IN MEMORIAM
FLOR1AN CAJORI
£K^^
-^ OW"1"
— w)
HOPKINS AND UNDERWOOD'S NEW
ARITHMETICS
ADVANCED BOOK
THE MACMILLAN COMPANY
NEW YORK • BOSTON • CHICAGO
ATLANTA • SAN FRANCISCO
MACMILLAN & CO., LIMITED
LONDON • BOMBAY • CALCUTTA
MELBOURNE
THE MACMILLAN CO. OF CANADA, LTD.
TORONTO
HOPKINS AND UNDERWOOD'S
NEW ARITHMETICS
ADVANCED BOOK
BY
JOHN W. HOPKINS
SUPERINTENDENT OF THE GALVESTON PUBLIC SCHOOLS
AND
P. H. UNDERWOOD
TEACHER OF MATHEMATICS, BALL HIGH SCHOOL, GALVESTON, TEXAS
gorfe
THE MACMILLAN COMPANY
1908
All rights reserved
K7-3
COPYRIGHT, 1903, 1907,
BY THE MACMILLAN COMPANY.
Set up and electrotyped. Published August, 1903.
New edition revised. Published December, 1907.
CAJOR1
NorfoooU
J. 8. Gushing Co. — Berwick & Smith Co.
Norwood, Mass., U.S.A.
PREFACE
THIS book assumes a working knowledge of the four
fundamental rules as applied to integers and United
States Money. It contains the essentials of practical
arithmetic arranged by topics in conformity with the
courses of study in some of the best school systems, each
chapter representing a year's work commencing with the
fifth grade.
It aims to teach principles rather than rules. As the
unitary method is the one most natural to the young
learner, the first two chapters give prominence to this
style of reasoning. Chapters III and IV give a thorough
review of arithmetic principles and practice. In these
chapters the method of ratio is brought into prominence.
Science to be of value must be more or less deductive in
form.
Characteristic features of the book are the early intro-
duction of Decimals, the large number of problems based
on the industrial resources of our country, the clearness
and directness with which problems are illustrated, and
the omission of complicated problems of doubtful utility.
The aim of teaching arithmetic is culture, accuracy,
and rapidity in the computation of problems arising in
actual life, and the acquisition of correct methods of
reasoning. This aim is always kept in view. However,
as students possess the power of learning readily to work
processes, and, furthermore, as the practice of arithmetic
918235
vi PREFACE
is of more importance to the majority of people than the
theory, attention is paid especially to the art of compu-
tation.
The book contains a short introduction to the method
of obtaining approximate results correct to any required
degree of accuracy. This matter is new, but it is believed
that it is well worthy of consideration.
Chapters III and IV contain all the arithmetic and men-
suration that is most needful to be known, and, in fact,
will be found comprehensive enough to suit the require-
ments of pupils taking a survey of commercial arithmetic.
JOHN W. HOPKINS,
P. H. UNDERWOOD.
GALVESTON, TEXAS,
September 9, 1907.
CONTENTS
PAGE
PREFACE v
CHAPTER I
NUMERATION AND NOTATION 1
DECIMALS 5
ADDITION AND SUBTRACTION 7
MULTIPLICATION AND DIVISION 11
BILLS 22
MEASURES AND MULTIPLES 24
TESTS OF DIVISIBILITY 27
GREATEST COMMON DIVISOR 30
LEAST COMMON MULTIPLE 31
FRACTIONS 33
REDUCTION OF FRACTIONS 35
ADDITION OF FRACTIONS 39
SUBTRACTION OF FRACTIONS 41
CANCELLATION 42
MULTIPLICATION OF FRACTIONS 44
DIVISION OF FRACTIONS. RATIO 48
DECIMALS 53
MULTIPLICATION AND DIVISION BY POWERS OF TEN . . 54
ADDITION 57
SUBTRACTION 57
MULTIPLICATION 59
DIVISION o 61
vii
viii CONTENTS
PAGE
REDUCTION OF FRACTIONS TO DECIMALS AND REDUCTION OF
DECIMALS TO FRACTIONS ....... 66
MISCELLANEOUS EXAMPLES 67
COMPLEX FRACTIONS 70
AREAS OF RECTANGULAR FIGURES 71
COMPUTATION ON THE BASIS OF 100, 1000, 2000 ... 73
PERCENTAGE 76
INTEREST AND PROPERTY INSURANCE . . ... 79
CHAPTER II
COMPOUND QUANTITIES . 81
REDUCTION OF COMPOUND QUANTITIES 82
ADDITION . . . 93
SUBTRACTION ... . . . . . . .96
MULTIPLICATION . .99
DIVISION . . . . 100
EXPRESSION OF ONE QUANTITY AS A FRACTION OF ANOTHER
QUANTITY . . . 102
MEASUREMENTS . ... . . . . . . 103
VOLUMES OF RECTANGULAR SOLIDS . . . . . . 105
AREAS OF PARALLELOGRAMS, TRIANGLES, AND TRAPEZOIDS . 107
BOARD MEASUREMENT ........ 110
MASONRY AND BRICKLAYING 112
CARPETING . . 113
MISCELLANEOUS EXERCISES 115
REVIEW OF FRACTIONS 120
PERCENTAGE . . . 122
To FIND A NUMBER WHEN A PER CENT OF IT is GIVEN . 124
TO FIND WHAT PER CENT ONE NUMBER IS OF ANOTHER
NUMBER . . . . 127
COMMERCIAL DISCOUNTS 130
PROFIT AND Loss 133
CONTENTS ix
PAGE
COMMISSION AND BROKERAGE . • . . . * . . . . 140
INTEREST . . 144
SPECIFIC GRAVITY . 148
RATIO . . ... . ... . . . 154
PROPORTION . . . . . 156
REVIEW . . . . < . • 158
CHAPTER III
GENERAL REVIEW BY TOPICS 163
ADDITION . . . 163
SUBTRACTION .... . . . 168
MULTIPLICATION 172
PARTICULAR SHORT METHODS OF MULTIPLICATION . . 173
DIVISION . . 177
LONGITUDE AND TIME . . ... . . . 181
STANDARD TIME 188
APPROXIMATIONS. CONTRACTED PROCESSES, GENERAL
METHODS OF SOLUTION .... . . . 190 '
LANGUAGE OF MATHEMATICS 196
s\
'RATIO 200
COMPOUND PROPORTION . . .... . . 205
PARTNERSHIP . .... . . . . . 208
PERCENTAGE . . 210
INTEREST . . . 217
EXACT INTEREST . . 218
INVERSE QUESTIONS IN INTEREST . 219
REVIEW .222
REVIEW QUESTIONS . 224
PROMISSORY NOTES . . . 225
BANKERS' INTEREST 228
COMMERCIAL DISCOUNT 231
PARTIAL PAYMENTS. UNITED STATES RULE . . . . 232
X CONTENTS
PAGK
MERCHANTS* RULE 234
EXCHANGE 236
VALUE OF FOREIGN COINS . . . . . . . . . 241
ENGLISH MONEY 242
FOREIGN EXCHANGE 244
STOCKS AND BONDS 246
CUSTOMS AND DUTIES 252
CHAPTER IV
INVOLUTION 255
EVOLUTION 259
PROBLEMS INVOLVING SQUARE ROOT ..... 264
AREAS OF PLANE TRIANGLES 267
MENSURATION OF THE CIRCLE 269
SIMILAR FIGURES 275
SURFACES OF PRISM, PYRAMID, CYLINDER, CONE, AND
SPHERE 277
VOLUMES OF SOLIDS 280
MEASURE OF TEMPERATURE 282
METRIC SYSTEM OF WEIGHTS AND MEASURES . . . 284
ANNUAL INTEREST 291
COMPOUND INTEREST 292
MISCELLANEOUS TOPICS 295
WORK AND TIME 295
MOTION IN THE SAME OR OPPOSITE DIRECTIONS . . . 298
CLOCKS 299
MISCELLANEOUS EXAMPLES BY TOPICS (A) .... 303
MISCELLANEOUS EXAMPLES (B) 318
TABLES . , 325
HOPKINS AND UNDERWOOD'S NEW
ARITHMETICS
ADVANCED BOOK
ADVANCED BOOK OF AEITHMETIC
CHAPTER I
Arithmetic is the science of numbers and the art of com-
putation.
The fundamental operation in arithmetic is counting.
The result of counting is a number.
Any one of the natural numbers, one, two, three, etc., is
called an integer, or -whole number.
A unit is one thing, or a group of things regarded as a
single thing.
NUMERATION
The number names are one, two, three, four, five, six,
seven, eight, nine, ten, eleven, twelve, thirteen, fourteen,
fifteen, sixteen, seventeen, eighteen, nineteen, twenty,
thirty, forty, fifty, sixty, seventy, eighty, ninety, hun-
dred, thousand, million, billion, trillion, etc. By combin-
ing these number names all numbers may be expressed in
words.
Ones, tens, hundreds, thousands, ten-thousands, are re-
spectively called units of the first order, units of the
second order, units of the third order, units of the fourth
order, units of the fifth order.
In our system of naming numbers ten units of any
order are equal to one unit of the next higher order. On
this account our system is known as the decimal system.
Numeration is the naming of numbers.
2 ADVANCED BOOK OF ARITHMETIC
NOTATION
Every number can be expressed by one or more of the
following figures, sometimes called Arabic numerals : 1, 2,
3, 4, 5, 6, 7, 8, 9, 0. The first nine of these figures are
called digits, or significant figures.
Tens are written in the second place ; hundreds, in the
third place ; thousands, in the fourth place ; millions, in
the seventh place. For example, four thousand seven
hundred eighty-nine is written 4,789. This number may be
read four thousands seven hundreds eight tens nine ones.
The nine ones occupy the first place ; the eight tens, the
second place ; the seven hundreds, the third place ; the four
thousands, the fourth place. To read a number expressed
by more than three figures, begin at the right, that is, with
units of the first order, and mark off with commas the
figures in groups of three each. The three places, or
orders, in which units of the first order occur constitute
what is called the units' period ; the next three places, the
thousands' period ; the next three places, the millions' period;
the next three, the billions' period, etc. As an illustration
take the number 1734902309 ; marking this number off
into periods, it becomes 1,734,902,309; this is read one
billion seven hundred thirty-four million nine hundred
two thousand three hundred nine. Observe that each
period has its hundreds, tens, and units. The periods
most used are the units, thousands, and millions. The bil-
lions period is rarely used. The billionth part of a great
circle of the earth is less than 2 inches. The names of a
few of the succeeding periods are trillions, quadrillions,,
quintillions, and sextillions.
Notation is the expression of numbers by means of
symbols.
NOTATION 3
EXERCISE 1
Express in words :
1. 45289. 10. 910003. 19. 8307308.
2. 90208. 11. 728000. 20. 8000401.
3. 75307. 12. 400098. 21. 7000014.
4. 392394. 13. 902023. 22. 6100079.
5. 738211. 14. 630006. 23. 3927173.
6. 328993. is. 1000000. 24. 5009020.
7. 401012. 16. 2227001. 25. 8000904.
8. 300287. 17. 3456000. 26. 6203003.
9. 200020. 18. 9287003. 27. 9000090.
28. What is the largest whole number expressed by
two figures?
29. What is the smallest whole number expressed by
two figures?
30. What is the largest whole number expressed by
three figures?
31. What is the smallest whole number expressed by
three figures?
32. Write the largest number expressed by the figures
0, 4, 5.
33. Write the smallest number expressed by the figures
0, 4, 5.
34. Write three numbers expressed by the figures 2, 3, 4.
35. Write four numbers expressed by the figures 7, 6, 8.
36. What is the largest number expressed by the figures
7, 3, 2, 8?
37. What is the smallest number expressed by the
figures 2, 5, 3, 4?
4 ADVANCED BOOK OF ARITHMETIC
EXERCISE 2
Write in figures :
1. Four thousand, eight hundred twenty-seven.
2. Nine thousand, seven hundred one.
3. Sixty-eight thousand, four hundred fifty-two.
4. Forty-seven thousand, three hundred eight.
5. Ninety thousand, six hundred four.
6. Eighty-seven thousand, one hundred one.
7. Twenty-two thousand, three hundred eleven.
8. Twelve thousand, fifteen.
9. Eighteen thousand, eighteen.
10. Fourteen thousand, thirty-four.
11. Thirteen thousand, five.
12. Ninety thousand, nine.
13. Fifty-four thousand, eleven.
14. Seventy-three thousand, one.
15. Six hundred four thousand, two hundred one.
16. One hundred sixty-three thousand, ten.
17. One hundred one thousand, three hundred.
18. One hundred thousand, seven.
19. Four hundred ten thousand, one hundred twenty,
seven.
20. Five hundred four thousand, three hundred eight.
21. Five hundred thousand, eleven.
22. Six hundred thousand, seventeen.
23. Nine hundred ninety thousand, fifteen.
24. Two hundred one thousand, one.
25. Seventy-two thousand, four.
READING DECIMALS 5
DECIMALS
NOTE. Pupils should draw and measure distances such as 9.3 centi-
meters, 2.87 inches.
The law pervading the decimal system of notation is
the value of a digit in any place is always ten times the
value of the same digit written in the next place to the
right. A familiar illustration of this law is the notation
of United States money. For example, $5.55. The 5 on
the left is ten times the value of the second 5, and the second
5 is ten times the value of the 5 on the right. The period
separating dollars and cents is called the decimal point.
The first place to the right of the decimal point is
called the tenths' place ; the second place, the hundredths'
place ; the third place, the thousandths' place ; the fourth
place, the ten-thousandths' place ; the fifth place, the hun-
dred-thousandths' place; the sixth place, the million ths'
place.
READING DECIMALS
Since .47 = 4 tenths 7 hundredths =
Therefore, .47 is read forty-seven hundredths.
Since .372 = 3 tenths 7 hundredths 2 thousandths = ^
+ To -o + To2o o = iVoV Therefore, .372 is read 372 thou-
sandths.
In general a decimal is read by regarding it as a whole
number and adding the name of the place the right-hand
digit occupies. In reading decimals and should not be
used except to connect the integral and decimal parts of
the number. For example, 500.005 is read five hundred
and five thousandths. .505 is read five hundred five
thousandths. 8.0379 is read eight and three hundred
seventy-nine ten-thousandths. .8379 is read eight thou-
sand three hundred seventy-nine ten-thousandths.
6 ADVANCED BOOK OF ARITHMETIC
EXERCISE 3
Read :
1. 6.2. 7. 6.201. is. 5.0067.
2. 7.9. 8. 4.027. • 14. 7.0123.
3. 8.4. 9. 5.029. is. 9.1238.
4. 4.32. 10. 9.001. 16. .0003.
5. .12. 11. 6.034. 17. .0054.
6. 5.17. 12. 8.295. is. .4008.
Write :
1. 33 hundredths. 3. 329 thousandths.
2. 2005 "thousandths. 4. 101 thousandths.
5. Two hundred three thousandths.
6. Seven hundred and four thousandths.
7. Nine hundred three thousandths.
8. Nine hundred and three thousandths.
9. Six thousand seven hundred ten-thousandths.
10. Six thousand seven hundred and one ten-thou-
sandth.
11. Five hundred ninety ten -thousandths.
12. Five hundred and ninety ten-thousandths.
13. Six thousand one ten-thousandths.
14. Seven hundred ten-thousandths.
15. Seven hundred ten thousandths.
16. Five hundred thousandths.
17. Five hundred-thousandths.
18. Two hundred seven hundred-thousandths.
19. Two hundred and seven hundred -thousandths.
20. Five thousand two hundred two hundred-thou-
sandths.
ADDITION AND SUBTRACTION 7
21. Six thousand four hundred-thousandths.
22. Three thousand ten thousandths.
23. Three thousand one ten-thousandths.
24. Five hundred seventeen ten-thousandths.
25. One hundred eleven hundred-thousandths.
26. Seventy-eight ten-thousandths.
ADDITION AND SUBTRACTION
The result of combining two or more numbers into a
single number is called the sum.
Addition is the process of finding the sum of two or
more numbers.
Only numbers of the same kind can be added.
The symbol for addition is +, and it is read plus.
The symbol = is read equal, or equals ; thus, 5 + 8 =
13 is read five plus eight equal thirteen.
The difference between two numbers is the excess of
one over the other.
Subtraction is the process of finding the difference be-
tween two numbers.
The number subtracted is the subtrahend. The num-
ber from which the subtrahend is taken is the minuend.
The result of subtraction is called the remainder, or
difference.
The sign of subtraction is — , and is read minus. Thus,
8 — 5 = 3 is read eight minus five equals three.
ILLUSTRATIVE EXAMPLE. From 913 take 537.
913 800 + 100 + 13 = 913
537 500+ 30+ 7 = 537
376 300 + 70 + 6 = 376
7 from 13 leaves 6 ; 3 from 10 leaves 7 ; 5 from 8
leaves 3. This is the usual explanation given by teachers.
8
ADVANCED BOOK OF ARITHMETIC
When a figure in the subtrahend cannot be taken from
the corresponding figure in the minuend, a unit of the
next higher order in the minuend is changed to ten units
and then added to the figure in the minuend. A better
way of subtracting is : 7 and 6 are 13. Write 6, carry
1. 1 and 3 are 4; 4 and 7 are 11. Write 7, carry 1. 1
and 5 are 6 ; 6 and 3 are 9. Write 3.
EXERCISE 4
The following table gives the number of children of
school age, number enrolled, average daily attendance,
and total expenditures for the public schools by states and
territories for the school year 1904 :
STATE
OR
TERRITORY
NUMBER
OP
CHILDREN
NUMBER
ENROLLED
AVERAGE
DAILY
ATTENDANCE
TOTAL
EXPEND-
ITURES
North Atlantic Division.
Maine
163,931
131,176
98,257
$ 2,080,109
New Hampshire
91,847
65,673
48,673
1,376,899
Vermont
81,358
66,535
48,845
1,176,784
Massachusetts
673,690
494,042
391,771
16,436,668
Rhode Island
108,471
70,843
51,692
1,804,762
Connecticut
223,174
163,141
123,317
3,795,260
New York
1,859,824
1,300,065
963,780
43,750,277
New Jersey
514,585
352,203
239,505
8,838,515
Pennsylvania
1,782,740
1,200,230
900,234
26,073,565
South Atlantic Division.
Delaware
50,695
36,895
25,300
453,670
Maryland
347,594
209,978
130,065
2,755,288
District of Columbia
64,766
49,789
39,300
1,576,354
Virginia
611,555
375,601
224,769
2,137,365
West Virginia
319,874
244,040
158,264
2,531,655
North Carolina
666,782
491,838
318,055
2,075.566
South Carolina
490,214
292,115
214,133
1,191,963
Georgia
789,939
502,014
310,400
2,240,247
Florida
180,501
122,636
83,631
945,848
ADDITION
South Central Division.
Kentucky
700,272
501,482
309,836
$2,662,863
Tennessee
678,782
502,330
344,882
2,602,141
Alabama
652,518
365,171
240,000
1,252,247
Mississippi
563,019
403,647
233,175
•1,868,544
Louisiana
483,967
208,737
155,794
1,551,232
Texas
1,128,934
722,904
461,938
6,200,587
Arkansas
467,821
339,542
212,131
1,729,879
Oklahoma
164,882
152,886
93,495
1,359,624
Indian Territory
162,641
38,422
23,053
643,616
North Central Division.
Ohio
1,151,007
835,607
618,495
15,802,002
Indiana
732,172
550,732
416,047
9,363,450
Illinois
1,428,613
978,554
783,563
21,792,751
Michigan
684,369
497,299
388,092
9,158,014
Wisconsin
658,474
461,214
288,300
7,885,050
Minnesota
566,397
423,663
272,500
8,073,323
Iowa
672,271
545,940
373,023
10,696,693
Missouri
965,598
731,410
464,706
9,878,198
North Dakota
110,938
95,224
58,442
2,316,346
South Dakota
130,844
106,822
73,700
2,239,135
Nebraska
321,822
278,930
180,771
4,774,146
Kansas
455,943
390,236
270,878
5,684,579
Western Division.
Montana
63,106
44,881
31,471
1,236,253
Wyoming
24,960
14,512
9,650
253,551
Colorado
145,799
134,260
95,117
3,984,967
New Mexico
64,094
39,704
29,582
353,012
Arizona
35,365
21,088
13,022
438,828
Utah
98,762
75,662
56,183
1,657,234
Nevada
9,013
7,319
5,182
257,501
Idaho
54,700
54,480
39,817
1,001,394
Washington
147,302
161,651
110,774
4,053,468
Oregon
118,977
103,877
72,464
1,803,339
California
363,846
299,038
222,182
9,401,465
Find the totals for each of the above divisions. Also
find the total in each instance for the entire country.
10 ADVANCED BOOK OF ARITHMETIC
EXERCISE 5
The following table gives the gross and net earnings of
the principal railroads of Texas for the eleven months
ending May 31, 1907 :
RAILROAD GROSS EARNINGS NET EARNINGS
1. C. R. I. and G. $3,082,314.24 $ 895,096.73
2. F. W. and D. C. 4,094,588.28 1,345,055.55
3. G. H. and S. A. 11,130,030.96 2,536,237.69
4. F. W. and R. G. 1,075,725.13 342,223.14
5. H. and T. C. 6,572,660.00 2,242,022.12
6. G. C. and S. F. 12,510,936.83 3,365,234.82
7. H. E. and W. T. 1,284,929.31 511,161.15
8. I. and G. N. 8,204,579.38 2,079,764.06
9. M. K. and T. 9,989,708.55 2,239,523.41
10. St. L. S. W. of T. 3,936,613.38 506,646.38
11. S. A. and A. P. 3,518,565.80 1,517,563.03
12. T. and N. O. 4,103,849.13 968,031.21
13. T. and P. 15,456,714.54 5,427,188.11
14. Texas Central 1,149,069.36 489,109.71
Find the operating expenses (difference between gross
and net earnings) of each of the above roads.
15. The increase in net earnings of the G. C. and S. F.
road for the eleven months ending May 31, 1907, over that
of the corresponding months of the preceding year was
$1,361,052.87. Find the net earnings for the eleven
months ending May 31, 1906.
16. A like increase for the I. and G. N. road was
$1,235,847.06. Find its net earnings for the eleven
months ending May, 1906.
MULTIPLICATION AND DIVISION 11
MULTIPLICATION AND DIVISION
97 + 97 + 97 + 97 + 97 = ? If these numbers are
added, the sum will be 485. In examples of this character
the usual process is as follows :
97 Five 7's are 35. Write 5, carry 3. Five 9's are
_5 45, and 3 are 48. The result is 485.
485 This short method of adding is called multiplication.
Multiplication is a short method of addition when the
numbers to be added are all the same. The number to be
repeatedly added is the multiplicand. The number in-
dicating how often the multiplicand is to be taken as an
addend is the multiplier. The result of a multiplication
exercise is the product. The multiplicand and multiplier
are factors of the product. Since the multiplier denotes
number of times, it must always be a pure number, or an
abstract number. The multiplicand may be either a con-
crete or an abstract number. The product is concrete or ab-
stract according as the multiplicand is concrete or abstract.
The sign of multiplication is x , and is read multiplied
by, or times. Thus, $ 34 x 7 means $ 34 is to be multi-
plied by 7. 7 x $ 34 means 7 times $ 34.
Is 7x8 = 8x7? Is 9x4 = 4x9?
Is 7x3x8 = 3x8x7? Is 9x6x5 = 6x9x5?
The order in -which numbers are multiplied is immaterial.
(a) How many times is $4 contained in $24 ? (5) What
is the sixth part of $24 ?
These two examples illustrate the two kinds of division;
the first is to determine the number of times one number
or quantity is contained in another number or quantity.
This is often called measuring. The second is to deter-
mine a part of a number or quantity, and is often called
parting, or dividing.
12 ADVANCED BOOK OF ARITHMETIC
Division is the process of determining how often one num-
ber is contained in another, and also of determining any given
part of a number. The first number is the divisor; the
second is the dividend ; the result is the quotient.
When divisor and dividend are concrete numbers, the
quotient is abstract. When the dividend is a concrete
number, and the divisor an abstract number, the quotient
is a concrete number.
The signs of division are -*-, and a horizontal stroke,
the dividend written above, the divisor below. Thus,
27 -5- 3, and -2JL indicate that 27 is dividend and 3, divisor.
ILLUSTRATIVE EXAMPLES
(a) The area of Massachusetts is 8,040 square miles,
and the average number of inhabitants per square mile,
for the year 1900, was 348.9. Find its population.
348.9
8040 One square mile averages 348.9 inhab-
139560 itants ; therefore, 8,040 square miles will
27912 have 8,040 times 348.9 inhabitants.
2805156.0
(5) The area of the United States, exclusive of posses-
sions, is 3,026,000 square miles, and the estimated popu-
lation for the year ending June 30, 1905, was 83,143,000.
Find the average number of inhabitants per square mile.
27.4
3026)83143. Observe the area is 3,026 thousands
6052 of square miles and the population is
22623 83,143 thousands of inhabitants. Hence,
21182 the required quotient will be obtained
14410 by dividing 83,143 by 3,026.
MULTIPLICATION 13
EXERCISE 6
1. An office desk costs $25. How much will 3 such
desks cost? 8 desks? 36 desks? 49 desks?
2. Eggs sell for 26 / per dozen. Find the cost of 8
dozen ; 18 dozen ; 94 dozen.
3. There are 5,280 feet in a mile. How many feet are
in 19 miles? in 76 miles?
4. How many days are in 39 weeks?
5. A contractor pays in wages $78 a day. How much
will he pay in 78 days?
6. How many hours are in 89 days?
7. A train travels at the rate of 34 miles an hour.
How far will it run in 47 hours?
8. How many acres are in a ranch containing 98
sections of land? (1 section = 640 acres.)
9. A degree on a meridian of the earth is about 69
miles. How many miles are in 17 degrees ?
10. A cubic foot of rock weighs 148 pounds. How
many pounds do 3,297 cubic feet weigh?
11. The rent of a dwelling is $28 per month. Find
the rent for 3 years.
12. A gallon of water contains 231 cubic inches. How
many cubic inches are in 368 gallons?
13. A book has 360 pages, each page has 32 lines, and
each line averages 9 words. How many words are in the
book?
14. A carpenter earns $3.20 a day. At this rate, how
much wages will he receive in 299 days?
15. A brick mason earns $ 5.60 a day. How much will
he earn in 310 days ?
14 ADVANCED BOOK OF ARITHMETIC
EXERCISE 7
1. A city block is 100 yards long and 90 yards wide.
Find its area.
2. Find the area of a square whose side is 84 yards.
3. Find the area of a square having 320 rods for a side.
4. Find the area of a rectangle, the length being 140
yards and the width 84 yards.
5. Find the area of a rectangle 238 yards long and 96
yards wide.
6. A farm, rectangular in shape, 440 yards long and
380 yards wide, contains how many square yards ?
7. Find the area of a rectangle 75 rods long and 63
rods wide.
8. Find the area of a grass plot 240 feet by 84 feet.
9. A sheet of paper 18 inches long and 14 inches wide
contains how many square inches ?
10. A township is 6 miles long and 6 miles wide. How
many square miles does it contain ?
11. A county, having the shape of a rectangle, is 24
miles long and 18 miles wide. How many square miles
are in its area ?
12. A street is 1760 yards long and 23 yards wide.
How many square yards does it contain ?
13. A garden is 50 yards long and 44 yards wide. How
many square yards are in its area ?
14. A city lot is 43 feet wide and 124 feet deep. How
many square feet are in its area ?
15. A window is 60 inches by 48 inches. How many
square inches are in its area ?
16. A yard is 36 inches. How many square inches are
in a square yard ?
MULTIPLICATION 15
EXERCISE 8
Find the product :
1. $ 79.94 x 8. 12. $ 79.29 x 7.
2. $ 32.20 x 7. 13. | 29.97 x 8.
3. $ 79.49 x 6. 14. $179.38 x 6.
4. $128.29 x 7. is. 1373.39 x 5.
5. $399.39 x 9. 16. $799.94 x 8.
6. $454.59 x 12. 17. $822.50 x 9.
7. $729.38 x 11. 18. $998.78 x 11.
8. $237.38 x 9. 19. $778.75 x 12.
9. $720.99 x 7. 20. $732.75 x 9.
10. $285.68 x 8. 21. $928.34 x 7.
11. $ 51.33 x 7. 22. $653.82 x 5.
23. When shoes sell for $3.90 a pair, how much will 24
pairs of shoes cost ?
24. If overcoats sell for $7.98, find the price of 20
overcoats.
25. Mackintoshes sell for $6.95 apiece. How much
will 27 mackintoshes cost?
26. When wheat is 84^ per bushel, how much will 384
bushels bring ?
27. Find the price of 349 bushels of corn at 56^ a
bushel.
28. Cheese costs 13^ per pound. Find the price of 54
pounds.
29. Find the cost of 325 pounds of sugar at 6? per
pound.
30. An acre of land is worth $60.75. Find the value
of 100 acres.
16 ADVANCED BOOK OF ARITHMETIC
EXERCISE 9
On the map of the United States published by the
General Land Office, Department of the Interior, 1 inch
represents 37 miles. On this map the distances in inches
between the cities named are given below :
1. New Orleans to Chicago 22.75.
2. Savannah to Indianapolis 16.6.
3. Mobile to Toledo 21.89.
4. Richmond to St. Louis 19.2.
5. Washington to San Antonio 37.88.
6. Boston to Jackson 34.6.
7. Atlanta to Des Moines 20.4.
8. Newport to St. Louis 27.9.
9. New York to Lincoln 32.2.
10. Chicago to San Francisco 50.8.
11. St. Louis to Portland, Oregon, 47.1.
12. Memphis to Seattle 51.2.
Find the distance in miles between each of the above-
named cities.
Find the number of inhabitants in the states named:
STATE AREAS IN SQ. Mi. NUMBER OF INHABIT-
ANTS PER SQ. Mi.
13. Georgia 58,980 37.6
14. Iowa 55,475 40.2
15. Illinois 56,000 86.1
16. Louisiana 45,420 30.4
17. Michigan 57,430 42.2
18. New Jersey 7,525 250.0
19. Ohio 40,760 102.0
20. Pennsylvania 44,985 140.0
DIVISION 17
EXERCISE 10
Divide and prove your answers correct :
1. 77,354 -;- 16. 9. 99,392 - 36. 17. 828,374 + 56.
2. 79,358 -*- 18. 10. 59,738 + 35. is. 528,739 ^ 64.
3. 97,854 H- 20. 11. 49,399 - 40. 19. 629,394 -*- 72.
4. 92,738 ^ 21. 12. 99,988 -f- 42. 20. 273,579 -5- 81.
5. 100,000 + 24. is. 68,698 -*- 48. 21. 179,246 -*- 84.
6. 73,948 -5- 25. 14. 123,456 -^ 49. 22. 739,264 -r- 90.
7. 69,593-^23. is. 876,543-^-50. 23. 543,293 -f- 56.
8. 85,376 -v- 32. 16. 789,295 -r- 54. 24. 665,670 + 81.
EXERCISE 11
1. When sugar sells for 6 cents per pound, how many
pounds can be bought for 84 cents ?
2. If a boy walks at the rate of 3 miles per hour, how
long will it take him to walk 87 miles ?
3. In a peck there are 8 quarts. How many pecks are
there in 3000 quarts ?
4. If a bicyclist rides 9 miles an hour, how many hours
will it take him to go from St. Louis to Indianapolis, a
distance of 265 miles? After riding 19 hours, how far
from Indianapolis will he be ?
5. When coal costs 9 dollars a ton, how many tons can
be bought for 3456 dollars ?
6. A teacher receives a salary at the rate of 4 dollars a
day for every day he teaches. His yearly salary is 716
dollars. How many days are in the school year ?
7. A brick mason receives 4 dollars a day for every day
he works. How many days must he work to earn 900
dollars ?
18 ADVANCED BOOK OF ARITHMETIC
8. How many feet are there in 2,500 inches ?
9. How many weeks are there in 364 days ?
10. If a dozen penknives cost 9 dollars, how many dozen
penknives can be bought for 126 dollars ?
11. Plows cost 12 dollars a piece. How many can be
bought for 192 dollars ?
12. A box of oranges is worth 3 dollars. How many
such boxes can be bought for 111 dollars ?
13. When a barrel of pork sells for 12 dollars, how
many barrels must be sold to realize 5004 dollars ?
14. A section foreman rides on a velocipede at the rate
of 11 miles an hour. How long will it take him to go
from Cincinnati to Cleveland, a distance of 264 miles ?
15. Divide 1000 dollars among 8 persons, giving to each
the same sum of money.
16. A flock of sheep sells for 966 dollars. How many
sheep are in the flock, if each sheep sells for 6 dollars?
17. A man has 795 dollars in 5-dollar gold pieces. How
many coins has he ?
18. Hogs sell for 8 dollars apiece. At this price how
many can be purchased for 360 dollars ?
19. A box of soap is listed at 4 dollars. How many
such boxes can be purchased for 980 dollars ?
20. How many barrels of flour can be bought for 1002
dollars, when flour sells for 6 dollars a barrel ?
21. Oyster crackers cost 5 cents a pound. How many
pounds can be bought for 95 cents ?
22. By buying horses at 75 dollars each and selling
them at 84 dollars each, a jobber makes a profit of 324
dollars. How many does he sell ?
DIVISION 19
EXERCISE 12
1. How many bags of Rio coffee can be bought for
882 dollars, if one bag costs 21 dollars ?
2. Currants sell for 14 dollars a barrel ; at this price
how many barrels can be bought for 546 dollars ?
3. Granulated sugar is worth 16 dollars a barrel.
How many barrels must be sold to realize 1264 dollars ?
4. There are 36 inches in one yard. How many yards
are in 100,000 inches ?
5. There are 32 quarts in one bushel. How many
bushels are in 7712 quarts ?
6. How many days are in 3000 hours ?
7. A degree on a meridian of the earth's surface is 69
miles long. Two places on the same meridian are 2484
miles apart. How many degrees apart are they ?
8. How many square yards are in 3276 square feet ?
9. A gallon contains 231 cubic inches. How many
gallons are in a barrel containing 8316 cubic inches ?
10. Oolong tea costs 15 dollars a chest. How many
chests can be purchased for 495 dollars ?
11. A barrel of sugar weighs 325 pounds. How many
barrels are in 105,625 pounds ?
12. How long will it take a train, rate 30 miles an hour,
to go from New York to San Francisco, a distance of 3270
miles, if 5 hours are allowed for stops ?
13. There are 10 square chains in an acre. How many
acres are in 10,000 square chains ?
14. Rhode Island contains in round numbers 800,000
acres. Find its area in square miles. (640 acres = 1 square
mile.)
20 ADVANCED BOOK OF ARITHMETIC
EXERCISE 13
1. If land is worth $68 an acre, how many acres can
be bought for $4,624?
2. A tract of land is sold for $4,795.50 at the rate of
$69.50 an acre. How many acres are in the tract ?
3. When horses sell for $85.40 apiece, how many can
be bought for $36,465.80 ?
4. If the price of wheat is 76^ per bushel, how many
bushels can be bought for $4,043.20 ?
5. When cans of asparagus sell for 35^ each, how many
can be bought for $85.75 ?
6. If a pair of patent leather shoes sells for $3.85, how
many pairs must be sold to bring $1,482.25 ?
7. A clothier invests in men's trousers $392. 04. How
many does he buy, supposing each pair to cost $1.98 ?
8. If a keg of pickles cost $1.70, how many kegs can
be purchased for $28.90 ?
9. Chipped beef is bought at 17^ a pound. At this
rate, how many pounds can be bought for $361.25?
10. A 12-pound sack of flour retails at 45^. How
many sacks can be bought for $322.65?
11. When a can of sardines retails for 27^, how many
cans will $218. 70 buy?
12. A farmer gets for his apples $816.35 at the rate of
$1.45 per barrel. How many barrels does he sell ?
13. Corn is worth 45^ per bushel. How many bushels
can be bought for $7,876.35 ?
14. Oats are worth 36 ^ per bushel. How many bushels
can be bought for $142.56 ?
15. A share of railway stock is quoted at $78.50.
How many shares must be sold to realize $3,061.50 ?
DIVISION
21
EXERCISE 14
The following table gives the area and population of
some of the principal countries :
COUNTRY
1. Austria-Hungary
2. Belgium
3. Denmark
4. France
5. German Empire
6. Italy
7. Japan
8. Netherlands
9. Russia
10. Spain
11. United Kingdom
AREA IN SQ. Mi.
241,300
11,370
15,360
207,050
208,800
110,600
147,700
12,560
8,660,000
194,800
121,370
POPULATION
47,355,000
7,161,000
2,574,000
39,300,000
60,478,000
33,604,000
47,975,000
5,592,000
141,000,000
18,618,000
43,221,000
Find the population per square mile of each of the above
countries.
12. Reduce 1 to a decimal.
17.00000
2.83333
.40476
Reduce to decimals :
42 can be resolved into two factors, 6
and 7. The simplest way of dividing by
42 is to divide by the factors 6 and 7.
; H;
rife;
"• M; H; i
15- ft; W; A; IJ; iflh
Reduce to decimals, correct to four figures :
16- i 5 T ; i ; T\ ; iV > yV 5 i1! 5 iV 5 iV-
17* 25Y ; M ; A 5 FT 5 II 5 yV'
22
ADVANCED BOOK OF ARITHMETIC
SPECIMEN BILL
GALVESTON, TEXAS, Jan. 31, 1907.
MR. A. B. C.
In account with K. M. & CO.,
DEALERS IN
FURNITURE, CARPETS, RUGS, &c.
Jan.
2
3 Chairs @ $2.25
$6
75
10
1 Library Table
25
00
15
3 Rugs @ $6.75
20
25
20
40 yd. Matting @ 45 $
18
00
24
2 Wardrobes @ $17.50
35
00
$105
00
Paid
Feb. 1, 1903.
K. M. & CO.
Per M.
\
MR. P. Q. R.
DALLAS, TEXAS, Feb. 5, 1907.
Bought of M. R. S.,
RETAIL GROCERS.
Jan.
5
3 Ib. Tea @50$
$1
50
a
28 Ib. Sugar @5\$
1
4^
it
3pks. Potatoes @ 40$
1
20
((
7 Ib. Bacon @ 15$
1
05
10
8 Ib. Butter @ 35$
•2
80
12
3 cans Salmon @ 17 $
51
13
6 Ib. Sausage @ 12$
72
® o
$>y
25
Paid
Feb. 8.
M. E. S.
Per X.
DECIMALS 23
EXERCISE 15
Make out the following bills and receipt them :
1. Mr. John Rye bought of William Merchant,
12 yd. Calico ...... @ 9 1
15 yd. Sheeting ..... @ 7 ^
11 yd. Flannel ..... @ 35 f
2 Hats ....... @ 13.75
18 yd. Carpet ..... @ 75 1
3 Smyrna Rugs ..... @ 110.50
2. Mr. VJ. Hill bought of F. Warner & Co.,
5 Stoves ....... @ $6.50
3 doz. Knives ..... @ 14.80
2 Saws ....... @ $1.50
5 Iron Beds ...... @ 115.75
6 Wrenches ...... @ $1.25
3. H. Van Oppen bought of Hegel & Co.,
2 bu. Potatoes ..... @ $1.50
5 Ib. Tea ....... @ 75 ^
2 boxes Herring . . . . @ $1.95
25 Ib. Ham ...... @ 15 f
45 Ib. Sugar ...... @
4. Mr. James Kay bought of Simpson, Perdue & Co.,
50 Ib. Sugar ...... @ 4%f
15 cans Tomatoes . @ 13 ^
27 cans Corn ...... @ 16 ^
10 packages Breakfast Food . @ 12J ^
8 cans Salmon ..... @ 18 ^
5 gal. Maple Sirup . . . @ $1.30
25 Ib. Butter ...... @ 37^
61b. Y. H. Tea. . . . . © 75^
24 ADVANCED BOOK OF ARITHMETIC
MEASURES AND MULTIPLES
NOTE. Measures and Multiples in this book have reference only to
numbers which are both integral and abstract.
A number is prime, or is said to be a prime number,
when it is exactly divisible by only itself and unity.
Thus, 1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, etc., are prime
numbers, or prime factors.
A number which is exactly divisible by other numbers,
as well as by itself and unity, is called & composite
number.
Thus, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, etc., are compos-
ite numbers.
An even number is one which is exactly divisible by 2.
Thus, 4, 6, 8, 10, 12, etc., are even numbers.
An odd number is one which is not exactly divisible
by 2.
Thus, 3, 5, 7, 9, 11, etc., are odd numbers.
One number is said to be a measure of another number
when it is contained an exact number of times in that
other number.
Instead of the word " measure," factor, divisor, and sub-
multiple are often used. Example: 5 is a measure, factor,
divisor, or submultiple of 10, 15, 20, 25, etc.
By Greatest Common Measure of two or more numbers
is understood the greatest measure which these numbers
have in common.
Thus, 6 is the greatest common measure of 12, 18, 30.
Greatest Common Measure is usually designated by the
letters G. C. M. It is called also Greatest Common Divisor
(G. C.D.).
If a number measures each of two or more numbers, it
is said to be a common measure of those numbers.
SURES AND MULTIPLES
25
Thus, 2, 3, and 6 are common measures of 12, 18, 30.
Two or more numbers are prime to each other when they
have no common measure but 1.
Thus, 8 and 9 are prime to each other; so also are 15
and 28 prime to each other. These numbers, while prime
to each other, are not themselves prime numbers.
The result obtained by multiplying a number by an
integer is called a multiple of the number.
Thus, the multiples of 8 are 8, 16, 24, 32, 40, 48, etc.
The Least Common Multiple of two or more numbers is
the least number which is a multiple of each of the num-
bers. In other words, the Least Common Multiple of two
or more numbers is the least number which is exactly
divisible by each of the two or more numbers.
Least Common Multiple is denoted by L. C. M.
Example 1. What is the L. C. M. of 8 and 12 ?
Writing the multiples of 8 and 12, we have:
8 16 24 32 40 48 64 72 80
12 24 36 48 60 72 84
Notice that 24 is a common multiple of 8 and 12. So
also are 48 and 72 common multiples. The L. C. M. is 24.
Example 2. What is the L. C. M. of 12 and 18 ? Here
the second multiple of 18 contains 12 as a factor. Hence
2 times 18 is the L. C. M.
Example 3. A man buys two kinds of sugar dorie up
in 4-pound bags and in 5-pound bags. What is the least
number of pounds he can buy so as to have the same
number of pounds of each kind ?
Here the answer is obviously a multiple of 4 and 5.
The L. C. M. of 4 and 5 is 20. Hence he buys 20 pounds
of each kind.
26 ADVANCED BOOK OF ARITHMETIC
EXERCISE 16
1. A person has equal sums of money in dimes and in
25-cent pieces. Find the least amount he can have.
2. Fence posts in two fences are respectively 14 feet
and 21 feet apart. What is the smallest distance corre-
sponding to an exact number of feet in both fences ?
3. A man earns $4 a day. How many days must he
work so as to be paid in 10-dollar notes ?
4. If a person earns $6 a day, how many days must
he work to be paid in $ 20 bills ?
5. A housewife puts her flour into 10-pound and 6-
pound sacks, and has the same quantity in the 10-pound
sacks as she has in the 6-pound sacks. What is the least
quantity of flour she can have ?
6. A man buys two grades of sheep at $4 and $6
a head respectively. He spends the same amount in
the purchase of the two grades of sheep. What is the
smallest amount he can spend ? How many sheep can he
buy?
7. A boy buys oranges at 3^, 4^, 5^ apiece. He spends
the same amount on each kind of oranges. What is the
least amount he can spend on each kind ? How many
oranges does he buy ?
8. A person spends the same amount of money on
eggs at 15^ a dozen and at 20^ a dozen. What is the
smallest amount he can spend on each kind ?
9. Three bells toll at intervals of 4, 5, and 6 seconds
respectively. If they start at the same time, after how
many seconds will they toll again at the same instant ?
10. Find in feet and inches the least distance that will
be measured exactly by a 15-inch and an 18-inch rule. .
TESTS OF DIVISIBILITY 27
TESTS OF DIVISIBILITY
A number is exactly divisible by 2 when its units'
figure is exactly divisible by 2. Thus, 196 is exactly
divisible by 2 since 6 is divisible by 2.
A number is exactly divisible by 5 if its units' figure
is 5 or 0.
A number is exactly divisible by 3 when the sum of its
digits is exactly divisible by 3. Thus, 735 is exactly
divisible by 3 since the sum of its digits, 15, is exactly
divisible by 3.
A number is exactly divisible by 6 when it is even and
the sum of its digits is divisible by 3. Thus, 624 is ex-
actly divisible by 6 since it is an even number and the
sum of its digits, 12, is a multiple of 3.
A number is exactly divisible by 9 when the sum of
its digits is exactly divisible by 9. Thus, 765 is exactly
divisible by 9 because the sum of its digits, 18, is a multi-
ple of 9.
A number is exactly divisible by 11 when the differ-
ence between the sums of its digits in the even and odd
places is 0 or a multiple of 11. Thus, 94,853, is exactly
divisible by 11 since the difference between the sums 3 4-
8 + 9 and 5 -h 4 is a multiple of 11.
A number is exactly divisible by 25 when the number
formed by its two right-hand digits is exactly divisible
by 25. Thus, 1,275 is exactly divisible by 25 because 75
is exactly divisible by 25.
A number is exactly divisible by 8 when the number
formed by its three right-hand digits is exactly divisible
by 8. Thus, 19,256 is exactly divisible by 8 because 256
is divisible by 8.
The same rule holds for 125,
28 ADVANCED BOOK OF ARITHMETIC
NOTATION
22 is a short way of writing 2x2.
23 is a short way of writing 2x2x2.
24 is a short way of writing 2x2x2x2.
25 is a short way of writing 2x2x2x2x2.
The result of taking a number any number of times as
factor is called a power of the number. Thus, 74 = 7 x 7 x
7x7=2,401.
2,401 is the 4th power of 7.
The 4 written to the right of 7 and slightly above it is
called the index or exponent of the power.
Example l. Resolve 1,001 into its prime factors. 1,001
*s no^ Divisible by 2 because its units' figure is
not exactly divisible by 2. It is not divisible by
3 because the sum of the digits, 1, 1, is not
divisible by 3. It is not divisible by 5 because
its units' figure is not 0 or 5. 7 is contained in 1,001, 143
times. 143 is divisible by 11 because the sum of the
digits, 1, 3, equals 4, the digit in the even place. Hence
the prime factors of 1,001 are 7, 11, 13. Hence, 1,001 =
7 x 11 x 13.
Example 2. Resolve 5,040 into its prime factors, and
express 5,040 as the product of prime numbers.
5040
2520
Divide by 2 as often as possible. Since 315
1250 ends in 5, 5 is a factor of 315. Divide next by
3 as often as possible.
The prime factors of 5,040 are 2, 2, 2, 2, 3, 3, 5, 7,
5040 = 2x2x2x2x3x3x5x7
= 24 x 32 x 5 x 7.
630
63
-
PRIME FACTORS 29
EXERCISE 17
Resolve into prime factors and express each number as
the product of its prime factors :
1. 8, 12, 16, 18, 20, 24, 27, 28, 30, 32, 36, 39, 40, 42.
2. 45, 48, 49, 50, 56, 60, 65, 69, 72, 75, 77, 80, 84, 88,
92.
3. 98, 99, 111, 117, 119, 120, 124, 128, 132, 133, 135,
140, 144.
4. 240, 720, 343, 512, 216, 729, 736, 608, 544.
5. 1,331, 11,011, 1,309, 858, 1,274, 891, 3,575.
6. Write all the measures of each of the following
numbers: 36, 360, 200, 567, 576, 448.
7. Write all the common measures of: (a) 36, 24 ;
(6)18,27; (c?)48,72; (d)21,63; 0) 32, 96; (/) 18, 72.
When several numbers are to be taken as a whole and
made the subject of an operation, they are inclosed in a
sign, or symbol, known as a parenthesis, ( ). Thus, 3 + (9
— 2) signifies that 3 is to be added to the difference of 9
and 2. A number written immediately to the left of a
parenthesis denotes multiplication. Thus, 7x4+ 3(8
+ 5) means 7 times 4 is to be added to 3 times the sum
of 8 and 5.
A composite number can be resolved into only one set
of prime factors. Thus, the prime factors of 36 are 2, 2,
3, 3. 36 = 22 x 32. The product of no other prime num-
bers will give 36.
If a number is prime to each of two other numbers, it is
prime to their product.
ILLUSTRATION. If 7 is prime to 207 and to 8, then 7 is
prime to 8 x 207. For 7 does not appear among the
prime factors of the product.
30 ADVANCED BOOK OF ARITHMETIC
Example 1. Find the G. C. M. of 48, 120, 168.
Expressing these numbers as products of their prime
factors,
48 = 24 x 3.
120 = 23 x 3 x 5.
168 = 23 x 3 x 7.
2 is contained 3 times as a factor in 168, 3 times as a fac-
tor in 120, and 4 times as a factor in 48 ; 3 is contained
once as a factor in each of the numbers. Hence, the
G. C. M. =23X3 = 24.
To find the G. C. M. of two or more numbers, express
each of the numbers as the product of its prime factors,
then take the product of the prime factors common to all
the numbers, each factor being taken the least number of
times it occurs in any of the numbers.
EXERCISE 18
Find the G. C. M. of:
1. 16, 24. 12. 26, 117. 23. 64, 80, 96.
2. 24, 32. is. 57, 76. 24. 63, 84, 105.
3. 18, 27. 14. 115, 161. 25. 64, 96, 224.
4. 24, 36. 15. 144, 264. 26. 72, 108, 180.
5. 45, 60. 16. 140, 252. 27. 88, 132, 220.
6. 75, 90. 17. 20, 30, 40. 28. 126, 189, 252.
7. 54, 72. is. 30, 75, 105. 29. 144, 240, 336.
8. 108, 180. 19. 36, 60, 84. so. 162, 270, 378.
9. 84, 96. 20. 39, 65, 91. 31. 168, 224, 392.
10. 120, 156. 21. 60, 84, 132. 32. 252, 420, 588.
11. 91, 105. 22. 54, 90, 108. 33. 264, 360, 600.
LEAST COMMON MULTIPLE 31
LEAST COMMON MULTIPLE
Since the L. C. M. of two or more numbers is exactly
divisible by each of the numbers, it follows that the
L. C. M. contains all the prime factors of each of the given
numbers.
This fact suggests a method of finding the L. C. M. of
two or more numbers.
Example. Find the L. C. M. of 48, 60, 72.
48 = 24 x 3.
60 = 22x3x5.
72 = 23 x 32.
Any multiple of 48 must contain 2, 4 times as a factor.
Any multiple of 72 must contain 3 twice as a factor.
Hence, the number 24 x 32 x 5 = 720 contains all the fac-
tors of the three numbers 48, 60, 72. Therefore the
L. C. M. of 48, 60, 72, is 720.
To find the L. C. M. of two or more numbers, resolve
each of the numbers into its prime factors, then find the
product of all the prime factors of the given numbers,
taking each factor the greatest number of times it occurs
in any of the numbers.
Another Method
Example l. Find the L. C. M. of 48, 60, 72.
2
2
2
3
48
60
72
24
30
36
12
15
18
6
15
9
253
L. C. M. =3x5x2x3x2x2x2 = 720.
32 ADVANCED BOOK IN ARITHMETIC
Step 1. Arrange the numbers in a horizontal row.
Step 2. Divide by a prime factor common to two or
more of the numbers. Set down the quotients and the
undivided numbers.
Step 3. Treat the second horizontal row in the same
manner, and so on until a horizontal row is obtained
which contains numbers prime to one another. If, at any
stage of the process, a horizontal row contains a number
which is a factor of some other number in that row, then
strike out such factor.
The continued product of the numbers in the last row
and of the divisors will be the L. C. M.
EXERCISE 19
Find the L. C. M. of:
1. 16, 20. 16. 60, 96. 31. 15, 20, 25.
2. 21, 14. 17. 84, 108. 32. 12, 18, 20.
3. 18, 60. is. 55, 77. 33. 45, 63, 70.
4. 18, 45. 19. 54, 90. 34. 14, 35, 40.
5. 21, 49. 20. 72, 108. 35. 12, 16, 18.
6. 28, 70. 21. 75, 125. 36. 14, 24, 40.
7. 42, 56. 22. 36, 54, 72. 37. 15, 24, 25.
8. 36, 54. 23. 36, 90, 60. 38. 77, 143, 22.
9. 34, 51. 24. 48, 64, 36. 39. 18, 20, 45.
10. 48, 72. 25. 12, 15, 18. 40. 30, 70, 105.
11. 96, 120. 26. 14, 21, 35. 41. 30, 40, 48.
12. 28, 30. 27. 30, 35, 21. 42. 21, 28, 35.
13. 32, 80. 28. 28, 42, 70. 43. 12, 18, 27.
14. 26, 39. 29. 32, 35, 150. 44. 12, 15, 16, 18.
is. 48, 84. 30. 30, 45, 48. 45. 36, 40, 45.
FRACTIONS 33
FRACTIONS
If 23 be divided by 4, the process is indicated %£- ; the
quotient is 5-|. This means 4 is contained in 23, 5 times
and 3 remains to be divided by 4. From questions of this
character the term fraction (Latin fractus, broken in pieces)
has arisen. -^-, 5f, f, are all fractions.
A Fraction is an indicated Division.
For purposes of instruction fractions are regarded from
another point of view.
If the rectangle ABOD is divided into four equal parts
by lines having the same direction as AB, one of these
parts is called one fourth of the whole rectangle ; two
of the parts are called two fourths of the rectangle ;
three of the parts are called three fourths of the rec-
tangle ; and four of the parts are called four fourths
of the rectangle. In general, if any one thing is divided
into four equal parts, one of the parts is called a fourth ;
two of the parts are called two fourths ;
three of the parts, three fourths, etc.
Similarly, if any thing is divided into
five equal parts, one of the parts is
called one fifth ; two of the parts are
called two fifths ; three of the parts,
three fifths, etc.
In the above rectangle, if the line EF is drawn so as to
divide AB and CD each into two equal parts, the whole
figure will be broken up into eight rectangles; one of
these rectangles is one eighth of the whole ; two of them
are two eighths ; three of them, three eighths, etc. Divide
AE and also EB into three equal parts and draw through
the points of division lines parallel to BO. What part of
ABOD is one of the small rectangles ? two of them ? etc.
34 ADVANCED BOOK OF ARITHMETIC
NOTATION OF FRACTIONS
1 eighth is written -|. 2 eighths is written f .
3 eighths is written f . 4 eighths is written |.
5 eighths is written f . 6 eighths is written |, etc.
How many thirds are in 1 thing ?
How many fourths are in 1 thing ?
How many sevenths are in 1 thing ?
How many eighths are in 1 thing ?
How many tenths are in 1 thing ?
In the notation of fractions, what does the number be-
low the line indicate ?
What does the number above the line indicate ?
If a unit quantity is divided into any number of equal
parts, one of these parts is called a fractional unit, or unit
fraction.
A fraction is one fractional unit, or two or more fractional
units of the same denomination.
A fraction is expressed by two numbers, one number
being written above a horizontal line and the other num-
ber being written below the same horizontal line.
The number above the horizontal line is called the
numerator, because it numbers the parts taken, i.e. tells
how many fractional units there are.
The number below the line is called the denominator :
it names the fractional unit, and indicates how many frac-
tional units there are in the unit quantity from which the
fractional unit is derived.
Thus, | signifies the unit quantity is broken into 4
equal parts and 3 of these parts are taken. Here the
fractional unit is \ (one fourth) ; 3 is the numerator, and
4 is the denominator.
FRACTIONS 35
The numerator and denominator are called the terms of
the fraction.
A proper fraction is one whose numerator is less than
its denominator. Thus, £ is a proper fraction because 4
is less than 7.
An improper fraction is one whose numerator is greater
than or equal to its denominator. Examples : -^-, -£.
A mixed number is a number, part integral and part
fractional. Thus, 4| is a mixed number.
Read and explain what each represents :
*• i f < I' f ' I' I' f' I -¥-> ¥> fr *. l< f ' ¥-' f < J. *. V- f
f , f v-, ¥' |, l, |, |, V-. -¥-' iV A- H' ft-
Is ^ of 1 week equal to ^ of 2 weeks ?
Is |^ of 1 week equal to ^ of 3 weeks?
Is ^ of 1 week equal to ^ of 4 weeks?
Isf of $1 equal to i of $2?
Is | of $1 equal to £ of f 3?
Is | of $1 equal to 1 of $4?
Is | of 1 foot equal to \ of 3 feet ?
The above illustrations show that a fraction may be
read in two ways. For example, f may be read two fifths,
or two divided by five ; ^ may be read four sevenths, or
four divided by seven.
REDUCTION OF FRACTIONS
Is If feet = | of 1 foot ? Is 1| = | ?
Is f of 1 hour = 41 hours ? Is | = 4£ ?
Is -|- of an apple = | of an apple ? Is ^ = f ?
Changing the form of fractions without changing their
values is called reduction of fractions.
36 ADVANCED BOOK OP ARITHMETIC
EXERCISE 20
Reduce to integers or mixed numbers :
1. Jf. 10. -%°-. 19. -2f . 28. ff.
2. -\3-. 11. -£g9-. 20. f|. 29. -££.
3. \t. 12. 1|. 21. ||. 30. \°-.
4. ^. 13. If 22. ff. 31. If.
5. ^g1-. 14. -4^. 23. ff. 32. ||.
6. \2-. 15. 1£°. 24. If. 33. -Ug8--
7. ^-. 16. -1^°-. 25. ff. 34. If-.
8. ^°-. 17. \22°-' 26' if 35« H^'
9. Y-- 18' T|- 27> '/• 36- -W-
Reduce to an improper fraction 4|.
1 = 6 sixths.
4 = 24 sixths.
4| = 29 sixths = 2/-.
Another Method
What number divided by 6 gives 4 as a quotient and
5 as a remainder ?
The answer is 6 times 4 -f the remainder, 5. There-
fore, 4 1 = %£-.
EXERCISE 21
Reduce to improper fractions :
1. 3|. 7. 8|. 13. 19f 19. 201f 25.
2. 3|. 8. 5|. 14. 18|. 20. 16^f. 26.
3. 5f. 9. 9T\. 15. 27T\. 21. 14|. 27. 14|.
4. 9f. 10. IQlf 16. 38-^. 22. 15f. 28. 37f.
5. 9f. 11. 11TL. 17. 39f 23. 19|. 29. 44T\.
6. 10T\. 12. 10^. 18. 19f. 24. 291. 30.
FRACTIONS
37
How many squares are in the rectangle ABCD?
How many are in | of it ? in -J- of it ?
In | of it? in \ of it? in f of it?
In 1 of it? in \ of it? in \ of it?
555
In | of it? in \ of it? in f of it?
In TL of it? in ^ of it? in -^ of it?
In T9o of it? in ^ of it? in ^ of it?
How many squares are in -$-§ of the rectangle ABCD?
How many squares are in -| of the rectangle AB CD ?
How do the fractions f and -f$ compare ?
How may the fraction -f$ be obtained from ^ ?
How many squares are in % of the rectangle ABCD ?
How many squares are in J|- of the rectangle ABCD?
How do !"! and ^ compare ?
How may ^ be obtained from J-| ?
From the above rectangle can it be shown that :
The terms of a fraction may be multiplied or divided by the
same number and the value of the fraction remains unchanged.
38 ADVANCED BOOK OF ARITHMETIC
Example. Reduce f- to fourteenths.
f = -£%. The result is obtained by multiplying the terms
of f by 2.
EXERCISE 22
Reduce :
1. | to 9ths, to 15ths; to 24ths; to SOths; to 36ths.
2. f to 8ths; to 16ths; to 24ths; to 32ds; to 40ths.
3. | to lOths; to 20ths; to 25ths; to 35ths; to 40ths.
4. | to 14ths; to 21sts; to 28ths; to 35ths; to 49ths.
5. f to 16ths; to 24ths; to 40ths; to 48ths ; to 64ths.
6. f to 32ds ; to 48ths ; to 56ths ; to 72ds ; to SOths.
7. I to 18ths ; to 27ths ; to 45ths ; to 63ds ; to 72ds.
8. -^ to 20ths; to 50ths; to 70ths; to SOths; to
90ths.
9. ^ to 24ths; to 48ths; to 72ds; to 84ths; to 96ths.
10. 3 to 7ths ; to lOths ; to 12ths ; to 20ths.
11. 5 to 4ths; to 9ths; to 14ths; to 25ths.
12. 7 to 3ds ; to 8ths ; to lOths ; to 12ths.
Reduce f to 28ths. f = f|. Do this by multiplying
the terms of the fraction ^ by 7. Conversely, reduce
||^ to | by dividing the terms of the fraction |-^ by 7.
A fraction is said to be in its simplest form when its
terms are integers prime to each other.
A fraction, in its simplest form, is also said to be in its
lowest terms.
Example. Reduce to lowest terms
1^0 = If = yf = •£. Dividing the terms of the fraction
by 2, the result is |~|. Dividing the terms of this fraction
by 4, the result is if. Dividing the terms of ^| by 3, the
result is -|.
FRACTIONS 39
EXERCISE 23
Reduce to lowest terms :
i- T62< A> ii' If A' if I*. fi I f •
2. if. iMMMMMt' t£-
3. M, If, *f , If, f I, It If, i\V
*. Afe T4oV T63°5> tt< At At '«|, ffl*
ADDITION
Example l. Add J, J.
In adding fractions, select, as a matter of convenience,
the lowest denomination common to the fractions.
2 6 "
1—2
3~6-
Hence, i+i = | + | = |.
Example 2. Add |, |.
Here the lowest denomination common to both fractions
is 12ths. Notice, 12 is L. C. M. of 4, 6.
f = T92-
f-lf
Hence, f + f = & + {% = || = 1 A-
Example 3. Add 4^, 2f, 1^.
First add the fractions J, f , y5^. To do
12~ this, find the L. C. M. of the denominators.
: The L- c- M- °f 2' 6' i2' is 12-
the sum is f|- = | = lf.
Write | in the sum and carry 1. Add next the integers
4, 2, 1, and the 1 carried. The sum is 8f .
40 ADVANCED BOOK OF ARITHMETIC
To add fractions, first reduce them to equivalent fractions
having the same denominator. Then add the numerators,
and underneath the sum write the common denominator.
If the resulting fraction is not in its simplest form,
reduce it to its simplest form.
To add mixed numbers, first add the fractions, and to
this sum add the sum of the integers.
EXERCISE 24
Find the sum of :
1. 1 ,J,1. 15. J,J,i,ft. 29. f,^, f.
2. M,f. I*' M.J.&- 30. M.&.H-
3. l,i,f. 17. TVM- 31, 1J,2J,8|.
4. 1 f, |. 18. I|< f, f 32. 21 5J, 7&.
5. J, |, J_. 19. f,TVlf 33. 31, 21 2f
6. $,£,£. 20. f,T\,M- 34.
7. 1 ^i- 21. |,^,H- 35' 9t.7|,10
8. J5, |, J. 22. J, |, ft. 36. 10&, 9f,
9- f iJ- 23. i,|,TVH. 37. 91
10. f,i,iV 24. |,^, If 38.
11. iVi'f 25. J,J,Tk&. 39.
12. TV|,1 26. |,|,|. 40. 41,51,711.
13. |,f. 27. |,1,^. 41.
14. 5V,^,f. 28. |,|,TV 42.
43. A boy has $ £ and f |. What part of $ 1 has he ?
44. How much is | of an hour ? ^ of an hour ? ^ of an
hour?
45. Which is the largest and which the smallest of the
three fractions, f » f > f ?
FRACTIONS 41
SUBTRACTION
Example 1. From | take f .
_ is
— Tl
tf-tt-*
Example 2. From 17^ take 12|.
Reduce the fractions to 20ths. |-| cannot be taken
from £$. Take it from 12% ; that is, from — ^t — . Carry
^jU
I. 1 and 12 are 13 ; 13 and 4 are 17. The remainder is
4H-
EXERCISE 25
Find the value of :
1. f- i 15. 5-3f. 29. 31Jf-20TV.
2. | - i. 16. 6 - 4^. 30. 40f - 30|i.
3. f-1 17. 11 -f 31. 41 -If.
4. § - |. 18. 13 - |. 32. 7| - 3|.
5. &-J. 19. 14-51. 33. 91-5TV
6. | - f . 20. 15 - B-^. 34. 101 - 9f .
7. T%-f. 21. 28-21TV 35. 16|-9|.
8. 11 -|. 22. 17|-101. 36. 41-1^.
9. ^-f 23. ISf-llf 37. 9f-2TV
10. y-|. 24. 5|-2TV 38. 81-3-^.
II. if-f. 25. 29|-24f. 39. 9f-4li.
12. |-TV 26. 33f-171. 40. 18J-9f.
13. 3 - 11. 27. 9^ - 3f 41. 28J - 8|.
14. 4 -If. 28. 32||-30TV 42. 17i-92V
42 ADVANCED BOOK OF ARITHMETIC
43. What number must be added to 1| to make 7-|?
44. A man buys a suit of clothes for $12-| and gives 3
five-dollar bills in payment. How much change should
he receive?
45. From a piece of cloth containing 17f yards 14|
yards are sold. How many yards are left?
46. A boy buys two books costing $ f and $-|. How
much change should he get out of a $ 2^ gold piece ?
MULTIPLICATION OF FACTORS. CANCELLATION
Is 2x3x4x9= (2 x 3) (4 x 9) = (2 x 9) x (3 x 4) ?
The product of any number of factors, no matter how the
factors are grouped, is the same. This is the Associative
Law.
Is 5 x (2 x 3 x 4 x 9) == 10 x 3 x 4 x 9>= 2 x 15 x 4 x 9
= 2x3x20x9 = 2x3x4x45?
Is (2x3x4x9) -2 = 2x3x^2x9 = 3x4x9?
A continued product is multiplied by a number if one of
its factors is multiplied by the number.
A continued product is divided by a number if one of its
factors is divided by the number.
Cancellation is the shortening of the process of division
by dividing dividend and divisor by the same factor or
factors.
Find by cancellation the quotient :
18x27xl6 = 2x9x2
9x8x3 Ixlxl*
Dividing 9 into 18, 8 into 16, and 3 into 27, the
quotient is 2 x9 X 2 = 36.
Ixlxl
FRACTIONS 43
EXERCISE 26
Find by cancellation the quotient :
4 x 18 x 3 x 24 22 x 88 x 15
9 x 4 x 144 132 x 4
18 x 90 x 105 1760 x 99
2. — — — — — — — • 9.
14 x 25 x 3 4 x 88 x 165
27 x 64 x 8 5280 x 14
o. — ^^ — - — • 1O.
108 x 32 176 x 84
343 x 125 1728 x 34
35 x 35 27 x 136 x 8
16 x 16 x 8 x 81 640 x 5200
64 x 32 x 3 125 x 512 x 13
225 x 216 2380 x 104
o.
75 x 9 x 12 119 x 8 x 13
108 x 27 x 121 111 x 39 x 12
22 x 33 x 18 " 74 x 27 x 13 *
15. A farmer exchanged 320 acres of land worth $50
an acre for 25 city lots. Find the price of a lot.
(price of a lot) x 25 = $50 x 320.
Hence, price of lot = $5Qx32° = $640.
2o
16. How many cows at $40 a head cost as much as 15
horses at $64 a head?
17. How many dozen eggs at 35^ a dozen must be sold
to pay for 7 barrels of apples at $2.10 a barrel ?
18. A laborer receives $3.20 a day. How many days
must he work to pay for 6 tons of coal at $8 per ton ?
19. A bicyclist rides at the rate of 9 miles an hour.
How long will it take him to travel as far as a train goes
in 6 hours at the rate of 33 miles an hour ?
20. How many cattle at $42 a head must be sold to pay
for 11,200 bushels of wheat at 75 j* a bushel ?
44 ADVANCED BOOK OF ARITHMETIC
MULTIPLICATION
Multiplication of a fraction by an integer.
Example 1. Multiply f by 18.
I x 18 = 3 fourths x 18 = 54 fourths = *£ = 13| = 131-
Example 2. Multiply 3T3^ by 14.
3T3_ x 14 = (3 x 14) + (^ x 14) =42 + 4i = 46f
To multiply a mixed number by an integer, first multiply
the fractional part of the mixed number by the multiplier,
next multiply the integral part by the multiplier ; add
the two results for the final product.
EXERCISE 27
Find the value of :
1. I x 18. 9. T3¥ x 24. 17. If x 10. 25. 7T\ x 33.
2. | x 14. 10. T5¥ x 40. is. 1| x 12. 26. 8T\ x 19.
3. I x 7. 11, T9g x 34. 19. 3f x 10. 27. 9T52 x 80.
4. -& x 18. 12. 11 x 42. 20. 5^ x 54. 28. 6ii x 102.
5. T^ x 16. is. if x 44. 21. 6| x 36. 29. 7T7e x 60.
6. f X 12. 14. If X 50. 22. 91 X 16. 30. 11^ X 15.
7. |x!9. is. i£x45. 23. 7f x44. 31. 12^x18.
8. | X 28. 16. If X 9. 24. 10T*g X 24. 32. lOf X 13.
33. Find the cost of a dozen cans of baking powder at
37^ cents a can.
34. A pail of mackerel cost 21 dollars. Find the cost
of 20 pails.
35. Find the cost of a barrel of sugar weighing 325
pounds, if the cost per pound is 411/.
36. When starch sells for 3^ a pound, find the price
of 15 pounds of starch.
FRACTIONS 45
37. When wheat is 79| cents a bushel, how much will
164 bushels bring ?
38. Find the cost of a 75-pound chest of Hyson tea at
42| ^ per pound.
39. If a dozen cakes of yeast cost 42|^, find the cost of
9 dozen cakes of yeast.
