IN MEMORIAM
FLOR1AN CAJO
StAA^
^v?
COMPLETE ARITHMETIC
BY
SAMUEL HAMILTON, Ph.D.
*»
SUPERINTENDENT OF SCHOOLS, ALLEGHENY COUNTY, PA.
AND AUTHOR OF "THE RECITATION"
NEW YORK •:• CINCINNATI •:■ CHICAGO
AMERICAN BOOK COMPANY
Copyright, 190S, 1900, by
SAMUEL HAMILTON.
Entered at Stationers' Hall, London.
HAM. COMPLETE ARITH.
w. p. 3
ORI
PREFACE
A complete arithmetic should meet all the ordinary de-
mands of the elementary school. It may omit that which
is non-essential and all matter that properly belongs to text-
books for secondary schools ; but it should include a full
treatment of all important topics taught in the elementary
school, and a limited treatment of those of minor importance.
This book is intended for a complete arithmetic, to be
used either with or without the author's Elementary Arith-
metic. The work is divided into three parts. Part One,
after giving a complete treatment of the fundamental opera-
tions, covers the work ordinarily found in the -sixth year.
Part Tioo, after reviewing the subjects of Bills and Accounts,
Denominate Numbers, and Practical Measurements, covers
the work of the seventh year. Part Three covers the work
of the eighth year.
The aim of this book is threefold :
(1) To give the pupil skill in the art of computation.
(2) To make him a good mathematician.
(3) To give him a working knowledge of modern business
methods.
The first necessarily suggests an abundance of graded
work.
The second requires both inductive and deductive thought.
The method, therefore, is inductive in the development of
all mathematical principles, and deductive in their applica-
tion. It requires also that all solutions shall be clear and
918185
4 PREFACE
concise, and that the statements of all definitions, rules, and
principles shall be as brief and comprehensive as possible.
The third aim demands a practical treatment of all sub-
jects from the standpoint of actual business methods.
With these ends in view, attention is invited to the
following :
1. The large number of graded problems under each sub-
ject and in the reviews.
2. The abundance of exercises for oral drill.
3. The study of problems and processes.
4. The treatment of Fractions, Practical Measurements,
and the Comparative Studies in Percentage.
5. The problems arising out of business conditions.
6. The treatment of Promissory Notes, Banking, Com-
mercial Discount, Exchange, and Stocks and Bonds accord-
ing to the actual methods of modern business.
The author gratefully acknowledges his indebtedness to
many prominent educators and friends for valuable aid, dis-
criminating criticisms, and helpful suggestions.
SAMUEL HAMILTON.
CONTENTS
PART I — SIXTH YEAR
Fundamental Processes . . .
Arabic System of Notation
and Numeration . .
Roman System of Notation
and Numeration ....
United States Money . . .
Addition
Subtraction
Multiplication
Division
Combining Processes . . .
Factors and Divisors ....
Tests of Divisibility . . .
Factoring
Greatest Common Divisor
Least Common Multiple . .
Cancellation
PAGE
7
8
11
12
13
19
23
29
36
37
38
39
40
41
42
Fractions 44
Fractional Units .... 44
Reading and Writing Frac-
tions 40
Reduction 47
Addition and Subtraction . 52
Multiplication 58
Fractional Parts of Fractions 05
Division 08
Complex Fractions .... 75
Fractional Relations ... 70
Review 78
Problems for Analysis ... 84
Decimal Fractions 90
Notation and Numeration . 91
Common Fractions and Deci-
mals 94
Addition and Subtraction . 90
Multiplication 98
Division 101
Changing Fractions to Deci-
mals 100
Review 107
Business Applications of
Decimals 108
Simple Accounts
Denominate Numbers . . .
Standard Units
Reduction
Addition and Subtraction
Multiplication and Division .
Review of Denominate Num-
bers
Practical Measurements . . .
Length
Surface
Painting and Kalsomining .
The Right Triangle . . . .
Volume
Practical Applications
Lumber ....
Review Problems .
Percentage ....
Commission . .
Commercial Discount
112
115
115
110
119
121
122
123
123
124
128
128
130
132
134
138
141
147
148
Interest 151
Review of Percentage and In-
terest
Receipts and Checks . . . .
General Review
155
157
159
PART II — SEVENTH YEAR
Bills and Accounts . . . . 166
Receipts 166
Ordering Goods 1H7
Receipted Bills 108
Accounts 170
Ledger Accounts .... 173
6
CONTENTS
PAGE
Denominate Numbers . . . 175
Reduction 175
Foreign Money 18 i
Addition and Subtraction . 184
Multiplication and Division . 186
Review Problems 188
Practical Measurements . . . 192
Length and Surface . . . 192
Lines and Angles . . . . 194
Triangles 196
Quadrilaterals 198
Rectangles 199
Plastering and Painting . . 201
Roofing and Flooring . . . 2<>2
Papering and Carpeting . . 2i)4
Areas of Triangles, Quadri-
laterals, and Circles . . . 2(16
Solids 212
Lumber 217
Concrete, Stone, and Brick-
work 220
The Cylinder 221
Bins, Tanks, and Cisterns . 223
PAGE
Approximate Measurements . 224
Review Problems .... 225
Analysis 231
The Equation 231
Percentage 237
Review 247
Gain and Loss 249
Review ........ 253
Commission and Brokerage . 255
Insurance 259
Commercial Discount . . . 264
Commercial Bills .... 2i>8
Local and State Taxes . . . 269
Duties or Customs .... 272
Interest 275
Simple Interest 275
Problems in Simple Interest . 283
Annual Interest 286
Exact Interest 287
Compound Interest .... 288
Savings Accounts .... 289
Investments 291
Promissory Notes .... 292
Partial Payments of Notes . 298
PART III — EIGHTH YEAR
Banks and Banking .... 303
Bank Discount 307
Exchange 315
Stocks and Bonds 324
Stocks 324
Bonds 331
Test Problems in Percentage . 335
Ratio and Proportion .... 337
Ratio 337
Simple Proportion .... 338
Partitive Proportion and Part-
nership 341
Problems for Oral and Written
Analysis 345
Longitude and Time .... 349
Government Land Measures . 356
Powers and Roots 358
Extracting Roots .... 360
Square Root
Mensuration
Regular Polygons ....
Solids
Similar Surfaces
Similar Solids
Specific Gravity
Review
Metric System
Agricultural Problems . . .
Test Problems
General Review
Optional Subjects
Present Worth and True Dis-
count
Foreign Exchange ....
Compound Proportion . . .
Cube Root
Reference Tables
Index
361
367
367
368
376
378
879
381
382
392
401
406
421
422
425
426
433
437
COMPLETE ARITHMETIC
PART I — SIXTH YEAR
FUNDAMENTAL PROCESSES
NOTATION AND NUMERATION
A unit is a single thing ; as, one, one cent.
A number is a unit or a collection of units.
Numbers are used to tell how many ; they are expressed
by figures or letters. The figures we now use are of Hindu
origin, but the Arabs were the first people to introduce them
into Europe. They are
Naught One Two Three Four Five Six Seven Eight Nine
012 34567 89
These ten figures are, therefore, called Arabic numerals.
The figure o is called naught, zero, or cipher, and has no
value.
The Arabic notation is a method of expressing numbers
by means of figures.
Numeration is a method of reading numbers expressed by
means of figures or letters.
Arithmetic is the science of numbers and the art of com-
puting by them.
7
8 FUNDAMENTAL PROCESSES
ARABIC. SYSTEM OF NOTATION AND NUMERATION
Any number containing but one figure, simply stands for
so many ones; thus, 9 stands for 9 units, or ones.
In any number containing two figures, the first place at the
right is called ones ; the second place is called tens ; thus,
in 25 there are 2 tens, or 20 ones, plus 5 ones, or 25 ones.
25 is read, "twenty-five."
1. Name the ones and tens in the following numbers and
then state how many ones each number equals :
15 25 30 75 82 60 72 45 70 99 20 10 39 47 90
In any number containing three figures, the third place
from the right is called hundreds; thus, in 325, the 3 stands
for 300 ones. 325 is read, "three hundred twenty-five."
2. Name the ones, tens, and hundreds in the following and
then state how many ones each number equals:
125 329 879 801 600 650 803 132 400 ' 904
109 705 105 550 900 901 502 999 570 809
In any number containing four figures, the fourth place
from the right is called thousands. 4635 is read, " four thou-
sand, six hundred, thirty-five."
3. Name the places in each number and then read:
2135 6005 6910 5604 5000 4025 2684 8709 7009 8900
For convenience in reading large numbers, in the Arabic
system, the figures are generally separated by commas into
groups of three figures each, called periods.
The first period, counting from the right, is units; the
second, thousands; the third, millions; the fourth, billions;
the fifth, trillions ; etc.
NOTATION AND NUMERATION
9
The following table shows the arrangement of these
periods, and the three orders of figures in each period:
TRILLIONS'
BILLIONS'
MILLIONS'
THOUSANDS'
UNITS*
PERIOD
PERIOD
PERIOD
PERIOD
PERIOD
O
U
(A
c
c
c
<tf
o
o
o
n
V*
3
73
~ y>
zz
«
«
(/>
o
C
I o
15
c
o
E
c
o
of
(A
T3
to
*U —
w»
*D
-o
— *
TO
j*
C
"O
c
o
2
T3
15
c
o
0)
E
c
o
0)
(4
to
3
E c
~
C
c
™
C
C
■f™
c
C
O
c
2 S
3 0,
"C
3
<u
~
3
0)
3
©
J=
D
o c
X H
1-
X
h-
00
I
1-
2
X
H
h-
X
1- o
1
1 .
3
4
5 ,
6
4
2
, o
1 ,
3
4 6
The number in the table is read, " 101 trillion, 345 billion,
642 million, 1 thousand, 346."
The ones are not named in reading numbers.
Since ten ones make 1 ten, and ten tens make 1 hundred,,
etc., our system of writing numbers is called a decimal sys-
tem. The word decimal comes from the word decern, mean-
ing ten.
Name the places and periods, then read :
4.
129,475
8.
1,700,425 pounds
12. 400,000,000
5.
407,575
9.
9,609,500 tons
13. 709,050,050
6.
600,905
10.
5,505,608 yards
14. 8,000,000,000
7.
790,505
11.
40,000,905 dollars
15. 9,075,074,093
1
Copy, point off, then read:
16.
700102
23. 5050504
30. 60414
17.
6067004
24. 12126012
31. 102365
18.
8011100
25. 87649101
32. 14763 1ST
19.
90314607
26. 104706950
33. 243540038
20.
910723586
27. 120243(U14
34. 2452603467
21.
837421012
28. 3183456109
35. 8703005020
22.
987654321
29. 7891234560
36. 6010100100
10 FUNDAMENTAL PROCESSES
Writing Numbers
Write :
1. 25 million, 161 thousand, 104.
2. 2 million, 12 thousand, 8.
3. 36 million, 1 thousand, 109.
4. 300 billion, 304 million, 100 thousand, 40.
5. 16 billion, 9 million, 70 thousand, 700.
6. 26 million, 18 thousand, 9.
7. 7 billion, 46 million, 900 thousand, 90.
8. 74 billion, 7 million, 46 thousand, 809.
9. Two hundred six thousand, eight.
10. Twenty-five million, six hundred.
11. One-thousand thousand.
Write each of the numbers 12 to 23, first, when 10 is
added, and second, when 10 is subtracted from each number.
12. 80,900 15. 60,804 is. 497,842 21. 50,001
13. 67,895 16. 50,000 19. 109,090 22. 70,080
14. 45,101 17. 70,800 20. 290,009 23. 59,009
24. One million, two hundred thousand.
25. Nine billion, six-hundred million, seven.
26. Six billion, six thousand, six hundred six.
27. Seven hundred nine thousand, two.
28. Sixty-nine million, eight thousand.
29. Nine hundred two million, forty-two thousand, sixty-two
30. Thirty-two thousand, thirty-two.
31. Six-hundred forty-four million, six hundred four.
32. Ten thousand, ten.
33. Fifteen hundred million.
34. Twenty-eight million, eight thousand, eight.
ROMAN NOTATION
11
ROMAN SYSTEM OF NOTATION AND NUMERATION
The seven letters used in Roman notation are :
I V X L C D M
1 5 10 50 100 500 1000
The other numbers are represented by combinations thus :
I. When a letter is followed by the same letter or by one
of less value, the values of the letters are to be added. Thus,
XX represents 20 ; XI represents 11.
II. When a letter is followed by one of greater value, the
value of the smaller is to be subtracted from that of the greater.
Thus, IV represents 4; IX represents 9.
III. When a letter is placed between two letters of greater
value, the value of the smaller is to be subtracted from the sum
of the other two. Thus, XIV represents 14; XIX represents
19.
A bar placed over a letter multiplies its value by 1000. Thus,
V represents 5000.
The following table further illustrates the system :
1,1
VIII, 8
XVI, 16
LXXX, 80
DCC, 700
11,2
IX, 9
XX, 20
XC, 90
DCCC, 800
111,3
X, 10
XXX, 30
C, 100
CM, 900
IV, 4
XI, 11
XL, 40
CCC, 300
M, 1000
V,5
XII, 12
L, 50
CD, 400
MCM, 1900
VI, 6
XIV, 14
LX, 60
D, 500
V, 5000
VII, 7
XV, 15
LXX, 70
DC, 600
M, 1,000,000
Read :
1. XLIII CDXLIX MDII MCDXCII
2. XCIX MCMVIII MDLXXVI MDCCCLXI
3. Express in Roman notation: 41, 63, 84, 99, 107, 218,
572, 735, 996, 1907, 1564, 1616, 1000, 260,000.
12 FUNDAMENTAL PROCESSES
UNITED STATES MONEY
10 mills = 1 cent
10 cents = 1 dime
10 dimes = 1 dollar
10 dollars = 1 eagle
1. From the above table, tell why our money is a decimal
system of money.
The dollar sign is $ ; it is placed before the number of
dollars. The sign for cent or cents is $ ; it is placed after
the number of cents.
When dollars and parts of a dollar are written as one
number, a period, called a decimal point, separates the dollars
from the cents ; thus, 8 dollars, 15 cents is written $8.15.
Parts of a dollar may be written in three ways ; thus, 15 cents may be
written 15?, §0.15, or $.15.
The first two places to the right of the decimal point are
for cents.
2. Read in as many ways as you can :
15^ 18.32 5/ 105^ 10.99 $.25 $0.50
900/ $.01 $0.35 $0.05 100^ 1101/ $9.90
The first place to the right of the point is for dimes,
or tenths of a dollar ; the second, for cents, or hundredths
of a dollar ; the third, for mills, or thousandths of a dollar.
$.025 is read, "two cents, five mills."
3. Read :
$.02 $8.06 $.901 $.515
8.022 $9,055 $.80 $.005
$.255 $1,005 $.801 $.50
4. Read, then write from dictation, using the dollar sign:
$2.50 275 dollars, 5 cents 100 cents, 8 mills
80.75 89 dollars, 2 mills 525 cents, 5 mills
85.05 10 dollars, 1 cent 875 cents, 3 mills
ADDITION
13
ADDITION
1. Count by 2's to 100 ; by 3's to 00 ; by 4's to 100.
2. Count by 6's to 102; by 7's to 105 ; by 8's to 104.
3. Count by 9's to 108; by ll's to 143; by 12's to 168.
Addition is the process of uniting two or more numbers
to form one number.
The sign +, called plus, indicates addition; the sign =,
called equal or equals, indicates equality.
Announce the sums at sight:
4.
5.
. 6.
7.
8.
9.
5 + 7
9 + 9
1 + 3
8 + 6
2 + 2
2+8
1 + 6
1 + 4
2 + 9
9 + 8
4 + 6
4 + 2
3 + 3
6 + 2
9+1
7 + 2
6+7
9 + 4
+ 6
9 + 3
7 + 3
8 + 7
8 +
+ 9
6 + 6
2 + 1
8 + 5
4 + 4
2 + 3
3 + 4
1 + 8
5 + 4
7 + 2
5 + 6
6 + 3
8+4
7 + 9
5 + 2
5 + 5
+ 4
7 + 7
3 + 8
5 + 3
1 + 1
9 + 5
7 + 4
8 + 8
5 + 1
Announce the sums at sight, first in rows across, then in
columns :
10.
11.
12.
13.
14.
15.
16
17.
154
534
561
147
964
275
784
368
371
826
729
684
837
984
926
574
723
943
358
493
496.
532
578
397
In like manner, announce tli
e sums :
18. 19. 20.
21. 22. 23.
4356
97t;5
3742
8674
7248
5632
8547
6934
9746
7897
6754
9985
7961
9847
6957
8753
3976
7987
5943
41
)57
6582
7541
)
6<
)37
9147
14
FUNDAMENTAL PROCESSES
The addends in addition are the numbers to be added.
The sum or amount is the result of addition.
Like numbers are numbers that express the same units ;
as, 3 pounds, 5 pounds, or 3, 5.
Unlike numbers are numbers that express different units ;
as, 7 days, 1 3.
Only like numbers can be added.
24. Name and add the like numbers in the following:
$5, 5 ft., 4 oz., $4, 3 oz., 3 gal., 6 gal., 6 ft.
25. Name the addends ; then give the sum or amount :
8753 4 12 5 7 12
1404 11 99 Oil
5 2 7 9 7 10 8 12 5
26. Add 45 and 23, thus : 45 + 23 = 45 + 20 + 3 = 68.
Add in like manner :
27. 55 and 28 = ? 31.
28. 29 and 47 = ? 32.
29. 31 and 42 = ? 33.
30. 57 and 32 = ? 34.
34 and 47 = ?
69 and 28 =
49 and 35 =
44 and 57 =
Add the two columns, as in problem 26.
39.
5
10
11
27
4
12
40.
7
15
20
4
9
12
41.
3
10
15
11
9
7
42.
9
8
16
4
7
3
43.
11
12
14
10
5
7
44.
10
12
7
14
9
7
35.
54 and 36 =
36.
72 and 19 =
37.
37 and 47 =
38.
46 and 53 =
45.
15
10
4
9
7
10
46.
11
7
4
12
9
6
47.
26
37
12
29
17
13
Note. — Practice on similar work, until pupils can add rapidly and
accurately.
ADDITION
15
Seeing Groups
Add, observing the groups that make 10, 15, etc.
20
5.
8
9
3
5
4
6
7
9
8
9
6
6.
3
9
8
2
7
8
5
4
2
1
3
7.
2
6
4
8
4
6
5
7
8
1
9
Written Work
l. Find the sum of 325, 436, 285.
Write the numbers in their order, as
indicated. In adding the first column, we
think 5, 11, 16 ones; then write the 6 ones
and carry the 1 ten to the tens' column.
9, 12, 14 tens ; then write the 4 tens and
carry the hundred to hundreds' column.
3, 7, 10 hundreds. The sum is 1046.
325
Addends 436
285
Sum
1046
Test by adding downwards
Practice until you can add these problems rapidl)
2.
3.
4.
5.
$729.15
206793
534891
726432
804.90
429874
567321
874921
728.16
358079
623413
785341
574.34
427451
347621
634541
420.85
874297
983122
442211
345.78
298743
246893
532891
16
FUNDAMENTAL PROCESSES
In adding two or more columns, business men frequently
write the sum of each column separately on a slip of paper,
and then add the separate sums as illustrated in problem 6.
In case of interruption or error it is then not necessary to go
over all the work.
6.
7.
8.
9.
10.
11.
279
224
746
581
910
375
874
921
317
841
725
421
398
234
567
789
234
291
569
321
104
902
803
123
768
572
571
846
931
146
343
937
847
641
261
143
41
921
803
731
283
197
39
321
830
450
847
190
28
731
476
903
275
913
3231
Axld these
; problems and test the work in
9 minutes :
12.
13.
14.
15.
16.
17.
5608 1
6720
2954
3783
1751
5068
2094
>12 4604
3261
5427
7173
7504
8675
7259
5050
7891
5009
8795
6985
> 10 8753
7406
5030
4287
6474
3745"
6758
2834
6793
5783
2758
5268
-20 4326
6498
7406
7205
6471
7777
6734
5065
5872
4987
2050
5989
7583
8439
3458
6034
6579
7481
! 3586
9823
7295
2985
2068
5479
t 10 2734
7984
8376
3046
6579
8705
1 4725
2030
2794
1154
2068
3074.
1 6050
5984
6384
3683
5432
5547
1 q 7438
3749
6589
4594
4280
6875
| 12 5006
5308
7405
5181
5683
ADDITION
17
Making Change.
Business men make change by the adding method. Thus,
if a purchase is made for 11.57, and $2.00 is given in pay-
ment, the clerk will probably say : " One dollar fifty-seven
cents, sixty, seventy, seventy-five, two dollars," laying down
each time the piece of money that makes the sum named.
Acting as clerk when the following purchases and pay-
ments are made, give the exact language you might use, if
the purchaser were actually present to receive the change:
Cost of
Purchase
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
$1.34
2.95
2.11
3.17
1.15
4.12
2.24
3.54
1.12
4.02
2.79
2.36
3.09
2.71
3.00
Amount
Given
Cost of
Purchase
$2.00
16.
$1.95
5.00
17.
7.23
3.00
18.
3.G7
4.00
19.
2.85
1.50
20.
5.02
4.50
21.
.91
3.00
22.
.79
3.75
23.
G.01
1.50
24.
7.11
5.00
25.
3.95
4.00
26.
2.01
5.00
27.
0.79
4.00
28.
7.04
3.00
29.
8.31
3.50
30.
2.98
Amount
Given
$2.00
10.00
5.00
4.00
6.00
5.00
5.00
10.00
8.00
5.00
10.00
7.00
8.00
10.00
4.00
HAM. COM PL. A KITH. 2
18 FUNDAMENTAL PROCESSES
Written Work
1. Chicago is 468 miles, by rail, west of Pittsburg, and
New York is 445 miles east of Pittsburg. What is the rail-
road distance from Chicago to New York, via Pittsburg?
2. A man purchased a farm for $6500 ; built a barn on it
at a cost of $1980; a house for $1825; and spent on im-
provements on the land, $971. What was the cost of the
farm and improvements ?
3. The area in square miles of the main body of the
United States is 3,088,519; the outlying possessions are
Alaska, 590,884 square miles; Hawaii, 6449; Porto Rico,
3435 ; Philippines, 115,026 ; other outlying possessions, 761
square miles. Find the total possessions in square miles.
4. The average annual production of corn in the United
States for a number of years was $854,000,000; of hay,
$467,000,000; the production of cotton, $406,000,000; of
wheat, $395,000,000; and of oats, $254,000,000. Find the
total average annual value of these five productions.
5. The values of the ten leading exports of the United
States for the year 1906 were: cotton, $413,137,936; meat and
dairy products, $208,586,501 ; iron and steel and machinery,
$172,555,588 ; copper, $90,773,151 ; petroleum, $85,738,866 ;
wood, $77,255,225; flour, $58,399,727 ; corn, $52,840,269;
wheat, $49,158,650 ; livestock, $45,614,748. What was the
value of all these products ?
6. The values of the ten leading imports for the same year
were: silk and silk manufactures, $100,052,211; raw and
manufactured fibers, $99,635,731 ; hides, $83,884,981 ; sugar,
$79,015,471;* chemicals, $78,647,978; coffee, $72,252,465;
cotton goods, $68,911,371; wool including woolen goods,
$61,040,335; india rubber, $58,664,651; jewelry, $46,047,021.
Find the value of all these imports.
SUBTRACTION 19
SUBTRACTION
1. Subtract by 2's from 97 to 1 ; by 3's from 75 to 9.
2. Subtract by 4's from 78 to 2 ; by 6's from 98 to 2.
3. Subtract by 7's from 101 to 3 ; by 8's from 106 to 2.
Subtraction is the process of finding the difference between
two numbers, or of taking one number from another.
The minuend is the number from which we subtract.
The subtrahend is the number to be subtracted.
The difference, or remainder, is the result of subtraction.
The difference added to the subtrahend equals the minuend.
The sign — , called minus, indicates subtraction.
Only like numbers can be subtracted.
Give differences at sight :
4.
5.
6.
7.
8.
9.
3-2
12-7
11-9
10-4
11-2
14-3
4-1
9-4
7-4
19-3
16-8
13-9
5-4
9-8
9-5
17-9
15-6
16-7
7-3
11-4
17-8
11-3
14-5
15-7
8-4
9-7
12-6
12-9
12-8
13-8
7-2
5-1
11-5
16-9
14-7
18-9
10. Subtract each of the following numbers from 20 ;
then, from 30, 40, etc.
4 6 11 17 13 14 7 8 12 10 15 14 16
11. Subtract each of the following numbers from 100.
40 70 60 25 45 75 44 64 84 37 57 68
12. Take 27 from 65, thus
Subtract in like manner:
13. 72 14. 84 15. 91
48 36 45
65-20 = 45; 45-
7 = 38.
16. 63 17. 48
18. 82
24 32
59
20
FUNDAMENTAL PROCESSES
A clerk's sales book shows the following sales and amount
given by customers. Give differences at sight :
Sales
19.
$1.55
20.
1.15
21.
3.17
22.
1.78
23.
3.15
24.
1.79
25.
2.34
Amount
GIVEN
$2.00
1.50
4.00
2.00
3.50
5.00
3.00
Sales
26.
$ .49
27.
1.23
28.
1.06
29.
2.14
30.
.99
31.
1.29
32.
.87
Amount
given
$ 2.00
5.00
2.00
3.00
2.00
10.00
5.00
Written Work
1. From 632 take 374.
Minuend 632 = 500 + 120 + 12 = 5 hundreds, 12 tens, 12 ones
Subtrahend 374 = 300 + 70 + 4 — 3 hundreds, 7 tens, 4 ones
Difference 258 = 200 + 50 + 8 = 2 hundreds, 5 tens, 8 ones
Since 4 ones cannot be. taken from 2 ones, take 1 ten = 10 ones,
from 3 tens; 10 ones + 2 ones = 12 ones; 12 ones — 4 ones = 8 ones.
Since 7 tens cannot be taken from the 2 tens remaining, take 1 hun-
dred = 10 tens, from the 6 hundreds; 10 tens + 2 tens = 12 tens;
12 tens — 7 tens = 5 tens. 5 hundreds — 3 hundreds = 2 hundreds.
Test. — 374 + 258 = 632. By adding the difference to
the subtrahend, pupils can quickly discern whether the
answer is correct.
Explain the steps in finding each remainder :
2. 800 3. 9004 4. 9080 5. 7001 6. 9040
594 7907 5987 4908 5879
206 1097
3093 2093 3161
Subtract :
7. 30984 8. 54009
9. 81704 10. 41711 11. 50000
24987 31047
54270 39111 42001
SUBTRACTION
21
The following methods of subtraction are also convenient:
I. By adding (the method used in making change).
l. From 842 take 385.
Think: AVhat number added to 5 will make 12?
(7.) Write down 7; carry 1 to 8 tens in minuend.
What number added to 9 (8 + 1) will make 14?
(5.) Write down 5; carry 1. What number added to
4 (3 + 1) will make 8 ? (4.) Write down 4.
842
385
457
II. By subtracting from 10.
2. From 653 take 378.
Borrow 10, subtract 8, and add 3, thus :
10 — 8 in subtrahend = 2 ; 2 + 3 in minuend = 5.
10 — 7 in subtrahend = 3 ; 3 + 4 (5 — 1) in minuend = 7.
5—3 = 2. The steps in the general method of sub-
traction are borrow, add, subtract. In this they are
borrow, subtract, add.
Write, subtract, and test four problems in 3 minutes :
85980 9. 1070.01 15. 590680 21. 6459871
71409 340.97 289796 2987598
753
378
275
4.
57004
20098
10.
11.
12.
13.
14.
1590.10
210.89
16.
17.
18.
19.
20.
998076
433011
22.
23.
24.
25.
26.
5798371
3099384
5.
39702
21308
1953.01
391.54
598801
303397
8342901
5217809
6.
70001
39005
* 401.97
207.58
743019
556601
7654321
3456780
7.
98235
60104
8701.49
511.10
831001
397018
6543903
4239001
8.
80021
51037
* 800. 67
610.34
458995
233450
4932459
2013307
22 FUNDAMENTAL PROCESSES
PROBLEMS
1. The minuend is 6389, and the difference is 4360
Find the subtrahend.
2. From the sum of 3645 and 5796, subtract their dif-
ference.
3. In 1900 the population of New York State was
7,268,894 ; of Pennsylvania, 6,302,115. How much greater
in population was New York than Pennsylvania?
4. In 1906 Iowa raised 373,275,000 bushels of corn ; Mis-
souri, 228,522,500 bushels. How much did the corn crop in
Iowa exceed the crop in Missouri ?
5. A and B together owe me $7650; B owes me $4675.
After each pays me -1*1600 on account, find the amount each
one still owes me.
6. A retail merchant bought goods to the amount of
61457. After selling stock from these goods to the amount
of $975, he found that the remainder of the goods unsold
had cost him 6473. How much had he gained or lost ?
7. Mr. Adams bought a farm for $8670; he expended
in improvements on barn and house, 81790; on stock and
farming utensils, $2080. How much more did he pay for
the farm than for improvements, live stock, and utensils?
8. The surface of the earth contains 196,907,000 square
miles, of which 144,500,000 square miles are water. How
much of the surface of the earth is land ?
9. A father divided 623.675 among his sons, giving
to James 66750 and to Henry 6 5000 less than the part
remaining after James was paid ; to Frank he gave the
remainder. How much did each receive ?
MULTIPLICATION
23
MULTIPLICATION
1. Count to 72 by 2's; by 3*s ; by 4's; by 6's; by 9's.
2. Count backwards from 48 by 2's; by 3's; by 4's;
by 6's ; by 8's.
3. Count forwards to 96 by 3's ; by 4's ; by 6's.
4. Count to 25 by 5's ; to 60 by 4's ; to 99 by 9's.
5. Build the multiplication tables by addition, thus :
2
2 2
2 2 2 [2 times 2 = 4
2 2 2 ; then write it in this form • 3 times 2 = 6
4 times 2 = 8
6. Drill on these tables until pupils thoroughly know them :
1
2
3
2's
3's
4's
5's
6's
7's
8's
9's
10' s 11' s
12s
4
6
6
9
12
8
12
16
20
24
10
12
14
21
16
24
32
40
48
56
64
•72
80
88
96
18
27
36
45
54
63
72
81
90
99
108
20
30
40
50
22
24
15
20
25
30
35
40
45
50
18
33
44
55
36
48
60
72
84
96
108
120
132
144
4
8
24
30
36
42
48
54'
60
66
72
28
35
42
49
56
63
70
77
84
5
10
15
6
12
18
60
70
80
90
100
110
66
77
8*
00
110
121
7
14
21
28
32
36
8
10
24
27
9
18
10
20
as
40
11
22
44
55
60
12
24
36
IS
120 132
The fust row of figures at the top stands for the different tables. By
multiplying each of the numbers in the first left-hand row by each of the
numbers in the top row, the tables can all be made. Thus, in the table
of the twos, the products are directly below the number of the table, etc.
24 FUNDAMENTAL PROCESSES
7. How many units are there in 4? 4x15 means that we
are to take $5, four times to find the product; this may be
found in two ways : $5 + $5 + $5 + #5 = $20, orby multipli-
cation, which is a short form of addition ; thus, 4 x §5 = $20.
Multiplication is the process of taking one number as many
times as there are units in another number.
The multiplicand is the number multiplied.
The multiplier is the number by which we multiply.
The product is the result of multiplication.
The sign x indicates multiplication ; it is read, " times,"
when the multiplier precedes the sign, and, " multiplied by,"
when the multiplier follows the sign.
Read each statement and then give products.
8. 4x$12 10. 12 x 7 yards 12. 11 x 4 pounds
9. 9 x 6 horses 11. 5x8 ft. 13. 7x8 bushels
14. In the above statements, how many times are $12
taken? 8 bushels? 7 yards?
15. Name the multiplicands in the above statements ; the
multipliers.
A concrete number is a number used with reference to a
particular object ; as, 5 days, 10 pounds, 8 inches.
An abstract number is a number used without reference to
a particular object ; as, 5, 8, 20.
Name the abstract and the concrete numbers in the follow-
ing statements and then give products :
16. 12 x 7 days 18. 11 x $7 20. 6 x 11
17. 15x10 19. 9 x 12 ft. 21. 12x5^
The multiplier is always regarded as an abstract number. The
multiplicand may be either abstract or concrete.
22. In problems 16-21 are the products like the multipli-
cand or the multiplier ?
The product and the multiplicand are like numbers.
MULTIPLICATION 25
Oral and Written Analysis
1. How many eggs are there in 6 dozen?
Since there are 12 eggs in one dozen, 1 doz. = 12 eggs;
in 6 doz. there are 6 x 12 eggs=72 eggs. 6 doz. = 6x 12 eggs=72 eggs.
2. How many trees are there in an orchard if then; are
11 rows and 10 trees in each row ?
3. James raised 7 bushels of potatoes on an average from
each of 10 rows. How many bushels did he raise?
What is the cost of :
4. 10 quarts of cherries at 8^ per quart?
5. 9 quarts of milk at 7^ per quart?
6. 8 bushels of apples at $2 per bushel?
7. A twelve-pound cheese at 12^ per pound?
8. 3 pecks of apples at 25^ per peck?
9. How far does a boy ride on his automobile in 4 hours
at the rate of 9 miles per hour?
10. How many miles are there in 4 streets, if the streets
average 12 miles ?
11. There are 32 quarts in a bushel. Find the number
of quarts in 13 bushels.
12. How far does an automobile run in 4 hours, if it
averages 14 miles per hour?
13. Find the cost of posting 18 letters at 2^ each.
14. Find the cost of a 7-pound turkey at 13^ per pound.
15. A lady purchased 2 dozen oranges at 40^ per dozen.
How much did they cost?
16. It takes John 15 minutes to walk to school. How
many minutes will be required to walk to school 60 times?
17. Frank used 12 tablets, at 10^ each, in a school term.
How much did they cost?
26 FUNDAMENTAL PROCESSES
Written Work
1. Multiply 146 by 3.
Multiplicand 146 3x6 ones = 18 ones, or 1 ten and 8
Multiplier 3 ones. Write 8 in ones' place and
Product 438 carry the 1 ten. 3x4tens = 12 tens ;
„ +. n - . n. -.~ . ~~ 12 ten s+ the 1 ten, carried from ones'
Test.-146 + 14b + 146 = 438 place = 13 tenB , or j himdred and 3
tens. Write 3 in tens' place and carry the 1 hundred. 3x1 hundred = 3
hundreds ; 3 hundreds + the 1 hundred = 4 hundreds. The product is 438.
Find products :
2. 139 6. 674 10. 307 14. 137 18. 427 22. 507
3 3 5 6 7 6
-
3. 135
7. 278
li. 342
15.
673
19. 784
23. 196
_2
J
6
5
7
8
4. 603
8. 147
12. 281
16.
901
20. 249
24. 379
4
5
4
7
8
7
5. 205
9. 219
13. 309
17.
419
21. 907
25. 583
2
4
_6
_6
2
_8
26. Multiply $1.25 by 3.
$1.25
3
Place the decimal
under the decimal poir
point in the product directly
it in the multiplicand.
$3.75
27. $2.53 30. $8.09 33. $3.27 36. $2.41 39. $3.19
9 9 8 10 9
28. $6.08 31. $2.25 34. $1.04 37. $3.74 40. $8.92
_9 10 11 12 10
29. $9.09 32. $1.29 35. $3.05 38. $5.05 41. $9.08
9 11 10 12 12
Ul'LTIl'LK ATIOX
27
Multiplication by Larger Numbers
Multiplying integers by 20. 100. 1000. etc.
ln x 12=12<); 100x2=200; 1000x2 = 2000.
Any integer may he multiplied by 10, 100, 1000, etc., by an-
nexing to the integer as many naughts as there are naughts in
the multiplier.
Multiply each number by 10, by 100, by 1000, writing
only the products : 16, 409, 290, 301, 205, 250, 175, 791.
Written Work
l. Multiply 72
by
36.
Multiplicand
72
72 In practice,
Multiplier
36
qq the in the
1 l * 1
1st partial product
2d partial product
432 =
2160 =
6 x72
30 x 72
.on second partial
product is
- jXU omitted, and
Entire product
2592 =
36 x 72
2592 21 (JO is written
as 216 tens.
Find products :
2. 150 x 40
4. 805
xl6
6. 304 x 71
3. 107 x 35
5. 500
x70
7. 691 x 74
Multiply and test :
8. 6425
■ a.
245
Form 100 problems by mul-
9. 1024
b.
344
tiplying each multiplicand by
10. 8720
0.
564
each of the multipliers, thus:
n. 9652
d.
746
8 a.
245 x 6425 = ?
12. m\o
. by .
e.
804
8 5.
344 x 6425 = ?
13. 7894
/■
961
15 i.
968 x 7695 = ?
14. 8465
9-
869-
Write, solve, and test each
15. 7695
16. 8425
h.
i.
796
968
problem in 1| minutes.
17. 9476,
■J-
898
28 FUNDAMENTAL PROCESSES
18 How much will 20,000 bricks cost at $7.75 per thou-
sand ?
Suggestion : 20,000 = 20 thousand = 20 M.
19. A ranchman sold 125 head of cattle at an average of
$42.75 per head, and 625 sheep at $3.85 per head. Find
the total amount of his sales.
20. The cost of drilling an oil well was 35/ per foot for
drilling and 65^ for the tubing. If the well was drilled
1177 feet and tubed 700 feet, find the total cost.
21. The freight rate on corn in car-load lots from Omaha,
Neb., to New York City is 20^ per hundred pounds. Find
the freight on a car of 42,000 lb.
22. A freight train of 32 cars is laden with corn. The
cars contain an average of 700 bushels of 56 lb. each. Find
the weight of the corn in pounds.
23. A commission merchant sold 1275 barrels of apples at
the rate of $3.25 per barrel, charging $.325 per barrel for
selling. How much was realized from the sale of the apples
after the charges for selling were deducted?
24. A man owned a farm of 142 acres, worth $72.50 per
acre ; 5 city lots worth $ 1875 per lot ; and a business house
worth $6350. Find the value of his entire property.
25. Frank lives 87 rods from the schoolhouse. How
many rods does he walk in going to school 140 days, if he
returns home each day for his dinner ?
26. Two trains leave a station at the same time. One
travels west 38 miles per hour ; the other travels east 45 miles
per hour. How far apart are they in 9 hours ?
27. A dealer bought a car load of coal, 42,000 lb., at $1.90
per ton of 2000 lb. If the freight was 70^ a ton, and he
retailed the coal at $3.25 a ton, find his profit.
DIVISION 29
DIVISION
1. How many times is the number 6 contained in 24?
Division is the process of finding how many times one
number is contained in another, or of separating a number
into equal parts.
The dividend is the number to be divided.
The divisor is the number by which we divide.
The quotient is the result of division.
The remainder is the part of the dividend remaining when
the quotient is not exact.
The sign + indicates division, and is read, " divided by."
Give quotients :
24 + 6 84 + 7 64 + 8 49 + 7 99 + 11
72 + 9 108 + 9 42 + 6 68 + 9 72+12
81 + 9 72+8 90 + 9 96 + 8 77 + 11
Division is indicated in three ways : 14 + 2; 2)14; and-M-.
How many times are :
2. 2 inches contained in (may be taken from) 12 inches?
3. 4 yards contained in (may be taken from) 12 yards?
If both the dividend and divisor are concrete, they must be
like numbers.
Compare 18 + 2 with J of 18 ; 15 + 3 with } of 15.
How many :
4. Cents are | of 48 f ? 48^+8=— cents.
5. Cents does one orange cost if 4 oranges cost 20^?
In separating a number into equal parts, the divisor is always
an abstract number and the quotient is like the dividend.
This kind of division is called partition.
In the following, point out the problems in partition :
6. $120 + 10 7. \ of 40^ s. 24 ft. +2 ft.
30
FUNDAMENTAL PROCESSES
Remainder in Division
34-r-5 = 6, and 4 remaining. $39 -4- 5 = $7, and $4 re-
maining.
Give quotients and remainders :
$26 -=-6; 79 -h 8; $48 -4- $9; 37-*- 4; 84^-*- 8^; 49-4-7.
Written Work
l. Divide 236 by 3.
Divisor 3)236 Dividend
78 Quotient
Remainder 2
Test. — 3 x 78 = 234 ;
234 + 2 = 236
23 tens -=-3 = 7 tens, and 2 tens
(20 ones) remaining. Write the 7
in tens' place.
20 ones + 6 ones = 26 ones ; 26 ones
-4-3 = 8 ones, and 2 ones remaining.
Quotient 78; remainder 2.
We think: "3 in 23, 7 times, and 2 remaining; 3 in 26, 8 times, and
2 remaining."
Find quotients :
2. 344 -*- 3 3. 763 ft. -4- 6 ft. 4. 466 i -4- 9
Divide $6.48 by 3.
3 )$ 6.48 Place the decimal point in the quotient directly
$2.16 under the decimal point in the dividend.
Divide and test :
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
$203,751
$678.34
$209.07
$390.08
$720.93
$379.38
$297.34
$427.84
$918.07
$847.12
by
a.
2
b.
3
c.
4
d.
5
e.
6
/•
7
9-
8
L
9
i.
10
•
11
Form 100 problems by divid-
ing each dividend by each of
the divisors, thus :
5 a. $203.75 -f- 2= ?
5 6. $203.75-*- 3 = ?
9e. $720,934-6 = ?
Write, solve, and test two
problems in 1 minute.
LONG DIVISION 31
Dividing by Larger Numbers
1. Divide 50, 90. 150, 600, 1 by 10.
2. Name the quotients when 130, 170, 1200, 2000, 100, is
each divided by 10.
3. Divide 500, 600, 1500, and 2500 by 100.
Removing one naught from the right of. a number divides the
number by 10 ; removing two naughts, divides it by 100; remov-
ing three naughts divides it by 1000 ; etc.
4. Divide 225 by 20, thus : 2 1 )22 '5 o tens j s contained in
1115 22 tens 11 times, with 5
20 remaining. 5 -=- 20 = 5 %.
5. Divide 2375 by 20 ; by 200.
Divide each number by 20 ; by 50 ; by 80 ; by 500.
6. 37,845. 8. 90.200 10. 409,805. 12. 390,075.
7. 50,240. 9. 74,079 11. 790,086. 13. 985,000.
LONG DIVISION
1. Divide 4310 by 21. Steps
205 1. Divide 43 by 21. Write the quotient
91 VPvFT) figure 2 over the figure 3 of the dividend.
' 2. Multiply 21 by 2.
_ 3. Subtract 42 from 13.
HO 4. Bring down the next figure. Is 21
105 contained an integral number of times in
5 remainder H? Write O in the quotient.
Test: 21 x 205 = 4305 5> Brin S dowu the next fi S ure and P ro "
," _ '* - ,01 n ceed as before. Write 5 in the quotient.
4305 + o = 4310
Divide and test :
2. 252-21 7. 2214 -s- 21 12. 1326-*- 51
3. 525 -j- 21 8. 4601-4-22 13. 1922-62
4. 724-22 9. 1271-f-31 14. 2193-5-51
5. 642-31 10. 1344-42 is. 7010-4-91
6. 345-4-31 11. 1024-*- 32 16. 6874-81
32
FUNDAMENTAL PROCESSES
Divide and test
17.
1364 by i
>2
25. 6207 by 76
33.
8538 by 94
18.
1395 by 31
26. 6572 by 68
34.
7646 by 87
19.
1728 by 42
27. 7010 by 91
35.
8544 by 79
20.
2193 by 51
28. 7284 by 92
36.
9584 by m
21.
2583 by 63
29. 6874 by 81
37.
7001 by 84
22.
3034 by 74
30. 6986 by 83
38.
8200 by 77
23.
4345 by 65
31. 7044 by 86
39.
7909 by 96
24.
5072 by 59
32. 8406 by 92
40.
8549 by 78
41.
1009
42
$11.17
715)* 7986. 55
43.
544
395)398555
805)437920
395
715
4025
3555
836
3542
3555
715
1215
3220
3220
Since 35 does not coi
itain 715
3220
395, the second figure in
the 5005
quotient is 0.
5005
Divide and test :
44.
6464341
' a. 268 Form
100 problems by di-
45.
7846760
b. 354 viding each dividend by each
46.
5864548
c. 676 of the divisors, thus :
47.
8645341
d. 758 44 a. 1
3464341
-h 268 = ?
48.
9624872
1
e. 865 44 b. I
3464341
-f- 354 = ?
49.
7784100
■ by ■
/. 984 49 c. '
r784100
-676 = ?
50.
6810404
g. 789 Write
', solve, and test each
51.
7904025
A. 897 problem
in 2 minutes.
52.
4867045
i. 509
53.
3234567
J. 890
MULTIPLICATION AND DIVISION 33
Problems of Two or More Operations
1. If 48 barrels of flour cost $324, how much will 275
barrels cost?
Cost of 4S bbl. = $ 324.00 Study of Problem
Cost of lbbL = 1824.00 -«-48 = * 6.75 i. What is given in
Cost of 275 bbl. = 275 x $6.75 = $1856.25 this problem?
a. Number of barrels in each purchase, b. Cost of 48 bbl.
2. What is required? a. Cost of 1 bbl. b. Cost of 275 bbl.
3. How do you find what is required? a. Divide cost of first pur-
chase by the number of barrels, b. Multiply the cost of 1 bbl. by the
number of barrels purchased.
Note. — The purpose of these studies is threefold :
1. To train the pupil to see and understand the conditions of a problem.
2. To give that logical, analytic grasp of conditions that forms the
basis of all mathematical power.
3. To direct the teacher in his efforts to attain these ends.
2. If 2675 bushels of wheat cost $2728.50, how much are
196 bushels worth?
3. A water tank holds 8640 gallons. If it receives 728
gallons per hour by one pipe and discharges 512 gallons by
another, in what time will it be filled ?
4. Two steamers sail towards each other from opposite
sides of the Pacific Ocean. If the distance across is 9872
miles, and one sails at the rate of 285 miles a day, and the
other 332 miles per day, in how many days will they meet ?
5. The daily pay of a railway conductor is $3.45. If he
works 310 days in a year, and spends on an average $65
per month, how much has he left at the end of the year?
6. The receipts of a street railway for 365 days were
$119,685.23. Find the average daily profits if the total
expenses were $96,478.02.
7. A and B divide an estate of $9875 between them. If
A receives $275 more than B, how much does each receive?
34 FUNDAMENTAL PROCESSES
8. A man sold 128 acres of land at #70 an acre, and 96
at $90 an acre. He invested the money in town lots at $550
each. How many did he buy ?
Study of Problem
1. What is given in the problem?
$8960 value of 1st farm 2 What is required?
$8640 value of 2d farm 3. What is the first step in the
$17600 value of both solution? the second? the third?
the fourth?
$17600 -r-$ 550 = 32, no. of lots bought.
9. If it costs 40 cents to ship a 10-gallon can of milk
from Hickory to Pittsburg, how much does the railroad
realize in 5 days, from a shipment of 135 cans per day?
10. A shipper pays 20 cents per barrel, per month, cold
storage charges on apples, and 15 cents per firkin on butter.
Find the charges for three months on 45 firkins of butter
and 328 barrels of apples.
11. A locomotive in making a certain trip uses 18 tons of
coal. If a trip is made in 2 days, how much coal will the
engine consume in 190 days?
12. A dealer buys three boxes of oranges for $3.50, $2.75,
and $2.50, respectively. If he sells 10 dozen at 50 cents per
dozen, 9 dozen at 40 cents per dozen, 12 dozen at 35 cents
per dozen, and the remaining 5 dozen at 25 cents per dozen,
find his gain.
13. An opera sale of tickets is as follows: 450 @ $1.50;
380 @ $1.00; 520@$.75; 310@$.50; and 240 @ $.25.
Find the total sale of the tickets, and the average cost of
each ticket.
14. A steamboat consumes 23 tons of coal per day. Find
the cost of the coal, at $5.85 per ton, for a trip of 39 days.
COMPARISON
COMPARISON
Comparison, as here used, indicates the relation of two simi-
lar numbers, expressed by the quotient of the first number
divided by the second.
1. Compare 10 and 5; 12 and 4; 16 and 8; 20 and 5;
24 and 6.
2. 20 is how many times 4? 30 is how many times 6?
How does 40 compare with 4?
3. What is the quotient of 48 divided by 8? by 6? by 4?
4. Compare 200 and 50; 400 and 100; 500 and 250.
5. 125 is what part of 250? of 500? of 375? of 625?
6. Compare 48 feet and 2 yards: 75 feet and 5 yards.
Xote. — Change yards to feet, or feet to yards ; then compare.
7. 6 feet is what part of 3 yards? of 10 yards?
Written Work
1. When 4 pounds of butter cost 80 cents, how much will
12 pounds cost?
Xote. — 12 pounds = 3x4 pounds: hence, 12 pounds will cost 3 x 80
cents.
2. Find the cost of 10 barrels of apples, when 2 barrels
cost 64.50.
3. At 3 pounds of coffee for $1, how much will 15
pounds cost ?
4. How much will 30 yards of silk cost when 3 yards
cost 63.75?
5. When 2 doz. oranges are selling for 60 cents, how
much will 8 dozen cost?
6. Find the cost of 20 barrels of cement, when 5 barrels
cost 66.25.
7. How much will ••'><> dozen eggs cost, when •*• dozen sell
for si?
36 FUNDAMENTAL PROCESSES
COMBINING PROCESSES
A parenthesis ( ) or a vinculum ' " groups together
several numbers and shows that the operations within the
groups are to be performed first ; thus, 6 — (3+ 2) = 6 — 5 = 1 ;
(5 + 3) x 2= 8x2= 16; 5 + (3x2) = 5 + 6=ll; 32-^4 + 3
= 8 + 3 = 11; 4x2-3=8-3 = 5.
When no parenthesis or vinculum is used, the signs x and
-=- indicate operations that are to be performed before those
indicated by either + or - ; thus, 4 + 8 x 3 = 4 + (8 x 3), or
28 ; 5 + 12 -=- 6 = 5 + (12 -*- 6), or 7.
In an expression like 12 -s- 6 x 2, mathematicians are not
agreed as to which sign shall be used first. To avoid ambi-
guity, the parenthesis should be used in such expressions.
Thus, (12 -s- 6) x 2 =4 ; but 12 + (6 x 2) = 1.
Find the value of :
!. 4 x 12 - 16 + 4. 5. (240 + 98) x (688 - 425).
2.7 + 8x7-26. 6. (56-18) x 11 + 4-6 x 4.
3 . (14 + 8 - 6) x 9. 7. (84 - 7 x 6 + 9 x 4 - 6) + 9.
4 . (87-65 + 96) x 24. 8. (56 + 7) x 12 + 97-7 x 9.
9. 6+10x5 + 8 + 2-4-2 + 8.
10. 7 x 5~+4 + 8^T6 + 2 - 3 x 4.
11. (6-+ 2 x 3) + 4 + (3 x 6) + 2 + 2 x (3 + 5 - 2).
12. 36- 6x4 + 2x6+ (40 + 5) +9 + 3x6.
13. 10 + 20-5 x 3 + 6x2-=- 3 + 5x6.
14. 3x(4+5-2) + 4+5x(4x5 + 2)+5.
15. 3 x (6 + 8) + 7 x (8 + 2) - 3 x (6 + 3) + 15 - 7.
16. 175 - 8 x (19 - 10) - 25 -*- 5 + 6^7 -9 + 3.
FACTORS AND DIVISORS
1. What two numbers will give 6 as a product? 8 as a
product? 10 as a product?
2. What are 2 and 3 in relation to 6? 4 and 2 in relation
to 8? 5 and 2 in relation to 10?
An integer or an integral number is a whole number.
The factors of a number are the integers whose product is
the number ; thus, 5 and 2 are factors of 10.
3. Name two factors that produce 24, 32, 40, 56, 49, 72, 96.
A factor of a number is an exact divisor of the number ;
that is, it is contained in the number an integral it umber of
times.
4. Name the exact divisors of 54, 81, 48, 36, 66, 64, 63.
2 x2 = 4
5. Observe the two equal factors that pro- 3 x 8 _ 9
duce 4; 9; 16. 4 x 4 = 10
2x2x2=8
6. Observe the three equal factors that pro- 3 x 3 x 3 = 27
duce 8 ; 27 ; 64. 4 x 4 x 4 = 64
Instead of repeating a factor, a small figure called an
exponent may be written to the right and a little above the
number to show how often it is used as a factor ; thus,
33 =3x3x3 = 27: 2 4 =2x2x2x2 = 16.
7. What number will divide 9 and 10? 21 and 25?
Numbers are prime to each other when they have no com-
mon factor ; thus, 9 and 10 are prime to each other.
37
38 FACTORS AND DIVISORS
Even numbers are numbers that contain the factor 2.
Odd numbers are numbers that do not contain the factor 2.
8. What are the factors of 7? of 11? Observe that 7
and 11 have no exact divisors except themselves and one.
A prime number is one that has no exact divisor except
itself and one ; thus, 5, 2, and 3 are prime numbers.
9. Name all the prime numbers to 31.
10. What are the factors of 15? Observe that 15 can be
divided by 3 and 5. It is composed of other factors than
itself and one.
A composite number is one that has other exact divisors
than itself and one ; thus, 6 and 10 are composite numbers.
11. Name all the composite numbers to 50.
TESTS OF DIVISIBILITY
1. Divide 12, 24, 26, 38, and 50 each by 2. What is the
ones' figure in each of the dividends? Divide other num-
bers ending in 2, 4, 6, 8, or by 2.
A number is divisible by 2, if the ones' 1 figure is 2, 4, 6, 8,
or 0.
2. Divide 15, 25, 40, 125, 150 each by 5. What is the
ones' figure in each dividend? Divide other numbers end-
ing in 5 or by 5.
A number is divisible by 5, if its ones' figure is 5 or 0.
3. Divide 36, 69, 48, 72, 162, 369 each by 3. Notice
that the sum of the digits (that is, of the figures) in each
number is divisible by 3. Divide by 3, other numbers the
sum of whose digits is divisible by 3.
A number is divisible by 3, if the sum of its digits is divis-
ible by 3.
FACTORING 39
4. Divide 18, 27, 279, 819, 639 each by 9. Notice that
the sum of the digits in each dividend is divisible by 9.
Divide by 9, other numbers the sum of whose digits is divis-
ible by 9.
A number is divisible by 9, if the sum of its digits is divisible
by 9.
5. Select the numbers that are divisible by 2 ; by 3 ; by
5 ; by 9.
86 96 123 918 515 3672
94 72 321 819 450 1909
FACTORING
1. Give the two factors that produce 15.
2. If one of them is given, how may the other be found ?
To separate a number into two factors, take any exact divisor
for one factor and the quotient of the number by this factor for
the other.
Factoring is the process of separating a number into its
factors.
A prime factor is a prime number used as a factor ; thus,
3 and 5 are the prime factors of 15.
Written Work
1. Find the prime factors of 126.
Divide by the least prime factor ;
divide the quotient by the next smallest
prime factor, etc.. until the last quotienl
is a prime number. The divisors and
_ o o "_io£ the last quotient are the prime factors;
thus, 2, 3, 3, and 4 are the prime fac-
0*i tors of 126.
2 x3 2 x 7 = l:2i;
2
3
3
126
63
21
40 FACTORS AND DIVISORS
Find the prime factors of :
2.
125
6.
945
10.
2431
14.
25600
3.
210
7.
2934
li.
7200
15.
64640
4.
225
8.
4620
12.
7700
16.
97125
5.
400
9.
3822
13.
6525
17.
78000
GREATEST COMMON DIVISOR
1. Name a number that will exactly divide both 16 and
24 ; 15 and 25 ; 14 and 27.
A common divisor of two or more numbers is a number
that exactly divides each of them ; thus, 4 is a common
divisor of 16 and 24.
2. Is 4 the greatest number that will exactly divide 16
and 24? What is the greatest number that will exactly
divide 16 and 24 ?
The greatest common divisor (g. c.d.) of two or more
numbers is the greatest number that exactly divides each
of them; thus, 9 is the g.c.d. of 27 and 36.
3. Name the g. c. d. of 24 and 36 ; of 32 and 40.
Written Work
l. Find the greatest common divisor of 56, 98, 154.
56 98 154 As the g.c.d. of two or more numbers is the
28 4^> 77 product of all their common prime factors, divide
the numbers by their common prime factors.
In the same way divide the quotients until
they are prime to each other. The divisors
2 and 7 are all the common prime factors of the
numbers. Hence, the g.c.d. of 56, 98, and 154
is 2 x 7, or 14.
2
7
4 7 11
g. c. d=2 x7, or 14.
LEAST COMMON MULTIPLE 41
Find the p-.c.d. of:
2. 42, 63, 189 7. 84, 56, 210
3. 54, 216, 360 8. 22, 110, 132
4. 48, 60, 96 9. 42, 84, 175
5. 84, 252, 512 10. 17, 68, 85
6. 21, 48, 78 11. 432, 720, 864
LEAST COMMON MULTIPLE
1. Name a number that will exactly contain 6 and 9 ; 8
and 12; 7 and 9.
A common multiple of two or more numbers is a number
that is exactly divisible by each of them ; thus, 36 is a com-
mon multiple of 6 and 9.
2. Name the least number that is exactly divisible by 6
and 9; hy 8 and 12.
The least common multiple (l.c.m.) of two or more num-
bers is the least number that is exactly divisible by each of
them ; thus, 18 is the 1. c. m. of 6 and 9.
3. Name the 1. c. i». of 6 and 8 ; of 9 and 12 ; of 8 and 12.
Written Work
l. Find the 1. cm. of 18, 32, and 40.
18 = 2x3x3 The I. c. m. of two or more numbers is
QO_ O y •) x O x 9 x 9 the product of all their prime factors,
tn \ r. -^ r each factor being used as often as it
40 = x x 2 x 5
occurs in any number.
1. c. m. = i 5 x o 2 X 5, or 1440. o occurs 5 times as a factor in 32.
It must, therefore, be used 5 times in the 1. cm. 3 occurs twice as a
factor in 18; it must, therefore, be used twice in the 1. c. m. 5 occurs
once as a factor in 1<>; it must, therefore, be used once in the 1. c. m.
Hence, the 1. c. 111. of is, 32, and ID is L )5 x 3 2 X 5 = 1110.
42 FACTORS AND DIVISORS
2. Find the ]. c. m. of 12, 36, 54, and 63.
'** — - Since 12 is a divisor of 36 the I.e. m.
3 )1§ '4l §§ of 36, 54, and 63 is also a multiple of
o )b 9 ^1 i2. 12 may therefore be rejected from
2 3 7 the work.
1. c. m. = 2 2 x 3 3 X 7 = 756. Divide any two of the numbers by
a common prime factor. Then divide the quotients in like manner until
the quotients are prime to each other. The product of the divisors and
the last quotients is the 1. c. m.
Find the 1. c. m. of :
3. 24, 48, 72
10.
48, 64, 72
4. 36, 70, 105
li.
144, 180, 240
5. 32, 40, 48
12.
85, 51, 255
6. 25, 35, 56
13.
120, 225, 540
7. 30, 60, 105
14.
98, 42, 126
8. 32, 48, 96
15.
180, 216, 120
9. 45, 70, 90
16.
100, 110, 440
CANCELLATION
144 _:- 36 = 4. We may separate the dividend 144 into
the factors 9 and 16, and the divisor 36 into the factors 9
and 4. We may, therefore, write 144 h- 36 = 4 as follows :
(9x16) -(9x4) = 4.
By striking out the common factor 9 in both dividend
and divisor, the problem is: (16 -j- 4) = 4.
Striking out equal factors from both dividend and divisor
does not change the quotient.
When the product of a number of factors is to be di-
vided by the product of another set of factors, the usual way
is to write the dividend above a line and the divisor below,
and strike out equal factors :
CANCELLATION 43
Thus, ^=2xl = Z = 2i.
Cancellation is the process of shortening operations in
division by striking out equal factors from both dividend and
divisor.
Written Work
1. Divide 3 x 6 x 8 x 20 by 11 x 4 x 20.
2
3x6x8x2p_36_ os Write the dividend above and the divi-
11 X 4 X 20 ~ 11 ~ ! T ' sor helow a line. First cancel the 20 from
dividend and divisor. Then cancel the
factor 4 from 8 in the dividend and from 4 in the divisor, leaving 2 in the
dividend and 1 in the divisor. As there are no other factors common to
dividend and divisor, you have 3x6x2, or 36, divided by 11, or ff,
which equals 3 T 3 T .
Note. — When equal factors in the terms are canceled, the factor 1
always remains, but as it does not affect the product, it need not be
written.
Divide :
2. 27 x 56 x 38 x 50 by 19 x 35 x 40
3. 5 x 51 x 36 x 63 by 17 x 9 x 54 x 10
4. 25 x 72 x 64 x 28 by 40 x 96 x 21 x 4
5. 69 x oG x 45 x 27 by 23 x 45 x 63 x 9
6. 72 x 48 x 84 x 28 by 24 x 48 x 42 x 14
7. 148 x 64 x 57 x 12 by 114 x 32 x 48
8. By selling butter at 24 ^ per pound a lady receives
enough money to buy 48 pounds of coffee at 20^ per pound.
How many pounds of butter does she sell ?
9. A man worked 16 days of 10 hours each at 20^ per
hour, and spent the money he received for corn at 40 ^ per
bushel. How many bushels of corn did he get ?
FRACTIONS
FRACTIONAL UNITS
When we say 8, 6 ft., $2, 6 rd., 7 mi., 5 in., what are the
units of measure ?
Observe that in each case the number and its unit of
measure are of the same denomination ; thus, 1 ft. is the
unit of measure in 6 ft.
A unit, therefore, is any single quantity with which
another quantity of the same kind is measured or com-
pared ; as, 1 is the unit of 10 ; 1 ft. is the unit of 8 ft. ; 1
yd. of 2 yd. ; 1 mi. of 12 mi. ; 1 acre of 5 acres, etc.
A fractional unit is one of the equal parts into which an
integral unit has been divided ; as, J^, -|, ^, ^, y 1 ^, etc.
A fraction is one or more fractional units ; as, |, f , |, |,
•|, etc.
The terms of a fraction are the numerator and the denomi-
nator.
The denominator indicates the size of the fractional unit ;
it is written below the line, and shows into how many parts
the integral unit has been divided. Thus, in the fraction |,
5 is the denominator, and shows that some unit has been
divided into 5 parts.
The numerator indicates the number of fractional units;
it is written above the line, and shows how many parts are
taken. Thus, in the fraction |, 4 is the numerator, and
shows that 4 parts have been taken.
44
FRACTIONAL UNITS 45
1. What is the fractional unit in jj ? { ? | ? f ?
2. Read the following fractional units in order of their
size, beginning with the largest: -, 1 ,., |, J, y 1 ^, £, |, and .,'j.
3. The use of the numerator and the denominator in
T 9 2 yd. may be explained thus, -^ yd. = 9 x ^ 2 - yd.
As the integral unit is the basis by which we measure
whole numbers, so the fractional unit is the basis by which
we measure fractions of the same kind.
4. Name the unit of 4 ft.; 5 mi.; f; §.
5. Which is the larger, | or 1 ? | or 1 ? | or 1?
Explain how much larger in each case.
6. What is the difference in value between the fraction |
and an integral unit ?
A common fraction is a fraction that has both terms ex-
pressed; as, f, f, \.
A proper fraction is a fraction less in value than 1 ; as, ^,
7 3 4 1 1 3 pfp
JT 4' 5' 9 1 T 1 3' euo "
An improper fraction is a fraction equal to or greater in
value than 1 ; as, |, f, |, f , f , ^g -, etc.
A mixed number is a number expressed by a whole number
and a fraction ; as, 3^, 12|.
Change each of the following to integral units, or to
mixed numbers. Thus, | = 1^.
10. ^
11. V
12. 17-
7.
10.
8.
f
9.
6
13.
¥
16.
¥
14.
9
17.
¥
15.
¥
18.
¥
46
FRACTIONS
READING AND WRITING FRACTIONS
Read
l.
2.
3.
4.
5.
!
6.
15
40
8
9
7.
4
10
1 1
12
8.
45
ft
1
6
9.
38.
110
H
10.
1 1 5
21?
11.
12.
13.
14.
15.
s$12§
To bu -
6| bbl.
1 Qfi 6
1 oz. = 7 4iii °f a ton.
Write in figures :
One fourth.
Three fifths.
Six ninths.
Three fourths.
Seven tenths.
1.
2.
3.
4.
5.
11.
12.
13.
14.
15.
16.
17.
18.
6. Eighteen twentieths.
7. Eight thousandths.
8. Sixty seventieths.
9. Nineteen forty-thirds.
10. Eighty-nine thousandths.
Eighty-nine three hundredths.
Twelve and three fourths.
Six and three fourths.
Five and one half.
Five ninths of three fifths.
One thousand ninety-fourths.
Nine hundred three thousandths.
Ten and three fourths.
19. Four hundred ninety and six thousand twenty-four
ten-thousandths.
Write in words :
20.
21.
22.
23.
4
5
L
9
11
1^
i
24.
25.
26.
27.
A
35
¥6
11
85
1 05
ITS
28.
29.
30.
31.
45- 9 -
"22 2
- UU 1000
- LVU1 2000
KEDICTIOX OF FRACTIONS
REDUCTION OF FRACTIONS
Reduction is the process of changing the form of a number
without changing its value.
1. Divide both terms of the fraction | by 2. How does
| compare in value with l ?
2. Multiply both terms of the fraction | by 2. How-
does | compare in value with | '! How may we obtain | or
•^2 from | ?
Multiplying or dividing both terms of a fraction by the same
number does not change its value.
Changing a fraction to higher terms.
1. Explain why a fraction is expressed in smaller fractional
units when it is changed to higher terms.
2. Explain why changing a fraction to higher terms does
not change the value of the fraction.
Change :
3. i to 12ths 6. ^to^ths 9. lJto72d&
4. | to 24ths 7. | to 56ths 10. | to 63ds
5. f to 18ths 8. f to 81sts n. | to 96ths
12. f =^ = A 16 '
13 -9_ — _?_ = J- 17
J-«. in — en — on ■*■'•
5 _ J_ - f_
6 "~ 36 " Y2
- •> 9
TO - 60 — 90 ■ L/ *8 "4'0 ~6¥
* -io K ? ?
14. A = A =A 18-
■ i
IT — I? _ 58" *""' 12 — 96 — 7 2
15. n = st = TTfO ^•"
_6_ —
9 — 8T ~ 10¥ "• 11 ~" 132 ""99
Written Work
l. Change f to 27ths.
27 -i- 9 = 3 Since multiplying both terms of a frac-
5 5x3 15 ti° n by the same number does not change
q = q S = 27 ^ vame ' multiply both terms of the frac-
tion by the quotient of 27 -*■ 9, or 3.
48
FRACTIONS
Change
2.
3.
4.
5.
6.
7.
8.
| to 20ths
I to 56ths
l| to 96ths
T 9 3 to 78ths
11 to 276ths
9.
10.
11.
12.
if to 132ds 13.
l| to 72ds 14.
-^ to 135ths 15.
Changing a fraction to lower terms or to lowest terms.
1. Explain why a fraction is expressed in larger fractional
units when it is changed to lower terms. Explain also why
changing a fraction to lower terms does not change its value.
Change :
ff to 12ths
l| to 275ths
if to 372ds
f! to 494ths
!! to 765ths
fi to 415ths
|f to 315ths
2. A to 4ths
6.
7.
12
|| to Cths
|f to 8ths
|f to lOths
£f to 12ths
|§ to 8ths
8.
9.
10.
11.
12.
13.
|| to Gths
ff to 9ths
ff to lGths
4
50
? _ 4 9
12 — 8¥
14.
IS.
16.
17.
18.
19.
_?_
21
?
9
9
8
9
t
?
9
21
63
_4_8_
10 8
13^
144
24
64
35_
5 6
40
Y2
Written Work
A fraction is expressed in its lowest terms when the nu-
merator and the denominator are prime to each other.
l. Change || to lowest terms.
Since dividing both terms of a fraction by the same
number does not change its value, we may reject by
cancellation all the factors common to both terms, leav-
ing the factors 7 and 9. Hence, ££ = J.
Or we may, in one step, divide both terms of the frac-
tion by their g. c. d., 6.
m_
n_
7
u
~%7l
: 9
Or
g.c.d
6
42
■f-6
7
54
*-6
9
REDUCTION OF FRACTIONS 49
2. Change |§£ to lowest terms.
357 + 3 = 119 + 7 = 17 r ff cd-21 35T - 21 17
483 + 3 161 + 7 23 ur 'S- c - a --^ 483 + 21 = 23
Cancel all the factors common to both numerator and denomi-
nator. Or, divide both numerator and denominator by their
greatest common divisor.
Change to lowest terms :
3.
4.
5.
6.
7.
8.
9.
10.
11.
Changing a mixed number to an improper fraction, or an
improper fraction to a mixed number.
18
24
12.
1 2 1
"13 2
21.
125
325
30.
4£5
53
39.
4 14
999
25
55
13.
54
12
22.
3&5
605
31.
75
S2 5
40.
1 2ii
189
42
49
14.
18
28
23.
480
66
32.
615
94 5
41.
4 3 5
630
72
81
15.
42
4 8
24.
18 2
196
33.
4i2
504
42.
in
1 96
21
36
16.
3£
42
25.
264
333
34.
67 2
936
43.
2 1 6
270
.24
28
17.
58
T4
26.
315
34 5
35.
256
92 4
44.
546
5 88
^5
18.
128
176
27.
200
450
36.
551
62 1
45.
396
4 :; 2
22
Y2
19.
94
144
28.
£28
624
37.
294
476
46.
561
783
84
96
20.
81
96
29.
288
444
38.
322
504
47.
837
945
Change to an improper fraction at sight :
1.
If
4. 31
7.
u 10
2.
n
5. 5|
8.
3-2-
°15
3.
01
~3
6. 2|
9.
8 to 12ths
11AM.
1 "Mil.. AJOIH.
— 4
10. 1 2 to 3ds
11. 8 to 6ths
12. 10 to 5ths
50 FRACTIONS
Written Work
l. Change lllf to an improper fraction.
1 _ 5. In 1 there are § ; in 111 there are 111
-i-ii iii x JL - - JL5.5. times |, or £§ s , which added to f equal
£56 I 8 -- JL5 8 " 5 "- Hence ' * 11 * equals^.
J,- "T" 6 — 5~
In small numbers the work may be done mentally, only
the result being written.
Change to improper fractions :
2.
12f
8.
™H
14.
268Jft
20.
391ft
3.
15^
9.
9*
15.
324f§
21.
18**
4.
22|
10.
103f|
16.
502*
22.
901^
5.
80ft
11.
118|i
17.
109^
23.
100^
6.
48 T i
12.
1,,J 66
18.
600JJ
24.
390||
7.
56i|
13.
* dit 'l0 5
19.
305f
25.
231H
26. Change -1| 4 to a mixed number.
In 1 there are ?, and in if 8 there are as
12 8 _ 128 -7- 7 = 181 many times 1, as 7 is contained times in
7 128, or 18f. Hence, if* is equal to 18$.
Every fraction is an indicated division.
Change to integers or mixed numbers :
27.
31
15
33.
w
39.
34
45.
2 3^8.
15
28.
_5_2JL
11
34.
J70
35
40.
876
38
46.
14 4
75
29.
955.
7
35.
18
41.
24
47.
5286
48
30.
13JL
16
36.
625
"2 5
42.
862
"84
48.
4646.
50
31.
261
8
37.
,5 7 6
24
43.
534
96'
49.
2200
12
32.
w
38.
.240
16"
44.
121A
24
50.
4 03 2
"36
REDUCTION OF FRACTIONS 51
Changing to least similar fractions.
A common denominator of two or more fractions is a num-
ber that contains all the denominators of the fractions an
integral number of times ; thus, 24 is a common denominator
of ^, ^, and J.
The least common denominator (Led) of two or more
fractions is the least number that contains all the denomina-
tors of the fractions an integral number of times ; thus, 12 is
the Led. of }, ^, and J.
The 1. c. d is the least common multiple of the denominators.
Similar fractions are fractions that express the same unit
value. They must therefore have a common denominator.
By inspection :
1. Change |, |, and ^ to similar fractions having the least
common denominator.
It is evident by inspection that 18 is the least common
multiple of 2, 3, and 9. It is therefore the least common
denominator of the given fractions. Changing the given
fraction to ISths. we find that \ = T \ ; t=\t\ and
s = jo. Hence, the fractions h §, and | may be changed
to the similar fractions ^, ff, and i§.
Change to least similar fractions
2.
3.
4.
5.
6.
1 x 0_
2x9
9
: 18
2x6.
3x6
12
= 18
5 x 2 _
9 x 2~
1°
18
t
3'
1 5
4' 12
7.
1
5'
3 7_
10' 40
12.
2
3"
1 19
6" 36
17.
1
6'
3 JL
7-14
2
3'
1 2_
5' 15
8.
1
3'
A»A
13.
1
3"
1 1
2* 6
18.
1
3'
1 • • 2
h
3' 1?
9.
2
5'
3- 9_
"8"' ¥5
14.
3
7-
1 1
14' - -
19.
2
5'
2 3_
;;• -JO
i
6'
3 17
- • 21
10.
1
9'
1 -5_
6' 18
15.
3
5 7
If 1 ',
20.
4
7"
5 1
8"' 5 6
1
9'
_4_ _§_
18' 3 6
11.
3
5'
7 JL
10' 2
16.
1
1
i r 53
21.
■
l 3
52 FRACTIONS
By factoring the denominators :
l. Change |, f , and ^ to similar fractions having the least
common denominator.
4=2x2
In finding the I.e. m., use each factor
O = -i X o as fteu as it occurs in any one number.
10 = 2 x 5
1. c. m. = 2x2x3x5 = 60
3 x 15 = 45
4 X 15 60 ,The least common multiple of the denominators is
5 x 10 50 ^' w °i cn * s the l eas t common denominator of the
£ 1 ft == ~fU\ gi yen fractions. Changing the given fractions to
o X a n i 60ths ' we find that I = M; I =»5 and & = M-
9 x 6_54
10 x 6~60
Change to least similar fractions :
2 4 2 3 1 o 2 19 _5_ U 11 _L L J 29
*' 4' 5' 8' 2 °' 5' 20' 12' 30 x *' 10' 12' 20' 3~0
3 2 _5_ _7_ 1 o 1 1 _9 11 is 3 _9_ Q3 9H
°* 3' 16' 12' 4 *' 7' 2' 28' 18 xa " 1?' 16' °8' ^2 1
4_7_i _8_ 9. in 1 _9_ _8_ _3_ ic J_ 11 3 5 1
*' 10' 5' 15' 20 1Ul 4' 10' 35' 16 - LO- 2 1' 35' Y0' 120
5 2 4 15 -n _7_ J>_ 11 li 17 5 8 _&_ 31
a * IT' 5' 8' 6 xx * 18' 15' J 9' 30 - L/< 6' 9' 2 1' 63
6 & 12 11 io 1 _5_ _9_ 31 in 8 14 Q 7
° - 9' 2 5' 18f ""' 18' 2 1' 2 8' 54 xo - 9' 2 5' °T5
7 2 5 _8_ 13 31 11 IS 12 iq 9 16 19 GJ
'• 7' 9' 11 ■ LO - °2' 18' A 6' 24 x *' 22' 3^' 66' 132
ADDITION AND SUBTRACTION
Size and kind of fractional units.
1. Why is it not possible to add 4 ft. to 5 oz. ?
2. What is the sum of \ and | ? of | and § ?
3. What is the size of the fractional unit of £ and | ?
4. What is the kind of fractional unit in each ?
5. Are the fractional units of | and | alike in size and
kind ?
ADDITION AND SUBTRACTION 53
6. What is the size of the fractional unit in $| and | ft.?
What is the kind of unit in each ?
7. Why can we add or subtract $ £ and $-§, or £ ft. and
2 ff 9
8. What kind of fractions can be added or subtracted ?
Before fractions can be added or subtracted they must be ex-
pressed in similar fractional units.
Fractions having the same kind of units or having related units can
be added; thus, f and J, or f yd. and \ ft. (= T \ yd.) can be changed to
similar fractions and then added. But fractions whose units are unrelated,
as f yd. and \ oz., cannot be added, because they cannot be changed to
similar fractions.
9. 5.] ft. and 2 yd. = ft. Why must you change
yards to feet before adding ?
10. Add the fractions in the following list that are similar:
\ da.
|rd.
3
8
2
3
hr.
ft.
3 fin -9-
tV min - i
min.
ft.
S in--
ird.
Add quickly :
11 1 + 1
■"■■ 2^4
15.
£_i_ 1
5^2
19. £ + |
23.
5 4-1
6 ' 3
12 2.4-1
±di - 3 ' 6
16.
4.1
3^2
20. f+1
24.
1 4- 8
3 • 6
13 3 _i_ 1
17.
2 . 1
3^9
21 -Ui
**■■ 5^2
25.
Bj.5
4 + ff
14 1 4_ 2
J-*. 4 -t" 3
18.
