizer $1.86, preparation of the
soil $1.62, planting $.42, cultivating $1.80, harvesting
$3.00, and other expenses $1.76?
21. What is the total cost of raising an acre of upland
cotton, allowing for the rent of land $3.25, fertilizer
$2.46, preparing the soil $3.00, seed $.21, planting $.28,
cultivating $2.31, picking $3.37, ginning and pressing
$1.65, wear of tools $.62, marketing $.64, and other
expenses $1.42?
22. A man owns 5 horses. The first is worth $90, the
second $125, the third $175, the fourth as much as the
first and second, the fifth .as much as the second and
fourth. What was the value of the 5 horses?
26
PRACTICAL ARITHMETIC
23. The number of miles of railroad in the world in
1900 was : North America 216,000, Europe 173,000, Asia
36,000, South America and West Indies 28,000, Aus
tralasia 15,000, Africa 12,000. What was the total
mileage ?
Alabama, S 1,828,697 The adioininsT table
Arkansas, S 1,311,564 . ^, "^ ^ , .
California, P 1,485,053 gives the population
Colorado . . . 539,700 of the States of the
Connecticut, N.E. . . . 908,420
Delaware, M 184,735 United States in
Florida, S 528,542 . Q^n
Georgia, S 2,216,331 ^^""•
Idaho 161,772 „, tji j ^i. ^
Illinois, E.C 4,82i;550 ^4. Fmd the popula
Indiana, E.C 2,516,462 tion of the New Eng
Iowa,W.C 2,231,853 , ^. . ■ ^ . j
Kansas, W.C 1,470,495 land States, indicated
Kentucky, E.C 2,147,174 bv N F
Louisiana, S 1,381,625 ^
Maine, N. E 694,466 , rfj ^Iio T^r.T.^^
Maryland, M 1,188;044 ^^ ^^"^^ *^^ P^P^"
Massachusetts, N.E. . . 2,805,346 lation of the Middle
Michigan, E.C 2,420,982 oj. 4. ■ j ^ j i, ht
Minnesota, W.C. . . . 1,751,394 States, indicated by M.
Mississippi, S 1,551,270 „_ t^ i .i ^
Missouri, W.C 3;i06,665 ^6. Find the popula
Montana 243,329 tion of the Western
Nebraska, W.C. . . . 1,066,300 ^ , , ^ ■ ■,.
Nevada 42,335 Central States, indi
New Hampshire, N.E. . 411,588 patpri Vnr "W P
New Jersey, M.. . . . 1,883,669 ^^^^'^ '^^ ^■^•
New York, M 7,268,894 ,? Find the nonula
North Carolina, S. . . . 1,893,810 ^ ^^^ ^^^ popuia
North Dakota, W.C. . . 319,146 tion of the Southern
Ohio, E. C 4,157,545 q. , • j 4. j u
Oklahoma, S 790^391 States, indicated by
Oregon, P 413,536 S.
Pennsylvania, M. . . . 6,302,115
Rhode Island, N.E. . . . 428,556 28. Find the popula
South Carolina, S. . . . 1,340,316 . ^ ^
South Dakota, W.C. . . 401,570 tion of the Pacifio
Tennessee, S 2,020,616 States indioatfld bv
Texas, S 3,048,710 ^^^^^s, inaicacea Dy
Utah 276,749 P.
ADDITION 27
Vermont N.E 343,641 29. Find the popula
Virgima, M 1,854,184 . . ,
Washington, P 518,103 tioii of the Eastern
West Virginia, M. . . . 958,800 Central Statp^i inrli
Wisconsin, E.C 2,069,042 ^^nrrai States, indi
Wyoming 92,531 cated by E.C.
30. The remaining States are Plateau States. What is
the population ?
Except in leap year the days in the months number :
January 31 31. How many days are there
J, , „^ in a year that is not a leap
April 30 year?
M*y 31 32. The twentysecond of
j^, gj February is how many days
August 31 after January first ?
September .... 30 33^ ^he fourth of July is
October 31 •'
November .... 30 '^ow many days after January
December .... 31 first?
34. The twentyfifth of December is how many days
after January first?
SOME IMPORTANT DATES
33. From the discovery of America by Columbus in
1492 to the founding of St. Augustine 73 years elapsed.
From the founding of St. Augustine to the settlement
of Jamestown 42 years elapsed. What were the dates of
these two settlements ?
36. From the discovery of America in 1492 to the
founding of New Amsterdam 122 years elapsed. From
the founding of New Amsterdam to the landing of the
Pilgrims 6 years elapsed. What were the dates of these
two settlements ?
28
PRACTICAL ARITHMETIC
37. From the first General Assembly in 1681 to the
first Colonial Congress 84 years elapsed. From the first
Colouial Congress to the second Colonial Congress 9 years
elapsed. What were the dates of these two Congresses?
Signing the Declaration of Independence
38. From the first General Assembly in 1681 to the
Declaration of Independence 95 years elapsed. From
the Declaration of Independence to the adoption of the
Articles of Confederation 5 years elapsed. What were
the dates of these two events?
39. From the first General Assembly in 1681 to the
adoption of the Constitution 106 years elapsed. From
the adoption of the Constitution to the election of the
first President of the United States 2 years elapsed.
What were the dates of these events?
SUBTRACTION
32. One dairy cow produces 9 pounds of butter in a
week, another 5 pounds. How much more does one pro
duce than the other ?
The process of finding how mucli greater one number
is than another, or finding the difference between two
numbers, is called Subtraction.
33. The larger number, or the number from which
another is subtracted, is called the Minuend.
34. The smaller number, or the number subtracted, is
called the Subtrahend.
35. The result obtained in subtracting is called the
Remainder or Difference.
36. The sign of subtraction, — , is called Minus. When
it is placed between two numbers, it means that the second
number is to be subtracted from the first, e.g., 8 — 5=3
is read, eight minus (less) five equals three.
EXERCISE 17. — ORAL
11= 22= 33=
21= 32= 43=
31= 42= 53=
41= 52= 63=
51= 62= 73=
61= 72= 83=
• 29
30 PRACTICAL ARITHMETIC
71 =
82 =
93 =
81 =
92 =
103 =
91 =
102 =
113 =
101 =
112 =
123 =
4:4: =
55 =
66 =
54 =
65 =
76 =
64 =
75 =
86 =
74 =
85 =
96 =
84 =
95 =
106 =
94 =
105 =
116 =
104 =
115 =
126 =
114 =
125 =
136 =
124 =
135 =
146 =
134 =
145 =
156 =
77 =
88 =
99 =
87 =
98 =
109 =
97 =
108 =
119 =
107 =
118 =
129 =
117 =
128 =
139 =
127 =
138 =
149 =
137 =
148 =
159 =
147 =
158 =
169 =
157 =
168 =
179 =
167 =
178 =
189 =
Note. This subtraction table should be thoroughly studied
by each pupil, and drill should be continued until the answers
can be given quickly and correctly.
SUBTRACTION 31
EXERCISE 18. — ORAL
Practice subtracting these numbers until the differ
ences can be told instantly.
a
b
c
d
e
f
g
h
i
1.
4
7
6
2
4
5
6
1
8
1.
1
1
_2
2
_3
J[
3
1
1
2.
2
3
9
10
3
5
4
6
10.
2.
1
3
J.
2
J.
_2
2
1
1
3.
7
10
5
3
4
9
6
11
8
3.
2
3
3
_2
_£
2
_4
2
2
4.
9
10
7
9
5
7
11
5
8
4.
_3
5
3
4
A
5
_3
_5
4
5.
6
8
12
13
8
7
9
6
8
5.
_6
6
_3
4
JT
4
5
5
3
6.
10
7
10
9
11
8
9
7
14
6.
_4
6
8
7
4
5
6
7
6
7.
11
14
13
8
12
12
10
9
13
7.
5
6
5
_8
4
JT
6
8
7
8.
11
15
10
12
12
15
16
16
12
8.
8
8
7
9
5
_9
7
_8
_8
9.
12
13
18
15
17
15
11
14
11
9.
6
9
9
6
9
7
_6
_9
7
10.
10
14
14
16
13
17
11
14
13
10.
9
_7
_5
9
_8
8
_9
_8
_6
a
6
c
d
e
/
S'
h
i
32 PRACTICAL ARITHMETIC
BXBBCISB 19. — ORAL
1. If a twig grew 6 ins. in 1906, and 4 ins. in 1907,
which year produced the greater growth? How much
greater ?
2. What number taken from 10 leaves 2 ?
3. What number with 6 makes 10 ?
4. If a strawberry plant has 12 leaves and 4 are
injured by insects, how many sound leaves are there on
the plant ?
5. If a limb on the north side of a tree measures 9 ft.,
and one on the south side measures 11 ft., which is the
longer ? How much longer ?
6. If the number of rainy and cloudy days in Septem
ber were 17, how many clear days were there in Sep
tember ?
7. Subtract by threes from 99 to 0.
8. Subtract by fours from 100 to 0.
9. Subtract by sixes from 96 to 0.
10. Subtract bj' sevens from 98 to 0.
11. If the highest temperature for July in the Middle
Atlantic States was 99 degrees, and the lowest tempera
ture 57 degrees, what was the range in temperature?
12. The highest soil temperature at which wheat will
grow is 104 degrees ; squash and corn grow at 115
degrees. At how much higher temperature will squash
and corn grow than wheat ?
13. Melons grow best at a soil temperature of 99
degrees, and clover at 70 degrees. How much higher
temperature is required for melons than clover ?
SUBTRACTION 33
14. The boiling point of water is 212 degrees, the freez
ing point 32 degrees. What is the difference in degrees
between the boiling and freezing points?
15. A man wishes to start a small fruit garden. He
has 115. to invest. The trellis and lattice cost 1 10.
How much is left for the plants and trees ?
16. If 2 pear trees cost 35 cts., 2 apple trees 25 cts.,
how much is left for smaller plants?
17. If 6 doz. strawberry plants cost 25 cts., and 4 rasp
berry plants cost *.25, how much is still left of the
original sum ?
18. A normal child 6 yrs. old weighs 45 lbs., measures
44 ins. in height, and has a chest measurement of 23 ins.
A normal child of 9 yrs. weighs 60 lbs., measures 50 ins.
in height, and has a chest measurement of 25 ins. What
is the increase in weight in 3 yrs.?
19. What is the increase in height in 3 yrs. ?
20. What is the increase in chest measurement in
3 yrs. ?
37. From 875 take 446.
875 Since we cannot take 6 iinits from 5 units we
446 add one of the 7 tens to the 5 units, making 15
429 units. We now have 6 units to be subtracted from
15 units, which leaves 9 units to be written under the
units' column. Since we have already taken 1 ten from the 7
tens we have 6 tens remaining in tens' column. 4 tens from 6
tens leaves 2 tens, which is written under tens' column; 4
hundreds from 8 hundreds leaves 4 hundreds. The remainder,
therefore, is 4 hundreds, 2 tens, 9 units, or 429.
34 PRACTICAL ARITHMETIC
38. From 584 take 296.
584 Since we cannot take 6 units from 4 units we must
296 take one of the 8 tens, which added to 4 units makes
288 14 units. 6 units from 14 units leaves 8 units. Since
we have already taken 1 ten from the 8 tens, we
have 7 tens left. We cannot take 9 tens from 7 tens, so we
must take 1 hundred from 6 hundreds, which added to 7 tens
makes 17 tens. 9 tens from 17 tens leaves 8 tens. Since
we have already taken 1 hundred from the 5 hundreds we
have 4 hundreds left. 2 hundreds from 4 hundreds leaves
2 hundreds. The remainder, therefore, is 2 hundreds, 8 tens,
8 units, or 288.
39. Add the remainder to the subtrahend in each of
the last two examples. What do you observe ?
The sum of the remainder and the subtrahend is equal
to the minuend. Hence, to test the subtraction add the
remainder and subtrahend together. If the sum equals
the minuend, the work is correct.
40. Rule for Subtraction. Write the subtrahend under
the minuend, placing units under units and tens under
tens, etc. Begin at the right and subtract each figure of
the subtrahend from the corresponding figure of the min
uend, and write the remainder underneath.
If any figure of the subtrahend is greater than the
corresponding figure of the minuend, increase the figure
of the minuend by taking one of the next higher order,
which will be an increase of ten. Then diminish by 1 the
order of the minuend from which the 1 was taken, and
subtract.
EXERCISE 20. — WRITTEN
Subtract and prove :
1. 5226. 2. 3728.
SUBTRACTION
35
3. 9458.
4. 9325.
5. 9268.
6. 8118.
7. 7036.
8. 9027.
9. 5013.
10. 9547.
11. 786235.
12. 598213.
13. 647238.
14. 321216.
15. 976247.
16. 876381.
17. 358149.
18. 467248.
a
b
c
1
18729
2628
278
2
28694
3979
379
3
39231
8642
428
4
47326
3981
471
5
22821
7986
879
6
32542
7842
642
7
73625
3765
839
8
39832
8429
341
9
32421
3764
768
10
31321
3428
429
The pupil should subtract sufficient of these problems to
become accurate and rapid.
1928. From the first number in column b take each
number in column c.
29118. From each other number in column h take
each number in column c.
119128. From the first number in column a take each
number in column h.
129218. From each other number in column a take
each number in column h.
SUBTEACTION OF UNITED STATES
MONEY
exercise 21. — oral
Makixg Change
1. I buy a book for 42 cts. ; how mucb change shall I
receive from 50 cts. ? Make change in two ways.
2. A yard of ribbon costs 12 cts., a spool of thread
5 cts., a paper of pins 5 cts. How much change do I
receive from a fiftycent piece ? Make change in three
ways.
3. A yard of silk costs 63 cts., 1 doz. buttons 10 cts.,
a bolt of tape 5 cts. Make change for $1 in two ways.
4. A copy of "Arabian Nights" costs $1.10. Make
change for 12.00.
5. Make change for $2.00 when you owe $1.75; $.15.
6. Make change for $5.00 when you owe $1.49; $2.58.
7. Make change for $5.00 when you owe $3.18; $2.16.
8. Make change for $2.00 when you owe $1.23; $1.87.
9. A pair of shoes costs $2.25, hat $1.00, tie 50 cts.,
4 handkerchiefs 72 cts. How much change does the
merchant return for $5.00?
10. Make change for $ 10.00 when you owe $ .81 ; $ 7. 84.
41. Subtract $45,755 from $90.20.
*Q0 200 Since the subtrahend has mills and there are no
.rrjrr mills lu the minuend, a cipher is added to fill the
4u. i DO , „ ..
— — — — ^ place 01 mills. The subtrahend is written under
the minuend so that units of the same order shall
36
SUBTRACTION 37
stand in the same column, and the decimal points be in a ver
tical line. Subtract as integers. The decimal point in the
remainder should stand directly under the decimal points of
the minuend and subtrahend.
EXERCISE 22.— WRITTEN
Subtract and prove:
1. $4.24111.10. 11. 11.001.75.
2. 13.86 #1.25. 12. 111.00 $3.25.
3. 14.50. ¥3.20. 13. $90.99180.25.
4. 110.75 $8.41. 14. $77.42 $65.94.
5. $6.54 $4.37. 15. $93.20 $84.25.
6. $9.27 $8.16. 16. $525 $177.02.
7. $12.50 $8.25. 17. $197 $184.09.
8. $17.2818.25. 18. $3.333 $2,999.
9. $18.24 $9.16. 19. $16.725 $.50.
10. $20.50 $10.25. 20. $28.07 $.125.
EXERCISE 23.— WRITTEN
1. Ill a day's milking of 769 lbs. of milk from a herd
of Jersey cows there are 661 lbs. of water, 69 lbs. of
solidsnotfat, and the remainder is butterfat. How
many pounds of butterfat in the day's milking?
2. In a day's milking of 769 lbs. from a herd of Hol
steinFresian cows there are 675 lbs. of water, 69 lbs. of
solidsnotfat, and the remainder is butterfat. How
many pounds of butterfat in the day's milking? How
many pounds less of butterfat than in problem 1 ?
3. The 3d of February 1907 is the 34th day of the
38 PRACTICAL ARITHMETIC
year ; the 5th of June is the 156th day of the year. How
many days between these dates ?
4. The 10th of September is the 283d day of the year ;
the 10th of June is the 191st day of the year. What is
the difference in dates?
5. The 22d of April is the 142d day of the year ; the
31st of October is the 303d day of the year. What is the
difference in dates?
6. Jamestown, Va., was settled in 1607; the James
town Exposition, near Norfolk, Va., occurred in 1907.
How many years elapsed between the dates ?
7. The boiling point of water is 212 degrees. Water
is at simmering temperature at 180 degrees. What is the
difPerence in temperature?
8. If a farmer kills a hog that weighs 369 lbs. when
alive, how much will the dressed carcass weigh if it weighs
82 lbs. less than the live weight ?
9. A butcher killed a beef animal of good quality
that weighed 1148 lbs. and an inferior one that weighed
1179 lbs. What is the difference in the weight of the
dressed carcasses of these beeves if the loss in weight of
the first animal was 397 lbs. and of the second animal
529 lbs.?
10. If 4000 lbs. of ordinary fertilizer contains plant food
as follows : phosphoric acid 320 lbs., nitrogen 80 lbs., and
potash 80 lbs., how many pounds of plant food are there?
How much that is not plant food ?
11. In 4000 lbs. of cottonseed hulls there are 444 lbs.
of water and 2152 lbs. of indigestible matter. The re
mainder is digestible. How much is digestible ?
SUBTRACTION 39
12. In 4000 lbs. of corn there are 436 lbs. of water and
408 lbs. of indigestible matter. The remainder is digest
ible. How many pounds of digestible matter are there?
13. A man took 1500 lbs. of seed cotton to the gin and
received a bale of lint weighing 493 lbs. How many
pounds of seed should he also receive, all of the seed cot
ton that is not lint being seed?
14. If to cultivate an acre of corn costs f 4.50, the
fertilizers for it $3, harvesting $3.50, and other expenses
f 1.00, and 35 bus. of corn worth §17.50, and a ton and
a quarter of stover worth i 5, are produced, what is the
farmer's profit?
15. If 1083 lbs. of cotton are harvested from a fertil
ized field planted with selected seed, and 670 lbs. are
harvested from an unfertilized field where unselected
seed was used, what is the increase in yield due to fertili
zation and seed selection ?
16. To grow a ton of oats requires 464 tons of water.
A ton of corn requires 271 tons, potatoes require 385 tons,
and clover 577 tons. How much more water is required
by potatoes than by corn ? By clover than by oats ? By
clover than by corn ? By clover than by potatoes ? By
oats than by corn ? By oats than by potatoes ?
17. If pat straw uses 163 lbs. of potash, 56 lbs. of ni
trogen, and 28 lbs. of phosphoric acid, and oat grain uses
48 lbs. of potash, 176 lbs. of nitrogen, and 68 lbs. of
phosphoric acid in every 10,000 lbs. of yield, which uses
the most of each plant food, the straw or the grain? How
much more of each ?
18. In Michigan, soy beans with tubercles on their
40 PRACTICAL ARITHMETIC
roots yield 113 lbs. of nitrogen to the acre ; without
tubercles 76 lbs. of nitrogen to the acre. How many
pounds' increase in nitrogen was there when the tubercles
were present ?
19. Cowpeas with tubercles on their roots yielded
139 lbs. of nitrogen and without tubercles 118 lbs. to the
acre. What was the gain when tubercles were present ?
20. A good clay soil contains 12,760 lbs. of potash,
a good sandy soil 4840 lbs. of potash, to the acre, in a
layer one foot deep. How much more potash is there in
clay soil than in sandy soil per acre one foot deep ?
HEIGHTS OF MOUNTAINS
The heights of some of the highest mountains are : Mt.
Aconcagua 28,082 ft., Mt. Blanc 15,744 ft., Mt. Everest
29,002 ft., Mt. McKinley 20,464 ft., Mt. Mitchell 6711 ft.
21. Mt. Mitchell is the highest mountain in the eastern
United States. How much higher is Mt. McKinley than
Mt. Mitchell ?
22. Mt. Aconcagua is the highest mountain in the
Americas. How much higher is Mt. Everest than Mt.
Aconcagua ?
23. Mt. Blanc is the highest mountain in the Alps.
How much higher is Mt. Everest than Mt. Blanc ?
24. How much higher is Mt. Aconcagua than Mt.
McKinley ?
25. The greatest known depth of the ocean is 27,930 ft.
How many feet less is this than the height of Mt. Everest ?
MULTIPLICATION
42. If a blackbird destroys 3 cabbage worms in 1 hr.,
at tlie same rate how many cabbage worms will it destroy
333333 in 2 hrs.? In 3 hrs.? In 4 hrs.?
3 3 3 3 3 3 In 5 hrs.? In 6 hrs.? In 7 hrs. ?
3 3 3 3 3
3 3 3 8
3 3 3
3 3
3
6 9 12 15 18 21
When several equal numbers are to be added, it is much
shorter to obtain the result by the process known as Mul
tiplication. To multiply, however, we must learn the
sums of the most common numbers added to themselves
definite numbers of times. Thus in the problem above
we see that 3 taken 2 times gives 6, taken 3 times gives
9, etc.
43. The number to be repeated is called the Multipli
cand.
44. The number which indicates how many times the
multiplicand is to be repeated is called the Multiplier.
45. The result obtained by multiplying one number b}?
another is called the Product.
46. The sign of multiplication is an inclined cross X .
When placed between two numbers it is read " times," or
41
42 PRACTICAL ARITHMETIC
"multiplied by." When the multiplier precedes the mul
tiplicand with the multiplication sign x between them,
the sign is read " times " ; when the multiplicand pre
cedes, the sign is read "multiplied by," e.g., 3 x 8 = 24, is
read 3 times 8 equals 24, or, regarding 8 as the multiplier,
the expression is read, 3 multiplied by 8 equals 24.
47. A number that is applied to any particular object
is called a Concrete number, e.g., 1 bird, 3 caterpillars,
48 hrs.
48. A number that is iiot applied to a particular object
is an Abstract number, e.g., 1, 3, 48.
49. In multiplication the multiplier is always an ab
stract number.
50. The most useful products are shown in the follow
ing tables. They should be committed to memory.
1x1= 1
1x2= 2
1x3= 3
2x1= 2
2x2= 4
2x3= 6
3x1= 3
3x2= 6
3x3= 9
4x1= 4
4x2= 8
4x3 = 12
5x1= 5
5x2 = 10
5 X 3 = 15
6x1= 6
6x2 = 12
6x3 = 18
7x1= 7
7 X 2 = 14
7 X 3 = 21
8x1= 8
8x2 = 16
8 X 3 = 24
9x1= 9
9x2 = 18
9 X 3 = 27
10 X 1 = 10
10x2 = 20
10 X 3 = 30
1x4= 4
1x5= 5
1x6= 6
2x4= 8
2x5 = 10
2x6 = 12
3 X 4 = 12
3x5 = 15
3 X 6 = 18
4x4 = 16
4 X 5 = 20
4 X 6 = 24
MULTIPLICATION 43
5x4 = 20
5 X 5 = 25
5x6 = 30
6 X 4 = 24
6 X 5 = 30
6x6 = 36
7x4=28
7 X 5 = 35
7 X 6 = 42
8x4 = 32
8 X 5 = 40
8 X 6 = 48
9 X 4 = 36
9 X 5 = 45
9 X 6 = 54
10 X 4 = 40
10 X 5 = 50
10x6 = 60
1x7= 7
1x8=8
1x9= 9
2 X 7 = 14
2 X 8 = 16
2x9 = 18
3 X 7 = 21
3 X 8 = 24
3x9=27
4x7 = 28
4 X 8 = 32
4 X 9 = 36
5 X 7 = 35
5 X 8 = 40
5 X 9 = 45
6x7 = 42
6 X 8 = 48
6 X 9 = 54
7 X 7 = 49
7 X 8 = 56
7x9 = 63
8 X 7 = 56
8 X 8 = 64
8 X 9 = 72
9 X 7 = 63
9x8 = 72
9 X 9 = 81
10 X 7 = 70
10 X 8 = 80
10 X 9 = 90
1x10= 10
1x11= 11
1x12= 12
2x10= 20
2x11= 22
2x12= 24
3x10= 30
3x11= 33
3x12= 36
4x10= 40
4x11= 44
4x12= 48
5x10= 50
5x11= 55
5x12= 60
6x10= 60
6x11= 6Q
6x12= 72
7x10= 70
7x11= 77
7x12= 84
8x10= 80
8x11= 88
8x12= 96
9x10= 90
9x11= 99
9x12 = 108
10 X 10 = 100
10x11 = 110
10 X 12 = 120
EXERCISE 24. — ORAL
1. If it costs f 6 to spray 1 acre of potatoes, how much
will it cost to spray 8 acres ?
44 PRACTICAL ARITHMETIC
2. If it costs f 3 to harvest an acre of corn, how much
will it cost to harvest 7 acres ?
3. If 1 acre of corn produces 9 tons of silage, how
many tons will 6 acres produce ?
4. If it costs 4 cents an acre to treat seed oats to pre
vent oat smut, what will it cost to treat the seed for 8
acres ?
5. If an acre of unfertilized land with unselected seed
will produce 1 bale of cotton, how many bales will 9 acres
produce ?
6. If 1 acre of fertilized land with very carefully se
lected seed will produce 2 bales of cotton, how many bales
will 9 acres produce ?
7. At §50 a bale, what will be the value of the cotton
in problem 5 ?
8. At $50 a bale, what will be the value of the cotton
in problem 6 ?
9. A square field is 12 rods on a side. What is the
distance around the field, or the perimeter?
10. If it takes 7 lbs. of ordinary seed corn to plant an
acre, how many pounds will it take to plant 8 A. ?
11. What is the cost of 6 tons of coal at i 8 a ton ?
12. If the school day is 5 hours long, how many school
hours are there in a school week ?
13. If a child sleeps 10 hours each night, how many
hours does it sleep in 1 week ?
14. If 10 hours are spent in sleep and 5 hours in school
in each day of 24 hours, how many hours in a school week
are left for play?
MULTIPLICATION 45
15. There are 12 inches in 1 foot. How many inches
are there in 3 feet ? What name is given to a measure 3
feet long ?
16. If a man walks 3 miles an hour, how far will he go
in 9 hours ?
17. If a horse travels 6 miles an hour, how far will he
go in 9 hours ?
18. If a steamboat goes 9 miles an hour, how far will
it go in 9 hours ?
19. If a man on a bicycle rides 12 miles an hour, how
far will he go in 9 hours ?
20. If a freight train averages 15 miles an hour, how
far will it go in 9 hours ?
21. If an ocean liner goes 17 miles an hour, how far
will it go in 9 hours ?
22. If an automobile travels 20 miles an hour, how far
will it go in 9 hours ?
23. If an express train averages 35 miles an hour, how
far will it go in 9 hours ?
24. To properly cook a ham it should be kept at sim
mering heat 30 minutes for every pound of weight after
first being plunged into boiling water and kept boiling for
10 minutes. How long will it take to cook a 9lb. ham ?
25. With spool cotton at 6 cents a spool or 50 cents a
dozfen, how many cents are gained by buying by the dozen
rather than by the spool on a purchase of 5 dozen spools ?
51. Find the product of 856 x 4.
Write the multiplier 4 under the multiplicand 856, placing
the units of the multiplier under units of the multiplicand, and
begin at the right to multiply.
46 PRACTICAL ARITHMETIC
r^in 4 X 6 = 24. The 4 is written in units' column. The
2 tens are to be added to the product of tens. 4x5
tens = 20 tens. 20 tens plus the 2 tens, carried
*'*^* from the multiplication of units, gives 22 tens, or 2
hundreds and 2 tens. 2 tens is written under tens' column and
2 hundreds are to be added to hundreds' product. 4x8
hundreds = 32 hundreds, which with 2 hundreds makes .34
hundreds. 34 hundreds is written under hundreds' column.
The product is 3424.
EXERCISE 25. — WRITTEN
Find the product of :
1. 2 X 365. 18. 3 X 1845. 35. T x 37,863.
2. 4 X 197. 19. 5 X 7543. 36. 6 x 31,245.
3. 3 X 754. 20. 6 X 1896. 37. 8 x 43,036.
4. 5 X 863. 21. 4 X 9806. 38. 5 x 68,734.
5. 7 X 189. 22. 3 X 5431. 39. 2 x 81,896.
6. 6 X 275. 23. 2 X 9864. 40. 8 x 52,783.
7. 4x986. 24. 4x1896. 41. 9x28,357.
8. 5 X 184. 25. 5 X 8652. 42. 7 x 48,021.
9. 7 X 689. 26. 8 X 6541. 43. 8 x 52,163.
10. 8 X 173. 27. 9 X 1864. 44. 4 x 59,136.
11. 9 X 602. 28. 8 x 8250. 45. 2 x 24,386.
12. 3 X 456. 29. 9 X 3475. 46. 9 x 12,854.
13. 6x629. 30. 4x8364. 47. 7x47,829.
14. 8 X 371. 31. 6 X 7928. 48. 9 x 79,836.
15. 7x864. 32. 3x6471. 49. 8x64,281.
16. 9 X 298. 33. 7 X 5498. so. 3 x 97,645.
17. 3x672. 34. 9x3762. 51. 6x58,792.
MULTIPLICATION 47
52. What effect does it have upon a number to move it
one place to the left in the period?
Moving a figure one place to the left has the same effect
as multiplying it by 10, e.g., 84 x 10 = 840.
To multiply by 10, place a cipher at the right of the
multiplicand, thus moving each figure one place to the
left and increasing its value 10 times. To multiply by
100, place two ciphers at the right of the multiplicand.
To multiply by 1000, place three ciphers at the right of the
multiplicand.
EXERCISE 26.— WRITTEN
1. Multiply 42 by 10, by 100, by 1000.
2. Multiply 24 by 10, by 100, by 1000.
3. Multiply 93 by 10, by 100, by 1000.
4. Multiply 930 by 10, by 100, by 1000, by 10,000.
5. Multiply 860 by 10, by 100, by 1000, by 10,000.
53. Find the product of 842 x 40.
Q J o The product is found by multiplying the multi
.f. plicand by 4 of the multiplier as if it stood alone
and increasing the product ten times by placing a
cipher at the right.
33680
EXERCISE 27. — WRITTEN
1. Multiply 35 by 40, by 400, by 4000, by 400,000.
2. Multiply 350 by 40, by 400, by 4000, by 400,000.
3. Multiply 3500 by 3, by 30, by 300, by 3000, by
30,000.
4. Multiply 3500 by 6, by 60, by 600, by 6000, by
60,000.
48 PRACTICAL ARITHMETIC
54. Rule for multiplying when either multiplier or
multiplicand ends in ciphers. Multiply the multiplicand
by the multiplier without regard to the ciphers, and annex
as many ciphers at the right of the product as are found
at the right of the multiplier and multiplicand.
55. Multiply 4280 by 200.
4280 Multiply as if the problem read 428 X 2, secur
n/^^ ing the product 856. In thus omitting the
856000
ciphers the multiplicand is decreased tenfold
and the multiplier one hundred fold, and the
product is therefore decreased 10 X 100 or 1000 fold. The
product then is 1000 x 856 or 856,000.
EXERCISE 28. — "WRITTEN
Multiply :
1. 876,420 by 3600. 7. 690,000 by 36,420.
2. 960 by 4600. 8. 86,290 by 720.
3. 87,640 by 300. 9. 370 by 6700.
4. 79,842 by 34,000. lo. 296,380 by 3000.
5. 88,967 by 360. li. 28,460 by 7200.
6. 37,900 by 67,000. 12. 67,981 by 37,100.
56. Find the product of 627 x 5864.
To find the product of 5864 multiplied by 627 we must
think of 627 as 6 hundreds plus 2 tens plus 7 units, or
600 + 20 + 7. Multiplying 5864 by each of these numbers
separately as in a, h, and c, we obtain the three partial products
41,048, 117,280, and 3,618,400. The sum of these products is
3,676,728. This method of securing the partial products by
separate multiplications is needlessly long. Since we know
that in multiplying by a number having ciphers at the right
MULTIPLICATION
a. 5864
I. 5864 d. 5864
600
20 627
3518400
, 5864
7
41048
117280
41048
11728
35184
3676728
49
we may disregard
the ciphers, we may
here write the par
tial products directly
under each other as
in the second opera
tion shown at d.
The products obtained by the separate multiplications
are called Partial Products.
Multiply :
EXERCISE 29.— WEITTEN
1.
842 by 56.
9.
3063 by 538.
2.
517 by 75.
10.
3769 by 645.
3.
829 by 88.
11.
8035 by 928.
4.
562 by 94.
12.
2380 by 763.
5.
755 by 48.
13.
4938 by 529.
6.
946 by 67.
14.
9123 by 372.
7.
258 by 99.
15.
5073 by 418.
8.
657 by 46.
16.
6392 by 239.
57
Find the product of 6004 x 1281.
1281
6004 Since the products corresponding to the zeros
5124 in the multiplier will be zeros, they need not be
7686 written in the partial products.
7691124
EXERCISE 30. — WRITTEN
Multiply :
1. 8375 by 206.
3. 2684 by 702.
2. 8295 by 2006.
4. 9367 by 9006.
60 PRACTICAL ARITHMETIC
5. 2473 by 906. 9. 7876 by 903.
6. 2482 by 7002. lO. 8345 by 9008.
7. 2954 by 906. ii. 5878 by 809.
8. 4375 by 603. 12. 7854 by 508.
58. Rule for Multiplication. — Write the multiplier
under the multiplicand, units under units, tens under
tens, etc; Multiply the multiplicand by each figure of
the multiplier. Place the righthand figure of each par
tial product under the figure of the multiplier used to
obtain it. Add the partial products.
Since regarding either number as multiplier or multi
plicand does not affect the product, in practice the smaller
is used as the multiplier.
EXERCISE 31. —WRITTEN
Multiply :
1. 7564 by 73. 14. 8957 by 79. 27. 3264 by 287.
2. 8715 by 86. is. 6428 by 64. 28. 8163 by 799.
3. 4781 by 86. 16. 6753 by 97. 29. 2345 by 986.
4. 3692 by 90. 17. 8429 by 68. 30. 9543 by 576.
5. 5878 by 46. 18. 8867 by 29. 31. 7432 by 438.
6. 4689 by 76. 19. 8456 by 25. 32. 6473 by 823.
7. 5873 by 256. 20. 6397 by 86. 33. 9761 by 82.
8. 6381 by 634. 21. 9876 by 38. 34. 8472 by S4.
9. 9537 by 752. 22. 2785 by 89. 35. 9781 by 73.
10. 2175 by 396. 23. 5432 by 92. 36. 9999 by 99.
11. 7009 by 438. 24. 7654 by 47. 37. 8756 by 65.
12. 8254 by 576. 25. 8765 by 59. 38. 9522 by 76.
13. 7826 by 86. 26. 3528 by 463. 39. 7543 by 57.
MULTIPLICATION
51
The pupil may use the following problems sufficiently
to gain skill in multiplying.
4055. Multiply the number at small a in the diagram
by each number in the larger circle.
• 56—295. Multiply each of the other numbers in the
smaller circle by each number in the larger circle.
Note. The teacher may assign these problems by chance by making
similar circles, pasting them upon pasteboard, and mounting the smaller
one upon a pin so that it may revolve freely.
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3 \
y
MULTIPLICATION OF UNITED STATES
MONEY
59. Multiply 195.35 by 25.
195.35
25 Multiply as in integral numbers, and point off
47675 in the product as many places for cents as there
19070 ^re places for cents in the multiplicand.
12383.75
EXERCISE 32. — WRITTEN
Find the product of :
1. 2. 3. 4.
165.83 189.57 148.75 $46.98
9 8 28 35
5. 6. 7. 8.
1640.25 $542.40 $864.19 $506.69
.89 127 .95 1588
Multiply :
9. $43.95 by 875. • 14. $675.30 by 981.
10. $640.50 by 846. is. $1250.25 by 806.
11. $981.23 by 852. is. $3540.30 by 905.
12. $300.20 by 945. 17. $2859.65 by 79.
13. $756.25 by 372. is. $875.90 by 350.
52
MULTIPLICATION 53
EXBBOISE 33. — WRITTEN
1. Since each State is represented in Congress by 2
senators, how many senators are there in Congress ?
2. A mechanic earns $2.75 a day. What will be his
wages for a week ? For a month of 26 days ?
3. A man earns $3.50 a day. What will be his wages
for a week ? For 26 days ?
4. If a hunter's coat and shoes cost him 16.50 each
and his gun costs 4 times as much as his coat, what is
the cost of his outfit ?
5. If it requires 23 yards of carpet for a floor, what
will be the cost at 11.25 a yard ?
6. What will be the cost of 18 cords of wood at f 3.25
a cord ?
7. What will be the cost of 3 barrels of sugar weigh
ing 284 lbs. each at 6 cts. a pound ?
8. How many feet of picture moulding will be required
for a room 12 ft. long, 15 ft. wide ?
9. What will moulding cost for the above room if two
sizes are used, one at ceiling at 2 cts. a foot and a plate
rail at 7 cts. a foot ?
10. A storm moving eastward across the United States,
travelling at the rate of 36 miles an hour, moves how far
in 3 days ?
11. If 1 cow drinks 73 lbs. of water daily, how many
pounds of water will 16 cows drink ? 28 cows ? 69 cows ?
12. If it costs 18.13 for labor and $7.39 for material
for each acre, to spray a vineyard 6 times, what will it
cost to spray 7 acres 6 times ?
54 PRACTICAL ARITHMETIC
13. If 100 gallons of Bordeaux mixture are sprayed upon
a melon field to prevent blight, and to make this mixture
6 lbs. of blue stone, at 7 cts. a pound, and 12 lbs. of lime,
at 1 ct. a pound, are used, and the cost of labor is 33 cts.,
what will 1 spraying cost ? What will 6 sprayings cost ?
14. If unsprayed melons all die, and sprayed melons
yield 150 baskets to the acre, at 76 cts. a basket, what
is the gain by sprajdng 8 acres of melons?
15. If unsprayed grapevines yield 1 lb. for each vine,
arid sprayed vines yield 4 lbs. each, what is the gain by
spraying an acre which yields 376 lbs. when not sprayed?
16. If spraying potatoes 5 times increases the yield
68 bushels per acre, and spraying 3 times increases the
yield 32 bushels, what is the increase from the last 2
sprayings? With potatoes at 65 cts. a bushel, what are
the 2 extra sprayings worth to the farmer?
17. If the best selected potato seed yields an average
of 11,100 lbs. per acre and poor seed yields 8034 lbs. per
acre, how much more will 9 acres yield when planted
with good seed than with poor seed?
18. If a dairy cow eats 40 lbs. of silage each day, how
many pounds of silage will it take to feed 17 cows 180
days?
19. One field of wheat of 9 acres yields 22 bushels per
acre, and another of the same size, due to better prepara
tion, yields 31 bushels an acre. What is the difference in
the value of the wheat produced on the two fields when
wheat is worth 87 cts. per bushel?
20. One sevenacre field of cotton produces 215 lbs. of
lint cotton and 430 lbs. of seed on each acre, while
MULTIPLICATION 55
another field of the same size produces 500 lbs. of lint and
1000 lbs. of seed per acre. What is the difference in the
value of the crop from the two fields when cotton is worth
11 cts. a pound and seed 1 ct. a pound?
21. If an unclean sample of clover seed contains 990
weed seeds in each pound, how many weed seeds are
there in each bushel weighing 60 lbs. ?
22. A sample of unclean clover seed was found to con
tain 27,600 weed seeds in each pound. If 15 lbs. were
sowed to an acre, how many weed seeds were planted on
each acre?
23. A sample of clover seed offered for sale contained
338,300 weed seeds in each pound. How many weed
seeds were there in each bushel of 60 lbs.?
24. What is the value of an acre of celery yielding
1500 dozen stalks at 26 cts. a dozen? If the annual grow
ing expense is $* 259, what is the profit per acre ?
25. There are 21,780 cubic feet in the upper 6 inches
of an acre of soil. If this soil weighs 79 lbs. per cubic
foot, how much will the upper 6 inches of soil on an acre
weigh ?
26. A man bought a carload of cattle, 25 in number,
each animal weighing 961 lb., at 4 cts. a pound. After
each had gained 889 lbs. in weight, they were sold for
6 cts. a pound. What was his profit or loss if the feed
consumed was worth $977, and the cost of labor to do the
feeding 1125?
27. A man bought 1728 acres of land at 167 an acre.
He spent $3600 on improvements and then sold 79 acres at
56 PRACTICAL ARITHMETIC
$170 an acre, 160 acres at $165 an acre', 215 acres at
$148 an acre, 450 acres at $132.18 an acre, and the rest
at $45 an acre. How much did he gain or lose ?
28. A common housefly lays her eggs in broods of
about 120 each. At this rate, how many flies may be
expected from 29 overwintering flies during the first
warm days of spring ?
29. Mosquitoes lay eggs in masses of about 350 each.
How many mosquitoes may be expected at the end of
the hatching period from 9 such masses ?
30. The white ant often lays as many as 80,000 eggs
in a day. How many eggs may be expected from 30 lay
ing ants at tlie end of 4 days' laying ?
31. A man in each of his working months earns $67.50
and spends $28.37. What are his savings yearly, allow
ing one month of vacation during which he spends $56.97
more than usual ?
32. A man bought 39 strips of carpet, each 17 yards
long, at 86 cts. a yard. What did they cost ?
33. A teamster hauling 4 loads of sand a day, of 27
cubic feet to the load, hauls how many cubic feet in 12
days?
34. Five boys purchase a camp outfit at the following
prices: a 9ft. tent for $5.80, two hammocks at 55 cts.
each, five bathing suits at 55 cts. each, 1 clothes bag at
60 cts., 1 junior camp stove at $1.74, one boat at $19,
2 pairs of oars at $1 a pair, 4 fishing outfits at $1.89
each. What was the cost of the outfit? Two boys paid
toward the outfit $12.22. What was left for the other
three to pay ?
DIVISION
60. If 3 gallons of milk yield 18 ounces of butter, how
many ounces will 1 gallon yield?
To solve this problem we must think of 18 ounces as
separated into 3 equal parts. Separating 18 ounces into
3 equal parts will give 6 ounces in each part.
61. The process of separating a number into equal parts
is called Division.
62. The number to be divided, or the number to be
separated into equal parts, is called the Dividend.
63. The number that indicates into how many equal
parts the dividend is to be separated, or the number by
which the dividend is to be divided, is called the Divisor.
64. The result obtained by Division is called the
Quotient.
To find the number of ounces of butter contained in 1
gallon of milk we divided 18 ounces by 3 (not by 3 gal
lons) which gave 6 ounces. Three, the divisor in this
problem, is an abstract number, and the term 3 gallons
serves only to indicate the number of groups into which
18 ounces is to be separated.
65. When the dividend is concrete and the divisor is
abstract, the quotient is like the dividend, e.g., 18 oz. f 3
= 6 oz.
57
58 PRACTICAL ARITHMETIC
66. When the dividend and divisor are concrete, they
must be alike and the quotient will be abstract, e.g.^
18 oz. = 6 oz. = 3.
67. Division is indicated by the sign, ^, or by writing
the dividend above the divisor with a line between, thus
18 = 3 = 6, or JgS. = 6, is read, 18 divided by 3 equals 6.
68. When the division is not exact, the part of the
dividend remaining is called the Remainder, e.g., 15 r 2 =
7, with 1 as a remainder.
69. When two or more numbers are multiplied together
to produce a product, e.g., 3 x 6 = 18, the numbers so
multiplied are called the Factors of the product.
70. In the problem 18 h 3, observe that we have the
product 18 and one factor 3, to find the other factor. We
may therefore think of division as the process of finding
one factor when the product and the other factor are
given.
EXERCISE 34. — ORAL
Find the quotients of tlie following and prove the divi
sion correct by multiplying the factors together :
1.
42 =
because
2x
= 4.
2.
63 =
because
3x
= 6.
3.
8h4 =
because
4x
= 8.
4.
18 =9 =
because
9x
= 18.
5.
24 =6 =
because
6x
= 24.
6.
427 =
because
7x
= 42.
7.
40 =8 =
because
8x
= 40.
8.
72^6 =
because
6x
= 72.
DIVISION
9. 72^9 =
because 9 x
= 72,
10. 63 f 7 =
because 7 x
= 63.
59
EXERCISE 35. — ORAL
1. The year contains 4 seasons. How many months
in each season ?
2. Alfred the Great divided the day into 3 equal
periods, for sleep, work, and recreation. How many
hours were there in each period ?
3. If f 1 is changed to tencent pieces, how many are
there ?
4. If $1 is changed to fivecent pieces, how many are
there ?
5. If f 1 is changed to twenty fivecent pieces, how
many are there?
6. If fl is changed to twentyfivecent, tencent, and
fivecent pieces, in how many ways can the change be
made?
7. With lettuce plants set 11 in. apart, how many will
be required for 1 row in a cold frame 88 in. long ?
8. The rows are 10 in. apart. How many rows are
there in a cold frame 60 in. wide ?
9. In a fruit garden 165 ft. long, how many peach
trees will be required to set a row, placing them 15 ft.
apart ?
10. How many rows can be set in a garden 90 ft. wide ?
11. An English shilling equals nearly 25 cts. in our
money? How many shillings in $1.50?
60 PRACTICAL ARITHMETIC
12. A French franc equals about 20 cts. in our money.
How many francs in $3.00?
13. A German mark equals about 24 cts. in our money.
How many marks in 2.88?
14. An Italian lira equals about 20 cts. in our money.
How many lire in $5.60?
15. A Dutch florin equals about 40 cts. in our money.
How many florins in $3.60?
16. An Austrian florin equals about 36 cts. in our
money. How many florins in $1.80?
17. At $ 72 a dozen for suits for sixteenyearold boys,
what is the cost of each suit?
18. There are 7 days in 1 week; how many week's in
28 days ? in 42 days ? in 98 days ?
19. If a man earns $ 8 while a boy earns $ 3, how much
will the boy earn while the man earns $48?
20. If an adult man eats 360 grams of carbohydrates,
120 grams of protein, and 60 grams of fats each day, how
much does he average per meal ?
21. If a laboring man eats 150 grams of protein, 99
grams of fats, and 516 grams of carbohydrates each day,
what is the average per meal ?
22. If a man can do as much work as 3 boys, and it
requires 48 boys 3 days to build a fence, how long will it
require 8 men to build it ?
23. If a man can hoe twice as much as a l)oy, and 4
boys and 3 men hoe a field of 5 acres in a day, what
amount does each hoe ?
SHORT DIVISION
71. When the divisor is so small that the work can be
performed mentally, the process is called Short Division.
72. Divide 9672 by 4.
4)9672 The divisor is written at the left of the dividend
2418 with a curved line between them and a line under
neath the dividend. To divide 9672 by 4, begin at
the left and find how many times the divisor, 4, is contained
in the first figure of the dividend. 4 is contained in 9 two
times, with a remainder 1. Reduce the 1 to the next lower
order, making 10, which with 6 makes 16 ; 4 is contained in 16
four times. 4 is contained in 7 one time, with a remainder
3. This remainder reduced to the next lower order makes 30,
which with 2, the next figure of the dividend, makes 32 ; 4 is
contained in 32 eight times. The quotient, therefore, is 2418.
73. Divide 3651 by 7.
7)3651 7 is not contained in the first
521 with 4 remainder, figure of the dividend, 3, so the
commonly written 4. 3 must be reduced to the next
lower order, making 30, which
with 6 makes 36. 7 is contained in 36 five times, with 1 re
mainder. Reduce 1 to next lower order, making 10, which
with 5 makes 15. 7 is contained in 15 two times and 1 remain
der. 1 is reduced to next lower order, making 10, which with
1 makes 11. 7 is contained in 11 one time, with 4 as remain
der. This final remainder is indicated with the quotient.
The quotient, therefore, is 621, with 4 remainder.
61
62
PRACTICAL ARITHMETIC
Find the
1. 376
2. 695
3. 459
4. 376
5. 714
6. 628
7. 984
8. 594
9. 4895
10. 9656
11. 7985
12. 7983
13. 8457
14. 2708
EXERCISE 36. — WRITTEN
quotient of :
h2. 15. 8736 T 7. 29. 56,216^12.
16. 7086^6. 30. 83,542 f 15.
17. 82324. 31. 64,90211.
18. 8361^9. 32. 85,563 =14.
19. 5643 ^ 8. 33. 37,642 ^ 12.
20. 3716^7. 34. 48,527^15.
21. 5472^6. 35. 87,5859.
22. 7483 f 5. 36. 96,464 =22.
23. 8721 f 7. 37. 83,691 r 12.
24. 9212^9. 38. 77,585^11.
25. 73,236411. 39. 23,07514.
26. 24,631412. 40. 89,576 V 11.
27. 75,477^12. 41. 98,25415.
28. 36,286^21. 42. 61,082421.
45.
49.
45.
44.
47.
48.
47.
=6
^9
6
74. How many tens in 20? How many hundreds in
200? How many thousands in 2000?
To divide by 10, 100, 1000, etc., set off as many figures
at the right of the dividend as there are ciphers in the
divisor. The figures thus set off are the remainder.
The other figures are the quotient, e.g., 45 ^ 10 = 4, with 5
as remainder. 468 4 100 = 4, with 68 as a remainder.
EXERCISE 37. .ORAL
1. 57,683 = 100.
2. 76,4934100.
3. 38,425^100.
4. 54,580100.
DIVISION 63
5. 42,676^1000. 8. 57,826  10,000.
6. 26,257^1000. 9. 32,81410,000.
7. 14,637^1000. 10. 82,740^10,000.
75. Divide 4560 by 200.
200 )4560 Cut off ciphers at the right of the 200, also
221^ two figures at the right of the dividend. Divid
ing 45 by 2 gives 22, with 1 remainder. The
1, whicli represents 100, added to the 6, wliich represents 60,
gives 160 remainder.
When the divisor ends in one or more ciphers, cut these
off, and also cut off an equal number of figures from the
right of the dividend. Then divide by the figures re
maining. Place the figures cut off from the dividend
at the right of the remainder, if there is a remainder, to
form the true remainder.
EXERCISE 38. — WRITTEN
Find the quotient of :
1.
376^
20.
12.
7534^
300.
2.
285^
50.
13.
35,456 r
500.
3.
653^
60.
14.
96,464 f
800.
4.
981^
40.
15.
58,775^
700.
5.
9212
80.
16.
28,976 i
600.
6.
9064^
70.
17.
92,123^
400.
7.
887^
60.
18.
82,601 ^
700.
8.
954
50.
19.
42,563 ^
800.
9.
3730
90.
20.
30,600 =
300.
10.
3645 =
200.
21.
67,332 
200.
11.
4328=
300.
22.
85,563 H
400,
LONG DIVISION
76. Long division is the same as short division, with the
exception that all the processes are written in full. The
quotient is written over the dividend, the first figure of
the quotient being written over the righthand figure of
the dividend used in obtaining it.
77. Divide 32,962 by 49.
It will be found helpful in the beginning tabi.e ^p Peoditots
of the study of long division to form a table 1 Xt;9 = 49
of products of the divisor with numbers from 2 x 49= 98
1 to 9. The problem is arranged as in short 3 x 49 = 147
division, with the exception that the quotient 4. v 49 = 196
is written either above (or at the right of) . aq^qa^
the dividend. Since 49 is not contained in ~
32, we must use the first three figures of the o X "iJ = 94
dividend, 329, for the first partial dividend. By 7 X 49 = 343
consulting the table of products, we find that 8 x 49 = 392
294 is the largest of the products that does 9 x 49= 441
not exceed the partial dividend, 329.
294 is the product of 6 x 49 ; hence,
49 is contained in 329 six times. 6 is
49)32962 written over the 9 of the dividend.
^^^ The product of the 6 x 49, 294, is sub
'^^^ tracted from 329, v/hich leaves as a
343 remainder 35. The next figure of the
132 dividend, 6, is annexed to form the sec
98 ond partial dividend. 343 is the largest
34 remainder, of the products that does not exceed
this partial dividend, 366. 343 is the
0,4
DIVISION 65
product of 7 X 49 ; hence, 49 is contained in 356 seven times.
7 is written over the 6 of the dividend and 343, the product of
7 X 49, is subtracted from 356, which leaves 13 remainder.
The next figure of the quotient, 2, is annexed to form the third
partial dividend, 132. 98 is the largest of the products that
does not exceed this partial dividend, 132. 98 is the product
of 2 X 49 ; hence, 49 is contained in 132 two times. 2 is writ
ten over 2 of the dividend and 98 is subtracted from 132, leav
ing 34 the final remainder.
78. Divide 1,270,563 by 396.
32081^5 '^fter some practice the quotient figures
^^^ can be estimated without making the table
^ of products. The first partial dividend is
il££_ 1270. 396 is contained in 1270 three times.
'^•^ 3x396 = 1188. 1188 subtracted from 1270
'^^ leaves 82 remainder. Annexing 6 gives
3363 the second partial dividend, 825. 396 is
3168 contained in 825 two times. 2 x 396 = 792.
195 792 subtracted from 825 leaves 33 re
mainder. Annexing 6 gives the third par
tial dividend, 336. 396 is not contained in 336, so a cipher is
written over the 6 of the dividend and the next figure of the
dividend, 3, is . annexed to 336 to form the next partial divi
dend, 3363. 396 is contained in 3363 eight times. 8 x 396 =
3168. 3168 subtracted from 3363 gives 195, the final remainder.
79. Rule for Long Division.
I. Write the divisor at the left of the dividend with a
curved line between them. For the first partial dividend
take the fewest number of figures at the left of the divi
dend that will contain the divisor, and write the partial
quotient over the righthand figure of the partial dividend.
II. Multiply the divisor, by this quotient, and write the
product under the partial dividend used.
66 PRACTICAL ARITHMETIC
III. Subtract this product, and to the remainder annex
the next figure of the dividend for the second partial
dividend.
IV. Divide as before, and continue the process until all
the figures of the dividend have been used to make partial
dividends.
V. If there be a remainder, write it with the quotient.
Proof. Find the product of the divisor and quotient,
and to this product add the remainder, if any. If the
work is correct, the result will equal the dividend.
EXERCISE 39.
— WB
IITTBN
Find the quotient of :
1.
528012.
17.
89,314 ;
86.
2.
1728^12.
18.
54,963 ^
863.
3.
1607 f 19.
19.
33,765 
542.
4.
5347  21.
20.
84,568 i
827.
5.
3987 ^ 94.
21.
74,938 ^
357.
6.
678473.
22.
861,618^
843.
7.
6548  89.
23.
98,125 =
563.
8.
878965.
24.
84,629 
189.
9.
5498 = 98.
25.
54,825 J
^254.
10.
3850 ^ 63.
26.
46,376 
308.
11.
3987 = 91.
27.
879,384 H
508.
12.
478843.
28.
343,861 H
948.
13.
340281.
29.
324,924 
357.
14.
6281^71.
30.
672,425 H
135.
15.
3485873.
31.
861,254
h927.
16.
6842  78.
32.
938,764
h879.
DIVISION
67
a
6
c
1
3762894
42896
329
2
3862847
76298
269
3
9683921
42546
873
4
1897892
79832
784
5
3984291
62891
672
6
8996823
79684
981
7
8672984
62341
679
8
6293478
98641
837
9
7986277
82439
568
10
1291382
87897
674
The pupil should practise dividing sufficiently to become skilful.
3342. Divide the first number of column h by each
number of column e.
43132. Divide each other number in column h by each
number in column c.
133142. Divide the first number in column a by each
number in column h.
143232. Divide each other number in column a by
each number in column h.
DIVISION OF UNITED STATES MONET
80. Divide $741.32 by 86.
$8.62
oo^ijj l^l.oz Divide as in integral numbers, writing the
688 first figure of the quotient over the righthand
533 figure of the first partial dividend. Place the
516 decimal point in the quotient directly over the deci
JJ2 ™^ point in the dividend.
172
81. Divide $46.75 by $.25.
187
25)4675
25 Change the dividend and divisor to cents which
2^7 gives 4675 cents to be divided by 25 cents. The
onA answer is 187.
~175
175
BXEECISB 40. —WRITTEN
Find the quotient of :
1. $17.28^12. 6. $52.56^16.
2. $17.28 ^$12. 7. $52.54 $16.
3. 17.28^.12. 8. $52.54 .16.
4. 856.90^41. 9. $438.9021.
5. 856.90 ^$41. 10. $438.90 $21.
68
DIVISION 69
EXERCISE 41. — WRITTEN
1. If 8 bushels of rye weigh 448 lbs., what is the
weight of 1 bushel ?
2. If 19 bushels of oats weigh 608 lbs., what is the
weight of 1 bushel of oats ?
3. If 27 bushels of wheat weigh 1620 lbs., what is the
weight of a bushel of wheat ?
4. If 32 bushels of corn weigh 1792 lbs., what is the
weight of a bushel of corn ?
5. If 37 bushels of peas Aveigh 2220 lbs., what is the
weight of 1 bushel of peas ?
6. How many boys' sweaters at 95 cts. can be bought
for 16.65?
7. A double plow harness costing $11.48 and a
double carriage harness costing $45.92, how many plow
harnesses can be had at the cost of 1 carriage harness ?
8. If a beef animal weighs 960 lbs. at the beginning
of the feeding period, and after being fed 162 days
weighs 1284 lbs., how much did it gain a day ?
9. If 40 lbs. of silage is the ration for a dairy cow,
how many acres of corn will it take to produce silage for
15 cows for 150 days, 1 acre of corn producing 18,000 lbs.
of silage ?
10. A baseball outfit for the Asheboro High School
Nine consists of the following articles : an American
Association ball 70 cts., 9 Junior League bats at 25 cts.
each, 1 catcher's mit fl.OO, 3 fielder's gloves at 55 cts.,
1 firstbaseman's mit 90 cts., 4 infielder's gloves at
30 cts., 1 amateur mask 50 cts., 9 pairs shoe plates
70 PRACTICAL ARITHMETIC
10 cts. per pair, catcher's breast protector fl.lO, 9 uni
forms at $1.25 each, 9 sweaters at $1.25 each. What
part of the entire cost should each member pay ?
11. The football eleven of Shelby ville Academy is
equipped with the following articles : boy's Rugby foot
ball with bladder $1.50, inflater 15 cts., 11 jackets at
60 cts. each, 11 pairs of pants at $1.00 eash, 9 pairs
shin guards at 45 cts. per pair, 6 pairs shoulder pads at
30 cts. per pair, 1 head harness at 75 cts. What part of
the cost of the outfit' does each member pay ?
12. The Asheboro High School Girls' Tennis Club
ordered a tennis outfit as follows : 2 hardwood poles at
75 cts. a pair, guy ropes and pins 23 cts. per set, court
marker 95 cts., net $2.00, 2 backstop nets $2.15 each.
What will it cost to equip three courts ? What part will
each member pay if the club is composed of 40 members ?
13. The Croquet Club has 2 croquet sets at $2.50 a
set. There are 16 members. What part of cost does
each pay?
14. If a man buys 1 ton of commercial fertilizer for $24
and puts it on 4 acres, and another ton for $ 20 and puts
it on 5 acres, how many additional pounds of cotton to
the acre must he produce on the first field at 10 cts. a
pound to pay for the greater cost of the fertilizer used
thereon ?
15. The following results for a period of 5 years'
spraying Irish potatoes to prevent disease are recorded in
New York : gain per acre due to spraying every 2 weeks,
123, 118, 233, 119, 63 bushels; gain per acre due to 3
sprayings, 98, 88, 191, 107, 32 bushels. What was the
DIVISION 71
average gain for the 5 years with 3 sprayings ? What was
the average gain from spraying every 2 weeiis ? If potatoes
average 57 cts. a bushel and spraying cost on an average
77 cts. an acre each time, what was the average gain from
spraying 3 times?
16. If 5 gallons of spraying mixture be used to each
tree in an apple orchard of 1000 trees and Paris green at
20 cts. a pound be used, 1 lb. to 150 gals, of water, what
would be the cost of the Paris green for 2 sprayings?
What would it cost per tree?
17. A pig weighing 50 lbs. cost 14. After eating 8 bus.
of corn at 50 cts. a bushel and 400 lbs. of wheat middlings
at 11 a hundred pounds, it weighs 250 lbs. At how much
a pound must it be sold to make a net profit of f 3 ?