40. Pepper sells for 14|^ a pound. Find the cost of 2
bags, each containing 120 pounds.
41. A pound package of chocolate costs 31J^. Find
the cost of 25 such packages.
42. A square rod equals 30^ square yards. Reduce 160
square rods to square yards.
43. A link of a surveyor's chain is 7|| inches. If the
chain contains 100 links, how many inches long is the
chain ?
44. When silk sells at $ |-J a yard, what is the cost of
14 yards of silk ?
45. A degree on a meridian of the earth's surface is
about 69^ miles. How many miles are in 15 degrees ?
in 40 degrees ? v
46. A person fails for $ 9,800. His creditors receive $|
on every dollar that he owes. How much in all do they
receive ?
47. A mass of copper and lead weighs 2,240 pounds; f
of the mass is copper. How much copper and how much
lead is in the mass ?
48. A man invests $2,300. At the end of a year his
gain is ^ of his investment. Find his gain and the value
of the investment at the end of the year.
46
ADVANCED BOOK OF ARITHMETIC
MULTIPLICATION OF A FRACTION BY A FRACTION
Multiplication of fractions extends the meaning of the
term "multiplication."
^ X f , or | of £, means 2 times ^ of -|.
•J X f , or | of |, means 3 times -| of J.
Example l. What is the area of a rectangle whose
length is |- of an inch and width | of an inch ?
D _ E C Take AB 1 inch. Let it be divided
into five equal parts. Construct upon
it a square ABCD. Divide AD into
three equal parts. Draw lines through
the points of division. KNED will
have for its dimensions |- of an inch
and J of an inch. By counting the
small rectangles in KNED the number
is found to be eight (4 x 2), and the number in ABCD
is fifteen (5x3). Hence, the area of KNED is T8^ of a
square inch.
1 of f = iV Therefore, 2 times J of | = ^. (Show
by figure.)
PBOCESS. x = £-'
Example 2. Find the area of a rectangle 3^ by 2T3^ feet.
To multiply a fraction by a fraction, take the product of
the numerators for the numerator of the product, and the
product of the denominators for the denominator of the
product.
To multiply mixed numbers, reduce them to improper
fractions, and then apply the rule for multiplication of
fractions.
FRACTIONS 47
EXERCISE 28
Multiply :
1. f f . 15. T%, ||, _V 29. 71 7J, ^.
2. f |. 16. ^, |f |J. 30. 3f 3f &.
3- |, if "• if T\°r If- 31. 51 5f T\.
*• f A- 18- ii & if- 32- 7I' 3^' &•
5. |, If 19- 1, i ^T- 33. If £, | .
6. £, J|. 20. |, f, Iff. 34. 7|, If ||.
7. 1|, ff 21. If, If. 35. 41, 4J, 3f .
8. H, Vft. 22. 1|, 1J, 1J. 36. If If f.
9. ^, J&. 23. 2f ^, f . 37. 20J, 20}.
10. If, |f. 24. 3|, T\, f 38. 4f 5f 1T3T.
U- M' *• 25- 2f' 2!< A- 39. 6|, If ^.
12. if, ^ |. 26. 7f 1|, If. 40. 2|, 1TV If
13. f, if A. 27. 2f, 3f, If 41. 2f, 8f,
14. f 15, |f 28. 41, 2|, |f 42. 6f 4|,
43. Find the value of 63| acres of land at $55^ an a,cre.
44. Find the cost of 12 J pounds of meat at 15^ cents
per pound.
45. Find the price of 4J bushels of wheat at 81^ cents
per bushel.
46. A speculator buys 10,000 bushels of wheat at 79|
cents per bushel and sells it when wheat is 81^ cents per
bushel. Find his profit.
47. Coal cost $9J a ton. Find the price of 5J tons.
48 ADVANCED BOOK OF ARITHMETIC
DIVISION AND RATIO
In division the product of two numbers and one of the
numbers are given, and the other number is sought.
Example. Divide 1^ by J.
f of the quotient = 1^ = -J^6-.
Therefore, | of the quotient = -|- of -1^6-.
Therefore, f of the quotient = | of ^6 .
Therefore, the quotient = f of -1/ = 2f .
Observe that -^ is divided by f by multiplying by |.
Hence the rule for division: Invert the terms of the
divisor and then proceed as in multiplication.
If the product of two numbers is unity, either is called
the reciprocal of the other.
ILLUSTRATIONS
7 x 7 = 1. The reciprocal of 7 is ^, and of ^- is 7.
| x f = 1. The reciprocal of 4J is |, and of | is 4J.
185 x ~V~ = !• The reciprocal of T8^ is -^-, and of -^5-, or 1|,
Hence the rule for division may be briefly stated :
Multiply the dividend by the reciprocal of the divisor.
By the ratio of one number to another number is meant
the quotient of the first number by the second. Thus the
ratio of 9 inches to 12 inches is 9 mches = | .
12 inches
The ratio of 9 inches to 12 inches is briefly indicated
9 in.: 12 in.
Example. Find the value of the ratio 4 days : 7^ hours.
4 days = 4 x 24 hours = 96 hours.
96 hours -f- 7-J hours = 96 -?- 7£ =12.8.
FRACTIONS 49
EXERCISE 29
Divide:
1.
if by 27.
25.
12by|.
49.
33 by ^T.
2.
1 by A-
26.
9A by 45.
50.
7T93 by 80.
3.
1 by TV
27.
4* by 9.
51.
1 by 7f
4.
f o by 16.
28.
14 by f
52.
40 by ^«
5.
3f by 10.
29.
16| by 63.
53.
5lf by 50.
6.
T3obyTV
30.
4f by If.
54.
1 by 9f .
7.
& by 12.
31.
16 by f
55.
60 by f .
8.
21 by 3.
32.
181 by 26.
56.
8| by 46.
9.
A by if
33.
8f by 2f-.
57.
lOf by 2f .
10.
if by 16.
34.
18 by f
58.
11 by |.
11.
3f by 8.
35.
17f by 75.
59.
A by f .
12.
fbyf
36.
4| by 9|,
60.
14| by 111.
13.
it by 15.
37.
21 by f.
61.
Ibyi.
14.
4T\ by 14.
38.
lOf by 48.
62.
ibyTV
15.
T9*byTV
39.
H by 31
63.
Mbyi^.
16.
ii by 25.
40.
25 by f .
64.
Ibyf.
17.
5 ft by 21.
41.
9f by 52.
65.
T9oby^.
18.
A*y&-
42.
4TV by 17f
66.
i|bylT\.
19.
IA by 21.
43.
26 by f
67.
ibyA-
20.
9^ by 46.
44.
9^ by 75.
68.
ibyf.
21.
A by if
45.
3*byf
69.
9JT by 111
22.
M by 36.
46.
26 by if.
70.
lby,V
23.
9^by77.
47.
81 by 15.
71.
fbyf
24.
22lf by 8.
48.
1 by 4|.
72.
7{ by 41
50 ADVANCED BOOK OF ARITHMETIC
EXERCISE 30
1. What is the ratio of 6 inches to 12 inches?
2. What is the ratio of 1 foot to 1 yard ?
3. What is the ratio of 1 square foot to 1 square yard ?
4. There are 30| square yards in 1 square rod. What
is the ratio of 1 square yard to 1 square rod ?
5. What is the ratio of 3 weeks to 10 days ?
6. What is the ratio of 1 hour to 1 minute ?
7. What is the ratio of 4 days to 15 hours? of a
minute to an hour ?
8. What is the ratio of 325 to 100 ?
Find the values of the following ratios:
9. 2:J. 15. 6:f. 21. 3f:2|. 27. lOf : 71.
10. 3:1. i6. 2l:3f. 22. 9|:31J. 28. 7^: 6J.
11. 4 : \. 17. 7-| : 21|. 23. 5J : 7f . 29. 41 : 7 \.
12. 4 : f . 18. 5| : 4|. 24. 4£ : 21. 30. 7| : 7^.
13. 5 : 1. 19. 9| : 7£. 25. 51 : 11. 31. 81 : If.
14. 7:f. 20. 3f:5f. 26. 61:31. 32. 6-/T : 41.
Example l. Find the price of 2,000 pounds of wheat at
84 / a bushel.
To solve this question there are two steps to take.
Step 1. Find the number of bushels by dividing the
number of pounds by the number of pounds in one bushel.
Step 2. Multiply the price of one bushel by the num-
ber of bushels.
SOLUTION. Number of bushels =
60
14
Price of the wheat » 84X x
*
60
FRACTIONS 51
Example 2. Three fifths of a man's money is $2,437.
How much money has he?
| of his money = 12-437.
I of his money = ^|^I Or £ of $2437.
f of his money = x 5 or | of $2437.
3
Hence, his money = $4,061.67 (to nearest cent).
This method of solving a problem is known as the
analytical method. It is called also the unit method,
because the value of the unit of the quantity under con-
sideration is first sought and from this the value of any
number of units is then obtained.
Note that the answer is obtained by multiplying $ 2437
by f , the reciprocal of -|. In this problem there are given
the product of two factors and one of the factors. The
other factor is sought. The problem is therefore one of
division.
EXERCISE 31
1. Find the price of 78 acres of land if 25 acres are
worth $1,375.
2. When 18 pounds of sugar sell for $1, find the cost
of 45 pounds.
3. When 7 bushels of wheat sell for $5.95, how much
can a person get for 255 bushels ?
4. If 5 bushels of barley sell for 12, how much will
343 bushels sell for ?
5. If 6 barrels of flour are sold for $45, at this rate
how much will 84 barrels sell for ?
52 ADVANCED BOOK OF ARITHMETIC
6. Seven barrels of pork sell for $80.50. Find the
cost of 50 barrels of pork.
7. Nine barrels of salt cost $11.70. Find the cost of
19 barrels of salt.
8. Eleven bushels of oats are sold for $4.51. Find
the value of 168 bushels.
9. Six barrels of lard bring $115. How much will
46 barrels bring ?
10. When 7 yards of sheeting cost 50 fa find how much
must be paid for 98 yards.
11. Six yards of cambric sell for 75^. How much must
be given for 34 yards of cambric ?
12. Four yards of flannel cost $1.16. How much will
29 yards of flannel cost ?
13. Eight yards of gingham cost 60 ^. How much will
103 yards cost ?
14. Nine yards of cotton fabric cost 75^. How much
will 69 yards cost ?
15. Six yards of cotton cheviot cost $1. How much
will 81 yards cost ?
16. Five-eighths of a man's money is $75. How much
money has he ?
17. Three-fourths of the length of a pole is 81 feet.
Find the length of the pole.
18. The eighth and the twelfth of a number are 15.
What is the number ?
19. A dealer sold -| of his coal and had 170 tons left.
How many tons had he at first ?
20. The fourth part and the sixth part of a number are
25. What is the number ?
DECIMALS 53
DECIMALS
It is well to fix in mind the following facts:
Tenths occupy the first place to the right of the decimal
point ; hundredths, the second place ; thousandths, the
third place ; ten-thousandths, the fourth place ; hundred-
thousandths, the fifth place ; millionths, the sixth place.
Read 22.234. Twenty-two and two hundred thirty -four
thousandths.
Write twenty-four tenths.
Write 24 as if it were an integer. Tenths occupy the
first place to the right of the decimal point. Hence, 24
tenths is written 2.4.
Write 2,304 hundredths.
Write 2,304 as if it were an integer. Beginning at the
right, point off two places for hundredths. Hence, 2,304
hundredths is written 23.04. If ^ff^- be reduced to a
mixed number, it becomes 23^^ ; that is, 23.04.
Write 11 hundred-thousandths.
Write 11 as if it were an integer. Beginning at
the right, point off five places for hundred-thousandths.
Hence, 11 hundred-thousandths is written .00011. Ob-
serve that places having no digits are filled in with
ciphers.
Write five hundred and five thousandths.
First write five hundred, and then write five thousandths.
Hence, five hundred and five thousandths is written
500.005.
Write seven hundred eight thousandths.
Here the number of units is 708; the denomination is
thousandths. As thousandths occupy the third place to
the right of the decimal point, hence 708 thousandths is
written .708.
54 ADVANCED BOOK OF ARITHMETIC
MULTIPLICATION AND DIVISION BY POWERS OF TEN
Consider the two numbers,
(«) 320.12,
(6) 3,201.2.
Both are expressed by the same figures written in the
same order. The number (5) can be obtained from the
number (a) by moving each figure one place to the left.
But moving a digit one place to the left makes its value
ten times as great, and, hence, moving several digits each
one place to the left makes the number they represent ten
times as great.
The number (5) can also be obtained from (a) by mov-
ing the decimal point in (a) one place to the right. Also
(a) can be obtained from (5) by moving the decimal point
in (5) one place to the left.
To multiply a number by 10, move the decimal point in
the number one place to the right.
To divide a number by 10, move the decimal point in the
number one place to the left.
Consider the numbers,
(a) 320.12,
(J) 32,012.
The number (5) is obtained from (a) by moving each
digit in (a) two places to the left. This multiplies each
digit by 100.
(6) may also be obtained from (a) by moving the deci-
mal point in (a) two places to the right; also (a) from
(6) by moving the decimal point two places to the left.
To multiply a number by 100, move the decimal point in
the number two places to the right.
DECIMALS 55
To divide by 100, move the decimal point in the dividend
two places to the left.
Consider the numbers,
(a) 320.12,
(6) 320,120.
(b) is here obtained from (a) by moving each digit in
(a) three places to the left. It can also be obtained from
(#) by moving the decimal point in (a) three places to
the right.
To multiply a number by 1,000, move the decimal point in
the number three places to the right.
To divide a number by 1,000, move the decimal point in
the number three places to the left.
The rules for multiplying by 10,000, 100,000, are left
for the reader to determine.
Example 1. Multiply 86.4 by 10,000. Moving the
decimal point four places to the right, the number be-
comes 864,000.
Example 2. Divide 12.3 by 100,000. Moving the deci-
mal point five places to the left, the number becomes
.000123.
EXERCISE 32
Multiply by 10 :
1. 120, 14.2, .1431, .00012, 1.7320, .01234.
Multiply by 100 :
2. 173, 172.8, 19.23, .001237, 8,654, 17.1.
Multiply by 1,000 :
3. 1156, 32.5, 7.123, .93891, .01275, .00011.
56 ADVANCED BOOK OF ARITHMETIC
Multiply by 10,000 :
4. 345, 34.25, 5.1739, 6.001, .01793, .12.
5. Divide each of the following numbers by 10; by
100; by 1,000; by 10,000 ; by 100,000 :
32,734
9,285.
773
3,745.3
325.
298
928.49
127
72,173.5
325
12.792
17
99,999.9
18.
326
3,728.3
7.
294
12.7564
670
1,201
1,000
3,450
7,100
Find the values of the
following ratios :
6.
22.3 : .223.
22.
.001:
10.
7.
3.74 : .374.
23.
.005 :
100.
8.
173.2 : 1.732.
24.
9.265
: 926.5.
9.
7.3 : .073.
25.
12.325: 1,232.5.
10.
1.25 : .0125.
26.
1.534
: 153.4.
11.
9.28 : .00928.
27.
1,001
: .1001.
12.
11.34 : .01134.
28.
54 : .054.
13.
7.04 : .0704.
29.
792 :
.0792.
14.
100 : .01.
30.
113:
.0113.
15.
1,000 : .001.
31.
79.28
: .7928.
16.
.012 : .12.
32.
6.45 :
6,450.
17.
1.24 : 124.
33.
99.29
: 99,290.
18.
9.53 : 9,530.
34.
7.35 :
73,500.
19.
7.1 : 7,100.
35.
9.24 :
92,400.
20.
6.5 : 65,000.
36.
8.123
: .008123.
21.
11.79 : 11,790.
37.
.04567 : 45.670.
DECIMALS 57
ADDITION"
Find the sum of 3.4, 2.38, 5.005, 6.2374, 11.1.
3.4 Write the numbers so that units of the same
2.38 denomination stand in the same vertical column.
5.005 Then add as integers are added.
6.2374 Write the decimal point in the sum in the same
11.1 vertical line with the decimal points in the
28.1224 addends.
EXERCISE 33
Add:
1. 2.2, .025, 37.3, 5.284, 6.294, 538.1, 77.77.
2. 3.5, 7.12, .339, 47.35, 39.28, .123, 54.275.
3. 9.28, 11.18, .999, 39.28, 7.451, 94.354, 98.76.
4. 12.49, 1.492, 38.75, 53.41, 98.69, 845.5, 892.9.
5. .009, 5.976, 40.99, 6.385, 9.278, 8.239, 64.271.
6. .098, 9.853, 19.47, 17.392, 28.394, 8.01, 77.47.
7. .285, 11.95, 29.99, 94.931, 1.732, 64.6, 78.75.
8. 11.4, 17.5, 99.37, 15.273, 9.394, 71.3, 92.95.
9. 1.21, 12.1, .121, 8.295, 7.777, 68.7, 78.28.
10. 15.9, 9.158, 91.58, 9.158, 2.293, 84.5, .139.
11. 98.5, 11.667, 66.66, 8.394, 9.928, 76.8, 9.359.
12. 77.8, 88.88, 99.99, 6.325, 7.384, 94.9, 1.798.
SUBTRACTION
Find the difference between 4,001 and 1.7003.
Arrange the numbers so that units of the
4001.0000 same denomination stand in the same vertical
1.7003 column. Ciphers may be inserted after the
3999.2997 decimal point in the minuend. Proceed next
as in the subtraction of integers.
58 ADVANCED BOOK OF ARITHMETIC
EXERCISE 34
Find the remainder and verify your answer in each
case:
1. 7.73-6.78. 14. 10.1-7.325.
2. 9.29-3.47. 15. 9.24-5.3481.
3. 6.34-1.95. 16. 8.73-4.4444.
4. 9.82-7.78. 17. 12.32-5.6741.
5. 7.45-3.59. 18. 19.33-6.2734.
6. 10.71-7.79. 19. 9.271-4.3847.
7. 8.94-3.95. 20. 3.213 -.9875.
8. 5.012-2.9. 21. 4.321 -.73201.
9. 10.943-7.97. 22. 5.204-1.3256.
10. 8.325-4.378. 23. 8.731-5.4557.
11. 8.924-5.938. 24. 9.21-7.2349.
12. 7.312-2.7. 25. 7.29-3.4551.
13. 9.419 - 5.57. 26. 6.001 - 5.112.
27. From seven hundred four thousandths take two
hundred five ten-thousandths.
28. From five hundred ten thousandths take five hun-
dred ten-thousandths.
29. From two thousand take two thousandths.
30. How much does one thousandth exceed one hundred-
thousandth ?
31. Find the difference between a hundred and a hun-
dredth.
32. From 39 tenths take 39 thousandths.
33. From 100 hundredths take 100 ten-thousandths.
34. How much must be added to one and five-tenths to
make ten ?
DECIMALS 59
35. By how much does 175 hundredths exceed 175
hundred-thousandths? What is the ratio of the first
number to the second?
36. By how much does $1 exceed 1 mill ?
37. By how much does $2 exceed 15 mills?
MULTIPLICATION
Example l. Multiply 3.23 by 25.
3.23 = 323 hundredths.
323 hundredths x 25 = 8075 hundredths = 80.75.
Example 2. Multiply 3.23 by .25.
Since the multiplier is y^g- of 25, the product 3.23 x
.25 =1^ of 3.23 x 25. ^ of 80.75= .8075.
The mechanical work of multiplying may be performed
as follows :
3 23
Multiply as if both numbers were integers, and
' point off in the product, commencing at the right,
as many places as there are decimal places in both
.8075
multiplicand and multiplier.
Example 3. Multiply .32 by .018.
Point off five places.
Another ^Explanation
x TUo = nfiflW = -00576-
To square a number means to multiply the number by
itself or to take the number twice as a factor.
To cube a number means to take the number three times
as a factor.
60 ADVANCED .BOOK OF ARITHMETIC
EXERCISE 35
1. Find .04 of $108; .05 of $274; .06 of $720; .07 of
1144.
2. Find .09 of $34.50; .3 of $75.30; .08 of $75.80;
.07 of $84.70.
3. Find .4 of $29.75; .5 of $69.48; .6 of $68.32; .1
of $328.50.
4. Find .125 of $80.80; .75 of $54; .6 of $300.50;
.25 of $98.84.
5. Find .625 of $688; .875 of $792.80; .375 of
$900.80.
6. Find .375 of 84 acres; .0625 of 64 acres; .3125 of
96 acres.
7. Find .1 of .1; .3 of .4; .3 of .3; .01 of .2 ; .01 of 1.2.
8. Multiply 27.9 by 18. 23. 1.18 x .1695 = ?
9. Multiply 1,327 by 1.6. 24. .97 x .97 = ?
10. Multiply 3,927 by .46. 25. .68 x .68 = ?
11. Multiply 120.01 by 3.6. 26. .373 x .373 = ?
12. Multiply 25 by .017. 27. .901 x .901 = ?
13. Multiply 37.5 by .07. 28. .803 x .803 = ?
14. Multiply 11.9 by 2.4. 29. .693 x .693 = ?
is. Multiply 182.54 by 1.49. 30. .1 x .1 x .1 = ?
16. Multiply .286 by 1.96. 31. .3 x .3 x .3 = ?
17. Multiply 92.24 by 2.7. 32. .4 x .4 x .4 = ?
is. .148x1.15 = ? 33. .7x.7x.7 = ?
19. .82x.51 = ? 34. 1.04x1.04x1.04 = ?
20. 1.875 x. 32=? 35. 1.06x1.06x1.06 = ?
21. 1.78x1.89 = ? 36. 1.08x1.08x1.08 = ?
22. 18.24 x. 95 = ? 37. .25x.25x .25 = ?
DECIMALS 61
38. .7645 of the asphalt found in West Virginia is com-
posed of carbon, .0783 is hydrogen, .1346 is oxygen, and
the remainder is ash. How much of each constituent is
in 254 tons of asphalt ? Check your answers.
39. .7217 of the asphalt found in Oregon is composed
of carbon, .079 of hydrogen, .1461 of oxygen, and the
remainder of ash. Find the amount of each in 385 tons
of asphalt. Check your answer.
40. Multiply the square of 14 by .7854.
41. The area of the surface of a sphere is obtained by
multiplying the square of the diameter by 3.1416. Find
the area of the surface of the earth, taking the diameter
to be 7,920 miles. Compare your answer with the area
given in your geography.
42. The moon is nearly 2,200 miles in diameter. Find
the area of its surface in square miles.
43. The velocity of the earth in its orbit is 18.5 miles
per second. How far does it go in 1 minute? in 1
hour?
44. A hurricane moves at the rate of 146.6 feet per
second. How far does it travel in 1 minute ? in 1 hour ?
45. One meter = 39.37 inches. Find in inches the dif-
ference between 64 meters and 70 yards.
DIVISION
Before undertaking Division, it may be well to lay
stress on the fact that numbers in the decimal system of
notation may be read in many ways. Thus, 32.25 may
be read, (a) 32 and 25 hundredths ; (5) 3,225 hundredths ;
O) 32,250 thousandths; (d) 322,500 ten-thousandths;
(e) 322.5 tenths; (/) 3.225 tens.
62 ADVANCED BOOK OF ARITHMETIC
Example l. Divide 1.293 by 8.
8")1 293000 ^ *nt° ^ tenths gives 1 tenth, with a re-
161625 mainder 4 tenths. 4 tenths = 40 hundredths ;
40 hundredths and 9 hundredths = 49 hun-
dredths. 8 into 49 hundredths gives 6 hundredths, with
a remainder 1 hundredth. Change 1 hundredth to thou-
sandths, and proceed as before.
Example 2. Divide .01234 by 4.
4). 012340 The work calls for no explanation.
.003085
EXERCISE 36
Divide:
1. 73.21 by 8. 9. 8.218 by 7. 17. 5.472 by 6.
2. 3.45 by 4. 10. 3.942 by 6. is. 8.2548 by 9.
3. 19.362 by 6. ii. 6.475 by 7. 19. .34794 by 9.
4. 1.791 by 9. 12. 9.143 by 8. 20. .67356 by 9.
5. 4.564 by 5. 13. .1234 by 5. 21. .999999 by 7.
6. 3.927 by 8. 14. .73206 by 6. 22. 7.3745 by 7.
7. .015 by 5. 15. 1.1466 by 7. 23. 6.2676 by 6.
8. 8.846 by 6. 16. 6.2751 by 8. 24. 1.7346 by 7.
Find the difference between .07858 and .078; also
find the difference between .07858 and .079.
Hence .07858 is nearer to .079 than
.07858 .07900 it is to .078. If, therefore, one is
.078 .07858 asked to give the value of .07858
.00058 .00042 correct to three figures, write for an-
swer .079.
Express .73948 correct to three figures. Ans. .739.
Express .25764 correct to three figures. Ans. .258.
Whenever asked to give a decimal correct to any num-
ber of figures, discard the remaining figures if the first
DECIMALS 63
one of them is less than 5; if it is 5 or more than 5, in-
crease the last figure by 1.
Example l. Divide .0732 by .8.
Make the divisor an integer by moving the decimal
point one place to the right. Make a corresponding
change in the dividend. This change is equivalent to
multiplying divisor and dividend by 10.
8). 7320
.0915
Example 2. Divide 12 by .125.
Move the decimal point in the divisor and in the
dividend three places to the right, i.e. multiply each by
i'000' 96
125)12000
1125
750
750
Example 3. Divide 3.274 by 6.25.
.523+
625)327.400
3125
1490
1250
2400
1875
525
"Whenever the divisor is a decimal, make it an integer by
moving the decimal point to the right. Make a corresponding
change in the dividend. After doing this, proceed in exactly
the same manner as in long division of integers. Write the
decimal point in the quotient in the same vertical line -with
the decimal point in the dividend transformed.
64 ADVANCED BOOK OF ARITHMETIC
EXERCISE 37
Divide:
1. 2.34 by .8. 26. 5 by .004.
2. .012 by .5. 27. .1 by .0001.
3. 3.475 by .4. 28. .04 by .0008.
4. 1.2348 by .6. 29. .32 by .00128.
5. .1798 by .5. 30. .45 by .0018.
6. 3.144 by 1.2. 31. .078 by .00312.
7. 5.96 by 1.6. 32. .067 by .0268.
8. 3.2903 by 1.3. 33. .01 by .8.
9. .27 by .2. , 34. .002 by 1.6.
10. 5.376 by 1.6. 35. .018 by 45.
11. 9.4851 by 1.5. 36. .54 by 81.
12. 3.2 by 6.4. 37. .243 by 1.944.
13. 20 by .5. 38. .216 by 1.44.
14. 10 by .16. 39. 5.12 by .16.
is. 40 by .32. 40. 7.29 by 270.
16. 56 by 1.12. 41. 34.7231 by .713.
17. 84 by 5.6. 42. 31.8791 by 3.97.
is. 392 by 7.84. 43. .267584 by 2.96.
19. 100 by .625. 44. .348336 by .492.
20. 100 by .008. 45. .190256 by .188.
21. 400 by .05. 46. 59.4204 by 5,860.
22. 144 by .288. 47. 55.9911 by 108.3.
23. 15.4 by .616. 48. .575484 by 54.6.
24. .096 by .192. 49. .461071 by 122.3.
25. 1 by .001. 50. 4.50775 by 123.5,
DECIMALS 65
EXERCISE 38
The mileage and valuation by counties in Texas of the
St. Louis and San Francisco Railway as given by the Texas
Railroad Commission for the year 1906 are as follows:
COUNTY MILEAGE VALUATION
1. Collin 19.51 $346,538.13
2. Dallas 2.7 53,300.16
3. Denton 9.99 188,311.64
4. Grayson 27.44 843,427.59
5. Hardeman 8.68 183,997.77
6. Tarrant 4.56 191,208.29
7. Wilbarger 12.77 192,843.01
Find the valuation per mile in each of the above counties.
8. On July 16, 1907, a contract for paving Broadway,
Denver, Colorado, was awarded on the following itemized
specifications and prices:
3,050 ft. 6" x 18" stone curb @ $ 1.05*
2,750 yd. brick gutter @ $ 2.25
22,900 yd. street asphalt pave-
ment @ $ 2.25
704 ft. oak header @ $ .50
945 ft. 27" pipe sewer @ $ 2.40
580 ft. 24" pipe sewer @ $ 2.00
580 ft. 21" pipe sewer @ $ 1.75
580 ft. 15" pipe sewer @ $ 1.10
398 ft. 12" pipe sewer @ $ .86
516 ft. 10" pipe sewer @ $ .75
12 manholes @ $45.00
17 catch basins @ 165.00
10 M ft. lumber @ $30.00
Find the total cost.
* 6" x 18'' means 6 inches by 18 inches.
66 ADVANCED BOOK OF ARITHMETIC
REDUCTION OF FRACTIONS TO DECIMALS AND REDUC-
TION OF DECIMALS TO FRACTIONS
Example l. Reduce -| to a decimal.
8)7.000
.875
Example 2. Reduce -fa to a decimal.
11)7.00000
.63636+
Example 3. Reduce -£$fa to a decimal.
=.016125.
Divide numerator and denominator by 1,000 by moving
the decimal point three places to the left; then divide the
numerator by 8.
A fraction in its lowest terms having for denominator a
number whose prime factors are 2's or 5's or both can always
be exactly expressed as a decimal.
A fraction in its lowest terms having for denominator a
number containing prime factors other than 2's and 5's will
give rise to a decimal which never terminates.
EXERCISE 39
Reduce to decimals:
1 3 5 JL _9_ 11 13 15 3_
*• 8' ¥' 16' 16' 16' 16' 16' 16*
2- A' A' ii- it' if' lV ii-
3- T30> iVk T^OU' TITOO
Q 11 9Q 1Q Q*7 Q1 ^Q
4' ¥0' H' ft' 2-0' f 0' 60' It'
e- f f iV' ^ ft' A' ii' if-
7< t' it' ill' 94090'
DECIMALS 67
Example l. Reduce .0625 to a common fraction. .0625
is read 625 ten-thousandths ; ^ff f ^ is read in the same
way.
•0625 = -
EXERCISE 40
Reduce to common fractions:
1. .3, .8, .25, .125, .1875.
2. .07, .0125, .00875, .0625, .0075.
3. .009, .0225, .1125, .0275.
4. .072, .0104, .035, .0119, .0375.
5. .144, .0504, .0768, .162, .0112.
6. .288, .0176, .0325, .0175, .425.
7. .2875, .3375, .5125, .7375.
EXERCISE 41
1. A man walks 3 miles an hour. At this rate, how
long will it take him to walk 12 miles ?
2. A train goes 25 miles an hour. How long will it
take it to go 300 miles at this rate ?
3. A bicyclist travels at the rate of 9 miles an hour.
How long will it take him to go 60 miles ?
4. How would you find the time to go any given dis-
tance, if you knew the distance gone in a unit of time ?
5. A man walks 3.5 miles an hour. At this rate, how
long would it take him to go 49 miles ?
6. The distance from London to Glasgow is 401.5
miles. An express train goes this distance in 8 hours.
Find its rate per hour.
7. From London to Edinburgh is 393.5 miles. The
daily mail train takes 7.75 hours to go this distance.
Find its rate per hour.
I 7
68 ADVANCED BOOK OF ARITHMETIC
8. The Empire State Express goes from New York
City to Buffalo, a distance of 440 miles, in 8.25 hours.
Find its rate per hour.
9. The mail train from Paris to Bayonne goes 486.25
miles in 8.983 hours. Find its rate per hour.
10. The distance from New York City to Cleveland is
568 miles. A train goes this distance in 19.5 hours.
Find its average speed.
11. A steamer goes from New York City to Bremen, a
distance of 4235 miles, in 7.75 days. Find its rate per
day. Also its rate per hour.
12. The earth moves in its orbit at the rate of 1110
miles a minute. How many times faster does the earth
move than a train which goes 54 miles an hour ?
13. A city lot is worth $1800. If this sum is .75 of
the value of the house on it, what is the value of the house ?
14. If .7 of a sum of money is $ 196, what is the sum of
money ?
15. Cast iron is 7.2 times as heavy as water. How
many cubic feet of cast iron weigh as much as 6120 cubic
feet of water ?
16. Coal is 1.3 times as heavy as water. How many
cubic feet of coal weigh as much as 546 cubic feet of water?
17. There are 231 cubic inches in a gallon. How many
gallons are in 1 cubic foot ? (1 cu. ft. = 1728 cu. in.)
18. If 2000 pounds of coal cost 18.75, find the price of
8750 pounds of this kind of coal.
19. If 3.5 yards of cloth cost $12.25, find the price of
7.5 yards of this cloth.
20. If 1.6 yards of velvet cost $2.88, find the price of
9.75 yards of velvet.
FRACTIONS 69
EXERCISE 42
1. What fraction of a yard is 1 foot? What fraction
of a yard is 2 feet ?
2. What fraction of 1 foot is 1 inch? 3 inches?
4 inches ? 5 inches ? 7 inches ? 8 inches ? 9 inches ? 10
inches ?
3. What fraction of 1 yard is 1 inch ? What fraction
of a yard is 2 inches ? 3 inches ? 4 inches ? 5 inches ? 6
inches ? 9 inches ? 12 inches ? 16 inches ? 17 inches ? 19
inches ? 24 inches ? 27 inches ?
4. There are 8 quarts in 1 peck. What fraction of
a peck is 1 quart ? What fraction of a peck is 2 quarts ?
3 quarts ? 4 quarts ? 5 quarts ? 6 quarts ?
5. What fraction of a square yard is 2 square feet ?
3 square feet ? 4 square feet ? 5 square feet ? 6 square
feet ? 7 square feet ?
6. What fraction of 10 is 2 ? What fraction of 10 is 7 ?
7. What fraction of 11 is 4 ? What fraction of 13 is 9 ?
8. What fraction of 100 is 80 ?
9. Which of the four fundamental rules enables us to
solve a problem of this character : What fraction of a
number is some other number ?
10. If 4 men can do a piece of work in 7 days, how
long will it take 1 man to do the same work ?
11. If a team of horses can plow a 40-acre lot in 16
days, how long will it take 4 teams, working together, to
plow the same lot ?
12. If a man can do a piece of work in 9 days, what
fraction of the work can he do in 1 day ? in 2 days ? in
3 days ? in 4 days ? in 6 days ?
70 ADVANCED BOOK OF ARITHMETIC
COMPLEX FRACTIONS
A complex fraction is a fraction one or both of whose
terms contain one, or more than one, fraction.
2i 3 1+1-4.
Ihus, -£i jyi — 22' are complex tractions.
' ^ 5 "5 T" 3"
21
Example 1. Simplify ^-|-
SOLUTION. 2f + If = f x T6T =
Or, multiply numerator and denominator by any number
which will make the terms of the fraction integers.
2| __ 2| x 6 _ 16 _
If If x6 11" ir
Example 2. Simplify '2J ~ *•
~~~
Step 1. Simplify the numerator.
Step 2. Simplify the denominator.
Step 3. Divide the result of Step 1 by the result of
Step 2.
EXERCISE 43
, m. 10. js. 13. ?Jtzi
8 u
9| ' 40|
9
' -~ '
I- + 11— 1£ 4 + | of 11
16. L_t_i. ,, ^1^. Z8. -
19. HiH+l2A. 20. i^l±}l.xljx:
AREAS OF RECTANGULAR FIGURES
71
AREAS OF RECTANGULAR FIGURES
EXERCISE 44
1. The dimensions of a room are 40 feet by 30 feet,
and 18 feet high. How many square yards are in its walls
and ceiling ?
2. Find the area, in square yards, of the walls and
ceiling of a room 24 feet by 16 feet, and 12 feet high.
3. ABQD is a rectangular plot of ground 400 feet by
160 feet. Surrounding it is a road 15 feet wide. Find
the area of the road.
4. A rectangular park, 600 feet long by 560 feet wide,
has a road surrounding it. Find the area of the road if
its width is 24 feet. Suppose the road is fenced in, how
many feet of wire will it take to go once round ?
5. A rectangular grass plot 252 feet by 180 feet has a
walk around it. The width of the walk is 9 feet. How
many flags, 9 inches square, will be required to flag the
walk?
6. Find the area of each of the following rectangles, in
square feet, correct to two decimal figures:
(a) 136 feet 8 inches by 115 feet 4 inches.
(J) 225 feet by 93 feet 10 inches,
(c) 78 feet 5 inches by 56 feet 6 inches.
25 feet 9 inches by 50 feet 2 inches.
72 ADVANCED BOOK OF ARITHMETIC
(e) 104 feet 2 inches by 153 feet 11 inches.
(/) 203 feet by 53 feet 9 inches.
(#) 223 feet 10 inches by 78 feet.
(A) 618 feet 1 inch by 130 feet 7 inches.
HINT. Reduce the inches in each example to the fraction of 1 foot.
7. Find the area of the following rectangles, giving
the results in square yards, correct to two decimal figures :
(0) 84. 5 feet by 76.75 feet.
(6) 90.67 feet by 84.33 feet.
0) 96. 34 feet by 85. 28 feet.
(d) 177.33 feet by 82.54 feet.
0) 129.55 feet by 79.63 feet.
8. A cornfield is 213^ rods long and 96 rods wide.
How many bushels of corn will it produce at 32 bushels
to an acre ? Find the value of the crop at $ .48| per
bushel.
9. A city block is 110 yards long by 90 yards wide.
How many acres are in a park which extends 7 blocks one
way and 5 blocks the other way ?
10. A 'street is 1,760 yards long and 20 yards wide.
How many thousand bricks, 8 inches by 4 inches, will be
needed to pave it ?
11. How many square tiles, 4 inches on a side, will be
required to tile a hall 60 feet by 16 feet ?
12. The dimensions of a room are 16 feet by 12 feet,
and 10 feet high. How many square yards are in the
four walls of the room ? How many square yards are in
the walls and ceiling ?
13. When the pressure per square foot of a hurricane
is 19.47 pounds, find in tons the total pressure exerted
against the side of a building 50 feet long 45 feet high.
COMPUTATION 73
COMPUTATION ON THE BASIS OF 100, 1,000, AND 2,000
Example l. Find the cost of transporting 5 bales of
cotton weighing respectively 510 lb., 515 lb., 508 lb.,
496 lb., 487 lb., at 46^ per 100 lb.
510 + 515 + 508 + 496 + 487 = 2516
= 25.16 x $ .46 = $11.5736.
Am. $ 11.57.
Example 2. Find the cost of shipping 7 head of cattle,
average weight 1,089 lb., at 97 ^ per 100 lb.
7 * J^89 x $ .97 = 7 x 10.89 x $ .97 = $73.9431.
Ana. $73.94.
10.89 The shortest way to multiply by .97 is
7 to take .03 of the multiplicand from itself.
76.23
2.2869 =.03x76.23
73.9431
Example 3. Find the value of a car load of coal weigh-
ing 43,275 lb. at $4.80 per ton of 2,000 lb.
x $4.80 = x $4.80 = 43.275 x$ 2.40
2000 2
= $103.86.
Example 4. How much will it cost a man a year to in-
sure his life for $8,750 if the annual premium is $32.80
per 1 1,000 ?
8750
1000
x $ 32.80 = 8.75 x $ 32.80 = $287.00.
NOTE. In the above examples the sign x is to be interpreted as mean-
ing times.
74 ADVANCED BOOK OF ARITHMETIC
EXERCISE 45
The following rates in cents per 100 Ib. are taken from
the annual Report of the Railroad Commission of the state
of Texas for the year 1906.
Find the cost of shipping:
1. 5 bales cotton, average weight 503 Ib., @ 45^.
2. 12 bales cotton, average weight 496 Ib., @ 48^.
3. 15 bales cotton, average weight 490 Ib., @ 8^.
4. 130 bbl. flour, 200 Ib. to the barrel, @ 16 f.
5. 124 bbl. flour, 200 Ib. to the barrel, @ 17 i.
6. 1 carload grain, weighing 27,500 Ib., @ 14^.
7. 256 sacks flour, 98 Ib. to the sack, @ 12 £
8. 32,800 Ib. grain @ 7j£
9. 1 carload cotton seed products, weighing 23,800 Ib.,
10. 1 carload cotton seed hulls, weighing 28,600 lb.,@
J*
11. 1 carload cotton seed meal, weighing 42,000 Ib.,
@
12. 1 carload cotton seed oil, weighing 43,600 Ib., @ 5^.
13. 1 carload brick, weighing 45,000 Ib., @ 5| ^.
14. 1 carload fire brick, weighing 27,000 Ib., @ 14|^.
15. 1 carload common brick, weighing 47,000 Ib., @
t.
16. 1 carload mules, weighing 29,000 Ib., @ 23^.
17. 1 carload cattle, weighing 25,000 Ib., @ 14^.
18. 1 carload sheep, weighing 15,500 Ib., @ 15/.
19. 1 carload crude petroleum, weighing 42,000 Ib., @
COMPUTATION 75
20. 1 carload asphaltum, weighing 27,000 lb.,
21. 1 carload melons, weighing 20,500 lb., @
22. 5,880 lb. molasses @ 7j£
23. 19,200 lb. sugar @ 48^.
24. The freight rate on coal in cents per ton of 2,000
lb. from Eagle Pass to the points named is :
Weimer 138^ Flatonia 127^ Columbus 140^
Beaumont 217^ Gonzales 121^ Schulenburg 134^
Find the cost of shipping 1 carload of coal, weighing
39,000 lb., from Eagle Pass to each of these points.
25. Find the cost of shipping 105,000 lb. gravel from
Austin to San Antonio at 60^ per ton of 2,000 lb.
26. Find the cost of shipping 116,000 lb. crushed rock
from Clay Quarry to Houston at 67-| ^ per ton of 2,000 lb.
27. Find the cost of shipping 130,000 lb. crushed rock
from Jacksboro to Fort Worth at 50 / per ton of 2,000 lb.
28. Find the cost of shipping a carload of sand, weigh-
ing 50,000 lb., from Kingsbury to San Antonio at 40^ per
ton of 2,000 lb.
29. Find the premium on a $5,500 life insurance policy
at $21.50 per 11,000.
30. Find the premium on a life insurance policy for
$4,500 at $25.30 per $ 1,000.
31. What is the premium on a life insurance policy of
$6,500 at $19.92 per $1,000 ?
32. Find the premium on a life insurance policy for
$10,500 at $29.80 per $1,000.
33. Find the premium on a life insurance policy for
$8,500 at $51.20 per $1,000.
34. A man insured his life for $9,450. Find the
annual premium at $62.40 per $1,000.
76 ADVANCED BOOK OF ARITHMETIC
PERCENTAGE
Per cent means by the 100, or on the 100.
Thus, 6 per cent means 6 on 100, or 6 out of 100.
25 per cent means 25 on 100, or 25 out of 100.
6 per cent is written 6%; 25 per cent is written 25%.
If a man invests $100 and gains on his investment $100,
he makes a profit of 100%. Therefore,
100% of a number = the number.
50 % of a number = J of the number.
25 % of a number = ^ of the number.
20 % of a number = ^ of the number.
16| % of a number = ^ of the number.
7 % of a number = ^fa of the number.
The following per cent equivalents should be remem-
bered :
100 % = 1 50 % = 1 331 % = 1
20 % = | 40 % = f 60 % = f
80 %=| 16|% = i
12-| <j0 = i 37-1 ^ = 3
87i%=| 150 % = 1| 175%=1|
Example l. In a city school system there are 5250
children in attendance. If 84 % are promoted, how many
are promoted ?
5250 x j^r = 5250 x .84 = 4410.
Example 2. Find 58J % of 3880.
581 = 175 = 7
100 300 12'
3880 x ^ = 22631.
PERCENTAGE 77
EXERCISE 46
Find:
1. 9 % of $ 84. 14. 7 % of $1250.
2. 8% of 1425. 15. 25% of $4840.
3. 6 % of $800. 16. 30 % of $3290.
4. 5% of $2000. 17. 40% of $4500.
5. 8% of $3250. 18. 50% of $3250.
6. 7% of $4500. 19. 75% of $4000.
7. 10% of $2250. 20. 70% of $3500.
8. 11% of $4000. 21. 80% of $2450.
9. 12% of $7250. 22. 100% of $7800.
10. 4% of $3600. 23. 16% of $3200.
11. 5% of $983. 24. 18% of $9200.
12. 8% of $750. 25. 125% of $4000.
13. 6 % of $850. 26. 225 % of $5400.
27. A man whose salary is $750 a year saves 35% of
it. How much does he save ?
28. In a city school system there are 8250 children ;
54% of this number are girls. How many girls are in
these schools? How many boys?
29. A farm of 175 acres has 24% woodland. How
many acres of woodland are in the farm?
30. A house costs $4740. The lot on which it is built
cost 32 % of the value of the house. Find the cost of the
lot.
31. In a certain year the number of rainy days was
20 % of the number of days in the year. How many
rainy days were there? How many fair days?
32. A lawyer charged 6 % for collecting a debt of $ 3720.
Find his fee. How much did he remit to his client?
78 ADVANCED BOOK OF ARITHMETIC
EXERCISE 47
Find:
1. 331% of $9600. 10. 81% of $5640.
2. 66f%of$3240. 11. 411% of $9120.
3. 25% of $4920. 12. 58|% of $7560.
4. 20% of $4500. is. 6f%of$4515.
5. 16-f%of$636. 14. 131% of $4845.
6. 831% of $792. 15. 26f%of$3900.
7. 12|% of $3280. is. 46|%of$2400.
8. 37|% of $4640. 17. 5f%of$3600.
9. 621% of $5720. is. 116|%of$672.
19. A man sells his house for $1800. If he paid for
it 83^ % of the price at which it was sold, what did the
house cost ?
20. A shoe dealer sold $720 worth of shoes. The shoes
cost him 66|% of the selling price. Find the cost price
of the shoes.
21. In an apple orchard of 840 trees 58| % bore fruit.
How many trees were fruit- bearing ?
22. A ranchman lost during a blizzard 16|% of his
sheep. If the number in his flock was originally 960, how
many did he lose, and how many were left ?
23. If the area of a county is 1230 square miles, and 75 %
of it arable land, how many square miles are arable land ?
24. Piles used in the construction of a railroad bridge
are 42 ft. long, and 83| % of their length is beneath the
water. Find the length in the water.
25. The railroad mileage of the United States in the year
1904 was 212,578. Of this the railroad mileage of Florida
was 1| %. Find the railroad mileage of Florida in 1901.
INTEREST AND PROPERTY INSURANCE 79
INTEREST AND PROPERTY INSURANCE
Interest is money paid for the use of money.
The sum loaned is the principal.
Interest is always reckoned as a rate per cent of
the principal. The rate per cent is for one year unless
otherwise stated.
Property insurance is idemnity against loss of property
and is reckoned as a rate on the basis of $100 valuation.
The sum paid for insurance is the premium.
The written contract between the assured and the
insurance company is called the insurance policy.
Example. What is the premium on an insurance
policy of $15,350 at $1.35 per $100?
x $1.35 = 153.5 x $1.35 = $207.225, or
15350 x $.0135 = $207.225.
In the first solution, the number of 100\s is multiplied
by the rate on $100. In the second solution, the number
of dollars is multiplied by the rate on $1.00.
EXERCISE 48
Find the interest on:
1. $600 for 1 yr. at 4%; for 1 yr. at 5%; for 1 yr. at
6%; for 1 yr. at 8%.
2. $850 for 1 yr. at 7%; for 1 yr. at 8%; for 1 yr.
at 9%.
3. $950 for 1 yr. at 3%; for 1 yr. at 4%; for 1 yr.
at S%.
4. $982 for 2 yr. at 4%; for 3 yr. at 5%; for 1 yr.
6 mo. at 6%.
80 ADVANCED BOOK OF ARITHMETIC
5. $738 for £ yr. at 5 % ; J yr. at 6 %.
6. $920 for \ yr. at 6 % ; i yr. at 7 %.
7. $1200 for 4 mo. at 5%; 4 mo. at 6%.
8. $1100 for 6 mo. at 7%; 6 mo. at 4%.
9. $1280 for 3 mo. at 8%; 3 mo. at 6%.
Find the premium for insuring dwellings against loss
by fire at the rates specified per $100:
10. $2500 at $1.30. 26. $22,500 at $1.10.
11. $2000 at $1.15. 27. $18,250 at $1.20.
12. $4500 at $1.50. 28. $2400 at $1.80.
13. $3000 at $1.25. 29. $9300 at $1.50.
14. $2500 at $1.90. 30. $8500 at $1.60.
15. $5500 at $1.70. 31. $9450 at $1.50.
16. $6500 at $1.50. 32. $6500 at $1.90.
17. $4000 at $1.80. 33. $5400 at $1.60.
18. $5400 at $1.70. 34. $9500 at $1.90.
19. $3300 at $1.80. 35. $12,000 at $1.75.
20. $7500 at $1.60. 36. $18,000 at $1.75.
21. $7250 at $1.40. 37. $200,000 at $1.25.
22. $10,500 at $1.30. 38. $15,500 at $1.60.
23. $19,250 at $1.25. 39. $16,200 at $1.60.
24. $16,450 at $1.60. 40. $1800 at $1.90.
25. $7900 at $1.35. 41. $1750 at $1.75.
42. A man insures his residence, valued at $5000, at |
of its value at the rate of $1.20 on the $100. Find the
premium paid.
43. A jobber insures a quantity of cotton, worth $ 30,000,
at | of its value at the rate of 75^ on the $100. Find his
premium.
CHAPTER II
COMPOUND QUANTITIES
CONCRETE quantities of the same kind, but consisting
of units of different denominations, are called compound
quantities.
Seventeen days, 10 hours, and 30 minutes is a compound
quantity. Here, we have three units of measurement;
namely, a day, an hour, and a minute. These units are of
different denominations, but each is of the same kind,
inasmuch as it stands for a definite portion of time.
Compound quantities are also called compound denominate
quantities. Quantities composed of units of one denomi-
nation are generally called simple quantities.
AVOIRDUPOIS WEIGHT
16 ounces (oz.) = 1 pound (Ib.)
100 pounds = 1 hundredweight (cwt.)
20 hundredweight, or 2000 pounds = 1 ton (T.)
2240 pounds = 1 long ton
Avoirdupois Weight is used in weighing all commercial
quantities excepting the precious metals, jewels, and drugs,
when sold by retail druggists.
The unit in Avoirdupois Weight is the pound of 7000
grains. One cubic inch of distilled water weighs in vacuo
252.286 grains, of which 7000 weigh 1 pound.
The long ton is used in the United States custom
houses, and in weighing coal and mineral products at the
mines.
G 81
82 ADVANCED BOOK OF ARITHMETIC
The process of reducing units of any given denomi-
nation to units of higher denomination is called reduction
ascending.
The process of reducing units of a higher denomination
or of higher denominations to units of lower denomination
is called reduction descending.
Example. Reduce 1,000,201 oz. to higher denominations.
Divide by 16 to get the num-
16 1000201 ber of pounds. Divide by 100
100
20
62512 Ib. 9 oz. to get the number of hundred-
625 cwt. 12 Ib. weights. Divide by 20 to get
31 T. 5 cwt. the number of tons. Ans. 31
T. 5 cwt. 12 Ib. 9 oz.
Weights are generally expressed in tons or in pounds.
EXERCISE 49
Reduce to higher denominations :
1. 7800 oz. 3. 75,497 oz. 5. 7987 Ib.
2. 9763 oz. 4. 1,000,000 oz. 6. 32,721 Ib.
7. How many ordinary or short tons in 100 long tons ?
8. Reduce 10,000 Ib. to long tons.
Example. Reduce 7 T. 3 cwt. 12 Ib. 10 oz. to ounces.
T. CWT. LB. OZ.
7 3 12 10
20
143 cwt. Reduce the 7 T. to hundredweights by
100 multiplying by 20. Add 3 cwt. to the prod-
14312 Ib. uct and get 143 cwt. Multiply 143 by 100
16 and add 12 Ib. to the product. This gives
85882 14,312 Ib. Multiply this by 16, adding 10
14312 oz., when the first figure is multiplied by 6.
2 29002 oz.
COMPOUND QUANTITIES 83
EXERCISE 50
1. Reduce 19 T. to pounds.
2. Reduce 14 T. 4 cwt. to pounds.
3. Reduce 17 T. 3 cwt. to pounds.
4. Reduce 25 T. 2 cwt. to ounces.
5. Reduce 3 T. 15 cwt. 2 Ib. to pounds.
6. Reduce 4 T. 11 cwt. 58 Ib. to pounds.
7. Reduce 8 T. 2 cwt. 73 Ib. to pounds.
8. A dealer buys 50 long tons of coal and sells it by
the short ton. How many short tons does he sell?
9. A dealer buys 100 long tons of coal at $6.75 per
ton. He sells it by the short ton at $6.75 per ton. How
much profit does he make ?
10. Convert 784 short tons into long tons.
11. Convert 550 long tons into short tons.
12. Three horses together weigh 2 T. 4 cwt. 91 Ib.
Find in pounds the average weight of the horses.
LINEAR OR LONG MEASURE
12 inches (in.) = 1 foot (ft.)
3 feet = 1 yard (yd.)
5£ yards = 1 rod (rd.)
320 rods = 1 mile (mi.)
6080 feet = 1 knot, geographical or nautical mile
3 knots = 1 marine league
1 mi. = 320 rd. = 1760 yd. = 5280 ft.
The unit of length is the yard.
The yard in the United States is denned as f f $$ of the
meter.
The standard yard of this country has been adopted
since 1893.
84 ADVANCED BOOK OF ARITHMETIC
EXERCISE 51
1. Reduce 4 yd. 2 ft. to inches.
2. Reduce 110 yd. 1 ft. to inches.
3. Reduce 5J mi. to yards.
4. Reduce 7 mi. 120 rd. to yards.
5. Reduce 10 mi. 110 rd. 4 yd. to yards.
6. Reduce 445| mi. to yards.
7. Reduce 7.74 mi. to yards.
8. Reduce 8.35 mi. to rods.
9. Reduce 238 rd. to feet.
SQUARE MEASURE
144 square inches (sq. in.) = 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.)
1 acre = 4840 square yards
1 section = 1 square mile
36 sections = 1 township
Square measure is used to measure the areas of surfaces.
A cube is a solid bounded by six plane surfaces, each of
which is a square.
A solid having the shape of a box or of an ordinary
room, i.e. one bounded by six plane surfaces, each of which
is a rectangle, is called a rectangular solid.
The volume of a solid means the amount of space it
occupies. This is measured by the number of times the
solid contains the unit of measurement.
The unit of volume is a cube having for an edge the
linear unit. The cubic unit from which all others are
derived is the cubic yard.
COMPOUND QUANTITIES 85
CUBIC OR SOLID MEASURE
1728 cubic inches (cu. in.) = 1 cubic foot (cu. ft.)
27 cubic feet = 1 cubic yard (cu. yd.)
MEASURES OF CAPACITY
There are. two measures of capacity in general use;
namely, Liquid Measure and Dry Measure.
LIQUID MEASURE
4 gills (gi.) = 1 pint (pt.)
2 pints = 1 quart (qt.)
4 quarts = 1 gallon (gal.)
A gallon contains 231 cu. in.
DRY MEASURE
2 pints (pt.) = 1 quart (qt.)
8 quarts = 1 peck (pk.)
4 pecks = 1 bushel (bu.)
One bushel contains 2150.42 cu. in. It is the volume
of a cylindrical vessel 18 J in. in diameter and 8 in. deep.
REDUCTION DESCENDING
Example. Reduce 5 gal. 2 qt. 1 pt. 2 gi. to gills.
5 gal. 2 qt. 1 pt. 2 gi.
A
20 = number of quarts in 5 gal.
2
22 = number of quarts in 5 gal. 2 qt.
' _2
44 = number of pints in 5 gal. 2 qt.
_1
45 = number of pints in 5 gal. 2 qt. 1 pt.
4
182 = number of gills in 5 gal. 2 qt. 1 pt. 2 gi.
86 ADVANCED BOOK OF ARITHMETIC
EXERCISE 52
Reduce :
1. 1 sq. mi. to sq. rd. 14. 20.25 cu. yd. to cu. in.
2. 2| sq. mi. to A. 15. 2 gal. 2 qt. to qt.
3. 12 A. to sq. ft. 16. 5 gal. 3 qt. to qt.
4. 27 sq. rd. to sq. ft. 17. 3 gal. 1 pt. to pt.
5. 3 mi. 50 rd. to ft. 18. 7 gal. 1 pt. to pt.
6. 8 mi. 40 rd. to ft. 19. 19,25 gal. to pt.
7. 2| mi. to yd. 20. 4 bu. to qt.
8. 3.75 mi. to yd. 21. 3| bu. to qt.
9. 2.125 mi. to ft. 22. 3.625 bu. to qt.
10. 25 cu. yd. to cu. ft. 23. 7 pk. to qt.
11. 38 cu. yd. 20 cu. ft. to cu. ft. 24. 7.375 pk. to pt.
12. 171 cu. yd. to cu. ft. 25. 18J bu. to pt.
13. 18.75 cu. yd. to cu. ft. 26. 13 bu. 3 qt. to pt.
27. How many feet are in | mi. ? in | mi. ? in -fa mi. ?
28. How many yards are in ^ mi. ? in ^ mi. ? in J mi. ?
29. What fraction of a mile is 440 yd. ? 176 yd. ? 88 yd. ?
30. How many square yards are in ^ of an A. ? in J A. ?
31. What part of a township is 1 sq. mi. ?
32. How many square rods are in £ A. ?
33. How many square rods are in .7 A. ? in. .9 A. ?
34. How many square feet are in f sq. rd. ?
35. How many cubic inches are in 1 pt., Dry Measure ?
36. How many cubic inches are in 1 pt., Liquid Measure?
37. How many quarts are in f pk. ?
38. How many gallons are required to fill 10 bu. measures?
COMPOUND QUANTITIES 87
REDUCTION ASCENDING
Example l. Reduce 85 pt. to higher denominations.
85 pt.
42 qt. 1 pt.
10 gal. 2 qt. 1 pt.
Example 2. The length of one degree of latitude at
40° north is 364,280 ft. Express this length in miles.
There are 5280 ft. in 1 mi. The fac-
80
6
11
364280 ft.
4553 g ' tors of 5280 are 80, 6, and 11. (A num-
ber is divided by 80 by dividing by 8
and writing each quotient figure one
place to the right.) Ans. 68.992 mi.
EXERCISE 53
Reduce to higher denominations:
1. 234 pt. (Liquid Measure). 5. 2000 pt. (Dry Measure).
2. 47,385 cu. in. 6. 393,000 cu. in.
3. 3456 pt. (Liquid Measure). 7. 20,000,000 A.
4. 10,240 rd. 8. 15,000 sq. in.
9. The equatorial diameter of the earth is 41,852,404 ft.
Express this distance in miles and the decimal of a mile
correct to two decimal figures.
10. The polar diameter of the earth is 41,709,790 ft.
What is the polar diameter of the earth in miles correct
to two decimal figures ?
11. By how many miles does the equatorial diameter
exceed the polar diameter ?
12. Light takes 8 min. 18 sec. to come from the sun to
the earth. The mean distance of the sun from the earth
is 92,790,000 mi. Find the velocity of light per second.
88
ADVANCED BOOK OF ARITHMETIC
CIRCULAR ARC MEASURE
A circle is a plane figure bounded by a line called the
circumference, every point of which is equally distant from
a point within the figure called the center.
A straight line from the
center to the circumference
is called a radius.
A straight line drawn
through the center and ter-
minated by the circumference
is called a diameter.
The lines AB and CD are
diameters.
Any portion of a circum-
ference is called an arc. mn
is an arc.
An arc equal to one half of a circumference is called a
semicircumference.
An arc equal to one fourth of a circumference is called
a quadrant.
60 seconds (") = 1 minute (')
60 minutes = 1 degree (°)
360 degrees = 1 circumference
ANGULAR MEASURE
An angle is a figure formed by two straight lines pro-
ceeding from a point. Its magnitude depends upon the
amount of turning necessary to bring one side into co-
incidence with the other.
NOTE. Beginners should be provided with a protractor and they
should draw and measure angles. To learn things by actual trial and not
by mere hearsay is to educate.
COMPOUND QUANTITIES
89
If one straight line meets another straight line so as to
make the adjacent angles equal to each other, each angle
is called a right angle.
S
If the lines MN, ST meet in B so as to make the angles
, SBM equal, then each angle is a right angle.
The unit of angular measure is 1 degree.
60 seconds (") = 1 minute (')
60 minutes = 1 degree (°)
90 degrees — 1 right angle
2 right angles = 1 straight angle
EXERCISE 54
1. What part of V is 1" ?
2. How many seconds are in | minute ?
3. Reduce 1' 30" to seconds.
4. What part of a straight angle is a right angle ?
5. What part of a right angle is an angle of 45° ? 30° ?
15°? 18°? 60°? 75°?
6. How many degrees are in 1 straight angle ? in | of
a straight angle ?
7. What part of a straight angle is an angle of 15° ? 24° ?
30° ? 45° ? 60° ? 80° ? 1006 ? 105° ? 120° ? 135° ? 150° ?
90 ADVANCED BOOK OF ARITHMETIC
TIME MEASURE
60 seconds (sec.) = 1 minute (min.)
60 minutes = 1 hour (hr.)
24 hours = 1 day (da.)
7 days = 1 week (wk.)
365 days = 1 common year (yr.)
366 days — 1 leap year (yr.)
100 years = 1 century
There are twelve calendar months in a year.
The following lines will enable one to remember the
number of days in each month :
" Thirty days hath September,
April, June, and November,
February twenty-eight alone,
And all the others thirty-one ;
But leap year, coming once in four,
Gives February one day more."
A day is the standard unit of time. It is of the same
duration at all places. It represents the period of time
that elapses between two successive passages of the sun
across the meridian of any place.
The length of a year is 365 daj^s, 5 hours, 48 minutes,
46 seconds. The common year has 365 days. The differ-
ence in length between the common year and the actual,
or solar year, gave rise to the introduction of leap years.
Centennial years are leap years when the number of the
year is exactly divisible by 400. Thus, the year 2000 is
a leap year because 2000 is divisible by 400. All other
years are leap years when their numbers are exactly
divisible by 4. The year 1907 is not a leap year, as the
number 1907 is not exactly divisible by 4. The year
1828 was a leap year, as 1828 is exactly divisible by -4.
COMPOUND QUANTITIES 91
MISCELLANEOUS MEASURE
1 bushel of barley
= 48 Ib.
1 bushel of wheat
= 60 Ib.
1 bushel of oats
= 32 Ib.
1 bushel of rye
= 56 Ib.
1 bushel of potatoes (Irish)
= 60 Ib.
1 bushel of potatoes (sweet)
= 55 Ib.
1 bushel of buckwheat
= 48 Ib.
1 bushel of beans
= 60 Ib.
1 bushel of shelled corn
= 56 Ib.
1 bushel of peas
= 60 Ib.
1 bushel of clover seed
= 60 Ib.
1 barrel of flour
= 196 Ib.
1 barrel of pork or beef
= 200 Ib.
1 cental of grain
= 100 Ib.
NUMBERS PAPER MEASURE
12 units = 1 dozen (doz.) 24 sheets of paper = 1 quire
12 dozen = 1 gross 20 quires = 1 ream
12 gross = 1 great gross 2 reams = 1 bundle
20 units = 1 score 5 bundles = 1 bale
TROY WEIGHT
24 grains (gr.) = 1 pennyweight (pwt.)
20 pennyweights = 1 ounce (oz.)
12 ounces = 1 pound (Ib.)
1 pound Troy = 5760 grains
Troy weight is used in weighing precious metals and
jewelry.
(The measures in this paragraph are inserted merely for
reference.)
92 ADVANCED BOOK OF ARITHMETIC
EXERCISE 55
Reduce to seconds :
1. 18° 20' 20". 3. 12° 5' 10". 5. 120.3°.
2. A quadrant. 4. 7|°. 6. 45° 30' 20".
Reduce to minutes:
7. 14i°. 9. 254.125°. 11. 18f°.
8. 75.75°. 10. 4f°. 12. 13|°.
13. Reduce a common year to minutes.
14. Find the number of minutes in the years 1903,
1904, 1905.
15. Find the number of minutes in February, 1904.
16. Find the number of minutes in the first three
months of the year 1903 ; also in the first three months of
the year 1904.
17. Find the number of seconds in a solar year, con-
sisting of 365 da. 5 hr. 48 min. 46 sec.
18. The pulse of a healthy person beats 70 times a
minute. At this rate, how many times will it beat in a
leap year ? How many times will it beat in the four suc-
cessive years, beginning 1904?
19. Reduce 30 wk. 6 da. 12 hr. to minutes.
20. Reduce 1| common years to days.
21. Reduce 5f wk. to hours.
22. Reduce 20.4 yr. to hours, allowing five of them to
be leap years.
23. How many days are there between Jan. 30, 1902,
and Jan. 30, 1910 ?
24. Reduce T2^ of a circumference to degrees.
25. Reduce ||- of a straight angle to degrees.
COMPOUND QUANTITIES 93
ADDITION
In adding compound quantities, proceed as follows :
Step 1. Arrange the quantities so that units of the
same denomination stand in the same vertical column, the
highest denomination being written first, the next to
the highest second, and so on.
Step 2. Beginning with the right-hand column, add the
numbers in it, divide their sum by the number of units
which makes one unit of the next higher denomination.
Write the remainder in the right-hand column, and carry
the quotient to the next column.
Step 3. Treat the next column to the left in the same
manner. The remaining columns are dealt with in the
same way.
LINEAR MEASURE
EXERCISE 56
Add: (1) (2) (3)
YD. FT. IN. YD. FT. IN. YD. FT. IN.
908 26 16 51 11
11 2 4 33 2 6 312
61 10 20 10 413
506 70 1 9 11 2 5
SQUARE MEASURE
(*) (5) (6)
A. SQ. BD. A. SQ. BD. A. SQ. BD.
76 144 33 79 127 38
85 131 173 27 192 99
37 33 254 28 238 77
63 99 45 53 413 25
94 ADVANCED BOOK OF ARITHMETIC
CAPACITY
7. Add: 2 gal. 3 qt. 1 pt., 3 gal. 2 qt. 1 pt., 5 gal. 2
qt. 1 pt., 4 gal. 2 qt. 1 pt.
8. Add: 7 gal. 2 qt. 1 pt., 9 gal. 3 qt., 4 gal. 1 qt. 1
pt., 6 gal. 3 qt. 1 pt., 9 gal. 1 pt., 7 gal. 1 pt.
9. Add: 3 bu. 3 pk. 5 qt., 4 bu. 2 pk. 4 qt., 9 bu. 2
pk. 7 qt., 9 bu. 7 qt., 8 bu. 2 pk. 3 qt., 6 bu. 3 pk. 2 qt.
10. Add : 4 bu. 7 qt., 3 bu. 4 pk. 6 qt., 7 bu. 2 pk. 6
qt., 8 bu. 3 qt., 9 bu. 2 pk. 3 qt., 4 bu. 3 pk. 2 qt.
11. Add : 17 gal. 1 pt., 14 gal. 2 qt. 1 pt., 2 gal. 2 qt. 1
pt., 15 gal. 1 pt., 13 gal. 1 qt. 1 pt., 14 gal. 3 qt. 1 pt.
12. Add: 14 bu. 2 pk. 7 qt., 29 bu. 3 pk. 5 qt., 23 bu.
2 pk. 6 qt., 39 bu. 6 qt., 28 bu. 3 pk., 17 bu. 2 pk. 5 qt.
13. Add : 38 bu. 3 pk. 2 qt., 16 bu. 2 pk. 1 qt., 28 bu.
3 pk. 7 qt., 3 bu. 7 qt., 5 bu. 3 pk., 24 bu. 2 pk. 2 qt.
14. Add : 15 bu. 5 qt., 12 bu. 3 pk., 17 bu. 7 qt., 18 bu.
6 qt., 29 bu. 2 pk. 3 qt., 71 bu. 3 pk. 2 qt., 18 bu. 3 pk.
AVOIRDUPOIS WEIGHT
is. Add: 20 T. 215 lb., 18 T. 425 lb., 17 T. 328 lb.,
92 T. 411 lb.
16. Add: 384 lb. 12 oz., 125 lb. 15 oz., 82 lb. 14 oz.,
73 lb. 11 oz.
17. Add: 425 lb. 10 oz., 17 lb. 14 oz., 30 lb. 12 oz., 72
lb. 9 oz.
18. Add: 15 T. 290 lb., 17 T. 184 lb., 12 T. 127 lb.,
15 T. 9 lb., 18 T. 18 lb.
19. Add : 18 lb. 8 oz., 64 lb. 7 oz., 82 lb. 6 oz., 90 lb.
5 oz., 16 lb. 13 oz.
20. Add: 16 T. 175 lb., 71 T. 29 lb., 28 T, 245 lb.,
97 T. 159 lb., 13 T. 1300 lb.
COMPOUND QUANTITIES 95
TIME
21. Add: 5 da. 4 hr. 15 min., 17 da. 17 hr. 17 min.,
92 da. 14 hr. 14 min., 27 da. 23 hr. 12 min., 29 da. 16 hr.
14 min., 45 da. 15 hr. 18 min.
22. Add : 4 wk. 5 da. 7 hr., 9 wk. 6 da. 11 hr., 18 wk.
5 da. 12 hr., 23 wk. 11 hr., 28 wk. 4 da. 4 hr., 73 wk.
6 da. 19 hr., 82 wk. 5 da. 21 hr.
23. Add : 20 hr. 30 min. 18 sec., 17 hr. 45 min. 37 sec.,
14 hr. 18 min. 18 sec., 14 hr. 12 min. 12 sec., 9 hr. 48 min.
48 sec., 8 hr. 39 min. 39 sec.
24. Add: 12 da. 17 hr. 44 min., 15 da. 18 hr. 18 min.,
31 da. 19 hr. 19 min., 33 da. 21 hr. 27 min., 12 da. 12 hr.