5 _i_ 3
1+2
22 I 1 4- 2
12 ' 3
26.
i+^
Written Work
l. Find the sum of f, |, and |.
The least common denominator is 36. Chang-
ob = 1. c. d. j n g tj ie gj ven fractions to 36ths, we find that
| = || ; 3 = |i ; | = 3|, The sum of these fractions
2
3
24
!
27
8
9
32
— 83 — 011
-T5 36"
8S — Oil
1. What is the first step in the work?
2. "What is the second step ?
3 6 3 6 3. What is the third step?
54
Add:
o 5. 7 A 7 3 2 _9_ _9 3
2. g> $, f /• ■$, ^ 10 •"• ^' 12' 32' 4
FRACTIONS
3
4'
2
"5"'
9
10
9
T'
9
14
3
1 4
2
5'
3
7'
1
9
2
3'
3
5'
1 5
4' 6
1
' 6'
2
3'
J7_ 14
12' 15
o 3 9 1 o 2 9 3 13 1 _5_ _3_ 11
3. y, -J4, 2 **• Y' 14' 4 A 8' 12' 16' 2 4
a. 5 11 3 a 2 3 1 14 JL 2 11 25
4 - Y' "2 5' f 9 5' 7' 9 ■'■*• 14' 7' 3 5' 4 9
c 2 1 1 in 2 3 1 5 K 1 1 J_ 12
5 - 31 ?i 2 - 10- 3' 5' 4' 6 4' 5' 10' 2 5
c 5 7 5 -n 12 7 14 16 1 -3 1 -7_
6 - 8' 12' 6 ■"" 6' 3' 12' 15 ±0 - 4' 8' 5' 12
17. Find the sum of 3^, 2$, and 7ft.
-jqq_] (J Since the numbers are mixed numbers, the integers
and fractions are added separately, and their sums
are united. The sum of \, f, and T 4 o is f^,or l T 3 gV
The sum of 3, 2, and 7 is 12. The sum of 12 and
8i
45
100
72 i T s^ i s 13/^.
12 + ftj = 12 + lftft = 13^
Add :
18. 3f, 7f, 9ft 24. 20f, 12Jf, 5|f
19. 18ft, 3y, 9ft 25. 14ft, 32^, 23ff
20. 8f, 10ft 16ft 26. 2f, 7f, lift, 14ft
21. 7f , 12ft, 24ft 27. 90ft, 60ft, 73ft
22. 16|, 30f,45ft 28. 3-i, 4f, 6f, 8ft
23. 50ft, 48ft, 16J| 29 - 84 T2' 36 i 33 f' 39 f
30. A bicycler rode 8| miles the first hour, 7| miles the
second hour, 6ft miles the third hour, and 8^ miles the
fourth hour. How many miles did he ride in the four
hours ?
31. Find the distance around a field 80| rods long and
60| rods wide.
32. Find the sum of the improper fractions f , ff , ft' io-
33. What number is that from which if 32ft is taken, the
remainder is 23y 9 ft ?
ADDITION AND SUBTRACTION
55
Subtracting fractions.
What kind of whole numbers can bo added ? what kind
of fractions ?
In adding like fractions we find the sum of the numer-
ators ; in subtracting like fractions we iind the difference
of the numerators.
l. From | take T 3 g
72 = 1. c. d.
J
63
3
18
12
5 1 _
1 7
~7 2 -
2 4
Written Work
Since fractions must be made similar
before they can be subtracted, f and -Ag are
changed to 72ds. The difference between
$| and \\ is f|, or \\.
1.
2,
3.
Observe the ihret steps in subtraction of fractions:
Change the fractions, if necessary, to like fractions.
Take the difference of their numerators.
Change the difference to its simplest form.
Find differences
2.
3.
4.
5.
I _ 3
9 4
JL _ 11
10 1^
J 5L
12 18
2 5
27
11
12
6.
7.
8.
9.
5 5
96
1 1
— 24
1 3
15
-H
2:?
2 4
14
15
3
108
_ 1 3
36
10.
11.
12.
13.
i_i _ in
12 13
2 3
36
13
2 5
21 _ 12
63 3 5
15
5 6
1 S
12
14. A boy wishes to buy a pair of skates costing 8^, but
he has only 6f. What part of a dollar is lacking?
15. From | take f.
16. What fraction added to | will give || ?
17. From -| of an acre of land, subtract | of an acre.
18. What part of a teacher's salary remained after he had
spent 1, A^-. and 1 of it ?
7
56 FRACTIONS
19. The minuend is |f, and the remainder f. What is
the subtrahend?
20. If a boy spends I of his money one day, \ of it the
next day, and has $1£ left, how much money had he at first:
Subtracting mixed numbers.
Written Work
1. From 1\ take 3$.
7\ = 6 + | + \ = 6|. The integers and fractions are
36 =1. C. d. subtracted separately. The least common denomi-
74 = 6| 45 nator is 36. Changing the fractions to 36ths we find
3 5 20 that f= H, and $=${}. tt ~ ** = H- 6-3 = 3,
o^ — |J which added to f£ = 3f|. Hence, the difference
3B between 7J and 3f = 3f£.
7w subtracting mixed numbers subtract the integers and frac-
tions separately.
Find differences :
2. 7f-3 T \ 8. 6311-24^ 14. 60^-35^
3. 9f-2f 9. 71^-19* 15 - 71 f!- 54 yf
4. 18H-8| io. 78JJ-18H 16. 39if-18ii
5. 30i-20if 11. 92||- 29 il 17 - 82 H- 45 M
6. 45i-24f 12. 82f|-29f| 18. 29^-llfi
7 . 50^-llf 13. 95|f-47ii 19. 20fi-15fi
20. If I pay a grocery bill of $221 a wa ter bill of $3£, and
a gas bill of $5§, how much shall I have left from 2 twenty-
dollar bills ?
21. The sum of three numbers is 150. The least number
is 151 and it is 63| less than the greatest. Find the other
number.
ADDITION AND SUBTRACTION 57
22. What fraction added to the sum of £, f, and -^g will
make f ?
23. If 5 is added to each term of the fraction f, is the
value of the fraction increased or diminished, and how
much?
24. Two boys undertake to save $50 apiece. When one
of them lacks $8$ of having $50, both together have
$ 84|. How much has each ?
25. A traveling man's grips, when starting out, weighed as
follows: 12$ pounds and 19| pounds. Find the weight of
both, and the difference in their weight.
26. James lives 1 1 miles east of the schoolhouse, and
Harry l^g miles west of the schoolhouse. Find the sum
of the distances walked by both each day and the distance
James walks farther than Harry.
27. Eight women do different parts in making a finished
garment. The cost of the different parts is 1}^, 2|^,
41 £ 3|£ 41^, 4f| £ l£ and T 7 ^. Find the cost of making
one garment.
28. Four automobiles finish a race in the following time:
8 T 4 5 hours, 8| hours, 7| hours, and 9^0 hours. Find the dif-
ference between the time of the winner and each of the others.
29. The average cost per mile of a fleet of freight boats on
the Great Lakes was : coal 13f £ crew 12f £ repairs 17f£
supplies 7^. The average cost per mile of a freight train
hauling the same freight was: coal 74 §£ crew 37^, repairs
19-^6 supplies 4|^. Find the saving per mile by water.
30. A drayman hauls freight by the ton from the depot
to a village. Find the amount hauled in 8 loads weighing
respectively : 1| tons, ^ tons, If tons, 2y\ tons, 2f tons,
If tons, 21 tons, 1^ tons.
58 FRACTIONS
MULTIPLICATION OF FRACTIONS
To multiply a fraction by an integer.
1. 1 + 1 + 1 + 1 = how many whole units ?
2. ^ + ^4-^ + 1 = how many fractional units ?
3. How many, then, are 4x1? 4x|?
4. Does it make any difference in the process of multi-
plication in problem 3 whether the multiplicand is a whole
number or a fraction ?
5. What term of the fraction A is multiplied by 4 ?
6. | may be multiplied by 4, thus : 4 x ^ = — - — , or - = 2| .
o o o
Give products :
7. 6 x § li. 5 x f 15. 5 x y\ 19. 16 x I
8. 10 X | 12. 12 X f 16. 15 X f 20. 14 X f
9. 9xf 13. 7x| 17. 12 x I 21. 13 xf
10. 12 x I 14. 7 x I 18. 8 x I 22. 11 x T \
23. In multiplying the above fractions by a whole num-
ber, did we increase the size or the number of the fractional
units ?
24. How, then, may any fraction be multiplied by
an integer without increasing the size of its fractional
units ?
25. 2 x | of a square = how many eighths of the square?
How many fourths of the same square ? Draw figures to
illustrate.
26. How does | of a square compare in size with | of the
same square ? Draw figure to illustrate.
MULTIPLICATION 59
27. 3 x | of a square = how many ninths of the square?
how many thirds of the same square ? Draw figures to
illustrate.
28. How does | of a square compare with § of the same
square ?
29. How much larger is the fractional unit in fourths
than in eighths ? in thirds than in ninths ?
30. How can we increase the size of the fractional unit
in | without decreasing the number of fractional units?
in | ?
31. How, then, may any fraction be multiplied by an in-
teger without increasing the number of its fractional units ?
Then < 3x l=db'°4
In what two ways, then, may we multiply a fraction by
an integer ?
Multiplying the numerator or dividing the denominator
of a fraction by a number multiplies the value of the fraction
by that number.
Give products :
32.
8 x |
40.
15 xf
48.
25 x A
33.
9x^
41.
11 x A
49.
15 x A
34.
11 x f
42.
9 x*
50.
18 x fl
35.
6x?k
43.
16 x &
51.
12 x Hh
36.
10 x ^
44.
7vU
1 A u:i
52.
36 x }|
37.
9xf
45.
3 x&
53.
42 x Jf
38.
12 xf
46.
4 x |f
54.
39 x*f
39.
18 x&
47.
5. x -y-
55.
64 x A
60 FRACTIONS
Written Work
Since multiplying the numerator of a
1. Multiply g^ by 5. fraction multiplies the fraction, 5 times
A v JL. — 15 — 1 _3_ _7_ _ 3 5 _ 1 a_
A 32 — 32 — 32 S2 — 32 — *52-
When possible, use cancellation.
Multiply :
2. - 2 8 T b7 7 11. |Jby72 20. fi by 96
3. f by 10 12. ii by 108 21. fe by 54
4. f£byl5 13. T ^by57 22. j* by 28
5. f by 8 14. II by 144 23. ^ by 105
6. J{by27 15. ft by 135 24. ^ by 96
7. 11 by 35 16. ^ by 21 25. { ° by 121
8. 11 by 28 17. £ by 16 26. if by 48
9. ff by 144 18. T faby72 27. &-by69
10. H by 70 19. T Ve by 24 28. J| by 126.
Finding fractional parts of an integer.
8 x I means that f is to be taken as an addend as
many times as there are units in the multiplier. Thus,
3_i_.a_i_3i.a_i3i3_i_jj_i_3
4 ' 4 ' l+l + l ' 4T4T4'
| of 8 means that 8 is to be divided into 4 equal parts
and 3 of these parts are to be taken.
What is the first step in rinding the fractional parts of the whole num-
ber? the second step?
While the process of finding fractional parts of a whole number is
classed as multiplication, the multiplicand at no time is taken as an ad-
dend, but is partitioned, that is, divided into equal parts, and a certain
number of these parts is taken.
The sign x is read " of " when the number preceding it is
a simple fraction ; as, § x $6 is read "| of $6."
MULTIPLICATION 61
| x 12 mo. means, therefore, 3 x (J of 12 mo.) or 3 x
3 mo. = 9 mo.
3
Or, - x 12 mo. = — t^- mo. = 9 mo.
4 £
Find :
1. j of #80 6. ^ofGOhr. 11. ].]xl24
2. | x42 7. {^ of 30 da. 12. ^ x 175
3. | x 63 8. I of 47 13. |$ x 450
4. fg x 48 9. I x 100 14. fj x 720
5. | x 20 10. | x 95 15. |f x 108
Written Work
1. If the Welsh mills turned out in a certain year 576000
tons of tin plate and the American mills || as much, find
the output of the American mills for that year.
2. A certain post office in one year handled 53678996
pieces of mail, of which f were letters. Find the number of
letters handled at that post office.
3. The records of a blast furnace show 1179360 tons of
iron made during the year. If | of this iron is sold as No. 1
iron, find the number of tons of No. 1 sold.
Multiplying a whole number by a mixed number.
Written Work
l. Find the cost of 48| gallons of molasses @ $.88.
$.88 qp.88
48f=40 + 8 + f 48|
.66 = | of .88 ^
7.04 =8x.88 704
35.20 =40 x .88 352
#42.90 =48| x.88 #42.90
62 FRACTIONS
Find the cost of:
2. 27§ tons of hay at #12 a ton.
3. 31| yards of cloth at $ .24 a yard.
4. 10| ounces of gold at # 18.75 an ounce.
5. 196 T 9 7 ounces of silver at $.46 an ounce.
6. 97| bushels of apples at $.70 a bushel.
7. 68| bushels of berries at #2.65 a bushel.
8. 147^ bushels of potatoes at $.85 a bushel.
9. 84^ pounds of prunes at $.12 a pound.
10. 257 1 feet of curbing at $.38 a foot.
11. A street-car conductor collects on an average $3.60
per hour. How much does he collect in 11| hours?
12. In 4 days a carpenter worked 7| hours, 8 hours, 7^
hours, and 6| hours. How much did the work cost at 45^
per hour ?
13. A ditch costs $.27 a rod. Find the cost of 227J rods.
14. The "Pennsylvania Special" averages 58 miles per
hour. How far does it travel in 16| hours?
15. A roller in a steel mill rolls 118| tons, 221^ tons, 193^
tons, and 180| tons in four days. If he is paid $.12 per ton,
how much should his pay envelope contain for the four days'
work?
16. A car load of corn contains 821| bushels. How much
is it worth at $.57 a bushel?
17. Three men in a day cut respectively 2^ cords, If
cords, and 2| cords of wood. How much does each receive
for his work at $1.25 per cord ?
18. At $26 per ton, how much will the rails for 18 miles
of railroad cost, if it takes 158| tons per mile ?
MULTIPLICATION
63
19. Two men work respectively 23] days .and 27| days in
a month. How much more does one earn than the other, it'
each receives $2.25 per day ?
Multiplying a mixed number by an integer.
In the problem 9 x 3|, which number is the multiplier?
the multiplicand ? The multiplier usually precedes the
sign X •
Give products :
1. 9 x 9i-
8.
14x3$
15.
11 x7f
2. 6x|
9.
11 x5|
16.
25x4§
3. 8 x 5J
10.
6x8|
17.
13x31
4. 7 x 8f
11.
18x3f
18.
12 x 15|
5. 9 x 8£
12.
20x4f
19.
15 x 7f
6. 10 x 10 T 3 o
13.
16x3|
20.
8x9|
7. 15x31
14.
9xll-J
21.
9 x6ft .
Written Work
1. Multiply 14§
by 9,
143
9
9x
t = ¥
= 3f.
3| = 9x|
9x14 = 12U.
3f + 126 = 129f.
126 =9x14
. 129f = 9 x 14
.3
8
Multiply:
2. 12f by 28
8.
44|1 by 14
14.
47f by 96
3. 11$ by 45
9.
78| by 63
15.
609f by 21
4. 14 A by CO
10.
89-^ by 105
16.
105| by 49
5. 25 11 by 75
11.
715f by 45
17.
290 T % by 45
6. 85f| by 68
12.
101$ by 63
18.
21 3f by 78
7. 64 \l by 56
13.
205f by 75
19.
735| by 21
64
FRACTIOXS
Find products:
20. 213 x 609f
25. 596 x 56f
30.
379 x 49 T ^
21. 612 x 48§
26. 972 x 32f
31.
49 x 465 1|
22. 842 x 95|
27. 96 x 325f
32.
786 x 49|
23. 728 x 34 T %
28. 856 x 98|
33.
9872 x 36^-
24. 96 x 207 §
29. 54 x 657|
34.
4398 x 94|
35. There are 16^ ft. in a rod. How many feet are there
in 12 rods ?
36. Find the cost of 35 tons of railroad iron at $38| a
ton.
37. At i$lf each, how much will 40 "General Histories"
cost?
38. If I sell 7 apples for 10 cents, how much shall I re-
ceive for 7 dozen apples ?
39. When oranges are sold at 3 for 10 cents, how much
will 3 crates carrying 180 each cost ?
40. Stephenson's locomotive weighed 4| tons. Find the
weight of a modern freight mogul weighing 40 times as
much.
41. A mail boy averaged 13-|^ per hour for 117 hours
worked during the month. Find his earnings for the month.
42. A residence is lighted by 21 incandescent lights.
The cost of each light per day is 1 f ^. Find the total
electric light bill for a year of 365 days.
43. The average shipment of ore from a mine per week of
6 days is 347 tons. If $12^ is the average profit realized
from each ton of ore, find the net profit for 40 weeks.
44. A mail carrier's deliveries average 23 1 pounds. How
many pounds of mail does he deliver in 2 trips each day for
312 days?
MULTIPLICATION 65
45. Iii a certain shoe factory a pair of shoes is finished
every 3| minutes. Find the number of days of 10 hours
each required to make 4600 pairs.
46. An establishment consumed in one year 8450 pounds
of twine at 9^ per pound ; 36| dozen bottles of ink at $1.90
per dozen ; 6 gross pens at 90 ^ per gross ; 500 pads of
paper at f $ per pad. Find the total cost of the purchase.
47. A real estate agent sold 6 pieces of land containing
respectively : 10§ A., 121 A., 18f A., 26j% A., 30§ A., 3| A.,
at 1250 per acre. Find the amount of the sale.
48. A town has a population of 3600. If the average
amount of water used by each person is 6| gallons per day,
find the number of gallons of water used in 90 days.
49. If a motorman receives 9\ per hour, how much does
he earn in 7 weeks of 6 days each, working 10 hours per day ?
Finding fractional parts of a fraction.
1. What is i of 9 hours ? § of 9 feet ?
2. What is i of 9 tenths ? § of 9 tenths?
3. 1 of 9 tenths means that we are to take l of 9 parts of
a unit that has been divided into 10 equal parts.
4. | of T 9 7 = how many tenths ?
5. If ^ of t 9 q = ^, how much is § of T 9 a ?
Find mentally :
6. loft
11.
7 of £1
9 OI TS
16.
2
3
Of -9-
01 16
7. fof^
12.
11 of £±
12 U1 33
17.
5
off
8 4 of 14
o. 8 ui 21
13.
_9_ of 4
10 U1 ?3
18.
1 1
12
off
9. foflf
14.
_4. Of A5
15 Ui 5 3
19.
:»
1 D
of|
10. * Offl
15.
3 of H
8 Ui 3 5
20.
B
8
Of ft
HAM. COMPL
. A KITH.
— 5
66 FRACTIONS
21. What does the expression ^ mean ?
22. Can T 9 7 be separated into 3 equal parts ; thus, T ^-, T 3 ^,
^, without changing the size of the fractional unit ?
23. To what fractional unit must we change f before we
can separate it into 4 equal parts ; that is, take \ of | ?
24. Is |§ the same in value as f ?
25. When || is separated into 4 equal parts, how many
fractional units are there in each part ?
26. Then 1 of | = how many twentieths ?
27. Observe that - of ? = L*!* or A.
4 5 4x5' 20
28. What 18 f Of |? Ifl0ff=,3_,3 0ff = 3x JL or JL
29. Observe that - of - = !L*J*, or — .
4 5 4x5' 20
A fractional part of a fraction equals a fraction times a
fraction.
Written Work
l. Find | x |f.
7 X 16 7x 16 _ 112 _ 2 To find I x if means to find J of |f.
8 x 21 " 8 x 21 == 168 "~ 3 * of H = A and J of h = 7 x A = W,
or*.
Or,
2
? y=?
$ x &l 3
3
A fractional part of a fraction is found by multiplying the
numerators for the numerator of the product and the denomina-
tors for the denominator of the product.
Indicate the operation, and cancel when possible.
A fractional part of a fraction is sometimes called a com-
pound fraction. Thus, \ of \ is a compound fraction.
MULTIPLICATION 67
An integer may be expressed in fractional form, thus, —
2
8 = 8. |of8 = - 3 off = | = 6.
Find products :
r> 3 v IB a 2r ' v 14 lO ?5 y ij 14 %k X ^M
o 5 v 38 7 21 v 33 in 21 y 15.5 15 :; s X MS4I
3 * 19 X 3 5 7- 44 X 5 6 ■ LJ " 6 2 X 162 XO ' 43 A 130
d. 15 v 14 o IS v (!5 12 1 3 y 3 4 ic 54 3 x 102
4 - 2% X 2 5 B - 2 6 X 7 2 X ^' 68*39 xo * 5 12 A 2 4 5
e 5 1 v 8 Q 1 1 v 4 6 i -a 7 1 y 3 7 17 111 y 110
5 - 6 4 X IT 9 - 6 9 X 121 13- 74 X 38 L/ ' 220 x 4 1 1
18. Find I of 12|.
Note. — Change the mixed number to an improper fraction.
Find :
19. § of 7j 23. 3| X 4 5 27. 2| x 7 1
20. f 0fl2| 24. 7| X| 28. 8| x5|
21. I ofl3f 25. 5 T %x{| 29. 7| x3|
T r of 10$ 26. 8| x I 30. 4 T ^ x 3§
Find the value of :
31. 4$ x 6f
8 5 6
A Q /-A 3$ 48 30 ^^ QA
4fx6f = |-xy = T , or 30.
32. 6| x2 T 3 T x7| 38. 35^x22^x12
33. 8^ x2| x2| 39. 51fx 19^x121
34. 101 x 11 x 7} 40. 6f x 6| x I x 2&
35. 14j x 17| x 8| 41. 20) x 20f x 20{i x 10
36. 27*fx42fx6J 42. 7§.]xl5 x2|£xl2$
37. 2i x 3| x 4| x 5| 43. 172.] x 2./ 5 x 3£
22
68 FRACTIONS
44. The schedule of a train between two cities is 12|
hours, and the train's speed is 35^ miles per hour. Find
the distance between the places.
45. Find the number of tons of sugar cane on 19 1 acres,
if the average number of tons per acre is 11 § tons.
46. An automobile's average rate of speed is 25 ^ miles
per hour. How far does it travel in 3| hours?
47. Estimate the value for one season of a Vermont
sugar camp at <$ J^- per pound if 6-| pounds are obtained on
an average from each of 1275 maple trees.
DIVISION OF FRACTIONS
Dividing a fraction by an integer.
1. 15 -5- 5 means that 15 is to be separated into 5 equal
parts of 3 units each ; and one of the equal parts taken ;
thus, 15-^5 = 3. Explain in the same way what 18 -*- 9
means.
2. \^ -5- 5 means that ^| is to be separated into 5 equal
parts of j 2 g each, and one of the equal parts taken ; thus, ||
-i- 5 = T 2 g. Which term of the fraction was divided by the
integer?
A fraction may be divided by an integer by dividing the
numerator of the fraction by the integer.
Divide :
3. § by 4
5.
15 j- 5
2 6 * °
•>■ IfbyS
4. if by 6
6.
-3-S-^ 9
3T • v
8. |f by 7
9. In | -s- 5, can you separate | into 5 equal parts in
which the fractional unit is thirds?
DIVISION 09
10. Can you divide § by 5 by dividing the numerator by
the integer an integral number of times ?
11. | -i-5 means that | of | is to be taken. Since -\ of
2 — - 2 - then 2- — 5 = - ? -
12. | = yV Can {5 be separated into 5 equal parts of T 2 ^
each ? Then § -h 5 = ^. Observe that multiplying the
denominator of | by 5 changes the fractional unit from
thirds to fifteenths and takes one of the 5 equal parts into
which § has been changed.
A fraction may be divided by an integer by multiplying the
denominator by the integer.
Solve the following problems by the more convenient
method and give your reasons :
13. fft + 7 16. J^6 19. T %-*-25
14. 1-8 17. T %% -=- 9 20. iff +- 15
is 25. _s_ 1 fl la _fi_ -s- 3 21 P -*- 7
Written Work
1. Divide || by 8.
04 o " o 1° a the division is performed by divid-
er. - — = — ing the numerator by 8.
25 25
~ . ^ . „ In b the division is performed by multi-
b. — — - — =— = — plying the denominator by 8, and changing
ZQ X o 200 ^5 tn e fraction to its lowest terms.
Q
In c the division is indicated and the quo-
c.
3
= — tient is found by cancellation.
25 x 8 25
A fraction may be divided by an integer either by divid-
ing the numerator or by multiplying the denominator of the
fraction by the integer.
70
FRACTIONS
Find quotients:
2. | + 5 11.
3. | -5- 9 12.
4. ft + 7 13.
5. f^l8 14.
6. 11 -=- 5 15.
7. T 8 2°o-25 16.
8. 1| -4- 7 17.
9. II -5- 8 18.
10. T 9 Y -=- 5 19.
29. Divide 3| by 5.
21^«
65 ■ °
31 • y
_8_ ^_ 24
2 1 '
12
13
16
19 • "'
16 ^40
36 ^_ 19
37 * Xii
12
ii
18
3 5 ^.42
4 3 * *^
31 = 25
°8 8 '
25 .
8
5
25
8xp
5
8
20.
21
28
-18
21.
12
83
-30
22.
96
113
-64
23.
51
61
-38
24.
132
16 3
-48
25.
111
12 7
-52
26.
11 1
112
-37
27.
256
"2 7 5
-80
28.
180
19T
-HOO
Note. — Small mixed numbers are frequently reduced to improper
fractions and then divided by the same principle as proper fractions.
30.
31.
32.
33.
34.
35.
2f-6
41-1-7
4 T « T h, 13
922
^2 5
51 3
48
23
17J + 8
36.
37.
38.
39.
40.
41.
28f - 25
284 h- 18
32- 8 -
21| -4- 25
181 + 26
19J T V45
21
48. Divide 1286f by 9.
42.
43.
44.
45.
46.
47.
10 T 2 3 + 24
29f + "12
22ft
30|
191
58
23
26
211 _=_ 56
9)1286f
142f|
Note. — The division may be performed by changing both numbers
to sevenths, but it is a much shorter process to perform the work as
indicated, changing the remaining mixed number, 8f, to an improper frac-
tion, V, and dividing it by 9. Thus, \ of - 6 ? 2 - = If-
DIVISION
49.
25681^ -s-7
54.
8002 fa
-12
50.
9863^ -=- 6
55.
1428*1
-5-12
51.
6532^h-8
56.
10935|
-4
52.
6879^ -5- 5
57.
9720|
-5-11
53.
36370|^-11
58.
7090^
-5-9
71
Dividing fractions by first making them similar.
1. What is the unit of measure in eaeh of the following :
8 ; 6 yd. ; 5 ft. ; 4 in. ; 8 rd. ; 5 oz. ; 2 lb. ; 10 mi.?
2. What kind of units may be added or subtracted ?
3. Since both the dividend and the divisor of concrete
numbers must represent like units, to divide 24 yd. by 3 ft.
we must either change yards to feet, thus, 72 ft. -5- 3 ft. ; or
feet to yards, thus, 24 yd. -=- 1 yd.
4. In | -5- 1 are the fractional units alike in size and
kind? Then how often is | contained in |?
5. Since | is contained 3 times in |, how can we divide
one fraction by another when the denominators are alike?
A fraction may be divided by another fraction by chang-
ing both fractions to a common denominator and dividing the
numerators.
6. Divide f by f .
3 _ JL . 5 _ 1
¥ — 12' 6 — 12
JL _=_ 1 _ Q _:_ 1 () — JL
12 * 12 — V ' ±XJ — i7
Find quotients :
8. X -=-1 12.
o
9
. n
• 6
T 5 2-
!_ 2
• 3
9
_ 3
5
1 3
15
3
JL
11
16- tf + J
17 5. _;_ 1
- 1,3 - 16 • 3 ■ L/# 1? ' 9
10. U + f !*•♦ + ♦ 18 - H + l
72
FRACTIONS
By inverting the terms of the divisor and multiplying.
1. How many half inches are there in 1 in.? in 2 in.?
3 in. ? 4 in. Then how many times is ^ contained in 1? in
2? in 3? in 4?
2. If 1 -j-'J = 4, what does 2 + \ equal ? 3 - \ ? 12 + \ ?
3. What does 1 -f- l equal ? 3 -=- l ? 6 -=-
i ?
To divide any number by a fractional unit multiply the num-
ber by the denominator ; thus, 12 h- \ = 12 x ^ = ^ or 48.
4.
5.
6.
13.
14.
15.
16.
12 — 1
15
16
?
1 _ 9
3 — •
7.
8.
9.
10 -^ - 1 = ?
io. 20-^i = ?
11.
25h-£ = ?
9 - iW
9-h T V = ? 12. 18-^1 =
1 -s- 1 = how many ?
1 -=- | = how many ?
1 -s- -Jj = how many ?
1 -=- | = how many ?
17. 1 -=- j 5 q = how many ?
18. 1 -f Jj = how many ?
1 -=- § = how many ?
1 -=- | = how many ?
19.
20.
Observe that the number of times each of the above frac-
tions is contained in 1 equals the number of times the
numerator is contained in the denominator.
The number of times a fraction is contained in 1 is called
the reciprocal of the fraction. Thus, f is contained in 1, |
times. Hence, | is the reciprocal of §.
1 divided by any fraction equals the fraction inverted.
Find quotients :
21.
8^| 23. 4 + |
25. 10-5-1
27. 25
22.
6H-| 24. 15-^-f
26. 12 -f
28. 14
29.
| -=- | = how many ?
2 _t
3 •
3 — 2 v 4 _ 8 Since 1
" 4 — 3 X 3 — 9 ° U
. 3 _ 4 2 ^ 3 _
• ? — 3> 3 • ¥ ~
| of |, or |.
DIVISION
7:;
Any number is divided by a fraction by inverting the terms
of the divisor arid multiplying.
30. Work if -r- | by both methods. What principle does
the second method introduce that the first does not?
Note. — The process should be shortened by cancellation when possible.
Written Work
Change the mixed numbers to
improper fractions before
dividing. Divide :
1.
18 by f
20.
13 _
18
. 26
■ 2 7
39.
119. 7
T3 8 • US'
2.
12 by |
21.
5
2 2
3
5
40.
85 .51
87" ' 1J5
3.
25 by|
22.
2 7 _
32
l. 9
' 16
41.
3 9 .13
6 4 *24
4.
32 by if
23.
2 5 _
3 6
. 15
16
42.
993 . 95
^"4 * -8
5.
40 by f §
24.
28
39
_ 42
65
43.
45| -4- 5«
6.
48 by 1|
25.
65 _
7 2
5
8
44.
6* + 8 |
7.
172-
8
9
26.
13 3 _
1 4 4
95
192
45.
128 ■*- 17/2
8.
235-
4
5
27.
99 _
12 5
209
2 r.
46.
160 by f of |
9.
770-
- 11
15
28.
81 _
1 12
45
"72
47.
| of 640 by 6£
10.
882-
_ 3
Y
29.
_4_9_ _
1 II s
91
132
48.
198 by 12f
11.
1035-
_ 5
6
30.
243
2 56
189
32
49.
8| + 4J
12.
984-
_ 3
31.
1 96 _
22 5
_ 56
3T5
50.
10| -*- 6f
13.
3 _
I
2
3
32.
185 _
22 4:
111
896
51.
101^31
14.
2
3
_ 3
1
33.
175 _
282
125
"375
52.
(f Of f) -4- (J Of |)
15.
1
8
_ 3
4
34.
23 _
231
2
33
53.
(f of £) -*-(f of f)
16.
9 _
10
_ 3
5
35.
143 _
lYO
l. 6 5__
' 136
54.
(4 ( > f «) + (A
17.
13 _
1?
6
Y
36.
114
-A
« " f H)
18.
7 _
12
2
3
37.
8 3
S4
_ 5
5 6
55.
(H^W+H
19.
15 _
16
_ 1 1
24
38.
55 _
7 9 "
22
56.
(«ofH)+(jofft:
74 FRACTIONS
Miscellaneous
1. At $1|- per day how long will it take a laborer to
earn $671?
2. Divide | of | of J by f of * of Jf
3. A man who owned T 7 y of an estate sold | of his share.
What part of the estate did he then own ?
4. A man who owned f of a store sold | of his share for
$1406.25. What was the value of the store? What part
had he left ?
5. One man has $35| ; another has $62|. If each gives
to the other § of what he has, how much more will the one
then have than the other ?
6. If | of 10 bushels of oats cost |3f, how much will 20|
bushels cost ?
7. What number must be multiplied by £ of 3| to give
a product of 32 -^ ?
8. A storm moves eastward at the rate of 18^ miles per
hour. In how many hours after the storm is first observed
in Chicago should it be due in Pittsburg, 468 miles east ?
9. Kerosene oil weighs 6| pounds to the gallon. Find
the number of gallons in a tank car of oil weighing 39200
pounds.
10. A steam threshing machine averages If bushels of
wheat per minute. How many minutes at the same rate
will it take to thresh 358| bushels?
11. The average cost of the education per pupil in a
certain city is $51. If the total cost is $16480, find the
number of pupils attending school.
12. A contractor employs 6 men for 27| days and pays
them $363. Find the average daily wages.
COMPLEX FRACTIONS 75
COMPLEX FRACTIONS
A complex fraction is a fraction which has a fraction or a
mixed number in either or both of its terms.
Thus, -2 , — , £, !i2 are complex fractions.
' o' 3' 4' 41 r
8 5 ^?
Such examples are simplified by the principles of division
of fractions.
Thus, £ = - ■*■ 3 = -. They rarely occur except in ad-
o — O
vanced courses of study.
Written Work
l. Find the quotient of 2J divided by § .
3
6 2
Simplify :
1 4 of 21 ,„ |x6|
o 9 7 5 5 12. A_ a
9 7. 5 *" -5 12.
?i if
3
_8.
°3 • 8
11 , , 44
3. 1| 8. | X | 13. |0f-£
4 W 9 . _LJ 14. A Of!
5 S + 2i 10 Six I' 8-gbtt)
6 (-fe-Dx(t + |) u (i°f8j)-j 16 (21x31)-,- If
76 FRACTIONS
FRACTIONAL RELATIONS
Finding what part one number is of another.
1. What part of 12 is 4? What part of 18 is 5? Ex-
press the answers also in the form of division. Thus,
T 4 2=4-12; t \=5h-18.
2. What part of | is f ? of | is f ? How, then, can we
find what part one number is of another?
Divide the smaller number by the larger number.
Written Work
1. What part of 208 is 96?
cm • 908 96 -5- 16 _ 6 We divide the smaller number 96
" 208 — 16 " 13 ky 208, and reduce the resulting frac-
tion 2 9 jfg to its lowest terms, T 6 j.
2. What part of | is -^ ?
3 7 _ . 3 §_3 We divide r 3 g by J by inverting the
16 8 ~ 16 7" 14 divisor of | and multiplying. The result
2 shows that r \ is fa of |.
What part of
3. 90 is 16? 6. | is |? 9. 4 is 31?
4. 120 is 50? 7. lis^? 10. 13|islf?
5. 200 is 18? 8. \l is if? li. 12| is If?
12. 250 pupils belong to a school, and only 128 are
present. What part of the whole number is present?
13. One year the population of a city was 220,000, and
the next year 250,000. What fraction of the second year's
population was the first year's?
14. The product of two numbers is 29,160, and one of the
numbers is 27. What part of the other number is 27?
FRACTIONAL RELATIONS 77
Finding a number when a fractional part of it is given.
l. If | of a number is 30, what is the number?
If three fourths of the number is 30, one fourth of the number is one
third of 30, or 10, and four fourths of the number, or the whole number,
is 4 x 10, or 10. Hence, 30 is | of 40.
Written Work
1. 360 is T 6 y of what number ?
i if aon en Since 36 ° is rV 0T a number,
T V of the number = \ of 350, or 60 , . . . . .
J! , , , ,„ _,, „ v A- of it is 1 of 360, or 60, and H
j I ot the number = 17 x 00, or 1020 T I .. . , , " , .,„. . ' 17
17 of it is V oi »360, or 1020.
2. If | is | of a number, what is the number ?
Since £ of the number is J,
\ of the number = \ of |, or 5 7 ¥ , £ of it is \ of |, or ^V- aQ d ? of
| of the number = Ix 2 7 ? = |f, or li the number is 1 x 2 7 ? , which
equals §£, or 1\.
Find the number of which :
3. 84 is | 5. 3| is I
7 -Z is -L
/. 9 it> 16
4. 196is T 4 T 6. 12|isi§
1 2 jo 6
8. T g 1» y
9- H i8 H
10. 5fis^
11. If T 9 3 of a man's salary is $900, what is his salary ?
12. A man lost in speculation $2700, which was -j 3 g of his
entire fortune. What was his fortune?
13. If I.]- of the number of books in a library is 9922,
how many books are there in the library ?
14. There are 30,205 women in a certain town, which is
y 7 ^ of the number of men. How many men are there ?
15. An author spent $2100 for a piece of property, which
was y 7 ^ of what he was paid for a novel. How much did
lie receive for the novel?
78 FRACTIONS
REVIEW OF FRACTIONS
1. | of 30 is f of what number ?
2. If 6 is added to both terms of the fraction f, will the
value be increased or diminished, and how much ?
3. Change iff to its lowest terms.
4. The sum of two numbers is 43^. One of the num-
bers is 18|. What is the other ?
5. If 51 tons coal cost $28.27, find the cost of 12§ tons.
6. If 2£ acres of land cost $110, how much will 12|
acres cost at the same rate ?
7. The dividend is 165, and the quotient is 6|. What
is the divisor ?
8. Five tubs of butter contain, respectively, 27^ lb.,
30| lb., 24 Jg lb., 32| lb., and 34| lb. How many pounds
are there in the five tubs ?
9. Multiply the sum of § and f by their difference.
10. If 10 men can build a wall in 35 days, how long will
it take 25 men to do the work ?
11. There are 27 1\ square feet in 1 square rod. How
many square rods are there in 43560 square feet ?
12. If a man travels 2|- miles in f of an hour, at the same
rate how far could he travel in 1\ hours ?
13. A man owning -^ of a mine sold his interest for
$48700. Find the value of the mine at that rate.
14. Find the difference between 1\ x 3| and 2\ -4- 3|.
15. A merchant owned f of a store, and sold | of his share
for $5760. Find the value of the whole store at that rate.
16. How much will 18000 stamped envelopes cost at
$21^ per thousand ?
REVIEW OE FRACTIONS 70
17. When oysters yield 1{ gallons to the bushel, how
many bushels will be required to till a 10-gallon tub ?
18. One buyer offered | of the cost of a property, another
| of the cost. The difference in their offers was $186. Find
the cost of the property.
19. Find the cost of 18 yards of cloth if 3| yards cost $9.
20. A clothier paid $180 for 12 suits of clothing, and
sold them at $19|- a suit. How much did he gain ?
21. E, who owns § of a factory, sells | of his share for
$3560. What is the value of the factory ?
22. What is the distance from Pittsburg to Philadelphia
if | of the distance is 265^ miles ?
23. The product of two fractions is | ; one is ^, what is
the other ?
24. If C's wages are $3| a day, and his daily expenses
$1|, how many days must he labor to save $28 ?
25. A traveler walked 25| miles the first day, 15| miles
the second day, 19| miles the third day, 20| miles the fourth
day, and 22| miles the fifth day. How far did he travel in
the 5 days, and what was the average rate per day ?
26. A field is 40| rods long, which is f of its width.
What is its width and what is the distance around the field ?
27. Find the cost of 10A cords of wood at $3 a cord, and
8| cords at $4 a cord.
28. If 12| bushels of apples cost $5, how much will 15|
bushels cost ?
29. A farmer raised 22f> bushels of potatoes. He sold
| of them to one merchant, and \ of the remainder to another.
Find the number of bushels he had left.
30. Reduce \% to a fraction whose denominator is 320.
lb
80 FRACTIONS
31. Find the value of a mill if f of f of it is worth 13750.
32. If % the trees in an orchard are peach trees, | apple
trees, % cherry trees, and the remaining 21, plum trees, how
many trees are there in the orchard ?
33. What number is it whose f exceeds its § by 40 ?
34. Divide into 5- equal parts the product of the sum and
difference of 1^ and 1|.
35. A real estate dealer sold some lots for §12360, gain-
ing 1 of the cost. If he had sold them for $10500, would he
& 5
have gained or lost, and how much ?
36. A piece of land was sold at $ 90 an acre, which was a
gain of | of the cost. How much did the land cost per acre ?
37. My board and room cost $32 per month. What does
each cost if § of the cost of the room equals § of the cost of
the board ?
38. A father left $30000 to his two children, giving the
daughter | as much as the son. What was the share of
each?
39. The owner of % of a mill sold | of his share for $4800.
How much at this rate would a man who owns ^ of the mill
get for 2 3 o of his share?
40. One fourth of a certain number minus 3| equals -fa.
What is the number ?
41. A carpenter, working 9| hours a day, built a shed in
16 days. How many hours a day, at the same rate, must he
work to build it in 18 days?
42. A merchant sold 10 dozen hats of a certain kind at
$2.75 each, and a number of dozens of another kind at $2
each, receiving $474 for all. How many dozen did he sell
at $ 2 each ?
REVIEW OF FRACTIONS 81
43. A man engaging in trade lost | of the money invested,
then gained $495, after which he had $2551. How much
was his capital at first ?
44. Simplify ^ + (£+1).
5 Ui 9
45. A hardware merchant bought a bill of hardware at
auction for \% of its value, and retailed it for f of its value.
If his gain was $48.75, how much did he pay for it?
46. A boy bought lemons at the rate of 4 for 5 cents, and
sold them at the rate of 3 for 5 cents. If he made $6 in 2
weeks of 6 days each, what were his daily average sales ?
47. A path leading to the top of a hill has a rise of 9
inches in 90 feet. What is the elevation of the hill in feet,
if the path is f of a mile long ?
48. A teacher taught 8* months, and after spending f of
his salary for board had left $204. How much did he earn
per month?
49. Divide § of 9 by A of 8|.
50. Two men, A and B, each bought farms, A's farm cost-
ing 2| times as much as B's. Find the cost of each, if both
cost $58,000.
51. Divide § of | of | of 3| by f of ^ of 8^.
52. A ship is worth $120000, and the owner of f of it sells \
of his share. Find the value of the part he has remaining.
53. Find the value of (18| - |) - (20^ - 1$).
54. I purchased 160 acres of land at $60 an acre, and sold
| of it at $70 an acre. Find the number of acres I had left,
and my gain on the number of acres sold.
55. A real estate agent bought land for $7200, and sold it
so as to gain ^ of the cost. If the gain was $6 per acre,
how many acres did he buy ?
HAM. COMPL. ARITH. — O
82 FRACTIONS
56. A man paid § of his indebtedness the first year, § of
the remainder the second year, f of what then remained the
third year, when he found that he still owed $ 1296. Find
the amount he owed at first.
57. If a man can do a piece of work in 6| days by work-
ing 9 hours a day, how long will it take him to do it,
working 8 hours a day ?
58. Mr. Williams has ^ of his money in government bonds,
| in the bank, and $ 520 in cash. How much is he worth ?
59. A real estate agent bought 2 houses ; his income
from the one was $480, which was | of the income from the
other. How much was his income from both houses ?
60. Four brothers agreed to pay the mortgage on their
father's farm. The first payment made was ^ of the mort-
gage, the second ^ of the remainder, and the third | as much
as the first and second. If the difference between the first
and third payments was $300, how much was the mortgage?
61. A merchant bought 250 barrels flour at $4.80 per
barrel. He sold 10 barrels which were damaged for | of
the cost. On the sale of 150 barrels he gained 15 cents per
barrel. If his total gain was $28.50, at what price did he
sell the remaining number of barrels ?
62. A merchant buys 36 pairs men's shoes, at $2^ per pair ;
24 pairs women's shoes, at $1| per pair; and 30 pairs slippers,
at $| per pair. What is the amount of his bill ?
63. If the men's shoes in example 62 are sold for $3
per pair, the women's shoes for $2| per pair, and the slip-
pers for $1| per pair, how much does the merchant make ?
64. A cubic foot of water weighs 62| pounds. A barrel
contains about 4* cubic feet. Find the weight of a barrel
of water.
REVIEW OF FRACTIONS 83
65. A student, in a half year at college, spent his money
as follows: tuition, |50; books, $9| ; 18 weeks' boarding, at
$2| per week; incidental expenses. $41|. What were his
expenses for the half year?
66. T 9 Y of a barrel of oil, containing 51 gallons, was sold
at 13 cents per gallon; the remainder of the barrel at 14
cents a gallon. If the oil cost 11| cents per gallon, what
was the gain ?
67. A merchant bought a stock of goods for 16000. He
sold | of it at a gain of l of the cost. \ of it at a gain of \ of
the cost, and the remainder at a loss of \ of the cost. How
much did he gain or lose ?
68. A hall is 21f feet long and 10] feet wide. At 8£
cents per foot, how much will it cost to put a molding
around this hall?
69. Find the perimeters of each of five rooms, the dimen-
sions being as follows : 221 ft. x 16 ft., 14f ft. x 15* ft.,
12§ ft. x 17 ft., 181 ft. x lof ft., m ft. x 24 ft.
70. From a certain number 18| + '2~^ was subtracted,
leaving a remainder of 9 T 5 g. What was the number?
71. From New York to Chicago by the Baltimore and
Ohio railroad the distance is 1052 miles. From Chicago to
El Paso by the Santa Fe railroad it is 1630 miles, and from El
Paso to Mexico City by the Mexican Central railroad it is
1224^ miles. What is the distance from New York to
Mexico City, and how long would such a journey take,
traveling 31 1 miles per hour ?
72. A freight train runs from Kansas City to St. Louis,
288 miles, traveling 16 miles per hour. How long does it
take it to make the trip? How long would it take a pas-
senger train, running \ as fast, to make the same trip?
PROBLEMS FOR ANALYSTS
Pupils should be taught :
(1) To express themselves accurately and rapidly ; (2) to
do this work without the aid of pencil and paper ; (3) to
give clear analytic statements in the solution of a problem.
1. The coal for the school buildings in a certain town
cost $94.50. How many tons were purchased at $3 per ton ?
2. A shoe dealer buys shoes at $21 per dozen pairs and
retails them at $2.50 a pair. How much does he gain on
each pair ?
3. A laborer earns $18 in 12 days. At that rate how
much can he earn in 80 days ?
4. What is f of f of 24 ?
5. A lady buys 10 yards of ribbon and uses 6| yards.
What part of the ribbon has she left ?
6. If eggs are bought at the rate of 4 dozen for $1.08
and sold at 30 cents per dozen, what will be the gain on 3
dozen ?
7. A man buys a farm of 102 acres and divides it into
lots of 6 to the acre. How many lots are there if £ of the
farm is laid out in streets ?
8. How many sheep, at $5 per head, may be purchased
with the money from the sale of 10 head of cattle at $42 per
head?
9. A contractor buys 8000 bricks at $12.50 per thousand.
Find the amount of his bill.
84
PROBLEMS FOR ANALYSIS 85
10. If I pay 6 cents for the use of §1 for one year, how
much should I pay for the use of §150 for - years/
11. If I buy goods at $1.37 per yard and sell them at
11.50 per yard, how much do I gain on 12 yards?
12. A boy earns 75 cents per day and pays f of his wages
for board. At that rate how much can he save in 26 days ?
13. My father pays §252 per year rent. How much is
that per month ?
14. A lady purchased 2 handkerchiefs at 35 cents each ;
6 yards of ribbon at 12 cents per yard ; 6 yards of cloth at
§1.12 per yard. How much change should she receive from
a ten-dollar bill ?
15. A man buys a farm for §2000 and pays | of the cost,
giving his note for the balance. For how much does he
give his note ?
16. A farmer buys fertilizer at §28 per ton and retails
it at §1.90 a hundred pounds. How much does he gain on
35 tons ?
17. At 16 cents per pound, how many pounds of steak
does a woman get if the amount of the purchase is 83 cents ?
18. A huckster bought 6|- pounds of butter at 16 cents
per pound, and 6| dozen eggs at 18 cents- per dozen. How
much did he pay for both?
19. A huckster buys chickens at 8^ cents per pound and
sells them at 12 1 cents per pound. How many pounds must
he purchase and sell in order to gain §25?
Solution. — On each pound he gains 4£ cents.
To gain $1 he must sell 24 pounds.
To gain $25 he must sell 25 times 24 pounds, or 600
pounds.
20. A street car conductor earns 23 ^ cents per hour.
How much does he earn in 10 hours ?
86 PROBLEMS FOR ANALYSIS
21. If the conductor averages 10 hours per day, how
much will he earn in 80 days ?
22. If a box of 40 dozen oranges is purchased for $5
and retailed at 40 cents a dozen, what is the entire gain ?
23. What fraction of a gallon is a pint ? a gill ? a quart ?
24. A man sold | of his farm for $1800. At the same
rate, how much would he receive for the whole farm ?
Solution. — § of the amount received for the farm =$1800.
£ of the amount received for the farm = \ of $1800, or
$900.
f , or the amount received for the farm = 3 times $ 900,
or $2700.
25. In a certain school there are 40 pupils in the grammar
grade, which are f of the number in the intermediate grade.
How many pupils are there in both grades ?
26. A man sold a piano for $360, which was | of what it
cost him. How much did it cost him ?
27. There are 112 cubic feet in | of a cord of wood. How
many cubic feet are there in 1 cord ?
28. A farmer sold | of his farm for $1521. At that rate,
what was the value of § of the farm ?
29. There are 198 cubic inches in $ f a gallon. How
many cubic inches are there in 2 gallons ?
30. If | of a ton of hay costs $16, how much will 2| tons
cost?
31. A lady paid 84 cents for | of a yard of silk. At the
same rate, how much would she pay for |- yard ?
32. If ^ of an acre produces 160 bushels of potatoes, how
many bushels will an acre produce ?
33. A man saves $220, which is f of his yearly salary.
What is his yearly salary?
PROBLEMS Foil ANALYSIS ST
34. If it takes 4 men 9 days to do a piece of work, how
long will it take 6 men, at the same rate, to do the work ?
35. If | of a farm is valued at $ 1240, what is the value of
^ of the farm ?
36. A lady spends § of her income for hoard, and 1 of the
remainder for clothes and travel. If she saves $160 per
year, what is her income ?
37. A farmer has 600 bushels of wheat, which is \ of |
of what his neighbor has. How many bushels has his
neighbor ?
38. A house was sold for 11800, which was § of its cost.
What was the loss ?
39. If 20 men can dig a ditch in 25 days, how long will it
take 5 men ?
40. A farmer sold 36 head of cattle, which was 6 more
than ^ of all he owned. How many had he remaining ?
41. If a man works | of a day and receives $1.50, how
much should he receive for f of a day ?
42. I purchased an overcoat for $45, and found I had % of
my money left. How much had I at first ?
43. James and John have together $40, and John has
seven times as much money as James. How much does
each have ?
Solution. — Once James's money = James's money.
7 times James's money = John's money.
8 times James's money = amount of both, or
Once James's money = $5.
7 times James's money = 8-35, or John's money.
44. There are two numbers whose sum is 40; one number
is A of the other. What are the numbers?
88 PROBLEMS FUR ANALYSIS
45. A man invests \ of his money in a mill, | in a farm,
and the remainder, which is $900, he deposits in a bank.
How much is he worth ?
46. A town has 3000 people. The number of pupils in
the school is \ of the remaining population. How many
pupils are there in the school ?
47. In a farm of 500 acres the woodland is § of the cleared
land. How many acres are there of each ?
48. A miller has 360 bushels of wheat, § of the number
of bushels of wheat equals | of the number of bushels of
corn. How much is the corn worth at 50 cents per bushel?
49. If | of a clerk's salary for the year is $800, how much
is I of his salary ?
50. | of 21 is -^ of what number ?
51. § of 20 is T 4 3 of what number ?
52. | of I of a number is 16. What is the number ?
53. I of C's farm equals ^ of D's, and C has 100 acres.
How many acres has D ?
54. There are two numbers: § of the first is f of the
second. The first number is 18. What is the second ?
55. A and B agree to do a piece of work for $80. If A
works 7 days and B 9 days, how much should each receive?
Suggestion. — Together they work 16 da. Hence, A should receive
r \ of $ 80, and B T 9 S of $ 80.
56. Sarah earns $10 a week, and ^ of what Sarah earns is
§ of what Edna earns. How much does Edna earn ?
57. | of T 3 ^ of 160 is T 8 T of what number ?
58. If | of | of a farm costs $2500, how much does the
farm cost ?
PROBLEMS FOR ANALYSIS 80
59. I of my money is invested in coal land, \ of the
remainder in building lots, and the remainder, which
amounts to $800, is in the bank. How much have I
invested in coal lands ?
60. A man sold a horse for 8120, winch was \ less than
the horse cost him. Find the cost.
61. If 24 is | of a number, 21 is what part of that number ?
62. I spend \ of my monthly salary for board and room,
\ of the remainder for clothing, and save the remainder,
which is $20. What is my salary?
63. If f of a piece of work can be done in 9 days, how long
will it take to complete the work after f has been done ?
64. By selling land at $120 an acre, I gain J of the cost.
Find the cost.
65. If | of the value of a farm is $000, what is the value
of | of the farm ?
66. If 4| tons of coal cost $27, how much will 3 tons cost ?
67. What is the difference between \ of \ and § of 1 ?
68. If I can do a piece of work in 7 days, what part can I
do in 1 day? in 3 days? in § of a day? in 1| days?
69. A fruit grower sold 200 bushels of apples, which was
A of his crop. How much did he realize from the sale of
his crop at $1.20 per bushel ?
70. An heir gets f of an estate and invests f of his share,
and still has $1600. What is the value of the estate ?
71. A tank holds 120 gal. and is f full. § of the quantity
is drawn off. How many gallons will it take to fill the tank ?
72. One man bids ^ of tlie cost of an article ' another man
bids § of the cost of the article. The difference between
their bids is 70 cents. Find the cost of the article.
DECIMAL FRACTIONS
1. What is the largest common fractional unit ?
2. Show that an integral unit may be divided into any
number .of fractional units.
3. Name the different fractional units from \ to 2V in order
of their size.
4. What fractional unit divides the integral unit into 10
equal parts ? into 100 equal parts ? into 1000 equal parts ?
The divisions of an integral unit into lOths, lOOths,
lOOOths, etc., are called decimal divisions.
There are three ways by which decimal divisions may be
expressed :
(1) by words, as nine tenths ;
(2) by common fractions, as j 9 q, y| 7 ;
(3) by decimals, as .9, .75.
A decimal fraction is any number of lOths, lOOths, lOOOths,
etc. of an integral unit. When expressed after a decimal
point and without a written denominator, it is usually called
a decimal.
A decimal point is a period placed after ones' place and
before tenths' place.
5. What is the largest decimal unit? the second largest?
the third largest?
In any decimal system 10 x 1 unit in any place =1 unit of
the next higher place.
6. Show that United States money is a decimal system.
90
NOTATION AND NUMERATION 91
NOTATION AND NUMERATION OF DECIMALS
1. Since the first decimal division of an integral unit is
tenths, what is the first place to the right of the decimal point?
2. What is the second place called ? the third place ?
3. Five tenths is written .5 ; five hundredths is written
.05 ; five thousandths is written .005, etc. Write 7 tenths,
6 hundredths, 8 thousandths.
4. Express decimally ^, ^, TT ^ iooo' Too' Tooo-
Every decimal contains as many decimal places as there are
naughts in the denominator of the equivalent fraction.
Table of places and names of integral and fractional units :
CO
a
©
s
H
a
o
5 3
CO
.3
CO
*j
13
-a
a
GO
3
d
GO
CO
S
CO
+3
CO
3
CO
J3
O
a
.«
CO
J3
.4
c«
CO
CO
tn
o
-
-a
i
so
T3
3
a
T3
H
•C
— *
13
+J
£3
CO
o
c*
CO
eS
CO
co
=s
o
CO
2
•r*
"2
—
1
CO
C5
-5
K
CO
S
'3
XI
■o
CO
©
"3
•3
.2
a
8
co
H
X
a
O
co
Q
CO
H
3
S
J3
r £
8
§
CO
H
5
7
8
4
5
5
•
7
4
8
9
8
5. In .555 what figure stands for tenths? for hun-
dredths ? for thousandths ? It is read 555 thousandths.
6. Is the decimal point named in reading a decimal ?
Observe that the decimal is read as an integer and that
the last figure is given the required denomination.
Read :
7. .25 li. .101 15. .60745
8. .05 12. .0045 16. .678705
9. .005 13. .4045 17. .0065
10. .375 14. .0002 18. .60005
92 DECIMAL FRACTIONS
19. 50.0745 is read 50 and 745 ten-thousandths. How
are both the integer and the decimal read ? How is the deci-
mal point read ? What name is given to the last decimal
place ?
20. When is the decimal point read ? When is it not read ?
21. What determines the value of any figure in a decimal ?
22. In writing .5, .45, .075, .0075 as common fractions,
what figure in each decimal tells us the size of the denomi-
nator ?
A mixed decimal is a whole number and a decimal ; as,
4.625.
23. What number in a mixed decimal is always read first?
24. How do the number of places in any decimal compare
with the number of naughts in the denominator when the
decimal is expressed as a common fraction ?
Read :
25. 45.075 28. 72.003745
26. 50.3007 29. 1001.1001
27. 290.25387 30. 794.3085
34. 5 thousandths calls for how many decimal places?
What part of the decimal (5 thousandths) stands for the
numerator of the fraction? what part of this decimal stands
for the denominator?
35. Name the numerator and the denominator in the fol-
lowing decimals: .05, .0006, .000025, .045.
In writing a decimal write the numerator, and point off
from the right as many decimal places as there are naughts
in the denominator.
31.
.00875
32.
.008090
33.
2.004890
DOTATION AND NUMERATION 93
Written Work
Write :
1. 34 hundredths.
2. 675 ten-thousandths.
3. 16 and 75 millionths.
4. 400 and 45 thousandths.
5. 6006 and 66 ten-thousandths.
6. 89 and 5 thousandths.
7. Seven hundred forty-six ten-thousandths.
8. Nine hundred and 84 millionths.
9. 5 million 9 and 4 hundred 9 ten-millionths.
10. 5095 millionths.
11. 8 and 17 ten-thousandths.
12. 125 millionths.
13. 896 and 301 hundred-thousandths.
14. One thousand and one thousandth.
15. 18051 and 957 thousandths.
16. 97 and 3 ten-thousandths.
17. 9864 millionths.
18. '2135 and 32 millionths.
19. One and one millionth.
20. One million and one tenth.
21. 90 thousand and 71 thousandths.
22. 1830 and 11684 hundred-thousandths.
23. 429 thousand and 46 ten-thousandths.
24. 7035 and 97 hundredths.
25. 67375 and 35 hundred-thousandths.
26. 5815 hundred-thousandths.
27. 375 and 69 thousandths.
94 DECIMAL FRACTIONS
28. 419863 and 23456 millionths.
29. 81 and 921 hundred-thousandths.
30. 2986 and 298643 ten-millionths.
31. 3020 and 302 hundred-thousandths.
32. 70 and 7 hundredths.
33. 8 thousand and 8 thousandths.
34. 645 million and 9 millionths.
COMPARISON OF COMMON FRACTIONS AND DECIMALS
2. Then observe that 5 tenths = 50 hundredths = 500
thousandths. .5 = .50 = .500.
3. 50 hundredths may be written .50, or .500. Does
adding naughts to the right of a decimal change the value of
the decimal?
4. What is the difference in value between $.5 and 1.50?
In writing decimal parts of a dollar, we always write two
places for cents even if the last place is a naught.
5. Compare in value \ and T 5 ^, T 5 ^ and -^$q.
6. Does canceling the same number of naughts from
both numerator and denominator change the value of a
fraction ?
7. Since .5 = .50 = .500, does canceling naughts from the
right of a decimal change the value of the decimal ?
8. Observe that canceling naughts from the right of a
decimal really means canceling naughts from the numerator
and the denominator.
Thu S ,.50 = ir.5 P = l|.
COMMON FRACTIONS AND DECIMALS 95
9. .400 is read 400 thousandths. How else may it be
read ?
10. Is the unit in .4 the largest decimal unit in which
.400 can be expressed ?
Read first as given, then as if the naughts at the right of
the decimal were canceled :
11.
.040
13.
.7500
15.
10.0057
12.
26.0050
14.
8.0090
16.
20.0900
Changing decimals to
common
fractions.
Write as common fractions and change to lowest terms :
l.
.25
3.
.045
5.
.50
7.
.0025
2.
.7
4.
.025
6.
.75
8.
.0775
9. Give the steps in changing a decimal to its fractional
equivalent.
A complex decimal is a decimal and a fraction united;
thus, .16 1 is read 16 1 hundredths.
* 3== 100x3 300 6'
10. Explain why multiplying both terms of the fraction
i-^ bv 3 does not change the value of the fraction.
100 J °
Change to common fractions:
11.
.371
15.
.831
19.
•62£
23.
lOl
12.
.41 1
16.
• 06£
20.
•33£
24.
.14f
13.
.66|
17.
.04 \
21.
.31^
25.
•58i
14.
.241
18.
.081
22.
•03£
26.
.88J
96 DECIMAL FRACTIONS
ADDITION AND SUBTRACTION OF DECIMALS
1. How must integers be written before they can be
added '? subtracted ?
2. What change must be made in \ and | before they can
be added or subtracted ?
In adding or subtracting decimals, tenths must be placed
under tenths, hundredths under hundredths, thousandths
under thousandths, etc.
Written Work
3. Add .8085 and .005. 4. Subtract .005 from .8085.
.8085 .8085
.005 .005
.8135 .8035
Add as indicated and test :
5. 6. 7. 8.
9. .45 + 72.5 + 8.557 + 87. = —
10. 8.07 + 5.07 + 0.039 + 6.5 = —
11. 62.093 + 89.09 + 16.909 + 11.9 = —
12. 40.937 + 20.98 + 41.005 + 0(15 = —
13. 4- + + = —
Subtract examples 14 to 17 and add the remainders :
14. 15. 16. 17.
40.275 9.0098 219.75 28.7
39.009 6.7849 8.95 12.5
18. + + + = —
Add:
19. 3.7 20. .001 21.- .65
5.06 12.3 .001
8.023 15.0248 10.1
9.04 18.0149 25.004
ADDITION AND SUBTRACTION 9'
Add:
22. 1.1
23.
120.2601
24.
36.15
4.01
230.81002
9.00999
1.0101
.05673
128.37
5.055
26.
3.7
27.
16,08753
25. 166.6
.27
185.057
7.0425
.0616
127.0348
28.318
.010912
216.253
142.0101
1.940054
456.03456
Add :
28. 12.015, 26.01102, 126.0592, 134.00876.
29. 100.001, 9.99, 149.0492, 7.077.
30. 2.2, 28.18, 140.027, 284.0295.
31. 318.003, 33.33, 495.0485, 12.0012.
32. Find the weight of four silver bars weighing as fol-
lows : 15.75 pounds, .125 pounds, 14.3125 pounds, and
16.875 pounds.
33. Find the number of acres in four fields containing,
respectively, 4.125 acres, .3125 acres, 8.8 acres, and 9.85
acres.
34. Find the sum of one hundred twenty-five and seven
hundredths, eighty-nine and two hundred thirty-five thou-
sandths, one hundred twenty-seven ten-thousandths, and
sixteen and four tenths.
35. A farm cost * 4225.50; stock, $745.25; buildings,
■$1825.75; and implements, $358.45. What was the total
cost?
36. How many square feet are there in four floors
measuring, respectively, 245| square feet, 278| square feet,
174.375 square feet, and 168.3125 square feet?
HAM. COMPL. A KITH- 7
98 DECIMAL FRACTIONS
Find differences :
37.
.75
.1825
38.
.3216
.275
39.
4.205
1.7856
40.
15.
5.007
41.
38.
18.276
42.
$45.67
12.09
43.
1251
87.432
44.
249|
178.625
45.
230.4897
116.5988
46.
100.001
99.9
47.
1001.101
900.909
48.
105.55
79.067
49.
1.1
.999
50.
5.05
.6565
51.
8.25
.0085
52.
1000.00
999.99
53. (9.5- 2.25) + (15.28-12.056) + (22.089- 19.063).
54. (11.001- 1.99) + (17.0107-14.014) + (29.3-23.2867).
55. The difference between two numbers is 1001.101, and
the greater is 1101.011. What is the smaller number?
56. A is 35.875 years old, B is 48.25 years old, and C's
age is 25.5 years less than the age of A and B combined.
How old is C ?
57. To the sum of .808 and 80.8 add their difference.
MULTIPLICATION OF DECIMALS
1. Multiply .05 by .5.
2. What is the numerator in .05 ? in .5?
3. What is the denominator in .05 ? in .5 ? What shows
the denominator in a decimal ?
4. Multiply the numerators in .5 x .05 ; thus, 5x5 = 25.
5. Multiply the denominators in .5x.05; thus, 10x100
= 1000.
6. Write the result as a common fraction ; thus, T f -| ^ .
7. What term of the fraction is expressed by the decimal
point ?
MULTIPLICATION 99
Observe that there are as many decimal places in each deci-
mal as there are naughts in the denominator of the equiva-
lent common fraction.
Find the product of the following decimals by multiplying
the numerators and the denominators, separately, and ex-
pressing the result as a decimal ; thus,
• 04x - 7 =i^ x ii=iIro=- 028
9. .2x.06 12. .35x.05 15. .02 x .42
10. .04 x .5 13. .07 x .05 16. .06 x .004
11. .25 x .05 14. .03 x .02 17. .05 x .009
The product of two decimals contains as many decimal places
as the sum of the decimal places in both factors.
Written Work
l. Multiply .25 by 5.
f a \ 5 x 5 hundredths = 25 hundredths or 2 tenths and
5 hundredths. Write the 5 in hundredths' place,
•"" and carry the 2. 5x2 tenths = 10 tenths; 10
2. tenths + 2 tenths = 12 tenths, or 1 aud 2 tenths.
1.25 Hence, 5 x .25 = 1.25.
2. Multiply .26 by .12.
(h)
cyo 1. What is the sum of the decimal places in
the two factors ?
2. The product, then, must contain how many
.12
52 places?
26
.0312
When the product has not enough decimal places, supply the
deficiency by prefixing naughts.
100 DECIMAL FRACTIONS
Find products :
3. .25 x .22 20. .1232 x .961 37. .4986 x .086
4. .17x.28 21. .2592x8 38. .006x20
5. .027 x .03 22. 65.65 x .65 39. 38.2 x .75
6. 27 x .12 23. 75.002 x 16.04 40. .0045 x .05.
7. .35x42 24. 275 x .007 41. 9.876 x .786
8. 8.7x9.22 25. .0018x720 42. 362.9 x .0076
9. .085x50 26. 1500 x .004 43. 119.8x2.74
10. .027 x 18 27. 124 x .064 44. 20.08 x .006
li. 1.005 x .011 28. 326 x .096 45. .375 x 2.027
12. 26.8 x 34 29. 627 x .78 46. 98.64 x 4.096
13. 28.25 x 12 30. 246 x .3 47. .069 x 8.92
14. 324.6 x 81 31. 29.4 x .08 48. 6.34 x 2.34
15. 39.10 x 18.4 32. 9.86 x 3.8 49. 12.34 x .004
16. .0214 x .016 33. 39.75 x .27 50. 6.08 x .0001
17. 12.134 x. 0025 34. 8.708x6.8 51. 1.002x1.004
18. 15.684 x 8 35. 368.9 x 8.5 52. .05 x .005
19. .1232 x 345 36. 2009 x .006 53. 6.876 x 4.37
54. How much must be paid for 85 acres of land at
$45.75 per acre ?
55. Three brothers divided an estate worth 19600. The
first received .125 of it, the second .375 of it, and the third
the remainder. How much did each receive ?
56. A contractor furnished 2,626,000 bricks at $7.75 a
thousand, and a laborer for 65 days at $2.75 a day. What
was the amount of his bill ?
57. If there are 39.37 inches in a meter, how many inches
are there in 12 meters ? how many yards ?
DIVISION 30]
Multiplying by moving the decimal point.
1. Multiply 6.385 by 10; by 100 ; by 100TT.
10 x 6.385 = 63.85 1- How do you murtfpjj a ribjrnWi 6j 10?
inn a oor />oc c -. How may you multiply it decimal b<'
100 xb.38o = 638.5 U)? by lu() , by U)()(( ,
1000 x 6.38o = 638o 3 How is the Ya]ue of a nuillber a ff ec ted
by moving the decimal point one place to the right? two places? three
places ?
Moving the decimal point one place to the right multiplies
the number by 10 ; two places by 100 ; three places by 1000.
2. Multiply 6.1234 by 10 ; by 100 ; by 1000.
3. Multiply .0342 by 10 ; by 100 ; by 1000.
4. Multiply 1.3412 by 10; by 100 ; by 1000.
DIVISION OF DECIMALS
Dividing a decimal or a mixed decimal by an integer.
1. Find J of 48 hundredths; of 64 hundredths.
2. Find \ of .25; of .35; .45; .75.
3. Find I of 6 and 36 hundredths; 12 and 24 hundredths.
4. Find 1 of 12.36; 24.42; 48.06; 54.06.
Observe that in each problem a decimal or a mixed decimal
when divided by an integer is simply separated or partitioned.
Give quotients at sight :
5. J of .16 9. i of 6.6 13. 6.42-6 17. .006-5-3
6. iof.25 io. J of 8.08 14. 12.04-4 18. .024-5-6
7. iof.08 li. -J of 10.10 15. 15.05-S-5 19. .008-5-4
8. J of .04 12. i of 12.06 16. 24.18-5-3 20. .105-5-5
102 DECIMAL FRACTIONS
Written Work
l. Divide 12t).685 by 15. 2. Divide 174.44 by 28.
_ 1 379 How many times is 15 con- 6.23
15)20.685 tainedin20? in 5.6 ? in 1.18? 28)174.44
15. in .135? ' 168 '
~~F~f> In practice we divide as in
the second example, placing the "
•" decimal point directly above 56_
1.18 the point in the dividend, be- 84
1.05 fore beginning to divide, and $4
J35 dividing as in integers.
.135
When the divisor is an integer, division simply separates or partitions
the dividend into equal parts. Thus, ^ of 20 and 085 thousandths
(20.085) = 1 and 379 thousandths (1.379).
A decimal or a mixed decimal is divided by an integer by
'placing a decimal point above or below the decimal point in the
dividend, before beginning to divide, and dividing as in the
division of integers.
3.
96.16-4-8
16.
12.312-4-27
4.
849.6-^-6
17.
2.25-4-15
5.
72.84-^-12
18.
809.6-4-16
6.
22.5-15
19.
256.25-4-25
7.
80.96-4-16
20.
96.064-4-32
8.
2.5625 -f- 25
21.
7010.5-4-35
9.
.96064-4-32
22.
61.472-4-68
10.
701.05-4-35
23.
27.8142-4-307
11.
2.268-4-27
24.
425.92-4-605
12.
2.867-4-47
25.
901.57-4-97
13.
36.54-4-42
26.
.2322-4-86
14.
.666-4-74
27.
34.356-4-409
15.
6.675-4-89
28.
45.76-4-650
DIVISION 103
Making the divisor an integer.
1 fi 25 _i_ 1 25 = 5 Study of Problems
2. 62.5 -s- 12.5 = 5 1- What is the first quotient? the secoml ?
3. 625.-5- 125. = 5 the third?
2. What was done to the first problem to
make the second? to the second to make the third?
3. How is a decimal affected by moving the decimal point one place
to the right ? two places ?
4. How did moving the decimal point to the right the same number of
places in both dividend and divisor of each problem affect the quotient?
Multiplying both dividend and divisor by the same number
does not change the quotient.
Written Work
Since multiplying both dividend and divisor by the same
number does not change the value of the quotient, make the
divisor an integer before beginning to divide.
1. 6.48 -4- .4 = 64.8 ■*■ 4. 1- Make the divisor an integer by
±\a\ a moving the decimal point one place
^ ' ' to the right in both dividend and
divisor.
2. Show that this does not affect the quotient.
3. Solve, placing the decimal point directly below the point in the
dividend, before beginning to divide.
2. 57.6 -r- .024 = 57600 -5- 24. 1. Make the divisor an in-
2400. teger by moving the decimal
24^57600 point three places to the right
' ,' in both dividend and divisor.
4o
— — 2. Solve, placing the deci-
°" mal point directly above the
96 point in the dividend, before
000 beginning to divide.
Divide as in integers, placing the decimal point directly
above or below the decimal point in the dividend, before begin-
ning to divide.
104 DECIMAL FRACTIONS
The use of the caret in division of decimals.
Many teachers prefer to mark off by a caret as many deci-
mal places from the right of the decimal point in the dividend
as there are decimal places in the divisor, and divide as in
integers, placing the decimal point directly below or above
the caret in the dividend. Thus,
. 8)5.68 = .8 )5.6 A 8
7.1
It is evident in the above problem that if both the divi-
dend and divisor were changed so as to make the divisor a
whole number, the decimal point in the dividend would be
in the place occupied by the caret, and that the decimal point
would be placed in the quotient immediately after the num-
bers to the left of the caret had been used in the process of
division.
The use of the caret determines the position of the decimal
point in the quotient, and at the same time retains the iden-
tity of the problem. Thus,
l. Divide 96.8 by .004. 2. Divide 1.2864 by .032.
.004 )96. 800 A .032)1.286,4(40.2
24 200. 1 28
64
64
Mark off by a caret the same number of decimal places from
the right of the decimal point hi the dividend as there are
decimal places in the divisor. Divide as in integers, placing
the decimal point in the quotient immediately after all the
numbers to the left of the caret have been used in the process
of division.
DIVISION
Find quotients :
3.
4.05 -.27
19.
1000 - .001
4.
.252-=- .14
20.
.2375 -j- .095
5.
8.398 -h 3.8
21.
177.8028-72.87
6.
2.173 -s- 1.06
22.
145.908 -=- 1.26
7.
144 -.12
23.
.0656 -=-.004
8.
.144-12
24.
.1701-63
9.
31. 36 -.056
25.
85. 75 -=- .0049
10.
.41912 -.338
26.
.025641 -.7.77
11.
3. 125 -.25
27.
.0022-200
12.
.3105-15
28.
222 -.002
13.
.5 -.625
.025 -=-.00025
14.
6.705-^.009
30.
.0003-1.5
15.
139.195 -*- 14.35
31.
$1. 05-1.005
16.
46.5 -.1875
32.
.685-5-500
17.
.00522 -.29
33.
.01058-5-46
18.
.001705.-5-. 31
34.
125.625-5-1.005
105
The division is frequently not exact. In such cases the sign +
may be placed after the decimal to show that the division is not com-
plete; thus, 1-7 = .142 +
Find the sum of the quotients :
35.
36.
37.
1- .1 =
3-
- .03 =
.18-=- 72 =
.1- 10 =
30-
- .3 =
.04-=- 50 =
.25-5- 50 =
6-
=-.006 =
2 -.025 =
2.5 -5- .5 =
16-
- .04 =
20 -.002 =
25-4-2.5 =
60-
- 300 =
200-12.5 =
.15-=-. 15 =
.6-:
- 30 =
64 -.016 =
1.5 h- 15 =
.66-
- 1.1 =
64- 160 =
.15-5-2.5 =
.9-
-.009 =
.4h- 400 =
106 DECIMAL FRACTIONS
38. Divide $.10 by 1100.
39. If a stone cutter earns $3.75 a day, how many days
will it take him to earn $311.25 ?
40. If 4275 acres of land cost $1731.375, what is the
price per acre ?
41. At $. 22 a dozen, how many dozen eggs can be bought
for $19.47?
42. If 16 stoves are sold for $292, what is the average
price per stove ?
43. Divide $.18 by $20.
44. If the wheel of a bicycle is 9.25 feet around, how
many times does it turn in going a mile ?
45. The product of two numbers is .9375. One of them
is .75. What is the other ?
46. There are 30 \ square yards in one square rod. How
many square rods are there in a plot containing 559.625
square yards ?
47. A merchant, in closing out his stock of goods, sold
.37^ of the stock the first month, .35 the second month, and
the remainder, $5500 worth, the third month. What was
the value of his stock of goods ?
Changing a common fraction to a decimal.
Written Work
Since A = 4 -=- 5, to change a fraction to a decimal,
consider it a problem in division of decimals. Thus, -| =
4-3-5 = 5)4.0
0.8
Change to decimals and test :
■■■•8 *' 12 5< ff 7 - T6 9 ' 16
2-3 4-5- fill «19 ir»17
*' J *' 16 ** 25 8 32 10 - 2~5
REVIEW OF DECIMALS 1Q7
When the division does not terminate, the quotient may he
shown as a complex decimal. Thus, ^ = 7)3.000 or as an
0.4281
7)3 000 u.-*z82-
incomplete decimal, J '
0.428
11.
t*t
12.
9
31
13.
45
7 3
14.