18. What is the average of 79, 83, 160, 74, and 62?
19. At three places in the Florida peninsula the num
ber of partly cloudy days in the year was 233, 145, 113,
the number of cloudy days was 59, 74, 81. What was the
average number of cloudy, partly cloud}', and clear days
as shown by these observations?
20. A man bought 782 acres of land for $98,762 and
sold it at $18 an acre. Wliat was the average price paid?
What was the average gain or loss per acre ?
21. The deaths from consumption in the United States
in 1900 were 111,059. W^hat was the average number of
deaths for each day in the year (365 days) ?
22. The deaths from typhoid fever in the United States
for 1900 were 35,379. What was the average number
daily ?
CANCELLATION
82. How many times is 3 x 6 contained in 12x6?
How many times is 3 x 10 contained in 6 x 10? How
many times is 6 x 7 contained in 12 x 7 ?
83. Divide 24 x 12 by 6 x 12.
24x12 = 288 6x12 = 72 288 h 72 = 4.
What factor is common to both dividend and divisor?
If the factor 12 is struck out, or cancelled, from both divi
dend and divisor, does it affect the quotient?
2iiil = 24 or 24^6 = 4.
6 X ;i;2 6
Divide 5 x 27 x 4 by 18 x 2.
3 3 Cancelling the common factor 2 from the
5x^7x^ ^1 4 of the dividend and the 2 of the divisor
78x2 ~ '^' gives 2 in the dividend. Cancelling the
2 common factor 9 from 18 gives 2; from 27
gives 3. Cancel 2 from both divisor and
dividend. Multiplying the remaining factors of the dividend
gives the quotient, 15.
84. Cancellation is the process of shortening work in
division by removing or cancelling equal or common
factors from the dividend and divisor.
EXERCISE 42. — WRITTEN
Solve by cancellation :
1. Divide 7 x 6 x 16 by 6 x 8 x 7.
72
CANCELLATION 73
2. Divide 11 x 27 x 30 by 9 x 15 x 3.
3. Divide 15 x 48 x 70 x 11 x 40 by 30 x 16 x 7 x 22 x 50.
4. Divide 84 x 18 x 5 by 91 x 4 x 15.
5. Divide 27 x 12 x 35 x 14 by 9 x 3 x 5.
6. Divide 48 x 35 x 42 x 54 by 12 x 7 x 6 x 9.
7. Divide 63 x 36 x 48 x 96 x 27 by 81 x 9 x 12 x 48.
8. Divide 420 x 68 x 88 x 22 by 210 x 11 x 44.
9. Divide 42 x 35 x 56 x 4 x 12 by 28 x 49 x 14 x 10.
10. Divide 17 x 9 x 12 x 11 x 28 by 34 x 6 x 72 x 6 x 22.
11. Divide 24 x 15 x 8 x 4 x 7 by 14 x 8 x 6 x 4.
12. Divide 20xl6x5x3x6by5x8x3xl0x6.
13. 18 cows, each eating 4 lbs. of cottonseed meal a
day, can be fed how long on 360 lbs. of meal?
14. 72 quarts of berries at 9 cts. a quart equal in value
how many pounds of sugar at 6 cts. a pound?
15. How long will it take a horse travelling 8 miles an
hour to go as far as an express train goes in 3 hours at
32 miles an hour ?
16. If 42 bushels of wheat make 9 barrels of flour, how
many bushels will it take to make 27 barrels?
17. How many bushels of corn worth 49 cts. a bushel
must be given in exchange for 63 bushels of oats worth
35 cts. a bushel?
18. How many bushels of oats worth 28 cts. a bushel
must be grown on an acre to equal in value a crop of 56
bushels of corn worth 42 cts. a bushel?
19. A lumber mill cuts 60,000 ft. of lumber in 6 hours.
How many feet will it cut in 3 days of 8 hours each ?
74 PRACTICAL ARITHMETIC
exercise 43. — "written
Review Problems
1. The population of Chicago in 1840 was 4470, in
1870 it was 300,000, in 1900 it was 1,698,675. What was
the yearly increase in population from 1840 to 1870?
What was the yearly increase from 1870 to 1900 ? What
was the daily increase ?
2. The value of the hay crop raised in the United
States for five years was as follows :
What was the total value
for five years ? What was the
average value ? How much
below the average was the
value of the smallest crop ?
3. The value of the corn exported from the United
States for five years was as follows :
1902 116,185,673. What was its total value?
1903 140,540,673. What was its average value?
1904 $30,071,334. What was the difference be
1905 147,446,921. tween the largest and the
1906 162,061,856. smallest amount exported?
4. A farmer has a small herd of 5 dairy cows which
produce as follows:
No. 1 = 396 lbs. butter a year, at 26 cts. a pound, cost of feed $59.00
No. 2 = 323 lbs. butter a year, at 26 cts. a pound, cost of feed $57.50
No. 3 = 257 lbs. butter a year, at 26 cts. a pound, cost of feed $55.67
No. 4 = 176 lbs. butter a year, at 26 cts. a pound, cost of feed $52.38
No. 5 = 147 lbs. butter a year, at 26 cts. a pound, cost of feed $49.84
1902 .
. . 1542,036,364,
1903 .
. . 1556,376,880.
1904 .
. . $529,107,625.
1905 .
. . $515,959,784.
1906 .
. . $592,539,671.
REVIEW PROBLEMS 75
What is the yearly profit on the herd ? What is the profit
or loss ou each cow?
If he sells Nos. 4 and 5 and buys instead 2 other cows
which produce as follows:
Xo. 6 = 298 lbs. butter a year at a cost for feed of $57.23
Xo. 7 = 276 lbs. butter a year at a cost for feed of $54.50
what will be his yearly profit ou the herd? on each cow?
5. The value of the cotton exported from the United
States for 5 years was as follows :
1902 .... 1291,598,350.
xA/ n«!>+ TUT" n CI fhci ^*»fril iTolna r\t
1903 . ,..,_._.
t^of.) OAQ Of? I the exported cotton? What
1904 .
1905 .
1906 .
. !iS381,398,939. '^^' ^^^'^ ^""''""^^ ^"^"^^
. 1401,005,921. ^^^"^^
6. A farmer delivered to a creamery during May
3674 lbs. of milk, during June 4876 lbs.', July 3929 lbs.,
August 3167 lbs., September 3067 lbs., October 2913 lbs.
What was the total number of pounds during the half
year?
7. During November a farmer delivered to the cream
ery 2974 lbs. of milk, during December 2984 lbs., Janu
ary 2798 lbs., February 2890 lbs., March 3043 lbs., April
3364 lbs. What was the total number of pounds deliv
ered during the latter half of the year? How much less
was this than in the warmer half of the year?
8. What number multiplied by 256 with 23 added to
the product will give 5399?
9. Light travels 186,680 miles a second. How far
will it travel in 29 seconds?
76 PRACTICAL ARITHMETIC
10. Sound travels 1130 feet in a second. How far
will it travel in 29 seconds ?
11. It is, in round numbers, 25,000 miles around the
earth. How many times will a light wave go round the
earth in a second?
12. What is the number nearest to 3980 which is di
visible by 234 without a remainder?
13. In the United States in 1899, $49,099,936 was
spent for fertilizers. This being used on 4,970,129 farms
was an average of how many dollars' worth on each farm ?
14. In 1880 the expenditure of the United States for
fertilizers was $28,500,000, in 1890 it was $38,500,000,
in 1900 it was $54,750,000. How much more was spent
for fertilizers in 1900 than in 1880? in 1900 than in
1890? in 1890 than in 1880? How much less in 1900
than in 1880 and 1890 together?
15. The shrinkage in weight of corn due to drying be
tween December and May is about 9 lbs. for every hun
dred pounds. What would be the loss in weight of 900
bushels (1 bu. = 56 lbs.)? The price of corn advances on
an average 4 cts. a bushel between December and May.
Will this advance in price compensate for the loss due to
shrinkage with corn in December at 57 cts. per bushel?
16. The expenses of raising an acre of white pine from
the seed may be stated as follows :
Cost of seedlings, $2.00. Cost of land, $6.00. Trans
planting to nursery, $1.21. Transplanting to field, $5.45.
Taxes, first decade, $1.69. Taxes, second decade, $3.04.
Taxes, third decade, $4.39. Taxes, fourth decade, $5.74.
What is the total cost ?
REVIEW PROBLEMS
77
CUTS OE BEEF
The above illustration shows butchers' cuts, their rela
tive weights, and Western retail prices ; live weight of
animal 1550 lbs., dressed weight of carcass 1046 lbs.
Cheap Cuts VALirABLE Cuts
No. 1. Neck, 40 lbs. @ 4 cts. No. 3. Prime of Rib, 117 lbs. @ 16
cts.
No. 2. Chuck, 237 lbs. @ 7 cts. No. 4. Porterhouse Steak, 103 lbs.
@ 22 cts.
No. 8. Shank, 30 lbs. @ 2 cts. No. .5. Sirloin Steak, 87 lbs. @ 18
cts.
No. 9. Flank, 52 lbs. @ 5 cts. No. 6. Rump, 36 lbs. @ 10 cts.
No. 10. Plate, 138 lbs. @ 5 cts. No. 7. Round Steak, 183 lbs. @
No. 11. Shin, 23 lbs. @ 4 cts. 10 cts.
17. The difference between the live weight and the
dressed weight is the waste in slaughtering. How much
was it in the animal illustrated ?
78 PRACTICAL ARITPIMETIC
18. The parts of the carcass numbered 1, 2, 8, 9, 10,
and 11 are in the forward and lower parts of the animal,
and are less valuable. How many pounds are there in
all these cheaper parts ? What is their total value ?
19. The parts numbered 3, 4, 5, 6, and 7 are in the
hind quarters and upper portion of the body, and are more
valuable. How many pounds are there in all these parts?
What is their total value ?
20. What is the difference in the total weights of the
cheaper and the more expensive cuts ? What is the dif
ference in their values ?
21. What is the average value per pound of the entire
carcass? What is the average value per pound of the
cheaper outs? What of the more expensive cuts?
22. One animal weighing 1550 lbs. produces a dressed
carcass of 1046 lbs. that sells for 9 cts. a pound ; an in
ferior animal of the same live weight produces a carcass of
990 lbs. that sells for 8 cts. a pound. How much greater
is the value of the better animal ?
POULTRY
23. If 1 gallon of water glass mixed with 16 gallons
of water makes a solution which will preserve 50 dozen
eggs, what is the gain in putting up 25 dozen fresh eggs
in summer when they are worth 15 cts. a dozen and keep
ing them until winter when they are worth 25 cts. a dozen?
24. If a lot of 8 dozen eggs that weighs 9 lbs. sells for
27 cts. a dozen, how much do they bring a pound? What
will be the gain in selling another lot of 8 dozen eggs
that weighs 14 lbs. for the same price per pound instead of
the usual way of selling them at the same price per dozen ?
EEVIEW PROBLEMS 79
25. A flock of hens averages 12 dozen eggs for each hen
yearly. If they each consume $1.45 worth of feed, what
is left to pay for their care, losses from death, and profit,
if the eggs bring an average price of "22 cts. a dozen?
26. If to raise 73 chicks to 8 weeks of age costs : for
eggs set $2.50, heat for incubators and brooders fl.OO,
feed $3.87, what is the average cost per chick?
27. If a flock of hens fed dry mash and grain produces
216 dozen eggs at a cost of 11 cts. a dozen, and a second
flock of the same number fed wet mash and grain pro
duces 163 dozen eggs at a cost of 10 cts. a dozen, how
much greater is the profit from the first flock if eggs sell
for 22 cts. a dozen ?
28. If the total weight of the chicks in problem 26 be
109 lbs., what is the average cost per pound ?
29. If they are sold at 23 cts. each, what remains to
pay for the labor and profit?
30. By inquiry find the local prices and costs in your
neighborhood, and substitute them in problems 25 and 26.
80 PRACTICAL ARITHMETIC
PREVENTION OF PLANT DISEASES
Total Yield of Marketable Potatoes from Two Rows Sprayed
Total Yield of Marketable Potatoes from Two Rows not Sprayed
31. In treating seed oats to prevent oat smut, 1 ounce
of formalin is used to every 3 gallons of water, and 1 gal
lon of this mixture suffices for 1 bushel of oats. How
many ounces of formalin will be needed to treat 24 bushels
of oats?
32. Using 3 bushels of seed oats to the acre, what will
be the cost of sufficient formalin to treat seed for 80 acres
at 35 cts. for each 16 ounces ?
33. The Bordeaux mixture consists of blue stone 5 lbs.,
lime 5 lbs., and water 60 gallons. With blue stone costing
7 cts. a pound and lime at 1 ct. a pound, what will the
materials for 150 gallons of Bordeaux mixture cost ?
34. It requires about 150 gallons of Bordeaux mixture
to spray an acre of Irish potatoes once. What will the
materials for 3 sprayings cost?
REVIEW PROBLEMS 81
35. Spraying apple trees twice with BordeauxParis
green mixture to prevent worminess costs 13 cts. for each
tree. If this increases the value of the yield f 1.37 for
each tree, what will be the gain by spraying an orchard of
87 trees ?
36. The cost of labor and materials for spraying an
acre of grapes to prevent the black rot being as follows,
what is the cost of each spraying and the total cost of
labor and materials for the six sprayings?
Cost op Material Cost of Labor Total Cost
1st spraying .... $0.45 $1.13
2cl spraying 0.68 1.13
3d spraying 0:68 1.10
4th spraying 1.86 1.47
5th spraying 2.06 1.65
6th spraying 2.06 1.65
Total I^ IZI
37. The average cost of spraying Irish potatoes to pre
vent blight being $5.18 an acre, and the average increase
in value being $19.07 an acre, what is the average profit
from spraying ?
38. Six unsprayed apple trees yielded 188 sound apples
and 4244 rotten apples. Six similar trees spra3'ed yielded
8674 sound apples and 989 rotten ones. What was the gain
for each tree in sound apples from spraying ? What was
the decrease in rotten apples for each tree ?
39. What would it cost to spray the potatoes and
apples and to treat all the oat seed planted within a mile
of your school?
40. What would the profit be if it equalled per acre
the increase indicated in the above problems ?
82 PRACTICAL ARITHMETIC
A Bountiful Harvest
FARM CROPS
41. The average yield of wheat for the United States
for each acre in 1906 was about 15 bus., the price was
67 cts. a bushel. What was the value of the yield per
acre?
42. The average yield in some States is about 10 bus., in
others 32 bus. How much greater value per acre is pro
duced in the latter than in the former?
43. If it costs $2.16 for fertilizer, 96 cts. for seed,
37 cts. for housing, $1.20 for threshing, 76 cts. for
marketing, and 12.81 for rent per acre in the United
States, what is the average left to pay for labor and
profit ?
44. What is left for labor and profit in the lowyielding
States?
45. What is left for labor and profit in the higher
yielding States?
46. If one farm yields 40 bus. an acre and another farm
16 bus. an acre, each at a cost of $9.67, how much greater
KEVIEW TROBLEMS 80
is the value of the yield on the former farm, supposing the
farms to consist of 79 acres each? How much greater
is the profit?
47. If a 3000lb. crop of cowpea hay per acre is grown
on these fields, the stubble and roots will add 28 lbs. of
nitrogen, worth 19 cts. a pound. What is the value of
the nitrogen added?
48. Find by inquiry the average yield of each of the
above crops in your locality, and substitute your local
value in the problems, and solve.
SHIPPING
Goods are shipped by express or freight, the former
commonly being used for lighter articles or when speed
is necessary. The freight rate varies with the articles
shipped, there being some twelve or more classes. In the
following table the class is indicated in parenthesis after
the name of the article. The smallest charge made is usu
ally equal to that on 100 lbs. of the class of goods shipped.
Empty barrels (6) Furniture, old, carload lots
Trunks of baggage (1) (6)
Trunks limited to 1 5 a Vegetables : potatoes,
hundred (D 1) onions, cabbage, etc. (6)
Clothing (1) Bicycles, set up (1)
New furniture, set up (1) Bicycles, not set up (2)
New furniture, not set up Vegetables, canned, less
(2) than carload (2)
Furniture, old, value limited Vegetables, canned, in ear
to $ 5 per 100 (4) load lots (4)
84 PRACTICAL ARITHMETIC
Grapes (1) Flour (6)
Apples and pears in bas Baled hay (D)
kets (2) Plows, set up (2)
Apples and pears in sacks Plows, not set up (4)
or barrels (6) Power cutter, set up (1)
Grain in bulk (D) Power cutter, not set up (3)
The freight rate between two cities per 100 lbs. being :
1st class 11.03, 2d 92 cts., 3d 79 cts., 4th 65 cts.,
5th 54 cts., 6th 43 cts., A 33 cts., B 39 cts., D 33 cts.,
E 54 cts., H 66 cts.
Solve the following problems :
49. What will be the freight on a shipment of bicycles,
weight 200 lbs., set up ? Not set up ? In a shipment of
bicycles weighing 500 lbs., what is the saving in freight if
they be not set up ?
50. How much cheaper is it to ship 1500 lbs. of new
furniture not set up than to ship it set up ?
51. How much cheaper is it to ship 1500 lbs. of old
household furniture, value limited to 15 per hundred
pounds in case of loss, than to ship the same weight of
new furniture set up ? Than to ship the same weight
of new furniture not set up ?
52. What is the cost of shipping one carload (20,000
lbs.) of old furniture? How many pounds of the same
goods shipped by the hundred will it take to cost the
same for freight ?
53. What is the difference in freight cost between 200
lbs. of empty barrels and the same weight of trunks of
KEVIEW PROBLEMS 85
54. What is the difference in cost of freight on a ton
(2000 lbs.} of canned vegetables and the same weight not
canned ?
55. What is the difference in cost between a carload
(24,000 lbs.) of canned vegetables and the same weight
sent in two separate shipments, i.e., by the hundredweight?
56. What is the difference in freight cost between
1900 lbs. of apples in baskets and the same weigiit in
barrels ?
57. Which costs the most, the freight on 1700 lbs. of
flour or on 1700 lbs. of wheat ?
The express charge between the points for which freight
rates were given above is 25 cts. for less than 1 lb., 35
cts. from 1 to 2 lbs.,, 45 cts. from 2 to 3 lbs., 55 cts. from
3 to 4 lbs., 60 cts. from 4 to 5 lbs., 70 cts. from 5 to 7 lbs.,
75 cts. from 7 +n 10 lbs., 85 cts. from 10 to 15 lbs., fl.OO
from 15 to 20 Ib^., fl.lO from 20 to 25 lbs., 11.15 from
25 to 50 lbs., f 2.30 from 50 to 100 lbs., and i2.30 per
hundred for weights greater than 100 lbs.
58. What will be the difference of cost for a 200lb.
trunk shipped by express and by freight ?
59. What will be the difference between freight and
express charges on 300 lbs. of apples in baskets ?
60. Which will be the cheaper way to ship a suit of
clothes weighing, when boxed, 11 lbs.? How much
cheaper ?
61. Pupils may add problems concerning produce
shipped to or from their nearest freight and express
offices.
DIVISOKS AND MULTIPLES
85. Name two factors of 18, 25, 32, 81, 120.
Name three factors of 18, 30, 45, 50, 66.
Name a factor common to 12 and 36.
If 3 is taken as one of the factors of 18, what is the
other factor ? How is the second factor found ?
The process of separating a number into its factors is
called Factoring.
86. An exact Divisor of a number is a factor of that
number.
87. A factor or a divisor that is common to two or
more numbers is called a Common Divisor.
EXERCISE 44. — ORAL
Find a common divisor :
1. 35, 45, 60. 5. 63, 72, 81. 9. 32, 48, 64.
2. 21,35,70. 6. 45,24,54. lO. 27,36,75.
3. 12, 24, 36. 7. 15, 30, 36. ii. 12, 18, 22.
4. 36, 28, 72. 8. 18, 24, 54. 12. 72, 81, 96.
88. Name all the factors or exact divisors of 3, 7, 19.
A number that has no factors or divisors except itself and
1 is called a Prime Number, e.^., 7, 11, 19, are prime
numbers.
89. Factors that are prime numbers are called Prime
Factors.
DIVISORS AND MULTIPLES 87
90. A number not a prime number is called a Com
posite Number.
91. Which of the following numbers are exactly divisi
ble by 2 or have 2 as a factor : 2, 4, 6, 9, 11, 13, 14, 15,
16, 17, 21, 22, 27, 30 ?
92. Every number which contains the factor 2 is called
an Even Number.
93. Numbers that are not divisible by 2 are called Odd
Numbers.
94. Numbers that have no common factors are said to
be prime to each other.
EXERCISE 45. — WRITTEN
1. Write a list of all prime numbers below 100,
2. Write a list of all odd numbers below 100.
3. Write all the exact divisors of all the numbers from
1 to 50.
4. Separate the following into prime factors : 4, 5, 7,
8, 10, 12, 13, 14, 16, 18, 21, 24, 25, 30, 34, 36.
5. Separate the prime, composite, even, and odd numbers
in the following, and give reasons for your answers : 1, 6,
7, 10, 11, 12, 14, 19, 21, 24, 26, 27, 32, 33.
6. Write a list of the numbers from 2 to 50 that con
tain 2 as a factor. What do you note regarding the units'
figure of each number?
7. Write a list of the numbers from 5 to 100 that con
tain 5 as a factor. What do you note regarding the units'
figure in each case?
88 PKACTICAL ARITHMETIC
8. Write a list of the numbers from 3 to 60 which
contain 3 as a factor. What do you note regarding the
sum of the figures or digits of each number?
9. Write a list of the numbers from 9 to 90 that con
tain 9 as a factor, and find whether 9 is exactly contained
in the sum of the digits of each of these numbers.
95. A number is divisible :
by 2 if the units' figure is 2, 4, 6, 8, or ;
by 3 if the sum of its digits is divisible by 3;
by 4 if the number represented by the two righthand
figures is so divisible ;
by 5 if the units' figure is 5 or ;
by 8 if the number represented by the three righthand
figures is so divisible ;
by 9 if the sum of its digits is so divisible.
96. Find the prime factors of 720.
5 )720
According to divisibility test 5 is a factor. 2 is a
second, third, fourth, and fifth factor, ,3 is the last
factor. Hence, the prime factors of 720 are 5, 2, 3,
2, 2, 3, and 3.
EXEECISB 46.—
Find the prime factors of :
1. 670.. 5. 420. 9.
2. 981. 6. 462. 10.
3. 310. 7. 741. 11.
4. 2650. 8. 1575. 12.
2)144
2)
72
2)
36
2)
18
3)
9
WRITTEN
385.
13.
321.
297.
14.
335.
2430.
15.
378.
1215.
16.
10,935.
DIVISORS AND MULTIPLES 89
97. The greatest factor or divisor that is common to
two or more numbers is called the Greatest Common
Divisor (G.C. DO
SS. Find the G. C. D. of 24 and 30.
04 _ 9 y ^ V 4 ^® ^'^^ ^^^^ *^® common factors of 24
and 30 are 2 and 3. Multiplying these
' common factors gives the Gr. C. D. There
fore, 2 X 3 = 6 is the G. C. D.
99. Rule. To find the G. C. D. of two or more num
bers, separate the numbers into their prime factors and
find the product of the prime factors that are common
to all the numbers.
EXERCISE
47.— WRITTEN
Find the G. C. D. :
1. 14 98,112.
6. 13, 91, 136.
2. 60, 120, 150.
7. 32, 48, 128.
3. 28, 42, 36.
8. 45, 72, 81.
4. 24, 30, 36.
9. 24, 80, 96.
5. 21, 28, 77.
10. 44, 77, 121.
100. When it is required to find the G. C. D. of two or
more numbers that cannot readily be separated into prime
factors, the following method may be employed.
Find the G. C. D. of 63 and 217.
63)217(3
189
''^ff The G. CD. is 7.
56
7)28(4
28
90 PRACTICAL ARITHMETIC
101. Rule. Divide the greater number by the smaller
and the divisor by the remainder; continue the process
until there is no remainder. The last divisor will be the
G. C. D.
BXBECISB 48.
 WRITTEN
Find the G. C.
D.
of:
1.
144, 576.
8.
6004, 3318.
2.
720, 144.
9.
1820, 3367.
3.
98, 112.
10.
1485, 1155, 1750.
4.
720, 1728.
11.
1254, 2361, 8163.
5.
820, 697.
12.
125, 175, 1792.
6.
1086, 905.
13.
1024, 1280, 1792.
7.
1220, 2013.
14.
315, 2267, 9012.
102. If 6 is multiplied by 3, the product is 18; 18 is
called a Multiple of 6 and 3.
EXERCISE 49. — ORAL
Name tvi^o multiples of :
1. 7 and 5. 4. 8 and 2.
2. 3 and 7. 5. 9 and 3.
3. 6 and 3. 6. 10 and 4
103. A multiple of each of two or more numbers is
called a Common Multiple of the numbers.
104. Of the common multiples of two or more numbers
the least is called the Least Common Multiple (L. C. M.),
e.g.^ 18 is a common multiple of 3 and 6, but 12 is the
least common multiple of these numbers.
7. 7 and 6.
10. 3, 6, and 9.
8. 8 and 5.
11. 8, 6, and 4.
9. 9 and 2.
12. 9, 2, and 8.
DIVISORS AND MULTIPLES 91
105. Find the L. C. M. of 30 and 70.
on _ y g „ r The prime factors of 30 are 2x3x5.
70—0 r. 7 '^^® prime factors of 70 are 2x5x7. To
obtain a number which will contain both 30
and 70, the factors 2, 3, 5, 7, are selected, which multiplied to
gether give 210, the L. C. M.
106. The following method of finding the L. C. M. of
two or more numbers is also used. Find the L. C. J\I. of
30, 70, 18.
By multiplying together the final quotients
and the divisors, 2x5x3x7x3, we get
630, L. C. M.
107. Rule. To find the L. C. M. of two or more num
bers, separate each number into its prime factors, and tak
ing each factor the greatest number of times that it appears
in any one of the given numbers, find the product of these
factors.
2)30
70 18
5)15
35 9
3) 3
7 9
EXERCISE 50.
— WRITTEN
Find the L.C.M.
of:
1.
5 and 15.
9.
21, 24, 26, 28.
2.
9 and 12.
10.
9, 10, 14, 15.
3.
21 and 36.
11.
7, 14, 56, 84.
4.
8, 16, 64.
12.
72, 66, 111.
5.
24, 30, 36.
13.
11, 22, 55, 110.
6.
9, 12, 50.
14.
8, 21, 28, 35.
7.
36, 54, 63,
15.
13, 15, 26, 39.
8.
17, 34, 51 .
16.
4, 21, 42, 63.
DECIMAL FRACTIONS
EXERCISE 51. — OEAL
1. Name the smallest common silver coin in our money;
name the largest silver coin.
2. The smallest silver coin is equal to what part of the
largest ?
3. Name the smallest coin in our money. The smallest
coin is equal to what part of the smallest silver coin?
Of the largest silver coin ?
4. Read the following as dollars, dimes, and cents:
13.33, 14.44, $5.55, 16.66, f 7.77, 19.99.
5. Read the following as dollars, tenths, and hun
dredths of a dollar: $8.33, $4.44, $5.55, $6.66, $7.77.
$9.99.
EXERCISE 52. — WRITTEN
Write the following as cents, using the dollar sign :
1. 55 hundredths of a dollar.
2. 90 hundredths of a dollar.
3. 36 hundredths of a dollar.
4. 34 hundredths of a dollar.
5. 25 hundredths of a dollar.
6. 10 hundredths of a dollar.
7. 1 dollar and 10 hundredths.
8. 5 dollars and 15 hundredths.
92
DECIMAL FRACTIONS 93
108. As dollars may be divided into tenths and hun
dredths, so units of anything may be divided.
How many tenths of a thing are equal to 1 unit?
To 10 units? To 100 units?
109. The division of units into tenths, hundredths, or
thousandths is called Decimal Division.
110. The part of a unit obtained by decimal division
is called a Decimal Fraction. Decimal fractions are
commonly called Decimals.
111. The period placed at the left of tenths, e.g., .2, is
called the Decimal Point.
112. Decimal fractions may also be expressed thus :
.2 or twotenths. It may be also written ■^^. When so
written, the decimal fraction is expressed in the form of
a Common Fraction.
113. The following table gives the names of the deci
mal places :
•a
.9 " ^ o
5 °
a ""Ho
2 j= S cs S
no = J S cS 3 oj
•OS * il "O Sj, si
4672.198462
EXERCISE 53. — OBAIj
Read the following :
1. 1.7. 4. 3.87.
7. 2.9.
10. 3.80.
2. 6.12. 5. 84.67.
8. 0.90.
11. 24.06.
3. 31.73. 6. 90.01.
9. 6.7.
12. 300.20,
94 PKACTICAL ARITHMETIC
EXERCISE 54. — ORAL
Name the place at the right of tenths, at the right of
hundredths, at the right of thousandths, the fourth place,
the fifth, the sixth, the seventh.
Read the following :
1. 15.004. 4. 0.0001. 7. 50.504. lo. 98.7256.
2. 25.102. 5. 1.4111. 8. 75.6281. ii. 100.2376.
3. 30.675. 6. 10.1063. 9. 86.5467. 12. 105.20068.
EXERCISE 55. — "WRITTEN
Write in decimal form :
1. Fivetenths, threetenths, fourtenths, one and one
tenth .
2. Six and eight thousandths, eight and fortytwo
thousandths, two tenthousandths, ten millionths.
3. One hundred fortyfive thousandths, three hundred
eightyone thousandths, four tenthousandths, two hun
dred and two hu.ndredths.
4. Five and sixtenths, nine and onetenth, seventyfive
hundredths, four hundred and four thousandths.
5. Write these common fractions as decimal fractions :
_3_. _9_ _e^ _i _1_ 6 s
10' 10' iTT' 10T5' 100' lOOO'T'OOOO
6. Distinguish between 0.300 and 0.00003.
7. Express as common fractions : 0.25, 0.250, 0.2500.
8. Annexing a cipher to a whole number increases its
value how many times ?
9. Does annexing a cipher to a decimal affect its value?
ADDITION OF DECIMALS
114. Add: 25.725,62.8,909.003,4.23681.
25.725
g2.8 ^^^ decimals as in the addition of United
States money. Place the decimal points in a
vertical line, and add as in integral numbers.
909.003
4.23681
EXERCISE 56. — WRITTEN
Find the sum of :
1. 5.47 2. 7.87 3. 112.63 4. 41.039
16.62 0.125 6.87 7.537
5.097 19.00 152.00 5.03
25.608 5.03 7.75 0.107
5. 19.083, 0.96, 5.03, 27.107, 5.1464, 0.905.
6. 489, 0.16, 0.49, 7.07, 5.0909, 0.0008.
7. 37.204, 3.1459, 143.59, 3415.18.
8. 6.2525, 62.525, 625.25, 6252.5.
9. Express as decimals and add : ^% ^\%, ^%\, j;^\^o'
h% 30^, 300^^0^.
EXERCISE 57.— WRITTEN
1. The amount of protein in pounds contained in 10 cts.
worth of each of the following kinds of beef is as follows :
tenderloin steak .064, sirloin steak .081, loin roast .090,
rib roast .088, round steak, first cut., .130, round steak
95
96 PRACTICAL ARITHMETIC
.135, chuck .129, rumiJ .114, shoulder .155, round, second
cut, .205, neck .207, brisket .20, plate .230, flank .284,
shank .256. I buy 10 cts. worth of each. Find the
total amount of protein purchased ; how much money
do I spend?
2. The amount of protein found in 10 cts. worth of
each of the following kinds of pork is : smoked ham .071,
bacon .065, smoked shoulder .108, fresh ham .112, fresh
shoulder .120, ribs and loin .134, fat salt pork .019. If
I buy 10 cts. worth of each, how much protein do I buy?
At what cost?
3. The amounts of protein in pounds found in 10 cts.
worth of veal are as follows : cutlet .089, loin and rib
.093, leg .098, shoulder and breast .18, chuck and neck
.133, knuckle or shank .346, flank .424. Find the total
amount of protein purchased if I buy 10 cts. worth of each
of these cuts of veal. What does such a purchase cost ?
4. The monthly rainfall at Davenport, la., a place of
moderate rainfall, expressed in inches, was: 1.6, 1.6, 2.2,
2.7, 4.4, 4.1, 3.7, 3.6, 3.2, 2.4, 1.8, 1.6. What was the
rainfall for the year ?
5. The monthly rainfall at IMobile, Ala. (heavy), was :
4.8, 5.3, 7.4, 4.5, 4.2, 6.1, 6.7, 6.9, 4.9, 3.2, 3.5, 4.6.
What was the yearly rainfall ?
6. The monthly rainfall at Winnemora, Neb. (scant),
was : 1.1, 0.9, 0.8, 0.9, 1.0, 0.6, 0.2, 0.2, 0.3, 0.5, 0.7, 1.2.
What was the annual rainfall ?
7. The monthly rainfall at Darjiling, India (excessive),
was: 1, 1.5, 2, 6, 8, 16, 16, 16, 16, 6, 0.5, 0.5. What was
the annual rainfall ?
SUBTEACTION OF DECIMALS
115. From 9.25 take 7.075.
9.250 ^^ there are more decimal places in the subtra
_ „_ hend than in the minuend, annex ciphers until they
— ^ have the same number of places.
EXERCISE 58. — WRITTEN
1. From 1.25 take 0.15. 7. From 25.25 take 2.525.
2. From 7.75 take 1.95. 8. From 37.18 take 9.189.
3. From 14.2 take 4.92. 9. From 1.25 take ■^^.
4. From 14.2 take 4.92. lo. From ^ take ^.
5. From 39. take 21.689. ll. From ^f^^ take j^^.
6. From 500 take 9.32. 12. From ^i^ take ^7.
EXERCISE 59. — WRITTEN
1. In 100 lbs. of wheat bran there are 15.4 lbs. of pro
tein, 62.9 lbs. of carbohydrates, 4 lbs. of fats, 5.8 lbs. of
ash, and the balance is water. How much water is there ?
2. If in 100 lbs. of milk there are 4.32 lbs. of fat, 3.34
lbs. of protein, 5.7 lbs. of sugar, .74 lbs. of ash, and the
balance is water, how much water is there?
3. If in 100 lbs. of cream there are 23.8 lbs. of fat,
4.12 lbs. of protein, 3.92 lbs. of sugar, 0.53 lb. of ash, and
the balance is water, how much water is there ?
97
98
PRACTICAL ARITHMETIC
4. If in 100 lbs. of timothy hay there are 13.2 lbs. of
water, 4.4 lbs. of ash, 7.4 lbs. of carbohydrates, 2.5 lbs.
of fats and the remainder is protein, how much protein
is there ?
5. If in 100 lbs. of redclover hay there are 15.3 lbs.
of water, 6.2 lbs. of ash, 62.9 lbs. of carbohydrates, 3.3
lbs. of fats, and the rest is protein, how much protein is
there ?
Nuts contain the following amounts of refuse and
water in each pound. The refuse represents the shell and
lining
Refuse
Watek
Almonds 45 lb.
.027 lb.
Beechnuts
.408 lb.
.023 lb.
Butternuts
.864 lb.
.006 lb.
Cocoanuts
.488 lb.
.072 lb.
Hickory nuts
.622 lb.
.014 lb.
Pecans
.532 lb.
.014 lb.
Walnuts .
.741 lb.
.006 lb.
Peanuts .
.245 lb.
.069 lb.
6. Which nuts contain most refuse ? Ho,w much more
than those of least refuse ?
7. Which nuts contain most water ? How much more
than those of least water ?
8. Which contain most nutritive matter ? The amount
not water and refuse is nutritive matter. How much more
than those of least nutritive matter ?
9. Find the amount of nutritive matter in each kind
of nut.
MULTIPLICATION OF DECIMALS
EXERCISE 60. — ORAL
1. Read the following: .225, 2.25, 22.5, 225. How
does moving the decimal point one place to the right
affect the value of a number ?
2. Read: .3720, 37.20, 3720. How does moving the
decimal point two places to the right affect the value?
3. Read: .973, 973. How does moving the decimal
point three places to the right affect the value ?
116. Moving the decimal point one place to the right
has the effect of multiplying tlie number by 10, two places
by 100, three places by 1000, etc.
117. To multiply by 10, 100, 1000, etc., move the deci
mal point as many places to the right as there are ciphers
in the multiplier, annexing ciphers at the right to com
plete the required number if necessary.
EXERCISE 61. — ORAL
1. Read the following: 675, 67.5, 6.75, .675. How
does moving the decimal point one place to the left affect
the value of the number ?
2. Read: 22.70, 2270. How does moving the decimal
point two places to the left affect the value ?
3. Read: 325, .325. How does moving the decimal
point three places to the left affect the value ?
99
100 PRACTICAL AKITHMETIC
118. Moving the decimal point one place to the left has
the effect of multiplying a number by onetenth (0.1),
two places by one onehundredth (0.01), three places by
one onethousandth (0.001), etc.
119. To multiply by 0.1, 0.01, or 0.001 move the deci
mal point as many places to the left as there are decimal
places in the multiplier, prefixing ciphers if necessary to
complete the required number of decimal places.
EXERCISE 62. — ■WRITTEN
Multiply :
1. 49.68 by 10, by 100, by 1000.
2. 6297.3 by 10, by 100, by 1000.
3. 9.6847 by 10, by 100, by 1000.
4. 429673.0 by 10, by 100, by 1000.
5. 84910.0 by 10, by 100, by 1000.
6. 49.68 by 0.1, by 0.01, by 0.001.
7. 6297.3 by 0.1, by 0.01, by 0.001.
8. 9.6847 by 0.1, by 0.01, by 0.001.
9. 0.429673 by 0.1, by 0.01, by 0.001.
10. 84910 by 0.1, by 0.01, by 0.001.
120. Multiply 32.482 by 3.
2 thousandths x 3 are 6 thousandths, the 6 is
32.482 written under the thousandths. 8 hundredths X 3
3 are 24 hundredths, which equals 2 tenths and 4
97.446 hundredths. The hundredths are written under
hundredths and the 2 tenths are to be added to
tenths. 4 tenths x 3 are 12 tenths, which with 2 tenths are 14
tenths, which equal 1 unit and 4 tenths. 2 units x 3 are 6
units, which with 1 unit equal 7 units. 3 tens x 3 are 9 tens.
DECIMAL FRACTIONS 101
121. Multiply 'S2A82 by 0.3.
Multiply by 3 as if it were a whole number,
32.482 pointing off 3 places in the product for the
0.3 thousandths in the multiplicand and an addi
9.7446 tional place to indicate that the multiplicand has
been multiplied by tenths. The product then has
4 decimal places.
122. To multiply decimal fractions multiply as with
whole numbers. Point off as many places as there are
decimals in the multiplier and multiplicand together. If
the product does not contain as man}'^ decimal places as
are required, prefix enough ciphers to make the required
number.
EXERCISE 63. — WRITTEN
Multiply :
1. 96.0 by 0.3. 8. 0.236 by 8.93. 15. 184.2 x 0.098.
2. 9.6 by 0.3. 9. 0.259 by 0.247. 16. 214.86x45.64.
3. 0.96 by 0.3. 10. 349 by 0.46. 17. 37.55 x 0.00025.
4. 132 by 2.47. ii. 4.39 x 0.74. 18. 873.0 x 0.675.
5. 13.2 by 24.7. 12. 5 .6 x .056. 19. 214.76 x 1.25.
6. 0.132 by 247.0. 13. 35.16 x 5.75. 20. 87.136 x 0.0042.
7. 9.06 by 1.24. 14. 50.05x0.0095. 21. 897.28x2.009.
EXERCISE 64. — WRITTEN
1. If a bushel of corn on the ear weighs 70 lbs. and of
each pound 0.2 is cobs, what is the weight of a bushel of
shelled corn?
2. If in 1 lb. of corn stover there is 0.6 lb. of stalks,
102 PRACTICAL ARITHMETIC
0.3 lb. of leaves, and 0.1 lb. of shucks, how many pounds
of each are there in 2000 lbs. of stover?
3. If in 1 lb. of commercial fertilizer there is 0.08 lb. of
phosphoric acid, 0.025 lb. of nitrogen, and 0.035 lb. of
potash, how many pounds of each are there in 100 lbs.
of such commercial fertilizer ? How many pounds of each
in 2000 lbs. ?
4. A housekeeper's Saturday grocery order was as
follows: 6 lbs. roast at f0.18 a pound, 3 lbs. rice at.
$0.0825 a pound, 1 pk. sweet potatoes at $0.25 a peck,
.5 lb. tapioca at $0.08 a pound, 1.25 lbs. cheese at
$0.19 a pound, 10 lbs. graham flour at $0.03 a pound.
Find total cost.
5. The average wheat yield to the acre in Great Britain
is 33.9 bus., in the United States 14.5 bus., in India 9.2
bus., in Russia 10.2 bus. Russia grows 39,215,686 acres,
the United States 33,766,233 acres, India 16,847,826 acres.
If the yield per acre in the United States and Russia
were brought up to that of Great Britain by skilful seed
selection and tillage, what would be the increase to the
world's wheat crop ?
6. If the rainfall in India were sufficient, the yield per
acre there might be equal to that in Great Britain. What
would the total yield of India then be ?
7. What does it cost to travel 198 miles at 2.5 cts. per
mile? At 2.25 cts.? At 3.75 cts.?
8. The distance round a wheel is 3.1416 times its
height. What is the distance round a wheel 4.75 feet
high ? Round a 28in. wheel ?
DIVISION OF DECIMALS
123. Divide $40.00 by 110.00, 140.00 by 10 cts.,
$75.00 by 115.00, $75.00 by 15 cts. Compare the
results.
We have seen that United States money is written as a
decimal fraction, e.^., $1 and 10 cts. is written $1.10.
In the division of United States money when the
divisor is represented by cents, both divisor and dividend
are changed to cents, and the division is performed as
with integers.
124. Divide 8.75 by 2.5.
3.5 2.5 multiplied by 10 is changed to the integer 25.
25)87.5 8.75 multiplied by 10 is changed to 87.5. Divid
'j'g ing as in United States money, 87 = 25 = 3, with a
remainder of 12. Adding .5 to the remainder gives
12.5 for a new partial dividend. 12.5 = 25 = .6.
Therefore, the quotient is 3.5.
To divide a decimal fraction, multiply both dividend
and divisor by 10, or such multiple of 10 as shall make
the divisor an integer; then divide as in United States
money.
EXERCISE 65.— WRITTEN
Find the quotient of :
1. 7.75 =25. 3. 7.75 =.25. 5. 7.75 .0025.
2. 7.75^2.5. 4. 7.75^.025. 6. 7.75^.00025.
103
12.5
12.5
104 PRACTICAL ARITHMETIC
7. 172804125. 9. 17280 ',T 1.25. xi. 17280 h .001 25.
8. 17280^12.5. 10. 17280 4 .125. 12. 17280 ^ .0125.
125. Divide 10.10 by 10, divide 100.100 by 100, divide
1000.1000 by 1000. Compare these results.
In preceding pages you have seen that moving a figure
one place to the right in its period decreases its value
tenfold. In like manner, the removal of the decimal
point one place to the left decreases a number tenfold, or
divides the number by 10. Hence, to divide a decimal by
10, 100, 1000, etc., remove the decimal point as many places
to the left as there are ciphers in the divisor. When
necessary, add ciphers to complete the required number
of places.
BXEBCISB 66. —WRITTEN
Find the quotient of :
1. 3725.4 470. 4. 810.18 49000.
2. 309.45 41500. 5. 810.18 40.009.
3. 132.4^4000. 6. 7325.1 4 1.045.
126. When there is a remainder after using all the
figures of the dividend, annex ciphers to the dividend
and continue the division. For ordinary affairs in busi
ness it is not necessary to carry the division further than
four or five decimal places.
EXERCISE 67. — WRITTEN
Find the quotient of :
1. 92323.15 f 6.275. 4. 281.85 43.85.
2. 281.8585 48.85. 5. 87.912 44.07.
3. 725.406 46956. 6. 0.375 5 .25.
DECIMAL FRACTIONS 105
7.
317.25 =75.
12.
600982  3.2909.
8.
0.0125^2.5.
13.
7.847^.03962.
9.
1361,51500.
14.
849.27 =38.0099.
10.
50 ^ .25.
15.
3.42981^2.86008.
11.
874.298^62.
85.
16.
8498.762 678.9084,
EXERCISE 68. — WRITTEN
1. If 100 lbs. of milk yield 5.452 lbs. of butter, how
much butter will 1 lb. yield?
2. If a gallon of milk weighs 8.6 lbs., how much but
ter will 2 gallons of milk of the quality mentioned in the
last problem produce?
3. If the corn plants on an acre weigh 7450 lbs., and
in each pound there are .417 lb. of ears and .583 lb. of
stover, how many bushels of ear corn, allowing 70 lbs. to
the bushel, are there, and how many pounds of stover?
4. Seven pigs, averaging 28.10 lbs., gained in 6 weeks
503 lbs. per acre of peanut pasture used. What was the
gain of each animal for each day? If the increase was
worth 6 cts. a pound, what value per acre was derived
from the pasture ?
5. Five pigs, aggregating 895 lbs., pastured 20 days
on Spanish peanuts, weighed at the end of that time 1124
lbs. What was the gain for each animal in the 20 days?
What was the gain for each day ? At 6 cts. a pound,
what was the increase in value ?
EXERCISE 69. — WRITTEN
The pupil should practise with decimals enough to
attain proficiency.
106
PRACTICAL ARITHMETIC
a
6
c
d
e
/
9
1
379.421
2.469
24.378
23.428
2.267
1.479
h
2
387.463
4.28
368.0
7.29
0.39
0.009
i
3
289.4
0.796
398.42
97.0
78.4
0.379
i
4
63.0
62.97
798.4
629.18
82.007
3.6298
k
5
78.4
17.843
6829.45
43.978
67.08
1.982
I
6
8.629
0.08976
0.7842
3692.0
0.0079
3.479
m
7
0.0078
0.42963
67.0
47.961
7900.0
0.062
n
8
76.24
0.0098
8.0009
692.0
6239.5
0.398
9
872.98
 7.627
764.02
798.42
78.42
3.429
P
10
642.87
8.429707
7.9628
68.394
86.39
6.729
Q
11
12
13
14
15
16
16. Add columns a, b, c, d, e,f.
716. Add the numbers in line 1, 2, etc., to 10.
1727. Subtract from the first number in column d each
number in column/.
28116. Subtract from each other number in column d
each number in column/.
117216. Multiply each number in column o by each
number in column e.
217316. Divide each number in column a by each num
ber in column b.
exercise 70. — written
Review Problems
1. If a bushel of corn contains 5.88 lbs. of protein,
40.15 lbs. of carbohydrates, and 3.024 lbs. of fats, what is
a bushel of corn worth for feeding at 3.7 cts. a pound for
protein, .5 ct. a pound for carbohydrates, and 3.1 cts. a
pound for fats.
REVIEW PROBLEMS 107
2. The different items of cost to build a stave silo 12 ft.
in diameter and 21 ft. high are as follows : hauling rock
f 3, hauling sand $1, cement 4.75 barrels at f 2.85 per
barrel, putting in concrete foundation, 1 man, 4 days at
11.25 a day, lumber 5, staves 128.84, hauling lum
ber $3, nails 25 lbs. at 3 cts., nails 10 lbs. at 3 cts.,
hoops $21.95, bolts 45 cts., small bolts 30 cts., tar,
gasoline, and brush !i5l.40, tar paper 2 rolls at •tl.37o,
carpenter 6 days at #1.75 a day, labor 2 days at $1 a
day. What was the total cost of building the silo?
3. If a field of cotton yields 4793 lbs. of seed cotton
and each poiuid yields .314 lb. of lint, how many pounds
of seed and how many pounds of lint are there ?
4. To make 1 lb. increase in weight in pigs pastured
on peanuts required 1.77 lbs. of grain, when pastured on
chufas 2.3 lbs. of grain, when on cowpeas 3.07 lbs. of
grain, when on sweet potatoes, 3.13 lbs. of grain, when
on sorghum 3.70 lbs. of grain. How many more pounds
of grain were required when on chufas than when
on peanut pasturage? How many more when on cow
peas than Avhen on peanuts? When on sorghum than
when on peanuts ? When on sorghum than when on
sweet potatoes?
5. The following is the average yield of wheat per
acre in the principal wheatgrowing nations of the world :
Great Britain 33.9 bushels, Germany 28.6, France 20.8,
Hungary 18.4, Austria 19.6, United States 14.5, and Rus
sia, 10.2. How many acres would it require in each of
the other countries mentioned to produce as much wheat
as is produced on 1 acre in Great Britain?
108
PRACTICAL ARITHMETIC
6. The following shows the cost and benefit from spray
ing Irish potatoes for several years in New York :
Year
Total Ackbs
Sl'RAVBD
Increase in Tield,
Bushels pee Acre
Cost op Spraying
PER Acre
Pkofit from
Speating
1903'
1904
1905
1906
61.2
180.0
160.7
225.6
57.0
62.2
46.5
42.6
$4.98
4.98
4.25
5.18
—
Total
What was the average cost of spraying per acre?
Valuing potatoes at 57 cts. a bushel, what was the aver
age profit per acre from spraying?
7. The items of expense for spraying 10.4 acres of
potatoes 5 times are : 234 lbs. of copper sulphate at 7 cts.,
195 lbs. of prepared lime at 1.5 cts., 90 quarts of arsenite
of soda solution at 25 cts., 70 hours' labor for man and
horse at 30 cts., wear on tools, 16.50. What is the cost
of spraying each acre for each application ?
8. In 100 lbs. redclover hay there are the following
constituents: 15.3 lbs. water, 6.2 lbs. ash, 62.9 lbs. car
bohydrates, 3.8 lbs. fats, and the balance is protein. In
100 lbs. of timothy hay there are 13.2 lbs. water, 4.4
lbs. ash, 74 lbs. carbohydrates, 2.5 lbs. fats, and the
balance is protein. How many more pounds of protein
are there in 2000 lbs. of clover hay than in 2000 lbs. of
timothy hay ?
9. If 100 lbs. of green cowpeas contain 1.7 lbs. of
ash, 2.4 lbs. of protein, 11.9 lbs. of carbohydrates, .4 lb.
REVIEW PROBLEMS 109
of fats, and the balance is water, how many pounds of
water are there in 1000 lbs. of fresh cowpeas ?
10. In buying redtop grass seed at 13.7 cts. a pound
when only 77 lbs. in every 100 lbs. is good live seed,
what is the price paid per pound for good seed ?
11. With redtop seed at 8.54 cts. a pound containing
11 lbs. of good seed in every 100 lbs., what is the price
per pound of the good seed?
12. With bluegrass seed at 14 cts. a pound containing
60 lbs. of good seed in every 100 lbs., what is the price
per pound of the good seed ?
13. With bluegrass seed at 10 cts. a pound containing
.4 lb. of good live seed to the hundred pounds, what is
the price per pound of the good seed ?
14. With timothy grass seed containing 96 lbs. per
hundred of good seed bought at $1.60 a bushel, what is
the price paid per bushel for good seed?
15. If a cow produces 7446 lbs. of milk in a year and
1 lb. of milk produces .059 lb. of butter, how many pounds
of butter does the cow produce, and what is it worth at
27 cts. a pound?
16. How much does the butter cost per pound in value
of feed, if she ate 12.4 cts. worth each day?
17. Another cow gave 3400 lbs. of milk, and each pound
of milk produced .0406 lb. of butter. Did this cow
make a profit with butter at 27 cts. if the cost of feed
was 11.2 cts. a day?
18. If 1 pound of cottonseed meal contains .0618 lb.
of nitrogen, .018 lb. of potash, and. .028 lb. of phosphoric
110 PRACTICAL ARITHMETIC
acid, how much of each of these fertilizing materials does
a ton, or 2000 lbs., of cottonseed meal contain? "What is
the ton of cottonseed meal worth for fertilizing at 18 cts.
a pound for nitrogen, 4 cts. for phosphoric acid, and 5 cts.
for potash?
19. If 1 pound of cottonseed contains .031 lb. of nitro
gen, .013 lb. of phosphoric acid, .012 lb. of potash, what
is the ton of seed worth at the same prices for fertilizing
materials as in problem 18?
20. The following are the plantfood constituents which
a farmer sells from his farm in the products named :
Nitrogen
PuosPiroRic Acid
Potash
3000 lbs. milk .
. 15.9 lbs.
5.7 lbs.
5.4 lbs
liO lbs. butter .
. 02.8 lbs.
0.12 lb.
0.35 lb.
500 lbs. cotton lint .
. 01.7 lbs.
0.5 lb.
2.3 lbs
1000 lbs. cottonseed
. .31.0 lbs.
13.0 lbs.
12.0 lbs
20 bus. wheat .
28.32 lbs.
9.48 lbs.
6.0 lbs
40 bus. corn
. 40..>7 lbs.
15.68 lbs.
8.96 lbs
1.5 tons timothy hay
. 37.8 lbs.
15.9 lbs.
27.0 lbs.
Valuing nitrogen at 18 cts. a pound, phosphoric acid at
4 cts. a pound, and potash at 5 cts. a pound, what is the
value of the fertilizing materials removed from the farm
by the sale of each of these products in the amounts
indicated ?
COMPOSITION OF FOODS
21. If for 10 cts. I can purchase .064 lb. of protein in
tenderloin steak, how much protein do I obtain in 50 cts.
worth of tenderloin ?
22. If for 10 cts. I can purchase .185 lb. of protein in
REVIEW PROBLEMS
111
round steak, in which do I get most protein for the money,
in tenderloin or in round steak ? How niucli more ?
A Good Loin Cut op Beef
23. If for 10 cts. I can purchase .03 lb. of protein in
oysters, how much protein do I obtain in 70 cts. worth of
oysters ? In which of the above articles do I obtain most
protein for my money ?
24. In the following dairy products these amounts of
protein may be purchased for 10 cts.: butter .004 lb.,
cheese .1631b., whole milk .110 lb., skimmed milk .20:5
lb., cream .034 lb. How much protein in each may I
obtain for 75 cts.?
LEGUMES
25. If each pound of vetch hay contains .17 lb. of nitro
gen, how many pounds of nitrogen are there in 2879 lbs.
of vetch hay? Each pound of alfalfa contains .143 lb,
of nitrogen. How many pounds are there in 2879 lbs. of
alfalfa? Each pound of red clover contains .123 lb. of
112 PRACTICAL ARITHMETIC
nitrogen. How many pounds are there in 2879 lb. of red
clover? Each pound of cowpea hay contains .166 lb. of
nitrogen. How many pounds are there in 2879 lbs. of cow
pea hay? What is the value of the nitrogen in each of
the above instances at 19 cts. a pound?
26. If 5953 lbs. of velvetbean hay are grown on an
acre, and in every pound of this hay there is .0221 lb. of
nitrogen, how many pounds of nitrogen are gathered by
the crop, and what is it worth at 18 cts. a pound?
27. If there is .024 lb. of nitrogen in 1 lb. of cowpea
hay, and 3000 lbs. of h.a,j are made on an acre, what will
be the value of the nitrogen collected by the crop, with
nitrogen worth 18 cts. a pound, if for each pound of ni
trogen in the hay there is left in the stubble and roots
.3 lb. of nitrogen?
28. Crown Jewel potatoes dug in Virginia 80 days
after planting yielded 170 bus., dug at 93 days after
planting, the same variety gave 255 bus. What was the
average increase daily during the additional period ?
29. Beauty of Hebron potatoes dug in Virginia at 101
days after planting showed an increase of 136 bus. over
those dug 80 days after planting. What was the average
increase per day during the additional time?
30. Using nitrate of soda on clover, 300 lbs. to the
acre, at §2.66 a hundred pounds, the yield was increased
from 2.09 to 2.8 tons per acre, valued at 19 a ton in the
field. Did the use of the fertilizer pay ? What was the
gain or loss per acre ?
COMMON FRACTIONS
EXERCISE 71. — OEAL,
1. How many weeks in a month ? One week is equal
to what part of a month ? Two weeks are equal to what
part of a month? Three weeks are equal to what part of
a month ?
2. One week is equal to what part of two months?
One week is equal to what part of three months ?
3. Two weeks are equal to what part of two months ?
Two weeks are equal to what part of three months ?
4. Three weeks are equal to what part of two months ?
Three weeks are equal to what part of three months ?
5. If a unit is divided into two equal parts, what is one
part called? If a unit is divided into three equal parts,
what is one part called ? If into four, six, ten ?
127. One or more of the equal parts of a unit is called
a Fraction.
128. A fraction is expressed by two numbers, one writ
ten above the other, with a line between them ; e.g., one
fourth is expressed thus, \.
129. Read the following : , \, \, Jg, ^^q, gV What
part of these fractions shows the number of parts into
which the unit is divided?
The number which shows into how many parts a unit
113
114 PRACTICAL ARITHMETIC
is divided is called the Denominator ; e.g., in the fraction
f, 6 is the denominator and shows that a unit has been
divided into six equal parts.
130. Which is greater, ^ or  ?
What does the number above the line indicate ?
The number which shows how many parts are taken
is called tlie Numerator ; e.g., in the fraction ^, 3 is the
numerator and shows that the fraction contains three
of four equal parts.
131. The numerator and denominator are called the
Terms of a fraction.
132. A fraction whose numerator is less than its
denominator is called a Proper Fraction; e.g., , ^, , ■^,
are proper fractions.
A proper fraction is always less than a unit.
133. A fraction in which the numerator is equal to or
greater than the denominator is called an Improper Frac
tion; e.g., , , ^, are improper fractions. An improper
fraction is always equal to or greater than a unit.
134. A whole number and a fractional number written
together is called a Mixed Number ; e.g., 3, 1^ , are mixed
numbers.
BXBECISE 72.— OBAL
Select proper fractions, improper fractions, and mixed
numbers from the following :
^* 3' F' 6' 9" *• ''e' 2'' '■"2' 5" 10' 3' 2' 9
2 t 1 34 1^ 5 5 la 8. J n 20 15 Q 1 5 6
"*■ 4' 5' '^¥' 14" ^ 4' 16' 5' 25 **• 19' ? ' ^^2' IV
3 15 IS 3 1 c 2_5. Q2 IflS 9 a 2_8 19 .£;i 16
"*• 2 0' 15^' f' 3 ° '#' ^Y' •''"?' TO ^ ■?9' '5"' ''2' 24
COMMON FRACTIONS 115
EXERCISE 73. — WRITTEN
Write as common fractions or mixed numbers :
1. Thirt3^one tenths.
2. Fiftysix elevenths.
3. Eightnineteenths.
4. Sevenfifteenths.
5. Eight onehundredths.
6. Ninety and threefourths.
7. One hundred and fortyfive fortysixths.
8. Seventyseven and sixtenthsi
9. Five hundred tentlis.
10. Twentyfive tliirtysixths.
11. Nineteen and seven twentyfirsts.
12. Twentyone and eighteen nineteenths.
13. One hundred twentyfive and one hundred twenty
four one hundred twentyfifths.
EXERCISE
74.
— WRITTEN
Write in words :
1 f^f A
6. 666f,25f
2. if, If if
7. 8 7.121, $12.25f
3 ihihi
8. $ 266. 26, $19,181
4. l,25f,^i,. 9. If, _^, ^^a lo^oji.
5. 9^^, 150f 10. 3Vlf\sl9f
135. An improper fraction may be regarded as an indi
cated division, e.g., ^^ indicates that each of several units
has been divided into 4 parts ; 13 of these parts are here
1. V
4. iLI.
7 If.
10.
ii
2. ^.
5 ¥/•
8. 3^1.
11.
W'
3. 1/.
6. \3..
9. ^M.
12.
6 93
l20'
116 PRACTICAL ARITHMETIC
represented. To find how many units are represented by
3^5., "vve may regard 13 as the dividend and 4 as the divisor;
13 T 4 = 3 whole units and ^ of a unit.
136. To change an improper fraction to a whole or
mixed number, divide the numerator by the denominator.
EXERCISE 75. — ■WRITTEN
Change to whole or mixed numbers :
13. ^f
14. 111.
15. 5^a.