36 min., 34 da. 20 hr. 23 min.
25. Add: 3 wk. 5 da. 23 hr., 8 wk. 6 da. 16 hr., 9 wk.
5 da. 18 hr., 4 wk. 4 da. 14 hr., 10 wk. 5 da. 13 hr.
26. Add : 14 hr. 14 min. 14 sec., 9 hr. 54 min. 38 sec.,
11 hr. 12 min. 19 sec., 4 hr. 31 min. 27 sec., 5 hr. 45 min.
43 sec., 8 hr. 41 min. 42 sec.
VOLUME
27. Add : 4 cu. ft. 1421 cu. in., 9 cu. ft. 294 cu. in.,
18 cu. ft. 998 cu. in., 7 cu. ft. 778 cu. in., 9 cu. ft. 499
cu. in., 15 cu. ft. 498 cu. in.
28. Add: 27 cu. yd. 19 cu. ft., 84 cu. yd. 24 cu. ft.,
87 cu. yd. 19 cu. ft., 16 cu. yd. 22 cu. ft., 55 cu. yd.
17 cu. ft., 34 cu. yd. 16 cu. ft.
29. Add : 37 cu. yd. 13 cu. ft., 38 cu. yd. 26 cu. ft.,
49 cu. yd. 25 cu. ft., 62 cu. yd. 26 cu. ft., 77 cu. yd.
17 cu. ft., 94 cu. yd. 28 cu. ft.
30. Add: 15 cu. ft. 578 cu. in., 18 cu. ft. 902 cu. in.,
18 cu. ft. 978 cu. in., 15 cu. ft. 293 cu. in.
96 ADVANCED BOOK OB^ ARITHMETIC
SUBTRACTION
From 19 sq. yd. 5 sq. ft. 20 sq. in. take 14 sq. yd.
7 sq. ft. 45 sq. in.
SQ. YD. SQ. FT. SQ. IN.
19 5 20
14 7 45
4 6 119
Step 1. Write the quantities so that units of the same
denomination are in the same column.
Step 2. Find what concrete quantity added to 45 sq. in.
will give 1 sq. ft. 20 sq. in., i.e. 164 sq. in. Write the
remainder, 119 sq. in., in the column for square inches.
Carry 1 sq. ft.
Step 3. Find what concrete quantity added to 8 sq. ft.
will give 1 sq. yd. 5 sq. ft., i.e. 14 sq. ft. Write the re-
mainder, 6 sq. ft., in the column for square feet. Carry
1 sq. yd.
Step 4. Find the difference between 15 sq. yd. and 19
sq. yd. and write it in its proper place.
EXERCISE 57
CIRCULAR ARC OR ANGULAR MEASURE
1. Subtract 5° 12' 13" from 84° 14' 30".
2. Subtract 19° 14' 14" from 27° 15' 10".
3. Subtract 38° 15' 45" from 90° 10' 10".
4. Subtract 54° 14' 54" from 172° 0' 19",
5. Subtract 84° 5' 15" from 90°.
6. Subtract 113° 13' 54" from 180°.
7. Subtract 94° 53' 50" from 180°,
COMPOUND QUANTITIES 97
8. Subtract 87° 15' from 133° 12'.
9. Subtract 119° 54' 17" from 180°.
10. Subtract 15° 14' 17" from 94° 14' 7".
11. The tropic of Cancer is 23° 27' 6" north of the equa-
tor ; the Arctic circle is 23° 27' 6" south of the north pole.
Find the width of the north temperate zone.
CAPACITY
12. From 3 bu. 2 pk. 4 qt. take 1 bu. 2 pk. 5 qt.
13. From 12 gal. 3 qt. 1 pt. take 4 gal. 3 qt.
14. From 11 gal. take 4 gal. 3 qt. 1 pt.
15. From 17 gal. take 11 gal. 1 qt. 1 pt.
16. From 37 bu. 2 pk. 4 qt. take 17 bu. 3 pk. 7 qt.
17. From 29 bu. 1 pk. 2 qt. take 19 bu. 3 pk. 5 qt.
18. From 37 gal. take 17 gal. 1 qt. 1 pt.
19. From 134 gal. take 112 gal. 3 qt. 1 pt.
20. From 1J gal. take f gal. and express the result in
quarts.
21. From If bu. take | bu. and express the result in
quarts.
22. From 5^- bu. take If bu. and express the result in
quarts.
TIME
23. From 3 da. 4 hr. 11 min. take 1 da. 7 hr. 14 min.
24. From 11 da. 5 hr. 10 min. take 4 da. 11 hr. 19 min.
25. Almanacs give the time of sunrise in Florida,
Louisiana, and Texas on March 5 as 6.22 A.M., and
that of sunset as 6.2 P.M. Find the length of the
day.
98 ADVANCED BOOK OF ARITHMETIC
26. On April 1, 1903, the moon rose at 10.28 P.M. On
April 4 following, it rose at 12.28 A.M. How many
hours and minutes earlier did it rise on April 1 than on
April 4 ?
Time between events happening in two different years.
Example. How many years, months, and days were
between Aug. 27, 1880, and Jan. 22, 1901 ?
YB. MO. DA. Since January is the first month of
1901 22 the r and August is the eighth
1 &&0 & 97
100U - £i month of the year, we write 1 instead
25 of January and 8 instead of August.
In finding the difference, a month is taken as 30 days.
The work is then performed as in the subtraction of
compound quantities.
EXERCISE 58
1. The battle of New Orleans was fought on Jan. 8,
1815. Find the time from that date to the present
day.
2. The first telegraph message was sent by Professor
Morse on May 24, 1844. Find the time from that date
to the present day.
3. The Spanish fleet under Cervera was destroyed near
Santiago on July 3, 1898. Find the time from that date
to Feb. 1, 1903.
4. The Mecklenburg Declaration of Independence was
signed May 20, 1775. Find the time from this date to
the surrender of Cornwallis, Oct. 19, 1781.
COMPOUND QUANTITIES
99
5. The following named men were born and died on
the dates specified. Find how long each lived.
John Milton ....
Alexander Pope . .
William Shakespeare .
Edmund Burke
Robert E. Lee . . .
U. S. Grant ....
Oliver Goldsmith .
Benjamin Franklin
Alexander Hamilton .
H. W. Longfellow .
J. H. Newman . . .
W. E. Gladstone ,
BORN
Dec. 9, 1608.
DIED
Nov. 8, 1674.
May 21, 1688.
May 30, 1744.
April 23, 1564.
April 23, 1616.
Jan. 12, 1730.
July 9, 1797.
Jan. 19, 1807.
Oct. 12, 1870.
April 27, 1822.
July 23, 1885.
Nov. 10, 1728.
April 4, 1774.
Jan. 17, 1706.
April 17, 1790.
Jan. 11,1757.
July 12, 1804.
Feb. 27, 1807.
March 24, 1882.
Feb. 21, 1801.
Aug. 11, 1890.
Dec. 9, 1809.
May 19, 1898.
MULTIPLICATION
Multiply 5 yd. 2 ft. 10 in. by 7.
YD. FT. IN. Multiply 10 in. by 7 and get 70 in.
5 2 10 = 5 ft. 10 in. Write 10 in. and carry
I 5 ft. 7 times 2 ft. are 14 ft. 14 ft. and
10 5 ft. = 19 ft. = 6 yd. 1 ft. Write 1 ft.
Carry 6 yd. 7 times 5 yd. are 35 yd. 35 yd. and 6 yd.
= 41 yd.
EXERCISE 59
Multiply :
1. 4 yd. 2 ft. 3 in. by 9.
2. 6 yd. 1 ft. 9 in. by 11.
3. 9 yd. 2 ft. 11 in. by 8.
4. 3 bu. 2 pk. 7 qt. by 7.
5. 4bu. Ipk. 6qt. by 12.
6. 9 gal. 3 qt. 1 pt. by 6.
7. 6 gal. 2 qt. 1 pt. by 12.
8. 3 bu. 3 pk. 7 qt. by 7.
100 ADVANCED BOOK OF ARITHMETIC
9. 12 T. 400 Ib. by 12. 13. 16° 38' 32" by 15.
10. 13 T. 387 Ib. by 9. 14. 64 A. 150 sq. rd. by 12.
11. 17 T. 254 Ib. by 10. 15. 15 A. 27 sq. rd. by 11.
12. 5° 29' 28" by 16. 16. 18° 9' 54" by 14.
17. Multiply | T. by 9 and express the result in pounds.
18. Multiply ^ mile by 19 and give the result in feet.
DIVISION
Divide 97° 10' 50" by 8.
n,Q7o 1(V r0,, The eighth part of 97° is 12°, with
12° 8' 511" a remainder of 10- 10 10' = 70'- The
4 eighth part of 70' is 8', with a remainder
of 6'. 6' 50" = 410". 8 into 410 goes 51 J times.
EXERCISE 60
Divide :
1. 21 yd. 2 ft. 3 in. by 9. 4. 34 yd. 2 ft. 8 in by 6.
2. 93° 15' 15" by 7. 5. 77 yd. 2 ft. 4 in. by 7.
3. 84° 14' 14" by 12. 6. 13 bu. 3 pk. 4 qt. by 6.
7. How often is 231 cu. in. contained in 1 cu. ft. 582
cu. in. ?
8. How often is 7 yd. 1 ft. contained in 1 mi. ?
9. A meter is 39.37 inches. How many meters equal
1 mile ?
10. The planet Mercury revolves around the sun in 88
days. Find in degrees, minutes, and seconds its daily
progress.
11. Civil engineers use a chain 100 feet long. How
many of these chains make 5 miles ?
COMPOUND QUANTITIES 10J
Example l. Reduce .875 yd. to feet and inches.
.875 yd.
3_
2.625 ft. .625 ft. = .625 x 12 in.
12 = 7.5 in.
7.500 in.
.875 yd. = 2 ft. 7.5 in.
Example 2. Reduce TV bu. to lower denominations.
£ bu. = J_ Of 4 pk. = I pk. = 2i pk.
1 pk.= t ofSqt. = fqt. = 2f qt,
| qt. = | of 2 pt, = | pt. = 11 pt.
Hence, TV bu. = 2 pk. 2 qt. 1^ pt. '
Example 3. Express -^ A. in square yards.
7 v 4840
^ of 1 A. = TV of 4840 sq. yd. = LJ^^^L sq. yd.
12
sq. yd. = 2823^sq. yd.
EXERCISE 61
Reduce :
1. % T. to pounds. 6. . 375 bu. to quarts.
2. .15° to minutes. ' 7. 18.4 mi. to feet.
3. ^ da. to hours and minutes. 8. -f^ mi. to yards.
4. .2345 T. to pounds. 9. .1875 mi. to rods.
5. .95 da. to hours and minutes. 10. 15|° to minutes.
11. | of a common year to days and hours.
12. .3125 common years to days, hours, and minutes.
13. .45 bu. to a compound quantity.
14. | gal. to a compound quantity.
15. .85 A. to square rods. 17. l| bu. to quarts.
16. II A. to square yards. 18. ^ gal. to pints.
1Q2 ADVANCED BOOK OF ARITHMETIC
EXPRESSION OF ONE QUANTITY AS A FRACTION OF
ANOTHER QUANTITY
Example 1. Express 27 rd. 4 yd. 2 ft. as a fraction of
1 mi.
27 rd. 4 yd. 2 ft.
52 1 mi = 5280 ft.
.
139 27rd.4yd. 2 ft. = of 1 mi.
1- yd. 919 . -
_ =i0560°flmi-
4591 ft.
To express one quantity as a fraction of another quantity,
reduce both to the same denomination, and divide the first
quantity by the second.
Example 2. Express 2 yd. 2 ft. 8 in. as a decimal of a
mile.
Step 1. Express 2 yd. 2 ft. 8 in. as a fraction of 1 mi.
Step 2. Reduce this fraction to a decimal.
Reduce: EXERCISE 62
1. 4400 ft. to the decimal of 1 mi.
2. 293 yd. 1 ft. to the decimal of 1 mi.
3. 117 yd. 1 ft. to the decimal of 1 mi.
4. 1 qt. 1 pt. to the decimal of a gal.
5. 1 pk. 6 qt. to the decimal of a bu.
6. 2 pk. 2 qt. to the decimal of a bu.
7. 3 pk. 1 qt. 1 pt. to the decimal of a bu.
8. 1^ in. to the decimal of 1 ft.
9. 2.34 in. to the decimal of 1 ft.
10. 3° 15' to the fraction of a right angle.
11. 1 da. 18 hr. to the decimal of 1 wk.
MEASUREMENTS
103
MEASUREMENTS
EXERCISE 63
1. Find the number of acres in the area of a rectangle
whose dimensions are 360 ft. and 121 ft.
2. Find the area of a rectangle 1331 ft. by 720 ft.
3. Find, in acres, the area of a rectangular plot of
ground 201 yd. by 10 rd. 5 yd.
6
5
4
3
2
1
7
8
9
10
11
12
18
17
16
15
H
13
19
20
21
22
23
24
30
29
28
27
26
25
31
32
33
34
35
36
FIG. 1.
4. A railway company acquires the right of way
through a territory 154 mi. long, and fences in a strip
80 ft. wide. How many acres does it thus inclose, and
how much does it pay for the land at $ 25 an acre ?
104
ADVANCED BOOK OF ARITHMETIC
5. Find the area of a park 396 ft. by 396 ft. Give
your answer in acres.
6. How many acres are in a rectangular farm 1.5 mi.
long by 1| mi. wide ? Find the value of the farm at $49
an acre.
A township is a tract of land 6 mi. square, and it is
divided into 36 sections each 1 mi. square. The sections
are numbered as in Fig. 1.
NJ/20f
N.W. y4
S. W. t/4 of
S. E. 1/4 of
N. W.i/4
N. W.1/4
s. w. y4
FIG. 2.
A section is subdivided as indicated in Fig. 2. There
are two divisions, E. | and W. J. The E. J is divided
into two equal squares called N.E. \ and S.E. \. The
W. J is divided into two equal squares called N.W. ^
and S.W. \. These are again subdivided as shown.
MEASUREMENTS
105
EXERCISE 64
1. Draw a figure and locate S.W. ^ of S.E. J; N.
W. \ of S.E. | ; N.E. 1 of S.W. J; S.E. J of S.
W.I
2. Locate N.E. J of the N.E. \ ; S.W. £ of N.E. -| ;
How many acres are in N.W. \ of S.W. | ? How many
acres are in N.W. J ? in S.E. * ? in S. J of S.W. £?
3. A man buys | of the S. J of N.W. ^ and also | of the
N.W. l of S.W. \ at the rate of $.20 an acre. Find the
cost of his purchase.
VOLUMES OF RECTANGULAR SOLIDS
The volume of a rectangular solid is obtained by taking
the product of its three dimensions expressed in units of
the same denomination.
EXERCISE 65
1. Find the volume of a box 12
ft. by 7 ft. by 6 ft.
2. Find the cubical contents of
a room 18 ft. by 12 ft. and 9 ft.
high.
3. Find the number of cubic
feet in a room 32 ft. by 24 ft. and 12 ft. high.
4. How many cubic yards of earth must be removed
for the foundation of a house 75 ft. by 54 ft., if the earth
to the depth of 21 ft. is removed ?
5. A cistern, in the shape of a rectangular solid, is 22
ft. by 14 ft. and 6 ft. deep. How many gallons of
water does it contain?
106 ADVANCED BOOK OF ARITHMETIC
6. A bin is 8 ft. by 3 ft. and 6 ft. high. How many
bushels does it hold ?
7. In order to build a concrete wall, earth is removed
to the depth of 6 ft. If the wall is 210 ft. long and 12 ft.
wide, how many cubic yards of earth must be removed ?
8. How many cubic yards of gravel are required to
fill, to the depth of 6 in., a street 1 mi. long and 36 ft.
wide ?
9. How many cubical boxes 2 ft. each way would a
storeroom 18 ft. by 12 ft. and 10 ft. high hold ?
10. The Sault Ste. Marie Canal is 1.6 mi. long, 160 ft.
wide, and 25 ft. deep. Express in cubic yards the volume
of water required to fill it.
11. A block of marble is 4 ft. by 3 ft. and 24 ft. long.
How many tons does it weigh, if a cubic foot of marble
weighs 170 Ib. ?
12. How many pounds does a cedar beam 14 in. by 10
in. and 40 ft. long weigh, if a cubic foot of cedar wood
weighs 38.1 Ib. ?
13. A cubic foot of clay weighs 75 Ib. Find, in tons,
the weight of a clay bank 10 ft. by 4 ft. and 80 ft. long.
14. A box 9 in. by 8 in. and 6 in. deep is filled with
mercury. Find its weight in pounds if a cubic foot of
mercury weighs 13,570 oz.
15. How many 3-in. cubes are required to fill a cubical
box each of whose edges is 1 yd. ?
16. A pile of 4-ft. wood 8 ft. long and 4 ft. high con-
tains a cord. How many cords of wood are in a pile of
4-ft. wood 120 ft. long and 12 ft. high ?
17. Find the weight of the water covering an acre to
the depth of 4 inches. 1 cu. ft. of water weighs 1000 oz.
MEASUREMENTS
107
FIG. 1.
FIG. 2.
A triangle is a portion of a plane bounded by three
straight lines. ABC is a triangle.
Two straight lines are parallel if
they can never meet no matter how
far they may be produced.
A quadrilateral is a portion of
a plane bounded by four straight
lines. Figure 2 represents a
quadrilateral.
A quadrilateral having its
opposite sides parallel is called
a parallelogram. Figure 3 is a
parallelogram.
The sides AB, CD are paral-
lel. Also the sides AD, BO
are parallel. t
Consider next the
parallelogram A B CD M_ D
(Fig. 4) and the rec-
tangle ABKM. By
actual trial the tri-
angle BKC is equal
to the triangle AMD.
Take the triangle
AMD from the figure ABCM, the parallelogram remains.
If the triangle BKQ is taken from the figure ABCM, the
rectangle remains. Hence, the rectangle equals the paral-
lelogram in area. But the area of the rectangle is obtained
by taking the product of its two dimensions, i.e. AB and
BK. Therefore, the area of the parallelogram is equal to
AB x BK. AB is called the base of the parallelogram ;
BK, i.e. the distance between the parallel sides, is called
the altitude, or height, of the parallelogram.
FIG. 3.
K
FIG. 4.
108
ADVANCED BOOK OF ARITHMETIC
FIG. 5.
Consequently, the area of a parallelogram equals the
product of its base by its altitude.
A quadrilateral having two sides parallel is called a
trapezoid. ABCD (Fig. 5) is a trapezoid, having AB
parallel to DC. AB
and CD are respec-
tively the lower and
upper bases of the
trapezoid.
Take a piece of
paper and make a
trapezoid. Make an-
other trapezoid just
equal to it.
^ Place them as shown
in Fig. 6. You then
have a parallelogram KBCL. Its area is equal to
-f- AB) x height of the parallelogram, i.e. equal to
AB) x height of the parallelogram.
The trapezoid is -|- of KBCL.
.-. the area of the trapezoid ABCD equals | of (DC +
AB) x the height of the trapezoid.
Consequently, the area of a trapezoid equals one half
the sum of its parallel sides multiplied by the distance
between them.
If one of the bases, e.g. DC, of a trapezoid were to
become smaller and smaller, the fig-
ure would ultimately be a triangle.
Hence, the area of a triangle equals
one half of its base multiplied by its
height.
This may also be seen readily from -
Fig. 7. FIG. 7.
MEASUREMENTS 109
EXERCISE 66
Find the areas of the following parallelograms :
1. Base 10' 6", altitude 6' 4".*
2. Base 17' 3", altitude 9' 8".
3. Base 12' 6", altitude 8' 3".
4. Base 15' 5", altitude 9' 4".
5. Base 36' 9", altitude 8' 4".
6. Base 40' 3", altitude 7' 6".
7. Base 27' 9", altitude 8' 7".
8. Base 28' 4", altitude 6' 3".
Find the areas of the following triangles :
9. Base 50' 3", altitude 23' 9".
10. Base 60' 4", altitude 42' 8".
11. Base 75' 6", altitude 35' 8".
12. Base 48' 4", altitude 29' 4".
13. Base 56' 9", altitude 27' 4".
14. Base 82' 6", altitude 64' 2".
15. Find the area of the upper surface of a board in the
form of a trapezoid whose parallel sides are 12' 6" and 5'
6" and height 8' 6".
16. Find the area of a trapezoid whose parallel sides
are 18' 4" by 12' 8" and 24' apart.
17. Find the area of a field whose parallel sides are
20 rd. and 36 rd. and width 18 rd.
18. Find the area of a lot whose shape is a trapezoid
having for parallel sides 60 ft. and 40 ft. and length 84 ft.
* 6 ft. 4 in., a notation used by engineers, architects, and mechanics.
110 ADVANCED BOOK OF ARITHMETIC
19. Find the area of a right triangle whose base is 26
ft. and altitude 19 ft. (A right triangle is a triangle
having a right angle.)
20. Find the area of a right triangle whose base is 96 ft.
6 in. and altitude 84 ft.
21. Find the area of a right triangle having for base
72 ft. 6 in. and altitude 63 ft. 9 in.
BOARD MEASURE
Lumber is measured by the board foot. A board foot is
a rectangular solid 12 in. by 12 in. and 1 in. in height.
If a board is 1 in. thick, the number of board feet in it
is measured by the number of square feet in its upper or
lower surface. Thus, a board 8 ft. long, 8 in. wide, and
1 in. thick contains 8 x f board feet. If a board is more
than 1 in. thick, the number of board feet in it is measured
by the area, in square feet, of its upper or lower surface
multiplied by the number denoting the thickness in inches.
Thus, a board 9 ft. by 14 in. and 2|- in. thick contains
9 x if x 21 board feet.
The number of board feet in a board less than 1 in. in
thickness is measured by the number of square feet in its
upper or lower surface. Thus, a board 10 ft. by 15 in.
and |- in. thick contains 10 x ^f board feet.
EXERCISE 67
1. How many board feet are in a board 8 ft. by 1J ft.
and 1 in. thick ?
2. How many board feet are in a board 9 ft. by 16 in.
and | in. thick? Find the cost of the board at 3|^ per
board foot.
MEASUREMENTS 111
3. How many board feet are in a plank 24 ft. by 15
in. and 2 in. thick ? In a board 20 ft. by 12 in. and 3 in.
thick ?
4. How many board feet are in a railroad tie 24 ft.
by 8 in. and 6 in. thick?
5. Find the number of board feet in each of the follow-
ing pieces of lumber:
(a) 18 ft. by 16 in. and 1J in. thick.
(5) 12 ft. by 8 in. and 6 in. thick.
0?) 24 ft. by 9 in. and 3 in. thick,
(d) 16 ft. by 8 in. and 4 in thick.
00 18 ft. by 12 in. and 3 in. thick.
(/) 21 ft. by 16 in. and 3 in. thick.
(jf) 30 ft. by 14 in. and 21 in. thick.
(i) 28 ft. by 15 in. and 3J in. thick.
6. Find the cost of 480 boards, each 11 in. by 10 in.
and 16 ft. long, @ 127.50 per M. ("Per M" means
" by the 1000 " board feet.)
7. Find the cost of 840 boards, each If in. by 10 in.
and 12 ft. long @ 1 25 per M.
8. How many board feet are in a board 24 ft. long, 12
in. wide at one end and 16 in. wide at the other end, and
2 1 in. thick ?
9. How many board feet are in a board 18 in. wide at
one end, 12 in. wide at the other end, 2J in. thick, and 28
ft. long ?
10. How many board feet are in a cubical block of
wood, each of whose dimensions is 2 ft.?
11. How many board feet are in a beam 9 in. by 9 in.
and 54 ft. long ?
12. How many board feet are in 16 railroad ties 8" by
6" and 8' 6" long ?
112 ADVANCED BOOK OF ARITHMETIC
13. How many cubic feet are equivalent to 120 board
feet?
14. How many board feet are in a beam 11 in. by 12 in.
and 36 ft. long ?
15. A piece of lumber contains 2980 cu. in. Express
this in board feet, understanding that the piece is more
than 1 in. in thickness.
16. Find the number of board feet in a beam 12 in. by
6 in. at one end, 9 in. by 6 in. at the other end, and 27 ft.
long.
MASONRY AND BRICKLAYING
Stone work is estimated by the perch, also by the cubic
foot.
A perch of masonry is 24f cu, ft. It is a wall 1 rd.
long, 1|- ft. wide, and 1 ft. high. Perch is, in Great
Britain, another name for rod. Of the 24| cu. ft. in a
perch, 22 cu. ft. are allowed for stone and 2| cu. ft. are
allowed for mortar, i.e. eight ninths for stone and one
ninth for mortar.
In estimating the number of perches of masonry in the
walls of a building, the outside dimensions of the walls
are taken. This method of reckoning counts the corners
twice. In estimating the amount of material, the com-
putation should be exact, inside and outside dimensions
being reckoned.
Twenty-two common bricks, i.e. bricks 8 in. by 4 in. by
2 in., are reckoned as a cubic foot.
EXERCISE 68
l* How many perches of masonry are in a wall 84 ft.
long, 16 ft. high, and 1J ft. thick ? How many common
bricks are required to build this wall ?
MEASUREMENTS
113
2. A cellar is 36 ft. long, 18 ft. wide, and 8 ft. deep.
How many cubic feet of masonry are in its walls if they
are 2 ft. thick ? What is the actual number of cubic feet
in the walls ?
3. How many bricks are necessary to build a wall 25
ft. long, 12 ft. high, and 24 in. in thickness ? What is
the cost of the bricks at $9 per M ?
4. How many cubic feet of mortar are required to build
a wall 600 ft. long, 10 ft. high, and 18 in. in thickness ?
How many common bricks are required to build this wall ?
5. At $9.50 per M, what is the cost of the bricks re-
quired to build a house 40 ft. by 36 ft. and 40 ft. high to
the eaves, the highest point of the gable end being 52 ft.
above the ground, and the walls 1^ ft. thick ? Allow 270
cu. ft. for openings.
CARPETING
1. How many yards of carpet are needed for a room
18 ft. by 12 ft., if the carpet is 27 in. wide ?
18ft.
12ft.
SOLUTION. (1) Let the carpet be placed crosswise.
Number of strips = 18 ft. -^ 27 in. = 18 ft. -*- 2£ ft. = 8.
.• . Number of yards = 12 x 8 -r- 3 = 32. Ans. 32 yd.
114 ADVANCED BOOK OF ARITHMETIC
(2) Let the carpet be placed lengthwise.
g ft Number of strips :
12ft. -5- 21 ft. =5^.
Here 6 strips are
needed. The fraction
12 ft. of a strip may be either
cut off or turned under.
.-. Number of yards
= 6 yd. x 6 = 36 yd.
Ans. 36 yd.
To find the number of yards of carpet required to carpet
a room:
1. Draw a diagram of the room.
2. Find the number of strips.
3. Multiply the number of strips by the number of
yards in one strip.
EXERCISE 69
1. How many yards of carpet are required to cover a
room 18 ft. by 16 ft. with carpet 30 in. wide, if the strips
are laid crosswise ?
2. How many yards of carpet f of a yard wide are
needed to carpet a room 24 ft. by 18 ft., if the carpet runs
lengthwise ?
3. How many yards of matting 36 in. wide are required
to cover a room 16 ft. by 12 ft., the matting being laid
crosswise ?
4. How many yards of carpet 30 in. wide are needed
to carpet a room 24 ft. long, 20 ft. wide, if the carpet is
laid crosswise ?
5. Find the cost of carpeting a room 16 ft. by 14 ft.
with carpet 27 in. wide at 90 ^ per yard, the strips being
laid lengthwise.
MISCELLANEOUS EXERCISES 115
MISCELLANEOUS EXERCISES
EXERCISE 70
1. How long will it take a person to earn $44.40, if
he earns 11.85 a day ?
2. If | of a quantity of coal is 71 J tons, how many
tons of coal are there in all ?
3. A train runs at the rate of 33^ mi. per hour.
How many hours will it take the train to run 350 mi.?
4. A man being asked his age replied, " Three elev-
enths of my age is 13 J years." Find the man's age.
5. A's share is f of the joint capital, or $2540. What
is the joint capital ?
6. A dealer sells a piano for $240, thereby gaining ^ of
the cost of the piano. What did the piano cost ?
7. If a dealer sold a piano for $240 and lost thereby -J
of the cost price, what did the piano cost ?
8. If the dividend is 18f and the quotient is 10J, what
is the divisor ?
9. The sum of two numbers is 25, and one of the num-
bers is 1^- times the other number. Find the numbers.
10. A farmer sells | of his cattle to one jobber and -| to
another jobber. If he keeps the remainder, what part .
does he keep ? Suppose the number remaining is 27, how
many head of cattle had the farmer originally ?
11. A dealer sells | of his coal, and then ^ of it, and has
12|- tons left. How many tons had he originally ?
12. If the ^ part of a number and the 1 part of the same
number together make 60J, what is the number ?
13. If .85 of A's money is $ 289, how much has he ?
116 ADVANCED BOOK OF ARITHMETIC
14. By gaining .15 of his outlay a man's property
amounts to $7245. What was his outlay?
15. Five sevenths of a farm is sold for $1385. What
is the value of the farm ?
16. How many times will a wheel 12 ft. 4^ in. in cir-
cumference revolve in going 3| mi. ?
17. How long will it take a person to go 10| mi. at
the rate of 6^ mi. per hour ?
18. A train travels at the rate of 52^ mi. in If hr.
Find its rate per hour. How many hours and minutes
will it take this train to travel 208 mi. ?
19. From the ground to the first floor of a house is -^
of the height of the house; if the first floor is 10 ft. above
the ground, how high is the house ?
20. What part of 4f is J- of 14-| ?
21. If .625 of a gallon of maple sirup cost $1.25, what
will 1 gal. of maple sirup cost ?
22. When 3 oranges sell for 5^, how many oranges can
I buy for 80^?
23. If 8^ yd. of calico cost 66^, how many yards can
I buy for $3.30?
24. A merchant, by selling tea at 69^ per pound, gains
2^- of the cost of the tea. Find the cost per pound of the
tea.
25. By selling cloth at 78^ per yard a clothier loses ^
of the cost of the cloth. Find the cost price per yard of
the cloth.
26. The rent of a house for 17 mo. is $225. Find the
rent of this house for 12 mo.
27. Lead pencils cost 3^ each. At this price, how
many can be bought for $1.40?
MISCELLANEOUS EXERCISES 117
28. A laborer gets 4 ^ per cubit foot for digging a cel-
lar. At this rate ho^ much will he get if the cellar is
15 ft. by 12 ft. and 6 ft. deep?
29. If 3 qt. of oil cost 13 J ^, at this rate how much will
1 gal. 2 qt. cost ?
30. If 1 Ib. 8 oz. of cheese cost 42 ^, find the cost of 4 Ib.
31. How many tiles 6 in. square will be needed to pave
a hearth 5 ft. by 2 ft. ?
32. Express ^ + -| -f ^ as a decimal.
33. Reduce 5.875 hr. to seconds.
34. Find the cost of a car load of coal weighing 35,880
Ib. at $5.50 per ton.
35. A man having a salary of $1500 a year spends
25% of it for board, 10% for clothing, and 15% for other
things. How much money does he spend ?
36. Reduce 2876 in. to yards.
37. Find the value of 32 tubs of butter each weighing
56 Ib. at 37| # per pound.
38. The wages of a motorman on the Metropolitan
West Side Elevated Railway, Chicago, are 30^ ^ per hour.
How much does he earn in ten weeks, working 8 hr. a
day ?
39. How many feet are in 87-| % of 1 mile ?
40. How many square yards are in 62|-% of 1 A. ?
41. A farmer having 320 bushels of apples sells 75% of
them. How many bushels does he sell ?
42. How much will a man earn in 10 weeks at $1.50
per day, Sundays excepted ?
43. A cubit foot of air weighs .08073 Ib. Find the
weight of the air in a room 15 by 16 ft. and 9 ft. high ?
44. How many grains are in 83^% of 1 Ib. Troy ? ,
118 ADVANCED BOOK OF ARITHMETIC
45. A railroad train runs 1 mi. in 1 min. 15 sec. Find
its rate per hour.
46. Find the number of acres in a field 90 rd. by 40 rd.
47. Find the cost of 18,750 bd. ft. of lumber at $30 per M.
48. Find the commission on sales amounting to $ 4750
at 2%.
49. A man borrows money on April 1, and agrees to
pay it in 90 days. On what date should he pay it ?
50. How many square inches are in the surface of an
18-in. cube?
51. Find in square rods 15% of 1 A.
52. Out of a class of 64 pupils 12J% failed to be pro-
moted. How many were promoted ?
53. A lot is 42 ft. by 120 ft. Find the cost of fencing
it at 85 ^ per yard.
- 54. Find the cost of making a concrete sidewalk 63 by
12 ft. at $ 1.75 per square yard.
55. Express in minutes .075 of 1 day, 16 hours.
56. A railroad train runs 72 ft. in one second. At
this rate how far will it run in one hour ?
57. Find the cost of plastering the walls of a room 18
ft. by 20 ft. and 10 ft. high at 25^ per sq. yd.
58. A watch loses 1 min. and 5 sec. in 3 da. At this
rate how much will it lose during the month of April ?
59. How many pounds avoirdupois are in 420 Ib.troy ?
60. A tank is 8 ft. by 6 ft. and 7 ft. deep. How many
gallons does it hold? (1 cu. ft. = 7J gal. nearly.)
61. Find the cost of 24 yd. of cloth at f 1.87J- per yd.
62. How many cords of wood are in a pile 50 ft. long,
12 ft. wide, and 8 ft. high ?
MISCELLANEOUS EXERCISES 119
63. When coal sells for $ 7.00 per ton, what is the cost
of a sack of coal weighing 200 Ib. ?
64. A bin is 18 ft. long, 6 ft. wide, and 4 ft. deep. How
many bushels will it contain? (1 bu. = 1-J- cu. ft.)
65. Change to decimals J£, ||, ^f 7, f .
66. Change to fractions in lowest terms .875, .00525,
.66|.
67. Express in simplest form
7 x 3.04 x 10,000 - 125,000.7.
68. A man rents a house for $ 550 a year. His rent the
previous year was 9% less. What was his rent last year?
69. A grocer having on hand 15 gal. 2 qt. 1 pt. of oil
buys 20 gal. 2 qt., and sells 30 gal. 2 qt. 1 pt. How
much has he left ?
70. Find %\% of 1 cu- ft. Give result in cubic inches.
71. The following is the lowest bid in detail for improv-
ing Lick Run Pike, Cincinnati, Ohio, submitted July
19, 1907 :
20,000 cu. yd. embankment @ $ .018J
500 cu. yd. excavation @ $1.00
1530 cu. yd. stone @ $2.25
265 cu. yd. screening @ $2.25
100 cu. yd. cement @ $8.00
400 ft. 12" pipe @ $ 1.00
10 ft. 12" slant @$. 80
30 ft. 4" box culvert concrete @ $ 6.00
60 ft. 5" box culvert concrete @ $7.00
2350 ft. 7" box culvert concrete @ $16.00
650 ft. 8" box culvert concrete @ $18.00
18,400 sq. yd. rolling @ 3^
6 manholes® $40.00
Find the amount of the bid.
120 ADVANCED BOOK OF ARITHMETIC
REVIEW OF FRACTIONS
EXERCISE 71
Add:
1> 6' A' ITP lV 7<
2. 2f,324,4ix. 8.
3- i> iV' A' iV 9- I' I3!' TO'
4. If, iWi'liV 10. ^,3T
6. 2i|, 2||, 42V 12. 2TV, 3if, 21 if .
Find the difference between :
13. 4T\ and If. 22. J| and if
14. l^andTW 23. Handlf
15. l^andTV 24. 7f and 311
16. -3*5- and T^. 25. 8 1 and 711.
17. 1 and -^L. 26. 9T52 and 6if .
18. 1 and ^g-. 27. 5T7g and 2||.
19. -^ and f . 28. 6T^ and
20. fandf 29.
21. 11 and if 30. 12^- and 8|f.
31. What number must be added to 3f to give 5f ?
32. A man's capital amounts to $ 1727J. By how much
must he increase his capital so that it may amount to
$3000?
33. What number must be taken from 30| to leave 14|?
34. By how much does 84^ exceed 17|?
35. Of the weight of the earth's atmosphere y%3oV ^s
oxygen. What fraction of the weight of the earth's
atmosphere do its other constituents aggregate ?
REVIEW OF FRACTIONS
121
Simplify :
1. foffof2f
2. f ofT^of4f
3. I of 1| of £$.
b£4f
EXERCISE 72
TT
xT3Tx4|.
4.
5.
6.
7.
8.
9.
10. 2|x3^x
11 ^1 V ^1 V
-I.J.. t-JS" /N '•^9' ^
12. 2|x2|x
13. 74-xiSx
14. 9^
15. 1\
16. (i
18.
19.
of 4| of If 20.
v £\1 v 1 1 0 O1
S\ ^^ /N J-"5-5". <iJ..
of 1| of
xlfxlfxl2V
22. 2ix(lf + 2i + l£)
23.
24.
25.
26.
EXERCISE 73
Divide :
l. 25fbylf.
2. 3lby2|.
3. IQi by Iff.
4. 7T5g by 2921
9- tobyT5iofll.
10. 2l| by 1|| of If.
ll. If by f of If
12. Sy3^ by Iy8y times
5. 521 by 1311 13. (T9__l)by(| + l).
6.
7.
of6f.
by t of
14.
15.
BiV-^bySl
8. l by f of If 16. lllby(l| + ll).
17. If the dividend is 6J and the quotient 1|, find the
divisor.
18. What must 10| be divided by to give as quotient 2|?
122 ADVANCED BOOK OF ARITHMETIC
PERCENTAGE
The result of taking a rate per cent of a quantity is
percentage. A percentage of a number is simply a fraction
of the number. The central fact in percentage is that
100 % is equivalent to 1, or 1 % is equivalent to yj^-.
What per cent is -|| equivalent to ?
^ofl = |f of 100% = 95%.
Example 1. Find 2f % of 1789.
17.89=1% of 1789.
_ _
35.78 2 times 17.89.
8.945 i of 17.89.
4.4725 of 17.89.
49.1975 = 2| times 17. 89.
Example 2. Find 3.6 % of 2992.
2992 x M = 2992 x .036 = 107.712, or
1 % of 2992 = 29.92 ; 29.92 x 3.6 = 107.712.
EXERCISE 74
Find:
1. 7% of 184. 10. 10% of 7850.
2. 9% of 275. 11. 11% of 983.
3. 6% of 213. 12. 12% of 2570.
4. 8 % of 534. is. 12| % of 928.
5. 9% of 3280. 14. 231% Of 5220.
6. 8% of 3297. is. 16f % of 733.
7. 4% of 615. 16. 37J%of828.
8. 7 % of 2630. 17. 62| % of 9200.
9. 9% of 4280, 18. 60% of 2855.
PERCENTAGE 123
19. 4-J% of 2280. 22. 5f % of 10,656.
20. 3f % of 3066. 23. |-% of 1690.
21. 6J% of 1820. 24. 1% of 8124.
25. Express the following fractions as per cents:
i' i> i> i> iV» A' iV-
26. Express each of the following decimals as a per
cent:
.04, .08, .121, .0165, .002, .006, .0024.
27. The composition of a piece of coal taken from the
Texas and Pacific Coal Company's mine is given as fol-
lows: moisture, 5.46% ; combustible matter, 35.66% ;
fixed carbon, 49.17% ; ash, 9.71%. Find the amount of
each constituent in 2000 Ib. of coal. Check your answer.
28. The analysis of a specimen of lignite is given as
follows: moisture, 29.07%; combustible matter, 28.96% ;
fixed carbon, 24.47% ; ash, 17.50%. Find the amount of
each constituent in a ton of lignite. Check.
29. Distilled water is composed of two gases, H|% by
weight being hydrogen, and 88f % by weight being oxy-
gen. Find the weight of each gas that can be obtained
from 10 Ib. of water.
30. A bookkeeper receives a salary of $ 1800 per annum.
If he spends 62 1% of his salary, and saves the remainder,
how much does he spend ? How much does he save ?
31. An auriferous ore contains .5% of gold. How
many pounds avoirdupois of gold would 2240 Ib. of this
ore yield ?
32. A copper ore contains 5J% of copper. Find the
number of pounds of copper in 500 Ib. of this ore.
33. A owns 16|% of a boat valued at $12,300. What
is the value of A's share of the boat ?
124 ADVANCED BOOK OF ARITHMETIC
34. The estimated value of the exports of the United
States for 1899 was $1,275,000,000. The percentages of
exports from United States ports for that year are given
as follows: New York, 37.4%; Boston, 10.43%; Phila-
delphia, 5.05% ; Baltimore, 8.9%; New Orleans, 6.4%;
Galveston, 7.17%. Find the value of the exports from
these cities.
Given a number as a per cent of some other number,
to find the other number.
Example 1. If 15.5% of a number equals 22.785, what
is the number ?
15.5% of the number = 22. 785
22.785
15.5
1 % of the number =
.-. 100 % of the number = x 100 = 147.
15.5
/. the number is 147.
Example 2. If 3| % of a number is 2934, what is the
number ?
.
100 400 80
-g^j- of the number = 2934.
.*. -g1^ of the number = 978.
.-. fj. of the number = 78,240.
/.the number is 78,240.
EXERCISE 75
1. If 5 % of a number equals 185, what is the number ?
2. If 12| % of a man's salary is $156, what is the man's
salary ?
3. If 37^ % of a man's property is valued at $324, what
is the value of his property ?
PERCENTAGE 125
4. The number 360 is equal to 5% of what number?
6% of what number? 8% of what number? 9% of what
number ?
5. The number 120 is equal to 3% of what number?
4% of what number? 5% of what number? 6% of what
number? 8% of what number? 12% of what number?
6. $750 is 6% of what sum of money? 6|% of what
sum of money? 7-|-% of what sum of money? 12|% of
what sum of money ?
7. $108 is 30% of what sum? 25% of what sum?
331% of what sum? 44|% of what sum?
8. $450 is 6^% of what sum? 6|% of what sum?
8J% of what sum? 12^% of what sum? 16 f% of what
sum ?
9. $420 is 37^% of what sum? 62|% of what sum?
87^% of what sum?
10. In 1900 the commercial value of silver was 3% of
the value of gold. Find the number of ounces of silver
equivalent in value to 126 ounces of gold.
11. Of the population of the United States in 1900,
27,849,760 were married. This number was 36.5% of the
population. Find the population in 1900.
12. The census of 1900 gives the number of married
men in the United States as 14,003,798. This number
was 35.9% of the number of males. Find the male popu-
lation.
13. The census of the same year gives the number of
married women in the United States as 13,845,963. This
number was 37.2% of the entire number of females.
Find the female population.
126 ADVANCED BOOK OF ARITHMETIC
14. Of the number of illiterates above 10 years of age in
the United States 15.5%, according to the census of 1900,
can neither read nor write. This number is 955,840.
Find the number of illiterates above 10 years of age in the
United States in 1900.
15. Of the average value of the raw cotton exported
from the United States in 5 years 50.4% went to Eng-
land. If the export of raw cotton to England amounts in
value to 1107,500,000, find the average value of the raw
cotton exported.
16. Of the sheep exported from the United States
86% are shipped to England. If the value of the sheep
shipped to England in a certain year was $1,685,800,
find the total value of the export of sheep for that
year.
17. Of the pupils attending school in a certain city .6%
are in the senior class of the high school. If the senior
class numbers 21, find the number of pupils attending
school in that city.
18. In a certain city the number of pupils promoted
at the end of the scholastic year was 2765. This
number was 79% of the number of pupils in school.
Find the number of pupils attending school in that
city.
19. The foreign -born population of a city is 10,668.
This number is 3^% of the population of the city. Find
the population of the city.
20. Of the water of the Dead Sea 22.8?T% is saline
material. If a quantity of Dead Sea water is evaporated,
and the saline material left behind weighs 914.28 lb., what
is the weight of the water before evaporation ?
PERCENTAGE 127
To find -what per cent one number is of another.
Example 1. What per cent of 12 is 5 ?
5 is of 12.
Example 2. The value of the property of the Fort
Worth and Denver City Railroad in Texas, as ascertained
by the Railroad Commission of Texas, for the year 1906
was $5,771,600, and the income from its operation was
$1,178,040. Find the rate per cent of income from opera-
tion.
4=20.4,,
To find what per cent one number is of another, find what
fraction the first number is of the second and multiply by
100%.
EXERCISE 76
1. What per cent of 15 is 12?
2. What per cent of 20 is 4 ? is 7 ? is 11 ? is 13 ?
3. What per cent of 40 is 2 ? is 8 ? is 12 ? is 17 ?
4. What per cent of 90 is 9 ? is 12 ? is 27 ? is 11| ?
5. What per cent of 120 is 15 ? is 18 ? is 45 ? is 80 ?
6. What per cent of 480 is 28.8? is 33.6? is 40 ?
7. What per cent of 1728 is 345.6 ? is 155.52 ? is 288 ?
8. What per cent of 231 is 21 ? is 77 ? is 34.65 ?
9. What per cent of 5280 is 88 ? is 440 ? is 330 ?
10. What per cent of a bushel is a quart ?
11. What per cent of a mile is 8 rd. ?
12. What per cent of an acre is a square rod ?
13. What per cent of a chain is a yard ?
14. What per cent of 1 sq. ch. is 1 sq. rd, ?
128
ADVANCED BOOK OF ARITHMETIC
15. What per cent of 1 gal. is 1 pt.?
16. What per cent of 1 sq. rd. is 7 sq. yd. 5 sq. ft.
9 sq. in.?
17. What per cent of a rod is 1 yd. 2 ft. 6 in.?
18. What per cent of 1 mi. is 1 knot?
19. What per cent of 1 mi. is an arc of 1' measured on
the 40th parallel of latitude ? (!' = 4670 ft.)
20. What per cent of 1 mi. is 22 yd.? is 176 yd.?
21. What per cent of 1 sq. mi. is the S. W. 1 of N.E.
-| of a section of land ?
22. What per cent of a common year is 73 da. ? is 219
da.? is 292 da.?
23. A meter is 39.37 in. What per cent of a meter is
1 yd. ? What per cent of 1 yd. is 1 meter?
24. A kilometer is 1000 meters. What per cent of
1 mi. is 1 km.?
25. What per cent of 1 Ib. avoirdupois is 1 Ib. troy?
26. What per cent of 1 oz. avoirdupois is 1 oz. troy?
27. What per cent of the area of each of the following
states consists of irrigated land?
ACRES IRRIGATED
AREA IN SQUARE MILES
(a) California .
1,446,114
158,360
(b) Colorado
1,611,270
103,925
(c) Louisiana .
201,685
48,720
(d) Montana .
951,054
146,080
(e) Nevada
501,168
110,700
(/) Oregon .
388,110
96,030
(</) Utah ....
629,290
84,970
(Ji) Washington
135,470
69,180
(0 Wyoming .
605,230
97,890
PERCENTAGE
129
28. Find the increase per cent in the population of each
of the following cities for the ten years from 1890 to 1900:
POPULATION IN 1890
POPULATION IN 1900
Mobile .....
31,076
38,469
Little Rock ....
25,874
38,307
Los Angeles ....
50,395
102,479
Denver
106,713
133,859
Pensacola ....
11,750
17,747
Savannah ....
43,189
54,244
Springfield ....
24,963
34,159
Evansville ....
50,756
59,007
Dubuque ....
30,311
36,297
Kansas City, Kan.
38,316
51,418
Lexington ....
21,567
26,369
Kansas City, Mo. .
132,716
163,752
Minneapolis
164,738
202,718
29. The total production of butter in the United States
in 1899 is estimated at 1,430,000,000 Ib. Of this quantity
16,002,000 Ib. was exported. Find the per cent of the
total production exported.
30. For the year 1899 the total production of cheese in
the United States is estimated at 300,000,000 Ib. Of this
quantity 27,203,200 Ib. was exported. Find the per cent
of cheese exported.
31. The number of farms in the United States June 1,
1900, was 5,739,657. The number of farms operated by
owners was 3,713,371. The number of farms operated by
tenants was 752,920. The number of farms operated
by share tenants was 1,273,366. Find the per cent oper-
ated by owners, by tenants, and by share tenants respec-
tively.
130 ADVANCED BOOK OF ARITHMETIC
COMMERCIAL DISCOUNTS
Commercial or trade discount is an allowance made upon
the list price of goods, or on the amount of a bill.
Discounts are reckoned upon the basis of 100 ; in other
words, as so many per cent.
Example. If watches are listed at $35 each, and a
discount of 10 % is allowed, what is the cost of one of the
watches ?
SOLUTION. 10 % of $35 = $3.50 = the discount.
$35 -$3.50 = $31. 50.
.-. the cost is $31.50.
EXERCISE 77
Find the cost when the list prices and rates of discount
are:
1. $60, 20% off. 16. $175, 2% off.
2. $75, 30% off. 17. $213, 4% off.
3. $15, 25% off. 18. $723, 6 %off.
4. $14, 20% off. 19. $3280, 8% off.
5. $24, 10% off. 20. $2712, 7% off.
6. $32, 15% off. 21. $5350, 8J% off.
7. $68, 121% Oif. 22. $2490, 6-|% off.
8. $78, 16|%off. 23. $3778, 9% off.
9. $92, 331% off. 24. $5062, 12% off.
10. $80, 37|% off. 25- $885, 15% off.
11. $152, 121% off. 26. $363, 12% off.
12. $176, 25% off. 27. $689, 8% off.
13. $88, 5% off. 28. $2034, 7% off.
14. $96, 5% off. 29. $992, 7-|% off.
is. $125, 15% off. 30. $2572, 4J% off.
COMMERCIAL DISCOUNTS 131
COMMERCIAL DISCOUNTS WHEN TWO OR MORE ARE
ALLOWED
In some cases several discounts are allowed. If there
are two or more discounts, the first is reckoned on the list
or catalogue price ; the next is reckoned on the remainder
after deducting the first discount ; the third is reckoned
on the second remainder ; and so on.
Example 1. What is the cost, if the list price is $ 750,
and discounts of 20%, 15%, and 10% are allowed?
SOLUTION. First discount = 20% of $750 = | of $750
= $ 150. .-. the first remainder = $ 750 - $ 150 = $ 600.
The second discount = 15 % of 1600 = $90.
$600 -$90 = $510.
The third discount = 10 % of $510 = $51.
$510 -$51 = $459. Or
100 % - 20 % = 80 % ; 100 % - 15 % = 85 % ;
100% -10% =90%.
90 % of 85 % of 80 % of $750 = $459.
Example 2. What single discount is equivalent to the
three discounts in the above example ?
SOLUTION. 80% x 85 % x 90% = .8 x .85 x .9 = .612
= 61.2%.
100 % - 61.2 % = 38.8 % Ana.
EXERCISE 78
1. A suit of clothes is marked $70, and is sold with
discounts of 25 % and 10 % for cash. Find the selling
price.
2. On a bill of $ 900 two discounts of 20 % and 15 %
are allowed. What is the net amount of the bill?
3. If goods are marked $175 and sold for $122.50,
what is the discount ?
132 ADVANCED BOOK OF ARITHMETIC
4. On a bill of $1500 discounts of 25 /o, 15 /o, and
are allowed. Find the net cash amount of the bill.
5. A piano is listed at $450, with discounts of
12J-^, and 10/>. Find the cost price to the purchaser.
6. Find the cash value of a bill of $320 with discounts
of 15/o, 10/o, and 5/o.
7. Suppose you were offered a single discount of 45/>,
or two discounts of 30 /> and 20 Jfc, which would you take ?
What would be the difference in a bill of $1000 ?
8. A dealer buys a quantity of goods marked $550,
with a discount of 20/>. If he sells the goods at 6jfc above
the marked price, what is his gain per cent ?
9. If I buy goods listed at $150, writh a discount of
10/>, and sell them at 12 fi above the marked price, find
my gain per cent.
10. A bookseller buys 100 books marked $1.50 each, at
a discount of 20 /> and sells them at the marked price.
What is his gain per cent ?
11. A bookseller bought 100 books at $1.25 each. He
make a profit of 20 /> after giving a discount of ^. At
what price did he mark each book, and what was his profit ?
12. Find the cost price in each case, if the list price and
the rates of discount are as follows :
LIST PRICE DISCOUNTS
O) $480 20/o, 80 #, 10 /o
(6) $1000 40/o,5^,4/o
0) $775 25/o, I
(i) $880 12i/o,4/o
00 $720 £, 16/o
(/) $960 12i?o, 12 Jb
(#) $1200 10^, 8/0, 5 Jfe
Qi) $1760
PROFIT AND LOSS 133
PROFIT AND LOSS
In actual business, gains and losses are reckoned as a
per cent of the cost price.
Example l. How much does a person gain by buying
360 yd. of cloth at $1.30 per yd. and selling it at a
profit of 15 % ? What is the selling price per yard ?
SOLUTION. 360 x $1.30 x 15 % = $70.20, gain.
$1.30
.13 =10% of $1.30
.065= 5% of $1.30
$1.495 = selling price per yd.
Observe, selling price per yd. is 115 % of cost price.
Example 2. A dealer buys apples at $1.75 per barrel,
and sells them at $2.10 per barrel. Find his gain per cent.
SOLUTION. $2.10 - $1.75 = $.35.
j|p~ = 1. The gain is £ of the cost. | = 20%.
EXERCISE 79
1. Find the selling price of articles, the cost prices
and rates per cent of profit being given as follows :
COST PRICE RATE PER CENT OP PROFIT
(a) $150 12%
(i) $75 25%
00 $31 7i%
(cT) $215 16|%
O5 $540 27%
GO $318 S£%
(<7) $512 18|%
(A) $234 15%
(0 $457
134 ADVANCED BOOK OF ARITHMETIC
2. Find the rate per cent of profit or loss, if the cost
prices and selling prices are given as follows :
COST PRICE SELLING PRICE
(a) $ .20 9 .25
(5) $ .22 $ .20
0) $ .90 $1.50
(df) $2.10 $1.40
0) $125 $160
Given selling price and rate per cent of profit or loss, to
find cost price.
Example l. A piano was sold for $450 at a profit of
121 % . Find the cost price.
(100 % + 12i %) of cost = 1121 cf0 of cost = f of cost.
| of cost = $450.
Therefore, cost =$450 •*- § = $400.
Example 2. A dealer sells goods for $200.56 at a loss
of 8%. Find the cost price.
(100 % - 8 % ) of cost = 92 % of cost.
92% of cost = $200. 56.
... cost = »~ 1218.00.
EXERCISE 80
1. Find the cost price, the selling price and rate per
cent of profit being given as follows :
SELLING PRICE RATE PER CENT OF PROFIT
(a) $63 25%
(6) 1143 8£%
O) $54 8%
(e) $205
(/) $315 5%
PROFIT AND LOSS 135
2. Find the cost, if the selling price and the rate per
cent of loss are given :
SELLING PRICE RATE PER CENT OF Loss
(a) $41.30 10%
(8)' $87.50 20%
(<?) $90 30%
(d) $45.50 50%
O) $59.90 331%
(/) $33.60 16|%
(#) $55.80 20%
(Ji) $253 12|%
Example l. A man sold a horse for $126, thereby los-
ing 20%. What should have been the selling price to
make a profit of 15 % ?
SOLUTION. The selling price is 80 % of cost.
To make a profit of 15%, the selling price should be
115 % of cost. In this problem 80 % of a number is
given and 115 % of it is required. Hence,
115 x r= $181.125= required selling price.
80
HJxample 2. How should goods be marked so that a
dealer may give a discount of 20 % off and still make a
profit of 15 % ?
SOLUTION. (100 % — 20 %) of marked price = 80 % of
marked price.
Cost + 15 % of cost = 115 % of cost.
% of marked price = 115 % of cost.
1 % of marked price = '° of cost.
80
.-. 100 % of marked price = L x 100 = 143| % of cost.
80
The goods must be marked 43| % above cost.
136 ADVANCED BOOK OF ARITHMETIC
EXERCISE 81
1. A sold a lot for $3835 at a gain of 18%. Find
the cost.
2. By selling a piano for $270, a dealer loses 10%.
Find the cost of the piano.
3. By gaining 25 % of his capital, a merchant increases
his capital to $4550. What was his original capital ?
4. If goods are bought for $20 and sold for $22.50,
what is the gain per cent ?
5. If the cost is $48 and the selling price is $52, what
is the gain per cent ?
6. If the cost is $190 and the selling price is $152,
what is the loss per cent ?
7. If the cost is $145 and the selling price is $159.50,
what is the gain per cent?
8. If the cost is $118 and the selling price is $128.62,
what is the gain per cent?
9. If the selling price is $106.70 and the gain is 10 %,
what is the cost ?
10. If the selling price is $75.25 and the loss is 121 %
what is the cost ?
11. If the selling price of a rug is $90 and the gain is
20 %, what is the cost of the rug?
12. If the selling price is $89.25 and the loss is 15%,
what is the cost ?
13. If the selling price is $95 and the gain is 18|%,
what is the cost? What is the profit ?
14. By selling a horse for $168, a man gains 40% of
the cost. What is the cost of the horse ?
15. By selling velvet at $4.55 a yard, a clothier makes
a profit of 8-| % . Find the cost per yard of the velvet.
PROFIT AND LOSS 137
16. At an auction a dealer buys goods at 20 % below
the market price. If he sells these goods at the market
price, what is his gain per cent ?
17. A lawyer collects a debt. He charges 3 % for col-
lection, and remits to his client, after deducting his fee,
$952.54. Find the amount of the debt.
18. If property was sold for $11,778, at a loss of 2|%,
what was the value of the property ?
19. Two horses were sold for $ 200 each, one at a gain
of 20 % and the other at a loss of 20% . Did the seller gain
or lose by the transaction, and how much ?
20. A sells goods to B at a profit of 10%, and B sells
them to C at a profit of 15%. If C paid $253 for the
goods, find A's cost price.
21. A merchant increases his capital 18|%, and at the
end of a second year he also increases his capital 18| % .
If his capital is then $ 14,440, what was it at first ?
22. The first year A adds 25 % to his capital, the next
year he adds 25 % to the capital of the previous year, the
third year he loses 40% of his capital, and is then worth
$10,800. How much capital did he begin with ?
23. If it takes $81,415 for the running expenses of the
schools of a city, and if 95 % of the tax levied for school
purposes goes to the support of the schools, what should
be the amount levied for school purposes ?
24. By selling tea at 65^ per pound a merchant gained
$118.75. If his gain was 62|- %, how many pounds of tea
did he sell ?
25. By selling turkeys at $2.50 each a dealer makes a
profit of 60%. What would have been his gain per cent
had he sold the turkeys for $1.75 each?
138 ADVANCED BOOK OF ARITHMETIC
26. By selling wine at $2.10 a gallon, a merchant makes
a profit of 20 %. What would be his gain per cent should
he sell the wine for $2 a gallon ?
27. A merchant mixes tea which cost him 60^ per pound
with tea which cost him 70^ per pound in the proportion
of 5 Ib. of the former tea to 6 Ib. of the latter. If he
sells the mixture at 80^ per pound, find his gain per cent.
28. A horse is sold for $145.50, at a loss of 3%. At
what price should the horse have been sold so as to make
a profit of 10 % ?
29. If oranges are bought at 5 for 3^ and sold at 3 for
5^, what is the gain per cent ?
30. If 325 Ib. of sugar are bought for 1 13, at what
price per pound must it be sold to make a profit of 25 % ?
31. If 15 yd. of silk are bought for $26.25, at what
price per pard should it be sold to make a profit of 20 % ?
32. By selling cloth at the rate of 17 J^ a yard a
clothier loses 12|%. What should the selling price per
yard be so as to make a profit of 20 % ?
33. A merchant buys butter at 18^ per pound and sells
120 Ib. for $25.20. What is his gain per cent ?
34. A grocer buys cheese at 9^ per pound and sells it
at the rate of 8 Ib. for $1. What is his gain per cent ?
35. A merchant mixes two kinds of tea in the ratio 5 : 2.
If the teas are worth respectively 68 ^ and 75^ per pound,
what should be the selling price per pound so as to make
a profit of 20 % ?
36. If I buy a horse for $80 and sell him for $71, what
is the loss per cent ?
37. By selling a horse for $83.30 I lost 15%. What
did the horse cost ? '.
PERCENTAGE 139
38. Two men, A and B, buy two horses, each paying the
same price. A sells his horse for $90 at a loss of 14| %.
B sells his horse at a loss of 6 % . Find the selling price
of B's horse.
39. A merchant buys 70 yd. of cloth at 50 ^ a yd. and
sells it at a gain of 15 %. He buys also 70 yd. of silk at
90 / a yard. On both he gains 10 %. At what price per
yard does he sell the silk ?
40. I buy a quantity of barley and intrust it to an agent
to sell. The agent sells it at an advance of 25 % on the
cost, and after deducting a commission of 2% he remits
the balance, $539. How much did the barley cost me ?
41. A fruit dealer buys a crate of oranges for $2.50.
He sells them at 2/ each, making a profit of 60 %. How
many oranges are in the crate ?
42. If goods are bought at 25 % below the retail price
and sold at the retail price, what is the gain per cent?
43. A coal dealer buys 120 T. of coal at $4 a ton. He
sells ^ of the coal at an advance of 25 %, ^ of it at an ad-
vance of 50%, and the remainder at an advance of 10%.
Find his entire profit and his gain per cent on the coal.
44. (a) How should goods be marked so as to make a
profit of 12 % after deducting 20 % from the marked price ?
(&) How should they be marked so as to make a profit
of 17 % after deducting 10 % from the marked price ?
(e) How should they be marked so as to make a profit
of 20% after deducting 25% from the marked price?
(d) How should they be marked so as to make a profit
of 10 % after deducting 10 % from the marked price ?
(e) By taking 30 % off the marked price a merchant
neither gains nor loses. How were the goods marked ?
140 ADVANCED BOOK OF ARITHMETIC
COMMISSION AND BROKERAGE
A very large proportion of the buying and selling of
the produce of the country is done through commission
merchants and brokers. A commission merchant is
usually intrusted with the goods bought or sold. A
broker merely buys and sells.
The fee which a commission merchant charges for his
services, or which an agent who buys and sells land, or
collects rents, charges, is called a commission. The fee
which a broker charges is called brokerage.
Commission and brokerage are usually reckoned as a
rate per cent of the buying price when goods or other
commodities are bought, or of the selling price when they
are sold.
The person who employs another person to buy, to sell,
or to collect money is called a principal. The person who
transacts business for a principal is called an agent. Com-
mission merchants and brokers act in the capacity of
agents.
Example l. A commission merchant sold 400 boxes of
oranges at $2.75 per box, charging 6% commission.
What was his commission, and how much did he remit ?
The price of 400 boxes of oranges @ $2.75 = $1100.
5 % of $1100 = $55, the commission.
$1100 - $55 = $ 1045, amount remitted.
Example 2. A commission merchant charges $94.50
for selling 4500 bushels of wheat. Rate of commission
2|%. What is the price of wheat per bushel ?
21 % = ^. Selling price = $ 94.50 x 40 - $ 3780.
Selling price of one bushel = $||f§ = $.84.
COMMISSION AND BROKERAGE 141
Example 3. A real estate agent remits to his principal
$9788.75, being the amount of the sales of four city lots
after deducting a commission of 4 J % . Find the selling
price of the four lots and the agent's commission.
SOLUTION. The selling price — 4 £ % of the selling
price = 95-| -% of the selling price.
95J % of the selling price = $9788.75.
ffi» Q7QQ 7^
.-. 1 % of the selling price = — — -^ —
t/O^
.-. 100 % of the selling price = $9788.75 x 100 _ $i0,250.
.-. $10,250 - $9788.75 = $461.25, the commission.
EXERCISE 82
1. What is the commission for selling 200 A. of land
at $40 an acre, the rate of commission being 4 % ?
2. An agent charges 6 % for selling real eatate. If he
sells 350 A. of land at $50 an acre, find his commission
and the sum he remits to his principal. What per cent of
the selling price does the principal receive ?
3. If 200 boxes of oranges are sold at $3.75 a box by a
commission merchant who charges 6%, what is the com-
mission ? How much is remitted to the principal ?
4. If 800 bu. of wheat are sold on a commission of J^
per bushel, what is the commission ?
5. How many bushels of barley at 45^ per bushel
must a commission merchant sell at 2J % commission to
make an annual salary of $1080 ?
6. If 480 bbl. of flour at $3.90 a barrel are sold on
commission at a rate of 3%, find the commission. What
is the net amount realized from the sale ?
142 ADVANCED BOOK OF ARITHMETIC
7. A broker sells 2500 Ib. of beef at 12** a pound.
What is his brokerage at \\% ?
8. Thirty-six hundred gallons of oil are sold by a
broker at 45^ per gallon. If he charges 3^% brokerage,
find his brokerage and the amount remitted.
9. A commission merchant sells 1500 T. of hay at
$15 per ton, and charges 5% commission. Find his com-
mission. How much does he remit ? What per cent of
the selling price does he remit ?
10. A coal dealer receives a commission of 90^ a ton
for selling anthracite coal, and 80^ a ton for selling bitu-
minous coal. If he sells an equal quantity of each kind
of coal, -and if his entire commission is $1530, how many
tons of each kind does he sell ?
11. A broker charges 35^ for selling a bale of cotton.
How many bales must he sell to realize for himself
$295.40? If a bale contains 500 Ib., and the brokerage
is equivalent to |%, find the price of cotton per pound.
12. A commission merchant sells 1200 doz. eggs at 18 $
per dozen. Find his commission at 8%.
13. If a commission of $70.98 is paid for buying 40
A. of land at $54.60 an acre, what is the rate per cent
of commission ?
14. If $284 is received for selling grain on a commission
of ^ per bushel, how many bushels of grain were sold ?
15. If an agent charges 3% for collecting debts, and in
one month his commission from this source is $126, what
is the amount collected ?
16. A real estate agent's commission at ^\% is $351.
Find the amount of his sales and the amount he remits to
his principal.
COMMISSION AND BROKERAGE
143
17. A broker buys 10,000 bu. of wheat at 79|^, and
sells the wheat the next day at T9-|^, charging ^ a bushel
for buying and ^f for selling. Find his commission.
18. An agent remits to his principal $9184.50 as the
net proceeds from the sale of 12,000 bu. of wheat. Find
the agent's commission if he charges at the rate of 2|-%.
Find also the selling price of wheat per bushel.
19. After deducting a commission of 3%, an agent
remits $1813.90 from the sale of oats at 46|^ per bushel.
How many bushels of oats were sold?
20. When the market price of pork is 13.6^ per lb.,
how many pounds can be bought for $520.20, if 2% is
charged for brokerage ?
21. How many acres of land can be bought for $4635
at $ 30 an acre, by a real estate agent who charges a com-
mission of 3 % ?
22. Find the commission and amount remitted to the
principal on the sale of the following articles :
09
1 urkeys
1750
lb.
@
9 $%
commission
(J)
Oranges
275
boxes
@
$2.
95
5%
commission
00
Apples
580
bbl.
@
$1.
85
8%
commission
09
Oysters
390
bbl.
@
$1.
45
5%
commission
(0
Potatoes
1270
bu.
@
$1.
05
1\%
commission
C/)
Celery
560
bunches
@
$ .
85
10%
commission
(.9)
Onions
240
bu.
@
11.
25
8%
commission
(A)
Beans
960
bu.
@ $2.15
5%
commission
(0
Rice
3500
lb.
@
* 5%
commission
0')
Chickens
984
lb.
@
4
^ 9%
commission
(*)
Cotton
287
bales
@
$49.
3%
commission
(0
Peaches
315
boxes
@ $1.35
9%
commission
144 ADVANCED BOOK OF ARITHMETIC
INTEREST
By amount is meant the sum of principal and interest.
To find the simple interest on a sum of money, Multiply
the principal by the rate to get the interest for 1 year, and
this product by the time expressed in years.
Example l. Find the interest and the amount of $1780
for 5 mo. at 7%.
Solution: 11780 x .07 x ^ = $51.92, interest for 5 mo.
$1780 + 151.92 = $1831.92, amount.
A concrete quantity which is contained an exact number
of times in another concrete quantity is called an aliquot
part of that quantity.
Thus, 21 yd. is an aliquot part of 10 yd.; $2.75 is an
aliquot part of $11.
Example. Find the interest on $780 for 1 yr. 2 mo.
10 da. at 7 %.
SOLUTION BY ALIQUOT PARTS.
$780
.07
$54. 60 = int. for 1 yr.
2 mo. = 1 of 1 yr. 9.10 = int. for 2 mo.
10 da. = I of 2 mo. 1.52 = int. for 10 da.
$65.22 = int. for 1 yr. 2 mo. 10 da.
EXERCISE 83
Find the interest on :
1. $700 for 1 yr. at 3 % ; for 1 yr. at 5 % .
2. $278 for 1 yr. at 6%; for 1 yr. at 7%.
3. $598 for 1 yr. at 9%; for 1 yr. at 8 %.
4. $289 for 2 yr. at 4 % ; for 2 yr. at 5 %..
INTEREST 145
5. $1000 for 1 yr. 3 mo. at 6 % ; for 1 yr. 5 rno. at 6 %.
6. $1200 for 1 yr. 4 mo. at 8 % ; for 1 yr. 5 mo. at 7 %.
7. $1500 for lyr. 7 mo. at 9%; for 1 yr. 8 mo. at 8%.
8. 11600 for 2 yr. 3 mo. at 5 % ; for 1 yr. 7 mo. at 6 %.
9. $ 2000 for 1 yr. 6 mo. at 4 % ; for 1 yr. 9 mo. at 5 % .
10. $3500 for 10 mo. at 7 %\ for 7 mo. at 8%.
