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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 loosely-connected 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 every-day 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, thirty-six, 36 ;
seventy-two, 72.

20. To express numbers greater than ninety-nine, three or
more figures are written side by side ; as, one hundred eighty-five,
185 ; two thousand six hundred twenty-four, 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, twenty-seven 27

Three tens, thirty . ; 30

One hundred, one ten and seven, one hundred seventeen. . . . 117
Three hundred, five tens and nine, three hundred fifty-nine. . 359
Six thousand, five hundreds, nine tens and two, is read, six

thousand five hundred ninety-two 6592

NOTE. In reading whole numbers the word AND should not be used.
Thus, seven hundred fifty-four, not seven hundred and fifty-four.

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 forty-six.
14- Seven hundred ten.

15. Six hundred three.

16. Two hundred ninety.

77. Five hundred thirty-eight.

18 Three thousand seven hundred nineteen.

19. Six thousand nine hundred twenty-seven.

20. Four thousand sixty-four.

21. Seven thousand four hundred one.

22. Five thousand forty.

23. Nine thousand six hundred ninety-six.
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 t-i 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. Ninety-seven.

14. Three hundred sixty-eight.

15. Two thousand four hundred seventy-five.

16. Thirty-seven thousand one hundred ninety-six.

17. One hundred thirty-six thousand three hundred

twenty-seven.

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. Twenty-three.
8. Fifty-eight.

4. Ninety-nine.

5. Eighty-four.



6. One hundred eighty-eight.

7. One hundred ninety-nine.

8. Five hundred seventeen.

9. Six hundred forty-five.
10. Seven hundred sixty-one.



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 ten-thousand and 2 thousands; write

the 2 thousands under the column of thousands and the 1
ten-thousand in the place of ten-thousands. 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
right-hand 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 forty-nine, seventy-six, three hundred twenty-five,
nine thousand six hundred thirty-three, five thousand one hun-
dred ten and sixty-two thousand four hundred eleven.

21. Find the sum of three hundred seventy, two thousand
eighty-one, seven thousand four hundred sixteen, fifty thousand
one hundred twenty-nine and four hundred forty-four thousand
six hundred ninety-three.

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, dry-goods
$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



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