THE
NEW BUSINESS
ARITHMETIC
REVISED EDITrON
LIBRARY
OF THE
UNIVERSITY OF CALIFORNIA
GIFT OF
Class
THE NEW
Business Arithmetic
A TREATISE ON
Commercial Calculations
REVISED EDITION
J. A. LYONS & COMPANY
NEW YORK CHICAGO
6
APR 4 19H
GIFT
COPYRIGHT. 1906,
BY
POWERS & LYONS
PREFACE
Since the great majority of those who study Arithmetic need
to use it in the transaction of the practical affairs of life, a text
book on the subject, should be especially practical. It has been
the aim in the preparation of this work to represent the business
methods of the times. While a few problems are intended to
illustrate some principle, by far the greater number show the
application of Arithmetic to the actual business transactions of
the day. In this practical character we believe the book will be
found unequaled.
The principles of each successive topic are carefully developed
by appropriate exercises, so graded that the mind of the student
must inevitably grasp the relations of the whole subject, and
when the work is completed, comprehend it, not as a mass
of looselyconnected details, but as a unified whole.
Since good methods economize time and energy, secure rapid
ity and accuracy in calculation, and strengthen the reasoning
powers, the aim has been to present each subject in the most
clear and concise manner, showing the reason for every operation
performed, in order that the student may learn to rely upon the
principle involved and not merely seek for a result.
The treatment of the fundamental rules of Arithmetic has
been made simple, and is free from all effort to exalt these rules
into complex and difficult propositions. A clear explanation,
followed by an abundance of well graded problems, will enable
the pupil to readily master the foundation work of Arithmetic.
The work in Fractions is made plain by giving brief, clear
and accurate definitions, and simple, concise and logical solutions
of concrete problems. Compound Numbers are explained by
showing their relation to simple numbers, while the exercises
and problems deal with facts found in everyday life.
The subject of Percentage and its applications is made most
3
 .    :
4 PREFACE
thorough and practical. A system of analysis is employed which
must inevitably fix the principles clearly in the mind of the
learner.
Equation of Payments, Averaging Accounts and Partnership
have been emphasized according to the demand of business.
That the reasoning powers of the pupil may be strengthened
and his ability to think independently of pencil and tablet may be
increased, several hundred problems for oral solution have been
added. The work undoubtedly now contains an abundance of
material, not only for giving facility in computations, but for
correct training in arithmetical thought.
We have made an earnest effort to present such a work to
teachers and students as will meet with their approval and suit
their wants. We believe by a thorough study of the work young
men and women will go out into the business world intelligent
persons with ability to apply their knowledge.
THE AUTHOR.
CONTENTS
PAGE:
DEFINITIONS 7
NOTATION AND NUMERATION 8
ADDITION 13
SUBTRACTION 18
MULTIPLICATION 22
DIVISION 30
FACTORING 37
CANCELLATION ' 38
GREATEST COMMON DIVISOR 40
LEAST COMMON MULTIPLE 41
FRACTIONS 44
Reduction 46
Addition : 52.
Subtraction 55
Multiplication 58
Division 64
DECIMAL FRACTIONS 74
Reduction 77
Addition 79
Subtraction 81
Multiplication 82
Division 83
COMPOUND NUMBERS 85
UNITED STATES MONEY 86
Addition 87
Subtraction 89
Multiplication 90
Division 91
SHORT METHODS 92.