11. $156.40 for 1 yr. at 4J % ; at 5| %.
12. $185.50 for 1 yr. at 4J % ; at 7 %.
13. $375.60 for 5 mo. at 6 %; for 4 mo. at 5%.
14. $928.40 for 2 mo. at 8 % ; for 3 mo. at 6%.
15. $735.60 for 3 mo. at 1% ; for 2 mo. at 8%.
16. $1200 for 2 mo. at 8% ; for 5 mo. at 6%.
17. $1350.50 for 5 mo. at 5% ; for 4 mo. at 7 %.
18. $393.80 for 7 mo. at 4 % ; for 5 mo. at 3 %.
19. $385.40 for 8 mo. at 5 % ; for 7 mo. at 6 %.
20. $934.54 for 9 mo. at 8 % ; for 8 mo. at 9 %.
21. $2713.64 for 10 mo. at 9% ; for 7 mo. at 8%.
22. $3800 for 11 mo. at 7 % ; for 10 mo. at 6%.
23. $2825 for 10 mo. at 7% ; for 9 mo. at 5%.
24. $2700 for 1 yr. 1 mo. at 6 % ; for 14 mo. at 5 %.
25. $3280 for 1 yr. 3 mo. at 5 % ; for 8 mo. at 4 %.
26. $4500 for 8 mo. 15 da. at 6 % ; for 7 mo. at 7 %.
27. $329.50 for 7 mo. 10 da. at 8%.
28. $982 for 9 mo. 18 da. at 6 %.
29. $545 for 10 mo. 25 da. at 8 %.
30. $775.24 for 11 mo. 24 da. at 1%.
146 ADVANCED BOOK OF ARITHMETIC
Example. Find the interest on $384.42 from Jan. 11,
1907, to April 30, 1907, at 7 %.
SOLUTION. First, find the difference between the two
dates.
YR. Mo. DA.
1907 4 30
1907 1 11
3 19
$384.42
.07
$26.9094 = int. for 1 yr.
3 mo. = J of 1 yr. " 6.727 = int. for 3 mo.
15 da. = £ of 3 mo. 1.121 = int. for 15 mo.
3 da. = 1 of 15 da. .224 = int. for 3 da.
1 da. = 1 of 3 da. .075 = int. for 1 da.
$8.147 = int. for 3 mo. 19 da.
Ana. $8.15.
EXERCISE 84
Find the interest on :
1. $450 from Jan. 12 to April 18 at 6 %.
2. $783 from Feb. 14 to April 24 at 8%.
3. $2385 from Mar. 6 to Nov. 11 at 6%.
4. $3950 from July 4 to Dec. 6 at 6 %.
5. $4280 from Aug. 31 to Nov. 18 at 8%.
6. $7335 from July 5 to Sept. 14 at 7 %.
7. $3280 from Jan. 8 to July 5 at 4%.
8. $4592.40 from Jan. 10 to May 4 at 3%.
9. $384.75 from Mar. 10 to Aug. 14 at
10. $327.50 from April 6 to July 3 at 9 %.
11. $935 from Jan. 1 to April 30 at 4J%.
12. $3540 from Jan. 10 to Oct. 5 at 4|-%.
13. $1382.60 from Feb. 8 to Nov. 1 at 5
INTEREST 147
EXERCISE 85
Find the amount of :
1. $800 for 1 yr. 2 mo. 15 da. at 4 %.
HINT. Find the interest and add it to the principal.
2. $580 for 3 yr. 3 mo. at 6 %.
3. $750 for 10 mo. 15 da. at 8%.
4. $327.60 for 11 mo. 12 da. at 9%.
5. $326.54 for 8 mo. 8 da. at 5%.
6. $739.90 for 5 mo. 11 da. at 4%.
7. $843.90 for 6 mo. 14 da. at 8%.
8. $325 for 9 me 12 da. at 4-|-%.
9. $982 for 8 mo. 16 da. at 5%.
10. $ 375 for 7 mo. 14 da. at 6 %.
11. $1280 from Jan. 10 to July 14 at 8 %.
12. $3580 from Jan. 14 to Aug. 18 at 8%.
13. $1500 from March 11 to July 19 at 6 %.
14. $4350 from Aug. 14 to Dec. 17 at 7 %.
15. $1200 from Jan. 1 to July 8 at 7%.
16. $480 for 9 mo. 15 da. at 5%.
17. $800 for 6 mo. 18 da. at 6%.
18. $550 for 3 mo. 15 da. at 8%.
19. $650 for 5 mo. 12 da. at 6%.
20. $850 for 8 mo. 15 da. at 5 %.
21. $980 for 7 mo. at 6%.
22. $2329 for 1 yr. 7 mo. 16 da. at 5%.
23. $3278 for 2 yr. 10 mo. 11 da. at 7 %.
24. $2594 for 1 yr. 5 mo. 10 da. at 3%.
25. $978 for 1 yr. 1 mo. 20 da. at 6%.
26. $1857 for 1 yr. 5 mo. 18 da. at 5%.
27. $903.53 for 1 yr. 3 mo. at 5 %.
148 ADVANCED BOOK OF ARITHMETIC
MEASUREMENTS — SPECIFIC GRAVITY
The ratio of the weight of any given volume of a sub-
stance to the weight of an equal volume of another sub-
stance taken as a standard is called the specific gravity of
that substance.
The standard taken for solids and liquids is water.
Specific gravity, then, simply means how many times
as heavy as water a substance is. Thus, cast iron is 7.21
times as heavy as water, and hence its specific gravity is
7.21. Cork is about one fourth as heavy as water, and
hence its specific gravity is |. 1 cu. ft. of water weighs
1,000 oz.
TABLE OF SPECIFIC GRAVITIES
Ash ... .84 Ebony . 1.33 Steel . . 7.83 Clay . . 1.2
Beech . . .85 Glass. . 2.89 Copper . 8.95 Mercury . 13.57
Brass . . 8.40 Gold . . 19.26 Silver . 10.47 Bar iron . 7.79
Butter . . .94 Granite . 2.78 Lead . . 11.38 Platinum 21.5
Ice ... .92
Example. Find the weight of 28 cu. in. of mercury.
SOLUTION.
-^-r x 1000 oz. == weight of cu. in. of water.
28
x 1000 oz. = weight of 28 cu. in. of water.
28
.Ul
12
6
6
:
A 1728 A
13.57 x
j u/j. — WClgil
7 x 1000 oz.
94990 oz.
94990
432
The factors
432
of 432 are 12, 6, and 6.
7915.833
1319.305
L6 |219.88 oz.
13 lb., 11.
88 oz.
MEASUREMENTS 149
EXERCISE 86
1. Find the weight of 1 cu. ft. of steel ; 1 cu. ft. of
glass ; 1 cu. ft. of clay.
2. Find the weight of 1 cu. in. of water. Find the
weight of 1 gal. of water.
3. Find the weight of 1 bushel measure filled with
water.
4. A cubic foot of marble weighs 2700 oz. Find the
specific gravity of marble.
5. A cubic foot of sea water weighs 64| Ib. Find the
specific gravity of sea water.
6. A cubic foot of goat's milk weighs 65 Ib. Find
how many times as heavy as water goat's milk is.
7. The mercury in the barometer exactly counterbal-
ances the pressure of the atmosphere. If the barometer
is 30 in. high, find the pressure of atmosphere upon every
square inch of surface.
8. A swimming pool of fresh water is 25 ft. by 16 ft.
and 5 ft. deep. Find the weight of the water it contains.
9. A block of granite is 6 ft. by 4 ft. and 2 ft. thick.
Find its weight in tons.
10. Find the weight of 1 cu. in. of gold; of 1 cu. in.
of silver; of 1 cu. in. of platinum ; of 1 cu. in. of lead.
11. A block of ice 3 ft. by 2 ft. and 1 ft. thick weighs
how many pounds ?
12. What is the weight in tons of 1 cu. yd. of clay ?
13. Find the weight of the butter required to fill a box
16 in. by 9 in. by 8 in.
14. A cellar is 18 ft. by 12 ft. and 8 ft. deep. Find
the weight of the water required to fill it.
150 ADVANCED BOOK OF ARITHMETIC
15. Find the weight of the air in a hall 27 ft. by 24 ft.
and 15 ft. 6 in. high. 1 cu. ft. of air weighs .08073 Ib.
16. A cubic foot of coal weighs 81| Ib. Find the
specific gravity of coal.
17. How many cubic inches of copper weigh just as
much as 1 cu. in. of platinum?
18. Find the weight of a block of ebony 4 ft. long, 9 in.
wide, and 8 in. thick.
19. Find the weight of a block of ash 12 in. long, 8 in.
wide, and 6 in. thick.
20. How many times as heavy as glass is mercury?
21. What is the weight of a bar of iron 2 in. by 2 in.
and 8 ft. long ?
22. Find the weight of a beam of beech wood 8 in. by
6 in. and 12 ft. long.
Example l. How many sq. yd. are in 1 sq. mile ?
1 sq. mi. = 1760 x 1760 x 1 sq. yd. = 3,097,600 sq. yd.
Example 2. A lot in the form of a rectangle contains
16 A., and is 48 rd. wide. Find its length.
SOLUTION. 48 x length = 16 x 160. (160 sq. rd.= 1 A.)
Therefore, length = 16 * 16Q = = 53J. An*. 53£ rd.
48 o
Example 3. A rectangular plot of ground 55 yd. by 11
yd. produces 2 bu. of buckwheat. How much will 1 A.
produce at this rate?
SOLUTION. 55 x 11 sq, yd. produce 2 bu.
2
1 sq. yd. produces - — bu.
55 x 11
4840 sq. yd. produce 4840 x 2 bu = 16 bu.
55 x 11
MEASUREMENTS 151
EXERCISE 87
1. The area of a rectangle is 12,500 sq. ft. and one
side is 100 ft. Find its other dimension.
2. The area of a rectangle is 30^ sq. yd. and its length
is 5J yd. Find its width.
3. A lot contains 16 A., and its length is 64 rd. Find
its width.
4. The area of a room is 252 sq. ft., and its length is
18 ft. Find its width.
5. A rectangular piece of ground 15 yd. by 12 yd.
yields 6 bu. of potatoes. Find the yield of an acre.
6. A rectangular tract of land 30 yd. by 11 yd. yields
2 bu. of oats. At this rate how much will 1 A. yield ?
7. A rectangular tract of land 50 yd. by 15 yd. yields
3 bu. of corn. At this rate how much will 1 A. yield ?
8. The base of a triangle is 84 yd. and its area is 840
sq. yd. Find its height.
9. Find the altitude of a triangle having for base 34
yd. and area 289 sq. yd.
10. Given the area of a triangle 1024 sq. yd. and base
64 yd., find its altitude.
11. The area of the walls of a room is 560 sq. ft. and
the dimensions of the room are 16 ft. by 12 ft. Find the
height of the room.
12. The width of a rectangle is 3.9 ft. and its area is
17.55 sq. ft. Find the length.
13. The area of a hall is 497 sq. ft., its length is 71 ft.
Find its width.
14. A street 1 mile long contains an area of 5J acres.
Find the width of the street in feet,
152 ADVANCED BOOK OF ARITHMETIC
EXERCISE 88
1. The inside dimensions of a box car are 36 ft. by 8 ft.,
and 8 ft. 6 in. high. Find the number of cu. ft. in the
car, the number of bushels it will hold allowing 1^ cu. ft.
to the bushel, and the weight of the carload of corn.
SOLUTION. 36 x 8 x 8|- = 2448 = number cu. ft. in car.
2448 -r- 1| = 2448 x .8 = 1958.4 = number bu. in car.
1958.4 x 56 Ib. = 109670.4 Ib. = weight of corn in car.
2. The dimensions of a box car are 32 ft. by 8 ft. and 7
ft. high. Find the number of cu. ft. in the car, the num-
ber of bushels it will hold allowing 1^ cu. ft. to the bushel,
and the weight of the carload of oats.
3. A refrigerator car is 28 ft. 9 in. long, 7 ft. 6 in. wide,
and 8 ft. high. Find the number of cu. ft. in it. The
capacity of this car is 64,000 Ib. How many dressed
turkeys, averaging 12| Ib. each, will the car hold, allow-
ing 4000 Ib. for ice ?
4. A car 100,000 Ib. capacity, length 40 ft., width 8 ft.
6 in., height 8 ft., inside dimensions, is loaded with wheat.
Find the number of cu. ft. in the car. Find the weight
of the wheat that will fill it. What fraction of this weight
is the capacity of the car ?
5. A car 40 ft. long, 8 ft. 6 in. wide, 8 ft. high, is
loaded with 4-ft. wood. How many cords does this car
contain? What is the value of the wood at $4.75 per
cord? What will it cost to ship the wood from Sublime
to San Antonio at $1.50 per cord?
6. The electric railroad from Rochester to Avon, New
York, is 19 miles long. The rails used in its construc-
tion weigh 80 Ib. to the yard. Find in tons the weight of
the rails of this railroad.
MEASUREMENTS 153
EXERCISE 89
Example 1. Find the weight in pounds avoirdupois of
5000 silver dollars. A silver dollar weighs 412| grains.
SOLUTION. 1 dollar weighs 41 2| gr.
5000 dollars weigh 5000 x 412| gr.
= ^00^±12i- lb_
7000
2. Find the weight in pounds troy of 1,000,000 silver
dollars. Find also the weight in pounds avoirdupois.
3. A silver dollar is 90 % silver. Find the weight of
pure silver in one silver dollar.
4. The commercial value of a silver dollar is the value
of the pure silver it contains. What is the commercial
value of a silver dollar when pure silver is worth 66^ per
oz. (troy oz.)?
5. A 10-dollar gold piece weighs 258 grains. Find the
weight of 10,000 dollars in gold. Find also the weight of
1,000,000 dollars in gold.
6. Gold coins contain 90 % pure gold. Find the weight
of pure gold in a 10-dollar gold piece.
7. The alloy in a gold coin neither adds to nor takes
from the value of the coin. Compute the value of 480
grains of pure gold. (Remember that 90 % of 258 gr. of
gold is worth $10.)
8. The bullion value of the pure silver in a silver dollar
at the average price of silver for the year 1905 was $.472.
Find the value of 1 oz. troy of pure silver.
9. The value of a fine ounce of silver for the year 1906
was $.6769, and the value of a fine ounce of gold the same
year $20.67. Find the commercial ratio of the value of
gold to the value of silver for that year.
154 ADVANCED BOOK OF ARITHMETIC
RATIO
The first number in a ratio is called the antecedent, and
the second number is called the consequent.
Since a ratio is a quotient, we may multiply antecedent
and consequent by the same number without affecting the
value of the ratio. It follows also that the terms of a ratio
must be both abstract, or both concrete, and that the value
of a ratio is always an abstract number.
Example l. Divide 343 in the ratio 4 : 3.
SOLUTION. Divide 343 into 7 equal parts and put 4 of
these parts in one group and 3 of them in another.
.-. first part = f of 343.
.*. second part = f of 343.
Example 2. Divide 136 into parts proportional to 2, 3, 4.
SOLUTION. Divide into 9 equal parts and group, as in
Example 1. 2 + 3 + 4 = 9.
. • . first part = f of 136 = 30f .
. • . second part = f of 136 = 45£.
. • . third part = | of 136 = 60$.
Example 3. Divide 529 proportional to the numbers f,
i.A-
SOLUTION. Multiply each of the fractions by 16, the
L. C. M. of 8, 2, and 16. The results are 6, 8, and 9. The
number may, therefore, be divided proportional to the
numbers 6, 8, and 9.
6 + 8 + 9=23.
. • . first part = /% of 529 = 138.
. •. second part = fa of 529 = 184.
. •. third part = fa of 529 = 207.
CHECK. 138 + 184 + 207 = 529.
138 + f = 184 - i = 207 -f- T9B.
RATIO 155
EXERCISE 90
1. Divide $130 between two persons in the ratio 8:5.
2. Divide $985 in the ratio 8 : 2.
3. Divide $3240 in the ratio 4 : 5.
4. Divide $7128 proportional to 1, 2, 3.
5. Divide $7225 proportional to 4, 6, 7.
6. Divide 2916 proportional to 7, 9, 11.
7. Divide 4745 proportional to 5, 6, 7, 8.
8. Divide 6728 proportional to 13, 14, 15, 16.
9. Divide 5040 proportional to 21, 22, 23, 24.
10. Divide 2592 proportional to 15, 17, 19, 21.
11. Divide 95 in the ratio | : 1.
12. Divide 1736 in the ratio J :-J.
13. Divide 365 in the ratio 21 : 3f .
14. Divide 1521 in the ratio 7| : 4f
15. Divide 4225 proportional to 1^, If, 2J.
16. Divide 2189 proportional to 1£, 2£, 3yL.
17. A, B, and C enter into partnership, A contributing
$2450, B, $3500, and C, $4000. They gain $1990.
Find each person's share of the profits, supposing each
person's gain is proportional to his investment.
18. Divide $2700 among A, B, and C, so that A shall
receive $4 as often as B receives $5 and C receives $6.
19. Three men rent a ranch for $900. The first man
puts in 100 head of cattle, the second 85, and the third
115. How much rent should each pay ?
20. Two men buy two lots for $10,000. One man
pays 25% more for his lot than the other man pays.
How much does each pay?
156 ADVANCED BOOK OF ARITHMETIC
PROPORTION
Two equal ratios constitute a proportion.
Thus, | = ^ or 2 : 3 = 6: 9 is a proportion.
The first and fourth terms of a proportion are called
extremes; the second and third are called means.
The law of a proportion is: The product of the extremes
equals the product of the means.
The simplest method of working examples in proportion
is the analytical or the unit method.
Example 1. If 17 bu. of wheat cost $15.30, find the
cost of 15 bushels.
SOLUTION. 17 bu. cost $15.30.
$15.30
1 bu. costs — ij-y —
15 bu. cost 15 x®15,80 =$13.50.
Example 2. If 25 men do a piece of work in 18 days,
how long will it take 20 men to do the same work ?
SOLUTION. 25 men take 18 da. to do the work.
.•.1 man takes 18 da. x 25 to do the work.
. •. 20 men take ^ of 18 da. x 25 = 22 J- days.
EXERCISE 91
Solve analytically:
1. If 19 bu. of wheat cost $14.25, find the price of
11 bu. of wheat.
2. Seven men can dig a treaich in 16 da. How long
will it take 10 men to do the same work ?
3. If 11 sheep cost $71.50, find the cost of 17 sheep at
the same rate.
4. If 14 men can do a piece of work in 11 da., how
long will it take 21 men to do the same work ?
PROPORTION 157
5. The earth revolves on its axis 15° in 1 hour.
Through how many degrees does it revolve in 23 minutes?
6. Twelve horses plow a field of 47 acres in 7 days.
How many acres will 8 horses plow in the same time?
7. A pole 30 ft. high casts a shadow 24 ft. long. Find
the length of the shadow cast by a pole 70 ft. high.
8. How long will it take 20 men to pave a street
which 15 men pave in 15 days?
9. A sum of money yields $40.30 interest in 125 days.
What interest will the same sum yield in 75 days?
10. A train runs 40 miles in 1 hr. and 20 min. How
far will it run in 2 hr.?
11. If 3| acres of land are worth $259, what is the
value of 2J acres?
12. If | of an acre of land is worth $89, how much is a
tract measuring 2^ acres worth?
13. A train runs 45 miles an hour. Find how many
feet it will go in 1 minute ; in 1 second.
14. When the rate of a train is 36 miles an hour, what
is its rate in feet per second?
15. Sound travels 1 mile in 5 seconds. How long will
it take sound to travel 4400 feet?
16. Light travels from the sun to the earth, a distance
of 92,790,000 miles, in 8 minutes and 18 seconds. Find
its rate per second.
17. If | of a clerk's yearly salary is $900, what is his
salary per month?
18. If 11 J yards of carpet can be bought for a certain
sum of money, how many yards can be bought for the
same sum when the price of carpet falls 8 % ?
158 ADVANCED BOOK OF ARITHMETIC
REVIEW
EXERCISE 92
1. Multiply the sum of J, ^, |, by 1T2^, and divide the
21
product by ^*
%
2. What number taken from 20^ leaves as a re-
mainder 4| ?
3. The difference between two numbers is 1 J, and the
less is 4|. Find the other number.
4. Which is the greater, f of 4$, or f of 4-f ?
5. From £ of the sum of J, |, |-, ^-, take the sum of
A> iV 6T-
6. From £ of (1 _ i + 1) take ( J + | - £).
7. Express in pounds the difference between .0125 of a
ton and | cwt.
8. What fraction having 48 for denominator is equiva-
lent to .1875?
9. Find in feet the value of l| of a mile.
10. Express f yd. as the decimal of a rod.
11. Express f of 95 Ib. as the decimal of 1^ cwt.
12. Express 27| Ib. as the decimal of a ton.
13. Take -| T. from 1J T., and express your result in
pounds.
14. From | of a right angle take 33° 45'.
15. From ^ of a circumference take ^ of the circum-
ference and express your answer in degrees.
16. There are two numbers in the ratio of 3 to 5.
What fraction of their. sum is their difference?
REVIEW 159
17. An estate is left to A, B, and C. A gets |- of the
estate, B, ^ of the estate, and C, the remainder. What
part of the estate does C get ? If C's share is $450, what
is the value of the estate ? Find A's share and B's share.
18. A man spends f of his salary on board, ^ on cloth-
ing, -^ on rent. He saves the remainder, amounting to
$125. Find his salary.
19. Subtract -| from 2.1, and divide the remainder by
.25.
20. The dividend is 3.562 and the quotient is .3125.
Find the divisor.
21. Express $5.24 as a decimal of $100.
22. After giving away |, -j^, and J£ of his money, a
man has left $392.95. How much money had he at first ?
23. The third part of a number exceeds the fifth part
of the same number by 15. What is the number ?
24. The sixth part and the eighth part of a number
together make 66 J. What is the number?
25. If | and ^ of a farm are together worth $1650,
what is | of the remainder worth?
26. Subtract the product of | and f from their sum.
27. What number divided by 3| gives ly1^ for the
quotient ?
28. Find the value of % of 1^ of $19.80.
29. A man owns J- of a boat and sells f of his share
for $750. At this rate, find the value of the boat.
30. If 6 men do a piece of work in 9 days, how long
will it take 4 men to do the same work ?
31. If 15 men pave a street in 16 days, how long
will it take 40 men to pave the same street ?
160 ADVANCED BOOK OF ARITHMETIC
32. If 18 men remove an embankment in 12 da., how
long will it take 24 men to remove the embankment ?
33. Two trains start at the same time from two sta-
tions 840 mi. apart, and travel toward each other, one
train going at the rate of 35 mi. an hour, and the other
of 25 mi. an hour. In how many hours will they meet ?
34. If a man performs ^ of a piece of work in 15 da.,
in how many days more will he complete the work ?
35. Add |, -£, •§, -j6g, and ^. Express the sum as a
decimal. Check by reducing to decimals and adding.
36. What is the smallest number which, added to
the sum of ^, ^, and -|, will make the final result an
integer ?
37. If -| of a barrel of sugar is sold and afterward
40 Ib. are sold, how many pounds of sugar were in the
barrel originally, supposing it still contains 90 Ib.?
38. By selling a piano at -^ of its cost, a dealer loses
$98. Find the cost of the piano.
39. If 45 sq. rd. of land cost $ 18, find the cost of
1 A. Find also the cost of 3| A.
40. If | of a clerk's salary per year is $675, find his
salary per month. If his expenses average $48.89 per
month, how much will he save per year ? How long will
it take him to save $652.75 ?
41. If 31 A. of land are worth $119, find the value
of 2|- A. How much is a rectangular strip of this land
| of a mile long and 33 feet wide worth ?
42. How much is a plot of ground 80 ft. by 60 ft.
worth, if an acre is worth $55 ? If an acre is worth $121 ?
REVIEW 161
43. Find the weight of a piece of coal in the shape of a
rectangular solid, if its dimensions are 1J ft. by 1J ft. by
10 in. A cubic foot of coal weighs 81-| Ib.
44. A man's property is assessed at $4550. If he pays
40^ on every $100 for school tax, how many dollars
school tax does he pay?
45. How much taxes will be paid on real estate worth
$9580, if the tax is at the rate of $1.27 on $100?
46. A man invests $4800 and gains $540. How much
does he gain on every dollar invested? How much does
he gain on every $100?
47. If an investment of $9600 produces a gain of
$1056, find the gain on $1 ; also on $100.
48. When $8400 produces a profit of $1092, how much
does $ 1 produce ? $ 100 ?
49. The school tax in a city is 2 mills on the dollar.
The assessed valuation of the property is $17,294,000.
Find the total tax levied for school purposes. If .95 of
this total is collectible, find the amount 3ollected.
50. Find the tax on $33,254,000 at 4 mills on the dollar.
51. A man insures his dwelling for $5450. If he pays
$11.50 on every $1000, how much does he pay altogether?
52. Find the insurance on a house valued at $7840, if
the rate of insurance is $1J on every $100?
53. If $47.50 is paid to insure a boat valued at $9500,
how much is paid on $1? on $100.
54. The distance from New York City to Plymouth,
England, is 2962 knots. The steamship DeutscUand
sailed from Plymouth to New York in July, 1900, in 5
da. 15 hr. 45 min. Find, in knots, its rate per hour.
Find also its rate in miles per hour.
162 ADVANCED BOOK OF ARITHMETIC
55. In March, 1902, the steamship La Savoie made
the voyage from Havre to New York in 6 da. 10 hr. The
distance from Havre to New York is 3170 knots nearly.
Find, in knots, the rate per hour. Express the rate also
in miles per hour.
56. The steamer Kronprinz Wilhelm made, in Septem-
ber, 1902, a voyage from Cherbourg to New York in 5 da.
11 hr. 57 min. Find its rate per hour, the distance from
Cherbourg to New York being 3184 knots.
57. In May, 1900, a passenger train ran from Burling-
ton to Chicago, 205.8 mi., in 3 hr. 8 min. 30 sec. Find
its rate per hour.
58. The fastest time on record by a passenger train for
a distance over 450 miles was made in October, 1895, on
the Lake Shore and Michigan Southern Railroad, from
Chicago to Buffalo, a distance of 510 mi., in 8 hr. 1 min.
Find its rate per hour.
59. The run from London to Edinburgh, 393 J mi., has
been made in 7 hr. 45 min. Find the speed per hour.
60. The market quotations, Feb. 19, 1903, were :
wheat, 78|^ per bushel; corn, 45 1^ per bushel; oats,
34^ per bushel. Find the price of 100 Ib. of each of
these commodities.
61. Market quotations of live stock sales are in dollars
per 100 Ib. Find the cost of :
(a) 44 cattle, average weight 1121 Ib., @ $4.80.
(6) 132 cattle, average weight 1018 Ib., @ $4.80.
O) 24 cattle, average weight 915 Ib., @ $4.20.
(d) 23 cattle, average weight 1060 Ib., @ $3.90.
(e) 133 heifers, average weight 862 Ib., @ $4.00.
(/) 69 calves, average weight 201 Ib., @ $5.00.
(jgf) 82 hogs, average weight 188 Ib., @ $6.20.
CHAPTER III
GENERAL REVIEW BY TOPICS r
ADDITION
To add numbers is to find a single number equivalent
to the numbers jointly. The result is the sum.
Only numbers of the same kind can be added. Thus,
5 yd. and 7 yd. may be added; but 5 yd. and $ 7 cannot
be added.
5 ft. and 7 in. may be added provided the 5 ft. is
changed to inches, or the 7 in. changed to feet. The sum
in one case is 67 inches, in the other case, 5^ feet.
Example 1. Add 279, 514, 928, 763.
The process is 3, 11, 15, 24 ; 24 units = 2 tens
279 " and 4 units.
514 Write the 4 units and carry the 2 tens.
928 2, 8, 10, 11, 18; 18 tens=l hundred and 8
763 tens.
2484 Write 8 and carry 1.
1, 8, 17, 22, 24 ; write 24.
Example 2. Add $2.79, 15.14, 19.28, $7.63.
12.79
5.14
9.28 The process is the same as in Example 1.
7.63
$24.84
163
164
ADVANCED BOOK OF ARITHMETIC
Example 3. Add 6 ft. 8 in., 3 ft. 6 in., 5 ft. 4 in.
FT. IN.
6 8 The process is 4, 10, 18 ; 18 in. = 1 ft.
366 in. Write 6 in., carry 1 ft.
5 4 1, 6, 9, 15. Write 15 ft.
15 6
^
Example 4. Add 6f , 3J, 51.
Change f , |, and J to equivalent fractions having
12 for denominator.
21 l_8 + 6+4_18
- 3+2+3~ ~12~ "12 *'
152 Write |-, carry 1. 1, 6, 9, 15. Write 15.
Observe the same principle pervades the four examples.
Units of the same name are placed in the same column,
and the columns added separately.
An example in addition may be checked by adding the
columns in reverse order.
EXERCISE 93
Add:
(i)
(2)
(3)
00
(5)
4792799
7998992
5998476
5879824
6876894
8399384
8409499
9208503
9473473
9219479
7207383
7294792
8392393
7777888
7328337
9476583
8514798
6555777
6456789
8474494
4728737
9998777
7918924
9875874
9318327
9219777
6666784
5729998
8294295
6438444
6444673
8542728
6329888
7318392
7473478
8299299
5293294
7774673
9299497
9888777
9218288
4445454
6428427
6713729
6666679
7666729
7778898
9444779
9873876
9218289
ADDITION
165
6. Add vertically and horizontally, arid finally sum
the vertical and the horizontal totals :
392
876
543
878
929
965
329
707
538
928
873
393
427
925
599
307
448
388
394
222
399
937
542
234
7. The mileage of railroads in operation in the several
states is given for the years 1903, 1904, and 1905 as
follows :
GROUP AND STATE
1903
1904
1905
NEW ENGLAND
Maine ......
2,004.79
2,029.89
2,091.12
New Hampshire ....
Vermont .....
Massachusetts ....
Rhode Island ....
Connecticut . . .
1,191.42
1,057.84
2,117.41
209.84
1,025.90
1,191.77
1,056.96
2,110.81
209.84
1,020.12
1,191.77
1,063.20
2,104.87
209.84
1,020.12
Total . . . . .
MIDDLE ATLANTIC
New York .....
8,180.85
8,167.21
8,212.12
New Jersey
Pennsylvania .
Delaware .....
Maryland .....
District of Columbia
2,242.56
10,784.54
333.63
1,368.98
24.70
2,266.64
10,991.97
334.86
1,364.45
24.70
2,269.61
11,161.45
333.60
1,406.81
24.70
Total
CENTRAL NORTHERN
Ohio
Michigan
Indiana
Illinois ......
Wisconsin .....
9,023.61
8,459.65
6,834.75
11,502.38
6,921.40
9,163.97
8,467.76
6,863.03
11,742.10
7,014.78
9,243.26
8,521.46
7,046.90
11,959.09
7,188.18
Total
166
ADVANCED BOOK OF ARITHMETIC
GROUP AND STATE
19O3
19O4
1905
SOUTH ATLANTIC
Virginia
3,833.09
3,823.67
3,862.11
West Virginia ....
2,565.49
2,820.82
2,966.05
North Carolina ....
3,790.73
3,913.86
4,015.58
South Carolina ....
3,112.48
3,146.24
3,184.19
Georgia
6,109.21
6,298.97
6,516.61
Florida
3,469.92
3,585.83
3,635.38
Total
GULF AND MISSISSIPPI VALLEY
Alabama .....
4,442.69
4,590.89
4,758.57
Mississippi .....
3,156.56
3,367.23
3,541.04
Tennessee .....
3,355.19
3,484.92
3,606.88
Kentucky .....
3,193.31
3,261.56
3,355.07
Louisiana
3,419.38
3,592.68
3,764.17
Total
SOUTHWESTERN
Missouri
7,316.62
7,797.18
7,859.57
Arkansas .....
3,651.28
3,946.54
4,165.72
Texas . . . .
11,308.05
11,614.13
11,949.02
Kansas ......
8,810.50
8,841.09
8,874.58
Colorado
4,852.44
4,989.85
5,093.20
New Mexico
2,450.02
2,441.93
2,596.64
Indian Country ....
2,320.02
2,585.69
2,686.47
Oklahoma .....
2,359.52
2,635.64
2,836.19
Total
Find the total mileage for each group of states as
indicated.
Add:
891 Q3 ^5
' %> "*¥' °6*
9.
10. 3i,2|,5f2.
11. 54, 9A, 2A.
12. If, 51 21.
13. 7f,6&,»&.
14. 9&,5|, 7|.
is. 12|, llf , 61
ADDITION 167
16. 2° IV 50", 7° 24' 30", 9° 27' 37", 128° 14' 43".
17. 2 ft. 9 in., 7 ft. 3 in., 9 ft. 11 in., 15 ft. 7 in.
18. 8 qt. 1 pt., 9 qt. 1 pt., 15 qt., 12 qt. 1 pt.
19. 4 gal. 2 qt., 7 gal. 3 qt., 9 gal. 1 qt., 8 gal. 3 qt.
20. 5 pk. 7 qt., 9 pk. 3 qt., 12 pk. 5 qt., 13 pk. 4 qt.
21. 3 bu. 3*pk., 9 bu. 2 pk., 7 bu. 1 pk., 4 bu. 3 pk.
22. 12 hr. 15 min., 15 hr. 8 min., 17 hr. 42 min., 5 hr.
13 min.
23. 5 da. 12 hr., 18 da. 17 hr., 13 da. 18 hr., 5 da. 3 hr.
24. 15 yd. 2 ft., 25 yd. 1 ft., 32 yd. 2 ft., 9 yd. 1 ft.
25. How many times does a clock strike in 24 hours?
26. If 4 jars contain 3.92 liters, 7.84 liters, 9.57 liters,
and 6.3 liters respectively, how many liters are in the four
jars?
27. The dimensions of a table are 8 ft. 3 in. by 3 ft.
7 in. How many feet are in its perimeter?
28. The Galveston Sea Wall was constructed by Gal-
veston County and the United States Government; the
former built 3.5 miles, and the latter .87 mile. Find the
total length of the sea wall.
In its construction there were used 1150 carloads of
cement, 6100 carloads of crushed rock, 1400 carloads of
round piling, 475 carloads of sheet piling, 4300 carloads
of riprap, and 6 carloads of reenforcing rods. How many
carloads of material were used in its construction ?
How many miles would the cars extend if placed end
to end, allowing 39.6 ft. to a car?
29. The decapod locomotives operating between Clarion
Junction and Freeman, Ohio, weigh 268,000 Ib. each.
Express this weight in tons.
168 ADVANCED BOOK OF ARITHMETIC
SUBTRACTION
Subtraction is the inverse of addition.
To subtract 7 from 16 is to find a number which added
to 7 will make 16.
Example l. Subtract 63 from 92. %
92
no PROCESS. 3 and 9 are 12 ; write 9, carry 1. 1 and
on 6 are 7, 7 and 2 are 9 ; write 2.
Example 2. Subtract 6.3 from 9.2.
9.2
6.3 The process is the same as in Example 1.
2^9
Example 3. Subtract 6 hr. 3 min. from 9 hr. 2 min.
HR. MIN. PROCESS. 3 min. and 59 min. make 1 hr. and
2 2 min.; write 59 min., carry 1 hr. 1 hr. and
6 _ § 6 hr. are 7 hr. 7 hr. and 2 hr. are 9 hr. ; write
2 59 2hp-
Example 4. Subtract 6| from 9|.
92.
PROCESS. | and -| are 1^ ; write -|, carry 1. 1 and
-~ 6 are 7, 7 and 2 are 9 ; write 2.
Example 5. From 75,218 take the sum of 4799, 3928,
9476, 8873.
75218 PROCESS. 3, 9, 17, 26 ; 26 and 2 are 28. Write
4799 2, carry 2. 2, 9, 16, 18, 27 ; 27 and 4 are 31. Write
3928 4, carry 3. 3, 11, 15, 24, 31 ; 31 and 1 are 32.
9476 Write 1, carry 3. 3, 11, 20, 23, 27; 27 and 8
8878 are 35. Write 8, carry 3. 3 and 4 are 7. Write
48142 4 The remain(ier is 48,142.
This example shows the practical value of this method
of subtraction. (Austrian Method.)
SUBTRACTION
169
EXERCISE 94
l. Exports of domestic manufactures from the United
States for the years ending June 30, 1897, and 1907 :
ARTICLE 1897 1907
Iron and steel, manufactures of ... $57,497,872 $181,530,871
Copper, manufactures of 31,621,125 88,791,225
Wood, manufactures of 35,679,964 79,704,395
Oils— mineral, refined 56,463,185 78,228,819
Leather and manufactures of .... 19,161,446 45,476,960
Cotton, manufactures of 21,037,678 32,305,412
Agricultural implements 5,240,686 26,936,456
Naval stores 9,214,958 21,686,752
Carriages, cars, and other vehicles . . 9,952,033 20,513,407
Chemicals, drugs, dyes, and medicines . 8,792,545 18,220,630
Instruments and apparatus 3,054,453 14,661,455
Paper and manufactures of 3,333,163 9,856,733
Paraffin and paraffin wax 4,957,096 9,030,992
Fibers, manufactures of ...... 2,216,184 3,308,112
India rubber, manufactures of .... 1,926,585 7,428,714
Furs and skins 3,284,349 7,139,221
Books, maps, engravings, etc. .... 5,647,548 5,813,107
Tobacco, manufactures of 5,025,817 5,735,613
Brass and manufactures of 1,171,431 4,580,455
Gunpowder and other explosives . . . 1,555,318 4,082,402
Paints, pigments, and colors 944,536 3,391,988
Soap 1,136,880 3,806,097
Musical instruments 1,276,717 3,252,063
Nickel and manufactures of 726,789 3,218,862
Clocks, watches 1,770,402 3,160,272
Coke 547,046 3,013,088
Glass and glassware 1,208,187 2,604,717
All other articles 19,799,642 47,295,739
Find the increase in the exports of each of the above
articles, or group of articles, and verify your work.
170 ADVANCED BOOK OF ARITHMETIC
Find the difference between :
2. 200 and .02. 10. $403.05 and 192.89.
3. 400 and 1.37. 11. 160.52 and $23.87.
4. $75 and 73^. 12. 100 and .01.
5. $700 and $2.84. 13. 6.29 and 2.9924.
6. $100 and $1.75. 14. 5.001 and 4.0073.
7. $1000 and 5^. 15. 7.2 and 2.77.
8. $324.80 and $100.99. 16. 11 and 1.5.
9. $70.73 and $19.94. 17. 17.3 and 11.9.
18. The square of 6.715 and the square of .285.
19. 7f and 41J. 25. 9T^ and 3J.
20. 18f and 7f . 26. 6T9g and 3|.
21. 9| and 4|. 27. 10T\ and 1\.
22. 21 1 and 11 U. 28. 19| and 84.
o 1 o o o
23. 7T3T and 2J. 29. 12^ and 9^-.
24. 8T3g and 5T%. 30. 23{£ and 12 Jf .
31. 5 ft. 7 in. and 4 ft. 9 in.
32. 17 ft. 3 in. and 12 ft. 8 in.
33. 19 ft. 1 in. and 9 ft. 4 in.
34. 27 ft. 3 in. and 18 ft. 4 in.
35. 9 Ib. 2 oz. and 4 Ib. 7 oz.
36. 17 Ib. 6 oz. and 5 Ib. 11 oz.
37. 33 Ib. 2 oz. and 18 Ib. 8 oz.
38. 12 hr. 10 min. and 9 hr. 24 min.
39. 90° and 34° 14' 15".
40. 180° and 115° 4' 50".
41. 180° and the sum of 56° 16', and 92° 18'.
42. 15 pk. 3 qt. and 3 pk. 7 qt.
SUBTRACTION 171
43. 23 pk. 5 qt. and 13 pk. 6 qt.
44. From 40,000 take the sum of 3211, 4711, 5283,
9438.
45. From 50,580 take the sum of 19,311, 12,218, 1273,
5559.
46. From 18,900 take the sum of 3419, 3428, 4584, 2293.
47. A man owns two houses worth respectively $2390
and $4575 ; he has deposited in the bank $3280 ; he owes
two notes for $783 and $870. How much is he worth ?
48. The area of the British Isles is 120,975 square
miles; the area of Texas is 265,780 square miles. By
how many square miles does the area of Texas exceed the
area of the British Isles ?
49. The population of the Chinese Empire is 433,553,000;
of the British Empire, 363,900,000 ; of the Russian Em-
pire, 141,000,000; of the United States, exclusive of
colonial possessions, 84,150,000 ; of Germany, 60,478,000.
How many more people are in the United States than in
Germany ? In the British Empire than in Russia, United
States, and Germany combined? By how many does the
population of China exceed the population of Russia,
United States, and Germany together?
50. The areas of Maine, New Hampshire, Vermont,
Massachusetts, Rhode Island, and Connecticut in square
mrles are respectively : 33,040, 9305, 9565, 8315, 1250,
4990. The area of California is 158,360 square miles.
By how many square miles does the area of California
exceed the area of the six New England states?
51. In going from Galveston to Chicago by rail, a dis-
tance of 1410 miles, a man travels the first day 345 miles ;
the next day, 201 miles ; the third day, 290 miles. How
far is he from Chicago at the end of the third day ?
172 ADVANCED BOOK OF ARITHMETIC
MULTIPLICATION
If one factor of the product is multiplied by a number,
and the other factor divided by the same number, the product
will be unchanged.
Thus, 84 x 20 = 1680.
420 x 4 = 1680. Here 84 is multiplied by 5, and
20 is divided by 5.
Example 1. Multiply 3782 by 234.
3782 or 3782
234 234
15128 = 4 x 3782 7564 = 200 x 3782
11346 = 30 x 3782 11346 = 30 x 3782
7564 = 200 x 3782 15128 = 4 x 3782
884988 = 234 x 3782 884988 = 234 x 3782
To multiply integers, write multiplicand and multiplier so
that units of the same name stand in the same column, then
multiply the multiplicand by each digit of the multiplier,
placing the first figure of each partial product directly under
the digit of the multiplier producing it, and add the partial
products.
.Example 2. Multiply 17.32 by .47.
17.32 PROCESS. The numbers are multiplied as if
.47 both were integers ; then beginning at the right
12 124 of the product four places are pointed off, that is
69 28 the number of decimal places in multiplicand and
8.1404 multiplier combined. This may be readily seen
by multiplying the multiplier by 100 and dividing the
multiplicand by 100.
Compare with explanation page 59.
MULTIPLICATION 173
Example 3. Multiply 4J by 2|.
2|=|
o o
13
Therefore, 4| x 2| = 3$. x f = ^ ^| = 13.
EXPLANATION. If the first factor is multiplied by 8,
the result is 39, and if the second factor is multiplied by
3, the result is 8. Hence,
8 x 3 x required product = 39 x 8.
Therefore, required product = —
o X o
Compare with explanation on page 46.
Example 4. Multiply 5 gal. 2 qt. 1 pt. by 9.
PROCESS. 9 times 1 pt. = 4 qt. 1 pt. ; write
GAL. QT. PT.
5 2 1 * pk* carry 4 qt. 9 times 2 qt. = 18 qt. 18
9 qt. + 4 qt. = 22 qt. = 5 gal. 2 qt. ; write 2
5Q 2 I *$"> carry 5 gal. 9 times 5 gal. = 45 gal. 45
gal. -f- 5 gal. = 50 gal.
PARTICULAR SHORT METHODS OF MULTIPLICATION
5 = 1 of 10 75 = 100 - \ of 100
25 = £ of 100 875 = 1000 - £ of 1000
125 = | of 1000 99 = 100-1
.16| = | 97 = 100-3
Example 1. Multiply 97.3 by 125.
125 x 97.3 = \ of 1000 x 97.3 = \ of 97300 = 12162.5
Example 2. Multiply 29.374 by 993.
29374. =1000x29.374
205.618= 7x29.374
29168.382= 993x29.374
174
ADVANCED BOOK OF ARITHMETIC
EXERCISE 95
1. Multiply each of the following numbers by 10 :
234, 350.2, 25.07, .127, .0788, 1.003.
2. Multiply the following numbers by 100 :
505, 67.5, 27.28, 5.347, .07954, .00392.
3. Multiply the following numbers by 1000 :
728, 96.4, 12.87, 1.732, .0139, .00782.
4. Multiply the following numbers by 10,000 :
318, 25.4, 19.96, 18.832, 27.796, .012.
5. Find in the shortest possible way the following
products :
O) 2780 x 99 ; 9218 x 998 ; 7215 x 999.
(5) 2.79x25; 3.18x125; 243x875.
O) 78 x .16|; 90 x .331; 297 x 9998.
6. Multiply 5280 by 5280 ; 1020 by 1020.
7. Multiply 7309 by 256 ; 9417 by 735.
8. The estimated production and value of the following
cereal crops as given in the Annual Report of the Depart-
ment of Agriculture for the year 1906 are as follows :
CEREALS
YIELD PER ACRE
VALUE PER BUSHEL
Corn
bushels
30.3
cents
39.9
Wheat
15.5
66.7
Oats
31.2
31.7
live .
16.7
58.9
Barley
28.3
41.5
Buckwheat
18.6
59.6
Find the value of the yield per acre of each of these
cereal crops.
MULTIPLICATION
175
9. The number of bales of cotton produced in Texas
in the season 1904-05 was 2,598,949, and in 1903-04,
3,214,133. Allowing 500 Ib. to a bale, how many more
pounds of cotton were produced in the latter year than
in the former?
10. The estimated production and value per ton of the
hay crop for the year 1906 are as follows :
STATE
YIELD PER ACRE
PRICE PER TON
New Hampshire ....
tons
1 15
$12 50
Massachusetts ....
1 31
17.00
Connecticut
1.17
15.00
New York
1 28
12.10
New Jersey
1.32
15.95
Pennsylvania
1 30
13.40
.Maryland
1 26
13.50
Virginia
125
15.50
South Carolina
146
1525
Georgia
1 65
15.75
Alabama
1.95
13.30
Louisiana
1.93
11.50
Tennessee
1 51
13.45
Kentucky
1.35
13.25
Illinois
.98
12.50
1.70
5.50
Kansas
1.28
6.25
Colorado
2.50
9.50
Utah
4.00
7.50
Idaho
2.95
8.00
1.85
11.25
Find the value of the yield per acre in each of the above
states.
176 ADVANCED BOOK OF ARITHMETIC
11. A piece of coal taken from the mine at Coos Bay,
Oregon, had the following composition by weight:
Moisture =.1042
Combustible matter = .4221
Fixed carbon = .4318
Ash = .0419
Find the amount of each in 87 tons of this coal: in 783
tons. Check your answers.
12. Find to the nearest cent the value of each of the
following articles:
(a) 25J bu. corn @ 42| ^ per bu.
(6) 12T^ bu. wheat @ 69 J^ per bu.
0) 28| bu. oats @ 25J per bu.
(d) 16f bu. rye @ 60| $ per bu.
O) 201 bu. barley @ 47f ^ per bu.
(/) 4| Ib. wool @ 61 ^ per Ib.
(#) 497 Ib. cotton @ llf $ per Ib.
(A) 512 Ib. cotton @ 10{ f per Ib.
13. The inside dimensions of the floor of a box car are
40 ft. -| in. by 8 ft. 6 in. Find the perimeter of the floor.
14. The inside dimensions of the floor of a refrigerator
car are 28 ft. 9| in. by 8 ft. 1^ in. Find its perimeter.
15. Multiply 5 yd. 2 ft. by 8 ; 9 ft. 8 in. by 7.
Find the product of :
16. 9 Ib. 4 oz. by 5 ; 16 Ib. 11 oz. by 9.
17. 3 hr. 20 min. 30 sec. by 6 ; 7 hr. 17 min. by 9.
18. 53° 12' by 10 ; 68° 12' 18" by 5.
19. 4 pk. 7 qt. by 6; 7 pk. 3 qt. by 8.
20. 5 gal. 2 qt. by 9; 9 gal. 3 qt. by 12.
DIVISION
177
DIVISION
Division is the inverse of multiplication.
To divide 84 by 7 means to find a number which multi-
plied by 7 gives 84.
If the divisor and dividend are both multiplied by the
same number, the quotient remains unchanged.
Thus, 96 - 8 = 12.
(96 x 6) - (8 x 6) = 12.
Example l. Divide 2483 by 7.
7)2483 PROCESS. 7 is contained in 24 hundreds 3
354|- hundred times, remainder 3 hundreds ; 3 hun-
dred = 30 tens. 30 tens and 8 tens = 38 tens. 7 is con-
tained in 38 tens 5 tens times, remainder 3 tens ; 3 tens =
30 units. 30 units and 3 units = 33 units. 7 is contained
in 33 4 times, with a remainder of 5.
Example 2. Divide .437 by 1.92.
.2276+
PROCESS. Move the decimal point two
places to the right in divisor and dividend ;
this multiplies both by 100. Then write
each quotient figure directly above the
right-hand figure of the partial dividend
which produces it. Write the decimal
point in the quotient above the decimal
point in the dividend.
192^43 7
QQ 4
1 344
Example 3.
Divide 3| by 6£.
| =(3| x 3 x 2)
(61 x 3 x 2) =
Multiply divisor and dividend by 3 x 2.
divisor and dividend both whole numbers.
This makes
178
ADVANCED BOOK OF ARITHMETIC
Example 4. To how many long tons are 3.30693 short
tons equivalent?
100
3.30693 x nn _ 330.693 __
112
To check an example in division, multiply the quotient
by the divisor.
EXERCISE 96
l. The estimated acreage, production, and value of the
potato crop by states for the year 1905 are as follows :
STATE
ACREAGE
PRODUCTION
FARM VALUE
New Hampshire
acres
19,700
bushels
2,367,000
dollars
1,704,000
Rhode Island
6,490
811,200
722,000
7,680
714,000
421,200
25,900
1,993,000
1,355,000
Florida
4,110
308,200
369,900
5,860
644,900
548,200
34,400
3,025,100
1,754,500
242,000
16,203,000
9,073,700
149,000
11,186,000
7,494,600
86,100
7,059,000
3,882,614
25,400
2,415,000
917,800
Nevada
2,800
336,700
276,100
Find the number of bushels yielded per acre in each
state, and the average price per bushel in cents.
Divide correct to four decimal places:
2. 128.016 by 420. 6. .02734 by .044.
3. 2.3774 by 7.8. 7. .035936 by .0888.
4. 10.4987 by 3.2. 8. 1.57899 by .639.
5. .77087 by .479. 9. 60.247 by 78.8.
10. 5.0748 by 3.88.
DIVISION 179
Divide :
11. 21 by 1^T. 16. 93f by 62f .
12. 48 by 2f . 17. 17f by 9fi.
13. 42bylTV 18.
14. 72 by 3f. 19.
15. 72lby4f 20. 2ilby2ff.
Find the value of :
SOLUTION. fxfx|xf=
22. £ X If -«- 2£ X TV 24. | X 2f X 1* -«- If.
23. | X If -*- 1J -H 8f 25. 1^-8J-!.3JXJ.
26. If -I- (I -8- 41) X 4f .
27. | Of If Of 31 - I Of If
HINT, i x f x I -i- if.
Observe the divisor is f of 1-J.
28. | of If of 2| - f of If.
29. 11 of If of TV^ A off.
30. |of4|X29o-^f|.
31. If Of 3f Off-*-! Off.
Express in long tons :
32. 3.36 T., 6.6139 T., .00992 T., 4.4092 T.
Express in short tons :
33. 1.9684 long T., 6.8894 long T., .004921 long T.
Express in troy pounds :
34. 5 Ib. avoirdupois, 13.228 Ib. avoirdupois, 6613.87
Ib. avoirdupois.
180 ADVANCED BOOK OF ARITHMETIC
Express in avoirdupois pounds :
35. 7 Ib. troy, 13.396 Ib. troy, 5.358 Ib. troy.
36. Express 1 ft. as a decimal of 1 mi.
37. Express 1 rd. as a decimal of 1 mi.
38. Express 1 sq. rd. as a decimal of 1 A.
39. Express 1 A. as a decimal of 1 sq. mi.
40. Express 1 sq. yd. as a decimal of 1 A.
41. Express 1 Ib. as a decimal of 1 ton.
42. A lot is 40 by 120 feet. How many such lots make
40 acres?
43. How many barrels of 31 J gallons each will a rec-
tangular tank 12 ft. by 8 ft. and 5 ft. deep hold ? (Allow
7^ gal. to a cubic foot.)
44. The weight of a half dollar is 12J grams. How
many half dollars can be made out of 7500 grams of stand-
ard silver ?
45. Find the cost of boring an artesian well 1400 feet
deep at $4 a linear foot for the first 900 feet, $4.50 per
linear foot for the next 200 feet, $ 5 per linear foot for the
next 100 feet, $ 5.50 per linear foot for the next 100 feet,
and $ 6 per linear foot for the remainder.
46. The rails of the Great Western. Railway, England,
weigh 97| Ib. per yard. Find in tons the weight of the
rails required to construct 1 mile of this railway.
47. Express in the ordinary decimal notation :
3.27 x 106, 17.45 x 109, 9.4 x 103, 7.3 x 106.
48. Given 10-i = TV 10-2 = _i_, 10-3 = _i_
= TWO 0> 10~5 = TFoVoT' 1°~6 = TOOOOOO-
Express in the ordinary decimal notation :
(a) 3.2 x 10-3, 4.71 x 10~6, 9.83 x 10'3.
(6) 4.98 x 10-5, 9.371 x 10"6, 4.329 x 10-4.
LONGITUDE AND TIME 181
LONGITUDE AND TIME
A meridian is an imaginary line running due north and
south from pole to pole.
Longitude is the distance, expressed in circular arc
measure, east or west from the prime or standard me-
ridian.
The meridian through any particular place may be used
as the prime meridian. The meridians through the ob-
servatories of Greenwich, Washington, Paris, Madrid,
Rome, Stockholm, Pulkova, and Lisbon have been used
as prime meridians by the nations to which these cities
belong. The International Geodetic Congress, which met
at Washington in 1884, recommended that the meridian
passing through the observatory at Greenwich, a suburb
of London, be the prime meridian. This recommendation
is now generally adopted by the great nations of the world.
The meridian of Greenwich is taken as prime meridian in
this book.
Longitude is reckoned in either direction halfway around
the earth from the prime meridian. The greatest longi-
tude a place can have is 180° E. or 180° W. The meridian
180° E. or 180° W. of the prime meridian is a continuation
of the prime meridian on the other side of the earth, and
forms with the prime meridian what is called a great circle
passing through the poles.
The earth rotates on its axis from west to east. Con-
sider two places not on the same meridian ; for example,
New York City and St. Louis. New York being farther
east will come the sooner under the influence of the sun's
rays. Therefore, when it is noon in New York City it is
before noon in St. Louis. Since the earth's motion is
uniform, and furthermore, since
182 ADVANCED BOOK OF .ARITHMETIC
in 24 hr. the earth rotates 360°,
/.in 1 hr. the earth rotates 15°;
.'.in 1 min. the earth rotates 15' ;
/.in 1 sec. the earth rotates 15".
A difference of 15° of longitude corresponds to a differ-
ence of 1 hr. of time. A difference of 15' of longitude
corresponds to 1 min. of time. A difference of 15" of
longitude corresponds to 1 sec. of time.
Hence, to convert difference of longitude into difference
of time, divide by 15.
EXERCISE 97
1. When it is noon at London, what is the time at New
Orleans, 90° W. ?
2. When it is 9 o'clock A.M. on the meridian 75° W.,
what is the time on the meridian 90° W. ?
3. The longitude of Denver is 105° W. When it is
3 o'clock P.M. in Denver, what is the time in London?
4. Two places differ in longitude by 20°. What is
their difference in time ?
5. A person travels east 15°. What change must he
make in the time indicated by his watch so that it may
indicate local time ? Supposing he goes the same distance
west, what change must be made in the time indicated by
his watch ?
6. When it is noon, in London, what is the longitude
of the places in which it is 4 o'clock P.M. ? 5 o'clock A.M.?
7. When it is 2 o'clock P.M. in Washington, what is
the time in places 30° W. of Washington ? in places 75° E.
of Washington?
8. What is the difference in longitude between places
which differ in time by 2 hr. 30 min. ? by 4 hr. 10 min. ?
LONGITUDE AND TIME 183
9. If a person travels from Denver to New York, will
his watch be fast or slow when he reaches New York, and
how much ?
10. To how many hours "does a difference of 80° in
longitude correspond?
11. What difference in longitude corresponds to a dif-
ference of 4 hr. 20 min. in time ?
12. A person living on the 90th meridian W. wishes
to send a telegram to a bank in New York City, directing
the bank to pay on the same day a sum of money. Up to
what hour in the afternoon may he do this, allowing 30
minutes for the transmission of the telegram, taking the
longitude of New York as 75° W. ? (New York banks
close at 3 P.M.)
13. At places on the same parallel of latitude the sun
rises at the same instant local time. How many minutes
earlier does the sun appear to a person who travels 1°E.?
LONGITUDES OF CITIES REFERRED TO IN THIS CHAPTER
Austin,
Baltimore,
Bangor,
Bismarck,
97°
76°
68°
100°
44' W.
37' W.
47' W.
47' W.
Galveston,
Havana,
Honolulu,
Louisville,
94° 47'
82° 21'
157° 52'
85° 46'
W.
30"
W.
W.
W.
Boston,
71°
3'
50"
W.
Melbourne,
144°
58'
32" E.
Brisbane,
153°
2'
E.
Manila,
120°
58'
3
n
E.
Buenos Ayres,
58°
22'
14"
W.
Mexico City,
99°
6'
39
n
W.
Charleston,
79°
52' 58"
W.
Montreal,
73° 33'
4
n
W.
Chicago,
87°
40'
W.
New Orleans,
90°
3'
28
11
W.
Cincinnati,
84°
24'
W.
New York
>
74°
0'
24
n
W.
Constantinople,
29°
0'
50"
E.
Norfolk,
76°
17'
22
n
W.
Detroit,
83°
3'
W.
Paris,
2°
20'
15
n
E.
Dublin,
6°
20'
30"
W.
Pekin,
116°
29'
E.
184 ADVANCED BOOK OF ^ARITHMETIC
Pensacola, 87° 16' 6" W. St. Petersburg, 30° 19' 40" E.
Philadelphia, 75° 9' 3"W. San Francisco, 122° 24' 32" W.
Portland, 122° 40' W. Savannah, 81° 5'25"W.
Providence, 71° 24' 20" W. Tientsin, 117° 11' 44" E.
Borne, 12° 28' 40" E. Tokyo, 139° 44' 30" E.
St. Louis, 90° 16' W. Washington, 77° 0' 36" W.
In recent years scientific publications often give longi-
tudes in terms of time, the -f sign denoting west and the
— sign denoting east.
H. M. S. H. M. S.
Harrisburg, +5 7 32 Adelaide, - 9 14 2
Milwaukee, + 5 51 37 Omaha, -f- 6 23 46
Example 1. Find the difference between the longi-
tudes of Austin and Honolulu.
SOLUTION. Honolulu, 157° 52' W.
Austin, 97° 44' W.
60° 8'
.-. Honolulu is 60° 8' farther west than Austin.
Example 2. Find the difference between the longi-
tudes of Galveston and Constantinople.
SOLUTION. Galveston, 94° 47' W.
Constantinople, 29° 0' 50" E.
Here, the places are on opposite sides of the prime
meridian. By going east from Galveston 94° 47', one
arrives at the prime meridian, and by going 29° 0' 50"
still farther east, he arrives at the meridian of Con-
stantinople. Hence, the difference between the longitudes
is (94° 47' + 29° 0' 50") = 123° 47' 50".
To find the difference in the longitudes of two places :
(l) Subtract their longitudes, if the places are on the same
side of the prime meridian. (2) Add their longitudes, if
the places are on opposite sides of the prime meridian.
LONGITUDE AND TIME 185
EXERCISE 98
Find the difference in longitude between :
1. Baltimore and Bismarck.
2. Bangor and Detroit.
3. Boston and Havana.
4. Buenos Ayres and Chicago.
5. Charleston and Constantinople.
6. Cincinnati and Honolulu.
7. Cincinnati and Melbourne.
8. Havana and Rome.
9. Louisville and St. Petersburg.
10. Constantinople and Tientsin.
11. Paris and Pekin.
12. Norfolk and Paris.
13. Montreal and Mexico City.
14. Pensacola and Portland.
15. St. Louis and St. Petersburg.
16. Savannah and Dublin.
17. San Francisco and Dublin.
Example 1. Find the difference in local time between
Boston and Portland, Ore.
Portland 122° 40' W.
Boston 71° 3' 50" W.
15)50° 36' 10"
3 22 25 Ans. 3 hr. 22 min. 25 sec.
Example 2. Find the difference in local time between
Washington and Manila.
Washington 77° 0' 36" W.
Manila 120° 58" 3" E.
15)197° 58' 39"
13 11 54.6 Ans. 13 hr. 11 min. 54.6 sec.
186 ADVANCED BOOK OF ARITHMETIC
EXERCISE 99
Find the difference in the local time of :
1. Mexico City and Montreal.
2. Philadelphia and San Francisco.
3. Philadelphia and Dublin.
4. Norfolk and Tientsin.
5. Chicago and Tokyo.
6. St. Louis and Rome.
7. Austin and St. Petersburg.
8. Savannah and Paris.
9. Washington and Brisbane.
10. Cincinnati and Manila.
11. Havana and Louisville.
12. Rome and Manila.
13.. New Orleans and Portland.
14. Providence and St. Petersburg.
15. Montreal and Tokyo.
16. Bangor and Melbourne.
17. Baltimore and Buenos Ayres.
Example l. When it is noon, February 22, in St. Louis,
it is 15 min. 6 sec. past three o'clock A.M., Feb. 23, in
Adelaide, Australia. Find the longitude of Adelaide.
SOLUTIOIST. The time difference between St. Louis and
Adelaide is
15 hr. 15 min. 6 sec.
Multiply by 15, 15
228° 46' 30' '
.-. Adelaide is 228° 46' 30" E. of St. Louis. Longitude
of St. Louis is 90° 16' W. .-. longitude of Adelaide =
(228° 46' 30" - 90° 16") E. = 138° 30' 30" E.
LONGITUDE AND TIME
187
EXERCISE 100
Calculate the longitude of each of the following cities,
the time difference between New York City and each of
them being given :
1. Berlin, 5 hr. 49.5 min.
2. Brussels, 5 hr. 13.4 min.
3. Calcutta, 10 hr. 49.2 min.
4. Edinburgh, 4 hr. 43.2 min.
5. Hamburg, 5 hr. 35.8 min.
6. London, 4 hr. 55.9 min.
7. Madrid, 4 hr. 41.1 min,
8. Vienna, 6 hr. 1.2 min.
9. The time difference between London and Amherst,
Mass., is 4 hr. 50 min. 3 sec. Find the longitude of
Amherst.
10. Find the difference in the time of sunrise between
two points in the same latitude and which differ in longi-
tude by 39° 20'.
oJfe^ncT/^ B«*l3Sfc*
K-r ir""iS^^^^BSfi^
188 ADVANCED BOOK OF ARITHMETIC
STANDARD TIME
Standard time is the time of a fixed meridian, generally
a multiple of 15°. It was established in the United States
in 1883 primarily for the convenience of railroads. It is
now adopted generally throughout the civilized world.
STANDARD MERIDIANS AND PLACES USING THEM
0°. Great Britain, Spain, Belgium, Holland.
15° E. Germany, Austria, Italy, Denmark, Norway.
30° E. South Africa, Egypt, Turkey.
821° E. British India (since July 1, 1905).
971° E. Burma (since July 1, 1905).
120° E. West Australia, eastern coast of China, Phil-
ippine Islands.
135° E. Japan.
142J° E. South Australia.
150° E. Victoria, Queensland, New South Wales.
172|° E. New Zealand.
60° W. Newfoundland and Eastern Canada.
75° W. Eastern belt of the United States.
90° W. Central belt of the United States.
105° W. Mountain belt of the United States.
120° W. Pacific belt of the United States.
135° W. Alaska.
150° W. Tahiti.
1571° W. Hawaiian Islands.
France uses Paris time, Ireland uses Dublin time.
STANDARD TIME 189
EXERCISE 101
1. Mariners carry on board ships chronometers which
keep Greenwich time. When it is noon, local time, the
chronometer indicates 4 hr. 48 min. P.M. What is the
longitude of the ship ?
2. When it is 10 o'clock P.M., March 2, in Washing-
ton, what is the standard time in Manilla? Melbourne?
Berlin?
3. When it is 2 o'clock A.M., standard time, in Den-
ver, what is the standard time of London? Manchester?
Glasgow? Tientsin? Constantinople?
4. A telegram is sent from Madrid to Washington at
9 o'clock A.M. Allowing 1 hr. for transmission, when
will it reach Washington?
5. At noon, local time, a chronometer indicates 11
o'clock P.M. What is the longitude?
6. A telegram is sent from Galveston to London at 10
o'clock P.M. When will it be received, allowing 2 hr.
for transmission?
7. When it is 2 o'clock A.M. in Washington, standard
time, what is the time in New Zealand? Tahiti? British
India?
8. The San Francisco earthquake occurred April 18,
1906, at 5 A.M. When should the news have reached
London? Berlin? Tokyo? Adelaide? (allowing 1 hour for
transmission).
9. When it is noon in Paris, France, what is the time in
Denver? Natal? Calcutta? Wellington (New Zealand)?
10. When it is 9 o'clock A.M. in Madras, what is the
time in St. John's, Newfoundland? Chicago? Sitka?
11. When it is noon in the Hawaiian Islands, what is
the time in Cairo (Egypt) ? Perth (Western Australia) ?
190 ADVANCED BOOK OF ARITHMETIC
APPROXIMATIONS. CONTRACTED PROCESSES.
GENERAL METHODS OF SOLUTION
In business problems results of computation are gen-
erally required to be correct to not more than two decimal
places. For example : The interest on $79.50 for 4 months
at 7% is $1.855. From a business point of view the
answer is $1.85.
In all practical measurements of length it requires skill
and long practice to get results correct to more than three
figures. For example : A surveyor measures the length
of a field and finds it to be 3729 feet. It is extremely
probable that the last figure in this result is not correct.
As the results of measurement are correct to only three
or four figures, hence it is useless in computation to give
results to more than three decimal places.
Before undertaking to show how results may be obtained
correct to any given number of figures, it is well to fix in
mind the following facts :
Tenths multiplied by tenths give hundredtks.
Tenths multiplied by hundredths give thousandths.
Tenths multiplied by thousandths give ten-thousandths.
Hundredths multiplied by hundredths give ten-thousandths.
Example 1. Multiply .0537928 by 43.27.
Move the decimal point one place to the
.537928 left in the multiplier.
4.327 Move the decimal point one place to the
2.151712 right in the multiplicand.
1613784 These changes make no change in the
1075856 product.
3765496 Suppose it be required to get the product
2.327614456 correct to two decimal figures. The answer
would be 2.33.
APPROXIMATIONS 191
Write the units' figure of the multi-
CONTRACTED PROCESS piier under the third decimal figure of
.537982 the multiplicand. Multiply 4 by 7
4.327 and carry 4 from 4 multiplied by 9 be-
2.152 cause 36 is nearer 40 than 30. Multi-
161 ply the remaining figures to the left by
11 4 in the usual manner.
4 As tenths multiplied by thousandths
2.328 give ten-thousandths, multiply 3 by 3
2.33 to the left of 7 and carry 2 from 3 times
7. 3 times 5 are 15 and 1 make 16.
As hundredths multiplied by hundredths give ten-thou-
sandths, multiply 2 by 5 and carry 1 from 2 times 3. As
thousandths multiplied by tenths give ten-thousandths, 7
is multiplied by no figure of the multiplicand, 4 is carried
from 7 times 5.
Compare the two processes.
Example 2. Multiply 253.7 by .079 correct to two deci-
mal figures.
Move the decimal point in the multi-
2.537 plier two places to the right. Move the
7.9 decimal point in the multiplicand two
17.759 places to the left. These changes make
2.283 no change in the product. Multiply by
20.04 7 in the usual manner. Multiply by 9
beginning with 9 times 3, and adding 6
to the product which is the figure carried from 9 times 7.
Move the decimal point in the multiplier so that it con-
tains one integral figure. Move the decimal point in the
multiplicand the same number of places in an opposite direc-
tion. Place the units' figure of the multiplier under the third
place of the multiplicand, if a product to two decimal figures
192 ADVANCED BOOK OF ARITHMETIC
is required. If a product to three decimal places is required,
place the units' figure of the multiplier under the fourth deci-
mal place of the multiplicand. Then multiply as indicated
in the above examples.
Example 3. Multiply .732 by .864 correct to two deci-
mal figures.
Begin multiplying by 8 by taking
.0732 the product 8x3, carrying 2 from 8 .0732
8.64 x 2. Begin the multiplication by 468
.586 6 with 6x7, carrying 2 from 6 x 586
44 3. Begin the multiplication by 4 44
3 with 4x0, carrying 3 from 4x7. 3
.63 The arrangement in the right mar- .63
gin conserves energy, for the multi-
plication by each figure of the multiplier is begun with
the figure directly above it.
Example 4. Divide 120.005 by 17.293 correct to three
decimal figures.
6.939+
17293)120005
103758
162470
155637
68330
51879
164510
155637
8873
The answer correct to three decimal figures is 6.940 as
the next will be 5.
APPROXIMATIONS 193
CONTRACTED PROCESS jn this example the quotient is
6.939+ required correct to four figures.
17293)120005 The divisor contains five figures.
103758 Whenever the divisor contains one
16247 or more figures than are required in
15564 the quotient, a figure may be struck
683 off the divisor in place of annexing
519 or taking down a figure, as is usu-
164 ally done in getting each figure of
155 the quotient.
