UNIVERSITY OF CALIFORNIA.
A PRACTICAL ARITHMETIC
MAY 29 i9ii
COPYRIGHT, 1909, BY
CHARLES SCRIBNER'S SONS
THE primary object of arithmetic is to enable the
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 subject-matter 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
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
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, upon 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
drawn 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
RALEIGH, N.C., November, 1908.
NOTATION AND NUMERATION 1
The Arabic System 1
The Roman System 9
United States Money 11
Addition of United States Money .... .19
Subtraction of United States Money 36
Multiplication of United States Money . . . .52
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-
DECIMAL FRACTIONS 92
Notation and Numeration 92
Subtraction. .......... 97
Multiplication ......... 99
Division . . .. . . . . . . . 103
REVIEW PROBLEMS ......... 106
COMMON FRACTIONS . . . 113
Multiplication . . . . . . . .125
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
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
Commercial Discount 263
Insurance . . . . 265
Stocks and Bonds . 285
Bank Discount . . . . . 293
Partial Payments . . .296
The Nutritive Ratio, Specific Gravity.
Levers, Compound Proportion.
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.
NOTATION AND NUMERATION
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 ten-thousands, thousands, hundreds,
tens, and ones in :
50207 34291 23845 10205 23814
35842 78354 91846 35841 87961
26459 52796 87964 88249 18462
11. How many thousands in this number, 100000?
12. How many hundred-thousands, ten-thousands, thou-
sands, hundreds, tens, and ones in:
1. Hundreds, tens, and ones written together form a
group or period, called Units' Period.
2. Hundred-thousands, ten-thousands, 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 :
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?
Moving a figure one place to the left increases its
value tenfold. Moving 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
14. St. Paul,
3,329 15. Jefferson City, 9,664
508,957 16. Madison, 19,164
39,441 17. Lansing, 16,485
560,892 18. Columbus, 125,560
19,632 19. Springfield, 34,157
175,597 20. Raleigh, 13,643
3,319 21. Jackson, 7,816
2,305 22. Tallahassee, 2,981
40,167 23. Phoenix, 5,544
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 :
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
12. The process of reading numbers is called Numeration.
13. Numbers may be expressed by Figures, Letters, or
EXERCISE 3. WRITTEN
Write these numbers in figures, using the comma to
separate periods :
1. Six hundred seventy -five.
2. Two hundred thirteen.
3. Four hundred ninety-six.
4. Two hundred twenty-nine.
5. Four hundred eight.
6. One thousand, three hundred fifty-two.
7. Six thousand, forty.
8. Eighty thousand, eighty.
9. Seven thousand, three hundred.
10. Thirteen thousand, four hundred fifty.
11. Ninety-nine thousand, nine.
12. Forty-four thousand, sixteen.
13. Four hundred six thousand, one hundred fifty.
14. Three thousand, fourteen.
15. Nine thousand, seventy-seven.
16. Fifty thousand, sixty-eight.
17. Eleven thousand, nine hundred seventy-three.
18. Seven hundred eighty-five thousand, two.
6 PRACTICAL ARITHMETIC
19. Ninety-two thousand, one hundred six.
20. One million, three hundred ninety-seven thousand.
21. Eight thousand, four hundred eighty- two.
22. Nineteen million, one hundred fifty-six thousand.
23. Eight million, six.
24. Five thousand, one hundred thirty-one.
25. Sixty-three million, sixty-eight thousand, seven.
14. The art of writing numbers is called Notation.
EXERCISE 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 seventy-one thou-
sand, two hundred fifty.
2. Alabama, five hundred sixty-eight thousand, seven
3. Montana, thirty-five million, eight hundred fifteen
4. Florida, three hundred sixteen thousand, six hun-
5. Wyoming, thirty-two 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,
NOTATION AND NUMERATION 7
NUMBER OF HOGS IN 1907
8. New York, six hundred seventy-five thousand, five
hundred forty -five.
9. Iowa, eight million, five hundred eighty-four thou-
sand, five hundred.
10. Pennsylvania, nine hundred eighty-nine thousand,
six hundred eighty-five.
11. Kentucky, one million, two hundred thirteen thou-
sand, three hundred eighty.
12. Illinois, four million, four hundred forty-nine thou-
sand, seven hundred five.
13. Texas, two million, eight hundred sixty thousand,
eight hundred seventy-nine.
14. Ohio, two million, four hundred thirty-six thousand,
seven hundred ninety-seven.
15. Alabama, one million, two hundred fifty-one thou-
sand, two hundred fifty-one.
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., eighty-one
18. From St. Paul, Minn., to Portland, Ore., two
thousand fifty-three miles; from Cleveland, O., to Cin-
cinnati, O., two hundred sixty-three miles.