BILLS 97
REDUCTION OF DENOMINATE NUMBERS 105
United States Money 105
Canada Money 105
English Money 106
Avoirdupois Weight 108
Trov Weight \ 110
Apothecaries' Weight Ill
Comparison of Weights 112
Long Measure 113
Surveyors' Long Measure 115
Square Measure 115
Board Measure 118
Surveyors' Square Measure 119
Cubic Measure 123
Liquid Measure 125
Apothecaries' Fluid Measure 126
Dry Measure 126
Comparison of Dry and Liquid Measures 127
Time 128
Circular Measure 130'
REDUCTION OF DENOMINATE FRACTIONS 134
6
6 CONTENTS
DENOMINATE NUMBERS Addition 139
Subtraction 141
Multiplication 143
Division 144
LONGITUDE AND TIME 146
RATIO 154
Proportion 155
Compound Proportion 158
MEASUREMENTS USED IN BUSINESS 161
PERCENTAGE 167
PROFIT AND Loss 183
MARKING GOODS 190
TRADE DISCOUNT 193
BILLS 197
COMMISSION 205
INSURANCE 213
Fire Insurance 214
Marine Insurance 217
Life Insurance 219
INTEREST 225
Sixty Days Method 232
Six Per Cent Method 233
Cancellation Method 235
Common Bankers' and Exact Interest Compared 236
Annual Interest 243
Compound Interest 245
COMMERCIAL PAPER 250
Partial Payments 255
Annual Interest, with Partial Payments 260
TRUE DISCOUNT % 261
BANK DISCOUNT 264
STOCKS AND BONDS 268
EXCHANGE 277
Domestic Exchange 278
Foreign Exchange 281
BANKS AND BANKING 284
National Banks 284
Savings Banks 287
TAXES 292
CUSTOMS OR DUTIES 295
EQUATION OF ACCOUNTS 299
CASH BALANCE 311
PARTNERSHIP 313
INVOLUTION 328
EVOLUTION. 329
Square Root 330
Applications of Square Root 332
Cube Root 334
Applications of Cube Root
MENSURATION 338
Plane Figures 339
Solids 341
Metric System 345
Value of Moneys 350
STATUTORY WEIGHTS! . ... 352
ARITHMETIC
DEFINITIONS
1. Arithmetic is the science of numbers and their use.
2. A Unit is a single thing ; as, one, one man, one horse.
3. A Number is one or more units ; as 1, 3, 9, 6 boys.
4. The Unit of a Number is one of the kind expressed by
the number. The unit of 9 is 1, the unit of 20 feet is 1 foot.
5. An Integer is a whole number.
6. Like Numbers are those which are composed of the
same kinds of units. Thus 25 and, 34 ; 3 yards and 10 yards.
7. An Abstract Number is one used without reference to any
particular thing or quantity. Thus 15; 64; 280.
8. A Concrete Number is one used with reference to some
particular thing or quantity. Thus 25 dollars ; 14 days ; 100 men.
Concrete numbers are called denominate numbers because the de
nomination or kind is named.
9. A Sign is a character used to indicate an operation, or ex
press the qualities or relations of numbers.
A Solution is a process of computation used to obtain a re
quired result.
10. A Problem is a question for solution.
llT An Example is a problem solved, illustrating a principle
or rule.
12. A Principle is a truth upon which the solution is founded.
13. An Analysis is a statement of the successive steps in a
solution.
14. An Explanation is a statement of the reasons for the
manner of solving a problem.
15. A Rule is a direction for solving problems.
7
NOTATION AND NUMERATION
16. Notation is the art of writing numbers by means of
characters.
17. Numeration is the art of reading numbers written by
characters.
Two systems of Notation are in general use: the Roman and
the Arabic. The Roman is supposed to have been invented by
the Romans and employs seven capital letters to express num
bers. The Arabic is said to be derived from the Arabs and em
ploys ten characters called figures.
ARABIC NOTATION
18. Figures are characters used to represent numbers.
There are ten figures :
0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
Zero, One, Two, Three, Four, Five, Six, Seven, Eight, Nine.
The Figure 0, Zero or Cipher, expresses no units, or nothing,
when standing alon,e. The other nine figures express the num
ber of units shown by. their names. These figures are called
digits.
19. To express numbers greater than nine and less than one
hundred, two figures are written side by side ; as, thirtysix, 36 ;
seventytwo, 72.
20. To express numbers greater than ninetynine, three or
more figures are written side by side ; as, one hundred eightyfive,
185 ; two thousand six hundred twentyfour, 2624.
21. When figures are written side by side, the one at the right
expresses units or ones, the next tens, the next hundreds, the
next thousands, etc.
22. The Simple Value of a figure is the value it expresses
when standing alone, or in unit's place ; as 3, 7, 9.
23. The Local Value of a figure is the value it expresses
when used with other figures to represent a number,
8
NOTATION AND NUMERATION 9
In the number 345, the figure 5 expresses a simple value, and
the figures 3 and 4 express local values.
24. The Order of a Unit takes its name from the place it
occupies. A figure in the first place expresses units of the first
order ; in the second place, units of the second 'order, etc.