9 Compare the two processes.
GENERAL METHOD
Example a. If 3 acres of land are worth $ 129, how
much are 5 acres worth at the same rate per acre ?
Example b. If 8 masons build a wall in 18 days, how
long would it take 9 masons to build the wall ?
Example (a) The cost of 3 acres = $ 129.
The cost of 1 acre = $ of $ 129.
The cost of 5 acres = f of 9 129 = f 205.
Example (5) The time 8 masons take = 18 da.
The time 1 mason takes = 8 x 18 da.
The time 9 masons take = | x 18 da. = 16 da.
The answer in Example (#) is a fraction of $129.
The answer in Example (6) is a fraction of 18 days.
The solution of examples of this character consists in
multiplying the quantity of the same kind as the answer
by a fraction.
If the answer is to be greater than the given quantity,
form the fraction so that the numerator is greater than the
denominator. If the answer is to be less than the given
quantity, form the fraction so that the numerator is less than
the denominator.
194 ADVANCED BOOK OF ARITHMETIC
Example l. If a dealer sells a piano for $425, thereby
losing 15 %, what should he have sold it for to make a
profit of 15 % ?
In this example 85 % of cost is given, and 115 % of cost
is sought. The answer will be obviously more than
$425.
Hence, iff- x $425 = $ 575, Ans.
Example 2. A kilometer is very nearly equivalent to
^ of a mile. Express a mile in kilometers.
5 eighths of 1 mile is given and 8 eighths is sought.
Hence, f of 1 kilometer = 1.6 kilometers.
EXERCISE 102
Solve by the above method.
1. If 6 horses plow a field in 9 days, how long will it
take 9 horses to plow the same field ?
2. If a train runs in 3| hours between two stations at
the rate of 18 miles an hour, how long will it take a train
whose speed is 30 miles an hour to make the same run?
3. If 5 acres of land sell for $ 423, at this rate what
will be the selling price of 7 acres ?
4. If 22 yd. of cloth are bought for a sum of money,
how many yards may be bought for the same sum when
the price falls 12 % ?
5. Eight horses consume a quantity of corn in 24 days.
How long should the same quantity of corn last 12 horses?
6. The minute hand of a clock goes 360° in 1 hour.
How many degrees does it go in 22 minutes ?
7. An arc of 75° is 4 ft. 6 in. How many feet are in
the circumference of the circle ?
APPROXIMATIONS 195
8. If 2^ of the number of miles from Paris to Turin is
27|, what is the entire distance separating the cities?
9. If ^ of the number of miles from New York City
to Panama is 1727, how far is Panama from New York?
10. Given .9 of the distance from London to Constan-
tinople as 1827 mi., how many miles is it from the former
to the latter?
11. If |~| of the distance from Hamburg to Vienna is
143 mi., find the distance between these cities.
12. In the year 1902, l^ of the United States internal
revenue receipts from tobacco amounted to $22,852,687.
Find the total internal revenue receipts from tobacco for
that year.
13. In the year 1902, ^ of the excise tax in the United
States on gross receipts under the War Revenue Law of
1898 amounted to $117,221. Find the total tax on gross
receipts in 1902.
14. In the year 1902, -fa of the United States internal
revenue receipts from the tax on oleomargarine amounted
to $1,325,021.40. Find the total receipts from this source.
15. The mark is the unit of money in Germany ; f^ of
its value in our currency is 42 mills. Express the value
of a mark in dollars.
16. The yen is the standard of value in Japan ; -^ of
its value is equivalent to 4 cents and 2 mills. Express in
dollars the value of the yen.
17. In Venezuela, the monetary unit is the Bolivar ;
| of its value is equivalent to $.1158. Find its value in
cents.
18. Thirty-two thirty-fifths of a meter is very nearly
equivalent to 1 yd. Express the value of a meter in yards.
196 ADVANCED BOOK OR ARITHMETIC
THE LANGUAGE OF MATHEMATICS, RATIO, PROPOR-
TION, PARTNERSHIP
By mathematics is understood those branches of knowl-
edge which deal with quantity. Arithmetic, algebra,
geometry, surveying, etc., are included in the term mathe-
matics.
Mathematics has a language of its own.
The word eight conveys a definite idea to the mind ; the
sign or symbol 8 conveys the same idea. The words
eight squared convey a definite idea to the mind ; the
symbol 82 conveys the same idea. The words three fourths
of sixteen convey an idea ; the symbols f X 16 convey the
same idea. Similarly, the words the quotient of seventy-
two divided by eight convey an idea ; the symbol ^- con-
veys the same idea.
Letters may represent numbers. Thus, a, 6, c, x, y, z,
etc., may each represent any number whatever. The
product of the numbers represented by letters is indicated
by writing the letters in succession, one after the other.
Thus, abc implies the continued product of a, 6, and c. lip
stands for principle, r for rate, t for time in years, and i for
interest, the rule for computing interest is given by the
relation prt = {
In like manner the rule for computing the area of a rec-
tangle may be expressed by the relation
F=la,
F stands for area, b for base, and a for altitude. A num-
ber written before a letter indicates multiplication.
Thus, 5 a means 5 times a. 5 a is then a short way of
writing a + a + a + a + a.
46 is a short way of writing b -f b + b + b.
THE LANGUAGE OF MATHEMATICS 197
02 is a short way of writing a x a or aa.
03 is a short way of writing a x a x a or aaa.
04 is a short way of writing 0x0x0x0 or aaaa.
05 is a short way of writing 0x0x0x0x0 or aaaaa.
The number denoting how many times a number is
added is called a coefficient. The coefficient of the expres-
sion 9 b is 9.
The expression a + b stands for the sum of any two
numbers. The expression a — b stands for the number
which when added to b gives 0, or in other words, the
remainder obtained when b is subtracted from a.
Example 1. If a = 5, b = 3, what is the value of 0 + 6 ?
a-b = ? 4a = ? 36 = ? 20-36=?
SOLUTION. 0 + 6 = 5 + 3 = 8. 0-6 = 5-3 = 2.
40 = 4x5 = 20. 36 = 3x3=9. 20-36 = 2x5-3
x3 = l.
Example 2. If a = 7, what is the value of a2? 03? 3 a2?
4 a3?
a2 = axa==7x7 = 49. aB==axaxa==j x 7x7 =343.
3 02 = 3x0x0= 3x7 x7=147. 403 = 4x0x0x0
= 4x7x7x7 = 1372.
EXERCISE 103
If 0=4, what is the value of 40? 70? 110? 130?
170? 190? 270? |0? ^0?
If 0=5, what is the value of 02? 03? 04? 202? 302?
203? 402? 0 + 02? 02 + 03?
If 0 = 3, 6 = 2, what is the value of 0 + 6? 0 — 6?
20-f6? 0+26? 20-6? 20-36? 30-26?
If 0 = 6, 6 = 3, what is the value of 60 + 36? 30 + 56?
60-36? 50-106? 70-56? 302? 02+62? 02-62?
If # = 5, y = 6, what is the value of xy? 2xy? oxyl
*V? xy? Ja?
198 ADVANCED BOOK OF^ ARITHMETIC
If x = 9, y = 4, what is the value of 2 x* - y* ?
? 2z2 +3 2/2? ?/2 -^?
If a = 10, 6 = 7, find the value of ab, 5 «6, a6 + a2, 2 a6
+ 62, ab + 2b*.
EXERCISE 104
1. What is the sum of two times a number and three
times the same number? What is the sum of 2x and 3#?
2. What is the sum of 4 x and 3 x? of 8 a and 3 a? of
56 and 26? of 66 and 4 6?
3. What is the difference between 8x and 3x? 6x and
2x? 116 and 76? Saanda?
4. Add 5 x and 7 # ; 4 # and 9 # ; 96 and 66; 10 y and
6 y ; 12 x and 4 #.
5. Subtract 4 # from 9 x ; 8 # from 14 #; 9#froml6#;
7 x from 13 x ; 5 a6 from 8 ab.
6. Find the difference between 11 y and 2y; 5a6 and
a6 ; 7 ab and 4 a6 ; 12 a6 and 2 #6.
Every sentence conveys a thought. (1) The sum of 3
and 4 is 7. This sentence is expressed in the language
of mathematics as follows : 3 + 4 = 7. (2) Write another
sentence : The difference between 18 and 7 is 11. This
sentence, written in mathematical language, is 18 — 7 = 11.
(3) Write a third sentence : Two thirds of 27 is 18. In
mathematical language this sentence is written | x 27= 18.
The statements (1), (2), (3), are called equations. An
equation is a statement in symbols that two expressions
are equal to each other.
The part of an equation to the left of the sign of equal-
ity is called the first member of the equation ; the part of
an equation to the right of the sign of equality is called
the second member of the equation.
THE LANGUAGE OF MATHEMATICS 199
What is the product of a and a? a x a = a2.
What is the product of a and a2? a x a2 = a x a x a
= a3.
What is the product of a2 and a3? #2 = ax#
a x a x a.
.-. a2 x a3 = (a x a) x (a x a x a) = a5.
What is the product of a4 and a3 ?
a4=axaxaxa.
a3 = a x a x a.
.'. a4 x a3 = (a X a x a x a) x (a x a x a) = a7.
What is the product of 4 a2 and
4 a2 = 4 x a x a.
.'.4a2x5a3=:4xaxax5x#x « x a
=4x5xaxa x a x a x a
= 20 a5. (Associative Law.)
EXERCISE 105
1. a x 2 a = ? 7. 3 a x 5 a3 = ? 13. 4 62 x 3 b = ?
2. 2axa2=? 8. 9a2x2a3=? 14. 553x453 = ?
3. 3 a2 x a = ? 9. 4 a3 x a2 = ? 15. 2 b2 x 5 6 = ?
4. 3a2x2a2=? 10. 5^x4z = ? 16. 4*/2x3#3=?
5. 4 a x a3 = ? 11. 6 x2 x :r3 = ? 17. 7 z2 x 5 a = ?
6. 7 a2 x 2 a2 =? 12. 5 5 x 62 = ? 18. 4 y x 8 53 = ?
What is the quotient when a3 is divided by a?
o a3 a x a x a o
ad -^ a = — = -- = a x a = a*.
a a
Here cancellation is utilized.
What is the quotient of 8 a3 by 2 a?
4
o 8 a3 $ x a x a x a 4
2a= --- = r— - = 4xaxa
2a $ xa
200 ADVANCED BOOK OF ARITHMETIC
EXERCISE 106
Find the following quotients:
i *± 6 ?1^ u 22o_6 16 §5^ 60^
2 a' 7 a2' 2 a2' Sz3' 10 fc5'
9a 28a* 16 a» 39^ gg 26^
3 ' ' 14 a2' 4a3' 13 a;' 13 a;'
12 a 18 a* 32a« 42 «» 33ft2
3. — 8. — — -. 13. . 18. s • 2d. .
4 o a6 Ib a* ba^* 116
16 a2 24 a4 24 a5 45 z4 48 J4
400 Q3 O 1^J7»
tt o a o d \j x J-O o
12 a3 11 a4 25 «4 50 x5 46 55
5. 10. . 15. -=— x-. 20. -. 25.
23 62'
RATIO
The ratio of one number a to another number b is the
quotient obtained by dividing a by b.
The ratio of a to b is written a : b. When the quotient
a -r- b is written -, the expression - is a fraction.
b b
The ratio b : a is called the inverse ratio of a to b.
EXERCISE 107
1. What is the ratio of 2 ft. to 6 ft.?
2. What is the ratio of 4 in. to 1 yd.? of 3 in. to 1 yd.?
of 1 yd. to 1 rd.? of l rd. to 1 rd.?
3. What is the ratio of 80 A. to 1 sq. mi.? of 120 A.
to 1 sq. mi.? of l A. to 2 A.?
4. What is the ratio of the distance traveled by two
trains in the same time, if the rate of the first train is 20
mi. per hour, and the rate of the second train is 30 mi.
per hour ?
THE LANGUAGE OF MATHEMATICS 201
5. If A walks at the rate of 2J mi. per hour, and B
walks at the rate of 5 mi. per hour, what is the ratio of
A's time to B's time in going any given distance ?
6. What is the ratio of the time that 8 men take to do
a piece of work to the time that 6 men take to do the same
piece of work ?
7. If you ride in a carriage at the rate of 7 mi. an hour
and walk back the same distance at the rate of 3 mi. an
hour, what is the ratio of the time in the carriage to the
time walking ?
8. What is the ratio of the price of 7 Ib. of sugar to the
price of 10 Ib. of sugar of the same kind ?
9. What is the ratio of the work done by 6 men to the
work done by 9 men ?
10. What is the ratio of the time that 9 men take to do
a piece of work to the time that 6 men take to do the
same work ?
11. I can buy two kinds of matting for 40^ and 50^ a
yard respectively. If I spend the same amount of money
in the purchase of the two kinds of matting, what is the
ratio of the number of yards of matting of the first kind
to the number of yards of the second kind bought ?
12. Divide 15 in the ratio 2 : 3.
13. Divide 20 in the ratio 3 : 7.
14. Divide f 1 in the ratio 18 : 7.
15. Divide 1 mi. in the ratio 7 : 9.
16. Divide 1 in the ratio 9 : 11.
17. Divide 1 gal. in the ratio 1 : 3.
18. Divide $'1 in the inverse ratio 9 : 16.
19. Divide 22 yd. in the inverse ratio 3 : 8.
20. Divide $1000 in the inverse ratio 3 : 5.
202 ADVANCED BOOK OF ARITHMETIC
PROPORTION
A statement indicating that two ratios are equal is
called a proportion.
Illustrations, 2:3 = 4:6. (1)
9:15 = 12:20. (2)
Statement (2) is a proportion because the value of the
first ratio is f , and the value of the second ratio, i.e. 12 : 20,
is also ^.
Statement (2) may read 9 is as large compared with
15 as 12 is compared with 20.
The first and fourth terms of a proportion are called
the extremes, and the second and third terms are called
the means, of the proportion.
In a proportion the product of the extremes is equal to the
product of the means.
Let a : b = c : d be any proportion whatever. Then
ad = be.
PROOF. 7 = -;- Multiply each member by Id and get
b d
abd cbd *, •,-, ,. 7 z
— ;— = . *. by cancellation ad = be.
b d
This property of a proportion enables us to find any
term of a proportion, if three of the terms of the propor-
tion are known.
The proportion a:b = c:d is sometimes written a: b
: :c: d. The double colon used as a sign of equality is
now rapidly becoming obsolete.
Example 1. Find x in the proportion x : 4 = 9 : 6.
SOLUTION. The product of the extremes is equal to
the product of the means.
.•.63=36. 3=6.
PROPORTION 203
Example 2. Find x in the proportion 10 : 35 = x : 42.
SOLUTION. Since the product of the means is equal to
the product of the extremes,
35 x = 10 x42.
Two numbers which vary directly are said to be directly
proportional. Two numbers which vary inversely are said
to be inversely proportional.
EXERCISE 108
If x stands for the unknown term in each of the follow-
ing proportions, find it :
1. 2:3=6: x. 13. 57 : 133 = x : 126.
2. 3: 4= 6: a?. 14. 68 : 85 = x : 75.
3. 15:25 = 12:z. 15. 36 : x = 52 : 65.
4. 12:20 = 18:z. 16. 28:^=36:63.
5. 14:21 = ^:27. 17. 27: a = 15: 50.
6. 21: 27 = a;: 45. 18. 15:^=21:77.
7. 35: 84 = z: 72. 19. 28: a =36: 81.
8. 20:48 = z:96. 20. 25: x = 45: 72.
9. 16:24 = ^:33. 21. 35:^=30:48.
10. 20: 32 = x: 72. 22. x: 81 = 16: 72.
11. 25:45 = ^:99. 23. x: 99 = 26: 117.
12. 45 : 126 = x : 154. 24. x : 65 = 24 : 52.
25. x : 112 = 45 : 144.
Example 1. If 7 bu. of wheat cost $5.25, find the cost
of 11 bu. of wheat at the same rate.
SOLUTION. It is reasonable to assume that the price of
11 bu. of wheat is greater than the price of 7 bu. of wheat.
.-. the price of 11 bu. of wheat = -U of $5.25= $8.25.
204 ADVANCED BOOK OF^ ARITHMETIC
Example 2. If 12 men pave a street in 15 da., how
long will it take 9 men to pave a street of the same area ?
SOLUTION. It will take 9 men longer than it takes 12
men. /. the time 9 men take = -^ of 15 da. = 20 days.
To solve a problem in proportion, find first the relation
of the answer to the quantity of the same kind as the
answer given in the problem. Second, multiply this
quantity by a fraction, proper or improper, according as
the answer is less or greater than it.
EXERCISE 109
1. If 20 men earn $450 in a given time, how much will
30 men earn in the same time ?
2. If 15 bu. of corn cost $7.20, what will 48 bu. of
corn cost?
3. If 12 A. of land cost $456.90, what will 16 A. of
the same land cost ?
4. If 4 men can do a piece of work in 15 da., how long
will it take 6 men to do an equal amount of work ?
5. If 18 head of cattle cost $1450, what will 27 head
of cattle cost at the same rate ?
6. If a train goes 400 mi. in 12 hr., how long will
it take to go 560 mi. ?
7. If 8 masons build a wall in 15 da., how long will
it take 6 masons to build a wall of the same size ?
8. If 18 horses consume 14 bu. of corn in a week,
how much will 24 horses consume in the same time ?
9. If 18 horses plow a tract of land in 13 da., how
long will it take 26 horses to plow the same tract ?
10. How long will it take 126 sheep to eat a quantity
of feed which will last 105 sheep 30 da. ?
COMPOUND PROPORTION 205
11. A garrison consisting of 1200 men has provisions
for 16 da. How many men must be sent away so that
the provisions may last 24 da. ?
12. A garrison consisting of 1400 men has provisions
for 27 da. If the garrison is reenforced by 400 men,
in how many days will the provisions be consumed?
13. If I can buy a dozen turkeys for $20.50, how
many turkeys can I buy for $30.75?
14. If the interest on $750 for 4 mo. is $12.50, what
is the interest on $39.60 for the same time?
15. If an arc of 12" on the 40th parallel of latitude is
933.92 ft., find the length of 1° on the 40th parallel of
latitude.
16. If an arc of 30' on the circumference of a wheel is
1^ in., find the length of the circumference of the wheel.
17. A fly wheel 63 ft. in circumference makes 150 revo-
lutions per minute. Find the velocity of its rim per second.
18. A train is running at 50 miles an hour. This speed
is 25% greater than usual. Find its usual speed.
COMPOUND PROPORTION
If the product of the corresponding terms of two or
more ratios are taken, the ratio of the resulting products
is called the ratio compounded of these ratios. For
example, the ratio compounded of the ratios 2 : 3, 4 : 5,
7:8, is the ratio 2x4x7:3x5x8, or 56 : 120, or
7 : 15.
A proportion in which the final result depends upon
a ratio compounded of two or more ratios is called a
compound proportion.
A concrete example may give a clearer conception of
compound proportion than any formal definition.
206 ADVANCED BOOK OF ARITHMETIC
Example l. If 15 men mow 90 A. in 12 da., how
many acres will 12 men mow in 14 da.?
SOLUTION. The 12 men in a given time will mow less
than 15 men in the same time. .*. the 12 men in 12 da.
will mow If of 90 A. But the 12 men in 14 da. will
mow more than this quantity. /. 12 men in 14 da. will
mow if of if of 90 A. = if of 90 A. = 84 A.
Example 2. If 24 men build a house in 18 da. of 10 hr.
each, how many men will it take to build the same house
in 30 da. of 8 hr. each?
SOLUTION. Step. 1. It will take fewer men to build
a house in 30 da. than it will take to build it in 18 da.
of the same length.
.*. the number of men it will take to build the house
in 30 da. of 10 hr. each = if of 24 men.
Step 2. More men are needed when they work 8 hr. a
day than when they work 10 hr. a day.
.*. the number of men, in 30 da. of 8 hr. each, required
to build the house = -^ of ^| of 24 men= 18 men.
EXERCISE 110
1. If 12 horses plow 84 A. in 6 da., how many acres
will 16 horses plow in 4^ da.?
2. If 14 men pave a street 200 ft. long in 8 da., how
many feet will 12 men pave in 7 da. ?
3. If a man earns $117 in 3 mo. working 6 hr. a day,
how much will he earn in 5 mo. working 8 hr. a day ?
4. A garrison of 3650 men consumed in 30 da. 82.3 T.
of food. How much food would be required for 7500
men for 1 yr. at the same rate ?
COMPOUND PROPORTION 207
5. If 8 masons build in 2 da. a wall 40 ft. long and
6 ft. high, what height of wall 30 ft. long can they build
in 5 da. ?
6. If 21 men complete a piece of work in 8 da. of 7|-
hr. each, in how many days of 10 hr. each can 18 men
do the same work ?
7. A wall is to be built in 10 da. by 30 men. After
2 da. 10 men are dismissed. In what time will the
remaining 20 men finish the work ?
8. If 4 men or 6 boys dig a trench in 12 da., in what
time can 2 men and 9 boys dig it ?
9. If 12 men mow 30 A. in 3 da. of 8 hr. each, how
many hours a day must 16 men work to mow 48 A.
in 4 da. ?
10. If the interest on $100 for 1 yr. is 16, find the
interest on 1840 for 2 yr. 3 mo.
11. If 12 men working 7 hr. a day earn 1227.50 in 20
da., how much will 15 men earn in 20 da., working 9 hr.
each?
12. If 6 men mow f of a meadow in 4J da., how long
will it take 8 men to mow the remainder ?
13. In 10 da. of 8 hr. each 9 horses can plow f of a
field. In how many days of 9 hr. each can the remainder
of the field be plowed by 15 horses ?
14. A marble block 3 ft. by 4 ft. and 5 ft. in length
weighs 5.1 T. Find the weight of a marble block 7 ft.
by 3 ft. and 10 ft. long.
15. A mason can build 3 yd. of a wall in 15 hr. How
long will it take 9 masons to build 24 yd. of a wall whi a
is one-third higher than the other wall ?
208 ADVANCED BOOK OF ARITHMETIC
PARTNERSHIP
NOTE. Partnerships are rapidly becoming a thing of the past. Those
partnerships that still survive are conducted on somewhat different
principles from the partnerships that existed prior to the introduction of
the telegraph, telephone, and modern means of rapid transit.
Example. A, B, and C enter into partnership. A puts
in $840, B puts in $350, and C puts in $2000. A with-
draws from the concern in 5 mo., C in 7 mo., and at the
end of 8 mo. the profits are divided. If the entire profit
is $450, how shall this be divided among A, B, and C?
SOLUTION. A has $840 in the concern for 5 mo. This
is equivalent to $4200 for 1 mo.
B has $350 in the concern for 8 mo. This is equiva-
lent to $2800 for 1 mo.
C has in the concern $2000 for 7 mo. This is equiva-
lent to $14,000 for 1 mo.
The profits will be divided in proportion to the num-
bers 4200, 2800, 14,000, or in proportion to the numbers
3, 2, 10, since 1400 divides each of them.
3 + 2 + 10 = 15.
.•. A's share = -^ of the profits = -f% of $450 = <
B's share = T2^ of the profits = T2^ of $450 = I
C's share = ^ of the profits = jf of $450 = $300.
EXERCISE 111
l. A, B, and C enter into partnership with capitals of
$3000, $3750, and $4500 respectively. At the end of
the year they divide among themselves a profit of $3000.
Find each person's share.
PARTNERSHIP 209
2. Two partners, A and B, invest $600 and $1125.
A's money remains in the business 6 mo., and B's 8 mo.
If they make a profit of $2100, find each person's share.
3. Two men rent a pasture for $171 ; one puts in the
pasture 30 cattle for 30 da., and the other 45 cattle for
18 da. How much rent should each pay ?
4. A and B enter into partnership. A's capital is
$200 more than B's. Out of a profit of $640, B gets
$280. Find A's and B's capital.
5. A and B enter into a partnership, A contributing
$6400 and B $7200. At the end of 3 mo. A withdraws
$1600, and at the end of 5 mo. B withdraws $1440. C
then enters into the partnership with a capital of $4800.
Seven months later a gain of $2154 is divided among
them. Find each person's share.
6. A, B, and C enter into partnership. A puts in
$1000, B $1200, and C $1800. At the end of 3 mo. C
withdraws, and at the end of 10 mo. B withdraws. At
the end of a year the profits are divided. If C gets $135,
how much do A and B receive ?
7. Two men form a partnership. Their capitals are
in the ratio 2 : 3. After 6 mo. the first man increases his
capital by J of itself, and the second man diminishes his
capital by J of itself. After 6 mo. more they divide their
profits, amounting to $1450. Find each partner's share.
8. A cistern 66' x 27' 6" x 10' will hold enough water to
irrigate 2J A. of land to the depth of 2 inches. How many
acres will a cistern 77' x 41' 3" x 12' 6" irrigate to the
depth of 3J inches ?
9. If A pays |- of the cost of irrigation when the rate
charged is $4 an acre to the depth of one inch, find his
share of the cost.
210 ADVANCED BOOK OF ARITHMETIC
PERCENTAGE
A per cent of a number implies a fraction of the num-
ber having 100 for denominator. Thus, 5 per cent, 5%,
yj-g-, and .05 are four ways of expressing the same fact.
The per cent equivalents of the following fractions
should be thoroughly fixed in mind :
i' i* f> i> I' i' t> t' i> <b f> i> t> f' s> iV-
Example l. The total value of imports into this coun-
try through the Atlantic ports for the year 1906 was
$974,562,800; of this 75.35% came through New York
City. Find the, value of the imports through this city.
SOLUTION. $974,562,800 x 7-^~ = $9,745,628 x 75.35
= $734,333,069.80, value of imports through New York.
As 75.35% is correct to four figures only, the result is
not likely to be correct to more than four figures. To
get four figures multiply 97.456 millions by 7.535 by the
contracted process explained on page 191.
EXERCISE 112
1. Write the equivalent per cents of the following
decimals :
.04, .08, .075, .0525, .1666f.
2. Express as decimals the following per cents :
41%, 15%, 121%, 621%, 6i%, 3|%.
3. Find 5 % of each of the following numbers :
2151, 366.7, 689.5, 7.188, 12.469.
4. Find 6 % of each of the following numbers :
5262, 520.7, 2.66, 3.097, 6.41, .783.
5. Find 4| % of each of the following numbers :
4150, 1418, 7120, 43.43, 53.17, 2.42.
PERCENTAGE 211
6. The Engineer's Year Book for the year 1906 gives
the cost of railway construction in England as $194,660
per mile. The per cents of cost were as follows :
Land 10 Permanent way 11|
Fencing 1J Sidings 3
Earthworks 24 Junctions 1
Tunnels 12 Stations 6-|
Viaducts and bridges 17 Maintenance J
Accommodation works 2 Legal and engineer-
Culverts 5 ing expenses 6
Find the cost of each of the above items of expense.
7. The value of the total imports to the United States
for the year 1906 was $1,226,560,000. Of this value
79.45% came through the Atlantic ports, 4.42% through
the Gulf ports, 1.38% through the Mexican border ports,
5.41% through the Pacific ports, 7.97% through the
northern border ports, 1.37% through the interior ports.
Find the value of the imports through each of these
divisions.
8. The value of the total exports of the United States
for the year ending June 30, 1906, was $1,743,860,000.
The per cents of total value by principal customs districts
were as follows :
New York 34.81 Savannah 3.72
Boston 5.66 Puget Sound 2.82
New Orleans 8.63 Detroit 2.02
Galveston 9.54 Buffalo Creek 1.72
Mobile 1.25
Philadelphia 4.73 Newport News 1.15
Baltimore 6.31 Wilmington 1.06
San Francisco 2.29 Pensacola 1.06
Find the values of the exports through these cities.
212
ADVANCED BOOK OF ARITHMETIC
Given a quantity, to find its value when decreased by a
per cent of itself.
Example 1. In the year 1906 the state of Ohio produced
11,562,500 Ib. of wool ; this shrunk 50 % from scouring.
Find the number of pounds of scoured wool.
SOLUTION. 100 % - 50 % = 50 % = J.
11,562,500x1=5,781,250.
Am. 5,781,250 Ib.
EXERCISE 113
l. The wool production and per cent of shrinkage from
scouring for the year 1906, as given by the Bulletin of
National Association of Wool Manufacturers, for the
states named are as follows :
STATE
NUMBER POUNDS
UNWASHED
PER CENT OF
SHRINKAGE
Michigan
9,450,000
50
2,450,000
52
568,750
40
35,815,000
65
Wyoming
32,849,750
68
Idaho
16,905,000
67
Oregon
15,300,000
70
13,125,000
67
Utah
12,350,000
65
New Mexico
15,950,000
62
Colorado
9,450,000
67
Arizona
4,420,000
66
Texas
9,360,000
66
4,887,500
70
Find the number of pounds of scoured wool produced
in each of the states.
PERCENTAGE 213
Given a per cent of a number, to find the number.
Example l. During the month of January the average
daily attendance of a school was 414. This number was
92 % of the school enrollment. Find the number enrolled.
SOLUTION. 92 % of enrollment is given.
100 % of enrollment is sought.
.-, enrollment = 414 x -^ = 450, or
if x stands for enrollment,
414
.9 a = 414, therefore x = - ^ = 450.
. y
Example 2. A dealer sells an article for $ 522 at a gain
of 16%. Find the cost price.
SOLUTION. 116 % of cost price is given.
100 % of cost price is sought.
/. cost price = $522 x — = 1450, or
1.16s = $522.
.'.x = $522 -s-1.16 = $450.
EXERCISE 114
1. Find the number of which 79 is 4 % .
2. In a certain town 60 % of the grown people are mar-
ried. If there are 2394 married people, how many grown
people are in the town ?
3. A man spends $320 for board. This sum is 40% of
his income. Find his income.
4. A man spends 83% of his salary and saves 1170.
What is his salary ?
5. A lot is sold for $3380 at a gain of 12f %. Find
the cost of the lot.
6. After a discount of 16f % is given, a man pays $84
for a bill of goods. Find the amount of the bill.
214
ADVANCED BOOK OF ARITHMETIC
7. The total levies of ad valorum taxes and tax rate
per cent of assessed valuation are as follows in the states
named :
STATE
LEVY
RATE PER CENT
Maine .
$ 6,855,776
1.95
Pennsylvania
58,269,455
1.49
South Carolina .... ...
3 736 344
1 91
Kansas
14 847 136
4.09
Tennessee . .
7,626 068
1.88
9,002,727
3.45
Texas
13,683,526
1.34
Find the assessed valuation of property in each of these
states.
To express one number as a percentage of another number.
Example 1. The foreign population of Danish extrac-
tion according to the United Census of 1890 and 1900 was
132,543 and 153,805.
Find the increase per cent during the ten years.
SOLUTION. 153805 - 132543 = 21262, increase.
21262
132543
21262
132543
= fraction the increase is of
population in 1890.
£,=16.04%. In-
16.04
crease per cent.
132^)2126200
132543
80077
79526
551
530
As the divisor contains 6 figures and
the quotient is required to 4 figures,
for each quotient figure cut off one
from the divisor instead of annexing a
cipher.
PERCENTAGE
215
Example 2. A dealer buys goods at a discount of 40 %
off the list price, and sells them at 16 % off the list price.
Find his gain per cent.
SOLUTION
Cost price to dealer = 60 % of list price (100 % - 40 %).
Selling price of dealer = 84 % of list price (100 % - 16 %).
Gain = 24 % of list price.
|-^ x 100 % = rate per cent of gain. Ans. 40 °/0.
EXERCISE 115
1. The railway mileage of the world January 1, 1906,
as given by a German statistician was as follows :
COUNTRY
MILES
COUNTRY
MILES
Europe
192,251
North America
253,098
Asia
50,593
South America . .
32,859
Africa
16,538
Australasia ....
17,441
Find the per cent of the total railway mileage in each
of the six continents.
2. The foreign-born population of the United States
by countries for the years 1890 and 1900 was as follows :
COUNTRY
1890
1900
COUNTRY
1890
1900
Austria . .
England . .
France . .
123,270
909,090
113,174
275,910
840,513
104,197
Germany
Ireland . .
Scotland
2,785,000
1,871,500
242,200
2,663,000
1,615,500
233,500
Find the rate per cent of increase or decrease.
3. A dealer buys goods at a discount of 40 % off the
list price, and sells them at 2 % below the list price. What
per cent of profit does he make?
216 ADVANCED BOOK OF ARITHMETIC
4. Eggs are bought at the rate of 5 for 4^, and sold
at the rate of 4 for 5^. What per cent of profit is made?
5. A lot is sold for $1560 at a profit of $120. Find
the rate per cent of profit.
6. Meat is sold at 18^ per pound at a profit of 20%.
Find the cost price per pound.
7. If the butcher has to pay 1^ per pound more for the
meat, how must he sell it to make a profit of 25 %?
8. A piano is sold for $470 at a loss of 6%. What
would the gain per cent have been if the piano had been
sold for $520?
9. A tradesman buys at a discount of 10%, and sells
at an advance of 15 % on the nominal cost price. Find his
rate per cent of profit.
10. A book costs the publisher 60^ for printing and
publishing. At what price should he sell the book in
order that he may make a profit of 20%, after paying the
author 10% on the selling price?
11. What should be the selling price of an article which
costs $15, so that a profit of 20 % may be made after giv-
ing the dealer a discount of 10 % ?
12. A tradesman marks his goods at 25 % above cost,
but allows the customer 6 % discount. What per cent of
profit does he make ?
13. Tea is sold at 60^ per pound at a profit of 33J%.
If the total gain is $15, how much tea is sold ?
14. A man buys a house for $4000 which he rents for
$40 per month; his taxes are 3 % on a valuation of
$ 3000. What per cent does his money yield ?
15. A merchant marks his goods 20 % above cost.
What discount does he give if he sells at cost ?
INTEREST 217
INTEREST
Example 1. Find the interest on $ 670 at 5% from
Jan. 14 to Aug. 10. .
MO. DA.
SOLUTION. $670 Aug. 10 = 8 10
.05 Jan. 14 = 1 14
6 mo. = £ of 1 yr. 2)133 50 = int. for 1 yr. 6 26
16.75 = int. for 6 mo.
20 da. = 1 of 6 mo. 1.861 = int. for 20 da.
5 da. = I of 20 da. .465 = int. for 5 da.
1 da. = | of 5 da. .093 = int. for 1 da.
$19.17 = int. for 6 mo. 6 da.
EXERCISE 116
Find the interest and amount of :
1. 1728 for 1 yr. 6 mo. at 5%.
2. $670 for 1 yr. 6 mo. at 7 %.
3. $1260 for 1 yr. 3 mo. at 8%.
4. $385 for 1 yr. 4 mo. 12 da. at 1%.
5. $2750 for 1 yr. 8 mo. at 3 %.
6. $3345 for 1 yr. 4 mo. at 6%.
7. $783 for 1 yr. 1 mo. 10 da. at 4 %.
8. $597 for 1 yr. 4 mo. 24 da. at 5 %.
9. $3000 for 3 mo. 6 da. at 7%.
10. $940 for 1 yr. 4 mo. at 3%.
11. $1800 for 1 mo. 15 da. at 4%.
12. $ 2100 for 2 yr. 9 mo. at 4%.
13. $960 for 8 mo. 17 da. at 1%.
14. $2911.25 for 1 yr. 7 mo. 16 da. at 4%.
15. $1857 for 1 yr. 5 mo. 18 da. at 5%.
16. $2775 from May 1 to Dec. 19 at 4 %.
218 ADVANCED BOOK OF ARITHMETIC
17. $1770 from Jan. 10 to Oct. 5 at 4J %.
18. $1975.14 from Feb. 8 to Nov. 1 at
19. $1218 from March 6 to Nov. 1 at
20. $1788 from Feb. 14 to Dec. 20 at 7-| %.
EXACT INTEREST
Interest reckoned on the basis of 365 days to the year
is called exact interest. Exact interest is used by the
United States government and sometimes in business
transactions.
Example. Find the exact interest on $2384.50 from
Jan. 12 to July 5 at 5%.
SOLUTION. From Jan. 12 to July 5 there are (19 -f 28
+ 31 + 30 + 31 + 30 + 5). days = 174 days.
$2384.50 x .05 x'i£f = exact interest.
$2384.50 x. 05x174
-— — - = $56.84, nearly.
obo
EXERCISE 117
Find the exact interest on :
1. $913 from Jan. 4 to Feb. 4 at 5%.
2. $731.11 from Jan. 14 to Jan. 28 at 7%.
3. $52.50 from Jan. 1 to April 28 at 7 %.
4. $2745 from Feb. 1 to April 6 at 5 %.
5. $1095.80 from March 6 to June 7 at 5%.
6. $1911.17 from March 1 to May 11 at 7 %.
7. $1464.98 from Jan. 4 to May 30 at 6 %.
8. $10565.65 from May 13 to June 25 at 4 %
9. $834 from Feb. 5 to July 12 at 11 %.
10. $3561.50 for 81 da. at 5%.
INVERSE QUESTIONS IN INTEREST 219
INVERSE QUESTIONS IN INTEREST
Example l. What principal will produce $78.75 interest
in 75 days at 1\%1
Let x denote the principal.
.•. $# will produce 178.75 x -y~- in 1 year.
= 78.75 x
Example 2. In what time will $840 produce $57.40 in-
terest at 5 % ?
Int. on $840 for 1 yr. at 5 % = $42.
$57.40 •
11 of 1 yr. = 11 of 12 mo. = 4f mo.
I of a mo. = | of 30 da. = 12 da.
The time is 1 yr. 4 mo. 12 da.
Example 3. At what rate percent will $720 produce
$42.50 interest in 1 yr. 2 mo. 5 da. ?
$720 will produce $42.50 -*- (1 + 1 + ^|^) in 1 yr.
$42 50
$ 1 will produce -in- -*-(!+£+ 3 to) in * Jr-
$100 will produce 100 x'^-(l + H si o)inlJr- = *5-
.•. the rate is 5%.
Example 4. What principal will amount to $136.27 in
1 yr. 3 mo. 15 da. at 5 %?
The interest on $1 for 1 yr. 3 mo. 15 da. is $.0645|.
.-. the amount of $1 for 1 yr. 3 mo. 15 da. is $1.0645f.
. \ the number of dollars in principal = $136.27 -*-
$1.0645f = $128, nearly.
220 ADVANCED BOOK OF ARITHMETIC
EXERCISE 118
What principal will produce:
1. $60 in 11 yr. at 8%?
2. $120 in 2yr. at 5 % ?
3. 1135 in 1 yr. 6 mo. at 9%?
4. $36 in 3 yr. at 5%?
5. $144 in 1 yr. 4 mo. at 4J%?
6. $12 in 1 yr. at 4%?
7. $21 inl yr. at 3| %?
8. $84 in 3yr. 6 mo. at 3%?
9. $16.90 in 2 yr. 2 mo. at 4 % ?
10. $ 42 in 2 yr. 4 mo. at 4 % ?
11. $25.50 in 6 mo. at 5 %?
12. $5.40 in 4 mo. at 5 %?
13. $6.75 in 9 mo. at 3%?
EXERCISE 119
In what time will:
1. $1088.75 produce $87.10 interest at 8 % ?
2. $144 produce $21.60 interest at 5 % ?
3. $215 produce $6.45 interest at 5 % ?
4. $ 1160 produce $278.40 interest at 6 % ?
5. $810 produce $56.70 interest at 7 % ?
6. $312.50 produce $43.75 interest at 8 % ?
7. $2220 produce $216.45 interest at S%?
8. $1400 produce $78.75 interest at 5%?
9. $480 produce $85.50 interest at 9| % ?
10. $3835 produce $345.15 interest at 8%?
11. $1380 produce $88.55 interest at 3|%?
12. $5400 produce $267.75 interest at 7 % ?
13. $7630 produce $1335.25 interest at 6 % ?
INVERSE QUESTIONS IN INTEREST 221
EXERCISE 120
Find the rate per cent when the interest on —
1. $750 for 1 yr. is $45.
2. 1928 for 1 yr. is 1 64. 96.
3. $880 for 1| yr. is $79.20.
4. $945 for 6 mo. is $37.80.
5. $828 for 8 mo. is $38. 64.
6. $1200 for 1 yr. 3 mo. is $90.
7. $1800 for 9 mo. is $67.50.
8. $2400 for 8 mo. is $64.
9. $2500 for 9 mo. 18 da. is $100.
10. $3000 for 7 mo. 12 da. is $90.
11. $3750 for 4 mo. 15 da. is $112.50
12. $2754 for 2 mo. 20 da. is $55.08.
13. $4846 for 6 mo. 20 da. is $121.15.
14. $1440 for 7 mo. 10 da. is $52.80.
Find the rate per cent when —
15. $1080 amounts to $1123.20 in 8 mo.
16. $1200 amounts to $1270 in 10 mo.
17. $1600 amounts to $1640 in 6 mo.
18. $2460 amounts to $2574.80 in 9 mo. 10 da.
19. $92 amounts to $102.12 in 2 yr.
20. $324 amounts to $333.72 in 8 mo.
EXERCISE 121
What principal will amount to —
1. $840 in 1 yr. at 5% ? 4. $903 in 1| yr. at 5 % ?
2. $749 in 1 yr. at 7% ? 5. $414 in 3 yr. at 5% ?
3. $645 in 1 yr. at 7-| % ? 6. $255.30 in 2 yr. at
222 ADVANCED BOOK OF ARITHMETIC
7. $12,540.45 in 2 yr. 3 mo. at 4 %?
8. $168.35 in 7 mo. 6 da. at ( % •
9. $618.67 in 1 yr. 4 mo. at 5%>
10. $646.80 in 8 mo. at 4%?
11. $776.07 in 1 yr. 1 mo. at 4| %?
12. $481.50 inlyr. at 7 %?
13. $432.55 in 11 mo. at 8 %?
14. $282.75 in 2 yr. 4 mo. 15 da. at 7| %?
15. $2090.07 in 1 yr. 1 mo. 15 da. at 8 %?
16. $2067.75 in 1 yr. 5 mo. at 10|%?
17. $268.28 in 1 yr. 7 mo. at 6%?
18. $254.25 in 7 mo. 15 da. at 9|%?
19. $25,346.25 in 1 yr. 3 mo. at 10%?
20. $843.70 in 8 mo. 3 da. at 7|%?
EXERCISE 122
REVIEW
1. Find the interest on $4000 for 13 mo. 2 da. at 9%.
2. Find the interest on $256.30 for 4 mo. 9 da. at 1%.
3. Find the interest on $30.85 for 11 mo. 6 da. at 5%.
4. Find the interest on $653 for 2 mo. 16 da. at 4%.
5. Find the interest on $2105.60 for 84 da. at 5%.
6. Find the amount of $805 for 10 mo. at 8%.
7. Find the amount of $507 for 1 yr. 12 da. at 8%.
8. What principal will produce $20.83 interest in 5
mo. at 5%?
9. What principal will produce $17.50 interest in 9
da. at 5%?
10. Find the rate of interest when $500 produces $2.92
interest in 1 mo.
REVIEW 223
11. Find the rate of interest when $250 produces 17.30
in 5 mo.
12. How much must I invest at 5% interest to have an
annual income of $1200 from my investment ?
13. A man buys a house and lot and rents it for $40 a
month. Taxes and insurance cost him $120 a year. If
his net receipts give him a profit of 6% on his investment,
find the cost of the house and lot.
14. For how long a time must $3000 be loaned at 5%
to produce $20 interest?
15. If I borrow $2400 at 1% interest and pay in princi-
pal and interest $2456, how long did I keep the money?
16. For how long a time must a sum of money be
loaned at simple interest at 8% to produce in interest
-1 of itself ?
17. For how long a time must a sum of money be
loaned at simple interest at 6% to produce in interest
•jV of itself?
18. Find the exact interest of $1200 for 292 da. at 5%.
19. Find the exact interest of $7300 for 146 da. at 7%.
20. The exact interest of $10,800 at 5% is $324.
Find the time.
21. In what time will $260 amount to $262.60 at 5% ?
22. What sum must be deposited in a savings bank
which pays 3-|% interest to produce semiannually $8.75?
23. A man deposits his money in two banks. In one
bank he has $572 which pays 3^%. The other bank gives
4% interest. If he receives as interest the same amount
from both banks, how much money has he all together?
24. If I invest half my money at 6% and the remainder
at 4%, and derive an income of $650 annually, how much
money have I invested ?
224 ADVANCED BOOK OF ARITHMETIC
REVIEW QUESTIONS
1. Define principal, rate, per cent, interest, amount.
2. How does exact interest differ from interest accord-
ing to the common use of the term? What is the dis-
tinction between simple interest and annual interest ?
3. How do you find the interest of a sum of money at
a given rate and for a given time ?
4. If you were given the interest, the rate, and the
time, how would you find the principal ?
5. Given the principal, interest, and time, how would
you find the rate per cent? How would you find the
rate ?
6. Given the principal, the rate, and the interest, how
would you find the time ?
7. Given the principal, the amount, and the rate, how
would you find the time ?
8. Given the principal, the amount, and the time, how
would you find the rate ?
9. Given the amount, the rate, and the time, how
would you find the principal ?
10. If you knew the interest of a sum of money for a
given time at 6%, how would you find the interest for the
same sum for the same time at 5% ? at 4% ? at 8% ?
11. If you knew the interest at 4%, how would you find
the interest of the same sum at 1% ? at 3% ? at 5%|? at
3i%?
12. If }^ou were given the interest of a sum of money
for a number of days, how would you determine from this
the exact interest of the same sum for the same number of
days?
PROMISSORY NOTES 225
PROMISSORY NOTES
A written promise by one person to pay another person
on demand, or after a specified time, a sum of money is
called a promissory note.
The following are promissory notes written in standard
form :
Galveston, Texas, ?Wcvi&h 7, 1907.
^t&v dat& c/ promise to pay to
the order of
--&/iis&& fvwndA&d, &LqhX/u ________ — Dollars
I / 100
at ______ t/i& ofi/i&t ofa£u>w&C joa/wk ________________________
Value received, w-UJ^ imteAs&^t at 6 %.
No.
Dallas, Texas, Tn^eA V, 1907.
dewuwvcL c/ promise to pay to the order of
—Dollars
at _____________ tfi
Value received, w£(A iM,t&^&&t at 7 % .
Due.
The first of the above promissory notes is called a time
note; the second is called a demand note.
Q
226 ADVANCED BOOK OF ARITHMETIC
The person who promises to pay is called the maker.
John Mosley is the maker of the first note above.
The person to whom the money is to be paid is called
the payee. The person who has legal possession of a note
is called its holder.
The sum specified in a note is called its face. A time
note is legally due on the date indicated. In some states
3 days more than are indicated in the note are allowed
before the note is legally due. These days are called
days of grace. The day on which a note is legally due is
called the day of maturity.
A note made payable to the order of a person, or a note
made payable to the bearer is negotiable, i.e. it may be
transferred from one person to another person.
A note made payable to the payee only is non-negotiable.
When a note payable to the order of the payee is trans-
ferred, every holder before parting with it must indorse it,
i.e. write his name on the back of it. Every indorser
thus becomes liable for the payment of the note, if the
maker fails to pay it. The holder in whose possession
the note is at maturity presents it to the maker for pay-
ment. If the maker refuses to pay it, the holder engages
a Notary Public to give to the indorser, or indorsers, a
written notice of its non-payment. This notice is called
a protest. A protest must be sent on the date of maturity ;
otherwise the indorsers are not held responsible for the
payment of the note.
An indorser who writes over his signature the words
ivithout recourse is not held responsible for the payment of
the note.
A note made payable to the bearer is negotiable without
indorsement. In some states a note must contain the
words value received in order to be legal.
BANK DISCOUNT 227
If the words with interest are not in a note, no interest
is charged. If, however, the note is not paid on the date
of maturity, interest at the legal rate may be charged. If
a note contains the words with interest, and no rate is speci-
fied, it is then understood that the note bears the rate of
interest usually charged in the state where it is made.
When the time of payment is indicated in months, cal-
endar months are understood.
A note drawn March 6, and payable two months after
date, matures on May 6 in states where days of grace are
not allowed, and on May 9 in states where days of grace
are allowed. About one half of the states and territories
allow 3 days of grace.
BANK DISCOUNT
New Orleans, La., <$c,6.. /P, 1903.
W.OO.
c/t/^£y cLaAf& a^tsA* cla,t& c/ promise to pay to
the order of ________________ fotefiA, (^o-an _________________
fauncU&ci fvfty __________________________ —Dollars
Value received.
No. 33. Due ftfalt 15 1 18,
The above time note is negotiable when indorsed.
Supposing the payee, Joseph Coan, needs money, he can
sell the note to a bank. The sum the bank gives him for
the note is called the proceeds of the note. The differ-
ence between the proceeds and the face of the note is
called the bank discount.
228 ADVANCED BOOK OF ARITHMETIC
The bank discount is always a rate per cent of the value
of the note 011 its day of maturity, reckoned from the date
of the sale of the note to the day of maturity.
The bank discount is then the interest on the maturity
value of the note computed from the date of discount to
the date of maturity. This time is called the term of
discount.
The maturity value of the note minus the bank discount
is the proceeds of the note.
From the computer's point of view the essential features
of a note are the face, maturity value, date of drawing, date
of sale, or date of discount, rate of interest the note bears,
rate of interest charged, known as rate of discount, and
date of maturity.
BANKERS' INTEREST
Banks charge interest for the exact number of days
between dates, allowing 30 days to a month. Banks
usually draw notes for 30, 60, or 90 days.
Example. Find the discount and the proceeds of the
above note, if it was discounted at 8%, March 1, 1903.
SOLUTION. The bank charges discount from March 1,
to April 18.
From March 1 to April 18 is 48 days.
$350. = maturity value.
.08
28.00 = int. for 1 yr.
45 da. = £ of 1 yr. 3.50 = int. for 45 da.
3 da. = T^ of 45 da. .23 = int. for 3 da.
$3.73 = int. for 48 da.
= bank discount.
$350 - $3.73 = $346.27 = proceeds of the note.
COMPUTING DISCOUNT 229
EXERCISE 123
Find the bank discount and the proceeds of the follow-
ing indicated notes, allowing 3 days of grace in examples
1, 2, 10, 11, 12, 13, and no grace in the remaining: —
DATE
TIME
FACE
Dis-
RATE OF
C'TI
SD
DISC'T
1.
Jan.
12,
60
da.,
$600,
Feb.
13,
8%.
2.
July
4,
60
da.,
$800,
Aug.
3,
8%.
3.
Mar.
3,
90
da.,
$500,
Mar.
3,
6%.
4.
April 5,
60
da.,
$700,
April 5,
6%.
5.
May
7,
60
da.,
$600,
May
10,
10%.
6.
May
9,
20
da.,
$900,
May
24,
8%.
7.
May
30,
90
da.,
$750,
July
1,
6%.
8.
June
5,
60
da.,
$450,
July
5,
6%.
9.
Aug.
11,
30
da.,
$800,
Aug.
11,
6%.
10.
Sept.
9,
30
da.,
$350,
Sept.
12,
10%.
11.
Oct.
4,
60
da.,
$800,
Oct.
7,
10%.
12.
Oct.
14,
60
da.,
$500,
Nov.
16,
12%.
13.
Nov.
10,
90
da.,
$600,
Nov.
25,
8%.
14.
Dec.
4,
90
da.,
$750,
Feb.
5,
6%.
15.
Jan.
10,
45
da.,
$650,
Feb.
9,
7%.
16.
Jan.
5,
60
da.,
$850,
Feb.
5,
9%.
17.
Jan.
30,
75
da.,
$950,
Feb.
28,
9%.
18.
July
10,
3
mo.,
$380,
July
12,
9%.
COMPUTING DISCOUNT ON INTEREST-BEARING NOTES
Find the bank discount and the proceeds of a 90-day
note for $250, dated Portland, Me., June 9, 1907, bearing
interest at 6%, and discounted July 8, 1907, at 8%.
230 ADVANCED BOOK OF ARITHMETIC
Step 1. Find the maturity value of the note. Maine
allows 3 days of grace. Hence, the interest will be com-
puted for 93 days.
$250
~ 3.87= interest at 6% for 93 days.
$253.87 = maturity value of the note.
Step 2. Find the term of discount (exact number of days
from July 8, to Sept. 10, the date of maturity).
Step 3. Find the bank discount. This is reckoned on
the maturity value of the note.
Int. on $253.87 for 64 days at 8% =$3. 61.
$254.87 - $ 3.61 = $250.26, proceeds of note.
EXERCISE 124
Find the discount and the proceeds of the following in-
dicated notes, allowing 3 days of grace in examples 2, 3,
6, 9, 10, 13, and no grace in the remaining : —
FACE
DATE
TIME
J.W1.J. K,
OF INT.
J.V.A. J. Hi \J
DISC'T
DISC'T
1.
$350,
Jan.
1,
45
da.,
10%,
12%,
Feb.
1.
2.
$395,
Jan.
10,
3
mo.,
6%,
10%,
Jan.
20.
3.
$450,
Feb.
1,
30
da.,
6%,
8%,
Feb.
18.
4.
$600,
Mar.
1,
60
da.,
8%,
12%,
Mar.
31.
5.
$500,
Apr.
2,
60
da.,
6%,
8%,
Apr.
17.
6.
$900,
Apr.
10,
30
da.,
6%,
8%,
Apr.
10.
7.
$1000,
Apr.
15,
60
da.,
8%,
10%,
Apr.
15.
8.
$750,
May
4,
60
da.,
6%,
6%,
May
5.
9.
$800,
July
10,
30
da.,
6%,
8%,
July
26.
10.
$400,
Aug.
15,
60
da.,
6%,
6%,
Aug.
15.
11.
$850,
Nov.
11,
90
da.,
1%,
10%,
Nov.
12.
12.
$900,
Dec.
12,
45
da.,
6%,
6%,
Jan.
13.
13.
$1200,
Dec.
5,
3
mo.,
6%,
10%,
Jan.
15.
COMMERCIAL DISCOUNTS 231
COMMERCIAL DISCOUNTS
Commercial, or Trade Discount is a reduction from the list
price of goods, or the amount of a bill.
If two or more discounts are allowed, the first is
reckoned on the list price, the next on the remainder after
deducting the first discount, the third is reckoned on the
second remainder, etc.
Example. Find the cost price of a bill of goods, if the
list price is $690 and discounts of 25%, 10%, and 5% are
allowed.
SOLUTION. $690
.$172.50 = 25% of $600.
$517.50 .= first remainder.
$ 51.75 = 10% of $517.50.
$465.75 = second remainder.
$ 23.287 = 5% of $465.75.
$442.46 = cost of the goods.
EXERCISE 125
1. Find the cost if the list price is $350, and the
discount 20%.
2. Find the cost when the list price is $823, and the
discount 12|%.
3. What is the cash value of a bill of goods listed at
$937.50 when a discount of 16|% is given?
4. A suit of clothes is marked $17.50, and is sold for
$12.00. What is the rate of discount ?
5. A bookseller buys 60 books marked $2.10 each at a
discount of 16|%, and sells them at the marked price.
What is his gain per cent, and how much profit does he
make on the sale of the books?
232 ADVANCED BOOK OF ARITHMETIC
6. A dealer buys goods at a discount of 16| % and sells
them at 5% above the list price. What is his gain per
cent?
7. Find the cost price in each case, if the list prices
and rates of discount are as follows :
COST DISCOUNTS
(a) $700 20%, 121%, 10%
(5) $7000 10%, 20%, 30%
(<?) $3000 15%, 12%
(rf) 19690 |%,33i%
(e) $5000 43%, 10%
(/) $3000 30%, 20%, 20%
(#) $ 4000 20%, 10%, 5%
Qi) $6000 46%, 45%
(l) $2760 30%, 15%
PARTIAL PAYMENTS
When payments are made on a note, these payments
are known as partial payments. These payments and their
dates of payment are written on the back of the note.
There are several methods of computing the amounts
due on such notes. The best known and most widely
used are the United States Rule and the Merchants' Rule.
UNITED STATES RULE
Find the amount of the principal until the time of the first
payment, or until the sum of two or more payments equals or
exceeds the interest.
Subtract the payment, or the sum of the two or more
payments from the amount.
Proceed with the remainder as a new principal. Continue
in this manner until the date of settlement.
PARTIAL PAYMENTS
233
A note for $600.00 dated Aug. 10, 1902, was
indorsed as follows :
1902, Dec. 15, $100.
1903, Feb. 12, $150.
1903, March 15, $150.
Find the amount due April 1, 1903.
SOLUTION.
DATES OF INDORSEMENT
yr. mo. da.
1902 8 10
1902 12 15
1903
1903
TIME BETWEEN DATES
yr. mo. da.
4 5
2
3
4
12
15
1
27
3
16
PAYMENTS
$100
$150
$150
1903
$600 = first principal, $600 x .06 x -| = $12.50.
12.50 = int. for 4 mo. 5 da.
$612.50 = amt. for 4 mo. 5 da.
$100 = first payment. _, Q
$512.50 = second principal. $512.50 x.06 x -^ = $4.87.
4. 87 = int. for 1 mo. 27 da.
$517.37 = amt. for 1 mo. 27 da.
150. = second payment. ^ _,
$367.37 = third principal. $367.37 x .06 x ^ = $2.02.
2.02 = int. for 1 mo. 3 da.
$369.39 = amt, for 1 mo. 3 da.
150. = third payment. ^ „
$219. 39 = fourth principal. $219.39 x .06 x^ = $-59.
.59 = int. for 16 da.
$219.98 = amt. for 16 da. $219.98. Am.
NOTE. Problems of this type are not as common in business as they
were some years ago. Payments are now usually made at equal intervals
of time.
234 ADVANCED BOOK OF ARITHMETIC
EXERCISE 126
1. A man borrows from a loan association $3000 at
8% interest. If he pays $ 100 at the end of every 3 mo.
for 1 yr., find the amount due at the end of the year.
2. A man borrows from a building and loan company
$2000 at 8% interest. He pays in monthly installments
of $50 each. How much does he owe at the end of 6 mo. ?
3. If you borrow $1800 at 6 % interest and pay
annually $ 600, how much do you owe at the end of
3yr.?
4. If you borrow $1500 at 8% interest and pay semi-
annually $300, how much do you ..owe at the end of
2yr.? "
5. If $2400 is borrowed at 8% interest, and paid in
quarterly installments of $150 each, how much is due at
the end of two years ?
6. A man borrows $ 3000 at 8 % interest, and pays in
semiannual installments of $500 each. How much is due
at the end of three years ?
7. A man borrows $1800 at 6%, and pays $400 per
year. How much is due at the end of five years ?
8. If I borrow $2500 at 5% interest, and pay $500 a
year, how much do I owe at the end of five years?
9. How much is due at the end of two years on a loan
of $2500 at 5 % interest when paid in semiannual install-
ments of $600 each?
10. A note dated Jan. 15, 1904, for $4000 had the fol-
lowing indorsements : July 10, 1904, $700 ; Dec. 24, 1904,
$600; June 18, 1905, $800; Nov. 13, 1905, $500; March
17, 1906, $900; Aug. 10, 1906, $400; /Feb. 12, 1907,
$400. How much was due May 28, 1907, at 6%?
PARTIAL PAYMENTS 235
THE MERCHANTS' RULE
The Merchants' Rule for computing interest on partial
payments is used when settlement is made within one
year from the date of the note. The rule is as follows :
I. Compute the amount of the face of the note until the
date of settlement.
II. Compute the amount of each indorsement from its date
until the date of settlement, and add these amounts.
III. Subtract the sum of the amounts of the indorsements
from the amount of the face of the note. The remainder -will
be the amount due on the date of settlement.
Example. A note for $500, with interest at 8%, dated
Jan. 10, 1902, had the following indorsements : April 1,
1902, $100; May 10, 1902, $200; July 1, 1902, $100.
Find the amount due Nov. 1, 1902.
SOLUTION. From Jan. 10 to Nov. 1 is 9 mo. 21 da.
From April 1 to Nov. 1 is 7 mo.
From May 10 to Nov. 1 is 5 mo. 21 da.
From July 1 to Nov. 1 is 4 mo.
$500 for 9 mo. 21 da. amounts to $532.33
$100 for 7 ino. amounts to $104.67
$200 for 5 mo. 21 da. amounts to 207.60
$100 for 4 mo. amounts to 102.67
The payments amount to 414.94
Balance due Nov. 1, $117.39
EXERCISE 127
l. A note for $500, with interest at 6 %, dated Jan. 2,
1902, had the following indorsements : March 2, 1902,
$100; May 5, 1902, $150; July 10, 1902, $200. Find
the amount due Nov. 1, 1902.
236 ADVANCED BOOK OF ARITHMETIC
2. A note for $600, with interest at 8%, dated Feb. 1,
1902, was indorsed as follows: March 11, 1902, $200;
May 15, 1902, $100; July 3, 1902, $100. Find the bal-
ance due Aug. 1, 1902.
3. A note for $800, with interest at 8%, dated Feb. 10,
1902, was indorsed as follows; March 15, 1902, $200;
April 10, 1902, $100; June 3, 1902, $200. Find the
amount due Nov. 1, 1902.
4. A borrows $1500 on Jan. 1, 1907, at 6% interest,
and pays $450 each quarter. How much does he owe
when three payments are made ?
EXCHANGE
The written order of one party to another, requesting
the payment of a specified sum of money, is called a draft.
The parties to a draft are drawer, drawee, and payee;
The following are forms of drafts:
SIGHT DRAFT
New Orleans, La., mwieh fO, 1903.
/ 600.00
, pay to the order of ___________________
jla&ok s'i&cf&t ____________ the sum of
Dollars.
Value received and charge to the account of
TO
EXCHANGE 237
A time draft payable after sight matures as a promis-
sory note.
Bank drafts form one of the most important mediums
of exchange. If a merchant owes money in New York,
Chicago, or in any other place, he can always purchase a
draft from his home bank, and by transmitting this through
the mails settle his indebtedness.
TIME DRAFT
Louisville, Ky., fl/ay. /O, 1902.
$1500.00
da,t& pay to the order of
_ .<3:vft&&yv Aat ndb&d .Q Dollars.
Value received and charge to the account of
To
No. 8. TniwttaotU, Tninn. tfet.
BANK DRAFT
Galveston, Texas, 7MaA*A, 10, 1903.
No. 7^.
f^utcfjmp, ^ealg ^ Co., Bankers,
Pay to the order of. ______
f 300.00 ___________ (&6AM kwvicii/£cC) __________ -^Dollars.
-^
Cashier.
To
238 ADVANCED BOOK OF ARITHMETIC
The price of exchange is a matter of supply and de-
mand. If, for instance, the Galveston banks freely offer
New York or Chicago exchange, a purchaser will very
probably get it for less than its face value. If exchange
is scarce, its price is high. When exchange costs more
than its face value, it is said to be at a premium ; when it
costs less than its face value, it is said to be at a discount.
Exchange is at par when the cost of a draft is its face
value. Exchange is always quoted as a rate per cent,
or as so many dollars on a $1000. The exchange is
always reckoned on the face value of the draft. Thus
\c/0 premium, or $1.25 premium, means that the cost of a
draft for $100 is $100|, and the cost of a draft for $1000
is $1001.25. The rate of exchange is generally a fraction
of 1%.
Express companies charge for money orders to any part
of this country or Canada not over 30^ for $100. It is
reasonable to assume that a person will not pay much
more than this rate for exchange.
EXERCISE 128
1. What is the cost of exchange on a sight draft for
$400 at 1% premium?
2. What is the cost of exchange on a sight draft for
$300 at |% premium?
3. Find the cost of a draft on New York for $200 at
\°/0 premium.
4. Find the cost of a demand draft for $500 at \%
premium.
5. Find the cost of a sight draft for $600 at
premium.
EXCHANGE 239
6. Find how much must be paid for a sight draft for
$1000 at $1.25 premium.
7. Find how much must be paid for a sight draft for
12000 at $2.50 premium.
8. Find the cost of remitting $800 from Dallas to
Chicago by means of a bank draft when exchange is ^ %
premium.
9. Find the cost of a sight draft on Chicago for $700,
exchange being at | % discount.
10. Find the cost of a sight draft on New York for
$400 when exchange is -|% discount.
11. Find the cost of exchange on a draft for $1000,
exchange being | % premium.
12. Find the cost of an express order for $1200 at the
rate of 30^ per $100.
13. If you had to remit $200 to Chicago when exchange
is \ % premium, which would you prefer, to buy exchange
or to buy an express money order at the rate of 30^ for
$100?
14. What is the cost of an express money order for
$1000?
15. What is the cost of a sight draft for $3000 at
$1.50 premium ?
16. What is the cost of a sight draft for $3000 at $2.50
premium ?
Example. What is the cost of a draft on Buffalo for
$800, payable in 30 da. at 6% interest, exchange being
at the rate of ^ % premium ?
SOLUTION :
The interest on $800 for 30 da. at 6% =$4.00.
Cost of exchange = ^ % of $800 = $2.00.
The cost of the; draft .= $800 4- $2 - $4 = $798. /i
240
ADVANCED BOOK OF ARITHMETIC
EXERCISE 129
Find the cost of the following sight drafts :
l.
2.
FACE
$2720,
$3480,
3. $5080,
4. $6290,
5. $8290,
$9980,
$5493,
8. $5280,
9. $6040,
10. $6090,
$5400,
$9870,
$4500,
$6920,
$7780,
16. $1234,
6.
7.
11.
12.
is.
14.
is.
RATE OF EXCHANGE
\% premium.
i% premium.
^°fo premium.
$2.50 premium.
$1.25 discount.
\% discount.
$1.50 discount.
$1.50 premium.
$2.00 discount.
$1.25 premium.
$1.25 discount.
1% discount.
\% premium.
discount.
\°/o premium.
\°/o discount.
17. Find the cost of an express money order for $1400
at 30^ for $100.
18. Find the cost of a draft for $10,000 at \% discount.
19. Find the cost of a draft for $600 payable in 30 da.
without grace, the rate of interest being 6%, exchange
being |% premium.
20. Find the cost of a draft for $ 1000 payable in 30 da.,
exchange being at par and the rate of interest being &%.
21. Find the cost of a 45 da. draft for $ 900, exchange
being at \% discount, and the rate of interest being 6%.
EXCHANGE
241
VALUE OF FOREIGN COINS IN UNITED STATES MONEY
(Proclaimed by the Secretary of the Treasury Oct. 1, 1906)
COUNTRY
STANDARD
MONETARY UNIT
VALUE IN
U. S. GOLD
DOLLAR
Argentine Republic . .
Austria-Hungary . . .
Belgium
Gold
Gold
Gold
Silver
Gold
Gold
Silver
Gold
Silver
Gold
Gold
Gold
Gold
Gold
Gold
Gold
Gold
Gold
Gold
Gold
Gold
Gold
Gold
Gold
Gold
Gold
Gold
Gold
Gold
Gold
Gold
Gold
Gold
Gold
Gold
Gold
Peso
Crown
Franc
Boliviano
Milreis
Dollar
Peso
Peso
( Shanghai
Tael < Haikwan
( Canton
Dollar
Colon
Crown
Sucre
Pound (100 piasters)
Franc
Mark
Pound sterling
Drachma
Gourde
Pound Sterling
Lira
Yen
Peso
Florin
Dollar
Crown
Balboa
Libra
Milreis
Ruble
Peseta
Crown
Franc
Piaster
Peso
Bolivar
$ 0.965
.203
.193 *.
.485
.54(5
1.00
.485
.365
.726
.808
.792
1.00
.465
.268
.487
4.943
.193
.238
4.8665
.193
.965
4.8665
.193
.498
.498
.402
1.014
.268
1.000
4.8665
1.08
.515
.193
.268
.193
.044
1.034
.193
Bolivia
Brazil
Canada
Central America . . .
Chile
China
Ecuador
E°°VDt
France
Germany
Great Britain ....
Greece
Haiti
India
Italy
Japan . .
IVIexico . . . .
Netherlands ....
Newfoundland ....
Norway
Panama
Peru
Portugal
Russia . . .
"•"•""Spain
Sweden
Turkey
Uruguay
The Circulars of the Secretary of the U. S. Treasury,
issued every 3 months, fix the legal equivalent of the
monetary units in the principal countries. In countries
which have adopted the silver standard monetary units
fluctuate in value, owing to changes in the market value
of silver.
242 ADVANCED BOOK OF ARITHMETIC
Example l. Express $100 in florins (Netherlands).
$ .402 = 1 florin.
... $ 100 - - of 1 florin = 248.76 florins.
EXERCISE 130
1. Express the value of <£! in French currency; in
German currency ; in Austrian currency.
2. Change to United States currency (a) 450 francs,
(6) 550 pesetas, (c) 280 Norwegian crowns, (ci) 900 marks,
(e) 200 colons, (/) 720 Austrian crowns, (#) 450 Danish
crowns.
3. Find in United States currency the difference in
value between 1000 francs and f 198.
4. Find the value of $1000 (a) in the money of Costa
Rica; (6) in Austrian currency; (<?) in Swedish currenc}7".
5. What is the equivalent in United States money of
200,000 bolivars (Venezuela) ?
6. What is the equivalent in United States money of
10,000 Mexican pesos ?
7. Express $1 in Peruvian monetary units.
8. Express $1 in Norwegian monetary units.
9. Find in United States currency the difference be-
tween the value of 10,000 Japanese yen and $5000.
10. What is the value of 10,000 Portuguese milreis?
ENGLISH MONEY
4 farthings = 1 penny (c?.)
12 pence = 1 shilling (.s.)
20 shillings = 1 pound sterling (£)
£1 = 20s. = 240(7. = 960 farthings
The abbreviations «£,«., c?., are the initial letters of the
names of the Roman coins, libra, solidus, denarius.
EXCHANGE 243
EXERCISE 131
1. What is the equivalent of 7s. 5|c?. in U. S. cur-
rency ?