137. A whole number or a mixed number may be
represented in the form of an improper fraction, e.g., 8
may be expressed in halves, as J^ ; 8 may be expressed as
halves, thus, ^^^.
Since one unit contains 2 halves, 8 units contain ^. ^
and J equals i^.
138. To change a mixed number to an improper frac
tion, multiply the whole number by the denominator and
add the numerator ; write the sum over the denominator.
EXERCISE 76. — WRITTEN
Change to improper fractions :
9. 17^^. 13. 45^.
10. 12lf 14. 17f.
11. 72f 15. 167f.
12. 25^. 16. 167.
1. 121.
5.
10^,
2. 15f
6.
25i.
3. 18.
7.
331
4. 45^.
8.
19i.
COMMON FRACTIONS 117
EXERCISE 77. — ORAL
1. How many twelfths in one unit ?
2. How many twelfths in two units ?
3. How many twelfths in onehalf unit ?
4. How many twelfths in onethird unit ?
5. What is true of the value of ^2 ^"^ 2 ^ What is
true of the terms of the second fraction ?
6. Does changing ^^ *o ^^^ lower terms, ^, change the
value ?
EXERCISE 78. — WRITTEN
Change the following to lower terms :
"• 21' 2 4' ^•
7 IS li. 24
'• 2T' 60' Tf
R 19 6JL la.
**• ¥"8'' 102' "rt
9 10 8 li l±
^ 111' ^1' YO
TO IS. 22 11
•^"" ?t' 6 6' 9 6*
139. A fraction is reduced to its Lowest Terms when
the terms are prime to each other.
To reduce a fraction to lowest terms, select factors com
mon to both terms, and cancel.
1.
6' iV' A' iV
2.
9' if' 15"
3.
]%' 1' 2V
4.
^IS' 1' 1*6
S.
■^(7' A' A
EXERCISE 79.
— WRITTEN
Reduce to lowest terms :
1 II
5. m
9 il
13.
^%
2 i.
6. If.
10. ,^.
14.
lit
3 U
7. ^Vl
11 Hf
15.
MI
*• u
8 ff
12. 78^.
16.
MI
h
h
4.
hh
7.
h i\
10.
hi
h
tV
5.
h "S
8.
hh
11.
1 1
3' 6
h
h
6.
hh
9.
¥' 9"
12.
f' 9'
118 1>RACTICAL ARITHMETIC
EXERCISE 80. — ORAL
1. Express ^ as 4ths. 6. Express ^, ^ as 8ths.
2. Express f as Sths. 7. Express , , ^^ as12ths.
3. Express  as 12ths. 8. Express ^, ,  as 12ths.
4. Express  as 14ths. 9. Express , ,  as 24ths.
5. Express f as 28ths. 10. Express ^, , ^^ as 20ths.
EXERCISE 81. — AVRITTEN
Express as fractions with the same denominator :
1.
2.
3.
140. If several fractions have the same denominator,
they are said to be Similar Fractions, and the denominator
is called a Common Denominator.
141. If the common denominators are the smallest pos
sible, the fractions are said to have the Least Common
Denominator (L. C. D.), e.ff., the fractions ^^ '^^^^ 1^2 ^^'^^
a common denominator 12, but they may be reduced to 
and , 4 being the L. C. D.
142. Reduce ^, , , to similar fractions with the least
common denominator.
The common denominator of ^, f, if, must be the least com
mon multiple of the denominators 3, 8, 12.
We find that the L. C. D. of these nnmbers
is 24. Change each fraction to 24ths by mul
tiplying each denominator by the factor that
will give 24, and multiplying the numerator by
the' same factor, thus :
2)3
8 12
3)3
4 6
2)1
4 2
1
2 1
COMMON FRACTIONS 119
To change 1 to 24ths, multiply both numerator and de
1x8 8
nominator by 8 =
To change  to
nominator by 3 =
To change ^ to
nominator by 2 =
3x8 2i'
To change  to 24ths, multiply both numerator and de
5 X 3 ^ 15
8x3 24'
To change 1 to 24ths, multiply both numerator and de
10 X 2 ^ 20
12 X 2 24'
143. In reducing fractions to a common denominator
they are changed to Higher Terms.
EXERCISE 82.— "WRITTEN
Reduce to similar fractions :
1 1 4 3 9 ii 5 5 _9_ n 3 J:
■^ 2'? ■*■ 21' ^0" ° S' 10 '• t' TT
^ 12' IZ *■ 7' 2 1 ° 9' S ° "5' 1(J' 20
EXERCISE 83. — WRITTEN
Reduce to fractions having a L. V. D. :
1 1 1 3_
^ ?' ^' 10
"• 2' S"' 12
o 3 4 2
"*■ 5' T' 3
*• 8' 5' ?0
K 6 5_ _e
^ 'SO' 15' 15"
EXERCISE 84. — WRITTEN
Change to improper fractions and reduce to L'. C. D.
1. 121, 18f 5. 682^ 45}1.
2. 2V„36,V 6. 42l, 61ff.
3. 72^, 26^\. 7. Sh%,2A^\.
4. 24, 3(11.?. 8. 16g\, 2111.
6.
12 1
14¥' 132
7.
2 22' ?^A
8.
9 7
25' "55
9.
9 8 5
IT' 9 9' T2 1
10.
A' i^' H
ADDITION OF FRACTIONS
KXEROISE 85. — DEAL
1. How many tenths in ^ + ^^? How many fifths?
2. How many twelfths in ^^ + 1^2 ' How many
fourths ?
3. How many sixths in ^ + ^? How many ones?
4. To find the sums of these fractions, what terms of
the fractions are added?
5. How much is 1 + 1? 1 + 1? l + J? 1+1?
3 + 2 •
6. To what kind of fractions must those above be
changed before they can be added?
144. Add,f,3^.
First find the L. C. D. The L. C. D. is 5 x 2 x 3 x 2 = 60.
Eeducing each fraction to higher terms with 60 for a cohj
mon denominator, we have,
4 X 12 ^ 48 6 X 10 ^ 50 9x5 ^45
5x12 60 6x10 60 12 x 5 60
ff + 1^ + 11 = Jg^. Eeducing J^ to a mixed number gives
145. Add 31 + 6f
The sum of 3 + 6 = 9.
By finding the L. C. T>. of ^ and J and changing these frao
tions to higher termsj we get J^j + i = ff
120
COMMON FRACTIONS 121
Changing J to a mixed number gives 1^^.
Uniting the sums, 9 + l^j = lO^ij.
146. To add fractions, reduce to similar fractions hav
ing the L. C. D., and add the numerators, placing the sum
over the common denominator. The answer should al
ways be reduced to lowest terms.
EXERCISE
86.
 WRITTEN
Find the sum of :
1 hi
5. 6,2f
2 i,.
6. 9,2.
3 hhi
7. I, 31 5f .
*■ f'i'/r
8. 2, 9jL, S^^j,
EXERCISE 87. — WRITTEN
1. A boy spends f of his money for a suit of clothes,
\ for an overcoat, ^ for a pair of shoes, and ^^ for a hat.
What part of his money has he spent ?
2. After making his purchases, he has left J of his
money, which is i 4. How much had he at first?
3. A girl's dress skirt is 26^ ins. long when finished;
if \ in. is allowed for gathers at the top and 2^ ins. for the
hem, how long must the material be cut for making?
4. If she wishes to add a ruffle 3 ins. wide, with an
inch hem at the bottom, and a fin. hem at the top, how
wide must she cut the material for the rafSe?
5. How many yards of fencing will be required to fence
an irregularly shaped school yard 23 yds., 42 yds., 40
yds., 27f yds. on the sides?
SUBTRACTION OF FRACTIONS
147. How much is f + ^? How much is  — ^?
How much is f + ^^ ? How much is  — ^■^?
How much is ^ + ^^ '■ How much is  — yj^?
To subtract fractions they must, as in addition, be
changed to similar fractions. Subtract the numerator
of the subtrahend from the numerator of the minuend to
obtain the numerator of the difference, e.g., f — iV = t^
 iV or 1^
EXERCISE 88.
ORAL
Find the value
of:
1 ll
*• HtV
7.
A
i
2 fi
5 T^i
8.
A
tV
3 lf
6 Hl
9.
A
f
EXERCISE 89. — WRITTEN
Find the value of :
■h
5
15"
5. ^,h
6.
7.
8.
9.
10.
f^
11
16
12
^•
. 1
1
2
2"S"
11.
12.
13.
14.
15.
2 5
^ IJ
3'B ~ 13'
2i_ a
3 1 8 •
A2._ 21
49 '58"
EXERCISE 90. — ORAL
What fraction does x stand for in the following?
1. i + 2;=f. 2. # + a; = 4.
3. iX=l.
122
COMMON FRACTIONS 123
4.
ix = i.
7.
 + :r=i^.
10.
^i = f
5.
i + ^ = A
8.
I*'=1^
11.
•if=6
6.
A = f
9.
41=^
12.
^^=
148. Find the value of 13  9^.
As in addition of fractions, the whole numbers and the
fractions may be subtracted separately, but in the above
problem we observe that the fraction ^ cannot be sub
tracted from the fraction ^ of the minuend.
The problem may be restated thus, 12— 9^. 12—9 = 3.
I = if 2 = i\ tI"  T^o = iV '^^^^ remainder, there
fore, is 3 and ■^^, ©r 3 j^.
EXERCISE 91. WRITTEN
Add or subtract as indicated :
1. 21. 6. 5f + 6^3^V
2. 3^111 7. 525150if
3. 94. 8. 4:2 + dl + 10{^.
4 1 + 1 + A 9 9tV + 3A + 2.
5. 5h6. 10. 413.
EXERCISE 92. — WRITTEN
1. A boy spends ^ of his school days in the common
school and  in the high school ; how much more of his
boyhood does he spend in the common school than in the
high school?
2. If a boy sleeps I of his time and studies  of his
time, how much time has he left for play?
3. What fraction added to i^ f  will make 2 ?
124 PRACTICAL ARITHMETIC
4. A normal child weighs at birth 7 lbs., at 1 year 20
lbs., at 2 years 26J lbs. What is the difference between
its weight at birth and at end of first year ? What is the
difference between its weight at end of first year and end
of second year ?
5. At birth a normal child measures in height 20 ins.,
at 1 year 29 ins., at 2 years 32 ins. What is the differ
ence between its height at birth and at end of first year ?
What is the difference between its height at birth and at
end of second year?
6. The organs of the human body are composed of
water, as follows : bones ^, muscles ^y^', brain and spinal
cord 1^, lungs r^^^^. How much more water is there in
muscle than in bone ? In brain than in muscle ? In lungs
than in muscles?
7. The organs of the human body are composed of
mineral matter, as follows : bones \^, muscles 2 o' ^^^S^
loro' brain yJ^. How much more mineral matter is there
in bone than in each of the other organs named ?
8. The head of a normal child at 6 years of age
measures 20 ins., at 3 years 19 ins., at 2 years 19^ ins.
Find the difference in measurement between the head of a
child 6 years old and that of a child 3 years old ; between
a 6yearold child and a 2yearold child; between a 3
yearold child and a 2yearold child.
9. The average boy of 8 years is ■ISy'g i'^^ ^^^^' ^^^^
average girl of 8 is 48^^^ ins. What is the difference in
their heights?
10. The average girl of 14 weighs 100^ lbs., the average
boy 99^. What is the difference ?
MULTIPLICATION OF FRACTIONS
EXERCISE 93. — ORAL
1. How much is ^ of 6 inches ? ^ of 9 dollars? ^ of 10
cents ? I of I ?
• 2. How much is i of 12? ^of? iofif?
3. How many inches are there in a foot ?
4. How many inches in  foot? In J of  foot?
5. In example 4 the result is what part of a foot?
6. How much is I of  yard? I of  dollar?
7. How much is  of ^ dollar? ^ of ^g dollar?
8. How much is  of  ?
149. Multiplying any number by another number larger
than 1 increases its value. When one fraction is multi
plied by another, the number of parts as shown by the
denominator is increased, but the value of the fraction is
decreased, e.ff., 2x3 = 6. ^ x ^ = J of ^ = ^.
150. Multiply I by 1 = 1 of .
iof = i.
I of f = 2 X ^ or . Hence,  multiplied by  = .
151. Multiply I by  = f off
1 of 7 _ i of + 2 = ?~
I of Jj = 5 times Jj, or f .
152. To multiply a fraction by a fraction, find the
125
EXERCISE 94.
ORAL
id the pi
roduct of :
ix.
6.
xf.
11.
ix.
*x.
7.
ixf^.
12.
A X ^5^
ix,V
8.
fxf.
13.
x.
ix.
9.
xif
14.
fxf.
fxf.
10.
fxf
15.
xi
126 PRACTICAL ARITHJIETIC
product of the numerators for the Numerator and the
product of the denominators for the Denominator.
1.
2.
3.
4.
5.
153. Instead of multiplying the terms of the fraction
together for the product, the process may be indicated, and
the factors cancelled.
Multiply I by .
2
6 5 ^ x^ 3'
3
EXERCISE 95. — WRITTEN
By cancellation find the product of :
1 i V fi 5 5 V 12 Q J!_ V _S_
J.. 9 A ^. o. jg A 25 »• 13 A 21'
3. J X ^^. 7. J2 X^f X ^. 11. gj X ;j9.
4 11 V 21 R 11 X 3 T3 fi V 41
*■ 12 X JS 8. 21 X 2 5 "• g X gy.
154. Find the product of  x 14.
I of 14 = I of Jj4 = \2 = \i = 101, or by cancellation :
7
3X14_21_
ix 1~2~'"*
COMMON rRACTIONS 127
155. To find the product of a fraction and a whole
number, regard tlie wliole number as an improper frac
tion with 1 as its denominator, and multiply the terms
together ; or indicate the multiplication and cancel the
common factors.
EXERCISE 96. — ORAL
Find the product of :
1.
15 xi.
7.
27 Xf.
2.
25 xi
8.
1 X 20.
3.
10 X .
9.
1 X 16.
4.
21 x.
10.
f x20.
5.
12 xf.
11.
f of 30.
6.
16 X .
12.
f of 27.
13.
^\ Of 22.
14.
f of 12.
15.
18 xf.
16.
V X 9.
17.
i X 12.
18.
1 of 15.
156. Find the product of : 3x2; 3 x ^ ; 3 x 21.
rind the product of 6 x 3i = 6 x 3 = 18, 6 x i = f = f ,
18 +  = 18f.
To multiply a mixed number by a whole number, mul
tiply the whole and fractional parts separately, and add
the products.
157. Find the product of 2f x 1^.
o2v1i— syS — i4 = 33, or 34
To multiply a mixed number by a mixed number, re
duce each to an improper fraction and multiply the terms.
EXERCISE 97. — WRITTEN
Find the product of :
1. 10x2i. 3. 351x27. s. 16 x 121.
2. 81 X 62f . 4. 331 X 181. 6. 18 x 331.
128 PRACTICAI, ARITHMETIC
7.
27 X 421
12.
81 J^ X 561
17.
100 X 6^.
8.
16 X 28f .
13.
34 X 121
18.
86 X 41.
9.
9 X lOli.
14.
81x,V
19.
Ill X 111.
10.
35 X 16.
15.
If of 512.
20.
35 X 8f .
11.
631 X 1.
16.
64 X 8f
21.
100 x5jV
EXERCISE 98. — WRITTEN
1. In 1 pint of whole (vinskimmed) milk ^^ is water,
jlg fat, ^^ protein, gV carbohydrates. How much of each
food constituent is there in 8 pints or 1 gallon ?
2. In 1 pint of buttermilk ^^^ is water, ^^^ fat, yj
protein, gV carbohydrates. How much of each food con
stituent is there in 8 pints or 1 gallon ?
3. In 1 pint of skim milk ^ is water, ^^ protein, gV
carbohydrates. How much of each food constituent is
there in 1 gallon of skim milk?
4. If a child of 4 to 6 years of age requires daily 1^
grams of protein, 1 grams of fats, and 5 grams of carbo
hydrates for each pound of weight, how much of each
of these food constituents will be required for a child of 6
years weighing 45 lbs. ?
5. How much of each for a child of 4 years weigh
ing 35 lbs. ?
6. If a boy of 10 or a girl of 8 years requires daily 1
grams of protein,  gram of fat, and 4^ grams of carbohy
drates for each pound of weight, how much of each con
stituent will be required for a boy of 10, weighing 67 lbs.?
7. How much of each for a girl of , 8 years weighing
55 lbs. ?
DIVISION OF FRACTIONS
158. How many times is ^ contained in 1? How
many times is  contained in 1?
Since ^ is contained in 1, or , four times,  is contained in
1, or I, ^ of four times, or t times.
How many times is  contained in 1 ?  is contained
in 1, or , onethird of 4 times, or  times.
How many times is ^ contained in 2?
How many times is  contained in 2?
Since ^ is contained in 2, or , eight times,  is contained
in 2, or ,  of 8 times, or f times.
How many times is f contained in 2?  is contained
in 2, or , ^ of 8 times, or  times.
EXERCISE 99. — DEAL
1. How many times is ^ contained in 1?  in 1?
2. How many times is  contained in 1?  in 1?
f inl?
3. How many times is ^ contained in 2 ? J in 2 ?  in 2 ?
4. How many times is ^ contained in 3? f in 3?
159. When the product of two numbers is equal to 1,
each of the two numbers is called the Reciprocal of the other,
e.g., 3 X 1^= 1. Hence, 3 is the reciprocal of ^, and ^ is
the reciprocal of 3. Again, ^ Xy =  = l. Hence,  is
the reciprocal of , and  is tlie reciprocal of .
129
130
PRACTICAL ARITHMETIC
160. To multiply by the reciprocal of a number is the
same as to divide by that number. Hence to divide a
whole number by a fraction, or a fraction by a whole
number, or a fraction by a fraction, multiply by its re
ciprocal.
161. Divide  by f .
Since, as we have seen, j is contained in 1, seven times,
f is contained in 1, i of seven times, or J times. If i is con
tained in 1,  times, it will be contained in , ^ of f, or ^ or
IJj or 1 times.
162. Mixed numbers should be reduced to improper
fractions before performing the division.
When possible, use cancellation in the process.
BXEECISE 100,
Find the quotients of :
OEAIi
1.
16 ^i
8.
1^8.
15.
f^l
2.
12^1
9.
,«^^100.
16.
Af
3.
8f
10.
ll
17.
^\f
4.
144 ^i.
11.
I'
18.
Ifi
5.
if
12.
21 =5.
19.
8^1.
6.
1^^
13.
63^1
20.
17 ^ 11.
7.
lG.
14.
3(5 ^.
21.
36^1
EXERCISE 101. — WRITTEN
1.
81^1
6. ,3
_i_ 1
T • S
11. 4
121.
2.
l?^f
7. 2
1^6 5.
12. If 4 30.
3.
U^i
8. 2
H^3G^6.
13. I
of
4.
144132
9. 8
^f
14. 1
off^l.
5.
3.23^1.
10. 9
f
15. i
^lofi
COMMON FRACTIONS 131
16. 3 of 71^61 off 18. 240Jj^of.
17. 9^J^off^i. 19. 3ofl^iofi.
EXERCISE 102. — WRITTEN
1. If a normal child of 7 years weighs 49^ lbs., and a
child of 8 years weighs 54 lbs., what is the average
monthly increase in weight?
2. If a child of 7 years has a chest measurement of
23J ins., and a child of 8 years lias a chest measurement
of '2l ins., what is the average monthly growth in chest
measurement?
3. If a child of 1 year weighs 20 lbs., and a child of
10 years weighs 6Q^ lbs., what is the average yearly increase
in weight?
4. A merchant buys a bill of goods : cotton goods
$250.50, silks ^5^125. 75, notions ¥75.80. He receives a
discount of ^ of the bill for cash. What does he pay for
the goods?
5. A man at hard muscular labor requires 1^ times the
food of a man at moderate muscular work. If a man at
hard work consumes 11 ounces of roast beef at a meal,
how much roast beef is required for a man at moderate
work?
6. If a man at moderate work requires  the amount of
food required for a boy from 10 to 12 years of age, and
the food of a man at moderate work for supper consists of
3 ozs. of bread,  oz. of butter, 3 ozs. of bananas, and
2 ozs. of cake, how much of these foods is required by
a boy of 12 years of age?
7. If the man requires 5 times the amount of food
132 PRACTICAL ARITHMETIC
required by a child of 6 years of age, how much of these
foods is required for the 6yearold child ?
COMPOUND AND COMPLEX FRACTIONS
163. An indicated multiplication of a fraction is called
a Compound Fraction, e,.g.^ f of f or  x f is a compound
fraction.
164. An indicated division of a fraction is sometimes
2 3
called a Complex Fraction, e.g., 2 and I are complex frac
tions, and are read f divided by ^ and  divided by 5.
They are solved as are other examples in division of frac
tions. '
EXERCISE 103. — "WRITTEN
Reduce to simple fractions :
12
1. ^
\'
3.
ii.
5.
12
4.
1
6'
6.
15
¥ofi%
iof2i
11.
1^.
of6i
31 of 91
12.
2i
13.
10. ^, TiTT 12. ^^i y^ 14.
7.
M.
27
8.
15^
10
^Ofll,^^
M °t' n%
Hi
8
29J^
6
FRACTIONAL RELATIONS OF NUMBERS
165. What part of 6 is 3 ? What part of 10 is 5 ?
Whatpartof 63 is9? 9 is g% of 63, or f
To find what part the second of two numbers is of the
first, divide the second by the first.
COMMON FRACTIONS 133
EXERCISE 104. — OEAL
Find what part the second number is of the first :
1.
16,8.
6.
^%l■
11.
,12.
2.
82,9.
7.
101 51.
12.
A' 5.
3.
36,4.
8.
n^ i
13.
9' 9*
4.
74, 18.
9.
8!, 2.
14.
hi
S.
2h H
10.
6, 3.
15.
i,7.
166. 6 is I of what number?
Since 6 is f of the number, ^ of the number is ^ of 6, or 3.
If 3 is ^ of the number, the number must be 3 times 3, or 9.
Hence, 6 is  of 9.
167. To find a number when another number and its
fractional relation to the unknown number are given,
divide the given number by the numerator of the fraction
and multiply by the denominator.
EXERCISE 105. — ORAL
1. 99 is I of what number? 7. 16 is  of what number?
2. 8 is f of what number ? 8. 24 is  of what number ?
3. 16 is I of what number ? 9. 20 is  of what number ?
4. 15 is I of what number? 10. 21 is  of what number?
5. 72 is 1^ of what number? li. 64 is  of what number?
6. 81 is ^^ of what number? 12. 54 is f of what number?
13. Tomatoes at 11.75 per dozen cans cost how much
per can ?
14. Corn at 90 cts. per dozen cans costs how much per
can?
134 PRACTICAL ARITHMETIC
15. Dried beef at #1.80 per dozen cans costs how much
per can?
16. A boy has 50 cts. He spends 30 cts. for a book, and
10 cts. for a tablet. What part of his money has he left?
17. A boy's spending money is $1.60. He spends  of
it for a fielder's glove. What does the glove cost?
18. If a halfton of coal costs $4. 50, what will 5 tons cost?
19. If the food required by a man at moderate work is
expressed as 100 parts, how many parts are required by
the following?
a, man at light work requiring ^^ of the food of man at
moderate work.
h, woman at moderate work requiring ^ of the food of man
at moderate work.
20. How many parts are required by the following?
a, child 6 to 9 years requiring  of the food of a man at
moderate work.
b, child 2 to 5 years requiring ^ of the food of a man at
moderate work.
c, child under 2 years requiring ^^ of the food of a man
at moderate work.
d, man at hard muscular work requiring 1^ of the food of
a man at moderate work.
21. A boy can walk  of a mile while an automobile
goes 4 miles. How far will the automobile have gone
when the boy has walked 5 miles?
22. If four boys can mow the school lawn in 1^ hours,
how long will it take one boy? How long will it take
two boys?
COMMON FRACTIONS 135
168. How many cents in ^ of a dollar? J of 1 dollar is
16 cts., therefore,  of a dollar multiplied by any number
is the same as 16 cts. multiplied by that number.
169. How many cents in ^ of a dollar? ^ of a dollar is
12 cts., therefore, ^ of a dollar multiplied by any number
is the same as 12 cts. multiplied by that number.
170. The part of a number which will exactly divide it
is called an Aliquot Part of that number.
It is often easier to multiply or divide by ^ dollars, ^
dollars, or ^ dollars than to multiply or divide by 12^, 16,
or 334 cts.
EXERCISE 106. — ORAL
1. What will 18 pineapples cost at 16^ cts. apiece?
2. At 12 cts. a yard, what will 12 yards of gingham
cost?
3. At 33J cts. apiece, what will 15 blank books cost ?
4. At 25 cts. each, what will 16 pairs of stockings cost?
EXERCISE 107. — WRITTEN
1. 40 is what part of 100? 7. 621 is what part of 100?
2. 60 is what part of 100? 8. 871 is what part of 100?
3. 80 is what part of 100? 9. 66 is what part of 100?
4. 75 is what part of 100? 10. 83 is what part of 100?
5. 37 is what part of 100? ii. 41f is what part of 100?
6. 31^ is what part of 100? 12. 58^ is what part of 100?
REDUCTION OF DECIMAL FRACTIONS
171. Express as common fractions and change to lowest
terms: .4, .04, .004, .0004.
136 PRACTICAL ARITHMETIC
Express as common fractions and reduce to lowest terms .■
.25, .025, .0025.
Express as common fractions and reduce to lowest terms :
.300, .00003.
172. To reduce a decimal to a common fraction, write
the figures of the decimal for the numerator, with 1 and
as many ciphers as there are decimal places in the decimal
number for the denominator, and reduce this fraction to
lowest terms.
EXERCISE 108. — WRITTEN
Reduce to common fractions in lowest terms or to mixed
numbers:
1.
0.4.
6.
0.7. 11. 0.55.
16. 1.25.
2.
0.5.
7.
0.135. 12. 1.05.
17. 17.875.
3.
0.115.
8.
0.008. 13. 5.125.
18. 25.135.
4.
0.125.
9.
0.375. 14. 10.50.
19. 2.9375.
5.
0.6.
10.
0.10. 15. 2.75.
EXERCISE 109. — OEAL
20. 7.875.
1.
Express
as a
, decimal : l , ^^.
2.
Express
as a
, common fraction : 0.2,
0.4, 0.5, 0.1.
3.
Express
as decimals : , , , , , 
173. Change §
3 3.0
8 8
to a decimal :
8)3.000.
0.375
1 = 0.375.
174. To reduce a common fraction to a decimal, divide
the numerator by the denominator, placing the decimal
point and annexing as many ciphers as are necessary to
complete the division.
COMMON TRACTIONS
137
175. If the denominator of a common fraction has other
prime factors than 2 or 5, the division of the numerator
cannot be completed. In such cases it is customary to
carry out the decimal to the fifth place and to place a plus
sign after the decimal, thus 24.28978 +.
^
""^
~X'
■
"7\
/^
^
61
7^
/
r
>.
.
/\
100.001
___^
/
169
7
12
/\
/ \
/ 26.78 ";:
<
\ 1.45
6 \
11 \
i /
1
8
>
/
\
27.1 \
/m ^H
t
61
82
11
18
\o\P
/
9
16^^
3
11
\
42§ d\
\l 7^
h
3
8
6 /
\
)29
9
29
21
32
y
/ 786.42 /
/
\ 1.00!
\k
)3 yc
8.4291
1 "*
26 \
3? \
\
\
>
H 7
V
^
6^
2^
y
■___I___^
H
^
>^
EXERCISE 110. — WRITTEN
116. Add the number found in e to each number in
the outer circle.
1732. Subtract the number in b from each number in
the outer circle.
138 PRACTICAL ARITHMETIC
3348. Subtract the nuinbei* in p from each number in
the outer circle in all cases where it is possible.
49112. Multiply the numbers at b, e, g, I by each num
ber in the outer circle.
113176. Divide each number in the outer circle by the
numbers at 6, c, g, and i,
EXEECISE HI. — WRITTEN
Express as decimals, carrying the division to the fifth
place in cases of inexact decimals :
1.
2.
3.
4.
5.
EXERCISE 112. — WRITTEN
REVIEVi^ PROBLEMS
1. 160 is f of vs^hat number?
2. 144 is j^ of what number?
3. 160 is 1^ of what number?
4. 180 is  of wliat number?
5. 364 is ^ of what number?
6. In repairing the waste pipes of a house 4 pieces are
needed, — 3^ ft., 9 ft., 41 ft., and 17 ft. long. What is
the total number of feet needed?
if
6.
JL.
II'
11.
u
16.
A'
f
7.
iV
12.
3f
17.
A'
9"
8.
_3_2_
111
13
37.
18.
i
M
9.
5A
14.
1261^
19.
if
lOA^
10.
\i
IS.
4H
20.
f
REVIEW PROBLEMS
139
7. A sixsided chicken yard is 3 rods on one side, 41
on another, 51 on another, 1 yL on another, 2 on the fifth
side, and 4 rods on the sixth side. How many rods of
fence will be required to enclose it ?
8. If an apple tree bears 2678 apples and J drop, and
^ of the remaining are wormy, what fraction of the apples
originally present is good? How many apples are good?
If the trees are sprayed to prevent worms, only about ^
will be wormy. What will the fractional gain from spray
ing be? How many apples are gained?
9. What is the cost of spraying 197 apple trees for
black rot at S^ cts. a tree for each application, spraying
3 times? Adding 1 cts. to cover cost of arsenite of lead
used to prevent the codling moth, what is the cost?
10. According to the Arkansas Experiment Station,  of
an acre of peanut pasture produced 312 lbs. of pork, and
the same area in corn produced 104 lbs. What fraction
of the value of peanuts as a pork producer is possessed by
corn ?
11. Corn at different stages of growth contains water
and dry matter in each ton, as follows :
Fully tasselled ....
Fully silked
Kernels watery to full milk
Kernels glazed ....
Ripe
COKN PER
100 Acres
90 tons
129 tons
163 tons
161 tons
112 tons
Water per
100 Acres
82 tons
113 tons
140 tons
125 tons
102 tons
Dry Matter
PER 100 Acres
8 tons
15 tons
23 tons
36 tons
40 tons
What fraction of the corn is dry matter in each period?
140 PRACTICAL ARITHMETIC
How much greater is the fraction representing dry matter
" ripe " than " fully tasselled " ?
12. A crib of corn holds 10,976 lbs. of ear corn. How
many bushels of shelled corn will it yield if ^ of the ear
corn is grain and 56 lbs. of shelled corn make a bushel?
13. If it requires 6 acres of corn to fill a 93ton silo,
how many acres will it require to fill a 60ton silo?
14. It is estimated that good roads to a farm increase
its value about f 9^ an acre. What would this amount to
on a 67acre farm?
15. Divide the number 81 into 2 parts such that ^ of
the first shall be f of the second.
16. A man by working 7 hrs. a day can complete a
piece of work in 11^ days, how many hours must he work
per day so that he will complete the work in 15 days?
17. If a boy can do  as much as a man, how many days
will he require to complete the work in the above problem,
working the same number of hours as the man?
CREAMING
18. In skimming cream from shallow pans in the usual
way, about ^500 of the skimmed milk is butter fat. Milk
before skimming usually contains j^^q butter fat. What
fraction of the original butter fat is lost in skimming?
19. If a cow produces 256 lbs. of butter fat in a year,
how much is lost in skimming? What is its value at
23 cts. a pound?
20. In setting milk in deep pails and using better skim
ming methods, the skimmed milk is only g^^ butter fat.
EEVIEW PROBLEMS
141
What fraction of the original butter fat is lost? How
many pounds? What value?
21. When a hand separator is used, skimmed milk is
only gj^ butter fat. What fraction of the butter fat is
lost? How many pounds? What is its value?
22. What fraction is saved by the separator that is losti
by the shallowpan method? What fraction is thus saved
that is lost by deep setting?
23. If a cow in a year gives 6278 lbs. of milk of which
^ is butter fat, and ^^^ of this amount be lost in sepa
142 PRACTICAL ARITHMETIC
rating, and there be an increase of J in weight in the
amount remaining (due to water, salt, etc., used in mak
ing the butter), what will be the butter yield?
24. How many cows will a dairyman need in order to
save enough butter in a year, by the use of a separator,
to enable him to pay for a 65 separator? Use the facts
given in problems 18, 19, and 21.
25. In one herd the average of butter fat per cow
was 285.62 lbs. If this were all converted into butter
containing ^ fat, what would be the amount? The best
cow in the herd gave 439.37 lbs. of butter fat. How
much butter does that equal?
26. Assuming that in problem 23 I of the whole milk
becomes skimmed milk and that it be ^q^q q" butter fat, and
that f of the cream becomes buttermilk containing ^^ of
butter fat, what is the loss in butter fat? What is its
equivalent in butter ?
27. The cost of feed to produce 100 lbs. of milk in New
York was as follows from 19 different cows: f .62, .61,
$.46, 1.55, 1.49, $.89, $.82, f.62, $1.48, $.77, $.70, $1.07,
$.74, $.85, $.75, $.81, $.59, $.53, $.44. What was the
average cost per pound ?
28. The cost for each pound of butter fat was $.115,
$.155, $.18, $.225, $.175, $.16, $.13, $.16, $.17, $.14, $.12,
$.26, $.125, $.14, $.185, $.21, $.27, $.15, $.225. What
was the average cost of butter fat per pound ?
29. If a tree 9 ins. in diameter yields only  as much
lumber as one 18 ins. in diameter, and the lumber of
the smaller tree is worth only  as much per foot as that
of the larger tree, what is the difference in the value of
KEVIEW PROBLEMS 143
two such trees, if the smaller one yields 132 ft. of lumber
and it is worth 1 cts. a foot?
30. If 1 lb. of butter is composed of j.^^^ watei, 
butter fat, gf^^ curd, and the balance salt, how much salt
is there in 16 lbs. of butter?
31. A hayseed mixture consists of 3 parts red clover
and 7 parts tall oat grass. How many pounds of each are
needed for 27 acres, using 35 lbs. to the acre?
32. Another mixture is red top 3 parts, orchard grass
4 parts, meadow fescue 2 parts, and red clover 1 part.
Using 40 lbs. to the acre, how many pounds of each kind
of seed are needed for 29 acres?
33. A good pasture mixture is Kentucky bluegrass 2^
parts, white clover 1 part, perennial rye 3 parts, red fescue
1 part, red top 2^ parts. Sowing 35 lbs. to the acre, how
many pounds of each are needed for 26 acres ?
34. If, in a score card for judging butter, ^^ of the total
points are given for perfect flavor, ^\ for grain, ^g for
color, and J^ for salting, what will be the numerical value
assigned to each in a score card of which the total number
of points is 100 ?
35. A hog when alive weighs 312 lbs. and the dressed
carcass weighs  less ; what is the weight of the dressed
carcass ?
36. What is the difference in cost of a 2250 (2 lbs.
of bluestone, 2 lbs. of lime, 50 gallons of water) and a
6450 Bordeaux mixture with lime at 1^ cts. a pound,
bluestone at 6 cts. a pound ? Using 150 gallons to the
acre, what is the difference per acre in the cost of these
two mixtures?
144 PRACTICAL ARITHMETIC
37. In an Arkansas orchard 6 apple trees sprayed to
prevent the bitter rot yielded 8674 sound apples and 989
diseased ones; on 3 unsprayed trees 188 sound apples
and 4244 diseased ones. What fraction was sound in each
case? Reduce to similar fractions for comparison. How
many times greater is the yield of sound apples when
sprayed than when not sprayed ?
38. Two cows, Glista and Belva, were very similar in
outward appearance, age, and weight. Glista required
21 cts. worth of feed to produce 1 lb. of butter fat.
Belva produced 1 lb. for 15 cts. What fraction of the
feed required by Glista was sufficient for Belva ?
39. Seven samples of clover seed sold per bushel (60 lbs.)
for: 15.50, §5.25, 15.00, 14.75, $4:.15, .$4.00, and 13.50.
They contained, respectively: 55^, 4^5^^^, 55y2_.^ 55^, 48,
52^3j_ and 27^\ lbs. of good seed to the bushel. What
was the price paid for a bushel of good seed in each case?
40. Concrete is made of cement, sand, and aggregate
(coarser material, gravel, crushed stone, etc.) in quanti
ties of 1 part to 2 parts to 4 parts. How many pounds
of each material are required for 4000 lbs. of concrete ?
41. Another mixture often used is 1 to 2J to 5. Em
ploying this mixture, how much of each material is needed
to make 4000 lbs. of concrete?
42. Using the quantities 1 to 3 to 6, how many pounds
of each ingredient are needed to make 4000 lbs. of con
crete ?
43. If the entire corn plants on an acre weigh 6800 lbs.,
what is the weight of grain, cobs, stalks, leaves, and
shucks, if 2^0" of the plant be ears, and of these  be grain
KEVIEW PROBLEMS
145
May 17, tgoS.
New Vork .Ive.
West 33d Street
liiberty Street , . .'
Phlla. (MUi&Cliestiiiit Sts.)..
[lhavb
Baltlmoro ((!uQ(lnSts.)+ .
Relay .....,....>
Hanover ..**
DoTScy ^
Jessup
An
n»poll« June $ ■ . .arr.
Annapclls+ .arr,
Annapolis Ive,
46. Ill nailing a i%in.
piece to a fin. piece, with a
3penny nail, how much will
the nail extend beyond the
wood?
AnnapolisJunctlon.lve,
Savage
Laurel £
Muirkirlc. 5
Beltsville
Branchvtlle
H/attsvllle
wash I ng jKev Hsioii) g ar.
ton X Station J+lv.
Rockville
Gaithersburg
Boyd ".
WashlnEton June ...
BrunswicK +..
Weverten
Harper's Fernr+ ....
Shenandoah Juno+ .
Kemeysville
Martinsburg + . . v. . . . $
North MoUDtsun
Cherry Run+
Hancock + $
Sir John's Run 5
Woodmont
Orleans Road
Baird.... <
Magnolia
Paw Paw
Okonoko
Green Spring
Patterson Creek
Cumberland + S
{arrive
Mis.
9.0
11.5
iS8
17.9
A M
tsoo
516
523
528
532
538
17.9
19.4
ai.3
249
27.1
299
334
4ao
40.0
565
fits
69.1
82.8
S9.8
92.7
9S.6
103.0
106.5
114.2
121.6
127.6
1370
142.6
147s
'52.9
1598
163.1
168.0
172.8
178.3
185.1
192.2
1*1
and \ cobs ; and \\ be stover, and of this  is stalk, ^
leaves, and J^ shucks ?
44. A 3penny standard wire nail is 1\ ins. long, a
4penny nail IJ ins., a 5penny nail 1 ins., a 6penny
nail 2 ins., an 8penny nail
2 ins., a 10penny nail 3 ins.,
and a 20penny nail 4 ins.
What fraction of an inch in
crease in length is there for
every pennyweight increase
between 3 pennyweights and
10 pennyweights ?
45. If it is desired to nail
a in. board to a IJin.
board with as long a nail as
possible but not allow the
nail to come nearer than ^^g
in. of going through both
pieces, how long a nail can
be used ?
5 38
542
546
5 54
V^
003
6
625
AM
IS
A M
fsao
644
653
714
729
740
752
807
8 14
82s
836
848
903
9 20
A H
A M
•6.51
t7io
725
738
804
820
A M
47. In nailing a Jin. piece
to a 1in. piece, with a 5
penny nail, how much will the nail lack of reaching
through the wood ? In using a 6penny nail, how much
will it project beyond the wood? In each of the
146 PRACTICAL ARITHMETIC
above cases what will be the result if the nail be coun
tersunk jJg in. ?
48. The table on page 145 is a reproduction of a
Baltimore and Ohio timetable showing the distances
between stations in miles.
How far is it from Baltimore to Washington? From
Annapolis Junction to Washington ? Washington to
Harper's Ferry? Harper's Ferry to Cumberland? Mar
tinsburg to Cumberland? Annapolis Junction to Mag
nolia ?
49. What ■ would be the cost for each of these dis
tances at the following rates per mile, which are charged
by some railroads: 2 cts., 2^ cts., 2 cts., and 3^ cts. ?
50. If a farm is ^ in corn, ^f in wheat, ^^ in clover,
1^ in oats, ^^ in pasture, and the remaining 5 acres is occu
pied by houses, other buildings, etc., what is the size of
the farm, and how many acres are in each crop?
51. A man left to his son ^ of his estate, to one
daughter , to the other daughter the remainder, amount
ing to f 3268. What was the value of the estate?
52. Walton caught  as many fish as Clara, who
caught j as many as Vernon, whose catch is 63. How
many fish did Walton catch?
53. A horse is worth 7 times the buggy. What part
of the value of the horse is ^^ the value of the horse and
buggy?
54. It is estimated that about 1700,000,000 are lost
annually in the United States through insect and fungous
diseases ;  of this could be prevented by spraying.
What amount could be saved by spraying?
ACCOUNTS AND BILLS
176. A Debt is that wliich one person owes to another.
177. A Debtor is a person who owes.
178. A Credit is that wliich is due one person from
another.
179. A Creditor is tlie person to whom the debt is due.
180. An Account is a record of debits and credits.
181. A Balance of an account is the difference between
the sums of the debits and credits.
182. A Bill describes the goods sold by giving quantity
and price.
183. The Footing of a bill is the total cost.
184. A Receipt acknowledges the payment of a bill.
A bill is receipted by the creditor's writing the words
" Received Payment " and his signature.
185. ORDERING GOODS
Hendersonville, S.C,
Sept. 10, 1908.
Chisholm, Ward & Co.,
Market and Madison Sts.,
New York, N.Y.
Dear Sirs : Please find enclosed 18.37, for which send
me by express the following items :
147
148
PRACTICAL ARITHMETIC
Catalogue No.
QCTANTITY
Articles Wanted
Price Each
C. S. 40805
B3137
C6490
B4692
1
1
1
1
A^'ickless 2burner Kerosene Stove
Utility Wasliiiigmachine
Eclipse Food Chopper
Empire Clothes Wringer
$3.13
2.65
.70
1.89
Yours truly,
(Mrs.) James A. Monroe.
186. An itemized bill of the above form is received by
Mrs. Monroe with the full amount of the cost of the arti
cles stated and the words, "Received Payment," with the
signature of the firm.
EXERCISE 113. — WRITTEN
Raleigh, N.C.
Jan. 7, 1908.
Adams,
Bought
of John G. Stroud.
5 lbs. Coffee
@$.30
1
20 lbs. Sugar
@ .05
3 doz. Eggs
@ .23
1 lb. Cheese
@ .15
2 lbs. Butter
@ .28
Compute the amount of each item, place in a column at
the right of the double line, add, and make a receipt.
Mrs. C. J. Adams, wishing to stock her pantry for the
first time, orders the supplies mentioned below. Make
out each number as a separate order to any firm you wish,
ACCOUNTS AND BILLS 149
complete it as a bill, and write a receipt indicating its
payment.
2. Cereals: 1 bbl. flour 7 ; 10 lbs. graham flour @ 3
cts. ; 10 lbs. corn meal @ 3 cts. ; 10 lbs. hominy @ 3J cts. ;
4 pkgs. breakfast foods @ 10 cts. ; 1 pkg. corn starch 10
cts. ; 10 lbs. rice @ 9 cts. ; 2 lbs. macaroni @ 15 cts. ; 3
lbs. tapioca @ 7 cts.
3. Sugars : 25 lbs. granulated sugar @ 6 cts. ; 5 lbs.
cut loaf sugar @ 10 cts. ; 2 lbs. pulverized sugar @ 10
cts. ; 10 lbs. brown sugar @ 5J cts.
4. Canned vegetables : corn, 6 cans@ f 1.50 a dozen;
peas, 6 cans @ $ 1.50 a dozen ; tomatoes, 12 cans @ $ 1.35
a dozen.
5. Canned fruits : peaches, 6 cans @ f 3 a dozen ; cher
ries, 2 cans @ 25 cts. ; plums, 2 cans @ 25 cts.
6. Dried fruits : raisins, 2 lbs. @ 15 cts. ; currants,
2 lbs. @ 13 cts. ; prunes, 5 lbs. @ 15 cts.; evaporated
apricots, 2 lbs. @ 19 cts.
7. Dried vegetables : dried black beans, 1 quart, 10
cts. ; dried Lima beans, 5 quarts @ 15 cts. ; dried white
beans, 1 peck, 75 cts. ; split peas, 2 quarts @ 10 cts.
8. Canned meats : ham, 2 lbs. @ 30 cts. ; tongue, 2
lbs. @ 30 cts. ; salmon, 2 cans @ 20 cts.
9. Beverages : coffee, 1 lb., 35 cts.; tea, 1 lb., 75 cts.;
cocoa, ^ lb. @ 88 cts.; chocolate, 1 lb., 35 cts.
10. Sundries : 1 gal. vinegar, 25 cts. ; 1 gal. New
Orleans molasses, 60 cts. ; baking soda, 2 lbs. @ 8 cts. ;
cream of tartar, 1 lb., 50 cts.; baking powder, 1 lb., 54
cts.; salt, 25 lbs., 25 cts.; white pepper, i lb. @ 50 cts.
DENOMINATE NUMBERS
187. How many inches are there in 1 foot ? Express
2^ feet as feet and inches.
A concrete number wliich represents one kind of unit is
called a Simple Number ; e.g., 2^ ft. is a simple number.
188. A concrete number which is expressed in differ
ent units is called a Compound Number, e.ff., 2 ft. 6 ins. is
a compound number.
189. Concrete numbers denoting units of measure are
called Denominate Numbers, e.g., 2 ft. is a denominate
number.
190. A denominate number composed of units of one
denomination is called a Simple Denominate Number, e.g.,
2^ ft. is a simple denominate number.
191. A denominate number composed of units of more
than one denomination is called a Compound Denominate
Number, e.g., 2 ft. 6 ins. is a compound denominate
number.
192. A unit of measure from which other units are
derived is called a Standard Unit; e.g., the dollar is a
standard unit of money, the pound is a standard unit of
weight, the yard is a standard unit of measure of length.
193. The process of changing the unit of a denominate
number from one denomination to another without chang
150
DENOMINATE NUMBERS
151
ing the value is called Reduction,
e.g., changing 1 yd. to 36 ins.
194. If the change be from a
higher to a lower denomination,
the process is called Descending
Reduction ; if from a lower to a
higher, it is called Ascending Re
duction, e.g., changing 1 yd. to
36 ins. is descending reduction ;
changing 24 ins. to 2 ft. is ascend
ing reduction.
195. UNITS OF LENGTH
A line has length only. Meas
ures that are used in measuring
lines are called Linear measures.
196. _ TABLE OF LINEAR
MEASURES
12 inches (ins.) = 1 foot (ft.)
3 feet or 36 ins. = 1 yard (yd.)
5^ yards, or 1Q\ ft. = 1 rod (rd.)
320 rods, or 5280 ft. = 1 mile
(mi.)
The foliowing abbreviations are
also commonly used : ' to represent
feet and " to represent inches.
EXERCISE 114. — ORAL
1. How many inches are there
in 5 ft.? In 10 ft.?
2. How many feet are there in
3 yds.? In 7 yds.?
—
«
.. —
M —
CO =
^ =
01 =
0) =
^ =
00 =
—
—

—
—
lO
—
—

—
Tfl
—

—
CO
(0 =
M —
— .

^
(M
—
10 =
CO =
01 =
—

—
—
iH
—
—

—
152 PEACTICAL ARITHMETIC
3. How many feet are there in 2 rds. ? In 4 rds. ?
4. How many inches are there in 2 yds. ? In 2 ft. 3 ins. ?
197. Reduce 12 yds. 2 ft. 8 ins. to inches.
Since 3 ft. = l yd., in 12 yds. there are 12 x 3 ft. = 36 ft.
36 ft. + 2 f t. = 38 ft.
Since there are 12 ins. in a foot, in 38 ft. there are 38 x 12
ins.=456 ins. 466 ins. +8 ins. =464 ins. Hence 12 yds. 2 ft.
8 ins. =464 ins.
198. To reduce a compound denominate number to
lower units, multiply the number of the highest denomi
nation by the number of units required to change to the
next lower denomination, and to the product add the
number of units of this denomination given. Proceed in
like manner with this result and each successive result
obtained until the number is reduced to the required
denomination.
EXERCISE 115. — WRITTEN
Reduce to inches: Reduce to feet :
1. 2 yds. 2 ft. 6 ins. 5. 2 mis. 15 rds. 6 ft.
2. 4 rds. 2 yds. 1 ft. 6. 3 mis. 30 rds. 15 ft.
3. 25 rds. 12 ft. 4 ins. 7. 25 rds. 10 ft. 4 ins.
4. I rds. 8. 3.875 mis.
9. Measure the length, width, and height of your school
room, and express as feet, as inches, as yards, as rods, and
as fraction of a mile.
EXERCISE 116. — ORAL
1. How many feet in 24 ins. ? In 36 ins.? In 18 ins. ?
2. How many yards in 30 ft.? In 48 ft. ? In 36 ft. ?
DENOMINATE NUMBEKS 153
3. How many rods in 33 yds.? In 66 yds.? In 77 yds.?
4. How many rods in 11 yds.? In 23 yds.? In 99 yds.?
199. Reduce 250 ins. to yards, feet, and inches:
Since 12 ins. equal 1 ft., J^ of 250 ins. = 20 = tlie number of
feet, with 10 ins. remainder. Since 3 ft. = 1 yd., { of 20 =
6 = number of yards, witli 2 ft. remainder. Hence, 250 ins.
= 6 yds. 2 ft. 10 ins.
200. To reduce a denominate number to an equivalent
number of higher denomination, divide the given num
ber by the number representing the units of the next
higher denomination. Proceed with this and each succes
sive quotient in like manner until the required denomi
nation is reached. The last quotient with the remainders
will be the required compound denominate number.
EXERCISE 117. — WRITTEN
1. Reduce 950 ins. to feet.
2. Reduce 496 ins. to feet,
3. Reduce 8966 ins. to yards.
4. Reduce 375 ft. to rods and yards.
5. Reduce 42,560 yds. to rods and miles.
6. Reduce 35'6^ rds. to miles.
201. METRIC UNITS OF MEASURE
In many foreign countries and to an increasing ex
tent in our own country, another system gf measures called
the Metric System is used. This system is founded upon
10, as are decimal fractions, and is simpler and easier for
purposes of computation than are other systems.
154
PRACTICAL ARITHMETIC
202. METRIC LINEAR MEASURES
10 millimeters (mm.) = 1 centimeter <^cm.)
10 centimeters (cm.) = 100 mm. = 1 decimeter (dm.)
10 decimeters = 100 cm. = 1000 mm. = 1 meter (m.)
1000 m. = 1 kilometer (km.)
The meter is the unit of measure corresponding to the
yard. It is 39.37 ins. long. 1 km. = .62138 mi.
EXERCISE 118. ORAL
1. What part of a meter is 1 centimeter? 3 cm.?
2. What part of a meter is 1 millimeter? 6 mm.?
3. What part of a meter is 100 millimeters? 500 mm.?
4. Measure the length, width, and height of your school
room, and express as meters ; as centimeters.
203. REPRESENTATION OF MAGNITUDES
A clear conception of the relative sizes of numbers
is often gained by representing the numbers by lines, let
ting the length of the line bear direct relation to the size
of the number, e.g., the approximate amount of wheat
produced in 1900 was: United States 520,000,000, Russia
400,000,000, France 300,000,000 bushels. Letting a line
\ in. long represent 50,000,000 bushels, a line 1^ ins. long
would represent the yield of France, 2 ins. that of Russia,
2 ins. that of the United States, thus:
50 100 150 800 260 300 350 400 4B0 500
UNITED STATES
RUSSIA
FRANCE
DENOMINATE NUMBERS
155
EXERCISE 119. — WRITTEN
100
90
1. Represent the coal production of the following coun
tries in metric tons. Let { in. equal 2n,000,000 metric
tons. The production was as follows : The United States
225,000,000, the United Kingdom 225,000,000, Germany
155,000,000, AustriaHungary 37,500,000, and France a
trifle over 25,000,000 metric tons.
2. Represent the silk pro
duction of the following
countries, letting ^ in. rep
resent 1,000,000 pounds:
China 14,000,000, Japan
7,000,000, Italy 7,000,000,
Turkey, France, Spain, and
India together 3,000,000 lbs.
3. Represent the cotton
production, letting { in.
equal 1,000,000 bales :
The United States 10,250,
000, India 2,250,000, China
1,250,000, Egypt 1,165,000.
4. Represent the corn
production in 1898, ^ in.
representing 100,000,000 ssMay 83
bushels. The production
of the United States was 2,100,000,000
countries about 700,000,000.
204. Change in a quantity from time to time may be
clearly shown by arranging, side by side, the lines repre
50
40
SO
20
10
34
Fig. a
that of other
166
PRACTICAL ARITHMETIC
senting the varying magnitudes so that they may be
readily compared ; thus, the highest temperature for May
22d, at a given place, was 75 degrees, the 23d 80 de
grees, the 24th 85 degrees, the 25tli 96 degrees, and the
26th 76 degrees. Rejiresenting the temperature by ver
tical lines, 1 degree = 1 mm., at equal distances apart,
and connecting the tops of these lines, we have figure A
on the preceding page. This is called a Curve. The
vertical lines are unnecessary, the curve alone being suf
ficient to picture the facts.
I860
soot
1870
1880
1890
190O
iu 700
I
«D
= 600
O 500
400
g 300
^ 200
1 100
Fig. B. — Curves showing the relative increase in population and in wheat
production in the United States, wheat and population both expressed in
millions.
EXERCISE 120. — "WRITTEN
1. Interpret in words the meaning of the curves shown
in the above figure.
2. The maximum price of corn per bushel in Chicago
in May during 9 years was as follows : In 1893, 38 cts.,
in 1894, 55 cts., in 1895, 29 cts., in 1896, 25 cts., in 1897,
37 cts., in 1898, 35 cts., in 1899, 40 cts., in 1900, 58 cts..
DENOMINATE NUMBERS 157
in 1901, 65 cts. Show these values hj a curve, letting
2 mm. represent 1 cent. Place the lines representing years
1 cm. apart.
3. The maximum v^holesale price of wheat per bushel
in Chicago in 1905 was as follows: Januarj^ 11.21, Feb
ruary 11.24, March 11.19, April fl.l8. May 11.14,
June 11.20, July 11.20, August 11.15, September 10.95,
October 10.92, November !0.92, December 0. 90. Show
the changes in price by a curve, letting 1 cm. represent 1
cent. Place the lines representing months 1 cm. apart.
SURFACE MEASURE
205. A surface has two dimensions. Length and Width.
206. If a surface is flat and has four square corners, it
is called a Rectangle.
207. A rectangle with four equal sides is called a
Square.
208. The unit of surface is a square. A square inch
is a square each side of which is 1 in. long. A square
foot is a square each side of which is 1 ft. long. A
square yard is a square each side of which is 1 yd. long.
209. The Area of a surface is expressed by the number
of square units it contains.
BXEECISB 121. — ORAL
1. A rectangle is 6 ft. long and 1 ft. wide. How
many square feet are there in its surface? How many
square feet if it is 2 ft. wide? 3 ft. wide? 6 ft. wide?
158
PRACTICAL ARITHMETIC
2. How many square inches are there in a square
whose sides are 5 ins.? 6 ins.? 8 ins.? 10 ins.? 12 ins.?
3. How many square centimeters are there in a square
whose sides are 6 cm. long? 7 cm. ? 9 cm. ? 18 cm. ?
4. How many square rods in a school lot whose sides
are 8 rods ? 9 rods ? 11 rods ? 13 rods ?
210.
SQUARE MEASURE
144 square inches (sq. ins.) = 1 square foot (sq. ft.)
9 square feet = 1 square yard (sq. yd.)
30^ square yards = 1 square rod (sq. rd.)
160 square rods = 1 acre (A.)
640 acres = 1 square mile (sq. mi.)
METRIC SQUARE MEASURE
100 square rnillimeters (sq. mm.) = 1 square centi
meter (sq. cm.)
100 square centimeters = 1 square deci
meter (sq. dm.)
100 square decimeters = 1 square meter
(sq. m.)
10,000 sq. m. = 1 hectare = 2.4711 A.
DENOMINATE NUMBERS 159
EXBBCISB 122. — WRITTEN
Reduce to square inches : Reduce to square yards :
1. 13 sq. yds. 5 sq. ft. 6. 130 sq. rds. 12 sq. yds.
2. 19 sq. rds. 7 sq. ft. 7. 2 sq. mis. 80 sq. rds.
3. 26 sq. rds. 6 sq. ft. 8. 8 A. 5 sq. rds. 29 sq. ins.
4. I sq. rd. 9. 5.47 A.
5. ^ sq. m. to sq. cm. 10. 86.4 sq. m. to sq. mm.
11. Measure the area of tlie floor, and eacli of the four
walls of your schoolroom, and express as square feet ; as
square yards.
EXERCISE 123. — WRITTEN
Reduce to higher denominations :
1. 6740 sq. ins. 4. 22,160 sq. ft. 7. 12,346 sq. rds.
2. 984 sq. ins. 5. 9678 sq. yds. 8. 39,271 sq. rds.
3. 540 sq. ram. 6. 6782 sq. cm. 9. 3240 sq. cm.
SURVEYORS' MEASURES
211. Surveyors in measuring use a Chain consisting
of 100 links ; its length is 4 rds., or 66 ft.
212. TABLE OF SURVEYORS' LINEAR MEASURE
7.92 inches =1 link (1.)
25 links = 1 rod (rd.)
4 rods = 100 1. = 1 chain (ch.)
80 chains = 1 mile (mi.)
160 PRACTICAL ARITHMETIC
213. TABLE OF SURVEYORS' SQUARE MEASURE
16 square rods = 1 square chain (sq. oh.)
10 square chains = 1 acre (A.)
640 acres = 1 square mile (sq. mi.)
1 square mile= 1 section (sec.)
36 sections = 1 township.
BXEECISE 124. — WRITTEN
1. Reduce 1560 sq. yds. to chains.
2. Reduce 960 sq. chs. to acres.
3. Reduce 67,820 ins. to chains.
4. Reduce  mi. to chains.
5. Reduce 60 chs. to feet ; to inches.
6. Measure the distance around your school yard. Ex
press in chains ; in links ; in feet ; in rods ; in yards. If
rectangular, how many square chains does it equal ? What
fraction of an acre ?
MEASURES OF VOLUME
214. A body that has length, width, and thickness is
called a Solid.
215. If the rectangles forming the faces of a solid are
squares, the solid is called a Cube.
216. The number of solid units that a body contains is
called the Solid Contents or Volume.
EXERCISE 125. — ORAL
1. How many cubic inches are there in a rectangular
solid 1 in. long, 1 in. wide, and 1 in. thick ?
DENOMINATE NUMBERS
161
2. How many cubic feet are there in a solid 1 ft. long,
1 ft. wide, and 1 ft. thick ? How many cubic inches ?
What is the name of such a solid?
3. How many cubic yards are there in a solid 1 yd.
long, 1 yd. wide, and 1 yd. thick? How many cubic
feet ? Name such a solid.
IIIIIIIIIII'
217. UNITS OF VOLUME, OR CUBIC MEASURE
1728 cubic inches (cu. ins.) = 1 cubic foot (cu. ft.)
27 cubic feet = 1 cubic yard (cu. yd. )
1^ cubic feet = 1 bu. (approximately)
218.
METRIC CUBIC MEASURE
1000 cubic millimeters = 1 cubic centimeter (cc.)
1000 cubic centimeters = 1 cubic decimeter (cdm.)
1000 cubic decimeters = 1 cubic meter (cbm.)
EXERCISE 126. —WRITTEN
Reduce to cubic inches :
1. 10 cu. ft. 14.5 cu. ins. 3. ^ cu. yd.
2. 46 cu. ft. 149 cu. ins.
4. I cu. ft.
162 PRACTICAL ARITHMETIC
5. 45 cu. yds. 18 cu. ft.
6. 8 cu. yds. 16 cu. ft. 425 cu. ins.
7. 29 cu. yds. 12 cu. ft. 1J cu, ins.
8. 15 cu. yds. 1234 cu. ins.