STREET SCENE IN ATLANTA, GA.
From a photograph.
Copyright, 1907, by Underwood & Underwood.
19. From Chattanooga, Teim., to New Orleans, La.,
iour hundred ninety-two miles ; from Nashville, Tenn., to
New Orleans, La.,
six hundred twenty-
four miles ; from
New Orleans, La.,
to Atlanta, Ga., four
20. By water from
New York to Liver-
pool, three thousand
fifty-eight miles ;
from San Francisco
to Yokohama, four
thousand, seven hundred ninety-one 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 eighty-five miles ; to San Francisco, three
thousand two hundred fifty miles.
24. The railroads of the United States aggregate one
hundred ninety-three thousand miles, bearing thirty-eight
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.
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 follows :
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, CO denotes 200, MMM
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
greater value, its value is taken from that of the letter
which follows it, e.g., XIX represents 10 and 9, or 19;
CXC represents 100 and 90, or 190.
EXERCISE 5. WRITTEN
Express in Arabic notation :
EXERCISE 6. WRITTEN
Express in Roman notation :
UNITED STATES MONET
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 money 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 ten-cent pieces equal one dollar. Ten one-
cent pieces equal a ten-cent 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 one-tenth of a dollar. One cent is one
one-hundredth of a dollar. One mill is one one-thou-
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 mills as they are written:
"3 33 S3 00
^ "3 > '3 .33
M 4S p< -g oj A'STS
o g s
12 PRACTICAL ARITHMETIC
EXERCISE 7. ORAL
Read as dollars and cents :
1. $ 3.24 5. $ 25.06 9. $349.99 13. $ 542.89
2. $ 8.72 6. $ 91.07 10. $698.42 14. $ 560.90
3. $ 9.87 7. $ 92.09 11. $100.10 is. $1845.24
4. $10.25 8. $900.09 12. $ 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; thirty-four dollars
and five cents.
2. Eighteen dollars and thirty-five cents ; ninety-five
dollars and twenty cents ; thirty-one dollars and sixty
cents ; one hundred twenty dollars and four cents.
3. One hundred dollars and ten cents ; fifty-four dol-
lars and nineteen cents ; fifty-three dollars and fifty-five
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 thirty-three dollars and thirty-three cents.
5. Five hundred dollars and three-tenths ; two hundred
dollars and thirty-three hundredths ; one hundred dollars
and three hundred thirty-three thousandths.
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
26. The sign of addition, + , is called Plus. When
placed between numbers it means that they are to be
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 =
3 + 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 =
1 + 2 =
5 + 2 =
1 + 8 =
5 + 5 =
4 + 6 =
3 + 6 =
2 + 5 =
8 + 1 =
6 + 2 =
14 PRACTICAL ARITHMETIC
6+3= 4+4= 2+8= 1+3=
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. ORAL
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 Ibs. ;
cotton-seed meal, 4 Ibs.; hay, 20 Ibs. 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
8. If a fruit cake requires 3 Ibs. of currants, 2 Ibs. of
raisins, and 1 Ib. 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
13. A ration for a cow is 15 Ibs. of hay, 30 Ibs. of
silage, 4 Ibs. of cotton-seed meal, 3 Ibs. of wheat bran,
and 3 Ibs. 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 83 and 85 respectively
a day. How much did the two earn together ?
20. Two unskilled laborers earned respectively 81.25
and 81.50 a day. How much did the two earn ?
21. The cultivation of an acre of corn costs 86, the fer-
tilizers 8 3, harvesting and other expenses 83. What is
the cost of the crop per acre ?
22. A ration for a horse weighing 1000 Ibs. when doing
moderately hard work is 6 Ibs. of corn, 8 Ibs. of oats, and
15 Ibs. of hay. What is the weight of the ration?
23. A ration for a fattening beef animal weighing
1000 Ibs. is 30 Ibs. of corn silage, 12 Ibs. of corn stover,
5 Ibs. of cotton-seed meal, and 4 Ibs. of cotton seed.
What is the weight of the ration?
24. If a man wishes to seed an acre for a meadow and
uses 11 Ibs. of timothy, 6 Ibs. of red top, and 5 Ibs. of
clover seed, how many pounds of seed does he sow on the
25. If 1000 Ibs. of an average mixed stable manure
contain 5 Ibs. of nitrogen, 6 Ibs. of potash, and 3 Ibs. 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
Ibs. of nitrogen, 37 Ibs. of potash, and 11 Ibs. 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 used in growing 11 Ibs,
of nitrogen, 23 Ibs. of potash, and 4 Ibs. 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 Ibs. of
nitrogen, 9 Ibs. of potash, and 13 Ibs. of phosphoric acid,
how many pounds of these materials were used ?