7s. 5jd. = (l + — V = 7.479166s. = £ .373958.
V 12 /
$4.8665 x .37396 = 11.82.
2. The average freight rate on wheat per bushel for
the year 1906 from Chicago to New York was :
By lake and canal 5.94^.
By lake and rail 6.48^.
By all rail delivered to steamer 8.10/, and from New
York to Liverpool l^gC?.
Find the cost of shipping 5 carloads of wheat averaging
79,000 Ib. from Chicago to Liverpool.
(a) By lake and canal to seaboard and thence by
steamer.
(5) By lake and rail to seaboard and thence by steamer.
(<?) By rail to seaboard and thence by steamer.
3. Find the cost of shipping from New York to Liver-
pool 12,000 bu. of wheat at l-f^d. per bu.
4. Find the cost of shipping 50 tons of sacked flour
from Chicago to Liverpool at lO.llcZ. per 100 Ib.
5. Find the cost of shipping from Galveston to Liver-
pool : —
(a) 10,884,000 Ib. of cotton at Is. 6%d. per 100 Ib.
(5) 19,490 bu. of wheat at M. per bu.
((?) 37,920 bu. of corn at 5d. per bu.
6. Find the cost of shipping from St. Louis to Liver-
pool via New Orleans : —
(a) 6475 bbl. of flour at Is. 8%d.
(>) 97330 bu. wheat at 7fd.
In examples 5 and 6 take £ 1 = $4.80.
244 ADVANCED BOOK Ol? ARITHMETIC
FOREIGN EXCHANGE
The settling of outstanding indebtedness between par-
ties in different parts of this country by means of drafts
is known as domestic exchange. The settling of debts
between parties in this country and parties in foreign
countries by means of drafts is called foreign exchange.
Foreign drafts are known as bills of exchange. These are
issued in duplicate, i.e. two are drawn exactly alike. The
two bills are sent by different mails. As soon as one is
paid, the other is void.
Usually exchange is a little above or a little below the
par of exchange, i.e. intrinsic value of foreign coins.
Exchange on Great Britain is quoted at the number of
dollars to <£!; exchange on France is quoted at so many
cents to the franc, or as so many francs to the dollar ; ex-
change on Germany is quoted as so many cents to 4
marks, or as so many cents to a mark.
BILL OF EXCHANGE
No. 24-89 6 New York, N.Y., fan. V-, 1903.
At sight of this first of exchange (second of
the same date and tenor unpaid) _______________________
________________________________________________ .pay to the
order of. ____________ , ____ 1
Value received and ^arge the same to the account of
To
FOREIGN EXCHANGE 245
EXERCISE 132
1. Find the cost of a draft on London for £1000, ex-
change being 4.872.
2. Find the cost of a bill of exchange for £550 15s.,
exchange being 4.85.
3. Find the cost of a bill of exchange for 5750 francs,
exchange being 5.15, i.e. 5.15 francs = $1.
4. Find the cost of a draft on Manchester for £3500
6s. Sd. when exchange is quoted at 4.87J.
5. Find the cost of a draft on Berlin for 1480 marks
when exchange is quoted at 95|, i.e. 4 marks = 95 J^.
6. Find the cost of a draft on Vienna for 8500 crowns,
rate of exchange being 1 crown = $.202.
7. Find the cost of a draft for 1500 lira when exchange
is quoted at 19.3, i.e 19. 3 / = 1 lira.
8. Find the cost of each of the following drafts :
FACE WHERE PAYABLE RATE OF EXCHANGE
(a) £928 10s. Liverpool 4.85
(6) £ 850 6s. 8 d. Belfast 4.86J
O) 2500 marks Berlin 24
(d) 9280 francs Havre 5.16
(e) 8500 lira Naples 19 J
(/) £584 13s. 4<Z. Dublin 4.85f
Example. What is the face of a New York draft on
Liverpool, which costs $7297. 50, when exchange is quoted
at 4. 861- ?
SOLUTION. $4.865 = £ 1.
£l
17297.50 = : = £ 1500.
246 ADVANCED BOOK OF ARITHMETIC
EXERCISE 133
Find the face of each of the following drafts :
COST WHERE PAYABLE RATE OF EXCHANGE
1. $5835 London 4.86J
2. $1218.75 Newcastle 4.87J
3. 19063.05 Berlin 95.2
4. 13724.90 Paris 19.3
5. 1718.24 Christiana 26.8
6. $2366.82 Belfast 4.86J
7. $1037 Paris 5.18J
8. $5664.80 Hamburg 23.8
STOCKS AND BONDS
A company or corporation is a number of persons asso-
ciated under a state law for the purpose of transacting
business.
The modern company, or corporation, has supplanted
in a great degree the old-time partnership. Nowadays
business is conducted on so extensive a scale that a com-
pany often numbers several thousand persons.
The money which a company invests in business is
called capital, or stock.
The stock of a company is usually divided into shares
of $ 10, $ 50, or $ 100 each. The face value of a share in
most large companies is $ 100.
These shares, as a rule, can be bought and sold. The
buying and selling is usually done by agents called stock-
brokers. The usual fee of a stockbroker is at the rate of
l % of the par value for buying a share and \ % of the
STOCKS AND BONDS 247
par value for selling a share. A broker's fee is called
brokerage.
If the business of a company is prosperous, a share may
sell for more than its face value. The stock is then said
to be at a premium. If the business is not prosperous,
the value of a share in the market is likely to be less
than its face value. The stock is then said to be at a
discount.
When a share sells in the market for its face value, it is
said to be at par. A quotation " 10 % premium " means
the same as 110% of the face value. A quotation "10%
discount " means the same as 90 % of the face value.
The profits of a company, usually distributed to stock-
holders annually or semiannually, are called dividends.
The dividend is generally so many dollars on a share,
or a percentage of the face value of the stock. Thus,
"4% dividend" means f 4 on a share whose face value
is $100.
Stock is of two kinds, common and preferred. The pre-
ferred stock is guaranteed a specified dividend whether the
business is profitable or not. The common stock earns
what is left over after expenses and the dividends on the
preferred stock are paid.
A bond is a written instrument made by a government,
municipal, state, or national, or by a corporation, for the
purpose of borrowing money.
Bonds are of two kinds, registered and coupon. A
registered bond is one which is recorded by number and
by name of the owner, and it is not transferable except in
writing and at the office of the treasurer. Coupon bonds
are so called because they have interest slips attached to
them, which are cut off as the interest falls due. These
interest slips are payable to the bearer.
248
ADVANCED BOOK tfF ARITHMETIC
The following is quoted from a daily newspaper
the New York stock market March 11, 1903.
Stocks
giving
CLOSING
Atchison ....
SALES
30,200
HIGHEST
82
971
93
LOWEST
801
97i
BID
974
o
92$
Atchison pf. .
1,800
Baltimore & Ohio .
. 14,400
Canadian Pacific
. 13,800
1291
127|
1291
Manhattan L .
. 17,300
142|
141f
142f
Metropolitan St. Ry.
. 21,100
1351
131f
134
Missouri Pacific .
. 30,800
1081
106$
108f
Pennsylvania .
St. Paul ....
. 53,400
, 58,500
. 54,400
144|
168$
1421
1671
61i
144
168J
62$
Southern Pacific .
Bonds
U. S. new 4s registered, 135 U. S. old 4s registered, 103
U. S. new 4s coupon, 136 Baltimore & Ohio 4s, 102|-
U. S. old 4s coupon, 109J- Chicago B. & Q. new 4s, 9S|
Notice that in stock quotations the fractions used are
halves, quarters, eighths. Notice further that the varia-
tions in the prices of stock in the course of a day are
quite considerable. There are instances where stocks
have fallen in one day 50% of their face value. Why the
prices vary so much would take the proverbial Phila-
delphia lawyer to explain. In the above quotations the
first column shows the number of shares sold ; the second
column gives the highest prices paid for the shares; the
third gives the lowest prices paid ; and the fourth gives
the closing bids. In the course of the day above referred
to there were several other prices than those given.
STOCKS AND BONDS 249
Example. Find the cost of 10 shares of stock at 107 J.
SOLUTION
Market price = $107|.
Brokerage =$|.
Cost of 1 share = |107f (i.e. $107J + f J).
Cost of 10 shares = $107* x 10 = $1076.25.
o
EXERCISE 134
1. Give the premium or discount of each of the stocks
quoted March 11, 1903.
2. What is the cost of 20 shares of Atchison at 80 J ?
at 81£?
3. What is the cost of 40 shares of Baltimore & Ohio
at 92| ?
4. What is the cost of 200 shares of Canadian Pacific
at 145 J ?
5. What is the cost of 25 shares of stock at 118| ?
6. Find the cost of 50 shares of Missouri Pacific at
106f.
7. Find the cost of 100 shares of each of the stocks
quoted March 11, 1903, at the prices indicated in the
fourth column.
Example 1. What sum should be received from the sale
of 40 shares of Pennsylvania at
SOLUTION
Market value of 1 share of stock =
Brokerage = §\.
Sum received from 1 share = $144.
Sum received from 40 shares = 1144 x 40 = 15760.
250 ADVANCED BOOK OF ARITHMETIC
Example 2. What profit is made by purchasing 50
shares of stock at 142 J and selling them at $ 144 ?
SOLUTION. Cost of 1 share = $1421 + $£ = $ 142f.
Receipts from 1 share = $144 — $£ = (J143J.
Gain on 1 share = $ 1^.
Gain on 50 shares = $1 J x 50=$ 75.
EXERCISE 135
1. What should a person receive from the sale of 100
shares of Manhattan at the closing bid above given ?
from the sale of 100 shares at the lowest bid ?
2. How much should a person receive from the sale
of 100 shares of Atchison preferred at 97 J ?
3. How many shares of Pennsylvania at 142^ can be
bought for $14,237.50?
4. What profit is made by buying 100 shares of Bal-
timore & Ohio at 91 J-, and selling them at 92|-?
5. If 100 shares of St. Paul are bought at 167| and
sold the same day at 168-|, what is the gain?
6. If 200 shares of New York Central are bought at
165 and sold the following day at 167^, what is the gain?
7. A speculator bought Missouri Pacific at the lowest
price quoted March 11, 1903, and sold at the closing
bid of the same day. If his gain was $300, how many
shares did he buy ?
8. If a speculator buys Metropolitan Street Railway
at 131 1 and sells on the same day at 134, find the num-
ber of shares bought to realize a profit of $225.
9. How many shares of Atchison bought at 80-| and
sold at 81f realize a profit of $1000?
10. How many Chicago B.&Q. new 4s at 93^ can I
buy for $18,650?
STOCKS AND BONDS 251
Example l. By investing money in 4 % stock a man
realizes a profit of 5 % on the money invested. Find
the price of the stock.
SOLUTION. The gain on $100 invested is $5.
The gain on $20 invested is $ 1.
The gain on $80 invested is $4.
The stock cost $80 a share.
The market price of the stock is $80 - $ 4 = $ 79 J.
A O O
Example 2. How much money must be invested in
United States old 4s registered at 103 to produce an
annual income of $600?
SOLUTION. United States old 4s registered pay a divi-
dend of $4 a share.
The number of 4s in 600 is 150.
One hundred and fifty shares produce $600.
The cost of 1 share is 103 + £.
The cost of 150 shares is'103£ x 150 = $15,462.50.
EXERCISE 136
1. If stock paying a dividend of 8 % gives an income
at the rate of 5 % on the money invested, what is the
market value of the stock?
2. If 6 % stock produces a gain of 4 % on the money
invested, find its market value.
3. How much money must I invest in Western Union
at 92|, paying a dividend of 5%, to derive an annual
income of $1000?
4. How much must be invested in Pullman Palace
Car at 218, paying a yearly dividend of 8 %, to derive an
income of $1200 a year ?
5. If I buy 4| % stock at 97£, what rate per cent do I
get for my money ?
252 ADVANCED BOOK OF ARITHMETIC
6. What must I pay for 5 % stock so as to make a
profit of 8% on my investment?
7. How much must be invested in Northwestern, pay-
ing 6%, at 206, to derive an income of $900?
8. Which is the more profitable investment, 5 % stock
at 119 1 or S% stock at 219 J?
9. How much must be paid for a $1000 bond at 87 J?
10. City bonds bought at 89 J pay 5 %. What rate of
interest do they pay?
11. If I have $ 5000 stock in United States 4s at 102,
what annual dividend do I receive ? If I sold my stock,
what should it bring?
12. How much should be realized by selling $10,000 of
each of the stocks quoted above at the highest prices
paid March 11, 1903?
13. Which is the better investment, Illinois Central,
paying 6% dividend, at 137|, or United States 4s at
109f?
14. How many shares of stock must a broker sell to
make $54.75 brokerage?
CUSTOMS AND DUTIES
Taxes levied on imported goods are called duties.
Duties are of two kinds, specific and ad valorem. The
latter is a percentage of the foreign value of the goods;
the former is a definite amount of money on some stand-
ard quantity, such as a yard, gallon, pound, etc. Some
articles are subjected to both specific and ad valorem
duties. Duties are collected by United States customs
officers at places known as ports of entry.
The United States revenue from duties for the year
ending June 30, 1906, amounted to $300,251,877.79,
CUSTOMS AND DUTIES 253
EXERCISE 137
1. Find the duty on 3200 Ib. of refined sugar at 1.95^
per pound.
2. Find the duty on 2000 bu. of potatoes at 25^ per
bushel.
3. Find the duty on 1800 Ib. of salt at 12^ per 100 Ib.
4. Find the duty on 5 doz. parasols at 4s. Qd. each at
40 % ad valorem.
5. Find the duty on 12 microscopes invoiced at £8
each, the rate of duty being 45 % ad valorem.
6. The duty on books printed in English is 25 % ad
valorem. If I import the following books at the prices
specified, how much duty in United States money will I
have to pay ?
Casey's Elements of Euclid .... 3s. 9d.
Casey's Analytical Geometry .... 10s. Od.
Leatham's Spherical Trigonometry . . 4s. Od.
Lamb's Infinitesimal Analysis . . . 12s. Qd.
7. Find the duty on the Clarendon Press edition of
Shakespeare's plays, invoiced at 24s., at 25% ad valorem.
8. A hardware merchant imported 80 doz. razors at
2s. Qd. each, duty $1.75 per dozen and 20% ad valorem.
Find the total duty paid.
9. What is the duty on 100 yd. of treble ingrain
carpet valued at 90^ a square yard, the duty being 22^ a
square yard and 40 % ad valorem ?
10. What is the duty on 40 clocks valued at 14.50
each, at 40 % ad valorem ? If the clocks are retailed at f*
profit of 30%, find the selling price of these clocks.
254 ADVANCED BOOK OF ARITHMETIC
11. Find the duty on 120 doz. linen cuffs at 7s. 6d. a
dozen, the duty being 40^ a dozen and 20% ad valorem.
12. Find the duty on a fur rug valued at $50, duty
35% ad valorem. At what price should the rug be sold
so as to make a profit of 20 % ?
13. Calculate the duty levied on 5 doz. straw hats
valued at $30 a dozen, the duty being $1 a dozen and
20 % ad valorem.
14. What is the duty on 800 yd. of silk valued at $1.15
a yard at 60 % ad valorem ? At what price per yard
should the silk be sold to make a profit of 20 % ?
15. Calculate the duty on 12 horses valued at $250
each at 25 % ad valorem.
16. An Axminster carpet, 18 ft. by 12 ft., valued at
$1.50 per square yard, is imported. Find the duty at
60^ per square yard and 40 % ad valorem.
17. What is the duty on 5 doz. opera glasses valued at
$3.50 each at 45% ad valorem?
18. What is the duty on 500 Ib. of glue valued at 50^
per pound, the rate of duty being 15^ per pound and 20 %
ad valorem ?
19. A suit of clothes imported from England costs a
merchant $38.80. The duty is 60% ad valorem. Taking
<£! = $4.85, find the invoice price in pounds sterling.
20. Find the duty on 1200 yd. of linen invoiced at l^d.
per yard at 45 % ad valorem. At what price per yard
should it be marked so that the dealer may give a dis-
count of 10 % and make a profit of 20 % ?
CHAPTER IV
INVOLUTION
IN the language of mathematics, 3 x means three times
the number x ; i.e. 3 x may stand for three times any
number whatever.
a -f b stands for the sum of any two numbers, a and b.
a — b stands for the difference of two numbers, a and b. In
other words, a — b is the number which when added to b
gives a for sum.
ab stands for the product of two numbers, a and b.
- stands for the quotient obtained by dividing a by b. In
other words, 7 is that number which when multiplied by
b gives a for product.
a2 stands for the square of a. a3 stands for the cube of a,
(a + ft)2 stands for the square of the sum of two numbers,
a and b.
(a — 5)2 stands for the square of the difference of two
numbers, a and b.
lab stands for seven times the number ab, which is
itself the product of two numbers, a and b.
— n ! ^— stands for one half the sum of two num-
bers plus one half their difference.
(a + b)x stands for the product of a number, #, by the
sum of two numbers, a and b.
255
256 ADVANCED BOOK OF ARITHMETIC
(a — b*)x stands for the product of a number, a?, by the
difference of two numbers, a and b.
EXERCISE 138
Add:
1. 2z
2. ly 3.
4 a 4.
9b 5. 4 z
6. 11 a
3 x
8y
9 a
Ib Iz
la
Bx
9y
11 a
Sb 9z
5 a
7. 4 ab
8. 9 2^
9. 4 fo
10. 12 ab
11. 9 ab
Bab
Ixy
lie
IBab
lab
7 ab
8xy
11 be
10 ab
Bab
12. T J<? 13. aa; 14. 10 x 15. 11 6 16. 12 m 17. 11 a?
8 be bx ax ab am bx
9 be
EXERCISE 139
1. 7a;-4aj=? 7. 15 a- 11 a = ? 13. 2 xy - xy = ?
2. 6rr-a;=? 8. 17 a -12 a=? 14. 14 a# - 10 xy = ?
3. lla:-8a;=? 9. 18a-13a = ? 15. 17a:y-9ajy = ?
4. 12^-2a; = ? 10. 12aJ-3a6 = ? 16. 19xy-7xy = ?
5. 5^-3o; = ? 11. Ha6-4aJ=? 17. ax-bx=?
6. 12a — a = ? 12. 5ab-ab = ? 18. cx—dx=l
19. ma? — nx — ?
Example 1. Multiply 4 a? by 5 #.
SOLUTION. 4#x5# = 4x#x5xa; = 4x5xa;X£
= 20 z2.
Example 2. Multiply 7 x^y by 3 or?/2.
SOLUTION. 7 o% x 3 ^2 = (7 x x x x x y) x (3 x a; x
INVOLUTION
257
Example 3. What is the square of a + ft?
a + 5 Multiply a -f- 5 by a, and write the
a 4- b result, a2 + aft. Next multiply a + ft by
a2 + ab b and write the result, ab + ft2. Then add.
+ aft + ft2 The following is a graphical representation
a2 + 2aft + ft2 of the result :
ab b
H
62
a2 a
ab
A a L b
Let AL= a,
Then
Let ABOD be the square on AB, ALHGr be the square
on AL. Then HYCKis the square on ft, and G-HKD =
LBYH, because their dimensions are equal to each other.
Now, ABOD = ALE a + GHKD + HYCK+ LBYH =
ALHG- + HYOK+ 2 G-HKD.
.*. (a + ft)2 = a2 + ft2 + 2 aft. Hence, we have the follow-
ing important conclusion:
The square of the sum of two numbers equals the square
of the first number plus the square of the second number plus
twice the product of the two numbers.
The process of finding the power of a number is called
involution.
258 ADVANCED BOOK OP ARITHMETIC
Example 1. Square x + 9.
SOLUTION, (x + 9)2 = x2 + 92 + 2 x 9 x a? = a? + 81 +
18^ or a? +18 a; + 81.
Example 2. What is the square of 43 ?
SOLUTION. 432 = (40 + 3)2 = 402 + 32 + 2 x 40 x 3 =
1600 + 9 + 240 = 1849.
EXERCISE 140
Find by Statement (3) the square of:
1.
10 +
x.
7.
70 +
b. 13.
29.
19.
72.
25.
89.
2.
20 +
x.
8.
80 +
C. 14.
37.
20.
76.
26.
92.
3.
30 +
X.
9.
90 +
y. 15.
47.
21.
78.
27.
93.
4.
40 +
X.
10.
14.
16.
54.
22.
79.
28.
67.
5.
50 +
X.
11.
15.
17.
62.
23.
84.
29.
98.
6.
60 +
a.
12.
24.
18.
68.
24.
87.
30.
55.
EXERCISE 141
1. Find the square of .1, .2, .3, .4, .5, .6, .7, .8, .9.
2. Find the square of the reciprocals of the numbers
from 1 to 20 inclusive.
3. Find the squares of f , f , f , f , ^, 11 if, £.
4. Find the squares of f, |, f 11, If, If, 21 6|.
5. How does the square of a fraction compare in, value
with the fraction itself?
6. Find the cubes of the reciprocals of the numbers
from 1 to 20 inclusive.
7. Find the value of I3 + 23+ 33 + 43 + 53 + 63 + 73 + 83
+ 93 + 103.
8. Square each of the following numbers and give
the product correct to four decimal figures: (a) 1.732,
(6) 9.256, O) 5.401, (d) 8.129, (e) .6834, (/) .7609,
(#) 9.482, (K) .7071, (i) .7746, (j) .9487.
EVOLUTION 259
EVOLUTION — SQUARE ROOT
By the square root of a number is meant that number
which when squared produces the given number. Thus,
4 is the square root of 16, since 42 = 16. 7 is the square
root of 49 for a similar reason.
The square root is defined also as one of the two equal
factors of a number. Thus, 7x7 = 49. One of the
factors is the square root of 49.
The symbol for square root, V> is called the radical
sign. It is a degenerate form of the first letter of the
word radix, the Latin word for root. The exponent |- is
also used as a sign for the square root. V49, 49* are the
two ways of indicating the same process, namely, the
extraction of the square root of 49. The number written
under the radical sign is called the radicand.
Students should fix firmly in mind : —
102 = 100 402 = 1600 702 = 4900
202 = 400 502 = 2500 802 = 6400
302 = 900 602 = 3600 902 = 8100
Example l. What is the square root of 5329?
SOLUTION. By trial the square root of 5329 is more
than 70 and less than 80.
.'.5339 = 4900 +140 x+x*
Subtract 4900, and get,
140 x + x2 = 439.
.'. (140 -f- z> = 429.
Since 140 is contained in 429 3 times, try 3 as a value of x.
(140 + 3)3 = 429.
.-. V5329 = 70+ 3=73.
260 ADVANCED BOOK ^OF ARITHMETIC
Example 2. V9025 = ?
SOLUTION. V9025 = 90 + x.
.-. 8025 = 8100 + 180 x + x*.
.-. 180 x + a^ = 925.
.-. (180 + s> = 925.
Since 180 is contained in 925 5 times, try 5 for the next
figure of the root.
(180 + 5)5 = 925.
.-. V9025 = 90+5 = 95.
In practice, the work is contracted as follows : Beginning
at the decimal point, point off the figures of the number
in periods of two figures each. By trial find
9 5 the greatest digit whose square is contained
90.25 in the number denoted by the period to the
81 left. Write it as the first figure of the root,
185
9 25 and write also its square. Subtract the latter
_9_2t> from the period to the left and bring down
the next period. Double the part of the root
just found for trial divisor. Find next the number of
times the trial divisor is contained in the number denoted by
the remainder and the period brought down. Write this
result in the quotient and in the divisor and then multiply.
EXERCISE 142
Find the square root of :
1.
169.
7.
2809.
13.
7056.
19.
7921.
2.
441.
8.
3844.
14.
7569.
20.
6084.
3.
625.
9.
4225.
15.
8464.
21.
4761.
4.
961.
10.
5329.
16.
9216.
22.
3481.
5.
1024.
11.
5776.
17.
9604.
23.
2401.
6.
1849.
12.
6724.
18.
9801.
24.
3364.
EVOLUTION
261
Example.
3 5
12 74 49
9
65
3 74
3 25
4949
hence
\/127449 = ?
SOLUTION. Divide the figures of the
number into periods of two as in the pre-
vious exercises. Then proceed to extract
the square root of the number denoted by
the two periods to the left.
The answer is obviously 350 + some
number. Let x represent this number;
350 + x = V127449.
.-.(350 + *)2 = 127449.
.-. 122500 + 700 x + x* = 127449.
.-. 700 x+x* = 4949.
... (700 + x)x = 4949.
Since 700 is contained in 4900 7 times, try 7 as a value of
x.
(700 + 7)7 = 4949.
.-. V127449 = 350 + 7 = 357.
Notice the trial divisor is twice the part of the root
found. Therefore in a problem in square root where the
radicand is an integer consisting of five or six figures,
proceed in exactly the same way as has been done in a
problem consisting of four figures. Beginning once more,
the solution in its contracted form stands as follows :
357
12 74 49
9 The trial divisor is always twice the part
of the root already found.
65
707
3 74
325
4949
4949
262 ADVANCED BOOK OF ARITHMETIC
EXERCISE 143
Extract the square root of:
1.
100,489.
7.
229,441.
13.
474,721.
2.
110,224.
8.
277,729.
14.
501,264.
3.
120,409.
9.
310,249.
15.
654,481.
4.
171,396.
10.
354,025.
16.
772,641.
5.
190,096.
11.
391,876.
17.
819,025.
6.
199,809.
12.
456,976.
18.
826,281.
Memorize : —
(.I)2 = .01 (.5)2 =.25 (.9)2 =.81
(.2)2 =.04 (.6)2 =.36 (.ll)2 =.0121
(.3)2 =.09 (.7)2 = .49 (.12)2 =.0144
(.4)2 = .16 (.S)2 =.64 (.Ol)2 = .0001
To extract the square root of a decimal, begin at the deci-
mal point, and proceeding to the right, point off the
figures in periods of two ; next proceed as if the number
were an integer. Thus, in taking the square root of
.0225, first point off in periods of two figures each. This
gives .02 25. Next extract the root of the number de-
noted by the figures 225. The result is 15. Hence, the
required root is .15.
To extract the square root of a number part integer and
part decimal, begin at the decimal point, and proceeding to
the left, point off the integral part in periods of two
figures each; next point off the decimal part in periods of
two figures each, beginning at the decimal point. If there
are not enough figures in the decimal part to make an
exact number of periods, annex a cipher or as many
ciphers as are necessary to make the required number of
periods.
EVOLUTION
263
Example. Extract i\
1. 3 0 3 8 4
1. 70 00 00 00 00
1.
ie square root of 1.7.
Double 1 for the first trial divi-
sor. Double 13 for the next trial
divisor. Then find the next figure
of the root is 0. Write it in the
root and in the trial divisor.
Then annex two more ciphers,
and find that the next figure of
the root is 3, and so on.
23 70
69
2603
26068
26076
10000
7809
219100
208544
4 1055600
1043056
EXERCISE 144
Extract the square root of :
1. .150932.
2. .246016.
3. .3448.
4. .2909.
5. .2632.
6. .4616.
7. .5319.
8. .61575.
9. .784.
10. .083.
11. .062.
12. .0037.
To extract the square root of a fraction when its numer-
ator and denominator are perfect squares is a simple
matter. Thus, the square root of || is f; the square root
of IW, or |4, is |, or 1J-.
b4:~ 84' o" O
To get the square root of a fraction, take the square root
of the numerator, and the square root of the denominator,
and then write the former result for numerator and the
latter result for denominator. The fraction thus found is
the required square root.
Example 1. What is the square root of -|| ?
SOLUTION. Vl7 = 4.123; V36 = 6.
264 ADVANCED BOOK OF ARITHMETIC
Example 2. What is the square root of || ?
SOLUTION. V23 = 4.796;
This is a roundabout way to take the square root of |~|.
A shorter and better way is to reduce the fraction to an
equivalent decimal, and then to extract the square root of
this decimal.
To extract the square root of a fraction whose denomi-
nator is not a perfect square, reduce the fraction to an
equivalent decimal and then extract the square root of
this decimal.
EXERCISE 145
Extract, to three decimal figures, the square root of :
1. 1.2. 4. 5.2. 7. 3i. 10. 41 13. f.
O 4 O
2. 4.25. 5. 3.3. 8. 9J. 11. 2|. 14. f.
3. 1.1. 6. 5|. 9. If. 12. TV 15. TV
EXERCISE 146
PROBLEMS INVOLVING SQUARE ROOT
1. The area of a square field is 1 A. Find the length
in yards of one of its sides.
2. The area of a square field is 12 A. Find the length
in yards of one of its sides.
3. The dimensions of a rectangle are 289 yd. and 196
yd. Find the side of an equivalent square.
4. The dimensions of a rectangle are 1| mi. and .7 mi.
Find, correct to four decimal figures, the side of an equiva-
lent square.
EVOLUTION
265
5. Find in rods the perimeter of a square field whose
area is ^ of a square mile.
6. The area of a rectangle whose length is twice its
breadth is 10 A. Find its dimensions in yards.
HINT. Draw a diagram ; divide it into two equal parts by a line paral-
lel to its width. Notice what each part is.
7. The area of a rectangle whose length is three times
its width is 20 A. Find its dimensions in yards.
8. A square and a rectangle have the same area, namely
40 A. If the length of the rectangle is twice its width,
find in rods the difference between their perimeters.
The side of a right triangle opposite the right angle is
called the hypothenuse. The other two sides are called the
legs of the right triangle. One of the legs is called the
base of the right triangle, and the other leg is called
the altitude of the right triangle.
In a right triangle the square on the hypothenuse is equal
to the sum of the squares on the two legs. This is the
famous Pythagorean Theorem.
Designate the sides of
the right triangle AB 0 by
the letters #, 5, c. (a -f 5)2
= a2 + 52 + 2 ab. But
(a -f- 6)2 = c2 -f- 4 triangles,
each having for its
base and altitude a and
b. AMNR = c2 + 4 x J ab
-f 52 + 2 ab = c2 + 2 ab.
266 ADVANCED BOOK > OF ARITHMETIC
Example. In a right triangle the legs are 7 and 24.
Find the hypothenuse.
SOLUTION. a2 + b* = c2.
.-.49 + 576 = <?2.
.-. £ = V625=25.
EXERCISE 147
1. In a right triangle, given a = 6, 5 = 8, find c.
2. In a right triangle, given a = 5, 5 = 12, find c.
3. In a right triangle, given a = 8, b = 15, find <?.
4. In a right triangle, given a = 20, 5 = 21, find c.
5. In a right triangle, given a =• 56, 5 = 90, find c.
6. In a right triangle, given a = 20, b = 99, find <?.
7. In a right triangle, given a = 17, b = 144, find c.
8. In a right triangle, given a = 39, 6 = 80, find c.
9. In a right triangle, given a— 51, b = 140, find c.
10. In a right triangle, given a = 44, b — 52.5, find c.
11. In a right triangle, given a = 87, 5 = 416, find c.
12. In a right triangle, given a = 136, 5 = 273, find c.
13. In a right triangle, given a = 145, b = 408, find c.
14. In a right triangle, given a = 207, b = 224, find c.
15. A ladder is placed 14 ft. from a wall 48 ft. high.
How long must the ladder be to reach to the top of the
wall?
16. Find the length of the diagonal of a square if one
side of the square is 10 rods.
17. Find the length of the diagonals of a rectangle, the
dimensions of the rectangle being 17 rd. and 25 rd.
AREAS OF PLANE TRIANGLES 267
Example. If the hypothenuse of a right triangle is
493 and one leg is 468, find the other leg.
SOLUTION. Let the required leg be a.
Then, a2 + 4682 = 4932.
.-. a2 + 219,024 = 243,049.
Subtract 219,024 from each member of this equation.
.-. a2 =24,025.
/. a = 155.
EXERCISE 148
1. Hypothenuse = 377, base = 345, find the altitude.
2. Hypothenuse = 545, base = 513, find the altitude.
3. Hypothenuse = 449, base = 351, find the altitude.
4. Hypothenuse = 5.05, base =4.56, find the altitude.
5. Hypothenuse = .461, base = .38, find the altitude.
6. Hypothenuse = .481, alt. = .36, find the base.
7. Hypothenuse = .641, alt. = .609, find the base.
8. Hypothenuse = .773, alt. = .195, find the base.
9. Hypothenuse = .697, alt. = .528, find the base.
AREAS OF PLANE TRIANGLES
The following rule gives the area of any triangle :
(1) Add the three sides and take half the sum.
(2) Subtract each side separately from the half sum.
(3) Find the continued product of the three remain-
ders and the half sum.
(4) The square root of this product is the area.
The proof of this rule is too difficult to be given in an
elementary arithmetic. This rule enables one to find the
area of a tract of land such as a farm.
268 ADVANCED BOOK OF ARITHMETIC
Example. Find the area of a triangle whose sides are
34 ch., 65 ch., and 93 ch.
SOLUTION
34 96 96 96
65 34 65 93
_93 62 31 3
2)192
"96 Area = V96 x 62 x 31 x 3 = V553535 = 744.
.*. area = 744 sq. ch. = 74.4 A.
EXERCISE 149
Find the area of each of the following triangles :
l. Given the sides, 13, 20, 21.
2. . Given the sides, 13, 30, 37.
3. Given the sides, 33, 34, 65.
4. Given the sides, 35, 52, 73.
5. Given the sides, 29, 60, 85.
6. Given the sides, 140, 143, 157.
7. Given the sides, 507, 603, 721.
8. Given the sides, 46 rd., 75 rd., 109 rd.
9. Given the sides, 40 rd., 51 rd., 77 rd.
10. Given the sides, 3.5 ch., 10 ch., 11.7 ch.
11. Given the sides, 5.6 ch., 6.1 ch., 7.5 ch.
12. Find area of a triangle, each side being 10 rd.
13. Find area of a triangle, each side being 50 rd.
14. Find the area of an isosceles right triangle if the
hypothenuse is 27 inches.
15. Find the area of a square whose diagonal is 72 feet.
16. Find the side of a square equivalent to the difference
of two squares whose sides are 89 feet and 68 feet.
1
MENSURATION OF THE CIRCLE 269
MENSURATION OF THE CIRCLE, ETC.
Take a string and nnv +^* length of the circumference
of a circle. Take another string and find the length of the
diameter of the circle. Divide the former result by the
latter to get the ratio of the circumference of the circle to
its diameter.
Let O = area of circle.
r = radius.
c = circumference.
d = diameter.
The ratio of the circumference of a circle to its diameter
is approximately 3.14159265. This ratio is denoted by the
Greek letter TT (Pi). In cases where the numbers involved
are not very large, or where extreme accuracy is not de-
manded, 3^ is a sufficiently accurate approximation of TT.
The ratio 355 : 113 is a close approximation to the value
of TT. 3.1416 is generally taken as the value of TT. In
this chapter we shall consider TT = 3.1416.
Since - = TT, . •. c = ird = TT (2 r) = 2 TTT.
d
Express in words the relation c — jrd.
Express in words the relation c = 2 Trr.
EXERCISE 150
Find the circumference when:
1. The diameter is 22. 6. The radius is 67,
2. The diameter is 46. 7. The radius is 86.
3. The diameter is 150. 8. The radius is 3.6.
4. The diameter is 164. 9. The radius is 5.9.
5. The diameter is 196. 10. The radius is 7.3.
270 ADVANCED BOOK OF ARITHMETIC
11. Find d when c = 320.44. 15. Find r when c = 377.
12. Find d when c = 477.52. 16. Find r when 0= 53.41.
13. Find c? when c = 24.50. ' V/. Find r when c = 60.319.
14. Find d when c = 41.7. 18. Find r when e = 42.097.
19. The diameter of the front wheel of a carriage is
3 ft. 6 in. How many times does the wheel revolve in
going 1 mi. ? Take 3| as the value of IT.
AREA OF A CIRCLE
A sector of a circle is a portion of a circle bounded by
two radii and their included arc.
If the arc of a sector is very small, the sector will not
sensibly differ from a triangle. Hence, the area of a sec-
tor equals J the arc multiplied by the radius, and as a cir-
cle may be divided into a large number of sectors, hence,
C= J- cr, but c = 2 TIT.
.-. (7=1 x 27rrxr = 7rr2.
Also, since r2 = (l <T)2 = | d2,
.-. 7rr* = ^cP.
4
Hence, the three rules for finding the area of a circle :
(1) Multiply one half the circumference by the radius.
(2) Multiply the square of the radius by TT.
(3) Multiply the square of the diameter by ^ IT.
There is a fourth rule for finding the area of a circle.
<?= 2?rr. .•. r = -
2
7T
... 7rr2==7rx __= xc2 = .07958^.
4?r2 4?r
That is, the area of a circle equals the square of its circum-
ference multiplied by .07958.
AREA OF A CIRCLE 271
EXERCISE 151
1. Given r = 14, find 0. ll. Given d = 74, find C.
2. Given r=22, find 0. 12. Given d=92, find 0.
3. Given r=36, find (7. 13. Given c = 100, find (7.
4. Given r= 4.7, find O. 14. Given c? = 7 8, find (7.
5. Given r = 6.5, find (7. 15. Given <?=83, find (7.
6. Given r=8.6, find (7. 16. Given c=93, find (7.
7. Given r = 9.7, find (7. 17. Given <?=8.7, find (7.
8. Given d=78, find (7. 18. Given <?=6.9, find (7.
9. Given d= 64, find <7. 19. Given <?= 9.8, find (7.
10. Given d=96, find- O. 20. Given c = 10.8, find (7.
HJxample 1. Given the area of a circle equal to 535.08,
find the radius of the circle.
SOLUTION. ?rr2 = 0.
.-. 3.1416 r2 = 535.08.
... ,«=f|||-no.si.
.*. r = 13.05, nearly.
Example 2. Given the area of a circle equal to 658.98,
find the circumference of the circle.
SOLUTION. .07958 c2= (7.
.07958c2=658.98
.*. <?=91, nearly.
EXERCISE 152
1. Given O.= 3019.1, find r.
2. Given (7=907.9, find r.
3. Given 0= 3421. 2, find r.
4. Given 0= 5541.8, find r.
5. Given (7=21.24, find r.
272
ADVANCED BOOK OF ARITHMETIC
6. Given (7=40.715, find d.
7. Given (7=66.476, find d.
8. Given (7= 75. 43, find d.
9. Given (7=109.36, find d.
10. Given (7=141.03, find d.
11. Given (7=4.5964, find c.
12. Given (7=6.883, find c.
13. Given (7= 779. 94, find c.
14. Given (7=57,495, find <?.
15. Given (7= 33,621, find <?.
16. How long must a rope be so that by tying one end
of it to a stake driven into the ground and fastening the
other end to a cow's horn, the cow
may graze over 1 A. ?
17. ABOD is a square described
in a circle ; MNRS is a square de-
scribed about the circle. If the radius
of the circle is 12, find the areas of
ABOD, MNRS, and of the circle.
18. If the diameter of a circle is
34 in., find the difference between the
area of the circle and of the square described in the circle.
19. If the diameter of a circle is 88 in., find the differ-
ence between the area of the square de-
scribed about the circle and the area of
the circle.
20. A regular hexagon is a figure of
six sides each of the same length, and
its six angles are equal to one another.
A hexagon may be broken up into six
equilateral triangles.
If the side of a regular hexagon is 10, find its area.
AREA OF A CIRCLE
273
21. If a regular hexagon is described in a circle, its side
equals the radius of the circle. If the radius of a circle is
16 in., find the difference between the areas of the circle
and the regular hexagon described in the circle.
22. A regular hexagon and a square have each a perim-
eter of 60 in. Find their areas.
23. The circumference of a circle equals the perimeter
of a square. Find which has the larger area.
Take any angle, J., formed by two radii in a circle, and
take another angle, B, formed by two radii of the same
circle, then designate
the subtended arcs by
arc A and arc B, then
A : B = arc A : arc B.
The reason for this is :
If A were twice B,
then arc A would be FIG. i. FIG. 2.
twice arc B. If A were three times B, then arc A would
be three times arc 5, and so on for any number of
times (Fig. 1).
A _ arc A
B arc B
or A : B = arc A : arc B. (a)
If two diameters, mn, sr (Fig. 2), are drawn at right
angles, the four angles at the center are all right angles.
If each of the right angles at 0 is divided into 90 equal
parts, each part is equal to 1°. The arcs these angles
intercept on the circumference are all equal, and as the
360th part of the circumference is called a degree on the
circumference, hence (a) the number of degrees in an
angle at the center is equal to the number of degrees in
274 ADVANCED BOOK OF ARITHMETIC
its arc on the circumference. This fact is generally stated
as follows : A central angle is measured by its intercepted
arc.
Example. The radius of a circle is
15 in. Find the length of an arc of 37°
30' of the circumference of this circle.
SOLUTION. 360° : 37° 30' = length
of the circumference : length of the
arc mn.
.-. c=2x 3.1416 x 15 in. = 94.248 in.
.-. 360 : 37 J = 94.248 in. : arc mn.
.-. 360 x arc mn= 94.248 x 37J in.
94.248 x 37J in. Q0 .
.*. arc mn = — - =9.82 in., nearly.
360
EXERCISE 153
1. The radius of a circle is 37 in. Find the length of
an arc of 72° of this circle.
2. The radius of a circle is 94 in. Find the length of
an arc of 30° of this circle.
3. What is the length of an arc of 1° on a circle whose
radius is 58 ft. ?
4. What angle does an arc of 40.212 ft. subtend at the
center of a circle whose radius is 64 ft. ?
5. What angle does an arc of 6.032 ft. subtend at the
center of a circle whose radius is 96 ft.?
6. The distance of the moon from the earth is 239,000
mi., and the diameter of the moon is 2170 mi. To an
observer on the earth, what angle does the moon's diame-
ter subtend ?
SIMILAR FIGURES
275
SIMILAR FIGURES
Similar figures are figures having the same form.
Examples. The triangles ABO, LMN, are similar
triangles.
All regular polygons of the same number of sides are
similar figures. The drawing which a surveyor makes of
a tract of land is similar to the tract of land.
It is shown in geometry that corresponding dimensions
of similar figures have the same ratio; also that the areas
of similar figures are to each other as the squares of their
corresponding dimensions.
Example l. When a pole 6
ft. high casts a shadow 5 ft.,
how high is a steeple whose
shadow is 90 ft.?
SOLUTION. Let AB repre-
sent the pole, AO
its shadow, x the ,
steeple, and MR its
shadow. Then
5 : 90 = 6 : x. 6
.-. 3=108. Ana.
108 ft.
A
276 ADVANCED BOOK OF ARITHMETIC
Example 2. The area of a triangle, one of whose sides
is 5 rd., is 11 sq. rd. Find the corresponding side of a
similar triangle whose area is three times as great.
SOLUTION. Let X equal the required side.
The area of the first triangle : the area of the second
triangle = square of the side of the first triangle : square
of the side of the second triangle.
.e.
JT = V75 = 8.662. Ans. 8.662 rd.
EXERCISE 154
1. When a tree 90 ft. high casts a shadow 75 ft. long,
find the length of the shadow cast by a pole 24 ft. high.
2. How high is an object which casts a shadow 110 ft.
when a pole 8 ft. high casts a shadow 5 ft. ?
3. A map is drawn to a scale of 40 mi. to 1 in. On
this map two cities are 2J in. apart. How many miles
are there between these cities?
4. In a map of a city two public buildings are 9J in.
distant. If the map is drawn to the scale of 1 in. to f of a
mile, how far is it from one of these buildings to the other ?
5. The area of a triangle is 15 sq. ft., and one of its
sides is 10 ft. Find the corresponding side of a similar
triangle five times as large.
6. The altitude of a triangle is 10 ft. If the triangle
is divided into two equal parts by a line parallel to its
base, how far from the vertex must this line be drawn ?
SURFACES OF SOLIDS
277
7. Corresponding sides of two similar quadrilaterals are
in the ratio of 4 to 11. Find the ratio of their areas.
8. The diameters of two circles are 12 and 18 in.
Find the ratio of their areas.
9. The distance between two cities is 90 mi., and on a
map containing both cities their positions are distant 5f
in. What area is represented by a circle of J in. radius
on this map ?
SURFACES OF THE PRISM, PYRAMID, CYLINDER,
CONE, AND SPHERE
A right prism is a solid, two of whose faces are equal and
parallel polygons, and whose other faces are rectangles.
The upper and lower faces are called the bases, and the
other faces are called lateral faces.
PRISMS
PYRAMIDS
CONE
A right pyramid is a solid whose base is a regular poly-
gon, and whose other faces are triangles equal in area.
A cone is a solid made by the revolution of a right
triangle about one of its legs.
A cylinder is a solid made by the revolution of a
rectangle about one of its sides when that side is fixed.
278 ADVANCED BOOK OF ARITHMETIC
The following rules may be easily established experimen-
tally by paper cutting or other devices :
The lateral surface of a right prism equals the product of
the perimeter of its base by the height of the prism.
The lateral surface of a right pyramid equals one half the
perimeter of its base by the altitude of one of its lateral
faces.
The convex surface of a cone equals one half the circum-
ference of its base by its slant height.
The convex surface of a cylinder equals the circumference
of its base by its height.
The surface of a sphere equals four times the area of one
of its great circles, i.e. 4 irr2.
EXERCISE 155
1. Find the lateral surface of a quadrangular prism, the
dimensions of whose base are 16 ft. by 8 ft., and whose
height is 12 ft.
2. Find the area of the walls of a room, having given
the dimensions of the floor as 18 ft. by 16 ft., and the
height as 10 ft.
3. Find the height of a triangular prism, the sides of
its base being 5, 6, and 7 ft., and its lateral area being 190
sq. ft.
4. Find the height of a pentagonal prism whose lateral
area is 300 sq. ft., and each side of whose base is 8 ft.
* 5. The base of a square pyramid is 40 ft. long, and
the altitude of each of its triangular faces is 26 ft. Find
its lateral area. Find the cost of painting its lateral
surface at 2J^ per square foot.
SURFACES OF SOLIDS 279
6. Each side of a hexagonal pyramid is 14 ft. and its
slant height is 15 ft. Find the area of its lateral
surface.
7. A pyramidal tent whose base is a square 22 ft. on a
side has a slant height of 30 ft. Find the cost of the
canvas for the tent, at 18^ per square yard.
8. Find the number of square yards in the lateral
surface of a triangular pyramid, each side of the base
being 21 ft. and the slant height being 42 ft.
9. The radius of the base of a right cone is 49 in.
and the slant height is 50 in. Find its convex surface
in
the
hei
ne,
int
3 3 a
CAife. ^ -^ x«. ana. the
slant height 16 ft. ?
12. Find the convex surface of a cylinder the diameter
of whose base is 19 ft. and whose height is 50 ft.
13. Find the convex surface of a cylinder, the radius of
the base being 41 in., and the height, 60 in.
14. A standpipe has a diameter of 30 ft. and is 150 ft.
high. Find the cost of painting it at 25 ^ per square yard.
15. Find the surface of a sphere whose radius is 98 in.
16. Find the surface of a sphere if its diameter is 42 in.
17. The diameter of the planet Mercury is 3030 mi.;
find the area of the planet.
18. The diameter of the planet Venus is 7700 mi. ; find
the area of the planet.
280 ADVANCED BOOK OF ARITHMETIC
19. The diameters of the major planets are, respectively,
86,000, 73,000, 32,000, 33,000 mi. Find the number of
million square miles in the area of each of these planets.
20. The surface of a sphere is 10,568 sq. in. Calculate
its diameter.
VOLUMES OF SOLIDS
The volume of a rectangular prism is equal to the area of
its base multiplied by its height.
The volume of a cylinder is also equal to the product of
its base by its height.
The volume of a pyramid and that of a cone are each
equal to one third the product of the area of the base and
height.
The volume of a cylinder having r for radius of base
and 2 r for height is Tir2 x 2 r — 2 Trr3.
The volume of a sphere having r for radius is f of 2 Trr2
= |- Trr3, and since r — \ d, .'. rs = -| c?3 ;
...A7rr3 = A7rX| J3 = | C?3=. 5236^.
EXERCISE 156
1. Find the volume of a triangular prism, the sides of
the base being 11, 25, 30 in., respectively, and the height
of the prism being 40 in.
VOLUMES OF SOLIDS 281
2. Find the volume of a square pyramid, if the sides
of the base are each equal to 10 in., and the height is 21 in.
3. Find the volume of a cone, the radius of its base
being 12 in., and its height being 27 in.
4. Find the volume of a cone, if the radius of the base
is 25 in., and the height is 24 in.
5. Find the volume of a hexagonal pyramid, each side
of its base being 10 in., and its height being 30 in.
6. Find the volume of a sphere whose radius is 20 in.
7. Find the volume of a sphere, the radius being 8 ft.
8. How many gallons does a cylindrical cistern hold,
the diameter of its base being 9 ft. 4 in., and its height
8ft.?
9. How many gallons does a cylindrical cistern contain,
if the diameter of its base is 11 ft. and its height is 6 ft.
5 in.?
10. The diameter of the base of a cylinder is 10 in.,
and its height is 10 in. Find the ratio of the volume of
this cylinder to the volume of a sphere 10 in. in diameter.
11. The diameter of the base of a cone is 1 ft. and its
height is 1 ft. Find the ratio of the volume of this cone
to the volume of a sphere whose diameter is 1 ft.
12. The surface of a cube contains 84 sq. ft. 54 sq. in.
Find its volume.
13. The base of a pyramid is a triangle whose sides are
1 ft. 1 in., 3 ft. 1 in., 3 ft. 4 in., and whose volume is 1 cu.
ft. 1152 cu. in. Find its height.
14. The surface of a sphere equals 1257 sq. in. Find
its volume.
15. The surface of a hemispherical dome is 2513.5 sq.
ft. Find its diameter.
282
ADVANCED BOOK OF ARITHMETIC
16. The volumes of similar solids are to each other as
the cubes of their corresponding dimensions. How many
times as large as the earth is the sun ? The diameter of
the sun is nearly 888,000 mi., and the diameter of the
earth is nearly 8000 mi.
17. Find how many times as large as the moon is the
earth. The moon's diameter is 2200 mi., nearly.
18. How many times as large as the earth is Saturn?
The diameter of Saturn is 73,000 mi.
19. How many times as large as the earth is Jupiter?
The diameter of Jupiter is 88,000 mi., nearly.
120-
70
50
40
30
20
30
MEASURE OF TEMPERATURE
A thermometer is an instrument for meas-
uring heat. The principle of the thermom-
eter is that substances expand with heat,
according to a natural law.
There are two different styles of thermom-
eter in general use, — the Centigrade and the
Fahrenheit. The Centigrade thermometer
marks the melting point of ice 0°, and the
boiling point of water 100°. The interval be-
tween these points is divided into 100 parts,
or degrees, so that the change in the volume
of the mercury between any two consecutive
marks is T-^ of the change from 0° to 100°.
The Fahrenheit thermometer divides the
interval from the melting point of ice to the
boiling point of water into 180°. It marks
the melting point of ice 32°, and the boiling
point of water 212°.
MEASURE OF TEMPERATURE 283
Notation. 92° C. means 92 degrees on the Centigrade
thermometer.
45° Fahr. means 45 degrees on Fahrenheit's thermometer.
-f- 10° means 10 degrees above zero.
— 10° means 10 degrees below zero.
Verify by counting 32° backward.
20° - 32° = - 12°.
10° - 32° = - 22°.
- 2° - 32° = - 34°, etc.
(1) To change from degrees Fahrenheit to degrees Centi-
grade, subtract 32° and multiply the remainder by -|.
(2) To change from degrees Centigrade to degrees Fahren-
heit, multiply the number of degrees Centigrade by | and add
32 to the product.
Explanation of the rules:
(1) Suppose the temperature on a Fahrenheit ther-
mometer is n degrees. Subtract 32° to get the number of
degrees from 0. A difference of 180° Fahrenheit = a dif-
ference of 100° Centigrade. Therefore, a difference of 1°
Fahrenheit = a difference of |° Centigrade. Therefore, a
difference of (n - 32°) Fahrenheit = f (n - 32°) Centi-
grade, which symbolizes the first of the above rules.
(2) A difference of n° C. = a difference of | n° Fahren-
heit. Hence, n° C. = (f n° + 32°) Fahr.
EXERCISE 157
Express the following Fahrenheit temperatures on the
C. scale :
1. 86°.
4. 248°.
7. 38°.
10. -13°.
2. 77°.
5. 68°.
8. 23°.
11. -40°.
3. 203°.
6. 54°.
9. 15°.
12. -90°.
284 ADVANCED BOOK OF ARITHMETIC
Express the following C. temperatures on the Fahr.
scale :
13. 35°. 16. 20°. 19. -10°. 22. -18°.
14. 55°. 17. 18°. 20. -20°. 23. -24°.
is. 25°. is. 8°. 21. -14°. 24. -273°.
TABLE OF MELTING POINTS
Mercury .- 40° C. Lead . . 326° C. Gold .... 1035° C.
Sulphur. 113° C.- Zinc . . 415°C. Cast iron, 1100° to 1200° C.
25. Give the above table in the Fahrenheit scale.
26. Water attains its maximum density of 4° C. Ex-
press this temperature on Fahrenheit's scale.
THE METRIC SYSTEM OF WEIGHTS AND MEASURES
The metric system is now used by more than forty
countries, and it is the only system used in text-books of
science. Upward of twenty nations contribute to the sup-
port of the International Bureau of Weights and Measures
in Paris. For these reasons the Metric System deserves
to be called the International System.
The only great nations which have not adopted the
Metric System are the United States and Great Britain.
In the United States the system is legalized, and none
other is used in the Philippines and Porto Rico.
The Metric System is so called because the meter is the
basis of the system. The meter is the standard unit of
linear measure. Its length is the ten-millionth part of
the distance from the equator to the north pole measured
on the meridian of Paris. Its length in this country is
39.37 inches. In the United Kingdom the legal equivalent
of the meter is 39.370113 inches and on the continent of
Europe 39.370432 inches.
METRIC SYSTEM OF WEIGHTS AND MEASURES 285
The standard meter from which all others are derived
is a bar made of an alloy of platinum and iridium, kept in
the International Bureau of Weights and Measures in
Paris. Duplicates of this standard meter have been fur-
nished to all the nations of the world.
The Metric System is a decimal sj^stem. In it there
are no compound rules. It is the simplest and the most
perfect system ever devised.
The names for the multiples of the standard unit in the
Metric System are formed from the names of the standard
unit by means of prefixes derived from the Greek words
meaning ten, one hundred, one thousand, and ten thou-
sand, i.e. deka, hekaton, chilioi, murioi. The names for the
submultiples of the standard units are formed in a similar
manner from the Latin words meaning ten, one hundred,
one thousand, i.e. decent, centum, mille. Thus:
Dekameter means ten meters.
Hektometer means one hundred meters.
Kilometer means one thousand meters.
Myriameter means ten thousand meters.
Decimeter means one tenth of a meter.
Centimeter means one hundredth of a meter.
Millimeter means one thousandth of a meter.
The standard units are the meter, the liter, and the gram.
The gram being a very small weight, the kilogram is most
used in ordinary affairs. In the Metric System the meter,
the liter, and the kilogram serve for everyday trade in
exactly the same manner as the yard, the quart measure,
and the pound Avoirdupois in our system of weights and
measures. The international meter and the kilogram are,
since 1893, the fundamental standards of length and mass
in the United States.
286
ADVANCED BOOK OF ARITHMETIC
The gram is the weight of distilled water at 4° Centi-
grade, which fills a cubical vessel, one of whose edges is
one centimeter.
inn iiiiiMii iiiiiuii miiiiii iimim iiiiiiiii iiiniiii iiiunii iiiiiiin MINIMI
01 234 50 7 89 10 cm.
imliiiilmiimili
I I I I I I Ml
0
4 in.
COMPARISON SCALE: 10 CENTIMETERS AND 4 INCHES. (ACTUAL SIZE.)
LINEAR MEASURE
10 millimeters (mm.) = 1 centimeter
10 centimeters (cm.) — 1 decimeter
10 decimeters = 1 meter
10 meters (m.) = 1 dekameter
10 dekameters = 1 hektometer
10 hektometers = 1 kilometer
10 kilometers (km.) = 1 myriameter
The units in common use are the centimeter, meter, and
kilometer.
SURFACE MEASURE
100 square millimeters (qmm.) = 1 square centimeter
100 square centimeters (qcm.) = 1 square decimeter
= 1 square meter
= lar
= 1 hektar
= 1 square kilometer (qkm.)
An ar is the area of a square whose side is ten meters.
1 qm. is therefore equal to a centar (ca.). The areas of
small tracts of land, such as gardens, are expressed in ars.
100 square decimeters
100 square meters (qm.)
100 ars
100 hektars (ha.)
METRIC SYSTEM OF WEIGHTS AND MEASURES 287
The areas of larger tracts, such as farms, are expressed in
hektars. The areas of still larger tracts, such as countries,
are expressed in square kilometers.
CUBIC MEASURE
1000 cubic millimeters (cmm.) = 1 cubic centimeter
1000 cubic centimeters (ccm.) = 1 cubic decimeter, or liter
1000 liters (1.) = 1 cubic meter (cbm.)
The liter corresponds to our quart. The hektoliter
(hi.) corresponds to our bushel. The cubic meter is also
called a stere.
WEIGHT
1000 milligrams (mg.) = 1 gram
1000 grams (g.) =1 kilogram, or kilo
1000 kilograms (kg.) = 1 tonneau (T.), or ton
A gram represents the weight of 1 ccm. of distilled
water at 4° C.
A kilogram is the weight of 1 cu. dm. of distilled water
at 4° C.
A tonneau, or ton, is the weight of 1 cbm. of distilled
water at 4° C.
A quintal is 100 kg.
REDUCTION
^Example l. Reduce 75.623 m. to centimeters, and also
to millimeters.
SOLUTION, (a) Since there are 100 cm. in 1m., reduce
meters to centimeters by multiplying the number of meters
by 100. This is done by moving the decimal point two
places to the right.
(6) Since there are 1000 mm. in 1 m., therefore multiply
the number of meters by 1000. This is done by moving
the decimal point three places to the right.
288 ADVANCED BOOK OF ARITHMETIC
Ans. (a) 7562.3 cm.
(5) 75623mm.
Example 2. Reduce 85679 mm. to meters.
SOLUTION. This is done by dividing by 1000, i.e. by
moving the decimal point three places to the left.
Ans. 85.679 m.
EXERCISE 158
Add:
(1) (2) (3)
96458.23 in. 92435.9 cm. 924357.9 mm.
23458.98 m. 23456.8 cm. 8765.1 mm.
14329.09 m. 98239.2 cm. 986201.8 mm.
920184.60 m. 98209823.9 cm. 87623.7 mm.
1908.02m. 1980098.76cm. 987.9mm.
90327.92 m. 6543097.1 cm. 23456.2 mm.
1872095.72m. 7609.9cm. 10098.3mm.
902376.253 m. 980098.3 cm. 90098.9 mm.
999999.9 mm.
4. A man walks on four successive days. The first day
he walks 11.7 km. ; the second day, 984 m. ; the third
day, 2950 m. ; the fourth day, 12.8 km. How far does he
travel? Give the answer in meters.
5. From 25.724 km. take 6270 m.
6. Multiply 11.732 m. by 12.
7. How many times is 25 mm. contained in 24 m. ?
8. How many times is 7.03 m. contained in .0494209
km.?
9. How many times is 12 ca. contained in 6 ha. ?
10. How many farms of 8 ha. each can be made out of
a square whose side is 3 km. ?
METRIC SYSTEM OF WEIGHTS AND MEASURES 289
11. A vessel contains 250 ccm. How many such vessels
would hold 5 cbm.?
12. How many times is 800 ccm. contained in 50 1.?
13. How many times is 450 ccm. contained in 22.5 hi.?
14. A dime weighs 2.5g. How many dimes can be
coined from 3 kg. standard silver ? how many 25 ^ pieces,
weights being in proportion to values ?
15. How often is 1521 mg. contained in 5.9319 kg. ?
Example. Find the area of a rectangle, if its length is
425.8 m. and its breadth is .3256 km.
SOLUTION. First, reduce .3256 km. to meters.
.3256km. = 325.6 in.
Second, multiply in the usual manner 425.8
and get for product 138,640.48 qm. 325.6
Reduce this to ars by dividing by 100, 25548
and reduce the ars to hektars by divid- 21290
ing by 100. Both operations can be 8516
performed at once by moving the 12774
decimal point four places to the left. 138640.48 qm.
Am. 13.864048 ha.
EXERCISE 159
Find the area of each of the following rectangles :
LENGTH WIDTH LENGTH WIDTH
1. 625 m. 125 m. 4. 369.4 m. 184.7 m.
2. 305 m. 61 m. 5. 488.9 m. 244.45 m.
3. 338 m. 169 m. 6. 5767 m. 11.534 m.
7. Find the volume of a rectangular solid whose di-
mensions are 1.2 m., .9 m., and .75 m.
8. Find the area of a triangle whose sides are 14 m.,
48 m., and 50 m.
9. Find the area of a circle whose radius is .78 m.
10. Find the volume of a sphere whose radius is 18 cm.
290 ADVANCED BOOK OF ARITHMETIC
11. The legs of a right triangle are 56 cm. and 105 cm.
Find the hypothenuse.
12. Find the volume of a rectangular prism, the dimen-
sions of the base being 14 m. by 12 m., and height 9.5 in.
13. A boiler has 300 tubes 2.4 m. long, 7.5 cm. diameter.
What is the area of the tube heating surface ?
14. Find the weight of a spherical cast iron shell 32.5
cm. outside and 27.5 cm. inside diameter.
15. An iron plate 3 mm. thick weighs 1 km. What is
its area ?
The following approximations should be fixed in mind :
1 meter = 39 inches ; 1 kilometer = |- of 1 mile.
1 centimeter = | of an inch ; 1 hektar = 2| acres.
1 liter = 1 quart ; 1 kilogram = 2^ pounds Avoirdupois.
EQUIVALENTS OF COMMON UNITS IN METRIC UNITS
1 inch = 25.4001 mm. 1 sq. foot = .0929 qm.
1 foot = .304801 m. 1 sq. yard = .8361 qm.
1 yard = .914402 m. 1 A. .4047 ha.
1 mile - 1.60935 km. 1 sq. mi. = 2.59 qkm.
1 quart = .94636 1. 1 cu. inch = 16.3872 ccm.
1 gallon = 3.78543 1. 1 cu. foot = .02832 cbm.
1 bushel = .35239 hi. 1 cu. yard = .7646 cbm.
1 sq. inch = .6452 qcm. 1 Ib. Avoir. = .45359 kg.
EQUIVALENTS OF METRIC UNITS IN COMMON UNITS
1 meter =39.37 in. 1 sq. centimeter = .155 sq. in.
1 kilometer = .62137 mi. 1 sq. meter = 1.196 sq. yd.
1 hektar =2.471 A.
1 qkm. = .3861 sq. mi.
1 cu. centimeter = .061 cu. in.
1 cu. meter = 35.314 cu. ft.
1 liter = 1.0567 qt.* 1 kilogram = 2.20462 Ib.
1 hektoliter = 2.83774 bu. 1 tonneau, or ton = 2204.6 Ib.
* Liquid quarts, or 0.9081 dry quarts.
ANNUAL INTEREST 291
ANNUAL INTEREST
Lincoln, Neb.,
dewuvncL c/ promise to pay to __________________
f&tt ______________ or order
Eight hundred.. ...°° Dollars.
100
Value received, with, i^nt&^e^t awtt/uaJUA* at
No. <)2.
In some states if a note contains the words "with
interest annually," and if the interest remains unpaid for
a number of years, then the interest due at the end of
each year bears interest until the date of settlement.
The interest by this method of reckoning is called
annual interest. The interest we have up to this time
considered is known as simple interest.
Example. How much is due on a note for $ 800, dated
Jan. 5, 1898, and bearing interest annually at 8%, if left
unpaid, both in principal and interest, until March 10,
1903?
SOLUTION
YB. MO. DA.
Interest on $800 for 1 yr. at 8% = $64. 1903 3 10
Interest on $800 for 5 yr. 2 mo. 5 da. 1898 1 5
at S% = $331. 56. 525
The first year's interest is due Jan. 5, 1899, and bears
interest until March 10, 1903. The interest falling due
at the end of the second year bears interest until March
10, 1903, and so on.
292 ADVANCED BOOK OF ARITHMETIC
$64 bears interest for 4 yr. 2 mo. 5 da.
$64 bears interest for 3 yr. 2 mo. 5 da.
$64 bears interest for 2 yr. 2 mo. 5 da.
$64 bears interest for 1 yr. 2 mo. 5 da.
$64 bears interest for 2 mo. 5 da.
Adding, $64 bears interest for 10 yr. 10 mo. 25 da.
Interest on $ 64 for 10 yr. 10 mo. 25 da. at 8 % = $55.82.
Interest on $800 for 5 yr. 2 mo. 5 da. at 8 % = $331.56.
Amount due = $800 + $ 331.56 + $55.82 = $1187.38.
EXERCISE 160
1. Find the amount due April 1, 1903, on a note for
$900, dated July 15, 1899, and bearing interest annually
at 7%.
2. Find interest from March 6, 1898, to Jan. 1, 1903,
on note for $1400, interest payable annually at 8 %.
3. Find amount due Jan. 1, 1903, on a note for $1200,
dated July 1, 1897, interest payable annually at 6 %.
4. Find amount due Jan. 1, 1903, on note for $1000,
dated Jan. 1, 1898, interest payable semiannually at 5%.
5. Find the amount due at the end of 5 years on a
coupon note, interest payable semiannually, if the face of
the note is $700, and the rate of interest is 7%.
COMPOUND INTEREST
Savings banks and other banks which give interest on
deposits add the interest semiannually or annually to the
amount deposited. This interest added bears interest
until the next date of balancing the depositor's account
book. The next interest is added in the same way.
Suppose a person deposited in a savings bank $250 and
allowed it to remain there for 10 yr. The question might
COMPOUND INTEREST 293
be asked, how much will principal and interest amount to
at the end of that time ? The accruing interest in this
case is called compound interest.
Designate principal by p, rate of interest by r, time
in years by w, and amount by a.
When the rate of interest on $1 is $r per year, the
amount of $1 at the end of 1 yr. is $(1 + r).
.-. the amount of 1 1 in 1 yr. = f (1-fr),
/. the amount of $p in 1 yr. =$p(l + r).
To get the amount of any principal for 1 yr. or for
6 mo., or for any other period at simple interest, mul-
tiply the principal by the amount of $1 for that time.
Take now $ j?(l-fr) as principal, then the amount of
r) in 1 yr. = $^1 + r)(l + r) = $X1 + O2-
The amount of $p(l + r)2 in 1 yr. = $p(l + r)2(l + r)
Similarly, the amount of $p (1 + r)3 in 1 yr. = $ jt?(l + r)4.
At compound interest,
$p amounts in 2 yr. to |jp(l + r)2.
$p amounts in 3 yr. to
$p amounts in 4 yr. to
$jp amounts in 5 yr. to
Generally, $p amounts in nyr. to $/>(!
At compound interest,
The amount of $1 at 4% for 20 yr. is f (1.04)20.
The amount of $1 at 5% for 20 yr. is 9(1.05)*.
The amount of 91 at 6% for 20 yr. is f (1.06)20.
The labor of raising 1.06 to the 20th power is
considerable.
A student who has a working knowledge of logarithms
can do this by the aid of a table of logarithms in less than
a minute.
294
ADVANCED BOOK OF ARITHMETIC
The following table gives the amount of
pound interest :
at com-
YRS.
2%
2*%
3%
4%
5%
6%
1
1.020000
1.025000
1.030000
1.040000
1.050000
1.060000
2
1.040400
1.050625
1.060900
1.081600
1.102500
1.123600
3
1.061208
1.076891
1.092727
1.124864
1.157625
1.191016
4
1.082432
1.103813
1.125509
1.169859
1.215506
1.262477
5
1.104081
1.131408
1.159274
1.216653
1.276282
1.338226
6
1.126162
1.159693
1.194052
1.265319
1.340096
1.418519
7
1.148686
1.188686
1.229874
1.315932
1.407100
1.503630
8
1.171659
1.218403
1.266770
1.368569
1.477455
1.593848
9
1.195093
1.248863
1.304773
1.423312
1.551328
1.689479
10
1.218994
1.280085
1.343916
1.480244
1.628895
1.790848
11
1.243374
1.312087
1.384234
1.539454
1.710339
1.898299
12
1.268242
1.344889
1.425761
1.601032
1.795856
2.012196
13
1.293607
1.378511
1.468534
1.665073
1.885649
2.132928
14
1.319479
1.412974
1.512590
1.731676
1.979932
2.260904
15
1.345868
1.448298
1.557967
1.800943
2.078928
2.396558
16
1.372786
1.484506
1.604706
1.872981
2.182875
2.540352
17
1.400241
1.521618
1.652847
1.947900
2.292018
2.692773
18
1.428246
1.559659
1.702433
2.025817
2.406619
2.854339
19
1.456811
1.598650
1.753506
2.106849
2.526950
3.025599
20
1.485947
1.638616
1.806111
2.191123
2.653298
3.207136
Example l. Find the amount of 12500 at compound
interest for 12 yr. at 5 % .
SOLUTION, a = $2500 (1.05)12 = $2500 x 1.T95856 =
$4489.64.
Example 2. Find the amount of $5000 for 8 yr. at 4 %
compound interest, the interest being compounded semi-
annually.
SOLUTION. $5000 x (1.02)16 = $5000 x 1.372786 =
$6863.93.
\
MISCELLANEOUS TOPICS 295
EXERCISE 161
With the aid of the above table, find the amount at
compound interest of :
1. $2000 for 8 yr.at 4%. 3. $5000 for 12 yr. at 4%.
2. $3000 for 10 yr. at 3%. 4. $6000 for 10 yr. at 5%.