9. 72 cu. yds. 25 cu. ft.
10. 15 cu. yds. 16 cu. ft.
Reduce to cubic centimeters :
11. 15 cbm. 5 cdm. 10 cc. 13. 200 cbm. 125 cdm.
12. 300 cbm. 150 cdm. 14. 275 cbm. 165 cdm. 15 cc.
EXERCISE 127.— "WRITTEN
Reduce to units of liigher denominations :
1. 12,865 cu. ins. 4. 4256 cu. ins.
2. 924 cu. ft. 5. 98,764 cu. ins.
3. 6573 cu. ins. 6. 325 cu. ft.
7. How many cubic yards in your schoolroom ? How
many cubic feet? Cubic inches?
8. How many cubic centimeters in your schoolroom?
How many cubic meters ?
219. DRY MEASURE
2 pints (pts.)= 1 quart (qt.)
8 quarts = 1 peck (pk)
4 pecks = 1 bushel (bu.)
A bushel contains 2150.42 cubic inches.
The dry quart contains 67.2 cubic inches.,
A Heaped Bushel, which equals about 1^ Stricken Bushels,
DENOMINATE NUMBEKS
163
is used in measuring potatoes, apples, and other large vege
tables, also for lime, coal, and other bulky substances.
EXERCISE 128. — ORAL
1. How many pints in 5 qts.? In 7 qts.? In 2 pks. ?
In 1 bu.? In 2 bus.?
2. How many quarts in 3 pks.? In 3 bus.? In 1 bu. ?
In 12 bus.?
3. How many bushels in 20 pks.? In 25 pks.? In 36
pks.? In 16 qts.?
4. How many pints in 1 bu. 2 qts. 1 pt.?
EXERCISE 129. — WRITTEN
Reduce to pints : Reduce to higher units :
1. 5 bus. 3 pks. 5 qts. 4. 2254 qts.
2. 5 bus. 2 pks. 3 qts.  pt. 5. 3360 pts.
3. .625 pk. 6. 4800 pks.
U'A
PKACTICAL ARITHMETIC
Reduce the following to fractions of a bushel. Express
both as decimals and as common fractions :
7. f^ pt. 8. f qt. 9. I pk. 10. 2 pks. 8 qts.
11. Get a bushel of sand or grain and a quart measure,
and practise estimating quarts, pints, and pecks to see
who can make the most accurate estimates at sight.
220.
LIQUID MEASURE
4 gills (gis.) = 1 pint (pt.)
2 pints = 1 quart (qt.)
4 quarts = 1 gallon (gal.)
A gallon contains 231 cubic inches.
31^ gals, are usually •considered a barrel, and 2 barrels a
hogshead.
221. METRIC LIQUID MEASURE
10 milliliters (ml.) = 1 centiliter (cl.)
10 centiliters = 1 deciliter (dl.)
10 deciliters = 1 liter (1.)
1 milliliter = 1 cubic centimeter. 1 milliliter of water weighs
1 gram.
DENOMINATE NUMBERS 165
EXBBCISB 130. — ORAL
1. How many gills are there in 6 pts. ? In 10 pts. ?
2. How many gills are there in 5 qts. ? In 8 qts. ?
3. How many pints are there in 8 qts. ? In 12 qts. ?
4. How many quarts are there in 60 pts. ? In 45 pts. ?
5. How many milliliters are there in 3 1.? How many
cubic centimeters in 7 1. ?
EXEBCISE 131. — WRITTEN
Reduce to units of lower denomination :
1. 12 gals. 2 qts. 1 pt. 5. 14 gals. 2 qts. 1 pt. 1 gi.
2. 25 gals. 1 qt. 1 pt. 2 gis. 6. 45 gals. 2 qts. 1 pt. 3 gis.
3. I gal. 7. .225 gal.
4. I gal. 8. J_. gal.
Reduce :
9. 12 liters to deciliters. 10. ^ liter to milliliters.
With a pail of water and pint and quart measures, be
come familiar with all of these units.
EXERCISE 132. — WRITTEN
Reduce to units of higher denomination :
1. 145 pts. 8. I gi. to fraction of quart.
2. 424 gis. 9. 1 qt. 1 pt. to decimal of gallon.
3. 380 pts. 10. 786 qts. to bbl. and fraction.
4. 1984 gis. 11. 6.7 gis. to decimal of gallon.
5. 1286 qts. 12. 29 cu. ft. to gallons.
6. 1254 pts. 13. 1728 pts. to hogsheads and fraction.
7. j9j deciliter to 14. 7 gals, to decimal of barrel.
liter.
166
222.
PRACTICAL ARITHMETIC
MEASURES OF WEIGHT
Avoirdupois Weight
16 drams (drs.) = 1 ounce (oz.)
16 ounces = 1 pound (lb.)
100 pounds = 1 hundredweight (cwt.)
20 hundredweight .= 2000 lbs. = 1 ton (T.)
1 avoirdupois pound = 7000 grains.
A cubic foot of water weighs 62.42 pounds.
The ton of 2000 lbs. is sometimes called a Short Ton, the
Long Ton being 2240 lbs. The long ton is used in measuring
coal at the mines.
500 cu. ft. of hay in loads, or
400 cu. ft. of hay in the mow
= 1 ton (approximately)
223.
METRIC UNITS OF WEIGHT
Kilo
im^
10 milligrams (nig.) = 1 centigram (eg.)
10 centigrams = 1 decigram (dg.)
10 decigrams = 1 gram (g.)
1000 grams = 1 kilogram (kilo or k.)
DENOMINATE NUMBERS 167
A gram weighs 15.432 grains ; it equals the weight of a
cubic centimeter of water.
1 pound = .45+ kilogram.
EXERCISE 133. — DEAL
1. How many ounces are there in 6 lbs. ? In 9 lbs.?
2. How many ounces are there in 5 cwts. ? In 7 cwts. ?
3. How many ounces are there in T. ? In 8 cwts. ?
4. How many pounds are there in 44 ozs.? In 100 cwts.?
5. How many tons are there in 60 cwts. ? In 100 cwts.?
BXBECISE 134. — WRITTEN
Reduce :
1. 3 cwts. 21 lbs. 6 ozs. to ounces. .
2. 10 T. 50 cwts. 15 lbs. to pounds.
3. I T. to pounds. 4. 5 g. 10 dg. to milligrams.
5. .375 T. to hundredweights.
6. ^Q T. to pounds.
7. 17 cwts. 15 lbs. 3 ozs. to ounces.
8. 2 T. 10 cwts. 25 lbs. to pounds.
9. .675 cwt. to pounds. 11. 21 lbs. 6 ozs. to grams.
10. .5 T. to hundredweights. 12. 26 g. to grains.
EXERCISE 135. — WRITTEN
Reduce to units of higher denominations :
1. 2500 ozs. 4. 5275 lbs. 7. 15,674 cwts.
2. 9546 ozs. 5. 4250 ozs. 8. 2654 grs. to lbs.
3. 1056 mg. 6. 5420 mg. 9. 5460 grs. to g.
168 PRACTICAL ARITHMETIC
10. Tie up packages of sand or grain of various weights,
and practise lifting them until you can judge weight with
some accuracy.
224. MEASURES OF TIME
60 seconds (sees.) = 1 minute (min.)
60 minutes = 1 hour (1 hr.)
24 hours = 1 day (da.)
7 days = 1 week (wk.)
12 months (mos.) = 1 year (yr.)
" Thirty days hath September,
April, June, and November,
All the others thirtyone
Except the second month alone,
Which hath but four and twentyfour
Till Leap Year gives it one day more."
The common year has 365 days, or 52 weeks and 1 day.
The leap year has 366 days. Years whose numbers are divisi
ble by 4 and not by 100, or by 400, are leap years.
A decade consists of 10 years ; a score, 20 years, and a cen
tury, 100 years.
EXERCISE 136. — ORAL
1. How many seconds are there in 3 mins.? In 10
mins.? In 15 mins. ? In 15 mins. 20 sees. ?
DENOMINATE NUMBERS 169
2. How many minutes are there in 2 hrs. ? In 6 hrs. ?
In 6J hrs. ? In 8^ hrs. ?
3. Howmany hours inS days? In 4i days? In 5 days?
4. How many days in 2 wks. ? In 5 wks. ? In 5 wks. ?
In 5 days?
5. How many days in 2 years? In 2 leap years?
EXERCISE 137. — WRITTEN
Reduce to units of lower Reduce to units of higher
denomination : denomination :
1. 6 hrs. 25 mins. 10 sees. 5. 2460 mins.
2. 10 days 5 hrs. 12 mins. 6. 142.2 hrs.
3. f day. 7. 89,764 sees.
4. 867 days. 8.  hr. to fraction of a da}'.
9. Practise estimating a second and a minute until
you can do so with some accuracy.
ANGLE MEASURE
225. A Circle is a plain surface bounded by a curved
line, the circumference, every point of which is equally
distant from a point within, called the Centre.
226. A straight line drawn through the centre of a
circle from one point in the circumference to another is
called the Diameter of the circle.
227. A Section of the circumference is called an Arc.
228. The diameter exactly divides the circle.
229. Onehalf the diameter is the Radius.
170
PRACTICAL ARITHMETIC
A
C
230. The difference in the direction of two lines that
meet is called an Angle.
231. A line which cuts an
other line making with it two
equal adjacent angles is said to
be Perpendicular to the other
line, e.g., the line AB is per B
pendicular to the line CB. The two angles made by the
line AB with the line CB are equal. Each of the angles
so made is called a Right Angle, e.g., the angles ABQ
and ABB are right angles.
232. The circumference
of a circle is measured in
Degrees. There are 360
degrees in a whole circum
^08ti— J ^ I Lo a ference. Degrees are not
of uniform length, but
vary with the circumference of the circle, e.g., the cir
cumference of the circle A measures 360 degrees as does
also the circumference of the circle B. Degrees are
divided into minutes and seconds.
233. TABLE or ANGLE MEASURE
60 seconds (") = 1 minute (')
60 minutes = 1 degree (°)
360 degrees = 1 circle.
EXERCISE 138. — ORAL
1. How many degrees are there in a circumference?
cir
In a half circumference?
cumference?
In I circumference?
In
DENOMINATE NUMBERS 171
2. If one line is drawn perpendicular to another, how
many degrees are there in each angle so made? How
many degrees in the two angles?
3. How many degrees are there in a right angle?
What is the difference between a right angle and an
angle of 60°? Of 90°? Of 100°? Of 120°?
4. What part of aright angle is 45°? 30°?
5. How many minutes in 6°? 8°? 24°?
6. How many seconds in 3'? 5'? 1°, 6', 6"?
EXERCISE 139. — ■WRITTEN
1. Draw an angle of 45 degrees ; 90 degrees ; 120 de
grees ; 180 degrees.
2. Reduce 40° 20' 20 " to seconds.
3. Reduce 43,200" to degrees. Draw an angle having
the number of degrees found.
4. Determine the number of degrees of " pitch " of
some roof that is accessible to you.
234. COUNTING
12 things = 1 dozen (doz.)
12 dozen = 1 gross (gr.)
12 gross = 1 great gross (g. gr.)
20 things = 1 score.
24 sheets of paper = 1 quire.
EXERCISE 140. — ORAL
1. A crate contains 492 eggs; how many dozen?
2. A single card contains 24 hooks and eyes. How
many gross are there on 48 cards?
3. How many years are three score and ten?
172 PRACTICAL ARITHMETIC
ADDITION AND SUBTRACTION
EXERCISE 141. — ORAL
Add:
1. 12 ft. 1 in. 2. 7 days 1 hr. 3. 12 lbs. 5 ozs.
13 ft. 9 ins . 8 days 12 hrs. 16 lbs. 9 ozs.
4. 120° 14' 12" 5. 20 rds. 4 yds. 1 ft. in.
10° 15' 30'' 10 rds. 1 yds. ft. 6 ins .
235.
16 lbs. 4 ozs. Addition and subtraction of com
25 lbs. 3 ozs. pound numbers is performed in the
18 lbs 12 07S same manner as with simple numbers,
FQ 1, ^ — T^ ; the only difference being that units of
oy IDS. iv ozs.^ or Til • 1
60 lbs. and 3 ozs.
compound numbers have a varying scale
of value, while with simple numbers the
scale is uniformly ten.
Add the numbers of eacb denomination separately, then
reduce the sums to the highest units possible.
EXERCISE 142. — WRITTEN
Add:
1. 12 lbs. 12 ozs. 4. 13 bus. 3 pks. 5 qts.
18 lbs. 10 ozs. 25 bus. 4 pks. 4 qts.
9 bus. 1 pk. 1 qt. 1 pt.
54 bus. 2 pks. 1 qt.
5. 18 gals. 1 qt. 1 pt.
68 gals. 1 qt.
25 gals. 3 qts. 1 pt.
13 gals. 1 qt. 1 pt.
2. 73 rds.
5 ft.
18 rds.
9 ft.
3. 12 lbs.
8 ozs.
25 lbs.
6 ozs.
DENOMINATE NUMBERS 173
BXBRCISB 143. — OBAIi
Subtract :
1. 15 ft. 9 ins. 3. 150° 16' 16"
9 ft. 4 ins. 100° 8' 10''
2. 27 T. 12 cwts. 11 lbs. 4. 3 yds. 31 ft.
1 T. 10 cwts. 10 lbs . 1 yd. 2 ft. 6 ins.
236. Subtract:
45 bus. 3 pks. 1 qt. pt. In the first righthand col
39 bus. 1 pk. qt. 1 pt. ^^^ we have 1 pt. to be sub
tracted from pt., which is
impossible. Taking 1 qt. from the quarts column and chang
ing it to pints gives 2 pts. 1 pt. subtracted from 2 pts. leaves
1 pt. Completing the subtraction the remainder is 6 bus.
2 pks. 1 pt.
Subtract as simple numbers. Increase when necessary
the number representing any denomination by taking one
unit from the next higher denomination.
BXBRCISB 144.  WRITTBN
Subtract :
1. 12 hhds. 10 gals. 3 qts. 4. 76 mis. 8f rds.
10 hhds. 50 gals. 1 qt. 19 mis. 9 rds.
2. 50 yds. 8 ins.
30 yds. 10 ins .
3. 25 rds. 3 yds. li ft.
10 rds. 1 yd. 2 ft.
237. DIFFERENCE BETWEEN TWO DATES
Find the difference between the dates March 6 and
July 4:
5.
12 m. 6 dm. 4 cm.
9 m. 9 dm. 8 cm.
6.
14 1. 8 dl. 5 cl.
5 1. 9 dl. 8 cl.
Yes.
Mos.
Days
1871
4
1
1815
7
15
174 PRACTICAL ARITHMETIC
The number of days in March after the 6th is 25 ■
The number of days in April is 30
The number of days in May is 31
The number of days in June is 30
The number of days in July before the 5th is 4
Total is 120
The difference of dates is 120 days.
To find the difference between two short dates, the
exact number of days in each month is taken.
238. Find the difference between the dates April 1,
1871 and July 15, 1845 :
Write the dates as in subtraction of
compound numbers, representing the
number of years, months, and days in
columns.
25 8 16
To find the difference between two dates of longer period,
that is, covering a period of more than a year, 30 days are
considered a month.
BXEBCISE 145, — ■WRITTEN
1. Find the difference of dates between July 4, 1776
and May 20, 1776.
2. Find the difference between April 19, 1774 and
October 12, 1492.
MULTIPLICATION
BXEBCISE 148. ORAL
Multiply :
1. 4 ft. 2 ins. by 2. 3. 12 lbs. 4 ozs. by '3.
2. 6 yds. 1 ft. 3 ins. by 2. 4. 49 pks. 2 qts. by 4.
DENOMINATE NUMBERS 175
239. Multiply 30 gals. 3 qts. 1 pt. by 6.
30 gals. 3 Qts. 1 pt. 6 X 1 pt. = 6 pts., or 3 qts. 6 x 3
qts., or 5 gals, and 1 qt. 6 x 30 gals.
185 gals. Iqt. Opt. ^^g^ ^^^^^ ^gO g^jg. + g g^^l3_ ^
185 gals. The product is 185 gals, and 1 qt.
Multiply separately the number representing each de
nomination. Reduce each product to units of the next
higher denomination and add to the product of that
denomination.
EXERCISE 147. — WRITTEN
Multiply :
1. 15 bus. 3 pks. 2 qts. by 8.
2. 56 ft. 61 ins. by 96.
3. 4 T. 25 cwts. 10 lbs. 9 ozs. by 28.
4. 46 mis. 3 ft. by 27.
5. 121 lbs. 4 ozs. by 34.
6. 46 gals. 2 qts. 1 pt. by 12.5.
7. 15 m. 6 dm. 8 cm. by 25.
8. 25 1. 3 dl. by .34.
9. 18 g. 4 dg. 6 eg. by 12.
10. 2 sq. mis. 80 sq. rds. 6 sq. yds. by .245.
11. 3 sees. 50 A. 7 sq. chs. by 364.
12. 15 cu. yds. 19 cu. ft. 108 cu. ins. by 2.75.
DIVISION
EXERCISE 148. — ORAL
Divide:
1. 12 ft. 6 ins. by 2. 3. 54 gals. 3 qts. by 3.
2. 35 lbs. 14 ozs. by 7. 4. 60 ft. by 12 ft.
176 PRACTICAL ARITHMETIC
5. 108 yds. 3 ft. 6 ins. by 9.
6. 54° 20' 36" by 6.
7. 25 hrs. 30 mins. 20 sees, by 10.
8. 144 sq. ft. 72 sq. ins. by 9.
9. 48 bus. 2 pks. 3 qts. by 3.
10. 174 yds. by 6 yds.
11. 136 yds. 2 ft. by 5.
12. 154 bus. 3 pks. 2 qts. by 4.
240. Divide 25 gals. 2 qts. 1 pt. by 4.
4 )25 gals. 2 qts. 1 pt. \ of 25 gals. = 6 gals, with
6 gals. 1 qt. 1\ pts. 1 remainder. 1 gal. = 4 qts. 4
qts. + 2 qts. = 6 qts. ^ of 6
qts. = 1 qt. and 2 qts. remainder. 2 qts. = 4 pts. 1 pt. + 4
pts. = 5 pts. ^ oi 5 pts. = 1^ pts.
Divide separately the number representing each denomi
nation. If there is a remainder, reduce it to the next
lower denomination and add to the number representing
that denomination.
241. Divide 18 lbs. 6 ozs. by 8 lbs. 3 ozs.
■ 18 lbs. = 256 ozs. + 6 ozs. = 262 ozs.
8 lbs. = 128 ozs. + 3 ozs. = 131 ozs.
262 ozs. ^ 131 ozs. = 2.
In order to divide one compound number by another,
both compound numbers must be reduced to simple num
bers of the same denomination.
242. Observe that in division of denominate numbers
when the divisor and dividend are concrete, the quotient
is abstract, e.ff., 131 ozs. is contained in 262 ozs. two times.
DENOMINATE NUMBERS
177
EXERCISE 149. — WRITTEN
Divide :
1. 245 mis. 120 rds. by 8.
2. 945 ft. 6 ins. by 12.
3. 2 lbs. 7 ozs. 14 grs. by 5.
4. 12 yrs. 6 mos. 5 days by 2 yrs. 4 mos. 5 days.
5. 32 gals. 2 qts. by 5 gals. 1 qt. 1 pt.
6.
7.
8.
9.
10.
10 1. 3 dl. 5 cl. by 2 1. 1 dl.
15 m. 8 dm. 6 cm. by 10 m. 2 cm.
25 g. 3 dg. by 12 g.
6984 ins. by 127.
8 A. 25 sq. rds. 3 sq. ft. by 9.5.
11. 1155 sq. rds. 10 sq. Is. by 2.5.
EXERCISE 150. — WRITTEN
a
h
c
1
64
23
24
2
72
16
19
3
93
19
16
4
11
78
12
S
2
72
19
6
98
18
78
7
16
21
42
8
18
42
73
9
27
18
64
10
24
7
77
1. If the numbers in line 1 a, J, c, represent respec
tively tons, hundredweight, and pounds, how many pounds
178 PRACTICAL ARITHMETIC
and how many hundredweight do they equal ? How
many tons and fraction?
210. Solve problems based on lines 2 to 10 as in
No. 1.
1120. If the numbers in columns «, b, c represent
respectively miles, rods, and yards, how many yards do
the numbers in each line 'equal? Rods and fraction?
Miles and fraction ?
21. If the numbers in line 1 a, b. c represent yards,
feet, and inches, respectively, how many inches do they
equal ? Feet and fraction ? Yards and fraction ?
2230. Solve each other line as in No. 21.
31. If the numbers in line 1 a, 5, c represent bushels,
pecks, and quarts, respectively, how many quarts do they
equal? Pecks and fraction ? Bushels and fraction?
3240. Solve each other line as in No. 31.
41. If the numbers in line 1 a, 5, e represent respec
tively centuries, years, and days, how many days do they
equal? Years and fraction? Centuries and fraction?
4250. Solve each other line as in No. 41.
51. If the numbers in line 1 a, J, c represent respec
tively hours, minutes, and seconds, how many seconds do
they equal ? Minutes and fraction ? Hours and fraction ?
5260. Solve each other line as in No. 51.
61. If the numbers in line 1 a, i, e represent respec
tively barrels, gallons, and quarts, how many quarts do
they equal? How many barrels and fraction?
6270. Solve each other line as in No. 61.
REVIEW PROBLEMS 179
71. If the numbers in line 1 a, b, e represent respec
tively square miles, acres, and square rods, how many
square rods do they equal ? Acres and fraction ? Square
miles and fraction ?
7280. Solve eacli other line as in No. 71.
8190. If each number in a represents grams, how many
pounds does each equal ?
91100. If each number in a represents meters, how
many inches does each equal ? How many yards ?
101110. If each number in a represents yards, how
many meters does each equal ?
111120. If each number in a represents pounds, how
many grams does each equal?
EXERCISE 151. — WRITTEN
Review Peoblems
1. If a man bu3's 373 lbs. of seed corn at $2 a bushel,
and plants 4 qts. to the acre, how many acres can be planted
with this seed, and how much will it cost to the acre ?
2. A dairyman producing 80 gals, of milk delivers ^ of
it in pint bottles, ^ of it in quart bottles. How many
bottles of each size are needed for one delivery ?
3. Two bushels of peaches are preserved and kept in
quart cans. Allowing J for shrinkage, how many cans
are needed?
4. A barrel of vinegar is sold, half in pints, half in
quarts. How many of each are sold?
180 PRACTICAL ARITHMETIC
5. Twenty bushels of garden peas are sold in quart
boxes. How many boxes are required?
6. In building a fence 80 rds. long, 4 boards high, with
boards 16 ft. long and posts 8 ft. apart, using 2 nails at
every point where a board is nailed to a post, how many
pounds of tenpenny nails will be needed, allowing 69 nails
to the pound ?
7. Wind with a velocity of 1 mile an hour (barely
observable) exerts a pressure of .005 lb. to the square
foot. What is the pressure of such a wind on ' a tight
board fence 1 rd. long, 5 ft. high ?
8. With a velocity of 8 miles (a pleasant wind) the
pressure is .32 lb. to the square foot. What is the press
ure on the fence mentioned in the last problem ?
9. With a velocity of 80 mis. an hour (a hurricane) '
the pressure is 32 lbs. to the square foot. What is the
pressure on the fence of problem 7?
10. If each tea bush yields 3 ozs. of tea each year, how
many bushes will be required to yield 18 lbs.?
11. If a pound of tea will make 400 cups, how many
tea plants will be needed to furnish 1 cup each day for a
year for each member of a family of nine ?
12. A square piece of land is 81 chs. on a side. What
is its perimeter?
13. A city block is 625 ft. on a side. How many chains
on one side ? How many square chains in a block?
14. How many acres are there in a rectangular enclos
ure 1 mi. long and I mi. wide?
REVIEW PROBLEMS 181
15. A rectangular farm contains 100 A. It is 80 rds.
on one side. How long is the other side?
16. Reduce 2482 sq. rds. to square chains.
17. At i960 a mile, what is the cost of making a road
4 mis. and 43 chs. long?
18. At $4.90 a square chain, what is the cost of 27 A.
43 sq. rds. of land?
19. The pressure of the air is 14 lbs. per square inch.
What is the air pressure on a table top 38" x 76"?
20. The longest day in Stockholm is 18 hrs. 30 mins.,
in London 16 hrs. 32 mins., in Paris 16 hrs., in Boston 15
hrs. 16 mins., in Washington 14 hrs. 62 mins. How much
longer is the longest Stockholm day than that of each of
the other cities?
21. The shortest day in Stockholm is 5 hrs. 54 mins.,
London 7 hrs. 44 mins., Paris 8 hrs. 10 mins., Boston 8
hrs. 58 mins. How much shorter is the shortest Stockholm
day than that of each of the other cities ?
22. How many cubic feet of dirt must be moved in
digging a ditch 18" wide, 2' 6" deep, 79 rds. long? How
many cubic yards?
23. What will it cost to excavate a cellar 18' long,
14' wide and 6' 3" deep at f.42 per cubic yard of dirt?
24. How many bushels will a bin hold if it is 5' wide,
7' long, 4' deep ?
25. A wagon bed 10', 8" long, 3' wide, and 19" deep
holds how many bushels? How many bushels will it
hold with a 6in. topbox added? How much with a 9in.
topbox?
182
PRACTICAL ARITHMETIC
26. If a bushel of soft coal weighs 80 lbs., how deep
must such a wagon bed be to contain 1 ton?
27. How deep a bin 7' long, 4' 7" wide must be built
to hold 3 tons of soft coal?
28. A solid piece of timber 8" x 9" x 16' contains how
many cubic feet ?
29. A cubic foot of seasoned mahogany weighs 65.1 lbs.,
of hickory 58.81 lbs., of pitch pine 47.44 lbs., of cedar
42.7 lbs., of hemlock 25.53 lbs., of white pine 21.72 lbs.,
of ash 39.19 lbs. What is the weight of a piece of each
of the above woods of the size mentioned in problem 28?
30. If a cubic foot of soil weighs 87 lbs., what is the
weight of the soil of 1 sq. yd., 1 ft. in depth? Of 1 sq.
rd., 1 ft. in depth? Of 1 A., 1 ft. in depth?
31. The following table shows the number of pounds of
the three important elements of plant food in a ton of
four different soils :
Nitrogen
Phosphoric
Acid
Potash
Loam
Clay
Drift
Sand
7 lbs.
3 lbs.
3 lbs.
lib.
3 lbs.
3 lbs.
i\h.
2 lbs.
8 lbs.
15 lbs.
6 lbs.
5 lbs.
How much of each of these plant foods is there in an
acre 1 ft. in depth?
32. In each bushel of the following crops there is the
amount of the three chief plant foods indicated by the
following table :
REVIEW PROBLEMS
183
Plant Food
NiTROGKN
PlIOSlMIORKl AOID
Potash
Wheat
20 ozs.
8 ozs.
5 OZS.
Rye
17 ozs.
9 (izs.
5 OZS.
Shelled corn
14 ozs.
5 ozs.
3 OZS.
Barley
12 ozs.
6 ozs.
4 OZS.
Oats
10 ozs.
3 ozs.
2 ozs.
Potatoes
3 ozs.
1 oz.
4 ozs.
How many pounds of each of these is taken from the
field with 65 bus. of crop?
33. What fraction of an acre is a garden 8 rds. 4 ft.
long, 6 rds. 1 yd. wide?
34. How many square feet are there in a floor 15 ft.
3 ins. long by 12 ft. 9 ins. wide ?
35. How many square j^ards are there in a ceiling
27 ft. 4 ins. long, 16 ft. 7 ins. wide?
36. If corn is planted in rows 3 ft. 6 ins. apart and the
pla,nts are 13 ins. apart in the row, how many corn plants
will there be in a field 30 rds. 3 yds. 2 ft. 9 ins. long by
27 rds. 4 yds. 1 ft. 8 ins. wide?
37. How many square feet are allotted to each plant
in problem 36?
38. In 1856 the steamship Persia crossed the ocean
between Queenstown and New York, a distance of 2800
miles, in 9 days 1 hr. 45 mins. What was the average
distance per day ?
39. In 1894 the Lucania making the same trip crossed
in 5 days 7 hrs. 23 mins. What was the average rate per
hour? What was the gain in time of crossing over that of
the Persia?
184 PRACTICAL ARITHMETIC
40. Through how many degrees does the hour hand
of a clock pass from 6 A.M. to 3 p.m.? From 12 M. to
3.30 P.M. ? From 9 p.m. to 10 p.m.?
41. Through how many degrees does the minute hand
pass from 6.15 to 6.30? From 5.45 to 6?
THREAD AND CLOTH
42. A calico cloth 27 ins. wide is made with 64 length
wise threads (warp) to the inch. How many linear yards
of warp are there in a piece of calico 3 yds. long? In a
square yard of calico?
43. Cloth made with 84 crosswise (filling) threads to
the inch has how many yards of filling in 3 yds. of cloth
28 ins. wide?
44. A standard wrapping reel is 4 ft. 6 ins. in circum
ference. How many times will it turn in wrapping 840
yds. of thread?
45. A cloth 30 ins. wide with 80 " ends " (lengthwise
threads) to the inch has how many ends?
46. A cloth 30 ins. wide consists of alternate close and
open stripes. The close stripes,  in. wide, have 60 ends
in ^ in. The open stripes have 96 ends in a 1in. stripe.
What is the average number of ends per inch ? What is
the total number in cloth 28 ins. wide ?
47. Cloth made with 84 threads of filling to the inch
has how many to the yard ?
48. A gentleman's cotton handkerchief 18" X 18"
weighs I oz. How many can be made from 450 lbs. of
cotton, not allowing for waste?
REVIEW PROBLEMS 185
49. A lady's handkerchief is onefourth smaller. How
many can be made from 450 lbs. of cotton ?
50. A bale of cotton weighs 500 lbs. Bagging and
ties weigh 25 lbs. This cotton loses f^ during the
process of making the warp and filling threads. How
many handkerchiefs can be made from the bale if a
handkerchief weighs  of an ounce?
51. The warp threads in a handkerchief weigh .43 oz.
The filling threads in same handkerchief weigh .32 oz.
How many pounds of cotton will be used for warp threads
and how many pounds for filling threads ?
52. Wilbur Wright of Ohio is reported to have flown
on his aeroplane, in France, on Aug. 8, 1908, a distance
of 3 kilometers in 1 min. 46 sees. What was the rate in
miles per minute?
53. A man purchases 2520 lbs. of corn and feeds 2 qts.
and 1 pt. three times a day to his horse. How long will
the corn last? (56 lbs. of corn = 1 bu.}
54. A dairyman produces 50 gals, of milk. Half of
this is delivered to customers taking 1 pt. each, ^^ to cus
tomers taking 2 qts. each, the remainder to customers
taking 1 qt. each. How many customers are there?
55. A quart of beans weighs 80 ozs. How many pounds
are there in a bushel of beans ?
56. For a grain bin 6'xl5^'x5' how much carbon
disulfide will be required to kill weevils, allowing 1 lb.
to every 1000 cu. ft. of space? What will this cost at
29^ per pound ?
MEASUREMENTS
243. A plane figure having four straight sides is called
a Quadrilateral.
244. A plane figure having four straight sides and its
opposite sides parallel is called a Parallelogram.
A QUADRILATEBAL A PARALLELOGRAM A TRAPEZOID
245. A quadrilateral which has only one pair of its
opposite sides parallel is called a Trapezoid.
246. A parallelogram the angles of which are right
angles is called a Rectangle.
247. A plane figure bounded bj three straight lines and
having three angles is called a Triangle.
A Rectangle A Triangle Base and Altitude
248. The line upon which a figure seems to rest is
called the Base.
186
MEASURKMENTS 187
249. The perpendicular distance between the base and
the highest opposite point is called the Altitude.
250. The straight lines joining the opposite angles of a
parallelogram are called its Diagonals.
Diagonals Equilateral Triangle Isosceles Triangle
251. Polygon is the name given to any figure bounded
by straight lines.
252. The Perimeter of a polygon is the distance
around it.
253. A triangle whose three sides are equal, is called an
Equilateral Triangle.
254. A triangle, two of whose sides are equal, is called
an Isosceles Triangle.
Right Triangle Acute Angle Obtuse Angle
255. A triangle having one right angle is called a
Right Triangle.
256. An angle less than a right angle is called an Acute
Angle.
188
PRACTICAL ARITHMETIC
257. An angle greater than a right angle is called an
Obtuse Angle.
EXERCISE 152.— ORAL
1. What is the perimeter of a square lOJ ft. on a
side?
2. What is the perimeter of a rectangle two sides of
which are 12^ ft. and 6^ ft. respectively?
3. What is the perimeter of a quadrilateral whose sides
are 10, 12, 6, and 8 inches?
4. What is the perimeter of an equilateral triangle 25
ft. on a side ?
5. What is the perimeter of a triangle whose sides are
21ft., 3ft., and 6 ft.?
6. A triangle has a perimeter of 120 ins. ; one side
measures 30 ins., another IJ times this number of inches.
What is the length of the third side ?
7. The base of an isosceles triangle is 12 ft. ; the perim
eter is 42 ft. What is the length of each of the equal
sides ?
BXEECISE 153. — ORAL
1. How many square inches are there in a rectangle
10 ins. long and 1 in. wide?
How many rows of square
inches are there in the
figure ?
2. How many rows of
square inches are there in a
rectangle 10 ins. long and 2 ins. wide ? How many square
inches are there in a row ? In two rows ? How many
MEASUREMENTS
189
rows of square inches are there in a rectangle 10 ins. long
and 3 ins. wide? How many square inches in a row?
In three rows ?
3. How many square inches are there in a rectangle
10 ins. long, 5 ins. wide ? In a rectangle 10 ins. long, 10
ins. wide ?
258. In finding the area or surface of a rectangle, we
must think of the surface in rows of square inches, which
will give the number of times the square inches in one
row is to be taken, e.g., if a square measures 10 ins., there
are, as may be seen, 10 rows of square inches of 10 sq. ins.
each, or 100 sq. ins.
Hence, the area of a rectangle equals the product of the
length by the width.
EXERCISE 154. —ORAL
State the areas of the following rectangles :
7. 7 ins. X 9 ins.
5. 9 ft. X 61 ft. 8. 4 ft. X 6 ins.
3. 10 ft. X 4 ft. 6. 3 m. X 1^ m. 9. 1 yd. x 10 ft.
1. 5 ins. X 10 ins. 4. 8 ins. x 8 ins.
2. 4 ft. X 6i ft.
EXERCISE 155.
1. What is the re
ORAL
lation of triangle a to
rectangle A ?
2. What is the rela
tion of triangle h to
rectangle B ?
3. What is the relation of triangle c to (7?
From the above, what is true of the area of a triangle ?
190 PRACTICAL ARITHMETIC
4. Give the areas of the following triangles : base 4 ins.,
height 6 ins. ; base 9 ins., height 12 ins. ; base 3 ft., height
4ft.; base 3ft., height 3 ft.
259. The area of a triangle equals J the product of the
base by the altitude.
EXERCISE 156. — WRITTEN
State the areas of the following triangles :
1. Base 45 ft., altitude 30 ft.
■ 2. Base 16 ft., altitude 12 ft.
3. Base 87 ft., altitude 28 ft.
4. Base 40 ft., altitude 26 ft.
5. Base 67 ft., altitude 55 ft.
260. If you CLit off the triangle a from the parallelo
gram X and place the line ho so that
X \ ^^ exactly coincides with de, what kind
^e of figure have you ? What is true
of the area of the first figure as compared with the
second ?
EXERCISE 157. — ORAL
Give the area of the following parallelograms :
1. Base 12 ins., height 9 ins.
2. Base 14 ins., height 6 ins.
3. Base 18 ins., height 12 ins.
4. Base 15 m., height 6 m.
5. Base 17 ins., height 17 ins.
261. The area of a parallelogram is equal to the prod
uct of the base by the height.
MEASUREMENTS 191
EXERCISE 158. — "WBITTEN
Find the areas of the following parallelograms :
1. Base 18.4 ins., height 9.2 ins.
2. Base 17 ft., height 16.5 ins.
3. Base 34 m., height 20 cm.
4. Base 16 ft. 6 ins., height 12 ft. 4 ins.
5. Base 171.4 ft., height 34.9 ft.
6. Base 3.75 ft., height 4.25 ft.
7. Base 35 m., height 9 cm.
8. Base 345.2 rds., height 6 yds.
fj 9
X
d^
262. If you cut the trapezoid X through the line ef
and place the two sections so that the lines cf and df coin
cide, what kind of figure have you ?
Is the area of the second figure the same as the area of
XI How long is the figure made froin the trapezoid?
How high is it? To find the area of a trapezoid, what
dimensions must be given?
EXERCISE 159. — ORAL
Find the area of the following trapezoids :
1. The parallel sides are 10 ins. and 12 ins., height 6 ins.
2. The parallel sides are 6 ins. and 4 ins., height 4 ins.
3. The parallel sides are 12 ins. and 9 ins., height 8 ins.
192 PKACTICAL ARITHMETIC
4. The parallel sides are 16 ins. and 12 ins., height 9 ins.
263. The area of a trapezoid is equal to the sum of the
two parallel sides multiplied by J the height.
EXERCISE 160. — WRITTEN
Find the area of the following trapezoids :
1. Parallel sides, 96 rds. and 180 rds., height 108 rds.
2. Parallel sides, 45 rds. and 95 rds., height 64 rds.
3. Parallel sides, 120 ft. and 90 ft., height 40 ft.
4. Parallel sides, 60 m. and 90 m., height 44 m.
5. Parallel sides, 135.5 m. and 34 m., height 110 m.
6. Parallel sides, 62 yds. and 80 yds., height 112^ yds.
264. It has been found that the circumference of a
circle is approximately 8^ times the diameter.
Find the circumference of circles with the following
diameters :
1. 7 ins. 4. 21 ins. 7. 0.7 m. 10. 45 ft.
2. 14 ins. 5. 77 ins. 8. 14 cm. 11. 84 ft.
3. 49 ins. 6. 0.28 in. 9. 14.7 cm. 12. 105 ft.
265. To find the circumference of a circle, multiply the
diameter by 3^, or more accurately by 3.1416.
EXERCISE IBI. — WRITTEN
Find accurately the circumferences of the following cir
cles; also find their approximate circumferences. Note
the difference. in results by the two methods.
1. Diameter 16 ft. 4. Radius 9 ft.
2. Diameter 21 ft. 5. Radius 12 m.
3. Diameter 19 ft. 6. Radius 45 rds.
MEASUREMENTS
193
EXERCISE 162. —WRITTEN
Find the diameters of circles having the following
circumferences :
1. 426 ft. 4. 1400 rds. 7. 318 m.
2. 218 ft. 5. 1676 yds. 8. 164 m.
3. 670 ft. 6. 115.6 rds. 9. 92.45 ins.
266. If you divide the circle shown in the figure by
cutting the line ah, then cutting each half circle into many
small triangles by
cutting along
each radius, it
will be seen that
the many trian
gles joined, as is
suggested in figure y, will approximate in area a parallelo
gram with a length equal to onehalf the circumference
of the circle and a height equal to the radius of the circle.
The area of a circle is found by multiplying onehalf
the circumference by the radius, or by multiplying the
square of the radius by 3.1416.
EXERCISE 163. — WRITTEN
Find the areas of circles with a :
1. Circumference of 242 ft.
2. Circumference of 110 ins.
3. Circumference of 154 ft.
4. Circumference of 96 yds.
5. Circumference of 3.75 ft.
6. Circumference of 18 yds.
7. Diameter of 77 ft.
8. Diameter of 35 m.
9. Diameter of 49 rds.
10. Diameter of 146 ft.
11. Diameter of ^ ft.
12. Diameter of 54 rds.
194
PRACTICAL ARITHMETIC
EXERCISE 164. — WRITTEN
Find the areas of circles having a radiugr of :
1. 8 ins. 3. 4 ft. 5. 9 m. 7. 4 rds.
2. 10 ins. 4. 16 ft. 6. 8 m. 8. 5 rds.
SOLIDS
267. A solid has three dimensions, • — Width, Length,
and Thickness.
268. A solid that has six rectangular sides or faces is
called a Rectangular Solid.
269. A solid that has six square faces is called a Cube.
270. A cubic inch is a solid, the faces of which are each
one inch square. What is a cubic foot? What is a
cubic yard?
BXEECISE 165. — ORAL
1. In a solid 12 ins. long, 1 in. wide, and 1 in. thick,
how many rows of cubic inches are there? How many
cubic inches are there ?
SI
Figure 1
I'igure 2
Figure 3 Figure 4
2. In a solid 12 ins. long, 2 ins. wide, 2 ins. thick, how
many layers of cubic inches are there ?. How many rows
MEASUREMENTS 195
of cubic inches are there in one layer ? How many cubic
inches are there in one layer ? How many cubic inches
are there in 2 layers ?
3. In a solid 12 ins. long, 12 ins. wide, 2 ins. thick, how
many layers are there ? How many rows are there in one
layer ?. How many cubic inches are there in one row ?
How many cubic inches are there in one layer ? How
many cubic inches are there in two layers ?
4. In a solid 12 ins. long, 12 ins. Avide, 12 ins. thick, how
many layers are there ? How many rows are there in a
layer? How many cubic inches are there in one row?
How many cubic inches are there in a layer ? How many
cubic inches are there in 12 layers ?
5. In a solid 4 ins. long, 4 ins. wide, and 4 ins. thick,
how many cubic inches are there in the length of the
solid? How many rows of cubic inches in the width?
How many cubic inches in each layer ? How many
layers in the solid? How many cubic inches in the solid?
In examples 6, 7, 8, answer all the questions made in
example 5.
6. In a solid 4 ins. by 5 ins. by 6 ins. ?
7. In a solid 2 ins. by 3 ins. by 12 ins. ?
8. In a solid 6 ins. by 8 ins. by 10 ins. ?
271. To find the volume of a cube, we must think of it
iis a solid having a length equal to the number of units in
its length and a width equal to that of the rows of cubic
units in its width, and a thickness equal to the number
of layers of cubic units in its thickness.
196
PRACTICAL ARITHMETIC
272. The volume of a rectangular solid equals the prod
uct of the three dimensions similarly expressed.
BXBBCISB 166. — WRITTEN
1. How many cubic inches are there in a box 16 ins.
long, 8 ins. wide, and 9 ins. deep?
2. How many cubic feet of masonry are there in a wall
120 ft. long, 2 ft. wide, and 4 ft. high?
3. How many gallons of water will a tank 5 ft. long,
4 J ft. wide, and 2^ ft. deep contain?
4. How many cubic yards of soil will be excavated in
making a ditch 12 ft. long, 2 ft. wide, and 3 ft. deep?
5. How many liters of air are there in a room IS^ m.
long, 9^ m. wide, and 3 m. high?
6. The box of a onehorse express wagon is T 2" by 3'
2" by 8". Find the capacity.
7. A twohorse farm wagon is 11' 6" x 3' x 10^".
Find the capacity.
THE CYLINDER
273. A cylinder is a solid
having for its ends two equal
circles joined by a uniformly
curved surface, called its lat
eral surface.
274. A cylinder may be
seen in the form of a rectangu
lar solid with a length equal
to the length of the cylinder, a width equal to onehalf
MEASUREMENTS
197
the circumference of the circular base, a thickness equal
to the radius of the circular base.
275. Hence, to find the volume of a cylinder, multiply
the number of square units in the base by the height.
vjy
276. The lateral surface of a cylinder
may be seen in the form of a rectangle,
with a length equal to the circumference
of the circular base and a width equal to
the height of the cylinder.
277. Hence, to find the area of the
lateral surface of a cylinder, multiply
the length of the circumference of the circular base by
the height of the cylinder.
EXERCISE 167.— WRITTEN
Find the volume of the following cylinders :
1. Area of base 126 sq. ins., height 16 ins.
2. Area of base 72 sq. ins., height 44.56 ins.
3. Area of base 81 sq. ins., height 16 fins.
4. Area of base 84 sq. ins., height 42 ins.
EXERCISE 168. — WRITTEN
Find the lateral surfaces of the following cylinders :
1. Circumference of base 40 ins., height 45 ins.
2. Circumference of base 24 ins., height 25 ins.
3. Circumference of base 66 ins., height 27 ins.
4. Circumference of base 14f ins. height 641 ins.
PEACTICAL MEASUREMENTS
PLASTERING, PAINTING, PAVING
278. In estimating labor of latliing, plastering, painting,
and paving, the square yard is taken as the unit. In mak
ing estimates for plastering, onehalf the areas of doors,
windows, and other openings is deducted from the total
area, and the result expressed to the nearest square yard.
EXERCISE 169. — ORAL
1. What is the perimeter of a room 12 ft. x 14 ft.?
2. If the room is 8 ft. high, what is the area of the
4 walls, making no allowance for the openings in them?
3. What is the area of the ceiling?
4. What will be the cost of plastering this room at
25 cts. a square yard?
EXERCISE 170. — WRITTEN
In this exercise exclude closets, and make no allowances
for windows or doors.
1. To plaster 100 sq. yds. of surface requires 1 barrel
of lime, 3 barrels of sand, and 1 bushel of hair. Find the
amount of these materials required to plaster each bed
room of the house shown in the two accompanying dia
grams, also living room, dining room, and kitchen.
2. What will it cost to lath and plaster the living
room, kitchen, and dining room of tlie lower floor with
pulp plaster at a cost of 60 cts. a square yard ?
198
PRACTICAL MEASUREMENTS
199
3. To lath and plaster the i bedrooms of the upper
floor with ordinary plastering will cost 25 cts. a square
yard. Estimate the cost.
Pitch 10 ft.
4. Estimate the number of gallons of paint required
to give two outside coats to a house 32 ft. long, 32 ft.
wide, 18 ft. high, the gable ends of the roof forming
700 sq. ft. of surface. One gallon of paint covers
300 sq. ft. of surface, 2 coats.
200
PRACTICAL AEITHMETIC
5. At f 1.25 a gallon, what would be the cost of paint
ing this house?
6. It requires 1.16 barrels of cement, 0.44 cu. yd. of
sand, 0.88 cu. yd. of gravel to make a cubic yard of con
Pitch 8 ft.
Crete. How much cement will it require to concrete the
basement floor of a house, 21 ft. x 18 ft., to a depth of
3 ins. ?
7. If 1 cubic yard of concrete costs $3.31, what will
be the cost of material in the last problem?
PRACTICAL MEASUREMENTS
201
CARPETING
279. Carpets are usually 1 yd. or  yd. wide.
Brussels and velvet carpeting is  yd. wide and is
usually sold with a border varying from 22 to 27 ins. in
width. Ingrain carpets are 1 yd. wide. Matting is
usually 1 yd. wide. Linoleum and oilcloth are sold by
the square yard. The short loops of wool seen in
Brussels carpet are made by the use of wires over which
the wool is thrown in weaving. The number of wires
(usually between 5 and 10 to the inch) used in weaving
varies with the price of the carpet and the fineness of the
wool.
EXERCISE 171. — OBAIj
1. How many loops of wool are there in a piece of 5wire
Brussels carpet  yd. wide and 5 ins. long?
2. How many loops of wool in an 8wire Brussels car
pet I yd. wide and 5 ins. long ?
280. In determining the number of yards of carpet
required for a floor, the
number of strips should be
found, the fractional part
of a strip being regarded as
a full strip. Often it is
necessary to allow for match
ing and shrinking; 3 inches
are also allowed on each end
of each strip for turning in. Borders are fitted at each
corner, hence the perimeter of the room indicates the
length of border required ; see figure.
I I
202 PRACTICAL ARITHMETIC
If a floor is 12 ft. wide and the carpet f yd. wide, it will
require 5^ strips. Since  strip cannot be bought, 6 strips
must be bought. If the room is 18 ft. long, it will require 6
strips 18 ft. long, or 36 yds. If 9 ins. must be allowed for
matching the figure, 9 ins. must be added to each strip, except
ing one, since it is not necessary to allow for matching the
first strip. Three inches additional must be allowed for turn
ing under. If a 22in. border is used, only 3^ strips of carpet
are actually used, but 4 strips must be bought.
BXBBCISE 172. — ORAL
1. How many strips of f yd. carpeting are required to
carpet a room 12^ ft. wide?
2. The room is 14 ft. long. How many yards, without
border, are required, allowing 12 ins. for matching?
3. A room is 12 ft. x 18 ft. How many strips of  yd.
carpeting are required?
4. How many yards allowing ^ yd. for matching?
EXERCISE 173. — WRITTEN
Find how many yards of carpet are needed for the fol
lowing rooms :
1. 15 ft. X 18 ft. 6 ins.: use 27in. carpet, allow 9 ins.
for matching and use border I yd. wide.
2. 16' X 18' : use carpet 1 yd. wide, and allow ^ yd.
for matching, turning in, and shrinkage.
3. The living room and hall of the house shown by
diagram on page 199 : use a 10wire body Brussels carpet
1^ yd. wide, with 22ins. border, which requires 9 ins. for
matching and turning in, and costs 11.50 a yard, with
10 cts. a yard extra for making.
PRACTICAL MEASUREMENTS 203
4. The dining room is covered with an Axminster rug,
8^ ft. X 10^ ft. at $1 a square yard. What is the price of
the rug? How much floor space is left at each side of
the rug when the rug is laid in the centre of the room ?
5. What will it cost to stain the floor around the rug
if 1 qt. of stain is sufficient for 125 sq. ft., the stain cost
ing 50 cts. a quart ?
6. The kitchen floor is covered with linoleum which
comes only in 2yd. and 4yd. widths at f 1 a square yard.
What is its cost?
7. The bedrooms are carpeted as follows : No. 1 with
plain ingrain carpet 1 j'd. wide, strips lengthwise. Al
lowing 6 ins. on each width for turning in and 3 ins. on
each width for shrinkage after it is cut, what is the cost
at 63 cts. a yard, allowing 3 cts. per yard additional for
making?
8. Bedroom No. 2 with China matting 1 yd. wide at
34 cts. a yard, strips lengthwise?
9. Bedroom No. 3 with Japanese matting 1 yd. wide
at 21 cts. a yard?
10. Bedroom No. 4 with Sanitas washable carpet 36 ins.
wide at 25 cts. a yard, strips lengthwise?
PAPERING
281. Wall paper is usually 18 ins. wide and is sold by
the roll. A single roll is 8 yds. long, a double roll 16
yds. Borders are from 3 ins. upward in width.
Deduction is not made for borders, since no allowance
is made for matching. Some paperhangers measure the
surface above the baseboard, and deduct for windows
204 PRACTICAL ARITHMETIC
and doors, allowing  roll for each door or window. It
is also customary to find the number of square feet in
the walls and ceiling, deduct for doors and windows, and
divide by 70, to find the number of double rolls required.
EXERCISE 174. — OBAIi
1. How many square feet of wall paper are there in a
single roll?
2. How many square feet of wall paper are there in a
double roll?
3. If a room measures 12 ft. x 12 ft. and 9 ft. high,
how many single rolls of wall paper will be required,
making no allowances?
4. How many double rolls are required for the same
room?
5. Allowing for 2 windows 6 ft. x 3 ft. and a door
7^ X 4 ft., how many square feet are there left to be
papered?
6. How many single rolls are required after making
this deduction, making no allowance for matching?
7. How many double rolls?
BXBECISE 175. — WRITTEN
1. How many rolls of paper will be required to paper
the living room of the house shown in the diagram on
page 199, the baseboard 9" wide, there being 3 windows
3' X 6', 1 window 6' x 6', one fireplace 7' x 6', and 1 door
7' x 4', alcove not papered ?
a. Estimate accurately by the number of square feet.
h. Estimate, allowing  roll for each opening.
PRACTICAL MEASUREMENTS 205
c. Estimate by dividing total number of square feet by
70, after deducting for the doors and windows.
2. How much will the paper cost at 50 cts. a double roll ?
(Use answer c above.) How much will it cost to paper
the ceiling at 20 cts. a double roll ? What will picture
moulding cost at 3 cts. a foot, no allowance for openings?
3. The dining room has 2 windows 6 ft. x 3 ft., one,
6 ft. X 6 ft., one opening 6 ft. x 7 ft., and two doors
7 ft. x 4 ft., buffet not papered. If the wall paper costs
30 cts. a double roll and the ceiling paper, 20 cts. a double
roll, with no border, but with picture moulding at 2J cts.
a foot, how much is the total cost?
Each bedroom has 2 windows 6 ft. x 3 ft. and 1 door
7 ft. X 3 ft., ceilings and closets not papered.
4. Find the cost of bedroom No. 3 with paper at 25 cts.
a double roll, picture moulding IJ cts. a foot.
5. Find the cost of papering room No. 4 with sanitary
wall paper at 30 cts. a double roll, 4in. border at 2 cts. a
yard, ceiling papered at the same price.
282. MASONRY AND BRICKWORK
Brickwork is usually estimated by the thousand brick.
The unit of measure for walls built of stone is the perch.
Twentytwo bricks of common size laid in mortar are
reckoned for each cubic foot of wall. Common brick
measure 8x2x4 ins. The mortar occupies \ in. in thick
ness between bricks. A perch measures 16^' x 1' X 1'
and equals 24 cubic feet. In practice a perch is under
stood to be 25 cubic feet,
206 PRACTICAL ARITHMETIC
It requires for each square foot of wall :
7 bricks, if the wall is 1 brick in thickness.
15 bricks, if the wall is 2 bricks in thickness.
22 bricks, if the wall is 3 bricks in thickness.
In estimating work, deductions are not usually made
for openings of less than 100 sq. ft. area. In estimating
materials, deduct for openings.
BXERCISB 178. — ORAL
1. How many bricks are required for a wall 10 ft.
long, 5 ft. high, 1 brick thick ? 2 bricks thick ? 3 bricks
thick ?
2. How many bricks are required for a wall 1 brick
thick, for a cellar 10 ft. x 12 ft., not allowing for open
ings? 2 bricks thick? 3
bricks thick ?
3. How manj'^ perch of
stone are required for a wall
50 ft. long, 10 ft. high, 1 ft.
4 ins. thick?
To find the number of bricks required for a wall, mul
tiply the number of square feet by 7, 15, 22, or 29 accord
ing as the wall is 1, 2, 3, or 4 bricks thick.
EXERCISE 177. — WRITTEN
1. If a cellar is 18' x 24' x 9', inside dimensions, how
many bricks are used in walling it 2 bricks thick?
2. How many perches of stone are required for the
walls of a cellar 18' x 24' x 9', 6", the wall 18" thick?
PRACTICAL MEASUREMENTS 207
3. Find cost of construction in problems 1 and 2 with
bricks at $8.50 per M and labor 13.00 per M, allowing no
deductions for openings, stone costing 118.00 per perch
and f 1.50 per perch for labor.
4. How many bricks will be required to wall a house
36' long, 30' wide outside, and 34' high, including the
cellar, the wall to be 2 bricks thick? No allowance for
openings.
5. The cellar and rear walls of the house of problem 4
are of common brick, the cellar dimensions being 36'
X 30', 6" X 9', the openings 1 door 6' x 3', 2 windows
3' X 1^', 2 windows 6' x 3'. What is the cost of the brick
at 18.45 per M ? Of the labor at $3 per M?
283, WOOD MEASURE
A cord of wood is 8 ft. long, 4 ft. wide, and 4 ft. high.
128 cu. ft. = 1 cord. 16 cu. ft. = 1 cord foot.
EXERCISE 178. — ORAL
1. A pile of wood is 8 ft. long, 4 ft. wide, 2 ft. high.
What part of a cord is it?
2. A pile of wood is
96 ft. long, 4 ft. wide, 4
ft. high. How many
cords are there?
3. Another pile is 12
ft. long, 4 ft. wide, 4 ft. high. How much wood is there
in the pile?
EXERCISE 179. — WRITTEN
1. How many cord feet are there in a pile of wood 96
ft. long, 16 ft, wide, 12 ft. high?
208 PRACTICAL ARITHMETIC
2. How many cords can be piled in a woodhouse 12 ft.
X 161 ft. X Hi ft.?
3. Find the number of cord feet in a pile of wood 100
ft. long, 25 ft. wide, 25 ft. high. Find the number of
cords.
BOARD MEASURE
284. Lumber is bought and sold by Board Measure.
285. The unit of Board Measure is the Board Foot,
which is 1 ft. long, 1 ft. wide, and 1 in. thick. Boards
less than 1 in. in thickness are reckoned as though they
were 1 in. thick. Boards more than 1 in. thick are
sold according to the number of board feet in them, e.g.,
a board 10' long, 1' wide, 2" thick, contains 20 board feet.
EXBBCISE 180. — OEAIj
Find the number of board feet in the following :
1. 10 boards 1" x 12" x 8'. 4. 10 boards 2" x 9" x 10'.
2. 10boardsl"x"xll'. 5. 10 boards 1^" x 6" x 8'.
3. 10 boards 1" X f" X 18'. 6. 10 boards If" x 9" x 14'.
EXERCISE 181. — "WRITTEN
Find the number of board feet in the following :
1. 12 planks 3" x 8" x 12'.
2. 14 planks 2" x 9" x 18'.
3. 18 planks 1^" X 12" x 12'.
4. 24 planks If" x 26" x 16'.
5. 60 planks If" x 4" x 12'.
6. How much flooring 18' long, 4" wide, If" thick
PRACTICAL MEASUREMENTS 209
must be bought to floor a room 18' x 12' ? What will
the flooring cost at 60 per M ?
ROUND LOGS
286. Round logs are sold by the number of board feet
that can be cut from them. To find the number of board
feet in a log, subtract twice the diameter in inches from
the square of the diameter ; ^ of the remainder will be
the number of board feet in 10 linear feet of the log.
Find the number of board feet in a log 18' long, 12" in
diameter.
122 _ (2 X 12) = 144  24 = 120
l of 120 = 63
If of 63 = 113.4 board feet.
EXERCISE 182. — WRITTEN
Find the number of board feet in the following :
1. In a log 12' long, 10" in diameter.
2. In a log 18' long, 15" in diameter.
3. In a log 14' long, 18" in diameter.
4. In a log 15' long, 22" in diameter.
MEASURING TEMPERATURE
287. Temperature is commonly measured in this coun
try in degrees by Fahrenheit's thermometer. The 32
degree mark is placed at the freezingpoint of water,
the 212degree mark at the boilingpoint of water. The
interval between the freezingpoint and the boilingpoint
is divided into 180 equal degrees. Another thermometer
used in many other countries and in most scientifc work
210
PRACTICAL ARITHMETIC
is the Centigrade, with zero at the freezingpoint, 100
degrees at the boilingpoint. Below zero is indicated by
the — sign. Above zero is indicated by the
( sign or by no sign.
EXERCISE 183. — ORAL
Unless otherwise stated, Fahrenheit de
grees are to be understood in the following
problems :
1. When the temperature of the air is 40
degrees, how much above freezing is it?
2. When water is at 180 degrees, how
much below boiling is it?
3. The body temperature of a person is
normally 98.6 degrees. A temperature of 99
degrees or more indicates fever. What is
the range between normal temperature and
fever temperature?
4. A body temperature of 108 degrees usually indicates
approaching death. What is the range between normal tem
perature and the death point? Only rarely does the body
temperature fall below 92 degrees and life continue. What
is the range of temperature ?
5. How many degrees Centigrade equal 180 degrees
Fahrenheit?
6. How many degrees Fahrenheit equal 1 degree Centi
grade ?
7. How many degrees Centigrade equal 1 degree
Fahrenheit?
PRACTICAL MEASUREMENTS 211
BXEECISE 184. — 'WRITTEN
1. Change 10° F. to C; change 10° C. to F.
2. Devise a rule for changing from Fahrenheit to Centi
grade.
3. Devise a rule for changing from Centigrade to
Fahrenheit ?
LONGITUDE AND TIME
BXEECISE 185. — ORAL
1. The earth appears to be flat. If it were really flat,
what would be true concerning the time of sunrise on
every part of the earth's surface?
2. What would be true of sunset under the same con
ditions? What are the facts concerning the time of sun
rise and sunset upon different parts of the earth's surface?
3. Which city has sunrise first, New York or Chicago?
Why? Chicago or San Francisco? Why?