29
53
13. .45^ + 0.42^ 5 4- 8.7£ + .95^ =
14. 86.55 +9.05^4- 9.87| +.00-| =■
15. .875 + 0.75 + 7.7 + . 41 g =-
16. + + +
1
6(J
Change to complex or incomplete decimals of not more
than four places :
15. lj 19. |f 23.
16. A »■ !i 24. &
17. V- 21. £ 25. 2 7
18. ff 22. £- 26 . is
REVIEW OF DECIMALS
1. Addf, .045, .12|, 18f, .675.
2. Subtract f from .5 of 3|.
3. Multiply (36.7 - 4|) by 6.7.
4. Subtract 6| from 11.065.
5. Take .0031 f rom 6.
6. Divide .047f by 2.3|.
Add as indicated and test :
7. 8. 9. 10.
11. 8.375 4- .025 + 6.24f +.87] =—
12. .05f +6.041 +98.005| +.05| =-
108
DECIMAL FRACTIONS
BUSINESS APPLICATIONS OF DECIMALS
In all business transactions three things must be considered :
(1) The quantity of the commodity bought or sold.
(2) The price per unit at which it is bought or sold.
(3) The total amount paid or received for the commodity.
Quantity is measured by standard units established by
custom or law ; thus, the pound is a unit of weight ; the foot
or the yard, a unit of length ; and the gallon or the barrel, a
unit of liquid measurement.
The price per unit is the amount of money paid or re-
ceived for a standard unit of the commodity ; thus, when
butter is sold at 25 ^ per pound, the standard unit is the
pound and the price is 25 cents.
1. What standard is used in measuring grain ? butter ?
eggs ? milk ? cloth '? hay ? oil ?
2. What unit is used in measuring values in money ?
How many cents are there in $1 ? in $% ? •$ \ ? $>§ ? 50 ^ is
what part of $1 ? 20 f is what part of ffl ? 25 i ? 10 ^ ? 5 1 ?
Parts of $1 that Should be Known
U =T*o<>f$l
25^ =iof$l.
2^ -^ of$l
33^ = £of$l
2^ = T V of$l
37.^ = fof$l
4^ =£, of$l
40? =fof$l
5^ =^ of $1
50^ =,Vof$l
6^ = T V of$l
62^ = | of $1
Sh? = T \ of$l
66|^ = |of$l
10^ =1 V of$l
75^ =fof$l
12^= | of$l
80^ =iof$l
16^=i of$l
83^ = £<>f$l
20^ = i of $1
87^ = |of$l.
BUSINESS APPLICATIONS OF DECIMALS L09
Finding the total cost when the quantity purchased is given
and the price of a unit is an even part of $1.
Written Work
Business computations may be shortened by knowing the
relation that the price of a unit bears to $1 or to $100.
1. How much will 44 bushels of potatoes cost at $.25 per
bushel ?
Decimal Method Short Method
$ .25 = price ±)$U
44, no. of bushels "$TT"
1 qq At $ 1 each, 44 bu. would cost $44.
$iT00 = total cost Atneach,the y costiof«44,or«ll.
Find the cost of the quantity at $1. Divide this by the
quantity that can be purchased for $ 1.
Find cost of :
2. 60 bu. apples at 33^ per bushel.
3. 25 lb. butter at 25^ per pound.
4. 960 yd. calico at 6^ per yard.
5. 50 lb. lard at 121^ per pound.
6. 80 lb. rice at 12 1 -fl per pound.
7. 120 yd. ribbon at 371** per yard.
8. 500 books at 40^ each.
9. 1200 doz. eggs at 25^ per dozen.
10. 600 bu. oats at 33^ per bushel.
11. 1600 bu. coal at 6|^ per bushel.
12. 86 qt. cherries at 6| ^ per quart.
13. 2500 bu. corn at 40^ per bushel.
14. 160 lb. beef at 10 4 per pound.
110 DECIMAL FRACTIONS
15. A merchant buys
240 lb. coffee at 12| t per pound,
300 lamp chimneys at 8^ each,
6000 qt. milk at 41^ per quart,
560 bu. potatoes at 50^ per bushel.
Find total cost.
16. A farmer sells
1260 heads of cabbage at b$ per head,
250 bu. potatoes at 50^ per bushel,
2240 lb. beans at 6| $ per pound,
600 qt. cherries at 8^ per quart,
1200 qt. strawberries at 12|^ per quart.
Find total receipts.
17. A milk dealer bought
300 bu. corn at 50 ^ per bushel,
6000 lb. bran at \$ per pound,
6000 lb. hay at f>l per hundred pounds.
He sold 4000 gal. milk at 6\ tf per quart. How much did
he make ?
Finding the quantity purchased when the total cost is given
and the price of a unit is an even part of $1.
Written Work
1. How many yards of calico, at Q\$ per yard, can be
purchased for $100?
Decimal Method Short Method
100 x 16 yd. = 1600 yd.
1600 . since 6\f = $ f L, $ 1 pays for 16 yd.;
$.0625)!$100.0000 A and $100 pays for 100 x 16 yd., or
1600 yd.
Multiply the quantity purchased for $1 by a number equal
to the number of dollars invested.
BUSINESS APPLICATIONS OF DECIMALS 111
Find the quantity of each article if a grocer invested
2. $1G0 in sugar at Q\ f per pound.
3. $120 in sugar at 4^ per pound.
4. $6.00 in rice at 10 ^ per pound.
5. $100 in cloth at 50^ per yard.
6. $13.00 in gingham at 5^ per yard.
7. $3.00 in cheese cloth at 2^ per yard.
8. $ 800 in milk at G\ i per quart.
9. $ 100 in meat at 12| t per pound.
10. $4.00 in collars at 12| ^ each.
11. $1.00 in bananas at 20 ^ per dozen (find number of
bananas).
12. A hotel keeper purchased
$100 worth of sugar at G\ ^ per pound,
$250 worth of potatoes at 33£ ^ per bushel,
$60 worth of soap at 2\ ^a cake.
Find number of pounds, bushels, and cakes purehased.
13. A farmer sold to a merchant
10 bu. apples at 40 ^ per bushel,
20 qt. beans at 10 $ per quart,
16 bu. potatoes at 75 ^ per bushel.
He invested | of the proceeds in cloth at 25 9 per yard
and the balance in coffee at 12* ^ per pound. How much
of each did he purchase?
14. A grocer bought coffee at 12| ^ a pound, and sold it
for $39, thereby gaining $G.50. How many pounds did he
buy ?
SIMPLE ACCOUNTS FOR BOYS AND GIRLS
An account is a statement of the receipts and disburse-
ments of any person.
There are two sides to an account : the first, or debit side,
on which are entered all receipts; the second, or credit side,
on which are entered all disbursements, or amounts paid out.
Dr. indicates the debit side of an account ; Cr. indicates the
credit side.
The balance is the difference between the debit and credit
sides.
September 1, 1906
Cash on hand
Dr.
Cr.
Sept. 1
$12
10
Sept. 1
Note-book, $.15; pencil, if .05
$
20
Sept. 4
Arithmetic
50
Sept. 5
Geography
1
00
Sept. 7
Copy-book, $.10; ink and pens, $.08
18
Sept. 15
History
1
00
Sept. 17
Worked one day
1
00
Sept. 25
Car fare
50
Sept. 29
Tools
1
60
Sept. 30
Balance, Cash on hand
A
$13
8
12
10
$13
10
Continue the balance of each month through the following
months to September, 1907.
Note to Parents. — Children should be encouraged to keep their
own personal accounts.
112
SIMPLE ACCOUNTS FOR BOYS AND GIRLS 113
1. October. Oct. 3, Bought 1 pair of shoes, $2.50. 1
hat, $1.50. Oct. 8, Repairs to bicycle, $.75. Oct. 15, Earned
$1.50. Oct. 17, Worked for Mr. Black and received $.15.
Oct. 25, Saturday outing, $.60.
2. November. Nov. 5, Bought a sled, $.95. Nov. 9,
Bought a cap, $.75. Nov. 15, Shoveled snow off Mrs.
Graham's walk, $.30. Nov. 17, Sawed kindling wood for
Mr. Goff, $.50. Nov. 26, Bought a knife, $.25. Nov. 30,
Ran errands, $.35.
3. December. Dec. 3, Bought 1 pair of skates, $.75.
Dec. 10, Received from Mr. Black for work in store, $1.00.
Dec. 17, Expense for school supplies, $.17. Dec. 21, Received
from Mrs. Williams for carrying in load of coal, $.30.
Dec. 22, Bought Christmas presents, $3.75. Dec. 25, Christ-
mas gift from Uncle James, $1.00. Dec. 29, Expense for
having skates sharpened, $.10.
4. January, 1907. Jan. 5, Received from Mrs. Jones for
fixing doorbell, $.15. Jan. 8, Bought 1 pair mittens, $.50.
Jan. 15, Delivered bills around town for Mr. Black, $.50.
Jan. 25, Bought necktie, $.25. Jan. 30, Bought " History
of French Revolution," $.75.
5. February. Feb. 6, Worked on Saturday for Mr. Black,
$.75. Feb. 11, Shoveled snow from sidewalk for Mr. Hart,
$.25. Feb. 16, Ran errands, $.40. Feb. 20, Helped unload
car of feed, $1.00. Feb. 26, Copied 2 leases for Mr. Irwin,
$.75. Feb. 28, Bought pair of gloves, $1.25.
6. March. March 1, Cleaned yard for Mrs. Williams,
$.50. March 6, Bought 2 pairs of socks, $.30. March 11,
Bought new umbrella for mother, $1.75. March 15, Repaired
fence for Mr. Jones, $.25. March 27, Car fare, $.30. March
30, Sold my old bicycle fur $5.00.
HAM. COMPL. ARITH. -8
114 SIMPLE ACCOUNTS FOR BOYS AND GIRLS
7. April. Apr. 1, Burned paper and refuse for Mr.
Hart, $.25. Apr. 8, Made garden for Mrs. Black, $.50.
Apr. 10, Whitewashed cellar for Mrs. Goff, $.35. Apr. 15,
Wheeled load of coal for Mr. Brown, $.35. Apr. 25, Bought
4 collars and 2 pairs of cuffs, $.90. Apr. 30, Bought neck-
tie, $.25.
8. May. May 3, Bought straw hat, $1.00. May 7,
Mowed lawn for Mrs. Jones, $.25. May 13, Repaired
Mr. Brown's sidewalk, $.40; May 30, Bought baseball, $.50.
May 31, Received a reward of $5.00 for finding a pocket-
book containing $50, which I returned to owner.
9. June. June 1, Made $.20 selling papers. June 6, t
Worked a day for Mr. Black, $.75. June 10, Delivered
package, $.25. June 17, Bought ball bat, $.50. June 20,
Wheeled a trunk for Mr. Hart, $.25. June 29, Bought 1 pair
of baseball shoes, $1.00.
10. July. July 4, Fireworks, $.50. July 6, Received
from Mr. Black salary for week, $5.00. July 12, Bought
2 shirts, $1.50. July 13, Received week's salary, $5.00.
July 15, Bought outing suit, $6.50. J^ily 20, Received my
salary, $5.00. July 25, Expense for small articles, $.95.
July 27, Received my week's salary, $5.00. July 30, Re-
ceived for overtime, for month, $7.50.
11. August. Aug. 3, Salary, $5.00. Aug. 8, Bought 1
pair of tan shoes, $2.50. Aug. 10, Received salary, $5.00.
Aug. 15, Bought fishing tackle, etc., $3.75. Aug. 17, Re-
ceived week's salary, $5.00. Aug. 31, Expenses for 2
weeks' vacation, $15.75.
Sept. 1, Balance, Cash on hand, .
Make out a statement at close of year, showing total
receipts and disbursements, and proving final balance.
DENOMINATE NUMBERS
1. Write from memory the following tables :
Liquid Measures, Dry Measures, Avoirdupois Weight,
Time Measures, and Measures of Length or Distance.
2. 1 yr. = mo. = da. = hr. = niin. =
sec.
3. 1 mi. = rd. = yd. = ft. = in.
4. 1 T. = cwt. = lb. = oz.
5. 1 bu. = pk. = qt.
6. 1 gal. = qt. = pt.
The standard or principal units of measure are as follows :
Liquid — gallon. Length or distance — yard.
Dry — bushel. Avoirdupois — pound (i6oz.).
Time — day.
All other measures are determined from the above unit
measures. Thus, the ton is 2000 times 1 pound (16 oz.).
The hour is ^ of the day, the period of one revolution of
the earth on its axis.
A denominate number is a concrete number whose unit is a
measure established oy custom or law; as, 10 feet, in which
1 foot is the unit of measure, or 5 pounds, in which 1 pound
is the unit of measure.
A simple denominate number is a number of one denomi-
nation ; as, 12 rods, 2 ounces, 5 days, etc.
A compound denominate number is composed of two or
more concrete numbers that express one quantity; as, 6
yards, 2 feet, 4 inches. Here yards, feet, and inches are
used to express but one quantity.
ll .
116
DENOMINATE NUMBERS
REDUCTION OF DENOMINATE NUMBERS
Change :
1. 5^ yd. to feet.
2. 90 in. to feet.
3. 3 yd. 2 ft. to feet.
4. .5 rd. to inches.
5. 25 ft. to yards.
6. 5.5 hr. to minutes.
7. .5 mi. to rods.
8. 3.5 gal. to pints.
9. ^ day to minutes.
10. .25 bu. to quarts.
11. | pk. to quarts.
12. 2 lb. 8 oz. to ounces.
13. | cwt. to pounds.
14. .5 yd. 1 ft. to inches.
15. .75 mi. to rods.
16. .25 bu. to pints.
17. 3.5 pk. to quarts.
18. 2 yd. 1.5 ft. to inches.
19. 3.5 min. to seconds.
20. 48 qt. to pecks.
21. 64 pt. to bushels.
22. 64 oz. to pounds.
Written Work
l. Change 3 gal. 3 qt. 1 pt. to pints.
gal.
3
4
qt. pt.
3 1
12
+ 3
15, number of quarts.
_2
30
+ 1
31, number of pints.
Observe that 4 qt. is really the multi-
plicand and 3 the multiplier in finding the
first product; and that 2 pt. is really the
multiplicand and 15 the multiplier in finding
the second product. In considering the num-
bers abstractly, however, either factor may
be regarded as the multiplicand and the
arrangement as indicated saves time.
REDUCTION
11'
2. Change .875 gallon to pints, etc.
.875
3.500, number of qt.
1.00, number of pt.
Siuce there are 4 qt. in a gallon, in .875 of
a gallon there are .875 of 4 qt., or :5.."> qt.
Since there are 2 pt. in 1 qt., in .5 of a quart
there is 1 pt. The answer is 3 qt. 1 pt.
Change
3. 15 lb. 8 oz. to ounces.
4. 96 ft. 5 in. to inches.
5. 5.5 bu. to quarts.
6. 3.5 pk. to pints.
7. 18 cwt. 25 lb. to pounds.
8. 23 hr. 16 min. to minutes.
9. 8.3 mi. to yards.
.75 yd. to inches.
4| T. to pounds.
7| min. to seconds.
10.
11.
12.
13. 6.5 L. T. to pounds.
14. 63.5 gal. to pints.
15. £ bu. to quarts.
16. 10| bu. to pecks.
17. Change 266 quarts to bushels, etc.
8 )266 There are | as many pecks as quarts,
4 )33, 110. of pk. + 2 qt. that is, 33 pk. + 2 qt. There are \ as
8, no. of bu. -f- 1 pk. many bushels as pecks, that is, 8 bu. +
., , „ 1 pk. Hence, 266 qt. = 8 bu. 1 pk. 2 qt.
8 bu. 1 pk. 2 qt.
Change to higher denominations:
18. 312 inches.
19. 6625 yards.
20. 5281 feet.
21. 2043 seconds.
22. 1033 ounces Av
28. 43920 in.
29. 6875 sec.
30. 56.5 pk.
31. 684.5 rd.
23. 347 cwt.
24. 6095 pounds.
25. 16857 rods.
26. 11097 qt. (Dry).
27. 952 pt. (Liquid).
33. How many gallons of milk will a family consume in
75 days, if they use 2 qt. 1 pt. daily ?
34. How much is received for H bushels of chestnuts
at 8 cents a quart ?
32. 964| min.
118 DENOMINATE NUMBERS
35. How much will 15 turkeys, averaging 14^ lb. each,
cost at 18 cents a pound ?
36. If 100 tons of coal are bought by the long ton, at
12.24 a ton, and sold by the short ton at the same price, how
much is gained ?
37. At 20 cents an hour, how much will a man earn in 26
days, working each day from 8 A.M. to 5 p.m., allowing 1
hour for lunch ?
38. If a flour mill grinds wheat at the rate of 1 pint in 5
seconds, in how many hours and minutes will it grind 21,600
bushels ?
39. A train goes 104 miles in 3 hours and 15 minutes.
What is the rate per hour ?
40. At 2 cents a foot find the length in miles and rods
of a telephone wire that costs $4672.80.
41. If a man's step averages 2 ft. 6 in., how far will he
travel in taking 6600 steps ?
Relations of denominate measures.
1. | pk. is what decimal part of a bushel?
f pk. = 6 qt.
6 qt. = — bu. = .1875 bu.
4 32
2. 3 ft. 2 in. is what fractional part of a rod ?
3 ft. 2 in. = 38 in.
1 rod = 198 in.
3 ft. 2 in. = T 3 9 8 ? , or |f rd.
Find the fractional part :
3. 2| hr. is of 1 day. 6. 1| pt. is of 3 qt.
4. 71 ft. is of 1 rod. 7. 2-| in. is of 10 ft.
5. 3| qt. is of 1 gallon. 8. 1^ qt. is of 2 gal.
ADDITION' AND SUBTRACTION 119
Find the fractional part :
9. 15 hundredweight is of 1 ton.
10. 3.5 quarts is of 1 bushel.
11. 280 rods is of 1 mile.
12. 37 pounds 8 ounces is of 1 hundredweight.
13. 440 yards is of 1 mile.
Find the decimal part :
14. 16 hours 48 minutes is of 1 day.
15. 1 foot 8 inches is of 1 rod 2 in.
16. 180 pounds is of 1 ton.
17. 16 minutes 48 seconds is of 1 hour.
18. A machinist works 10 hr. per day in summer and
8| hr. per day in winter. If his wages in summer are $3.35
per day, at the same rate find his wages per day in winter.
ADDITION AND SUBTRACTION
1. Find the sum of 2 gal. 3 qt. 1 pt., 4 gal. 1 qt. 1 pt.,
7 gal. 1 pt., 5 gal. 3 qt.
gal. qt. pt.
^ 3 The sum of the pints = 3 pt. = 1 qt. and 1 pt.
4 11 The sum of the quarts + 1 qt. carried = 8 qt. =
7 12 gal. qt.
5 3 The sum of the gallons + 2 gal. carried = 20 gal.
20 o r
Add:
2. 14 bu. 2 pk., 5 bu. 6 qt., 7 qt. 1 pt., 9 bu. 6 qt.
3. 5 T. 11 cwt., 4 T. 15 cwt. 60 lb., 11 T. 80 lb., 19 T.
3 cwt. 64 lb.
4. 9 yr. 120 da. 8 hr., 12 yr. 104 da. 17 hr., 14 da.
5. 3 wk. 6 da. 15 hr., 4 wk. 3 da. 9 hr., 7 wk. 5 da. 14 hr.
7 3 3
3 15
1 P k.
6 qt,; 2
4 16
Subtract :
mi. rd.
yd. ft.
7. 80 120
12
57 215
14
120 DENOMINATE NUMBERS
6. From 7 bu. 3 pk. 3 qt. take 3 bu. 1 pk. 5 qt.
bu. pk. qt.
1 pk., or 8 qt., + 3 qt. = 11 qt.; 11 qt. - 5 qt.
6 qt.; 2 pk. - 1 pk. = 1 pk.; 7 bu. - 3 bu. = 4 bu.
gal. qt. pt.
8. 23 1
9 3
9. From 18 hr. take 9 hr. 16 min. 45 sec.
Finding the difference in time between two dates is the most
practical application of subtraction of denominate numbers.
10. Find the difference in time between November 15, 1903,
and August 12, 1905.
Aug. 12, 1005, is represented as the 12th day
yr.^ mo. da. of the 8th mouth of 1905, and Nov. 15, 1903,
1905 8 12 as the 15th day of the 11th month of 1903.
1903 11 15 1 mo., or 30 da., + 12 da. = 42 da.; 42 da. -
1 8 27 15 (la - = 27 c,a -' 1 vr -> or 12 mo -' + 7 mo - = 19
mo.; 19 mo. — 11 mo. = 8 mo.; 1904 yr. —
1903 yr. = 1 yr.
Subtract :
yr. mo. da. yr. mo. da.
li. 1908 7 12 12. 1905 9 1
1901 9 15 1890 8 15
13. How many years, months, and days old is each pupil in
the room?
14. General Robert E. Lee was born January 19, 1807, and
General Ulysses S. Grant April 27, 1822. How old was
each at the close of the Civil War, April 9, 1865? How
much older was General Lee than General Grant?
15. How old is a man to-day who was born July 3, 1882 ?
MULTIPLICATION AND DIVISION 121
MULTIPLICATION AND DIVISION
1. Multiply 3 \vk. 5 da. 9 hr. by 7.
wk. da. hr.
o r g 7 x fl hr. = 63 hr. = 2 da- and 15 hr.; 7x5 da.
= 35 da.; 35 da. + 2 da. = 37 da. = 5 wk. and 2
1 da.; 7 x 3 wk. = 21 wk.; 21 wk. + 5 wk. = 26 wk.
26 2 lo Hence, the answer is 26 wk. 2 da. 15 hr.
Multiply :
2. 3 gal. 2 qt. 1 pt. by 3.
3. 12 bu. 3 pk. 3 qt. by 6.
4. 15 T. 5 cwt. 12 oz. by 10.
5. 27 wk. 3 da. 14 hr. by 9.
6. 23 mi. 124 rd. 11 ft. 4 in. by 12.
7. Divide 54 T. 15 cwt. 72 lb. by 12.
54 T. -r- 12 = 4 T. and 6 T. remain-
ing; 6 T. = 120 cwt. ; 120 cwt. +
io\c\J Ye to 15 cwt. = 135 cwt.; 135 cwt. h- 12
= 11 cwt. and 3 cwt. remaining ; 6
cwt. = 300 lb.; 300 lb. + 72 lb. =
372 lb. ; 372 lb. -*- 12 = 31 lb.
T. cwt. lb.
11 31
Divide :
8. 18 wk. 5 da. 21 hr. by 5.
9. 188 gal. 1 pt. by 7.
10. 88 bu. 3 pk. 4 qt, by 9.
11. 61 yr. 11 mo. 18 da. by 11.
12. 86 T. 3 cwt. 44 lb. by 6.
13. Find the cost of 19 gross of pencils at 10^ a dozen.
14. A man digs 4 rods, 2 yards of ditch in a day. How
many rods, etc., can he dig in 6 days ?
15. How many packages, weighing 5 ounces each, can be
made from 5 pounds of candy ?
122 DENOMINATE NUMBERS
REVIEW
1. If a watch gains 18 seconds in a day, how much too fast
will it be in three weeks ?
2. How many barrels, each holding 2 bushels and 3 pecks,
will be required to pack 88 bushels of apples ?
3. How many bushels of potatoes are necessary to plant
8| acres, allowing 6 bu. 1 pk. to the acre?
4. A merchant sells linseed oil at 12 ^ a pint that cost him
56^ a gallon. Find his profits on 45 gallons 3 quarts.
5. 5 car loads of coal weigh : 57,698 lb., 49,875 lb.,
63,545 lb., 49,897 lb., and 54,273 lb. Find the number of
tons, hundredweight, and pounds in all.
6. 4 men buy a plot of land that has 222 feet 8 inches
street frontage. Allowing for an alley 20 feet in width in the
center, what is the width of each man's lot if they divide the
plot equally ?
7. A force pump in a coal mine lifts 76| gallons of water
to the surface per minute. Find the number of gallons
pumped out in one day.
8. If 3 pounds 4 ounces of coal are consumed in generat-
ing power to lift 5 gallons of water in problem 7, find the
number of tons of coal consumed each day.
9. A Kentucky farmer clipped 241^ pounds of mohair
from 70 Angora goats. Find the average clip from each goat
and its value at $.37*- per pound.
10. An automobile runs 2| miles in 5 minutes. At that
rate, find the distance in miles, rods, and feet it runs in 1
hour 35 minutes.
11. A pencil factory makes 6| gross of pencils per hour.
Find the number of dozen made in 26 days of 9^ hours each.
PRACTICAL MEASUREMENTS
MEASURES OF LENGTH
1. Measure the length of your desk ; the length of the
room ; the length of the blackboard ; the height of the
window from the floor.
2. In what are these short lengths measured?
To the Teacher. — Secure a tape measure 50 feet long.
3. Measure the distance around the schoolroom in feet
and fractions of a foot. How many yards is it around the
room ?
4. Measure the distance around the school grounds in
rods, feet, and inches.
5. Take 16| ft. of the tape measure and measure 10 rods
along the public road or street.
6. 320 x 16| ft, = how many feet?
1760 x 3 ft, = how many feet?
7. How many feet equal a mile ? how many yards ?
8. James walks 1| miles to school each day. How many
rods does he walk in going to and from school ?
9. How many rods equal 5280 ft, ? | of a mile? 3560 ft.?
10. Mary walks f of a mile to school each day. How
many miles does she walk in going to and from school in
180 days?
11. Henry walks .8 of the number of miles Mary walks.
Find the distance Henry walks in a term if he attends 160
days.
123
124
PRACTICAL MEASURED ENTS
MEASURES OF SURFACE
Observe that the straight lines AB and CD cannot meet,
however far they may be extended. Such lines are called
parallel lines.
A
C-
-B
■D
Lines that meet, making a square corner, form a right
angle.
A figure that has four straight sides and four right angles
is called a rectangle.
A rectangle having its four sides equal is called a square.
1. Name six different rectangles in the schoolroom. Are
the opposite sides of a rectangle parallel lines ?
2. How many dimensions has every rectangular surface ?
How does a surface differ from a line ?
3 in.
.c
1
V
f \
3. Draw, on a scale of £, a
rectangle 3 inches long and 2
inches wide. Divide it by lines
into square inches. How many
square inches are there in the
first row ? in the second ? in
the rectangle ? What is the
unit of measure in this surface ?
Observe that 2x3x1 sq. in. = 6 sq. in.
The area of a rectangle is found by multiplying its unit of
measure by the product of its two dimensions^ when expressed in
like units.
4. Draw a rectangle 2 ft. by 6 ft. Divide it into sq. ft.
5. Draw a square a foot on a side. Mark off the sides
into 12 equal parts and connect them by straight lines. How
many square inches equal a square foot?
MEASURES OF SURFACE 125
6. Draw oil the blackboard a line 4 feet long. From each
end draw lines in the same direction 3 feet in length, making
square corners with the 4-foot line. Connect by a straight
line the ends of the 3-foot lines.
7. Are the sides of the figure straight ? Are the corners
equal in size? Find the area of the figure.
8. What is a right angle ? a rectangle ? (p. 124).
9. Show by a diagram the number of square feet in a
square yard.
10. Draw a diagram on a scale of 1 inch to 3 feet to rep-
resent a rectangle 24 ft. long and 18 ft. wide. Find its area.
Draw diagrams on scales suitable to the size of your tablet
or slate and rind the surface of each of the following :
11. A rectangle 20 ft. by 24 ft,
12. A flower bed 16 ft. by 8 ft.
13. A floor 16 ft. long and 14 ft. wide.
14. A wall 15 yd. long and 5 yd. high.
15. By actual measurement find the number of square feet
in the floor, the door, the blackboard, and the walls of the
schoolroom.
16. In what denominations did we find the lengths and
widths of the problems just given?
Land is measured in acres, square rods, square feet, etc.
17. Measure a square rod on your playground. How
long is it ? how wide ?
18. Measure the length and width of your school grounds
in rods and feet.
19. Since 16^ feet equal 1 rod, how many yards equal 1
rod ? How many square yards equal 1 square rod ?
126 PRACTICAL MEASUREMENTS
20. Since 16 1 feet equal 1 rod, how many square feet equal
1 square rod?
21. A field is 70 rods long and 40 rods wide. How many-
square rods are there in it? how many acres?
22. Memorize this table :
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 A. = 160 sq. rd. =
4840
sq. yd. = 43,560 sq. ft.
Change :
23. 2700 sq. yd. to sq. ft. 26. If A. to sq. rd.
24. 50 sq. ft. to sq. in. 27. 800 sq. yd. to sq. rd.
25. 1600 sq. rd. to A. 28. 5f A. to sq. ft.
29. A farm is 90 rods long and 60 rods wide. Find the
number of acres in it. Find its cost at 160 per acre.
30. A lot 100 ft. square has a house 36 ft. by 42 ft.
located on it. The remaining space is lawn. Find the
number of square feet of lawn. Draw diagram.
31. A concrete sidewalk in front of the lot is 4 ft. wide.
Find its cost at 19 ^ per square foot.
32. Find the cost of a flagstone walk, 135 ft. long and 6
ft. wide, at 21 i per square foot.
33. City lots are sometimes sold by the square foot. Find
the cost of a lot in Pittsburg 21 ft. by 70 ft. at $27.50 per
square foot.
MEASURES OF SURFACE
127
34. A farm 160 rods long and 12<> rods wide is sold in
two pieces, § of it at 160 per acre, and the remainder at
$ 50 per acre. Find the amount of the entire sale.
35. An Iowa farmer owns a farm a mile square. How
many acres has he ? Find its value at $85 per acre.
36. A western wheat held 100 rods long and 80 rods wide
yields 880 bushels of wheat. Find the average yield per acre.
37. City lots are usually sold by the front foot. Find the
cost, at $20 per foot front, of a lot 25 ft. front by 120 ft.
deep. Find the cost per square foot.
38. A four-room school building has a slate blackboard
24 ft. by 4 ft. in each room. Find the total cost of the
blackboard at 23^ per square foot.
39. The area of a field in the
form of a rectangle is 8 acres.
If one side is 32 rods, what is
the other?
These diagrams represent pieces of
land. The dimensions are given in
rods, and the corners are all square.
40. Divide the first piece into
3 rectangles and find (1) how
many square rods there are in each ; (2) the perimeter of
each ; (3) the area of the entire
piece in acres.
41. Divide the second piece
into rectangular lots, and find
(1) the perimeter of each ;
(2) the area of each ; (3) the
area of the entire piece.
20
6
14
4
6
5
4
7
25
7
5 '°
13
5
a
9
7
12
5
3
6
7
128 PRACTfCAL MEASUREMENTS
PAINTING AND PLASTERING
Painting, plastering, and kalsomining are generally meas-
ured by the square yard. In some localities an allowance is
made for doors and windows, but there is no uniform rule
in practice.
1. How much will it cost to paint a ceiling 18 ft. long
and 15 ft. wide at 10/ per square yard?
2. How much will it cost to kalsomine a hall 30 ft. long,
9 ft. wide, and 15 ft. high, at 5/ per square yard ? (Observe
that the perimeter of the hall is 78 ft.)
3. How many square yards of plastering are there in a
room 21 ft. long, 18 ft. wide, and 12 ft. high, making no
allowance for openings?
4. How much will it cost, at 15/ a square yard, to plaster
a room 24 ft, x 191 ft. x 15 ft.?
5. A public hall is 120 ft. x G6 ft. x 22| ft. How much
will it cost to paint the walls and ceiling at 10/ per square
yard?
THE RIGHT TRIANGLE
1. Draw on the blackboard a rectangle 12 inches long and
8 inches wide. Connect the opposite corners by a straight
line.
This line is called the diagonal of the rectangle.
2. Into how many parts have we divided the rectangle?
Shade one of the parts with chalk. How many angles are
there in each part ? how many right angles ?
A triangle is a surface bounded by three straight lines.
THE RIGHT TRIANGLE
129
A right triangle is a triangle having one
right angle.
The base of a triangle is the side on
which it is assumed to stand.
The altitude of a triangle is the line
that meets the base line at a right angle.
To the Teacher. — As an aid in drawing have each pupil, if possible,
gei a right triangle as here shown.
3. Point out the base and
altitude in the triangles at the
right.
o
4. Fold a rectangular piece of paper, as ABCD, on its
diagonal. Observe :
(1) That the rectangle ABCD and
the triansfle ABB have the same base
and altitude.
(2) That the area of the triangle
is just I the area of the rectangle.
Hence, the area is | of 4 x 2 x 1 sq. in. = 4 sq. in.
m
c
*
■■"v.
n
InllTTThfc.
■o
IhTttk
c
»~
T->
ip ''M
ilUlHiit^
^
A Base 4 in. B
The area of a right triangle eqvals the unit of measure mul-
tiplied by \ the product of the base and altitude.
Draw on a scale suitable to your paper and find the ana
of the following right triangles in square inches :
5. Base 10 in., altitude 8 in. 7. Base 25 in., altitude 18 in.
6. Base 12 in., altitude G in. 8. Base St! in., altitude 21 in.
9. Find the area of a field in the form o( a right triangle
whose base is 80 rods and altitude 40 rods.
HAM. COMPL. AKITU. 9
130
PRACTICAL MEASUREMENTS
MEASURES OF VOLUME
1. How many dimensions has the cube ? Name them.
2. What dimensions has a line ? What dimensions has a
surface ? a solid ?
3. Name the different units of measure in which the
length of a line may be expressed.
m4. Name the
^-= — : -» different units of
square measure in
which surface may
be expressed.
5. What cubic
unit have we in
the first cube ?
6. If a cube is
1 foot on an edge,
what is the cubic
unit ?
7. Draw a square 1 foot on a side. Show that it contains
144 square inches.
MEASURES OF VOLUME
131
3 in.
8. Observe that 144 cubes 1 inch on an edge can be
placed on a surface 1 foot square. How many layers of such
cubes will it take to make a cube 1 foot on an edge ?
9. How many cubic inches equal 1 cubic foot ?
10. How many surfaces has a cube ?
11. Show that all the surfaces of an inch cube are the
same in area ; of a 2-inch cube ; of a 9-inch cube.
12. Examine carefully the fig-
ure. Observe :
(1) That the surface of the
face upon which it rests con-
tains 9 square inches.
(2) That the first layer of
units of volume contains 9
cubic inches.
(3) That the whole solid, if
6 inches high, contains 6x9
cubic inches, or 54 cubic inches.
13. How many 1-inch cubes
are there in the first layer ? how
many in the solid ?
14. What is the shape of the
surfaces of the solid? Is each surface a rectangle?
A rectangular solid is a solid whose surfaces are all rectangles.
15. Observe that the number of inch cubes in the solid is
equal to the product of its three dimensions.
16. What is the unit of measure in the solid ?
Observe that 3 x 3 x 6 x 1 cu. in. = 54 cu. in.
The contents, or volume, of a rectangular solid equals the
unit of measure multiplied by the product of its three
diynensions.
132
PRACTICAL MEASUREMENTS
To the Teacher. — Secure 144 1-inch cubes.
17. Build a cube 2 inches on an edge.
18. Build a cube 4 inches on an edge.
19. Compare the 4-inch cube with the 2-inch cube.
20. Give the different units of measure of surface ; of
length ; of contents.
21. Find the contents of a box 3 ft. long, 2 ft. wide, and
11 ft. high.
22. Observe the cube. What
is its length? width? height?
3ft. 23. How many 1-foot cubes
does it contain ?
A cube 3 ft. on an edge is
called a cubic yard.
3f 24. Memorize this table :
1728 cubic inches (cu. in.) = 1 cubic foot (cu. ft.)
27 cubic feet = 1 cubic yard (cu. yd.)
A cart load of earth is considered 1 cubic yard.
PRACTICAL APPLICATIONS
Excavations are estimated by the cubic yard.
l. Find the cost, at 30^ per cubic yard, of excavating a
cellar 36 ft. in length, 24 ft. in width, and 4 ft. in depth.
36 x 24 x 4 x 1 cu.fi. = 3456 cu. ft., the contents of the cellar.
3456 cu. ft. ^ 27 = 128, number of cu. yd.
128 x $.30 = S 38.40, cost of excavation.
PRACTICAL APPLICATIONS
133
This diagram shows the outline of a cellar 5 ft. deep. Its
dimensions are given in feet.
2. Find its
square feet.
area in
4
4
14
14
30
12
12
30
3. Find the length of
its walls.
4. What is the cost of
excavating it at 32 ^ per
cubic yard?
5. How much will it
cost to cement the floor at
$ .90 per square yard ?
6. If a boy inhales 24
cubic inches of air at a breath, how many times must he
breathe to inhale 1 cubic foot ?
7. 29 pupils and their teacher occupy a schoolroom 30 ft.
in length, 24 ft. in width, and 12 ft. in height. What is the
average number of cubic feet of air for each person ?
8. A city lot of 37| ft. by 120 ft. is to have a layer of earth
1 ft. thick over its surface. Find the number of loads
needed and its cost at 25 ^ per load.
9. A dining room is 13 ft. by 18 ft. and has a rug on it
9 ft. by 15 ft. Find the surface not covered by the rug.
10. If the rainfall on a certain day was 21 inches, find
the number of cubic inches that fell on a lot 25 feet wide
and 100 feet long. Find the number of gallons.
11. Find the cost of digging a ditch, 60 rods long, 3£ feet
wide, and 6 feet deep, at 60 ^ per cubic yard.
134 PRACTICAL MEASUREMENTS
12. Memorize :
1 gallon = 231 cu. in.
1 bushel = 2150.42 cu. in.
1 bushel = 1] cu. ft. (nearly)
13. Compare a 3-inch cube with a 4-inch cube. If a
2-inch cube weighs 6 ounces, how much will a 4-inch cube
of the same material weigh ?
14. A bin is 8 ft. long, 6 ft. wide, and 4 ft. deep. Esti-
mate quickly about the number of bushels of wheat or oats
it will hold.
15. A farmer has a tank 12 ft. long, 8 ft. wide, and 6 ft.
deep. How many gallons of water will it hold ?
16. How much larger is a farm 80 rods square than a
farm 60 rods square ? Draw diagrams on a suitable scale to
represent this.
17. The base of a rectangular tank is 48 sq. ft. and the
volume is 192 cu. ft. Find the height.
18. What is the area of each surface of a cube 8 ft. on an
edge ? the entire surface ?
MEASUREMENT OF LUMBER
A board 1 foot
\i K y ^? square and 1 inch
c ,- v ^ thick or less is a board
I in.
foot.
.{i 4 ^- A board foot is the
vnit in measuring
N lumber.
l. Draw on the blackboard a figure to represent a board
4 feet long, 1 fout wide, and 1 inch thick.
MEASUREMENTS 01 LUMBEB
!
4ft.
2. Show that this board contains 4 hoard feet.
3. How many board feet are there in a sill 4 ft. long,
1 ft. wide, and 4 in. thick?
Observe :
(1 ) That the sill is rft;
equal to 4 boards 1 ft.
long, 1 ft. wide, and
1 in. thick.
( -1 > That each board contains 4 hoard fe<
Hence, the -ill contains 1x4x1 board foot = 16 board feet.
27//? number of board feet in a piece of lumber equals the
number of hoard feet in one surf 'ace multiplied by the number
of inches in thickness.
Find the number of board feel in the following:
4. A plank 12 ft. long, 12 in. wide, and 2 in. thick.
5. A board 12 ft. long, 9 in. wide, and 1 in. thick.
6. A plank 15 ft. long, 12 in. wide, and 3 in. thick.
7. A plank 16 ft. long, 18 in. wide, and 2 in. thick.
8. A sill 20 ft. long, LO in. wide, and 6 in. thick.
9. A .-.ill 30 ft. long and 12 in. squar<
Buying and selling lumber.
Lumber is usually .sold at so much per 1000 (M.) board feet.
Find the cost of :
1. 5000 ft. poplar at 8 40 per M.
2. 500 ft. hemlock at 3 24 per M.
3. 10,850 ft Georgia pine at 8 24 per M.
4. 8000 ft. white pine at $50 per M.
Small bills of lumber are usually estimated al 90 much per
board foot.
136
PRACTICAL MEASUREMENTS
5. Show that lumber at $ 40 per M. = $.04 per board foot;
at $27 per M. = $.027 per board foot.
6. Make out a receipted bill to Henry James for the
following : 365 ft. hemlock at $ 25 per M., 780 ft. white pine
at $40 per M., 980 ft. yellow pine at $29 per M.
The dimensions 10 ft. by 6 in. by 10 in. are commonly
written 10' x 6" x 10".
Estimate the cost of the following at $28 per M. :
7. 4 sills 4" x 8" x 24' 11. 60 joists 3" x 8" x 20'
8. 6 girders 6" x 10" x 16' 12. 90 studding 2" x 6" x 16'
9. 2 posts 6" x 9" x 10' 13. 90 planks 2" x 10" x 14'
10. 8 beams 3" x 8" x 20' 14. 60 rafters 2" x 4" x 24'
15. Observe the dimen-
sions of the school build-
ing. What is the height of
the sides of the building ?
16. Find the number of
board feet of siding needed
for the sides and the two
ends of the same height as
the sides, making no allow-
ance for openings.
17. The triangular parts
at the top of the house, in
front and in back, are called gables. Each gable can be
divided by a line through the center of its base into two
right triangles. How many board feet of siding are neces-
sary for the two gables ?
18. Find the cost of painting the siding at 10 cents per
square yard.
S- 28ft.--
■-#
MEASURING WOOD
137
MEASURING WOOD
A pile of wood, of 4-foot sticks, 8 ft. in length and 4 ft. in
height, is called a cord of wood.
4x4x8x1 cu. ft. = 128 cu. ft. = 1 cord of wood.
1. How man) r cords are there in a pile of 4-foot wood,
160 feet long and 4 feet high ?
2. Two men cut several piles of 4-foot wood that measure
in all 640 feet in length and 4 feet in height. How many
cords do they cut and how much do they receive for the Avork
at $5.50 per cord?
Wood is frequently cut for house purposes into short
lengths from 16 inches to 2 feet. The price of such a cord
varies according to the length of the sticks.
The number of cords of short wood in a pile is found by
dividing the number of square feet in one side by 32.
3. At a school building there is a pile of 16-inch wood 80
ft. long and 4 ft. high. Find its cost at $1.50 per cord.
4. Two men cut 4 cords of 2-foot wood each day for 16
days. Find the cost of the cutting at 70 cents per cord.
5. One side of a pile of 2-foot wood contains 400 square
feet. Find the number of cords it contains.
138 PRACTICAL MEASUREMENTS
REVIEW OF PRACTICAL MEASUREMENTS
1. How many tiles 12 in. square will be required to lay
a floor 36 ft. by 15 ft. ?
2. What is the length of a board walk that is 4 ft. 8 in.
wide and contains 1350 sq. ft. ?
3. How many cubic yards of earth must be removed in
digging a cellar 36 ft. long, 26 ft. wide, and 8 ft. deep ?
4. Find the cost of covering the floor of a hall 45 ft. long
and 30 ft. wide with matting, a yard wide, at 70 cents a yard.
5. How many times will the wheel of an engine 9 ft. in
circumference turn in going 3000 miles ?
6. Find the cost of 30 boards 16 ft. long, 12 in. wide,
and 1 in. thick, at 5 ^ a board foot.
7. At $.80 a. bushel what is the value of a bin of wheat
16 ft. long, 8 ft. wide, and 4 ft. deep ?
8. What is the number of gallons in a tank 12 ft. long,
10 ft. wide, and 8 ft. deep ?
9. How much will it cost to cement the floor of a cellar
50 ft. long and 28 ft. wide at $1.08 a square yard?
10. At 7^ a square yard, how much will it cost to paint the
four sides of a building 50 ft. long, 20 ft. wide, and 15 ft.
high ?
11. My farm is in the form of a rectangle, and contains 40
acres. What is its width, if its length is 128 rods ?
12. What will be the cost of plastering the ceiling of a
room 22 ft. by 18 ft. at 11 ^ a square yard ?
13. A rectangular field contains 5 acres. If its length is
80 rods, what is its width ?
REVIEW OF PRACTICAL MEASUREMENTS 139
14. How many cakes of soap 4 in. by 3 in. by 2 in.
can be packed in a box whose inside dimensions are 2 ft.,
3 ft., and 4 ft. ?
15. Find the cost of digging a cellar 42 ft. long, 30 ft.
wide, and 6 ft. 3 in. deep, at 40 cents a cubic yard.
16. How much flooring 1 inch thick will be required to
lay the first floor of a house 22 ft. by 36 ft., no allowance
being made for waste, and how much will it cost at $30
per M. ?
17. The length of a field is 80 rods, and its width is 30
rods. How many acres are there in the field ?
18. What is the number of bushels in a bin 20 ft. long,
16 ft. wide, and 8 ft. deep ?
19. A tank 9 ft. square and 8 ft. deep contains how many
gallons ?
20. A building lot 100 foot front contains 15,000 sq. ft.
What is its depth ?
21. A baseball ground 160 yd. by 170 yd. has a tight
board fence around it 8 ft. high. How much will the paint-
ing of the outside of the fence cost at 5| cents a square yard ?
22. The area of a right triangle is 560 sq. ft., and its alti-
tude is 28 ft. What is the base of the triangle ?
23. How much will it cost to excavate a street 800 ft. long
and 50 ft. wide, to a depth of 18 in., at 36 cents a load ?
24. A plot of ground in the form of a square is 100 ft.
on each side. A straight walk 8 ft. wide divides it into 2
equal parts — a lawn for flowers and a garden for vege-
tables. In the lawn there is a flower bed 5 ft. by 8 ft.
Draw the plot.
140 PRACTICAL MEASUREMENTS
25. Find the perimeter of the plot ; of the lawn ; of the
garden ; of the flower bed ; of the walk.
Find the area in square yards :
26. Of the plot. 28. Of the flower bed.
27. Of the lawn. 29. Of the walk.
30. How much will it cost to fence the plot at $3f per
rod ?
31. How much will it cost to pave the walk at 11.55 per
square yard ?
32. How much will it cost to spade the flower bed at 5
cents per square yard ?
33. How much will it cost to sod the lawn, excluding the
flower bed, at $0.25 per square yard ?
34. A board 16 ft. long contains 9 sq. ft. Find its width.
35. A room is 20 ft. long, 16 ft. wide, and 10 ft. high.
How much will it cost to plaster the walls and ceiling at 20
cents a square yard ?
36. How many gallons of water are there in a tank 12 ft.
long, 8 ft. wide, and 6 ft. deep, if it is half full ?
37. Find the cost of 40 boards, each 14 ft. long, 18 in.
wide, and 1 in. thick, at 8 20 per M.
38. A city 5 miles long and 3 miles wide is equal in area
to how many farms of 160 acres each ?
39. How many sods 16 in. square will be required to turf
a lawn 106 ft. 8 in. long and 50 ft. wide ?
40. What will be the cost of painting the outside of a
house 48 ft. long, 30 ft. wide, and 20 ft. high, at 18 cents a
square yard ?
PERCENTAGE
Per cent means by the hundred or hundredths. The sign
for it is % .
We may express the per cent of a number either as a
common fraction or a decimal.
Thus, 6% = r fcj = -06 ; 6% of 500 means ^ of 500, which equals 30;
or, expressed decimally, .06 of 500 = 30.
2% of a number mean's T § ff , or .02, of the number.
25'/o of a number means r 2 ^,or .25, oi the number.
1. What term in common fractions corresponds to the
number before the sign % ? to the sign % ?
2. What expresses the numerator and what indicates the
denominator of the fractions represented by the following :
1%? 20%? 40%?
90%?
6%? 30%? 75%?
100 % ?
3. Find 6 % of 100.
jfo of 100 = 6 ; or .06 x 100 =
= 6.
4. 5% of 100
5. .05 of 100
6. T ^ of 100
7. 6 % of 150
8. .06 of 150
9. .10 of 100
10.
10 % of 100
16.
8% of 75
11.
25 % of 100
17.
.08 of 75
12.
.25 of 400
18.
i!o of 75
13.
3 % of 60
19.
331 of 300
14.
.03 of 60
20.
S3 1
ioo " f 300
15.
t§o of 60
21.
33$% of 300
111
142 PERCENTAGE
Changing per cents to equivalents.
Since 5% = .05 = T ^ = 2V these expressions may be called
equivalents.
l. Give the fractional and decimal equivalents for 10 % ;
6%; 4%; 20%; 25%.
Read the following equivalents :
2.^,20%, .20,1 s |7±, 87Wf .3TJ, I
3. 1Q0 , 12|%, .12J, J 6> ^ 80% , .80, I
4. ^, 40%, .40, f 7 . 87i 87 i % , , 87i , 1
8. Change the fractions -|, §, |, | to their equivalent
decimals and per cents ; also ^, |, |, |.
i= 5)1.00
.20, or 20%
I = 2 x .20 = .40, or 40%
| = 3 x .20 = .60, or 60%
| = 4 x .20 = .80, or 80%
*.=
! =
5 _
j —
7 _
j —
8)1.00
.12.1, or
3 x .12V = .
5 x .12* = .
7 x .12| = .
37£, or 37*%
62|, or 62J%
87*, or 87J%
Change to their equivalent decimals and per
cents :
9 l
v. 2
13. f
17.
3
10
21. |
25. f
10. £
14. i
18.
a
22. |
26. f
11. §
15. f
19.
■A
23. i
27. |
12. 1
16 - tV
20.
i
24. $
• 28 ' 16
Give the products rapidly:
29. 2x.33£
32. 5x
.121
35.
4x.l2i
38. 4x.04l
30. 5x.l6|
33. 7x
.12|
36.
6x.l2£
39. 3x.l6|
31. 3x.l2i
34, 3x
.81
37.
2x.l5
40. 4x.l6|
PERCENTAGE 143
Memorize the following table:
* = 50%
i = 20%
f = 83 : \%
rV = 8i%
i = 33) ; %
f = 40%
1=13*0
A = 41|%
f = 66f%
f = 60%
t = 37|%
xV = 6i%
i=25%
| = 80%
1 = 62-> %
^f = 5%
1 = 75%
* = 16f%
| = 87.V%
2^ = 4%
Name rapidly the fractional equivalents of the following
per cents :
41. 50% 46. 20% 51. 37|% 56. 90%
42. 331% 47. 40% 52. 621% 57. 12.1%
43. 66|% 48. 60% 53. 87 1 % 58. 16f%
44. 25% 49. 16% 54. 10% 59.
45. 75% 50. 83|% 55. 30% 60. 70%
Write the equivalents of the following in decimals, thus
1% = .01; 32% = . 32; |% = .00|; etc.
61.
1%
67.
1%
73.
50%
79.
13%
62.
32%
68.
3%
74.
1%
80.
131%
63.
1%
69.
11%
75.
6%
81.
1%
64.
2%
70.
\%
76.
\°Io
82.
100%
65.
16£%
71.
4%
77.
7%
83.
12S%
66.
«
72.
43£%
78.
\%
84.
127%
144 PERCENTAGE
Finding a given per cent of a number.
Recite the following thus : Look at " 66f%," think "§":
l. 66f % of 18.
12.
37|% of $7200.
2. 331% of 90.
13
121% of $6400.
3. 50% of $500.
14.
75% of $4800.
4. 25% of $2000.
15.
66|% of $999.
5. 75% of 16 inches.
16.
80% of 60 sheep.
6. 20% of 100 yards.
17.
60% of 75 horses.
7. 40% of 60 feet.
18.
40% of 90 miles.
8. 60% of 40 miles.
19.
871% of $160.
9. 80% of 75 gallons.
20.
621% of $240.
10. 16f% of $6000.
21.
37|% of $880.
11. 831% f) f $1200.
22.
121% f 24.
Written Work
1. A man had 100 cows
many did he sell?
25% =.25
and sold 25 % of them. How
100 cows
• 25 As 25% =
500 result is 25,
200
.25 we multiply 100 by .25. The
the number sold.
25.00 number sold
Find results decimally:
2. 50% of 750 4. 40% of 8.75 6. 32% of 1000
3. 25 % of 85.5 5. 28 % of 840 7. 75 % of 980
8. John earns $21.60 per month, and spends 75% for
clothes. How much do his clothes cost him ?
PERCENTAGE 145
9. There are 780 pupils in school, and 40% are males.
How many are males ? ■
10. If a man buys a horse for $150 and sells it at a
profit of 20 %, how much does he gain ?
In each of the preceding problems we have two terms, a per cent
and a whole or a mixed number. The per cent in each problem is the
multiplier, and is called the rate. The whole or the mixed number is the
multiplicand, and is called the base. The product is called the percentage.
The base is that number of which some per cent is to be
taken ; as, 5 % of $ 200 (base}.
The rate is the number of hundredths taken ; as, 5%
(rate) of 80 horses ; that is, ^^ of 80 horses.
The percentage of a number is the result obtained by tak-
ing any per cent of it ; as 10 % of 200 acres is ^q of 200
acres, or 20 acres (percentage).
11. What is 75% of 85.12?
Multiplier Multiplicand Product
Rate Base Percentage
75% of $5.12 = ( )
Decimal Method Study of Problem
75 % = .75. What is the base ? $5.12. What is the
$5.12 = base rate? 75 %-
-r To what do the base and rate correspond
— - — — in simple multiplication? Multiplicand
^" uv and multiplier.
3o84 To what does percentage correspond in
£3.8400 = percentage simple multiplication? Product.
How is the product found in simple
Fractional Method multiplication? Multiplier y. multiplicand.
75 % = f How is the percentage found? Rate X
| of $5.12 = ¥3.84 base '
The percentage of a number equals the product of the base
by the rate.
HAM. COMPL. A KITH. — 10
146 PERCENTAGE
Find :
12. 6 % of 1200. 17. 8 % of 400.
13. 33| % of 6 months. 18. 7 % of 400 horses.
14. 60 % of 30 days. 19. 3^ % of 99.
15. 1 % of 100 acres. 20. 6 % of 150 lb.
16. 5 % of 100 acres. 21. 1| % of $75.
22. 80 is the base, 25 % is the rate, find the percentage.
23. A house costs $2500, and the damage by fire is 8%.
Find the amount of the damage.
24. John owes his tailor $80, and pays 37 1 % of the debt.
How much does he still owe ?
25. Mary spells 90 % of 80 words correctly. How many
does she miss ?
26. A boy buys apples at $1 per bushel, and sells them at
a profit of 20%. How much profit is that per bushel ?
27. G| % of 3680 equals what number ?
28. A man bu} T s a farm for $2500, and sells it for 25 %
more than it cost him. For how much does he sell the farm ?
29. A man earns $180 per month, and puts 33^ % of it in
the savings bank. What is his deposit each month ?
30. If 37| % of a man's farm is in timber, and the total
area is 240 acres, how much timber land has he ?
31. A teacher who earned $1200 a year spent 66| % of her
salary. How much did she save ?
32. Mr. Scott's horse is valued at $250 and Mr. Hill's
at 60% of this. What is the value of Mr. Hill's horse ?
33. The population of a town of 9672 inhabitants increased
12-i-% [ n a vear# What was the increase in population?
COMMISSION 147
COMMISSION
An agent is a person who transacts business for another.
Commission is the sum charged by an agent or commission
merchant for his services.
The net proceeds is the sum left after the commission and
other expenses have been paid.
l. A real estate agent sold a house for 15000, retaining
5% of this sum for his services. How much did he receive?
How much did the owner receive ?
$5000 = selling price.
.05 — rate charged by the agent.
$250.00 = amount charged by the agent.
$5000 - $250 = $4750, amount received by the owner.
A commission merchant made the following sales. Find
his commission for each day at 5 %.
2. Monday, $1800 5. Thursday, 11400.80
3. Tuesday, $1594 6. Friday, $1528
4. Wednesday, $1954 7. Saturday, $2370.60
8. Find his total commission for the week.
9. A real estate agent sells a house and lot for $6750,
charging 2% commission. Find his commission and the
net proceeds.
10. A traveling salesman sold $50,000 worth of goods in
a year at a commission of 8%. If his expenses for the year
were $2200, how much had he left?
11. An agent rents 12 houses at $40 per month. If he
receives 5% for collecting the rents, how much is remitted
to the owners each month ?
148
COMMERCIAL DISCOUNT
COMMERCIAL DISCOUNT
Wholesale merchants and manufacturers usually publish
printed price lists of their goods. The prices in these lists
are higher than the wholesale prices and are subject to
deductions called trade discounts or commercial discounts.
Note. — A discount is any deduction from a fixed price.
Sometimes several discounts are allowed. The first is a
discount from the list price; the second, a discount from the
remainder, etc.
The net price is the price less all trade discounts.
Find the selling price of goods marked :
1.
$15, less 20%.
7.
$40, less 60%.
2.
$20, less 40%.
8.
$48, less 25%.
3.
$6, less 50%.
9.
$6.80, less 25%.
4.
$25, less 20%.
10.
$4.50, less 331%.
5.
$7.50, less 20%.
11.
$9.60, less 16§ %.
6.
$12.50, less 40%.
12.
$4.80, less 371%.
Written Work
Find the selling price of goods marked :
1. $168.75, less 25%.
2. $1374, less 16|%.
3. $1872, less 331%.
4. $278.40, less 371%.
5. $3030 less 40%.
Find the cost of :
Discount, 20%
11. 60 readers @ $ .40
150 geographies @ $ 1
78 grammars @ $ .60
6. $225.65, less 20%.
7. $875.50, less 30%.
8. $278.90, less 10%.
9. $2378.50, less 4%.
10. $6775.20, less 5%.
Discount, 4 %
12. 160 lb. rice @ $.06
300 lb. sugar @ $.041
200 lb. coffee @ $.16
COMMERCIAL DISCOUNT 149
13. Find the net price of a bill of goods for $ 75.40, trade
discounts 20%, 10%.
List price, $ 75.40
1 ess °0 °! 15 08 Observe that the second discount is
,,. . , — Ql . ., , reckoned on the first remainder. As there
tirst remainder, 00. o'l . ,. ,,
are only two discounts, the second reinam-
Less 10 %, 6.03 der is the net price.
Net price, $51.29
Find the net price of articles listed at :
14. $100, less 20%, 10%. 17. $ 10.75, less 40%, 5%.
15. 375.50, less 25 %, 5 %. 18. $ 6.80, less 25 %, 10 %.
16. 290.80, less 40%, 10%. 19. 112.75, less 331%, 10%.
Find the net price of the following bills of goods :
20. 36 dozen boys' caps @ $ 6, discounts 25 %, 20 %.
21. 50 buggies @ $ 120, discounts 20%, 15%.
22. 75 sets harness @ $ 40, discounts 30%, 10 %.
23. 25 grain drills @ $ 95, discounts 40 %, 5%.
24. 12 rubber hose, each 50 feet long at 15^ per foot,
discounts 30%, 15%.
25. Mr. Austin buys a wagon listed at $95, less 20% .
15%. Find the amount paid for the wagon.
26. A merchant buys 12 stoves listed at $45, less 40%,
10 %. Find the net amount of the bill. Compare this with
the net amount of the bill with only one discount of 50 %.
27. A hotel keeper buys 675 yards of carpet at $1.25,
less 20 %, 5 %. Find the cost of the carpet.
28. Compare the net price of an article listed at $500,
discounts of 20 %, 10 %, with the net price of a similar article
listed at $500, discounts of 10 %. 20 %.
150 COMMERCIAL DISCOUNT
Solve according to conditions :
29. The Packard Hardware Co. bought for cash from
Jas. M. Armstrong Co., Chicago, 111., 4 doz. Acme lawn
mowers @ $30 a dozen, 50 lb. lawn seed @ 15^ a pound, 2^
doz. brushes @ 40^ a dozen. Trade discounts: 20%, 10%.
Terms: 30 days net; 2% cash in 10 days.
Cost of bill of goods = ,1128.50.
■1128.50 less trade discount of 20% = 1102.80; $102.80 less trade dis-
count of 10% = f92.52, net price of bill if paid in 30 days. If the bill is
paid within 10 days from date of purchase, the buyer gets a further dis-
count of 2%. This is called a cash discount. $92.52, less 2% for cash
within 10 days, = $90.68.
30. Jamison and Redmond, South Bend, Ind., bought
for cash from the Acme Buggy Co., Cincinnati, O., 72
buggies @ $105, 50 sets harness @ $45, 15 sleighs @ $60,
40 robes @ $20. Trade discounts: 30%, 15%. Terms:
30 days net ; 3 % cash in 10 days.
31. James Cubbison, Greenville, O., buys for cash from
Arbuthnot, Stevenson & Co., Pittsburg, Pa., 5 doz. hand-
kerchiefs @ $3.60; 5 bolts muslin, 40 yd. each, @ 8^; 5
bolts prints, 42 yd., @ 7 ^. Trade discount : 33^%. Terms:
30 days net ; 2 % cash in 10 days.
32. S. H. Gardner Co., piano dealers, Detroit, Mich.,
order from the Harmonic Piano Co., Chicago, 111., 2 Har-
monic pianos #266 @ $600, less 40%, 10% trade discount.
Terms : 90 days net ; 10 % off 10 days. Find the cash price.
Find the net price if paid in 30 days.
Notk. — The sign #, when placed before a number, is read " number."
33. M. L. Smith, tailor, Brockton, Mass., orders from
Bender & Co., New York, importers, 3 pieces suiting, 22 yd.
each, @ $3.15. Terms: 30 days net; 2% off 10 days.
Find the net amount of bill if paid within 10 days.
INTEREST
1. Mr. Johnston pays the liveryman $6 for the use of a
horse and buggy for two days. What does he get in ex-
change for the $6?
2. Mr. Daniels pays $6 for the right to pasture his cow
in a field for two months. What does he get in exchange
for the $6?
3. Mr. Watson pays $6 for the use of #100 for one year.
What does he get in exchange for the $6 ?
4. In the first two examples money is paid for the use of
something that is not money. For what does Mr. Watson
pay the money in the last example ?
Interest is money paid for the use of money. Interest
corresponds to the percentage in percentage.
5. How much does Mr. Watson pay for the use of the
money? What is the $6 called ?
6. On what is the interest reckoned ? The $100 is called
the principal.
The principal is the sum on which the interest is paid.
The principal corresponds to the base in percentage.
The rate of interest is a certain number of hundredths of
the principal paid for the use of the principal for one year.
Time is always a factor in interest. Interest, then, is the
product of three factors: principal, rate, and time.
The amount is the sum of the principal and the interest,
151
152 INTEREST
Interest for Years and Months
1. What part of a year are 6 months ? 4 months ? 3
months ? 2 months ? 1 month ?
2. If the interest for a year is $100, what should it be for
6 months ? for 4 months ? for 3 months ? for 2 months ? for
1 month?
Written Work
1. What is the interest on -1200 for 2| years at 6 % ?
$200 principal . .
11 1 he interest for 1 year is .06
of the principal, or .$12. The in-
il2.00 interest for one year terest for 2\ years is 2\ x $12, or
21 $30.
$30.00 interest for 2| years
Multiply the 'principal by the rate and the product by the
number of years.
The year is usually considered as 360 days, that is, 12 months of 30
days each.
Find the interest on :
2. $300 at 5% for 1 year. 4. $150 at 6.} % for 3 years.
3. $800 at 8 % for 2 years. 5. $700 at 4| % for 4 years.
Find the interest of :
6. $250 fori | years at 4%. 11. $500 for 21 years at 4| %.
7. $75 for 2 years at 8 % . 12. $960 for 9 mo. at 6 %.
8. $100 for 3f years at 7%. 13. $900 for 2f years at 7 %.
9. $ 80 for 41 years at 5 % . 14. $654 for f year at 6 % .
10. $40 for 21 years at 6-1 % . 15. $ 220 for J year at 8 % .
INTEREST FOR YEARS, MONTHS, AND DAYS 153
Find the interest at 6 % on :
16. $100 for 6 months. 19. 1624 for 120 da.
17. $500 for 4 months. 20. $170 for 8 mo.
18. $150 for 2 yr. 2 mo. 21. $355 for 130 da.
Interest for Years, Months, and Days
1. What part of a month (30 days) are 15 days? 12 days ?
20 days ? 3 days ? what part is 1 day ?
2. If the interest for 1 year is $360, what is the interest
for 1 month ? If the interest for 1 month is $ 30, what is the
interest for 1 day ? for 15 days ? for 12 days ?
Written Work
l. Find the amount of $ 200 at 6 % interest for 2 yr.
7 mo. 12 da.
Principal = $200
Rate = .06
Int. for 1 yr. =$12.00
Int. for 2 yr. = 2 x $12.00, or $24.00
Int. for 7 mo. = T 7 5 of % 12.00, or 7.00
Int. for 12 da. = j$, or f, of $1.00. or .40
Int. for 2 yr. 7 mo. 12 da. = $31.40
Principal = $200.00
Amount for 2 yr. 7 mo. 12 da. = $231.40
Study of Problem
a. What is the first step in the work ? the second step?
b. How do we find the interest for 1 month? for 7 months? for 12
days?
c. What new term is introduced in interest? For what length of
time is rate of interest always considered?
151 INTEREST
Find the interest and amount of :
2. 1300 for 3 yr. 6 mo. at 6%.
3. 1 250 for 2 yr. 4 mo. at 7 %.
4. $ 160 for 1 yr. 3 mo. at 5 %.
5. $ 50 for 1 yr. 8 mo. at 5 % .
6. $ 800 for 3 yr. 2 mo. at 6 %.
7. $50.80 for 9 mo. at 10%.
8. $16 for 8 mo. at 6%.
9. $75 for 8 mo. at 6 %.
10. 1420 for 10 mo. at 10 %.
11. $40.50 for 1 yr. 1 mo. at 6 %.
12. $ 300.40 for 5 mo. at 7 %.
13. $ 100 for 7 mo. at 7 ft, .
14. $500 for 11 mo. at 6 %.
15. $1000 for 1 mo. at 6%.
16. $60.(30 for 8 mo. at 8%.
Find the interest and amount of :
17. $250 at 8 % for 3 yr. 5 mo. 20 da.
18. $75.80 at 5 % for 4 yr. 1 mo. 16 da.
19. $ 1500 at 6 % for 2 yr. 9 mo. 15 da.
20. $125.50 at 4 % f or 4 yr. 11 mo. 12 da.
21. $ 1140 at 5| % for 4 yr. 8 mo. 24 da.
22. $ 912.60 at 5 % for 2 yr. 10 mo. 11 da.
23. $ 3209 at 6 % for 3 yr. 7 mo. 21 da.
24. $634.50 at 8 % for 11 mo. 12 da.
25. Henry Boydson borrows $ 275 Sept. 1, 1906, at 6 %
interest, and settles the note Jan. 1, 1908. Find the amount
of the note at settlement.
REVIEW OF PERCENTAGE AND INTEREST
1. A boy has $30 in a savings bank and deposits a sum
equal to 10% of it. What is the total amount he has in
bank?
2. Mr. James's salary is $1200 per year and he saves
33^% of it. How much does he spend?
3. A boy spends $8 for an overcoat and 37.] % of that
sum for shoes. How much does he spend for shoes?
4. In a school of 45 pupils, 33^ % of the pupils are boys.
What is the number of girls ?
5. Find '2;')% of .05; of .5; of 5.5; of .25.
6. John earns $50 during his vacation, and Margaret
25% as much as John. How much does Margaret earn?
7. A farmer sold a horse that cost him $ 80 at a loss of
20%. Find the selling price.
8. What is the interest on $150 for 2] years at »!% ?
9. Find the interest on $100 for 00 days at 5%.
10. My father borrows $75 from his neighbor and promises
to pay it in 4 months at 0%. Find the amount my father
must pay at the end of four months.
11. A huckster buys eggs at $.20 per dozen. For how
much per dozen must he sell them to gain 20 % ?
12. If I borrow $50 from Mr. James for 6 months at 0%,
how much interest must I pay him?
155
156 REVIEW OF PERCENTAGE AND INTEREST
13. A grocer sold flour last week at $1.20 per sack and
this week at 10 % advance on last week's selling price. Find
the price of flour per sack this week.
14. A huckster buys 150 dozen eggs at $.20 per dozen and
sells them to a merchant at a gain of 25 %. The merchant
sells them at a gain of 20 %. How much does the merchant
receive for the eggs ?
15. If I buy cloth at $1.50 a yard, for how much must I
sell it to gain 33£ % '■
16. A grocer buys goods to the amount of $1200, 10 % off
for cash. He sells them for $1500 cash. How much does
he gain ?
17. A mother took two boys and a girl to a store to buy
clothes. The first boy's suit cost $10 less 10% for cash.
The second boy's suit cost 66| % of the cash price of the first
boy's suit. The girl's coat cost 331% f the money paid
for both boys' suits. How much did the mother pay for the
children's clothes ?
18. Find 25 % of 200, and divide the result by .00J.
19. A man buys a house and lot for $5000. It costs
every year $25 for repairs and $50 for taxes and insurance.
He rents the house for 8% of its cost. How much has he
left after paying expenses ?
20. A coal dealer bought 300 tons of coal for $600. The
freight, storage, and delivery cost 331 cj f the cost of the
coal. What was the retail price per ton if he sold it at a
gain of 12^% ?
21. A real estate agent purchased a house for $1250. For
how much per month must he rent the house to make 6 %
after paying each year $18 for taxes and insurance and $15
for repairs ?
RECEIPTS AND CHECKS
John Watson pays James Adams $ 35.50 for work for
one month, and asks Mr. Adams for a receipt. Write the
receipt to show that the money was paid by Mr. Watson
and received by Mr. Adams.
$ Rochester, jY. Y., fwyva /, 1907.
Eeceifceli from
Dollars
for
1. What must every receipt show?
2. Write the receipt your grocer would give you in pay-
ment of $18.50 on account by your father or mother.
3. Your school district pays the National Book Company,
New York, $ 25.75 for school books on Sept. 15, 1907. Make
out the receipt of the National Book Company.
4. Henry Smith received I 3.65 from James Brown for 3
months' water rent. Make out a receipt for the amount.
5. Ralph Taylor pays II. W. Henderson |5 for a month's
tuition. Write the receipt Ralph Taylor should receive.
157
158 RECEIPTS AND CHECKS
6. Write a receipt for $ 75 which Nelson Page paid Edgar
Poe for balance due on a buggy.
7. Make out and receipt the bill for the following articles
bought by James Thomas from Jos. Home & Co. :
3 shirts @$1.75 2 neckties @ $.75
6 collars @ .20 4 pairs cuffs @ .20
8. Presuming that you are a collector for the Gazette-
Times, Pittsburg, Pa., make out a receipt to a subscriber who
has paid you $ 2.60 in full of account.
A check is an order on a bank where a person keeps a
deposit, ordering the bank to pay money.
Stub Check
c/t*. 875 J No. 875
I Seattle, Wash., fan. fO, 1907
oo &\>t gufcon Rational Bank of Seattle.
PAY TO THE
i Order of- fame*, lO-avci
fan. /O, '07
/ &vxiu-4,&v-&K—~~~~~lJoUars
' 100
3ov jCa-6-cyv i Z0-. $. ?)1oov&.
1. Name the different things stated in this check.
2. Observe that this check is payable to the order of James^
Ward. He orders it paid by writing his name across the
back of it. This is called indorsing the check.
3. Write the check your father would give your teacher
in payment of $3.50 for September tuition.
4. Emil Smith borrows from Joseph McLean $240 to
attend school and pays the same in 2 yr. 4 mo. 18 da. at 0%.
Write the amount of the check that would pay the note.
GENERAL REVIEW
1. The remainder is 92,568 and the minuend is 202,660.
Find the subtrahend.
2. The dividend is 364,450 and the quotient is 9850.
What is the divisor ?
3. Add 3.5, .035, 45.006, and 2.06.
4. Write decimally twenty-five and sixty-one thousandths;
one hundred twenty-five and five tenths ; and three hundred
and two ten-thousandths.
5. What number multiplied by one hundred seventy-
nine is equal to 848,818 ?
6. From 2.0011 take 1.9892.
7. Explain the difference between \°J and \.
8. Add 1 2f, 1 ||, and 4f .
9. Find 1%, -i-, 1%, fa £>%, 50%, and ^ of 100.
10. The multiplicand is 1325 and the multiplier is .0416.
What is the product ?
11. If 38 dozen eggs cost $11.40, what is the cost per
dozen ?
12. A building is 46 ft. 3 in. wide, and twice as long as
wide. Find the distance around the building.
13. From 86 miles and 3 inches, take 46 miles and 8
inches.
159
160 GENERAL REVIEW
14. A man and his son together earn $72 per month. If
the man's earnings in 6 months amount to $300, how much
are the son's earnings in the same length of time ?
15. A man bought 48 head of cattle, at $36 per head, and
sold them at a gain of 25%. What was the total amount
received for the cattle ?
16. Find the interest on $370.50 for 4 yr. 8 mo. at 6%.
17. Divide 48| by 21f
18. Divide .65 by 6.5.
19. Reduce 187^ rods to the fraction of a mile.
20. How much will it cost to ship a car load of wheat
containing 42,000 lb. from Fargo, N.D., to Chicago, 111., if
the freight rate is $.06 per bushel ? (60 lb. = 1 bu.)
21. A train leaves Chicago at 8:15 a.m., and arrives at
Pittsburg at 8:20 p.m. The distance is 468 miles. Find
the number of miles per hour the train travels.
22. The steel rails on the Bessemer railroad weigh 100
pounds to the yard. Find the number of tons necessary to
lay 5 rods of single track.
23. How much does an architect receive, at 4* %, for the
plans of a house that cost $8350?
24. A man's salary is $150 per month. He spends 40 % of
it for clothing and other expenses. How much does he save
in a year ?
25. A man purchases 80 acres of land for $6400, and sells
them at 25% gain. How much does he receive per acre?
26. Frank Stewart borrows $250 Sept. 15, 1906, at 6%
interest. Find the amount of the note if paid March 15, 1908.
27. Find the area in acres of a street 7 miles long and
66 feet wide.
GENERAL REVIEW 161
28. A town lot is 43 ft. 3 in. wide and 120 ft. deep.
How much is it worth at 75 $ per square foot?
29. A western farmer harvests 8960 bu. wheat from a
field 320 id. long and 160 rd. wide. If he sells the wheat at
60^ per bushel, how much does he realize from each acre?
30. In 1 hour 20 minutes and 40 seconds, a train travels
60 miles. At that rate how long would the train be in
traveling 1200 miles?
31. The average wages of a steel mill employing 3000
men are $2.50 per day. If a 10% reduction in wages
is made, how much per day will the company's pay roll be
reduced?
32. In a certain class the salary of the teacher for a year
is $500. The books and supplies cost $90.65 ; fuel, $40 ,
repairs and other expenses, $75.30. There are 35 pupils in
the class. Find the average cost per pupil for the year.
33. Reduce to improper fractions: 6.25; 3.375; 4.66| ;
and 2.05.
34. If it costs $72 to carpet a room 18 ft. long and 18 ft.
wide, how much will it cost to carpet a room 36 ft. long
and 36 ft. wide, with the same quality of carpet?
35. Mt. Rainier is 14,363 ft. high. Reduce the height to
miles and the fractions of a mile.
36. How many cubic inches are there in a bin 9 ft. 7 in.
long by 8 ft. 3 in. wide and 4 ft. 9 in. deep? how many
cubic feet?
37. A grocer bought 225 bu. apples at $.50 per bushel.
He sold 150 bu. at $.75 per bushel. The remainder, which
were damaged, he sold at $ .40 per bushel. Did he gain or
lose and what per cent?
HAM. COMPL. ARITI1. 11
102 GENERAL REVIEW
38. What is 331% of 24? of 84.80? of 862.50?
39. A piece of land 30 rods wide and 480 rods long was
sold at 862.50 per acre. Find the amount of the sale.
40. Time, 3 months; rate of interest, 5 % ; money borrowed,
8100. Find amount to be paid.
41. If f of a bushel of potatoes cost 8.40, how much will
1\ bu. cost?
42. A piece of land 40 rods long in the form of a rec-
tangle contains 5 acres. Find its width in rods.
43. A farmer sold 12| acres of land at 855| per acre.
How much did he receive for the land?
44. Houser Brothers sold the following bill of goods to
William Pool :
12 lb. sugar
10 cans tomatoes
@ 8.061
@8.15
6 lb. rice
11 lb. prunes
2 pair boots
1 overcoat
1 pair shoes
@ I.07J
@ 8.071
@ 83.50
@ 813.50
@ 84.00
Pool at the same
time sold Houser Brotl
85 bu. potatoes
50 bu. corn
16 lb. butter
@8.65
@ 8.421
@$.24
10 lb. butter
@ 8.28
Houser Brothers gave Mr. Pool the balance in cash. Make
out the account.
45. A painter worked 17^ days. After spending |- of
his wages for board he had 815 left. Find his daily wages.
GENERAL REVIEW 163
46. I owe -Frank Morrison, the grocer, $32.50 and pay him
$23.75. Write the receipt that Mr. Morrison should give
me.
Note. — When a debt is not paid in full, the receipt should read " On
account."
47. A cellar 24 ft. by 32 ft. is to be excavated to an
average depth of 5| ft. Find the number of cubic yards to
be removed.
48. Express 22| yards as rods, feet, and inches.
49. The width of a rectangle is 20 rods and the area
is 560 square rods. Find the length.
50. What is the difference between a square and a rec-
tangle ?