5. $8000 for 8 yr. at 5%, interest compounded semi-
annually.
6. $2250 for 6 yr. at; 4%, interest compounded semi-
annually.
MISCELLANEOUS TOPICS
WORK AND TIME
Example l. A can do a piece of work in 6 da., and B
can do the same piece of work in 8 da. In what time can
A and B do the work together?
ANALYTICAL SOLUTION. A does l of the work in 1 da.
B does | of the work in 1 da.
.-. A and B together do (l -f |) of the work in 1 da.
/. A and B do ^ of the work in 1 da.
.*. A and B do ^ of the work in ^ of 1 da.
.-. A and B do || of the work in ^ of 1 da.
.•. A and B do the work in 3^ da.
Example 2. A cistern has three pipes. The first pipe
fills the cistern in 12 hr., the second, in 15 hr., and the
third empties it in 10 hr. In what time will the cistern
be filled, if all three pipes run together, and the cistern is
empty when the pipes start running?
ANALYTICAL SOLUTION. The first pipe fills -^ of the
cistern in 1 hr.
The second pipe fills ^ of the cistern in 1 hr.
The third pipe empties ^ of the cistern in 1 hr.
296 ADVANCED BOOK OF ARITHMETIC
/. the three pipes fill CiV + T5~~lV) °^ ^ne cistern in
Ihr.
.*. the three pipes fill ^ of the cistern in 1 hr.
/. the three pipes fill |§- of the cistern in 20 hr.
.*. the cistern is filled in 20 hr.
Example 3. A and B do a piece of work in 3^ hr. ;
A alone, in 7 hr. In what time does B do the work?
SOLUTION. A and B together do — - of the work in
1 hr. ; i.e. A and B together do -|J of the work in 1 hr.
But A alone does ^ of the work in 1 hr.
/. B alone does (^| — j-) of the work in 1 hr.
/. B alone does -| of the work in 1 hr.
/. B does the work in 8 hr.
Example 4. A and B can mow a field in 21 hr. A can
do | as much work as B. Find the time in which each
4 <, does the work.
SOLUTION. Represent B's work
by the rectangle mnst.
Represent A's work by the rec-
tangle pqxy.
A will do in J hr. the work A
and B do together in 1 hr.
/. A will do in Jhr. x 21 the
work A and B do together in
21 hr.
.'. A will do in 49 hr. the work A and B do together
in 21 hr.
B will do in | hr. the work A and B do together in 1
hr.
.*. B will do in | hr. x 21 the work A and B do together
in 21 hr.
m
V
/t-k
a?
MISCELLANEOUS TOPICS 297
/. B will do in 36 1 hr. the work A and B do together in
21 hr.
A's time, 49 hr.; B's time, 36| hr.
EXERCISE 162
1. If a person can do a piece of work in 7 da., what is
his day's work ? If he can do the work in 4^ da., what is
his day's work ?
2. If B can copy a manuscript in 5|hr., how much can
he copy in 1 hr. ?
3. John travels T5g of the distance between two cities
in 1 hr. How many hours will it take him to travel the
remainder of the distance ?
4. A can do a piece of work in 3 hr., B in 4 hr., and
C in 5 hr. How long will it take the three working
together to do the work ?
5. A can do a piece of work in 4 hr., B in 5 hr., and A,
B, and C together in 1J hr. How long would it take C
alone to do the work ?
6. A, B, and C can do a piece of work in 6, 8, and 10 da.,
respectively. If they begin the work together, what
part of the work remains to be done at the end of the
second day ?
7. A, B, and C can build a fence in 10, 15, and 20 hr. re-
spectively. They work together for 4 hr., when B quits.
In what time can A and C finish the work ?
8. A cistern has two pipes. One can fill it in 20 min.,
and the other can empty it in 30 min. If the cistern is
empty, in what time can it be filled, if both pipes begin to
flow at the same instant ?
298 ADVANCED BOOK OF ARITHMETIC
MOTION IN THE SAME DIRECTION, OR IN OPPOSITE
DIRECTIONS
Example 1. A starts to overtake B, who is 100 yd. ahead
of him. A travels 11 yd. to B's 9 yd. How far must A
travel in order to overtake B ?
SOLUTION. A gains on B 2 yd. in every 11 yd. he goes.
.'. A gains on B 1 yd. in every 5J yd. he goes.
.*. A gains on B 100 yd. in every 550 yd. he goes.
Ans. 550 yd.
Example 2. A freight train moving at the rate of 18
mi. an hour is 78 mi. ahead of a passenger train moving in
the same direction at the rate of 80 mi. an hour. Find
the distance the passenger train must run to overtake the
freight train.
SOLUTION. In 1 hr. the passenger train gains on the
freight (30 - 18) mi., i.e. 12 mi.
.*. in (78 -f- 12) hr. the passenger train will overtake the
freight train, i.e. in 6| hr.
In 6^ hr. the passenger train goes 30 mi. x 6-|- = 195 mi.
EXERCISE 163
1. Two ships leave New York for Glasgow, one on Mon-
day morning at 9 o'clock, and the other on the following
morning at 9 o'clock. Their. rates are 15 and 21 miles
an hour respectively. How far from New York City will
the second ship overtake the first ?
2. Dallas and Galveston are 315 mi. apart. A train
leaves Dallas for Galveston at 8 o'clock A.M. at the rate of
30 mi. an hour. At the same time a train leaves Galves-
ton for Dallas at the rate of 33 mi. an hour. How far
will the trains be from Dallas when they meet ?
CIRCULAR MOTION: CLOCKS
299
3. Paris, Texas, is 584 mi. from St. Louis. A passen-
ger train leaves Paris for St. Louis at 6.50 P.M. Three
hours later a freight train leaves St. Louis for Paris.
When and where will they meet, the rates being respec-
tively 24 mi. and 16 mi. per hour ?
4. A man walking at the rate of 4 mi. an hour is
overtaken by a train 88 yd. long, and is passed in 10 sec.
Find the rate of the train.
5. A train going at the rate of 40 mi. an hour passes
in 6 sec. a man walking in the same direction at the rate
of 4 mi. an hour. What is the length of the train?
6. Two trains start from the same station and travel
in the same direction. The first train leaves at 7 A.M.,
and the second train at 9 A.M. How many miles from
the station will the second train overtake the first if the
rate of the first train is 30 mi. per hour and the rate of the
second train is 45 mi. per hour?
CIRCULAR MOTION: CLOCKS
Example l. At what time between 4 and 5 o'clock are
the hands of a clock together?
SOLUTION. At 4 o'clock the
hour hand is 20 minute spaces
ahead of the minute hand.
In 1 hr. the minute hand goes
60 minute spaces.
In 1 hr. the hour hand goes 5
minute spaces.
.-. ratio of rates of motion of
minute hand and of hour hand is
60 : 5, or 12 : 1.
300
ADVANCED BOOK OF ARITHMETIC
.*. in 12 min. the minute hand gains on the hour hand
11 minute spaces.
.'.in IT*J min. the minute hand gains on the hour hand
1 minute space.
.'.in l^y min. x 20 the minute hand gains on the hour
hand 20 minute spaces.
But IJy min. x 20 = 21^ min.
/. the hands are together 21y9^ min. after 4 o'clock.
Example 2. At what time after 7 o'clock do the hour
and minute hands first point in
opposite directions ?
SOLUTION. They will point in op-
posite directions whenever they are
30 minute spaces apart.
At 7 o'clock the hour hand is 35
minute spaces ahead of the minute
hand.
.'. as soon as the minute hand
gains on the hour hand 5 minute spaces, they will point in
opposite directions.
But the minute hand gains on the hour hand 5 minute
spaces in l^y min. x 5, i.e. in 5T6T
min. (See Example 1.) Ans.
5^\ min. past 7 o'clock.
Example 3. When, between 5
and 6 o'clock, will the hour and
minute hands be at right angles ?
SOLUTION. The hour and min-
ute hands will be at right angles
when they are 15 minute spaces
apart. FlG- i-
/. they will be at right angles between 5 and 6 o'clock
CIRCULAR MOTION: CLOCKS
301
FIG. 2.
when the minute hand gains on the hour hand (25 — 15)
minute spaces, i.e. 10 minute spaces.
l^min. x 10 = 1014 min. Ans.
101-J- min. past 5 o'clock.
They will also be at right angles
when the minute hand gains on the
hour hand (25 + 15) minute spaces,
i.e. 40 minute spaces. (See Fig. 2.)
l^Y min. x 40 = 43TrT min.
the hands will be at right
angles to each other at 10^ min.
past 5 o'clock and at 43^ min. past
5 o'clock.
To the young learner the following suggestions may be
of use :
To solve a question in circular motion : first, draw a dia-
gram showing the things that move ; second, find the
ratios of the rates of motion of the things moving ; third,
having found the relative rates of motion of the objects,
proceed as in a simple exercise involving motion in a
straight line.
EXERCISE 164
1. What angle do the hour and minute hands of a clock
make with each other at 1 o'clock ? at 2 o'clock ? at 3
o'clock? at 4 o'clock? at 5 o'clock? at 6 o'clock? at 12
o'clock ?
2. What angle do the hour and minute hands of a clock
make when they point to positions 8 minute spaces apart?
12 minute spaces apart? 19 minute spaces apart? 23
minute spaces apart? 27 minute spaces apart? 50
minute spaces apart? At what time between 1 and 2
o'clock do the hands of a clock make an angle of 36 ° ?
302 ADVANCED BOOK OF ARITHMETIC
3. How many minute spaces apart do the hands of a
watch indicate when they make an angle of 36°? 66°?
84°? 114°? 126°? 144°? 162°? 174°?
4. Find at what time between the hours of 2 and 3
o'clock the hands of a clock are together ; between 3 and
4 o'clock ; between 5 and 6 o'clock ; between 7 and 8
o'clock; 9 and 10 o'clock ; 10 and 11 o'clock; 11 and 12
o'clock.
5. At what time do the hands of a watch point in
opposite directions between
(a) 1 and 2 o'clock? (cT> 9 and 10 o'clock?
(5) 3 and 4 o'clock? (V) 11 and 12 oclock?
0) 8 and 9 o'clock ? (/) 12 and 1 o'clock ?
6. At what time after 3 o'clock do the hands of a watch
first point in opposite directions ? When after 6 o'clock
do they first point in opposite directions? When after
10 o'clock ?
7. At what time or times are the hands of a watch at
right angles between (a) 2 and 3 o'clock? (6) 3 and 4
o'clock ? (<?) 4 and 5 o'clock ? (d) 6 and 7 o'clock ? 0) 8
and 9 o'clock? (/) 9 and 10 o'clock ? (#) 11 and 12 o'clock?
Qi) 12 and 1 o'clock ?
8. At what time after 4 o'clock do the hands of a watch
first point to positions 8 minute spaces apart ? 35 minute
spaces apart ? 24 minute spaces apart ? 50 minute spaces
apart ?
9. Find the time, between 5 and 6 o'clock, when the
minute hand is ^ of the circumference of the dial in
advance of the hour hand. The time between 9 and 10
o'clock when the minute hand is \ of the circumference in
advance of the hour hand.
MISCELLANEOUS EXAMPLES
(ARRANGED BY TOPICS)
NOTATION
1. Write in figures ten million ten.
2. Write in figures seven million two hundred five
thousand.
3. Write in figures one billion one million one.
4. Write in figures five million fifty thousand.
5. Write in figures ten billion ten million one hundred.
6. Write decimally : Twenty-five tenths. One hundred
twenty hundredths. One hundred and fifty-five hun-
dredths. Ten thousand ten hundred thousandths. One
million one ten-millionths. Fifty-five thousand two hun-
dred eighteen ten-thousandths. One million two hundred
thousand four ten-millionths. Five and five hundredths.
Six hundred six hundredths. Seven hundred ten thou-
sandths. Nine hundred and fifteen thousandths. Nine
million and nine millionths.
ADDITION AND SUBTRACTION
7. The following table of railroad mileage in the United
States is taken from the report of the Interstate Commerce
Commission :
YEAR
MILEAGE
YEAR
MILEAGE
1890
163,597.05
1898
186,396.32
1891
168,402.74
1899
189,294.66
1892
171,563.52
1900
193,345.78
1893
176,461.07
1901
197,237.44
1894
178,708.55
1902
202,471.85
1895
180,657.47
1903
207,977.22
1896
182,776.63
1904
213,904.34
1897
184,428.47
1905
218,101.04
303
304 ADVANCED BOOK OF ARITHMETIC
Find the number of miles built during each year, begin-
ning with 1860, up to 1905.
8. From 9000 take .009.
9. From 275 take .000275.
10. Add: 657,987,324,011
119,008,675,987
199,887,564,999
999,555,777,888
987,345,234,876
985,234,678,100
923,524,896,987
987,567,342,959
725,926,846,368
929,935,829,349
929,563,768,968
MULTIPLICATION
11. Multiply 10,500 by 60,600.
12. Multiply 15,010 by 50,080.
13. Multiply 1.01 by 7.07.
14. Multiply 9010.9 by 90.4.
15. A train travels at the rate of 30.25 mi. an hour.
How far will it go in 5£ hr. ?
16. Find the price of 87^ A. of land at $43.75 an acre.
17. Find the price of 4225 bu. of wheat at 84^ per
bushel.
18. Find the value of .1 x .2 x .3 x .4 x .5 x .6 x .7.
19. Find the value of (1.03)4.
20. The number of bales of cotton produced in Texas
in 1901-02 was 2,993,000, and in 1900-01, 3,550,000.
Allowing 500 Ib. to a bale, how many more pounds of
cotton were produced in the latter year than in the former?
MISCELLANEOUS EXAMPLES 305
DIVISION
21. A steamer's cargo consisted of 120,000 bu. of corn,
valued at $57,000; 13,541 bbl. of flour, valued at $47,499;
3050 bales of cotton, valued at $158,224. Find the value
of a bushel of corn, a barrel of flour, and a bale of cotton.
22. Divide 26.78508 by .072 (not by long division).
23. The annual consumption of sugar in a certain state
was, in 1890, 702,201 T., which was found to be 49.93 Ib.
per head of population. Find the population.
24. Make a column of eight numbers, the first of which
is 73,214, the second is f of the first, the third is f of the
second, and so on for the other numbers.
25. How many miles are in 278,784,000 ft. ?
26. Divide 1.1252- (.784)2 by 1.125 -.784.
27. Divide (.75)3 - (.26)3 by .75 - .26.
28. Divide 14.302 by 83.92, correct to four places.
29. Divide 24.619 by 56,000.
30. The length of a degree on the earth's surface is ap-
proximately 69.15 mi. Two places are on the same merid-
ian and 1000 mi. apart. Find, in degrees, the difference
in latitude.
31. Two places on the 60th parallel of latitude are 300
mi. apart. Find the difference of their longitudes. (1° =
183,085 ft.)
32. A bankrupt's liabilities are $47,875; his assets are
$38,650. How many cents on the dollar can he pay?
33. The product of two numbers is 642,978, and one of
the numbers is 5.67. Find the other number.
34. If the quotient is 24,400, the remainder is 15, and
the dividend is 6,100,015, find the divisor.
306 ADVANCED BOOK OF ARITHMETIC
35. The total amount of money in circulation in the
United States on March 1, 1903, was $2,353,738,834. The
per capita circulation in the United States on the same day
was $29.41. Find the population of the United States.
36. Divide the square of 1001 by 77 x 169.
37. When 450 Ib. of sugar cost $20.25, find the price
of 84 Ib.
38. Find the value of a rectangular plot of ground 726
yd. long and 240 yd. wide, at $50 an acre.
39. Find, in United States currency, the value of £79.
40. When 1.75 yd. of silk cost $3.85, find the cost of
14yd.
41. Divide 39.328 by .0032.
42. If . 6 of a yard of cloth cost 27 ^, find the cost of 45 yd.
43. Divide 1 by 1.732.
44. Divide the cube of 11.1 by 27 times 1369.
45. What is the ratio of 25 A. to 640 A.?
G. C. M. AND L. C. M.
46. Find the G. C. M. of 288 and 432.
47. Find all the common measures of 36 and 54.
48. Find the common divisors of 288 and 360.
49. Express 1110, 777, and 1001 as the products of prime
numbers. Find their L. C. M.
50. Find the G. C. M. of 208, 572, and 1326.
51. Find the L. C. M. of 26, 28, 48, 70, and 117.
52. Find the G. C. M. of 625 and 2525.
53. Find the G. C. M. and L. C. M. of 209, 304, and 380.
54. Find the prime factors of 80,850.
55. Two numbers have for their G. C. M. 101, and for
L. C. M. 27,573. Find the product of these numbers.
MISCELLANEOUS EXAMPLES 307
56. Eesolve 61,776 into its prime factors.
57. Two tracts of land, containing 1225 acres and 1675
acres, are divided into farms each containing the same number
of acres. What is the largest possible acreage of each farm ?
58. Telephone poles are 231 ft. apart. What is the smallest
number of poles which will correspond to an exact number of
half miles ?
FRACTIONS, DECIMALS, AND DENOMINATE NUMBERS
59. Arrange in order of magnitude f , •£, -^.
60. Find the difference between the greatest and the least of
the fractions -f , -f, -|-£, and \\ .
61. Add: 21 31 5f, 3&.
62. Eeduce to its lowest terms iViV
63. Express as decimals -||-, ^, -^f^.
64. Eeduce to common fractions .0375, .0175, .03125.
65. Simplify l x 15|.
66. Simplify 2f of ^| - ( j of 17| of f of 1|).
4¥
67. Eeduce 198 ft. tp the decimal of 1± mi.
68. Eeduce 2° 30' to the decimal of 90°.
69. Eeduce 3 pt. to the decimal of 5 gal.
70. Eeduce .375 of 16s. Sd. -f f of 15s. 6d, to the decimal
of £5.
71. Show that if our calendar had 8 leap years in every
33 yr., it would be more correct than it now is.
72. If our calendar were so arranged that 31 leap years
would occur in 128 yr., how many years would elapse before
the error would amount to 1 da. ?
308
ADVANCED BOOK OF ARITHMETIC
73. The following distances have been run by trains in
the times indicated. Find in each case the rate per hour.
EOUTE
DISTANCE IN
MILES
TIME
Jersey City to Oakland . . .
New York to Chicago . . .
Chicago to New York . . .
London to Aberdeen ....
3311
964
962
539.75
510.1
83 hr. 45 min.
19 hr. 57 min.
17 hr. 45 min.
8 hr. 32 min.
8 hr. 1 min. 7 sec.
Albany to Syracuse ....
Erie to Buffalo Creek ....
Camden to Atlantic City . .
Liberty Park to Absecon . .
Berlin to Absecon
147.84
86
58.3
49.8
35.6
2 hr. 10 min.
1 hr. 10 min. 45 sec.
45 min. 45 sec.
37 min. 30 sec.
25 min. 45 sec.
New York to Philadelphia . .
90
1 hr. 17 min.
74. Find the value of 25,000 bu. of oats at 46f ^ per
bushel.
75. The price of oats in June, 1900, was 26^ per
bushel, and in August it was 21^ per bushel. If a specu-
lator lost f 1050 by buying oats at the former price and
selling at the latter, how many bushels did he buy ?
76. A speculator in Chicago bought 10,000 bu. of corn
in February, 1901, at 38f ^ per bushel, and sold it in
December, 1901, at 69|^ per bushel. Find his profit.*
77. The total number of bales of cotton exported from
the United States for the season of 1901-1902 was
6,715,793, valued at $284,779,190. Find the average
price per bale, correct to the cent.
78. The total number of farms in Alabama is 223,220 ;
the total acreage of these is 20,685,427. Find, correct to
two decimal places, the average number of acres to a farm.
* Allow J $ per bushel brokerage for buying and for selling.
MISCELLANEOUS EXAMPLES 309
79. The total sugar production of California was, in 1902,
356,500 T., valued at $15,500,000. Find the average price
per 100 Ib.
80. According to the census of 1900, the number of persons
employed in manufacturing industries in Florida was 1778,
and the salaries paid amounted to $1,295,139. Find the
average salary received by each person, correct to the cent.
81. The total enrollment in the elementary and secondary
schools in the United States in 1901 was 15,603,451, and the
total number of teachers was 430,004. Find the average num-
ber of pupils to a teacher.
82. The total expenditure for higher education in Canada in
a recent year was $1,014,254. This expenditure was 19.5^
per capita of the total population. Find the population of
Canada.
83. The total expenditure for higher education in Germany
in a recent year was $ 7,450,366. The per capita expenditure
was 14.3^. Find the population of Germany.
The cost of higher education in Great Britain and Ireland
for a recent year was given as $8,353,655. The per capita
expenditure was 21.7 ^. Find the population of Great Britain
and Ireland.
84. The total amount of money in circulation in the German
Empire is, estimated in our currency, $1,080,100,000. The
per capita circulation is $19.53. Find the population of
Germany.
85. Express in feet .002357 of a mile.
86. Find the value of ^- of a ton -f- % of a hundredweight.
Give your answer in pounds.
87. How many times is 12 Ib. 8 oz. contained in 2 T. ?
88. Light travels at the rate of 185,000 rni. a second. How
long does it take a ray of light to pass from the earth to the
moon, a distance of 239,000 mi. ?
310 ADVANCED BOOK OF ARITHMETIC
89. How many cubic yards of sand are required to fill a
street 1^ mi. long, 40 ft. wide, to the depth of 5 in. ?
90. Express |- of a day as a decimal of a common year.
91. If T5g- of an acre of land is worth $ 23, find the value of
85 A. of land.
92. If .375 of an acre of land is worth $ 22, find the value
of 57 A.
93. Multiply 68.4 by .0027, and divide the product by
f of .96.
94. Find the value of .1875 of a guinea + f of £ 1 + .25 of
7s. Sd. Give your answer in pounds, shillings, and pence.
95. Eeduce 12s. 6cL to the decimal of £ 4 sterling.
96. (a) Divide $1293.46 by .00 J. (6) Divide $147.32 by
.00|. (c) Divide $ 3473.85 by .OOf . (d) Divide $ 3295 by .OOf .
97. (a) Divide $1456.77 by .OOf (6) Divide $3947.85
by .OOf.
98. How many acres in a field 160 ch. long, 40 ch. wide ?
99. A wheel is 12^ ft. in circumference. How many revolu-
tions will it make in going 6 mi. 80 rd. ?
100. How many bushels will a bin 7 ft. by 5 ft. and 4 ft.
deep hold?
101. Keduce ij to a fraction having 12 for denominator.
102. .08 of a boy's money is $6. How much money has
the boy ?
103. .875 of a man's property is valued at $21,700. What
is the value of the man's property ?
104. How many acres are in a square field whose side
is 40 rd. ?
105. A and B can mow a field in 7 da. A, B, and C can
mow the same field in 5 da. for $50. What should C
receive ?
106. Write decimally three-eighths of one hundredth, and
reduce it to a simple decimal.
MISCELLANEOUS EXAMPLES 311
107. Reduce |> ^, T\-, and ^ to equivalent fractions
having 100 for denominator.
108. Find the difference between "— - and — — •
16 .16
109. Reduce 16| to an improper fraction having 16 for
a denominator.
110. A rectangular field which is 18 rd. wide contains
6 A. How much will it cost to fence it at 75^ a rod ?
111. What decimal of 4 ft. 2 in. is 9 ft. 6 in. ?
112. Find the least fraction which added to J, |, ^, ^,
and |- will make the sum an integer.
113. Divide 27.8 of a yard by .00125 of a foot.
114. 8 cwt. 20 Ib. of sugar cost $41.42. What will 1
T. cost at the same rate ?
115. Find the least length which is a multiple of 1 ft.
3 in., 1 ft. 8 in., 2 ft. 1 in., and 2 ft. 6 in.
116. Twelve tenths of a number equals 42. Find it.
117. Divide 54,218 by 64, using the factors of 64.
118. A train 165 yd. long passes a telegraph pole in 12
sec. Find the rate of the train in miles per hour.
119. A city lot 42 ft. by 120 ft. is sold for $840. At
this rate, find the value of 1 A. of land in that city.
120. Find the greatest number which, when divided
into 1958 and 2741, will give for remainders 8 and 11
respectively.
121. By buying eggs at 25^ per dozen and selling them
at 60^ a score, a dealer makes a profit of $10.01. How
many eggs does he sell ?
122. Reduce •£$££$ to its lowest terms.
123. If a sum of money which will pay A's wages for
41| da. will pay B's wages for 55| da., for how long will
it pay both ?
312 ADVANCED BOOK OF ARITHMETIC
124. If gold weighs 19.3 times as much as water, and copper
8.9 times as much as water, how much heavier than water is an
alloy consisting of 16 parts of gold and 3 of copper ?
125. A rectangular tank is 18 ft. 8| in. long, 11 ft. 3f in.
wide, and contains 41 cu. yd., 6 cu. ft., and 34^ cu. in. Find
its depth. Find the area of each of its faces.
126. A tennis court is 42 yd. long and 20 yd. wide. It has
a walk around it 6 ft. wide. Find the cost of paving the walk
at $ 1.25 per square yard.
127. Telegraph poles along a certain railroad are 132 ft.
apart. Find the rate of a train, in miles per hour, which passes
18 poles in 24 sec.
LONGITUDE AND TIME
128. A train leaves New York City at 9 A.M., Apr. 1, 1903,
and arrives in Carson City, Nev., in 109 hr. 15 min. Find
the hour of the day, and day of the month, Standard time,
that it reaches its destination.
129. The time of mail transit between Chicago and Santa
Fe, N. M., is 60 hr. 55 min. " The California Limited " leaves
Chicago at 10 P.M. At what time, by the clocks in Santa Fe,
should " The California Limited " pass Santa Fe ?
130. The longitude of Cairo, Egypt, is 31° 21' E., and the
longitude of Savannah, Ga., is 81° 5' 30" W. Find the differ-
ence in time.
131. The longitude of Toulon is 5° 56' E. The time differ-
ence between Toulon and Halifax, N. S., is 4 hr. 38 min. 4 sec.
Find the longitude of Halifax.
132. The time difference between Toulon and Point Barrow,
Alaska, is 10 hr. 48 min. 44 sec. Find the longitude of Point
Barrow.
133. The time difference between Osaka and Point Barrow
is 19 hr. 26 min. 48 sec. Find the longitude of Osaka. (See
previous problem.)
MISCELLANEOUS EXAMPLES 313
134. (a) The difference between Standard and local time of
Portland, Me., is 19 min. Find the longitude of Portland, Me.
(6) The difference between Standard and local time of Fort
Wayne, Ind., is 20 min. Find the longitude of Fort Wayne.
(c) Cleveland, 0., uses Central time ; the difference between
its local and Standard time is 33 min. Find the longitude of
Cleveland, 0.
PERCENTAGE
135. The total sugar production of the world for the year
1902 was 9,635,000 T. The amount of sugar consumed in the
United States the same year was 2,372,000 T. What per cent
of the world's production was the amount consumed in the
United States?
136. The foreign-born population of New Orleans, according
to the census of 1900, was 30,325 ; of this number, 1262 came
from England, 4428 from France, 8733 from Germany, 5398
from Ireland. What per cent of the foreign-born population
of New Orleans came from England? from France? from
Germany ? from Ireland ?
137. The number of Canadians in Detroit, according to the
census of 1900, was 25,400; this number was 26.3% of the
foreign-born population. Find, correct to 100, the number of
foreign-born population of Detroit.
138. A horse is sold, at a loss of 15%, for $127.50. Find
the cost of the horse.
139. By selling silk at $1.60 per yard, a dealer makes a
profit of 25%. What would the selling price be if he made a
profit of 12 \% ?
140. When cloth is sold for $1.04 per yard, a clothier
makes a profit of 30%. What would his profit be if he sold
the cloth at 96^ per yard ?
141. A wholesale dealer makes a profit of 10% on canned
goods. The retail dealer makes a profit of 25%. Find the
original cost of canned goods which cost the consumer $11.
2i
314 ADVANCED BOOK OF ARITHMETIC
142. A coal merchant buys coal by the long ton at $ 4.50 a
ton, and sells it at the rate of $ 5 a short ton. Find his gain
per cent.
143. How much water must be added to a 25% wine mixture
to make it a 20% mixture?
144. A sells goods to B at a profit of 20% ; B sells them to
C at a profit of 20% on his outlay; C sells them to D for
$ 180, thereby losing 16|%. How much did the goods cost A ?
145. A merchant buys goods at 20%, and 10% off list price,
and sells them at the list price. Find his per cent of gain.
146. When 20 Ib. of tea are sold for what 22^ Ib. cost,
what is the gain per cent ?
147. (a) A vessel contains 31 gal. of wine and 17 gal. of
water. What per cent of the mixture is wine and what per
cent is water ? (5) How many gallons of water must be added
to this mixture to make a mixture containing 60% wine?
148. The following table gives the distances from Atlantic
to Pacific ports by the present routes :
New York to San Francisco 13,244 mi., nautical
New York to Sydney 14,560 mi., nautical
Charleston to San Francisco 13,180 mi., nautical
Charleston to Valparaiso 8,296 mi., nautical
New Orleans to San Francisco .... 13,644 mi., nautical
New Orleans to Melbourne 15,535 mi., nautical
Galveston to San Francisco 13,826 mi., nautical
Galveston to Wellington 14,182 mi., nautical
Liverpool to San Francisco 13,844 mi., nautical
Hamburg to Callao . 10,702 mi., nautical
Bordeaux to San Francisco 13,691 mi., nautical
The following table gives the distances from Atlantic to
Pacific ports via the Panama Canal route :
New York to San Francisco 5299 mi., nautical
New York to Sydney 9852 mi., nautical
Charleston to San Francisco 4898 ini., nautical
MISCELLANEOUS EXAMPLES 315
Charleston to Valparaiso 4229 mi., nautical
New Orleans to San Francisco .... 4698 mi., nautical
New Orleans to Melbourne 9826 mi., nautical
Galveston to San Francisco 4800 mi., nautical
Galveston to Wellington 8392 mi., nautical
Liverpool to San Francisco 8038 mi., nautical
Hamburg to Callao 6527 mi., nautical
Bordeaux to San Francisco 7938 mi., nautical
What per cents of the distances by the old routes are saved
by the Panama Canal route ?
INTEREST
149. Find the simple interest on $78 for 93 da. at 8%.
150. Find the simple interest on $98 for 63 da. at 7%.
151. Find the amount of $179 for 123 da. at 6%.
152. Find the simple interest on £ 324 7s. 9d. from June 12
to Dec. 7 following at 5%.
153. Find the simple interest on £1169 6s. Sd. from Jan.
25 to June 18 following at 9%.
154. What principal will produce $19.50 in 1 yr. at 6±% ?
155. What principal will produce $180 interest in 3 mo.
at 5% ?
156. What principal will amount to $ 412.50 in 7| mo. at 5 % ?
157. What principal will amount to $ 1219 in 3 mo. 5 da.
at 6% ?
158. What principal will produce $29.17 in 5 mo. at 7% ?
159. At what rate will $1000 produce $23.33 interest in
4 mo. ?
160. What principal will produce 75 ^ interest in 9 da. at 6 % ?
161. Find the exact interest on $73.15 from June 18 to
Aug. 1 at 1%.
162. A note for $3500 bearing interest at 8% and dated
Jan. 2, 1900, was indorsed as follows: June 1, 1900, $450;
316 ADVANCED BOOK OF ARITHMETIC
Aug. 2, 1900, $208; Jan. 2, 1901, $500; July 7, 1901,
Oct. 4, 1901, $500; Jan. 11, 1902, $300; Aug. 4, 1902, $700.
Calculate, by the United States Rule for partial payments, the
amount due on this note on Jan. 1, 1903.
163. A demand note dated Jan. 5, 1902, and drawn for
$575 was paid 6 mo. 18 da. later. Find the date of payment
and the amount of the note, the rate of interest being 7%.
BANK DISCOUNT
164. A note for 60 da. is drawn on Jan. 10, 1903. Find the
proceeds of this note, if its face is $ 150, the date of discount
Feb. 5, and the rate 6%. (Neglect days of grace.)
165. A note for $900, dated Mobile, Ala., Jan. 8, 1903, and
drawn for 90 da., is discounted March 1. Find the proceeds.
166. A 60-day note bearing interest at 8%, drawn Feb. 1,
1903, for $1000, is discounted Feb. 27 at 9%. Find the pro-
ceeds of this note.
167. A demand note was drawn Oct. 1, 1902, for $ 800, and
paid 5 mo. 10 da. later. Find the date of payment and the
amount of the note; rate of interest, 1%.
168. The proceeds of a note is $ 450 when the term of dis-
count is 93 da., and the rate of interest is 8%. What is the
maturity value of the note ?
MENSURATION
169. Find the area of a parallelogram if its base is 100 yd.
and its altitude is 75 yd.
170. Find the area of. a trapezoid if its parallel sides are 60
yd. and 80 yd., and its altitude is 50 yd.
171. A tract of land is sold for $ 3943.84 ; the land cost as
many dollars per acre as there were acres in the tract. Find
the cost per acre.
172. Find the number of square yards in the walls and ceil-
ing of a room 36 by 23, and 16 ft. high.
MISCELLANEOUS EXAMPLES 317
173. Find the perimeter of a square which contains 40 A.
174. A tract of land in the shape of a rectangle contains
320 A. ; its length is twice its width. Find its dimensions.
175. Find the area of an equilateral triangle one side of
which is 100 ft.
176. Find the area of a regular hexagon each side of
which is 50 ft.
177. Find the circumference of a circle whose radius is
56.5 in.
178. Find the area of a circle if its diameter is 20 in.
179. Find the surface of a sphere whose diameter is 20 in.
180. Find the volume of a cube one of whose dimen-
sions is 1 ft. 3 in.
181. The surface of a cube is 221.0694 sq. in. Find the
length of one edge of this cube.
COMPARISON OF PRICES
Express :
182. 6 francs per kilogram as dollars per pound Avoir-
dupois.
183. 5 francs per meter as dollars per yard.
184. 1.369 francs per liter as dollars per gallon.
185. 9 francs per hektoliter as dollars per bushel.
186. 7 marks per kilogram as dollars per pound Avoidu-
pois.
187. 4 marks per meter as dollars per yard.
188. 2 marks per liter as dollars per U. S. gallon.
189. 9 marks per hektoliter as dollars per bushel.
190. 13.785 marks per meter as dollars per yard,
191. $40 per acre as francs per hektar.
MISCELLANEOUS EXAMPLES (B)
(TAKEN FROM VARIOUS EXAMINATION PAPERS)
1. What fractional part of f of a gallon is -£-% of a pint ?
2. The difference in time between two places is 2 hr. 33 min. -
Find the difference in longitude.
3. A bicycle wheel measuring 88 in. in circumference r^ust
make how many revolutions a minute to run eighteen mileL ail
hour ?
4. A coal bin 16^- ft. long and 8 ft. 9 in. wide must be how
deep to contain 10 T. of coal, if one ton of coal occupy 40 cu.
ft. of space ?
5. Eeduce 2 yr. 21 da. to years and decimals of a year.
6. Eeduce .09625 bbl. to integers of lower denominations.
7. Find the value of a piece of land 64 ch. by 13^ ch. at
$48^ an acre.
8. Required the cost of 18 2^ in. plank 16 ft. long and
10 in. wide, and 33 pieces of scantling 2 in. by 4 in. 16 ft. long,
at $ 22 per M, board measure.
9. The average yield per bushel of wheat is 14 bu. 1 pk.
What will 7 bu. 3 pk. 2 qt. yield ?
10. What is the difference in weight, expressed in Avoirdu-
pois pounds, between 100 Ibs. Troy and 100 Ibs. Avoirdupois ?
11. Eeduce to simplest form 3_i-i-j — 2 — .
7-1 2 + *
12. If it cost $510 to fence a rectangular field 98 rd. by
72 rd., what will it cost to fence a square field of the same
area?
13. Express f as a decimal fraction.
318
MISCELLANEOUS EXAMPLES 319
14. What is the ratio of 32 ft. to 6 yd. ? Express the result
decimally.
15. What is the length of a plank 11 in. thick, 1 ft. 6 in.
wide, containing 36 board feet ?
16. When it is 12 M. in New York City (74° W.), what is
the time in Manila (120° E.) ?
17. If the value of | of f of an estate is $ 4500, what is the
value of T\ of ^ of it ?
18. At $ 16 per M, board measure, find the cost of 20 plank
2 in. by 8 in. 18 ft. long, and 30 plank 1-J in. by 6 in. 10 ft.
long.
19. A can do a piece of work in 6 da. and B can do the same
work in 8 da. How long will it take B to finish after they
have worked together two days ?
20. 20f is the product of three factors. Two of these factors
are If and 4|. Find the other factor.
21. How many bushels of wheat will a box 6 ft. by 3± ft. by
2 ft. 8 in. hold ?
22. How many yards are in .04675 mi. ?
23. If the dividend is 807 and the quotient 34^, what is the
divisor ?
24. How many rods of fence will inclose a square field
whose area is 20 acres ?
25. Coal sells at $ 5.75 per ton. What will be the cost of
2315 Ib. at this rate ?
26. How many gallons of water will a tank 5 ft. by 2 ft. by
2 ft. hold ?
27. What is the length of one side of a square piece of land
whose area is 538,756 sq. rd. ?
28. A room is 27 ft. by 22 ft. 6 in. How many yards of
carpet 27 in. wide will be required to carpet this room ?
29. A man is hired to dig a cellar 20 ft. by 15 ft. by 5 ft.
How much money will he receive at 30^ per cu. yd. ?
320 ADVANCED BOOK OF ARITHMETIC
30. How many days are there between Aug. 14 and Dec. 29 ?
31. Find the value of a car load of wheat, estimated at
21,643 lb., at 92^ per bushel.
32. Two persons travel in opposite directions from the same
point at the rate of 4^ and 7f mi. per hour, respectively. How
far apart are they after traveling 37 J hrs. ?
33. A man was born Nov. 22, 1861. What is his age to-day ?
34. Factor the following numbers and from these factors
determine the G. C. M. : 42, 112, 140, 308.
35. What will 75 boards 2 in. by 4 in. by 16 ft. long cost at
$ 12 per M board measure ?
36. 160 rd. of fence will inclose how many acres in the form
of a square ?
37. The difference in longitude between two places is 7° 42'
30". Find the difference in time.
38. How wide is a rectangular field containing 5 A., the
length of the field being 7 ch. 25 1. ?
39. A pavement is 5^ rd. long and 8 ft. 6 in. wide. What
did it cost at $ 1.40 per sq. yd. ?
40. Three men, A, B, and C do a piece of work ; A works
3 da. of 5 hr. each, B, 2 da. of 6 hr. each, and C, 7 da. of 3 hr.
each. At the same rate of wages, how should they divide
$ 120, the total amount received for doing the work ?
41. A miller charges y1^- for toll. How many bushels of
wheat must one take to mill to get 12 bbl. of flour, each con-
taining 196 lb., if a bushel of wheat makes 40 lb. of flour ?
42. The annual rainfall in a certain locality is 30 in. How
many tons of water fall on an acre of land in this locality, if a
cubic foot of water weighs 1000 oz. ?
43. How much does a man gain or lose on the sale of two
houses at $ 1200 each, if he gains ^ of the cost price on one,
and loses ^ of the cost price on the other ?
MISCELLANEOUS EXAMPLES 321
44. What is the ratio of 7 Ib. Troy weight to 10 oz. Avoir-
dupois ?
45. The divisor is 357, the quotient is 6f ; what is the divi-
dend?
46. A farmer had 28 A. of land left after selling \ of his
farm to one neighbor, f of it to another, and f of the remainder
to another. How large was his farm ?
47. Multiply 8.035 by .0035, add 3, and divide the sum by
.000625.
48. Divide $459.25 into three parts that shall be to one
another as f, f, and 3 respectively.
49. When it is two o'clock P.M. in Jerusalem, what is the
time in Cincinnati? The longitude of Jerusalem is 35° 12'
E., and of Cincinnati, 84° 26' W.
50. Find the exact number of days between Dec. 23, 1902,
and to-day.
51. A man's farm is mortgaged for f of its cost ; he sells it
for $6000 which is 25% above its cost. How much money
will he have after paying the mortgage ?
52. A note for $ 600, dated Oct. 24, 1902, and due in 8 mo.,
with interest at 6% per annum, is discounted at bank Dec. 20,
1902. Find the proceeds.
53. A man sold two lots each for $ 600, gaining 20% on one,
and losing 20 % on the other. What was his gain or loss ?
54. A man buys a book the list price of which is $ 7.20, at a
discount of 16|%, and sells it for $7.50. What is his gain
per cent ?
55. What principal at interest for 1 yr. 3 mo. will amount
to $506, the rate of interest being 8% per annum?
56. Twelve per cent of 90 is what per cent of 100 ?
57. At the following rates per annum of simple interest,
what time is required for the accruing interest to equal the
principal: 6%, 8%, 9^% ?
322 ADVANCED BOOK OF ARITHMETIC
58. What is the exact interest on $10,000 from Jan. 18,
1903, to May 6, 1904, at 3±% ?
59. A 30-day note, without interest, is discounted at a bank
at 8% for $350. What is the face of the note ?
60. Bonds bearing 5% interest are bought at 120. What is
the rate of income on these bonds ?
61. An agent buys sugar at 4|^ per pound ; his commission
at |-% is $25. How many pounds of sugar does he buy ?
62. The discount of a note, discounted at bank, for 3 mo. 18
da. at 5% is $4.20. Find the proceeds.
63. What single discount is equivalent to trade discounts of
25%, 10%, and 5% on the list price of an article ?
64. The property in a school district is assessed at $ 196,000.
What rate of taxation would be required to provide about
$ 800 annually for the improved maintenance of the schools ?
What annual tax would a man pay on this account whose prop-
erty is assessed at $ 1200 ?
65. A man sold two horses at $ 80 each. On one he gained
20%, on the other he lost 20%. Find the gain or loss.
66. What must I ask for an article worth $36 that, after
falling 20%, I may gain 25% on the value ?
67. A school district advertised for bids to build a school-
house, the lowest bid being $21,049. If it costs 3% to collect
the money, how large a levy should be made, supposing 29%
of it to be non-collectible ?
68. WThat must a man pay for 4% stock to get 5% on his
investment ?
69. If you buy United States 3's at 110, what per cent per
annum would your investment pay ?
70. A merchant's expenses average 10% of his sales. At
what per cent advance on cost must he sell his goods to clear
20% profit?
MISCELLANEOUS EXAMPLES 323
71. A merchant sold goods to the amount of $ 760.95, thereby
losing 11%. What did he pay for the goods ?
72. A ship is insured for half its value for $374. If the
rate is 2f %, what is the value of the ship ?
73. A carriage dealer sold 16 buggies at $ 200 each ; on one
half he gained 10%, and on the other half he lost 10%. Find
his net gain or loss.
74. What principal will amount to $1253.86 in 2 yr. 11 mo.
13 da., interest at 5% ?
75. How do you find the rate per cent per annum when the
principal, interest, and time are given ?
76. How do you find the principal when the rate per cent
per annum, time, and interest are given ?
77. How do you find the time when the principal, rate per
cent per annum, and interest are given ?
78. The list price of office desks is $ 15, but 12 desks are sold
for $ 126. What rate of discount is allowed ?
79. In a certain time $650 will amount to $713.05 at 6%
simple interest. Find the time.
80. A note for $ 500, due in 3 months, is discounted at bank
at 6%. Find the proceeds.
81. 2361 is what per cent of 78£ ?
82. A man insures his life, paying a premium of $ 28, which
is at the rate of -|% on the amount of his insurance. Find the
face of the policy.
83. If 25% of the selling price of an article is profit, what
is the per cent of gain on its cost ?
84. A man fails in business; his assets amount to $2100,
his liabilities to $ 6000. What per cent will his creditors
receive ?
85. What is the interest on $ 475 for 1 yr. 3 mo. 24 da. at
324 ADVANCED BOOK OF ARITHMETIC
86. A man bought four loads of hay, each weighing 2750 lb.,
at $ 20 per ton ; he gave in payment his note, without interest,
at 60 da. What are the proceeds of this note, discounted at
a bank at 6% ?
87. What per cent of -J- is -|- ?
88. An agent's commissions at 5% amount to $37.65. Find
the amount of his sales.
89. The tax on property assessed at $ 8500 is $ 48.37. What
is the rate on $ 1000 ?
90. Find the date of maturity of a note made and dated
Sept. 11, 1902, and payable 90 da. after date.
91. Find the cost of 87 shares of stock at 76J, brokerage |
per cent.
92. A New York sight draft was sold in Atlanta, Ga., for
$3542, exchange being at f% premium. What was the face
of the draft ?
93. What per cent of 5 lb. is 3 oz. Avoirdupois ?
94. An agent's commission for renting a house is $13.25;
his rate of commission is 2^%. What is the yearly rent of the
house ?
95. A man pays a premium of $ 150 for insuring his house
for -| of its value ; the rate of premium is 1^ per cent per
annum. What is the value of the house ?
96. A building worth $6000 is insured for f of its value
at 75 ^ on the $ 100. In case of the destruction of the building
by fire, what will be the owner's loss, including premium ?
97. What per cent of 1 bu. is 3 qt. ?
98. A merchant can buy flour on six months' credit at $ 8
per barrel, or for cash at $ 7.50 per barrel. He buys 100 bbl.,
paying cash, but borrows the money at 8% to pay for it. Is
this better than to buy on credit, and how much better ?
99. A man sells 16 shares of bank stock at 127f, brokerage
i-%. How much does he receive for his stock ?
S EXAMPLES
325
ocks at 20 % premium and
discount. What per cent
>
^s books show sales during
me montn ot March amounting to f 1000. One half of
his sales are at a profit of 25 % on the cost, and the other
half a loss of 16| % on the cost. Find the cost of the
gcods sold during the month.
102. A merchant failing in business paid his creditors
$3874.75, which was at the rate of 55^ on every dollar of
his indebtedness. Find his indebtedness.
103. The list price of a mower is $38 ; the retail dealer
is allowed discounts of 20%, 5%, and 3%. What does
he pay for mowers ? If the retailer sells these mowers at
a profit of 50 %, what does the farmer pay for these
mowers ?
104. A certain stock, selling at 121|, pays a semiannual
dividend of 4%. What is the rate per cent per annum
on an investment in this stock ?
TABLES
APOTHECARIES' WEIGHT
20 grains (gr.) = 1 scruple (3)
3 scruples = 1 dram (3)
8 drams = 1 ounce ( § )
12 ounces = 1 pound (fib)
LIQUID MEASURE
4 gills (gi.) = 1 pint (pt.)
2 pints = 1 quart (qt.)
4 quarts = 1 gallon (gal.)
31| gallons = 1 barrel (bbl.)
2 barrels = 1 hogshead (hhd.)
LONG MEASURE
12 inches (in.) =1 foot (ft.)
3 feet = 1 yard (yd.)
5^ yards = 1 rod (rd.) , or pole
40 rods = 1 furlong
8 furlongs =1 mile (mi.)
TROY WEIGHT
24 grains (gr.) = 1 pennyweight
(pwt.)
20 pennyweights = 1 ounce (oz.)
12 ounces =1 pound (Ib.)
326
ADVANCED BOO
DRY MEASURE
2 pints (pt.)
8 quarts
4 pecks
= 1 quart (qt.)
= 1 peck (pk.)
= 1 bushel (bu.)
NUMERICAL MEASURE
12 articles = 1 dozen
12 dozen = 1 gross
12 gross = 1 great gross
20 articles = 1 score
AVOIRDUPOIS WEIGHT
16 drams (dr.) = 1 ounce (oz.)
16 ounces = 1 pound (Ib.)
25 pounds
100 pounds
CIRCULAR ivu&Asuivr,
60 seconds (") = 1 minute (')
60 minutes = 1 degree (°)
30 degrees
12 signs
= 1 sign (S.)
= 1 circle (C.) or
circumference
= 1 circumference
20 cwt.
2240 pounds
= 1 quarter
= 1 hundredweight
(cwt.)
= 1 ton (T.)
= 1 long ton
360 degrees
NAUTICAL MEASURE
6 feet = 1 fathom
608 feet = 1 cable length
10 cable lengths = 1 nautical mile
(6080 feet)
The following denominations are also used :
1.152 statute miles = 1 geographic mile, or knot
3 geographic miles = 1 league
60 geographic miles, or )
69.1 statute miles } = 1 Ae^* °f latltude on a meridlan
360 degrees = the circumference of the earth
4 inches
= 1 hand
SURVEYORS' AND LAND MEASURE
9 inches
= 1 span
7.92 inches = 1 link (1.)
21.888 inches
= 1 sacred cubit
25 links = 1 rod
3 feet
= 1 pace
4 rods = 1 chain (ch.)
TIME
MEASURE
10 square chains = 1 acre
60 seconds (sec.
) = lminute(min.)
640 acres = 1 square mile
60 minutes
= 1 hour (hr.)
625 square links (sq. 1.) = 1 pole (P.)
24 hours
= 1 day (da.)
16 poles = 1 square
7 days
= 1 week (wk.)
chain
4 weeks
= 1 lunar month
30 days
= 1 commercial
APOTHECARIES' FLUID MEASURE
month
60 minims (m.) = 1 fluidrachm(f 3)
12 months
= 1 year
8 fluidrachms — 1 fluidounce(f J )
365 days
= 1 common year
16 fluidounces = 1 pint (O)
366 days
= 1 leap year
8 pints = 1 gallon (Cong.)
APPENDIX 327
CUBIC MEASURE
1728 cubic inches (cu. in.) — 1 cubic foot (cu. ft.)
27 cubic feet = 1 cubic yard (cu. yd.)
16 cubic feet, or i , , A , A \
> = 1 cord of wood (cd.)
8 cord feet /
24} cubic feet = 1 perch of stone or masonry (pch.)
SQUARE MEASURE
144 square inches (sq. in.) = 1 square foot (sq. ft.)
9 square feet — 1 square yard (sq. yd.)
30£ square yards = 1 square rod or perch (sq. rd. or sq. pch.)
40 square rods = 1 square rood (sq. R.)
4 roods = 1 acre (A.)
640 acres — 1 square mile (sq. mi.)
SPANISH LAND MEASURE
In Texas, California, New Mexico, and other parts of this
country which were formerly parts of the Spanish empire, the
vara, the unit of linear measure, is still used in connection with
original grants of land. In Texas, the value of the vara is
33^ in. In California and New Mexico it is usually considered
33 in.
1,000,000 square varas = 1 labor = 177.136 acres
25,000,000 square varas = 1 league — 4428.4 acres
3,612,800 square varas — 1 square mile = 640 acres
1,806,400 square varas = J square mile = 320 acres
903,200 square varas = J square mile = 160 acres
451,600 square varas = J square mile = 80 acres
225,800 square varas = TJ? square mile = 40 acres
5645 square varas = 1 acre
ANSWERS
Exercise 4. — North Atlantic Division, 5,499,620, 3,843,908, 2,866,074,
$ 105,332,839. South Atlantic Division, 3,521,920, 2,324,906, 1,503,917,
$15,907,956. South Central Division, 5,002,836, 3,235,121, 2,074,304,
$19,870,733. North Central Division, 7,878,448, 5,895,631, 4,188,517,
$ 107,663,687. Western Division, 1,125,924, 956,472, 685,444, $24,441,012.
Totals, 23,028,748, 16,256,038, 11,318,256, $273,216,227.
Exercises. — 1. $2,187,217.51. 2. $2,749,532.73. 3. $8,593,793.27.
4. $733,501.99. 5. $4,330,637.88. 6. $9,145,702.01. 7. $773,768.16.
8. $6,124,815.32. 9. $7,750,185.14. 10. $3,429,967. 11. $2,001,002.77.
12. $3,135,817.92. 13. $10,029,526.43. 14. $659,959.65.
15. $2,004,181.95. 16. $853,917.00.
Exercise 6. — 1. $ 75, $ 200, $ 900, $ 1225. 2. $ 2.08, $ 4.68, $ 24.44.
3. 100,320ft.; 401,280ft. 4. 273 da. 5. $6084. 6. 2136 hr.
7. 1598 mi. 8. 62,720 A. 9. 1173 mi. 10. 487,956 Ib. 11. $1008.
12. 85,008 cu. in. 13. 103,680 pages. 14. $956.80. 15. $1736.
Exercise 7.— 1. 9000 sq. yd. 2. 7056 sq. yd. 3. 102,400 sq. rd.
4. 11, 760 sq. yd. 5. 22,848 sq. yd. 6. 167,200 sq. yd. 7. 4725 sq. rd.
8. 20,160ft. 9. 252 sq. in. 10. 36 sq. mi. 11. 432 sq. mi.
12. 40,480 sq. yd. 13. 2200 sq. yd. 14. 5332 sq. ft. 15. 2880 sq. in.
16. 1296 sq. in.
Exercise 8. — 1. $639.52. 2. $225.40. 3. $476.94. 4. $898.03.
5. $3594.51. 6. $5455.08. 7. $8023.18. 8. $2136.42. 9. $5046.93.
10. $2285.44. 11. $359.31. 12. $555.03. 13. $239.76. 14. $1076.28.
15. $1866.95. 16. $6399.52. 17. $7402.50. 18. $10,986.58. 19. $9345.
20. $6594.75. 21. $6498.38. 22. $3269.10. 23. $93.60. 24. $159.60.
25. $187.65. 26. $322.56. 27. $195.44. 28. $7.02. 29. $19.50.
30! $6075.
Exercise 9. — 1. 841.75 mi. 2. 614.2 mi. 3. 809.93 mi. 4. 710.4 mi.
5. 1401. 56 mi. 6. 1280.2 mi. 7. 754.8 mi. 8. 1032.3 mi. 9. 1191.4mi.
10. 1,879.6 mi. 11. 1742.7 mi. 12. 1894.4 mi. 13. 2,217,648.
14. 2,230,095. 15. 4,821,600. 16. 1,380,768. 17. 2,423,546.
18. 1,881,250. 19. 4,167,520. 20. 6,297,900.
329
330 ANSWERS
Exercise 10.— 1. 4834, rem. 10. 2. 4408, rem. 14. 3. 4892, rem. 14.
4. 4416, rem. 2. 5. 4166, rem. 16. 6. 2957, rem. 23. 7. 2485, rem. 13.
8. 2668. 9. 2760, rem. 32. 10. 1706, rem. 28. 11. 1234, rem. 39.
12. 2380, rem. 28. 13. 1431, rem. 10. 14. 2519, rem. 25. 15. 17,530,
rem. 43. 16. 14,616, rem. 31. 17. 14,792, rem. 22. 18. 8261, rem. 35.
19. 8741, rem. 42. 20. 3377, rem. 42. 21. 2133, rem. 74. 22. 8214,
rem. 4. 23. 9701, rem. 37. 24. 8217, rem. 3.
Exercise 11. — 1. 14 Ib. 2. 29 hr. 3. 375 pk. 4. 29 hr., 4 mi.
rem. ; 94 mi. 5. 384 T. 6. 179 da. 7. 225 da. 8. 2083, 4 in. rem.
9. 52 wk. 10. 14 doz. 11. 16. 12. 37 boxes. 13. 417 bbl.
14. 24 hr. 15. $ 125. 16. 161 sheep. 17. 159 coins. 18. 45 hogs.
19. 245 boxes. 20. 167 bbl. 21. 19 Ib. 22. 36 horses.
Exercise 12. —1. 42 bags. 2. 39 bbl. 3. 79 bbl. 4. 2777yd.,
28 in. rem. 5. 241 bu. 6. 125 da. 7. 36°. 8. 364 sq. yd.
9. 36 gal. 10. 33 chests. 11. 325 bbl. 12. 114 hr. 13. 1000 A.
14. 1250 sq. mi.
Exercise 13. — 1. 68. 2. 69. 3. 427. 4. 5320. 5. 245.
6. 385. 7. 198. 8. 17. 9. 2125. 10. 717 sacks. 11. 810.
12. 563 bbl. 13. 17,503. 14. 396. 15. 39.
Exercise 14. — 1. 196.2. 2. 629.8. 3. 167.6. 4. 189.8. 5.289.6.
6. 303.8. 7. 324.8. 8. 445.2. 9. 16.3. 10. 95.6. 11. 356.1.
13. .5, .75, .8, .3125, .24, .21875, .171875, .6171875. 14. .85, .38, .1025,
.415, .536, .348. 15. 2875, .425, .45, .74, .206, .02725. 16. .333, .1429,
.111, .0909, .0833, .0769, .0714, .0667, .0588. 17. .1852, .4783, .3214,
.1290, .4571, .0959.
Exercise 15.— 1. $58.48. 2. $136.15. 3. $ 16.35. 4. $31.42.
Exercise 16. — 1. $1. 2. 42ft. 3. 5 da. 4. 10 da, 5. 60 Ib.
6. $24, 5 sheep. 7. 60^ on each kind, 47 oranges. 8. 60^. 9. 60 sec.
10. 7ft. 6 in.
Exercise 18. — 1. 8. 2. 8. 3. 9. 4. 12. 5. 15. 6. 15. 7. 18*
8. 36. 9. 12. 10. 12. 11. 7. 12. 13. 13. 19. 14. 23. 15. 24.
16. 28. 17. 10. 18. 15. 19. 12. 20. 13. 21. 12. 22. 18. 23. 16.
24. 21. 25. 32. 26. 36. 27. 44. 28. 63. 29. 48. 30. 54. 31. 56.
32. 84.
Exercise 19.— 1. 80. 2. 42. 3. 180. 4. 90. 5. 147. 6. 140.
7. 168. 8. 108. 9. 102. 10. 144. 11. 480. 12. 420. 13. 160.
14. 78. 15. 336. 16. 480. 17. 756. 18. 385. 19. 270. 20. 216.
21. 375. 22. 216. 23. 180. 24. 576. 25. 180. 26. 210. 27. 210.
28. 420. 29. 16,800. 30. 720. 31. 300. 32. 180. 33. 630. 34. 280.
35. 144. 36. 840. 37. 600. 38. 2002. 39. 180. 40. 210. 41. 240.
42. 420. 43. 108. 44. 720. 45. 360.
ANSWERS 331
Exercise 21. — 1. |. 2. y. 3. y. 4. -V5. 5. -*f-. 6. -VT3-. 7. Y-
15. -3T°T2- 16- W- 17- ^*-. 18. V- I9- -T¥- 20. ^. 21. 1J*.
22. -if*. 23. I}*. 24. *fi. 25. -3T2^. 26. J^. 27. 1^. 28. *f*.
29. -Vr8- 30. -3TV.
Exercise 22. - 1. f , j§, J|, if It- 2. f , Jf , if, If, «• 3. T%, JJ,
if, H, «• 4, A, i2, J}, H, jj. 5. I?, if, f5, fj, f|. 6. H, if, J}, »},
If J' if «, if *f M- 8- 18. if ?f if tt^ »- if if *f I!' if
Exercise 23. —1. J, f |, |, f, f, f, f, f. 2. |, f, |, f
3. f , f f , f, f, f, f f - 4. f, f, f, A,. A, Ai A, A- 5.
T96, i & A-
Exercise 24.— 1. lrV 2. 1JJ. 3. 1TV 4. 1}. 5. If.
6. 1J. 7. f. 8. 1TV 9. If. 10. If. 11. If. 12. 1J.
13. If. 14. 1J}. 15. 1}J. 16. 2-}i. 17. 1A- 18- 2A-
19. 2^. 20. 1}}. 21. 2T%. 22. 1}§. 23. 1T%. 24. 1}}.
25. |f. 26. 2 A- 27. 1J}. 28. 1J}. 29. 2J. 30. 1J}.
31. 7A- 32. 15J. 33. 8f. 34. 25^. 35. 26}J. 36. 22f
37. 17f 38. 12J}. 39. 19^. 40. 17}. 41. 25^. 42. 14}|.
43. $ A- 44. 1-A hr. 45. f is largest ; § is smallest.
Exercise 25.— 1. J. 2. J. 3. }. 4. }. 5. |. 6. }.
7. ^. 8. }. 9. J. 10. A- 11- i- 12- }• 13- 1}.
14. 2i.
20. 11 A-
26. 16}.
32. 3i|. 33.
38. 4|.
44. $2}. 45.
Exercise 26.
7. 27. 8. 55. 9. 3. 10. 5. 11. 2. 12. 4. 13. 20.
14. 2. 16. 24 cows. 17. 42 doz. 18. 15 da. 19. 22 hr.
20. 200.
Exercise 2 7. — 1. 12. 2. 10}. 3. 5f 4. 16J. 5. 11}.
6. 4}. 7. 11}. 8. 24}. 9. 5f ^ 10. 12}. 11. 19}.
12. 38}. 13. 35}. 14. 47*. 15. 38J. 16. 15}. 17. 16f
18. 22}. 19. 33}. 20. 301}. 21. 247}. 22. 152. 23. 335}.
24. 247}. 25. 243. 26. 160A- 27. 753}. 28. 705}.
29. 446J. 30. 166}. 31. 221|. 32. 132|. 33. $4}.
34. $42}. 35. $15.23TV 36. 52} j*. 37. $129.76}. 38. $31.87}.
39. $3.84f. 40. $35,40, 41. |7.81J. 42. 4840 sq. yd. 43. 792 in.
if. 16. IA-
51. 6TV • 22. 7}.
'. 6}. 28. 2&-
17. 10}. 18. 12}.
23. 7A- 24. 3}.
29. 11}}. 30. 10&.
19. 8f.
25. 51.
31. 2f
\. 3}}. 34. A
35. 6 A
36. 2A-
37. 6A-
4f 40. 8H-
41. 19f
42. ?A.
43. 5f
i. 2i}. 46. $1.15.
1.— 1. 1. 2. 162.
3. 4.
4. 35. 5. 27.
6. 6.
332 ANSWERS
44. $11.90. 45. 1036 J mi.; 2764* mi. 46. $7350. 47. 1792 Ib.
copper, 448 Ib. lead. 48. Gain $805, investment at end of year $3105.
Exercise
28. — 1.
J- 2
• J-
3. T«r.
4. «
5- iVo-
6.
»
7.
ft.
8
. f.
9. f.
10.
A«
11. f.
12. f.
13.
A-
14.
i-
15. A-
16. f.
17.
TV
18. TV
19- rVV
20.
i-
21.
2}-
s
52. 3f.
23. f.
24. f.
25. f.
26. 7.
27.
9.
28.
lOf.
29. 15.
30.
8J.
31.
22.
32.
26H-
33.
1-
34.
14A-
35. 64
|. 36. 1.
37.
420£.
38. 311.
39.
7.
40.
2.
41. 272*J$. 42. $398|f.
43. $35241. 44.
$ 1.9
3|.
45.
$3.65f. 46. $162.50.
47.
, $52J.
Exercise
29. —1
. A-
2. £
*•
3. If.
4. T*8-
5.
1-
6.
&
7. A-
8- «-
9.
10. A-
11. f-
12.
13.
A'
14. A-
15.
1A.
16.
A-
17.
if.
18.
3i.
19.
*V
20. A-
21.
1-
22.
A-
23.
7%.
24. 2
25.
18.
26. A-
27.
A-
28.
49.
29.
T43-
30.
2f
31.
28.
32. A-
33.
iff.
34.
21.
35.
A-
36. ^
IT?-
37.
28.
38. &.
39.
2A-
40
, 30.
41.
A
42.
/f
43.
321.
44. A-
45. 4f.
46,
, 28.
47
. f.
48.
T9-
49.
601.
50. A-
51.
A-
52.
96.
53.
17-
54.
A
55.
671.
56. A-
57
. 4.
58.
49^.
59.
f.
60. 7
61.
2.
62. f.
63.
i-
64. 1J.
65.
I Jf.
66.
H-
67.
51.
68. |f.
69.
ft. 70. 2f. 71. A- 72.
If
Exercise
30. — 9.
4. 10. 9.
11
. 16.
12.
5}.
13.
25.
14.
81.
15. 9.
16. |.
17.
1. 18. 11.
19. li.
20. i
If-
21.
If
22. JW-
23.
1-
24. 5
2A- '
25.
*^
26. 1
H-
27. IfJ-. 28. 1TV 29. TV 30. 1J. 31. 6. 32.
Exercise 31. — 1. $4290. 2. $2|. 3. $216.75. 4. $137f
5. $630. 6. $575. 7. $24.70. 8. $68.88. 9. $881|. 10. $7.
11. $4.25. 12. $8.41. 13. $7.72f 14. $5.75. 15. $13J.
16. $120. 17. 108ft. 18. 72. 19. 255 T. 20. 60.
Exercise 33. — 1. 666.973. 2. 151.987. 3. 261.304. 4. 1943.232.
5. 135.148. 6. 160.687. 7. 282.238. 8. 317.187. 9. 176.483.
10. 212.728. 11. 281.308. 12. 377.077.
Exercise 35. — 1. $4.32, $13.70, $43.20, $10.08. 2. $3.105, $22.59,
$6.064, $5.929. 3. $11.90, $34.74, $40.992, $32.85. 4. $10.10, $40.50,
$180.30, $24.71. 5. $430, $693.70, $337.80. 6. 31.5 A., 4 A., 30 A.
7. .01, .12, .09, .002, .012. 8. 502.2. 9. 2123.2. 10. 1806.42.
11. 432.036. 12. .426. 13. 2.625. 14. 28.56. 15. 271.9846.
16. .56056. 17. 249.048. 18. .1702. 19. .4182. 20. .6. 21. 3.3642.
22. 17.328. 23. .20001. 24. .9409. 25. .4624. 26. .139129.
ANSWERS 333
27. .811801. 28. .644809. 29. .480249. 30. .001. 31. .027.
32. .064. 33. .343. 34. 1.124864. 35. 1.191016. 36. 1.259712.
37. .015625. 40. 153.9384. 41. 197,061,258. 42. 15,205,344.
43. 1110 mi., 66,600 mi. 44. 8796 ft., 527,760 ft. 45. .32 in.
Exercise 36. — 1. 9.16125. 2. .8625. 3. 3.227. 4. .199.
5. .9128. 6. .490875. 7. .003. 8. 1.47433. 9. 1.174. 10. .657.
11. .925. 12. 1.142875. 13. .02468. 14. .12201. 15. .1638.
16. .7843875. 17. .912. 18. .9172. 19. .03866. 20: .07484.
21. .142857. 22. 1.0535. 23. 1.0446. 24. .2478.
Exercise 37. — 1. 2.925. 2. .024. 3. 8.6875. 4. .2058.
5. .3596. 6. 2.62. 7. 3.725. 8. 2.531. 9. 1.35. 10. 3.36.
11. 6.3234. 12. .5. 13. 40. 14. 62.5. 15. 125. 16. 50.
17. 15. 18. 50. 19. 160. 20. 12,500. 21. 8000. 22. 500.
23. 25. 24. .5. 25. 1000. 26. 1250. 27. 1000. 28. 50.
29. 250. 30. 250. 31. 25. 32. 2.5. 33. .0125. 34. .00125.
35. .0004. 36. .00666+. 37. .125. 38. .15. 39. 32.
40. .027. 41. 48.7. 42. 8.03. 43. .0904. 44. .708. 45. 1.012.
46. .01014. 47. .517. 48. .01054. 49. .00377. 50. .0365.
Exercise 38. — 1. $17,762.08. 2. $19.740.80. 3. $18,850.01.
4. $30,737.16. 5. $21,197.90. 6. $41,931.64. 7. $15,101.25.
8. $69,022.28.
Exercise 39. — 1. .375, .625, .4375, .5625, .6875, .8125, .9376, .1875.
2. .2667, .4667, .7333, .8667, .9333, .5833, .9167. 3. .3, .79, .087,
.0183, .2779. 4. .0375, .1375, .3625, .95, .926, .6167, .8833. 5. .03125,
.09375, .15625, .28125, .40625, .59375, .96875. 6. .2857, .7143, .0769,
.3077, .2143, .6429, .7857, .9286. 7. .5556, .1919, .1471, .0544, .0274, .0569.
Exercise 40. -1. &, f, J, J, A- 2. TJ^, ^ *fo» A» dhr- 3- To9o<r>
9 911 A 9 13 7 119 3 K 1 8 fi 3 48 « 1
¥00"' S7> 407' *• T25» T2^> ^¥tf» T~QQ^Qi SO' »' T25' Ittfr 6^3' 5frO>
vh- 6. T\<V, 6%, A3o, *fo, H- 7. ||, jj, f J, U-
Exercise 41. — 1. 4 hr. 2. 12 hr. 3. 6.6667 hr. 5. 14 hr.
6. 50.1875 mi. per hr. 8. 53.33 mi. per hr. 9. 54.13 mi. per hr.
10. 29.13 mi. per hr. 11. 546.45 mi. per da., 22.77 mi. per hr.
12. 1233.3 times. 13. $2400. 14. $280. 15. 850 cu. ft.
16. 420 cu. ft. 17. 7.48 gal. 18. $38.28. 19. $26.25. 20. $17.55.
Exercise 43. -1. &. 2. 1 3. |. 4. TV 5. &. 6. f.
7. Iff. 8. If. 9. f. 10. tV 11- f- 12- it- 13- rf-
14. if. 15. 5, 16. "H. 17. 1J. 18. Iff. 19. Jo- 20. 1|.
Exercise 44. — 1. 413.33 sq. yd. 2. 149.33. sq. yd. 3. 1966J
sq. yd. 4. 6442J sq. yd., 2512 ft. 5. 14,400. 6. (a) 15,762.22 sq. ft.,
(5) 21,112.5 sq. ft., (c) 4430.54 sq. ft., (d) 1291.79 sq. ft., (e) 16,032.99
334 ANSWERS
sq. ft., (f) 10,911.25 sq. ft., (g) 17,459 sq. ft., (ft) 80,711.38 sq. ft.
7. (a) 720.6 sq. yd., (ft) 849.58 sq. yd., (c) 912.88 sq. yd., (d) 1626.31
sq. yd., (e) 1146.23 sq. yd. 8. 4096 bu., $1996.80. 9. 71. 6 A.