4. What places upon the earth have sunrise at the
same time? What places have noon at the same time?
What places have midnight at the same time?
288. A meridian is an imaginary line running north
and south from pole to pole. All places upon the same
meridian have their midday or noon at the same moment,
i.e., they are touched by the vertical rays of the sun at
the same time.
289. The distance east or west from a given meridian
is called Longitude.
212 PRACTICAL ARITHMETIC
EXERCISE 186.— ORAL
1. Through how many degrees does the earth rotate
from sunrise until the sun is in the zenith?
2. Through how many degrees does the earth rotate
from sunrise to sunset ? Through how many degrees does
a point upon the earth pass in a complete rotation ?
3. Since the earth turns on its axis through 360 degrees
in 1 day of 24 hours, through how many degrees does it
turn in 1 hour ?
290. Longitude is measured by degrees. The merid
ians from which longitude is commonly reckoned are
two, one of which passes through Washington, D.C., one
through Greenwich, England.
291. The given meridian from which longitude is gen
erally reckoned is called the Prime Meridian. Longitude
is reckoned east and west of the prime meridian to 180
degrees. West longitude is designated by the letter
"W." and east longitude by the letter "E."
EXERCISE 187. — ORAL,
1. When it is sunrise at Baltimore, how long will it be
before it is sunrise at a place 15 degrees west of Balti
more? 30 degrees west? 45 degrees west?
2. When it is noon at St. Louis, how long will it be
before it is noon at a place 15 degrees west of St. Louis?
3. If I travel eastward, will my watch become too slow
or too fast ? If I travel westward, will my watch become
too slow or too fast ?
PRACTICAL MEASUREMENTS 213
292. Since the earth turning upon its axis once in 24
hours, passes through 360 degrees in that time, the follow
ing table may be deduced :
Longitude Timb
360° corresponds to 24 hours.
15° corresponds to 1 hour.
15' corresponds to 1 minute.
15" corresponds to 1 second.
1° corresponds to 4 minutes.
1' corresponds to 4 seconds.
293. Two places are 40° 15' 30" apart. What is the
difference in time between them ?
Since places 16° distant
15 )40° 15' 30" from each other have a dif
2 hrs. 41 mins. 2 sees, ference of 1 hr. in time, and
since places 15' apart have a
difference of 1 min. in time,
and since places 16" apart have a difference of 1 sec. in time,
jig of the numbers representing degrees, minutes, and seconds
will give the difference in time in hours, minutes, and seconds.
BXBRCISB 188. — WRITTEN
1. The difference in longitude between two places is
46 degrees, 20 minutes, 35 seconds. What is the difference
in time?
2. Washington is 77° 3' from Greenwich. What is
the difference in time?
The longitude of the following cities is :
Washington, 77° 3' 0" W.
New York, 74° 3" W
Paris, 2° 20' 22" E.
214 PRACTICAL ARITHMETIC
San Francisco, 122° 24' 15" W.
Peking, 116° 27' 30" E.
Constantinople, 28° 59' E.
Berlin, 13° 23' 43" E.
Find the difference in time between :
3. Washington and Peking.
4. New York and San Francisco.
5. Washington and Constantinople.
6. New York and Paris.
7. San Francisco and Berlin.
294. The difference in time between two places is 5 hrs.
30 mins. 25 sees. What is the difference in longitude?
ri on ■ oc Since there are 15 times as many
5 hrs. 30 mms. lo sees. , . ■, , ^.
. r degrees, mmutes, and seconds of
' longitude as there are hom's, min
utes, and seconds of time, 15 times
the number representing the difference in time will give the
difference of longitude in degrees, minutes, and seconds.
BXBBCISE 189. WRITTEN
1. The difference in time between two places is 10 hrs.
16 mins. 24 sees. What is the difference in longitude?
Find the difference in longitude between the following
places and Greenwich, the difference in time being:
2. Athens, Greece, 1 hr. 34 mins. 54.9 sees. E.
3. Calcutta, 5 hrs. 53 mins. 20.7 sees. E.
4. Chicago, 5 hrs. 50 mins. 26.7 sees. W.
5. St. Petersburg, 2 hrs. 1 min. 13.5 sees. E.
6. What is the difference in longitude between St.
Petersburg and Chicago?
PRACTICAL MEASUREMENTS 215
7. When it is noon at Greenwich, it is 6.09 A. jr. at
Chicago. What is the longitude of Cliicago?
8. Wlien it is 6.09 a.m. at Chicago, it is 3.50 a.m. at
San Francisco. What is the difference in longitude?
STANDARD TIME
295. To use the exact sun time for each place brings
about so many complications that another method has
been devised by which the world is divided into 24 time
Standard Time Map
belts, all places within the same belt using the same time.
The time meridians of the timebelts of the United States
are those of 75°, 90°, 105°, and 120°, as is shown in the
accompanying map. Accordingly, the United States and
Canada are divided into four belts extending north and
south. All places in the same belt have the same time
regardless of their exact longitude. In practice, these
216 PRACTICAL ARITHMETIC
belts do not have regular boundaries ; but the points of
change are determined rather by the position of important
cities. Time determined in this manner is known as
Standard Time.
EXERCISE 190. — OEAL
1. When it is noon at Greenwich, what time is it at
New York, estimating the longitude at New York as 75"?
2. When it is 6 a.m. at New York, what time is it at
Chicago ? ' At Denver ? At San Francisco ?
EXERCISE 191.— WRITTEN
1. What is the difference between standard and local
time at Chicago ?
2. The longitude of Pittsburg is 80° 2' 0". What is
the difference between standard and local time there?
3. What is the difference between standard and local
time at New Orleans, longitude 90° 3' 28.5" ?
4. The longitudes of St. Louis, Richmond, Denver, and
Boston are respectively : 90° 15' 15" W., 77° 26' 4" W.,
104° 59' 33" W., 71° 3' 30" W. At what time, standard,
will an electric time signal sent from Washington at
noon reach them ? At what local time ?
REVIEW PROBLEMS
EXERCISE 192. — WRITTEN
1. How many barrels of water will a trough hold, if it
is 7 ft. long, 2 ft. wide, 16 ins. deep?
2. If a windmill pumps 2 gals, and 8 qts. each minute
and pumps 4 hrs. in a day, how many cows will it supply
if each cow consumes 1.5 cu. ft. of water daily?
REVIEW PROBLEMS 217
3. How many pounds of water does each cow consume
if one gallon weighs 8.35 lbs.?
4. A pound of timothy seed conta,ins 1,170,500 seeds.
Sowing 16 lbs. to the acre, how many seeds are sown to
the acre ? How many seeds are there in an ounce ?
5. What is the value of a rectangular field 23 chs. 2 rds.
17 Iks. 6 ins. long by 17 chs. 3 rds. 18 Iks. 7 ins. wide at
137.50 an acre?
6. There are 15 steps on the hall stairs. The tread is
12", the rise 7". What Avill sufficient stair carpet cost
at 62 cts. a yard, allowing Jq extra carpet for the pro
jection of the tread ?
7. In raising Irish potatoes three grades of seed were
used : 1st, seed from the best hills to be found ; 2d, seed
from ordinary hills ; 3d, seed from the very poor hills.
The yields for each 100 hills planted with these seed were:
ordinary seed 136 llw. 1t ozs., best seed 172 lbs. 8 ozs.,
poor seed 75 lbs. 10 ozs. What was the average yield per
hill? What was the increase in yield from seed No. 1 over
that of No. 2? Of seed No. 2 over seed No. 3 ?
8. Sprayed grapes yielded 4 lbs. 5.8 ozs. to the row.
Unsprayed grapes yielded 1 lb. 1.5 ozs. to the row. What
was the gain per row in pounds from spraying?
9. Sprayed grapes yielded 3 lbs. 4.3 ozs. more per row
than unsprayed grapes. What was the difference in
yield on 17 rows ?
10. Western Yellow Pine in Colorado, 20 years old,
measured 1.2 ins. in diameter ; at 80 years, 2.9 ; at 40, 4.9 ;
at 50, 6.8; at 60, 8.6; at 70, 10.2; at 80, 11.7; at 90,
12.8; at 100, 13.8; at 110, 14.7; at 120, 15.5. What
218 PRACTICAL ARITHMETIC
was its circumference at each of these ages? How much
did the diameter increase from the fourth to the fifth
decade ? How mucli the circumf ereiice ? How much did
both the diameter and the circumference increase from the
eleventli to the twelfth decade?
IX. An ordinary milk pail is llj ins. in diameter at the
top; another pail has an opening only 7 ins. in diameter.
What is the difference in the areas of the openings? If
seven billion bacteria fall into the milk in the first pail in
15 mins., at the same rate how many would fall into the
other pail in the same time ?
12. Planting 7 ozs. of tomato seeds to the acre, how
many pounds are needed for 19 A. ?
13. Sowing 3 pks. of cowpeas to the acre, how many
bushels of seed are needed for 40 A. ?
14. Bordeaux mixture consists of 5 lbs. of bluestone,
5 lbs. of lime, and 50 gals, of water. Applying this to
potatoes at the rate of 150 gals, of mixture to the acre,
for each application, and making 3 applications: a. How
many gallons will be needed on 40 A. ? 6. How many
barrels? c. What will the bluestone cost at 6 cts. a
pound? d. What will the lime cost at \\ cts. a pound?
15. To produce 1 T. of oats requires 376 T. of water.
How many cubic feet of water are needed? How many
barrels? How many hogsheads? How many quarts?
16. To produce a ton of wheat requires 338 T. of water.
Solve for all the items of the last problem.
17. To produce a ton of dry matter, the average crop
plant requires 325 T. of water. Solve as in problem 15.
REVIEW PROBLEMS 219
18. A man sells 7 customers 1 pt. each of syrup at 5
cts. a pint, 9 customers 1 qts. each at 10 cts. a quart, and
13 others 1 gal. each at 35 cts. a gallon. How much a
gallon did he average for his syrup?
19. A horse moving at 2:40 gait, moves how far in 1
sec. ? At a 2: 10 gait? At a 3nnn. gait? At a 4min. gait ?
20. What is the cost of materials for 1 cu. yd. of con
crete made of 1.16 bbls. of cement @ 2, 0.44 cu. yd.
of sand @ 75 cts., 0.88 cu. yd. of gravel @ 75 cts. ? If
20 posts 6" X 6" at the bottom, 6" x 8" at the top, and 7'
long, can be made from 1 cu. yd. of concrete, what will
the materials cost per post?
21. Adding 6 cts. for 28 ft. of 0.16in. steel wire @ 3
cts. a pound, what is the total cost for materials per post?
22. A well is 7 ft. in diameter. Wliat is its circumfer
ence? If it contains 6 ft. of water, how many gallons
are there?
23. A tree has a circumference of 12 ft. What is its
diameter? If 8 ft., what is the diameter?
24. The triangular end of a house gable is 37 ft. on
its base, and 16 ft. from the base line to apex. What is
its area in square feet?
25. A triangular garden has a base of 90 ft. and an
altitude of 75 ft. What fraction of an acre is it?
26. A cistern is 12 ft. deep and has a diameter of 8 ft.
What is the volume of the cistern? How many gallons
of water will it hold?
27. The water pipes on a house measure 13 ins. outside
220 PRACTICAL ARITHMETIC
circumference. There are 200 ft. of piping on the house.
What is the surface of this piping ?
28. How many cubic feet are there in a cylindrical
tank 12 ft. in diameter and 20 ft. deep? How many
gallons? How much does the water in it weigh when it
is full? How many square feet in its outer surface, ex
clusive of bottom? How many gallons of water does it
contain for each foot in depth?
29. How many feet of picture moulding are needed for
a room 17' 6" long and 9' 9" wide?
30. AVhat is the length of the tire of a wheel 4 ft. in
diameter? Of a wheel 3^ ft. in diameter?
31. Mr. Akers owned a rectangular piece of land, the
north and south sides of which were each 42 rds. long.
The east and west sides were each 38 rds. long. Two
railroads cut off portions of it ; one took all west of a line
beginning at the N.W. corner and running S.E. to a
point 14 rds. east of the S. W. corner. The other took
all east of a line beginning at a point 12 rds. south of the
N. E. corner and running S. W. to a point 8 rds. west of
the S.E. corner. Draw a map of the land as cut off by
the railroads. The railroad owners bought the land cut
off by their roads at $150 an acre. How much did it
cost ? How much land was left in the tract ?
32. How much lumber will be required for 80 rds. of
fence 4 boards high, each board 6 ins. wide ? How much
lumber will be required to make a tight board fence the
same length and 5 ft. high, nailing the boards to two 2"
X 4" scantling? No allowance for waste in either case.
33. How many yards of 27in. velvet carpet with 22in.
REVIEW PROBLEMS 221
border will be required to carpet a society hall 42 ft.
long, 36 ft. wide, having one rectanglar alcove at each
end? Each alcove is 5 ft. deep and 20^ ft. wide. Esti
mate the amount with strips of carpet running lengthwise.
34. Estimate with strips running crosswise.
35. What will carpet and border cost at $ 1.25 a yard ?
36. How much paper will be needed to paper the walls
and ceiling of this hall, there being 4 windows 9 x 6 ft.
and the pitch of the ceiling being 14 ft.? The alcove
ceilings are papered, but the alcove walls are finished in
hardwood panels, and require no paper.
37. The paper for walls costs 95 cts. per double roll,
for ceiling 25 cts. per double roll, picture moulding 7 cts.
a foot, no moulding being used in the alcove. What is
the cost ?
38. Estimate in meters the height of Mt. Everest 29,002
ft.. Pikes Peak 14,111 ft., Mt. Blanc 15,744 ft., and Mt.
Aconcagua 23,082 ft.
39. What number of bricks is needed for a wall 4^ ft.
high, 3 bricks thick, around an enclosure 14 x 23 ft.?
40. How many cubic yards of concrete are required to
make a circular wall 8 ins. thick around the top of a well,
the wall to be 29 ins. high and 5 ft. 3 ins. inside diameter ?
41. How many cubic yards of concrete are needed to
make a semicircular walk in front of a schoolhouse, the
walk to be 4 ft. 3 ins. wide, 27 ft. long on its shorter side,
and the concrete to be laid 4 ins. thick ?
42. A farmer uses a potato crate 19" x 13" x 12" to
contain a bushel of potatoes, 60 lbs. How much does
222 PRACTICAL AKITHMETIC
this bushel of potatoes exceed in size the standard bushel
of dry measure ?
43. A crate 22" x llf" X 14" contains how many
such bushels ? How many standard dry bushels ?
44. A tank of a spraying machine is in. the form of a
halfcylinder resting on its convex side, 5 ft. long and 2 ft.
6 ins. in diameter, inside measurement. What is its
capacity? If it were shortened 5 ins. and increased in
height 3 ins. without further curving of the sides, what
would the capacity be ?
45. The usual 40qt. milk can is 12 ins., inside di
ameter. How many inches tall need it be if of uniform
diameter ? What fraction of an inch in depth equals 1 qt.?
46. In the usual 50qt. milk can, 1 qt. occupies .375
in. in depth. What is the diameter of the can ?
SILOS AND SILAGE
47. The base of a, silo is 16 ft. in diameter. What is
its circumference? If 26 ft., what is its circumference?
48. Find the capacity, in tons, of a silo 10 ft. in diame
ter and 20 ft. high, if a cubic foot of silage from such a
silo weighs 30 lbs.
49. How many tons of silage will this silo contain, if,
after it has settled, the silage is 5 feet from the top ?
50. Find how many tons of silage a silo will hold, that
is 20 feet in diameter and 32 feet deep, if in such a silo
a cubic foot of silage weighs 40 pounds.
51. How many tons of silage will there be in this silo,
if, after it has settled, the silage is 7 feet from the top?
REVIEW PROBLEMS
223
52. If 43 COWS be fed 37 lbs. each of silage per day,
how long will the contents of this silo last ?
53. In order that the silage may settle sufficiently to
insure its preservation, a silo should not be less than
30 ft. deep. What di
ameter must it have to
hold enough silage to
feed 25 cows 40 lbs. per
day for 185 days, if the
silage weighs 38 lbs. to
the cubic foot?
54. How many acres
of corn will it take to
furnish a feed of 35
pounds per day each to
a herd of 32 cows for
150 days, each acre yield
ing 12 tons of silage ?
What must be the diameter of the silo necessary to hold
this silage, if the height of the silo is 32 feet, and 1 cubic
foot of silage weighs 40 pounds?
55. The corn on a field of 18 acres when ready for
cutting arid shocking or for putting in the silo weighs
9 tons per acre, of which ^ is water. If by cutting and
shocking the corn, there is a loss in dry matter of ^^^, and
by putting it in a silo there is a loss of dry matter of Jg,
what is the value of the feed gained by putting the crop
in the silo, if the dry matter in silage is worth .71 of a
cent per pound ?
56. How many acres of corn will it require to produce
224 PRACTICAL ARITHMETIC
the silage to feed 18 cows 37 pounds per day for 184
days, if each acre produces 13 tons of silage corn?
DRESSMAKING
57. How many yards of 36in. material will be required
to make a shirtwaist, if the tucked front measures 24 ins.
before tucking and the two backs 12 ins. before tucking ;
the front length from shoulder measuring 18 ins. and two
backs 16 ins.; sleeves 16 ins. long before finishing and
18 ins. wide in the widest place? Cuffs, collars, belts are
found in pieces cut from strips. See diagram.
58. How many yards of insertion will be required for
this waist for 4 strips the length of the front, 4 strips
for the back, one piece for the neck 12 ins., and two
pieces for the sleeves 8 ins. each?
59. How many yards will be required for a ninegore
skirt of the same linen if gores average 5 ins. wide at the
top, and 15 ins. wide at the bottom, the length of
each gore being 38 ins. when finished with a 3in. hem,
and allowing 1 in. at the top for finishing?
60. How much insertion will be required for two rows
5 and 8 inches from the bottom of the skirt, if the gores
for the bottom row average 13 ins. and second row 12 ins. ?
61. At 50 cts. for linen and 15 cts. a yard for insertion,
1 doz. buttons at 10 cts., 1 bolt of tape at 5 cts., what
will the dress cost?
62. The lining and finishings for a dress are called the
findings. What will findings consisting of the following
items cost : 2 J yds. of waist lining at 15 cts., 6 yds. skirt
lining at 25 cts., 2 spools silk at 10 cts., 1 spool cotton
REVIEW PROBLEMS
5 cts., 1 bolt braid 10 cts., 1 card hooks
and eyes 10 cts., 1 bolt silk binding 10
cts., 1 yd. featherbone 5 cts., 1 pr.
shields 35 cts. ?
63. Find the cost of a Panamacloth
dress ; 8 yds. of material at 89 cts. a
yard, with findings at price given in
last problem.
64. What will the dress cost if a
professional shopper charges 10 cts.
on each dollar for buying the above
materials and a dressmaker charges f 8
for making the dress?
65. If, for the cotton skirt lining,
a silk lining be substituted which will
require 10 yds. of silk at 59 cts. a yard,
what will the dress cost? What will
be the shopper's fee?
66. Races are run at the
Olympic games at Athens
over distances of 100 meters,
400 meters, 800 meters, 1500
meters. How many j'ards
in each of these distances ?
C7. The best time for
these races prior to 1908
was respectively 10 sees.,
49^ sees., 1 min. 56 sees.,
and 4 mins. 5 sees. What
was the speed per yard in
each race ?
FOLD OF GOODS
226 PRACTICAL ARITHMETIC
68. Some races run in the Public Schools' Athletic
League, high school events, are 100 yds., 440 yds., 880
yds., and 1 mi. What is the diiference between these dis
tances and those mentioned in problem 66 ?
69. The time for these races was respectively 10 sees.,
53 sees., 2 mins. 7 sees., and 4 mins. 59 sees. What
was the speed per yard ?
70. Some Olympic records are : running long jump
24 ft. 1 in. ; running high 6 ft. 2^ ins.; pole vault 11 ft.
6 ins. P. S. A. L. high school records for the same events
are respectively: 21 ft., 5 ft. 6 ins., 9 ft. 9 ins. By what
fraction does the P. S. A. L. high school record fall short
of the Olympic record in these events ?
71. The steamship Lusitania, in July, 1908, made an
average speed of 25.01 knots an hour. One knot equals
6086.7 ft. What was the average speed in miles per day?
72. The Lusitania covered the distance from Daunt's
Rock to Sandy Hook lighthouse in 4 das. 20 hrs. and 15
mins. What is the distance?
73. What is the speed per second of the circumference
of a grindstone 4 ins. in diameter, run at 30 revolutions
per minute?
74. Allowing 5 sq. ft. of floor space to each fowl, what
may the dimensions of a henhouse be to accommodate 100
fowls? If one side be 14 ft. what will the other dimen
sion be? With lumber at $15 per M, what will the
boards for sides and ends cost if 5 ft. 6 ins. high when
built square ? When built 14 ft. wide? How much does
the extra light and ventilation secured by the narrower
form of house cost?
PERCENTAGE
296. If a spraying mixture contains 98 parts water, 1
part lime, and 1 part bluestone, how many parts are there
in the mixture? How many hundredths of the mixture
are water? How many hundredths are lime? How many
hundredths are bluestone?
Of a number of seeds tested, one out of every ten fails
to grow. How many fail to gi'ow out of every hundred?
How many hundredths fail to grow?
The number of hundredths of a number is commonly
expressed by the term per cent.
Out of 100 oak leaves examined, 60 were injured by
insects. What per cent were injured by insects ? What
per cent were uninjured?
297. The term per cent is expressed by the sign %, e.g.,
8 % is read 8 per cent.
298. A given per cent, or a given number of hun
dredths, of a number may be expressed as a whole number
with the per cent sign, as a decimal, or as a common frac
tion, e.g., 1 per cent may be written 1 %, .01, or ^Jg. 8
per cent may be written 8 %, .08, or jfiy
EXERCISE
193
.  WRITTEN
Express as
decimals :
1. 4%.
4. 15%.
7. 6f%.
10. 621%.
2. 6%.
5. 75%.
8. 5f%.
11. 831%.
3. 5%.
6. 41%.
9. mo.
12. 115%,
227
228 PRACTICAL ARITHMETIC
13. 125^%. 16. 2.25%. 19. .4%.
14. 250%. 17. 6.4%. 20. .04%.
15. 631%. 18. 9.34%. 21. .004%!
299. Express 25%, 2.5%,  %, as common fractions in
lowest terms :
25%=TVu = i
1
1 pi„ ~i 1
'i JO — iTo ~ TiriT
EXBBCISB 194. — "WRITTEN
Express as common fractions in lowest terms:
1. 12%, .12%. 6. 16f%, 1%. 11. 150%, .15%.
2. 25%, .25%. 7. 3.31%, 1%. 12. 625%, 6.25%.
3. 42%, .42%. 8. 621%, 1%. 13. 375%, 3.75%.
4. 55%, .55%. 9. 83^%, f%. 14. 1%, 1.2%.
5. 75%, .75%. 10. 121%,!%. 15. 1%, .05%.
300. Express  %, ^%, j^^%, as decimals of a per cent,
and as decimals :
i%=.6%=.005.
i % = .2 % = .002.
t\%=.2%=.002.
exercise 195. — written
Express the following as decimals, and as decimals of a
per cent :
10. ^%.
11. 1%.
12. ^%.
1 \1o.
4 \1o.
7. 1%
2. \1o.
5 \1o.
8. i%
3. \1o.
6. 1%.
9 \1o
fERCENTAGE 229
13. ^%. 15. f%. 17. f%. 19. 1%.
14. i%. 16. T^%. 18. 1%. 20. 1%.
301. Important per cents to be remembered :
1 = 100%. 1 = 15%. i=m%. tV = 6%.
1 = 50%. i = 20%. 1 = 371%. ^, = 5%.
i = 33i%. 1 = 60%. 1 = 621%. 2^ = 4%.
f=66%. i = 16f%. t = 87i%. ^V = 2%.
i = 25%. 1 = 831%. J^=10%. 1^0 = 1%.
EXERCISE 196. — ORAL
Find:
1. 10 % of 50. 6. 371 % of 48. ii. 16 % of 66.
2. 25 % of 80. 7. 871 % of 72. 12. 621 ^^ of 64.
3. 60 % of 300. 8. 5 % of 125. 13. 60 % of 10.
4. 121 % of 72. 9. 2 % of 200. i4. 83^ % of 42«
5. 16 % of 90. 10. 831 % of 96. 15. 5 % of 120.
EXERCISE 197. — WRITTEN
Find:
1. 75 % of 1280. 8. 66f % of 5432.
2. ^% of 5420. 9. 4.2% of 2437.
3. 2.5% of 2655. 10. 42% of 8432.
4. 871 % of 6424. 11. § % of 9812.
5. 4% of 2254. 12. 381% of 5431.
6. 1% of 8243. 13. 4.5% of 5240.
7. 831% of 9846. 14. 8.9% of 1872.
302. The number of which so many hundredths or a
certain per cent is to be taken is called the Base.
230 PRACTICAL ARITHMETIC
303. The number indicating how many hundredths of
the base are to be taken expresses the Rate or Rate Per
Cent.
304. The result obtained by taking the number of hun
dredths of the base indicated by the rate is the Per
centage.
Find 9 % of 50 :
50 is the base.
.09 is the rate.
4.5 is the percentage.
305. Three kinds of problems occur in percentage:
1. Those in which base and rate are known and per
centage is to be found.
2. Those in which base and percentage are known and
rate is to be found.
3. Those in which rate and percentage. are known and
base is to be found.
306. Case One. Given the base and the rate to find
the percentage.
In testing a certain ore it was found that 25 % of it was
iron. How much iron was contained in 496 lbs. of ore? Since
25<fo =^, the percentage is most readily obtained by taking 
of 496, which is 124. Hence, 124 lbs. was iron.
In testing an ore it was found that 29 % of it was iron.
How much iron was there in 469 lbs. of ore?
Since 29% or ^ is not conveniently used as a common
fraction, it is best to express it as a decimal, .29. 469 x .29 =
136.01 lbs. Hence 136.01 lbs. was iron.
PERCENTAGE 231
Knowing the base and rate to find the percentage.
Multiply the base by the rate expressed either as a com
mon or as a decimal fraction.
EXERCISE 198. — ORAL
1. Find 5 % of 80 ; 120 ; 40 ; 200 ; 180 ; 160 ; 20.
2. Find 4% of 75; 125; 150; 175; 50; 25.
3. Find 10 % of 40 ; 110 ; 140 ; 90 ; 70 ; 860 ; 10.
4. Find 20% of 30; 75 ; 90; 65; 95; 100; 25; 5; 10.
5. Find 25 % of 20 ; 40 ; 80 ; 60 ; 12 ; 8 ; 10 ; 4.
6. Find I % of 100 ; 200 ; 50.
EXERCISE 199. — WRITTEN
Solve the following, using the rate both as a decimal
and as a common fraction :
1. Find 7% of 425; of 67.3; of 526; of 9642; of 87.9.
2. Find 14% of 48.8; of 68; of 125.6; of 7981.
3. Find 6^% of 32; of 96.32; of 128.64; of 842.
4. Find 19 % of 38 ; of 99 ; of 132 ; of 869 ; of 7684.
5. Find 72 % of I ; of fy ; of J^ ; of ^\ ; of If.
6. What is 15% of $5.20? of *13.50? of 75? of
$8764.75?
7. What is 4 % of  of an acre of land ?
8. What is 13% of 96 sq. ch^. 48 sq. rds. of land?
9. What is 37% of 1 sq. mi. 168 sq. rds. of land?
10. Find 129% of 76; of 128; of 7842.1.
11. Find 300 % of 84 ; of 78.2 ; of 3.
12. Find .3% of 782; of 6498.
232 PRACTICAL ARITHMETIC
13. Find .17% of 6842; of 17; of 386.7.
14. Find .06% of 98; of 782; of 6428.
15. A owned f of a mill and sold 33^% of his share.
What part of the mill did he sell, and what part does
he still own?
307. Case Two. Given the base and the percentage to
find the rate.
What per cent of 85 is 17?
Dividing 17 by 85, we have ^ = \.
^ of any number is 20 % of that number, therefore 17 is
20 % of 85.
Since the percentage is found by multiplying the base
by the rate, the rate may be found by dividing the
percentage by the base.
_ What per cent of 85 is 17.85?
^
85)17.85 Dividing 17.85 by 85, we have .21 = 21 %.
I'^QO Therefore, 17.85 is .21, or 21 hundredths, or
85 21 % of 85.
85
BXEECISB 200. — ORAL
1. What per cent of 50 is 25? Is 10?
2. What per cent of 1^.00 is 125? Is $2.50?
3. Whatper cent of 112.50 is $3.00? Is 12.75?
4. What per cent of 172.50 is $15.50? Is $17.40?
3. What per cent of 96 is 31? Is 27?
6. What per cent of 122 is 6.1? Is 9.76?
PERCENTAGE 233
BXBECISB 201. — WRITTEN
1. What per cent of 86 is 24? Is 8? Is 1? Is ^?
2. What per cent of 7862 is 18? Is 986? Is 12? Is 7?
3. Wliatper cent of 7847 is 67? Is 7614? Is 96?
4. Whatper cent of 848is 424? Is 212? Is 106?
5. What per cent of 1 is ^ ? Is f ? Is .08?
6. A man whose salary is f 182 a month pays 126.40
for board, and $5.28 for amusement. What per cent of
his salary remaining after his board is paid, does he pay
for amusement?
7. A workingman's day is 10 hours long. He spends
^ of the remaining hours in sleep. What per cent of his
time is given to sleep? What per cent to other pursuits?
8. A farmer held cotton bought of a renter; the price
declined 12%, then rose 15% ; he then sold. What per
cent did he gain on the transaction ?
9. If a merchant's scales weigh 14 ozs. for a pound,
what per cent does the purchaser lose?
10. If the retail merchant's fair profit is equivalent to
3 ounces and the scales are as in the above example, what
per cent more does the buyer pay than if he were to buy
from wholesale houses?
11. What per cent of a number is 33^% of 6 % of it?
308. Case Three. Given the rate and the percentage
to find the base.
29 is 20 % of what number ?
20 % = TW) or h 29 is therefore i of the base ; the base is
234 PRACTICAL ARITHMETIC
29 H ^ = 29 X f , or 29 multiplied by 5, or 145, or expressing
20 % as a decimal = .20 = .2, and dividing we have,
■2 )29.0
145.
The second method is preferable when the rate cannot
readily be expressed as a fraction.
Since the percentage is the product of the rate and the
base, the base may be found by dividing the percentage
by the rate expressed as a decimal.
EXERCISE 202. — OBAL
1. 8 is 5% of what number? 10% of what number?
2. 8 is 20% of what number? 25% of what number?
3. 12 is 12^ % of what number? 75% of what number?
4. 12.50 is 16% of what? 33^% of what?
5. 6 ft. 10 ins. is 66 % of what? 331% of what?
6. 21 ft. 7 ins. is 871% of what?
7. 16 gals. 4 qts. 2 pts. is 87i% of what?
BXBECISB 203.— "WRITTEN
1. 44.1 is 105% of what number? 90% of what
number ?
2. 60 is 125% of what number? 75% of what number?
3. 96 is 30% of what number? 18% of what number?
4. 47 is 8% of what number? 20% of what number?
5. 776 is 60% of what number? 83% of what number?
6. 6 is 16% of what number? 90% of what number?
7. 87 is .05 % of what number? 178 % of what number?
PERCENTAGE 235
309. When the base is added to the percentage, the sum
is known as the Amount.
310. When the percentage is subtracted from the base,
the remainder is called the Difference.
311. The rent of a farm is §68.20 and this is an advance
of 10 "Jo over the previous year. What was the rent the
previous year?
100 "Jo = rent the previous year.
10 % = advance this year.
110 % = r? ()8.20.
1% = .62.
100% = $62.
Hence, % 62 was the rent of the farm for the previous year.
A regiment returns from battle with 1197 men, whicli
is 5 % less than it started with. How many men were in
the regiment when the battle began?
100 % = nmnber started with.
6 % = number lost.
95% =1197.
1%=12.6.
100 % = 1260.
To find the base when the rate and amount are given,
divide the amount by 1 plus the rate expressed as a
decimal.
To find the base when the rate and difference are given,
divide the difference by 1 minus the rate expressed as a
decimal.
EXERCISE 204. — ORAL
1. Amount 110, rate 10 ; find the base.
2. Amount 150, rate 50 ; find the base.
23G
PRACTICAL ARITHMETIC
3. Amount 75, rate 50 ; find the base.
4. What number plus 7 % of itself equals 214 ?
5. What number minus 10 % of itself equals 90?
EXERCISE 205. — WRITTEN
Base
Peecentase
Eate
Amount
DiFFEEEHOE
1
17.62
.75
7%
647
784
2
1784.2
1.50
5i%
829
679
3
368.0
7.84
27%
36.89
87.24
4
42.879
26.75
219%
726.47
629.84
5
6784.0
179.50
0.6%
99i
88J
6
20090.0
16A
r/o
642^
79if
7
6425.7
A
0.08%
i
t\
8
84
6f
0.009%
0.06
0.098
9
9
0.074
0.01
8.09
9.99/t
10
.0086
6.28
0.6
7.09
6.4J
The pupil should practise with these problems enough to
attaiu skill, accuracy, and certainty.
110. Using the base in line 1, find the percentage with
the rate given in each line from 1 to 10.
1140. Using bases in lines 2, 3, and 4, find percent
ages with each rate given.
4180. Using the percentages in lines 4, 5, 6, and 7,
with each rate given, find the bases.
81120. Using the bases given in lines 3, 4, 5, and 6,
with each percentage given, find the rates.
121160. Using the amounts given in lines 6, 7, 8, and
9, find the bases with each rate given.
161200. Using the differences given in lines 1, 2, 3, and
4, find the bases with each rate given.
PERCENTAGE 237
EXERCISE 206. — WRITTEN
FARMING IN THE UNITED STATES
1. In 1900, 7.1 % of the farms, or 407,012 farms, were
between 10 and 20 A. in area. How many farms were
there in the United States then ?
2. Of 29,285,922 working inhabitants in the United
States in. 1900, 10,438,217 were engaged in agriculture,
7,112,987 in manufacturing, and 1,264,735 in professional
service. What per cent was engaged in each of these
pursuits ?
3. $3,560,198,191, the value of the farm buildings of
the United States, is 21.4 % of the total farm value. What
is the total farm value?
4. Of the 5,739,657 farms in the United States in 1900
54.9% were worked by their owners, 13.1% by cash
tenants, and 22.2% by share tenants. How many farms
were worked in each of these ways?
5. In 1900, 1,366,167 of the 5,739,657 farms of the
United States were between 50 and 100 A. in area.
What per cent of farms were of this size ? 47,276 were over
1000 A. in area. What per cent did they constitute?
6. In 1900 24.8% of the farms in the United States
were between 100 A. and 175 A. in area. How many
farms of this size were there? 21.9% were between 20
and 50 A. How many were there of this latter class?
7. In 1900 there were 9,349,922 men and boys engaged
in farming in the United States, 20.1 % more than in 1890.
At the same rate of increase, how many farmers will there
be in 1910?
238 PKACTICAL ARITHMETIC
8. In 1894 there were 2712 Agricultural College
students in the United States; in 1899, 5035. What was
the per cent of increase for these five years?
9. The value of the fertilizers used in the United
States in 1899 was 149,099,939. If by home mixing of
fertilizers and more intelligent use 23 % of this could be
saved, what would be the saving to the farmers of the
United States?
IMPORTANCE OF GOOD SEED
10. A certain number of tobacco plants raised from
heavy seed produced 12.5 lbs. of tobacco; an equal num
ber of plants raised from light
seed gave only 6.4 lbs. What
was the per cent of gain by the
use of heavier seed?
11. With ordinary tobacco
seed yielding 816 lbs. of tobacco
per acre, and heavy seed yield
ing 29 % more, what would be
the number of pounds of gain
on an acre, if heavy seed were used? Tobacco selling at
8 cts. a pound, what is the gain in value ?
12. If a machine to separate heavy seed (see picture)
costs 18 and the labor of such separation costs 5 cts. for
seed for an acre, what would be the profit per acre the
first year, supposing 7 A. were to be raised?
13. What would be the per cent of profit the second
year on the same acreage? There is no cost for the
machine the second year.
TERCENTAGE 239
14. It is estimated that by using only the best varieties
of corn and selecting the seed by the bestknown methods,
the corn crop of the United States might be increased 10 %,
or 1116,662,647, in value. What is the present value of
the corn crop?
15. What could be its value if the best methods of seed
selection were used?
16. Light and heavy seeds were tested for germinating
power. It was found that with lettuce, 44 % of the light
seed germinated, 88 <fc of the heavy seed germinated ;
onions, 38 % of the light seed germinated, 85 % of the
heavy. What was the per cent of increase of the heavy
seeds over light seeds in germinating power in each case ?
17. If selected heavy cottonseed yield 8.25% more
than ordinary cottonseed, what will be the value of the
increase in yield on 600 A., averaging  of a bale to the
acre with the unselected seed, when cotton sells at $56 a
bale?
18. If cowpea stubble plowed under increases the cot
ton crop of the following year 47%, and the cotton crop
was originally  of a bale to tlie acre, what is the money
value of the stubble to the farmer, witli cotton at $55 a
bale?
19. Heavy cottonseed produces better plants than does
light seed. On 20 rows at Lamar, S.C., heavy seed
yielded 1047 lbs. of cotton ; ordinary seed yielded 944
lbs. What was the per cent of gain by the use of heavy
seeds ?
20. On another trial, heavy seed gave 1164 lbs. of
240 PRACTICAL AKITHMETIC
cotton; ordinary seed 1075 lbs. What would the gain
from the use of heavy seed amount to on a 500acre plan
tation averaging ^ of a bale per acre ? (500 lbs. equals 1
bale.)
21. What would be the value gained, with cotton at
11^ cts. per pound ? What could the planter afford to
pay to have seed separated, and still make 12 % upon the
investment ?
22. Which is most economical to buy, a. Red clover
seed at 15.20 per 100 lbs. containing only 48.06% of real
clover seeds and only 38 <fo of them alive ; or i. Seed at
$6.10 per 100 lbs. with 46.24% of real clover seed, 27.5%
of which are alive ; or c. Seed at 17.20 per 100 lbs. with
73.8% real clover seed, 89% of which are alive?
23. One lot of redtop seed at f 5.00 per bushel con
tained 77.4% of good seed; another lot at fl.lO per
bushel contained 10.48% of good seed. Which was the
cheaper? How much was paid per bushel for good seed
in each case?
PREVENTION OF PLANT DISEASES
24. In using 1 oz. of formally to 3 gals, of water as
a steep for oat seed to prevent smut, what per cent of
formalin is used? (1 gal. of water weighs 8J lbs.)
25. How many pounds, of formalin will be needed for
25 A., allowing 1 gal. of mixture to each bushel of oats
and 2 bus. of oats to the acre ? At 38 cts. a pound, what
will it cost ?
26. Formalin, being 40% formaldehyde, what per cent
of formaldehyde is there in a loz.to3gal. mixture?
PERCENTAGE
241
27. The average net profit from spraying potatoes in
experiments carried out during 1906 was 113.89 per
acre ; the net profit from similar experiments in 1905 was
120.04, in 1904 $24.06, in 1903 123.43. What was the
average benefit for all these years? What would such
amount to on 40 A.?
28. Apple trees sprayed and not sprayed for the pre
vention of worminess yielded as follows :
Total
Yield
Windfalls
Picked Fruit
Total
Number
OF
Apples
Per
cent
OP
Sound
Fruit
Wormy
Not
wormy
Total
Wormy
Not
wormy
Total
Sprayed r
Treel
Tree 2
Tree 3
Bushels
U
13.25
13.75
No.
20
11
26
No.
No.
188
102
78
No.
153
129
206
No.
1754
1605
1.562
No.
Total
Unsprayed :
Treel
Tree 2
Trees
11.75
6.875
5.50
464
224
315
502
248
404
1258
697
564
383
488
428
Total
Copy table and fill each blank.
29. A block of 69 sprayed trees yielded : merchantable
fruit, 255 bus. ; culls, including windfalls, 36.5 bus. What
was the per cent of merchantable fruit?
30. The crop from 10 unsprayed Winesap trees was :
salable fruit, 6.75 bus. ; culls, including windfalls, 10.25
bus. What was the per cent of salable fruit?
31. The use of Bordeaux mixture to prevent the downy
mildew of cucumbers in New York State at an expen
diture of 19.50 an acre increased the profit 1163.50
242 PRACTICAL ARITHMETIC
an acre. What was the per cent of gain on the amount
invested?
32. Three trees affected with bitter rot yielded 21.1
bus. of apples, 188 sound apples and 4244 diseased apples.
Six similar trees, sprayed to prevent the rot, yielded 101.3
bus., 8674 sound and 989 diseased. What was the per
cent of increase in bushels by spraying? In sound apples?
Spraying costs each time 3 cts. per tree. They were
sprayed 3 times. What was the cost of spraying? What
was the profit with apples at 85 cts. a bushel?
33. A man plants two fields of 10 A. each with corn.
In A he uses seed corn selected in the field ; in B he
uses seed selected in the barn. The cost of fertilizers
and labor for each field was flO an acre. Field A
produced 45 bus. and field B, 36 bus. an acre. The man
sold the corn at $.60 a bushel. Allowing f4 an acre for
rent of land, what was the profit on each field? What
per cent was gained by selecting seed in the field?
34. Two farmers plant 5 A. each in peanuts, using
the same kind of seed. Mr. A spends f 1.50 for lime,
$3.00 for commercial fertilizer, and $2.00 for land plaster
to the acre. Mr. B uses the same as the above, except the
lime. Rent of land and cost of labor was $7.00 an acre for
each. Mr. A made 1500 lbs. and Mr. B 1000 lbs. of
peanuts an acre, which they sold at 3 cts. a pound.
How much more did Mr. A get for his crop than Mr. B ?
What per cent was made on the money invested in lime?
ROOT TUBERCLES
35. Cowpeas with tubercles upon their roots yield
139.29 lbs. of nitrogen, without tubercles 118.45 lbs.;
PERCENTAGE
243
soy beans with tubercles 113.55 lbs., without 75.98 lbs.
What is the per cent of additional nitrogen with each of
these crops when tubercles are present?
36. If 100 lbs. of cowpea tops contain: nitrogen 1.84
lbs., phosphoric acid .67 lbs., potash 1.29 lbs.; and 100 lbs.
of cowpea roots contain: nitrogen 1.47 lbs., phosphoric
acid .67 lbs. and potash 1.43 lbs., — what per cent of each
of these foods is in the root ? What per cent in the tops ?
In a harvest of 3.2 T. per acre, how much nitrogen is re
moved ? How much potash ? How much phosphoric acid ?
37. The following amounts of fertilizers in each 100
lbs. of dry substance were found in leguminous crops.
Plant and
Part
NiTEOOEN
Phosphoric
AniD
Potash
Red clover
Tops
Roots
Alfalfa
Tops
Roots
Crimson clover ....
Tops
Roots
Cowpea
Tops
Roots
Pounds
2.28
2.74
2.89
2.04
2.72
1.50
2.79
1.46
Pounds
0.72
0.84
0.53
0.43
1.10
0.47
0.57
0.16
Pounds
1.40
0.82
1.46
0.48
1.56
1.02
2.00
.77
In each crop what per cent of the top is nitrogen?
What per cent phosphoric acid? What per cent potash?
What per cent of the roots is nitrogen? What per cent
potash? What per cent phosphoric acid? By what per
cent does the nitrogen of red clover exceed that of cow
peas
244
PRACTICAL ARITHMETIC
COMMERCIAL FERTILIZERS
312. Commercial fertilizers are used for the nitrogen,
phosphoric acid, and potash they contain. The nitrogen,
phosphoric acid, and potash are
obtained from different sub
stances. Some of these sub
stances contain one, some two,
and some all of these plant
foods. The substances used
for supplying nitrogen, phos
phoric acid, and potash in com
mercial fertilizers contain dif
erent per cents of these plant
foods. The composition of
some of these substances is
quite uniform or constant, while in others the per cent of
nitrogen, phosphoric acid, and potash varies considerably.
200 POUNDS
AMMONIATED^
FERTIUZER
MANUFACTURED BY
JOHN DOE i. CO
ATLANTA GA.
GUARANTEED ANALYSIS^
AVAILABLE PHOSPHOR ^
IC ACID 8 PERCENT
NITROGEN 3
POTASH 3
EXERCISE 207.— WRITTEN
1 If ammonia is 82.4% nitrogen, how many pounds
of nitrogen are there in a ton of fertilizer that has 2% of
ammonia ?
2. If there is 1 lb. of nitrogen in 1.214 lbs. of ammonia,
how many pounds of nitrogen will there be in a ton of
commercial fertilizer which has 6.07% of ammonia?
3. What per cent of nitrogen has a fertilizer which has
3.642% of ammonia?
4. What per cent of ammonia has a fertilizer that has
2 % of nitrogen ?
PERCENTAGE 245
5. If a ton of cottonseed contains 60 lbs. of nitrogen,
what per cent of cottonseed is nitrogen '?
6. If cottonseed meal contains 6.2% nitrogen, 2.8%
phosphoric acid, 1.8% potash, how many pounds of each
are there in a ton of cottonseed meal ? What is the ton
worth for fertilizer, valuing nitrogen at 19 cts. per pound,
phosphoric acid at 4 cts., and potash at 5 cts. ?
7. If cottonseed meal has 7.5% of ammonia, how
many pounds of nitrogen will there be in a ton of cotton
seed meal? How much will the nitrogen be worth at 18
cts. per pound? How much, when the cottonseed meal
has 8% ammonia?
8. If there are 124 lbs. of nitrogen in a ton of
cottonseed meal, what per cent of cottonseed meal is
nitrogen ?
9. If there is 15.8% of nitrogen in nitrate of soda,
how many pounds of nitrogen are there in a ton of
nitrate of soda. What will the ton of nitrate of soda be
worth, valuing nitrogen at 18 cts. per pound?
10. If a ton of cottonseed contains 26 lbs. of phosphoric
acid, what per cent of cottonseed is phosphoric acid?
11. If there are 14 lbs. of nitrogen in 17 lbs. of
ammonia, and 15.8 lbs. of nitrogen in 100 lbs. of nitrate
of soda, how many pounds of ammonia will it take to
furnish as much nitrogen as 1 T. of nitrate of soda?
12. If a farmer mixes a fertilizer so as to contain 800
lbs. of a 16 % acid phosphate to the ton, what per cent of
phosphoric acid will his fertilizer contain ?
13. A farmer is offered a ton of 16 % acid phosphate
246 PRACTICAL ARITHMETIC
at 112.75 a ton, 14% acid phosphate at $12.00 a ton, or
a ton of 12% acid phosphate for $11.00. Which is the
cheapest, and liow much cheaper, valuing phosphoric acid
at 4 cts. a pound?
14. Which is cheaper, a ton of a 282 fertilizer (i.e.,
one containing 2 % of nitrogen, 8 % of phosphoric acid, and
2 % of potash) at $18, or a ton of a 383 fertilizer (i.e., one
containing 3 % of nitrogen, 8 % of phosphoric acid, and 3 %
of potash) at $22, valuing nitrogen at 20 cts. per pound,
phosphoric acid at 4 cts. per pound, and potash at 5 cts.
per pound?
15. If there are 250 lbs. of potash in a ton of kainit,
what per cent of potash is there in kainit?
16. If a ton of cottonseed contains 24 lbs. of potash,
what per cent of cottonseed is potash?
17. Sulphate of potash contains 4 times as much potash
as kainit contains. What per cent of sulphate of potash is
potash, if there are 48 lbs. of potash in 384 lbs. of kainit?
18. If there are 1000 lbs. of potash in a ton of muriate
of potash, what per cent of muriate of potash is potash? ,
19. If there are 36 lbs. of potash in a ton of cottonseed
meal, what per cent of cottonseed meal is potash?
20. If kainit contains 12^% of potash, and muriate of
potash contains 50 % potash, how many pounds of kainit
will it take to supply as much potash as there is in 40 lbs,
of muriate of potash?
21. If a ton of dried blood contains 280 lbs. of nitrogen,
what per cent of dried blood is nitrogen?
PERCENTAGE 247
22. If a ton of fish scrap contains 180 lbs. of nitrogen,
wliat per cent is nitrogen?
23. If fish scrap contains 8.25% of nitrogen and 6% of
phosphoric acid, what is a ton of fish scrap wortli, allow
ing 20 ots. per pound for nitrogen and 4 cts. per pound
for phosphoric acid?
24. If a ton of wood ashes contains 120 lbs. of potash,
what per cent of wood ashes is potash? If a ton contains
130 lbs. of potash, what per cent of wood ashes is potash?
25. If there are 140 lbs. of phosphoric acid in a ton of
fish scrap, what per cent of fish scrap is phosphoric acid?
26. If there are 320 lbs. of phosphoric acid in a ton of
acid phosphate, what per cent of acid phosphate is phos
phoric acid? If there are 280 lbs., what per cent is phos
phoric acid? If 240 lbs., what per cent is phosphoric
acid? If 200 lbs., what per cent is phosphoric acid?
27. If there are 56 lbs. of phosphoric acid in a ton of
cottonseed meal, what per cent of cottonseed meal is
phosphoric acid?
28. What is a ton of acid phosphate analyzing 16 %
phosphoric acid worth, when phosphoric acid is worth
4 cts. a pound?
29. How much is a ton worth when it analyzes 14%,
12%, 10%, and 8% respectively?
30. If there are 329.4 lbs. of nitrogen in a ton of
nitrate of soda, what per cent of nitrate of soda is
nitrogen ?
The composition of many fertilizing materials varies
considerably, but the following is a fair average:
248
PRACTICAL ARITHMETIC
COMPOSITION OF FERTILIZING MATERIALS
Acid phosphate
Acid phosphate
Acid phosphate
Giound phosphate rock . . .
Tobacco stems
Sulphate of potash (high grade)
Muriate of potash
Nitrate of potash
Kainit .■
Wood ashes (unleached) . .
Cottonseed meal
Cotton seed
Tankage (concentrated) . . .
Dried blood (high grade) . .
Fish scrap
Nitrate of soda
Sulphate of ammonia ....
Per Cent of
Per Cent of
Nitrogen
Potash
1.5
5
50
50
13
45
12.5
6
6.2
1.8
3
1.2
12
14
9
15.6
20.5
Per Cent op
Phosphoric
AoiD
16
14
13
32
2
1.5
2.8
1.3
1.5
31. If in making a ton of fertilizer containing 3.2235%
of nitrogen, 9.099% of phosphoric acid, and 3.5815% of
potash, 100 lbs. of nitrate of soda are used, what number
of pounds of cottonseed meal, 16% acid phosphate, and
of muriate of potash must be used ?
32. What per cent of nitrogen, phosphoric acid, and
potash will there be in a ton of fertilizer composed as
follows: 1000 lbs. of 16% acid phosphate, 700 lbs. of
cottonseed meal, 100 lbs. of nitrate of soda, and 200 lbs.
of muriate of potash?
33. What will be the per cent of nitrogen, phosphoric
acid, and potash in a fertilizer composed of the following
PERCENTAGE
249
materials : 800 lbs. of cottonseed meal, 800 lbs. of 16 %
acid phosphate, 400 lbs. of kainit ?
34. How many pounds of nitrogen, phosphoric acid, and
potash will the ton of fertilizer mentioned in the last prob
lem contain, and what will it be worth at 18 cts. a pound
for nitrogen, 4 cts. a pound for phosphoric acid, and 5 cts.
a pound for potash, adding 5 for mixing, bags, and
freight ?
FEEDS AND FEEDING
313. If corn contains 10.3 % of protein and 76 % of this
protein is digestible, what per cent of digestible protein is
there in corn?
If corn contains 72.5% of carbohydrates and 92% of
these are digestible, what per cent of digestible carbohy
drates is there in corn?
If 5 % of corn is fats and 86 % of these fats are digest
ible, what per cent of digestible fat is there in corn?
314. Table of digestible nutrients in certain feeds.
Feeds
Corn
Oats
Wheat bran . .
Wheat middlings
Cottonseed meal
Timothy hay . .
Redclover hay
Cowpea hay . .
Alfalfa hay . . .
Corn stover . . .
Corn silage . . .
Skim milk . . .
Per Cent op
Protein
Per Cent op
Carbohydkates
Per Cent
OF Fats
7.8
9.22
12.2
12.8
37.2
2.8
6.8
10.8
11.10
1.7
0.9
90
66.7
47.3
39.2
53.0
16.9
43.4
35.8
38.6
39.6
32.4
11.3
5.2
4.3
4.2
2.7
3.4
12.2
1.4
1.7
1.1
1.2
0.7
0.7
0.3
250
PRACTICAL ARITHMETIC
315. A Ration is the amount of feed given to an animal
during 24 hrs.
316. A Balanced Ration is one which contains the dif
ferent nutrients in sucli amounts as best to meet the needs
of the animal being fed.
317. Scientists have formulated balanced rations or the
nutritive requirements of the various kinds of farm ani
mals at different stages of growth and when being fed
for different purposes, calculated for 1000 lbs. live
weight. These are known as Feeding Standards, and are
used as guides in practical feeding in compounding
rations.
318. The following table shows the amounts of digest
ible nutrients in feeding standards, calculated for 1000
lbs. live weight :
FEEDING STANDARDS
Kind of Animal
DiGESTiiiLE Nutrients
Protein
Carbohy
drates
Fats
Fattening cattle (first period) . . .
Fattening cattle (second period) . .
Fattening cattle (third period) . .
Horse (heavy work)
Horse (light work)
Dairy cow (giving 16i lbs. milk daily)
Dairy cow (giving 22 lbs. milk daily)
Fattening swine (first period) . . .
Fattening swine (second period) . .
Fattening swine (third period) . .
Growing cattle (6 to 12 mos. old)
Growing swine (3 to 5 mos. old) . .
Poimds
2.5
3.0
2.7
2.5
1.5
2.0
2.5
4.5
4.0
2.7
3.5
5.0
Pounds
15.0
14.5
15.0
13.3
9.5
11.0
13.0
25.0
24.0
18.0
12.8
23.1
Pounds
0.5
0.7
0.7
0.8
0.4
0.4
0.5
0.7
0.5
0.4
1.5
0.8
PERCENTAGE • 251
When feeds rich in protein are high in price, the
amount of protein may be reduced 10 % without injury
to the ration.
319. How nearly will the following ration meet the
" Feeding Standard " requirements of a dairy cow giving
22 lbs. of milk daily?
Ration No. 1. 35 lbs. corn silage, 8 lbs. cowpea hay, 6
lbs. corn stover, 3 lbs. cottonseed meal, 4 lbs. corn.
To find total protein in ration :
By reference to table on page 249 it is found that corn
silage contains .9 per cent of digestible protein. If there is .9
per cent of protein in corn silage, there is .9 lb. in 100 lbs. ; there
fore, in 1 lb. of silage there is .9 = 100 = .009 lb. If in 1 lb.
of silage there is .009 lb. of protein, in 35 lbs. there is 35 x .009
= .315 lb. of protein.
Ration :
35 lbs. silage = 35 x .009 = .315 lb. protein
8 lbs. cowpea hay = 8 x .108 = .864 lb. protein
6 lbs. corn stover ' = 6 x .017= .102 lb. protein
3 lbs. cottonseed meal = 3 x .372 = 1.116 lbs. protein
4 lbs. corn = 4 x .078 = .312 lb. protein
Total = 2.709 lbs. protein
By reference to the table of feeding standards, page 250, it
is found that a dairy cow giving 22 lbs. milk a day should
receive 2.5 lbs. of protein. In the ration above we have found
that she receives 2.709 lbs. The ration therefore contains
.209 lb. more protein than is required.
To find total carbohydrates in ration :
By reference to table (page 249), it is found that corn silage
contains 11.3 per cent of digestible carbohydrates. If there is
11.3 per cent of carboliydrates in corn silage, there are 11.3 lbs
252 • PRACTICAL ARITHMETIC
in 100 lbs. ; therefore, in 1 lb. of silage there are 11.3 f 100 =
.113 lb. of carbohydrates. If in 1 lb. of silage there is .113 lb.
of carbohydrates, in 35 lbs. of silage there are 35 X .113=3.995
lbs. of carbohydrates.
35 lbs. silage = 35 x .113 = 3.995 lbs. carbohydrates
8 lbs. cowpea hay = 8 x .386 =3.088 lbs. carbohydrates
6 lbs. corn stover = 6 X .324 = 1.914 lbs. carbohydrates
3 lbs. cottonseed meal = 3 x 169 = .507 lb. carbohydrates
4 lbs. corn = 4 x .667 = 2.668 lbs, carbohydrates
Total = 12.162 lbs. carbohydrates
By reference to the table of feeding standards (page 250),
it is found that a dairy cow giving 22 lbs. of fnilk a day needs
13 lbs. of carbohydrates. In the ration above she receives
12.162 lbs. The ration therefore lacks .832 lb. of carbo
hydrates.
To find total fats in ration :
By reference to table, it is found that corn silage contains
.7 per cent of digestible fats. If there is .7 per cent of fats in
corn silage, there is .7 lb. in 100 lbs. ; therefore, in 1 lb. of
silage there is .7 f 100 = .007 lb. of fats. If in 1 lb. of silage
there is .007 lb. of fats, in 35 lbs. there is 35 x .007 = .245 lb.
of fats.
35 lbs. silage = 35 x .007 = .245 lb. fats
8 lbs. cowpea hay = 8 x .011 = .088 lb. fats
6 lbs. corn stover = 6 x .007 = .042 lb. fats
3 lbs. cottonseed meal = 3 x .122 = .366 lb. fats
4 lbs. corn = 4 X .043 = ^172 lb. fats
Total = .913 lb. fats
By reference to table of feeding standards, it is found that
a dairy cow giving 22 lbs. of milk a day needs .5 lb. of fats.
In the ration above, there is .913 lb. of fats, or an excess of
.413 lb.
In a similar manner the amounts of protein, carbohydrates,
and fats in the different feeds in any ration may be computed.
PERCENTAGE 253
320. TABLE SHOWING PRICES OF FEEDS
Feeds
Estimated
Price
Local Price
Corn, per bu
Oats, per bu
^Vheat bran, per T. . .
Wheat middlings, per T.
Cottonseed meal, per T.
Timothy hay, per T.
Redclover hay, per T. .
Cowpea hay, per T. . .
AKalfa hay, per T. . .
Corn stover, per T. . .
Corn silage, per T.
Skim milk, per cwt. . .
I 0.49
0.37
18.00
19.00
30.00
12.00
12.00
12.00
12.00
5.00
3.00
0.20
Pupils should ascertain local prices of such feeds as are
used in their neighborhood, and complete column three.
EXERCISE 208.— "WRITTEN
1. Wherein will the following ration fail, according
to the feeding standard, in meeting the requirements of
a dairy cow giving 16.5 lbs. milk a day ?
Ration No. 2. 35 lbs. corn silage, 10 lbs. corn stover,
5 lbs. corn, 5 lbs. wheat bran.
2. Construct a ration from the feeds used in your
locality (if given in the table) that will better meet the
requirements of this cow, and calculate and compare the
cost of this ration with the one given.
3. Wherein and how much will the following rations
fail, according to the feeding standard, to meet the needs
of a horse doing hard work?
254 PRACTICAL ARITHMETIC
Ration No 3. 10 lbs. corn stover, 6 lbs. cowpea hay,
11 lbs. corn, 2 lbs. cottonseed meal.
Ration No. 4. 15 lbs. timothy hay, 14 lbs. corn.
Ration No. 5. 15 lbs. timothy hay, 16 lbs. oats.
Ration No. 6. 4 lbs. timothy hay, 10 lbs. alfalfa, 7 lbs.
corn, 8 lbs. oats.
4. It is seen that ration No. 3 is much nearer the
requirements of sach a horse than ration No. 4. Which
is cheaper?
5. "Wherein will the following ration fail, according
to the feeding standard^ in meeting the needs of a horse
doing light work ?
Ration No. 7. 7 lbs. corn stover, 8 lbs. alfalfa hay,
6 lbs. corn.