51. Give the rule for rinding percentage. On what is gain
or loss always reckoned ?
52. A man's farm and personal property cost $5600. The
first year he cleared 12| % of the money invested. The sec-
ond year, on account of floods, he lost 5 % of the cost of the
property. How much was his gain in the two years?
53. Of a bill of $155 sent to a collector, 80% was col-
lected and the collector retained $12.40. What per cent did
he charge for collecting?
54. A boy receives $1.20 per day and a man $2.50 per
day. How long will it take the boy to earn as much as the
man can earn in 30 days?
55. The perimeter of a rectangle is 72 rods. The width
is 12 rods. Find the length.
56. Estimating that 300 cu. ft. of air is required for each
pupil, how many pupils, including the teacher, should occupy
a room 40 ft, long, 30 ft, wide, and 12 ft. high?
164 GENERAL REVIEW
57. Divide one thousand and one thousandth by one
and one thousandth.
58. What is the interest on $375 for 270 days at 6 % ?
59. Reduce f of a mile to lower denominations.
60. A boy deposited half of his money in the savings
bank; ^ of the remainder he spent for clothes; and he had
$3 remaining. How much had he at first ?
61. Reduce .025 cwt. to lower denominations.
62. A man bought two city lots costing him -13500 and
$ 4100 respectively. He sold them at a gain of 25 %. What
was the gain in dollars ?
63. How many gallons of water will a tank contain that
is 11 ft. long, 3^ ft. wide, and 4 ft. deep ?
64. A barn floor is 20 ft. wide and 45 ft. long. How
much will it cost to cover it with plank 2 inches thick at 1 20
per thousand board feet ?
65. Divide j by f of f .
66. John and James have together 165 acres of land, but
James has twice as many acres as John. How many acres
has each ?
Suggestion. — 165 acres = twice John's + once John's or 3 times
John's.
67. What fractional part of a day are 10 hours, 50 min-
utes, 40 seconds?
68. Divide nine ten-thousandths by one hundred twenty-
five thousandths.
69. What is the interest on $ 180 for 4 yr. 8 mo. at 5f % ?
70. I made $1.95 by selling 15 dozen eggs at $.31 per
dozen. What was the cost of the eggs per dozen ?
GENERAL REVIEW 165
71. Find the net proceeds from the sale of 145 books, at
|2 each, on which a commission of 33^% is paid.
72. A father divided his farm of 202 A. 16 sq. rd. equally
among his four sons. How many acres did each receive ?
73. .21 of a mile is equal to how many feet?
74. An automobile that cost $2675 was sold at a loss of
28 %. For how much was it sold ?
75. What is the cost of 18 planks 20 ft. long, 12 in. wide,
and 2 in. thick, at 120 per M? ,
76. \ of 7 is what part of 9 ?
77. f of a farm is worth $7500. What is 20% of the
farm worth ?
78. What is the cost of a car load of bituminous coal weigh-
ing 84,000 pounds at $2.65 per ton ?
79. A farm in the form of a rectangle containing 120 acres
is 60 rods wide. How long is it ?
80. Express decimally the quotient of £ ■*- .35.
81. If | of a ton of hay is worth $12, how much are
33,000 pounds of hay worth ?
82. What is the value of a pile of 4-foot wood 48 ft. long
and 6 ft. high, at $4.50 per cord ?
83. A dairyman owns a cow that averages 3 gal. 2 qt.
1 pt. of milk daily. If he sells the milk at $.06 per quart,
how much will he realize from the cow during the month
of May ?
84. I can buy an automobile at one store for $3000, with
discounts of 25 %, 10 %; or at another store for $3000 with
only one discount of 35%. Which is the cheaper?
PART II — SEVENTH YEAR
BILLS AND ACCOUNTS
RECEIPTS
John Bentz rents a house in Boston, Mass., from James
Smith for one year for $240, rent payable the first day of
each month in advance.
1. Every receipt should state (1) the place and date of payment;
(2) who pays the money ; (3) who receives the money; (4) for what the
money is paid ; (5) the amount both in figures and in writing.
2. Every receipt in full should state in full to date.
Write the receipt given Mr. Bentz for September's rent.
1. Providing John Bentz fails to pay the rent for August
when due, but pays on September 1 the rent for both
August and September, write the proper receipt.
2. Write the receipt for the tuition for the term of your
school that any non-resident pupils would have to pay.
Write the receipt in full to date for each of the follow-
ing bills which I owe :
166
ORDERINCx GOODS 167
3. John Thompson for milk, $6.75.
4. Frank Jones for coal, $16.85.
5. Smith & Co. for books, $3.75.
ORDERING GOODS
These forms of orders should be studied carefully, as they
come into almost daily use in business life.
A
FURNEE AND KENNERDELL, KlTTANN.NG, Pa., fat. W, / <? 7 .
Booksellers.
(X %>vEA,Lean- fSaok, (HvwLJbaAvy,
/OO IM-a^kinato-n ^qioaA&, cftzw- 1fr>ik-
Jbe,av cfvi&:
ffl&a&t Qs/i ifi at &n&& 6-if S > &n / n^ / iflv-ayyvuv ^vt^Ujkt :
300 Rois, §'vlr,i&v&.
"Uoii 1 1 1 1 ;ily,
~j :<sWiz& V* fCzvm&'uLzli.
B
Sh^rCkiirv, i'a., fan. /, /907-
&0 BcKjCfQ, ¥ Buhl,
(luecjh&wif, <Pa.
fCvyidtu a&nd 6-1/ (I dam*, €<xJja,xz-io, th& foltoiv-tna,
4.&V &a'mAl&, and &kai<fb tk& <uvm,e, to- inAf amount.
fO ltd. ot vikkon, at $0.50
/2 ud. ot dve&i- cjoodos, at $/.2<5
(nil*.) &ka*. Ci. dtttte,.
a&
168
BILLS and accounts
1. Make out an order to each of the various schoolbook
companies for the books you are studying.
2. Make out an order to McCreery & Co. of New York,
for some goods to be sent to you C.O.D. (cash on delivery).
RECEIPTED BILLS
Study the following :
New Castle, Pa., ftfay /, /<?07.
TfK. yaAn&Q, &ux,nt,
2V- (HounUf Line, dt.
Bou^bt of JOHN KNOX & SON,
196 E. Washington St.,
FANCY GROCERS.
TERMS: &a^lh
/407.
/
20
25
27
KM. Mt, f/.50
20 16-. £/iM. nutated c/^^a-t, 6}£$
10 6-tc. gotaUe*, 60 1
6 10-. ftonesy, 20$
R&&&lv-uL S'axpn^eml , THaif, 10, 07-
jlo-Artv /Cru>?c If ofon.
Jch W-at^yyv.
//
/
6
/
50
30
00
20
fO
00
Observe :
1. The place and date of sale. 2. The names of the buyer
and the seller. 3. The name, quantity, and price of each arti-
cle. 4. The entire amount of each separate item. 5.
total amount of the bill. 6. The receipt of the bill.
The
RECEIPTED RILLS
it;o
A bill is a written statement in detail of goods sold or of
services rendered.
A bill is receipted when the words "Received payment"
are written at the bottom of the bill, either by the seller or
by some person authorized by him.
Note. — When the person authorized signs the name of the seller, he
should always write on the next line below the word " by " or " per " and
his own name or initials.
When a person purchases anything on time, the purchaser is called a
debtor.
When the seller extends the time of payment to any one, the seller is
said to give credit, and therefore is called a creditor.
Some abbreviations used in business:
Acct. %, account mdse.,
Amt.,
bal.,
Co.,
Cr.,
Dr.,
do. CO,
amount
balance
company
creditor
debtor
the same
No. (#),
paymt.,
pd.,
per,
pc,
rec'd,
merchandise
number
payment
paid
by
piece
received
The symbol # means pounds when placed after a number;
but number if placed before a number.
Thus 6 j means 6 pounds but *6 means Number 6.
Make receipted bills for the following transactions, per-
forming all necessary operations :
l. Carter Bros., Elkins, W. Va., purchase from Bindley
Hardware Co., Pittsburg, Pa., the following: 3 dozen
locks @ $4.80, 67 kegs of nails @ $4.10, 6 dozen lanterns
@$6.25, 1300 feet steel tracks @ 16^, and 7 lawn mowers
@ $4.25.
170 BILLS AND ACCOUNTS
2. John Dunn & Son, Akron, Ohio, bought from Thomas
Townsend & Company, Cleveland, Ohio, 36 barrels of flour
@ $4.80, 4 boxes of prunes @ $1.65, 500 pounds of coffee
@ llf^, 7 boxes of yeast @ 75^, 50 pounds of Huyler's
cocoa @ 32^.
3. James Brown, Lincoln, Neb., purchases from May &
Co., St. Louis, Mo., 22 bunches bananas @ $1.75, 32 boxes
oranges @ $3.15, 17 boxes lemons @ $'2.80, 29 crates cran-
berries @ $2.25, 6 boxes grape fruit @ $2.90, and 35 bbl.
apples @ $2.75.
4. James Sweitzer, Peoria, 111., bought from Swift & Co.,
Chicago, 1587 pounds of dressed beef @ 7-^, 267 pounds of
mutton @ 9^, 933 pounds of pork @ 5|^, and 180 pounds
of lard @ 12£
5. Lyle Bros. & Co., Dubuque, la., bought of the De-
laney-Brown Lumber Co., Grand Rapids, Mich., 28215 feet
oak boards at $32 per M., 147820 feet hemlock at $27 per
M., 92629 feet No. 1 white pine at $60 per M., 63605 feet
poplar boards at $35 per M.
Note. — $32 per M. equals $.032 per board foot.
6. Mrs. James Thorpe bought of B. Altman & Co., New
York, 1 pair gloves at $2.75, 5 yd. ribbon at 39^ a yard,
I dozen handkerchiefs at 25^ each.
ACCOUNTS
In the study of bills we simply found how much the
debtor owed to the seller, or what one party owed to another
for services rendered.
In an account we have a business transaction covering a
period of time in which there is both a debtor's bill and a
debtor's payments.
ACCOUNTS
171
Form of Account
Pittsburg, Pa., TTicwf /, / c /07.
Plv. cfawvu&f- BarucL,
U^/lt.alVna, Z&. Ucl:
Jo JOSEPH HORNE CO., Dr.
3v.
Ofa.
/
&o CLmawnt ieM,deA,&cL
$ffO
29
ti
fO
" 20 yd. ^Ifc @ $f. 50
SO
00
1 1
25
" 2 Ladies' &uvU @ ¥-5.00
qo
00
//-
28
7
80
238
oq
//
/¥■
ISu &<x&lv
fOO
00
//
2V-
6o
00
/60
00
09
78
Note. — If the above balance were paid in full May 1, the words
"Received Payment " would be written.
Joseph Horne Co.,
Per
It is customary for the creditor to send an itemized account
to the debtor. If it is not paid, another form of bill called
a statement is sent and contains only these words : " To
account rendered" or "To Mdse." followed by the amount.
In the above account, what shows that there was a previous
transaction ?
172 BILLS AND ACCOUNTS
Render the following statements:
1. Jan. 31, 1906, the debits and credits of George Weil in
account with John Wanamaker, Philadelphia, Pa., were as
follows :
Debits
Jan. 1, To account rendered, 1295.63,
Jan. 7, To 3 overcoats @ $32,
Jan. 12, To 7 yd. dress goods @ $4.75,
Jan. 20, To 1 suite of furniture, $185.
Credits
Jan. 5, By cash, $250,
Jan. 20, By note for 30 days, $200.
Find balance due Wanamaker.
2. Nov. 30, 1905, the debits and credits of R. D. McClurg,
with Stevenson & Co., Richmond, Va., were as follows:
Debits
Nov. 1, To account rendered, $86.25,
Nov. 10, To 12 cases corn @ $1.65,
Nov. 19, To 4 bbl. sugar, 1692 lb.. @ 4j£
Nov. 22, To 75 lb. dry beef at 19 £
Credits
Nov. 5, By cash, $85,
Nov. 30, By note for balance due.
3. On Oct. 31, the account of Win. B. Eager with H. A.
Soltori, New York, was as follows :
Debits Credits
Oct. 2, To mdse., $93.37, Oct. 10, By cash, $80,
Oct. 8, To mdse., $107.92, Oct. 20, By 30-day note, $100.
Oct. 21, To mdse., $21.58. Oct. 31, By cash, $25.
LEDGER ACCOUNTS
173
LEDGER ACCOUNTS
The orders or payments when received by a firm are first
put in a day book in the order of their arrival. Each person's
or firm's business is then placed in a book called the ledger,
which is ordinarily balanced each month or when an account
is paid.
A ledger account is headed by the name of the person and
arranged so that the purchases appear on the left side as debits,
and the payments or services rendered appear on the right side
as credits.
The statement of account as given on page 171 is simply a
copy of Mr. Bond's ledger account with Joseph Home Co.
The following form shows a balanced ledger account at the
close of the month with James Roberts, the balance being
brought down to continue the account into the next month.
Dr.
$awve<a/
Rok&itb
Cr.
1907
1907
Feb.
1
Bal. bro't f'w'd
$19
30
Feb.
6
Cash
6*
$30
00
it
2
Lumber
2*
19
80
u
11
Dray age
46
15
75
(i
4
Cement
8
40
50
(I
18
Dray age
60
8
50
u
5
Sand
Lfi
9
50
l(
22
Drayage
70
26
50
(<
9
Tile
30
15
70
11
25
Cash
71
82
50
((
13
Plaster
5ii
56
30
((
28
Balance
38
75
1(
18
Sewer pipe
60
20
90
l(
26
1
Lumber
Bal. bro't f'w'd
75
20
202
00
00
75
202
00
Mch.
38
l. What debts in this ledger account did Mr. Roberts
incur during the month ? What payments did he make ? .
* These numbers refer to the pages in the day honk in which the accounts
are found.
174 BILLS AND ACCOUNTS
2. Find the sum of the debits and the sum of the credits.
Does the difference equal the balance, §38.75?
3. Is the balance in favor of, or against Mr. Roberts ?
4. Had the balance been in favor of Mr. Roberts, on which
side would it have been entered ?
5. How do you determine on which side to enter the
balance ? on which side to bring down the balance ?
To foot a ledger is to add and set down (usually in pencil)
the total debits and credits of the accounts.
Test. — When the footing of one side equals the footing of the other
side, the account is in balance.
Written Work
The day book shows the following sales and receipts.
Make a ledger for the year, enter each item, foot and close
the accounts.
1. William Stone.
Debits. — Feb. 1, cook stove, $ 22 ; Feb. 4, 40 lb. nails,
$2.40; Feb. 5, heater, $75.70 ; Feb. 12, tin roofing, $79.08;
Feb. 19, hardware, $ 14 ; Feb. 23, lime and cement, $50.70;
Feb. 28, tile, $ 22.
Credits. —Feb. 1, sand, $15 ; Feb. 4, drayage, $9.50 ;
Feb. 12, cash, $50 ; Feb. 16, lumber, $74.25; Feb. 25, cash,
$20; Feb. 28, cash, $105.
2. Morris Brown & Co.
Debits. —Apr. 4, mdse., $15.90 ; Apr. 5, lumber, $190.72 ;
Apr. 11, mdse., $23.15; Apr. 12, wagon, $90; Apr. 27,
mdse., $20.70; surrey, $129.70; May 7, mdse., $40.05;
May 12, cash, $ 100 ; May 20,- lumber, $ 189.
Credits. —Apr. 10, labor, $129.71 ; Apr. 14, cash, $ 75 ;
Apr. 21, labor, $29.70 ; Apr. 28, cash, $147 ; May 7, labor,
$270.10; May 19, stone work, $175.39; May 28, cash, $70.
DENOMINATE NUMBERS
We measure the quantity of anything by finding how
many times it contains some unit of the same kind, called
the unit of measure.
Thus, the number of bushels in a load of apples is found by seeing
how many times it contains the unit of measure, 1 bushel.
A denominate number is a concrete number whose unit is
a measure established by custom or law ; as, 5 yards or 8
bushels. In these numbers 1 yard and 1 bushel are the
units of measure.
A simple denominate number is a number of one denomi-
nation ; as, 3 feet, 5 pecks.
A compound denominate number is a number composed of
two or more denominations that express one quantity; as, 8
yards, 2 feet, 3 inches (length).
REDUCTION
. Reduction of denominate numbers is changing their form
without changing their value ; thus,
2 bu. = 8 pk. = 64qt.
or, 16 qt. = 2 pk.= .5 bu.
l. Review thoroughly these tables: Liquid Measures;
Dry Measures ; Measures of Length ; Avoirdupois Weight ;
Troy Weight ; Time Measures ; Stationers Measures ;
Counting.
Note. — The other tables of denominate numbers are found on pages
433 to 436.
175
176 DENOMINATE NUMBERS
2. Change 61 qt. to bushels ; to pints.
3. How many pecks in \ bu. ? 3 bu. ? 1| bu. ?
4. How many days from June 28 to October 1 ?
5. Express 5 yards as feet; as the fraction of a rod.
6. How many rods equal 27| yards ? 49^ yards ?
7. Change $2.50 to mills; to dimes.
8. Change 1.3 T. to pounds.
9. Express 27 pecks as bushels ; as quarts.
10. How many ounces (avoir.) equal | lb. ? 2.5 lb. ? 4|
lb. ? 7 J lb. ?
11. If a ring is 18 carats fine, what part of it is pure gold ?
12. Express in minutes 1.5 hr. ; 1.75 hr. ; 3^ hr.
13. Express in feet 1| rd.; 1\ rd.; 8 yd.; and 5| yd.
14. How many minutes equal 720 seconds ? 3600 seconds'.''
15. How many days is it from January 28, 1908 to March
5, 1908 ?
16. If a man works 8 hours a day, how many minutes does
he work ? How many minutes equal 9 hours ?
17. How many sheets of paper are there in 2 reams of
paper ? in 5 reams ?
18. I bought 2 gross of lead pencils. How many lead
pencils did I buy ?
19. If a horse eats 10 quarts of oats a day, how long will
5 bushels of oats last ?
20. How many dozen equal 180 things? how many gross?
21. How many days is it from Memorial Day (May 30)
to the fourth of July ?
22. What is the perimeter of square that is 3.6 ft. on a
side ?
REDUCTION 177
23. How many pecks does a 3-bushel bag hold ? a 2-
bushel bag ? a 5-bushel box ?
24. A bushel of wheat weighs GO lb. How many bushels
are there in a ton of wheat ?
25. How many feet are there in 125 yards ? in 120
inches ?
26. Change 5 T. 4 cwt. to tons ; to pounds ; to hundred-
weight.
27. Find the weight in ounces of a dozen teaspoons, each
weighing 12 pennyweights.
Changing to smaller denominations.
Written Work
1. Change 7 gal. 3 qt. 1 pt. to pints.
Since there are 8 pt. in 1 gallon, in
7 gal. = 7 X 8 pt. = 56 pt. 7 ga l. there are 7 times 8 pt., or 56 pt.
3 qt. =3x2 pt. = 6 pt. Since there are 2 pt. in 1 qt., in 3 qt.
^ t _ ] pt, there are 3 times 2 pt., or 6 pt.
=^-i — 5 1 — 1 7TTT- — Hence, in 7 gal. 3 qt. 1 pt. there are 56
7 gal. 3 qt. 1 pt. = 03 pt. pt + 6 pt * l pt * or 63 pt
2. A dairyman delivered to his customers in one morning
19 gal. 3 qt. 1 pt. of milk. Find the number of pints delivered.
19, number of gal.
— The product is numerically the same
7" whatever number is the multiplier. Thus,
+ 3 4 X 19 = 19 x 4; hence to shorten the work
79, number of qt. 4 and 2 may be regarded as multipliers,
9 although in the explanation it must be re-
■z-^ membered that 4 qt. and 2 pt. are really
the multiplicands.
+ 1
159, number of pt.
MAM. COMPL. AR1TH. — 12
178 DENOMINATE NUMBERS
Reduce :
3. 28 bu. 3 pk. to quarts. 8. 16 T. 15 cwt. 85 lb. to
27 gal. 1 qt. to pints. pounds.
10 da. 7 hr. to hours. 9 ' 1 sokr ? ear to hours
4.
5.
6. 8 mi. 80 rd. to rods.
A , t n i on • +~ 11. 25 rd. 3 yd. 2 ft. to feet.
15 da. 9 hr. 20 mm. to J
10. 12 fathoms 5 ft. to inches.
7
minutes. 12. 5 knots 1050 ft. to feet.
13. 11 lb. 8 oz. to ounces (avoir.).
14 . _5_ f a ton to a hundredweight and pounds.
T 5, T. = T 5 ? of 20 cwt. = 6i cwt.
£cwt. = i of 100 1b.= 25 1b.
T 5 6 of a ton = 6 cwt. 25 lb.
Change:
15. If lb. to ounces.
16. 1| bu. to pints.
17. 2 mi. 4020 ft. to inches.
18. | mi. to rods.
19. | long ton to hundredweight and pounds.
20. | common years to days, hours, etc.
21. In an automobile race the fastest machine ran | of a
mile in 36 seconds. Find the number of feet it ran per
second.
22. Change .75 week to days, etc.
23. School is in session 4.25 hours. Find the number of
minutes it is in session.
24. James lives 1.85 miles from school. Find the number
of feet he walks to school.
REDUCTION 179
Changing to larger denominations.
Written Work
1. An ice cream dealer retailed in one day 127 pints of
ice cream. How many gallons, quarts, etc., did he sell ?
2 )127, no. of pt. Since 2 pt. = 1 qt., 127 pt. = 63
,. ,.,, <• , 1 qt. and 1 pt. over. Since 4 qt.
4)b3, no. of qt. + 1 pt. * 1 ^ ^ = lg ga] and g qt
15, no. of gal. + 3 qt. over Hence> 127 pt _ 15 gal# 3
117 pt. = 15 gal., 3 qt., 1 pt. q t. i p t.
Note. — The dividend and divisor are regarded as abstract numbers.
Do not read 127 pt. •*■ 2 = 63 qt. Such a statement would be absurd.
( hange:
2. 225 qt. to gallons, etc. 7. 5675 sec. to minutes, etc.
3. 2550 qt. to barrels, etc. 8. 75000 in. to miles, etc.
4. 1463 pk. to bushels, etc. 9. 36481.62 ft. to leagues, etc.
5. 15000 min. todays, etc. 10. 175680 oz. to long tons, etc.
6. 3184 in. to rods, etc. 11. 9049 in. to rods, etc.
12. 3 qt. 1 pt. to the decimal of a gallon.
2 )1.0 , no. pt.
• 5 Since 2 pt. = 1 qt., 1 pt. = .5 qt.; 3 qt.
3. + .5 qt. = 3.5 qt. Since 4 qt. = 1 gal.,
4 )^5, n o. qt. 3 - 5 ^ = - 875 S al -
.875, no. gal.
Change :
13. 45 yd. .6 ft. to the decimal of a mile.
14. 6 cwt. 8 oz. to the decimal of a ton.
15. | ft. to the common fraction of a yard.
1 ft. = 4 yd.
1 ft. = § of i yd., or f yd.
16. | in. to the fraction of a foot ; of a yard.
17. 12 oz. to the fraction of an avoirdupois pound.
180 DENOMINATE NUMBERS
18. 240 rd. to the fraction of a mile.
19. The winner in an automobile race won in 21 hr. 3
min. 3.6 sec. What decimal of a day did it take ?
20. How much did a merchant receive for 3 barrels (42
gallons each) 9 gallons 3 quarts of oil, retailed at 15 cents
per gallon ?
21. A grocer bought 150 bushels of potatoes at 60^ a
bushel. He lost ^ of them by freezing, and retailed the
remainder at 10^ a half peck. How much did he gain ?
22. How many boxes, each holding a quart, can be filled
from 3 bu. 1 pk. 7 qt. of blackberries ?
23. At $2.88 a bushel, how many quarts of chestnuts can
be bought for $13.50?
24. What is the profit on 9 quires of paper, bought at
$2.40 a ream and sold at a cent a sheet ?
25. A rural mail carrier's route is 21 miles 176 rd. 4 yd.
in length. Find the number of feet he travels in one delivery
of mail.
26. A fruit grower sold in one season 23 bu. crates of
cherries at 10 cents per basket; and 45 bu. crates and 17
baskets of strawberries at 13 cents per basket. Find the
amount of the sales.
A crate contains 32 baskets.
27. A milk dealer put his milk in pint bottles. Find the
number of bottles delivered in one evening if he sold 23 gal.,
3 qt., and 1 pt.
28. How many 4-ounce packages of soda can be put up
from IT., 3 cwt., and 75 lb. of soda ?
29. A huckster bought 3 barrels of apples, each containing
2| bushels, for $8.25 and retailed them at 15^ a half peck.
Find his profits.
FOREIGN MONEY 181
30. Find the length of ;i double-track railroad laid with
1640 rails, each 30 feet in length.
31. An ocean steamer in making a certain trip consumed
l'.>20 tons of coal. If the time was 6 days, 5 hours, and 8
minutes, find the average number of pounds consumed per
minute.
32. The report of a cannon was heard 1 minute 5 sec-
onds after it was discharged. If sound travels 1120 feet per
second, how many miles, rods, etc., was the hearer from the
cannon ?
33. A pupil pays 845 tuition in a term of 9 months of 20
days each, and is absent from school 16 days. Counting
6 school hours to a day, find the amount of tuition lost to
him by his absence.
FOREIGN MONEY
English Money
The standard unit of English money is the pound, $4.8665.
4 farthings (far.) = 1 penny (</.)
12 pence = 1 shilling (s.)
20 shillings = 1 pound (£), or sovereign
In writing pounds, place the sign first; thus, £5.
A farthing, like a mill, is not coined, but is expressed as a fraction
of a penny.
French Money
The standard unit of French money is the f ranc = $.193.
100 centimes = 1 franc (fr.).
The peseta of Spain and the lira of Italy are of the same
value as the franc. The franc is the standard unit of value
also in Belgium and Switzerland.
182 DENOMINATE NUMBERS
German Money
The standard unit of German money is the mark = 1.238.
100 pfennigs (pf.) = 1 mark (m.).
All systems of money, except English money, are decimal
systems. Canada has a decimal system of money like the
United States. The denominations are the dollar, the dime,
the cent, and the mill.
Memorize:
1 pound = $4.8665
1 franc = .193 Approximate Values
1 peseta =193 1 pound=$5.00
1 lira = .193 1 mark = .25
1 drachma = .193 1 franc = .20
1 mark = .238
Change:
1. £ 1 to 8. ; to d. 5. 615 centimes to francs.
2. 200 centimes to francs. 6. 8.5 m. to pfennigs.
3. £1 5s. to s. 7. G| francs to centimes.
4. 5 francs to centimes. 8. 15 shillings to d.
9. A lady paid <6d. a yard for cloth. Find the cost, in
shillings, of 20 yards.
10. An English merchant takes 1 pound and 5 half sov-
ereigns to a bank to exchange for pennies. How many
pennies does the merchant receive ?
11. The admission to a German show costs 50 pfennigs.
How many tickets can be bought for 5| marks?
12. In a French school a collection for charity contained
6 francs, 5 half francs, and 250 centimes. Find the value
in francs.
FOREIGN MONET 183
Written Work
Change to U.S. money, expressed in the nearest cent:
1. £37.5. 5. 575 m.
2. 675 lira. 6. 380 drachmas.
3. 579 peseta. 7. 786.8 fr.
4. £35. 8. £6.
9. Change £ 25 12s. Sd. to U.S. money. Change 12s. Sd.
to the decimal of a pound; thus, Sd. = ^j, or |s. ; 12s. 4-
12|
20
f s. = 12f «. 12§s. =X4^, or £.633+.
Hence, £25 12s. 8rf. = £25.633.
1 pound = $4.8665.
25.633 x $4.8665 = $124.74.
Change to U.S. money expressed to the nearest cent.
10. £10 5s. 6<*. 14. £87 10s. Hd.
11. £27 lid. 15. 769 lira.
12. £98 19s. bd. 16. 104 fr. 75 centimes.
13. 100 m. 60 pf. 17. 745 peseta.
18. Change $1685 to pounds, etc.
$1685 -s- $4.8665 = 346.244+, the number of pounds.
.244 x 20s. = 4.88s.
.88 x \2d. = 10.56rf. = to the nearest penny lid.
Hence, $ 1685 = £346 4s. lid.
19. The cost of a bill of goods in England was $ 780.50.
Express the value to the nearest penny in English money.
20. A German clock cost in Strasburg $ 119. Express
the value to the nearest pfennig in German money.
21. An Italian workman saved on an average $27.89 per
month. Find the value to the nearest lira.
184 DENOMINATE NUMBERS
ADDITION AND SUBTRACTION
In addition, subtraction, multiplication, and division of
denominate numbers, the work is performed just as with
other numbers. It is necessary, however, to bear in mind
how many units of any denomination equal one unit of the
next higher denomination.
Written Work
l. Add :
6 yd. 2 ft. 4 in. The sum of the inches is 18 in. or 1 ft. 6 in.
3 2 6 Write 6 in. and add 1 ft. to the feet. The sum of
9 „ the feet is 7 ft., or 2 yd. 1 ft. Write 1 ft. and add
2 yd. to the sum of the yards. The sum of the
yards is 15 yd. The sum is 15 yd. 1 ft. 6 in.
15 1 6
2. Subtract 54 lb. 9 oz. (av.) from 75 lb. 4 oz. (av.).
., 9 oz. cannot be subtracted from 4 oz. Chauge 75 lb. 4 oz.
lb. OZ. to 7i lb 20 Qz 2() oz _- 9 Qz _ n oz ^ wh j ch is wr itten in
"5 4 t h e remainder. 74 lb. left in the minuend - 54 lb. = 20 lb.
54 9 Hence, the remainder is 20 lb. 11 oz.
20 11
3. Add : 4. Add :
bu. pk. qt. hr. min. sec.
16 3 5 18 44 48
18 7 21 39 29
42 1 27 51 58
56 1 3 29 36 24
5. Subtract : 6. Subtract :
hr. min. yr. mo. wk. da.
23 42 21 5 3 4
17 56 9 7 3 6
ADDITION AND SUBTRACTION 185
The most important application of subtraction of denomi-
nate numbers is in rinding the difference between two
dates.
7. How long a time elapsed from April 19, 1775 to April
14,1861?
The later date is written in the minuend as the
yr. mo. da. Uth day of the 4th mon th of 1861 ; and the earlier
1861 4 14 date in the subtrahend as the 19th day of the 4th
1775 4 10 month of 1775. In borrowing, consider 30 days to a
— ^FTT — ly 7 month and 12 months to a year. The difference will be
the time in years, months, and days.
8. Washington was born February 22, 1732 ; he was in-
augurated president April 30, 1789. How old was he when
he became president ?
9. Find the time from the signing of the Declaration of
Independence, July 4, 1776 to the beginning of the Civil
War, April 14, 1861.
10. General Ulysses S. Grant was born April 27, 1822.
How old was he when the Civil War closed April 9,
1865?
11. Find the sum of the collections at an English theater
for four different evenings as follows : £ 479 10s. Sd. ; £ 531
15s. l\d. ; £ 594 9d. ; £ 401 lis. hd.
12. Find the difference between the largest collection in
problem 11 and each of the others.
13. A United States mail agent leaves New York Monday
on his trip to Buffalo at 9:35 a.m. and returns Tuesday at
6: 15 p.m. How long is he gone?
14. A coal miner's 4 wagons for the day weighed as
follows : 1 T. 6 cwt. 19 lb. ; 1 T. 7 cwt, 13 lb. ; 1 T. 5 cwt.
85 lb. ; 1 T. 4 cwt. 98 lb. Find the total weight.
186 DENOMINATE NUMBERS
MULTIPLICATION AND DIVISION
Written Work
1. A square field is 40 rd. 5 yd. and 2 ft. on a side. Find
the perimeter, or the distance around the field.
rd. yd. ft. 4 x o f t . _ 8 ft. or 2 yd. 2 ft.
40 5 2 4x5 yd. = 20 yd. ; 20 yd. + 2 yd.= 22 yd. or 4 rd.
4 4 x 40 rd. = 160 rd. ; 160 rd. + 4 rd. = 164 rd.
TgT n o The perimeter of the field is 164 rd. 2 ft.
2. Six English workmen divide their profits equally. Find
each one's share, if the total profits are X 44 17s. 6d.
£ 8. d. £ 44 - 6 = £ 7 and £ 2 remaining ; £ 2 = 40s. ; 40s.
6 )44 17 6 + ^ s ' = ^ s " ' *^ s ' ^ ^ — ® s - an d 3s. remaining ; 3s. =
— „ q „ 36rf. ; 36rf. + 6(7. = 42rf. 42rf. - 6 = Id. The share of
each workman is £ 7 9s. Id.
Division may be performed by changing all the denomina-
tions to a common denomination ; thus, 164 rd. 2 ft. -5- 40 rd.
5 yd. 2 ft. = 2708 ft, -s- 677 ft, = 4, the quotient ; or by ex-
pressing the common denomination as a mixed decimal ; thus,
172 ft. 3 in. -s- 6 ft. 6 in. = 172.25 ft. -*- 6.5 ft. = 26.5, the
quotient.
3. A gardener sold on an average 6 bushel crates and
17 baskets of blackberries each day for 8 days. Find the
total number of bushels and baskets sold.
4. 57 students at a ball game each wore a badge 5^ inches
long. How many yards and inches of ribbon were needed to
make the badges ?
5. A wire fence inclosing a square field 40 rd. 5 yd. on a
side has 4 wires. Find the total length of wire in feet.
6. A 45 horse-power automobile used on an average 3 gal.
3 qt, 1 pt. of gasoline daily on a certain trip of 11 days.
Find the number of gallons, etc, used.
MULTIPLICATION AND DIVISION 187
7. Aii English estate of X 8000 10s. Sd. was divided as
follows: the widow received \ of the estate and the re-
mainder was equally divided among 5 children. Find the
amount to the nearest cent in United States money that
each received.
8. A Greek confectioner bought 12 bushels of chestnuts
at 82.50 per bushel and retailed them at 5^ per pint.
Find his gain.
9. How many times will a wheel 10 feet 8 inches in
circumference turn in going 12 miles ?
10. A ball room is 46| feet long and 30| feet wide. Find
the cost of a picture molding around it at 9 ^ per foot.
11. A bicyclist traveled 63 miles 170 rods 2 yards in a
forenoon, and 30 miles 5Q rods less in the afternoon. Find
the distance he traveled that day.
12. A silver dollar weighs 4121 grains. Find in tons, etc.,
the weight of $5,600,000 silver dollars.
13. A horse is fed 1| pecks of oats per day. How much
will the oats for the horse cost at 40 per bushel for Decem-
ber and January ?
14. Find | of 275 ft. 4 in. ; f of 11 lb. 8 oz. ; j of 36 bu.
3 pk. 4 qt.
15. A dairyman's sales for each of 4 weeks were as fol-
lows : 1st week, 115 gal. 3 qt. 1 pt. ; 2d week, 105 gal. 3 qt.
1 pt. ; 3d week, 113 gal. 1 qt. ; 4th week, 103 gal. 1 qt.
1 pt. If ^ of the total sales were uncollectable, find the cash
amount of his sales for the four weeks at 7.] i per quart.
16. A city weighman one morning recorded the weight of
4 loads of hay as follows: 1 T. 5 cwt. 19 lb.; 1 T. 7 cwt.
29 lb.; 1 T. 98 11). : 1 T. 9 cwt. 3 lb. Find the total amount
of the sales at %\\\ per ton.
REVIEW PROBLEMS
L. What is the sum of $8.45, $.55, I.87J, $15,055, and
.45?
2. If .9 of a ton of structural iron is worth $23.40, how
much are 11.75 tons worth?
3. Express as a common fraction in its lowest terms
.15625.
4. A huckster sold 29| bushels of potatoes at $.80 a
bushel and bought apples with the proceeds at $2.36 a bar-
rel. How many barrels of apples did he receive ?
5. A carpenter worked 235| days for $2.75 a day. He
received $480.75. How much was still due him?
6. Divide .0001 by .00001, and multiply the quotient by
1000.
7. If shovels are worth $15 a dozen, how many dozen can
be bought for $37.50?
8. Find the sum of 85 ones, 85 tenths, 85 hundredths,
and 85 thousandths.
9. The fastest long-distance train in the world (1906)
runs from New York to Chicago, a distance of 979.52 miles,
in 18 hours. Find the average rate per hour that it runs.
10. The fastest English train runs for 8 hours at a speed of
50.18 miles per hour. What distance does it run ?
11. The regular fare from New York to Chicago is $21.50,
but on this fast train a total excess fare of $9 is charged.
What is the actual cost per mile to the passenger ?
188
REVIEW PROBLEMS 189
12. The train in problem 9 makes 6 stops, averaging 5
minutes each. At what rate must the train aetually run so
as to reach Chicago from New York in 18 hours ?
13. A gallon of water weighs 81 pounds. Cast iron is
7.08 times as heavy as water. How much would a piece of
iron equal in volume to 3.25 gallons of water weigh ?
14. Gold is 19.3 times as heavy as water. How much
would the same volume of gold weigh ?
15. A bushel of shelled corn weighs 56 pounds. How
many bushels are required to fill an ordinary freight car
whose capacity is 60000 pounds?
16. The freight on such a car shipped from Kansas City
to Albany, N.Y., was $75. How much was that per
bushel ?
17. This corn cost 56 $ per bushel in Kansas City. How
much did this car load of corn cost delivered at Albany?
18. The dealer's other expenses in handling this car of
corn were 127.50. He sold it for 70 cents per bushel. How
much did he gain ?
19. In an automobile race 297 miles were covered in
290.173 minutes. What was the time per mile?
20. The speed of one machine in this race was 67.63 miles
per hour. At this rate how far could the machine travel in
1.25 hours ?
21. How many rods of hedge surround a school ground
8.625 rods long and 5.75 rods wide?
22. If .76 of a pound of gunpowder is niter, 1425.76
pounds of niter is necessary in making how many pounds of
gunpowder?
23. \ - (.125 x 7) + (16 x .375) = ?
190 REVIEW PROBLEMS
24. If the price of gas is 1.27 per thousand cubic feet,
how much is the average gas bill per month, when 85620
cubic feet are consumed in 6 months?
25. Allowing 2.75 bushels to the barrel, how many barrels
of apples, at $.65 a bushel, can be bought for $178.75?
26. A teacher paid .35 of his salary for board, .18 for
clothing, .07 for travel, .1 for incidentals, and had $300 left.
How much was his salary?
27. From 3| take the sum of 3| thousandths and 3| mil-
lionths.
28. If seventy-five hundredths of a number is 372, what
is eighty-three and one third hundredths of it?
29. What is the value of (~ ~ ^? + -) x -03?
V.4 4.5 2/
30. A real estate agent bought 4 lots at $650.50 each.
He sold two of them at a loss of $67.25. If he sells the other
two at a profit of $75.75, how much does he gain on his
investment ?
31. A school board paid $144 for books : § of the amount
was paid for General Histories at $1.20 each; ^ of it for
Algebras at $1 each ; and the remainder for Rhetorics at $.96
each. How many books were purchased?
32. A man invested .32 of his money in mining stock, .48
in railroad stock, and had $12000 remaining. How much
had he at first?
33. An engineer took a contract to build a bridge for
$18500. The material cost $10575; he employed 40 men
7^ weeks of six days each at $2.25 per day, and 30 men for
41 weeks of five days each at $2.75 a day. How much did
he gain?
REVTEW PROBLEMS 191
34. A merchant bought 140 boys' suits at $6.75 a suit.
He sold | of them at 19.50 a suit and the remainder at
$8.50 a suit. How much did he gain?
35. Four men purchased an oil property, the first paying
for .3 of it, the second for .375 of it, the third for | of it,
and the fourth the remainder, which was $3000. How
much did the property cost them ?
36. A merchant bought 975 pounds of sugar for $48.75.
He sold | of it at $.055 a pound, £ of it at $.06 a pound, and
the remainder at $.065 a pound. Find his gain.
37. I invested .4 of my money in a farm and deposited .75
of the remainder in a bank. If the amount paid for the
farm was $300 less than the amount deposited in the bank,
how much money had I?
Find the sum of the quotients :
38.
40.
5 +
5
39. .6 +
.2
5 +
.5
66-
.22
5 +
.05
11-
.022
5 +
.005
.08 +
.2
.005 +
5
.088 +
22
.5-5-
5
.16 +
.004
500 +
.5
.078 +
.013
001 +.01
41.
3.24 + 18
42. .002 + 20
.72-5-. 004
.5+. 0125
81.11 + 6.1
096 -5-. 32
675 + .75
.003 + 37.5
198-6.6
9 + .225
16 + . 016
.05-=-. 125
.288 + 32
.4+250
8.1 + .09
100 + .001
40.04 + 1.43
216 -.036
39. 2 + .14
.576 + . 8
.8-160
4.4 + 55
43.3 + 100
PRACTICAL MEASUREMENTS
LENGTH AND SURFACE
To the Teacher. — Secure a 50-foot tape measure and have pupils
make as many actual measurements as possible.
1. Find the dimensions of the school ground.
2. If your school is in the city, measure the length and
width of some square near your school, or if in the country,
some field.
3. Measure | of a mile along a street or a public road.
Compare a mile with a rod ; with a yard ; with a foot.
After some practice in actual measurements, the pupils should be able
to give quite accurate estimates of short distances.
4. Have the pupils estimate the length and the height of
the schoolroom ; the height of the school building; the length
and the width of the playground, etc.
5. Have each pupil draw on the blackboard, without
the aid of a rule, an inch, a foot, a yard.
6. Show by diagram the number of square inches in a
square foot.
7. Show by diagram the number of square feet in a
square yard.
8. Draw a square rod on a scale of 3 inches to 1 yard.
9. Show by a diagram, on a scale of 1 inch to the yard,
the number of square yards or square feet in a square rod.
160 square rods of land is called an acre.
A square mile of land is called a section of land.
192
LENGTH AND SURFACE 193
10. How many yards are there in the perimeter of a field
a mile square ?
Note. The perimeter of a figure is the line that bounds it, or the
sum of its sides.
11. Name the different units of long measure and the
different units of surface measure.
12. What is the shape of the figure that represents a square
inch ? a square foot ? a square yard ? a square rod ?
13.' Draw two figures of different dimensions to represent
an acre. How do you show that 160 square rods equals
each surface?
14. What unit of surface measure is not a square unit?
15. Have each pupil draw a unit surface, without the aid
of a rule, to show a square inch, a square foot, a square yard.
Learn this table of surface or square measure :
144 square inches (sq. in.) =1 square foot
9 square feet (sq. ft.) = 1 square yard (sq. yd.)
30 4 S( l uare y ards fa- y d } = 1 square rod (sq. rd.)
272 \ square feet (sq. ft.) j
160 square rods (sq. rd.) \ =\ acre (A )
43560 square feet (sq. ft.) j
1 mile square = 1 section
640 acres = 1 square mile
36 square miles = 1 western township
100 square feet =1 square in roofing and flooring
Written Work
Change :
1. 3 sq. ft. 48 sq. in. to square inches.
2. 4 A. 35 sq. rd. to square rods.
3. 125 sq. rd. to a decimal of an acre.
4. .375 of an acre to square rods.
HAM. COMPL. A l; 1 111. — 18
194 PRACTICAL MEASUREMENTS
5. 4 A. to square feet.
6. | sq. ft. to square inches.
7. 4.5 sq. rd. to square inches.
8. 1\ sq. mi. to square rods.
9. 1800 sq. rd. to acres, etc.
10. 1584 sq. in. to square feet, etc.
11. 6.75 A. to square feet.
12. Mr. Jamison's farm contains 125 A. 120.8 sq. rd.
Three fourths of it is purchased at $312.50 per acre, to be
laid out in town lots. Find the number of acres sold and
the amount received from the sale.
LINES AND ANGLES
1. Observe the two lines. How do they
compare in direction ?
Parallel lines are lines that cannot meet however far they
may be extended.
An angle is the difference in direction of two lines that meet.
When two lines meet each other forming a square corner,
they form a right angle ; thus, |
Lines drawn at right angles to each other are .
perpendicular ; thus, AB and BD in the cut are |
perpendicular to each other. B
2. Draw a circle and divide it into 4 equal parts by
diameters perpendicular to each other.
3. How many angles are there at the center
of the circle ? What is each angle called ?
The circumference of a circle is the perimeter
or distance around it. The circumference of a
circle is measured in degrees; every circum-
LINES AND ANGLES
195
ference, whether large or small, is divided into 360 degrees
(written 360°;.
} of a circle is 90°. Observe that the angle between
two lines that meet at a point is measured l>y the part
of the circumference cut by the lines extended.
4. How do you explain that ^ of circum-
ference A contains as many degrees as ^ of
circumference B?
5. Observe the figure. Show
that the curved lines are simply
parts of circumferences of circles B
that could be formed about the
point B. How do you show that
each curved line measures an angle of 30° ?
gles on the figure.
Show the an-
6. Show that an angle is the difference in direction of two
lines that meet at a point and that the angle remains the
same, however far the lines may be extended.
Angular Measure
Angles are measured by an instrument called a protractor.
When the center of the
protractor is placed at the
vertex of the angle to be
measured, the size of the
angle may be seen on the
scale between the lines that A B
form the angle. Thus, BOO is an angle of 30°, and AOC is
an angle of 150°.
Every circumference contains 360 degrees (360°), each
degree, 60 minutes (60'), and each minute, 60 seconds (60").
196
PRACTICAL MEASUREMENTS
Table of Angular Measure
60 seconds ('
60 minutes
360 degrees
') = 1 minute (')
= 1 degree (°)
= 1 circumference (C)
The length of a degree at the equator is 691 miles.
Draw an angle of 90°; 45°; 60°; 120°; 30°.
Kinds of Angles
Which one of these angles is a right angle ? Why ?
Which is less than a right angle ? Which is greater than
a right angle?
A right angle is an angle of 90°.
An acute angle is an angle less than 90°.
An obtuse angle is an angle greater than 90°.
TRIANGLES
A triangle is a surface bounded by three straight lines.
(Tri means three.^)
A vertex of a triangle is a point where two sides meet.
The base of a triangle is the side on which it seems to
rest.
The altitude of a triangle is the perpendicular distance
from the vertex opposite the base to the base, or the base
extended.
T III AN(1 LESS
197
Triangles are named in two ways :
I. From their angles :
(1) Right-angled triangles. (One right angle.)
(2) Acute-angled triangles. (All angles less than a
right angle.)
(3) Obtuse-angled triangles. (One angle greater than
a right angle.)
Right-angled
Acute- angled
Obtuse- angled
II. From their sides :
(1) Equilateral. (Having three sides equal.)
(2) Isosceles. (Having two sides equal.)
(3) Scalene. (Having no two sides equal.)
Equilateral
Isosceles
Scalene
Measuring degrees and angles.
1. How many right angles are there in the
square ?
2. How many right angles are there in the
rectangle ?
3. Cut from paper a square. Fold it on
the line connecting the opposite corners, and
cut it into two triangles.
Square
Rectangle
4. How many degrees are there in each angle of each
triangle thus formed?
198
PRACTICAL MEASUREMENTS
5. Every right triangle contains how many degrees?
By Geometry it is shown that the sum of the angles in any
triangle is equal to 180°. This can also be shown by meas-
uring the angles with a protractor.
The sum of all the angles of any triangle is equal to two right
angles, or 180°.
The following numbers in each case represent the size of
two angles of a triangle. Find the size of the third angle :
6. 90° and 45°
10. 60° and 40°
7. 90° and 60°
11. 100° 45' and 37°
8. 120° and 30°
12. 75° 10' and 95° 30'
9. 1201° and 601°
13. 100° and 45° 40'
KEOIAJSULli
QUADRILATERALS
A quadrilateral is a surface having four straight sides.
(Quadrilateral means having four sides. )
l. Examine the quad-
rilaterals. What are the
essential features of A ?
A square is a quadri-
lateral having four equal
sides and four right angles.
2. What are the essential features of B ? In what way
does figure B differ from figure A ?
A rectangle is a quadrilateral having four straight sides
and four right angles.
3. Show that the opposite sides of a rectangle must be
equal and parallel. Is a square a rectangle ?
A parallelogram is a quadrilateral whose opposite sides are
parallel.
Square
AREAS OK RECTANGLES
199
D
liaoimoiu
4. Examine these quadrilat-
erals. Why are they parallelo-
grams ? How do the sides of
surface C compare in length ? **°**™
Show that its angles are not right angles.
A rhombus is a quadrilateral whose sides are equal, and
whose angles are not right angles.
5. Why is surface D a parallelogram ? Show that its
angles are not light angles. Show that its sides are not equal.
A rhomboid is a quadrilateral whose opposite sides are equal
and whose angles are not right angles.
6. Why is surface E
a quadrilateral ? Why
is it not a parallelo-
gram? How many of
TBAI'EZOID
Tkatezium
its sides are parallel ?
A trapezoid is a quadrilateral having but two sides parallel.
7. Why is surface F not a trapezoid ? What is its name ?
A trapezium is a quadrilateral having no two sides parallel.
8. Describe each of the six quadrilaterals named above
with reference to its sides and angles. How many of these
quadrilaterals are parallelograms ? Give reasons.
AREAS OF RECTANGLES
Finding the area of a rectangle.
Find the area of a rectangle 4 yd.
long and 3 yd. wide. How long is this
rectangle ? how wide ? What is
the unit of measure ? How many
such units are in the first row? in
the second ? in the entire surface ?
4yd.
*fsq.
yd
200 PRACTICAL MEASUREMENTS
If the length and width of a rectangle are expressed in inches, the
unit of measure is 1 sq. in.; if expressed in feet, the unit of measure is
1 sq. ft. ; if expressed in rods, the unit of measure is 1 sq. rd. If the length
and width are expressed in related units, as feet and inches, or yards,
feet, etc., the dimensions must be changed to like units before finding
the area, that is the number of square units it contains.
Written Work
1. Find the area of a flower bed 20 feet 8 inches in length
by 10 feet 6 inches in width.
Length = 20§ ft. ; width = 10| ft.
Area = 20§ x 10£ x 1 sq. ft., or
- 6 3 2 x ^ x 1 sq. ft. = 217 sq. ft.
Tlie area of a rectangle is found by multiplying its unit of
measure by the product of its two dimensions when expressed in
like units.
Find the areas of rectangles having the following dimen-
sions :
2. 20.5 ft. by 12 ft. 6. 115 ft. by 54 in.
3. 21 ft. by 6.9 ft. 7. 45 yd. by 7 ft.
4. 72 yd. by 401 yd. 8. 108 in. by 3 ft.
5. 6 yd. 1 ft. by 3 yd. 2| ft. 9. 54 ft. by 108 in.
10. How many square yards are there in a lawn 45 feet
long and 36 feet wide ?
11. A square ball-park 600 feet on a side is inclosed with
a tight board fence 9 feet in height. Find the outside sur-
face of the fence in square yards.
12. Compare in area a surface 8 inches square and a sur-
face 2 inches square ; a surface 20 rods square and a surface
40 rods square.
13. Bricks are generally 8 in. x4 in. x 2 in. in size. Esti-
mate the number necessary to lay a sidewalk 100 ft. long
and 5 ft. wide, if the bricks are laid on the flat side. Find
the cost of the bricks needed at $13.75 per thousand.
PLASTERING AND TAINTING 201
14. A surveyor finds a field in the form of a rectangle to
be 680 ft. long and 330 ft. wide. Find its area without
changing feet to rods.
15. A field in the form of a rectangle contains 1200 sq. rd.
and the length is 40 rods. Find the width.
16. How many lots, each 30 ft. by 120 ft., can be made
from a plot of ground 120 ft. in depth and containing
10800 sq. ft. ? (Make a diagram.)
PLASTERING AND PAINTING
In plastering, painting, and kalsomining, the unit of meas-
ure is the square yard.
In some localities an allowance is made for openings and baseboards,
but there is no uniform rule in practice. Any allowance should always
be specified in the contract.
There are either 50 or 100 laths in a bundle. A bundle
of 100 is generally estimated to cover 5 square yards of
surface.
Written Work
1. How many square yards of plaster are necessary to
cover the ceiling of your classroom ?
2. Find the cost of painting both sides of a tight board
fence, 150 ft. long and 8 ft. high, at 15 ^ per square yard.
3. Allowing nothing for openings, how much will it cost
to kalsomine the walls and ceiling of a room 20 ft. long,
16 ft. wide, and 12 ft. high, at 6^ per square yard ?
4. A store room is 75 ft. long, 20 ft. wide, and 15 ft.
from floor to ceiling. It has a door in the rear 7 ft. by
3£ ft., and a window 8 ft. by 3 ft. How many bundles of
laths, each containing 100, are required for the sides, rear,
and ceiling, making full allowance for openings ?
202
PRACTICAL MEASUREMENTS
5. How much will it cost to plaster the walls and ceiling
of a store room, 40 ft. by 18 ft. and 12 ft. high, at 6^ per
square yard for lathing, and 18^ per square yard for plas-
tering, deducting ^ the area of 2 doors, each 9 ft. by 4 ft.,
and of 4 windows, each 6^ ft. by 3| ft. ?
6. A building 90 ft. by 24 ft. contains 3 stories, each
13 ft. high. The first story is plastered on the sides and
rear. The second and third stories each have 3 windows
in the front, each 8 ft. by 3^- ft., and each 2 windows in the
rear, each 8 ft. by 3 ft. If the ceilings are sheet iron, find
the cost of the plastering, at 33^ per square yard, deducting
for all openings.
7. In modern business buildings metal laths are used.
Estimate the cost of metal laths, for the building in example
6, at 25^ per square yard.
ROOFING AND FLOORING
In roofing, tiling, and flooring, the unit of measure is the
square of 100 square feet.
Written Work
1. Each of the two slopes of a roof is 60 ft. long and 20 ft.
wide. Find the cost of covering them with tar paper at
•$5.60 per square.
2. The floor of a hallway 30 ft. by 12 ft. is inlaid with
2-inch square tile. Find the number necessary.
In roofing with slate,
each course of slate is part-
ly overlapped. Each slate
as here shown is 10 in. by
16 in. and has 4 in. ex-
posed to the weather.
nil
■hi
It
•!:?
Hi
Ujll*!'
t
lllp!
in
.
! 1
fe
:ii
ryi
|
If
1 1 r in
pywi
11
ROOFING AND FLOORING 203
3. How many square inches of each slate are exposed?
4. If a 10-inch by 16-inch slate is exposed 4 inches to the
weather, find the number of slates necessary to lay a square
(10 ft. by 10 ft.)-
5. If slate 10 in. by 16 in. is laid 6 in. to the weather, find
the number necessary to lay a square. Find the weight of a
square of slate at 4 ; ]- lb. per square foot.
6. Each slope of a roof is 40 ft. by 20 ft. Find the
number of slates, 10 in. by 16 in., exposed 4 in. to the weather,
required for this roof, allowing nothing for breakage. Find
the cost of the slates at $5.50 per square.
There are 250 shingles in a bunch.
Shingles average 16 inches in length and 4 inches in width.
The exposed surface of a shingle laid 4| inches to the weather
is, therefore, 18 square inches. Without waste 8 shingles
will lay one square foot, and 800 shingles will lay 100 square
feet, or 1 square. Allowing for waste, 4 bunches, or 1000
shingles are estimated to lay a square.
7. Allowing nothing for waste, how many bunches of
shingles are required to cover a barn roof 35 ft. in width on
each side and 70 ft. in length. Find the cost at $4.00 per
thousand shingles.
8. Adding \ for waste, estimate the cost at $3.50 per
thousand of 157 bunches of shingles required to cover the
roof in example 7.
Flooring is frequently estimated by the square.
9. How much will it cost, at $5.00 per square, to lay the
floor of a hall 30 ft. by 60 ft., adding l for waste?
10. Estimate the number of squares of flooring required
for two floors of a store room 25 ft. by 60 ft.
204 PRACTICAL MEASUREMENTS
PAPERING AND CARPETING
The unit of measure in wall paper is the single roll, which
is 8 yards in length and usually 18 inches in width. A double
roll is 16 yards in length.
In approximating the number of rolls, paper hangers generally deduct
from the perimeter of the room the width of the doors and windows.
The remaining number of feet divided by H ft. (18 in. = 1| ft.) gives the
number of strips required for the surface of the wall. Dividing the
total number of strips by the number that can be cut from a double
roll gives the number of double rolls required. Fractional parts of a roll
are not sold. The ends of the rolls are generally sufficient to paper the
surfaces above and below the doors and windows. Border is sold by the
linear yard.
Carpet, matting, and border are sold by the linear yard.
Oil cloth and linoleum are sold by the linear yard or by the
square yard. Ingrain carpets are usually 1 yard wide, other
carpets are generally 27 inches wide.
Liberal allowance must be made for loss in matching.
Written Work
1. Estimate the number of double rolls of paper required
for a ceiling 18 ft. by 22 ft., strips running lengthwise.
22 ft. = length of one strip.
16 yd. = 48 ft. ; 48 ft. -4-22 ft. = 2, the number of whole strips in a
double roll.
18 ft. -4- li ft. = 12, the number of strips required.
12 -=- 2 = 6, the number of double rolls required.
2. A dining room 15 ft. by 22 ft. is 11 ft. from baseboard
to ceiling. It has four openings 3^ ft. by 7 ft. Estimate
the paper required for it, strips on ceiling running lengthwise.
3. The dining room in problem 2 lias a plate rail extend-
ing around it between the openings. Find the cost of this
rail at 80^ per foot.
PAPERING AND CARPETING 205
4. How much carpet 27 in. wide, laid the long way of the
room, is required for a room 18 ft. long and 15 ft. wide,
allowing 12 in. on each strip except the fust for matching ?
6 yd. = the length of one strip.
27 in. = 2\ ft. ; and 15 ft. ■*- 2\ ft. = % therefore
7 = the number of strips.
7x6 yd. = 42 yd.
6 x 12 in. = 72 in., or 2 yd., the waste on 6 strip.'-.
42 yd. + 2 yd. = 44 yd. of carpet required.
5. Explain why it takes fewer yards of carpet to cover a
room 18 ft. by 27 ft. with ingrain carpet (1 yard wide)
than with Brussels carpet (27 inches wide). Laying the
carpet the long way of the room, how many yards of each
would it take, if 10 in. were allowed on each strip, except
the first, for matching?
6. The widths of certain floors are 15 ft., 13^ ft., 15| ft.,
18 ft., 16 ft, Estimate the number of strips of ingrain carpet
necessary to cover each room.
7. Estimate the number of strips of Brussels carpet
necessary to cover each room described in example 6.
8. Find the cost of covering a kitchen 13| ft, by 12 ft.
with linoleum at $1.60 per yard double width, if £ of a
yard is allowed for matching and the linoleum is laid the
long way of the room.
9. Estimate the difference in cost between covering a room
18 ft. by 20^ ft, with Axminster carpet 27 inches wide,
at $1.45 per yard, laid lengthwise, allowing 12 inches on
each strip except the first for matching, and covering the
room with ingrain carpet at 85^ per yard, laid in the same
way, allowing 12 inches on each strip, except the first, for
matching.
206
PRACTICAL MEASUREMENTS
ib.
>,
^. <st-
5:
■*&&
&
-%.
<b
AREAS
Finding the area of a right triangle.
1. Find the area of a right triangle whose base is 4 yards
and whose altitude is 3 yards.
4yd.
Observe : 1. That the diagonal divides
the rectangle into two equal right triangles.
2. That the unit of measure is 1 sq. yd.
3. That the area of one of the right tri-
angles is \ of the area of the rectangle ; that
is, A of 4 x 3 x 1 sq. yd., or 6 sq. yd.
DdSe
The area of a right triangle is found by multiplying the
unit of measure by half the product of the base and the altitude.
Name the unit of measure, and find the area of each of
the following right triangles :
5. Base 10 ft., altitude 7 ft.
6. Base 14 ft., altitude 10 ft.
7. Base 6 ft., altitude 20 ft.
Finding the area of any triangle.
2. Base 10 in., altitude 6 in.
3. Base 12 yd., altitude 8 yd
4. Base 9 ft., altitude 6 ft.
Written Work
1. Find the area of two right triangles, the base of one
being 2 ft. and of the other 4 ft. and the alt. of each 3 ft.
Draw the triangles as shown in the
figure.
Observe : 1. That the unit of measure is
1 sq.ft.
2. That the area of the right triangle N
4 ft. is equal to \ of 2 x 3 x 1 sq. ft., or 3 sq. ft.
3. That the area of the right triangle M is equal to \ of 4 x 3
x 1 sq. ft., or 6 sq. ft. Therefore, the area of N plus the area of M is
equal to \ of 6 X 3 x 1 sq.ft., or 9 sq. ft.
AREAS
207
2. Find the area of a triangle whose base is 6 ft. and
whose altitude is 3 ft.
Observe that the area of the trian-
gle in example 2 is equal to the area of
the two right triangles in example 1, and
is, therefore, equal to \ of 6 x 3 x 1 sq. ft.,
or 9 sq. ft.
Show by cutting and folding paper, as indicated in the following
figures, that the area of each triangle is equal to one half the area of
a rectangle, having the same base and altitude.
The area of any triangle is found by multiplying the unit
of measure by one half the product of the base and altitude.
Find the area of the following triangles :'
3. Base 20 ft., altitude 14 ft. 5. Base 10 ft., altitude 30 ft.
4. Altitude 8 ft., base 15 ft. 6. Altitude 50 ft., base 18 ft.
Finding the area of a parallelogram.
Find the area of a parallelogram whose base is 8 in.
and altitude 3 in.
Observe: 1. That the diagonal of the paral-
lelogram divides it into two equal triangles.
2. That the area of each triangle is equal to
\ of 8x3x1 sq. in., and the area of the
parallelogram is equal to |, or once the prod-
uct of the base and altitude ; that is, 8 x 3 x 1 sq. in., or 24 sq. in.
The area of a parallelogram is found by multiplying the unit
of measure by the product of the base and altitude.
din.
208 PRACTICAL MEASUREMENTS
Written Work
Find the area in acres of:
1. A parallelogram whose base is 140 rd. and altitude 60 rd.
2. A rhomboid whose base is 90 rods and altitude 50| rods.
3. A rhombus whose base is 120 rods and altitude 100 rods.
Find the altitude of :
4. A rhomboid whose area is 7.5 A., base 48 rd.
5. A rhomboid whose area is 6.125 A., base 140 rd.
6. Find the base of a parallelogram whose altitude is 60|
rods and whose area is 30.25 acres.
Finding the area of a trapezoid. •
Written Work
1. Find the area of a trapezoid whose parallel sides are 20
inches and 12 inches, and whose altitude is 8 inches.
Examine the trapezoid ABCD. Draw the
-—IHl-^C diagonal A C, dividing it into two triangles.
,--'c;i\ Observe: 1. That the area of the trapezoid is
CO; \ equal to the area of its two triangles ABC and
2 Q m " Ba CD.
2. That the area of triangle ABC equals \ of
20 x 8 x 1 sq. in.
3. That the area of the triangle A CD equals I of 12 x 8 x 1 sq. in.
4. That the area of the trapezoid equals | of (20+ 12) x 8 x 1 sq. in., or
128 square inches.
The area of a trapezoid is found hy multiplying the unit of
measure by the product of the altitude arid ^ the sum of the
parallel sides.
2. The parallel sides of a trapezoid are 38 inches and 62
inches respectively, and its altitude is 21 inches Find its
area.
AREAS
209
3. The area of a trapezoid is 2.5 acres. The sum of it.s
parallel sides is 80 rods. Find its altitude.
4. The area of a trapezoid is 4± A. If its altitude is 20
rd., and one of its parallel sides 38 rd., what is the other ?
Finding the area of a trapezium.
Written Work
1. Find the area of a trapezium whose diagonal is 30
inches, and whose altitudes are 12 inches and 8 inches
respectively.
Observe: 1. That the area of the trapezium equals
the area of its two triangles.
2. That the area of one triangle equals \ of 30 x 8
X 1 sq. in.
3. That the area of the other triangle equals h of
30 x 12 x 1 sq. in.
4. That the area of the trapezium equals \ of 30 x 20 x 1 sq. in., or
300 sq. in.
Tfie area of a trapezium is found Ig dividing it into tri-
angles and finding the sum of their areas.
2. The base line dividing a trapezium into two triangles
is 40 ft. The altitude of one triangle is 10 ft,, of the other
is 12 ft. Find the area of the trapezium.
3. A trapezium is divided into two triangles by a line 28
ft. long. Find the area of the trapezium, if the altitude of
one triangle is 8 ft. and of the other triangle 14 ft.
THE CIRCLE
Observe the figure. What is its shape? Observe
that its boundary line changes its direction regularly
at every point.
A circle is a plane figure bounded by a
curved line, every point of which is equally
distant from a point within called the venter.
c^eSsjh.
HAM. i OMPL. ARJTH.
■14
210
PRACTICAL MEASUREMENTS
The circumference of a circle is its bounding line.
A diameter is a, straight line passing through the center
with both ends terminating in the circumference.
A radius is a straight line extending from the center to
the circumference.
Measure carefully with a cord the distance around a circle
1 foot in diameter, and you will find it is about 3.1416 ft. in
circumference. This relation of diameter to circumference
is true of all circles.
The circumference of a circle is found by multiplying the
diameter by 3.1416. This ratio is represented by the symbol
ir (pi)-
The diameter of a circle is found by dividing the circumfer-
ence by 3.1416.
Written Work
Find the circumference if the diameter is:
lOf ft.
7. 40 ft. 4 in.
8. 30 in.
9. 8 yd. 2 ft. 4 in.
Radius = ?
Diameter = ?
1. 15 ft.
2. 25 ft. 5. 12 ft. 6 in.
3. 60 ft. 6. 25 yd.
10. Circumference equals 25.1328 ft.
11. Circumference equals 125.664 yd.
Finding the area of a circle.
1st Methodo
Examine the figure :
Observe: 1. That the circle may be considered as made up of
triangles whose bases form the circum-
ference.
2. That the radius of the circle is
equal to the altitudes of the triangles.
3. That the area of the circle is equal
to the areas of all the triangles, or £ of
the sum of their bases (circumference)
by their altitude (radius).
AREAS
211
The area of a circle is found by multiplying the circumfer-
ence by one half the radius.
2d Method.
Examine the circle inscribed in the square.
Observe : 1. That the diameter of the circle is just
equal to the side of the square.
2. That much of the surface of the square, but not
all of it lies within the circumference. Careful meas-
urement shows that about .7854 of the surface of any-
square lies within the circumference of the inscribed circle.
Tl\e area of a circle equals .7854 of the surface of the cir-
cumscribed square.
Written Work
Circumference = C. Diameter = 2). Radius = R. Area = A.
Find the area if :
4. 22 =18 ft.
5. D = 20 in.
6. 22 = 20 in.
Find the area to two decimal places if:
io. (7=3. 1416 ft. 13. Z> = 35yd.
Li. C=6.2832rd. 14.22 = 10 ft.
L2. C = 94.248 in. 15. D =10 yd.
19. A circle 20 ft. in diameter is inscribed in a square.
What is the area of one of the corners within the square,
but outside the limits of the circle ?
20. A circular fountain 20 ft. in diameter is surrounded
by a cement walk 4 ft. wide. How much will the walk
cost at $1.50 per square yard?
Xote. — Find the difference between the areas of the two circles,
the first bounded by the circumference of the fountain, and the second,
by the circumference of the walk.
1. 2> = 10 rd.
2. R = 10 rd.
3. 2> = 18 ft.
7. R = 40 rd.
8. 2> = 25 yd.
9. R = 40 ft.
16. R = 125 ft,
17. 2) =120 yd.
18. 22=19;yd.
212
PRACTICAL MEASUREMENTS
SOLIDS
1. How many faces has this solid?
What is their shape? How do they
compare in size ?
A cube is a solid bounded by six
equal square faces.
2. Every 1-inch cube rests on how
many square inches of surface ?
3. Show that 144 one-inch cubes
may be placed on 1 square foot of surface.
4. How many cubes would make 12 such layers?
5. Show that 1728 cubic inches equal 1 cubic foot.
6. How many 1-inch cubes can be put into a cubical box
whose edge is 3 inches?
7. How many 1-foot cubes can be placed on 9 square feet
of surface? (Make diagram.)
8. How many cubes are there in three such layers?
9. Show that 27 cubic feet equal 1 cubic yard.
Learn this table of solid or cubic measure :
1728 cubic inches (
cu. in
) = 1 cubic foot (cu.
ft.)
27 cubic feet (cu.
ft.)
= 1 cubic yard (cu
yd.)
128 cubic feet (cu.
ft.)
= 1 cord of wood or tanbark
100 cubic feet (cu.
ft.)
= 1 cord of stone
1 cubic yard of earth equals 1 load.
The unit of cubic measure is a cube whose edge is one of
the linear units ; thus, a cube each edge of which is one inch
in length is a cubic inch.
SOLIDS
213
A cubic foot is a cube whose edge is one foot.
A cubic yard is a cube whose edge is one yard.
A cord of wood or tanbark is a pile of 4-foot wood or tan-
bark 8 feet long and 4 feet high.
A cord of short wood is a pile of short lengths 8 feet long
and 4 feet high.
The number of cords of short wood in a pile is found by
dividing the number of square feet in one side by 32.
Surface of Rectangular Solids
How is the surface of each face
found ? How many faces has this
solid ? Show that the sum of the faces
in this solid is the surface of the solid.
A rectangular solid is a solid bounded by six rectangular
surfaces.
Written Work
Find the entire surface of :
Rectangular Solids
1. 12 ft. by 8 ft. by G ft.
2. 20 ft. by 10 in. by 10 in.
3. 1G ft. by 2 ft. by 1^ ft.
4. 10 ft. by 8 ft. by 7 ft.
5. G ft. by 5 ft. by 5 ft,
6. 13 ft. by 8 ft. by 3 ft.
7. 20 ft. by 9 ft. by 7 ft.
Cubes
8. 4 inches on an edge.
9. 12 inches on an edge.
10. 2 feet on an edge.
11. 121 inches on an edge.
12. 14 inches on an edge.
13. 11| feet oil an edge.
14. \\\ inches on an edge.
214
PRACTICAL MEASUREMENTS
Volume of Rectangular Solids
4 in.
Scale : \ inch= 1 in.
Each cube in the solid represents
one cubic inch. How man)" cubic
inches are there in the first layer ?
How many such layers does this solid
contain? How many cubic inches
does the solid contain ? Observe
that the product of the three dimen-
sions expresses the number of cubic
units.
The volume of a solid is the num-
ber of cubic units it contains.
If the dimensions are expressed in inches, the unit of measure is 1
cubic inch : if expressed in feet, the unit of measure is 1 cubic foot ; if
expressed in yards, the unit of measure is 1 cubic yard. If the dimen-
sions are expressed in related units, as feet and inches, or yards and
feet, they must first be changed to like units.
Written Work
1. Find the volume of a rectangular solid 8 ft. 6 in.
square and 12 ft. 4 in. in length.
Thickness = 8.5 ft. : width = 8.5 ft.; length = 12J ft.
Contents or volume = S.5 x 8.5 x 12£ x 1 cu. ft., or 891.08$ en. ft.
The volume of a rectangular solid is found by multiply ing
the unit of measure by the product of its three dimensions when
expressed in like units.
Find the contents or volume of the following solids:
2. 10 ft. by 6 ft. 3 in. by 4 ft. 5. 1 yd. by 2 ft. by 18 in.
3. 12 ft. by 9 ft. 6 in. by 6 ft. 6. 68 in. by 1 ft. by 10 in.
4. 10 ft. square and S ft. high. 7. 5 yd. by 1 1 yd. by 2 ft.
SOLIDS 215
8. How many loads of earth must be removed in excavat-
ing for a cellar 30 ft. by 24 ft. and 8 ft. in depth ?
9. Estimate the number of cakes of soap 3 inches by 2
inches by 2 inches that can be packed in a box 3 feet by 2
feet by 2 feet.
10. A schoolroom is 40 ft. by 28 ft. by 16 ft. How many
cubic feet of air space are there for each of 39 pupils and
their teacher ?
11. How many cords of 4-foot wood are there in a pile
40 ft. long and 4 ft. high ?
12. Estimate the number of cords of 18-inch wood in 3
piles each 60 ft. long and 4 ft. high.
13. How many cubical boxes 3 ft. 6 in. on an edge can be
placed in a storage room 14 ft. in length, width, and height ?
In a certain township, the piles of 20-inch wood in the
yards of four schools were as follows :
14. Sykes,
15. Graham,
16. Wilson,
17. Clark,
Estimate the number of cords at each school and the
value of the wood at $1.85 per cord.
18. Find the number of loads of earth removed in exca-
vating for a cellar 16 feet wide, 30 feet long, and 6 feet in
depth.
of Piles
Length of Piles
60 ft.
40 ft.
Height of Piles
4 ft.
6 ft.
<2
u
50 ft.
60 ft.
5 ft.
4 ft.
\\
72 ft.
45 ft.
5 ft.
6 ft.
\l
36 ft.
60 ft.
5 ft.
4 ft.
216 PRACTICAL MEASUREMENTS
19. A pile of tanbark is 8 ft. wide, 9 ft. high, and 100 ft.
long. Find the number of cords.
20. A cubical block of granite 2 ft. on an edge is what
part of a cubical block of granite 6 ft. on an edge ?
When possible, use cancellation in the following problems :
21. Cape Cod cranberries are shipped in a crate whose
inside dimensions are 20 in. x 10|^ in. x 6| in. How many
cubic inches are 'there in a crate ?
22. Sweet potatoes are sometimes sold in a box 19^ in. x
llf in. x 10 in. How much does this differ from a bushel
(2150.4 cu. in.)?
23. Colorado apples are sometimes shipped in a box 18 in.
X 11 J in. x 11 in. How many cubic inches more or less
than a bushel does such a box contain ?
24. Colorado apples are sometimes shipped in a box 16
in. x 11 1 in. x 8| in. How many cubic inches does such a
box contain ?
25. California celery is shipped in crates 24^ in. x 22 in.
X 20£ in. How many cubic inches are there in such a crate ?
26. California dates are sold in boxes 17| in. x 10 in. x
9^ in. How many cubic inches are there in such a box ?
27. Figs are packed solid in a box 12 in. x 9 in. x If in.
How many cubic inches are there in such a box ?
28. The standard California orange box is now 24 in.
x 11^ in. X 12 in. Tangerines are shipped in boxes 24 in.
x 12 in. x 6| in. Which is the larger, and how many cubic
inches larger is it ?
29. How many cubic inches are there in a box 5| in. x
8| in. x 2| in. ? Find the number of cubic inches in a box
81 in. x 12i in. x 44 in.
LUMBER
217
One Board Foot
LUMBER
Measurement of lumber.
Lumber is any kind of sawed timber as boards, planks,
sills, etc. The unit of lumber measure is the board foot; it
is a board 1 foot long, 1 foot wide, and 1
inch thick. Draw it.
Note. — Boards less than 1 inch in thickness are
measured as if they were 1 inch thick ; boards over
1 inch in thickness are measured by their actual
thickness in inches and fractions of an inch.
1. How many board feet are there in a board 1 foot
wide, 1 inch thick, and 3 feet long? 5 feet long? 9 feet
long ?
2. How many board feet are there in a board 6 inches
wide, 1 inch thick, and 3 feet long? 10 feet long? In a
board 6 inches wide, \ inch thick, and 12 feet long?
3. How many board feet are there in a sill 5 feet long, 1
foot wide, and 4 inches thick ?
5 ft.
Written Work
1. Find the number of board feet in a sill 18 ft. long,
10 in. wide, and 8 in. thick.
10 in.= | ft. One surface = 18 x | x 1 hoard foot, or 15 board feet.
The sill contains 8 x 15 board feet, or 120 board feet.
The number of board feet in a piece of lumber is found by
multiplying the number of board feet in one surface by the
number of inches in thickness.
218 PRACTICAL MEASUREMENTS
Find the number of board feet in the following:
2. 1 board, 10 ft. long, 11 ft. wide, and 1 in. thick.
3. 1 board, 16 ft. long, 11 ft. wide, and | in. thick.
4. 2 boards, each 16 ft. long, 1 ft. wide, and | in.
thick.
5. 6 boards, 15 ft. x 2 ft. x 1 in.
6. 4 boards, 16 ft. x 1| ft. x -|- in.
How many feet of lumber are there in :
7. 1 plank, 12 ft. long, 1 ft. wide, and 3 in. thick ?
8. 1 sill, 15 ft* long, 1^ ft. wide, and 8 in. thick ?
9. 4 planks, 12 ft. long, 1| ft. wide, and 2 in. thick ?
10. 2 pieces, 18 ft. by 1 ft. by 1 ft.?
Find the number of feet of lumber in :
11. 10 planks, each 8 ft. long, 1^ ft. wide, and 3 in. thick.
12. 12 sills, each 20* ft. long and 10 in. square.
13. 20 joists, each 12 ft. long, 12 in. wide, and 3 in. thick.
14. 3 beams, each 40 ft. long and 10 in. by 12 in.
15. 30 scantlings, each 16 ft. long and 2 in. by 3 in.