When a figure stands in the second place it represents tens;
in the third place hundreds ; in the fourth place thousands, etc.
One ten is written 10
One ten and four, fourteen 14
Two tens, twenty 20
Two tens and seven, twentyseven 27
Three tens, thirty . ; 30
One hundred, one ten and seven, one hundred seventeen. . . . 117
Three hundred, five tens and nine, three hundred fiftynine. . 359
Six thousand, five hundreds, nine tens and two, is read, six
thousand five hundred ninetytwo 6592
NOTE. In reading whole numbers the word AND should not be used.
Thus, seven hundred fiftyfour, not seven hundred and fiftyfour.
Copy and read the following numbers :
' 1. 297 5. 1790 9. 3096
2. 472 6. 4607 10. 7006
3. 685 7. 9218 11. 8200
4. 920 8. 2030 12. 6303
Write the following in figures :
13. One hundred fortysix.
14 Seven hundred ten.
15. Six hundred three.
16. Two hundred ninety.
77. Five hundred thirtyeight.
18 Three thousand seven hundred nineteen.
19. Six thousand nine hundred twentyseven.
20. Four thousand sixtyfour.
21. Seven thousand four hundred one.
22. Five thousand forty.
23. Nine thousand six hundred ninetysix.
24> Eight thousand eight hundred.
10
NEW BUSINESS ARITHMETIC
25. Seven units of the first order and two of the second.
26. Nine units of the fourth order, three of the third,
two of the second and one of the first.
27. Five units of the third order, one of the first.
28. Six units of the fourth order, seven of the second,
two of the first.
25. For convenience, figures are arranged in periods of three
places each ; the first three at the right being called units or one's
period; the next three the thousand's period; the next three the
million's period, etc.
Quadrillions. Trillions, Billions,
6th period. 5th period, 4th period,
TABLE
Millions,
3d period,
Thousands.
2d period,
546,
897,
to ti o
534,
,000 a
ffi HO
e rt c
3^3:
o p ct
a o
2 2. 3
s a
i
H W
3 g

3
2 6
05 Or
3* 3*
i!l
Units,
1st period, Periods.
9 8 ~~T? Figures.
% % Places.
*Z$
General Principles
1. Ten units of any order equal one unit of the next higher
order.. .Ten units equal one ten; ten tens equal one hundred; ten
hundreds equal one thousand, etc.
2. Removing a figure one place to the left increases its value
ten times. Removing a figure one place to the right decreases its
value ten times.
To Write Numbers
a. Begin at the left and write the figures belonging to the
highest period.
b. Write the hundreds, tens and units of each period in their
order, putting a cipher in the place of any order that is omitted.
NOTATION AND NUMERATION 11
To Read Numbers
a. Begin at the right and point off the numbers into periods
of three figures each.
b. Begin at the left and read each period separately, giving
the name to each period except the last.
Read the following:
1. 384 5. 136042 9. 147002001
2. 9328 6. 100420 10. 3073640240
3. 11765 7. 9793642 11 73260479142
4. 29470 8. 3106053 12. 48600752052
Write the following numbers :
13. Ninetyseven.
14. Three hundred sixtyeight.
15. Two thousand four hundred seventyfive.
16. Thirtyseven thousand one hundred ninetysix.
17. One hundred thirtysix thousand three hundred
twentyseven.
18. Five million three hundred six thousand five hun
dred three.
ROMAN NOTATION
26. Roman Notation employs seven capital letters to ex
press numbers, as follows :
Letters I, V, X, L, C, D, M. '
Values 1, 5, 10, 50, 100, 500, 1000.
These letters may be combined to express numbers according
to the following principles :
1. Repeating a letter repeats its value.
Thus, II represents 2 ; XX, 20 ; CCCC, 400 ; DD, 1000.
2. When a letter is placed after one of greater value, its value
is to be added to that of the greater.
Thus, VI represents 6 ; XV, 15 ; XXI, 21 ; DC, 600 ; DCX,
610.
3. When a letter is placed before one of greater value, its
value is to be taken from the greater.
Thus, IX represents 9 ; XL, 40 ; XC, 90 ; CD, 400.