6. If a draught horse weighing 1500 lbs. requires 37^%
more nutrients than one weighing 1000 lbs., what will be
the weight of the different nutrients required daily by
such a horse when doing hard work? (For nutritive
requirements for a 1000lb. horse, see table of feeding
standards, page 250.)
7. Construct a ration that will approximate the needs of
such a draught horse, and compare its cost with a similarly
composed ration for a 1000lb. horse.
8. Wherein will the following ration fail to meet the
requirements of a fattening beef animal during the third
portion of the feeding period ?
Ration No. 8. 35 lbs. corn silage, 15 lbs. corn stover,
4.5 lbs. cottonseed meal, 6 lbs. corn.
9. How nearly will the following meet his needs dur
ing the second part of the feeding period ?
PERCENTAGE 255
Ration No. 9. 15 lbs. corn stover, 10 lbs. clover hay,
1 lbs. cottonseed meal, 8 lbs. corn.
10. Wherein will the following ration fail to meet his
requirements during the first part of the feeding period?
Ration No. 10. 15 lbs. corn stover, 10 lbs. timothy
hay, lo lbs. corn.
11. Find the weights of the different nutrients that
would be required daily by a beef animal weighing 750
lbs. for each of the three fattening periods, if the same
amounts of nutrients are required for eachlOOlb. live
weight as for each 100 lbs. of a 1000lb. aifimal.
12. If pigs during the second part of the fattening
period require daily for 1000 lbs. live weight, 4 lbs.
protein, 24 lbs. carbohydrates, and .5 lb. fats, wherein
will the following ration fail to meet their needs?
Ration No. 11. 24 lbs. corn, 10 lbs. wheat middlings,
25 lbs. skim milk.
13. Wherein will 40 lbs. of corn fail to meet their re
quirements?
14. Wherein will 20 lbs. of corn,, and 20 lbs. of wheat
middlings fail to meet their requirements ?
15. Ascertain the rations being fed to the various kinds
of animals in your locality, and compare them with the
feeding standards given in table, page 250. Compare
cost at local prices.
16. Ascertain local prices of feeds and construct rations
that will come near to the requirements of the feeding
standards for each of the animals for which sample rations
have been given. Compute cost of each.
PROFIT AND LOSS
EXERCISE 209. — ORAL
1. I bought a book for $1 and sold it for 10% more
than I paid. What was the selling price?
2. When 10% is gained, what part is gained?
3. A grocer sold butter for 30 cts. a pound that cost
20 cts. What was his per cent of gain?
4. Damaged goods bought at 125 were sold for $100.
What was the per cent of loss?
5. If 1120 is the cost price and 16% is the gain, what
is the selling price ?
321. The per cent of gain or loss in a business transac
tion is always reckoned on the cost or the sum invested.
322. A merchant buys goods for $125 and sells them
for $110. Does he gain or lose by the transaction?
$ 125 = cost
$ 110 = selling price
$126 f 110= f 15 loss
$15 =$126= 12% loss.
•
EXERCISE 210. — "WRITTEN
1. A crate of berries, worth $6.40, was delayed in ship
ping and was sold at a loss of 25 % . What was the loss
on the crate.
256
PERCENTAGE 257
2. A man bought a team for $280 and sold it at 15%
profit. What did he gain?
3. A man bought merchandise for $472, and in sell
ing gained $67. What was the per cent of gain?
4. A storekeeper bought goods for $679.50 and sold
them for $665.00. What was the per cent of loss?
5. A quantity of dress goods sold for $67 at a loss of
15%. What did it cost?
6. A transaction nets 18 % profit. How much must
be invested to gain $650?
7. A steamboat sold for $117,600, 12% more than it
cost. What did it cost?
EARM PROFITS
8. With wheat yielding 20 bus. to the acre at 75 cts.
a bushel, the land, 66 A. at $40 an acre, costing in
rent 5 % of its value, and the cost of raising the crop
amounting to $6 an acre, what is the profit? What is
the per cent of profit ?
9. With potatoes yielding 200 bus. to the acre at 40
cts. a bushel, the land 12^ A. at $ 50, costing in rent
5 % of its value, and the cost of raising amounting to
$ 20 an acre, what is the profit ? What is the per cent
of profit ?
10. With onions yielding 500 bus. at 50 cts. a bushel,
the land 4 A. at $ 100 an acre, costing in rent 5 % of
its value, and the cost of raising amounting to $100 an
acre, what is the profit ? What is the per cent of profit ?
258 PKACTICAL ARITHMETIC
11. Suppose in problem 8 the crop weighs 4 T. ; in
problem 9, 40 T. ; in problem 10, 50 T. ; and the farm
being 5 mis. from town, marketing costs 12.50 a ton.
How much will the results in each of the last three prob
lems be changed, if you consider cost of marketing ?
12. Suppose wheat be raised under the conditions of
problem 8, on land valued as that in problem 10, but with
the yield increased 50 % above that in problem 8. What
Avill be the profit or loss ?
13. Suppose onions to be raised under the conditions
of problem 10, on land valued as that of problem 8, with
a production decreased 75 % below that of problem 10.
What will be the profit or loss ?
14. When wellsorted apples bring $2.00 a barrel and
poorly sorted apples f 1.50 a barrel, what will be the
profit on sorting 13 bbls. of apples, if in sorting, 2 bbls.
of culls worth f 1.00 a barrel be taken out ? Estimate the
labor of sorting at 75 cts. What is the per cent of profit
upon the investment ? (Note that the expense of sorting
is the only item of expense in the investment.)
15. Suppose there were no sale for the culls and they
are wasted. What will then be the profit of sorting ?
16. If a miller buys wheat at 86 cts. a bushel and in
grinding obtains 72 % of flour, what per cent profit does he
make in selling flour at $ 5.85 a barrel (196 lbs.), if the cost
of milling is ifl.Sl per barrel of flour and he gets 33 cts.
for the byproducts ?
17. At what per cent above cost must I sell  of a city
lot so that I may retain ^ of it for ray own use, free of cost ?
PERCENTAGE 259
18. With a farm worth $4000, buildings 1 1000, teams
and tools 11000, and with total sales amounting to $1500,
allowing for use of the money 5 %, depreciation in team
and tools 10 %, depreciation and repairs on buildings 5 %,
taxes and insurance f 50, labor f 200, supplies $ 200, what
is the profit ? What per cent is paid on the investment?
19. If by $300 additional labor the total sales can be
increased to $2000, what is then the net profit? What
the per cent of profit? What the net profit on the $300
additional labor ? What the per cent of profit on this ?
20. If I sell  of an article for the price which I paid
for the whole article, what per cent have I gained ?
21. A merchant marks his goods at 25 % above cost.
At what per cent below the marked price must he sell to
make 17^ % ?
22. At what price above cost must this merchant mark
his goods so that after he has deducted 15 % from the
marked price, he will still have remaining a profit of 121 ^?
23. A buys an engine for $125. He trades this for
another piece of machinery, and receives in addition 20 %
of the cost of the first in money. The second piece of
machinery is sold for $ 90. Has he gained or lost on his
investment, and what per cent ?
COMMISSION
EXERCISE 211. — ORAL
1. A trucker sliips 100 baskets of lettuce at fS a
basket, and payg 8 % to the commission merchant for
selling. What does he pay for selling?
2. What are the net proceeds from the shipment?
3. A real estate agent receives 3 % for collecting rents.
If his collections amount to #500 in a month, what is his
percentage ?
323. A person who buys or sells goods, or transacts
business for another, is called a Commission Merchant,
Agent, or Broker.
324. The pay is usually reckoned at a certain per cent
of the price, and is called a Commission or Brokerage.
325. The sum left after the commission and other
expenses have been paid is called the Net Proceeds.
EXERCISE 212. — "WRITTEN
1. What will be an agent's commission for collecting
rents amounting to $ 675 at 2 % commission ?
2. If an agent charges Sj'fo for handling supplies for
an orphanage, and his commission amounts to f 750, what
was the amount of supplies purchased?
260
PERCKNTAGE 261
3. A man has $1250.00 to invest with a land company.
He pays his agent $40.75 for making the investment.
Wliat was the rate charged by the agent?
4. A clerk in a city store receives a salary of $1200
a year ; in addition he receives a commission of 2 % on
all goods he sells. If he receives $1500 for the year, what
is the amount of his sales ?
5. An agent receives $67.84 commission on sales. At
5 ojo what was the amount sold ?
6. A lawyer collected a debt of $96 at 18% commis
sion. What did he receive?
7. What is the commission on 20 crates of eggs at
$4.50 a crate, commission 10^? On 250 T. of cabbages at
$3.30 a ton, commission 8% ?
8. If a farmer ships 38 baskets of beans to a commis
sion merchant in Baltimore, who sells them for $1.25 a
basket, how much money should the farmer receive if
the merchant charges 8% commission for selling, and
pays 40 cts. freight per basket?
9. A strawberry grower ships 27 crates of straw
berries to a commission merchant in New York City, who
sells them for $3.25 a crate. After deducting his com
mission, paying drayage of 3 cts. a crate, and refrigerator
car charges of 75 cts. a crate, he remits $59.67 to the
grower. What per cent commission did he charge?
10. A farmer ships 34 crates of cabbages to a commis
sion merchant in Philadelphia, who sells them for $1.50
a crate. How much should the merchant send the farmer
after deducting 8 % commission, paying drayage charges
of 5 cts. a crate and freight charges of 50 cts. a crate ?
262 PRACTICAL ARITHMETIC
11. A livestock dealer ships a carload of 27 cattle
weighing 1437 lbs. each to Chicago, and the commission
man sells them for $5.75 per cwt. After paying $1.00
switching charges on the car, $5.00 yardage, 600 lbs. hay
at iflO.OO a ton, 400 lbs. corn at 50 cts. a bushel (1 bu. =
56 lbs.), $59.00 freight, and charging 50 cts. a head for
selling, what will be the amount of net proceeds sent
to the shipper?
12. What per cent of the gross receipts are the total
charges connected with selling the cattle in problem 11 ?
13. If this same carload of cattle had been shipped to
Richmond, Va., and the freight charges and selling price
had been the same, what would have been the net pro
ceeds returned to the shipper, if the other charges had
been as follows? Commission for selling 4%, 600 lbs.
hay at $20 per ton, 400 lbs. corn at $1 a bushel, yard
age $10.
14. What per cent on gross receipts are the charges
connected with selling the cattle in problem 13 ?
15. If a grower ships 27 bbls. of potatoes to a com
mission merchant in Philadelpliia, who sells them for
$2.75 a barrel, pays 50 cts. a barrel freight, 5 cts. a barrel
drayage, and remits $53.46 to the grower, what per cent
commission did the agent charge for selling?
16. A publisher sells at auction for various customers
rare books which bring respectively: $125.50, $62.25,
$187.50, $285.00, $845.25, $35.50, $48.75, $88.00,
$672.00, and $950.50. If he charges 8% commission,
what does he receive for tlie transaction ?
COMMERCIAL DISCOUNT
EXERCISE 213. — ORAL
1. If a merchant buys a bill of goods marked f 500
and receives 10 % reduction for paying cash, what does
he pay?
2. A school liistory costs 50 cts. If 25 % is allowed
for an old textbook taken in exchange, how much cash
is paid for the new history?
3. If a grocer allows 10 % from his bill for cash, what
amount will he deduct from a bill of l25?
326. A deduction from the price or value of anything
is called a Commercial Discount.
Manufacturers and dealers in merchandise often issue
a Price List from which certain discounts are allowed.
327. Prices published in catalogues known as List
Prices are often subject to discounts.
328. Sometimes several discounts are allowed to the
buyer. In cases of this kind the first discount is to be
deducted, then the second computed upon the remainder
and deducted, and so on for each discount; e.g., when
40%, 10%, and 5% discounts are allowed from a bill of
goods, the 40% is first deducted, then 10% from the
remainder, and then 5% is deducted from the last re
mainder. The amount remaining after deducting the
discounts is the Net Price or Net Amount.
263
264
PRACTICAL ARITHMETIC
EXERCISE 214— WRITTEN
1. A chemist buys a bill of laboratory supplies from
A. H. Newsom & Co., New York. He is allowed 50%
and 10% discount from list price and 5% off for cash.
What is the discount on an original bill of f 350 ?
2. Mr. A's grocery bill for the month was $34.75;
his cash discount was 12.78.
was allowed for cash ?
What per cent of discount
3. A discount of 10 % is offered on all purchases dur
ing a certain season and 5 % additional for cash. What
is the discount on a bill of .$249.60? On $ 678.00 ?
4. The bill of a certain gas company reads :
" The following discount will be allowed if this bill is paid
at the office of the company on or before the fifteenth instant :
100 to 2000 cubic feet, monthly,
$ 1.26 per 1000 feet.
Over 2000 to 6000 cubic feet,
monthly, $ 1.25 per 1000 feet less
5%.
Over 5000 to 10,000 cubic feet,
monthly, $ 1.25 per 1000 feet less
10%.
Over 10,000 to 15,000 cubic feet, monthly, $1.26 per 1000
feet less 15 % .
Over 15,000 to 20,000 cubic feet, monthly, 1.25 per 1000
feet less 20%.
20,000 cubic feet or over, monthly, light and fuel, f 1.25 per
1000 feet less 25%."
What is the net cost of the following amounts of gas :
1990 cu. ft. ? 2100 cu. ft. ? 3500 cu. ft. ? 4900 cu. ft. ?
6100 cu. ft.? 9000 cu. ft.? 9900 cu. ft.? 10,000 cu. ft.?
INSURANCE
EXERCISE 215, — OBAL
1. What will be the cost of insuring a building worth
$6000 for  its value at 40 cts. on a hundred dollars?
2. A merchant insures his store valued at $5000 for 
its value at 1%. What does it cost him?
3. I pay i96 annually for a life insurance of $4000.
What is the rate per thousand?
329. Insurance is the promise of indemnity for personal
and property losses.
330. Insurance is of two kinds — Property: fire, hail,
tornado, etc. ; Personal : life, accident, health.
331. The written contract between the insurance com
pany and the person insured is called the Policy.
332. The Premium is the sum paid for the insurance.
EXERCISE 216. — WRITTEN
LIFE INSURANCE
1. What is the premium on a 14000 policy at $28 per
$1000?
2. If a $1000 policy of life insurance costs the insured
$32.16 each year, what will be the total cost of a $4000
policy for 20 yrs. ?
265
266 PRACTICAL ARITHMETIC
The rates per $ 1000 on a policy in a certain company
are : at 20 yrs. of age, 114.96; at 30, $19.06; at 40,
$26.07 ; at 50, $38.92 ; at 60, $63.42.
3. What would be the cost of a $4000 policy per year
if taken at 20 yrs. of age? What if taken at each of the
other years given?
4. If taken at 20 yrs., what would be the total cost if
carried 20 yrs. ? What would it be if taken at each of
the other ages and carried for 20 yrs. ?
5. What is the difference in premiums, per thousand,
between a policy for a man of 20 yrs. and a policy for a
man of 40? Of 60?
6. A participating 20yr. endowment policy is one
which pays the insured the face of the policy at the end of
20 yrs., and in addition thereto profits or dividends. If this
policy costs $48.10 each j'ear and the profit at the end of
the 20 yrs. is $316.00, what per cent of the total premiums
is this profit ?
PROPERTY INSURANCE
7. Insurance on a shingleroofed dwelling with a
water supply and fire protection costing 50 cts. on $100
for one year, what is the cost of insurance to the amount
of $4500?
8. If it costs twice as much to carry the insurance
3 yrs. as it does to carry it 1 yr., what will be the cost
per year in the last problem if the owner insures for a
period of 3 yrs. ?
9. If it costs three times as much to insure for a term
of 5 yrs. as for 1 yr., what is the rate per year in prob
lem 7 if taken for a term of 5 yrs. ?
PERCENTAGE 267
10. Each dwelling within 30 ft. adds 15 cts. per f 100
to tlie cost of insurance. What will the insurance in
problem 7 cost if there is a building 29 ft. 6 ins. to the
right of this dwelling and another 32 ft. to the left?
What will it be if there is a dwelling 30 ft. to the right
and another 30 ft. to the left?
11. Suppose the dwelling of problem 7 were slate in
stead of shingle roof. The rate then might be 40 cts. per
f 100. How much less would the insurance be?
12. Suppose the dwelling instead of being in a city
well protected from fire were in a city with no fire protec
tion, the rate might then be 75 cts. per $100 under the
conditions of problem 7. What would be the cost per
year ?
13. The rates on certain classes of unprotected property
are very high. Sawmills sometimes cost 14 % yearly.
What would be the premium for 1 yr. on a $3500 saw
mill insured for  value ?
14. What per cent of the face of the policy is the yearly
premium in problem 7 ?
15. The insurance rate in the country being 1 %, what
is the premium on the insurance of a barn with contents
valued at $2700, if the property is insured for  of its
value?
16. What is the premium on a country dwelling valued
at $6400 at the same rate, if the property is insured for J
of its value?
17. The rate being %, or 75 cts., on the hundred, if
the buildings are metal roofed, what is then the premium
in problem 16 ?
TAXES
EXEBCISB 217. — ORAL
1. If a man pays annually for public purposes 2% of
the value of his property, which is f 7000, what amount
does he pay ?
2. Mr. A pays for public purposes 2% on his real
estate, valued at $4000, 2J% on money in bank, which is
$1000. What amount does he pay for public purposes?
3. If a man pays annually $200 upon real estate for
public purposes, which is 2% of its value, what is the
value of his property?
333. Taxes are required from individuals for the sup
port of the government and for other public purposes.
334. Land, buildings, and other fixed property is
called Real Estate.
335. ^lovable property, as household goods, clothing,
jewelry, mortgages, livestock, machinery, etc., is Per
sonal Property.
336. The valuation of property for taxation is made by
Assessors, or Tax Listers. Taxes are levied at a certain
rate or per cent on the assessed valuation of the property.
337. A Tax Collector is an officer who collects the taxes.
He usually receives a percentage of the taxes collected.
268
PERCENTAGE 269
EXERCISE 218. — WRITTEN
1. A farm valued at $6750 near a city is assessed at 
its value. The tax rate being .7%, what are the taxes?
2. The city boundary is extended so as to include one
half of this farm. The city tax rate being 2.49%, what
is the increase in taxes?
3. A man owning $4200 worth of personal property,
living in the city, pays what taxes at the above rate?
4. With the same property what are his taxes if out
side the city limits?
5. An income tax of 1 % on all annual incomes over
#1000 is collected in some states. What is the income tax
of a man receiving a salary of $1200? Of $2400? Of
$3500? What is it if his salary be $2400 and he earns
$ 250 additional by extra labor ?
6. The rate being .6 %, what are the taxes on 160 A. of
timber land assessed at $9.38 per acre? At $16.88? At
$24.38? At $31.88?
7. If a man owns $8700 worth of property valued for
taxes at , and the rate for state and county taxes is 1.05 %,
the city or local rate is 1.25%, poll tax is $2.00, and he
has an income of $2700, on which he pays an income tax
of 1 % on all over $1000, what will be his taxes?
8. If a school district having an assessed property
valuation of $200,000, maintains a public school 7 mos.
at a cost of $720, and could maintain it 2 mos. more at
an additional cost of $180, what will be the increase in
the taxes of a man owning property valued for taxation
270 PRACTICAL ARITHMETIC
at $2600, if a special tax is voted for maintaining the
school 2 additional months?
9. A man owns property in the city worth $9750.
The valuation for taxation is  and the rate 2.15%. He
receives $776 rent for this property.
He also owns farm property of the same value, which is
valued the same for taxes, but the rate is .9%. The rent
obtained for this property is $682.50. On which does he
receive the larger net profit, and how much more ?
10. If a state having an assessed valuation of $393,571, 
982 increases its appropriation for the support of its Agri
cultural College from $100,000 to $150,000 per year, what
will be the increase in taxes of a man who owns property
valued for taxes at $3500?
11. If the assessed valuation of the property in a county
is $14,844,364 and the total state and county taxes amount
to $185,554.55, what is the tax rate, stated in per cent
and as mills on the dollar?
12. If the assessed valuation of the property in a county
is $15,452,776 and the total state and county taxes col
lected amount to $245,931.64, what is the rate of taxation
if there be 7552 polls on which there is a tax of $1.87
each?
DUTIES OR CUSTOMS
338. A charge is fixed by the government upon goods
imported from other countries. Such charges are called
Duties.
339. When the duty is a certain per cent of the cost
of the goods, it is called an Ad Valorem Duty.
PERCENTAGE 271
340. When the duty is fixed without regard to the
value of the goods, it is called a Specific Duty.
341. This table gives the customs duties according to
the Tariff Act of 1897 on several articles imported into
the United States :
Apples, 25 cts. per bu. Coffee, free.
Barley, 30 cts. per bu. Diamonds, 60 % ad val.
Beef, 2 cts. per lb. Eggs, 5 cts. per doz.
Bottled beer, 40 cts. per gal. Hay, $i per ton.
Bonnets, 60% ad val. Horses, f 30 per head.
Books, 25 % ad val. Jewelry, 60 % ad val.
Books for public libraries. Potatoes, 25 cts. per bu.
free. Wool, 11 cts. per lb.
Cheese, 6 cts. per lb. Woolen clothing 40 cts.
Cigars, $4.50 per lb. and per lb. and 60% ad val.
25% ad val.
EXERCISE 219. — WRITTEN
1. What is the dutj' on 360 lbs. of cheese ?
2. What is the duty on 400 lbs. of cheese costing 8 cts.
per pound?
3. What is the duty on 12000 worth of diamonds?
4. What is the duty on a consignment of 3 head of
horses valued at $350 each?
5. What is the duty on a carload of 9 T. of hay worth
18 a ton?
6. What is the duty on 4 lbs. of cigars valued at 4
a pound?
7. What is the duty on 4 lbs. of cigars valued at 8 a
pound?
PRACTICAL ARITHMETIC
8. Pupils should make ten similar problems concerning
the articles given in the table that are used in their homes.
342. COMPOSITION OF FOODS
Foods
Kefuse
Water
Peotei.n
Fat
Carbo
hydrates
Asir
Beef flank . .
Per Cent
5.5
Per Cent
56.1
Per Cent
18.6
Per Cent
19.9
Per Cent
Per Cent
0.8
Salt pork . . .
Pickled tongue .
Bacon ....
6
8.7
7.9
58.9
18.4
1.9
11.9
9.5
86.2
19.2
59.4
3.9
4.8
4.5
Veal cutlet . .
3.4
68.3
20.1
7.5
1
Fish . . .
44.7
40.4
10.2
4.2
0.7
Chicken . . .
25.9
47.1
13.7
12.3
0.7
Ham ....
12.2
35.8
14.5
33.2
4.2
Jlilk ....
87
3.3
4
5
0.7
Cheese ....
34.2
25.9
33.7
2.4
3.8
Butter ....
11
1
85
3
Eggs ....
Flour ....
11.2
65.5
12
13.1
11.4
9.3
1
75.1
0.9
0.5
Oatmeal . . .
7.3
16.1
7.2
67.5
1.9
Corn meal . .
12.5
9.2
1.9
75.4
1
Apples ....
Bananas . . .
25
35
63.3
48.9
0.3
0.8
0.3
0.4
10.8
14.3
0.3
0.6
Bread ....
35.3
9.2
1.3
53.1
1.1
Sugar ....
Potatoes . . .
20
62.6
1.8
0.1
100
14.7
0.8
Cabbage . . .
Lettuce . . .
15
15
77.7
80.5
1.4
1
0.2
0.2
4.8
2.5
0.9
0.8
Peas ....
74.6
7
0.5
16.9
1
String beans
7
83
2.1
0.3
6.9
0.7
Turnips . . .
Squash . . .
Dried beans . .
Canned tomatoes
Rice ....
30
50
62.7
44.2
12.6
94
12.3
0.9
0.7
22.5
1,2
8
0.1
0.2
1.8
(1.2
0.3
5.7
4.5
59.6
4
79
0.6
0.4
3.5
0.6
0.4
Sweet potatoes
Plain cake . .
20
55.2
19.9
1.4
6.3
0.8
9
21.9
63.3
0.9
1.5
PERCENTAGE
273
343.
TABLE OF DIGESTIBILITY OF FOODS
Meats and fish . . .
Eggs
Dairy products . . .
Cereals
Sugars
Dried peas and beans
Vegetables . . . .
Fruits
Protein
Ptr Cent
97
97
97
85
78
83
Fats
Per Cent
95
95
95
90
90
90
90
Carbo
hydrates
Per Cent
98
97
95
90
EXERCISE 220. — WRITTEN
1. The human body is composed of: ash 6%, protein
18 %, fats 15 %, carbohydrates about 1 %. The remainder
is water. A man weighing 150 lbs. has how many
pounds of water in his body tissues?
2. A man weighing 150 lbs. has how many pounds of
fat? The fat equals what per cent of the water?
3. A man weighing 150 lbs. has how many pounds of
protein? The fat equals what per cent of the protein?
4. The carbohydrates equal what per cent of the pro
tein ?
5. Since 97 % of the protein of meat and fish is
digestible, how many ounces are digestible in 2 lbs. of
beef flank?
6. Since 95% of the fats of meat and fish is digestible,
how many ounces are digestible in 1 lb. of fish?
7. Since 98 % of the carbohydrates of meat and fish is
digestible, how many ounces are digestible in 3 lbs. of veal?
274
PRACTICAL ARITHMETIC
Find the number of ounces of water, refuse, and digest
ible protein, fats, and carbohydrates in the following :
8. 1 lb. of bread. 9. 1 lb. of chicken. lo. 1 lb. of
milk. 11. 1 lb. of corn bread. 12. 1 lb. of potatoes.
13. 1 lb. of cabbage. 14. 1 lb. of turnips. 15. lib. of ham.
16. 1 lb. of rice. 17. 1 lb. of apples. 18. 1 lb. of cheese.
The following food materials were consumed in one
day by a family of four grown persons, two men and two
women, at moderately hard work:
Food Materials
BREAKFAST
Oatmeal .
Milk . .
Sugar .
Veal cutlet
Bread . .
Butter .
Coffee . .
DINNER
Roast beef (flank)
Potatoes . . .
Sweet potatoes .
Bread ....
Pounds
2
12
12
Food Materials
DINNER
Butter .
Rice pudding
Rice . .
Eggs . .
Milk . .
Sugar
Tea . . .
SL'PPER
Bread . .
Butter .
Bananas .
Cake . .
PoUN'lS
Ounces
4
4
6
3
12
3
12
8
19. Find the amount of digestible protein, carbohy
drates, and fats per person in the three meals.
20. Estimating 453.6 grams to a pound, compare with
food standards, as follows : man requiring protein 100
grams, carbohydrates 400 grams, fats 60 grams ; woman
requiring protein 90 grams, carbohydrates 350 grams, fats
40 grams. Wherein is the ration excessive ? Wherein
deficient ?
INTEKEST
What per cent of 100 is 6 ?
What per cent of f 1 is 6 cts. ?
If $6 is charged for the use of f 100, what per cent of
the sum loaned is the sum charged?
344. The money charged for the use of money is called
Interest.
345. The money loaned is called the Principal.
346. The per cent which the interest is of the princii^al
is the Rate of Interest.
347. The principal plus the interest is called the
Amount.
348. Many states have established a fixed rate of
interest, an excess of which it is unlawful to accept.
This fixed rate is known as the Legal Rate of interest.
349. Usury is interest charged in excess of that
allowed by law, or the legal rate.
350. Interest is usually calculated by taking a certain
rate per cent of the principal for one year.
351. The time for which interest is to be reckoned is
calculated in years and days. When over a year, both
years and daj^s are used, and when under a year, days
275
276 PRACTICAL ARITHMETIC
only. In both cases the exact number of days is counted,
if exact interest is desired.
352. In practice, however, when money is loaned for
less than a year interest is usually reckoned on a basis
of 30 days to the month, and 12 months, or 360 days,
to the year, instead of 365 days as when exact interest is
computed.
If 1500 is loaned from May 1, 1906, to Aug. 1, 1907,
the exact time is 1 year and 92 days, but in business,
interest is usually reckoned for 1 year and 3 months, or
1^ years, instead of for l/g^ years. The United States
government and some banks, however, use the exact
method, and in the above example would reckon interest
ior 1^6^^ years.
353. When money is loaned for a certain number of
months, or when interest is payable monthly, calendar
months are meant.
EXERCISE 221. — ORAL
Find the interest on the following sums for the times
and rates specified :
1. $5 for 1 year at 4% ; at 5% ; at 6%.
2. 125 for 1 year at 3 % ; at 6 % ; at 8 %.
3. 150 for 1 year at 4 % ; at 6 % ; at 7 %.
4. $300 for 2 years at 4 % ; at 7 % ; at 9 %.
5. $600 for 1 year and 6 months at 4% ; at 6 %.
6. 1800 for 1 year and 3 months at 6 % ; at 8 %.
7. $400 for 2 years at 6 % ; at 8 % ; for 2 years and
3 'months at 8 % ; at 10 %.
INTEREST 277
8. 11200 for 1 year at 8%; for 3 years at 7%; for
6 months at 9 % ; for 3 months at 5 %.
354. Find the interest on $375.25 at 6% for 2 years.
The interest for 1 year is 6% of 375.25, or, 375.26 X .06 =
22.515 = $22,515.
For 2 years the interest is $22,515 x 2 = $46.03.
Find the interest on $450 for 3 years, 4 months, and
15 days at 8% per year?
The interest for 1 year is 450 x .08 = 36.00 = $36.00
The interest for 3 years is $36 x 3 = $ 108.00
The interest for 4 months is j\, or , of $ 36 = 12.00
The interest for 15 days is ^f, or i) of J^ = ^^ of $ 36 = 1.50
' Therefore, the total interest is $ 121.50
355. To find the interest for a given time, find the
interest for 1 year and multiply by the number of years
or fraction of a year.
EXERCISE 222. — WRITTEN
Find the interest and amount :
1. 1 275.25 for 1 yr. 3 mos. 20 days at 8 %.
2. $345 for 2 yrs. 6 mos. 15 days at 6 %.
3. 1 435 for 3 yrs. 11 mos. at 5 %.
4. $ 850 for 1 yr. 8 mos. 15 days at 5 %.
5. I 340 from June 1, 1906, to Sept. 15, 1908, at 6 %.
6. $ 450 from Sept. 10, 1905, to Dec. 25, 1907, at 8 %.
7. I 815.27 from Jan. 10, 1905, to Oct. 1, 1908, at 10 %.
8. $427.25 from Nov. 5, 1906, to Jan. 20, 1908, at 6 %.
9. $ 650 from April 10, 1908, to Oct. 10, 1908, at 4 %.
10. 1 347.50 from Feb. 5, 1908, to Sept. 10, 1908, at 6 %.
1^78
PRACTICAL ARITHMETIC
BXEECISB 223. —WRITTEN
Find the exact interest on :
1. 1200 from July 1, 1905, to Aug. 7, 1907, at 6%.
2. 1450 from July 3, 1907, to May 31, 1908, at 8 %.
3. $775 from Dec. 9, 1905, to Feb. 17, 1908, at 6 %.
4. 1 2550 from May 21, 1907, to Dec. 13, 1907, at 4J %.
5. % 90 from Aug. 10, 1906, to Nov. 10, 1906, at 7 %.
6. 11750 from Jan. 24, 1906, to Feb. 16, 1907, at 5 %.
7. 1335 from July 1, 1900, to June 19, 1903, at 6 %.
8. 1 3750.50 from Jan. 29, 1899, to Feb. 29, 1908, at 6 /o.
356. For convenience in finding the correct or exact
time, the following table is frequently used by bankers and
others :
Jan.
Feb.
Mar.
April
May
June
July
Aug.
Sept.
.Oct.
Nov.
Dec.
January . .
365
31
59
90
120
151
181
212
243
273
304
334
"February
334
365
28
59
89
120
150
181
212
242
273
303
March . .
306
387
365
31
61
92
122
153
184
214
245
275
April . . .
275
306
334
.365
30
61
91
122
153
183
214
244
May . . .
245
276
304
335
365
31
61
92
123
153
184
214
June . . .
214
245
273
304
334
365
30
61
92
122
153
183
July . . .
184
215
243
274
304
335
365
31
62
92
123
153
August . .
153
184
212
243
273
304
334
365
31
61
92
122
September .
122
153
181
212
242
273
303
334
365
30
61
91
October . .
92
123
151
182
212
243
273
304
335
365
31
61
November .
61
92
120
151
181
212
242
273
304
334
365
30
December .
31
62
90
121
151
182
212
243
274
304
335
365
The exact number of days from any particular day of any
month to the corresponding day of any other month, within 1
year, is found opposite the first month and under the second.
If the day of the month is not the same in the two months,
INTEREST 279
the difference may be added or subtracted as the case may
require, e.g., from July 10, to Dec. 10 is 163 days. Erom
July 10 to Dec. 23 is 153 + 13 = 166 days. From July 31 to
Dec. 20 is 153  11 days = 142 days.
357. On many calendars used in business offices the
days of the years are numbered consecutively through the
months, and in calculating interest for periods less than a
year, business men frequently take the number of the day
on which interest begins and subtract it from the number
of the day when it is due ; e.g., June 18, 1908, is the 170th
day of the year, and Nov. 12 is the 317tli day, therefore
from June 18 to Nov. 12 is 317170 = 147 days.
SIX PER CENT METHOD
358. What is the interest on $1 at 6 % for 1 month?
For one day ?
If the interest on $ 1 for 12 mos. is $ .06, the interest on $ 1
for 1 mo. is j\ of $ .06 = $ .005, and the interest on $ 1 for
1 da. is ^ of $ .005 = $ .000^.
Thus:
The interest on f 1 for 1 yr. at 6 % = f .06.
The interest on $1 for 1 mo. at 6 % = f .005.
The interest on $1 for 1 da. at 6 % = f .OOOf
What is the interest on $360.40 at 6% for 2 yrs. 8 mos.
18 days?
The interest on f 1 for 2 yrs. is  .06 x 2 = f .12
The interest on $1 for 8 mos. is $.005x8= .04
The interest on $ 1 for 18 days is $ .000^ x 18 = .003
The interest on $ 1 for 2 yrs. 8 mos. 18 days = $ .163
The interest on $ 360.40 for 2 yrs. 8 mos. 18 days = $ 360.40
X .163 = $58.7452, or f 58.75.
280 PRACTICAL ARITHMETIC
359. To find the interest on any sum at 6 % for a given
number of years, months, and days, multiply 6 cts. by the
number of years ; . 5 cts. by the number of months ; and
1^ of a mill by the number of days. Find the sum of these
and multiply this sum by the principal.
EXERCISE 224.  "WRITTEN
Find the interest by the six per cent method on:
1. $475.40 for 1 yr. 2 mos. 6 days, at 6%.
2. 1820 from June 1, 1904, to Oct. 12, 1905, at 6%.
3. $528.60 for 3 yrs. 8 mos. 18 days at 6 %.
4. 1360.50 for 3 yrs. 9 mos. 24 days at 6 %.
5. $750 from May 12, 1901, to April 18, 1903, at 6%.
360. Shorttime loans are frequently made for 3, 4, 6,
8, or more months :
Since the interest on $1 for 1 mo. at 6 % is $.005, or
^ ct., to find the interest on any sum at 6% for time
expressed in months, move the decimal point two places to
the left in the principal and multiply by one half the
number of months.
The interest on $320.50 for 6 mos. is 3.205 x 3 = 9.615
= $9,615.
361. Loans made for three months or less are usually
made for 30, 60, or 90 days :
Since the interest on $1 for 1 da. at 6 % is ^ of a mill, to
find the interest on any sum for time expressed in days,
move the decimal point three places to the left in the
principal and multiply by  the number of days.
INTEREST 281
The interest on 1160.75 for 30 days is .16075 x } of 30
= .16075 X 5 = .80375 = $.80375, or 80 cts.
EXERCISE 225. — WRITTEN
Find the interest at 6% on :
1. 1325 for 8 mos. 6. 1 945 for 30 days.
2. $785 for 6 mos. 7. 11500 for 60 days.
3. $957 for 9 mos. 8. $525 for 90 days.
4. $2450 for 7 mos. 9. 1 343 for 30 days.
5. $375 for 11 mos. 10. $250 for 90 days.
362. The six per cent method may be used for finding
the interest at other rates, as follows :
2 % is f, or , of 6 %, therefore, take ^ of the interest at 6 %.
3 % is f, or ^j of 6 %, therefore, take J of the interest at 6 %.
4 % is ^, or f, of 6 %, therefore, take  of the interest at 6 %.
4i% is — 2^, or J of 6 %, therefore, take f of the interest at
6 %, or subtract I of the interest at 6 %.
5 % is f of 6 %, therefore, take f of the interest at 6 %, or
subtract ^ of the interest at 6 %.
7 % is I of 6 %, therefore, take ^ of the interest at 6 %, or
add ^ of the interest at 6 %.
■71
J i cf„ is ~^! or I of 6 %, therefore, take J of the interest at
6%, or add J of the interest at 6 %.
363. The six per cent method being based on 12 months,
or 360 days, to tlie year, of course, does not find tlie exact
interest.
EXERCISE 226.— WRITTEN
Find the interest by the six per cent method on :
1. $ 250 for 1 yr. 3 mos. at 2 %.
282 PRACTICAL ARITHMETIC
2. 1335 from July 10, 1903, to June 20, 1905, at 3%.
3. 1 945 for 2 yrs. 4 mos. 20 days at 4 %.
4. 11350 from Dec. 12, 1905, to Feb. 1, 1907, at 41%.
5. 1725 for 1 yr. 7 mos. 13 days at 5%.
6. $ 2450 from Aug. 15, 1905, to Nov. 20, 1908, at 7 %.
7. $ 183 for 2 yrs. 5 mos. 23 days at 7J %.
8. 1420 from July 20, 1898, to June 5, 1901, at 8 %.
364. It is seen that in computing interest there are four
factors involved : time, rate per cent, principal and inter
est, or principal and amount, or interest and amount.
Any one of these factors can be found if the other' three
be given, but in business the principal, rate per cent, and
time are usually given and the interest, or interest and
amount, are required to be found.
365. It may be necessary to find the principal, rate, or
time when only two of these and the interest, or the
amount, are given.
366. To find the principal, when the rate, time, and
interest or amount are given :
What principal Avill produce fl5 interest in 2 yrs.
6 mos. at 6 % ?
Interest on f 1 for 1 yr. at 6 % = $.06.
Interest on $ 1 for 2^ yrs. .at 6 % = f .06 X 2i = f .15
Hence, principal required = — — = $ 100.
.15
What principal at 6 9^ interest for 2 yrs. 6 mos. will
amount to f 115 ?
A principal of f 1 at 6 % for 1 yr. = $ .06 interest.
INTEREST 283^
A principal of $ 1 at 6 % for 2 yrs. = $ .06 X 2^ = $ .15
interest.
A principal of $ 1 at 6 % interest for 2i yrs. amounts to
principal 1 1 + interest $ .15 = $ 1.15.
$115
Hence, principal required = ~ = f 100.
$1.15
EXERCISE 227. — WRITTEN
Find the principal that will bring :
1. 1450 interest in 3 yrs. at 6 %.
2. 172.74 interest in 3 yrs. 9 mos. at 2^%.
3. 111.33 interest from July 5, 1904, to Feb. 9, 1905,
at 4%.
4. 118.72 interest in 5 mos. 27 days at 5^%.
EXERCISE 228. — WRITTEN
Find the principal that will amount to :
1. 11283.33 at 5 % interest for 3 yrs. 4 mos.
2. 1 1303.41 at 6 % interest for 60 days.
3. 1 25.76 at % interest for 33 days.
4. $ 94.31 at 6 % interest for 1 yr. 7 mos. 21 days.
367. To find the rate per cent, when the time, princi
pal, and the interest or the amount are given:
At what rate per cent will $100.00 produce 124.50
interest in 3 yrs. 6 mos. ?
The interest on f 1 for 1 yr. at 1 % = f .01.
The interest on f 1 for 3^ yrs. at 1 % = f .035.
The interest on 1 100 for 31 yrs. at 1 % = $ 3.50.
Hence, the rate is f2450 ^ ^ ^^
3.50
284 PRACTICAL ARITHMETIC
EXERCISE 229. — WRITTEN
Find the rate per cent :
1. When the interest on 1 2300 for 1 yr. is 1138.
2. When the interest on f 675.88 for 5 yrs. is $118.28.
3. When 16130 at interest from June 6 to Nov. 24
amounts to $6237.28.
4. When $1050 at interest from Sept. 21, 1904, to
March 5, 1905, amounts to $1069.15.
368. To find the time, when the rate, principal, and the
interest or the amount are given:
In what time will $100, at 6%, bring $21 interest?
The interest on $ 1 at 6 % for 1 yr. =  .06. The interest
on $ 1 for the unknown time = $ 21 ^ 100 = $ .21.
$.21
.06
Hence, the unknown time = '^^ = 3 yrs.
EXERCISE 230.— WRITTEN
Find the time in which :
1. The interest on $1500 will be $180 at 4%.
2. The interest on $8520 will be $1746.60 at 6%.
3. The interest on $17,040 will be $3493.20 at 6%.
4. $238.74 at 4i% will amount to $308.20.
369. In the foregoing solutions the unit in finding the
principal was $1 ; the unit in finding the time was 1 year,
and in finding the rate the unit was 1 per cent. Hence,
to find the principal, time, or rate, when the other factors
are given, divide the given interest by the interest ob
tained by using the unit as the required factor.
STOCKS AND BONDS
370. When two or more persons form an organization
under the laws of the state to conduct business as one
body, the organization is called a Company, Stock Company,
or Corporation.
371. The money contributed for the purpose of carry
ing on the business of a company is known as its Capital,
Capital Stock, or Stock.
372. Tlie stock of a company is divided into a number
of equal parts known as Shares, or Shares of Stock.
373. A person holding or owning one or more shares of
stock is known as a Stockholder.
374. The document or certificate issued to each stock
holder showing the number of shares he owns, is called a
Certificate of Stock.
375. A Bond is a written promise or obligation of a cor
poration, or a government, to pay a specified sum of
money at a certain time, with interest at regular stated
intervals and at a certain rate.
376. The original, certificate, or face value of a stock is
its Par Value. The price for which it sells is its Market
Value. When the market value is greater than its par
value, the stock is said to be at a premium, and when it
285
286 PRACnCAL ARITHMETIC
sells for less than the face or par value, it is at a Discount
or Below Par.
377. The profits of a company which are divided among
the stockholders according to the stock which they hold
are known as the Dividends.
378. The capital stock of a corporation is sometimes
divided into two kinds : Preferred and Common. A divi
dend is usually guaranteed on the preferred stock, and in
such case must be paid before any dividend is paid on the
common stock.
379. Persons who buy and sell stocks and bonds are
called Stock Brokers, or Brokers, and their commission is
called Brokerage. Brokerage is computed on the par
value of the stock.
380. Small certificates of interest, called Coupons, are
usually attached to bonds. These coupons are promises
to pay interest at a certain time at a specified rate. If
this interest is not paid when due, the interest itself draws
interest at the legal rate.
381. Large numbers of different stocks and bonds are
regularly on the market, and are bought arid sold exten
sively. The following are a few with the quotations
given in a daily newspaper June 27, 1908 :
The figures indicate the price in dollars per share or bond.
The par value is usually f 100, but in some cases it may be $50,
or some other value. In the following problems f 100 is the
par value of the stocks or bonds in question, unless otherwise
stated ;
INTEREST
287
Stocks
Bonds
L. & N. . .
. 103^
U. S. 4's ...
121i
Am. Copper .
■ 66f
Minn. & St. L. 4's
76
B. & 0.
. 86
Mo. K. & Tex. .
96i
Ch. & N. W. .
. 150
B. & 0. 4's . . .
98^
111. Cent. . .
. 130
Col. Mid. 4's . .
63
382. If I pay a broker ^ % commission to buy 9 shares
of L. and N. R.R. stock worth on the market 103^, what
will be the total cost?
$ 100 = par value of 1 share.
I % of f 100 = f .126 = commission on 1 share.
$ .125 + $ 103^ = f 103.625 = cost of 1 share.
$ 103.625 X 9 = $932,625 = cost of 9 shares.
383. Find the annual income from 327 shares of stock
which pays a semiannual dividend of 3 % ?
3^ % of $ 100 = $ 3.50 = semiannual interest on 1 share,
f 3.50 X 2 = $ 7.00 = annual interest on 1 share.
327 X f 7 = f 2289 = income on 327 shares.
384. What rate of interest on money invested shall I
receive if I buy S.A.L. R.R. bonds at 53^, which pay 4%
interest, brokerage ; % ?
4 % of f 100 = $4 = income from 1 share.
\% ot^ 100 = 1 .25 = brokerage on 1 share.
$ .25 + $ 53 = $ 53.75 = cost of 1 share.
$.53.75 invested brings $4 in interest.
$ 53.75 at 1 % = $ .5375.
$ 4 = .5375 = 7.5"^ = 7 % = rate of interest on money invested.
EXERCISE 231. — WRITTEN
1. Find the cost of 18 shares American Copper at 66^,
brokerage ^%.
•288 PRACTICAL ARITHMETIC
2. Find the cost of 27 shares of stock (par value $ 50)
at 34, brokerage ^%.
3. Find the amount realized from the sale of 32 shares
of B. & O. R.R. stock at 86, brokerage ^%.
4. Find the amount realized from the sale of 9 §500
U. S. bonds, at 113, brokerage ^%.
5. How many shares of stock having a par value of
$50 and a market value of $67.50, can be bought for
$4869, brokerage %?
6. How many U. S. $100 bonds must I sell at 107 to
raise sufficient money to pay a debt of $3432.80, broker
age %?
7. How much income will $3600 Standard Oil stock,
paying 36 % dividend, yield ?
8. What is the profit in buying 26 shares of Chicago
and N. W. R.R. stock at 150 and selling for 159, broker
age on each transaction ^ % ?
9. What is the gain in buying 37 shares of stock (par
value $50) at 49 and selling at 62^, brokerage on each
transaction j % ?
10. What sum must be invested in U. S. 4's at 118,
brokerage ^%, to yield an income of $160?
11. What sum must be invested in P. R.R. stock at
92 of par value, paying 3 % half yearly, to give an in
come of $ 630, brokerage ^%?
12. What income will be obtained from the investment
of $5287.50 in stock at 105J, paying 3% semiannually,
brokerage  % ?
13. A man bought Illinois Central R.R. stock at 137
. INTEREST 289
and sold it for 1421, making a profit of $512.50. How
many shares did lie buy, brokerage ^ % in each transaction?
14. A man buys 200 shares of Colorado Midland R.R.
stock, paying 4%, at 63 and holds it 2 years, receiving 2
dividends, and sells it at 78. Money being worth 6 % per
annum and brokerage ^^ on each transaction, how much
did he gain?
15. Which is the best investment : 7 % stock at 127^,
5 (Jo bonds at 82J, or money loaned at 6 % ?
16. If a man buys U. S. 4 % bonds at 116, what per
cent does he receive on his investment ?
17. Which pays better, a 5 % stock at 90 or a 7 % bond
at 130? How much better?
18. A man has 13800 to invest. Which wilt yield the
larger income, 5% stock at 95, or 5J% bonds at par?
How much larger ?
NEGOTIABLE PAPERS
385. A written order to a bank to pay money, from one
having money deposited in such bank, is called a Check.
Ji'o.337 J^ewYorh,...Ci'wc}. 6, ..,190 8
The Corn Exchange Bank
FIFTH AVENUE BRANCH
Pay to ^. (S, B'uyuyyi& or order
(Z. /"f. MaAJMi^ V &0
A Bank Check
290
PRACTICAL ARITHMETIC
386. A check may be made payable to Payee or to
Bearer, in which case it may be collected by auy one ; or
to Payee or order. If made payable to "order," it must
be indorsed by the person to whose order it is made pay
able. A check may be " indorsed in blank," when the
person to whose order it is made payable writes only
his name across the back of the check as in A ; or he
may make a special indorsement, as in B.
A. Indorsed in Blank
S'au to tk& ovcIea, ol
/'ftnvu j.akni'Cyn,
^eA>\a£, l/OLttia.yyh^
B. Special Indorsement
387. A Certified Check is one upon which the cashier of
a bank has written, in red ink, the word " Certified," with
date, name of bank, and his name as cashier. This makes
the bank responsible for its payment.
DRAFTS
388. A written order to one person to pay a specified
sum of money to another person at a certain time is called*
a Draft, or Bill of Exchange.
389. The person who makes the order is called the
Drawer of the draft ; the person who is ordered to pay is
the Drawee ; and the person to whom the money is to be
paid is the Payee.
INTEREST 291
390. A draft which is paj'able whenever presented to
the drawee is called a Sight Draft.
391. A draft payable after date or at a certain time
after sight is a Time Draft.
JVo. __f/., Atlanta, Ga.,..Mel. f,..190 8
Pay to the order of
3'k& ^ovn ^cK^kanaSt ffa,nA, ol c^e^w Z/oik $260
■c■^^^vc^•^^■>^^^c^•^i«>■ /iwncll&ct c/t/^i^^^i^T^^^^^^^^^r^ Dollars.
To T. T. Sands,
Jfeiv York.
A Draft
392. A time draft must be presented to the drawee or
person who is to pay, who must accept the draft by writ
ing across its face the word " Accepted," together with
the date and his signature.
393. A draft by one bank upon another in which it has
a deposit is a Bank Draft.
PROMISSORY NOTES
394. A written promise to pay a certain sum of money
at a specified time after date is called a Promissory Note.
395. The person who signs a note and thereby promises
to pay is the Maker of the note. The Payee is the person
to whom the promise to pay is made.
292 PRACTICAL ARITHMETIC
396. A note signed by one person only is called an
Individual Note.
397. A note signed by two or more persons is a Joint
Note.
398. A note is usually written so as to include the
payment of interest at a specified rate for the time of the
note.
3'kvt^ •yyuyyCth^ after date.. o^^.. promise to jiay to
tlze order of. /%5a^ra. ^ton& a/yicl /'f&vvi&k
at the Xational City Bank, of Jfew York, with interest
at six per cent.
Value received.
Jfo.. .■¥/. I>ue...fa,rv. /, /f^f_._
A Promissory Note
399. When no rate is mentioned, a note bears interest
at the legal rate of the state in which it is payable.
400. A note may be made payable to the payee or
bearer, to the payee only, or to the order of the payee.
Notes are indorsed in blank or to special persons, as with
checks. The indorser of a note is I'esponsible for its
payment unless the words " Without Recourse " precede
his signature.
INTEREST 293
401. The Day of Maturity of a note is the day on which
it becomes due or the last day of the time for which it is
drawn. In some states after the note becomes due, three
days are allowed for its payment. These three days are
known as Days of Grace.
BANK DISCOUNT
402. Bankers and others make a certain deduction
from a note or draft for paying it before it is due. This
deduction or charge is the simple interest on the face of
the note or draft from the date on which it is paid to the
date of maturity. This charge or deduction is known as
Bank Discount.
403. The Proceeds of a note or draft is the amount of
the note or draft less the discount and exchange.
404. The paying or transferring of money by means of
cheeks, drafts, or money orders is called Exchange. The
charge made by banks and other institutions for issuing
or cashing these checks, drafts, or money orders is also
called Exchange. The rate of exchange varies from ^ to
1^ of 1%, or may be a certain fixed amount for different
sums.
EXERCISE 232. — WRITTEN
1. A draft for 1 320 is sold by a bank in Raleigh, N.C.,
on a New York bank, rate ^fo. What is the exchange ?
2. A draft for $125 is sold by a bank in New Orleans,
La., on a bank in Chicago, 111., rate %. What is the
exchange ?
3. A United States postal money order for $76 costs
29i PKACTICAL ARITHMETIC
30 ets., and one for $100 can be bought at the same price.
What is the rate per cent on each ?
4. Which is cheaper, a draft on a New York bank for
f 40 at ^ %, or a U. S. money order for the same amount at
a cost of 15 cts. ?
5. If a bank in Chicago, 111., charges 15 cts. for a
draft for f 100 on a New York bank, what is the rate of
exchange charged?
405. A note for 100 at 6% interest drawn July 15,
1905, due 9 months after date, is presented to a bank Oct.
15, 1905, for discount. Find the discount and the pro
ceeds.
The discount is the simple interest on $ 100 at 6 % from
Oct. 15, the day it is discounted, to maturity. The note is
drawn July 15, 1905, for 9 months, and is, therefore, due April
15, 1906. From Oct. 15, 1905, to April 15, 1906 = 6 months.
The interest on $ 100 at 6 % for 1 year = f 6.
The interest on $100 at 6% for 6 months = ^ of 6 =
f 3 = the discount.
$ 100  $3 = $ 97 = the proceeds.
BXEECISB 233. — WRITTEN
1. Find the discount on a note for f 480 dated Sept.
15, 1905, and payable in 90 days, if discounted Sept. 30,
1905, at 5%.
2. Find the proceeds of a note for 1375, dated May 5,
1907, payable Aug. 20, 1908, at 6 % interest, if discounted
by a bank July 15, 1907.
3. Find the discount on a note for 12250, drawn
Jan. 1, 1905, for 4 months at 8% interest, discounted
Jan. 15, 1905.
INTEREST 295
Find the discount and the proceeds on the following :
4. 1342.00. Raleigh, N.C, Jan. 25, 1907.
Seven mouths after date I promise to pay to the order
of Samuel Wolfe three hundred forty two dollars, with
interest at. six per cent, for value received.
James Baird.
Discounted Feb. 10, 1907.
5. 1785.00. Atlanta, Ga., July 1, 1905.
Ninety daj's after date I promise to paj' John Williams,
or order, seven hundred eightyfive dollars, for value
received.
William Jones.
Discounted July 10, 1905, at 7%.
6. I wish to pay for a carload of cattle which cost 1860,
but must give my note to the bank for 90 days to obtain
the money. For how much must I draw the note to
obtain this amount if it is discounted at 10 per cent ?
7. I sell fifty acres of land for f 62.50 an acre, on a
cash basis, but the purchaser is not able to pay the money,
and offers instead his note for 6 months at 8 % interest.
For how much must the note be drawn that I may receive
the cash price after discounting the note at the bank?
8. I sell a carload of cattle, taking in payment a prom
issory note for 60 days. After discounting the note at
the bank at 5 per cent, I have 11487.50. For how much
did I sell the cattle?
9. Write a promissory note for 90 days, with yourself
as payee. Properly indorse it and find the bank discount
and proceeds at the legal rate of interest in your state
on the day it is drawn.
PAETIAL PAYMENTS
406. A payment made on a note which is not equal to
tlie principal and the interest then due is known as a
Partial Payment.
407. When such partial payments are made, the amount
paid and the date, with the signature of the person receiv
ing the payment, are indorsed on the back of the note.
Received on the within note :
$S20, March 1, 1904
jokn j. fcm&i,
$1SB, April 15, 1905
john f. jon&O'
$85, June 20, 1906
UNITED STATES RULE
408. The rule adopted by the United States Supreme
Court for finding the amounts due on notes when partial
payments have been made, has been adopted as the legal
method hy most states, and is as follows :
Find the amount of the principal to the time when the
payment or the sum of the payments equals or exceeds
the interest due ; from this amount subtract the payment
or the sum of the payments; treat the remainder as a
new principal, and proceed as before.
296
INTEEEST 297
409. A note for 1 935, drawn Jan. 1, 1903, due one year
from date, at 6 per cent interest, has the following in
dorsements on it: March 1, 1904, 1320; April 15, 1905,
135; June 20, 1906, 1 85.
What was the balance due Sept. 1, 1906 ?
Solution by United States Eule
Principal Jan. 1, 1903 = $ 935.00
Interest from Jan. 1, 1903, to March 1, 1904 = 65.45
Amount March 1, 1904 = $ 1000.45
First payment March 1, 1904 = 320.00
New principal March 1, 1904 = $ 680.45
Interest from March 1, 1904, to April 16, 1905 = 45.93
(Since the interest due April 15, 1905, is greater
than the payment made, no new principal is
calculated.)
Interest from March 1, 1904, to June 20, 1906 = 94.13
Amount June 20, 1906 = $ 774.58
Payments deducted June 20, 1906 ($ 35 + $ 85) = 120.00
New principal June 20, 1906 =  654.58
Interest from June 20, 1906, to Sept. 1, 1906 = 7.64
Amount due Sept. 1, 1906 = f 662.22
MERCHANTS' RULE
410. For calculating the amount due on accounts and
notes bearing interest, on which partial payments have
been made, merchants often use what is known as the
Merchants' Rule. It is not as accurate as the United
States method and is not legal, but for short periods, one
year or less, it gives nearly the same results, and is easier.
It is as follows: 1. Find the amount due on the original
principal to the time final settlement is made. 2. Find
the amount of each payment from the date it was made
298 PRACTICAL ARITHMETIC
to date of final settlement. 3. Subtract the sum of the
amounts of the payments from the amount of the original
principal at the time of final settlement.
If used for periods greater than one year, find the
balance due at the end of each year by the rule above
stated, and take this balance as a new principal.
411. Find the correct settlement on Feb. 15, 1908, on a
note for 100 dated March 1, 1907, at 6 % interest, with'
the following indorsements : May 1, 1907, $ 20 ; July 15,
1907,150; Jan. 1,1907,120.
Solution by the Merchants' Rule
Amount of principal for 11^ mo. = $ 105.75
Amount of $ 20 for 91 mos. = $ 20.95
Amount of f 60 for 7 mos. = $ 51.76
Amount of $ 20 for 1 mos. = $ 20.16
= $ 92.85
Correct settlement = $ 12.90
412. In a few states other rules are used. In such states
the teacher should furnish the pupils with the special rule
in use, and have the following problems solved by this
special rule as well as by the rules indicated.
EXERCISE 234. — WRITTEN
Find the balance due on date of settlement by the
United States Rule :
1. A note for ;$300 at 6% is dated March 1, 1905, and
has the following indorsements: July 1, 1905, f 50; Sept.
1, 1905, 130; Jan. 1, 1906, $100. How much is due
June 1, 1906 ?
INTEREST 299
2. A note for fSOO dated i\[ay 1, 1901, at 6% interest,
has f 50 partial payments indorsed on it every 6 months
from date to Nov. 1, 1905. What is the correct settle
ment Feb. 1, 1906?
3. A note is settled 4 years after date. Its face value
is $400, and it draws 8% interest. It has $150 paid on
it 1 year and 8 months before final settlement. How
much is required to settle the note?
4. A note for $800 dated April 10, 1901, has the follow
ing indorsements: June 20, 1901, $30; July 10, 1901,
$50; Dec. 1, 1901, $20; May 25, 1902, $25; Sept. 10,
1902, $100. How much was due Jan. 20, 1903, at 7%
interest ?
Find the amounts necessary to settle the following
notes, by both the United States and the Merchants' Rules,
and compare results :
5. A note for $320 dated July 15, 1906, at 6 % interest,
is indorsed as follows: Sept. 25, 1906, $80; Dec. 1, 1906,
$40; Jan. 1, 1907, $50; June 1, 1907, $10. How much
was due July 1,1908?
6. A note for $150 dated May 10, 1906, for one year,
at 6% interest, has indorsed on it payments of $25 every
three months. How much will be required to settle bal
ance of note on the date it falls due ?
7. An account amounting to $375, due Jan. 1, 1904, is
settled May 15, 1905. The following amounts had been
paid on it: March 1, 1904, $75; Jiine 15, 1904, $30; Sept.
10, 1904, $80; Jan. 15, 1905, $100. How much was re
quired to pay the balance, with interest at 6%?
300 PRACTICAL ARITHMETIC
8. On Jan. 1, 1904, A owes B 1650, on which he agrees
to pay 6 % interest and make a payment of f 75 every three
months and the balance Jan. 1, 1905. When the balance
is due, A is unable to pay it, and B charges him 8 % inter
est until paid. If A pays 185 Feb. 1, 1905, and f 135 on
July 10, 1905, how much money must he raise to pay the
balance Sept. 20, 1905?
COMPOtWD INTEREST
413. When the interest as it falls due is added to the
principal, and the amount forms a new principal on which
interest is paid, the owner receives Compound Interest.
414. Compound interest on notes is not now generally
allowed by law, but savingsbanks allow compound inter
est on balances remaining on deposit for a certain definite
interest term. Interest may be compounded annually,
semiannually, or quarterly. Unless otherwise stated, it
is usually compounded annually.