16. How much will the flooring for two rooms, each 18 ft.
x 20 ft., cost at $30 per M.?
Buying and selling lumber.
Lumber is bought and sold by the thousand board feet.
In practice the cost is computed at so much per board foot ,•
thus, $20 per thousand feet (M.) is $.02 per board foot.
Show that $35 per M. = $.035 per board foot.
$60 per M. = $.06 per board foot.
LUMBER 219
Written Work
1. Estimate the cost of 378 feet of oak at $26 per M.;
of 6389 ft. white pine at $48 per M.; of 972 ft. cherry at
$72 per M.; of 693 ft. white ash at $47 per M.
Find the cost at $35 per M. of:
2. 50 hoards, 16 ft. long, 12 in. wide, and 1 in. thick.
3. 60 hoards, 12 ft. long, 15 in. wide, and 1| in. thick.
4. 100 boards, 15 ft. long, 6 in. wide, and f in. thick.
5. 75 boards, 18 ft. long, 10 in. wide, and 1 in. thick.
6. 45 boards, 16 ft. long, 5 in. wide, and 1 in. thick.
Short forms are used by carpenters, architects, and mechanics ; thus,
one mark (') represents feet, and two marks (") represent inches.
Find the number of board feet:
7. 120 studding, 2" x 4" x 12'.
8. 400 planks, 2" x 1' x 16'.
9. 300 boards, 1" x 10" x 14'.
10. 600 boards, 1" x 6" x 16'.
li. 100 boards, f " x 12" x 16'.
12. 15 sills, 6" x 10" x 20'.
13. 250 joists, 2" x 8" x 24'.
14. 70 sills, 10" x 12" x 30'.
15. 125 sleepers, 3" x 10" x 28'.
16. 200 boards, J" x4'"x 16'.
17. 500 joists, 21" x 8" x 20'.
18. 325 planks, 3" x 14" x 16'.
19. 300 sills, 5" x 8" x 24'.
20. 50 posts, 10" x 12" x 14'.
21. 400 studding, 2" x 3" x 18'.
22. 500 beards, 11" x 10" X 16'.
220 PRACTICAL MEASUREMENTS
23. Estimate the cost of the planks in examples 8 and 18,
at $.027 per board foot.
24. Estimate the cost of the sills in examples 12, 14, and
19, at $.032 per board foot.
25. Estimate the cost of the studding in examples 7 and
21, at $.026 per board foot.
CONCRETE, STONE, AND BRICKWORK
Concrete work is estimated by the cubic yard.
Stone work is estimated by the perch, of 24.75 cu. ft., or
by the cord of 100 cu. ft. Stones are often sold by the pound.
3200 pounds are estimated to lay 1 perch.
In estimating either contract work or cost of labor, in concrete and
stone work, the distance around the wall is considered the length. In
cases where there are inside corners, however, as at a and b in the
figure on p. 226, add, for each inside corner, twice the thickness of the
wall. This measures all corners twice. In estimating material, deduct
for openings and measure the corners but once.
Range work and lintels are measured by the linear foot.
Brickwork is estimated by the thousand. Bricks vary in
size, but they are usually 8" by 4" by 2".
In estimating the number of bricks in a wall, measure the corners
once, deduct all openings, and multiply the number of square feet re-
maining in the surface by 7 when the wall is 1 brick thick; by 14 when
the wall is 2 bricks thick ; and by 21 when the wall is 3 bricks thick.
Written Work
1. Find the number of cords of stone in a breakwater
200 ft. long, 14 ft. wide, and 16 ft. high.
2. A building 150 ft. by 130 ft. has a concrete foundation
4 ft. in width and 10 ft. in depth below the structural iron.
Estimate the number of cubic yards of material used.
THE CYLINDER 221
3. If the cement, the gravel, and the sand are in the ratio
of 1, 5, and 2, find the number of loads of gravel and of sand
used in the construction of the foundation in example 2.
4. Estimate the contract cost of the concrete work at
$7.75 per cubic yard.
5. Estimate the number of ^^.^^..^..^.^ . ^....^^
cubic yards of concrete in this M?-_ . . . . — . : . ; . , . . .
retaining wall. ^;Y. ',' '•'.". . '-• / '.^v ■;•;• ■.]
6. The walls of a brick house ^gfc '.•'•'••; .':"•:! ^: ■■■'..
36 ft. long, 24 ft. wide, and 18 ft. 200'
high are 13 in. or 3 bricks thick.
Estimate the number of bricks required for the walls, allow-
ing for 11 windows averaging 3± ft. by 6 ft., and 2 doors
averaging 3|- ft. by 7 ft.
7. A house whose walls are 9 in. or 2 bricks thick is 40
ft. long, 30 ft. wide, and 24 ft. high. Estimate the number
of bricks required for the walls, allowing for 12 windows
3 ft. by 7 ft., and 3 doors 3| ft. by 8 ft.
8. The stone work for the foundation of a house 28 ft.
by 38 ft. is 1* ft. in thickness and 6 ft. in height to the
range work. Estimate the cost of the stone work at $6.30
a perch, and the range work along the two sides and the
rear at 60 cents per linear foot.
THE CYLINDER
Examine this solid. ^SlS=s
How many ends or bases has it? What is the shape
of each? Are the bases equal and parallel? Describe
the shape of the body.
A cylinder is a solid whose two bases are
equal and parallel circles and whose diameter is |
uniform.
000
— — —
PRACTICAL MEASUREMENTS
The convex surface of a cylinder is the lateral or curved
surface. The altitude is the perpendicular distance between
its two bases.
Examine this cylinder.
Observe: 1. That if a piece of
paper is fitted to cover its convex sur-
face and then unrolled, its form will
be that of a rectangle.
2. That the circumference of the
base is the length of the rectangle,
and the altitude of the cylinder is the
width of the rectangle.
The convex surface of a cylinder is found by multiplying the
unit of measure by the product of the circumference and
the altitude.
The entire surface of a cylinder is found by adding the area
of the bases to the convex surface.
•
Written Work
Find the convex surface of a cylinder:
1. D. 10 in., height 24 in. 4. D. 20 in., height 4 ft.
2. D. 15 in., height 30 in. 5. D. 8 in., height 4 ft.
3. D. 2 ft., height 10 ft. 6. D. 6 ft., height 15 ft.
Find the entire surface of :
7. A water tank 12 ft. in diameter and 12 ft. in height.
A steam boiler 15 ft., long and 3 ft. in diameter.
Find the volume of a cylinder 3 ft. in diameter and 5 ft.
high.
Observe: 1. That the area of the base is 3 2 x .7854
x 1 sq.ft., or 7.0686 sq. ft.
2. That the first row of cubic units contains 7.0686
cu. ft.
3. That the cylinder contains 5 times 7.0686 cu. ft., or
35.343 cu. ft.
8.
9.
BINS, TANKS, AND CISTERNS 223
The volume of a cylinder is found by multiplying the unit
of measure by the urea of the base and this product by the
height of the cylinder.
Find the volume of a gas tank, silo, cistern, etc. :
10. D. 15 ft., height 18 ft. 14. R. 2 ft., depth 8 ft.
11. D. 25 ft., height 30 ft. 15. R. 8 ft,, height 30 ft.
12. D. 16 ft., height 20 ft. 16. D. 1 ft., length 16 ft.
13. D. 20 ft,, depth 15 ft. 17. D. 5 ft., length 12 ft.
BINS, TANKS, AND CISTERNS
Wheat and other grains are generally sold by weight, but
the capacity of bins is often estimated in bushels. The
capacity of tanks and cisterns is estimated in gallons or
barrels.
Note. — The standard bushel in the United States contains 2150.42 cu-
bic inches, stricken measure, and 2747.71 cubic inches heaped measure.
231 cu. in. = 1 gal. 31 \ gal. = 1 bbl. when estimating contents.
Written Work
Find contents in bushels of :
1. A bin 20 ft. by 10 ft. by 5 ft.
2. A box 12 ft. x 9 ft. x 6 ft.
3. A metal trough for watering cattle is 12 ft. long, 3 ft.
wide, and 20 in. deep. Estimate the number of gallons it
holds.
4. A cistern tank for a windmill pump is 8 ft. in di-
ameter and 10 ft. in depth. Estimate the number of barrels
of water it holds.
5. The rainfall on a certain day was 1\ inches. Find
the number of barrels of water that fell on Mr. Anderson's
flower plot which is 20 ft. long and 10 ft. wide.
224 PRACTICAL MEASUREMENTS
APPROXIMATE MEASUREMENTS
Approximate equivalents of the following measures are
1 bu. shelled grains
=
1] cu. ft.
1 bu. apples, coal, roots, corn in ear, etc.
=
If cu. ft.
1 bbl. in estimating contents
=
4i cu. ft.
1 cu. ft. of water
=
62] lb.
1 gal. of water
=
81 lb.
1 cu. ft. of any liquid
=
7.i gal.
1 ton of hay well packed
=
450 cu. ft.
1 ton of clover hay
=
550 cu. ft.
1 ton of bituminous coal
=
42 cu. ft.
1 ton of hard coal
=
35 cu. ft.
Written Work
1. A water meter registered 900 gallons of water con-
sumed in a month. Estimate the weight of the water used,
and its volume in cubic feet.
2. The inside measurement of a wagon box is 12 ft. 4 in.
by 3 ft. 6 in. by 16 in. Estimate the number of tons, etc.,
of anthracite coal it would contain; the number of tons, etc.,
of soft coal.
3. The rainfall on a roof 20 ft. by 30 ft. during April
and May was 9.5 in. Find the weight of the water that fell
on the roof during that time.
4. A swimming pool is 80 ft. long, 60 ft. wide, and 5 ft.
deep. Estimate the number of barrels of water in the pool.
5. Estimate the number of bushels in an oat bin 14 ft.
long and 10 ft. wide if the bin is filled with oats to a depth
of 6 feet.
REVIEW PROBLEMS 225
6. I Tow many tons of hard coal are there in a bin 16 ft. x
12 ft., when the pile is 1 ft. high ?
7. There are 5 ft. of water in a cistern 4 ft. in diameter.
How many gallons of water are there in the cistern?
8. Find the weight of the water in a railroad tank 12 ft.
in diameter and 16 ft. in depth, if the tank has 12 ft. of
water in it.
9. How many bushels of wheat can be shipped in a car
whose inside measurements are 36 ft. by 8 ft. 6 in. by
8 ft. ?
10. Estimate the number of tons of clover hay in a mow
60 ft. by 18 ft. by 16 ft. Estimate the number of tons of
timothy hay in the same mow.
REVIEW PROBLEMS
1. A field containing 20 acres is 61 rods long. How wide
is it?
2. The area of the floor of a schoolroom contains 1120 sq.
ft. The air in the room occupies 16800 cu. ft. What is the
height of the ceiling ?
3. How many tiles 6 in. square are required for a hall 40
ft. by 20 ft. 6 in.?
4. The side of a square is 20 inches. Find its area; its
perimeter.
5. The edge of a cube is 18 inches. Find its surface; its
contents.
6. How many cubical boxes whose edges are 6 in. can be
put into a box 8 ft. 6 in. by 4 ft. 6 in. by 3 ft.?
7. How many cakes of soap 2 in. x 2 in. x 4 in. may be
packed in a box 2 ft. long, 1 ft. wide, and 1 ft. high?
HAM. t OMl'L. A HI III. — 15
226
PRACTICAL MEASUREMENTS
14
Elii'iiiii I*
Miniii nun
; O
8. The edges of two cubes are respectively 10 inches and
12 inches. How much more surface has one than the other?
9. Find the cost of a farm, 480 rods long and 320 rods
wide, at 860 per acre.
10. How much will it cost to put a wire fence around this
farm at 50 per rod ?
11. A ranchman bought one square mile of land at $10
per acre. He put a fence around it and then divided it by
fences into four equal square farms for his sons. Find the
entire cost if the fence cost $.40 per rod.
12. Estimate the contract price of
building this cellar wall 18 in. thick
and 6^ ft. in height at $5.95 a perch.
Distance around the wall = 164 ft.
Add to this distance twice the thickness of
the wall for each of the inside corners, a and
\^) 6; that is, twice 2 x 18 in., or 6 ft. Then,
the length of the wall with 8 corners counted
twice = 164 ft. + 6 ft. = 170 ft. The volume
of the wall = 170 x 6 J x U x 1 cu. ft. =
1657.5 cu. ft. The number of perch of stone
= 1657£ -h 24| = 66f| perch. The cost of the wall = 66ff x $5.95 =
$398.47.
Query. — Why are the 8 corners measured twice ?
13. Estimate the number of bricks necessary for a dwell-
ing erected on the foundation as given in example 12, if the
walls are 3 bricks (13 in.) thick and 20 feet in height, mak-
ing an allowance of 150 square feet for openings.
Query. — Why are the 8 corners measured once ?
14. Estimate the cost of the face brick for the above
dwelling at $16.50 per thousand and $9.00 per thousand for
laying.
Note. — Consider the face brick as a wall 4 in. thick and measure the
8 corners once.
11 IIIIIIITTTT
I 1 MM. I ,
40'
REVIEW PROBLEMS 227
15. A level lot 60 ft. by 120 ft. has erected on it a dwell-
ing otj ft. by 42 ft. If the excavating averages 5 ft. and the
removed earth is placed on the lot, to what height will it
raise the grade of the lot ?
16. A certain town has a cylindrical water tank 20 ft. in
diameter and 45 ft. in height. The gauge shows 30 ft. of
water in the tank. Estimate the weight of the water.
17. How many yards of carpet, 27 in. wide, are required
for a room 24 ft. by 20 ft. 3 in. ? Strips are to run length-
wise.
18. How much will it cost to carpet a room 24 feet square
with carpet 27 inches wide, at $1.25 per yard, allowing 10
inches on each strip except the first for matching ?
19. How much will it cost to plaster a room 20 ft. by 16
ft., and 12 ft. to the ceiling, at 20^ per square yard, allowing
for one door 3 J ft. by 7 ft., and 2 windows, each 4 ft. by
6 ft. ? 700 ft.
20. This plot of ground is 700 ft. long | Qak'St_
and (500 ft. wide. Find the cost of grading
streets 40 ft. in width run through the cen-
ter each way, as here shown, at $1.90 per
linear yard. Find the amount from the sale
of lots 30' x 140' facing on Oak, Center,
and Clark Streets, at $ 20 per front foot. ..C/arkSt,.
21. Each of the three sides of a triangle is 50 ft. What
is the size of each of its angles ? Draw the figure.
22. The sides of a triangle are 80 fr., 80 ft., and 30 ft.
respectively. If the angle opposite the short side is 24°,
what is the size of each of the other angles ? Draw the
figure.
Cet
itt
>rSt
228
PRACTICAL MEASUREMENTS
Oo
O
23. The angle opposite one of the equal sides of an isos-
celes triangle is 75°. Find the size of the other two angles.
24. One angle in a right triangle is 13.75°. Find the
other two angles.
25. The angle opposite the base in an isosceles triangle
is 18^°. What is the size of the other two angles ?
26. A railroad com-
pany owns a strip of land
in the form of a parallel-
ogram, 66 ft. wide and
92.78 + rd. long through
this farm. Find the
area of A and the area
of the part owned by the
railroad. How can the
area of B be found ?
27. A wheel is 3 ft.
in diameter. How many
revolutions will it make in moving forward 942.48 ft.?
28. The speed of a vessel for 5 hours was 23.17 knots per
hour. Find her average speed per hour in statute miles.
29. Lead is 11.35 times as heavy as water. Find the
value of a cubic foot of lead at 5 ^ per pound.
30. How many rolls of paper are required for a room 20 ft.
X 18 ft., and 13 ft. 3 in. from the top of the baseboard to
the ceiling, allowing for 2 windows 3^ ft. in width and one
door 3^ ft. wide (papering the ceiling lengthwise) ?
31. The base of a triangle is 30 ft. and its altitude 23 ft.
Find its area.
32. A barn is 80 ft. by 50 ft." It is 40 ft. to the base of
the gable and 58 ft. to the top of the gable. How much will
it cost to paint it at 8 ^ per square yard ? .
<. lOOrd.-
-65rd Z
w
<£>\ \
<s>\ \
,-. "—A \
B cp\ \
\ A
-v\
— ^ \
o-\
\l8rd.
REVIEW PROBLEMS 229
33. If the slope of the roof of the barn in problem 32 is
32 ft. long, and it projects 1| ft. at each end, how much
will it cost to roof it at $ 8 per square?
34. A schoolroom is 40 ft. long and 30 ft. wide. Estimating
450 cubic feet of air to each person, what should be the height
of the room to accommodate 39 pupils and their teacher ?
35. Find the cost of a stone wall 30 ft. long, 2| ft. thick,
and 6 ft. high, at $5.30 a perch.
36. How much will it cost to cement the floor of a cellar,
40 ft, by 20 ft., at 90^ a square yard?
37. A street 50 ft. from curb to curb is opened for a dis-
tance of 300 yards. How much will it cost to excavate it
to a depth of 1 foot at 40 ^ per cubic yard ?
38. How much will the curb of this street cost at 26^
per linear foot ?
39. The sidewalk on this street is 12 ft. wide, including
a curb of 8 inches. How much will the brick for the walk
cost, at $ 9 per thousand, if the exposed surface of a brick is
4 in. x 8 in. ?
40. A farmer built a circular silo 12 ft. in diameter and
24 feet high. Find its contents in cubic feet.
41. How many blocks of ice, 2 ft. x 1 ft. x 1 ft., can be
packed in a ear ft, x 8 ft. x 40 ft. ? Ice is .92 as heavy
as water. Find the weight of the ice if a cubic foot of water
weighs 62| lb.
42. A cistern is 4 ft, in diameter and 6 ft. deep. How
many barrels of water will it contain ?
43. Estimate the weight of the water in a tank 8 ft. long,
6 ft. wide, and 2 ft. deep.
44. A vault is 5 ft. square and G ft. deep. How much
will it cost to cement the sides and bottom at $ .50 per sq. ft. ?
230
PRACTICAL MEASUREMENTS
45. A circular amusement park is 80 rods in diameter.
Find the cost of the boards for a tight board fence 8 ft. high,
inclosing the park, at $ 20 per M.
46. A corner lot in Seattle is 25 ft. by 100 ft. At $ 25
per M., what will be the cost for 2-inch plank for a 10-foot
sidewalk in front and on the side, including the corner ?
Note. — Illustrate by diagram.
47. Mr. Ames owns a 50-ft. lot fronting on a street 60 ft.
wide from curb to curb. The law compels him to pay ^ of
the cost of paving the street in front of his lot. How much
will it cost at $ 2.90 per square yard ?
48. A tank open at the top is 50 ft. long, 4 ft. wide, and
3 ft. deep. How much will the lead for lining it cost, at 8 ^
per pound, estimating 4 pounds to a square foot ?
49. Clay weighs 1.2 as much as the same volume of water.
Estimate the weight of a load of clay.
j j' 50. Find the cost of
painting the sides and
ends of this hay barn
at 15^ per square yard,
and the cost of stain-
ing the roof at 12^ per
square yard.
51. In excavating for a cellar 60 ft. long, 30 ft. wide,
and 8 ft. deep, the material was evenly distributed over a lot
90 ft. by 40 ft. To what depth was the lot covered ?
52. A two-story school building has 8 rooms 30 ft. x 32 ft.
and a hallway 28 ft. x 15 ft., on each floor. How much
will the flooring for the building cost at $44 per M. ?
53. In digging a sewer 31 ft. in width and 8 ft. deep,
1244| cubic yards of earth were excavated. Find the length
of the sewer in feet.
Test:
8 + 4 =
-4
:12
-4
5. 8-
8
-2 = 6
= 12-
-4
ANALYSIS
THE EQUATION
l. 8=8 2. 8 + 4=12 3. 8 = 12-4
In example (1) we have an equal number on each side
of the equality sign. In example (2) we have 8 + 4 = 12 ;
but in example (3), in order to preserve the equality, when
we take 4 from the left of the equal-
ity sign in example (2), we must sub-
tract 4 from the number on the right
of the equality sign. Thus,8= 12 — 4.
4. 8 = 6 + 2
Observe that a number may be moved from one side of an
equation to another by changing its sign.
Written Work
Change the following so that the first number in each
problem will stand alone at the left of the equality sign :
6. 20 - 10 = 10 10. 75 - 20 = 55
7. 40-15= 25 n. 85- 5-10=70
8. 80 + 15 = 95 12. 90 - 10 + 5 = 85
9. 100 + 75 = 175 13. 100 + 10 - 20 = 90
14. First add, then subtract, 5 from each member of the
equation 10 = 10.
(a.) 10 = 10 (b.) 10 = 10
+ 5 = +5 -5 = —5
15= 15 5= 5
The same number may h? added to or subtracted from both
sides of an equation without dest roying the equality.
231
232 ANALYSIS
Factors and their Product
1. 5 times a certain number is 35. What is the number ?
Factors Product
5 x the number = 35
The number = 35 -=- 5 = 7
When the product of two factors is divided by one of the
factors, the quotient is the other factor. When one of the factors
is unknown, it may be found by dividing the product by the
known factor.
State the factors and solve :
2. 5 times John's money = $ 40. How many dollars has he?
3. 2 times A's sheep are 60. How many sheep has he ?
4. 6 times B's age is 360 years. How old is he ?
5. l of a number is 150. What is the number?
6. 1.25 times a number is 30. What is the number?
7. .75 of a number is 75. Find the number.
8. ^ of a number is 75. Find the number.
I of the number = 75
The number = 3 x 75 = 225
The work may be shortened by calling the unknown
factor x. For example,
9. Mr. Brown's profits equal 4 times Mr. Long's profits,
and together their profits are 8 125. Find the profits of each.
Let x = Mr. Long's profits.
4 x = Mr. Brown's profits
5 x = $125, or the profits of both.
x = 825, Mr. Long's profits.
4 x = $ 100, Mr. Brown's profits.
10. Mr. Byers and Mr. Boydson together have 240 acres
of land, and Mr. Byers has 40 acres more than Mr. Boydson.
How many acres has each?
THE EQUATION 233
Let x = the number of acres in Mr. Boydson's farm.
x + i() = the number of acres in Mr. Byers's farm.
2 x+ 40 = the number of acres in both farms, or 240 acres.
2 x = 240-40
2x = 200
x = 100, the number of acres in Mr. Boydson's farm.
x + 40 = 140, the number of acres in Mr. Byers's farm.
Solve first by written analysis, then orally :
11. Four times my money and $6 more is $50. How
much money have I ?
12. $80 is $5 more than twice the cost of a bicycle. Find
the cost.
13. Harry's age plus | his age plus 6 years equals 30
years. How old is he ?
14. 21 times the number of books in Henry's library, less
5, equals 70. How many books has he ?
15. James spent \ of his money for a top, | of it for a
ball, and had 10 cents remaining. How much money had
he at first ?
16. After paying | and ] of my debts, I still owed 125.
How much did I owe at first ?
17. A merchant lost \ of his capital, then gained | as
much as he had left, and then had $10800. How much was
his capital at first ?
18. Robert's money, diminished by | and 4 of itself, equals
$1.25. How much money has he ?
19. After a fruit dealer had sold § of his apples, and | of
the remainder, he had 12 bushels left. How many bushels
had he at first ?
20. If 1 of Wilbur's money is increased by \ of f of his
money, the sum will be $54. How much money has he?
234 ANALYSIS
21. A banker gave a f interest in a bank to one son, a \
interest to another son, and the remaining interest, valued
at $10000, to his wife. What was the value of the bank ?
22. A lot was sold for $360, which was f of what it cost.
Find the cost.
23. Mr. Amos sold his farm for $3300, which was § more
than it cost him. Find the cost.
24. A typewriter spends | of his income and saves $400.
How much is his income ?
25. A suit of clothes was sold for $18, which was \ less
than it cost. Find the cost.
26. A merchant sold apples at $1.80 a barrel, which was
•| more than they cost him. How much did they cost per
barrel ?
27. There are 1200 pupils in a certain school. The num-
ber of boys is f of the number of girls. How many girls are
there in the school ?
28. The united ages of Alice and Mary are 28 years ;
Alice is f as old as Mary. How old is Alice ?
29. A lady paid $30 for a watch, which was \ more than
it cost. Find the cost.
30. A house that cost $1200 was sold for \ more than the
cost. How much was gained ?
31. A has 45 cents, which is | more than B has. How
much has B ?
32. How much will a two-thirds interest in a store cost,
when a four-fifths interest sells for $6000?
33. There are 40 pupils in a school, and \ of them are
boys. How many girls are there in the school ?
34. If a man owns § of a mill, and sells § of his interest
for $3000, what is the value of the mill ?
THE EQUATION 235
35. A lady paid $35 for a cloak. £ <>f tin 1 cost of the
cloak was \ of what she paid for other clothing. How
much did all cost?
36. A house and lot cost $8000. The lot cost f as much
as the house. How much did the lot cost ?
37. A traveler went 30 miles in two days ; the first day
he went 1^ times as far as the second. How many miles
did he travel the first day ?
38. A sold a watch to B for \ more than it cost him ; B
sold it to C for $20, thereby losing 1 of what it cost him.
How much did A pay for it ?
39. The difference between two numbers is 36, and the
greater is three times the less. What are the numbers ?
40. If to | of Mr. Barnhart's salary you add $40, the sum
will be | of his salary. How much is his salary ?
41. A merchant sold a dry goods store, receiving | of the
price in cash. He invested | of the sum received in a jew-
elry store bought at $900. For how much was the dry
goods store sold ?
42. What is the value of f of a ship if | of it is worth
$48000?
43. A man invested | of his money in a lot. Had he paid
$100 more he would have invested | of his money. Find
the cost of the lot.
44. Two merchants had a profit of $9600. After paying
^ of it for rent, they divided the rest so that one received |-
as much as the other. How much did each receive ?
45. If Wayne can do a piece of work in 6 days, what part
of it can he do in 1 day ? If Ray can do the same work in
4 days, what part of it can lie do in 1 day?
236 ANALYSIS
46. What part can Ray and Wayne both do in 1 day ?
47. If Ray and Wayne can do T 5 ^ of it in 1 day, in how
many days can they do the whole work, working together ?
48. If 4 men can do a piece of work in 3 days, how long
will it take 1 man to do it ?
49. If one man can do a piece of work in 12 days, how
long will it take 2 men to do it ?
50. If 8 men can do a piece of work in 2^ days, how long
will it take 5 men to do it ?
51. A jeweler sold a watch for $60, and gained | of the
cost. What was the cost of the watch ?
52. A horse, sleigh, and harness cost $220 ; the sleigh cost
twice as much as the harness, and the horse cost 4 times as
much as the sleigh. Find the cost of each.
53. Ira can do a piece of work in 12 days; Baxter can do
it in 16 days. If Baxter's wages are $1.50 a day, how much
per day should Ira receive ?
54. A man has three houses which together are worth
$5700. The second house is worth twice as much as the
first, and the third is worth -^ as much as the other two.
How much is the third house worth ?
55. If A can do a piece of work in l-i days, B in 3 days, and
C in 4 days, in what time can they do it working together ?
Suggestion. — Since A does the whole work in § days, he does f of
it in a day ; B does i in a day, and C \ in a day. What part of the work
do they do together in a day? How long, then, will it take them to do
the whole work together?
56. A man bought three automobiles. The first cost
$1500, the second cost 1| times as much as the third, and
the third cost twice as much as the first. Find the cost of
the second and the third.
PERCENTAGE
The term per cent means hundredths or by the hundred.
The sign for it is %.
Thus, five hundredths may be written 1 ^ 5 , .05, 5 per cent, or 5%.
These are called equivalents.
Percentage is the process of computing by hundredths. It
is simply an application of decimal fractions.
Write both as a decimal and as a common fraction in its low-
est terms each of the following per cents; thus, 10% = .10 =
JUL — 1
too iU-
l. 5%
5. 371%
9. 120%
13. 75%
2. 6f%
6. 14|%
10. 250%
14. 300%
3. 81%
7. 331%
11. 43%
15. ■ 1%
4. 121%
8. 125%
12. 65 %
16. % %
Write the following decimals
as fractions and
per cents:
17. .05
21. .081
25. 1.50
29. .25
is. .20
22. 2.50
26. Ill
30. .16f
19. .331
23. 1.25
27. .031
31. .371
20. .50
24. 1.20
28. .06f
32. .45
Write the following as decimals and as per
cents :
33. 1
37. |
41 1
**■■ 9
45 -1-
12
34. \
38. 1
42 - lV
46. |
35. \
39. \
43 - iV
47. §
36. f
40. 1
44. }
48. |
237
238
PERCENTAGE
Memorize the following equivalents :
49. What is 50% of 100 ? 40 ?
10? 2? i? I?
50. What is 16f % of 90? 150?
120? 9? 12? 3?
51. What is 331% of 3000?
2000? 50? 75? 100?
52. What is 12|% f 200?
32 ? 96 ? 4 ? 6 ? 1 ?
53. Whatis37±% of 72? 56?
800 ? 2000 ? 40 ? 8 ?
54. What is 20 % of 400 ? 600 ?
l? i? 16? 20?
l% = iio
14f% = |
2% = 5 1 o
16|% = i
H% = £>
20% =i
4%= A
25% = J
5% = A
33|%=i
6i% = tV
37f% = |
6!% = iV
50% =i
8J% = A
62J % = |
n % = it
66}*-}
io%= T V
75% =|
in*=*
8^% = f
12i% = l
871% =|
55. Find 12|% of 16; of 48
800 ; of 220 ; of 404.
of 72; of 96; of 168; of
56. Find 5% of 25; of 50; of 75; of 100; of 125.
In each example in 56 we have two terms, a per cent and a number. In
each case we are to find 5 % of the number. The number of which we take
the y|^ (viz. 25, 50, etc.) is called the base. The number of hundredths
(5) to be taken is called the rate, and the number of hundredths actually
taken, that is, the answer, is called the percentage.
The base is the number on which the percentage is
computed.
The rate or rate per cent is the number of hundredths.
We generally express rate as a decimal.
The percentage is the product obtained by taking a certain
per cent of the base.
The sum or amount is the base plus the percentage.
The difference is the base minus the percentage.
PERCENTAGE 239
Finding a given per cent of any number.
l. What is 20% of 300? Think of 20% of 300 as \ of
300, or 60.
Find :
2. 2 % of 100 17. 81 % of 480
3. 5 % of 400 is. 9£ % of 660
4. 10% of 500 19. 111% of 729
5. 20% of 800 20. 121% of 648
6. 50% of 1200 21. 16|%of366
7. 40% of 1000 22. 331% of 333
8. 25 % of 360 23. 621 % f 864
9. 30% of 90 24. 66|% of 724
10. 3% of 420 25. 75% of 968
li. 2 % of 500 26. 871 oj of 568
12. 6% of 150 27. 6% of $60
13. 7% of 800 28. 8% of $560
14. 5% of 440 29. 10% of 350 acres
15. % of 550 30. 30 % of 960 sheep
16. 31% of 900 31. 62i % of 856
Written Work
l. What is 7 % of 245 ? 66f % of 300 ?
(a) 245 base 7<>/ ° of a number e q uals - 07 of ih
^ J n „ Therefore, 7% of 245 is .07 times 245,
—M rate or 17.15.
17.15 percentage
-« n a 662% f a number equals § of it.
( 5 > ? of £00 = 200 1 of 300 = 200.
^4. given per cent of any number is found by multiplying
the base by the rate.
PERCENTAGE
Find:
2.
4% of 328
9.
80% of 6.75
3.
9% of 1126
10.
331% of 75
4.
11% of 263
11.
60 % of f
5.
15% of 380
12.
14f% of 105
6.
24% of 165.5
13.
75% of f
7.
38% of $77.50
14.
87^ oj f 168
8.
72 % of 328
15.
331 cj of 336
16. In a school of 400 pupils, 45 % are girls. How many
girls are there? how many boys?
17. A clerk who received $50 a month had his wages
increased 15%. How much were his wages increased?
18. If iron ore yields 63 % of pure metal to the ton, how
much iron is there in 40 tons of ore?
19. A merchant, failing in business, paid 85% of his
debts. How much should a creditor receive whose claim is
$2850?
20. A bill of goods cost $137.50. How much was gained
by selling the goods at a profit of 12% ?
21. Compare 48% of $45 and 45% of $48.
22. I owe a debt of $246.50. If I pay 40% of it at one
time, and 50% of the remainder at another time, how much
do I still owe ?
23. A n automobile cost $ 3500 and the repairs for 2 years
were 10% of the cost. If the automobile was sold at 40%
reduction from the cost, find the entire loss.
24. The operating expenses of a factory are 45 % of the
sales. If the sales for a year amount to $650450, how much
are the operating expenses?
PERCENTAGE 241
25. 400 men were employed in a factory at daily wages
averaging $1.95. If 100 of these men received 33| % of the
entire daily wages, rind their average daily wages. Find
the average daily wages of the other 300 men.
26. Three newsboys, John, James, and Henry earned to-
gether $550 in a year. John earned 40%, James 60% of the
remainder, and Henry what remained. Find how much
each earned.
27. 40% of a Western farm containing 600 acres is in
wheat, 30% of the remainder in corn, 66|% of the remainder
in oats and grass. How many acres are there in each crop,
and how much remains not cultivated ?
Finding what per cent one number is of another number.
1. What part of $10 is $5? What % of $10 is $5?
Think $5 is \ of $10, or 50% of $10.
What % of:
2.
20 is 10 ?
8. 3| in. is 1^ in.?
3.
35 ft. is 7 ft.?
9. 25 gal. is 64^ gal.?
4.
100 is 16| ?
10. 60 rods is 20 rods?
5.
500 lb. is 100 lb.?
11. 1 mile is 80 rods?
6.
871 is 121?
12. 1 lb. (av.) is 1 oz. (av.) ?
7.
ij y d - is i y cL?
13. 1 dollar is 1 dime ?
Written Work
l. What per cent of 75 is 15 ?
The unknown number is the rate.
15 .j_ 75 = .20 = 20% Since the percentage equals the base multi-
plied by the rate, the rate must equal the
percentage divided by the base. 15 divided by 75 is .20, or 20%. Or, 15
is ff, or \, or 20%, of 75. Test : 20% of 75 = 15.
The rate equals the percentage divided by the base.
BAM. COMPL. IR1TH. — 16
242
PERCENTAGE
What per cent of :
2. 25 is 10 ?
32 is 12?
65 is 39?
196 is $72?
3.
4.
5.
6.
7.
8.
9.
18
621 a. is 6| A.?
-2- is -&?
9 i& 9 *
4isf?
125 yd. is 75 yd.?
10. 4 bu. is 1 pk.?
11. 40 is 6§?
12. 75 is 3.125?
13. | is .125?
14. I|is2i?
15. $18 is $45?
16. 10 qt, is 36 qt.?
17. $3 is 18 cents?
Out of 350 words, I spelled 315 correctly. What
per cent did I make in spelling?
19. From a farm of 160 acres, 24 acres were sold. What
per cent was sold?
20. If a man saves $262.50 out of his salary of $1250,
what per cent does he save ?
21. A farmer raised 150 bu. of potatoes from 6 bu. of
seed. What per cent of the crop was the seed ?
22. A merchant owes $8750 and his assets are $3675.
What per cent of his debts can he pay ?
23. A pupil misspelled 35 words out of 80. What per
cent did he spell correctly?
24. .875 is what per cent of .3125?
25. I paid $5.25 for the use of $75. What per cent did I
pay?
26. A son, on receiving $5000 from his father, bought a
farm for $2750, a store for $1875, and deposited the rest in
a bank. What per cent of his inheritance did he deposit ?
27. A house rents for $240 a year. The taxes and in-
surance are $30. If the property is valued at $3500, what per
cent does the owner realize on the value of the property ?
PERCENTAGE 243
Finding the number when a per cent of it is given.
1. If £ of a number is 10, what is the number? If 20%
of a number is 10, what is the number ?
2. If 33| % of a man's loss is $300, how much does he
lose?
3. If 87 ^ % of a man's gain is $70, how much does he
gain ?
What is the number of which :
4. 10 is 66| %? li. 40 is 16| % ? 18. $48 is 10 % ?
5. $100 is 331 % ? 12. lOt) is 12.] rr/ ? 19 . #12 is 16f % ?
6. 500 is 50 % ? 13. 300 is 37J % ■> 2 o. 30 is 331 % ?
7. 60 is 75%? 14. 500 is 62^ %? 21. 60 is 66f%?
8. $24 is 40 % ? 15. 700 is 874 % ? 22. 120 is 20 % ?
9. 300 is 60 % ? 16. 500 is 25 % ? 23. $10 is 831 % ?
10. 500 is 50 % ? 17. 600 is 75 % ? 24. $50 is 831 <p ?
Written Work
1. Find the number if 33 % of it is 3135.
(a) 3135 -T- .33 = 9500 The unknown number is the base. Since
the percentage equals the base multiplied by
the rate, the base equals the percentage divided by the rate, 3125 ■*- .33 =
9500, the number. Test : .33 of 9500 = 3135.
2. Find the number if 8| % of it equals 250.
si\ Q\tf _ i 8£% of the number equals J 2 of it.
OK(\ 3 12 ~" Vnnn H of the number is 12 x 250 > or 300 °-
zou x - r - duuu Test . A of 3000 _ 250
The base equals the percentage divided by the rate.
PERCENTAGE
?in
d the number if :
3.
8% of it is $2.40 8.
4.
12% of it is $3.60 9.
5.
12|% of it is $91 10.
6.
37|% of it is $27 11.
7.
331% of it is $42.50 12.
244
6% of it is $72
32% of it is $3.60
45% of it is $3.60
62-i-% of it is $35.50
12. 87|% of it is $28.28
13. A has $3612, which is 87^% of what B has. How
much has B?
87|%of B's = $3612, A's.
14. If $14 is 25% of A's salary, find his salary?
15. After a battle 70 % of a regiment, or 644 men, were
left. How many men were there in the regiment at first?
16. I drew from the bank $1500, or 83± % of my deposit.
How much was my deposit?
17. If a man rents a house for $752 per year, which is
16% of its value, what is the value of the house?
18. A teacher's expenses are $30 a month, and this
amount is 37| % of his salary. How much does he save?
19. The number of pupils in attendance at school in a cer-
tain town is 576, which is 96 % of the enrollment. What is
the enrollment?
Finding a number when the number plus the rate of increase
is given.
Written Work
1. What number increased by 17 % of itself equals 585?
100% of itself = the number
100% of itself + 17%of itself, or 117% of the number = 585, amount
1% of the number = T i y of 585, or 5
100% of the number = 100 X 5, or 500
Divide the sum by one plus the rate.
PERCENTAGE 245
What number increased by :
2. 8% of itself is 324? 6. 50% of itself is 69?
3. 30% of itself is 260? 7. 250% of itself is 105?
4. 37} % of itself is 550? 8. 16| % of itself is 1050?
5. | % of itself is 2011? 9 . 70% of itself is 510?
10. 1 gained 35% by selling an article for $4.05. How
much did it cost?
11. A laborer had his wages twice increased 10%. If he
now receives $2.42 a day, what were his wages before they
were increased?
12. A property sold for $4025, which was an increase of
15% of the cost. How much did the property cost?
13. A receives $1600 salary, which is 60% more than B
receives. What salary does B receive?
14. W. H. Richmond bought a jewelry store for a certain
sum and increased the stock 27 % of the purchase price. He
found that the whole investment amounted to $5969. What
was the purchase price of the store ?
15. The land surface of the District of Columbia is 60
square miles, which is 500 % more than the water surface.
What is the water surface?
Finding a number when the number minus the rate of de-
crease is given.
Writt&a Work
1. What number diminished by 16% of itself equals 168?
100% of the number = the number
100% of the number — 10% of the number,
or 84 % of the number = 168 (difference)
1 % of the number = & of 168, or 2
100% of the number, or the number = 100 x 2, or 200
Divide the difference by one minus the rate.
246 PERCENTAGE
What number diminished by :
2. 45% of itself equals 55? 6. 18f % of itself equals 325?
3. 18% of itself equals 246? 7. 23% of itself equals 308?
4. 621% ( ,f itself equals 27? 8. 95% of itself equals 25?
5. 50% of itself equals 22.5? 9. 10% of itself equals 4| ?
10. John has $35, which is 12| % less than his brother has.
How much has his brother?
11. Mrs. Lee spent $24 for a coat, which was 33| % less
than the cost of a suit. Find the cost of both.
12. After losing 8|% of his money, a man had $352 left.
How much had he at first?
13. What number decreased by 35 % of itself equals $1300 ?
$520? $6500?
14. A school enrolls 249 boys, which is 17 % less than the
number of girls it enrolls. How many pupils are there in
the school?
15. A lady when shopping spent $15 of her money for a
hat, which was 25 % less than the amount she spent for a
coat. How much did she spend for the coat ?
16. The fraction T 9 g is 40 % less than what fraction ?
17. If a certain number is increased 40 % of itself, and
this sum is diminished by 50 % of itself, the result is 700.
Find the number.
18. I sold two lots for $1200 each ; on one T gained 25%
and on the other I lost 25 % . Did I gain or lose and how
much?
19. The population of a town in 1906 was 14000, which
was 121% less than the population in 1907. What was the
population in 1907?
REVIEW OF PERCENTAGE 247
REVIEW OF PERCENTAGE
1. What is 50% of 200?
2. .05 is what per cent of .25?
3. .25 is 25% of what number?
4. I sold goods for $8.75. My actual loss was $1.25.
What was the cost and the per cent of loss ?
5. A man owns a farm valued at $9600. His annual
taxes are -$86.40. How much must he make each year to
clear 8 % on the cost of his property ?
6. Ten is what per cent of 20? 15 is 150% of what
number ?
7. What per cent of 10 days are 30 days? What per
cent of 30 days are 10 days ?
8. | of 72 is how many per cent of 120 ?
9. I paid $7200 for a house, $150 for repairs, and $350 for
delinquent taxes. I then sold it for $9000. What per cent
did 1 make on my money ?
10. In a business college 20 % of the students study book-
keeping, 60 % of them study typewriting, and the remaining
68 students study other courses. How many students are
there in the school ?
11. The income from an investment which pays 5.}% is
$220. What sum is invested ?
12. Mr. Wilson buys a house and lot for $6400. The
average expenses per year for taxes are $80; for insurance
$12; and for repairs $24. What must be the annual rent
that he may have an income of 6 % net on the original cost
of the property?
13. What fraction increased by 35% of itself equals J$?
248 PERCENTAGE
14. If my property sells for $7433.25 and I owe $8745,
what per cent of my debts can I pay ?
15. Ten thousand boxes of fruit were sold for $9450,
which was 33^% less than the cost. What was the cost?
16. If 12| % of 16| % of a number is 2J, what is the whole
number ?
17. A section of land was sold for $4000, which was 25%
more than it cost. How much did the land cost per acre ?
18. Express as a per cent : | ; fa ; |.
19. A horse and buggy cost $300. If the cost of the
horse was 200% of the cost of the buggy, what was the
cost of each ?
20. After paying 70 % of his debts, a man found that $3600
would put him out of debt. Find his original indebtedness.
21. Thirty-five per cent of 640 pounds is 5.6% of how
many tons ?
22. A bankrupt sold his property for $4100, which was
18 % less than its real value. If the property had sold for
$5250, what per cent above its real value would it have
brought ?
23. Two adjacent properties sold for $13200. 75 % of the
sale of one equaled 90% of the sale of the other. Find
the selling price of each.
24. A certain excavation cost $120. What would be the
cost of an excavation 20% wider? 25% deeper? 50% longer?
25. What per cent of 121.92 is 15.24 ?
26. What is the difference between \ % of $7000 and 25 %
of $7000?
27. I bought 100 bu. apples at 50^ a bushel, but lost 20 bu.
by freezing. At what price per bushel did I sell the remain-
der, if my entire loss was 4 % of the cost of the apples ?
GAIN AND LOSS 249
28. The net profits of a store in two years were 83483.
The profits the second year were 15% more than the first
year. How much were the profits the first year ?
29. An executor in settling an estate found 7| % uncol-
lectable, 12^ % invested in city lots, 40% in cash, 15% loaned,
and the remainder, 810,000, invested in the home. The
estate was equally divided among four sons. How much
did each receive ?
30. A merchant increased his capital the first year 331%,
and the second year 25 % of the capital at the end of the first
year. He lost 36% of his original capital the third year,
and had $11760 left. Was his original capital increased
or decreased, and how much ?
31. An estate was worth $8400. Had it been sold for
that amount, the creditors would have received 87| % of their
claims, but \ of the estate was sold at 18-| % below its value,
and the remainder at 12| % below its value. What per cent
of the debts did the estate pay?
GAIN AND LOSS
1. A dealer bought goods for 11000, and sold them at a
gain of 10%. What was the selling price? What was the
gain ?
2. Sugar that cost 5P per pound was sold at a gain of
20 % . What was the gain per pound ?
3. A huckster bought fruit for 820, but found he had to
sell it at a loss of 20 %. How much did he lose ?
4. A grocer sold butter that cost 20^ per pound, for
25 $. What part of the cost did he gain ?
5. Books that cost 81.50 wholesale were sold at a gain
of 10%. Find the selling price.
250 PERCENTAGE
Gain and loss are terms used to designate the profits or
the losses in business transactions.
The cost is the amount paid for an article ; the selling
price is the amount received for it.
The gross cost of goods is the original cost increased by
what is paid for freight, storage, etc.
The net proceeds is the amount received for goods after all
charges incident to the sale have been deducted.
The per cent of gain or loss is always reckoned on the cost or
on the sum invested.
Written Work
1. A merchant bought goods for $4500 and sold them at
a gain of 12 %. How much did he gain ?
Comparative Study
$4500, cost The amount bought or sold corresponds to what term
. 1 2 rate * n f ercentct ff e ?
% 5-1-0 00 m in ^ e S 3 * 11 or l° ss corresponds to what term in Per-
' ° l centage ?
Find the gain or loss:
2. $75, gain 20%. 6. $356, gain 21 %.
3. $96, gain 381%. 7 . $132.50, gain 28%.
4. $115, loss 15%. 8. $485.60, loss 5%.
5. $227, loss 19%. 9. $880.80, gain 12 1 %.
10. How much is gained by selling a property that cost
$3250 at a profit of 8% ?
11. A grocer bought 120 dozen eggs at 18 cents a dozen,
and sold them at a profit of 11^%. What was his gain?
12. A real estate dealer bought three lots for $1500, $1800,
and $2000 respectively. He sold the first at an advance of
8%, the second at an advance of 10%, and the third at an
advance of 12%. What was his gain ?
GAIN AND LOSS 251
Finding the gain or loss per cent.
Written Work
.1. I bought a piece of property for $2500 and sold it
for $2750. Find the gain or loss.
$■2730 - $2500= $ 250, gain
$250 h- $2500 = .10, or 10%, gain
Find the gain or loss per cent when the :
2. Cost is $100 and the selling price 1 105.
3. Cost is 1175 and the selling price $210.
4. Cost is $240 and the selling price $280.40.
5. Cost is $476.80 and the selling price $309.92.
6. Cost is $775.50 and the selling price $1008.15.
7. If flour is bought for $4.50 a barrel and sold for $6 a
barrel, what per cent is gained ?
8. A farm was bought for $3500 and sold for $4200.
What was the gain per cent ?
9. If hats are bought for $27 a dozen and retailed at
$2.75 each, what per cent gain is realized ?
10. A grain dealer bought 500 bushels of wheat at 84 cents
a bushel, 000 bushels at 80 cents a bushel, and sold it all at
82 cents a bushel. What per cent did he gain or lose ?
Finding the cost.
Written Work
l. What was the cost of a house if the owner, by selling
it at an advance of 25%, gained $900?
$900 = gain
$ 900 -- .25 = $3600, cost
252 PERCENTAGE
Find the cost when:
2. 5% loss is $25. 6. J% gain is $16.80.
3. 12± % gain is $37 J. 7. 44 % loss is $1100.
4. 150% gain is $750. 8. 35% gain is $12.25.
5. I % loss is $12±. 9. 16| % loss is $ 37.50
10. A dealer sold a buggy at 25% gain, and received $90
for it. How much did the buggy cost?
Selling price = £ of the cost
$90 = | of the cost
Cost = | of $90, or $72
Find the cost when selling price at:
11. 10 % gain is $ 220. 14. 37-| % loss is $600.
12. 18 % loss is $492. 15. 16| % gain is $1190.
13. 121 % g a i n i s $990. i 6 . 114% gain is $1284.
17. A merchant, after losing 25% of his goods by fire,
had $8700 remaining. What was the value of his goods at
first?
18. An attorney turned over to his client $1125 after re-
taining 10% for his services. What amount of money did
the attorney collect?
19. If 8% is lost by selling an article for $1.15, how much
did it cost?
20. If dress goods sold at $1.50 a yard yielded a profit of
20 %, how much did the goods cost per yard?
21. Mr. Rice sold his farm at a gain of 5 % and received
$ 4200 for it. What would the gain per cent have been had
he sold it for $4400?
$4200 - 1.05 = $4000, cost
$4400 - $4000 = $400, gain
- $4000 = .10, or 10%, gain
REVIEW PROBLEMS 253
22. I sold a horse for -1240 and thereby lost 20%. What
selling price would have given me a gain of 20%?
23. If a merchant loses 10% by selling goods at 45 cents
a yard, for what should they have been sold to gain 20 %?
24. A piece of cloth was sold for $32, which was at a loss
of 20%. What would have been the loss percent had it
been sold for $39?
REVIEW PROBLEMS
1. Find the selling price of goods on which there is a
loss of 2}, % which amounts to $106.25.
2. How many per cent above cost must an article be
marked in order to make 25% after a discount of 20% has
been given?
3. I sold | of an acre of land for what £ of it cost.
What per cent did I gain or lose?
4. At what price must an article that cost $60 be marked,
so that after deducting 25 % from the marked price, a profit
of 10% may be realized?
5. A sold a property to B and gained 20 % ; B sold it to
C and lost 16| % ; C sold it to D for $2800 and gained 12%.
How much did A receive for the property?
6. Find the value of a property that increased annually
4 % on the previous year's value and after three increases
was worth $11,248.64.
7. A note was bought for 5% less than its face and sold
for 2% more than its face. If $40.25 was gained, what was
the value of the note?
8. I bought a lot for $1600, which was 20% less than its
real value, and sold it for 20% more than its real value.
How much did I gain?
254 PERCENTAGE
9. If 80% of a car load of wheat is sold for what the car
load costs, what per cent is gained ?
10. A company sells sewing machines to a wholesaler at
33^% profit, the wholesaler sells to the retailer at 12i%
profit, and the retailer sells to the trade at a profit of 20 % .
Find the cost of a machine to the company when the retailer's
profit on the machine is $9.
11. A merchant marked his goods so as to gain 20 % ■ By
giving credit, 5 % of his sales were imcollectable. His gain
was $1050. What was the value of his goods?
12. A bankrupt can pay only 80 cents on the dollar.
What will be the gain or loss per cent to the retailer who
sold him a buggy at 30 % profit ?
13. A merchant marked a lot of goods costing $ 12.000, at
25 % above cost, but sold them at 10 % less than the marked
price. What per cent did he gain ?
14. A merchant sold silk at 45 cents a yard above cost,
and gained 20%. What was the selling price per yard?
15. Mr. McKay bought 150 shares of W. Va. Lumber Co.
stock at $ 140 a share, and 200 shares of telephone stock at
$120 a share. He sold the lumber stock at a loss of 10%.
For how much did he sell the telephone stock per share,
if he gained 10% on the transaction?
16. A and B invested an equal amount of money in busi-
ness ; A gained $2000 on his investment, and B lost $1500;
B's money was then 65% of A's. How much money did
each invest?
17. A bought a horse and sold it to B at a gain of 10% ;
B sold it to C and gained 10%. If C paid 131.50 more for
the horse than A, how much did the horse cost A ?
COMMISSION AND BROKERAGE 255
COMMISSION AND BROKERAGE
A person who buys or sells goods or transacts business
for another is called an agent, collector, commission merchant,
or commission broker, according to the nature of the business
transacted.
A commission merchant actually receives the goods which he buys
and sells for another. A commission broker simply makes the contract
between the buyer and the seller for whatever is to be bought or sold,
the goods being delivered directly from the seller to the buyer.
The commission or brokerage is a certain per cent of the
amount of money involved in the transaction.
A commission merchant gets a certain per cent on the
amount of his sales; a collector gets a certain per cent on
the amount collected ; a broker gets a certain per cent of the
cost or the selling price.
The net proceeds is the amount left after commission and
all other charges have been paid.
The one who sends the merchandise to be sold is the prin-
cipal, the shipper, or the consignor.
Selling or collecting through an agent.
Written Work
l. A Pittsburg commission house sold 275 barrels of apples
at $4 per barrel, on a commission of 10%. The freight
from Rochester, New York, was $67.50 and the drayage was
$11.25. Find the commission and the net proceeds of the
sale.
Amount of sale 275 bbl. at $ 4 $1100.00
Commission I0%of $1100 $110.
Freight 07.50
Drayage 11 ■-•"'
188.75 188.75
Net Proceeds $ 911-25
256 PERCENTAGE
Comparative Study
The amount bought, sold, or collected in Commission corresponds to
what term in Percentage?
The rate in Commission corresponds to what term in Percentage ?
The commission or brokerage in Commission corresponds to what
term in Percentage ?
The net proceeds in Commission correspond to what term in Percent-
age?
2. A real estate agent sold four lots for $ 250, $ 325, $ 395,
and $405 respectively. How much was his commission at
5%?
3. A commission merchant sold 320 barrels of apples at
$3.25 a barrel and 16 barrels of sweet potatoes at $4.80 a
barrel. Find his commission at 7%.
4. A lawyer collected an account of $385 for a client,
charging 5%. How much should he remit?
5. A cotton broker sold 200 bales of cotton of 225 pounds
each, at 12 J ^ per pound, charging a commission of 2| %.
Find the net proceeds.
6. A lawyer collects 80 % of an account of $ 1125. Find
his rate of commission if he charges $ 9.
7. A real estate agent sold 475 acres of coal at $ 85 an acre.
His commission was $1211.25. Find the per cent charged.
8. My agent bought 180 barrels of flour at $4.80 per bar-
rel. He paid $50 freight and $6 storage. I sent him
$ 937.28. What was his rate of commission?
9. An attorney succeeded in collecting 90 % of the amount
of a consignment of cotton sold for $6700. He remitted
$5874, retaining the balance to pay attorney's fees and
$5.25 freight charges. What per cent did he charge for
collecting?
COMMISSION AND BROKERAGE 257
10. My Chicago broker sells for me 82560 bushels of wheat
at 83£ ^ per bushel, charging \ # per bushel brokerage.
Find the amount remitted to me.
Find the selling price if a commission of:
li. 2% =$3.05 14. 41% = $56.25 17. 3f % =$36.25
12. i%= 3.12 15. ?>{% = 75.60 18. 1% = 7.59
13. \%= 1.54 16. 1|%= 61.50 19. 4 mills = 79.00
20. An agent for the Diamond Pneumatic Tool Company
received 20% on the sales of 13 pneumatic drills averaging
$737.50 each. If his expenses for traveling and freight on
machines were $697.50, find his net profits.
21. James Amidon & Co. offered one of their salesmen
$2400 per year, $1800 for traveling expenses, and 2%
on all sales over $40000; or 6% on all sales if he paid his
own expenses. He chose the former, and sold $72000
worth of goods in the year. Did he gain or lose, and how
much, by accepting the first offer?
22. For selling a house, a real estate agent received $ 96. 25,
which included $7.65 for advertising, and $27.90 for re-
pairs. Find the rate of commission, if the house was sold
for $3035.
Buying or investing through an agent.
Written Work
1. An architect charged 2} 2 % for plans and specifications
and 2^ % for superintending construction. His commission
amounted to $ 810. How much did the building cost ?
2. I telegraphed my agent at Chicago to buy me 10000
bushels of wheat at 79 cents or less. lie bought at 77|
cents and charged me | cent per bushel brokerage. Find
the amount of the check I should send him.
HAM. COM PL. Alt I'l'll. — 17
258
PERCENTAGE
3. An agent's investments in 1905 were $ 80500 and in 1906
$ 87650. His commissions for 1906 were $ 143 more than for
1905. Find his commission for each year.
4. My broker in New Orleans buys 50000 pounds -of cot-
ton at 11|^ per pound. His commission is | % and
freight, storage, and cartage amount to $ 95.80. How much
should I remit ?
5. A house and lot bought for $8500 was sold afterwards
for 80 % of the purchase price. If the agent received 2%
on each transaction, find his commission.
6. An agent buys a property for his principal for $36790
at 2% commission. The owner puts $674.20 in repairs and
afterwards sells the property through the same agent for
$43000, at 2% commission. Find the agent's commission,
and the principal's rate per cent of gain.
7. An attorney invested for his client $8750 in a mort-
gage. The owner of the property paid the attorney 2 %
commission for getting the money and $35.75 for examining
title and for docket fees. How much did it cost the mort-
gager to secure the money ?
REVIEW
Find the values of the missing terms :
1.
Gross Sales
EXPENSES
Commission
Rate of Commission
Net Proceeds
$675
$11.25
( )
8%
( )
2.
$550
$4.00
$22.00
( )
( )
3.
( )
$119.00
( )
20%
$681
4.
$560
00
$56.00
( )
( )
5.
( )
00
$48.00
( )
$552
6.
$1000
00
( )
4%
( )
7.
( )
( )
$134.00
2%
$6566
8.
$2500
$46.00
$50.00
( )
( )
INSURANCE 259
9. A fruit grower ships to his commission merchant 600
barrels of apples, which are sold at $ 3.50 per barrel. The
agent deducts 143.90 freight charges, $27.75 cartage, 12 ^
per barrel for cold storage, and 5 % commission. Find the
amount remitted.
10. My agent sends me a bill for $37800, which includes
the cost of some land bought at $100 an acre and his com-
mission of 5%. Find the number of acres purchased.
11. My purchasing agent in Chicago sends me a bill for
$6776.60, covering cost of mining machinery, commission of
2%, and $65 for extra expenses. Find his commission and
the cost of the machinery.
12. My agent retains %\°fo commission, pays freight
charges of $16.80, storage and drayage of $9.75, and remits
to me $1594.65. Find the amount of gross sales.
13. An investment broker in Denver receives a draft for
$ 23935. This includes a commission of 2 % for investing,
and an allowance of $ 175 for traveling expenses. Find the
amount invested.
INSURANCE
Insurance is security against loss or damage.
A merchant owns a store, uninsured, valued at $ 5000. If it is burned,
who will bear the loss ? An insurance company agrees to insure the store
for $4000 at 1 % annually. In case the store is totally destroyed by fire,
how much is the company expected to pay to the merchant? How much
must the merchant pay annually to the company to guarantee this loss?
The policy is a written contract between the person insured
and the insurance company.
The premium is the sum paid for the insurance.
The rate is a specified number of cents or dollars per $100
of insurance, or a certain per cent of the sum insured.
The term is usually a year or a period of years. Short
rates arc rates charged when the term is less than one year.
ofjO PERCENTAGE
Property Insurance
The principal kinds of property insurance are fire insur-
ance and marine insurance. Other forms are burglar insur-
ance, insurance against bad debts, etc.
l. A frame dwelling with a tin roof is insured for $2800
for one year at 1 %. Find the premium.
Written Work
Comparative Study
The amount insured corresponds to what
$2800, amount insured term in Percentage ?
^-. , The rate corresponds to what term in
' ' Perppntfuip ?
Percentage t
$28.00, premium -pj ie premium corresponds to what term
in Percentage f
2. A brick house is insured for ¥4000 at 60^ on the $100.
Find the rate of premium and the annual premium.
3. If the three-year rate is twice the rate for one year, find
the cost of insuring a brick dwelling for $6500 for 3 years
when the annual rate is 45^ per $100.
4. A store building is insured for $8500 and the annual
premium is $212.50. Find the rate per cent of premium
and the annual cost per $100 of insurance.
5. A school board pays annually $45 for $6000 of fire pro-
tection on a school building. Find the rate of premium.
6. The premium on a dwelling insured for $5500 is $38.50
for three years. Find the average rate for a year.
7. If the premium on a plate-glass policy is $9.50 and the
rate is |%, rind the face of the policy.
8. Mr. Lawrence wrote a check for $31.50 to pay the in-
surance on his dwelling for 3 years. If the house cost $2400
and was insured for I of its value, rind the rate for the term.
INSURANCE Jtil
9. A farmer insured his house for $2700 at 1]%. his bam
for $1200 at \%, and his furniture for $900 at 1%. What
premium did he pay ?
10. A drug store is insured for | of its value at 2%.
What is the value of the store if the premium is #192 ?
11. The premium on 8000 bushels of wheat, valued at 90^
per bushel and insured at \ of its value, is $57.60. Find the
rate of insurance.
12. A jewelry store is insured for $20000 and its con-
tents for $27000. The premium is $705. What is the rate
of insurance ?
13. A clothier insured his stock of goods, valued at
$12000, for 1 year at 1|%. At the end of 6 months he
surrendered his policy. If the " short rate " for G months
was 90^ per $100, how much premium was returned?
14. A farmer insured his buildings for $3500 at \\% for
a term of 3 years. After he had paid the premium for 4
terms the buildings were totally destroyed by fire. What
was the farmer's loss ? the company's loss ?
15. A vessel worth $27000 is insured for § of its value
at 3|%. In case of shipwreck, what is the company's loss ?
What is the owner's loss ?
16. How much insurance, at |%, can be placed on a build-
ing for $42 ?
17. A business block valued at $300000 was insured in
4 different companies, the rate of each being 1%. The
first company took $50000 ; the second, $60000 ; the third,
$90000 ; and the fourth the remainder. After the premiums
had been paid four times, the block was damaged by fire to
the amount of $120000. What was the loss of each com-
pany ?
262
PERCENTAGE
Personal Insurance
The principal kinds of personal insurance are life insurance
and accident insurance.
Kinds of life insurance policies :
1. A life policy is one that guarantees a fixed sum of money on the
death of the insured. The premiums on a life policy run for life.
2. A life policy with a twenty-year settlement is one that guarantees,
after twenty annual payments have been made, either a cash surrender
value, or a paid-up policy, payable at death.
3. An endowment policy is one in which the face of the policy and
the profits on the premiums are guaranteed to the insured if living at
the end of a specified time, or to his estate if his death occurs within the
time.
4. A term policy is one in which the face of the policy is paid, provid-
ing the insured dies within the time the policy runs. Otherwise nothing
is paid.
Most insurance companies pay dividends on the premiums already paid,
thus lessening the amount of the annual premium.
The premium is always so much on $ 1000 of insurance; thus, a
premium of f 23.40 means $23.40 on $1000.
The age of the insured is always reckoned according to the age at his
nearest birthday.
This table shows the annual premiums for each $ 1000 insurance in a
leading insurance company.
Age
Ordinary Life
20-Payment Life
•20-Year Endowment
•20-Year Term
20
$18.95
$27.64
$49.35
$ 12.48
25
21.14
30.05
45.98
13.34
30
23.96
32.98
50.74
14.61
35
27.63
36.62
51.88
16.70
40
32.48
41.18
53.69
20.15
45
39.02
47.09
56.70
25.85
50
47.79
54.98
61.75
35.00
55
60.33
65.81
70.02
60
77.48
81.09
INSURANCE 263
Dividends vary according to the number of premiums that have been
paid. It is fair to estimate that the dividends on an ordinary life policy
after 20 annual premiums have been paid, will amount to about 25% of
the sum of the annual premiums.
Written Work
Rates as given in the table on page 262.
1. What is the premium on an ordinary life policy of
$5000 at the age of 30?
2. A man at the age of 25 takes out a $ 2000 ordinary life
policy. If he dies after paying 16 premiums, what per cent
of the face of the policy has been paid in premiums ?
3. If, in example 2, the insured had taken a 20-payment
life policy, what per cent of the face of the policy would
have been paid in premiums ?
4. A young man at the age of 20 took out a 20-payment
life policy for 12000. The dividends at the end of 20 years
amounted to $142 per thousand. What was the net cost of
this insurance at the expiration of this policy, if the interest
on the premiums is not considered ?
5. What is the premium on a 20-year endowment policy
for $5000 at the age of 40 ?
6. The first annual premium on a 20-year endowment
policy for $8000 amounts to $453.60. What is the age of
the insured ?
7. If a man 25 years old takes out a 20-term policy for
$ 3000, how much will he have paid for his insurance at the
close of the term ?
8. A man at the age of 25 took out a 20-payment life
policy for $1000. His dividends for the 20 years amounted
to $150.45. What was the amount of his premiums, less the
dividend?
264 PERCENTAGE
COMMERCIAL DISCOUNT
1. I owe a bill of $50 clue in 60 days. As the creditor
needs the money he offers to take $40 if I pay at once. What
per cent is the reduction ? On what is the reduction reckoned ?
2. A catalogue lists goods at 11.00, -$.50, $.25, subject to
a discount of 20 %. Find the net price of each article.
3. In the above examples what numbers correspond to the
base in Percentage ?
The fixed or list price of an article or the amount of an
obligation is always considered the base.
Commercial discount is a reduction from the fixed or list
price of an article, or from the amount of a bill or obligation.
1. Trade discounts are reductions from the fixed or list price of an
article at the time of sale.
2. Time discounts are reductions from a bill or other obligation for
payment within a certain time.
3. Cash discounts are reductions made for the immediate payment of
a bill of goods sold on time.
The net price is the price after trade discounts have been deducted.
Written Work
1. Neckties listed at $6.00 per dozen are sold at 50 % dis-
count. Find the net price. What is the kind of discount ?
2. An agent buys $100 worth of goods on 60 days' time,
or 5% off, if paid immediately. What kind of discounts is
he offered ?
3. A merchant offers $100 worth of goods for $90. What
is the kind of discount ? What is the per cent of discount ?
4. I pay cash for a bill of goods sold on 60 days' time, and
thereby get 10% discount, or $12.50. Find the amount of
the purchase. Why is this a cash discount ?
COMMERCIAL DISCOUNT 265
5. A suit marked $40 is offered for $28. Find the per
cent of discount. What is the kind of discount ?
6. A merchant buys boys' suits at $60 per dozen list
price, less 20 %. What is the kind of discount ?
7. A dealer sells $75 worth of goods at 5 % discount, if paid
within 60 days, or 10 %, if paid in cash. Explain the different
kinds of discount, and the amount saved by paying cash.
8. Why does a cash discount always cut out a time dis-
count ? What discount is made without reference to time ?
Business houses print on their billheads their terms of
credit. For example:
1. "Terms: 60 days net; 3% off 10 da,"
2. "Terms: 00 days net; 60 days 2%; 10days5%."
3. "Terms: 30 days net ; cash 5%, etc."
9. Johnston Bros., Toledo, O., purchase of Amidon & Co.,
Chicago, $300 worth of merchandise. Terms: 60 da. net;
5% off 10 da. Find the discount if paid within 10 days.
10. I buy $600 worth of goods. Terms: 30 da. net;
5 % off for cash. Find the amount I save by paying cash.
11. Explain the meaning of a bill head which reads as
follows: "Terms: 60da.net; 10% cash."
12. $200 worth of goods are bought Aug. 10, 1905. Terms:
60 da. net ; 5 % off 30 da. ; 10 % off 10 da. Find the amount
saved by payment Sept, 5.
13. Jamison & Son, Baltimore, Md., bought $1500 worth
of merchandise from Brown & Co., Philadelphia. Terms:
90 days net; 2% off 60 da.; 5% off 30 da.; 10% off cash..
What was the amount of the bill if cash was paid ?
14. Which is the best discount on a bill of goods for $200
and how much: 10% for cash, 5% for 60 days, or 2% for
90 days? Explain why.
266 PERCENTAGE
Successive Trade Discounts
Wholesale merchants, publishers, and manufacturers
usually have fixed price lists for their goods from which the
retailer gets a certain trade discount. If a reduction from
these prices is to be made, an extra discount is taken from
the former discount price.
1. A music dealer buys an organ on 60 days' time, list
price $100 at 40 %, 20 % off. Find the net price.
.40 X $100= $40, 1st discount Observe that, when
$100 - $40 = $60, 1st discount price more than one discount
-.„. ,»..« rti t is given, each succes-
.20x$60 =$12, 2d discount . * ' . , ,
' " ^ w v ' sive discount is reckoned
$60 -$12 =$48, net COSt on the last discount price.
2. Providing the music dealer gets a further discount of
10 % for cash, find the price of the organ.
3. On what discount price was the last discount
reckoned?
N OTE . — The trade discount is first deducted ; then the cash discount
is taken from the remainder.
Find the net cost of articles listed at :
4. $90, discount 30%, 10%.
5. $120, discount 25 %, 20%.
6. $240, discount 10%, 331%.
7. $ 35, discount 20 % , 5 % .
8. $12.50, discount 20%, 20%.
9. $348, discount 20%, 10%, 5%.
10. $100, discount 10 %, 5 %, 2 %.
11. $425, discount 37^%, 16%.
12. $400, discount 20%, 10%, 5%.
13. What is the net price of a bill of goods for $5700
after discounts of 25 %, 20 %, and 5 % are allowed ?
COMMERCIAL DISCOUNT 26
>i
14. The amount of discounts at 20%, 5% off, is $42.
What is the list price of the bill ?
Suggestiox. — The total discount expressed in per cent is 20%+ (5%
of 80%),or 24 %. $42 is 24 % of what number ?
15. What is the cost of a bill of farming implements
listed at $480, discounts 50%, 5%, and freight $3.60 ?
16. Goods marked $50 were bought at 10 % trade discount,
2% off for cash. If sold at $55, what was the rate of gain ?
17. A dealer bought 50 gross of buttons for 25%, 10 % ?
5% off and sold them for $35.91, making a profit of 12%.
What was the list price of the buttons per gross ?
18. A grocer is offered a discount of 10 % from one firm
on a bill of goods of $1000, and two successive discounts of
5 % by another firm on the same bill. Which is the better
offer and how much ?
19. What is the difference between a discount of 15 %, 5 %,
on a bill of $2000 and a discount of 5 %, 15%, on the same
bill ? Show that the same result is obtained from any num-
ber of discounts on a bill in whatever order they are taken.
20. Three firms bid on the glass for a building as follows :
(1) $2000, 00%, 20%, off. (2) $2100, 70%, 10%. off.
(3) $2400, 80%, 10%, off. Which offer is the best and
how much ?
21. A jobber buys merchandise listed at $1500, at 20%,
15 % off, and sells at 15 %, 1 %, 5 % off the list price. Find
his profits.
22. I purchased $2000 worth of goods trade discount
20%, cash 5%. I pay cash plus $5.95 freight. What is
the entire cost of the goods ?
23. Find a single discount equal to successive discounts
of 40%, 20%, 10%.
268
PERCENTAGE
COMMERCIAL BILLS
1. On June 10, 1906, James Boydson, Sandusky, O.,
bought of J. M. Gordon & Co., Cleveland, O., 35 dozen
bronze locks at $5.50 per dozen, less 20 %, 10%. Terms:
30 days net; 5% off 10 days.
Form of Bill
Cleveland. O., fane /O, /<?06.
Bou<$r;t of J. M. GORDON & CO.,
Hardware Merchants
Terms : 30 cla,y& net ; 5 °Jo o-f^ /O cLx^i^.
Re*
,&iA>-&C
35 J&oa. fduyyvit, Lo-@£& @$5. 60
X&o-a* 5/ to
&aoA, t&QA> 5%
f. 711. gcyutcni V &0-.
53
50
<?0
60
$3
f/31
67
138
6
In the above problem J. M. Gordon & Co. received payment within
10 days, so a cash discount of 5 % was deducted from the $ 138.60, making
the payment 1131.67. Had the bill not been paid for 30 days, the net
amount would have been $ 138.60.
2. Harmon & Co., Seattle, Wash., order from Peabody &
Sons, Chicago, 111., 10 doz. Acme lawn mowers, list $10 each
less 40 %, 20%. Terms: 60 days net; 5% off 10 days.
Make out receipted bill if paid in 10 days.
LOCAL AND STATE TAXES 260
3. Howard Johnston, Johnstown, N.Y., orders from the
Acme Buggy Co., Cincinnati, O., 12 buggies, list 1100, less
40 %, 20 %. Terms : 30 days net ; 2 % off in 10 days. Make
out receipted bill if paid in 10 days.
4. Fisher Bros., Hagerstown, Md., order from the Fulton
Hardware Co., New York, N.Y., the following; discount 33^%:
1 Gas Range @ 8 32.00
100 ft. Hose @ .15
24 Garden Rakes @ .50
Terms : 30 days net ; 2 % off in 10 days. Make out receipted
bill if paid in 30 days.
LOCAL AND STATE TAXES
A tax is a sum of money levied on a person, his prop-
erty, or business for public purposes.
Cities, towns, townships, and counties levy and collect taxes annually
on all taxable property. States ordinarily levy taxes only on special
kinds of property.
A poll tax is a tax levied upon male citizens over 21 years
of age without regard to the property they own.
Real property, or real estate, is any fixed property ; as land
and buildings erected thereon.
Personal property is any movable property ; as money,
stocks, furniture, etc.
An assessor is a person elected or appointed to estimate
the value of property to be taxed.
The rate of taxation is a certain number of mills on each
dollar of assessed valuation, or a number of cents on a hun-
dred dollars of assessed valuation. Thus, a tax levy of 2
mills on the dollar, means 2 mills (or t 2 q%) on each dollar.
A collector is a person elected or appointed to collect the
tax. He receives either a salary or a commission.
270 PERCENTAGE
I. Find my taxes on property assessed at $5000 on which
I pay 5 mills on the dollar.
Find the tax on property assessed at :
2. $2000, tax levy 3 mills. 6. $3200, tax levy 2| mills.
3. $3000, tax levy 5 mills. 7. $5000, tax levy 3| mills.
4. $5000, tax levy 12 mills. 8. $12000, tax levy 5| mills.
5. $8000, tax levy 11 mills. 9. $20000, tax levy J mill.
How many mills on the dollar is paid if the assessment is :
10. $6000, taxes $60? 14. $600, taxes $7.20?
II. $1200, taxes $24? 15. $1000, taxes $45?
12. $8000, taxes $32? 16. $5000, taxes $10?
13. $ 6000, taxes $ 3 ? 17. $ 950, taxes $1.90?
Find the assessed valuation if the taxes are :
18. $4.00, rate 8 mills. 22. $60, rate 3 mills.
19. $12.00, rate 12 mills. 23. $200, rate 10 mills.
20. $6.50, rate 13 mills. 24. $180, rate 6 mills.
21. $20, rate 4 mills. 25. $200, rate 25 mills.
Written Work
1. A town whose property is assessed at $1750000 needs
$5150 for improvements; there are 620 persons who pay a
poll tax of $1.25 each. What is the rate of taxation, and
what is Mr. Randolph's tax, whose property is valued at
$6500 and who pays his own and one other poll tax?
620 x $ 1.25 = $775, the amount of poll tax.
$5150 - $775 = $4375, the amount of property tax.
$4375 -- $1750000 = .0025, the rate of taxation.
.0025 x $6500 = $16.25, Mr. Randolph's property tax.
2 x $1.25 = $ 2.50, Mr. Randolph's poll tax.
$16.25 + $2.50- = $18.75, Mr. Randolph's entire tax.
LOCAL AND STATE TAXES
271
Comparative Study
The assessed valuation corresponds to what term in Percentage t
Tax corresponds to what term in Percentage f
The rate of taxation corresponds to what term in Percentage f
Find the missing terms :
Estimated
Valuation
ASSESSED
Valuation
Rate
Taxes
Poll Tax
2.
$ 23000
$19000
.002
( )
3 polls, $1 each
3.
$147500
80%
.002
( )
2 polls, $2 each
4.
$ 120000
$95000
( )
$332.50
5.
125%
( )
.015
11.25
6.
150%
( )
.022
$3269.20
7. The assessed valuation of a town is $ 900000 and the
amount of taxes to be raised is $ 16200. What is the rate of
taxation and what is Mr. Owen's tax who owns property
assessed at $ 10000 and personal property assessed at $2500?
8. How much tax does a farmer pay who owns 80 acres
of land valued at $ 100 an acre, assessed at § of its value, and
personal property assessed at 11250, if the rate of taxation is
three mills?
9. Mr. Day's city tax, at the rate of 13 mills on the dollar,
is $84.50. What is the estimated value of his property, if
it is assessed at i of its value ?
10. A collector paid to the borough authorities 821898.50
after deducting his commission of 2k% for collecting. What
was the amount of taxes collected, and what were the collec-
tion fees?
Si <;t;i stion. —The proceeds of each dollar returned by the collector
was $.97£.
272 PERCENTAGE
11. The real estate of a town is valued at $985470 and
the personal property at $195645. A tax of $7866.69 is
to be raised. There are 780 persons, each assessed a poll
tax of $1. How much tax will Mr. Cowperthwait, a non-
resident (paying no poll tax), pay whose property is assessed
at $12460?
12. Mr. Arbuthnot's property has an actual valuation of
$4800. If he pays 17 mills city tax and 3| mills county tax
on a | valuation, in a certain year, find the amount of his tax
for that vear.