12
NEW BUSINESS ARITHMETIC
4 When a letter of any value is placed between two letters,
each of greater value, its value is taken from the sum of the other
two.
Thus, XIV represents 14 ; XIX, 19 ; LIX, 59 ; CXL, 140.
5. A bar placed over a letter increases its value one thousand
times.
Thus, X~ represents 10000 ;lCL, 40000; CD", 400000.
27. TABLE OF ROMAN NOTATION
Roman
Arabic.
Roman.
Arabic.
Roman.
Arabic.
Roman.
Arabic.
I,
1.
IX,
9.
XX,
20.
xc,
90.
II,
2.
x,
10.
XXI,
21.
c,
100.
III,
3.
XIII,
13.
XXX,
30.
ccc,
300.
IV,
4.
XIV,
14.
XL,
40.
D,
500.
V,
5.
XV,
15.
L,
50.
DCC,
TOO.
VI,
6.
XVIII,
18.
LX,
60.
M,
1000.
VIII,
8.
XIX,
19.
LXXX,
80.
MD,
1500.
28. Express by Roman notation :
1. Eighteen.
2. Twentythree.
8. Fiftyeight.
4. Ninetynine.
5. Eightyfour.
6. One hundred eightyeight.
7. One hundred ninetynine.
8. Five hundred seventeen.
9. Six hundred fortyfive.
10. Seven hundred sixtyone.
11.
12.
13.
14
15.
428.
975.
1116.
23480.
76103.
29. Express by Arabic notation :
1. XXIX.
2. LXVIII.
8. CLXIV.
4. CXXIV.
5. cccxxxm.
6. DCLVI.
7. MDLVIII.
8. CLIL
9. VDXXII.
10. DX.
11. CXIX.
12. XICCIV.
13. MMDCXVIIL
14. VDXLIV.
15. MDLXXII.
ADDITION
30. Addition is uniting two or more numbers into one num
ber.
31. The Sum or Amount is the number obtained by adding.
32. The Sign of addition is an upright cross +, and is read
plus. When it is placed between two numbers, it shows that
they are to be added. $3 + $2 is read 3 dollars plus 2 dollars,
and means that 2 dollars are to be added to 3 dollars.
The sign $ is used for dollars, c. or cts. for cents.
33. The Sign of equality is two horizontal lines =, and is
read equal or arc equal to. 2 \ 5 = 7 is read 2 plus 5 equal 7.
34. When the amount of each column is less than ten.
1. A farmer raised 232 bushels of corn, 142 bushels of wheat
and 223 bushels of oats; how many bushels did he raise in all?
Find the sum of each of the following :
1
2
3
4
5
6
232
323
245
312
437
1102
142
242
321
243
140
1312
223
324
132
412
321
4132
597
7. What is the sum of 321, 142 and 323?
8. What is the amount of 213, 152 and 401 ?
9. What is the sum of 3232, 2323 and 4102 ?
10. I paid $212 for a wagon, $150 for one horse, $210 for
another horse, and $11 for a set of harness. What did I pay
for all?
13
14 NEW BUSINESS ARITHMETIC
35. When the sum of any column is greater than 9.
1. Find the sum of 3164, 2247, 4234 and 3232.
EXPLANATION. The sum of the units 2, 4, 7 and 4 is 17
units or 1 ten and 7 units ; write the 7 units under the col
umn of units and add 1 ten to the column of tens. The
sum of the tens 1, 3, 3, 4 and 6 is 17 tens or 1 hundred and
2247 ^ tens; write the tens under the column of tens and add
4234 the * hundred to tne column of hundreds. The sum of
the hunci reds ! 2, 2, 2 and 1 is 8 hundreds; write under the
column of hundreds. The sum of the thousands 3, 4, 2 and
19877 ^ * S ^ thousands or 1 tenthousand and 2 thousands; write
the 2 thousands under the column of thousands and the 1
tenthousand in the place of tenthousands. The result
12877 is the sum required.
1. Units of the same order are written in the same column ; and when
the sum in any column is 10 or more than 10, it produces one or more
units of a higher order, which must be added to the next column. This
process is sometimes called "carrying the tens."
2. In adding, learn to pronounce the partial results without naming
the numbers separately; thus instead of saying 2 and 4 are 6 and 7 are 13,
simply pronounce the results 6, 13, 17, etc.