415. What is the compound interest on f 100 for 3
years at 6 %?
$ 100 == first principal.
f 100 X .06 = $ 6 = first interest.
$ 100 I $ 6 = f 106 = second principal at end of 1 year.
f 106 X .06 = $ 6.36 = second interest.
$ 106  $ 6.36 = $ 112.36 = third principal at end of 2 years. •
f 112.36 X .06 = $ 6.7416 = third interest.
$ 112.36 I I 6.7416 = f 119.10 = amount at end of 3 years.
Therefore $119.10  .f 100 = $19.10 = compound interest
on f 100 for 3 years at 6%.
INTEREST 301
EXERCISE 235. — WRITTEN
Find the compound interest and compound amount on
the following :
1. $375 for 4 yrs. at 4%.
2. $125 for 3 yrs. 6 mos. at 5 %.
3. $650 for 5 yrs, at 6 %.
4. $2300 for 3 yrs. at 6%.
5. $1875 for 4 yrs. at 8%.
6. $3250 for 6 yrs. at 6%.
7. $4325 for 3 yrs. at 5%.
8. $7575 for 7 yrs. at 5 %.
Find the amount and interest compounded semiannually
on the following :
9. $500 for 2 yrs. 6 mos. at 6 %.
10. $ 250 for 2 yrs. 6 mos. at 7 %.
11. $ 1750 for 1 yr. 9 mos. at 8 %.
12. $2225 for 3 yrs. 6 mos. at 5 %.
13. $ 3575 for 3 yrs. 9 mos. at 6 %.
14. If a man deposits $2500 in a savingsbank, which
pays 4% semiannually, and leaves it for 2 yrs. 6 mos.,
will he have more or less money than if he had loaned it
at simple interest for the same time at 4^ % ?
15. How much money did I have to my credit Jan. 1,
1908, if I made deposits as follows in a savingsbank, pay
ing 4 % interest and compounding it Jan. 1 and July 1 :
July 1, 1906, 1200; Aug. 1, 1906, $75; Dec. 1, 1906,
$150 ; May 10, 1907, $100; and July 1, 1907, $ 185.
EATIO
416. How many weeks are there in one month? What
part of one month is one week? What is the relation of
one week to one month ?
417. This relation is expressed in fractional form, ^.
It may also be expressed thus, 1: 4, which is read 1 to 4.
The relation of two numbers of the same kind as ex
pressed by division is called the Ratio of the first to the
second.
418. The two numbers of a ratio are called its Terms.
EXERCISE 236. — ORAL
Express the following common fractions in the form of
ratios :
1. i 4. 4. 7. I.
?• •*• 3
2. f 5. f. 8. 1.
3. i. 6. i. 9. 1.
1
Express the following ratios as fractions :
10. 3:10. 12. 8:9. 14. 1:15.
11. 10:3. 13. 12:11. 15. 15:1.
419. The first term of tlie ratio is called the Antecedent.
The second term is the Consequent.
302
RATIO 303
420. Two quantities of different kinds cannot form the
terms of a ratio. A ratio is always abstract, and the
terms may be written as abstract numbers.
421. Multiply, also divide, both terms of the ratio 4 :
6 by 2.
Multiplying, we have 8 : 12; dividing, we have 2:3. If we
express as a fraction the value of each ratio so obtained, we see
that multiplying or dividing has not changed the value of the
ratio.
Multiplying or dividing both terms of a ratio by the
same number does not change its value.
422. In order to compare readily two or more ratios, it
is convenient to reduce the ratios to such forms that the
first terms of the ratios to be compared shall be the same,
usually 1.
423. Reduce 7 : 28 to a ratio having 1 for its first term :
Dividing both terms by 7, we have 1 : 4. To obtain a
ratio with one for its first term, divide the second term by
the first term ; take the quotient as the second term and
1 as the first term of this new ratio.
EXBRCISB 237. — WRITTEN
Reduce each of the following to a ratio having 1 for its
first term :
1.
12 : 24.
6.
16;
;39.
11.
.1:3.
16.
1:2.
2.
3:9.
7.
19:
:72.
12.
.9:6.
17.
j:ll.
3.
6:60.
8.
11;
:23.
13.
.784:9.
18.
T^ = 24.
4.
7:21.
9.
24
:98.
14.
.62:11.
19.
j.?y:9642.
5.
9:81.
10.
96;
;700.
15.
.7:6.
20.
tV9=1>2V
304 PRACTICAL ARITHMETIC
The following numbers express the density of popula
tion per square mile in various countries. Compare the
population of the Uuited States with that of other coun
tries in the form of ratios, using the number represent
ing the population of the United States as the first term
in every case. Reduce all the ratios so that the first term
shall be 1.
United States 25, Great Britain and Ireland 344, France
188, Germany 288, Italy 293, Belgium 609, Switzerland
208, Canada 1.
THE NUTRITIVE RATIO
424. What is the ratio of 2.5 lbs. to 16.25 lbs., with the
first term reduced to 1 ?
What is the ratio between 1.5 lbs. protein and 10.5 lbs.
carbohydrates and fats ? Reduce to a ratio having 1 for
its first term.
425. Find the nutritive ratio of the following standard
ration for a dairy cow : 2.5 lbs. of digestible protein, 13 lbs.
of digestible carbohydrates, and .5 lb. of digestible fats :
The fats in a ration are 2.4 times more valuable than the
carbohydrates. To reduce the fats in this ration to the value
of carbohydrates, the number expressing the amount of fats,
.5, must be multiplied by 2.4. .5 x 2.4 = 1.2. Adding this to
the number expressing carbohydrates, we have 1.2 + 13 = 14.2,
the value of the carbohydrates and fats taken together. The
nutritive ratio of this standard dairy ration is, therefore, as
2.5 is to 14.2, or, reducing the first term to 1, we have 1 : 5.68.
The nutritive ratio is therefore 1 : 5.68.
426. The Nutritive Ratio of a ration is the ratio of
the weight of the digestible protein to the sum of the
RATIO 305
weights of the carbohydrates plus 2.4 times the sum of
the weights of the fats.
Digestible protein : (2.4 x fats) + carbohydrates.
427. A Balanced Ration is one having the correct nutri
tive ratio for the particular animal and purposes for
which it is being fed.
438. A Narrow Ration is one in which the proportion
of protein to carbohydrates and fats is greater than the
standard requirements.
429. A Wide Ration is one in which the proportion of
protein to the carbohydrates and fats is less than the
standard requirements.
EXERCISE 238. — WRITTEN
1. A ration consisting of 10 lbs. of corn and 20 lbs. of
timothy hay contains 1.35 lbs. of digestible protein, 15.35
lbs. of carbohydrates, and .71 lb. of fats. What is the
nutritive ratio?
2. A ration consisting of 10 lbs. of corn and 20 lbs. of
alfalfa hay contains 2.99 lbs. of digestible protein, 14.57
lbs. of carbohydrates, and .67 lb. of fats. What is the
nutritive ratio ?
3. A ration consisting of 8 lbs. of corn, 1 lb. of cotton
seed meal, 12 lbs. of alfalfa hay, and 20 lbs. of corn silage
contains 2.50 lbs. of digestible protein, 12.52 lbs. of digest
ible carbohydrates, and .75 lb. of fats. What is the
nutritive ratio ? Compare these three rations with the
feeding standard for a cow giving 22 lbs. of milk a day.
(See page 250.) Which is too narrow? Which is too
306 PRACTICAL ARITHMETIC
wide? Which most nearly meets the requirements of the
standard?
4. If a ration consisting of 10 lbs. of corn and 20 Ihs. of
redclover hay contains 2.15 lbs. of digestible protein,
13.83 lbs. of digestible carbohydrates, and .77 lb. of digest
ible fats, what is the nutritive ratio ?
5. If a ration for dairy cows, consisting of 7 lbs. of
corn and cob meal, 1.5 lbs. of cottonseed meal, 10 lbs. of
clover hay, and 40 lbs. of corn silage contains 2.14 lbs.
of protein, 13.02 lbs. of carbohydrates, and .93 lb. of fats,
what is the nutritive ratio?
6. A ration consisting of 20 lbs. of corn, 5 lbs. of tim
othy hay, and 5 lbs. of corn stover contains 1.8 lbs. of
protein, 17.13 lbs. of carbohydrates, and .96 lb. of fats.
What is the nutritive ratio? Is this too wide or too
narrow a ration for a fattening beef animal for the third
period? (See page 250.)
7. Determine the nutritive ratio of the feeding stand
ards in the table on page 250.
818. Find the nutritive ratio of rations Nos. 1 to 11,
on pages 251 to 255.
Find the nutritive ratio of the following rations:
19. Redclover hay, 12 lbs., corn stover, 15 lbs., corn,
12 lbs.
20. Corn silage, 40 lbs., redclover hay, 5 lbs., corn
stover, 5 lbs., corn, 6 lbs., cottonseed meal, 3 lbs.
SPECIFIC GRAVITY
430. The Specific Gravity of a substance is the ratio of
its weight to the weight of an equal volume of water.
RATIO
307
431. To find the specific gravity of a substance, state
the ratio of its weight to that of an equal volume of water,
reducing this ratio so that the second term shall be 1.
EXERCISE 239 — WRITTEN
Find the specific gravity of each of these substances, tlie
weight of which is given per cubic foot :
1. Water, 62.42 lbs.
2. Cast iron, 449.86 lbs.
3. Cast copper, 548.55 lbs.
4. Cast lead, 708.59 lbs.
Ice,
58.04 lbs. 10. Clay,
6. Cork, 14.9808 lbs.
7. Maple wood, 46.815 lbs.
8. Ebony, 83.0186 lbs.
9. Butter, 58.6748 lbs.
74.904 lbs.
Find the weight per cubic foot of the following sub
stances, the specific gravities of which are given :
11.
Ash wood,
.84.
24.
Paraffin,
.874.
12.
Mercury,
13.598.
25.
Glycerine,
1.26.
13.
Glass,
2.89.
26.
Milk,
1.031.
14.
Cider,
1.02.
27.
Petroleum,
.836.
15.
Sea water.
1.03.
28.
Olive oil.
.915.
16.
Silver,
10.47.
29.
Sulphur,
2.043.
17.
Gold,
19.26.
30.
Anthracite,
1.8.
18.
Steel,
7.816.
31.
Elm wood
.8.
19.
Tin,
7.291.
32.
Oak wood,
.845.
20.
Flint glass.
3.329.
33.
Yellow pine wood, .657.
21.
Platinum,
22.069.
34.
Poplar wood.
.389.
22.
Copper,
8.878.
35.
Beech wood,
.852.
23.
Aluminum,
2.68.
PROPORTION
432. When two ratios are equal, the four terms form
a Proportion, e.g.^ 2:4::8:16 are in proportion, since
f equal Jg.
433. A proportion, therefore, is an expression of equal
ity between two ratios and is written by placing the sign
of equality or the double colon between the two ratios.
The proportion 2 : 4 : : 8 : 16, or 2:4=8:16, is read, 2 is to
4 as 8 is to 16.
434. The first and last terms of a proportion are called
the Extremes, and the two middle terms are the Means.
435. When four numbers are in proportion, the product
of the extremes equals the product of the means. This
constitutes the test of the proportion.
EXERCISE 240. — ORAL
Complete the following statements of proportions :
1. 2: 4:: 3:? 5. 6:?:: 5: 10.
2. ?:4::4:8. 6. 2.8 : 5.6 :: ? : 4.8.
3. 6: 9:: 12:? 7. 9:24::?: 8.
4. 5: 15:: 6:? 8. 1.2 : ? : : 2 : 4.
436. Find the missing term in the following propor
tion :
? • 8 1 • • 1 9,^ ^ '^^^ product of the extremes equalling
the product of the means, 8.1 X 1.25 = 5
SOS
PROPORTION . 309
times the unknown number. The unknown number is
8.1 X 1.25
therefore — ^ — . Solving this example by cancellation,
o
.25
we have 8.1 x l?^ _2qk
The completed proportion, therefore, is 2.05 : 8.1 : : 1.25 : 5.
EXERCISE 241. — WRITTEN
Find the missing terms in the following proportions :
1. ?: 136:: 84: 336. 5. 17.4 : 191.1 :: 11.5: ?
2. 12: 144::?: 1728. 6. ?: 6.25 :: .75 : 8.36.
3. 15:?:: 6: 82. 7. ?: 18:: 45: 32.
4. 22: 5:: 35:? 8. 18.2 :?:: 7.3 : 9.1.
437. If 10 bus. of apples cost 17.60, what will 15 bus.
of apples cost ?
This problem may be stated and solved as a proportion
since we have two ratios, one between the number of bushels
of apples in the first instance and the number of bushels in the
second instance, 10 : 15 ; the other between the amount paid in
the first instance and the amount paid in the second instance
$ 7.50 and f ?. The price of apples remaining the same, these
two ratios are equal. Therefore, we have
10 bus. :15 bus.:: $7.50:$?
Since unlike quantities cannot be multiplied, we may substi
tute abstract numbers instead of the first two terms, which will
not affect the value of the ratio. We then have
10 : 15 : : f 7.50 : $ ? Solving, we have $12.25, the cost of
the 15 bus. of apples.
Arrange the three known quantities and the unknown
310 PRACTICAL ARITHMETIC
quantity of the problem in the form of two ratios which
shall be equal. Solve as a proportion.
BXEBCISB 242. — WRITTEN
1. If 25 lbs. of sugar cost $1.50, what will 75 lbs.
cost ?
2. If 6 tons of hay cost $ 72, what will 36 tons cost ?
3. If a railroad ticket good for 80 miles costs $1.80,
what will a ticket good for 60 miles cost ?
4. If a railroad ticket for 64 miles at 2 cts. a mile
costs $1.76, what will a ticket for 17 miles cost at the
same rate ?
5. If it costs 14.01 cts. to spray 3 acres 3 times, how
much will it cost to spray 21 acres 3 times ?
6. If the net profit from spraying 6 acres of potatoes
is $ 138, what is the profit from spraying 33 acres ?
7. If at a certain moment a post 32 ft. high casts a
shadow 48 ft. long, how long is the shadow of a tree which
is 48 ft. high?
8. At the same moment how tall is a tree that casts
a shadow 32 ft. long ?
9. Measure the height of a post in your neighborhood
and the length of its shadow; also, at the same time,
measure the length of the shadow of any tall object, and
by means of these measurements calculate the height of
the tall object.
10. The height of a tall object is also often measured,
as is shown in the accompanying figure, using a triangle
of which UD = DC. The sides of the triangle are in the
PROPORTION
311
proportion AB :GD:: BE : I)E. Exclusive of the height
of the triangle from the ground, what is the height of the
tree, if the distance EB
is 64 ft.?
H. If in 15 lbs. of
oats there are 13.35 lbs.
of dry matter, 1.395 lbs.
of protein, 7.125 lbs. of
carbohydrates, and .54 lb.
of fats, how many pounds
of each of these are there
in 2000 lbs. of oats ?
12. If in 5 lbs. of cow
pea vine hay there are
4.4 lbs. of dry matter, .465 lb. of protein, 1.92 lbs. of
carbohydrates, and .06 lb. of fats, how many pounds of
each are there in 2000 lbs. of cowpea vine hay?
13. If in alfalfa hay there is 10.6% of digestible pro
tein, how much protein is there in 2500 lbs. of alfalfa
hay?
LEVERS
438. In considering le
vers, three things must be
recognized : (1) the Power
applied to do the work or
overcome the resistance ;
(2) the Weight or the re
sistance to be overcome ;
and (3) the immovable
point on which the lever turns, called the Fulcrum.
312
PRACTICAL ARITHMETIC
In the figure on page 311 where is the weight ? Where
is the power ? What is the fulcrum?
439. A lever can always be divided into two parts :
(1) the distance from the weight to the fulcrum, called
the Weight Arm ; (2) the distance from the power to the
fulcrum, called the Power Arm.
In the figure how long is the power arm? How long
is the weight arm ?
440. The factors of a lever are always in proportion,
thus:
Power arm : Weight arm : : Weight : Power,
or, abbreviating.
Pa :Wa : : W: P
EXERCISE 243. — ORAL
1. In the figure on page 311 how many pounds must
the man exert to lift the stone if it weighs 30 lbs. ?
90 lbs. ?
2. If the man exerts a power of 3 lbs., how heavy a
stone can he lift? If he exerts 9 lbs. ? 100 lbs. ? 3^ lbs. ?
3. If the power arm is 9 ft. and the weight arm 1 ft.,
what will be the answers to No. 1? To No. 2?
4. In this figure where is the power? Where the ful
crum? Where the
weight (resistance)?
Will a wire be cut more
^ easily near the tip of the
'"""■' ~ ' shear blades, or near the
base ? Why ? What is the power arm ? The weight
arm ?
PROPORTION
5. In this figure what is the length of power arm?
weight arm ?
313
Of
6. With a pressure of 80 lbs., how much resistance
can be overcome ?
EXERCISE 244. — WRITTEN
1. In raising a stone weighing 1400 lbs. with a weight
arm 1 ft. long, how long must the power arm be to enable
a man who can exert only 200 lbs. of power to do the
work? If the weight arm is reduced to 9 ins., how long
need the power arm be ?
2. How much does the weight of a 156lb. man, on
the end of the power arm, fall short of raising a stone
weighing 1200 lbs. with a lever having a power arm of
4 ft. and a weight arm of 1 ft.?
3. With a wagonjack of the dimensions indicated, how
much power is needed to lift
the rear end of a loaded
wagon, the rear end, loaded,
weighing 1600 lbs.?
4. With steelyards as in
the figure on page 314, with
a weight (a) of 8 ozs., how
far from the fulcrum must this weight be to balance 18
lbs. ? 25 lbs. ? 16 lbs. ?
314
PRACTICAL ARITHMETIC
5. In fishing John lield the large end of the pole sta
tionary in the left hand. The right hand was 2 ft. 9 ins.
farther up the pole, which
was 12 ft. long. With his
right hand it took a force
of 8 ozs. to sustain the fish.
What did it weigh ?
Draw a diagram to aid in
stea solving this problem.
6. In drawing a nail with a hammer the distance from
the fulcrum to the nail is 2 ins., from fulcrum to hands is
11 ins. How much direct pull is exerted upon the nail if
it requires 95 lbs. of pull
upon the hammer handle to
extract it ?
7. A boy weighing 96
lbs. is swinging on a gate,
having 2 hinges, 12 ft. from
the hinges. It is 3 ft. 4 ins.
from one hinge to the other.
How ,much pull does the
boy's weight exert upon the upper hinge ? How much
does it push upon the lower hinge ? Draw a diagram
before attempting to solve. Compare with the last
problem.
8. Two men carry a weight of 195 lbs. suspended on
a pole between them. If the weight is 6 ft. from one
man and 9 ft. from the other, how many pounds does
each carry ? In order that one may carry ^ of the weight
where must the weight be hung?
PROPORTION 31c
COMPOUND PROPORTION
441. If 18 men working (j hrs. per day can dig a canal
50 ft. long in 25 days, how many men working 10 hrs. per
day can dig a canal 80 ft. long in 8 days ?
This problem can be separated into simple proportions
and solved as follows :
If 18 men can dig a canal in 25 days, how many men are re
quired to dig the same canal in 8 days ? This is expressed in
proportion. (1) Solving, we get ^j^ men. Tlie canal is, how
ziN a ok 18 9 ®^''^'^'' ^"^ ^^ ^°"S instead
' 50 80   of 50 ft. long. If...
)y^ 10 • 6 • • 90^ ■" ™^^ ^^^ ^^° ^ °^^^^^ ^^ ^*'
„ ' y ..y ^r m „, lo'ia; ill 8 davs, how many
8 X no X 10 18 X ifi X90 ■ A ^ A
— ^ ^, =: ^^^ ^ *^^^ ^ . ^ men are required to dig a
canal 80 ft. long in 8
days ? This is expressed
in proportion. (2) Solv
ing, we get 90 men. The
previous statements have been made under tlie assumption that
the men were to worlt 6 hrs. per day instead of 10 hrs. If 90
men can dig a canal 80 ft. long in 8 days, working 6 hrs. per
day, how many men will be required to dig a canal 80 ft. long
in 8 days, working 10 hrs. per day ? This is stated in pro
portion. (3) Solving, we get 54 men, which is tlie final answer.
This method of procedure may be shortened. IMultiplying the
completed proportions, 1, 2, and 3 together, term by term, we
obtain a new proportion, which, expressed as a ratio, is shown
in 4. We see that the answers obtained from the first two pro
portions cancel, leaving the second member a simple ratio. The
ratio may now be expressed as a proportion, as is shown in 5,
and solved, as follows :
10 3
23X80X0X18
18 : ? men.
8 X 50 X 10
: 54 men.
316 PRACTICAL ARITHMETIC
The fact that the first two answers cancel shows that it was
unnecessary to obtain them to arrive at the final answer.
Rule 1. Place the unknown quantity as the fourth
term of the proportion.
2. Place as the third term of the proportion the quan
tity given in the problem expressing the same kind of
thing as the unknown quantity.
3. Take each of the other ratios separately, and arrange
according to their relation to the ratio already stated.
4. The product of all the means divided by the product
of all the extremes, except the unknown one, will give the
answer.
442. The product of two or more simple ratios is a
Compound Ratio.
443. A proportion in which either or both ratios are
compound is a Compound Proportion.
EXERCISE 245, — WRITTEN
1. If 17 men working 7 hrs. a day can build a bridge
in 22 days, how many men working 10 hrs. a day will it
take to build the bridge in 4 days ?
2. If 3 men can milk 35 cows in 1.5 hrs., how many
men will it take to milk 65 cows in J hr. ?
3. If 3 teams working 5 hrs. a day can haul dirt as
fast as 5 men can excavate it, how many teams working
7 hrs. a day are required to haul dirt as fast as 15 men
can excavate it ?
4. If 6 men can draw and house 32 tons of hay in 2
days, how many men are needed to draw and house 14
tons in 6 hrs. ?
PROPORTION 317
5. If 2 men cut 8 cords of wood in 4 days, how long
will it take 12 men to cut 36 cords ?
6. If 4 men with a onehorse plow break 28 acres in
7 days, how many days will it take 3 men with twohorse
plows (a man with a twohorse plow doing twice as
much work as a man with a onehorse plow) to break
42 acres ?
7. If the eggs laid by 28 hens in 16 weeks are worth
132.50, what will be the value of the eggs laid by 50 hens
in 12 weeks ?
8. If it requires 35 cows giving 77 qts. of milk each,
per week, to supply 425 customers, how many cows giv
ing 270 qts. per month will be required to supply 125
customers ?
9. If 30 cows give 462 lbs. of milk in 21 days, how
many cows are required to give 1200 lbs. in 7 days?
10. If 4320 lbs. of silage last 30 cows 48 days, how
much silage is needed for 15 cows for 60 days?
11. If the contents of a tank of water 2 x 4 x 10 ft.
weighs 4994 lbs., what will the contents of a tank 7 x 12
X 19 ft. weigh?
12. If it takes 2 cu. yds. of concrete to make 40 posts
6" X 4J" X 7', how many yards will it take to make 678
posts 4" X 4" X 5'?
13. If the weight of a volume of water 1 x IJ X 9 ft.
is (pupil fill blank), what is the weight of a piece
of ebony  x  x 3 ft., ebony weighing 1.33 times as much
as water of equal volume?
POWEES
In the figure how many units are there on each side of
tlie square? How many in the whole
square ? There are 9 square units in a
square of 3 units on a side, therefore, 9
is said to be the Square of 3. Simi
larl}, a square witli 4 units on each
side has a total of 16 square units.
The Square of a number is the product of a number with
itself.
444. In the figure how many units are there on each
edge? How many cubic units in the cube? There being
27 cubic units in a cube with 3 units
on an edge, 27 is said to be the Cube
of 3.
The Cube of a number is the prod
uct of the number taken 3 times as
a factor.
y y /
y'/ /
/
/
/
/
445. Squares and cubes are called Powers of a number.
As the square and the cube are the second and third
powers of numbers, so taking the number 4 times as a
factor gives the fourth power, 6 times the fifth, etc.
446. The power of a number to be taken is indicated
by a small figure written above and to the right of the
318
POWERS 319
number, e.g., 3^ means the square of 3, 4^ means the cube
of 4, 6^ means the ninth power of 6, etc.
447. The figure used to indicate the power taken is
called the Exponent.
EXERCISE 246. — ORAL
Find the value of the following :
1. 62. 4. 13. 7. 19. 10. 122.
2. 92. 5. 82. 8. 102. 11. 252.
3. 202. 6. 112. 9. 34_ 12. 303.
EXERCISE 247. —WRITTEN
Find the value of :
1. 843. 3. 1.253. 5. (^V)2. 7_ (ll^)3.
2. 123. 4. .752. 6. (7)3. 8. 12.53.
448. We have seen that the square of a number is the
number multiplied by itself, e.g.., the square of 24 is
24 X 24. We may write this multiplication thus :
2 tens + 4 units
2 tens + 4 units
4 units X 2 tens 4 units^, result of multiplying
by 4 units.
2 tens' 4 units X 2 tens, result of multiplying
by 2 tens.
2 tens^ + 2(4 units X 2 tens) H 4 units', sum of partial products.
Since any number greater than 10 may be regarded as com
posed of the sum of two numbers, the square of the sum of any
two numbers is equal to the square of the first number + 2 X
(the product of the first x the second) ( the square of the last.
The same formula may be arrived at by considering the
square of 24 as representing an area 24 units on each edge, as
in the accompanying figure, and cutting the lines of the sides
320
PRACTICAL ARITHMETIC
into 20 units and 4 units to represent 2 tens and 4 units. It is
then seen that the whole square of 24 consists of ; the large
square a, which is 20^ and two times
the rectangle 6, which is 2 x (20 x 4),
and the small square c, which is 4^
Represent the square of the same
number, 24, in a diagram, similar
to that just used, but cutting the
lines into 18 and 6 units instead of
into 20 and 4. Solve as above.
Does this change affect the result?
We may represent the lines by letters n and s, as in the
figure ; then we have :
(n + s)2 = w2 + 2(w, X s) + s2.
By this formula determine the square of 63 :
63 = 6 tens and 3 units.
6 tens squared = GO x 60 = 3600
2 X (6 tens x 3 units) = 2 x 60 X 3 = 360
3 units squared = 3x3 = 9
3969
Si
a
o
6
«9
20
n
S4
S
6
Tl<
c
EXERCISE 248. — "WRITTEN
Find by the last method given above, the squares of :
1.
62.
8.
67.
15.
199.
22.
672.
2.
79.
9.
98.
16.
86.
23.
999.
3.
24.
10.
99.
17.
205.
24.
897.
4.
81.
11.
107.
18.
640.
25.
862.
5.
63.
12.
129.
19.
783.
26.
978.
6.
72.
13.
15.
20.
297.
27.
209.
7.
84.
14.
22.
21.
248.
28.
679,
EOOTS
449. "What number multiplied by itself will give 25 ?
16? 4? 100? What number used three times as a factor
gives 8? 27? 125? 1000? What number squared equals
25? What number cubed equals 8? 27?
450. A number which when squared equals a certain
number is said to be the Square Root of that number, e.g.,
5 is the square root of 25 ; 4 is the square root of 16.
451. A number which when cubed equals a certain num
ber is said to be the Cube Root of that number, e.g., 2 is the
cube root of 8 ; 3 is the cube root of 27.
452. The root of a number is indicated by the sign, V~,
known as the Radical Sign. V~indicates the cube root,
V~, the fourth root, etc.
453. What is the square of 1? Of 10? Of 100? Of
1000 ? Of 10,000 ?
How many figures does it take to express the square
root of a number of 1 or 2 figures? Of 3 or 4 figures? Of
5 or 6 figures? How many figures in the power equal 1
figure in the square root ?
454. If a whole number be divided into groups of 2
figures each, beginning at units' place, the number of
groups will equal the number of figures in the root.
455. Find the square root of 529 :
321
322 PRACTICAL ARITHMETIC
Separating into periods, we see that since there are two peri
ods, the root consists of two figures. The square of the tens
of the root must be contained in the second period, 6. The
greatest square in 5 is 4, the root of which is 2, which is
therefore the tens' figure of the root desired.
5'29^23 Subtracting the square of 2 tens or 400
^QQ from 529, we have 129. This remainder must
A.^y[9Q contain, 2 x (tens X units) + units^ (see para
— 19C)' graph 448). Two times the 2 tens = 4 tens.
— — ■ 4 is contained in 12, 3 times. We therefore
take 3 as the next figure of the root. To find whether the
remainder exactly contains 2 x (tens x units) + units ^, or
what is equivalent (twice the tens + the units) x units, add
the units to twice the tens and multiply by the units, thus :
(3 + 40) X 3, securing 129. 23 is therefore the square root
of 529.
To prove square root, multiply the square root by itself,
456. Find the square root of 15,713,296 :
) 15,713,296 )3964
9_
69)671
nni Proceed as in the preceding pro'b
_— — ■ lem. In each step consider the part
. 786)5082 Qf ^j^g j.jjjj^ already found as tens with
4716 relation to the next figure.
7924)31696
31696
457. Find the square root of 94,864 :
94864)308
g Since the divisor 6 is not contained in 4,
fin8"vd8RJ. ^^ placed in the root as well as in the divisor,
^ and the next group is brought down.
4864 ^
ROOTS 323
458. Rule for the extraction of square root.
1. Separate the number into groups of two figures
each, beginning at the decimal point.
2. Find the greatest square in the left period. Its
root is the first figure of the required root.
3. Subtract the square of this root from tlie first
period, and bring down the next period.
4. Divide the remainder by twice the part of the root,
already found, considered as tens, as a trial divisor, secur
ing the next figure of tlie root.
5. To the trial divisor add the new figure of the root,
then multiply by the last figure found, and subtract the
product from the last remainder.
6. Bring down the next period, and continue as above.
459. If the number is not a perfect square, add ciphers
to the number and continue the division, expressing the
result as a decimal.
460. What is the square of .25? Of .75? Of 1.25?
How many more decimal places are there in the square of
a decimal than in the decimal? How many decimal places
are there in the square root of a decimal?
The square root of a decimal has half as many decimal
places as the decimal itself.
Each group of the decimal must contain two figures.
Annex a cipher, if need be.
461. What is the square of J? Off? Off?
What is the square root of ^6? Of ? Of^?
To obtain the root of a fraction, extract the root of
both numerator and denominator separately.
324
PRACTICAL ARITHMETIC
]
3XBRCIS
9.
10.
11.
12.
13.
14.
15.
16.
E 249. WB]
V9433_.
vtiif.
V3.33.
TTEN
17.
18.
19.
20.
21.
22.
23.
24.
1.
V7569.
V8.7.
2.
V743044.
^■
3.
V6889.
V987.12.
4.
V283024.
vj.
5.
V7921.
V64289.
VS.
V61
V.
6.
V236089.
vn.
7.
V.9216.
Vfllf.
V.0178.
8.
V69284.7632
EXERCISE 250. — 'WRITTEN
In a right triangle tlie square of the hypotenuse, the
side opposite the right angle, is equal to the sum of the
squares of the other two sides.
1. Two sides of a right tri
?%_ angle are 76 and 84 ft. What
is the length of the hypote
nuse?
1% In ba.se ^s. . __, , 
; ^^ 2. Ihe hypotenuse oi a tri
angle is 82 ft. One side is 79 ft. What is the length of
the other side?
3. The base of a ladder is 12 ft. from the house. The
top touches the eaves 39 ft. high. How long is the
ladder ?
4. A pasture shaped as a right triangle is 80 rds. 3 yds.
4 ft. on its long side, 12 rds. 2 ft. on its short side. How
long is the other side ?
5. The area of a circle is 9678 sq. ft. What is its
radius ?
ROOTS
325
6. What is the diagonal of a rectangle 80 x 40 rds. ?
7. The base of a 32 ft. ladder is 7 ft. from the house.
How high on the wall can it touch?
8. In cutting across the diagonal of a lot twice dailj^ for
5 days, how much walking is saved if the lot measures
203' X 98"?
In carpenter work and other building it is frequently
necessary to know the hypotenuse of a right triangle. It
may be found with sufficient
accuracy for many such
purposes by use of the
square and rule.
9. What length of brace
is needed from an upright
post to a horizontal beam,
the two ends of the brace
/
iil'ill'iiiiir'iH'iiM;'Mji
i,i°'i.i^,i?i,Ki,i^i,m,!':i,i':hRi,r,Fa
to be 5 ft. and 4 ft. from the right angle ? Place the rule
on the square at 4 and 5 ins., as shown in the figure.
Read the length of the hypotenuse in inches. Let each
inch on the square and rule stand for a foot in the prob
lem. Solve the same problem by square root, and compaire
answers.
CUBE ROOT
Since the extraction of the cube root may be much more
easily understood after algebra has been studied, and
since it is customary to defer its study until then, it will
not be discussed here. The method of extracting the
cube root is given in the Appendix for the benefit of any
who may have occasion for its use.
326 PRACTICAL ARITHMETIC
EXERCISE 251
MiscELLAXEous Review Problems
1. What is the cost of spraj'ing 6 A. of cucumbers
6 times with Bordeaux mixture, applying 100 gals, per
acre each time, using 3 lbs. of bluestoue at 1^ cts. per
pound, 6 lbs. of lime at 1 ct. per pound in each 50 gals,
of mixture ? Allow 33 cts. for labor of making each 100
gals.
2. Some tall structures are: the Eiffel Tower 984 ft.,
the Washington Monument 555 ft., the Cologne Cathe
dral 521 ft., Cheops Pjramid 486 ft., Strasburg Cathe
dral 474 ft.. Bunker Hill Monument 221 ft. How much
higher is the Eiffel Tower than each of the other struc
tures named? How much higher is the Washington
Monument than each of the others, except the Eiffel
Tower ?
3. Alcohol is .83 as heavy as water. Copper is 8.8 times
as heavy. How much heavier is 1 cu. ft. of lead than
1 cu. ft. of alcohol, if 1 cu. ft. of water weighs 62.42 lbs. ?
4. What is the cost of: 1 smoothing plane, $2.40; 1
spoke shave, 1.90; 3 chisels at $.42; 1 gauge, 52 cts.;
1 claw hatchet, $.80; 1 ratchet brace, $1.08; 5 bits, at
$19, $.28, $33, $.41, and $.49; 1 screwdriver, $.16;
1 level, $2.00; 1 square, $1.50 ; 1 rip saw, $1.65; 1 hand
saw, $1.65?
5. From a roll of carpet 21^ yds. long there are cut
two pieces, one 1\ yds., one 17f yds. How much
remains ?
6. Allowing ^ in. per foot for shrinkage, how much
MISCELLANEOUS REVIEW PROBLEMS 327
larger and wider than the desired article must be the pat
tern for a castiron support 9' 11" long x 2' 1.3" wide?
7. Allowing the same shrinkage for lead, how long
must be the mould for a casting, to be, when finished,
2' 4" X 5"?
8. Allowing j^g" per foot for shrinkage in brass cast
ings, what would be the answer in the last two problems?
9. In an arithmetic class where term grade was valued
at 75% and the examination at 25%, one student with a
term grade of 84 made a final mark of 77. What was his
mark on examination?
10. On an examination paper of 16 questions one ques
tion was threefourths correct, three were onehalf right,
four were onethird correct, one was eightninths correct.
What was the grade?
11. If the same errors had occurred on an examination
paper of 12 questions, what would have been the grade?
If on a paper of 10 questions? If on a paper of 9
questions ?
12. If in feeding pigs, 1 bu. of corn produces 10.9 lbs.
of gain, what price per bushel is obtained for corn when
pigs sell for 5.75 cts. per pound?
13. If .92 A. of rape furnish feed for pigs equal to
1596 lbs. of corn and 796 lbs. of wheat middlings, what
is the feeding value of an acre of rape, when corn is
worth 45 cts. a bushel and wheat middlings 1^18 a ton?
14. Five pigs at 10 months of age average 243 lbs. in
weight and sell for 5J cts. per pound. If they consumed
98 lbs. of wheat bran at $ 18 a ton, 1862 lbs. of corn at
328 PRACTICAL ARITHMETIC
50 cts. a bushel, and grazed .25 A. of clover, .25 A. of sor
ghum, and .60 A. of peanuts, what was the combined
value per acre of these grazing crops for pig feeding ?
15. If 1 A. of peanuts and 37.8 bus. of corn pro
duce 1426 lbs. of gain in hogs and they sell for 5 cts. per
pound, what is the value of an acre of peanuts if corn is
worth 47 cts. per bushel?
16. The population of the United States in 1900 was
76,059,000. It is estimated that the number of births
exceeds the number of deaths by 15.2 per 1000 of population
every 10 yrs. Based on these estimates, what may the popu
lation of the United States be expected to be in 1910?
17. The increase in population of the United States
through immigration in 1903 and 1904 was 800,000 annu
ally. In 1905 and 1906 the increase was 1,000,000 annually.
Taking the average of these four years as the probable
increase through immigration for the next ten years, to
gether with the results of the last problem, what is your
estimate of the population of the United States in 1910?
18. Estimated in a similar way, basing each time upon
the estimated population of the preceding decade, what
will be the population in 1920? 1930? 1940 ? 1950?
19. Add each line without writing in columns :
a. 79, 46, 87, 93, 84, 72, 16.
I. 857, 965, 847, 964, 876.
c. 965, 864, 791, 862, 764, 968.
d. 6482, 9683, 7981, 8472, 6897, 9861.
e. 64,875, 89,672, 978,459, 679,821. >
MISCELLANEOUS REVIEW PROBLEMS 329
20. Estimate the average speed per mile of the trains
as given in the timetable on page 145.
21. It requires 151 days for the eggs of the honey bee
to develop into honey bees that are to become queens:
■^j of this period is required for the eggs to hatch;  of
this period is required to mature the larvae and pupse.
How many days are spent in the egg state? How many
in the larval and pupal states ?
22. It requires ^ more days for the eggs to develop
to become worker bees than to become queen bees : f of
this period is required for the eggs to hatch ; ^ to mature
the larvae and pupee. How many days are spent in the
egg state? How many in the larval and pupal states?
23. The plant louse often produces in 12 generations
in one season, 10,000,000,000,000,000,000,000 ofCspring.
These are each about ^^ of an inch long. If all should
live, how many miles long would such a procession be, ar
ranged single file? How many times would this proces
sion reach around the earth, considering the circumference
of the earth as 25,000 miles?
24. If on an average 1 cattle tick produces 1000 young
and there are four generations in one year, how many
ticks may be produced in the fourth generation ? If one
half these ticks are females, ^ inch in length, and onehalf
males, ^ of an inch long, arranged single file, how many
times would this procession encircle the earth ?
A family of two persons has an income of f 520. Their
expenses are as follows :
Eent $120
Food 210
Clothing $60
Fuel 30
330
PRACTICAL ARITHMETIC
Charity . . .
. $10
Church . . .
10
Eecreation . .
10
Incidentals
10
Emergencies .
14
Light $ 7
Insurance 24
Replenishing .... 10
Car fare 5
Literature 5
25. What per cent of the income is spent for rent?
26. What per cent is spent for food ?
27. What is the ratio of money spent for rent to that
spent for food?
A family of three persons —
years old — has an income of
penses are as follows:
Eent .
Food .
man, wife, and one child 6
f 780. The itemized ex
Clothing
Fuel .
Light .
Insurance
Eeplenishing
$120
320
120
40
10
40
20
Car fares .
$10
Literature
.
10
Church
V
15
Eecreation
.
20
Charity . .
.
10
Incidentals
20
Emergencies
.
25
28. If the food of the child is regarded as J of that of
the man, and the food of the wife ^ that of the man,
what is the cost of food for each?
29. What per cent of the money expended for food by
the first family is expended for food by the second family?
30. What is the ratio of the sum of the last six items
of the first family to the same items of the second family?
31. The five states of greatest population in 1900 were:
New York 5,997,853, Pennsylvania 5,258,014, Illinois
,3,826,351, Ohio 3,672,316, Missouri 2,679,184. How
MISCELLANEOUS REVIEW PROBLEMS 331
much does the population of the two latter together
exceed that of New York?
32. Five sugarproducing districts are : Cuba 1,664,
862,000 lbs., Louisiana 695,101,878 lbs., Hawaiian Islands
520,138,232 lbs., Philippine Islands 435,000,000 lbs.,
Porto Rico 122,000,000 lbs. How much does the produce
of the other four together exceed that of Cuba?
33. The five leading states in wheat production are :
Minnesota 142,345,672, Kansas $32,469,706, North Da
kota 128,383,767, Ohio 127,788,094, Indiana 124,208,
398. What per cent of the wheat produced by these five
.states is grown in each state?
34. The iron manufactured in five states was : Pennsyl
vania $264,571,624, Ohio $65,206,828, Illinois $39,011,
051, New York $15,849,531, New Jersey $11,018,575.
Find the total amount. The production of each state is
what per cent of the total?
35. The textile manufactures of five leading states
were: Massachusetts $184,938,074, Pennsylvania $132,
367,499, New York $86,171,293, Rhode Island $67,005,
615, New Jersey $52,831,023. The production of each
state is what per cent of the total of the five states?
36. The number of miles of railroad for every 100 sq.
mis. of area in five groups of states was : Middle Atlan
tic States 15.8, New England 11.72, Central 10.63,
Southern 5.31, Western 2.09. What was the average
mileage per 100 sq. mis. ? The mileage per 100 sq. mis.
for each section is what per cent of the total?
The following table shows the value of raw cotton and
332
PRACTICAL ARITHMETIC
its manufactured product for 6 years, with the number
of wageearners engaged :
Materials
Vat.ub of Pkoduct
Wageearners
Yeak
1447,546,540
$759,262,283
517,237
1890
303,709,894
532,673,488
384,251
1880
353,249,102
520,386,764
274,943
1870
112,842,111
214,740,614
194,083
1860
76,715,959
128,769,971
146,877
1850
11,540,347
15,454,430
1840
37. What was the per cent of increase in value of the
annual product each decade based upon the preceding
decade?
38. What was the per cent of increase in value of manu
factured cotton over the crude cotton each decade?
39. What was the per cent of increase in number of
wageearners in each decade since 1850?
40. At 12 cts. per pound, for how much will a 500lb.
bale of cotton sell?
41. A bale of cotton weighs 500 lbs. and sells for
i60. The bagging and ties (strap iron) used to wrap
it weigh 24 lbs., and are included in the total weight of
the bale. What is the actual cost of the cotton per
pound?
42. In making a 450lb. bale of cotton into thread for
cotton cloth with a loss of 17 % due to particles of leaf,
sand and other impurities, how much cotton is actually
used?
43. The warp threads in a sheet weigh 10 J ozs., the
filling threads weigh 8 ozs., 20,160 yds. of warp threads
MISCELLANEOUS KEVIEW PROBLEMS 333
weigh 1 lb., and 22,260 yds. of filling weigh 1 lb.
Find the number of yards of warp and filling used in
the sheet.
44. A stripe gingham cloth is made 28" wide, 56 ends
per inch. The colors arranged in the following order
form a pattern : 28 ends white, 16 ends blue, 8 ends
black, 4 ends red. How many times is the pattern re
peated in the cloth, and how many ends of each color are
there ?
45. A loom weaves 50 yds. of calico cloth in 10 hrs.
How many yards will be woven in 60 hrs. ?
46. If 1 lb. of cottonseed meal is equal to 1.75 lbs.
of corn for cattle feeding, what is the value of a ton of
cottonseed meal for cattle feeding when corn is worth
50 cts. a bushel? 1 bu. = 56 lbs.
47. If for cattle feeding 3 lbs. of cottonseed are equal
to 3.48 lbs. of corn, what is the value of cottonseed per
ton for cattle feeding when corn is worth 45 cts. a bushel?
48. Ten pigs weighing 56 lbs. each bought for f 50,
after feeding for 120 days weigh 224 lbs. each. They
then sell for $5,625 a hundred. What was the net profit
if it cost 4.25 cts. in feed to produce a pound of gain?
49. If it costs 4.07 cts. to produce a pound of gain
with hogs fed corn and wheat middlings, and 3.25 cts. to
produce a pound of gain with hogs fed corn and green
clover, how much less will it cost to grow 12 hogs from
50 lbs. up to a weight of 243 lbs. each on corn and clover
than on corn and wheat middlings, if no charge be made
for the clover pasture ?
334
PRACTICAL ARITHMETIC
50. A beef animal weighing 986 lbs. is bought for
3.875 cts. a pound and is fed 30 lbs. of silage worth 3
a ton, 10 lbs. of clovei hay worth !^8 a ton, 8 lbs. of
corn worth 41 cts. a bushel, and 3 lbs. of cottonseed meal
worth $32 a ton, per day for 180 days. What is the
profit on the feeding if the animal gains 1.95 lbs. a day
and sells for $5.37^ per hundredweight? If the animal
gains 2.1 lbs. a day and sells for $5.42 per hundred
weight ?
51. How much will it cost to fence each of the lots of
equal area, a, h, c, as chicken lots, if the figures showing
l§
8
20 West
12
40
East
ss
g
S
c
64
dimensions in the dia
grams be taken to indicate
feet: posts 8 ft. apart;
cost of posts 19 cts. each ;
cost of digging postholes
and setting posts 6 cts.
each ; cost of wire netting
5 ft. high, 7 cts. a linear
yard ; estimate exclusive
of cost of other labor ?
52. If the figures indicating dimensions in these dia
grams be taken to mean yards, what will be the cost of
fencing each lot with boards : placing posts 8 ft. apart ;
the posts costing 17 cts, each ; cost of digging postholes
MISCELLANEOUS REVIEW PROBLEMS 335
and setting posts 6 cts. each ; fence to be 5 boards high
at 11.2 cts. for each board 16 ft. in length. Estimate
exclusive of cost of nails and labor of construction.
53. If tlie figures indicate rods, what will be the cost
of fencing each with barbed wire: posts 16 ft. apart;
cost of posts Irt cts. cach ; cost of setting posts 1 cts.
each; fence 4 wires high costing $2.75 per hundred
pounds of 1480 ft. Estimate exclusive of cost of staples
and labor of construction.
54. Procure at home the prices of the materials used
in the last three problems and the cost of construction in
your community, and using such facts solve each of these
problems.
55. In plowing, a furrow one foot wide is turned.
How many times must the field a be crossed to plow
all of it ? How many turns must be made ? (Figures in
dicate rods.)
56; If each turn consumes 30 sees. , what time is lost in
turning in field a ? In field 5, plowing lengthwise ? In
field c, plowing from east to west? (Figures indicate
rods.)
57. What is the value of the time lost in turning in
plowing each field at % 3.50 per day of 10 hrs. ? (Figures
indicate rods.)
58. In raking the same fields with an eightfoot raks,
how many turns are made ? How much time is lost in
each field ? (Figures indicate rods.)
59. With a twelvefoot rake, what will be the number
of turns and the time lost in each field ? (Figures indi
cate rods.)
336 PRACTICAL ARITHMETIC
60. Two wagon tiies, one 22 ft. the other 17 ft. in
circumference, lie upon a floor in contact at one point.
How far apart are their centres ?
61. A window of tJie third story of a house is 32 ft.
from the ground. How long must a ladder be to reach
it, with the foot of the ladder 10 ft. from the building ?
62. At i 18 a thousand feet, what will be the value of
a piece of timber 18 ft. long, 10 ins. wide, and 8 ins. thick ?
63. How many tons of silage are there in a silo 16 ft.
in diameter, if the silage is 24 ft. deep in the silo and 1
cu. ft. weighs 33^ lbs. ?
64. How many barrels of water will a tank 8 ft. in
diameter and 8 ft. high hold ?
65. How many tons of silage will a silo 10 ft. in diame
ter and 20 ft. Iiigh hold, filled within 5 ft. of the top, if a
cubic foot of silage from such a silo weighs 30 lbs. ?
66. To insure the silage keeping well, a silo should not
be less than 30 ft. deep. AVhat must be its diameter to
hold enough silage to feed 25 cows 40 lbs. each a day for
185 days, if a cubic foot of silage weighs 37 lbs.?
67. When the temperature is 37.5° Centigrade, what
will a Fahrenheit thermometer indicate ?
68. If I pay f 3.50 a cord for wood and $.95 a cord for
sawing, how much will a pile of wood 29 ft. long, 8 ft.
wide, and 3 J ft. high cost me ?
69. If a ring is 18 karats fine, what per cent of it is
gold ? (1 karat = ^^.) If it is 14 karats fine, what per
cent of it is gold ?
70. A haberdasher buys hats at $ 18 a dozen and sells
MISCELLANEOUS REVIEW rROBLEMS 33T
them at a profit of 33^%. What is the gain? "What
is the selling price ?
71. A man invests $1250 so as to gain 12] %. What
is his income from his investment ?
72. If flour costs f 6 a barrel, at what mnst it be sold
to gain 8^ % ? To gain 16f % ?
73. A man purchases a hogshead of 12 gross of glass
articles for $45; 5% was broken in shipping. At what
price per dozen must the remainder be sold to gain 20 %
on the whole?
74> I buy a 30gal. barrel of vinegar at 25 cts. a gallon.
Upon examination it is found that 4 gals. 2 qts. have
leaked out. How must the remainder be sold to gain
36% on the whole?
75. A and B are engaged in different lines of business
with a capital of $4000 each. The first year A gains and
B loses 20 % of the investment. The second year A loses
and B gains 20 % on the capital each has then. Which
is now the better off?
76. A farmer losing 20 % of his tobacco crop by hail
received from the insurance company in which the crop
was insured for 75 % of its value, the sum of $450, cover
ing the loss. For what amount was the entire crop
insured ?
77. 10% of 5% of a number is 2.88. What is the
number?
CROP ROTATION
Systems of crop rotation prove to be more profitable
under most conditions tlian the continuous growing of
338 PRACTICAL ARITHMETIC
one crop. The conditions in one section of the South are
illustrated on the opposite page by the record of a 90acre
farm continuously cropped and that of a similar farm
divided into three 30acre fields and the crops rotated.
The cost and produce per acre for each field for a series
of years are given.
78. Find the profit on each 30acre field for each year
and the profit each year on the whole farm under rotation.
79. Find the profit of the whole farm each year under
continuous cropping.
80. Which is the more profitable for the first year?
How much more profitable than the other method?
81. Which is the more profitable for the last year?
How much more profitable than the other method?
82. How much does the profit of the last 3 years under
rotation exceed the profit of the last 3 years under con
tinuous cropping?
83. A farmer has land that, with $1.86 worth of fer
tilizer per acre and $6.85 per acre for labor and other
expenses, including taxes, will produce 56 bus. of corn
per acre. With corn at 50 cts. a bushel the land is draw
ing 5% on what capital?
84. L'nd that, with il.46 worth of fertilizer and
$12.50 for other expenses, will produce f of a bale of cot
ton, worth for the seed f5, for the lint 333 lbs. at 11 cts.
per pound, is drawing 4 % on what capital ?
85. Land that will produce 260 bus. of potatoes,
worth 50 cts. a bushel, with an outlay of $96 for all ex
penses, is paying interest at 4 % on what amount ?
MISCELLANEOUS REVIEW PROBLEMS
339
<
■O OS
1%
Corn 37 bus.
per A. at 50
Cost §11 per A.
Profit . . .
Oats 4 bus. per
A. at40)« .
Cost S 9 per A.
Profit . . .
■= u
CO
oi
CO
Cotton 550 lbs.
per A. at Wf
Cost $22. 50 per
A
Profit . . .
d
1
250 lbs. per A.
atl0>< . .
Cost $20 per A.
Profit . . .
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340 PRACTICAL ARITHMETIC
86. Land near a certain city produces 1300 worth of
garden truck per acre at a total expense of f 120 per acre.
Land of similar quality, but so located as to have no
ma,rket for truck, can raise of its most profitable crop, corn,
60 bus. per acre at 50 cts. per bu. at a cost of $8.50.
Disregarding taxes, estimate the capital or value of each
farm at 4% interest.
87. What is the increased value of the first farm, due
to its location?
88. A field produces 816 lbs. of tobacco averaging 8 cts.
per pound at a total cost of $47 per acre. The land is
drawing interest on what capital at 4 % ? Suppose the
tobacco wilt to take possession of the field and render to
bacco raising impossible and that the next most valuable
crop is corn, 23 bus. per acre, valued at 65 cts. per bushel,
costing to raise $9.20 an acre. What capital at the same
rate will the land then represent?
89. If the yellow yam sweet potato contains 16.64%
of starch, how many pounds are required to contain 2 tons
of starch ?
90. The Red Bermuda contains 2.59 % of sugar, how
much sugar is there in 3 cwts. ? How many tons are
required to contain 2.9 T. of sugar?
91. The average starch content of corn is 65 %, wheat
58%, rice 75%, Irish potatoes 18%, sweet potatoes about
21 % . What is the value of the starch in a ton of each of
these crops at the following prices : corn starch 1.93 cts.,
potato starch 3.57 cts., rice starch 6.53 cts., wheat starch
4.78 cts. per pound?
92. If there is 28% of starch in the Red Bermuda
sweet potato and 9.79 % in the Early Nansmond, how
MISCELLANEOUS REVIEW PROBLEMS 341
much more starch is there in 2 T. 3 cwts. of the first
variety than in 3 T. 7 cwts. of the second? If 58 % of the
starch present can be separated, how mucli can be had in
each case?
93. At 13.50 a day for a man and team, wliat is due
for 3 wks. 4 days 3 hrs. work?
94. At 19.00 a day for a man, team, and liarvester,
what is due for 3 days 7 hrs. work?
95. Ascertain the local price of such labor in your com
munity, and solve the last two problems, using these prices.
96. Drawing produce on a good road a farmer can haul
four loads daily, on a poor road only two loads daily.
How many loads less are hauled in a week because of poor
roads? In 6 weeks? Counting 4 loads to the day, what
is the loss in this time to one man with team ?
97. On poor roads a teamster can draw  of a ton at
each load, on a good road 1 T. Allowing four loads
to each daj'' on a good road and two loads a day on a poor
road, how much lost time is occasioned by the poor road in
drawing 1600 bus. of potatoes to market (1 bu. = 60 lbs.)?
How much is the loss at #3.50 a day?
98. If 3 fortyquart cans of milk containing 3 % of but
ter fat bring 6 cts. a quart, for what should 6 sixtyfive
quart cans of 4 % milk sell ?
99. Grafting wax is made of: resin 5 ozs., beeswax
5 ozs., tallow 3 ozs. State the composition in per cents.
100. If 12 % of the stalks of a cornfield producing 63
bus. per acre are barren, how many bushels are lost
through barren stalks?
34i
PRACTICAL ARITHMETIC
101. What is the weight of a 40 bu. wheat crop on au
acre if the grain weighs 2400 lbs. and the straw weighs
3600 lbs. ? What per cent of the total crop is grain?
102. The following amounts of water expressed as
inches in depth are required to mature wheat crops of
the following yields : 15 bus. to the acre require 4.498
ins. ; 20 bus. require 5.998 ins.; 25 bus., 7.497 ins.; 30
bus., 8.997 ins. ; 35 bus., 10.495 ins. ; 40 bus., 12 ins. An
acre of water 1 in. deep weighs 226,584 lbs. How many
more pounds are required to raise a 35bu. crop than a
15bu. crop? A 20bu. crop than a 15bu. crop? A 40bu.
crop than a 15bu. crop? Make other similar problems
from the data given.
A Week's Food foe Two Persons
Meats
2 lbs. flank of beef
1 lb. pickled tongue
\ lb. salt pork
\ lb. bacon . . .
3 lbs. veal cutlet .
1 lb. fresh fish . .
Cfroceries and Milk
qts. milk
2 lbs. butter ....
1 doz. eggs ....
 lb. of coffee . , .
^ lb. tea
6 lbs. flour ....
lbs. corn meal and 2
lbs. oatmeal . .
Cost
06^
08^
24^
18^
40/
20/
08/
08/
23/
20/
Groceries and Milk
2 lbs. sugar . .
baking powder .
seasoning . .
3 lbs. bananas .
5 lbs. apples
Vegetables
potatoes 1 pk. (15 lbs.)
cabbage 2 lbs. . . .
lettuce ^ lb
Cost
16/
05/
10/
15/
10/
15/
08/
05/
peas 1 lb 16/
string beans 1 lb. . . . 05/
turnips 5 lbs 05/
squash 5 lbs 08/
2 lbs. dried beans . . . 10/
2 lbs. canned tomatoes . 12/
MISCELLANEOUS REVIEW PROBLEMS 348
103. Find the total cost of supplies for a week.
104. At the same rate what will the food cost for the
. year ?
105. If the food costs ^ of the income, what is the
income ?
106. Find the amount of digestible protein in the meat
and fish consumed. (For data consult tables on pages
272 and 273.)
107. Find the amount of digestible protein in the
vegetables. Find the amount of digestible protein in the
groceries and milk.
108. Digestible fats in meat and fish.
109. Digestible carbohydrates and fats in groceries.
110. Digestible carbohydrates and fats in vegetables.
111. If in one day a man requires 100 grams of protein,
60 grams of fats, and 400 grams of carbohydrates, and a
woman requires 90 grams of protein, 40 grams of fats, and
350 grams of carbohydrates, how much does the diet for
a week exceed or fall short of this standard ? (See
preceding page.)
112. The cost of spraying cantaloupes being 4.47 an
acre per season, what is the cost for 12 A. ?
113. The net profit from spraying one acre of cucumbers
on Long Island being 1163.50, what would be the profit
at the same rate from spraying 14 acres?
114. The net profit from spraying an averagesized tree
from 12 to 20 years old throughout one season at a total
cost of 50 cts. is from 13 to 17. When apples are worth
11 per bushel, what are the highest and the lowest per
cents realized upon the investment?
344
PRACTICAL ARITHMETIC
115. Sprayed and iinsprayed apple trees yielding as is
shown in tlie table below, what is the total yield of the 6
sprayed and the 3 unsprayed trees, in bushels, in sound
apples and in diseased apples?
Date of Spraying and Tree Numher
Trees sprayed July 10 and 26 and Aug, 9 :
No. 1
No. 2
No. 3
No. 4
No. 5
No. 6
Trees not sprayed :
A
B
C
Biixli els
15.76
18.26
18.76
10.26
13.76
24.50
5.875
9.35
5.875
Sound
Apples
1278
1.563
1526
749
1394
2164
3
135
50
Diseased
Apples
Number
144
253
163
103
17
309
1276
1670
1298
116. The population of several American and foreign
cities for two decades is shown in the following table.
What was the per cent of increase in each city ? Was the
average increase greater abroad or in the United States ?
ISTO
1890
New York ....
950,000
1,51.5,.301
Berlin . .
800,000
1,576,794
Hamburg .
348,000
562,260
Boston . .
342,000
448,000
Cologne ,.
144,800
287,800
Buffalo . .
155,000
2.55,600
Magdeburg
97,500
202,000
jNIihvaukee
116,000
205,000 .
MISCELLANEOUS REVIEW PROBLEMS
345
117. The population of the United States in 1790 was
3;928,037 ; in 1800, 5,308,937 ; in 1810, 7,239,814 ; in 1820,
9,638,191; in 1830, 12,860,702; in 1840, 17,017,788; in
1850, 23,151,876 ; in 1860, 31,335,120 ; in 1870, 38,784,597;
in 1880, 50,152,866; in 1890, 62,622,260 ; in 1900, 76,059,
000. What was the increase in popuhition the first decade
given? The second? The third? For each succeeding
decade?
118. What was the increase from 1800 to 1900? How
mach greater was the increase during the last decade than
was the total population of 1790? Than of 1820?
119. According to the twelfth census the working popu
lation of the United States was engaged as follows :
Pursuit
Nr.MBER
Pee cknt of
Total Population
Agricultural pursuits
Professional seivine
Domestic service
Trade and transportation ....
Manufacturing and mechanics . .
10,438,219
1,264,735
5,691,746
4,778,233
7,112,987
Total
What per cent of the total working population was en
gaged in each pursuit?
120. The number of agricultural college students in the
same year was 5035. What per cent was this of the num
ber engaged in agricultural pursuits?