13. A borough bridge cost $4380.48. It was paid by a
tax on the assessed valuation of the borough property. If
4% of the tax was uncollectable, and the collector's fee was
2|%, what was the borough assessment?
14. A tax collector's report in a certain city, for Oct. 2,1906,
showed by the assessment in the tax book that $22742.90
was collected during September. If 5 % discount was allowed
on each one's taxes paid, and the collector's commission was
2 %, find the amount paid to the school treasurer.
DUTIES OR CUSTOMS
The revenues of our national government are of two
kinds : internal revenues and custom revenues, or duties.
Internal revenue is a tax on the manufacture or sale of
malt liquors, tobacco, etc.
Custom revenues, or duties, are taxes on goods imported
from foreign countries. These are collected at the custom-
houses, which are maintained by the government at various
ports of entry.
A tariff is a schedule of duties on imports fixed by our
government.
DUTIES OR CUSTOMS 273
All merchandise brought into our country is classified as
follows :
(1) Merchandise on the free list, that is, free of duty.
(2) Merchandise subject to an ad valorem duty, that is, a
certain per cent of the cost of the goods, as shown by the
invoice.
Invoices are statements showing the market price of goods, expressed
in the money of the country where the goods are bought.
(3) Merchandise subject to a specific duty, that is to a
certain amount per yard, pound, bushel, etc., without regard
to value.
(4) Merchandise subject to both an ad valorem and a
specific duty.
For example, the duty on Brussels carpet is 28 j* per square yard
specific and 40% ad valorem.
Duties are not computed on parts of a dollar. If the invoice shows
a number of cents less than 50, they are rejected in computation of
duties. More than 50 cents are counted as another dollar.
Tare is an allowance made for the weight of the boxes or bags
used in packing. It is deducted before computing duties.
Comparative Study
The cost of imported goods corresponds to what term in Percentage ?
The rate in duties corresponds to what term in Percentage f
The duty corresponds to what term in Percentage ?
Written Work
Find the duty on the following imports :
1. 1650 lb. of hops, specific duty 12 ^ per pound.
2. $6250 worth of clocks at 40 % ad valorem.
3. $ 10000 worth of uncut diamonds, ad valorem duty
60%.
4. 2400 pounds of butter at ^ per pound.
HAM. COMPL. AKITII. — 18
274 PERCENTAGE
5. 150 boxes plate glass (each 25 plates 16 in. by 24 in.),
duty $.08 per square foot.
6. 800 yd. Brussels carpet 27 in. wide, valued at $ 1.45
per yard. Duty 40% plus $.44 per square yard.
Explain the different kinds of duty in this problem.
7. 500 boxes cigars (each box weighing 1 lb. and con-
taining 100 cigars) invoiced at $3.50. Duty 25% plus
$4.50 per pound.
8. The specific duty on macaroni is 1| $ per pound. If
the duty is $57.90, find the number of pounds imported.
9. A department store in Buffalo imported from London
3i tons (1. T.) of woolen blankets invoiced at 40 cents per
pound, subject to a specific duty of 22 cents per pound and
an ad valorem duty of 30%. How much duty was paid?
10. A merchant imported from Sheffield, England, 12 gross
of table knives costing 10s. per dozen in Sheffield. If the
duty was $1.44 per dozen and 15% ad valorem, and trans-
portation $9.60, find the cost per dozen delivered.
11. A certain painting in Rome is purchased for 50000
lire ($ .193). Find the cost when delivered in New York, if
the duty is 20% and freight and insurance $49.75.
12. Find the entire cost of importing 3000 lb. of worsted
yarn, invoiced at £360, if the freight charges are $11.75;
duty 27|^ per pound ; and 40% ad valorem.
13. Brown & Co. import 1200 sacks of cocoa, each contain-
ing 90 lb. If the duty is 2±? per pound and tare 1 %, find
the duty on the goods when imported.
14. An importer paid $480 duty on lace on which he
paid a duty of 60 %. If the lace cost 50^ per yard, find the
number of yards imported.
INTEREST
Interest is money paid for the use of money.
The principal is the sum loaned.
The amount is the principal plus the interest.
The rate is the number of hundredths of the principal paid
for its use for one year.
Time is always a factor in computing interest.
SIMPLE INTEREST
Simple interest is interest allowed on the principal only.
Interest is generally computed on the basis of a year of
12 months of 30 days each, or 360 days. This is called
simple or common interest.
1. How much is the interest on $200 at 6% for 1 year?
for 1 month ? for 15 days ?
2. How much is the interest on $100 at 6% for 1 year?
for 1 month ? for 15 days ?
How much is the interest at 6 % on :
3. $100 for 11 years? 10. $200 for 120 days?
4. $200 for 21 years? 11. $60 for 1 year 1 month ?
5. $200 for 6 months? 12. $600 for 240 days?
6. $300 for 9 months ? 13. $300 for 30 days? 1 day ?
7. $200 for 60 days? 14. $500 for 11 mo. 12 da.?
8. $600 for 90 days 9 15. $700 for 6 mo. 15 da. ?
' 9. $500 for 4 months ? 16. $ 800 for 3 mo. 5 da.?
•27."»
276 INTEREST
Written Work
1. Find the interest on $600 for 2 yr. 9 mo. at 6%.
dh />aa i Comparative Study
$600 = principal , m ,
1. The principal corresponds to what
term in Percentage ?
$36.00 =illt. for 1 yr. o. The rate corresponds to what
2f =time in years term in Percentage f
$ 27 =int. for 9 mo. <*. The interest corresponds to what
72 =int. for 2 yr. term in Percenta ff e?
1, nn : — : — ;; tt- 1 t. 4. What term is found in interest
$99 =int. tor 2 yr. 9 mo. ,, , . , e , . n , 9
J that is not tound in Percentage ?
The interest equals the product of the principal, the rate, and
the time expressed in years.
Find the interest on :
2. $250 for 21 yr. at 6%. 7. $320 for 5 yr. at 5J%.
3. $708 for 4 yr. at 7%. 8. $600 for 3^ yr. at 6%.
4. $650 for If yr. at 4%. 9. $200 for 11 mo. at 6%.
5. $800 for 10 mo. at 6 %. 10. $570 for If yr. at 4%.
6. $260 for If yr. at 4* %. 11. $290 for 3 yr. at 6|%.
Method by Aliquot Parts for Years, Months, and Days
Written Work
1. Find the amount of $ 960 for 3 yr. 7 mo. 18 da. at 5 %
Principal = I960
Rate = .05
Int. for 1 yr. = $ 48.00
Int. for 3 yr. = 3 x $48, or $144.00
Int. for 6 mo. = * of $ 48, or 24.00
Int. for 1 mo. = j\ of $48, or 4.00
Int. for 15 da. = \ of $4.00, or 2.00
Int. for 3 da. = T \ of $4.00, or .40
Int. for 3 yr. 7 mo. 18 da. = $174.40
Principal = 960.00
Amount = $1134.40.
SIMPLE INTEREST 277
Find the amount of :
2. $1400 for 3 yr. 3 mo. 12 da. at 6%.
3. $975 for 5 yr. 8 mo. 24 da. at 4* %.
4. $360 for 4 yr. 5 mo. 10 da. at 5.] %.
5. $480 for 2 yr. 9 mo. 15 da. at 7%.
6. $2700 for 6 yr. 6 mo. 20 da. at o%.
7. $1040 for 4 yr. 9 mo. 18 da. at 8 % .
8. $176.45 for 3 yr. 7 mo. 25 da. at C %.
9. $1840 for 5 yr. 4 mo. 27 da. at 4|%.
10. 81875 for 7 yr. 2 mo. 6 da. at 8%.
11. 8200 for 2 yr. 1 mo. 15 da. at 6%.
12. $1200 for 1 yr. 8 mo. 10 da. at 7%.
13. $97.30 for 3 yr. 3 mo. 3 da. at 8%.
14. 83500 for 2 yr. 7 mo. 24 da. at ('.%.
Find the interest on :
15. 81575 at 5% from Jan. 3, 1907 to Sept. 5, 1909.
16. 81790.80 at 4* % from Sept. 8, 1906 to Dec. 10, 1910.
17. $2005 at 3.] % from June 12, 1906 to Oct. 8, 1909.
18. $4000 at 4| % for 5 yr. 8 mo. 21 da.
19. $4670 at 5| % from Oct. 10, 1906 to June 5, 1907.
20. $2890 at 6 1 % from April 3, 1905 to Oct. 1, 1908.
21. $290.75 at 6% from May 8, 1905 to Sept. 19, 1908.
22. 89S0.60 at 5' % from Aug. 7, 1906 to June 7, 1912.
23. $759.40 at 1% from May 9, 1907 to June 10, 1909.
24. $800.50 at 4 T \, % for 1 yr. 7 mo. 20 da.
25. 8200.80 at b\% from May 3, 1906 to Sept. 5, 1911.
278 INTEREST
The Sixty Day Six Per Cent Method
Since the interest on $100 at 6% for 1 yr. is $6, the in-
terest for 60 da. Q of a year) = jt of $6, or $1.
How much is the interest at 6 % for 60 days on :
1. 1200 3. $300 5. $150 7. $125
2. $ 50 4. $ 6 6. $250 8. $120
Does the interest in each problem equal j^ of the princi-
pal?
Written Work
1. Find the interest at 6% on $650 for 90 days.
Interest for 60 days (2 mo.) = $6.50
Interest for 30 days (1 mo.) = $3.25
Interest for 90 days (3 mo.) = $9.75
The interest of any principal for 60 da. (2 mo.) at 6 % is
found by moving the decimal point in the principal two places
to the left.
Find the interest at 6 % on :
2. $275 for 2 mo. 5. $198 for 3 mo.
3. $ 95 for 3 mo. 6. $475 for 3 mo.
4. $172 for 3 mo. 7. $280 for 3 mo.
8. Find the interest at 6% on $345 for 8 mo.
The interest for 2 mo. = $3.45, or T ^ 5 of the principal
The interest for 8 mo. = 4 x $3.45 = $13.80
Find the interest at 6% on:
9. $125 for 120 da. 13. $805 for 75 da.
10. $190 for 4 mo. 14. $425 for 2 1 mo.
11. $325.50 for 8 mo. 15. $500 for 30 da.
12. $ 62.50 for 240 da. 16. $280 for 45 da.
SIMPLE INTEREST 279
17. $ 600 for 6 mo.
18. $350 for 9 mo. (8 mo. + 1 mo).
19. $450 for 19 mo. (18 mo. + 1 mo.).
Find the interest at 6 <f on :
20. $550 for 22 mo. 30. 8845.50 for 1 yr. 6 mo.
21. $640 for 18 mo. 31. $850 for 1 yr. 2 mo.
22. 8435 for 180 da. 32. $392 for 1 yr. 4 mo.
23. $552 for 21 mo. 33. $362 for 1 yr. 2 mo.
24. $632 for 120 da. 34. $563.25 for 1 yr.
25. $562 for 16 mo. 35. $147.50 for 1 yr.
26. $335 for 8 mo. 36. $150 for 90 da.
27. $222 for 9 mo. 37. $650 for 120 da.
28. $375 for 1 yr. 4 mo. 38. $160 for 100 da.
29. $387.50 for 1 yr. 6 mo. 39. $60.20 for 10 mo.
40. Find the interest on $562.50 for 1 yr. 10 mo. 15 da.
at 6%.
The interest for 2 ino. = $ 5.625, or ^ of the principal
The interest for 20 mo. = 56.25, or 10 x 5.625
The interest for 15 da. = 1-106 , or \ of 15.625
Int. for 1 yr. 10 mo. 15 da. = *i>:J.2>>l
Find the interest at 6 $> on :
41. $ 500 for 6 mo. 15 da. 49. $175.50 for 105 da.
42. $250 for 8 mo. 20 da. 50. $ 150 for 9 mo. 12 da.
43. $360 for 5 mo. 10 da. 51. $387.50 for 6 mo. 25 da.
44. $475 for 90 da. 52. $125.50 for 10 mo. 21 da.
45. $900 for 6 mo. 25 da. 53. $345.50 for 6 mo. 15 da.
46. $125 for 4 mo. 19 da. 54. $ 755 for 1 yr. 9 mo. 6 da.
47. $ 325 for 6 mo. 23 da. 55. $ 544 for 5 yr. 3 mo. 5 da.
48. $25.50 for 3 mo. 29 da. 56. 8 80. SO for 2 yr. 15 da.
280 INTEREST
57. $5175 for 4 yr. 10 mo. (60 mo. — 2 mo.).
58. $640 for 3 yr. 2 mo. (40 mo. — 2 mo.).
59. $ 1240.60 for 2 yr. 9 mo. 15 da.
From the interest on any principal at 6 %, the interest at
other rates may be found by adding or subtracting aliquot
parts of the interest at 6 % ', thus,
4 % = 6 % - i of 6 % 7-1 % = 6 % + I of 6 %
41 % = 6 % - J of 6 % 8 % = 6 % + £ of 6 %
5 % = 6 % - 1 of 6 % 9 % = 6 % + 1 of 6 %
7 cj = 6 % + \ of 6 % 10 % = i of 6 % x 10
60. Find the interest at 5 % on $ 360 from May 1, 1905
to March 16, 1907.
Time, 1 yr. 10 mo. and 15 da. = 22| mo.
Principal = $380.00
Interest of $ 360 for 20 mo. at 6 % = $ 36.00
Interest of $360 for 2 mo. at 6 % = 3.60
Interest of $ 360 for j mo. at 6 % = .90
Interest of $360 for 22$ mo. at 6 % = $40.50
Less interest of $ 360 for 22} mo. at 1 % = 6.75
Interest of $360 for 22 h mo. at 5% = $33.75
Find the interest on :
61. $216 from July 25, 1891 to Sept. 10, 1893, at 6 %.
62. $348 from Jan. 16, 1893 to Feb. 4, 1896, at 7 %.
63. $ 650.40 from April 19, 1903 to March 4, 1906, at 4 %.
64. $1200 from Nov. 2, 1900 to Oct. 19, 1903, at 4| %
65. $1800 from Aug. 25, 1902 to June 1, 1906, at 4| %.
66. $476.25 from Aug. 19, 1902 to June 1, 1906, at 8 %.
67. $1600 from Sept. 25, 1902 to May 18, 1906, at 4| %.
68. $164.88 from July 3, 1903 to Dec. 31, 1905, at 5 %.
SIMPLE INTEREST 281
The One Dollar Six Per Cent Method
The interest on $1 for 30 da. (1 mo.) = $.005.
The interest on $ 1 for 1 da. (^ of $.005) = $ .0001.
• Change the time to months and days. Since the interest on
$1 for 1 mo. is | of a cent and for 1 da. ^ of a mill, the
interest on one dollar ivill be ^ as many cents as there are
months and ^ as many mills as there are days. Midtiply the
result by a number equal to the number of dollars in the
principal.
Written Work
1. What is the interest on $240.60 for 2 jr. 3 mo. 13
da. at 6%?
2 yr. 3 mo. = 27 mo.
Interest on $1 at 6 % for 27 mo. = $.135
Interest on $ 1 at 6 % for 13 da. = $.002$
Interest on $1 at 6% for 2 yr. 3 mo. 13 da. = $.137$
Interest on $240.60 = 240.60 x $.137 f, or $33.00
Find the interest at 6% on :
2. $ 7450 for 93 da. 14. $ 8790 for 5 mo.
3. $ 8400 for 65 da. 15. $ 8250 for 6 mo.
4. $ 9800 for 40 da. 16. $ 150 for 3 mo. 6 da.
5. $ 8440 for 72 da. 17. $ 180 for 5 mo. 9 da.
6. 8 5500 for 5 da. 18. #195 for 6 mo. 10 da.
7. $ 0750 for 8 da. 19. $ 250 for 8 mo. 12 da.
8. 1 4765 for 25 da. 20. $ 340 for 2 yr. 4 mo.
9. 8 6245 for 110 da. 21. $275 for 3 yr. 8 mo.
10. $8425 for 52 da. 22. $450 for 1 yr. 5 mo.
11. $ 5150 for 3 mo. 23. $ 675 for 3 yr. 7 mo.
12. * 8465 for 4 mo. 24. $64.60 for 2 yr. 9 rao.
13. 8 9640 for 7 1110. 25. $78.40 for 4 yr. 3 mo.
282 INTEREST
Find the interest :
26. At 5% on $ 237.50 from Jan. 3, 1906 to Sept. 11, 1908.
27. At 7% on $309.75 from May 5, 1905 to Jan. 12, 1907.
28. At 7| % on $ 7500 from June 12, 1907 to Nov. 1, 1909.
29. At 61% on $ 6225 from Oct. 11, 1907 to Mch. 1, 1910.
30. At A\% on $750 from Feb. 12, 1907 to Aug. 9, 1911.
31. At 8% on $2900 from July 1, 1906 to May 10, 1910.
32. At 4% on $3675 from June 4, 1907 to Apr. 1, 1910.
33. At 3% on $ 290.80 from Nov. 12, 1907 to Apr. 10, 1912.
34. At 5% on $875 from June 11, 1907 to Mch. 11, 1912.
35. At 5|% on $8000 from Apr. 1, 1907 to May 9, 1912.
Solve the following problems by both the six per cent
methods and compare results as to time and accuracy.
Find the interest from the following conditions:
Prin.
Time
Rate
Prin.
Time
Rate
36. $350
105 da.
1%
39. $129
23 mo.
9%
37. $685
4 mo.
8%
40. $750
13 mo.
n%
38. $850
87 da,
6%
41. $492
97 da.
H%
42. Fin
d the amount of
$1275.80 for 1 yr
. 7 mo.
24 da.
at 1%.
43. Find the interest of $4780 from April 1, 1907 to
Sept. 18, 1909, at 6|%.
44. On the 16th of September, 1907, I borrowed $3600 at
8 %. How much will settle the loan April 1, 1909 ?
45. July 28, 1907, a broker borrows $3200 at 5% in-
terest and on the same day loans it at 7| % interest. If full
settlement is made April 1, 1909, how much will the broker
make by reloaning the money ?
PROBLEMS IN SIMPLE [NTEREST 283
PROBLEMS IN SIMPLE INTEREST
Finding the principal.
Written Work
1. What principal invested at 4 % per annum will yield an
annual income of $200 ?
Since the interest on $ 1 for 1 year at 4% is $.04, as many dollars must
be invested to yield 8200 per year as $.04 is contained times in $200.
$200 h- $.04 = 5000. Hence, $5000 must be invested.
TJie principal equals the given interest divided by the interest
on $lfor the given time at the given rate.
2. What principal at 4£% will gain $213.75 interest in
4 yr. 9 mo.?
3. What principal at 5% will gain $120.70 interest in
3 yr. 6 mo. 18 da.?
4. What principal at 8% will gain an interest of $163.20
from Sept. 30, 1903 to June 12, 1905 ?
5. A man gave his note April 1, 1901, at 6%. When he
settled the note Aug. 13, 1903, he paid $195.25 interest.
What was the principal, or face of the note ?
Finding the rate.
Written Work
1. At what rate must $500 be invested to yield $75
interest in 2 yr. 6 mo. ?
Since the interest on $500 for 2\ yr. at 1% is $12.50, it will require
a rate of as many per cent to yield $75, as $12.50 is contained times in
$75, or 6%.
The rate equals the given interest divided by the interest for
the given time at 1 % •
2. The interest on $1125 for 3 yr. 4 mo. 24 da. is $229.50.
What is the rate per cent ?
284 INTEREST
3. The interest on $1800 for 4 yr. 8 mo. 16 da. is 1424.
What is the rate ?
4. At what rate will $2460 give $682.65 interest in 5 yr.
6 mo. 18 da.?
5. A note for $880 was given July 5, 1900, and settled
May 2, 1904, for $1081.96. What rate per cent was charged?
Finding the time.
Written Work
1. In what time will $450 at 0% yield $90 interest?
Since the interest on -1450 for 1 yr. at 6% is $27, the required time is
as many years as $ 27 is contained times in $90, or 3$ yr.
The time in years equals the given interest divided by the
interest at the given rate for 1 year.
2. In what time will $275 gain $55 interest at 6 % ?
3. In what time will any principal double itself (that is,
gain 100 % of itself) at 5 % ? at 6 % ? at 8 % ?
4. In what time will any principal treble itself (that is,
gain 200% of itself) at 5 %? at 6 % ? In what time will it
quadruple itself at 8 % ? at 10 % ?
5. A note of $500 at 5% interest was paid May 1, 1905,
the interest amounting to $63.75. When was the note
given?
REVIEW PROBLEMS
1. Find the interest on $80 for 6 yr. 6 mo. 18 da. at 6 %.
2. In what time will $180 at 5 % yield $22.50 interest?
3. In what time will $600 amount to $715.50, at 5|%
interest ?
REVIEW PROBLEMS 285
4. At what rate will $1650 gain #326.70, in 4 yr. 4 mo.
24 da.?
5. What sum of money loaned at 0% will give a semi-
annual income of $13.02 ?
6. Interest $31.80; time 3 yr. 6 mo. 12 da.; rate 5%.
Find the principal.
7. The interest of ^ of a principal for 3 yr. 6 mo. at 6%
is $19.25. Find the principal.
8. If $186 pays a debt of $150 which has been due for
4 years, what is the rate of interest ?
9. I borrowed $450 at 5% and kept it until it amounted
to $525. When did I settle the note ?
10. Dec. 1, 1901, 1 loaned $300 at 5* %. Find the amount
due March 16, 1905.
11. Find interest at 6% on $350 from June 1, 1898 to
Aug. 13, 1902.
12. If $300 was borrowed April 1, 1902, at 5%, when
should the principal and interest be paid that their sum may
be $357?
13. The amount of a certain principal at 6 % for a given
time is $780, and at 10% for the same time it is $900.
Find the principal.
14. If $148 is loaned April 1, 1902, at 5%, when will it
amount to $179.45 ?
15. What principal for 3 mo. at 8 r / c will yield the same
interest as $5100 for 5 yr. 8 mo. at 6 % ?
16. A farmer bought 75 acres of land at $ 50 an acre,
paying ^ cash, and giving his note for the balance, due in
3 yr. 6 mo., with interest at 6%. What was the amount of
the note at maturity ?
286
INTEREST
ANNUAL INTEREST, OR SIMPLE INTEREST ON UNPAID
INTEREST
In some states when a note reads with interest " payable
annually," simple interest may be collected upon the princi-
pal and upon each year's interest from the time it was due
until paid.
In most states annual interest is not collectable by law.
Interest payable annually is simple interest; but interest collected
on the principal and on the overdue payments of simple interest is
annual interest.
Written Work
1. James Brown borrows $1200 at 6% interest, "payable
annually." In case no interest is paid for 3 years, 6 months,
and 15 days, how much money is necessary to pay the debt ?
Simple interest on $ 1200, at 6%, for 3 yr. 6 mo. 15 da.
The interest for each year is $72.
The 1st annual int., $72, remains unpaid for 2 yr. 6 mo. 15 da.
The 2d annual int., $72, remains unpaid for 1 yr. 6 mo. 15 da.
The 3d annual int., $72, remains unpaid for 6 mo. 15 da.
Interest on $72 at 6% for . . . 4 yr. 7 mo. 15 da.
The annual interest on $1200 for 3 yr. 6 mo. 15 da.
The principal ... .....
The amount of $ 1200 at annual interest for 3 yr. 6 mo. 15 da.
255
19
274
1200
1474
98
98
00
98
Annual interest is the simple interest on the principal for
the given time plus the simple interest on each year's interest
for the time it remains unpaid.
2. Find the total interest due on a note of $ 675 for 2 years,
8 months, and 20 days at 6%, with interest payable annu-
ally, if no interest has been paid.
3. Find the amount of $6400 for 4 years, 5 months, and
15 days, with interest payable annually at 6 (f .
EXACT INTEREST lis;
4. An attorney collects a note of $3750 with annual in-
terest on it at 6% for 4 yr. 9 mo. 18 da. Kind the amount
collected and his commission on it at 10%.
EXACT INTEREST
Exact interest is simple interest on the principal reckoned
on the basis of 365 days to a common year and 366 days to a
leap year.
It is used in computing interest on all obligations by the United
States government ■; on all foreign securities; and to some extent by city
controllers and bankers.
Since common interest is computed on the basis of 12 months of 30
days each, or 360 'lays ; and exact interest is reckoned on the basis of 365
days to a common year, or 366 to a leap year, 1 day's exact iuterest
is jl-s of a year's common interest.
It is evident that the common and exact interest for 1 year are the
same. Thus, fff of one year's common interest equals one year's exact
interest. They differ only for parts of a year.
Written Work
1. Find the exact interest on $2400 for 95 days at 6%.
Exact interest for 1 year = 6% of $2400, or $144
Exact interest for 1 day = -353 of $144, or $.394.")
Exact interest for 95 days = 95 x $.3945, or $37.48
Exact interest is found by dividing the common interest at the
given rate for one year by 365 and multiplying the quotient by
the exact number of days.
Find the exact interest of:
2. 6800 for 78 days at 6 % . 6. $500 for 90 days at 9 %.
3. $2000 for 92 clays at 7%. 7. $1020 for 74 days at 10%.
4. 62400 for 115 days at 8%. 8. $6500 for 280 days at 6%.
5. 61775 for 100 days at 8|%. 9. $10000 for 61 days at 7%.
10. Find the exact interest on $1020 from Oct. 19, 1905
to April 1, 1907, at 6%.
288
INTEREST
Notk. — Why do we find exact interest for a fraction of a year only V
The exact number of days from Oct. 19, 1905 to April 1, 1907, is found
as follows : Oct., 12 da. ; Nov., 30 da. ; Dec, 31 da. ; Jan., 31 da. ; Feb.
28 da. ; March, 31 da. ; April, 1 da. Total, 164 days.
11. Find the exact interest on $1795.80 from July 7, 1901
to Sept. 1, 1907 at 7%.
12. The United States government paid exact interest at
4 (f on a warrant of $ 650000, 83 days past due. Compute
the amount paid.
COMPOUND INTEREST
Mr. Reed Colburn loans Robert Patterson $200 for 2
years at 6 % .
Suppose Mr. Patterson says to Mr. Colburn, at the end of the first
year : " I cannot pay you the $ 12 interest due, but will pay you interest
at 6 % on the $12 for a year." How much interest should Mr. Patterson
pay Mr. Colburn at the end of the 2 years? How does the 12-1.72 inter-
est differ from simple interest?
Compound interest is interest on both the principal and
the unpaid interest added to the principal when due.
Interest may be added to the principal annually, semiannually, or
quarterly, according to agreement.
Written Work
1. Find the compound
months at 0%.
Principal
Interest for 1st yr. atG %
Principal for 2d year
Interest for 2d yr. at 6 %
Principal for 3d yr.
Interest for 6 mo. at 6 %
Amount for 2 yr. 6 mo. at 6 %
Original principal
Compound interest for 2 yi
iterest on $200
for
2 years 6
$200.00
.
12.00
. ,
212.U0
. ,
12.72
. .
224.72
. .
6.74
6% •
231.46
.
► o
200.00
. G mo. at (
i% • ■
31.46
SAVINGS ACCOUNTS 289
Note. — 1. Unless otherwise stated in the agreement, interest is com-
pounded annually.
2. When interest is compounded semiannually, consider the rate
as A the annual rate, or if quarterly, £, etc.
2. Find the compound interest on $1000 for 2 years at
5%, with interest compounded semiannually.
3. Find the compound interest at 6% on $800 for 1 yr.
5 mo., interest payable quarterly.
4. Find the compound interest on $600 for 9 mo. at 6%,
interest payable quarterly.
SAVINGS ACCOUNTS
Compound interest is no longer allowed on notes. Its
only practical application for elementary schools is found in
computing interest on savings accounts.
Many banks to-day have a savings department. The amounts thus
deposited are not subject to check, but draw from 2% to 4% interest
which is usually compounded semiannually.
The interest periods are generally January 1 and July 1
of each year, although sometimes the interest is compounded
quarterly. Thirty days are reckoned to a month.
Interest on savings accounts is sometimes calculated from the 1st and
15th of each month succeeding the serend dejiosits. Thus, $ 10 deposited on
the 1st of any month would draw interest from date; but $10 deposited
on the 2c? of any month would draw interest from the loth: or money
deposited on the 16th w r ould draw interest from the 1st of the next
month. There is no fixed rule, however, as each bank determines for itself
when the interest date begins. No interest is allowed on a fractional
part of a dollar, and parts of a cent are omitted on all interest credits.
Most banks require notice from a depositor before a
savings account may be withdrawn. Amounts withdrawn
before the end of an interest period draw no interest for
that period.
HAM. COMPL. A KITH. — 19
290 . INTEREST
Written Work
1. On July 1, 1905, Raymond Wilkinson makes a sav-
ings deposit of $400 at 4% interest, payable semiannually.
If the interest at each period is added to the deposit, what
is the total amount in bank January 1, 1907 ?
Deposit July 1, 1905 $400.00
Interest on $400 at 4 % July 1, 1905 to Jan. 1, 1906 . .. 8.00
Amount in bank Jan. 1, 1906 408.00
Interest at 4 % on $408 from Jan. 1, 1906 to July 1, 1906 . 8.16
Amount in bank July 1, 1906 ...... 416.16
Int. at 4 % on $416 (why?) from July 1, 1906 to Jan. 1, 1907 8.32
Amount in bank Jan. 1, 1907 424.48
2. Find the difference between the simple interest on a
note of 8200 dated July 1, 1906, due in two years at 4| %,
and the interest on $ 200 deposited in a savings bank at 4 %,
compounded semiannually, for the same period.
3. A savings account of $150 deposited April 1, 1906,
at 3% interest, payable January 1 and July 1, is with-
drawn April 12, 1908. Find the amount withdrawn.
4. A savings bank pays 4% interest, calculated from the
1st and the loth of each month succeeding the several de-
posits. The deposits are Sept. 1, $20; Oct. 10, $15; Nov.
15, $20; Dec. 10, $25. Find the amount in bank the
following January 1, if the interest periods are January 1
and July 1.
5. The Lincoln School had on deposit in the Holmes Sav-
ings Bank Jan. 1, 1907, $495.80. The deposits were as fol-
lows: Feb. 1, $76.90; March 1, $105.05; April 1, $114.29;
May 1, $129.70; June 1, $98.75. Find the amount in the
bank Jan. 1, 1908, at 4 % interest, compounded the first of
January and July.
INVESTMENTS
291
Find the amount in bank from the following deposits
Deposit
Date
Rate
Int. Payable
Amount in Bank
6.
$200
Jan. 1, 1905
3%
Jan. 1 and July 1
July 1, 1906
7.
$150
Mar. 16, 1906
4%
Jau. 1
Jan. 1, 1908
8.
$875
May 29, 1906
21 %
Jan. 1
Jan. 1, 1908
9.
$1200
Aug. 10, 1906
2%
Jan. 1, Apr. 1,
July 1, Oct. 1
Jan. 1, 1908
INVESTMENTS
Compound interest tables are frequently used by insurance
companies, building and loan associations, and trust com-
panies, to calculate the income, from investments where the
interest is added each interest period to the amount invested.
The following table shows the amount of $1 at compound interest at
the given rates for 10 years.
Compound Interest Table
Yr.
n%
2%
n%
3%
n%
4%
1
1.0150 000
1.0200 0000
[.0250 in ii 10
1.0300 0000
1.0350 0000
1.0400 0000
8
1.0302 250
1.0404 0000
1.0506 2500
1.0609 0000
1.0712 2500
1.0816 0000
3
1.0456 784
1.0612 0800
1.0768 9062
1.0927 2700
1.1087 17S7
1.1248 6400
4
1 .0618 636
1.0824 3216
1.1088 1289
1.1255 0881
1.1475 2300
1.1698 5856
5
1.0772 840
1.1040 8080
1.1314 0821
1.1592 7407
t.1876 8631
1.2166 5290
6
1. 0984 488
1.1261 6242
1.1596 9342
1.1940 5230
1.2292 5533
1.2653 1902
7
1.1098 (50
1.1486 8567
1.1886 8575
L.2298 7887
1.2722 7926
1.S159 8178
8
1.1264 926
1.1716 5938
1.2184 0290
1.2667 7008
1.3168 0904
1.3685 6905
9
1.1488!
1.1951)9257
1.2488 6297
1.3H47 7318
1.3628 9735
1.4288 1181
10
1.1605 408
1.2189 9442
1.2800 8454
1.8489 1688
1.4105 9876
1.4802 (423
The compound interest on any amount for 4 years at 8% payable
semiannually is evidently the same as upon the same amount for 8 years
at 4 % payable annually.
The amount of any given principal for any given number
of years is found by multiplying the principal by the amount
of $1 at the given rate for the time as given in the table.
202 INTEREST
Written Work
1. Find the amount of $1200 invested for 7 years at 3^ %,
interest compounded annually.
2. Find the compound interest at 4 % on $ 10000 invested
for 9 years.
3. The amount of $12000, invested for 10 years at 31 %,
interest compounded annually, is divided equally among
3 sons. Find each one's share.
4. Find the amount of $1200 for 2 years and 6 months
at 4 %, compound interest payable semiannually.
PROMISSORY NOTES
Mr. James H. Ames, a grocer, Erie St., Buffalo, N.Y., has
an account of $52.00 against Robert Patterson for groceries,
and Mr. Ames asks Mr. Patterson to give him a note at 6 %
interest for the amount of the bill.
The note reads as follows :
$52.00 Buffalo, JV.T., Mv-. Bf, 1905.
c/ux, 'nvcyyttJt^ after date -_.c/--- promise to pay to the
order of jtawvea- /if. CL , wu&^
&C££y - 1 w- c i rrrrrrrrrr?ccrr^'rrrrrr^rcc'rrrr^^ / ars.
Value received, with interest at 6%.
f\ot>evt ^utt&VQycyyv.
1
A promissory note is a written promise to pay to a certain
person named in the note, or his order, a specified sum of
money at a specified time.
PROMISSORY NOTES 293
The Essentials of a Promissory Note :
1. It should state the place where and the time when given.
2. It should promise to pay to a certain person or to his order.
3. It should promise to pay a certain sum of money, expressed both in
figures and in writing.
4. It should state when the money is to be paid.
5. It should state by whom the money is to be paid.
6. It should state for value received.
(Xot absolutely necessary, but usually written in a note.)
7. It should state with interest and tlie rate, if it is an interest-bearing
note.
The promissor is called the maker of the note.
The person who is to receive the money is called the payee
of the note.
1. Who is the maker of the note on page 292?
2. Who is the payee of the note on page 292?
3. Find the amount to be paid when due.
4. The face of the note is the sum written in the note. What is the
face of the note on page 292?
5. This note reads "pay to the order of James H. Ames," and mean-
that Mr. Ames has the right to sell this note to any one by simply writing
his name across the back of the note and delivering it to the purchaser.
What words in the above note' give Mr. Ames the right to sell it?
When the owner of a promissory note writes his name
across the back of it, he is said to indorse the note.
If Mr. Ames indorses the note and then sells it to Mr. B.. and Mr. B.
indorses it and sells it to Mr. C, to whom does the note belong?
A promissory note, therefore, like any other property
may be bought and sold ; hence it is called negotiable paper.
When a note is made payable to a definite person, it cannot be trans-
ferred, and is therefore nut negotiable.
294
INTEREST
jlavi'&& CLnd&x^o-ru.
Indorsement in Full
Promissory notes may be indorsed as follows:
Indorsement in Blank
(1) In blank : In this form the
indorser simply writes his name
across the back of the note, thus
making the note payable to the
holder.
(2) In full : In this form the in-
dorser designates that the note is
to be paid to the order of a definite
person.
(3) In limited form : In this form
the indorser writes ''without re-
course " above his name. This
means that the holder cannot com-
pel him to pay if the maker fails
to do so. lAJ-iZkout \,e&cm,'v&&.
&aAf to- tk& o'uLeA, o
Limited Indorsement
/
Every indorser in blank, or in full,
makes himself liable for the amount of
the note if the maker and the previous
indorsers /a;7 to pay. Banks are required
to notify the indorsers in a manner prescribed by law in case the note is
not paid when due. This is called protesting the note. If the note is
not protested, the indorsers are released from the liability of payment.
Forms of promissory notes.
I. As to time :
1. If the words "on demand" are substituted for the words "six
months " in the note of Mr. Ames, page 2. c )2, it will then be a " demand
note " ; that is, the maker may be called upon to pay it at any time after
date.
2. The note of Mr. Ames is a "time note" because it is not to be
paid until a certain time named in the note.
The time of payment in a note must be definite. A promise to pay
" when able " is too indefinite, and not binding.
PROMISSORY NOTES 295
II. As to payees :
1. When a note is payable to the order of some particular person, he
alone can collect it, or sell the note by indorsement.
2. When a note is payable to some particular person, or bearer, the
holder can collect it, or sell it by indorsement.
III. As to the number of makers :
1. An individual note is a promise made by one person.
2. " A joint and several note " is a promise made by more than one
person. It contains the words " we, or either of us," and is signed by
the makers.
Maturity of Promissory Notes.
A note is said to mature on the last day of the time named in the note.
Some states allow 3 days, called " days of grace," from the time a note
matures before the payee can proceed to collect the note. In this case
three days are added to the time on which the interest is computed.
Days of grace are now abolished in most states. The note on page 2i)'J
matures May 21, 1906.
If a note falls due on Saturday, Sunday, or a legal holiday, it is usu-
ally payable on the next succeeding business day. Some states require
such notes to be paid on the preceding business day.
Interest on Promissory Notes.
If either a time or a demand note contains the words "with interest,"
the note bears interest from date at the legal rate in that state.
If the words "with interest" ai - e omitted from a time note, it bears
interest from the date of maturity until paid.
If the words " with interest " are omitted from a demand note, it bears
interest from the time payment is demanded until paid.
Written Work
1. Is the promissory note given by Mr. Patterson (p. 292)
a demand or a time note? an individual or a joint and sev-
eral note ? payable to order or bearer ?
2. Write a promissory note, in which you are the maker,
for $125 due in 6 months, payable to the order of Ellsworth
Slater, with the legal rate of interest in your state.
296 INTERp;ST
3. Mr. Slater sells this note to Herman Gross, and in-
dorses it in full. Write the indorsement on the note.
4. Write a joint and several note for 8250, dated Sept.
24, 1905, due on demand, with interest at 6%, payable to
the order of James Harbison. Your teacher and yourself
may sign this note as makers.
5. May the two names to the above note be written by
the same person ? Why not ?
6. In case James Harbison sells this note to James Brown,
but says to Mr. Brown, " I shall not be responsible for the
collection of this note," write the indorsement and explain
why you use that form.
7. Find the amount paid to the holder of Mr. Harbison's
note (Ex. 4), if settled Jan. 4, 1907.
Note. — Time from Sept. 24, 1905 to Jan. 4, 1907, 1 yr., 4 mo., 1 1 da.
8. Name the different kinds of negotiable notes :
(1) as to time ; (2) as to payee ; (3) as to number of makers.
9. Find the amount to be paid on the following note, if
settled March 1, 1907 :
The note is legally due June 1, 1906; therefore it bears interest at the
legal rate from that date.
$300.00 Chicago, 171., iHoA^k /, 1906.
&ki&& wuyn&kb after date ■.__ .J __ promise to pay to the
order of Qavi&a, 6vy&&
Value received.
PROMISSORY NOTES 297
10. Find the interest to be paid on the following note, if
settled Aug. 10, 1906, with interest at 6 % '
$/75.60 Boston, Mass., fam,. 20, I<?06.
Hrv d&wuwul after date— J— promise to pay to the
order of 3~kex>cLav& <Ao&l
dne, hxincUvbcl &&v&itty-'flv-& i^rrrrrrrrrrrccccc'rrrr: Dollars.
Value received, with interest.
Clxtkiov 71'lawn.
11. If the words " with interest " are not mentioned in a
time note, from what date does the note bear interest?
12. If a note reads " with interest," and no rate is men-
tioned, what rate per cent is to be taken?
13. If the words "with interest" are not written in a
demand note, does it ever bear interest?
14. If days of grace are allowed in your state, how do you
find the time on which the interest is to be reckoned?
15. How do you compute time on a promissory note?
16. Which is the safer form :
Pay to the order of Henry James, or
Pay to the order of Henry James or bearer? Why?
Note. — The note of Mr. Ames (p. 292) could have been made payable to
James H. Ames or bearer. Why is the form as written in the note better ?
It is safest to indorse a note in full, for the reason that if it is lost in
its delivery by mail or messenger, it can be collected only by the party
named on the back of it.
298
INTEREST
Write negotiable notes, observing the essential conditions
as given on p. 293, and adding days of grace if allowed in
vour state. Find the amount due at date of settlement. On
overdue, non-interest bearing notes, compute interest at 6%.
Date
Face
Time
Payee Maker
Int.
Rate
Settle-
ment
1.
•2/ 5 /05
$ 100 6 months
George Kimes Yourself
6%
Maturity
2.
4/21/06
250 On demand
A. J. Edwards James Clyde
8%
7/29/06
3.
6/10/05
500 1 year
John Dunn
John Grant
( )
6/15/07
4.
8/16/04
350 6 months
N. J. Noel
James Palm
( )
12/15/05
5.
7/12/05
125 On demand
James Bryce
Yourself
7%
1 / 2 /06
6.
5/15/05
1200
3 months
M. J. Boyce
B. J. Morrow
H%
Maturity
7.
10/10/05
300
6 months
Ralph George
Ben Jarrett
6%
9/12/06
PARTIAL PAYMENTS OF PROMISSORY NOTES
United States Rule
It is frequently inconvenient for the borrower to pay the
face of the note all at one time. He is sometimes permitted,
by special contract, to make payments at any time or at in-
terest bearing periods, until the note is paid. These amounts
are called partial payments and are credited on the back of
the note, together with the date of payment.
l. For example, a borrower gives the following note :
$200. Braddoch, Pa., c/tav-. 26, /qo6.
dirt, d&vyuwul, for value received, I promise to pay
tJaAYueAs ^we^^rC^^ivc^^^c^^^^rc^^vr^^orc^^^^^v^^^^or order,
^W ffundh&d a/ncL —t^r>c^z^^c^^r^r^^^r2^^r^CzDollars.
100
With interest ; at 6%. /ferny Bxow-n.
PARTIAL PAYMENTS OF PROMISSORY NOTES 299
The following payments are
indorsed on this note :
What amount is due March 2,
1908?
Nov. 26, 1907, $50.00.
Jan. 2, 1908, $25.00.
Solution :
Principal
Interest on $200 for 1 yr.
The amount of the note Nov. 20, 1907 .
Payment Nov. 20, 1907 ...
Balance = new principal due Nov. 26, 1907
yr.
mo.
da
1908
1
2
1907
11
26
$200.00
12.00
212.00
50.00
162.00
Interest on $ 1 62 for 1 mo. 6 da. .
Amount due Jan. 2, 1908
Payment Jan. 2, 1908 ....
Balance = new principal due Jan. 2, 1908
$ .97
162.97
25.00
137.97
1908
1908
3 2
1 2
Interest on $ 137.97 for 2 mo.
The amount due March 2, 1908
(I
1.38
$ 139.35
2. What was the amount due on the note on Nov. 26,
1907?
3. How much interest was due Nov. 26, 1907 ? What
payment was made ? How much greater was the payment
than the interest ?
4. How much was the new principal due Nov. 26, 1907,
after the payment of $ 50.00 ?
300 INTEREST
5. How much interest was due Jan. 2, 1908? How
much greater was the payment than the interest?
Observe : 1. That the interest was computed on the principal to the
time of the first payment ; then on the balance, as a new principal, to the
time of the second payment; then on the balance, as a new principal,
until March 2, 1908.
2. That the interest at each payment was first paid and the balance
of the payment was credited on the principal.
3. As the interest must^rs^ be paid, in case the payment does not equal
the interest, the interest must be computed until such time as the sum
of the payments equals or exceeds the interest.
This is the United States rule of partial payments, and is
the legal one in most states.
Find the amount of the principal to the time of the first pay-
ment, and from the amount subtract the first payment. Con
skier the remainder as a new principal and proceed as before
until the time of final settlement. If any payment does not-
equal or exceed the interest, then find the interest to the time
when two or more payments equal or exceed the interest.
The Supreme Court of the United States has decreed:
(1) That the payment on a note must first be applied to
cancel the interest then due, before the piincipal may be
diminished.
(2) That interest must not be charged upon interest.
Written Work
1. A note for $1800, bearing 6% interest, was given
April 1, 1903, and settled Oct. 22, 1906. On the back
of the note were these indorsements: May 10, 1904, $225;
June 16, 1905, $50; Sept. 28, 1905, $340; March 10, 1906,
$475. Find the balance due on the note at date of settle-
ment.
PARTIAL PAYMENTS OF PROMISSORY NOTES 301
Principal
Interest from April 1, 1903 to May 10, 190-1
Amount due .May 10, 1 : * < » i
First payment made May 10, 1901 .
l>a lance = new principal due May 10, 1904
Interest from May 10, 1901 to June 16, 1905
51800.00
119 .70
1919.70
225.00
1694.70
$111.85
The interest exceeds the payment and a new principal is not
formed.
Interest from June 16, 1905 to Sept. 28, 1905, . . 28.81
Interest from May 10, 1904 to Sept. 28, 1905 .... 140-66
Amount due Sept. 28, 1905 18:35.36
Sum of second payment June 16, 1905, and third payment
Sept. 28, 1905. $50.00 + 8340.00 • . = 3 90.00
Balance = new principal due Sept. 28, 1905
Interest from Sept. 28, 1905 to March 10, 1906
Amount due March 10, 1906 ....
Fourth payment made Marc h 10, 1906 .
Balance = new principal due March 10, 1906 .
Interest from March 10, 1906 to Oct. 22, 1906
Balance due Oct. 22, 1906 .
1445.36
39.024
1484.384
475.00
1009.384
37.347
$1046.731
2. A mortgage for $960, bearing 6% interest, was given
June 20, 1900, and settled Dec. 26, 1904. On the mortgage
wore these indorsements: Nov. 2, 1901, $140 ; Jan. 14, 1903,
1200; June 1, 1904, $30; June 20, 1904, $150. Find the
balance due on date of settlement.
3. On a claim of $850, dated May 2, 1901, interest 5%,
the following payments were made: Aug. 8, 1901, $200;
Dec. 14, 1901, $2<i0; April 26, 1902, $200. How much
was due at final settlement April 26, 1903?
4. A note of $1000, dated Aug. 1, 1903, bearing interest
at 6%, had the following payments indorsed upon it:
Dec. 20, 1903, $250.50 ; May 12, 1904, $300 ; Nov. 20, 1904,
$400. Find the amount due June 26, 1905.
302
INTEREST
5. June 1, 1900, a note was given for $1700, with interest
at 6%. The following payments were indorsed on this
note
$300.
300.
300.
15.
585.
Dec. 1, 1900
June 1, 1901
Nov. 1, 1901
April 1, 1902
May 1, 1902
Find the amount due July 1, 1902.
6. A note for $240 was made June 1, 1903, with interest
at 6 %. The following indorsements were made on the note :
Oct. 13, 1903, $120 ; Jan. 19, 1904, $60 ; June 1, 1904, $60.
Find the amount due on the note Dec. 16, 1905.
Merchants' Rule
When partial payments are made on mercantile accounts,
overdue, or on notes running a year or less, the interest is
often computed by the merchants' rule.
Find the amount of the principal from the time it begins to
hear interest to the date of settlement.
Find the amount of each payment from the time it was made
to the date of settlement.
From the amount of the principal, subtract the sum of the
amounts of the payments. The result will be the balance due.
Written Work
1. A note for $1200, dated July 15, 1905, has the following
indorsements: Sept. 25, 1905, $450; Jan. 1, 1906, $200;
March 9, 1906, $150. How much is due July 1, 1906, at
6 °Jo interest ?
2. A note for $2500, dated Jan. 1, 1907, has the following
indorsements: Feb. 1, 1907, $50; March 1, 1907, $75;
May 1, 1907, $100. How much is due Oct. 1, 1907 ?
PAET III EIGHTH YEAR
BANKS AND BANKING
A bank is an institution that receives and lends money.
A national bank may also issue notes that circulate as
money.
Anions: the various forms of banks in the United States
may be mentioned national banks, which are under control
of the Federal government ; state banks, which are under the
control of the state ; private banks ; and savings banks.
A trust company is an institution empowered by its
charter to accept and execute all kinds of trusts, to act as
executor, administrator, assignee, and receiver. In most
states it is also empowered to do a general banking
business.
Savings accounts have been treated under the head of Compound
Interest, as the computations involved are a direct application of it.
The chief business of banks is to receive deposits for safe
keeping ; to lend money on approved security ; and to collect
drafts and bills of exchange.
Discounting notes is simply lending money on approved
security.
Opening an account with a bank.
When a person opens an account with a bank, he first fills
out a deposit slip, as indicated on page 304, and gives it. to-
gether with the deposit, to the "cashier" or "receiving
teller."
303
304
BANKS AND BANKING
DEPOSITED WITH
American Rational IBanh
PITTSBURG, PA.
€tt. ?0,
1907
Bills
Dollars
Cents
50
Gold
60
Silver
20
Chech s
ENTER CHECKS SEPARATELY
/a-C cfiat. toaivk,
65
80
ZLnuyyv c/vu&t @.o.
/SO
fO
data I
325
qo
to pay a bill by check, lie fills
book, similar to the following :
Some banks require the name
of the bank on which the check
is issued to be written on the
deposit slip.
The depositor then
writes his name and ad-
dress in a book kept by
the bank, so that the bank
may have his signature
for identification.
He then receives a bank
book, which should always
be presented to the teller
when a deposit is made, in
order that the dates and
amounts of all the deposits
may be entered. He also
receives a check book, each
page of which has one or
more blank checks and
stubs. When he wishes
out a form from the check
Stub
Check
No. /y-O/
Pittsburg, Pa., fwn& 26, /<?07. No. 1^0/
Datefun&26;07
American National Bank
Payable to Qo/m
Pay to the
f\. <=J~/vo-ynfow>i
order of fo/m ft. 3 Ivo-yyv^o-n, f^OO.*-^-
For R&nt to- <r(ut&
c4vn& hurvclv&cL and _?£_~-^-^ -^^Do/lars.
100
Arn't $<?OO. s -£
^. d. cJ'^Wi.
BANKS AND BANKING 30.")
A check is a written order by a depositor in a bank, direct-
ing the payment of money.
The stubs remaining in a check book, after the checks are
torn out, give a complete record of the checks issued.
1. Who is the maker of the check on p. 304 ?
2. To whose order is this check written?
Observe : 1. The maker of a check is the one who signs the check.
2. The payee is the one to whose order the check is made payable.
:'>. This check is made payable to the order of John R. Thompson,
which means that in order to receive the money from a bank, or transfer
the check to another person, he must write across the back of the check
the na'mtT^John R. Thompson." This is called a blank indorsement,
because it (jojeajiot state to who m the cheek is made, payable. If John
R. Thompson should write across the hack of the check the following :
Pay to the order of
Marshall Field & Co.,
Chicago, 111.
John R. Thompson,
this would be known as a full indorsement, for no one but Marshall Field
& Co. could collect or indorse the check.
Checks, like promissory notes, may be written in different
ways, as follows :
1. Pay to bearer,
2. Pay to cash, collectable by bearer.
3. Pay to James Ogden, or bearer, .
4. Pay to self (collectable by maker only).
5. Pay to the order of self (collectable by indorsement of the maker).
6. Pay to the order of James Ogden (collectable by indorsement of
James Odgen only).
Note. — The last form of check is the one in general use.
3. How may the check on p. 304 be indorsed in blank ?
4. How may it be indorsed in full ?
5. Suppose Mr. Thompson wishes to send this check to
Sage, Allen & Co., Hartford, Conn., in payment of an
HAM. COMF1.. ARITH.— 20
306 BANKS AND BANKING
account: first, write the check as indorsed by Mr. Thomp-
son in blank ; second, write the check as indorsed by Mr.
Thompson in full.
6. Give reasons why it will be better for Mr. Thompson
to indorse the check in full.
7. When a check is indorsed and sent by mail, what form
of indorsement should always be used ? Why ?
8. Give the essentials of a check.
Balancing Accounts ; Depositing ; Checking on Accounts :
1. Your deposits in a bank for the month of September
are as follows:
Sept. 1, currency, $50; silver, $10; check, $15.
Sept. 6, currency, $20; silver, $10 ; check, $100 ; gold, $20.
Sept. 10, currency, $45; silver, $4.75.
Sept. 16, currency, $20; silver, $3.40; check, $80.
Sept. 25, gold, $40; check, $40; silver, $10.
Sept. 29, check, $80; currency, $80; silver, $35.
Make out deposit slips and find amount of deposits for
September.
2. Your check book shows the following :
Balance in bank Sept. 1, $847.10.
No. 1, Sept. 4, Keller Bros., for coal, $15.50.
No. 2, Sept. 4, Geo. K. Stevenson & Co., for groceries for
August, $49.50.
No. 3, Sept. 4, Dr. S. N. Pool, for services to date, $90.
No. 4, Sept. 5, cash, $55.
No. 5, Sept. 7, Jos. Home Co., for merchandise, $65.30.
No. 6, Sept, 11, Midland Lumber Co., for lumber, $93.75.
No. 7, Sept. 15, Johnson & Co., for repairs on automobile,
$29.35.
No. 8, Sept. 19, cash, $25.
BORROWING FROM HANKS 307
No. 9, Sept. 24, J. H. McFarland, for interest due on
note, 124.
Write the checks for the bills paid for September, and
find balance in bank.
3. Arriving at Chicago, I find in my mail a check from
the Keystone Lumber Co., Pittsburg, Pa., for $415.40, in
payment of my salary and expenses for September. I wish
to deposit the same to my account in the Colonial Trust Co.,
Pittsburg, Pa. How should I indorse the check before send-
ing it through the mail ?
BORROWING FROM BANKS AND COMPUTING BANK
DISCOUNT
Banks usually lend money on promissory notes drawn in
one of three forms :
1. The note is made payable to the indorser, who signs his name across
the back of it.
2. A joint and several note is made payable to the order of the bank
and signed by both parties as makers.
3. The note is made payable to the order of the bank. The security
in the form of stocks, bonds, mortgages, etc., is deposited as collateral.
$200.^. Pittsburg, Pa., ifefit. 8, 1905
3~kx,&& vio-nthfr after date J. promise to pay to
the order of. ft. A W-at&aw at the
Lincoln Rational Bank of Pittsburg
Zfw-o- f'ficncLi&cL avid — -^^o^^r^r^^^c^^ryr^^^yr^Dollars
too
without defalcation, for value received.
J. & &kandle,v.
308 BANKS AND BANKING
This note matures three months after Sept. 8, or Dec. 8. If the time
in the note were " ninety days " instead of " three months," the note would
mature ninety days after Sept. 8, or Dec. 7.
If Mr. Chandler wishes to borrow money at the bank, he may make
out a note as on p. 307 and get Mr. Watson to indorse it.
If both men are responsible from a financial point of view, the bank
will buy the note and give Mr. Chandler the difference between the value
of the note at its maturity and the interest on that value at the legal rate
for the exact number of days the bank is without the use of its money.
The value of Mr. Chandler's note is the amount the Lincoln National
Bank will receive from Mr. Chandler at its maturity. If Mr. Chandler
fails to pay. Mr. Watson will be held responsible.
The buying of notes by a bank is called discounting notes,
and the interest deducted is called bank discount.
The proceeds of a note discounted by a banker or a broker
is the value of the note at its maturity less the discount.
The term of discount is the exact number of days that
the borrower has the use of the money.
There are two methods, however, of reckoning this term: the first
method counts the day of maturity, but not the day of discount; the
second counts both : thus, by the first method Mr. Chandler had
the use of the money 22 days in Sept., 31 days in Oct., 30 days in Nov.,
and 8 days in Dec, or 91 days in all; by the second method he had the
use of the money 23 days in Sept., 31 days in Oct., 30 days in Nov., and
8 days in Dec, or 92 days in all.
Note. — Pupils should solve the problems according to the practice
in their vicinity. Answers are given for both methods.
When days of grace are allowed they are included in the term of dis-
count, but in this book days of grace are not reckoned.
Computing bank discount on note on p. 307 :
Date of maturity, December 8.
Term of discount. 91 days (not including day of discount).
Bank discount on 8200 for 91 days at 6% = $3.03.
l. How much does the bank pay to Mr. Chandler? How
much does Mr. Chandler pay to the bank at maturity?-
BORROWING FKo.U BANKS
SOP,
•^. If Mr. Chandler had borrowed $200 from Mr. Watson
for 3 months at 6%, how much would he have paid Mr.
Watson when the note became due?
3. How much would Mr. Chandler have received from
Mr. Watson at the time he borrowed the money?
4. Find the difference between the discount paid to the
bank and the interest he would have paid to Mr. Watson.
Comparative Study
Banks differ from individuals in lending money, as follows:
1. Banks require the interest on a note to be paid in advance ; indi-
viduals demand interest when the note is due, or annually, if for a longer
period than a year.
2. Banks compute interest for the exact number of days; individuals
compute interest by months and years.
3. Banks lend money for short periods, usually not exceeding four
months; individuals lend for longer periods, not exceeding five years in
most states.
4. Banks require the maker to give additional security ; individuals
may or may not demand security.
5. Interest is computed on the face value of a note; bank discount is
computed on the value of a note at its maturity.
Bank discount is the simple interest paid in advance on the
value of a note at its maturity for the exact number of days
the banker is without his money.
Given the dates and time of notes, to find the date of
maturity.
Find the date of maturity of the following :
Date
Time
Date
Time
1. June 1
2. July 3
3. Aug. 5
4. Sept. 10
2 months
50 days
100 days
1 month
5. Jan. 2
6. March 3
7. April 1
8. Ma\ 5
3 months
7~> days
70 days
4 months
810
BANKS AND BANKING
Find the date of maturity and the term of discount.
Date of
Time of
Date of
Date of
Time of
Date of
Note
Note
Discount
Note
Note
Discount
9. March 1
60 da.
April 1
14. Jan. 2, '08
90 da.
March 1
10. April 10
3 mo.
June 15
15. March 23
4 mo.
June 2
11. July 10
4 mo.
Sept. 30
16. Oct. 8
60 da.
Nov. 1
12. Mav 24
30 da.
June 1
17. June 5
90 da.
July 10
13. August 5
70 da.
Sept. 1
18. Sept. 24
30 da.
Oct. 1
Written Work
1. Write a promissory note for $ 300 payable to John
Jackson, dated Aug. 3, 1907, due in four months, with inter-
est at 6%, and signed by Glenn Campbell.
2. Mr. Jackson indorses the note in example 1, and Mr.
Campbell borrows the money from the Park National Bank.
Indorse the note in full and find the bank discount and pro-
ceeds.
The following notes are each discounted on the day of
issue. Find the date of maturity and the bank discount.
Date of Note
Time
Face
Rate of
Discount
3. Aug. 10, 1906
90 da.
9 150
6%
4. June 12, 1906
2 mo.
515
6J%
5. July 2, 1906
3 mo.
1000
u%
6. Jan. 8, 1906
4 mo.
625
n%
7. Mar. 5, 1906
50 da.
570
6%
8. May 8, 1906
70 da.
423.25
5*%
9. Sept. 1, 1906
1 mo.
1200
8J%
10. Dec. 1, 1906
100 da.
387.75
6%
11. Mar. 5, 1906
72 da.
1125
7%
DISCOUNTING NOTES 311
Discounting Interest and Non-interest bearing Notes.
Business men frequently take notes from their customers
due at a future date, and in case they need the money before
the notes become due, they sell them to a bank. The bank
deducts from the value of each note at maturity the interest
(bank discount) for the term of discount. These notes may
or may not bear interest. •
Mr. James Edwards has two notes that read as follows :
$<?0.^ Columbus, Ohio, 7n<M*A 2, 1907
3/i%&& vu>nth& after date J. promise to pay to
the order of <^^ryr*^r^^j!am-&a, £dnAXMj£&c^^c^r^c^^>-
c/tinetu and i ^^^^r^^o^^o^^c^^^r^^c^^rC^Z>oZZa;»5.
' 100
Value received.
ff&nvy (ZmaXaav.
$/50^- Columbus, Ohio, ?11awA fO, 1907
100
&ouv i yYuy)ith& after date J. promise to pay to
the order of^^yr^c^y^yc^^a-'viv&a, €dwavcU,-c^?^<r?crr^ry?^^'-
€n& f^vAvdv&d S^viUf and —*^^^y^^cY>?^c^^I)ollars.
Value received, with interest at 6%.
312 BANKS AND BANKING
1. What is the value of the first note at maturity?
2. What is the value of the second note at maturity?
3. Mr. Edwards gets both notes discounted April 20,
1907, at 6 %. Why is the discount on the first note computed
on $90? on the second note, on $153?
Banks always discount notes on the amount they are to
receive at maturity.
Discounting the first note on page 311 :
Maturity of note, June 2.
Value of the note at maturity . . . = $90.00, or face
Bank Di scount for 43 da. at 6 % . . = .65
Proceeds April 20 = 9 89.35
Discounting the second note on page 311:
Maturity of note, July 10.
Value of the note at maturity = % 153.00 ox face + interest for 4 months.
Bank Discount for 81 da, at 6% = 2.07
Proceeds April 20 ....=$ 150.93
Written Work
1. A 60-day note for $2500, without interest, dated Jan.
12, 1907, was discounted Feb. 12, 1907. Find the proceeds
of this note.
2. Find the proceeds of a 90-day note for $1560, with
interest at 6 %, dated March 8, and discounted April 12.
3. Find the proceeds of the note in example 2 without
interest.
4. A 90-day note for $ 4500, with interest at 6 %, is dis-
counted 30 days after date. Find the proceeds.
Suggestion. — Note the difference between a note for 90 days and a
note discounted for 90 days.
DISCOUNTING NOTES 313
5. Find the proceeds of the note in example 4 without
interest.
6. A 120-day note for $3500, without interest, dated
June 5, is discounted Aug. 10. Find the proceeds.
7. Find the proceeds in the note of example 6, if the note
bears interest at 6 %.
8. The proceeds of a 90-day note, without interest, dis-
counted 30 days after date, is $ 990. Find the face.
9. A note for $1200, bearing interest for 3 mo. at 6 %,
was dated Jan. 15 and discounted Feb. 20. Find the
proceeds.
10. Mr. Boyd gives his note Jan. 10, 1905, to William
Savers for $200, payable in 9 months, with interest at 6%.
What is Mr. Boyd's note worth on the day of issue? on
the day of maturity ?
11. Should Mr. Boyd get the note discounted July 5,
on how much money would the bank reckon the discount ?
12. Find the amount Mr. Boyd would receive July 5.
What is the money received by Mr. Boyd called with ref-
erence to the note ?
13. Write the note given by Mr. Sanders and transfer it
by indorsement in full to one of your local banks.
14. Face, $223.50 ; time, 90 days ; rate of interest, 6% ; term
of discount, 50 days; rate of discount, 8%. Find proceeds.
15. A business man's bank account is overdrawn ^381.50,
and he presents to the bank, May 1, two notes to be dis-
counted, at 6%, and the proceeds to be placed to his credit.
Face Date Time Rate of Interest
$290 Mar. 10 5 mo. Without interest
$355 Apr. 20 90 da. 6%
Find his balance.
314 BANKS AND BANKING
16. The discount on a note for $400 for 60 days, exact
time, is $6.00. Find the rate of discount.
17. A broker buys a $300 note, thirty days before matu-
rity, for $297. Find the rate of discount.
18. Mr. James sold a horse for $155, and took the pur-
chaser's note, dated Jan. 20, 1905, due in one year, with
interest at 4| %. Mr. James sold the note to the Farmers'
Bank, Oct. 10, at 7 % discount. How much did he realize ?
19. A note for $1800, at 8%, dated August 1, due in 3
months, was discounted October 6, at 6 °J • Find the proceeds.
20. What should the Merchants' National Bank pay for a
note of $1200, bearing &% interest, dated April 12, due in
4 months, if purchased June 1, at 6 % discount ?
21. What are the proceeds of a note for $2500, dated
February 10, 1907, and due in 4 months, without interest,
if discounted March 24, at 6 % ?
22. A merchant's bank book shows a balance of $1375.50,
and he presents at the bank four notes, which are discounted
June 1, at 6 %, and the proceeds placed to his credit :
Face Date
$600 March 4
$1375 April 2
$1050 March 19
$2000 May 29
Find his balance in bank then.
23. For how much must I give my note, discounted at a
bank for 60 days, at 6%, to realize $990 ?
Note. — The proceeds of $1, discounted for 60 days, at 6% = $.99.
24. For what sum must I draw my note so that when dis-
counted at 6% for 90 days I may realize $2758?
Time
Rate of Interest
90 days
6%
4 months
No interest
90 days
5%
3 mo.
No interest
EXCHANGE
Paying Bills at a Distance
What is meant by a debtor? by a creditor? How may a
debtor pay a bill in a distant city without the actual transfer
of cash ? How may a creditor collect a bill in a distant city
without the actual transfer of cash ?
Exchange is a method of paying or collecting bills at a
distance without the actual transfer of money.
There are several different ways in which bills may be paid
at a distance without the transmission of money:
(1) By a postal money order.
(2) By an express money order.
(3) By a telegraphic money order.
(4) By a personal check.
(5) By a bank draft (banker's check).
Paying by postal or express money order.
If you wish to order from Siegel, Cooper & Co.. Chicago, $15 worth of
merchandise, unless you have credit there, you will probably send them
(1) either a postal money order, or (2) an express money order. The
first will direct the postmaster at Chicago, the second some express agent
at Chicago, to pay to the order of Siegel. Cooper & Co. i 15.
The cost of either of the above orders is the same ; the only differ-
ence being that a postal money order is payable to the order of the
party or firm upon identification at the place named in the order, while
an express money order is payable to the party or tirin upon identifica-
tion at any office of the same company where orders are sold.
3 1 5
31G EXCHANGE
Money orders may be purchased for any amount up to $100, payable
to any person or firm in the United States, or foreign countries where
such orders are sold.
The rates charged 'in the United States are as follows :
$2.50 and under 3?
5?
8?
10?
12?
15?
18 ?
20?
25 (»
30?
Over $2.50 and not exceeding $5.00 .
Over $5.00 and not exceeding $10.00 .
Over $ 10.00 and not exceeding $20.00 .
Over $20.00 and not exceeding $30.00 .
Over $30.00 and not exceeding $40.00 .
Over $10.00 and not exceeding $ 50.00 .
Over $50.00 and not exceeding $ 60.00 .
Over $60.00 and not exceeding $75.00 .
Over $75.00 and not exceeding $ 100.00 .
The rates to foreign countries are from 10 ? to $ 1 for the same
amounts as domestic orders.
This fee of from 3 ? to 30? for domestic orders, and from 10? to $ 1
for foreign orders, to cover the cost of paying the bills at a distance,
is called the exchange for issuing the orders.
Paying by telegraphic money order.
Such orders are drawn by agents of the telegraph company, and
direct the agent at some designated office to pay to the person named in
the telegraphic message, upon identification, the sum specified.
Th3 present rates for sending money by telegraphic order are as follows :
For $25 or less, double the cost of a ten-word message, plus 25 ?.
Above $ 25, double the cost of a ten- word message, plus 1 % of the
amount of the order.
Paying by checks.
Business men find it necessary to pay bills in their vicinity or at a
distance, almost daily, and if their financial standing is good, their checks
are generally accepted in payment. In fact, most bills to-day are paid
by checks.
Sometimes the seller does not know the financial standing of the pur-
chaser, and therefore requires the check accompanying the order to be
certified ; that is, the cashier of the bank on which the check is drawn
stamps the word " certified," with the date and his signature, across the
face of the check. The check is thereafter the check of the bank, and
is good as long as the bank is solvent.
EXCHANGE 317
A certified check is a notice to the payee of the check that
the amount named on the face has been taken from the
maker's deposit and placed with the bank's funds for the
payment of the check when presented.
Certified checks are frequently demanded in payment of notes and
collections at banks, and in payments where the payee does not wish fco
take a personal check. Like other checks, they are mailed daily in
payment of bills in all parts of the country.
1. What is a check? What are the essentials of a check?
2. In buying a lot from James Carothers for $800, you are
asked to give your certified check for the amount. Write the
check on your local bank, yourself being the maker.
3. Moore & Co., Youngstown, Ohio, purchase 'IB 825 worth
of furniture at 20% and 10% off from James Boydson & Co.,
Detroit, Mich., 3% off for cash in 10 days. They send a
certified check within ten days on the Diamond Trust Co.,
of which James Patterson is secretary and treasurer. Write
the certified check in payment of the bill.
Note. — The secretary and treasurer of a trust company corresponds
to the cashier of a bank.
4. William Anderson, 7531 Hermitage Ave., Chicago, 111.,
receives a check on the First National Bank of Wilkins-
burg, Pa., from Freeman Lewis for $730.80 in settlement
of an estate. The Commercial National Bank of Chicago
charges Mr. Anderson $1.50 for collecting the check. This
fee is called the exchange for collecting.
A person who cashes a check at a bank in which he is not a depositor
is frequently charged an exchange of 10^ and upward, according to the
amount of the check.
Paying by bank draft.
A draft is a check drawn by one bank on another.
As New York City and Chicago banks collect exchange
on outside checks, nearly all banks keep deposits there, as
318
EXCHANGE
well as in most of the other large commercial centers, to
accommodate their depositors and others, who have occasion
to remit payment for bills in any part of the country.
Bauks usually charge exchange on drafts to cover the cost of keeping
funds on deposit at these commercial centers. This fee varies from -^%
to \% of the face. When the draft is less than $100, a fixed charge is
frequently made, varying from 10 f to 50^.
The custom of banks is not to charge depositors for drafts.
In issuing or collecting a draft, the exchange is either a
fee or a certain per cent of the face of the draft.
New York and Chicago drafts are usually cashed at any point in the
United States without exchange. Drafts on other large cities are cashed
without exchange in the territory contiguous to those cities.
Brown & Foster, Cleveland, O., buy $2500 worth of
merchandise from John Wanamaker & Co., New York ; and
$650 worth of machines from the Wheeler Wilson Co.,
Bridgeport, Conn. The business method of paying these
bills is either by a check or by a bank draft. The draft is
made payable to the order of the purchaser, who indorses it
in full to the payee. For example :
Cleveland National Bank
Cleveland, Ohio- ^we 2, 1097 , ;y a /o^-O
Pay to tlie order of Bvowi, V dfaat&v $2600^.
3w-&nty-(vv-E, /runoU&d V — -c^^c^^c^^r^vyc^DoUaj^s.
To (Efje iJHercanttle National Bank,
New York, JV*. Y.
d. ?Vl. /MwveA,..
Cashier.
EXCHANGE :;i'.t
This draft simply means that Brown & Foster purchased at the bank
where they kept their deposit a draft (banker's check) for the above
amount. The Cleveland National Bank had money on deposit at the
Mercantile National Bank, and simply checked on its deposit. Had
Brown & Foster not been depositors in the Cleveland National Bank,
they would probably have been charged T \% exchange. The draft
would then have cost them $2502.50.
The party who signs a draft is called the drawer of a
draft.
The party to whose order the draft is drawn is called the
payee of the draft.
The party who is to pay the money is called the drawee of
the draft.
Thus, in the draft on p. 318, the cashier of the Cleveland National
Bank is the drawer; Brown & Foster the payee; and The Mercantile
National Bank the drawee.
Written Work
1. Find the cost of a New York draft for $550.25 at ^ %
exchange.
2. Mr. Amidon buys $2500 worth of farm implements at
30% and 10% off, and pays by a Chicago draft at \% ex-
change. Find the face of the draft and the exchange.
3. Find the cost of sending $80 from Pittsburg to Chi-
cago by telegraphic money order, the rate being 25 f for
10 words.
4. I sent $75.80 to Chicago by express money order. How
much could I have saved by purchasing a bank draft at 15
cents exchange ?
5. A draft costs $1080, including the exchange at ^%.
Find the face.
Suggestion. — $ 1080 is 100^% of the face.
320 EXCHANGE
6. Write a draft for $2600, making' one of your local
banks the drawer, the First National Bank of Buffalo, N.Y.,
the payee, and the Niagara Falls Power Co. the drawee.
Indorse the draft in full to James Osborne & Co., Syra-
cuse, N.Y.
7. Mr. Madison had a note for 81000 discounted for 60
days at 6%, and with the proceeds bought a Chicago draft
at yo% exchange, which he mailed to Mandel Bros., Chicago,
to apply on account. Find the face of the draft and the cost
of exchange.
8. My settlement of an account in New Orleans gives
me $26785.50. After investing #13750 of this amount in
a land deal on which I pay my agent 2% commission, I pur-
chase a New York draft with the balance at ^% exchange.
Find the exchange, the commission, and the face of the
draft.
9. James Anderson & Son, Helena, Montana, order $790
worth of goods from a Boston firm, and send in payment a
New York draft at \ % exchange. Find the cost of the
draft.
10. A dealer in San Francisco buys $2000 worth of goods
at 30% and 10% off and sells them at an advance of 25%
on the cash price. After paying for these goods with a
Chicago draft at \ % exchange, find his profit.
Collecting Bills at a Distance
Bills are collected at a distance in two ways :
(1) By a sight commercial draft of a creditor on a debtor.
(2) By a time commercial draft of a creditor on a debtor.
Bills collected by a sight commercial draft.
Tf Letche & Co., grain dealers, Pittsburg, Pa., order from Harris
Bros., Chicago, 111., 1 car load of No. 1 oats, Harris Bros, will ship
EXCHANGE 321
the car load of oats to Pittsburg to tlic order of themselves and draw
a sight draft on Letche & Co., payable to the order of some Chicago
bank, and deposit it with the bill of lading for collection. The Chicago
bank will then mail the draft, together with the bill of lading, to some
bank in Pittsburg. The Pittsburg bank will notify Letche & Co. If
the car load of oats is accepted by Letche & Co., they will pay the draft
and receive the bill of lading which will entitle them to the oats.
This form of draft is commonly known as a commercial
sight draft and reads as follows :
$^0^. Chicago, III., fusne, 27, 1906
~Cy^yC^C^yC^C^^Clt av^/ito^yr^^^^^^^y^yc^c^-JPay to
the order of /at oAatio-na.1 Bank, <&,Aie-aao,
Sow, /funded <$£fty V ^y^^^^os^c^DoUars-
J'al//r received, and charge to account of
To JUUAe, V &». ) , • n
L Zi-aAsVva, Jovoa-.,
No. /#■ <t&6-Ul<p, <Pa. ) Chicago, III.
A bill of lading is a receipt given by the carrier to the
shipper. The goods shipped and their value are described
on its face, and on the back of the receipt is stated the
contract of shipment.
In case Letche & Co. refused to accept the oats, the draft would be
returned to the Chicago bank, which in turn would notify Harris Bros.
Creditors use sight drafts in the collection of debts due or
past due.
Bills collected by a time commercial draft.
Often a draft reads '••'!() to 90 days after sight." Such a draft is
called a time commercial draft.
HAM. COMPL. AIM I II. — 21
322 EXCHANGE
The method of collecting by a time commercial draft is as follows :
$/200^ Pittsburg, Pa., fun& 27, 1006
^v?cZu Gbxufr aft&v octant P a y to
tJie order of JClk&iVu cftatuyyuxl Bowk,
cfu<-&£ / v-& fi-wyictv&cL and — rrrrrrrr^rcccccc^yy^i'rrrrrLDoUars.
100
Value received, and charge to account of
To She, (Z&YYI& Buqqu &&., ) n . „„ ^
7// y favdan, V Ola.,
No. /32 , €-ln&Lnnatv. €.. ) Pittsburg, Pa.
The Cincinnati bank to which this draft is mailed immediately notifies
the Acme Buggy Co. and if the Company agrees to pay the draft, when
due, the following is written across the face of it :
" Accepted
Date
Acme Buggy Co."
If the Company refuses to accept the draft, the Cincinnati bank will
return it to the Liberty National Bank, and Jordan & Co. will be notified
that the goods are at Cincinnati at their risk.
The term of discount in a draft payable " after sight " begins to run
from the date of acceptance ; in a draft payable after date, from the date
of the draft.
In collecting by draft, the exchange is always collected on the face,
not on the proceeds, of the draft.
Written Work
1. If the draft of Jordan & Co. was accepted and dis-
counted June 29, 1906, find the term of discount. '
2. Find the proceeds remitted to Jordan & Co. if this
draft was discounted June 29 at 7 %, with | o/ exchange for
collecting.
EXCHANGE 323
3. The Jareki Mfg. Co., Sandusky, O., draw at sight on
James Howard, Canonsburg, Pa., for $159.70 through the
Erie National Bank, Sandusky, O. Write the draft.
4. Freeman Bros., Fargo, N.D., Jan. 2, 1907, sell on
60 days' time to the Standard Milling Co., Minneapolis,
Minn., 12000 bu. No. 2 wheat at 92|^ per bu., delivered
at Minneapolis, and draw a time draft which is accepted by
the Standard Milling Co. The Merchants National Bank of
Minneapolis buys this draft Feb. 1, 1907, at 7 % discount.