From the foregoing examples and illustrations we deduce the
following :
To Add Whole Numbers
a. Write the numbers so that figures of the same order are in
the same column.
b. Begin at the right and add each column separately.
c. When the sum of any column is greater than 9, place the
righthand figure of the result under the column added and add
the remaining figure or figures to the next column.
d. Write at the left the sum of the last column.
PROBLEMS
m
24
(8)
265
432
(5)
1362
to
3420
(V
9416
32
314
864
1487
1862
3624
46
286
526
4532
1425
1583
84
627
893
2386
6347
2436
ADDITION
15
,8)
(9)
(10)
(11)
(12)
(IS)
234
979
9140
94187
71758
986756
562
2864
6968
71849
3680
863694
846
52
8947
48197
797
387623
324
715
7968
89471
36425
890124
118
3680
5392
19478
943628
1369479
462
9289
18364
26480
102154
279562
367
360
27147
62849
864209
8325791
214
14006
38297
56783
579135
2345678
14. 128 + 324 + 116 + 893 + 246 + 427 = how many?
15. 1265 + 3482 + 2149 + 3625 + 1304 + 107 = how
many?
16. 28603 + 24567 + 39042 + 16841 + 40218 = how many ?
17. Find the sum of $347, $962, $375, $842 and $636.
18. What will be the amount of $3476, $1924, $4822, $3965
and $7180?
19. Add 8765 feet, 5678 feet, 6758 feet, 7685 feet and 3629
feet.
20. Add fortynine, seventysix, three hundred twentyfive,
nine thousand six hundred thirtythree, five thousand one hun
dred ten and sixtytwo thousand four hundred eleven.
21. Find the sum of three hundred seventy, two thousand
eightyone, seven thousand four hundred sixteen, fifty thousand
one hundred twentynine and four hundred fortyfour thousand
six hundred ninetythree.
22. A paid $762 for hogs, $1869 for cattle, $3796 for horses
and then had $9240 remaining. How much had he at first?
28. I sold six cows that weighed as follows: 1824 pounds,
1369 pounds, 964 pounds, 2217 pounds, 1746 pounds, 1940 pounds.
How many pounds did they all weigh ?
24. A farmer bought four farms. He paid $3221 for the first,
$5680 for the second, $4216 for the third and $2645 for the fourth.
How much did he pay for all ?
25. I paid $212 for a wagon, $154 for one horse, $210 for an
other horse and $65 for a set of harness. What did I pay for all ?
16 NEW BUSINESS ARITHMETIC
26. R. D. Lyman bought four lots. He paid $2232 for the
first, $3124 for the second, $1485 for the third and $2238 for the
fourth. Find the cost of the four lots.
27. A merchant paid $746 for calico, $294 for linen, $2864
for shoes, $212 for toys and $1169 for carpets. How much did
he pay for all ?
28. A farmer raised 1278 bushels of corn, 1642 bushels of
wheat, 765 bushels of oats, 367 bushels of rye, 93 bushels of bar
ley and 160 bushels of buckwheat. Find the number of bushels
of grain he raised.
(29) (SO) (31) (32) (33) (34)
476 + 908 + 126 + 443 + 180 + 1265 = x
390 + 371 + 324 + 298 + 976 + 3428 = x
915 + 569 + 503 + 876 + 209 + 1456 = x
207 + 245 + 891 + 569 + 314 + 9234 = x
841 + 703 + 736 + 137 + 563 + 1867 = x
632 + 421 + 5.17 + 910 + 842 + 2854 = x
234 + 127 + 143 + 347 + 175 + 3629 = x
143 + 354 + 274 + 256 + 224 + 2872 = x
536 + 781 + 531 + 324 + 135 + 3428 = x
245 + 436 + 275 + 463 + 253 + 9234 = x
x + *" + x + x + x + x = x
35. The proprietors of a college paid $2675 for rent, $6286
for teachers, $824 for school furniture, $269 for lights and $970
for fuel. Find the total expense.
36. A bankrupt firm's resources are cash $740, drygoods
$1965, boots and shoes $1647, Brown's note $1278, office furniture
$280 and real estate $2394. Find the total resources of the firm.