121. According to the census of 1899, 93 % of the people
of the United States lived on annual incomes of $400 for
three persons in a family. According to ■ statistics each
346 PRACTICAL ARITHMETIC
member of the average Pennsylvania farmer's family re
ceives 1198.26 yearly. What per cent above the average
income do these farmers receive?
122. In 1820, of the people engaged in commerce, manu
facturing, and agriculture,  were engaged in agriculture.
In 1900,  were engaged in agriculture. What fraction
less was there iia agriculture in 1900 than in 1820?
123. In 1870 for every 1000 persons engaged in agri
culture there were 1112 engaged in other pursuits : in
1900 for every 1000 in agriculture there were 1806 in
other pursuits. Show these relations by lines, letting 
in. represent 100 people.
124. The average value of farms in the United States
in 1900 was: total value 13574, buildings 1620, imple
ments and machinery $133, live stock $536. What per
cent of the value of the land is each of the three other
items given?
125. For each $1000 of total investment, how many dol
lars are invested in each of the other three items given?
126. For each f 1000 invested in land, how much is in
vested in each of the other three items?
127. If 1200 cu. ins. of air is rendered unfit for breath
ing by one person in one minute, what volume of air will
be required to last 100 persons an hour?
128. When the air is taken into the lungs (inspired), it
contains 20.81% oxygen, 79.15% nitrogen, .04% carbon
dioxide ; when it comes out of the lungs (expired), it has
16.035% oxygen, 79.557% nitrogen, and the balance is
carbon dioxide. What per cent of the oxygen is retained
MISCELLANEOUS REVIEW PROBLEMS
347
in the body, and what per cent of carbon dioxide has been
added ?
129. If the following be the pulse rate and rate of
breathing per minute, what per cent is the rate of breath
ing of the pulse rate for each animal?
Man .
Horse
Cow .
Sheep
Dog .
Kespiration
16
12
15
14
16
130. Compare the number of heartbeats of a man for
one day with that of a horse and a cow. Compare horse
with cow. Cow with sheep. Sheep with dog.
131. If the pulse (heartbeat) of a horse is 38 per min
ute, and that of a mule 46, and both are increased 50 %
by disease, what will then be the pulse rate of each ?
132. If the body temperatures of five horses are as fol
lows, what is the average body temperature ?
7 a.m.
12 m.
(1) 98.9°
(2) 99.6°
(3) 99.4°
(4) 98.8°
(5) 100. 1°
133. If the body temperatures of five cows were taken
as follows, what was the average temperature ?
8 a.m. 12 m. 6 p.m.
(1) 100.5° 100.8° 100.9°
99.3°
99.7°
100°
100.3°
99.8°
100.4°
99.6°
100.1°
100.3°
100.7°
318 PEACTICAL AKITHMETIC
(2) 99.8°
100.7°
101.2°
(3) 101.2°
101.1°
101.4°
(1) 100.8°
101.2°
101.8°
(5) 100.9°
101.3°
101.6°
134. A horse does its best woi'k at 2J mis. per hour,
pulling 150 lbs. It requires a pull of 8 lbs. to move 2000
lbs. on a level road on iron rails, 33 lbs. on a good pave
ment, 41 lbs. on a plank road, 65 lbs. on a macademized
road, 150 lbs. on gravel, 237 lbs. on an ordinary road,
and 457 lbs. on a loose, sandy road. How heavy a load
should be given a twohorse team on each of the above
roads to have the team do its best work ?
135. A horse working 10 hrs. per day, and travelling at
4 mis. per hour, pulls about 63 lbs. 'How heavy a load
should it have on each of the above roads at this rate ?
136. It is estimated that a poor road costs a farmer
about $15 a year for each horse. What is his loss in this
way if he keeps 5 horses ?
137. With hay at $15 per ton, wheat at 75 cts. a bushel,
pork at 5 cts. a pound, and butter at 25 cts. a pound, what
is the value of each per ton ? If it costs $ 2.50 a ton to
market produce, what per cent of the value of each of
these is consumed by the cost of marketing?
138. If a man spends f of a day worth 11.60 in market
ing  of a cord of wood worth $2.85 per cord, what is the
per cent of value consumed in marketing ?
139. If cottonseed is 54% as valuable for fertilizer as
cottonseed meal, and it costs i 2 a ton to haul the seed to
market and haul the equivalent in meal back to the farm,
how much meal should a farmer receive for 5 tons of seed.
MISCELLANEOUS REVIEW PROBLEMS 349
when meal is worth $28 a ton, to justify him in exchang
ing seed for meal ?
140. If the fare of 4 people, 96 mis., is 18.64, what will
be the fare of 12 people, 48 mis.? What will be the fare
of 8 people, with 4 chiklren Q fare), going 72 mis.?
141. If a payroll for 9 employees for 6 days is 172,
how much is the pay roll for 15 employees for 9 days ?
142. If it takes 8 boys, working 8 hrs. a day, 12 days to
make a tennis court, how long will it take 50 boys work
ing 2 hrs. daily ?
143. If 20 men can do a certain piece of work in 7 days
of 8 hrs. each, how many men will be required to accom
plish the same work in 12 days of 7 hrs. each?
144. If a grain bin 4 x 9 x 4 ft. holds 1262 bus. of
wheat, how many bushels will a bin 4^ x 15 x 6 ft. hold?
145. A truck grower ships 18 baskets of beans and 27
crates of strawberries. The freight is 40 cts. a basket
on the beans and the refrigerator charges 75 cts. a crate
on the berries. The beans sell for $1.35 a basket and
the berries for f 2.87^ a crate. How much will the grower
receive for his shipment if a commission of 8 % is charged
for selling?
146. If at the Chicago stockyards the usual charge
for selling beef cattle is 50 cts. a head, what is the per
cent of commission on 25 head of cattle weighing 1325
lbs. each and selling for 5.75 cts. a pound ?
147. What will two cattle weighing 1225 lbs. each bring
at i 5 per hundredweight ?
148. A farmer makes the following exchange with his
350 PRACTICAL ARITHMETIC
grocer: he gives 27^ lbs. butter at 221 cts a pound,
15 doz. eggs at 17 cts. a dozen, and 3J bus. potatoes at
45 cts. a bushel, and receives 20 lbs. sugar at 5J cts. a
pound, 3 lbs. cheese at 13 cts. a pound, and the balance
in cash. How much cash does he receive ?
149. If 10 lbs. of corn and 2 lbs. of cottonseed meal a
day for each horse may be substituted for 14 lbs. of corn,
what will be the saving in a year on a farm where 15
horses are being fed, if corn is worth 50 cts. a bushel and
cottonseed meal 1 28 a ton ?
150. If a 10acre field of corn can be cut, shocked, and
shredded for $4 an acre and" yields 2960 lbs. of stover
per acre, what is the value of the stover over the cost of
harvesting the corn, if a ton of stover is worth 1200 lbs.
of grass hay valued at $10 a ton?
151. If the following items represent the cost of pro
ducing an acre of potatoes, what will be the profit per
acre on a yield of 235 bus. worth 47 cts. per bushel ?
Cost of seed $3,939, plowing $1,607, dragging 92 cts.,
planting 60 cts., cultivating $1.709,. hoeing $2,865,
spraying $3.60, digging $4.35, wear of machinery $21.24,
and rent of land $5.00.
152. In Minnesota it costs as follows to cultivate an
acre of corn : manuring 57,6 cts., seed 21.3 cts., shelling
seed 2.3 cts., plowing $1,205, dragging 48 cts., planting
22.5 cts., cultivating $1,619, cutting 66.5 cts., shock
ing 52.6 cts., twine 46.7 cts., picking up ears 24.9 cts.,
shredding $3,794, wear of machinery $1,202, rent of land
$3.50. What is the profit on 40 acres yielding 53 bus.
per acre at 42 cts. a bushel, and 2964 lbs. of shredded
MISCELLANEOUS REVIEW PROBLEMS 351
stover at 13.00 per ton? What is the profit on one
acre?
153. If an acre of clay soil 1 ft. deep contains 6.38
tons of potash and grows a crop of 40 bus. of wheat hav
ing a ratio of weight between grain and straw of 2 : 3,
how many such crops would it take to contain as much
potash as there is in the soil if 1000 lbs. of wheat contain
6.3 lbs. of potash and 1000 lbs. wheat straw 5.2 lbs. ?
154. The average number of marketable potatoes from
each " seed " piece of several varieties was : variety No.
1, 3.14 potatoes, No. 2, 3.03 potatoes, No. 3, 3.28 potatoes.
No. 4, 3.29 potatoes, No. 5, 3.48 potatoes. No. 6, 1.91 po
tatoes, No. 7, 1.7 potatoes. What was the average num
ber of potatoes per seed piece of all of these varieties ?
By what per cent did the best exceed the poorest ? By
what per cent did the best exceed the average ?
155. In Virginia the Burbank potato yielded 230 bus.
of marketable potatoes per acre, the Crown Jewel 197 bus.
What would be the difference in value of yield on 40
acres of each of these two varieties at 55 cts. per bushel ?
156. Of 67 hens tested for egg laying ability for two
years, 10 laid more eggs the second year than the first
year, and 57 laid fewer eggs the second year. What per
cent laid more the second year ? What per cent laid fewer
the second year ?
157. A 10acre field continuously in corn yielded 357
bus. When crimson clover was used in a rotation, the
yield the year following the clover was 551 bus. What
was the per cent of increase in value with corn at 48 cts.
per bushel ? What was the per cent of increase in bushels ?
352 PRACTICAL ARITHMETIC
158. I buy a bill of hardware on which the list price
is $134. The discounts are 25%, 10%, and 5%. How
much do I have to pay?
159. If I bu} a bill of groceries on which there is a
discount of 15% and 5% for cash, how much must I
pay?
160. I am offered a quantity of tile for $390 with dis
counts of 20 % and 5 % off for cash. A second firm offers
me the tile at the same price with a discount of 15 % and
10% off for cash. From which firm shall I buy? Why?
161. A farmer sold a team of horses for i 350 on 1 year's
time. He refused an offer of §325 cash. Did he gain or
lose, and how much, by selling on time, money being
worth 6%?
162. A feed merchant mixed 62 bus. of oats at 34 cts. a
bushel, 1200 lbs. of wheat bran at 118 a ton, 35 bus. of
corn at 43 cts. a bushel, and 700 lbs. of cottonseed meal at
§28 a ton. How much must he charge per 100 lbs. in
order to make a profit of 20 % after giving a toll of 6 %
by weight for grinding the oats and 5 % for grinding the
corn ?
163. If when corn is cut and cured in the shock the
loss in dry matter is 31%, and when put into the silo
11 %, what is the total gain in dry matter by putting
the corn from 10 A., weighing 12 T. per acre, into a
silo, if, when ready for cutting, corn contains 73.2% of
water? How much is the dry matter saved worth if silage
containing 79.1 % water has a feeding value of $3 per ton?
164. If I buy corn at 42 cts. a bushel on 3 months' time
at 6 % interest, and the shrinkage in the weight of the corn
MISCELLANEOUS REVIEW PROBLEMS 353
is 10 %, at what price must I sell at the end of 3 months
to make 20 % on the transaction ?
165. The interest on a mortgage for 1320 at 8% for
one year is collected by an agent who receives 10 % com
mission for collecting. What does he receive? What
per cent is his commission of the principal?
166. A farmer has 1350 bus. of corn for which he is
offered 37 cts. a bushel on December 1. He keeps it until
March 1 and sells it for 43^ cts. a bushel. Did he make or
lose by holding the corn if it lost 8 % in weight, and
money was worth 8 % interest ?
167. If the farmer makes 12^% profit on his wheat, the
miller charges 20 % for grinding it into flour, and the
grocer receives 10 % for selling a barrel of the flour for
$5.65, what did it cost the farmer to produce the wheat
required to make a barrel of flour? \
168. A man pays 197 taxes on $9682. What is the
rate ?
169. What are the taxes on $19,862, the rate being
.00971?
170. A town raises $6859 on property assessed at
$684,936, there being 428 polls at $1.67 each. What
will be the rate ?
171. A town having 248 polls, property valuation
$868,730, raises $6482.94 taxes. What is the rate?
172. A note for $1575 dated April 9, 1903 was in
dorsed as follows: Oct. 15, 1909, $115; July 27, 1904,
$135; Jan. 21, 1905, $325. What amount was due
April 9, 1906 at 6% interest?
354 PRACTICAL ARITHMETIC
173. What must I pay, Sept. 17, 1907, to take up my
note for 268 given Jan. 25, 1905, at 8% interest?
174. A note dated Aug. 24, 1907, secured by a mort
gage, is given for 1400 at 8 % interest, but the money is
not procured until Oct. 9, 1907. What should be the
amount deducted from the first halfyearly interest be
cause of the delay in delivering the money?
175. What are the taxes on 17682 of real and 12986
of personal property valued at f, rate .9% ?
176. A man takes out life insurance for 13500 at 30
years of age for f 19.06 a year per thousand, and pays for
20 years. If he lives to 76 years of age and money is
worth 6%, what has his protection cost over and above
the $3500 which his estate receives?
177. If a farmer living 15 miles from market makes a
trip costing $2.50 in time, to deliver 50 lbs. of butter at
25 cts. a pound, what per cent of his receipts is consumed
in cost of marketing ?
178. If he delivers 650 lbs. instead, at a cost of $3
for time, what per cent of the receipts is the cost of mar
keting? By what per cent is the cost of marketing re
duced in the second instance ?
179. If it costs a strawberry grower living 5 miles
from market 12 in time to deliver a 32qt. crate of
berries at 11^ cts. per quart, what per cent of the value
of the crate is consumed in marketing?
180. If he markets 20 crates at a cost of 3 in time,
what per cent is so consumed ?
181. If it costs the Vermont farmer $1.93 to haul
MISCELLANEOUS REVIEW PROBLEMS 355
2321 lbs. of potatoes 7.1 miles, and the Connecticut farmer
$2.80 to haul 2500 lbs. of potatoes 6.7 miles, how much
more will it cost the Connecticut farmer to haul the crop
on 9 A. yielding 268 bus. per acre 5 mis. to market than
it would the Vermont farmer to haul a similar crop the
same distance? How much more per acre?
182. If it costs $2.11 to haul 3020 lbs. of corn 6.6 miles
in Pennsylvania and $2.76 to haul 1553 lbs. of corn 11.8
mis. in Georgia, how much more will it cost the Georgia
farmer to haul the corn from 25 A., averaging 32 bus.
per acre, 9 miles to market, than it will a Pennsylvania
farmer to haul a similar crop the same distance ?
183. If it costs 6 cts. to haul 100 lbs. of corn 6 miles to
market in Ohio, and 30 cts. to haul 150 lbs. of corn 13 miles
to market in Arkansas, the difference being due to better
roads and larger horses in Ohio, what is the gain per acre,
from good roads and large horses, to the Ohio farmer living
7 mis. from market on a crop of 42 bus. per acre ? If this
gain is taken to represent interest at 6 % on the increased
value of the land, what is the increase in the value of
the land per acre?
184. A dairyman has 2000 lbs. of milk containing 4 %
of butter fat. From 1000 lbs. of this he skims the cream
by the gravity settling method, which loses in the skim
milk 20 % of all the butter fat. The butter fat that he
does recover in this way he sells for 20 cts. per pound.
He separates the cream from the other 1000 lbs. of milk
with the centrifugal cream separator, which loses in the
skim milk 2 % of all the butter fat. The butter fat he
obtains in this way, on account of the superior quality.
356 PRACTICAL ARITHMETIC
he sells at 25 cts. per pound. How much greater was his
profit on the 1000 lbs. of milk skimmed by the separator
than on the 1000 lbs. skimmed by the gravity system?
185. If 70 gals, of milk containing 5 % of butter fat are
worth $28, what is the value of 100 gallons of milk con
taining 3.5% of butter fat?
186. If milk containing 5 9^ of butter fat is worth 10
cts. a quart, what is milk containing 3.5% of butter fat
worth ?
187. I have 1000 lbs. of milk containing 3.5 % of butter
fat. How much cream containing 25 % of butter fat must
I add to bring it up to a 4 % butter fat standard? If I
have no cream to add, how much skim milk containing
no butter fat must I remove to bring it up to the standard
of 4 % butter fat?
188. I have 1000 lbs. of 4.5 % milk and wish to reduce
it to a 4% standard. How much skim milk must I add?
If I have no skim milk to add, how much 25 % cream
can I take out of it ?
189. I wish to mix cream containing 35 % of butter fat
and milk containing 5 % of butter fat, so as to produce
140 lbs. of cream containing 25 % of butter fat. How
many pounds of each must I use in the mixture? .
190. I wish to obtain 200 lbs. of milk containing 4%
of butter fat from two lots of milk, one testing 3.8% of
butter fat and the other 4.9% of butter fat. How maily
pounds of each must I use?
191. If cream containing 20 % of butter fat is worth 60
cts. per gallon, what is cream worth which contains 25 %
of butter fat ?
MISCELLANEOUS REVIEW PROBLEMS 357
192. What is the price per pound of butter fat if cream
containing 20 % of butter fat and weiy:liing 8.3 lbs. sells
for 50 cts. a gallon ? At 32 cts. per pound of butter fat,
what would be the price per gallon of cream containing
27 % of butter fat ? Of cream containing 25 % of butter
fat? 20 % of butter fat? 15 % of butter fat?
193. If 5 qts. of milk containing 3.5% of butter fat
have an energy value as food of 3066 calories, and 1 pound
of rpund beefsteak has an energy value of 841 calories,
what should round steak sell for per pound Avhen milk
is worth 8 cts. a quart? 5 cts. per quart? 10 cts. per
quart?
194. A dairyman has 1000 lbs. of milk containing 5 "Jo
of butter fat. 500 lbs. of this he skims with the gravity
process, and obtains a 20 % cream. He, however, loses
in the skim milk 25 % of the butter fat. This skim milk
he sells at 15 cts. per 100 lbs. From the remaining 500
lbs. of milk he separates the cream with the centrifugal
separator, making a 50 % cream and losing but 1 % of the
butter fat in the skim milk. On account of its being
fresh and pure from the separator he sells this skim milk
for 25 cts. per 100 lbs. How much more did the dairy
man receive from his separator skim milk than from the
skim milk left from the gravity skimming process?
195. The maximum amounts of fertilizer removed by
various forms of farm produce is shown in the following
table. Placing nitrogen at 18 cts. a pound, phosphoric
acid at 6 cts., and potash at 5 cts., find the value of each
of these fertilizers removed in each crop given in the
table.
358
PRACTICAL ARITHMETIC
Produce
Pounds op
Kind
Nitrogen
Phosphoric
Acid
Potash
Corn, grain . . .
Corn stover . . .
Oats, grain . . .
Oat straw ....
AVheat, grain . .
Wheat, straw . .
Timothy hay . . .
Clover seed . . .
Clover hay . . .
Cowpea hay . . .
Alfalfa hay . . .
Apples
Apple leaves . . .
Apple wood growth
Potatoes . . . .
Sugar beets . . .
Fat cattle . . . .
Fat hogs . . . .
Milk
Butter
Cotton lint . . .
100 bus.
3 T.
100 bus.
2^T.
50 bus.
2iT.
3T.
4 bus.
4T.
8T.
8T.
600 bus.
4T.
^tree
300 bus.
20 T.
1000 lbs.
1000 lbs.
10,000 lbs.
500 lbs.
500 lbs.
100
48
66
31
71
25
72
7
160
130
400
47
59
63
100
25
18
57
1
1.7
17
6
11
5
12
4
9
2
20
14
36
5
7
2
13
18
7
3
7
0.2
.5
19
52
16
52
13
35
71
3
120
98
192
57
47
5
90
157
1
1
12
0.1
2.3
196. Of 4402 samples of commercial fertilizer analyzed
in Indiana from 1902 to 1907, 3158 were equal in value to
that guaranteed, 323 were not within 10 % of such value,
and 835 were with one or more of the ingredients 20 %
below the guaranteed value. What per cent was equal
to the guaranteed value ?
MISCELLANEOUS REVIEW PROBLEMS
359
197. What per cent was not within 10% of value?
What per cent fell in the last class named ?
TABLE OF ANALYSIS OF FERTILIZERS IN INDIANA
Class
1901
1904
1905
190T
1. Number of samples collected ....
2. Number equal to guarantee in every
particular
3. Number equal to guarantee in value
4. Number within 10% of guarantee . .
5. Number not within 10% of guarantee .
6. Number with one ingredient or more
20% below guarantee
7. Number with one ingredient or more
30% below guarantee
8. Number with one ingredient or more
50% below guarantee
592
281
469
85
38
103
679
335
564
93
22
112
674
286
492
139
43
138
648
248
451
148
44
132
65
21
734
812
528
158
48
148
77
21
879
374
642
176
61
136
64
25
793
265
481
210
102
177
75
29
What per cent of all samples analyzed in each year fell
in class 2 ? In class 3 ? In class 4 ? In class 5 ? In class 6 ?
In class 7 ?
198. The upper 6 ins. of soil weighs per acre 1,370,000
lbs. If the surface is cultivated thoroughly, it may contain
14.21% of moisture; if poorly cultivated, only 8.02%.
What will be the total amount of Avater in the soil under
poor cultivation? Under good cultivation? Obtain the
last answer by two different processes.
199. In Minnesota it was found that during 10 yrs.
of exclusive grain farming the per cent of nitrogen in a
soil had decreased from .601 % to .523%. If the soil to
a depth of 8 ins. per acre weighs 1282 T., how much
nitrogen was lost from the soil ?
360 PRACTICAL ARITHMETIC
200. How much nitrogen will 30 bus. of wheat per acre
annually take from the soil in 10 yrs., if 1 bu. of wheat
takes .625 lb. nitrogen from the soil?
201. What per cent of the total nitrogen lost from the
soil of problem 199 in the 10 yrs. would be taken out by
wheat as in the preceding problem?
202. After 10 yrs. under a system of mixed farming,
including rotation of crops, the growing of legumes, and
livestock husbandry, in another field, the nitrogen in the
soil had decreased only from .31% to .309%. If the soil
to a depth of 8 ins. weighs 58.9 lbs. per square foot of sur
face, how many pounds less nitrogen did this field contain
after 10 yrs. ? What was the difference in the loss of
nitrogen per acre from this field and that of problem 199
in the 10 yrs. ?
203. If it costs $29.52 to raise an acre of white pine
to 40 yrs. of age, and it yields on the stump $4 per cord,
what is the profit on 40 cords of boxboard timber ?
204. In 1871 a strip of forest land 40 mis. wide and 180
mis. long was devastated by fire. The loss is estimated at
4,000,000,000 board feet worth 110,000,000. What was
the loss per square mile in board feet ? In dollars ?
205. If one toad destroys in 30 days 720 cutworms,
600 thousandlegged Avorms, 720 sowbugs, 1080 ants, 120
weevils, 120 ground beetles, how ;^any of each of these
would 19 toads destroy? How many in a week? In 3
months ? Many gardeners pay their children for killing
cutworms at the rate of a penny each. What would 19
toads earn in a month at this rate ? In 3 months ?
206. Two boys were known to kill 19 toads. What
MISCELLANEOUS REVIEW PROBLEMS
361
was the loss in 3 mos., even if it required 19 toads to do
what it was shown one could do in the last problem?
207. Corn at different stages of growth contains dry
matter (useful) and water as follows, expressed in tons:
Fully tasselled
Fully silked
Kernels watery to full milk
Kernels glazed
Ripe
Corn
I'KR ACKR
9
12.9
16.3
16.1
14.2
Watbr
I' Kit ACKK
8.2
11.3
14
12.5
10.2
Dry Matteu
PER Acre
Complete column of dry matter. What per cent of water
and what per cent of dry matter are there at these different
periods?
208. If a good clay soil contains 12,760 lbs. of potash
per acre one foot deep, and a 40bu. crop of wheat uses
35.1 lbs. of potash, how many crops will it take to use
potash equal to that in the clay soil?
209. If, with the same field conditions, one variety of
corn yields 7 bus. per acre more than another variety,
what will be the gain in planting the better variety on
96 acres, if the poor seed costs 60 cts. per bushel and the
seed of the better variety $2.50 per bushel ? Allow 1 bu.
of seed corn to every eight acres planted and value the
corn of the crop at 49 cts. per bushel.
210. Eight bushels of clover seed containing 3 bus. of
dead seed were bought at I 3.50 per bushel. What was
the price paid for the live seed?
211. Fifteen bushels of clover seed containing 1 bu. of
362 PRACTICAL ARITHMETIC
dead seed were bought at 15.50 a bushel. What was
the price paid for the live seed ?
212. A variety of flax, improved by seed selection,
yields, on an average, 2.2 bus. per acre better than the
best common varieties, which yield 15 bus. What is the
per cent increase due to seed selection ?
213. If it takes 6 hrs. to test 9 bus. of seed corn for
germinating power, how long will it take to test seed
enough for 600 A., allowing 1 bu. to each 5 A. ?
214. A pound of cotton can be spun into 168 spools of
No. 40 sewing thiead, 200 yards to the spool. How
many yards of thread can be spun ?
215. A much finer thread can be made, of which the
number of j^ards in the last problem equals f^. How
many miles of this fine thread can be made from a pound
of cotton ?
216. A still finer thread can be made, of which the
number of yards in problem 214 equals only ^J j. How
many miles of this quality of thread can be made from a
pound of cotton ?
217. From one pound of cotton can be made 4 yards
of bleached muslin worth 8 cts. a yard. What amount
of lawn can be made from the same amount of cotton if,
for every yard of bleached muslin, 2\ yards of lawn may
be made ? The value of the bleached muslin is ^^ of the
value of the lawn. What is the value of the lawn ?
218. Four yards of sheeting can be made from a pound
of cotton with a value equal to  the value of the bleached
muslin. What is the value of the sheeting ?
MISCELLANEOUS REVIEW PROBLEMS 363
219. The number of handkerchiefs that can be made
from a pound of cotton is ^f of the number of yards of
calico that can be made. If the number of yards of calico
is J of the number of yards of bleached muslin and the
price of the calico is ^ of the price of the sheeting, how
many yards of calico can be made ? What is the value
per yard ? How many handkerchiefs can be made ?
What is the value of each, if one handkerchief equals 
the value of a yard of lawn ?
220. The value of the denim that can be made from a
pound of cotton bears the same ratio to the value of
gingham as the number of yards of lawn to the number
of handkerchiefs. \ as many yards of denim as gingham
can be made. With gingham worth 7 cts. a yard, what
is the denim worth per yard?
221. Arrange in tabulated form the products of a
pound of cotton from the smallest quantity of material
at lowest price to the most valuable product here con
sidered,
222. An untreated loblolly pine fence post, set, costs
about 14 cts. It lasts about 2 yrs. Compounding the
interest at 5 per cent, what is the annual cost of such a
post? If a preservative treatment, which costs 10 cts., in
creases the length of use of the post to 18 yrs., what
is then the total cost of a post, set? What does this
amount to annually, compounded as above ? What is the
saving due to treatment per year ? Assuming that there
are 200 posts to a mile of fence, what is the saving each
year for a mile ? This is the interest on what amount at
5 per cent?
364
PRACTICAL ARITHMETIC
223. The accompanying table shows the production in
thousand feet of the leading kinds of lumber for 3 yrs.
What was the total production for each year ? What was
the per cent of increase or decrease of each kind from
1899 to 1906? What the total per cent of increase or
decrease ?
Yellow pine
Douglas fir
White and Norway pine
Hemlock
Oak
Spruce
Western pine ....
Maple
Cypress
Poplar
Redwood
Red gum
Chestnut
Basswood
Birch
Cedar
Beech
Cottonwood ....
Elm
Ash
All others
Mfeet
9,658,923
1,736,507
7,742,391.
3,420,673
4,438,027
1,448,091
944,185
633,466
495,836
1,115,242
360,167
285,417
206,688
308,069
132,601
232,978
415,124
456,731
269,120
486,848
Mfeet
11,533,070
2,928,409
5,332,704
3,268,787
2,902,855
1,303,886
1,279,237
587,558
749,592
853,554
519,267
523,990
243,537
228.041
224,009
223,035
321,574
258,330
169,178
684,526
1906
Mfeet
11,661,077
4,969,843
4,583,727
3,537,329
2,820,393
1,644,987
1,386,777
882,878
839,276
683,132
659,678
453,678
407,379
376,838
370,432
357,845
275,661
263,996
224,795
214,460
936,555
224. A tree weighing 10,000 lbs. when dry is 50 per
cent carbon. How many pounds of carbon are there in
the tree?
MISCELLANEOUS KEVIEW PROBLEMS 365
225. Carbon dioxide being ^\ carbon, and all the carbon
of the tree being derived from carbon dioxide, how many
pounds of carbon dioxide are required to furnish the
carbon in the tree of the last problem ?
226. Air being .03^ % carbon dioxide, how many pounds
of air are required to furnish this amount of carbon
dioxide ?
227. Air weighing 31.074 grs. per 100 cu. ins., how
many cubic yards of air are required to furnish the
amount of carbon used in the growth of the tree men
tioned above ?
228. There are in the atmosphere of the earth about
6,000 billion pounds of carbon dioxide. How much carbon
does it contain? For how many trees like that of prob
lem 224 would this suffice ?
229. An adult exhales daily into the air about 245 g. of
carbon. Estimating the earth's population at 1400 mil
lion, how much carbon is thus restored daily to the air ?
230. Wood, coal, etc., in burning restore their carbon
to the air. One manufacturing works thus restores from
the coal burned about 5,100,000 lbs. of carbon.
A forest consisting of how many trees like the one men
tioned in problem 224 would be raised from this carbon?
231. One square meter of pumpkin or sunflower leaf in
a summer day of 15 hrs. makes 25 g. of starch which is
1^ carbon. How many cubic meters of air are required
to furnish the requisite carbon ? How long a room 3 m.
wide and 3 m. high would be required to contain it ?
232. Convert all the measurements of the last problem
into English measure, and solve.
366 PRACTICAL ARITHMETIC
233. If Alfalfa hay contains 10.44 per cent digestible
protein, 39.6 per cent carbohydrates, and 1.2 per cent fats,
and redclover hay contains 6.8 per cent protein, 35.8 per
cent carbohydrates, and 1.7 per cent fats, — what is the
difference in the feeding value of a ton of Alfalfa and a
ton of red clover, estimating digestible protein at 3 cts.
a pound, carbohydrates at 1 ct. a pound, and fats at 21 cts.
a pound ?
234. On land worth $65 an acre Alfalfa is sowed and
maintained for 4 yrs. at an expense of 130. The cost of
harvesting the hay is $1.25 a ton ; the crops are : 1st year
2.78 tons, 2d 3.15 tons, 3d 4.60 tons, 4th 4.28 tons.
What is the profit on 9 acres, allowing 10 per cent inter
est on the value of the land, $12 a ton for hay, and $3 a
ton for cost of baling and marketing ?
APPENDIX
SURFACES OF SOLIDS
The surface of a solid except its base or bases is called
the Lateral Surface. The Entire Surface includes its bases.
A solid, the base of which is a poly
gon and the sides of which are triangles
meeting at a point or vertex, is called a
Pyramid.
The distance from the vertex to the
side of the base is called the Slant Height.
If the sides and angles of the pyramid
are respectively equal and the apex is
directly over the centre of the base, the pyramid is said
to be regular.
The surface of a pyramid, as may be seen, is composed
of a number of triangles with an altitude equal to the
slant height of the pyramid and the bases
forming the perimeter of the solid.
A solid, the base of which is a circle, and
the surface of which tapers to a point or ver
tex, is called a Cone.
The lateral surface of a cone may be assumed
to be made up of a number of infinitely small
triangles.
Hence, to find the lateral surface of a pyramid or cone,
multiply the perimeter of the base by ^ the slant height.
The portion remaining after a part of the top has been
cut from a pyramid or cone is called a Frustum of a pyra
mid or of a cone.
The lateral surface of a frustum of a pyramid may be
regarded as composed of a number of trapezoids, the sum
of the parallel sides of which forms the perimeter of the
367
368
PRACTICAL ARITHMETIC
bases and the slant height of which equals the altitude of
the frustum.
The lateral surface of a frustum of a cone
may be considered as made of a number of
infinitely narrow trapezoids.
To find the lateral surface of the frustum
of a pyramid or of 'a cone, multiply half the
sum of the perimeters of the two bases by
the slant height.
A solid having equal polygons parallel to each other
for its two ends and parallelograms for its sides is a
Prism.
From the form of their bases, prisms
are triangular, quadrangular, etc.
The lateral surface of a prism may be
regarded as a series of parallelograms,
with their combined bases equal to the
perimeter of the two bases of the prism
and a height equal to the altitude of the
prism.
To find the lateral surface of a prism, multiply the
perimeter of the base by the altitude.
VOLUMES OF SOLIDS
The volume of a solid is the number of solid units it
contains.
To find the volume of a prism, multiply the area of
the base by the altitude.
A square prism has three times the
solid contents of a pyramid. In like
manner, the cylinder has three times the
solid contents of the cone.
To find the volume of a pyramid or
cone, multiply the area of the base by
onethird the altitude.
The frustum of a pyramid or cone is equal to three
pyramids or cones, the common altitude of which is the
altitude of the frustum and the bases of which are the
APPENDIX
369
upper base, the lower base, and a mean proportional be
tween them.
Hence, to find the volume of a frustum of a pyramid
ar cone, add to the sum of the areas of the two ends, the
square root of the product of these areas,
and multiply the result by onethird of the
altitude.
A solid, bounded by a curved surface,
every point of which is equally distant from
the point within, called the centre, is a
Sphere.
The Diameter of a sphere passes through the centre and
terminates at the circumference.
Onehalf the diameter is the Radius. The circumfer
ence of the circle, the radius of which is the radius of the
sphere, is the Circumference of
the sphere.
The surface of a sphere is
equal to the square of the di
ameter of the sphere multiplied
by 3.1416.
A sphere may be regarded as composed of pyramids,
the bases of which form the surface of the sphere and the
altitudes of which equal the radius of the sphere.
Hence, to find the volume of a sphere, multiply the
surface by onethird the radius, or multiply the cube of
the diameter by .5236.
THE EXTRACTION OF THE CUBE ROOT
Rule.
1. Beginning at units, separate the number into groups
of three figures each.
2. Find the greatest cube contained in the lefthand
group. Write its cube root as the first figure of the re
quired root.
3. Subtract this cube from the first period, and bring
down the next period.
4. Divide the number so found by three times the
370
PRACTICAL ARITHMETIC
square of the root already found, considered as tens, as a
trial divisor, to find the next figure of the root.
5. To this trial divisor add three times the product of
the two parts of the root plus the square of the second
part of the root, to make the complete divisor.
6. Multiply the complete divisor by the second figure
of the root ; subtract and bring down the next period.
7. Continue in a similar way until all periods have
been used.
For example, to find the cube root of 242,970,624, pro
ceed according to the rule as follows :
242'970'624(624 required root
216 cube of 1st figure of root
1st trial divisor 10800
3 X (60 X 2) 360
2^ 4
1st complete divisor 11164
2d tiial divisor 1153200
3 X (620 X 4) 7440
42 16
2d complete divisor 1160656
26970
22328 product of 2d figure
of root with 1st com
plete divisor.
4642624
4642624 product of 3d figure
of root with 2d
complete divisor.
1. Separate into periods.
2. We find by inspection or trial that the cube of 6 is the
largest cube contained in the first period, 242. 6 is put down
as the first figure of the root.
3. Subtracting the cube of 6, 216, from the first period, 242,
we have 26 ; bringing down the next period, we have 26,970.
4. The root already found considered as tens is 60. The
square of this is 3600; three times this is 10,800, the trial
divisor. This trial divisor, 10,800, is contained in the dividend,
26,970, two times. 2 is, therefore, set down as the next figure
of the root.
5. The two parts of the root already found are 60 and 2.
The product of 60 x 2 is 120; three times this is 360. The
square of the last figure found is 4. Adding 360 plus 4 to the
trial divisor, we have the complete divisor 11,164.
6. Multiplying the complete divisor by the second figure of
the required root and subtracting, we have 4642. Bring down
the next group, 624, giving 4,642,624 for the next dividend.
7. Proceeding as before :
APPENDIX 371
The root already found is 62 or, considered as tens, is 620. The
square of this is 384,400. This multiplied by 3 is 1,163,200,
the trial divisor. This trial divisor is contained 4 times in the
dividend. 4 is, therefore, set down as the next figure of the
root.
The two parts of the root already found are 620 and 4.
Their product is 2480; three time^ this is 7440. Adding thi.s
together with the square of the last number of the root, whicih
is 16, to the trial divisor, we have the complete divisor 1,160,656.
Multiplying this complete divisor by the last figure found,
we have 4,64/!,624, completing the problem.
The cube root of 242,970,624 is 624.
If the number of vt^hich the cube root is to be extracted
has decimal places, divide the figures at both sides of the
decimal point into periods of three figures each, annexing
ciphers to the last period of the decimal, if need be, to
give three figures.
There are as many decimal places in the cube root of
a decimal as there are periods of three figures each in the
decimal.
If the number is not a perfect cube, annex ciphers and
continue the process to as many decimal places as may
be desired.
The cube root of a fraction is found by taking the cube
root of its numerator and of its denominator, or by reduc
ing the fraction to a decimal and then extracting the root.
PROOF OF THE FUNDAMENTAL PROCESSES BY CAST
ING OUT NINES
Addition.
Add: 71287 7
67328 8
79816 4
42983 8
54631 1
316045 1
Add.
To test accuracy of the addition, cast out the nines from
each row and from the sum, setting down the remainders.
Thus in the first row drop 7 + 2, 8+ 1, set down 7; second
372 PRACTICAL ARITHMETIC
row drop 6 + 3 and 7 + 2, set down 8 ; third row drop 9 and
8 + 1, add 7 + 6 = 13, "drop 9 from 13, leaving 4, set down 4 ;
fourth line drop 4 + 2 + 3, drop 9, set down 8 ; fifth line drop
6 + 4 and 6 + 3, set down 1. Prom sum drop 6 + 3, 5 + 4, set
down 1. Add the iiumbers set down from the first five num
bers, cast out 8 + 1, add 4 + 8 + 7 = 19, cast out 2x9, leav
ing 1. The number remaining being 1 in both cases, the work
is presumably correct.
Subtraction.
Subtract: 3726813 3l
2619832 41 Subtract.
1106981 ^ sj 
Cast out the nines from both minuend and subtrahend ; sub
tract the number remaining from the subtrahend from that re
maining from the minuend, restoring one of the nines cast out,
if need be, in order to subtract. If the work is correct, the
number so found should equal the number left after casting
nines from the remainder.
Multiplication.
Multiply : 643 4
249 6
I Multiply.
6787 24
2572
1286
160107 6
Cast nines from multiplier and multiplicand. Find the prod
uct of the numbers remaining, and cast nines from it. The
number then remaining should equal the number left a,fter
casting nines from the product.
ARITHMETICAL PROGRESSION
An Arithmetical Progression is a. series of numbers which
increase or decrease by a common and constant difference,
e.g., 4, 8, 12, 16, 20, 24, etc., the common difference here
being 4.
The numbers of the series are called its Terms.
APPENDIX 373
To find any term of an arithmetical progression, multi
ply the common difference by a number one less than the
required term; add this product to the first term if the
series is increasing; subtract this product from the first
term if the series is decreasing.
Find the 17th term of the series 7, 14, 21 .
16x7 = 112. 112+7 = 119.
Find the 12th term of the series 217, 213, 209 .
11x4 = 44. 21744 = 173.
To find the sum of the terms of an arithmetical progres
sion, multiply the number of terms by the sum of the first
and last terms and divide by two.
What is the sum of the first 10 terms of the series
2,4,6 ?
9 X 2 = 18. 18 + 20 = last term.
(2 + 20) X 10 = 110 = sum of the terms.
2
GEOMETRICAL PROGRESSION
A Geometrical Progression is a series of numbers in
which any term is equal to the product of the preceding
term and a constant factor, e.g., 2, 4, 8, 16, 32, 64 — — .
To find any term of a geometrical progression, raise the
constant factor to a power one less than the number of
the required term, and multiply by the first term.
Find the 6th term, of the series 1, 2, 4, 8 .
The constant factor 2 to the fifth power = 32.
32 X 1 = 32. Ans.
To find the sum of the terms of a geometrical pro
gression, multiply the last term by the constant factor ;
subtract the first term and divide by the constant factor,
minus 1.
Find the sum of the series 1, 2, 4, 8 to and in
cluding the 6th term.
The last term = 32. The constant factor = 2.
32 X 2 = 64. 64  1 = Go. 63 h 1 = 63. Ans.
374
PRACTICAL AKITHMETIC
If the progression is decreasing, subtract the product
from the first term.
TABLES OF MEASURES
LENGTH
inches (ins.)= 1 foot (ft.)
feet = 1 yard (yd.)
1 = 1 rod (rd.)
= 1 mile (mi.)
12
3
16J feet, or
5J yards
820 rods
rds.
yds.
1 = 320 = 1760 = 5280 = 63,360
1 = 51 =i6i =198
1 = 3 ' =36
1 =12
= 1 inch, used by shoemakers.
= 1 common cubit.
= 1 sacred cubit.
= 1 hand, used to measure the
height of horses.
= 1 span.
= 1 fathom, used to measure depths
at sea.
~ ^ P 1 , used in pacing distances.
= 1 mile.
= 1 geographical nautical mile or
knot.
3 geographical miles = 1 league.
I of latitude on a meri
dian, or of longitude
on the equator.
3 barleycorns
18 inches
21.888 inches
4 inches
9 inches
6 feet
3 feet
5J paces
8 furlongs
1.15 statute miles
60 geographical miles 
69.16 statute miles
The length of a degree of latitude is commonly re
garded as 69.16 miles, and is that adopted by the United
States Coast Survey.
APPENDIX 375
ANGULAR MEASURE
60 seconds (") = 1 minute (')
60 minutes = 1 degree (°)
360 degrees = 1 circumference
SURFACE OR SQUARE MEASURE
144 square inches (sq. ins.) = 1 square foot (sq. ft.)
9 square feet = 1 square yard (sq. yd.)
30 square yards = 1 square rod (sq. rd.)
160 sqiiare rods = 1 acre (A.)
640 acres = 1 square mile (sq. mi.)
1 square mile = 1 section
36 square miles = 1 township
sq. mis. A. sq. rds. sq. yds. sq. ft.
1 = 640 = 102,400 = 8,097,600 = 27,878,400
1 = 1600 = 4840 = 43,560
1 = 30J = 272J
100 square feet = 1 square (in roofs, floors, etc.)
SOLID OR CUBIC MEASURE
1728 cubic inches (cu. ins.) = 1 cubic foot (cu. ft.)
27 cubic feet = 1 cubic yard (cu. yd.)
cu. yd. cu. ft. cu. ins.
1 = 27 = 46,656
WOOD MEASURE
16 cubic feet = 1 cord foot (cd. ft.)
128 cubic feet] ^ a ^ a \
8cordfeetl = l''°^''i(°^)
CAPACITY
Liquid Measure
4 gills (gi.) = 1 pint (pt.)
2 pints = 1 quart (qt.)
4 quarts = 1 gallon (gal.)
376 PRACTICAL ARITHMETIC
gal. qts. pts. gis.
1 = 4 = 8 = 32
1 = 2= 8
1 gallon = 231 cu. ins.
DRY MEASURE
2 pints = 1 quart
8 quarts = 1 peck (pk.)
4 pecks = 1 bushel (bu.)
2.5 bushels = 1 barrel (bbl.)
bbl.
bus.
pks.
qts.
pts.
1 =
2i
= 10 =
80 =
160
1
= 4 =
32 =
64
1 =
8 =
16
1 bushel
= 2150.42 cu.
ins.
1 heaped
bu.
= 1^ bus
s.
WEIGHT
AvoiKDUPOis Weight
16 drams = 1 ounce (oz.)
16 ounces = 1 pound (lb.)
100 pounds = 1 hundredweight (cwt.)
2000 pounds = 1 ton (T.) (short)
2240 pounds = 1 long ton
T. cwts. lbs. ozs.
1 = 20 = 2000 = 32,000
1 = 100 = 1600
The following deneminations are used in Avoirdupois
Weight
14 lbs.
= 1 stone
100 lbs.
butter
= 1 firkin
100 lbs.
grain or flour
= 1 cental
100 lbs.
dried fish
= 1 quintal
100 lbs.
nails
= 1 keg
196 lbs.
flour
= 1 barrel
200 lbs.
pork or beef
= 1 barrel
230 lbs.
salt at N.Y. works
= 1 barrel
APPENDIX
377
WEIGHTS OF PRODUCE
The following are minimum weights per bushel of cer
tain articles of produce according to the laws of various
States :
Wheat
Coru in the ear
Corn shelled
Rye
Buckwheat
Barley
Oats
Peas
White beans
White potatoes
Sweet potatoes
Onions
Turnips
Dried peaches
Dried apples
Clover seed
Flax seed
Millet seed
Hungarian grass seed
Timothy seed
60 lbs.
70 lbs., except in Miss., 72 lbs.;
in Ohio, 68 lbs.; in Ind. after
Dec. 1, and in Ky. after May 1,
following the time of husking,
it is 68 lbs.
56 lbs., except in Cal., 52 lbs.
56 lbs., except in Cal., 54 lbs.;
in La., 32 lbs.
48 lbs., except in Cal., 40 lbs.;
Ky.,56 1bs.; Ida., N.D., Okl.,
Ore., S.D., Tex., Wash., 42
lbs.; Kan., Minn., N.C., N.J. ,
Ohio, Tenn., 50 lbs.
48 lbs., except in Ore., 46 lbs.;
Ala., Ga., Ky., Pa., 47 lbs.;
Cal., 50 lbs.; La., 32 lbs.
32 lbs. except in Ida., and Ore.,
36 lbs.; in Md., 26 lbs.; in
N.J. and Va., 30 lbs.
60 lbs.
60 lbs.
60 lbs., except in Md., Pa., Va.,
56 lbs.
65 lbs.
57 lbs.
55 lbs.
33 lbs.
26 lbs.
60 lbs., except in N.J., 64 lbs.
56 lbs.
50 lbs.
50 lbs.
45 lbs., except in Ark., 60 lbs. ;
N.D., 42 lbs.
378 PRACTICAL ARITHMETIC
Blue grass seed 44 lbs.
Hemp seed 44 lbs.
Corn meal 50 lbs., except in Ala., Ark.,
Ga., 111., Miss., N.C., Tenn.,
48 lbs.
Bran 20 lbs.
TROY WEIGHT
For Precious Metals, Jewels, etc.
24 grains = 1 pennyweight (pwt.)
20 pennyweights = 1 ounce
12 ounces = 1 pound
iS7^ grains = 1 ounce I »
7000 grains = 1 pound J
480 grains = 1 ounce 1 rp
5760 grains = 1 pound] •'
APOTHECARIES' WEIGHT
20 grains = 1 scruple (sc. or 3)
3 scruples = 1 dram (dr. or 3)
8 drams = 1 ounce (oz. or 5 )
12 ounces! , , .,1 ,» .
5760 grains 1 = 1 P°^"^'l<^l^°'^^)
APOTHECARIES' LIQUID MEASURE
60 minims = 1 fluid dram (f 3)
8 fluid drams =1 fluid ounce (fS.)
16 fluid ounces = 1 pint (O.)
8 pints = 1 gallon (cong.)
COUNTING
12 things = 1 dozen (doz.)
12 dozen = 1 gross (gro.)
12 gross = 1 great gross (G. gr.)
20 things = 1 score
APPENDIX
379
24 sheets (paper) = 1 quire
20 quires, or 480 slieets = 1 ream
TIME
60 seconds (sees.)
60 minutes = 1
24 hours = 1
7 days = 1
2 weeks = 1
30 (31, 28, 29) days = 1
3 months, or 13 weeks = 1
12 months, or 365 days = 1
365 days 5 hrs. 48 mins. 49.7 sees. = 1
366 days = 1
10 years = 1
100 years = 1
VALUE
U. S. Money
10 mills = 1 ct. (ct., c, or ^)
10 cents =1 dime (di.)
100 cents or 10 dimes = 1 dollar (I)
10 dollars = 1 eagle
Canadian Monet
100 cents =1 dollar = 11
= 1 minute (min.)
hour (hr.)
day (da.)
week (wk.)
fortnight
month (mo.)
quarter
year (yr.) (common)
true or solar year
leap year
decade
century (C.)
English Money
12 pence (cZ.) = 1 shilling (s.) = *0.243+
20 shillings = 1 pound (£) = * 4. 8665
Ekench Money
100 centimes = 1 franc (fr.) = $0,193
German Money
100 pfennigs = 1 mark (i\I.)= 10.238
380
PEACTICAL ARITHMETIC
Russian Money
100 copecks = 1 ruble
AUSTEOHUNGAKIAN MONET
100 kreutzer = 1 florin
VALUE OF FOREIGN COINS IN UNITED STATES MONEY
(Proclaimed by the Secketaky of the Treasury)
Country
Money Unit
Yalue in United States
AustriaHungary . .
Belgium
Brazil
Canada ......
Central America . .
Chile
China
Denmark
Ecuador
Egypt
I ranee
Germany
Great Britain . . .
Greece
Hayti
India
Italy
Japan
]\lexico
Netherlands ....
Norway ■
Panama
Peru
Portugal
Russia
Spain
Sweden
Switzerland ....
Turkey
Crown
Franc
JNlilreis
Dollar
Peso
Peso
Teal
Crown
Sucre
Pound = 100 piasters
Franc
Mark
Pound
Drachma
Gourde
Pound
Lira
Yen
Peso
Florin
Crown
Balboa
Libra
Milreis
Ruble
Peseta
Crown
Franc
Piaster
$ .203
.193
.546
1.00
.'485
.365
.72 to .80
.268
.487
^ 4.943
.193
.238
4.8665
.193
.965
4.8665
.193
.498
.498.
.402
.268
1.00
4.8665
1.08
.515
.193
.268
.193
.044
APPENDIX
NEGOTIABLE PAPERS
381
Interest Laws
Statutes of Limitation
States and Terbitokies
,
Legal
Kate
Muximum
Contract
Kate
Grace
Judg
ments
Notes
Open
Accounts
Yetrm
Years
Years
Alabama
8
's
G
20
6
3
Arkansas
6
10
G
10
5
3
Arizona
6
Any
G
5
4
3
California
7
Any
5
4
2
Colorado
8
Any
20
6
6
Connecticut .
6
fi
6
Delaware
6
6
6
3
District of Columbia .
6
10
12
3
3
Florida ....
8
10
20
5
2
Georgia ....
7
8
G
7
6
4
Idaho ....
7
12
6
5
4
Illinois ....
5
7
20
10
6
Indiana ....
8
G
20
10
6
Iowa ....
6
8
G
20
10
5
Kansas ....
6
10
G
5
5
3
Kentucky
6
6
6
15
15
5
I^uisiana ...
5
8
G
10
5
3
Maine ....
6
Any
20
6
6
Maryland
6
6
12
3
3
Massachusetts
6
Any
20
6
6
Michigan
5
7
G
10
6
6
Minnesota .
7
10
G
10
6
6
Mississippi .
6
10
G
7
6
3
Missouri
6
8
G
10
10
5
Montana
■ 8
Any
10
8
5
Nebraska
7
10
G
5
5
4
Nevada ....
7
Any
G
6
4
4
New Hampshire .
6
6
20
6
6
New Jersey .
6
20
6
6
New Mexico .
6
12
G
7
6
4
New York . .
.^
6
20
6
6
North Carolina .
6
a
G
10
3
3
North Dakota
7
12
10
6
6
Ohio . , .
6
8
15
15
6
Oklahoma •.
7
12
G
5
5
3
Oregon ....
6
10
10
6
6
Pennsylvania
6
6
5
6
6
Rhode Island
6
Any
20
6
6
South Carolina .
7
8
G
20
6
6
South Dakota
7
12
G
10
6
6
Tennessee
6
(>
10
6
6
Texas ....
6
10
G
10
4
2
Utah ....
8
Any
8
6
4
Vermont
6
6
8
6
6
Virginia
6
6
20
5
2
3
Washington .
G
12
6
6
West Virginia
6
6
10
10
5
Wisconsin
6
10
20
ti
6
Wyoming
8
12
G
21
5
8
382 PEACTICAL AlilTHMETIC
TABLE OF COMPOUND INTEREST
Periods
% Per Cent
1 Per Cent
IJ Per Cent
H Per Cent
2 Per Cent
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
1.00750000
1.01505625
1.02266917
1.03033919
1.03806673
1.04585223
1.05369612
1.06159884
1.06956083
1.07758254
1.08566441
1.09380689
1.10301044
1.11027552
1.11860259
1.12099211
1.13544455
1.14396038
1.15254009
1.16118414
1.01000000
1.02010000
1.03030100
1.04060401
1.06101U05
1.06152015
1.07213535
1.08285670
1.09.368527
1.10462212
1.115668.34
1.12682503
1.13809328
1.14947421
1.16096895
1.17257864
1.18430443
1.19614747
1.20810895
1.22019003
1.01250000
1.02515625
1.03797070
1.05094533
1.06408215
1.07738318
1.09085047
1.10448610
1.11829217
1.13227082
1.14642421
1.16075451
1.17526389
1.18995469
1.20482913
1.21988949
1.23513811
1.25047734
1.26610830
1.28193466
1.015000
1.030225
1.045678
1.061364
1.077284
1.093443
1.10ti845
1.126493
1.143390
1.160541
1.177949
1.195618
1.213552
1.231756
1.250232
1.268985
1.288020
1.307341
1.326951
1.346855
1.020000
1.040400
1.061208
1.082432
1.104081
1.126162
1 .148686
1.171660
1.195093
1.218994
1.243374
1.268242
1.293607
1.319479
1.345868
1.372786
1.400241
1 .428246
1.456811
1.485947
Periods
2^ Per Cent
8 Per Cent
8J Per Cent
4 Per Cent
5 Per Cent
6 Per Cent
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
1.025000
1.050625
1.076891
1.103813
1.131408
1.159693
1.188686
1.218403
1.248863
1.280085
1.312087
1.344889
1.378511
1.412774
1.448298
1.484506
1.521618
1.559659
1.598650
1.638616
1.030000
1.060900
1.092727
1.125509
1.159274
1.194052
1.229874
1.266770
1.304773
1.343916
1.384234
1.425761
1.4685.34
1.512590
1.557967
1.604706
1.652848
1.7024.33
1.753506
1.806111
1.035000
1.071225
1.108718
1.147523
1.187686
1.229255
1.272279
1.316809
1.362897
1.410599
1.459970
1.511069
1.563956
1.618695
1.675349
1.73.3986
1.794676
1.857489
1 .922501
1.989789
1.040000
1.081600
1.124864
1.169859
1.216663
1.265319
1.315932
1.368569
1.423312
1.480244
1.539454
1.601032
1.665074
1.731676
1.800944
1.872981
1.947901
2.0258.7
2.106849
2.191123
1.050000
1.102500
1.157625
1.215506
1.276282
1.340096
1.407100
1.477455
1.551328
1.628895
1.710.339
1.795856
1.885649
1.979932
2.078928
2.182875
2.292018
2.406619
2.526950
2.653298
1.060000
1.123600
1.191016
1.262477
1.338226
1.418519
1.503630
1.593848
1.689479
1.790848
1.898299
2.012197
2.132928
2.260904
2.396558
2.540352
2.692773
2.854339
3.025600
3.207136
APPENDIX 383
METRIC TABLES
MEASURES OF LENGTH
10 millimeters (mm.)= 1 centimeter (cm.)
10 centimeters = 1 decimeter (dm.)
10 decimeters = 1 meter (m.)
10 meters = 1 dekameter (Dm.)
10 dekameters = 1 hektometer (Hm.)
10 hektometers = 1 kilometer (Km.)
20 kilometers = 1 myriameter (Mm.)
MEASURES OF SURFACE
100 sq. millimeters (sq. mm.) = 1 sq. centimeter (sq. cm.)
100 sq. centimeters = 1 sq. decimeter (sq. dm.)
100 sq. decimeters =1 sq. meter (sq. m.)
100 sq. meters = 1 sq. dekameter (sq. Dm.)
100 sq. dekameters = 1 sq. hektometer (sq.Hm.)
100 sq. hektometers = 1 sq. kilometer (sq. Km.)
MEASURES OF VOLUME
1000 cu. millimeters (cu. mm.)= 1 cu. centimeter (cc.)
1000 cu. centimeters = 1 cu. decimeter (cu. dm.)
1000 cu. decimeters = 1 cu. meter (cu. m.)
MEASURES OF CAPACITY
10 milliliters (ml.) = 1 centiliter (cl.)
10 centiliters =1 deciliter (dl.)
10 deciliters = 1 liter (1.)
10 liters = 1 Dekaliter (Dl.)
10 dekaliters = 1 Hektoliter (HI.)
10 hektoliters = 1 Kiloliter (Kl.)
384 PRACTICAL ARITHMETIC
MEASURES OF WEIGHT
10 milligrams (mg.) = 1 centigram (eg.)
10 centigrams = 1 decigram (dg.)
10 decigrams = 1 gram (g.)
10 grams , = 1 dekagram (Dg.)
10 dekagrams = 1 liektogram (Hg.)
10 hektograms = 1 kilogram ( K.)
10 kilograms = 1 myriagram (Mg.)
10 myriagrams = 1 quintal (q.)
10 quintals = 1 tonneau (T.)
METRIC EQUIVALENTS OF ENGLISH MEASURES
1 acre= .4047 hectare.
1 bushel = 35.24 liters.
1 cubic foot = 28.316 liters.
1 cubic inch = 16.39 cubic centimeters.
1 cubic yard = .7645 cubic meter.
1 foot = 30.48 centimeters.
1 gallon = 3.785 liters.
1 grain = .0648 gram.
1 inch = 25.4 millimeters.
1 mile = 1.609 kilometers.
1 ounce (avoirdupois) = 28.35 grams.
1 ounce (Troy) = 31.1 grams.
1 peck = 8.809 liters.
1 pint =.4732 liter. •
1 pound = .4536 kilo.
1 quart (dry) = 1.101 liters.
1 quart (liquid) = .9464 liter.
1 square foot = .0929 square meter.
1 square inch = 6.452 square centimeterSc
1 square yard = .8361 square meter.
1 ton (2000 lbs.) = .9072 metric ton.
1 ton (2240 lbs.) = 1.017 metric tons.
1 yard = .9144 meter.
APPENDIX
385
ENGLISH EQUIVALENTS OF METRIC MEASURES
1 centimeter = .3937 inch.
1 cubic centimeter = .061 cubic inch.
1 cubic meter = 35.31 cubic feet, or 1.308 cubic yards.
1 gram = 15.43 grains.
1 hectare = 2.471 acres.
1 kilo = 2. 205 pounds.
1 kilometer = .6214 mile.
1 liter = .9081 dry quart, or 1.051 liquid quarts.
1 meter = 3.281 feet.
1 millimeter = .0394 inch.
1 square centimeter = .155 square inch.
1 square meter = 1.196 square yards, or 10.76 square
feet.
1 metric ton = 1.102 short tons, or .9842 long ton.
CATTLEMEN'S NOTATION
Explanation of the Valuk of the Various Notches.
A notch in bottom of the
animal's left ear equals 1,
two notches equal 2.
A notch in top of left ear
equals 3, two notches equal
6, three notches equal 9.
A notch in bottom of
right ear equals 10, two
notches equal 20.
A notch in top of right
ear equals 30, two notches
equal 60, three notches equal
90.
A notch in end of left ear
equals 100.
A notch in end of right
ear equals 200.
A hole in end of left ear
equals 400.
A hole in end of right ear
equals 500.
A hole in bottom of left
ear equals 1000.
Numhers can thus he made from 1 to 1999.
200
SOOi
386 PRACTICAL ARITHMETIC
LUMBERMEN'S NOTATION
In marking lumber the following characters are used ^
A Al All Alii X X XI XII XIII yn
S 6 7 8 9 10 11 1« 13 14
X\ Al All Alll XIX X' /I X'll XIII
15 16 17 18 19 2(} 21 22 23
Xf X\ X\l XMI XXIil XMIII )f I
2t 25 26
25 26 27 28 29 30 40
50 60 70 80 90 100 200