If exchange is \%, rind the proceeds from the sale of the
wheat.
5. Charles Boyd, Fremont, O., owes Samuel Johnson,
Jacksonville 111., 1600 due June 10. 1907. Mr. Johnson
desires the money immediately, and draws on Mr. Boyd
March 24, 1907, through the Illinois National Bank, Jack-
sonville, a time draft due June 10, which Mr. Boyd accepts
March 30. The Fremont bank discounts the draft at 6%
on the day of acceptance. If the exchange is \°Jo-> how much
is remitted to Mr. Johnson ?
6. T. F. Bowman & Co., Chicago, sell to Speer Bros.,
Seattle, Wash., $4000 worth of merchandise. Terms: 60
days net; 6% off 30 days. Write the banker's check
given by the Seattle National Bank and indorsed in full by
T. F. Bowman & Co. Find the cost of the banker's check
at I % exchange, if paid within 30 days.
7. James Brown, Lansing, Mich., sells $2500 worth of
celery on March 1, 1908. to Grimm Bros., Boston, Mass.,
and draws a draft for 90 days after sight. The draft is
accepted March 18, and discounted the same day at 6%.
If the cost of collection is \ % exchange, find the proceeds
from the sale of the celery.
STOCKS AND BONDS
STOCKS
When an individual or a few persons do not wish to fur-
nish all the capital or money required for a business, or to
assume all the responsibility, they may secure a charter from
the state government to form a corporation or stock com-
pany, and choose a board of directors to transact the busi-
ness in the name of the firm designated in their charter.
A corporation is a company authorized by a charter to
transact business as an individual.
The capital stock of a company is the amount of stock
for which shares are issued. Thus, 1000 shares at $10 each
make a capitalization of $10000.
The par value of the shares in different corporations varies from % 1
to % 100. The persons who form the corporation determine the number
and par value of the shares. Observe that the certificate on p. 325
gives the number and value of the shares.
The par value of a share of stock is the amount written on
the face of the stock certificate.
What is the par value of the stock certificate on p. 325 ?
The market value of a stock is the price at which it is
selling.
A stock is selling at a discount when purchased for less than its pai
value, and at a premium when selling for more than its par value.
324
STOCKS 325
STOCK CERTIFICATE
Incorporated under the Laws of the State of Pennsylvania
<yfi>. 2 <
20 tf/iat&fr
Enorpcnocnt Iron OTompang of pttsimrg
This certifies that j/oont&o, W-oo-cL is the owner
of. Sw-tnty full paid shares
of the Capital Stock of One Hundred Dollars each of the
Independent Iron Company.
Transferable only on the books of the Company by
the holder in person or by an attorney upon the surrender
of this certificate.
j. &. 711&I/&I, President.
B. c/1 #W£A, Secretary.
Pittsburg June 1, 1906.
A stockholder is one who holds stock in a corporation.
An assessment is a sum levied on the par value of each
share of stock to defray expenses and losses when the earn-
ings are not sufficient.
A dividend is a part of the net profits divided among
the stockholders, in proportion to the par value of their
stock. These dividends are paid yearly, half-yearly, or
quarterly, as the board of directors may determine.
A stock broker is a person who buys and sells stocks for
another. The charge, called brokerage, is usually \ % to \ %
on a par value of a hundred dollars. Most brokers belong to
some stock exchange.
326 STOCKS AND BONDS
Par Value and Brokerage.
1. Mr. James buys 10 shares of railroad stock, par value
$100 per share, at $89 per share, brokerage ^%.
1. What is the par value of each share?
2. What is the market value of each share?
3. Show that each share costs Mr. James $ 89.12J.
In the study of this subject the following should always
be observed :
1. When the par value of a stock is not stated it is always regarded
as |100.
2. When a stock is quoted at 90, 110, 78, etc., it always means so
many % of the par value. A stock quoted above 100 is said to be above
par, or at a premium, and one quoted below 100, below par, or at a discount.
3. Brokerage is always reckoned on the par value and the broker
collects brokerage from both the buyer and the seller. \°/ brokerage
means $.12^ on a par value of $100, or $.06£ on a par value of $ 50.
To find the cost we add the brokerage to the selling price or pur-
chase price and then multiply that amount by the number of shares
bought or sold. Brokerage is not to be computed,_unless stated in the
problem.
Written Work
1. Find the cost of 48 shares of railroad stock bought at
95, brokerage \ % •
$95 + $ \ = |9oi = cost of 1 share.
48 x $95£ = 14566, cost of 48 shares.
Find the cost of :
2. 60 shares of stock at 101, brokerage | %.
3. 88 shares of stock at 102, brokerage \°Jo-
4. 104 shares of bank stock at 116J, brokerage \ %.
5. 120 shares railroad stock at 94|, brokerage \ % •
STOCKS 327
6. My broker sold for me 128 shares of mining stock at
156, brokerage |%. What sum should I receive?
Find the net amount received from the sale of the follow-
ing, including brokerage at \o/ :
7. 125 shares at 09|. 9. 500 shares at 132|.
8. 145 shares at 142|. 10. 1000 shares at 37|.
11. I bought 125 shares of Penn. R. R. stock at 123| and
sold it at 129|; brokerage |%. Find the net gain.
$ 129f - %\ = $129.25, amount realized from each share
$ 123J + $ I = $123.625, cost of each share
$5,625, gain on each share
125 x $5,625 = $703.13, net gain
Why do we subtract the brokerage when selling stock?
Why do we add the brokerage when buying stock ?
12. How many shares of railroad stock at 108 1, brokerage
l %, can be purchased for $26100 ?
13. How many shares of stock must be sold at 99|> brok-
erage | %, to pay a debt of $793 ?
14. I receive $1375 net profits on stock bought at 58 and
sold at 72, brokerage ^ % in each case. Find the number of
shares.
15. If I realized $1595 net from the sale of stock, broker-
age | %, rind the number of shares sold at $40 per share.
16. How many shares of stock selling at $45 per share,
brokerage \ %, can be purchased for $2000 ? (Parts of a share
are not sold.)
17. A note for $5000, with interest at 6 %, was paid 1 year
4 months and 18 days after date and the amount invested in
stock at 87|. Find the number of shares purchased.
328 STOCKS AND BONDS
Premium and Discount.
1. What is the cost of 1 share of stock, par value $100,
selling at 10 % discount ? at 40 % premium ? at 30 % discount ?
2. What is the cost of 60 shares of bank stock, par value
$50, at 5% premium, brokerage \ % ?
Par value of 1 share = $50.
Premium of 1 share = .05 x $50 = $2.50.
Brokerage on 1 share = .00^ x $50 = $ A2\.
Cost of 1 share = $50. + $2.50 + $.12|, or $52.62£.
Cost of 60 shares = 60 x $52.62| = $3157.50.
Note. — All premiums are reckoned on, and added to, the par value;
and all discounts are reckoned on, and subtracted from, the par value.
Find the cost of the following stock, brokerage ^ % :
3. 16 shares, par value $50, at 3% premium.
4. 42 shares, par value $50, at 10% discount.
5. 100 shares, par value $25, at 14 % discount.
6. 220 shares, par value $100, at \ % premium.
7. 150 shares, par value $50, at f % discount.
8. My broker purchased for me 256 shares of milling
stock, par value 100, at 4-|-% premium, brokerage \°/ .
How much did the stock cost me ?
9. How much will 120 shares of railroad stock cost at
114f , brokerage | % ?
10. How much is realized from the sale of 480 shares of
gas stock, par value $50, at 2% discount, brokerage \°Jo ?
11. How many shares of stock can be bought for $2475,
at 3 % premium, brokerage \ % ?
Par value of 1 share = $ 100
Premium on 1 share = $ 3.
Brokerage on 1 share = $ .125
Entire cost of 1 share = $ 103.125
$ 24750 - $ 103.125 = 240, number of shares
STOCKS 320
Find the number of shares bought for :
12. $5827.50, par value $50, at 3% discount, brokerage
Wo-
13. $11970, at 106|, brokerage |%.
14. $ 2165, par value $ 50, at 8% premium, brokerage \%.
15. $ 19677, at 116|, brokerage \%.
16. $10025, par value $50, brokerage \ %.
Dividends and Investments.
1. What is the income from $ 1000 loaned for 1 year at
6%?
2. What is the income from $ 1000 invested in a manu-
facturing plant that pays 8% in dividends each year?
3. Why is a $100 share of stock that pays $ 12 in divi-
dends each year worth more than $ 100 ?
4. Why is a $100 share of stock that pays only $2 in
dividends each year worth less than $ 100 ?
5. 1|% dividend payable quarterly is equivalent to what
per cent payable annually ?
6. $ 60 per year is the dividend on $ 1200 worth of stock
par value. Find the rate of dividend.
7. I receive $120 from a dividend of 6%. What is the
par value of my stock ?
8. Mr. Johnston owns 100 shares of stock in a company
whose capital is $200000. If a dividend of 8% is declared,
find the amount of the check that will pay the whole divi-
dend ; the amount of Mr. Johnston's share.
Observe: 1. Dividends are always declared on the par valne of a
stock.
2. Incomes are always reckoned on how much a stock costs.
330 STOCKS AND BONDS
9. A dividend of 6 % is declared on a stock, par value $100,
purchased at $ 120. What % is received on the investment ?
6% of $ 100 = $ 6, income on one share
<| 120 = cost of one share
$ 6 h-$120 = .05,or5%
10. A share of stock, par value $ 100, is sold at $ 250.
What is the per cent of income if an annual dividend of
10% is declared?
Find the rate of income when :
11. $150 is paid for 9% stock, par value 1 100.
12. $133^ is paid for 6% stock, par value $100.
13. $75 is paid for 8% stock, par value $50.
14. $ 80 is paid for 5% stock, par value $100.
15. $50 is paid for 4 % stock, par value $100.
16. Would a stock yielding 6% have to be purchased for
more or less than par value to yield 8% on the investment ?
Explain why.
17. Explain why a stock, par value $100, dividend 12%,
yields only 6% on the mone}'" invested when purchased at
$200.
18. A man buys 120 shares of stock at 137|, and receives
a 6% dividend. He sells it at 141f. Find his net profits
on the investment after paying brokerage at -| % each for
buying and selling.
19. Which yields the better income and how much, 6%
stock at $ 120 or 4% stock at $ 85?
20. The Amidon Asbestos Co. is capitalized at $80000.
The gross receipts for a year are $ 170000. The expenses,
material, and repairs amount to $ 14(3000. If $ 14000 is put
in the surplus fund, what dividend can be declared from the
balance ?
BONDS 331
21. A gas company declares a dividend of 8% which
amounts to $64000. What is its capitalization ?
22. A bank is capitalized at 1100000, and pays S%
dividend. How much is the dividend on 3(3 shares?
23. How much must be invested in 0% stock, at 108,
brokerage -|%, to yield an annual income of I 366?
Since $1 of stock
8 366 -=- $.06 = 6100; $6100, par value yields $ . 06 income, the
1.08^ x$ 6100 = $ 6595.63, sum invested par value to yield § 366
must be as many dollars
as $.06 is contained times in $366, or $6100. Adding brokerage, the sum
invested must be 1.08$ times $6100, or $6595.63.
24. If a stock paying 3%% semiannual dividend is quoted
at 120, how much must be invested in it to produce an
annual income of 81400, brokerage \°fo ?
What sum must be invested, at \ % brokerage, in :
25. 3i% stock, at 104, to yield an annual income of $245 ?
26. 5% stock, at 109, to yield an annual income of $1675 ?
27. 4^% stock, at 116 J, to yield an annual income of
$364.50^?
BONDS
When corporations need large sums of money to carry on
their business, instead of issuing more stock, they frequently
issue a series of bonds payable at some future date with in-
terest.
Bonds are written obligations, under seal, by which cor-
porations or governments bind themselves to pay specified
sums, at a fixed rate of interest, at or before the time speci-
fied in the bonds.
The bonds of a business corporation are secured by a
mortgage on its property. This mortgage authorizes the
332 STOCKS AND BONDS
sale of the property in case the conditions of the bonds are
not fulfilled. The bonds of governments are without
mortgage.
Bonds and stocks are at a premium when they sell for
more than the face, or par value ; at a discount when they
sell for less than the face, or par value.
Coupon
ACME GL2ASS COMPANY
will pay to Bearer at the
Colonial Erust GTo. of $ tttsuttrg, $a.
on the first day of June, A. D., 1907,
in United States Gold Coin, being six months' Interest
on Bond No. 501
fawve& &hui&, Treasurer.
A coupon bond is a bond with interest coupons attached.
These coupons are detached when the interest is due, and the
amount may be collected personally or through a bank.
Coupon bonds are payable to the bearer.
A registered bond is a bond registered on the books of
the corporation issuing it. The interest when due is sent
by check to the owner. Registered bonds are payable to the
owner or to his assignee.
The name of a bond often indicates its rate of interest
and the time when the bond is due. Thus, " U. S. 4's, 1907,"
are United States 4 % bonds, due in 1907 ; " Western Union
7's, 1920," are Western Union bonds, due in 1920, and
BONDS 333
bearing ~<f interest; " T T . S. Steel 7's," are United States
Steel bonds, bearing 7% interest.
Stock or bond quotations are the prices at which stocks and bonds are
selling. Thus, B. & O. t's quoted at 96 means that Baltimore & Ohio
bonds are selling at 96% of their par value The buyer pays 96 4- \
brokerage = 96|, or $96,125, per share. The seller receives 96 — £ broker-
age = 95|, or $95,875, per share.
Comparative Study
Stockholders are the otoners of corporate property; bondholders are
creditors who have loaned money to the corporation or government.
Bonds bear interest at a fixed rate and mature at a time specified in the
bond, stocks continue while the corporation exists and pay dividends
according to the earnings of the company.
Commission or brokerage in commission is reckoned on the actual
amount of goods sold, or the amount of money involved in a transaction ;
brokerage in stocks and bonds is always reckoned on the par value of the
stocks or bonds bought or sold.
Problems in Stocks and Bonds
1. Find the cost of 15 $1000 United States bonds at
103|, brokerage \%.
2. How many Mound City water bonds, at 4| % premium,
can be purchased for 85225, brokerage not included ?
3. Find the par value of 4% government bonds that
yield $1200 annually.
4. Find the face of a 3% bond, when an interest coupon
brings $60 annually.
5. A man desires to invest in 4% bonds sufficient to yield
an income of $1200 per year. If the bonds are selling at 118,
find the amount that must be invested, including brokerage.
6. A $2000 4% bond, interest payable semiannually, is
sold after one interest period at 10% premium. Find the per
cent of gain on the investment if the bond was bought at par.
334 STOCKS AND BONDS
7. $5 is the dividend on a 1100 stock bought at 180.
Find the rate of income on the investment.
8. Copper stock, par value $10 and paying 24% divi-
dend, is bought at $40. No allowance being made for
brokerage, find the rate of income from the investment.
9. I purchase 100 shares of stock, par value $50, at $65
and, after receiving 2 dividends of 2% each, sell at $78.
Find my gain, brokerage -| % in each transaction.
10. Compute the rate of income from 5% stock bought at
80 ; at 90 ; at 100 ; at 120 ; at 125.
11. My income from G %> bonds is $240 per year. How
much have I invested ?
12. I bought 24 shares of mining 'stock at 89 and, after
keeping it 3 years, sold it at 39. As no dividends were paid,
find my loss, money being worth 6% simple interest.
13. A $1000 5% bond, bought at par, after paying 6 annual
dividends, is sold for $.60 on the dollar. Find the loss,
money being worth 5% simple interest.
14. A certain stock bought in 1904 at 40^ per share
was sold in 1906 at $2.40 per share. Find the gain per
cent on the investment.
15. A $1000 5% bond due in 10 years was purchased for
81100. Find the average rate of interest on the investment,
if the bond is held to maturity.
The total income on the bond for 10 years at 5% = $500. Loss by
redemption of bond §1100 - 81000 = $100. Total income in 10 yr. =
$500 - $100 = $400. Average annual income = ^ of $400 = $40.
Average annual rate = $40 -*- $1100 = 3^%.
16. A §2000 4 % bond due in 5 yr. was bought for $1900.
Find average rate of interest, if bond is held to maturity.
Note. The purchaser gains $100 when the bond is redeemed.
TEST PROBLEMS IN PERCENTAGE
1. Mr. Byers's farm is valued at $18000. He pays 4-^j
mills taxes on an 80% valuation of it. Find his tax.
2. An agent buys 30 tons of fertilizer, at $1.50 per hun-
dred, 20%, 10% off. Terms: 30 days net, or 2% for cash.
If he pays cash, find the cost.
3. A western farmer sells 10000 bu. of wheat, \t per
bushel brokerage, at 89^. After deducting freight and
drayage of $>67A, find the net amount of the sale.
4. An Iowa farmer buys cattle for $1500, and sells them
for $2350. If the grazing and feeding are 20% of the cost,
find the per cent of profit on the sale.
5. A merchant has a note against Mr. Johnston for $500,
bearing 6 % interest, dated June 1, 1906, due in one year.
If he discounts the note at his bank March 1, 1907, at 7 %,
find the proceeds.
6. A farmer has the following annual insurance on his prop-
erty : house valued at $ 2000, insured at | % ; barn valued at
?2500, insured at T 9 o%; grain valued at $1000, insured at
|%; live stock $1200, insured at |%. If twice the annual
premium covers the cost of the insurance for 3 years, find
the yearly cost if his property was insured for 3 years.
7. Mr. Ames, who keeps a general store, sold goods in
one year amounting to $24000. If he has an average of
20% profit on the cost of the goods, find his profits for the
year.
335
336 TEST PROBLEMS IN PERCENTAGE
8. A manufacturer, owing to a depression in business,
offers goods at 12|% discount, but finally sells at a further
discount of 8%. Find the entire per cent of discount.
9. What per cent is gained by buying stocks at 15 °}
discount and selling at 5 % premium, brokerage \ °J ?
10. The tax rate in a certain city is 17 mills on the dollar
on a valuation of $66390. Find the tax on a property val-
ued at $12500.
11. A salesman who received a salary of $2400 and $1500
expenses, sold $75000 worth of goods. In addition he re-
ceived 2% on all sales over $60000. What per cent of the
selling price of the goods did it cost the firm to sell them ?
12. A man buys through his broker 10 shares of railroad
stock (par value $100) at $125 per share. After receiving
5 semiannual dividends of 3% each, he sells the stock at
$131^. Find the rate per cent of gain on the investment.
13. By selling a piano at 40 % above cost, a profit of $150
is realized. For how much must the piano be sold to realize
a profit of 56 (f ?
14. A collector is given a bill of $1500 to collect at 5%.
He succeeds in collecting 90 cents on the dollar. Find how
much is due his client and how much is the collector's com-
mission.
15. A house and lot cost $6000. The insurance averages
$14, taxes $50, and repairs $56 annually. For how much
must the house rent per year to realize 6% net on the
investment ?
16. A house rents for $40 per month, and it costs the
owner on an average $125 per year for insurance, taxes,
and repairs. If the property yields him 5% net on the
investment, find the cost of the house.
RATIO AND PROPORTION
RATIO
1. The quotient of 30 -f- 10 is 3. Compare 30 with 3 in
such a way as to show how many times 3 is contained in 30.
What, then, is the relation of 30 to 3?
Ratio is the relation of two similar numbers as expressed
by the quotient of the first divided by the second.
2. What is the ratio of 10 to 8 ? of 12 to 4 ? of 3 to 6? of
2 yd. to 8 yd.? of $12 to 83.
Since the division of two similar numbers gives an abstract
quotient, all ratios are abstract.
The sign of ratio is a colon : placed between the numbers.
Thus, the ratio of 12 to 3 is written 12 : 3. It is read, l » the
ratio of 12 to 3." It may be written also 12 ~- 3, or J^-.
The terms of a ratio are the numbers compared. The first
is the antecedent ; the second the consequent.
19 # r _ antecedent _ 19 _i_r -- dividend !- numerator
consequent divisor denominator
Since the antecedent of a ratio may be regarded as the
numerator, and the consequent as the denominator of a frac-
tion, both terms of a ratio may be multiplied or divided by the
same number without changing the value of the ratio.
Find the ratio of :
3.
10 to 5
7. 18 to 9
11.
40 to 10
15.
50 to 25
4.
5 to 15
8. 27 to 9
12.
8 to 24
16.
24 to 8
5.
8 to 2
9. 35 to 5
13.
2 to]
17.
4 to 1
6.
2 t0 i"
HAM. COHP1
10. J to 1
.. AKITH. — 22
14.
:;37
fto|
18.
ftoj
338 RATIO AND PROPORTION
Written Work
Find the value of the following ratios :
1.
125 : 25
5.
-2- to -1-
3 lo 12
9.
$225 to $2.25
2.
6.25 : 25
6.
(3.4 to 16
10.
3 yd. to 3 ft.
3.
1 toj
7.
37| to 200
11.
75% tol2|%
4.
i to H
8.
&2} to 500
12.
1 mi. to 1 rd.
SIMPLE PROPORTION
Proportion is an equality of ratios; thus, 12: 6 = 8: 4 or
-g 2 - = | is a proportion.
Proportion is generally indicated by the equality sign or
by a double colon : : between the ratios. Thus, 12:6 as
8 : 4, is written, 12 : 6 = 8 : 4, or 12 : 6 : : 8 : 4.
A proportion may be read in two ways ; thus, 12:6 = 8:4
is read, " The ratio of 12 to 6 is equal to the ratio of 8 to 4,"
or, "12 is to 6 as 8 is to 4."
The extremes are the first and the fourth terms of a pro-
portion ; the means are the second and the third terms.
In 15 : 5 = 12 : 4, the extremes are 15 and 4 ; the means, 5 and 12.
Find the product of the means ; then the product of the
extremes :
1. 8 : 4 = 10 : 5 3. 24 : 4 = 36 : 6 5. § = f
2. 15:3 = 30:6 4. | = T % 6. | = ^
Observe how the product of the extremes in each proportion
compares with the product of the means.
In every proportion the product of the means is equal to the
product of the extremes.
SIMPLE PROPORTION
:;:',!)
Written Work
Find the value of x, the unknown term :
1. 36: 0=24: x
Then, 30 times x, or 36 x = 144, and once x, or x = 4.
2. 15 : 25 = x : 40
Then, 25 x = 15 x 40, or 600, and x = 24.
8.
9.
10.
11.
12.
3.
60 : 15 = 75 : x
4.
75: a; =90: 18
5.
40: z=72:18
6.
x.: 30 = 8 : 48
7.
a: 45 = 7 : 63
3
5
5
8
2 _ q
3 — '
z= 25
< .5
: 1.5=2.5
6.25
: 2.5= x
60
150= 36
X
8
x
1
a;
13. If 8 sheep cost $48, how much will 20 sheep cost ?
8 sheep cost $48
20 sheep cost $ x
Since ratio is the relation of two similar numbers, 8 sheep and 20
sheep form one ratio, and $48 and $x, the other ratio.
Write as the second ratio $48 : $x. Since 20 sheep cost more than
8 sheep, $x represents a larger sum than $48. Therefore, as the larger
number is the consequent of the second ratio, the larger number must be
made the consequent of the first ratio. The proportion, therefore, is,
8 sheep : 20 sheep = $48 : $z
8x = $960
x = $120
14. It is estimated that 25 men can build a bridge in 18
days. How long at the same rate will it take 15 men to
build it ?
15. How much will 30 bushels of potatoes cost, if 70
bushels cost $42 ?
16. It is estimated that 90 men are necessary to grade
a certain street in 45 days. If only 81 men are hired to do
the work, how long will it take them?
340 RATIO AND PROPORTION
17. The ratio is }. The first term is | of (6x4). What
is the second terra ?
18. A bankrupt's debts are $32000, and his assets $10000.
Counting nothing for court costs, how much will be paid on
a claim of $6150?
19. If $2.25 is paid to clean 35|- square yards of paper, how
much at the same rate will it cost to clean 65| square yards ?
20. A bakery sells 5^ loaves weighing 6 oz. when flour is
$4. What size loaves should they sell at 5^ when flour is
$6 per barrel?
21. A map is drawn on a scale of 100 miles to | of an inch.
What distance is represented on the map by T 5 g of an inch ?
22. If the interest on $500 for 6 months is $15, how
much is the interest on the same sum for 1 year 4 months ?
23. If 6.5 tons of coal cost $55.90, how much will 9.25
tons cost at the same rate ?
24. A estimates that he can do a piece of work in 20 days,
working 8 hours per day. How long will it take him to do
\ of it, working 10 hours per day ?
25. Sound travels 1120 feet per second. How long will
it take the sound of a cannon to travel 8 miles ?
26. In a stamp canceling machine 1000 letters were can-
celed in one minute and 20 seconds. If the machine was in
operation for 5 minutes and 10 seconds, how many letters at
the same rate were canceled ?
27. A monument casts a shadow 150 feet long. At the
same time a post 3 feet in height casts a shadow 2 feet and
6 inches long. Find the height of the monument.
28. A machine for making pressed brick turns out 7500
brick in 6 days. How large an order for pressed brick can
be filled in 25 days ? *
PARTITIVE PROPORTION AND PARTNERSHIP ^41
29. An automobile passed 5 mile-posts in 8 minutes 10
seconds. How many miles per hour was it moving?
30. It is estimated that 24 men working 18 days can re-
pair a certain street. The contract calls for the work to be
completed in 8 days. How many extra men must be employed?
31. It is estimated that GO men can dig a sewer on Main
Street in 24 days. The contract time is 40 days. How
many men may be discharged and yet have the work com-
pleted within the contract time ?
32. A's property is assessed at $2750, on which $37.90
taxes are paid each year. How much tax should B pay on
his property assessed at $4375 ?
33. A train of 30 cars of ore contains 1200 tons. How
many cars must be added so that the train may carry 2700
tons ?
34. A city's assessed valuation is $5675000. There must
be raised in taxes on this valuation $ 85125. How much is
Mr. Templetons tax on a property assessed at $15750?
PARTITIVE PROPORTION AND PARTNERSHIP
Partitive proportion is the process of separating a number
into parts proportional to two or more numbers.
Written Work
l. Separate 180 into parts proportional to 1, 2, and 3.
Since the parts are in the ratio of 1, 2, and :5, then 1 + 2 + 3, or G
parts = 180.
The 1st number = 180 - 6, or 30
The 2d number = 2 x 30, or GO
The 3d number = 3 x 30. or 90
Test : 30 + 60 + 90 = 180 ; 30 : 60 : 90= 1 :2: 3.
342 RATIO AND PROPORTION
2. A man and two boys earn $162 and agree to divide it
as follows : 3 parts to the man, 2 parts to the first boy, and
1 part to the second boy. How much should each receive?
3. The receipts of a street railway in one month are
$15600, and the expenses are to the profits as 1 to 2. Find
the expenses and the net savings.
4. Four men own a gold mine valued at $805000. The
parts owned by each are in the ratio of 6, |, f, and T 9 g. Find
each man's share of the mine.
5. The cost of shipping a train load of 2000 tons of iron
ore from Duluth, Minn., to Bessemer, Pa., is $2800. If the
lake freight is to the railroad freight as 50 to 90, find each
one's share of the freight charges.
6. Two railroads, valued at $6900000 and $23000000,
share charges of $299000 for freight carried over both
roads in proportion to the valuation of each road. Find the
earnings apportioned to each road.
Partnership is the associating of two or more persons
who agree to combine their money, labor, goods, skill, or
" good will " in some enterprise, and to share the profits or
losses of the business in proportion to the interest each part-
ner owns.
The partnership is frequently called a firm, or a house, and derives
its name from the persons that compose it; as," Brown & Hamilton."
The capital of a partnership is the sum of the investments
of the partners. This capital may be money or anything that
has a money value, as skill, good will, experience, labor, etc.
Gains and losses in a common partnership are usually appor-
tioned in proportion to the amount of capital each partner invests
and the length of time such capital is invested ; but in case any
partner cannot pay his proportionate share of the loss, the re-
maining partners are liable for the whole loss.
PARTITIVE PROPORTION AND PARTNERSHIP 343
Written Work
l. A and B engage in business ; A furnishes $800 and B,
81200; they gain $500. What is each man's share?
$800 + $1200 = $2000, entire capital
SS,H)
$2000
= I, A's share of the capital
® UQ0 = §, B's share of the capital
$2000
| of $500 = 8200. A's .share of the gain
| of $500 = $300, B's share of the gain
Or, $2000 : $800 = $500 : $200
$2000 : $1200 = $500 : $300
The ratio of the whole capital to each partner's investment is equal to
the ratio of whole gain or loss to each partner's share of the gain or loss.
2. A, B, and C engaged in manufacturing iron. A in-
vested 842000, B $96000, and C his skill, valued at $72000.
Their profits the first year were $12600. How much was
each man's gain ?
3. M, N, and R formed a partnership; M furnished | of
the capital, N §, and R the remainder. They gained $7560.
What was each man's share of the gain ?
4. E, F, and G engaged in merchandizing with a capital
stock of $28000. E furnished 87000, F $6000, and G the
remainder. They gained 14f% on the investment. What
was each man's share of the gain?
5. The assets of a firm that failed in business were
$3750 ; their liabilities 8 22000. How much will two credit-
ors, to whom they owe $7800 and $5400 respectively, receive?
6. A storeroom belonging to Smith, Jones, & Brown
was entirely destroyed by fire. They received $9675 insur-
ance. What was each man's share, if Smith owned ^, Jones ^,
and Brown the remainder of the stock?
344 RATIO AND PROPORTION
7. A and B formed a partnership January 1, and each
invested 12500; May 1 A added $500, and B withdrew
$500. At the end of a year their gain was $1800. How
much should each one receive?
A's capital, $2500 for 4 mo. = $10000 for 1 mo.
A's capital, $3000 for 8 mo. = $24000 for 1 mo.
A's total capital
= $34000 for 1 mo.
B's capital, $2500 for 4
mo.
= $10000 for 1 mo.
B's capital, $2000 for 8
mo.
= $16000 for 1 mo.
B's total capital
= $26000 for 1 mo.
Total capital of both
= $60000 for 1 mo.
A , • 34000
As Gram = , or
fe 60000
?0>
of $1800 = $1020.
-o, ■ 26000
B s sain = , oi
& 60000
1 3
3T>>
of $1800 = $780.
$1020 -f $780 = $1800, total gain.
Test:
8. M and N formed a partnership for 2 years. M put
in $6400; N put in $3600 and at the end of 6 months
added $1400. Their settlement at the end of 2 years
showed $7956 profits. How should it be divided?
9. R and S began business as partners April 1, 1904,
each investing $5000. On July 1, 1904, R added $3000
and S, $2000. They dissolved partnership January 1, 1905,
sharing a profit of $3150. Find each one's share.
10. A, B, and C formed a partnership for 3 years. A put
in $10000, B $8000, and C $6000. A withdrew $2000 at
the end of 18 months. They dissolved partnership at the
end of 2 years with a loss of $4750. As nothing could be
collected from C, what proportionate share of the loss should
A and B pay?
PROBLEMS FOR ORAL AND WRITTEN
ANALYSIS
1. Two properties are valued at $1000; J of the value
of the first equals f of the value of the second. Find the
value of each.
2. At a certain election 1080 votes were cast for A and B ;
| of the votes cast for A equaled f of those cast for B. How
many votes were cast for each candidate ?
Solution. — f of A's vote = | of B's vote.
\ of A's vote = i of % of B's vote, or 1 of B's vote.
| or A's vote = 4 x 4, of B's vote, or 4. of B's vote.
| of B's vote = B's vote,
f of B's vote = A's vote.
| of B's vote = vote of both, or 1080 votes.
B's vote = 600.
A's vote = 480.
3. A real estate dealer paid 87200 for two city lots; f of
the cost of the first lot equaled ^ of the cost of the second.
How much did each cost?
4. A mill and machinery cost $27000; f of the cost of
the mill equaled f of the cost of the machinery. How
much did the machinery cost?
5. What per cent of a day are 12 hours? 6 hours? 36
hours?
6. | of Frank's money equals £ of Henry's, and Frank has
83 more than Henry. How much has each?
345
346 PROBLEMS FOR ORAL AND WRITTEN ANALYSIS
7. Walter and Philip bought sleds; f of the cost of
Walter's sled equaled | of the cost of Philip's; both sleds
cost $2.70. How much did each cost?
8. A pair of shoes that cost a dealer $2.50 were sold for
$3.50. What was his gain per cent?
9. An estate was so divided between two sons that the
share of the elder was to that of the younger as | to ^. If
the elder son received $1000 more than the younger, what
was the value of the estate ?
10. Brown and Long were partners in business ; Brown
furnished | as much capital as Long, and their profits for*
the first year were $2250, which was divided in the ratio of
the capital invested. What was the share of each ?
11. In a partnership A invested | as much as B, and C in-
vested | as much as B ; they shared a loss of $ 2000. How
much should C pay ?
12. A piano sold for $360, which was at a loss of 20 %.
What was the cost?
13. Moore, Silvens, and Rogers were partners in business
and made a profit of $ 4500. Moore owned y 4 ^ of the stock,
Silvens •§, and Rogers ^. What was each partner's share of
the total profit ?
14. A clerk's expenses are $30 a month, which is 66| %
of his salary. How much is his salary ?
15. A stone cutter received $4 a day for his labor and
paid $6 a week for his board. At the end of 16 weeks he
had saved $212. How many days did he work ?
16. If in an investment | of A's capital equaled | of B's,
and A received $900 for managing the business, how should
profits of $5100, including cost of management, be divided?
PROBLEMS FOR ORAL AND WRITTEN ANALYSIS 347
17. A wagon was sold for I 12, which was 12.] % less than
the price paid. What was the cost of the wagon?
18. A carpenter's wages were $3.50 a day, and he paid
50 cents a day for his board. If in 40 days he saved $>99,
how many days was he idle ?
19. A real estate dealer bought some lots at #150 each,
and twice as many at 8175 each. He sold them at $200
each, thereby gaining $800. How many did he buy in all?
20. A manufacturer pays boys $1, women $1.25, and men
$1.75 a day, and his weekly pay roll is $348. He employs
three times as many boys as men, and twice as many women
as men. How many persons does he employ ?
21. A farmer paid $7200 for two farms of equal size, pay-
ing $50 an acre for one and $40 an acre for another. How
many acres were there in both farms ?
22. A dealer bought 20 dozen glasses at 50^ per dozen.
At what price per dozen must he sell them to make a profit
of 20 % on the transaction ?
23. It is estimated that 80 men can make an excavation
for a public building in 30 days. After working 12 days,
\ of the men were discharged. In how many days could the
remainder finish the work ?
24. A father desires that the amount of $5000 for 6 years
at °J shall be divided between his son and daughter in the
ratio of 8 to 9. Find the share of each.
25. An architect gets a commission of 5% for drawing
plans and superintending the construction of a building cost-
ing 825000. How much is his commission ?
26. The interest on | of Robert's money and | of Samuel's
money for 4| years, at 6%, is $81 and $121.50 respectively.
How much money has each?
348 PROBLEMS FOR ORAL AND WRITTEN ANALYSIS
-
27. The sum of E's and F's money being on interest for
five years, at 5%, amounts to $3000. How much money has
each, if E's is f of F's?
28. The amount of a certain principal for 4 years at a
certain per cent is $620 ; and for 7 years, $710. Find the
principal and the rate per cent.
29. 30 is 6 % of what number? 25 is | % of what number?
30. It is estimated that 15 men can build an embankment
of earth in 20 days. If 5 additional men are employed, in
how many days can it be built?
31. Harley spent ^ of his money and $5 more for a suit of
clothes, and had $11 remaining. How much money had he
at first?
32. After paying 25 % of his debts, a merchant found that
$240 would pay the remainder. How much did he owe at
first?
33. How shall I mark goods that cost $750, so that I can
deduct 10 °J from the marked price, and yet make 20 % on
the cost?
34. A speculator bought wheat at 80^ per bushel and sold
it at 90^ per bushel. How many bushels did he buy if his
gain was $2000?
35. An agent remitted $95 as the proceeds of an account
he collected. How much did he retain if his rate of commis-
sion was 5 % ?
36. A bankrupt, who owed $12000, paid 60^ on the dol-
lar. Find A's claim, and his loss if he received $600.
37. 8 men take equal shares in an oil lease, agreeing to
give the owner of the land \ royalty on all oil produced.
How much greater interest in the oil has the owner than any
of the other men ?
LONGITUDE AND TIME
Meridians are im-
aginary lines passing
north and south from
one pole of the earth
to the other.
The equator is an
imaginary line pass-
ing around the earth
midway between the
poles.
These imaginary lines
aid in locating places on
the earth and in determin-
ing differences in time.
Observe that the equator is a circumference of a circle ; therefore
distances along it are measured in degrees.
The prime meridian is a meridian from which time and
place, east and west, are reckoned.
The meridian passing through the Royal Observatory at Greenwich,
England, is the prime meridian in common use.
Longitude is the distance east or west of this prime merid-
ian measured in degrees.
Places east of this prime meridian have east longitude; places west of
this prime meridian have ivest longitude.
From the time the sun's rays are vertical over any meridian
until they are vertical again it is 24 hours. Therefore, any
349
350 LONGITUDE AND TIME
point passes through 360° in one rotation of the earth on its
axis.
Since 360° of longitude pass under the sun's vertical rays
during 24 hours, how many degrees pass during 12 hours ?
1 hour ?
Since ■£% of 360° or 15° pass under the sun's rays in 1 hour,
then 1 hour of time corresponds to 15° of longitude.
Since 15° of longitude correspond to 1 hour of time, ^ of
15° or |°, or 15' of longitude, correspond to 1 minute of time,
and 15" of longitude to 1 second of time.
Table of relation between longitude and time :
360° of longitude correspond to 24 hours of time
15° of longitude correspond to 1 hour of time
15' of longitude correspond to 1 minute of time
15" of longitude correspond to 1 second of time
1° of longitude corresponds to 4 minutes of time
1' of longitude corresponds to 4 seconds of time
When the sun's rays are vertical on the 90th meridian,
all places on that meridian have noon.
The rotation of the earth from west to east makes the sun
appear to move from east to west. The Mercator's map on
p. 351 shows that when it is noon at Greenwich, it is before
noon or earlier at all places west, because the sun's rays are
not yet vertical on any meridian west of the prime meridian.
It is after noon or later at all places east, because the sun's
rays have already been vertical on all meridians east of the
prime meridian.
Examine the map. What time is it on the meridian of
Greenwich ? 45° east of Greenwich ? 45° west of Greenwich ?
In traveling from London to New York would a watch be
set forward or backward'? about how much? About how
much change in time must be made in traveling from Cal-
LONGITUDE AND TIME
351
cutta westward to Sun Francisco? from Honolulu east-
ward to Cape Town ?
135 105 75
45 15 15
45 75
105 135
East 160 150 I20i°"9 90 w«> 60
30
30 60 Long 90 East 120
Map showing Noon, February 1, at Greenwich
What is the difference in degrees between a place 30° east
longitude and a place 45° east longitude? What is the
difference in time and which has the earlier time ?
Table of Longitude of Some Important Places
London
0° 5' 48" W.
Cape Town
18° 28' 45" E.
New York
74° 0' 3" W.
Honolulu
157° 50' 30" W.
Pittsburg
80° 2' 0" \V.
Tokyo
139° 11' 30" E.
Washington
77° 3' 00" W.
Manila
120° 58' 0" E.
Chicago
87° 30' 42" W.
Canton
113° 10' 30" E.
San Francisco
122° 2.">' 42" W.
Berlin
13° 23' 44" E.
Boston
71° 3' 50" \V.
Rome
12° 27' 11" E.
1 >>• iiver
104° 58' ()" W.
Paiis
2° 20' 15" E.
Longitudes are given to the nearest seconds.
352
LONGITUDE AND TIME
Written Work
1. When it is noon, solar time, at Paris, what is the solar
time at New York ?
Since the earth rotates 15° in 1 hr., 15'
in 1 tnin., and 15" in 1 sec, the differ-
ence in time is as many hours, minutes,
and seconds as there are degrees, minutes,
and seconds in T x 5 of the difference in
longitude.
The difference in time is 5 hours, 5 minutes, 21| seconds. Since New
York is west of Paris, the time in New York is earlier ; that is, when it is
noon at Paris, 't is 6 o'clock, 54 min., and 38| sec. a.m. at New York.
2°
20'
15" E.
74°
0'
03" W.
15)76°
20'
18"
5°
5'
91 \U
* dl 5
5 hr. 5
min.
21isec.
5rir-.5niin. 2iy s .| 12 Noon
NewYork ' n lon 3 J6°20'\& Paris
What is the difference in longitude between the two places? the dif-
ference in time? In going west from Paris to New York would a trav-
eler set his watch forward or backward? how much?
Note. — Study this diagram and make a similar one for each problem.
Find difference in degrees, difference in time, and which
place has earlier time :
Places
2. 60° W. and 45° W.
3. 120° W. and 75° W.
4. 15° W. and 45° E.
5. 30° E. and 60° E.
6. 75° W. and 30° E.
Places
7.
120° W. and 30° E.
8.
90° W. and 30° E.
9.
135° E. and 30° E.
10.
45° W. and 60° E.
11.
45° E. and 15° E.
LONGITUDE AND TIME 353
When the sun's rays are vertical on the meridian of Wash-
ington, find the solar time in the following places :
12. Denver 15. Paris 18. Berlin
13. Chicago 16. Rome 19. New York
14. San Francisco 17. Honolulu 20. Pittsburg
21. When it is midnight (solar time) on the last day of
the year in Boston, how much of the year (solar time)
remains to the people of Honolulu ?
22. The first shock of the earthquake at Kingston, Jamaica
(long. 76° 47' W.), Jan. 14, 1907, occurred at 3:25 p.m.
What was the solar time at New York ? at Cape Town ?
23. A ship sets sail from Liverpool for New York, Jan.
10, 1907. When in longitude 34° 6' 10" W. its chronometer
reads 2:30 p.m. Jan. 15. Find the difference in the read-
ings between the ship's time and the meridian time of New
York.
24. Berlin meridian time is 6 hr. 44 min. and l\i sec. later
than Chicago meridian time. Find the longitude of Berlin.
101 26 difference in longitude
87 36 42 W. (Chicago)
101 26 13 23 44 E. (Berlin)
15 times the difference in time expressed in hours, minutes, and
seconds corresponds to the difference in longitude expressed in degrees,
minutes, and seconds.
Therefore, 15 x 6 hr. 44 min. \\\ sec. corresponds to 101° 0' 26" of
longitude.
This difference in longitude would not tell us whether Berlin is east or
west of Chicago, but as Berlin has faster time than Chicago, it must be
east of it. Chicago is 87° 36' 42" west of the prime meridian. There-
fore, Berlin must be 13° 23' 44" east of the prime meridian.
HAM. COMPL. AIMTII. — 23
hr.
min.
sec.
6
44
m
15
354 LONGITUDE AND TIME
25. The "Treaty of Portsmouth" between Japan and
Russia was signed at Portsmouth, N.H., Sept. 5, 1905, at 47
minutes past 3 p.m., 75th meridian time. What was the cor-
responding solar time at St. Petersburg, Russia, 30° 17'
51" E. and at Tokyo, Japan, 139° 44' 30" E. ?
Tokyo is east of the 75th meridian 214° 44' 30", therefore its time is
14 hr. 18 min. 58 sec. faster than Portsmouth, which has 75th meridian
time. Counting this time forward from 3:47 p.m. Sept. 5, gives 5
minutes and 58 seconds past 6 o'clock a.m. Sept. 6, Tokyo solar time.
26. The President of the United States takes the oath of
office at 12 noon, 75th meridian time. Find the solar time
and date in each of the following places for the inaugura-
tion of March 4, 1909 : Honolulu ; Berlin ; San Francisco ;
London.
International Date Line
The nations have agreed upon the 180th meridian, with slight
changes as shown on page 355, as the place where the new day always
begins. The calendar is set forward one day on ships crossing this line
sailing westward : the calendar is set back one day on ships crossing this
line sailing eastivard.
1. A ship sets sail from San Francisco for Manila, Oct. 9,
1906, at 9 A.M., 120th meridian time; it arrives at Manila,
Oct. 27, at 9 a.m., meridian time. How long is the voyage ?
2. The same ship sets sail from Manila, Nov. 3, 1906, at
3 p.m., meridian time, and arrives at San Francisco, Nov. 23,
at 3 p.m., 120th meridian time. How long is the voyage ?
Standard Time
The railroads of the United States in 1883 agreed upon
a system of standard time and divided our country into four
time belts, as shown on the map. In the eastern time belt-
all trains keep the time of the 75th meridian, known as
STANDARD TIME
355
eastern standard time. In the central belt all trains keep
the time of the 90th meridian, known as central standard
time. In the mountain time belt all trains keep the time of
the 105th meridian, known as mountain standard time. In
the Pacific time belt all trains keep the time of the 120th
meridian, known as Pacific standard time.
Each railway has selected the most convenient towns on its route, as is
shown on the map, to change from the standard time of one belt to the
standard time of another belt. The time in any belt is 1 hour faster
than the time in the belt west of it, or 1 hour slower than the time in
the belt east of it. Correct time is telegraphed each day to all parts of
the United States from the Naval Observatory at Washington.
/~*V. pr»ndca
; • .-• \ — Vandaa •
/ : u
Fort
Minneapolis '.
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'peJI"r." 1 .roag Pine _.'.--
. ! V North .'■ K^ l f
f ^*°\ oke f-.McCook V «
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\ > ' "— £. ^vpagc City* Hoieington
£ i
Dallaa '•
STANDARD TIME
/ BELTS
ISO
y - / '~"\
j, J^O^J ^few Orleans
Galveston
1. What is the difference in time between closing of the
election polls at 7 p.m. in New York and 7 P.M. Seattle, on
the day for choosing presidential electors ?
2. A telegraph message was sent from Philadelphia at 11
a.m. Oct. 12, l!»0*i, to San Francisco and delivered at 7:45
A.M. San Francisco time. Why could this be true?
GOVERNMENT LAND MEASURES
Surveyor's square measure is used by surveyors in measur-
ing and computing land areas.
16 square rods
= 1 square chain
10 square chains
= 1 acre
640 acres
= 1 square mile
36 square miles
= 1 township
The Gunter's chain for measuring land is gradually going out of use.
In its place surveyors use a steel tape 100 ft. long, divided into feet and
decimal parts of a foot. They find the number of square feet in the plot
to be measured, and change the result to acres by dividing by 43560, the
number of square feet in an acre.
The public lands of the United
States are surveyed by selecting
a north and south line, called a
principal meridian, and intersect-
ing this by an east and west
line, called a base line.
Range lines are lines running
north and south on each side of
the principal meridian, at dis-
tances of 6 miles. They divide
the land into strips 6 miles wide,
called ranges.
East and west lines parallel to the base line, and at dis-
tances of 6 miles, divide the ranges into townships. A range
is, therefore, a row of townships running north and south.
The townships in each range are numbered north and south
356
A Group of Townships
<;<)\T.KXMK\T I. AND MEASURES
357
from tlir base line, and the ranges are numbered cast and
west from the principal meridian.
A township is designated by its number and direction from
the base line, the number and position of its range, and the
name or number of the principal meridian. Tims, Township
A is 4 North, Range 5 West of Principal Meridian.
i
S
*
3
2
1
7
a
9
10
II
12
IB
17
76
IS
It
13
19
20
21
22
23
24
.30
29
28
27
25
25
31
32
33
34-
35
J6
W/2
section
(320 A)
n.e.v*
of
NE. '/*
S.E./*
of
NE.'/*
Eti
of
■.: .
of
S£3i
of
S.E.'/*
A Township
A Section
A township is 6 miles square and is divided into 30 sec-
tions each one mile square. Each section contains 640 acres.
1. W. \ Sec. 31, T. 22 N., 4 E. 3d P. M. is read west |
section 31, township 22 North, Range 4 east of third princi-
pal meridian.
2. Read: S. 1 of S.E. J, Sec. 31.
3. Read : NAY. \ of S.E. J, Sec. 31.
4. Read : N.E. \ of N.E. \, Sec. 31.
Locate the tract of land in sections and give number of
acres:
5. SAY. \, Sec. 5, T. 4 S., R. 3 W.
6. S.l of N.E. l Sec. 4, T. 15 N., R. 5 E.
7. S.E. \ of N.W. {, Sec.8, T. 125 S., R. 4 W.
8. Find the number of rods of fence required for the tract
in problem 7.
9. How many rods of fence are required for the tract
mentioned in problem ?
POWERS AND ROOTS
1. 3x3 = 9. 4x4 = 1G. 5x5x5 = 125.
2. Name the two equal factors that produce 9. 16.
3. Name the three equal factors that produce 125.
4. What two equal factors produce 25? 36? 49? 81?
5. What three equal factors produce 8? 27? 64?
6. How many times is 5 used as a factor in 5 ? 25 ?
125?
7. How many times is 3 used as a factor in 3 ? 9 ? 27 ?
A power of a number is the product obtained by taking
the number one or more times as a factor. Thus, 9 is a
power of 3, and 8 is a power of 2.
The first power of a number is the number itself. The
second power of a number is called the square of the number.
Thus, 16 is the square of 4. The third power of a number is
called the cube of the number. Thus, 64 is the cube of 4.
8. What is the square of 4 ? 5 ? 6 ? 7 ? 8 ? 9 ?
9. What is the cube of 2? 3 ? 5 ? 6 ? 7?
10. | x | = |. What, then, is the square of f ?
11. I x | x | = §£. What, then, is the cube of f ?
12. .3 x .3 = .09. .5x.5=.25. What, then, is the
square of .3 ? of .5?
The square of a fraction is found by squaring both terms, the
cube of a fraction by cubing both terms.
358
roots 359
13. 3x3 = 9. The square of 3 is indicated thus, 3 2 ;
4 x 4 x 4 = 64. The cube of 4 is indicated thus, 4 3 .
14. How much is 52? 6 2 ? 7 2 ? 5 3 ? 6 3 ? What is the
value of (f) 2 ? (§) 3 ? .4 2 ? .4 3 ? (2\f!
An exponent is a small figure placed at the right of, and a lit-
tle above the number, to indicate the number of times it is to
be taken as a factor ; thus, 4 3 = 4x4x4 = 64. Exponent, 3.
Written Work
1. Square 6, 7, 8, 9, 10, 12, 15.
2. Cube 3, 4, 5, 6, 7, 8, 0, 10, 12.
3. Square 30, 50, 60, 80, 120.
4. Cube 20, 30, 40, 50, 100.
5. Find the value of 6 2 , 8 2 , 9 2 , 5 3 , 6 3 , 7 3 .
6. Find the value of (f) 2 , (f) 3 , (f) 2 , (|) 3 , (f) 2 , (ll) 2 .
7. Square .3, .04, .05, .6, .06. Cube .4, .04, .6.
Find the value of :
8. 15 2 ll. 22 2 14. .75 2
9. 16 2 12. 25 2 15. (If) 2
io. 18 2 13. 6.5 2 16. (16i) a
Find the number of square units in a surface whose side is:
17. 15 in. 20. 8 ft. 6 in. 23. 10 yd.
18. 25 ft. 21. 5 in. 24. 5 yd. 2 ft.
19. 16 yd. 22. 8.5 in. 25. mi.
Find the number of cubic units in a volume whose edge is:
26. 8 in. 28. 3 ft. 3 in. 30. 1 yd. 10 in.
27. 2 ft. 29. 42 in. 31. 12 ft. 4 in.
360
POWERS AND ROOTS
EXTRACTING ROOTS
A root of a number is one of the equal factors that pro-
duce that number. Thus, 3 is a root of 9.
The square root of a number is one of its two equal factors.
Thus, 4 is the square root of 16.
The cube root of a number is one of its three equal factors.
Thus, 4 is the cube root of 64.
1. What is the square root of 25 ? 36 ? 49 ? 81 ? 100 ?
2. What is the cube root of 8 ? 64 ? 125 ? 216 ?
The root of a number is generally indicated by writing
the number under the radical or root sign -y/~ ' , and
placing a figure called the index in the angle of the sign ;
thus, a/27 denotes the cube root of 27. The square root is
indicated by V without the index.
The root of a fraction equals the root of the numerator di-
vided by the same root of the denominator.
Find the required root : Thus, V64 = V8 x 8 = 8.
/16
3.
4.
5.
6.
7.
\V27
a/25
aV«34
8.
9.
10.
11.
12.
v 100
V49
VI 25
v 125
a/81
13.
14.
15.
16.
17.
V216
a/1000
V343
V.064
18.
19.
20.
21.
22.
V512
V144
V169
a/2500
V8100
Some perfect powers and their roots.
Memorize:
VI=1 a/36 = 6
a/4 = 2 a/49 = 7
a^9= 3 a/64 = 8
Vl6 = 4 a/81 = 9
a/25 = 5 a/100 = 10 aVI25 = 5
aVI = 1
aV8 = 2
a/27 = 3
aV64 = 4
a/216 = 6
a/343 = 7
aV512 = 8
a/729 = 9
a/1000 = 10
SQUARE ROOT
361
Finding the root of a perfect square or cube by factoring.
Written Work
1. Find the square root of 1225.
When factored 1225 =5x5x7x7.
Arranged into two like groups 1225 = (5 x 7) x (5 x 7).
V1225 = 5 x 7, or 35.
Find the root of each number, as indicated, by factoring :
7. VT84 12.
13.
V225
V576
V441
4.
5.
6. V1600
V196
8.
9.
10.
11.
V1296
V17G4
^13824
V4096
V2304
a/42875
14.
15.
16.
V3136
a/5832
a/8000
17.
18.
19.
20.
21.
V5184
a/15625
a/32768
ID! Hi
a/19683
SQUARE ROOT
Comparing roots and periods.
The squares of the smallest and the largest integers com-
posed of one, two, and three figures are as follows :
!2 = i 10 2 = 1 00 100 2 = 10000
92 = 8 i 992 = 9801 999 2 = 998001
1. Separate each of these squares into periods of two
figures each, beginning at the right ; thus, 99' 80' 01.
2. How does the number of periods in each square compare
with the number of figures in the corresponding roots ?
The number of periods of two figures each, beginning at units,
into which a number can be divided equals the number of
figures in the root.
Note. — The left-hand period may contain only one figure.
3. How many figures are there in the square root of 4225?
of 12544? of 133225? of 810000?
362
POWERS AND ROOTS
Written Work
1. Square 25. 25 = 20 + 5, hence it may be squared in
two ways, thus:
25 = 20 + 5 The square of 25 has three partial
25 = 20 + 5 products :
125^ 20 x 5 + 52 f(l) 20 2 = 400 1
500= 2Q2+20x5 25 2 = (2) 2(5x20) = 200 \ =625
625 = 20 2 + 2 (20 x 5) + 5 2 [ (3)
2. Find the square root of 625.
5 2 = 25 J
6'25
20 2 = 4 00
Trial divisor, 2 x 20 = 40
_5
Complete divisor, = 45
225
225
20
_5
25
In practice:
6'25 J 25
4
45
225
225
mtm
m
i
Since 625 has two periods, its square root is composed of two figures,
tens and ones. Since the square of tens is hundreds, 6 hundreds must be
the square of at least 2 tens. Two tens or 20
squared is 400, as shown in figure A ; and 625 — 400
leaves a remainder of 225. The root 20, therefore,
must be so increased as to exhaust this remainder
and keep the figure a square.
The necessary additions to enlarge A, and keep
it a square, are the two equal rectangles B and C,
and the small square D.
B, C, and D contain 225 square units ; and since
rj the area of D is small, if 225 is divided by
40, the combined length of B and C, the
quotient will indicate the approximate
width of these additions. The quotient is
5; the entire length of B, C, and D is
20 + 20 + 5 = 45 units ; and the area of the
additions is 5 times 45 x 1 sq. unit, or 225 sq.
units. Since these three additions exhaust
the remaining 225 sq. units, and keep the fig-
ure a square, the side of the required square
is 25 units, and the square root of 625 is 25.
B
:M
±t
20
$z'z
SQUA1JK HOOT
363
3. Square 35(30 + 5); then find the square root of 12:1").
Partial
Products :
(1) 3o 2 =900
(2) 2(5x80) =300
(3) 5 2 = 25
12'25
30 2 9 00
Trial divisor, 2 x 30 = 60
_5
Complete divisor, 05_
► 1225
30
_5
S3
Study of Problem
1. How many
periods in 1225 ?
2. How many
figures, then, will
be in its root ?
3. What is the
largest partial
product in 1225 ?
4. What is the
square root of 9<>o?
5. What two
partial products are contained in the 325 remaining?
6. 325 is composed principally of which partial product? 2(5 x 30).
325 is composed principally of 2 times the first figure of the mot by the
second. Hence, if 325 is divided by (2 x 30), the quotient will give
approximately the second figure of the root.
7. What is the quotient ? What, then, is probably the second figure
of the root?
8. What must be added to the trial divisor (2 x 30) to form the com-
plete divisor?
Since 5 x 65 = 325, 5 is the second root figure, and 30 + 5, or 35, is the
square root of 1225.
9. How does (5 x 60) + (5x5) compare with 5 x 65?
4. Find the square root of 21.16. Separate the number
into periods, left and right from the decimal point.
. 40 2 =
'21.16'
16 00
4.0 + .6 =
4.6
In practice :
'21.16' | 4.6
Trial divisor, 2 x 40 = 80
5 16
5 16
16
_6_
Complete divisor, 86
86
516
5 16
Beginning <tt unit*, separate the number into periods of two
figures each.
364
POWERS AND ROOTS
Find the largest square in the first period on the left, and
write its root as the first figure in the required root. Subtract
its square from the period and annex the second period to the
remainder.
Double the root found for a trial divisor. Divide the re-
mainder, omitting the last figure, by this trial divisor, and
annex the quotient to the trial divisor and also to the root.
Multiply the complete divisor by the second figure of the root,
and subtract the product from the remainder.
Double the root already found for another trial divisor, and
proceed as above until all the periods have been used.
Note. — When a naught occurs in the root, annex a naught to the
trial divisor, bring down another period, and proceed as before.
In extracting the square root of a decimal or a whole number and a
decimal, point off into periods of two figures each, the whole number
toward the left and the decimal toward the right of the decimal point.
In extracting the square root of a fraction, extract the square root of
the numerator and of the denominator if both terms are perfect squares ;
or reduce the fraction to a decimal, and extract the root of the decimal.
Solve the following
5.
V484
9. V1089
13.
v 100
17.
V.0225
6.
V576
10. V1849
14.
Vi as.
v 225
18.
V4.41
7.
V676
V96I
11. V2601
15.
16.
v 324
V725
19.
20.
V6.25
8.
12. V3969
V.0625
Find the square root to the nearest hundredth of
:
21.
315
26. 178.25
31.
96.8256
36.
.8464
22.
525
27. 14884
32.
192ft
37.
.012996
23.
1156
28. 26569
33.
540JL
38.
.000566
24.
4356
29. 136161
34.
31921
39.
.43681
25.
210.25
30. 20.7936
35.
5643||
40.
112225
THE RIGHT TRIANGLE
365
The hypotenuse <>f a right tri-
angle is the side opposite the right
angle.
1. How many square units are
there in the square described upon
the hypotenuse '! in the square
described upon the perpendicular?
in the square described upon the
base ?
2. How do the number of square
units described upon the hypotenuse compare with the gum
of the square units described upon the other two sides ?
That this is universally true is shown by the following diagram :
iri
5
3
\ 4
Take any right triangle, as 1; lay it off on a piece of cardboard and
draw the square on its hypotenuse. Cut this square into the four equal
triangles 1, 2, 3, and 4, and the small square 5. as here shown.
By changing the position of the triangles 1 and 2 as indicated, we
change the first diagram into the second. But the first is the square on
the hypotenuse, and the second is the sum of the squares on the other two
sides. Since they are equal, the truth of the proposition is evident.
The square on the hypotenuse of a right triangle equals the
sum oj thesquar* * described on the other two sides.
366 POWERS AND ROOTS
Written Work
1. Find the hypotenuse of this
right triangle.
Hypotenuse 2 = W 2 + 60 2 , or 4756
Hypotenuse = V4756 = 68.9 + ft.
60 ft. ~
Draw figures to a convenient scale and find the unknown
side :
Base Perpendicular Hypotkxuse
2. 18 in. 27 in. (x)
3. (a;) 4 ft. 5 ft.
4. 24 ft. (a;) 40 ft.
5. 15 yd. 20 yd. (x)
6. (x) 40 in. 60 in.
7. Find the length of the longest straight line that can
be drawn on a table 8 ft. by 4 ft.
8. A has a field 40 rd. long and 30 rd. wide. B has a
square field whose side equals the diagonal of A's field.
What is the difference in the area of the two fields ?
9. Find the longest straight line in a room 16 ft. in
length by 12 ft. in width by 10 ft. in height.
10. Two automobiles, A and B, start from the same point.
A goes east 10 mi., then north 10 mi. ; B goes west 20 mi.,
then south 20 mi. Draw figure and find distance apart.
11. Find the length of the diagonal of a square field con-
taining 225 sq. rd.
12. Find the side of the largest square that may be in-
scribed in a circle 2 ft. in diameter.
13. A fireman's ladder just reaches a window 36 ft. above
the ground. How long is the ladder if its foot is 27 ft. from
the building?
MENSURATION
Rectangular surfaces, rectangular solids, and the cylinder
have been treated under Practical Measurements.
REGULAR POLYGONS
A plane is a surface such that a straight line joining any
two of its points lies wholly in the surface.
A polygon is a plane figure bounded by straight lines.
A regular polygon is a plane figure having equal sides and
equal angles.
Triangle
Square
Pentagon
Hexagon
Polygons are named from their sides. A regular polygon of three
sides is called an equilateral triangle; one of four sides, a square; one
of Jive sides, a pentagon ; one of six sides, a hexagon, etc.
Finding the area of a regular polygon.
Draw a circle. Divide the circumference into 6
equal parts by points marked at distances apart
equal to the length of the radius. Join these points,
thus making a hexagon. Connect the opposite
points by dotted lines, thus dividing the hexagon
into six equilateral triangles.
Show by folding together the opposite sides of
any equilateral, and the equal sides of any isosceles triangle, that each
may be divided into two equal right triangles.
The area of any regular polygon equals the sum of the areas
of the triangles composing it.
3G8
MENSURATION
• 1. Find the surface of the bottom of a hexagonal silo that
is 12 feet on a side, the distance from the middle point of the
side to the center of the bottom being 10.3 ft.
2. How far from the corner is the center of a square field
that is 40 rods on a side? (Draw the figure.)
3. Find the area of an equilateral triangular design that
is 15 inches on a side. (Divide into two right triangles.)
SOLIDS
A solid is anything that has length, breadth, and thickness.
The faces of a solid are the surfaces that bound it.
The lateral or convex surface of a solid is the area of its
sides, or faces.
The volume of a solid is the number of cubic units it con-
tains.
A prism is a solid whose ends are equal and parallel poly-
gons, and whose sides are parallelograms. Prisms are named
from their bases, as triangular, square, rectangular, pentagonal,
hexagonal, etc.
Triangular
Prism
Square
Prism
Pentangular Prism Rectangular Prism
A cylinder is a solid with circular ends and uniform
diameter. The ends are the bases, and the curved surface
is the convex surface.
The altitude of a prism or of a cylinder is the perpendicu-
lar distance between the bases.
SI'IMWCKS OF SOLIDS
3fi0
Cylinder
A pyramid is a solid whose
base is a regular polygon, and
whose faces are triangles that
meet at a point called the vertex.
Pyramids are named from their
bases, as, triangular, square, pen-
tagonal, etc.
Pyramid
The slant height of a pyramid is the altitude
of the triangles that bound it.
A cone is a solid whose base is a circle, and
whose convex surface tapers uniformly to a
point called the vertex.
The altitude of a pyramid, or of a cone, is the
perpendicular distance from the vertex to the
base.
The slant height of a cone is the distance between the
vertex and any point in the circumference of
the base.
A globe or sphere is a solid bounded by
a curved surface, every point of which is
equally distant from a point within, called
the center. sphere
Cone
SURFACES OF SOLIDS
Surfaces of Prisms and Cylinders
Observe : 1 . That if a piece of paper is
fitted to cover the convex surface of a prism
or a cylinder, and then unrolled, its form will
be that of a rectangle, as ABCD.
2. That the perimeter of the solid forms
one side of the rectangle, and the altitude of the solid the other
side.
HAH. COMPL. AKITII. — 24
370
MENSURATION
q The convex surface of a prism
or of a cylinder is found by multi-
plying the unit of measure by the
product of the perimeter and the
S altitude.
Find the convex surface of a regular prism of :
1. 5 sides ; 1 side 10 ft. ; height 5 ft.
2. 3 sides ; 1 side 20 in. ; height 42 in.
3. A steam boiler, diameter 3 ft. ; length 10 ft. Entire
surface = ?
4. A water pail, diameter 11 in. ; height 15 in. Entire
surface = ?
Surfaces of Pyramids and Cones
Observe : 1. That the convex surface of a pyramid is composed of
triangles.
Pyramid
Cone
2. That the convex surface of a cone may also be considered as made
up of small triangles.
3. That the bases of the triangles in both pyramid and cone form the
perimeter of the base of the figure, and the altitude of the triangles the
slant height. Hence,
TJie convex surface of a pyramid or of a cone is found by
multiplying the unit of measure by one half the product of the
perimeter and the slant height.
CYLINDER AND SPHERE
371
Find the convex surface of a pyramid or a cone if :
1. Diameter of base of cone = 9 ft. ; slant height = 12 ft.
2. One side of a square pyramid = 16 ft. ; slant height =
24 ft.
3. One side of a square pyramid = 5 ft.; altitude =
16 ft.
4. Altitude of square pyramid = 24 ft. ; one side = 14 ft.
5. A church spire is in the form of a hexagonal pyramid,
each side being 10 feet, and the slant height 65 feet. Find
the cost of painting it at 25^ per square yard.
6. A spire on the corner of a church is in the form of a
cone. Its base is 12 feet in diameter and its slant height 24
feet. Find the cost of covering it with tin at $13 per
square (100 sq. ft. = 1 square).
Comparative Surfaces of Cylinder and Sphere
Examine the solids. What is the height of the cylinder? What is
the diameter of the cylinder? What is the diameter of the sphere?
How does the diameter of each compare with the height of the cylinder?
Observe that the dimensions are equal.
Geometry shows that the surface of a sphere is equal to the convex
surface of a cylinder w T hose height and diameter are each equal to the
diameter of the sphere.
372
MENSURATION
To show this, wind a hard wax cord around a cylinder 1 in. in height
and 1 in. in diameter until its convex surface is covered. Unwind the
cord from the cylinder on to a sphere 1 in. in diameter as shown in the
illustration. When one half the surface of the sphere is covered with
the cord, one half of the convex surface of the cylinder is uncovered.
Hence,
The surface of any sphere equals the convex surface of a
cylinder of equal dimensions.
It may also be shown by geometry that
The surface of a sphere equals the square of the diameter
multiplied by 3.1416, or ttcP (representing the diameter by d
and 3.1416 by 7r).
Find the surface of :
1. A globe, D. 12 in.
2. A ball, R. l-i- in.
3. A sphere, Z>. 13 in.
4. A ball, I). 4 in.
5. How much will it cost to paint a dome in the form of a
hemisphere, 20 ft. in diameter, at 25 cents per square yard ?
VOLUME OF SOLIDS
Prisms and Cylinders
- N iif
Scale \ in. = 1 in.
Observe : 1. That the solids are all 4 in. high.
2. That the first row in the rectangular prism contains 4 cu. in.
3. That if the first row in each solid contains 4 cu. in." the volume of
each solid is 4 times 4 cu. in., or 16 cu. in.
PYRAMIDS AND CONES
373
The volume of a prism or of a cylinder is found by multiplying
the unit of measure by the product of the numbers corresponding
to the area of the base and the altitude.
Find the volume of :
1. A prism 4 inches square ; altitude 8 inches.
2. A square prism, side 12 in.; altitude 24 in.
3. A hexagonal silo is 25 ft. high, 12 ft. on a side, and
10.3 ft. from the middle point of a side (measuring at the
base) to the center of the base. Estimating 50 cu. ft. to a
ton of ensilage, how many tons will the silo contain ?
4. In the rotunda of a building there are 6 cylindrical mar-
ble columns, 18 in. in diameter and 18 ft. in height. Esti-
mate the number of cubic feet in all.
PYRAMIDS AND CONES
1. Fill a hollow pyramid with sand. Empty it into a
prism having the same base and altitude. How often must
the pyramid be filled and emptied to fill the prism ? The vol-
ume of a pyramid, then, is what part of the volume of the
prism ?
2. Measure in like manner with a cone the volume of a
cylinder having the same dimensions. The volume of the
cone is what part of the volume of the cylinder ?
374 MENSURATION
Observe : 1. That the dimensions of the pyramid and of the prism are
the same, and that those of the cone and of the cylinder are the same.
2. That the volume of the pyramid is \ that of the prism, and the
volume of the cone is | that of the cylinder.
By geometry, it is shown that the volume of a pyramid is
^ of that of a prism having an equal base and an equal alti-
tude. Hence,
The volume of a pyramid equals one third of the volume of a
prism of like dimensions.
The volume of a cone equals one third of the volume of a
cylinder of like dimensions.
But we have already learned that the volume of a prism
or of a cylinder is found by multiplying its unit of measure
by the product of the area of its base by its altitude. Hence,
The volume of a pyramid or of a cone is found by multiply-
ing its unit of measure by one third the product of the altitude
and the area of the base.
1. Find the volume of a cone whose altitude is 12 in. and
the diameter of the base 8 in.
2. How often can a conical cup 8 in. high and 6 in. in di-
ameter be filled from a cylindrical vessel 2 ft. high and 6 in.
in diameter?
3. Find the volume of a pyramid whose base is 12 in.
square and whose altitude is 30 in.
4. A square pyramid whose side is 18 in. is 32 in. high.
Find its volume.
5. Find the volume of a pyramid whose altitude is 12 ft.
and whose base is a square 8 ft. on a side.
6. Find the contents of a rectangular pyramid 15 ft.
high, the sides of whose base are 10 ft. and 12 ft. respectively.
7. A pile of grain in the form of a cone is 15 ft. in diameter
and G ft. high. How many bushels of grain does it contain ?
SPHERES 375
8. A concrete mixer, 6 ft. from base to apex, being coni-
cal in form, and measuring 3 feet across the base, is filled
six times an hour. How many cubic feet of concrete mate-
rial may be manufactured with it in a week of six working
days of 8 hours each ?
9. A wooden hopper supplying coal to a furnace is in the
form of an inverted pyramid. If it is 8 ft. deep and 6 ft.
square at the top, how many tons of hard coal will it contain ?
10. A square pyramid, the perimeter of whose base meas-
ures 64 inches, contains 2048 cubic inches. Find its altitude.
11. The contents of a cone are 471.24 cu. ft. ; the altitude
is 18 ft. Find the diameter.
SPHERES
Examine the figure:
Observe : 1. That the solids formed
by the dissected part of the sphere are
pyramids.
2. That the radius of the sphere is
the altitude of the pyramids.
3. That the combined bases of the pyramids form the convex surface
of the sphere.
The volume of a sphere is found by multiplying its unit of
measure by one third the product of the radius and its convex
surface.
It is also shown by geometry that
The volume of a sphere equals four thirds of the cube of the
radius multiplied by 3.1416, or {representing the radius by r
and 3.1416 by tt)^^.
Find the volume of :
1. A globe 12 inches in diameter.
2. A bowling ball with a radius of 4 inches.
3. A cannon ball with a diameter of 8.2 inches.
376
MENSURATION
Comparative Volumes of Cone, Sphere, and Cylinder
-/-"---
w. /'i — _J I /':
Compare the diameters of the bases and the altitudes of the cone and
the cylinder with each other, and with the diameter of the sphere. Ob-
serve that the dimensions are all equal.
By geometry it is shown that the volumes of these three
solids are in the ratio of 1, 2, and 3. The volume of the cone
is ^, and of the sphere § of that of the cylinder.
SIMILAR SURFACES
Similar figures are plane surfaces that have exactly the
same shape, but differ in size. Point out similar figures:
V
C\J
<M
2"
SIMILAR SURFACES
377
Kl
<M
St
Find the areas of the similar surfaces on p. 376. Square
the corresponding lines of the similar figures and express their
ratio. Compare the ratio of the areas of the similar figures
with the ratio of the squares of their corresponding lines.
In the similar figures observe :
1. That the cor-
responding sides
are proportional ;
that is, 2:4:: 3:6.
2. That the ra-
tio of their areas
equals the ratio of
the squares of their corresponding lines ; that is, 6 sq. in. : 24 sq. in. as
2 2 :4 2 , oras3 2 :6 2 .
Corresponding lines of similar plane surfaces are proportional.
The areas of similar plane figures are proportional to the
squares of their corresponding lines.
Written Work
1. If a rectangle is 20 ft. by 50 ft., what will be the
length of a similar rectangle 30 ft. in width ?
2. The side of a square field is 40 rods. Find the side of
a similar field that contains four times as many acres.
3. A lady buys two rugs, one 6 ft. by 9 ft. and a similar
rug 18 ft. in length. Find its width.
4. In East Park a circular
fountain is 40 ft. across and in
West Park a circular fountain
is 26 ft. across. The area of
the first fountain is how many
times the area of the second
fountain ?
5. Find the length of the
side marked x in the larger of these similar triangles.
4ft
378
MENSURATION
<■
6. It costs 119.50 to gild a sphere 18 inches in diame-
-^fe. ter. At the same rate estimate
the cost of gilding a sphere 12
inches in diameter. Why are the
surfaces of all spheres similar?
7. A telephone pole and a lamp
post cast shadows as shown in the
S8ft. fl [j figure. Find the height of the
24 ft telephone pole.
X
SIMILAR SOLIDS
Similar solids are solids that have the same shape but
differ in contents or volume ; thus,
Observe : 1 . That the length of the first solid is to the length of the
second as 1 : 2, and that the heights and widths of the solids are in the
same ratio.
2. That the ratio of their contents or volume equals the ratio of the
cubes of their corresponding lines ; that is, 2 cu. in. : 16 cu. in. as l 3 : 2 3 ;
or as 2 3 : 4 3 .
The corresponding lines of similar solids are proportional.
The contents or volumes of similar solids are proportional to
the cubes of their corresponding lines.
Written Work
1. Compare in volume a 5-in. cube with a 10-in. cube.
SPECIFIC GRAVITY 370
Compare in volume:
2. A (ij-imh globe with a 25-inch globe.
3. A 3^-inch cube with a 25-inch cube.
4. A l(>f-inch sphere with a 50-inch sphere.
5. What are corresponding lines in spheres? in triangles ?
6. The dimensions of a rectangular solid are 30 ft. by
20 ft. by 12 ft. What are the dimensions of a similar solid
20 ft. in length ?
7. A village has two similar cylindrical water tanks, one
15 ft. in diameter and 24 ft. high; the other 10 ft. in di-
ameter. Find the height of the second tank.
8. Make three problems with similar rectangular solids.
9. The volume of a sphere is 1600 cu. in. What is the
volume of a sphere with one half the diameter ?
10. A rectangular bin 8 ft. long contains 92 bu. of wheat.
How many bushels will a similar bin 10 ft. long contain ?
SPECIFIC GRAVITY
1. A piece of timber 12 inches square and 6 feet long
floating in water shows 2 inches of the log above water.
How many cubic feet of the log are under water ? How
many cubic feet of water are displaced by the log ?
2. A cubic foot of water weighs 62| lb. How many
cubic feet of water will a log displace that weighs 250 lb. ?
If the log in problem 1 is § as heavy as the same volume of
water, how much of the log is under water ?
Every ohject floating in water displaces its own weight of
water.
Every object that sinks in water displaces its own volume of
water.
380 MENSURATION
3. A piece of wood when floating is | under water. The
ratio of the weight of the wood to the weight of an equal
volume of water is therefore 1 ^ 2 or ^. What is the ratio
of the weight of objects to the weight of the same volume
of water if the displacement is \ ? § ? | ? f ? .5 ?
Copper is 8.9 times as heavy as an equal volume of
water. The ratio of the weight of copper to the weight of
an equal volume of water is, therefore, 8.9.
Specific gravity is the ratio of the weight of any substance
to the weight of an equal volume of water.
A cubic foot of water weighs 62^ pounds or 1000 ounces.
4. The specific gravity of ice is .92. Find the weight of
a block of ice 2' by 18" by 12".
5. The specific gravity of pure cows' milk is 1.03. Find
the weight of 50 gallons of milk.
6. The specific gravity of cork is .24. Find the weight
of 10 cu. ft. of cork.
7. The specific gravity of lead is 11.3. Find the weight
of 20 bars 16" long, 4" wide, and 2" thick.
Find the weight of 1 cu. ft. of each object whose specific
gravity is given :
8.
Silver
10.50
16.
Copper 8.90
9.
Milk
1.03
17.
Lead 11.30
10.
Ice
.92
18.
Sandstone 2.90
11.
Nickel
8.90
19,
Iron 7.80
12.
Granite
2.70
20.
Tin 7.29
13.