37. I bought four horses for $85 each. I sold the first for $12
more than cost, the second for $16 more than cost, the third for
$26 more than cost and the fourth for $41 more than cost. How
much money did I receive for all?
38. A, B, C and D form a partnership. A invests $2640, B
invests $3160, C invests $1125 more than A and B together, and
D invests as much as A. and C together. How much 4id they all
invest in the business ?
ADDITION
17
39. A stock dealer bought 218 sheep for $568, 319 hogs for
$1162, 123 calves for $2316, 24 oxen for $695 and 11 horses for
$957. How many head of stock did he buy and how much did
they cost?
40. I sold a house for $3278 and a lot for $1360. I lost $392
on the house and $125 on the lot. What did both cost me?
41. Find the sum of $618, $974, $1243, $7896, $20374,
$36345, $9289, $33696, $180, $49270 and $37025.
(42)
(4?)
(44) (45)
(46)
852 
f 895 +
967 + 58378
+ 47114
= x
734 
f 766 +
3833 + 64956
4 89725
= X
3383 
f 677 +
592 + 7895
+ 65836
= X
7930 
f 2814 +
5745 + 6384
+ 85684
= X
496 
f 5920 +
824 + 5463
+ 78912
' ===  X
757 
f 6782 +
978 + 981
+ 97865
= X
2183 
f 588 +
684 + 4752
+ 65438
= X
3652 
f 676 +
756 + 3946
+ 99914
= X
1138 
f 983 +
1492 + 895
+ 88827
= X
2795 
f 1495 +
767 + 1574
+ 77715
== X
676 
f 6 ?4 +
4543 + 6388
+ 66624
= X
764 
f 542 +
786 + 5946
f 55568
= X
. 842 
f 721 +
692 + 7892
+ 89735
= X
13798 
f 2987 +
370 + 1147
+ 97814
= X
X
+ x 4
x f x
i .,
= X
(47)
(48)
(49)
(50)
(5*1)
790
9999
49
123456
213579
965
8989
428
789012
486420
1208
7897
3695
654321
397531
9669
36925
16378
210987
124683
375
52963
875692
913579
610793
92648
13579
3346279
806421
239701
30245
97531
963015
793519
896543
89762
496894
97892
421608
528647
24689
345678
496835
988997
134569
765432
876543
9469358
657893
174682
234567
6543210
642086
798979
212345
98898
9876543
59371
397856
167890
SUBTRACTION
36. Subtraction is taking one number from another.
37. The Minuend is the number from which we subtract.
38. The Subtrahend is the number to be taken from the
minuend.
39. The Remainder or Difference is the number left or
remaining after subtracting.
40. The Sign of subtraction is a short horizontal line , and
is called minus; when placed between two numbers it shows that
the second is to be subtracted from the first. G  2 is read 6
minus 2, and means that 2 is to be subtracted from 6.
The minuend and subtrahend must be like numbers; thus, 5 dollars
from 9 dollars leave 4 dollars ; 5 apples from 9 apples leave 4 apples ; but
it would be absurd to say 5 apples from 9 dollars, or 5 dollars from 9
apples.
41. When each figure in the minuend is greater than its cor
responding figure in the subtrahend.
1. From 958 subtract 324.
SOLUTION
MINUEND 958
SUBTRAHEND 324
DIFFERENCE OR
REMAINDER 634
Find the difference or remainder in each of the following :
(2) (3) (4) (5) (6) (7)
67 98 86 876 676 925
35 26 31 334 415 213
8. Bought a house for $547 and sold it for $315. What was
my loss?
9. Bought a farm for $620 and sold it for $855. What was
my gain?
10. A and B together bought real estate for $6985. A paid
$4130. How much did B pay?
18
SUBTRACTION 19
11. A farmer had 4687 bushels of wheat and sold 2380 bush
els. How many bushels remained?
12. A man having 96489 bricks, sold 34375 of them. How
many had he left ?
13. In a factory 86955 yards of cloth were made in one week,
of which 36520 yards were sold. How many yards remained?
4:2. When the figures in the minuend are not all greater than
the corresponding figures in the subtrahend.
1. From 834 lake 378.
SOLUTION EXPLANATION. Since 8 units cannot be subtracted from
834 4 units, add 1 ten of 3 tens to units, thus leaving 2 tens