10. 1425.6 thousand. 11. 8640 tiles. 12. 62.2 sq. yd., 83.55
sq. yd. 13. 21.9 T.
Exercise 45.— 1. $11.32. 2. $28.57. 3. $5.88. 4. $41.60.
5. $42.16. 6. $38.50. 7. $30.11. 8. $24.60. 9. $28.56.
10. $41.47. 11. $67.20. 12. $21.80. 13. $24.75. 14. $39.15.
15. $25.85. 16. $66.70. 17. $35. 18. $23.25. 19. $37.80.
20. $40.50. 21. $38.95. 22. $4.41. 23. $92.16. 24. Weimer,
$26.91; Flatonia, $24.76; Columbus, $27.30; Beaumont, $42.31; Gon-
zales, $23.59 ; Schulenburg, $26.13. 25. $31.50. 26. $39.15.
27. $32.50. 28. $10. 29. $118.25. 30. $113.85. 31. $129.48.
32. $312.90. 33. $435.20. 34. $589.68.
Exercise 46. — 1. $7.56. 2. $34. 3. $48. 4. $100. 5. $260.
6. $315. 7. $225. 8. $440. 9. $870. 10. $144. 11. $49.15.
12. $60. 13. $51. 14. $87.50. 15. $1210. 16. $987. 17. $1800.
18. $1625. 19. $3000. 20. $2450. 21. $1960. 22. $7800.
23. $512. 24. $1656. 25. $5000. 26. $12,150. 27. $262.50.
28. 4455 girls, 3795 boys. 29. 42 A. 30. $ 1516.80. 31. 73,
292. 32. $223.20, $3496.80.
Exercise 47. — 1. $3200. 2. $2160. 3. $1230. 4. $900.
5. $106. 6. $660. 7. $410. 8. $1740. 9. $3575. 10. $470.
11. $3769.60. 12. $4410. 13. $301. 14. $646. 15. $1040.
16. $1120. 17. $200. 18. $784. 19. $1500. 20. $480.
21. 490 trees. 22. 160, 300. 23. 922.5 A. 24. 35 ft. 25. 3542.97 mi.
Exercise 48. — 1. $24, $30, $36, $48. 2. $59.50, $68, $76.50.
3. $28.50, $38, $76. 4. $78.56, $147.30, $88.38. 5. .$18.45, $22.14.
6. $13.80, $16.10. 7. $20, $24. 8. $38.50, $22. 9. $25.60, $19.20.
10. $32.50. 11. $23. 12. $67.50. 13. $37.50. 14. $47.50.
15. $93.50. 16. $97.50. 17. $72. 18. $91.80. 19. $59.40.
20. $120. 21. $101.50. 22. $136.50. 23. $240.62. 24. $263.20.
25. $106.65. 26. 3247.50. 27. $219. 28. $43.20. 29. $139.50.
30. $136. 31. $141.75. 32. $123.50. 33. $86.40. 34. $180.50.
35. $210. 36. $315. 37. $2500. 38. $248. 39. $259.20.
40. $34.20. 41. $30.62. 42. $45. 43. $150.
Exercise 49. — 1. 4 cwt. 87 Ib. 8 oz. 2. 6 cwt. 10 Ib. 3 oz.
3. 2 T. 7 cwt. 18 Ib. 9 oz. 4. 31 T. 5 cwt. 5, 3 T. 19 cwt, 87 Ib.
6. 16 T. 721 Ib. 7. 112 T. 8. 4Jf T.
ANSWERS 335
•
Exercise 50.— 1. 38,000 Ib. 2. 28,400 Ib. 3. 34,300 Ib.
4. 803,200 oz. 5. 7502 Ib. 6. 9158 Ib. 7. 16,273 Ib. 8. 56
short tons. 9. $81. 10. 700 long tons. 11. 616 T. 12. 1497 Ib.
Exercise 51.— 1. 168 in. 2. 3972 in. 3. 9680yd. 4. 12,980
yd. 5. 18,209yd. 6. 784,520yd. 7. 13,622.4yd. 8. 2672 rd.
9. 3927ft.
Exercise 52.— 1. 102,400 sq. rd. 2. 1600 A. 3. 522,720 sq. ft.
4. 7350f sq.ft. 5. 16,665 rd. 6. 42,900ft. 7. 4840yd. 8. 6600yd.
9. 11,220ft. 10. 675 cu. ft. 11. 1046 cu. ft. 12. 816,480 en. in.
13. 506J cu. ft. 14. 944,784 cu. in. 15. 10 qt. 16. 23 qt. 17. 25 pt.
18. 57 pt. 19. 154 pt. 20. 128 qt. 21. 124 qt. 22. 116 qt.
23. 56 qt. 24. 59 qt. 25. 1208 pt. 26. 838 pt. 27. 2640 ft.,
1320 ft., 480 ft. 28. 440 yd., 352 yd., 586f yd. 29. J, TV, ^V 30. 1210
sq. yd., 2420 sq. yd. 31. ¥V 32. 140 sq. yd. 33. 112 sq. rd.,
144 sq. rd. 34. 204r\ sq. ft. 35. 33.6 cu. in. 36. 28,875 cu. in.
37. 6 qt. 38. 93.1 gal.
Exercise 53. — 1. 29 gal. 1 qt. 2. 1 cu. yd. 729 cu. in. 3. 432 gal.
4. 32 mi. 5. 31 bu. 1 pk. 6. 8 cu. yd. 11 cu. ft. 744 cu. in. 7. 31,250
sq. mi. 8. 11 sq. yd. 5 sq. ft. 24 sq. in. 9. 7926.59 mi. 10. 7899.58
mi. 11. 27.01 mi. 12. 186,325 mi.
Exercise 55. — 1. 66,020". 2. 324,000". 3. 43,510". 4. 27,900".
5. 433,080". 6. 163,820". 7. 855'. 8. 4545'. 9. 15,247.5'.
10. 280'. 11. 1130'. 12. 832R 13. 525,600mm. 14. 1,578,240
min. 15. 41,760 min. 16. 129,600 min., 131,040 min. 17. 31,556,926
sec. 18. 36,892,800 times. 147,268,800 times. 19. 311,760 min.
20. 584 da. 21. 960 hr. 22. 178,826.4 hr. 23. 2922 da. 24. 48°.
25. 174°.
Exercise 56. — 1. 32 yd. 2 ft. 4 in. 2. 151 yd. 9 in. 3. 25 yd. 9 in.
4. 263 A. 87 sq. rd. 5. 506 A. 27 sq. rd. 6. 971 A. 79 sq. rd.
7. 16 gal. 3 qt. 8. 44 gal. 3 qt. 1 pt. 9. 42 bu. 3 pk. 4 qt. 10. 38 bu.
2 pk. 3 qt. 11. 77 gal. 3 qt. 12. 153 bu. 3 pk. 5 qt. 13. 117 bu.
3 pk. 3 qt. 14. 183 bu. 1 pk. 7 qt. 15. 147 T. 1379 Ib. 16. 667 Ib.
4 oz. 17. 546 Ib. 13 oz. 18. 77 T. 628 Ib. 19. 272 Ib. 7 oz.
20. 225 T. 1908 Ib. 21. 218 da. 18 hr. 30 min. 22. 241 wk. 6 da. 13
hr. 23. 85 hr. 14 min. 52 sec. 24. 141 da. 13 hr. 47 min. 25. 38 wk.
Oda. 12 hr. 26. 54 hr. 20 min. 3 sec. 27. 64 cu. ft. 1032 cu. in.
28. 307 cu. yd. 9 cu. ft. 29. 362 cu. yd. 30. 67 cu. ft. 1023 cu. in.
Exercise 57. — 1. 79° 2' 17". 2. 8° 0' 56". 3. 51° 54' 25".
4. 117° 45' 25". 5. 6° 54' 45". 6. 66° 46' 6". 7. 85° 6' 10".
8. 45° 57'. 9. 60° 5' 43". 10. 78° 59' 50". 11. 43° 5' 48".
336 ANSWERS
•
12. 1 bu. 3 pk. 7 qt. 13. 8 gal. 0 qt. 1 pt. 14. 6 gal. 0 qt. 1 pt.
15. 5 gal. 2 qt. 1 pt. 16. 19 bu. 2 pk. 5 qt. 17. 9 bu. 1 pk. 5 qt.
18. 19 gal. 2 qt. 1 pt. 19. 21 gal. 0 qt. 1 pt. 20. 3 qt. 21. 28 qt.
22. 120 qt. 23. 1 da. 20 hr. 57 min. 24. 6 da. 17 hr. 51 min.
25. 11 hr. 40 min. 26. 2 hr.
Exercise 58. — 3. 4 yr. 6 mo. 28 da. 4. 6 yr. 4 mo. 29 da.
5. Milton, 65 yr. 10 mo. 29 da. ; Pope, 56 yr. 0 mo. 9 da. ; Shakespeare.
52 yr. ; Burke, 67 yr. 5 mo. 27 da. ; Lee, 63 yr. 8 mo. 23 da. ; Grant, 63
yr. 2 mo. 26 da. ; Goldsmith, 45 yr. 4 mo. 24 da. ; Franklin, 84 yr. 3 mo.;
Hamilton, 47 yr. 6 mo. 1 da. ; Longfellow, 75 yr. 0 mo. 27 da. ; Newman,
89 yr. 5 mo. 20 da. ; Gladstone, 88 yr. 5 mo. 10 da.
Exercise 59. — 1. 42 yd. 2 ft. 3 in. 2. 72 yd. 1 ft. 3 in.
3. 79 yd. 2 ft. 4 in. 4. 26 bu. 1 qt. 5. 53 bu. 1 pk. 6. 59 gal. 1 qt.
7. 79 gal. 2 qt. 8. 27 bu. 3 pk. 1 qt. 9. 146 T. 800 Ib.
10. 118 T. 1483 Ib. 11. 171 T. 540 Ib. 12. 87° 51' 28".
13. 249° 38'. 14. 779 A. 40 sq. rd. 15. 166 A. 137 sq. rd.
16. 254° 18' 36". 17. 11,250 Ib. 18. 75,240ft.
Exercise 60. — 1. 2 yd. 1 ft. 3 in. 2. 13° 19' 19f". 3. 7° 1' 111".
4. 5 yd. 2 ft. 5i in. 5. 11 yd. 4 in. 6. 2 bu. 1 pk. 2 qt.
7. 10 times. 8. 240 times. 9. 1609.35. 10. 4° 5' 27T3r". 11. 264.
Exercise 61. — 1. 1750 Ib. 2. 9'. 3. 18 hr. 40 min. 4. 469 Ib.
5. 22 hr. 48 min. 6. 12 qt. 7. 97,152 ft. 8. 1120 yd.
9. 60 rd. 10. 945'. 11. 243 da. 8 hr. 12. 114 da. 1 hr. 30 min.
13. 1 pk. 6 qt. 3.2 gi. 14. 2 qt. 1 pt. 1J gi. 15. 136 sq. rd.
16. 4290 sq. yd. 17. 26 qt. 18. 3J pt.
Exercise 62. — 1. .8333. 2. .16667 nearly. 3. .06667. 4. .375.
5. .4375. 6. .5625. 7. 796,875. 8. .09375. 9. .195. 10. jfc.
11. .25.
Exercise 63. — 1. 1 A. 2. 22 A. 3. 2.492 A. 4. 1493 J A.
.$37,3331. 5. 3.6 A. 6. 1200 A. $58,800.
Exercise 64. — 3. 81200. 4. $5000.
Exercise 65. — 1. 504 cu. ft. 2. 72 cu. yd. 3. 9216 cu. ft.
4. 3150 cu. yd. 5. 13,824 gal. 6. 115.71 bu. 7. 560 cu. yd.
8. 3520 cu. yd. 9. 270 boxes. 10. 1,251,655$ cu. yd. 11. 24.48 T.
12. 1481} Ib. 13. 120 T. 14. 212 Ib. | oz. 15. 1728 boxes.
16. 45 cd. 17. 453.75 T.
Exercise 66. —1. 66J sq. ft. 2. 166f sq. ft. 3. 103 J sq.ft.
4. 143| sq. ft. 5. 306^ sq. ft. 6. 301J sq. ft. 7. 238T\ sq. ft.
ANSWERS 337
8. 177^ sq. ft. 9. 596|| sq. ft. 10. 1287^ sq. ft. 11. 1346T\ sq. ft.
12. 708| sq. ft. 13. 775r72 sq. ft. 14. 2646| sq. ft. 15. 76£ sq. ft.
16. 372 sq. ft. 17. 3.15 A. 18. 4200 sq. ft. 19. 247 sq. ft.
20. 4053 sq. ft. 21. 2310J-f sq. ft.
Exercise 67.— 1. 12. 2. 12,42^. 3. 60,60. 4. 96.
5. (a) 36, (6) 48, (c) 54, (d) 42J, (e) 54, (/) 84, (</) 87J,
(ft) 122J. 6. $264. 7. $367.50. 8. 77 bd. ft. 9. 87 J bd. ft.
10. 96 bd. ft. 11. 364£ bd. ft. 12. 544 bd. ft. 13. 10 cu. ft.
14. 396 bd. ft. 15. 20|f bd. ft. 16. 141| bd. ft.
Exercise 68. — 1. 81.45 perches, 44,352 bricks. 2. 1984 cu. ft.,
1856 cu. ft. 3. 13,200 bricks, $118.80. 4. 1000 cu. ft., 198,000
bricks. 5. $1909.84.
Exercise 69. — 1. 42| yd. 2. 64yd. 3. 24yd. 4. 66| yd. 5. $33.60.
Exercise 70. —1. 24 da. 2. 114 T. 3. 10 J hr. 4. 49£ yr.
5. $42331, $16931. 6. $200. 7. $300. 8. \\\. 9. 10 and 15.
10. i, 81 cattle. 11. 75 T. 12. 165. 13. $340. 14. $6300.
15. $1939. 16. 1600. 17. If hr. 18. 30 mi., 6 hr. 56 min.
19. 37ift. 20. i. 21. $2. 22. 48 oranges. 23. 41^ yd.
24. 60^perlb. 25. 91 j* per yd. 26. $158.82. 27. 42 pencils.
28. $43.20. 29. 27^. 30. $1.12. 31. 40 tiles. 32. .783.
33. 21, 150 sec. 34. $98.67. 35. $750. 36. 79f yd. 37. $672.
38. $170.80. 39. 4620ft. 40. 3025 sq. yd. 41. 240 bu.
42. $90. 43. 174.3768 Ib. 44. 4800 gr. 45. 48 mi. 46. 22.5 A.
47. $562.50. 48. $95. 49. June 30. 50. 1944 sq. in.
51. 24 sq. rd. 52. 56. 53. $91.80. 54. $147. 55. 180 min.
56. 49^ mi. 57. $21.11. 58. 10 inin. 50 sec. 59. 345.6 Ib.
60. 2520 gal. 61. $45. 62. 37 J cords. 63. 70?. 64. 345.6 bu.
65. .68, .6818, .012, .8571. 66. J, rffo, f. 67. 87799.3.
68. $500.50. 69. 5 gal. 2 qt. 70. 144 cu. in. 71. $60,138.75.
Exercise 71. — 1. ft. 2. 11JJJ. 3. fJJ. 4. 3J}J. 5. 1JJJ.
6- 9JH- 7. HJ 8- ISftV 9. iff. 10. 10&. 11. 8JJ.
12. 9&V- 13. 2JJ. 14. -ii 15. -&V 16- Th- W. A-
1ft 5 1Q 1 Oft 1 O1 1 OO V O1 1 Od. Q13
JLO. y2"« Av. g^-. <6U. "Jo". <wA. yj^* <w<w. 1^'Q' <6O. ~Q~§' *rx.* T¥*
25. ff. 26. 2f|. 27. 2jf. ' 28. f|. ' 29. Iff. 30. 3|.
31. 111. 32. $1272£. 33. 16^. 34. 66 A- 35. fffc.
Exercise 72. — 1. If. 2. 3. 3. J. 4. 1. 5. J. 6. 1JJ.
7. If. 8. 8. 9. J. 10. 20. " 11. 1. 12. 8TV 13. 2J.
14. 171. 15. 5. 16. TV 17. TV 18. |§f 19. ij. 20. 1^.
21. 12T3T. 22. 15i|. 23. 1^. 24. 0. 25. 0. 26. 3|.
z
338 ANSWERS
Exercise 73. — 1. 14. 2. 1J. 3. 6T6T. 4. ^. 5. f. 6. 2j.
7. 1. 8. TV 9. li|. 10. ljfl. 11. 3J. 12. Tjjfd. 13. J.
14. _3_. 15. i. 16. 31. 17. 82%. 18. 4.
Exercise 74. — 1. 12.88. 2. 24.75. 3. 12.78. 4. 42.72. 5. 295.2.
6. 203.76. 7. 24.0. 8. 184.1. 9. 385.2. 10. 785. 11. 108.13.
12. 308.4. 13. 116. 14. 1218. 15. 122 J. 16. 310.5. 17. 5750.
18. 1713. 19. 102.6. 20. 114.975. 21. 118.3. 22. 612.72. 23. 8.45.
24. 20.31. 25. 331%, 25%, 20%, 121%, 6J%, 15|%, 43}%. 26. 4%, 8%,
121%, 1.65%, i%, |%, .24%. 30. $1125. 31. 11.2 Ib. 32. 27£ Ib.
33. $2050. 34. N.Y., $476,850,000; Boston, f 132,982,500 ; Phila.,
$64,387,500; Baltimore, $113,475,000; N.O., $81,600,000; Galveston, .
$91,417,500.
Exercise 75. — 1. 3700. 2. 1248. 3. $864. 4. 7200,6000,4500,
4000. 5. 4000, 3000, 2400, 2000, 1500, 1000. 6. $12,500, $11,250,
$10,000, $6000. 7. $360, $432, $324, $243. 8. $7200, $6750,
$5400, $3600, $2700. 9. $1120, $672, $480. 10. 4200 oz.
11. 76,300,712. 12. 39,007,793. 13. 37,220,331. 14. 6,166,710.
15. $213,293,651. 16. $1,960,233. 17. 3500. 18. 3500. 19. 304,800.
20. 4000 Ib.
Exercise 76. -1. 80%. 2. 20%, 35%, 55%, 65%. 3. 5%, 20%,
30%, 42J%. 4. 10%, 13J%, 30%, 12 J%. 5. 121%, 15%, 37.]%, 60|%.
6. 6%, 7%, 81%. 7. 20%, 9%, 16|%. 8. 9^-%, 33*%, 15%. 9. If %, 8J%.
10. 31%. 11. 2J%. 12. f%. 13. 4^-%. 14. 6i%. 15. 12J%.
16.25%. 17.331%. 18. 115. 15% nearly. 19.88.44%. 20. 1^%, 10%.
21. 6i%. 22. 20%, 60%, 80%. 23. 91.44% nearly, 109.36% nearly.
24. 62.14% nearly. 25. 82f%. 28. 109f%. 27. (a) 1.43%, (6)2.42%,
(c) .65%, (d) 1.02%, (e) .71%, (/) .63%, (0) 1.16%, (ft) .3%, (Q .97%.
28. Mobile, 23.79%; Little Rock, 48.05%; Los Angeles, 103.35%;
Denver, 25.44%; Pensacola, 51.04%; Savannah, 25.6%; Springfield,
30.84%; Evansville, 16.26%; Dubuque, 19.75%; Kansas City, Kan.,
3-1.19%; Lexington, 22.27%; Kansas City, Mo., 23.38%; Minneapolis,
23.05%. 29. 1.12% nearly. 30. 9.07% nearly. 31. 64.69%, 13.12%,
22.19%.
Exercise 77. — 1. $48. 2. $52.50. 3. $11.25. 4. $11.20.
5. $21.60. 6. $27.20. 7. $59.50. 8. $65. 9. $61.33. 10. $50.
11. $133. 12. $132. 13. $83.60. 14. $91.20. 15. $106.25.
16. $171.50. 17. $204.48. 18. $679.62. 19. $3017.60.
20. $2522.16. 21. $4904.17. 22. $2334.37. 23. $3437.98.
24. $4454.66. 25. $752.25. 26. $319.44. 27. $633.88.
28. $1891.62. 29. $917.60. 30. $2456.26.
ANSWERS 339
Exercise 78. — 1. $47.25. 2. $612. 3. 30%. 4.
5. $283.50. 6. $232.56. 7. $10. 8. 32}%. 9. 24f%. 10. 25%.
11. $2, $25. 12. (a) $241.92, (6) $547.20, (c) $405, (d) $739.20,
(e) $403.20, (/) $739.20, (g) §943.92, (ft) $962.50.
Exercise 79. — 1. (a) $168, (6) $93.75, £c) $33.32, (d) $250.83,
(e) $685.80, (/) $344.50, (p) $608, (ft) $269.10, (i) $514.12. 2. (a) 25%,
(6) Wo, (^ 66|%, (d) 331%, (6) 28%.
Exercise 80. — 1. (a) $50.40, (6) $132, (c) $50, (d} $161.54,
(e) $200, (/) $300. 2. (a) $45.89, (5) $109.37, (c) $ 128.57, (d) $91,
(e) $89.85, (/) $40.32, (</) $69.75, (ft) $289.14.
Exercise 81. — 1. $3250. 2. $300. 3. $3640. 4. 12}%. 5. 8J%.
6. 20%. 7. 10%. 8. 9%. 9. $97. 10. $86. 11. $75. 12. $105.
13. $80, $15. 14. $120. 15. $4.20. 16. 25%. 17. $982. 18. 12,080.
19. $16.67 nearly. 20. $200. 21. $10,240. 22. $11,520. 23. $85,700.
24. 475 Ib. 25. 12%. 26. 14f%. 27. 22|%. 28. $165. 29. 177$%.
30. 5^. 31. $2.10 per yd. 32. 24^. 33. 16|%. 34. 38|%. 35. 84 J*
perlb. 36. 11 J%. 37. $98. 38. $98.70. 39. 96} 0. 40. 440.
41. 200. 42. 33i%. 43. $148, 30|%. 44. (a) 40% above cost, (6) 30%
above cost, (c) 60% above cost, (d) 22 1% above cost, (e)42f % above cost.
Exercise 82. — 1. $320. 2. $1050, $16,450. 3. $45, $705. 4. $4.
5. 96,000 bu. 6. $56.16, $ 1815.84. 7. $4.50. 8. $56.70, $ 1563.30.
9. $1125, $21,375, 95%. 10. 900 T. 11. 844 bales, 8^ per pound.
12. $17.28. 13. 3J%. 14. 227,200 bu. 15. $4200. 16. $ 7800, $ 7449.
17. $25. 18. $235.50,78]^. 19. 4000 bu. 20. 3750 Ib. 21. 150 A.
22. (a) $17.85, $205.27; (6) $40.56, $770.69; (c) $85.84, $987.16;
(d) $28.27, $537.23; (e) $100.01, $1233.49; (/) $47.60, $428.40 ; (0) $24,
$276; (7i) $103.20, $1960.80; (i) $ 10.06, $191.19. (j) $10.18, $101.98.
(A-) $421.89, $13,641.11. (Z) $38.27, $386.98.
Exercise 83.— 1. $21, $35. 2. $16.68, $19.46. 3. $53.82, $47.84.
4. $23.12, $28.90. 5. $75, $85. 6. $128, $119. 7. $213.75, $200.
8. $180, $152. 9. $120, $175. 10. $204.17, $ 163.33. 11. $7.04,
$8.60. 12. $8.35, $12.98. 13. $9.39, $6.26. 14. $12.38, $13.93.
15. $12.87, $9.81. 16. $16, $30. 17. $28.14, $31.51. 18. $9.19, $4.92.
19. $12.85, $13.49. 20. $ 56.07, $56.07. 21. $203.52, $ 126.64.
22. $243.83, $190. 23. $ 164.79, $105.94. 24. $175.50, $157.50.
25. $205, $87.47. 26. $191.25, $ 183.75. 27. $16.11. 28. $47.14.
29. $39.33. 30. $53.36.
Exercise 84. — 1. $7.?0. 2. $12.18. 3. $97.39. 4. $100.07.
5. $74.19. 6. $98.41. 7. $64.51. 8. $43.63. 9. $16.46. 10. $7.12.
11. $13.90. 12. $117.26. 13. $51.08. 14. $8.46. 15. $87.88.
16. $42.37.
340 ANSWERS
Exercise 85. — 1. $838.67. 2. $693.10. 3. $802.50. 4. $365.61.
5. §337.79. 6. §753.13. 7. $880.28. 8. $330.46. 9. $1016.92.
10. $389. 11. $1332.34. 12. $3750.25. 13. $1532. 14. $4454.04.
15. $1243.63. 16. $499. 17. $826.40. 18. $562.83. 19. $667.55.
20. $880.10. 21. $1014.30. 22. $2518.56. 23. $3935.15. 24. $2706.41.
25. $1044.83. 26. $1993.18. 27. $960.
Exercise 86. — 1. 7830 oz., 2890 oz., 1200 oz. 2. .579 oz., 8.355 Ib.
3. 77.78 Ib. nearly. 4. 2.7. 5. 1.036. 6. 1.04. 7. 14.7241b. 8. 62.5T.
9. 4.17 T. 10. 11.146 oz., 6.059 oz., 12.442 oz., 6.586 oz. 11. 345 Ib.
12. 1.0125 T. 13. 39| Ib. 14. 54 T. 15. 810.86 Ib. 16. 1.3.
17- 2.402 cu. in. 18. 166J Ib. 19. 280 oz. 20. 4.6955 times.
21. 108.194 Ib. 22. 212.5 Ib.
Exercises?. — 1. 125ft. 2. 5^ yd. 3. 40 rd. 4. 14ft. 5. 1611 bu.
6. 291 bu. 7. 19.36 bu. 8. 20 yd. 9. 17 yd. 10. 32 yd. 11. 10 ft.
12. 4.5ft. 13. 7ft. 14. 44ft.
Exercise 88. — 2. 1792 cu. ft., 1433.6 bu., 45,875.2 Ib. 3. 1725 cu. ft.,
4800 turkeys. 4. 2720 cu. ft., 130,560 Ib., fff. 5. 20 cords, $95, $30.
6. 2675.2 T.
Exercise 89.— 2. 71,614^ Ib. Troy, 58,928f Ib. Avoir. 3. 371.25 grains.
4.51.050. 5. 36£ Ib. Avoir, 3685 \ Ib. Avoir. 6. 232.2 grains. 7. $20.672.
8. 61^. 9. 30.5.
Exercise 90. — 2. $80, $50. 2. $591, $394. 3. $1440, $1800.
4. $1188, $2376, $3564. 5. $1700, $2550, $2975. 6. 756, 972, 1188.
7. 912.5, 1095, 1277.5, 1460. 8. 1508, 1624, 1740, 1856. 9. 1176, 1232,
1288, 1344. 10. 540, 612, 684, 756. 11. 57, 38. 12. 992, 744. 13. 140,
225. 14.975,546. 15.1170,1300,1755. 16.396,770,1023. 17. $490,
$700, $800. 18. $ 720, $ 900, $ 1080. 19. $300, $255, $345.
20. $4444f, $5555f.
. Exercise 91. — 1. $8.25. 2. 11.2 da. 3. $110.50. 4. 7J da.
5. 5f°. 6. 311 A. 7. 56ft. 8. 11J da. 9. $24.18. 10. 60 mi.
11. $185. 12. $333.75. 13. 3960 ft., 66 ft. 14. 52.8ft. 15. 4J sec.
16. 186,000 mi. nearly. 17. $112.50. 18. 12 J yd.
Exercise 92.— 1. If 2. 15f. 3. 6J. 4. The first. 5. 3^.
6. 0. 7. 50 Ib. 8. ?\. 9. 4950 ft. 10. .1212+ rd. 11. .456.
12. .013875 T. 13. 1400 Ib. 14. 45°. 15. 12°. 16. J.
17. .3, $1500, A's$750, B's $300. 18. $1875. 19. 7.9. 20. 11.3984.
21. .0524. 22. $1178.85. 23. 112.5. 24. 228. 25. $150. 26. J-i.
27. 3.9. 28. $18.70. 29. $1500. 30. 13J da. 31. 6 da.
32. 9 da. 33. 14 hr. 34. 20 da. 35. 1.7875. 36. JJ. 37. 325 Ib.
38. $326.67. 39. $64, $240. 40. $75. $313.32, 25 mo. 41. $89.25,
ANSWERS 341
$107.10. 42. $6.06. 43. 126 Ib. 15J oz. 44. $18.20. 45. $121.67.
46. 11J0, $11.25. 47. 110, $11. 48. 130, $13. 49. $34,588, $32,858.60.
50. $133,016. 51. $62.675. 52. $117.60. 53. 5 mills, 500. 54. 21.82
knots per hr., 25.126 mi. per hr. 55. 20.584 knots per hr., 23.703 mi.
per hr. 56. 24.13 knots per hr. 57. 65.5 mi. nearly. 58. 63.617 mi.
per hr. 59. 50.77 mi. per hr. 60. Wheat, $1.31J; corn, 81J0 ; oats, $1.06J
nearly. 61. (a) $2367.55, (b) $6450.05, (c) $922.32, (d) $950.82,
(e) $4585.84, (/) $693.45, (g) $955.79.
Exercise 93.— 1. 75,453,652. 2. 74,944,016. 3. 73,781,838.
4. 80,963,637. 5. 80,903,198. 7. New England, 7607.2, 7619.39,
7680.92 ; Middle Atlantic, 22,930.26, 23,149.83, 23,408.29 ; Central
Northern, 42,741.79, 43,251.64, 43,958.89; South Atlantic, 22,880.92,
23,589.39, 24,179.92; Gulf and Miss. Valley, 17,567.13, 18,297.28,
19,025.73; Southwestern, 43,068.45, 44,852.05, 46,061.39. 8. 12T^.
9. llf. 10. 111. 11. 17|7. 12. 9T\. 13. 23£J. 14. 22}i.
15. 30r78-. 16. 147° 24' 40". 17. 35 ft. 6 in. 18. 45 qt. 1 pt.
19. 30 gal. Iqt. 20. 41 pk. 3 qt. 21. 25 bu. 1 pk. 22. 50 hr. 18 min.
23. 43 da. 2 hr. 24. 83 yd. 25. 156 times. 26. 27.63 liters.
27. 23 ft. 8 in. 28. 4.37 mi. ; 13,431 carloads ; 100.7325 mi. 29. 134 T.
Exercise 94. — 2. 199.98. 3. 398.63. 4. $74.27. 5. $697.16.
6. $98.25. 7. $999.95. 8. $223.81. 9. $50.79. 10. $310.16.
11. $36.65. 12. 99.99. 13. 3.2976. 14. .9937. 15. 4.43. 16. 9.5.
17. 5.4. 18. 45.01. 19. 2|. 20. lOjf. 21. 5T^. 22. 9JJ. 23. 4-||.
24. 2|J. 25. 5-ij. 26. 2J|. 27. 3?V 28. 11^. 29. 2JJ. 30. 11^.
31. 10 .in. 32. 4 ft. 7 in. 33. 9 ft. 9 in. 34. 8 ft. 11 in. 35. 4 Ib. 11 oz.
36. 11 Ib. 11 oz. 37. 14 Ib. 10 oz. 38. 2 hr. 46 min. 39. 55° 45' 45".
40. 64° 55' 10". 41. 31° 26'. 42. 11 pk. 4 qt. 43. 9 pk. 7 qt.
44. 17,357. 45. 12,219. 46. 5176. 47. $8592. 48. 144,805 sq. mi.
49. 23,672,000, 78,272,000, 147,925,000. 50. 91,895. 51. 574 mi.
Exercise 95.— 6. 27,878,400 ; 1,040,400. 7. 1,871,104 ; 6,921,495.
8. Corn, $12.09; wheat, $10.34; oats, $9.89; rye, $9.84; barley,
$11.74; buckwheat, $11.09. 9. 30,759,200 Ib. 10. N.H., $14.37;
Mass., $22.27; Conn., $17.55; N.Y., $15.49; N.J., $21.05; Penn., $17.42;
Md., $17.01; Va., $19.37; S.C., $22.26; Ga., $25.99; Ala., $25.93;
La., $22.19; Tenn., $20.31; Ky., $17.89; 111., $12.25; Minn., $9.35;
Kan., $8.00; Col., $23.75; Utah, $30.00; Idaho, $23.60; Cal., $20.81.
12. (a) $10.84, (6) $8.97, (c) $7.24, (d) $10.09, (e) $9.79, (/) $2.90,
(g) $56.53, (A) $55.68. 13. 97 ft. J in. 14. 73 ft. 9£ in. 15. 45 yd.
1 ft. ; 67 ft. 8 in. 16. 46 Ib. 4 oz. ; 150 Ib. 3 oz. 17. 20 hr. 3 min. ;
65 hr. 33 min. 18. 532° ; 341° 1' 30''. 19. 29 pk. 2 qt. ; 59 pk.
20. 49 gal. 2qt. ; 117 gal.
342 ANSWERS
Exercise 96. — 1. N.H., 120.2 bu., 72^; R.I., 125 bu., 89^;
Del., 93 bu., 69^; N.C., 77 bu., 68^; Fla., 76 bu., 120^; Miss., 110 bu.,
85 jZ; W. Va., 88 bu., 68^; Mich., 67 bu., 56^; 111., 75 bu., 67^;
Mo., 82 bu., 55^; N.D., 95 bu., 38^; Nev., 120 bu., 82?. 2. .3048.
3. .3048. 4. 3.2808. 5. 1.6093. 6. .6214. 7. .4047. 8. 2.471.
9. .7646. 10. 1.3079. 11. 16J. 12. 21. 13. 32. 14. 21J. 15. 17J.
16. 1£. 17. If 18. ||. 19. If 20. J. 22. &. 23. Jf. 24. If.
25. jfo. 26. 45. 27. 2TV 28. 8J. 29. 3J. 30. 4f|. 31. 3f.
32. 3 long T. ; 5.905 long T. ; .00886 long T. ; 3.9368 long T. 33. 2.2046 T. ;
7.7161 T. ; .00551 T. 34. 6.0764 Ib. Troy ; 16.076 Ib. Troy ; 8,037.69 Ib.
Troy. 35. 5.76 Ib. ; 11.023 Ib. ; 4.409 Ib. 36. .0001894 mi.
37. .003125 mi. 38. .00625 A. 39. .0016625 sq mi. 40. .0002066 A.
41. .0005 T. 42.363. 43. 114$ barrels. 44.600. 45. $6160. 46. 171.6 T.
Exercise 98. — 1. 24° 10'. 2. 14° 16'. 3. 11° 17' 40". 4. 29° 17' 46".
5. 108° 63' 48". 6. 73° 28'. 7. 229° 22' 32". 8. 94° 60' 10".
9. 116° 5' 40". 10. 88° 10' 64". 11. 114° 8' 45". 12. 78° 37' 37".
13. 25° 33' 35". 14. 35° 23' 54". 15. 120° 36' 40". 16. 74° 44' 65".
17. 116° 4- 2".
Exercise 99 (answers correct to the second). — 1. 1 hr. 42 min. 14 sec.
2. 3 hr. 9 mm. 2 sec. 3. 4 hr. 35 min. 14 sec. 4. 12 hr. 53 min. 56 sec.
5. 15 hr. 9 min. 38 sec. 6. 6 hr. 50 min. 69 sec. 7. 8 hr. 32 min.
15 sec. 8. 6 hr. 33 min. 43 sec. 9. 15 hr. 20 min. 10 sec. 10. 13 hr.
41 min. 28 sec. 11. 13 min. 38 sec. 12. 7 hr. 13 min. 58 sec. 13. 2 hr.
10 min. 26 sec. 14. 6 hr. 46 min. 56 sec. 15. 14 hr. 13 min. 10 sec.
16. 14 hr. 15 min. 2 sec. 17. 1 hr. 12 min. 69 sec.
Exercise 100. — 1. 13° 22' 6" E. 2. 4° 20' 36" E. 3. 88° 17' 36" E.
4. 3° 12' 24" W. 5. 9° 66' 36" E. 6. 1' 54" W. 7. 3° 43' 64" W.
8. 16° 17' 36' E. 9. 72° 30' 45". W. 10. 2 hr. 37 min. 20 sec.
Exercise 101. — 1. 72° W. 2. 11 o'clock A.M. March 3 ; 1 o'clock
P.M. March 3 ; 4 o'clock A.M. March 3. 3. 9 o'clock A.M. in London,
Manchester, Glasgow ; 5 o'clock P.M. in Tien-Tsin ; 11 o'clock A.M. in
Constantinople. 4. 5 A.M. 5. 165° W. 6. 6 A.M. following day.
7. 6.30 P.M. ; 9 P.M. previous day ; 12.30 P.M. 8. 2 P.M. ; 3 P.M. ; 11 P.M. ;
11.30 P.M. 9. 4 hr. 50 min. 39 sec. A.M. ; 1 hr. 50 min. 39 sec. P.M. ;
6 hr. 20 min. 39 sec. P.M. ; 11 hr. 20 min. 39 sec. P.M. 10. 11 hr.
30 min. P.M. previous day ; 9 hr. 30 min. P.M. previous day ; 6 hr.
30 min. P.M. previous day. 11. 12.30 A.M. following day; 6.30 A.M.
following day.
Exercise 102.— 1. 6 da. 2. 2£ hr. 3. $592.20. 4. 25yd.
5. 16 da. 6. 132°. 7. 21.6ft. 8. 600 mi. 9. 2355 mi. 10. 2030 mi.
ANSWERS 343
11. 605 mi. 12. $51,937,925. 13. $730,377. 14. $2,944,492.
15. $.238. 16. $.498. 17. 19.3^. 18. 1.09375.
Exercise 106. — 1. 4. 2. 3 a. 3. 3 a. 4. 4. 5. 3 a2. 6. 3 a.
7. 2 a2. 8. 6 a. 9. 3 a2. 10. 11 a3. 11. 11 a3. 12. 4 a2. 13. 2 a2.
14. 3 a2. 15. 5 a2. 16. 7 x. 17. 3 a*. 18. 1 x. 19. 5x3. 20. 5x.
21. 6. 22. 2z4. 23. 36. 24. 363. 25. 2 63.
Exercise 108. —1. 9. 2. 8. 3. 20. 4. 30. 5. 18. 6. 35.
7. 30. 8. 40. 9. 22. 10. 45. 11. 55. 12. 55. 13. 54.
14. 60. 15. 45. 16. 49. 17. 90. 18. 55. 19. 63. 20. 40.
21. 56. 22. 18. 23. 22. 24. 30. 25. 35.
Exercise 109. — 1. $675. 2. $23.04. 3. $609.20. 4. 10 da.
5. $2175. 6. 16 hr. 48 min. 7. 20 da. 8. 18f bu. 9. 9 da.
10. 25 da. 11. 400 men. 12. 21 da. 13. 18 turkeys. 14. 66^.
15. 280,176ft. 16. 75ft. 17. 157.5ft. 18. 40 mi. per hr.
Exercise 110.— 1. 84 A. 2. 150ft. 3. $260. 4. 2057.5 T.
5. 20 ft. 6. 7 da. 7. 12 da. 8. 6 da. 9. 7 hr. 12 min.
10. $113.40. 11. $365.625. 12. 2J da. 13. 8 da. 14. 17.85 T.
15. 17£hr.
Exercise 111.— 1. A' s share, $ 800 ; B's share, $1000; C's share,
$1200. 2. A's, $600; B's, $1500. 3. A's, $90 ; B's, $81. 4. A's,
$900; B's, $700. 5. A's, $780; B's, $954 ; C's, $420. 6. A, $300;
B, $300. 7. $700, $750. 8. 3J A. 9. $12.50.
Exercise 112. — 1. 4%, 8%, 7}%, 6J%, 16 j%. 2. .045; .15; .125;
.625; .0625; .036. 3. 107.55; 18.335; 34.475; .3594; .62345.
4. 315.72; 31.242; .1596; .18582; .3846; .04698. 5. 186.75; 63.81;
320.4; 1.95435; 2.39265; .1089. 6. Land, $19,466; fencing,
$2919.90; earthworks, $46,718.40; tunnels, $23,359.20; viaducts and
bridges, $33,092.20; works, $3893.20; culverts, $9733; way,
$22,385.90; sidings, $5839.80; junctions, $1946.60; stations,
$12,652.90; legal expenses, $11,679.60; maintenance, $973.30. 7. Ans.
correct to one-tenth of one million. $974,500,000; $54,200,000;
$16,900,000; $66,300,000; $97,800,000; $16,800,000. 9. Ans. correct
to one-tenth of one million. N.Y., $607,000,000 ; Savannah, $64,900,000;
Boston, $98,700,000; Puget Sound, $49,200,000; New Orleans,
$150,000,000; Detroit, $35,200,000 ; Galveston, $166,400,000 ; Buffalo
Creek, $30,000,000; Philadelphia, $82,500,000; Mobile, $21,800,000;
Baltimore, $110,000,000; Newport News, $20,100,000; San Francisco,
$39,900,000; Wilmington, $18,500,000.
Exercise 113. - Michigan, 4,725,000 Ib. ; Minnesota, 1,176,000 Ib. ;
Alabama, 341,250 Ib.; Montana, 12,535,250 Ib. ; Wyoming, 10,511,920 Ib. j
344 ANSWERS
Idaho, 5,578,650 Ib. ; California, 4,331,250 Ib. ; Utah, 4,322,500 Ib. ; New
Mexico, 6,061,000 Ib. ; Colorado, 3,118,500 Ib. ; Arizona, 1,502,800 Ib.;
Texas, 3,182,400 Ib. ; Washington, 1,466,250 Ib.
Exercise 114. — 1. 1973. 2. 3990. 3. $800. 4. $1000.
5. $3004.44. 6. $100.80. 7. Maine, $351,577,436. S.Pennsyl-
vania, $3,910,701,678; South Carolina, $195,620,105; Kansas,
$363,010,660; Tennessee, $405,641,915; Washington, $260,948,609;
Texas, $1,021,158,657.
Exercise 115. —1. Europe, 34.16%; Asia, 8.99%; Africa, 2.94%;
North America, 44.97%; South America, 5.84%; Australasia, 3.10%.
2. Austria, 123.8%; England, 7.5%; France, 7.9%; Germany, 4.4%;
Ireland, 13.7%; Scotland, 3.6%. 3. 63J%. 4. 56£%. 5. 8J%.
6. 15^. 7. 20^perlb. 8. 4%. 9. 27J%. 10. 80^. 11. $20.
12. 17}%. 13. 100 Ib. 14. 9|%. 15. 16}%.
Exercise 116. — 1. $54.60; $782.60. 2. $70.35; $740.35.
3. $126; $1386. 4. $36.83; $421.83. 5. $137.50; $2887.50.
6. $267.60; $3612.60. 7. $34.80; $817.80. 8. $41.79 ; $638.79.
9. $56; $3056. 10. $37.60; $977.60. 11. $9; $1809. 12. $231;
$2331. 13. $47.97; $1007.97. 14. $189.56; $3100.81. 15. $136.18;
$1993.18. 16. $70.30; $2845.30. 17. $58.63 ; $1828.63. 18. $51.08;
$2026.22. 19. $43.73; $1261.73. 20. $ 113.98 ; $1901.98.
Exercise 117. — 1. $3.88. 2. $1.96. 3. $1.18. 4. $24.07.
5. $13.96. 6. $26.02. 7. $35.16. 8. $49.79. 9. $39.46.
10. $39.52.
Exercise 118. — 1. $500. 2. $1200. 3. $1000. 4. $240.
5. $2400. 6. $300. 7. $600. 8. $800. 9. $195. 10. $450.
11. $1020. 12. $324. 13. $300.
Exercise 119. — 1.1 yr. 2. 3 yr. 3. f yr. 4. 4 yr. 5. 1 yr.
6. 1 yr. 9 mo. 7. 3 yr. 3 mo. 8. 1 yr. 1 mo. 16 da. 9. 1 yr.
10 mo. 15 da. 10. 1 yr. 1 mo. 15 da. 11. 1 yr. 10 mo. 12. 8 mo.
15 da. 13. 2 yr. 11 mo.
Exercise 120. - 1. 6%. 2. 7%. 3. 6%. 4. 8%. 5. 7%.
6. 6%. 7. 5%. 8. 4%. 9. 5%. 10. 4ff%. 11. 8%. 12. 9%.
13. 4}%. 14. 6%. 15. 6%. 16. 7%. 17. 5%. 18. 6%.
19. 5}%. 20. 4}%.
Exercise 121. — 1. $800. 2. $700. 3. $600. 4. $840.
5. $360. 6. $230. 7. $11,505. 8. $162.50. 9. $580.
10. $630. 11. $740. 12. $450. 13. $403. 14. $240.
15. $1917.50. 16. $1800. 17. $245. 18. $240. 19. $22,530.
20. $803.05.
ANSWERS 345
Exercise 122. — 1. $392. 2. $6.43. 3. $1.44. 4. $5.51.
5. $24.57. 6. $858.67. 7. $548.91. 8. $999.84. 9. $14,000.
10. 7%. 11. 7%. 12. $24,000. 13. $6000. 14. 1 mo. 18 da.
15. 4 mo. 16. 2yr. 6 mo. 17. 1 yr. 8 mo. 18. $48. 19. $204.40.
20. 219 da. 21. 2 mo. 12 da. 22. $500. 23. $1072.50. 24. $13,000.
Exercise 123. —1. Discount $4.13. 2. Discount $5.87. 3. Dis-
count $7.50. 4. Discount $7.00. 5. Discount $ 9.50. 6. Discount
$1.00. 7. Discount $7.25. 8. Discount $2.25. 9. Discount $4.00.
10. Discount $2.92. 11. Discount $13.33. 12. Discount $5.00.
13. Discount $10.40. 14. Discount $3.37. 15. Discount $1.90.
16. Discount $6.16. 17. Discount $ 10.92. 18. Discount $ 8.55.
Exercise 124. — 1. Discount 01.65, Proceeds $352.72. 2. Dis-
count $9.25, Proceeds $391.87. 3. Discount $1.61, Proceeds $450.86.
4. Discount $6.08, Proceeds $601.92. 5. Discount $5.05, Proceeds
$499.95. 6. Discount $6.64, Proceeds $898.31. 7. Discount $16.89,
Proceeds $996.44. 8. Discount $7.45, Proceeds $750.05. 9. Discount
$3.04, Proceeds $801.36. 10. Discount $4.24, Proceeds $399.96.
11. Discount $21.38, Proceeds $843.49. 12. Discount $1.96, Proceeds
$904.79. 13. Discount $ 17.60, Proceeds $ 1201.
Exercise 125.— 1. $280. 2. $720.13. 3. $781.25. 4. 31$%.
5. 20%, $21. 6. 26%. 7. (a) $441, (6) $3528, (c) $2244,
(d) $4845, (e) $2565, (/)$1344, (gr) 2736, (ft) $1782, (i) $1642.20.
Exercise 126. — 1. $2835.13. 2. $1776.30. 3. $233.67.
4. $480.85. 5. $1524.53. 6. $479.46. 7. $153.97. 8. $427.89.
9. $268.02. 10. $122.08.
Exercise 127. — 1. $62.84. 2. $215.47. 3. $325.31. 4. $197.25.
Exercise 129. — 1. $2726.80. 2. $3497.40. 3. $5086.35.
4. $6305.72. 5. $8279.64. 6. $9955.05. 7. $5484.76. 8. $5287.92.
9. $6027.92. 10. $6097.61. 11. $5393.25. 12. $9771.30. 13. $4509.
14. $6913.08. 15. $7789.72. 16. $1232.46. 17. $4.20.
18. $9987.50. 19. $598.50. 20. $993.33. 21. $891.
Exercise 130. — 1. 25.215 francs, 20.45 marks, 23.973 crowns.
2. (a) $86.85, (6) $106.15, (c) $75.04, (d) $214.20, (e) $93,
(/) $146.16, (g) $120.60. 3. $5. 4. 2150.53 colons, 4926.11 crowns,
3731.34 crowns. 5. $38,600. 6. $4980. 7. .2055 libra. 8. 3.731
crowns. 9. $20. 10. $10,800.
Exercise 131. — 2. $582.94, $618.49, $725.14. 3. $349.78.
4. $205.00. 5. (a) $40,270.80, (6) $11(J9.40, (c) $3792. 6. (a) $2654.75,
(5) $14,842.82,
346 ANSWERS
Exercise 132. — 1. $4872. 2. $2671.14. 3. $1116.50.
4. $17,064.12. 5. $352.42. 6. $1717. 7. $289.50.
8. (a) $4503.22, (6) $4134.75, (c) $600, (<f) $1798.45, (e) $1636.25,
(/) $2840.02.
Exercise 133. —1. £1200. 2. £250. 3. 38,080 marks.
4. 19,300 francs. 5. 2680 krones. 6. £486 10s. 7. 5376.85
francs. 8. 23,801.68 marks.
Exercise 134. — 2. $1620, $1640. 3. $3720. 4. $29,075.
5. $2965.62. 6. $5337.50. '7. $8200 ; $9750; $9300 ; $12,962.50 ;
$14,287.50; $13,412.50; $10,850; $14,412.50; $16,837.50; $6300.
Exercise 135. — 1. $14,262.50 ; $14,125. 2. $9737.50. 3. 100
shares. 4. $125. 5. $125. 6. $400. 7. 200 shares. 8. 100
shares. 9. 1000 shares. 10. 200 shares.
Exercise 136. — 1. 159}. 2. 149}. 3. $18,550. 4. $32,718.75.
5. 4||o/o> 6 62a. 7. $30,918.75. 9. $875; no brokerage.
10. 5f%. 11. $200, $5093.76. 13. Central. 14. 438 shares.
Exercise 137. — 1. $62.40. 2. $500. 3. $2.16. 4. $26.28.
5. £43 4s., or $210.23. 6. $1.81. 7. $1.46. 8. $256.80.
9. $58. 10. $72, $8.19 each. 11. $91.80. 12. $17.50, $81.
13. $65. 14. $552, $2.208. 15. $750. 16. $28.80. 17. $94.50.
18. $125. 19. £5. 20. $82.12,29.4^.
Exercise 141. — 1. .01, .04, .09, .16, .25, .36, .49, .64, .81. 2. 1, J,
i» re? A» A» A> A» 8T> Trioi T?T> ii?> ri-g? rirs> si?» siff» *}?» *ii» *ii»
^o- 3. t, A, it, A, j&» ifj, m ^v 4. 2j, 5|, if i, IA, SA,
3ft, 4fj, 39 A- 6. 1, J, sV, ^, Ti5, ^ rfs^sh. 7l* ioW» TsVp
T?V^' ^1^7' ^TT¥? "SlVs' 4^^' ¥"^r"J' 3FS^» ^S^' S'TjW' ^' 3025.
8. (a) 3, (6) 85.6735, (c) 29.1708, (d) 66.0806, (e) .4670, (/) .5790,
(0) .8991, (ft) .5, (0 .6, (j) .9.
Exercise 142. — 1. 13. 2. 21. 3. 25. 4. 31. 5. 32.
6. 43. 7. 53. 8. 62. 9. 65. 10. 73. 11. 76. 12. 82.
13. 84. 14. 87. 15. 92. 16. 96. 17. 98. 18. 99. 19. 89.
20. 78. 21. 69. 22. 69. 23. 49. 24. 58.
Exercise 143. — 1. 317. 2.332. 3.347. 4.414. 5.436.
6. 447. 7. 479. 8. 527. 9. 557. 10. 595. 11. 626. 12. 676.
13. 689. 14. 708. 15. 809. 16. 879. 17. 905. 18. 909.
Exercise 144. — 1. .388. 2. .496. 3. .587. 4. .539. 5. .513.
6. .679. 7. .729. 8. .785. 9. .885. 10. .288. 11. .249.
12. .0608,
ANSWERS 347
Exercise 145. — 1. 1.095. 2. 2.062. 3. 1.049. 4. 2.28.
5. 1.817. 6. 2.291. 7. 1.768. 8. 3.028. 9. 1.173. 10. 2.121.
11. 1.696. 12. .764. 13. .816. 14. .632. 15. .7977.
Exercise 146. — 1. 69.57yd. 2. 241yd. 3. 238yd. 4. 1.025
mi. 5. 739 rd. 6. 311.13 yd., 155.56 yd. 7. 538.89 yd., 179.63
yd. 8. 19.41 rd.
Exercise 147. — 1. 10. 2. 13. 3. 17. 4. 29. 5. 10(3.
6. 101. 7. 145. 8. 89. 9. 149. 10. 68.5. 11. 425.
12. 305. 13. 433. 14. 305. 15. 50ft. 16. 14.14 rd.
17. 30.232 rd.
Exercise 148.— 1. 152. 2. 184. 3. 280. 4. 2.17. 5. .2P1.
6. .319. 7. .2. 8. .748. 9. .455.
Exercise 149. — 1. 126. 2. 180. 3. 264. 4. 840. 5. 522.
6. 9240. 7. 150,769. 8. 8.625 A. 9. 5.775 A. 10. 1.638 A.
11. 1.68 A. 12. 43.301 sq. rd. 13. 1082.53 sq. rd. 14. 182.25 sq. in.
15. 2592 sq. ft. 16. 57.42 ft.
Exercise 150. — 1. 69.12. 2. 144.51. 3. 471.24. 4. 515.22.
5. 615.75. 6. 420.97. 7. 540.35. 8. 22.62. 9. 37.07.
10. 45.87 11. 102. 12. 152. 13. 7.8. 14. 13.27. 15. 60.
16. 8.5. 17. 9.6. 18. 6.7. 19. 480 times.
Exercise 151. — 1. 615.8. 2. 1520.5. 3. 4071.5. 4. 69.4.
5. 132.73. 6. 232.35. 7. 295.59. 8. 4778.4. 9. 3217.
10. 7238.2. 11. 4300.8. 12. 6647.6. 13. 795.8. 14. 484.15.
15. 548.2. 16. 688.3. 17. 6.023. 18. 3.789. 19. 7.643.
20. 9.282.
Exercise 152. — 1. 31. 2. 17. 3. 33. 4. 42. 5. 2.6.
6. 7.2. 7. 9.2. 8. 9.8. 9. 11.8. 10. 13.4. 11. 7.6. 12. 9.3.
13. 99. 14. 850. 15. 650. 16. 39.25yd. 17. 288,576,452.4.
18. 329.9 sq. in. 19. 1661.9 sq. in. 20. 259.8. 21. 139.1 sq. in.
22. 259.8 sq. in., 225 sq. in. 23. Circle.
Exercise 153.— 1. 46.5 in. 2. 49.22 in. 3. 1.01ft. 4. 36°.
5. 3° 36'. 6. 31' .2.
Exercise 154. — 1. 20ft. 2. 176ft. 3. 110 mi. 4. 7J mi.
5. 22.36 ft. 6. 7.07 ft. 7. 16 : 121. 8. 4:9. 9. 186.96 sq. mi.
Exercise 155. — 1. 576 sq. ft. 2. 680ft. 3. lOf ft. 4. 7J ft.
5. 2080 sq. ft., $52. 6. 630 sq. ft. 7. $26.40. 8. 147 sq. yd. 9. 53.45
sq. ft. 10. 4021 sq. in. 11. 67 yd. 12. 2984.5 sq. ft. 13. 15,456.6
sq. in. 14. 8392.70. 15. 120,687 sq. in. 16. 5541.7 sq. in.
17. 28,842,700 sq. mi. 18. 186,265,000 sq. mi. 1'9. (1) Jupiter,
23,235 million sq. mi. ; (2) Uranus, 3217 million sq. mi. ; (3) Neptune,
348 ANSWERS
3421 million sq. mi. ; (4) Saturn, 16,741 million sq. mi. 20. 58 in.
Exercise 156. — 1. 5280 cu. in. 2. 700 cu. in. 3. 4071. 5 cu. in.
4. 15,708 cu. in. 5. 2598 cu. in. 6. 33,510 cu. in. 7. 2144.7
cu. ft. 8. 4094 gal. 9. 4562 gal. 10. 3:2. 11. 1:2.
12. 52 cu. ft. 1269 cu. in. 13. 3 ft. 14. 4189 cu. in. 15. 40 ft.
16. 1,367,631. 17. 48 times. 18. 760 times. 19. 1331 times.
Exercise 157. —1. 30° C. 2. 25° C. 3. 95° C. 4. 120° C.
5. 20° C. 6. 12f°C. 7. 3i°C. 8. - 5° C. 9. - 10° C.
10. -25°C. 11. -40°C. 12. -67J°C. 13. 95° Fahr.
14. 131° Fahr. 15. 77° Fahr. 16. 68° Fahr. 17. 64.4° Fahr
18. 46.4° Fahr. 19. 14° Fahr. 20. - 4° Fahr. 21. 6.8° Fahr.
22. 0.4° Fahr. 23. - 11.2° Fahr. 24. - 459.4° Fahr. 25. Mer-
cury, -40°; sulphur, 235.4°; lead, 618.8°; zinc, 779°; gold, 1895°;
cast iron, 2012° to 2102°. 26. 39.2°.
Exercise 158. — 1. 3,921,138.813 meters. 2. 107,934,859.86cm.
3. 3,131,587.7mm. 4. 28.434km. 5. 19454m. 6. 140.784m.
7. 960 times. 8. 7.03 times. 9. 5000 times. 10. 112|.
11. 20,000. 12. 62.5 times. 13. 5000. 14. 1200, 480.
Exercise 159. — 1. 7.8125 ha. 2. 1.8605 ha. 3. 5.7122 ha.
4. 6.82 ha. 5. 11.95 ha. 6. 6.65 ha. 7. .81 cbm. 8. 3.36 a.
9. 1.91 ca. 10. 24,429 c.cm. 11. 119 cm. 12. 1596 cbm. 13. 169.65 ca.
14. 51.08 km. 15. 462.3 qcm.
Exercise 160. — 1. $1156.43. 2. |622.91. 3. $1650.
4. $1278.12. 5. $983.59.
Exercise 161. — 1. $2737.14. 2. $4031.75. 3. $8005.16.
4. $9773.37. 5. $11,876.05. 6. $2853.54.
Exercise 162. — 3. 2J hr. 4. IJf hr. 5. 4T8g hr. 6. £J.
7. f hr. 8. 1 hr.
Exercise 163. — 1. 1260 mi. 2. 5 hr., 150 mi. 3. 10.38A.M.
4. 22 mi. per hr. 5. 105.6 yd. 6. 180 mi.
Exercise 164. — 1. 48, 72, etc. 2. 6 min. spaces. 3. lO^min.
past 2 o'clock, 16^ min. past 3 o'clock, 27T3T min. past 5 o'clock, 38r2T
min. past 7 o'clock, 49^ min. past 9 o'clock, 54T6T min. past 10 o'clock.
At no time. 5. (a) 38T2T min. past 1 o'clock, (&) 49Jr min. past 3
o'clock, (c) 10}£ min. past 8 o'clock, (d) 16^ min. past 9 o'clock,
(e) 27T3r min. past 11 o'clock, (/) 32T8T min. past 12 o'clock. 6. 49^
min. past 3 o'clock, 5j\ min. past 7 o'clock. 7. (a) 27^ min. past
2 o'clock, (&) 32T*r min. past 3 o'clock, (c) 5j5r min. past 4 o'clock,
38r2T min. past 4 o'clock, (d) 16r4T min. past 6 o'clock, 49T11 min. past
0 o'clock, (e) 27^ min. past 8 o'clock, (/) 32T8r min. past 9 o'clock,
ANSWERS 349
ft) 16T4r min. past 12 o'clock. 8. 13^-
min. past 4 o'clock, 5 o'clock, 48 min. past 4 o'clock. 9. 38T2r min.
past 5 o'clock.
MISCELLANEOUS EXAMPLES (A)
8. 8999.991. 9. 274.999225. 10. 8,447,537,940,492. 11. 636,300,000.
12. 751,700,800. 13. 7.1407. 14. 814,585.36. 15. 166.375 mi.
16. $3828.12. 17. $3670.12. 18. .000504. 19. 1.12550881.
20. 278,500,000. 21. (1) $.475, (2) $3.508, (3) $51.877.
22. 372.015. 23. 28,127,000 nearly. 25. 52,800 mi.
26. .341. 27. .8251. 28. .1704. 29. .000439625. 30. 14.461°.
81. 8° 39' 6". 32. 80.7 X'. 33. 113,400. 34. 250. 35. 80,032,000 nearly.
36. 77. 37. $3.78. 38. $1800. 39. $384.45. 40. $30.80.
41. 12,290. 42. $20.25. 43. .5774. 44. .037. 45. T|T, or .0390625.
46. 144. 47. 2, 3, 6, 9, 18. 48. 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72.
49. 1110 = 2 • 3 • 5 . 37 ; 777 = 3 . 7 . 37 ; 1001 = 7 • 11 • 13 ; L.C.M. 2 . 3 -
5 - 7 • 11 - 37. 50. 26. 51. 65,520. 52. 25. 53. 19 ; 66,880.
54. 2, 3, 52, 72, 11. 55. 2,784,873. 56. 2* .38 - 11 . 13. 57. 25 A.
58. 80 poles. 59. }, f , JJ. 60. fa. 61. 14 j} 62. T%.
63. 1.09375, .109375, .128. 64. J>ff, ^T, ^. 65. 3&. 66. 1^.
67. .025. 68. .027. 69. .075. 70. .16583. 72. Upwards of
80,000 yr. 73. (1)39.534, (2) 48.321, (3) 54.197, (4) 63.25,
(5) 63.615, (6) 68.234, (7) 72.932, (8) 76.459, (9) 79.68,
(10) 82.951, (11) 70.13. 74. $11,687.50. 75. 20,000 bu.
76. $3062.50. 77. $42.40. 78. $92.67. 79. $2.174. 80. $728.42.
81. 36.3 nearly. 82. 5,201,300. 83. 52,100,000 nearly. 84. 56,310,000.
85. 12.44496 ft. 86. 712.5 Ib. 87. 320. 88. 1.292 sec.
89. 4888fcu. yd. 90. .002055 nearly. 91. $6256. 92. $3344.
93. .2565. 94. 19s. 2J<2. 95. .15625. 96. (a) 147,824, (ft) 22,098,
(c) 463,180, (d) 823,750. 97. (a) 166,488, (ft) 526.380. 98. 640 A.
99. 2640 revolutions. 100. 112.5 bu. 101. ^. 102. $75.
103. $24,800. 104. 10 A. 105. $ 14f 106. (i) .00}, (ii) .00375.
107. (a) — , (ft) — , (c) J?t, (d) — .• 108. 99.99. 109. -2r6/.
100 ' 100 ' 100 100
110. $107. 111. 2.28. 112. iJ. 113. 66,720. 114. $101.02.
115. 25ft. 116. 35. 117. 847.15625. 118. 28J. 119. $7260.
120. 390. 121. 1092 eggs. 122. ^f. 123. 23f-. 124. 17.66.
125. 5.25ft., 212. 0045 sq. ft., 98.328 sq. ft., 59.427 sq. ft. 126. $330.
127. 67^ mi. 128. 7.15p.M., Apr. 5. 129. 9.55A.M. 130. 7 hr.
29 min. 46 sec. 131. 63° 35' W. 132. 156° 15' W. 133. 135° 27' E.
134. (a) 70° 15' W., (ft) 85° W., (c) 81° 45' W. 135. 24.62%.
350 ANSWERS
136. (i) 4.16%, (ii) 14.6%, (iii) 28.79%, (iv) 17.8%. 137. 96,600.
138. $150. 139. $1.44. 140. 20%, or 16? per yard. 141. $8.
142. 24f%. 143. }. 144. $150. 145. 38|%. 146. 12J%.
147. (i)64TV/0, (ii)3fgal. 148. (a) 59.99%, (6)32.34%, (c) 62.84%,
(d) 49.02%, (e) 65.57%, (/) 36.75%, (?) 65.28%, (ft) 40.83%,
(0 41.94%, 0) 39.01%, (fc) 42.02%. 149. $1.61. 150. $1.20-
151. $182.67. 152. £7 17s. 8d 153. £41 16s. Id. 154. $300.
155. $14,400. 156. $400. 157. $1200. 158. $1000. 159. 7%.
160. $500. 161. 62^. 162. $546.69. 163. July 23, $597.14.
164. $149.15. 165. $891.80. 166. $1004.72. 167. Mar. 11, 1903,
$824.89. 168. $459.50. ' 169. 7500 sq. yd. 170. 3500 sq. yd.
171. $62.80. 172. 301|sq. yd. 173. 1 mi. 174. 1 mi. long, J mi.
wide. 175. 4330 sq. ft. 176. 6495 sq. ft. 177. 355 in. 178. 314.16
sq. in. 179. 1256.64 sq. in. 180. 1.9531 cu. ft. 181. 6.07 in.
182. $.525. 183. $.882. 184. $1. 185. $.612. 186. $.756.
187. $.871. 188. $1.802. 189. $.755. 190. $3. 191. 512.12 francs.
MISCELLANEOUS EXAMPLES (B)
1. T|o. 2. 38° 15'. 3. 216. 4. 2.77ft. 5. 2.0575 yr.
6. 3 gal. 1.02 gi. 7. $4176. 8. $20.94. 9. Ill bu. 1 pk. 2J qt.
10. 17f . Ib. 11. IfjjJ. 12. $504. 13. 1.2857142. 14. 1.777+.
15. 16ft. 16. 12.56A.M. 17. $500. 18. $11.28. 19. 3J da.
20. 2ii|. 21. 45J bu. 22. 82.28 yr. 23. 23^. 24. 226.27+ rd.
25. $6.66. 26. 149.61+ gal. 27. 734 rd. 28. 90yd. 29. $16.67.
30. 137 da. 31. $331.86. 32. 4531 mi. 33. Answers will depend
on date. 34. 14. 35. $9.60. 36. 10 A. 37. 30 inin. 50 sec.
38. 6ch. 89.6+b. 39. §116.35$. 40. $37.50, $30, $52.50.
41. 62.72 bu. 42. 3403J T. 43. Neither gained nor lost. 44. 9r%.
45. 2295. 46. 140 A. 47. 4844.996. 48. $75.15, $83.50, $300.60.
49. 6 hr. 1 min. 28 sec. A.M. 50. Answer will depend on day calculation
is made. 51. $2800. 52. $604.86. 53. $50 loss. 54. 25%.
55. $460. 56. 10.8%. 57. 16 yr. 8 mo., 12 yr. 6 mo., 11 yr.
58. $453.561. 59. $352.35. 60. 4J-%. 61. 210,526.3 Ib.
62. 8275.80. 63. 35J%. 64. 410 on the $ 100, $4.92. 65. $6.67.
66. $56.25. 67. $30,563.38+. 68. 80. 69. 2X\%. 70. 33J%.
71. $855. 72. $27,200. 73. $32.32 loss. 74. $1092.56.
78. 30%. 79. 1 yr. 7 mo. 12 da. 80. $492.50. 81. 300%.
82. $3200. 83. 33 J%. 84. 35%. 85. $37.525. 86. $108.90.
87. 25%. 88. $753. 89. $5.69. 90. Dec. 13, 1902. 91. $6644.63.
92. 83515.63. 93. 3|%. 94. $530. 95. $15,000. 96. $1533.75.
97. 9f% 98. Yes, $20 better. 99. $2042. 100. 25%.
101. $1000. 102. $7045. 103. $28.01, $42.02. 104. 6}Jf
Tarr and McMurry's Geographies
A New Series of Geographies in Two, Three, or Five Volumes
By RALPH 5. TARR, B.S., F.G.S.A.
CORNELL UNIVERSITY
AND
FRANK M. McMURRY, Ph.D.
TEACHERS COLLEGE, COLUMBIA UNIVERSITY
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When ordering, be careful to specify the Book or Part and the Series desired,
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THE MACMILLAN COMPANY
64-66 FIFTH AVENUE, NEW YORK
BOSTON CHICAGO ATLANTA SAN FRANCISCO
FIRST BOOK OF
PHYSICAL GEOGRAPHY,
By RALPH S. TARR,
Professor of Dynamic Geology and Physical Geography
at Cornell University.
tamo. Illustrated. Half leather. $1.10, net.
M The style is simple, direct, and the illustrations helpful; the book,
indeed, being so attractive that one hopes it will inspire even in the
pupil who gives it briefest time a longing to know more of the marvels
of our world." — Providence Journal.
" Although intended for school use, there are few readers who will
not be profoundly interested in the volume, which is profusely illus-
trated. Technical terms are avoided as far as possible, and where they
are used they are clearly explained." — Boston Transcript.
" This book is packed with information needed by every grammar-
school pupil; but what signifies vastly more, the pupil gets this infor-
mation in a way that gives thorough discipline — in observation, careful
reading, discriminating thinking. This book is the best possible proof
of the statement that all new science work depends for its value upon
being rightly taught. This book is an admirable presentation of prac-
tical pedagogy." — Journal of Education.
"The style of Professor Tarr's book is literary, scholarly, and sane;
a pleasing relief from the disjointed paragraphs of some of his con-
temporaries. . . . This book will prove a formidable rival to the best
physical geographies now in the field. "— - Educational Review.
" No written description of the book can do justice to it. It will well
repay personal examination." — New York Education.
THE MACMILLAN COMPANY
66 FIFTH AVENUE, NEW YORK.
THIS BOOK IS DUE ON THE LAST DATE
STAMPED BELOW
AN INITIAL FINE OF 25 CENTS
WILL BE ASSESSED FOR FAILURE TO RETURN
THIS BOOK ON THE DATE DUE. THE PENALTY
WILL INCREASE TO SO CENTS ON THE FOURTH
DAY AND TO $1.OO ON THE SEVENTH DAY
OVERDUE.
OCT' 5 1936
._.„._::.•,_
OCT 6 1936
OCT 1.1.
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918235(54,03
//73
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