Mercury
13.59
21.
Steel 7.83
14.
Gold
19.30
22.
Marble 2.70
15.
Cork
.24
23.
Ivory 1.83
REVIEW OF MENSURATION :M
REVIEW OF MENSURATION
1. From an artesian well 2 inches in diameter the water
flows out at the rate of 1| ft. per second. Find the number
of barrels that flow out per hour.
2. One cubic inch of gold could be pounded into how
many square inches of gold leaf ^^ of an inch in thickness ?
3. A spherical ball must be how large in order that a cube,
each surface of which contains 36 square inches, could be cut
from it ?
4. A copper ingot containing 1 cubic foot is to be drawn
into a copper wire ^ of an inch in diameter. Find the length
of the wire when drawn.
5. A plowman found by measurement that he had plowed
a strip 4 rd. in width around a rectangular field 40 rd. long
and 20 rd. wide. Find the number of acres he had plowed.
Draw figure to illustrate.
6. A western farmer has a pile of corn in the ear, 400 ft.
long. The pile at the end is in the form of an isosceles tri-
angle, 12 ft. wide at the bottom, and the altitude of the pile
is 6 ft. Find the number of bushels, allowing If cu. ft. to a
bushel.
7. A solid ball 6 inches in diameter is in a cylinder 10
inches in diameter and 10 inches high. How many cubic
inches of water will the cyclinder contain ?
8. What is the convex surface of a piece of stove pipe
2 feet long and 8 inches in diameter ?
9. The inside diameter of a hollow globe is If ft. How
many gallons of water will it contain ?
10. The flow of water from the same source through two
different pipes depends upon the area of the cross section of
the openings. Cm pare the flow through a i-inch pipe with
the flow through a |-inch pipe.
METRIC SYSTEM OF WEIGHTS AND
MEASURES
By the United States system of money we may write 5
dollars, 9 dimes, and 7 cents thus, $5.97, because there is
a uniform ratio between the dollar and the dime, the dollar
and the cent, the dollar and the mill ; the dollar being 10
times the dime, 100 times the cent, and 1000 times the mill.
As a mill is y^j of a dollar, a cent T Jo of a dollar, and a dime ^ of
a dollar, United States money is based on a decimal system.
By the English long measure, 12 in. = 1 ft., 3 ft. = 1 yd.,
and 5^ yd. = 1 rd. ; thus, we see there is no uniform ratio
between the rod and the yard, the rod and the foot, and the
rod and the inch, the rod being 5| times the yard, 16| times
the foot, and 198 times the inch.
By the English measure of weights, 16 oz. = 1 lb., 100 lb.
= 1 cwt., 20 cwt. = 1 ton. The ton equals 20 times the
hundredweight, 2000 times the pound, and 32000 times the
ounce.
The metric system was devised by the French govern-
ment in an effort to establish a system of weights and meas-
ures that would be on a uniform decimal scale, so that a
unit of one denomination might be changed to a unit of
another denomination by simply moving the decimal point.
The meter is the fundamental unit of the metric system. Its length
(about 39.37 in.) was meant to be .0000001 of the distance from the
equator to the pole. Though an error has since been discovered in the
measurement of the distance from the equator to the pole, the standard
unit has not been changed. The original standard is a bar of platinum
39.37 inches in length, deposited in the archives in Paris.
382
METRIC SYSTEM 383
From the meter every other unit of measure or weight is
derived. Thus, the unit of weight is the gram, which equals
the weight of 1 cubic centimeter of pure water.
Draw a cube .01 of a meter on an edge and state the
length of the edge in inches.
The unit of capacity is the liter (leter), which contains 1
cubic decimeter.
Draw a cube .1 of a meter on an edge and state the length
of the edge in inches.
The metric system is now in use in most of the civilized countries
except Great Britain and the United States, and in the latter it is in
use in some of the departments of the government. It is the official
system adopted by Congress for our island possessions. It is univer-
sally used by scientists. The United States by a vote of Congress per-
mitted its use in I860.
Observe :
The meter measures length.
The square meter measures surface.
The cubic meter measures solids or volume.
The gram measures weight.
The liter measures capacity.
Latin prefixes.
To express .1 of a meter, .1 of a gram, and .1 of a liter, we
prefix deci to each of the words, meter, gram, and liter.
Thus, decimeter means -^ of a meter ; decigram, -^ of a gram ;
and deciliter, -^ of a liter.
To express .01 of a meter, gram, and liter, we prefix centi
to each of the words, meter, gram, and liter.
To express .001 of a meter, gram, and liter, we prefix milli
to each of the words, meter, gram, and liter.
Note. — From these Latin prefixes we get our words dime, cent, and
mill.
384
METRIC SYSTEM
Greek prefixes.
To express 10 times a meter, 10 times a gram, and 10 times
a liter, we prefix deca to each of the words, meter, gram,
and liter. Thus, decameter means 10 times a meter ; deca-
gram, 10 times a gram ; and decaliter, 10 times a liter.
To express 100 times a meter, gram, and liter, we prefix
hecto to each of the words, meter, gram, and liter.
To express 1000 times a meter, gram, and liter, we prefix
kilo to each of the words, meter, gram, and liter.
METRIC MEASURES OF LENGTH
Comparison of Fundamental Units of Measures of Length
i i I i i i I i i i \ 2
24
36
1 Yard.
hum
/
2
3
4
5
6
7
8
9
10
I Meter
Table of Long Measures
1 millimeter (mm.)
= .001 of a meter
1 centimeter (cm.)
= .01 of a meter
1 decimeter (dm.)
= .1 of a meter (nearly 4
in
)
1 meter
= 39.37 in.
1 decameter (Dm.)
= 10 meters
1 hectometer (Hm.)
= 100 meters
1 kilometer (Km.)
= 1000 meters (nearly .6
mi
.)
In metric long measure 10 times one unit of any denomi-
nation equals one unit of the next higher denomination.
The denominations most frequently used are given in
black-faced type.
MEASURES OF LENGTH 385
Approximately :
1 yard = \\ meter 1 mile = 1.6 kilometer
The kilometer is used for measuring long distances; the
meter, for short distances and for measuring cloth, etc. ; and
the millimeter is used in the sciences and to show veiy small
measurements, as the thickness of wire, etc.
Written Work
1. What decimal parts of a meter are expressed by the
Latin prefixes ? What multiples are expressed by the Greek
prefixes ?
2. Draw a line one meter long. Show the number of
decimeters in a meter ; the number of centimeters ; the
number of millimeters.
3. Explain why 5m. = 50 dm. = 500 cm. = 5000 mm.
4. In the metric S3'stem the fundamental operations are
decimal or multiple operations. . Thus, 8 m. 5 dm. G cm.
25 mm. are added in this manner.
Added in meters: Added in millimeters:
8 m. =8 m. 8 m. = 8000 mm.
5 dm. = .5 m. 5 dm. = 500 mm.
6 cm. = .06 m. 6 cm. = 60 mm.
25 mm. = .025 m. 25 mm. = 25 mm.
8.585 m. 8585 mm.
5. Add 1 m., 3 dm., 6 cm., 3mm. Add 6.5 m., .25 mm.,.
65 dm.
6. The distance between two towns is 5 Km. and 45.V m.
After a bicyclist has traveled 3 Km. 57.5 m., how much of
the distance remains to be traveled ?
7. The distance from Paris to Calais is 295.32 Km.
Express this distance approximately in miles.
HAM. COMPL. ARITH. — 25
386
METRIC SYSTEM
8. How many meters of ribbon are necessary to make
150 badges, each 25 cm. in length?
9. The distance from Erie, Pa. to Buffalo, N.Y. is
112.651 Km. Express the distance approximately in miles.
10. Reduce the decimal in the last problem to meters and
lower denominations.
11. The distance from New York to San Francisco is 4000
miles. Approximate this distance in kilometers and meters.
METRIC MEASURES OF SURFACE
Comparison of Fundamental Units of Square Measure
•
IS a. Yd
ISq.M
Table of Square Measures
1 sq. millimeter (sq. mm.)
1 sq. centimeter (sq. cm.)
1 sq. decimeter (sq. dm.)
1 sq. meter (sq. m.)
1 sq. decameter (sq. Dm.)
1 sq. hectometer (sq. Hm.)
1 sq. kilometer (sq. Km.)
.000001 sq. meter
.0001 sq. meter
.01 sq. meter
1.196 sq. yd.
100 sq. meters
10000 sq meters
1000000 sq. meters
(nearly .4 sq. mi.)
In metric measure of surface 100 times one unit of any
denomination equals one unit of the next higher denomination.
MEASURES OF VOLUME
387
Land Measure.
The standard unit used for measuring is the are (ar) con-
taining 100 square meters = 119.(3 square yards.
Table :
1 centare = 1 sq. meter.
1 are = 100 sq. meters (nearly 120 square yards)
1 hectare = 10000 sq. meters (nearly 2* acres).
The square meter is used in measuring ordinary surfaces,
such as are found in houses, lots, farms, etc. ; the square
kilometer for measuring areas of countries and their divi-
sions into states, counties, etc.
METRIC MEASURES OF VOLUME
Comparison of Fundamental Units of Cubic Measure
1 Ctbic Yard
1 Cubic Meter
Table of Solid or Cubic Measures
1 cu.
millimeter (cu. mm.) = .000000001 cu. meter
1 cu.
centimeter (cu. cm.) = .000001 cu. meter
1 cu.
decimeter (cu. dm). = .001 cu. meter
1 cu.
meter =1 308 cu. yd.
In metric measure of volume 1000 times <<>"' unit of any de-
, equals "/<>■ unit <>f th<< next higher denomination.
388 METRIC SYSTEM
In measuring wood 1 cu. meter is called a stere.
The cubic meter is the practical unit of measure of vol-
ume for all purposes.
Written Work in Square and Cubic Measures
1. Find the number of square meters in the floor of your
schoolroom.
2. At 27 cents per cubic meter, find the cost of excavating
a cellar 10 m. by 18 m. by 1| m.
3. How much will it cost to paint one side of a tight board
fence 25 meters long and 3 meters wide at $10 per square
decameter ?
4. How many square meters of linoleum will be required
to cover the floor of a hall 8 meters long and 3 meters
wide?
5. How many steres are there in a pile of wood 2 meters
high, 2 meters wide, and 6 meters long?
METRIC MEASURES OF CAPACITY
Comparison of Fundamental Units of Capacity
Quart
MEASURES (>F WEIGHT
:;*!>
Table of Measures of Capacity
1 millimeter (ml.) = .001 of a liter
1 centiliter (cl.) = .01 of a liter
1 deciliter (dl.) = .1 of a liter
1 liter = 1.0567 liquid quarts = .908 dry quart.
1 decaliter (Dl.) = 10 liters
1 hectoliter (HI.) = 100 liters (nearly 2.84 bu.)
In metric measure of capacity 10 times one unit of any de-
nomination equals one unit of the next higher denomination.
The liter is used for liquid and dry measures.
METRIC MEASURES OF WEIGHT
Comparison of Units of Weight
I Ounce I Gram
lib.
I Kilo
Table of Measures of Weight
1 milligram (mg.)
= .001 of a gram
1 centigram (eg.)
= .01 of a gram
1 decigram (dg.)
= .1 of a gram
1 gram
= .03527 of an oz Avoir.
1 decagram (Dg.)
= 10 grams
1 hectogram (Hg.)
= 100 grams
1 kilogram (Kg.)
= 1000 grams (nearly 2.2 lb.)
1 myriagram (Mg.)
= 1O000 grams
1 quintal (Q.)
= 100000 grams
1 tonneau (T.)
= 1000000 grams ( nearly 2205 lb.)
390 METRIC SYSTEM
In the metric measure of iv eight 10 times one unit of any de-
nomination equals one unit of the next higher denomination.
The gram is the weight of 1 cubic centimeter of water, the
kilogram, of 1 cubic decimeter, and the metric ton, of 1 cubic
meter; the gram is used by druggists and chemists ; the kilo-
gram (usually called the kilo) for weighing small articles;
and the metric ton for large, heavy articles.
Table of Metrical Equivalents
1 cu.
mm. of water weighs 1 mg. and measures .001 ml.
1 cu.
cm. of water weighs 1 g. and measures 1 ml.
1 cu.
dm. of water weighs 1 Kg. and measures 1 1.
1 cu.
m. of water weighs 1 T. and measures 1 Kl.
Written Work
Things sold in the United States and England by the quart
are sold in countries using the metric system by the liter.
1. Estimate the number of liters in a tank 2£ m. long,
| m. in width, and ^ m. in depth.
2. A cylindrical tank 6 m. in diameter and 10 m. in
height is § full. Estimate the number of liters it contains.
Estimate the weight in metric tons.
3. A Paris milkman retailed on an average 110 liters of
milk daily at 25 centimes a liter. Find the amount of his
sales in our own money for 30 days.
4. The rainfall in a certain place in one week was 1 dm.
Find the number of liters that fell on 3 hectares of land.
5. A horse eats 4 liters of oats 3 times a day. How
many hectoliters does it eat in 60 days ?
6. From an olive orchard, 4.5 Kl. of olive oil was put up
in bottles holding .5 1. How many bottles were used?
METRICAL EQUIVALENTS 391
7. Find the amount in United States money from the sale
of 2000 HI. of wheat at 10 francs per hectoliter.
8. A German ice dealer retails blocks of ice .8 m. in
length, .3 m. in width, and .2 m. in thickness. The weight
of ice is .92 that of the same volume of water. Find its cost
at 3.5 pfennigs per kilo. Find the amount in our money.
9. Change a cubic meter of water to liters.
10. Find the weight of a barrel of flour in kilos.
11. A stone 8 ft. by 3 ft. by 2 ft. contains how many
cubic meters ?
12. If stone is 2.9 times as heavy as the same volume of
water, find the weight of the stone in kilo*.
13. The Washington Monument is 555 feet high. Express
its height in meters.
14. Find the cost of laying a cement walk .025 Km. in
length and 1.5. m. in width, at $1.70 per sq. m.
15. Mr. James bought a tract of land in the Philippine
Islands, 3 Km. in length and 2.5 Km. in width, at $15.75
per hectare. Find the cost of inclosing this land with wire
fence at 10^ per meter.
16. A hallway is 12 m. in length and 5 m. in width. Esti-
mate the number of tiles 1 cm. square necessary to cover it.
17. A railroad in building a retaining wall used 52000 cu.
m. of stone. Find its weight in metric tons if stone is 2.7
times as heavy as the same volume of water.
18. A city sewer is 1.3 Km. in length, 1.2 m. in width,
and averages 3 m. in depth. Estimate the number of cubic
meters of earth removed.
19. A certain kind of cloth costs 90^ per meter + 25 %
ad valorem duty. For how much must it be marked in
United States mi y to gain 25% on tlie yard?
AGRICULTURAL PROBLEMS
FEEDING STOCK
Table of Digestible Nutrients This table shows
the number of pounds
of digestible nutrients
found in 100 pounds of
each kind of feed tab-
ulated.
A fair price for these
nutrients is as follows:
Protein, $4 per cwt.
Carbohydrates, 40 ^ per
cwt. Fat, $1 per cwt.
Find the total value
of the digestible nutri-
ents in 1 ton of the
following:
1. Timothy hay. 2.
Red clover hay. 3. Dry
corn fodder. 4. Corn
silage. 5. Wheat
straw. 6. Alfalfa hay.
7. Corn, grain. 8.
Oats, grain. 9. Wheat
bran. 10. Oil meal.
11. Oat straw. 12.
Cowpeahay. 13. Corn
fodder, green. 14. Red clover, green.
Protein is a muscle former, while carbohydrates and fats are fat formers.
The ratio of the muscle formers to the fat formers is called the nutri-
tive ratio. A nutritive ratio of 1 :3 is a narrow ratio, while one of 1 : 15
is a wide ratio.
A desirable nutritive ratio for a dairy cow is about 1 :5.7; for a work
horse, about 1:7; for swine, about 1 : 5.9.
392
Feeds
PRO-
TEIN
Carbo-
hydrates
Fat
Corn fodder, green .
1.1
12.1
.4
Red clover, green .
3.1
14.8
.7
Alfalfa, green . .
3.9
11.2
.4
Corn silage . . .
1.2
14.6
.9
Corn fodder, dry
2.3
32.3
1.1
Timothy hay . . .
2.9
43.7
1.4
Red clover hay . .
7.4
38.1
1.8
Alsike clover hay .
8.1
11.7
1.4
Alfalfa hay . . .
10.6
37.3
1.4
Cowpea hay . . .
10.8
38.4
1.5
Wheat straw . . .
.4
36.3
.4
Oat straw ....
1.2
38.6
.8
Corn, grain . . .
7.1
66.1
4.8
Oats, grain . . .
9.2
48.3
4.2
Wheat bran . . .
12.0
41.2
2.9
Oil meal ....
30.6
38.7
2.9
Cotton-seed meal .
37.0
16.5
12.6
Whole milk . . .
3.4
4.8
3.7
Skim milk . . .
3.0
5.0
.3
FEEDING STOCK 393
Fat, as a food constituent, is regarded as 2} times more valuable than
carbohydrates. Hence, the customary ride for rinding the nutritive ratio
is to add to the carbohydrates 2\ times the fat, and divide by the protein.
15. What is the nutritive ratio of green corn fodder?
Protein, 1.1; carbohydrates, 12.1; fat, .4.
»
Tims, .4 lb. fat = 2\ x .4 lb. carbohydrates = .9 lb. carbohydrates;
12.1 lb. carbohydrates + .9 lb. carbohydrates = 13 Lb. carbohydrates;
13 lb. -^ 1.1 lb. = 11.8 +. Hence, the nutritive ratio is 1 to 11.8 +.
16. Find the nutritive ratio of a feed composed of 100 lb.
timothy hay and 50 lb. wheat bran.
17. A farmer feeds his sheep 100 lb. of alfalfa hay to
10 lb. of bran. Find the nutritive ratio.
18. An Ohio farmer feeds his dairy cows 100 lb. red clover
hay to 20 lb. of wheat bran. Find the nutritive ratio.
19. The best authorities suggest that a dairy cow weighing
1000 lb. and giving daily 22 lb. of milk should be fed 2.5 lb.
protein, 13 lb. carbohydrates, and 0.5 lb. fat. Find the
nutritive ratio of this feed.
A balanced ration is a mixture of different kinds of feed according to
a nutritive ratio that will produce the desired end of the feeding, whether
that end be growth, fat, milk, or butter.
20. A farmer mixed a balanced ration of 3 parts by
weight red clover hay and 1 part oats grain. Find the
nutritive ratio.
21. A feeder prepared a balanced ration for his hogs witli
the following parts by weight: 1| parts corn grain, 1 part
oats grain. Find the nutritive ratio.
22. A balanced ration is as follows : 3 parts by weight
timothy hay and 1 part corn grain. Find the nutritive
ratio.
394 AGRICULTURAL PROBLEMS
23. A farmer prepared a balanced ration consisting of
150 lb. of alfalfa hay and 100 lb. corn grain. Find the
nutritive ratio.
24. Find the nutritive ratio of a balanced ration consist-
ing of 200 lb. corn silage and 100 lb. red clover hay.
25. What is the nutritive ratio of a balanced ration
that uses 200 lb. of dry corn fodder with 50 lb. of wheat
bran ?
26. A balanced ration for a dairy cow should be about 1
to 5.7. Is the nutritive ratio of the following balanced
ration too wide or too narrow for dairy feeding? 200 lb.
alsike clover hay ; 100 lb. corn grain.
27. A balanced ration for work horses should have a nutri-
tive ratio of about 1 to 7. Is the nutritive ratio of the fol-
lowing ration too wide or too narrow for work horses ?
200 lb. timothy hay ; 100 lb. oats grain.
28. A dairy man prepared a ration for his cows consisting
of 100 lb. corn silage, 50 lb. timothy hay, and 100 lb.
oats grain. Find the nutritiv-e ratio. Is this ration too
wide or too narrow?
29. A livery man feeds his horses on a ration consisting
of 50 lb. timothy hay, 25 lb. corn grain, and 25 lb. oats
grain. Find the nutritive ratio. Is this ratio too wide or
too narrow?
30. A stockman preparing cattle for market feeds 100 lb.
red clover hay to 30 lb. corn grain, 10 lb. oats grain, and 1
lb. oil meal. Find the nutritive ratio.
31. A Kansas farmer feeds his young cattle 100 lb. alfalfa
hay to 10 lb. corn grain and 2 lb. cotton seed meal. Find the
nutritive ratio.
FERTILIZERS
39:.
FERTILIZERS
Poinds of Fertilizing Constit-
uents in One Ton
This table shows the num-
ber of pounds of plant tood,
such as nitrogen, phosphoric
acid, and potash in one ton of
the article named. All other
constituents necessary to plant
food can be obtained from the
soil, if the water supply is suffi-
cient and the heat conditions
proper.
The leading artificial ferti-
lizers are rock phosphate,
ground bones, ammonia sul-
phate, nitrate of soda, animal
refuse called "tankage," fish
scrap, and potash salts, espe-
cially the sulphate and the
muriate of potash.
Lime is used on land to neu-
tralize the acidity of the soil.
Nitrogen in fertilizing material is sometimes, though not often, given
as ammonia. 14 lb. of nitrogen correspond to 17 lb. of ammonia.
Phosphoric acid contains 43.6% phosphorus! this constituent of ferti-
lizer is sometimes, though rarely, given as phosphorus. A ton of timo-
thy hay contains 16 lb. of phosphoric acid, which is equivalent to 7 lb.
of phosphorus.
Potash contains 83% potassium. This constituent is sometimes,
though rarely, given as potassium. A ton of timothy hay contains 17
lb. of potash, which is equivalent to 39 lb. of potassium.
The value of fertilizing constituents depends largely on the forms in
which they occur. An average value per pound is 15 cents for nitrogen.
3 cents for phosphoric acid, and 5 cents for potash.
The composition of farm produce varies greatly, so that the same kind
of crop may be almosl twice as rich in certain constituents in some cases
as it is in other ca-es. The table given here is intended to show the
am rage composition of each.
Materials
Nitro-
gen
I'llos-
PHORIO
A. 11.
P.)T-
A 11
Timothy hay
Clover hay .
Alfalfa hay
Com, grain
Corn, stover
Bran . . .
Oat straw .
Milk . . .
Butter . .
Farm animals
Farm manure
19
39
46
34
12
51
13
12
1.6
53
10
16
13
13
13
9
62
7
4
1
37
5
47
44
35
9
39
35
38
3
1
3
10
390 AGRICULTURAL PROBLEMS
Through the growth of the leguminous plants, such as clover, alfalfa,
beans, peas, etc., it is estimated that the farmer may obtain nitrogen
from the air at a cost of about 2| f per pound.
1. How much nitrogen is required to produce each of the
following : 6 tons clover hay ? 3000 lb. corn grain ? 5500
lb. timothy hay ? 4 tons alfalfa ?
2. How much potash did a farmer sell in 1500 lb. of but-
ter ? in 760 lb. milk ? in 5 steers averaging 1450 lb. each ?
3. A farmer cut on an average 3 tons per acre of clover
hay from a 5 acre field. How many tons of manure are
required for the field to supply the phosphoric acid in the
hay?
4. A farmer sold 50 tons of timothy hay in one season.
How many tons of farm manure will balance the loss in
potash ?
5. When nitrogen is worth 16^ per pound, phosphoric
acid 5^, and potash 0^, find the fertilizing value of 5 tons
of farm manure.
6. A commercial fertilizer when analyzed was found to
contain by weight 3.4% nitrogen, 6.6% phosphoric acid,
and 9% potash. Find its value per ton, when nitrogen
is worth 15^ per pound, phosphoric acid 3^ per pound and
potash 5^ per pound.
7. The analysis of a certain brand of bone meal showed
the following: 1.6% nitrogen, and 27.9% phosphoric acid.
What is its value per ton at 15 ^ per pound for nitrogen and
Z\$ per pound for phosphoric acid.
8. A farmer sold 500 bu. of corn grain (shelled corn 56
lb. to the bushel) from a certain field. How many tons of
farm manure will supply the field with the phosphoric acid
removed by the corn grain ?
sn;.\Y!\<; plants 39'
SPRAYING PLANTS
The Bordeaux mixture used for killing fun- us growths — such as Mack
rot and scab of apples, black rot and mildew of grapes, brown rot of
plums and cherries, etc. — was first discovered in Bordeaux. France. To
make a solution for spraying: first, dissolve 4 pounds of copper sulphate,
(blue vitrol) in 25 gallons of water; second, dissolve 4 pounds of freshly
slaked stone lime in 25 gallons of water until it forms a milk of lime;
third, pour the two solutions together and they are ready for use. The
solution should be applied to the fruit while small. In this way a coak
ing is formed over the fruit through which the fungus cannot grow.
Note. — In all directions for spraying, 50 gallons are considered a barrel.
1. For a vineyard of 25 acres of grapes suffering from
black rot of the fruit, how many pounds of sulphate of
copper and lime are necessary, if 100 gallons of Bordeaux
mixture will spray § of an acre ?
2. If sulphate of copper can be boughtat 6/ per pound,
and stone lime at \$ per pound, what is the cost of mate-
rials for 500 gallons of Bordeaux mixture ?
3. If a man with a team of horses at $3.50 per day, using
a geared sprayer, can spray 3 acres of vineyard in a day, and
Bordeaux mixture costs 27/ per barrel, using 10 barrels per
day, what is the cost of spraying 30 acres of grapes ?
4. 600 bushels of apples from 40 trees without spraying
for fungus growth were sold as follows : 100 bu. of perfect
fruit at 81 per bushel, and 500 bu. of scabby and wormy
fruit at 30/ per bushel. If spraying at 15/ per tree re-
versed the quantities of perfect and imperfect fruit, what
would be gained by spraying ?
5. The estimated value of a crop of grapes is $400 per
acre. Thorough spraying costs* $6 per acre. If without
spraying, fungus diseases destroy 30 % of the crop, what is
the net value of the spraying of 10 acres ?
HAM. SCH. ARITH. — 21
398 AGRICULTURAL PROBLEMS
6. The average yield from 4 acres of sprayed grapes was
4770 lb. per acre, and the yield from unsprayed grapes was
3108 lb. The grapes were sold for 2±^ per pound. The
cost of spraying was $8.60 per acre. What was the net
gain on the 4 acres as a result of spraying?
A Paris green solution is used for killing chewing insects that destroy
the plants by eating the leaves; and for destroying the codling moth to
prevent wormy fruit. A solution for this purpose is made by mixing i
ounces of Paris green with 50 gallons of water, a sufficient amount to
spray 20 apple trees. When fungus diseases are to be controlled also, the
Paris green is added to Bordeaux mixture.
7. What is the cost of materials to spray an apple orchard
of 250 trees, for the apple worm, with Paris green at 32^ per
pound, using a barrel of the solution to 20 trees ?
8. Mr. Wagner sprayed 8| acres of potatoes 4 times with
Bordeaux mixture containing Paris green to control blight,
rot, and insects. His expense account was as follows: 183
lb. copper sulphate at 8^; 204 lb. lime at #1.15 per hun-
dred; 10 lb. Paris green at 35 ^; 48 hr. labor for a man at
20^; 40 hr. labor for team at 15 ^; wear of sprayer, $1.50.
What was the cost of spraying per application per acre ?
9. The 8| acres mentioned above yielded 1567.5 bu. pota-
toes. A portion of the same field that was left unsprayed
yielded at the rate of 156 bu. per acre. Mr. Wagner sold
his crop for 55^ per bushel. What was his net profit per
acre resulting from spraying ?
10. Mr. Gould sprayed an orchard of 240 trees five times
during the season, using altogether 3013 gal. Bordeaux
mixture containing Paris green. The cost of materials was:
Copper sulphate, 256 lb. fit 8^; lime, 347 lb. at \t\ Paris
green, \b\ lb. at 35^; and the cost of applying was labor,
3 men, \\ days at $1.50; 1 team,4| days at $1.50; wear on
SPRAYING PLANTS 399
spray machinery, $5. Determine the cosl of materials and
the cost of applying per gallon and also per tree, including
wear on machinery.
11. Thirty-four apple trees sprayed to control scab and
codling moth yielded 90 bu. merchantable fruit and 32 bu.
culls and windfalls. In the same orchard 21 unsprayed trees
yielded 11 bu. merchantable fruit and 40 bu. culls and wind-
falls. The merchantable apples were sold for 67^ per bushel
and the culls and windfalls for 22^ per bushel. The cost of
spraying was 25^ per tree. What was the net gain from
spraying per tree?
Kerosene Emulsion is used to kill insects that suck the juices of plants.
A solution for this purpose is made as follows : first, dissolve 2 ounces of
soap in one quart of water by heating until the soap is dissolved ; second,
add two quarts of kerosene oil and stir for five minutes; third, add to
this mixture 17 quarts of water to make 5 gallons of a 10% mixture.
After thorough stirring, it is ready for use.
12. What would be the cost, at 12^ per gallon, for
enough kerosene to make 500 gallons of a 10 % kerosene
emulsion that might be used on plants in leaf ? To make
128 gallons of an 8 % emulsion ?
13. A farmer has b\ A. of cabbage. He estimates that
the yield will be 6500 heads per acre, and that the selling
price will be 1\t per head. Cabbage lice threaten a loss of
10%. If by applying kerosene emulsion at a cost for mate-
rials and labor of $7.25 per acre, the loss can be reduced to 2 %,
what will be the gain from applying the treatment to the b\ A.?
14. A nurseryman used 10% kerosene emulsion to kill
plant lice on his roses and young fruit stock. His expendi-
ture for kerosene was $4.50. If kerosene was 15^ per gal-
lon, how many gallons of the 10% mixture did he use?
How many pounds of suap were used in making it ?
400 AGRICULTURAL PROBLEMS
The San Jose" scale is a very small insect not larger than the head of
a pin. It injures the tree by sucking the juice of the bark. A lime-
sulphur solution for killing the insect is made as follows: hist, dissolve
20 pounds of freshly slaked lime and 15 pounds of sulphur in 12 gallons
of water, boiling the mixture 1 hour; second, add to this enough water
to make 50 gallons of spraying mixture. This forms a hard covering
over the tree, thereby preventing the scale insect from doing further
harm.
15. If sulphur is worth 3^ per pound and lime is worth
\t per pound, and the labor of making lime-sulphur solution
is worth 15^ per barrel, what will be the cost of making
enough solution to spray an orchard of 252 peach trees, 25
gallons being sufficient for 9 trees?
16. In the above mentioned orchard, the cost of applying
is 85 % of the cost of solution. How much does it cost per
tree to spray the orchard? If one application per year for
ten years prolongs the life of each tree four years and the
average annual yield of fruit is worth $1.25, what is the net
gain per tree due to prolonged life?
17. A lime-sulphur solution costs 75^ per barrel to make,
and a soluble oil preparation for killing scale insects costs
$1.25 per barrel. A barrel of the former will cover 18 trees
and a barrel of the latter 15 trees. What would be the sav-
ing from using a lime-sulphur solution on a ten acre orchard
of pear trees set 140 to the acre, the cost of applying being
equal and the solutions being equally effective insecticides?
18. The market value of a crop of peaches is $225 per
acre. The total cost of production, including interest on in-
vestment, cost of marketing, etc., is $175 per acre. What
would be the per cent of loss on the market value from fun-
gus diseases and insect injuries, if the market value is so
reduced as to equal the cost of production ?
TEST PROBLEMS
1. J. P. Black & Co.'s bank account is overdrawn $182.50.
They place to their credit a 90-day note for 11500 with in-
terest at 6%, discounted 30 days after date. They then
deposit $84.30 and check on their account to the extent of
$341.65. Find their balance in bank, if the custom of this
bank in discounting is to count both the day of maturity and
the day of discount.
2. A cow is tied in the corner of a square field by a rope
2 rods lonsr. Find the extent of the surface over which it
can graze.
3. A cold air register is 18 in. by 30 in. What are the
dimensions of a similar register to let in double the volume
of cold air?
4. A clothier sold a suit marked $24, at 10% off for cash,
making a profit of 20 %. How much did the suit cost?
5. There are 50 persons in a schoolroom 36 ft. long, 30 ft.
wide, and 15 ft. high. How many cubic feet of air space are
there for each person? The law fixes the minimum of fresh
air at 30 cubic feet per minute per person. How frequently
must the room be filled with air to meet this requirement ?
6. A municipality paved a street 48 ft. from curb to curb
at $1.60 per square yard, under an ordinance that assessed
property holders abutting on the street | of the cost-
Mr. James owns two lots each 25 ft. wide fronting on the
street. He puts in a curb at 75^ per linear foot and a side-
walk 12 ft. wide at $1.80 per square yard. How much is
his bill for the improvements?
lot
402 TEST PROBLEMS
7. A conical glass is 4 inches in diameter and 6 inches
high. How often can it be filled from a cylindrical vessel
4 inches in diameter and 12 inches high ?
8. An article was sold at 2") oj advance on the cost. The
proceeds were invested in a second purchase which was after-
wards sold for $240. The latter sale was at a loss of 20 c / .
Find the selling price of the first article.
9. It cost 1280 to fence a field 80 rods long and 60 rods
wide. How much less will it cost to fence a square field of
equal area with the same kind of fence?
10. How much will it cost to bronze a globe 12 inches in
diameter with gold leaf at 5^ per square inch ?
11. The diameter and altitude of a cone, the diameter of
the base and the altitude of a cylinder, and the diameter of
a globe are each 3 ft. Find the volume of each.
12. A town assessed at $2,450,000 must raise a tax of
$7766.25. The poll tax is $825. Find the tax rate, if the
collector is allowed 5 % for collecting.
13. When the Florida collided with the Republic off the
coast of Massachusetts, the wireless message summoned the
Baltic to the wreck from a point 62 miles north and 84 miles
east. What was the distance in a direct line ?
14. A merchant sold 20 lb. more than l of his butter.
He then reduced the price b$ per pound, thereby reducing
his profit $ 2. How many pounds had he at first ?
15. Mr. Adams sold 50 % of his stock at 20 % gain, and
80% of the remainder at 25% gain. What was his total
gain, if the stock unsold cost $1000?
TEST PROBLEMS 403
16. A cow and a horse cost 8190, and $ the cost of the
cow plus $12 equal -^ the cost of the horse. Find the cost
of each.
17. A piece of steel in the form of ;i cylinder is 4 ft. long
and 2 inches in diameter. How long is it when rolled into a
bar 1 inch square ?
18. A rectangular field that contains 40 acres is four times
as long as it is wide. Find its dimensions.
19. A man bought a tenement containing 8 apartments for
§35,000. His taxes, repairs, and insurance cost him 81157
annually. He rented 4 suites at 840 per month, 2 suites
at §35 per month, and 2 suites at §28 per month. What
per cent did he realize on his investment ?
20. If a piano is marked 80 % above cost, what per cent
discount can be allowed from the marked price to realize
20 % profit ?
21. The edge of a cube is 10 inches. Find its diagonal.
22. Find the shortest distance on the surface of the cube
mentioned in Ex. 21 between two diagonally opposite corners.
23. A rectangular box is 8 ft. long, 6 ft. wide, and 4 ft. high.
Find the dimensions of a similar box whose length is 12 ft.
24. What is the ratio of the volume of the two boxes
mentioned in Ex. 23 ?
25. Find the duty in U.S. money on an invoice of leather
from Paris, if the leather costs 12,350 francs and the duty is
35% ad valorem.
26. Rose Adhorn and Co. imported 10 cases of woolen
goods from England of 385 pounds each, invoiced at £ 408
per case. Find the total duty at 40^ per pound and 60 <f
ad valorem.
404 TEST PROBLEMS
27. A merchant sold goods for $1242; one half he sold
at an advance of 20 % on the cost, three tenths at an advance
of 16| %, and the remainder at cost. How much did he pay
for the goods ?
28. Two successive trade discounts of 20 % and 10 % re-
duced a bill $560. What was the original bill ?
29. I sold my horse for $180 and took in payment a 90-day
note bearing interest at 5%, dated Feb. 1, 1909. On April 5,
I had the note discounted at the bank at 6 %. What were
the proceeds of the note ?
30. I asked for a grain binder 40 °]o more than it cost. I
accepted ^ of what I asked, and gained $ 15. How much did
I ask?
31. I have a rectangular field, the perimeter of which is
240 rods. It is twice as long as it is wide. How many acres
does it contain ?
32. A, B, & C have 1400 acres of land. \ of A's share
is equal to \ of B's, and | of B's share is equal to | of C's.
How many acres has each ?
33. Mr. Fair bought five $1000, 5% San Francisco water
bonds, due in 6 years, at 104, brokerage \$o- He sold the
bonds one } r ear before they were due at 102, brokerage \ %.
Find his average annual rate of income.
34. A 90-day note for $ 640, without interest, is discounted
at the bank at 6 % on the day of issue. Find the proceeds
of the note. What would be the proceeds, if the note read
" with interest " ?
35. I buy furniture listed at $4400, getting trade discounts
of 20%, 10 %, and 5 % ; I sell at 40 % above cost, taking a
120 -day note in payment without interest. I have the note
discounted at 6 <f on the day of sale. How much do I gain ?
TEST PROBLEMS 105
36. A solid bar of silver weighs 6| lb. avoirdupois. Esti-
mate its weight by Troy, and its value at 53^ per ounce.
37. A steel ingot is 16 in. square and 8 ft. long. What
Lenerth of steel bar will it make 4 in. thick and 6 in. wide?
38. Find the cost at 27 ^ per load, to excavate the ground
for a cellar 2-1 ft. wide, and 42 ft. long, the ground being
excavated to a depth of 7^ ft. at one end and 2 ft. 4 iu. at
the other end.
39. Mr. Samuels bought eight $1000, 4%, ten-year bonds
when issued, interest payable semiannually, at 97|-, broker-
age |%. After keeping the bonds 3| years, he sold them
at 110], brokerage \ %, and loaned the money at 6% interest
for 2^ years. Find his average annual income on the original
investment for the 6 years.
40. A boy climbs a flag pole to the height of 40 feet. An-
other boy is standing on the ground 120 feet from the foot of
the flag pole. If the second boy is 165 feet from a ball on
the top of the pole, how far is the first boy from the ball ?
41. The product of two equal factors multiplied by a third
factor, 7, and that product by a fourth factor, 8, is 35,000.
What are the equal factors?
42. Mr. Adams buys two * 1000, 5-year, 5 % bonds, inter-
est payable semiannually, that have 3^- years to run. tor
*1020 each, brokerage \°/o- Find his average annual in-
come, if he keeps the bonds until due ?
43. Mr. Brown sells five ^1000, 10-year, 4% bonds, inter-
est payable semiannually, that have 4| years to run, at 90^
on the dollar, brokerage \%. Find the buyer's average rate
of income, if the bonds are redeemed when due without
brokerage.
GENERAL REVIEW
Oral Problems
1. A boy buys oranges at 25 cents per dozen and retails
them at 3 for 10 cents. Find his gain per cent.
2. Mr. Henderson was earning $3.50 per day of 10 hours.
His wages were reduced 20 % • Find the rate of wages per
hour he then earned and his daily wages.
3. A man purchased 4 % bonds at par. How many dol-
lars' worth did he purchase if his income from the bonds
was $1200 per year ?
4. A merchant buys potatoes at 60 cents per bushel and
retails them at 30 cents per peck. Find his gain per cent.
5. A can do | of a piece of work in 10 hours and B can
do | of it in 9 hours. In how many hours working together
can they do the work ?
6. If I sell | of an article for | of the cost, what per
cent do I lose ?
7. A book dealer's selling price is 25 % above cost. A
special discount of 10% is allowed to teachers. A teacher
pays him $14.85 for books. Find the dealer's profit.
8. A merchant buys goods at a discount of 20 %, 5 % off,
and sells them at the list price. Find the per cent of profit.
9. Read as per cents : ^, £, |, 1£, |, -^g, -^, .16|, .05, .06|,
.37£, .40, .25.
406
GENERAL REVIEW 407
10. M and N have a profit of $1700 in business, § of
iM's capital equals | of N's capital. How should the profit
be divided ?
11. A man sold \ of his farm to B, | of the remainder to
C, and the remaining 60 acres to D. How many acres were
in the farm at first ?
12. A coal merchant pays $2 per ton for coal. The
freight and delivery cost him 50^ per ton. If he retails
the coal at $3 per ton, find his per cent of profit on the
entire cost of the coal.
13. Wishing to find whether a corner was square, a man
measured 8 feet along one side from the corner and 6 feet
along the other side from the corner. What should be the
distance between the ends of the lines if the corner is square ?
14. Explain why, when it is 12 o'clock M. at Washington,
it is approximately 9 o'clock a.m. at San Francisco and 5
o'clock p.m. at London (standard time).
15. A commission house received a consignment of peaches
at 10% and retailed them at $1.50 per bushel crate. If the
commission was $45, how many bushel crates were sold ?
16. After depositing f of his month's salary, a man pays
with the remainder bills of $8, $12, and $25, and has $30
left. Find his monthly salary.
17. 30%, 10%, 5% off is equivalent to what single com-
mercial discount ?
18. 27 cents is | % of how many dollars ?
19. A retailer bought pencil tablets at $2.40 per hundred
and sold them at 5 cents each. Find the gain per cent.
20. An automobile was sold for $1200 at a loss of 25%.
What would have been the loss per cent if it had been sold
for $1500 ?
408 GENERAL REVIEW
21. The surface of a cube is 96 square inches. Find its
volume.
22. A house and lot cost §6000. It rents for 8360 a year.
The taxes average 1%, insurance |%, and repairs $30.
What is the rate per cent of net income ?
23. A, B, and C gain $2100 in business. A's share of the
profits is | of B's, and C's share of the profits is equal to A's
and B's together. Find the profits of each.
24. $2900 is divided between two partners in the ratio of
1| to 1|. Find each one's share.
25. A withdraws | of his deposit in a bank. He then de-
posits ^ as much as he has drawn out, and still has $2500 in
the bank. Find the amount in the bank at first.
26. A block of business houses was insured for -| of its
value at 3 %. If the premium was $600, what was the value
of the block ?
Written Work
27. Simplify: a. 4 1 * ^ ~ 5 * J. L!i±l
4 of 1 of A i n f 5 i 7
3 Vl 4 Ui 5 4 U1 6 + 8
28. A commission of $43.47 was charged for selling $1242
worth of goods. What was the rate of commission ?
29. How many yards are there in 3.1 mi.? in 7.9 Km.?
30. What are the proceeds of a note for $400, when dis-
counted for 95 days at 6 % ?
31. A lawyer's commission for collecting a bill, at 5 %,
was $125.50. Find the amount of the bill.
32. The square of a number is 1024. What is its cube ?
M Q . v.* 4 x 1 x 25 , /37! 2A\ U
33. kMmplif v : a. 5 6. — a. _= — a x _2.
" fx 11x10 U| 2iJ 21
GENERAL REVIEW 109
34 A bankrupt's liabilities are #15000 and his assets
#'.•000. How much does a creditor receive whose claim is
$6000, no allowance being made for court costs?
35. How many cakes of soap 4 in. long, 2 in. wide, and I. 1 ,
in. thick can be packed in a box 2 ft. long, 1 ft. wide, and
1 ft. high ?
36. Write in figures the number eleven million, eleven
thousand eleven, and eleven millionths.
37. What is the square root of 14^ ?
38. Find the sum of the quotients :
3-3 3 - .03 300 -=- 30
3 -*- .3 .003 -j- 3 30 -- 300
.03 -=- 3 30 - .03 .03 -*- 30
39. In a town 83500 was raised from a tax of 14 mills.
What was the assessed valuation of the property?
40. The perimeter of a square field is 320 rods. How
many acres does it contain ?
41. Reduce to decimals: f, |, ^ T 9 g, f, ^.
42. A man borrowed I860 September 1, 1903, and paid
the note Jan. 16, 1900, with interest at 7%. How much
was paid at settlement ?
43. Find the omitted term in each:
5: 8 = 15: () J:f =0 = f
():12= 5:4 1£ : ( ) = 3| : 4£
44. I pay $36 for insuring my house at | %. For how
much is the house insured ?
45. Find the exact interest of $180 from April 1, 1904 to
August 25, 1904, at 6%.
46. The assessed valuation of a town is #875000. What
rate must be levied to raise $10937.50?
410 GENERAL REVIEW
47. A 60-day note for $1000 is discounted at a bank at
8 % on the date of issue. Find the proceeds.
48. A speculator buys a tract of land 4^ miles square.
How many sections does it contain? How much will it cost
to put a fence around it at $0.50 per rod ?
49. Find the cost of the following bill of lumber:
30 pieces 30' x 6" x 12" at $30 per M.
60 pieces 24' x 6" x 8" at $24 per M.
120 pieces 18' x 4" x 6" at $20 per M.
150 pieces 16' x 3" x 4" at $18 per M.
50. A commission merchant sold 2240 pounds of butter at
24^ a pound. His commission for selling was 5% and he
paid freight charges amounting to $5.72. How much did he
remit to the shipper ?
51. A barn valued at $1800 is insured for | of its value,
at 1 1 % for a term of 3 years. Find the average annual cost
of insurance.
52. $200 was borrowed at 5% on Oct. 1, 1904. When it
was paid, it amounted to $217.50. On what date was it
paid ?
53. Two successive discounts of 20% and 15% reduce
a bill to $306. How much was the original bill ?
54. From a piece of land 45 rods square, I sold 145 square
rods. What is the value of the remainder, at $60 an acre ?
55. Simplify: a. » ' _ ^ * b. ^f- 8 'J
-t + 8 X 3 + 12 ' 2 X 8 ¥ X TT
56. A note for $320, dated June 1, 1902, falls due Septem-
ber 16, 1904. What amount will pay the note if it draws
4|% simple interest?
57. What is the tax on property assessed for $12480 at
$13.50 a thousand?
GENERAL REVIEW -111
58. A broker's purchase at 105, with his brokerage at \<? ,
was $6307.50. How many shares were bought?
59. In what time will any principal double itself at 5 % ?
at 6 % ? at 8 % ?
60. A street 40 rods long and 40 feet wide is to be graded
down on an average 1| feet. How much will the excavating
cost at 27 f a cubic yard ?
61. A coal dealer bought 200 tons of coal at $3.50 per
long ton and sold it at $4.20 per short ton. Find the gain.
62. A wholesale merchant imports 30000 yards of Brus-
sels carpet, 27 inches wide, purchased in Belgium at 40 ^ per
yard. The duty is 18 ^ per yard and 40 c/ ad valorem.
Find the wholesale price in the United States, if a profit of
25 <f on the yard is made.
63. I invested $5000 in bank stock at 156 and sold it at
167, brokerage \°/o in each case. Find the gain.
64. The specific gravity of oak is .934. Find the weight
of an oak sill 24 ft. x 1 ft. x 1 ft.
65. The residence of Mr. Daniels valued at $ 8000 was in-
sured for 3 years at 90 cents per $100 on 80% of its
valuation. How much was the average annual premium ?
66. A man 45 years old takes out a $ 5000 life insurance
policy payable in 20 years. If he pays an annual premium
of $36.87 on the $1000, and lives till the policy falls due,
how much will his insurance cost him, estimating that he
gets back in dividends an average of 20 % of the premiums
paid?
67. A train running 30 miles an hour is 54 minutes in
going from one city to another. If it makes 3 stops of 4
minutes each, how far apart are the cities ?
412 GENERAL REVIEW
68. A square field contains 10 acres. How much longer
is its diagonal than its side ?
69. A man desires to realize 6 % on his investment. How
much should he pay for borough bonds bearing 4| % interest?
70. In an examination 150 questions were asked each of
5 members of a class. The first answered 140, the second
135, the third and fourth 120 each, and the fifth 110. Find
the average per cent of the class.
71. A farm roller 8 ft. long and 2| ft. in diameter will
pass over how much surface in 100 revolutions ?
72. The floor of a room is 16 ft. x 131 ft. Find the cost
of carpeting it with carpet f of a yard wide, laid lengthwise
and costing $1.20 a yard, allowing 9 inches for matching on
each strip except the first.
73. A grocer pays $16 for 5 bushels of cranberries, and
sells them so as to gain 30%. What is the selling price per
quart ?
74. I loaned $450 at 5%, and received $502.50 when it
was paid. For how long was it loaned ?
75. The face of one side of a cube is a surface of 100
square feet. Find the volume.
76. A tax of $6181.40 is to be assessed upon a town.
There are 620 persons subject to a poll tax of $1.25 each.
The property assessment is $675800. Find the 'tax on Mr.
Anderson's property, which is assessed at $4750 if he pays
for one poll.
77. A man buys a house and lot for $3500 ; he pays $1000
cash and gives a mortgage for the balance at 6%. At the
end of 9 months he sells the house and lot for $4500, paying
the interest due on the mortgage at the time of sale. How
much does he realize on his investment ?
GENERAL REVIEW 413
78. I desire to invest #9000. I can buy 7% stock at
20% premium or loan the money at t>%. Which is the
better, both being safe investments?
79. Solve the following equations :
6% of $100 = 5% of ($ ).
37*% of 64 = ( %) of 120.
80. SI 650 yields 8530.75 interest in 5 years, 4 months,
10 days. Find the rate.
81. What sum of money, at 6% simple interest, will
produce in 2 years, 6 months, the same interest that $900
will produce in 3 years, 4 months, at 5 % ?
82. The slant height of a church spire is 48 feet, and its
base is a hexagon 6 feet on each side. Find the cost of
painting it at 40^ a square yard.
83. On a bill of goods amounting to $900, I am offered
a discount of 25%, or two successive discounts of 15% and
10 % . Which would be more advantageous for me to accept,
and how much more ?
84. My gas meter, Jan. 1, registered 11800 cu. ft.; Feb. 1,
35800 cu. ft. I paid the bill before Feb. 10, receiving a dis-
count of 2 ^ on the even thousand. At 27 / a thousand cubic
feet, how much did I pay for the gas used in January?
85. Simplify: a. JL^^g^ J. gf^ x ^
86. I imported from Canada 7500 yards of flannel valued
at 80^ a yard, and weighing 1480 pounds. Specific duty
22^ per pound, and ad valorem duty 30 % . Find the amount
of duty paid.
87. The taxes on a property last year were $42, which
were | less than this year. Find the per cent of increase in
taxes.
414 GENERAL REVIEW
88. A street £ mile long and 30 feet wide is paved and
curbed. The paving costs $ 3 a square yard, and the curb-
ing 30 ^ a linear foot. Find the entire cost.
89. Find the sum of the quotients :
.01-*-. 001 .1 -10 .001 + . 1
10+. 01 .01 + 10 .01 +.001
90. The specific gravity of milk is 1.032. Find the
weight of 18 gallons of milk.
91. A hotel and farm sold for $6000 each. The hotel
was sold at a gain of 20%, and the farm at a loss of 20%.
Find the gain or loss on both sales.
92. The cost of insuring a dwelling at f % is $33.75 a
year, and the cost of insuring the furniture at 1 % is $12.75.
Find the amount of each policy.
93. Which produces the greater per cent of income and
how much, 4 % bonds at 75 or 5 % bonds at 90?
94. Find the sum of the square roots of :
.5625 226.5025 110J
.0016 100.2001 148|i
95. I bought stock at 89 J, brokerage | %, and after receiv-
• ing a dividend of 4%, sold at 104|, brokerage \% clearing
$150. Find the amount of stock purchased.
96. A town expends for improvements $6894. The
assessed valuation is $480000. Find the rate levied to cover
the expense, including the collector's commission estimated
at $306.
97. A swimming pool is 30 meters long, 14 meters wide,
and averages 1.5 meters in depth. Find the number of
kiloliters of water it contains and its weight in kilograms.
GENERAL REVIEW U5
98. A Mexican ranchman purchased a tract of land in the
form of a rectangle 10 kilometers in length and 4 kilometers
in width. Find the number of acres in it.
99. The oxygen in the air is to the nitrogen as 21 to 79.
Find the number of cubic feet of each gas in a schoolroom
whose inside dimensions are 30 ft. x 24 ft. x 12£ ft.
100. What sum will cancel a note for $122.50 bearing in-
terest at 6%, dated April 10, 1903, and maturing September
4, 1905 ?
101. A garrison of 800 men have provisions for 90 days.
A reenforcement arrives at the end of 40 days, and the pro-
visions last only 40 days longer. Find the number of the
reenforcement.
102. Simplify: ^-A^xfofii-^J-
103. What decimal bears the same ratio to .05 that f does
toll?
104. A savings bank pays 4 % interest compounded semi-
annually, the interest periods being April 1, and October 1.
I deposited $100, April 1, 1903, and an equal amount semi-
annually to and including October 1, 1905. What amount
had I on deposit April 1, 1906?
105. An insolvent debtor pays 40 cents on the dollar.
How much will a creditor receive whose claim is $960, after
paying his attorney 10 % for collecting it ?
106. A rectangular field whose width is f of its length
contains 7| acres. Find the distance between the opposite
corners.
107. I loaned f of a certain sum at 6%, and the remain-
der at 5%. The entire income was $322.50. Find the sum
loaned.
416 GENERAL REVIEW
108. A gentleman wishes to invest in 4| % bonds, selling
at 102, so as to provide for a permanent income of -$1620.
How much should he invest ?
109. From one tenth take one thousandth ; multiply the
remainder by 10000 ; divide the product by one million, and
write the answer in words.
no. A drugget 9 ft. by 12 ft. covers 50% of the floor of
a room 13| ft. wide. Find the length of the room.
ill. A mechanic had his wages twice increased 10%.
Find his wages before the first increase, if he now receives
$4.84 per day.
112. A house which had been insured for 13000 for 9
years at If % for a term of three years was destroyed by fire.
How much did the money received exceed the premiums
paid ?
•113. A natural gas company declares a semiannual divi-
dend of 4 % on a capital stock of $150,000. Find the yearly
dividend of a stockholder who owns 36 shares, par value
$50 a share.
114. A merchant in Denver, Col., buys a New York draft
for $600, at \% exchange, and mails it in payment of a bill
in Memphis, Tenn. Find the amount paid for the draft.
115. How long must $120 be at interest at 6% to earn
$34.64?
116. If a clerk's wages are $48 a month, when he works
8 hours a day, how much should he receive for 9 months'
work, of 10 hours a day ?
117. Each side of a roof is 30 ft. long and 18 ft. wide.
How many shingles 16 in. long and 4 in. wide, laid \ to
the weather, will be required to cover the roof?
GENERAL REVIEW 417
118. City bonds bearing 4% interest are sold at 12%
premium. Find the rate per cent the buyer gets on his
investment.
119. The capital stock of a company is $50,000. There
is a deficit of $ 4000 in the earnings. I own 80 shares. Find
the amount 1 must pay if an assessment is levied.
120. A lawyer in collecting a note of $ 3000, compromised
by taking 80 % and charged 5 % for his fee. Find his com-
mission.
121. A man's income is *- of his capital. His taxes are
2| (f of his income. Find the amount of his capital if he
pays $ 24 taxes.
122. The specific gravity of iron is 7.80. Find the weight
of an iron bar 12 ft. long and 2 in. square.
123. Find the result of C°5 * 1.25+ .1875) x 96 _ 16>5#
(.4 -5 +.17)
124. The inside measure of a cubical box is 4 in. on each
side. A sphere 4 in. in diameter is placed in the box. Find
the per cent of space unfilled.
125. At $1£ a rod, it cost $240 to fence a square field.
Find the cost of fencing a rectangular field of equal area
wmose sides are to each other as 1 is to 4.
126. A piano listed at $ 650 was sold at a discount of 40 %
and 20 %. If the freight was $3.25 and dray age $5, what
was the net cost of the piano?
127. A merchant sold a bill of goods amounting to $ 3600
and took a 90-day note for it. Fifteen days later he sold the
note at a bank at 6 % discount. How much did he receive
for the note ?
128. A boat in crossing a river 400 feet wide, drifted with
the current 300 feet. How far did the boat go?
HAH. COM PL. Altllll. — 27
418 GENERAL REVIEW
129. Mr. Brown owed Mr. Smith $2000 which he was
unable to pay ; but he gave him two 90-day notes covering
the amount. One was for $1000 without interest; the other
with interest at 6%. Mr. Smith had both notes discounted
at a bank at 7 % on the day they were given. How much
cash did he receive ?
130. A merchant owing a bill of $1250 in New York is
asked to send a draft in settlement of the account. The
merchant has only $868 in bank and holds a note of $900
due in 30 days without interest. This he has discounted at
6 % for 20 days and the proceeds is placed to his credit. He
buys a draft at | % exchange, giving his check for the
amount. How much does he then have in bank?
131. Mr. Franks has a promissory note of $800 dated July
1, 1906, due in one year with interest at 6%, against Boyd
Emerson, on which are the following endorsements :
Jan. 1, 1907, $50.00 Jan. 1, 1908, $150.00
Dec. 1, 1907, 25.00 Apr. 1, 1908, 200.00
Write the note with the indorsements and find the balance
due Aug. 1, 1908.
132. A San Francisco banker discounts a draft for $3000
payable at Portland, Oregon, 90 days after sight. Exchange
-Jg %, discount 8 %. Find the proceeds.
133. The United States government pays exact interest
at 5 % from April 1 to Oct. 10 on a claim of $63500. Find
the interest the government pays.
134. Find the cost of carpeting a hall 30 ft. by 50 ft with
carpet 27 inches wide, laid lengthwise, at $1.10 per yd., sur-
rounded by a carpet border 18 inches wide, at $ 1.10 per yd.,
allowing 6 inches for matching on each strip except the first.
GENERAL REVIEW 419
135. A cable message was sent at 6:15 a.m. from New
York to London. It was delivered 28 minutes 20 seconds
after being sent. At what time was it delivered?
136. Mr. Coll leased some property for three years at
$2400 per year. His commission for leasing was 1 % of the
first year's rent and his commission for collecting 5 % of each
year's rent. Find the agent's entire commission for the
three years, if all the rent was collected.
137. A commission merchant was offered $1800 per year
salary and 2 % on all sales above $40000, or 5 % on all salts.
He chose the latter and sold $85000 worth of goods. Did
he gain or lose, and how much, by so doing ?
138. 2000 yards of silk when imported cost 215 ^ per
meter. If sold at $2.25 per yard, find the gain.
139. A railroad tank along the line of the Paris and Lyons
railway is 3.5 meters in diameter and 6 meters in height.
Find the number of kiloliters it contains.
140. The specific gravity of iron is 7.80. Find the weight
in kilograms of a bar of iron 1 meter long, 1 decimeter wide,
and 5 centimeters in thickness.
141. A bookkeeper's income is $2700 per year. His
expenses average $50 per month. If he deposits the bal-
ance every six months in a savings bank, how much will he
have in the bank, at 4 % interest, compounded semiannually,
after 3 deposits ?
142. A ship sets sail at Seattle, Nov. 15, at 1 p.m., 120th
meridian time, and arrives at Canton in 21 da. 5 hr. 18 niin.
Find the solar time of arrival in Canton.
143. Sound travels 1120 ft. per second. The thunder from
a flash of lightning was heard 8 seconds after the flash was
seen. How far distant was the cloud ?
420 GENERAL REVIEW
144. At what price must a bank stock paying 6 % annual
dividends be purchased so as to net the purchaser 5 % income
on his investment ?
145. An agent's commission at 2\ % is $57.85. What
must be the amount of a check mailed to cover the purchase ?
146. A railway company declared a If. % quarterly divi-
dend. How much did the purchaser pay for the stock, if it
yielded him 10 % on the amount invested ?
147. Ames Bros., brokers, bought for me 10 shares of
Delaware & Hudson, selling at 129| % premium, par $ 100.
Write the check in payment to Ames Bros, for the stock.
148. A clerk made the following deposits in a savings
bank, at 4 % interest, payable January 1 and July 1 : January
1, 1906, $500 ; July 1, 1906, $350 ; July 1, 1907, $200. On
January 2, 1907 he drew out $100. What was his balance
July 1, 1908?
149. In one section of the Bessemer Railroad there was
laid in one year 6 miles of double track. The rails weighed
100 pounds to the yard and the market price was $32.50
per long ton. The ties cost delivered 69 cents each and
were laid on an average of one tie to every two feet. Find
the cost of the rails and ties.
*
150. A commission broker was to receive 5 % on the first
$ 50000 from a sale of coal land and 2 % for the remaining
amount of the sale. His entire commission amounted to
$ 4000. Find the total amount of the sale.
151. Mr. Adams bought a property for $ 20000. He ex-
pended $4000 in improvements. The repairs each year
averaged $ 250, the insurance and taxes 2^ % on | of the
original cost of the property. For how much a year must
lie rent the property to realize 6 % net on his investment ?
OPTIONAL SUBJECTS
PRESENT WORTH AND TRUE DISCOUNT
A owes B $ 106, due in one year without interest. If A
pays the debt to-day, $100 will cancel it, since $100 at 6 %
will amount to $106 when the debt is due. $106 is the
debt; $100 is the present worth; and $106 minus $100, or
$6, is the true discount.
The present worth of a debt, due at a future time without
interest, is a sum of money which, at a given interest, will
amount to the debt when it becomes due.
The true discount is the difference between the debt and
its present worth. True discount is seldom used in business.
Written Work
l. Find the present worth and true discount of a debt of
$287.50, due in 2 yr. 6 mo. without interest, money being
worth 6 %.
$.15 = the interest of $1 for 2$ yr. at 6 %.
$1.00 + $.15 = $1.15 = the amount of $1 for 2£ yr. at 6%.
$287.50- $1.15 = 250, and
250 x $1.00 = $250, the present worth of $287.50.
Since the present worth of $1.15 is $1, the present worth of $287.50,
which is 250 times $1.15, is 250 times $1 = $250.
$287.50 - $250.00 = $37.50, the true discount.
Comparative Study-
Bank Discount is the interest paid in advance upon the value of a note,
or a debt, at maturity; true discount is the interest on the present worth of
the note, or debt, for the given time. In bank discount notes may, or may
not, bear interest ; in true discount debts are without interest.
The present worth corresponds to the principal; the true discount, to
the interest: the sum due at a future time, to the amount.
121
422 OPTIONAL SUBJECTS
2. What principal will, in 3 yr. and 6 mo., at 6$>, amount
to $344.85 ? (The principal is the present worth ; the interest
is the true discount.)
3. Find the present worth of 1517.50, due in 2 yr. and
6 mo., without interest, money being worth G %.
4. A merchant buys goods amounting to $355.25 and
agrees to pay for them in 3 mo. What cash sum will pay
the bill, money being worth 6 % ?
5. A farmer is offered $5000 for his farm, or $5600, pay-
able one half in cash and the balance in 1 yr. without interest.
How much is the second offer better than the first, money
being worth 6 % ?
6. Find the true discount of $1350, due in 1 yr., 4 mo.,
without interest, money being worth 6 %.
7. What is the difference between the true discount of
$575 due in 2-| yr. without interest, money being worth 0%,
and the simple interest of $575 for 2^ yr. at 6 % ?
8. A purchaser is offered a horse for $195 cash, or $206
due in 6 mo. without interest. Which is the better offer and
how much ?
FOREIGN EXCHANGE
Foreign exchange is a method of paying or collecting bills
in foreign countries without the actual transfer of money.
A bill may be paid in a foreign country by a postal money order, by
an international express money order, by a telegraphic money order, or
by a foreign draft, called a bill of exchange.
Bills of exchange are usually issued in duplicate and numbered first
and second of exchange. By sending them by different mails, the payee
is almost certain to receive one. Each contains a condition that it shall
be void after the other is paid.
Bills in foreign countries are collected by commercial drafts in the
same manner as in domestic exchange.
FOREIGN EXCHANGE 423
A foreign draft, or bill of exchange, is similar to a domestic l>ank
draft and is payable in the money of the country on which it is drawn.
'I'll us. a bill of exchange on Paris is payable in francs.
Premium and discount.
In domestic exchange, there is practically no premium or
discount on money, except during financial panics.
In foreign exchange, the premium or discount varies ac-
cording to the demand for, and the supply of, money.
English exchange is quoted as so many dollars to the pound. Thus,
a quotation of 4.91 means that a foreign draft for £ 1 will cosl s L91.
French exchange is quoted either at so many francs to the dollar, or at
so many cents to the franc. Thus, a quotation of 5.8 means that a
draft for $1 will purchase 5.8 francs ; or a quotation of 20 J T means that,
a draft for 1 franc will cost 2QJ s f.
German exchange is quoted at so many cents to the -i marks or to the
mark. Thus, a quotation of 98 means that 98 ^ will purchase a draft for
4 marks ; or a quotation of 24.2 means that a draft for 1 mark will cost
24.2 cents.
The par of exchange between two countries is the standard value of
the monetary unit of one expressed in that of the other. The English
par of exchange is $4.8665. A quotation of 4.90 is above par and one of
4.84 is below par. The French par of exchange is about 5.18^, or 19.3.
The German par of exchange is about 95.2, or 23.8.
The following is a newspaper quotation of commercial
foreign exchange :
60 Days Demand
Sterling 4.81 .... 4.87
Germany, reichsmarks .93f . . . .94|
France, francs 5.21 .... 5.18$
This means that a XI draft payable on demand will cost
8 4.87 ; or $4.81 payable in 60 days.
Written Work
1. Find the cost of a demand draft, at the quotation
given, for £ 2">.
Cos! of £ 1 =8 1.87; cosl of a £ 25 draft = 25 x 1 I. -7 = 9 121.7."..
424 OPTIONAL SUBJECTS
2. Find the cost of a 60-day draft for 4480 marks.
Cost of 4 marks = 93|^; cost of 4480 marks = ^^ x 934/ =$1050.
4
3. Find the cost of a 60-day draft for 3200 francs.
Cost of 5.21 francs = $1; cost of 3200 francs = ?=92 x $1 = $614.21.
5.21
Find the cost of a 60-day draft for :
4. £ 25 6. 6500 fr. 8. 635.5 M.
5. 275 M. 7. £ 255.5 9. 398.2 fr.
10. Find the cost of a sight draft on London for £ 400,
at the foreign quotation given.
11. Find the face of a demand bill of exchange on Paris
for $2500, at the foreign quotation given.
12. What is the cost of a demand draft on Hamburg for
1125 marks, exchange 94| ?
13. What is the cost of a draft on Lyons for 14,000 francs
at 5.19?
14. A merchant buys a London draft 60 days after
sight for £ 95. If exchange is 4.82, rind the cost of the
draft,
15. How large a bill of exchange on Berlin can be pur-
chased for $1590, exchange being 98 ?
16. A mechanic has $ 1500 with which to purchase a
London draft, If exchange is 4.845, how much is the face
of the draft in English money ?
17. An American tourist bought a letter of credit on
London for £ 300 at $4.88 and 1 % commission. How much,
in United States money, did this letter of credit cost him ?
If lie drew £50 in Paris, how many francs did he get in
exchange, the pound being valued at 25.2 francs?
COMPOUND PROPORTION 425
COMPOUND PROPORTION
A simple ratio is a ratio of two numbers ; thus, 9:3 is a
simple ratio.
A compound ratio is the produet of two or more simple
.. ,n qn ra on f- ] 6\ [9:31 .
ratios ; thus, (9:3) x (b : 2), or ( - x - j, or j ,, t9 l is a com-
pound ratio.
A compound proportion is a proportion in which one or
f 9*3]
both ratios are compound; thus, jV^[ = 18:2 is a com-
pound proportion.
Written Work
1. If 3 men earn $24 in 4 days, how much can 6 men
earn in 3 days?
Since the answer is to be dollars, the second ratio in this compound
proportion is $24 : $ x.
The first ratio in this proportion is compound. If 3 men earn $ 24 in
a certain time, G men can earn more in the same time; x, then, repre-
sents a larger sum than $24, and the Jim t simple ratio in the compound
ratio is 3:6. If a given number of men earn $24 in 4 days, in 3 days
they will earn less; x, then, represents a smaller sum than $24, and the
second simple ratio in the compound ratio is 4 : 3.
Combining these two ratios, the compound ratio is I . \ q 1 and the com-
, ,. . f 3 : 6 1 ^ oi m 6 x 3 x $24 «.,„
pound proportion is \ . . ._, j- = $24 : x. 1 hen, x = T~,~~Z =
2. If 6 men earn $75 in 5 days, how much can 12 men
earn in 3 days ?
3. If 8 men, in 10 days of 9 hours each, earn $280, how
much can 9 men earn in 5 days of 8 hours each ?
4. If 24 men dig a trench 72 rods long, 3 feet wide, and
5 feet deep in 12 days, how long a trench, '2\ feet wide and
3 feet deep, can 18 men dig in »i (lavs?
426 OPTIONAL SUBJECTS
CUBE ROOT
Comparing Roots and Periods
The cubes of the smallest and the largest integers com-
posed of one, two, and three figures are as follows :
l 3 = 1 103 = loco 100 3 = 1,000,000
9 3 = 729 99 3 = 970,299 999 3 = 997,002,999
1. Separate each of these perfect cubes into periods of
three figures each, beginning at the right; thus, 997 '002' 999.
2. How does the number of periods in each cube compare
with the number of figures in the corresponding roots ?
The number of periods of three figures each, beginning at
units, into which a number can be separated, equals the number
of figures in the cube root of the number.
Note. — The left-hand period may contain one, two, or three figures.
3. How many figures are there in the cube root of 46,656?
1,030,301? 12,326,391? -
4. Cube 25. 25 = 20 + 5 ; hence, it may be cubed in two
ays, thus :
25
20 + 5
25
20 + 5
125 =
(20 x 5 ) + 5 2
50 tens =
20 2 + (20x5)
625
20 2 +2(20x5)+5 2
25
20 + 5
3125
(20 2
X 5) + 2(20 x 5 2 ) + 5 3
1250 tens =
20 3 +
2(20 2
x5)+ (20x5 2 )
15625 = 20 3 + 3(20 2 x5) + 3(20 x5 2 ) + 5 8
Or, representing the tens by t and the units by u, we have
the formula:
( r" + u y = f + 3 f'u + 3 tu 1 + u\
CI BE HOOT
427
25 3 =
= 15,625
Observe thai 15,625, tin- cube of 25, is composed of four
partial products :
, 1 ) f 3 =2<>3=S
(2) Zfiu =3(202 x 5)= 6000
(3) 3fn a =3(20 x5 2 ) = 1500
(4) tf s = 5 8 = 125
5. Find tbe cube root of 15,625, or find the edge of a cube
whose volume is 15,625 cubic units.
15'625
20 3 = 8000
Trial divisor, 8x20 2 =l:i<"i
3 x 20 x 5 = 300
5 2 = 25
Complete divisor,
1525
7 625
7 625
20
5
2^>
Separate the num-
ber into periods of
three figures each.
Since 15,625 con-
tains two periods, its
cube root is com-
posed of two figures,
tens and ones. As
the cube of tens is
thousands, the largest cube found in 15 thousands is 2 tens, or 20 ones.
20 3 = 8000 (1st partial product, or t 3 ) as shown
in figure A; 15,625 — 8000 leaves a remainder
of 7625 (3 t-u + 3 tu 2 + u*). The root 20. there-
fore, must be so increased as to exhaust this
remainder,
and keep the
figure a per-
fect cube.
The neces-
sary additions to enlarge A and keep it
a cube are : First, the three equal square
solids B. C, and D (2d partial product,
or %Pu)\ second, the three equal rec-
tangular solids (p. £28) E. F, and G
(8d partial product, or 3 tu*)) and third,
the small cube (p. 428) H (4th partial product, or n 3 ).
The sum of these three additions is 7625 cubic units ; and, since the
square solids B, C, and D (2d partial product, or 3 t*u) contain the greatest
part of the additions, their volume is nearly 7r.-_>:> cubic units. If we
rr ^^^a-:;v:^25Sg^
-r- — — - — —
;
_ +j 1_ __^ —
e- — —
i , , , , . . ,
/>
a
. f f. - -- - shall find their
H I ■ ■_ is 9 i • b '.-- I - the surface of
--— j j i ".'-:.••-■
. iitions.
1. C.L. E F. G. aud H
v - ic each Sraatet] I : - i » ihnne " "
cubic Trnxte) vould be feimil* i than
H< . -■ ; -. - T
-r - - ' unite.
: •-■ -- " ii- - ..L"- - - _- . .-. i " - - r*
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=^ :ji anc ei e L._ . -. ;:-_.,-
' ami:- [ - - mite wide I -
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- -i«ft*iMw - i be UK, ^ ^^
• . ' - ■ - fcb±tf
T • - ..iditio: ..•-.••- 25,
- ' . ' .Are Tir
- ■ . - -■ ^^m
' .' ■ V: -.--
- - - .. ~ * =
uii"*rt c.'j'.- - : . - : naust - - "
■Bttfc. li> - - ". - ■ ^^^^^^^^^
- ' - - - ' ■
"'■—' ' - - - " ' I " i
I
- umbere eaiino; - -->••. be iato two or f/tr*^ - Kfan;
l ■ vd. art -
.' In -eacli case, one o:
. - root of
-.j* iiuiD' regarded aB
■e number of unit* iu tfce «<fpe o/
■ a ""Hy cif eubie units equai t>o the g
CUBE ROOT
129
6. Cube 35 (30 + 5) ; then find the cube root of 42875.
(1) f»=30 8 = 27000
Partial I (2) 3 fiu = 3 ( 31 > 2 x 5 ) = 1 3500
products: 1(3) 3f» 2 = 3(30 x 5 2 ) = 2250
(4) tf 3 = 5 3 = 125
42875
42'875
30
Trial divi- 30 3 =
= 27000
5
sor, 3 X 30 2 = 2700
15875
35
3 x 30 x 5 = 450
5 2 = 25
Complete divisor, 3175
15875
Study of Problems
1 . How many periods are there
in 42,875 ?
2. How many figures are in
its cube root?
3. The left period is 42 (42,000).
The largest cube found in it is 27
(27,000), the cube of the tens.
4. What, then, is the tens' figure of the root ? ( \/27 =3.)
5. Subtract; annex the next period, and the new dividend is 15.s7.">.
The cube, 30, or 3 tens, is to be so enlarged as to exhaust this remainder
and yet preserve it a cube.
6. This dividend is composed mainly of what partial product? (The
second, 3 fiu) = 3 (30 2 x 5) .
7. What factors in the second partial product are already known?
(3 x 30 2 .)
8. If 15,875 is divided by (3 x 30 2 ) as a trial divisor, the quotient
(5) will be (approximately) the other factor in the second partial product.
9. What, then, is probably the units' figure of the root? (5.)
10. Observe that the trial divisor, 2700, is equal to three times the
square of the root found, considered as tens.
11. The first addition to the trial divisor is 3 x 30 x 5, or 450 (three
times the root found, considered as tens). The second addition is 5 2 ,
or 25.
12. What, then, is the complete divisor?
3 p + 3 tu + «a = 3 (30)2 + (3 x 30 x 5) + 5 2 = 3175.
13. Multiplying 3175 by 5 gives 15,875 ; or, u(3 P + 3 tu + u 2 ) = 3:*u
+ 3 tu 2 + u*. This exactly exhausts the remainder of 15,875. Hence, the
unite' figure of the root is 5, and the cube root of 42.875 is 30 I •>,
or 35.
430
OPTIONAL SUBJECTS
7. Find the cube root of 34012.224. (Separate the num-
ber into periods, to the left and right from the decimal
point.)
34'012.'224' | 32.4
3 3 = 27
Trial divisor, 3 x 30 2 = 2700
3 x 30 x 2 = 180
2 2 = 4
Complete divisor,
2884
012
5768
Trial divisor, 3 x 320 2 = 307200
3 x 320 x 4 = 3840
4 2 = 16
Complete divisor,
311056
1244224
1244224
When the cube root consists of more than two figures, three times the
square of the root already found (considered as tens), is used as a trial
divisor in finding the next figure of the root.
8. Find the cube root of 5 to thousandth
IS.
5. | 1.709+
1 3 =
1
Trial divisor, 3 x 10 2 = 300
4000
3 x 10 x 7 = 210
7 2 = 49
Complete divisor, 559
3913
Trial divisor, 3 x 170 2 = 86700
87000
Trial divisor, 3 x 1700 2 = 8670000
87000000
3 x 1700 x 9 = 45900
9 2 = 81
Complete divisor, 8715981
78443829
When a ciDher occurs in the root, annex tiro more ciphers to the trial
divisor, bring down the next period, and proceed as before.
CUBE ROOT 4:11
Separate the number into 'periods of three figures each, to the
left and right from the decimal point.
Find the largest cube in the left-hand period, and write its
root as the first root figure sought. Subtract this cube from fin-
left-hand period and annex the next period to the remainder.
For a trial divisor, take three times the square of the root
already found, considered as tens, or three times the square
of the root with two naughts annexed. Divide the dividend by
it, and the quotient, or the quotient diminished, will be the
second part of the root.
To the trial divisor, add three times the product of the first
part of the root, considered as tens, by the second part, and
also the square of the second part. Their sum will be the
complete divisor. Multiply the complete divisor by the second
part of the root, and subtract the product from the dividend.
When other periods remain, take three times the square of
the root already found, considered as tens, for a trial divisor,
and proceed as before.
Note. — 1." When a number is not a perfect cube, annex periods of
naughts, and continue the work as far as desired.
2. Decimals are separated into periods of three figures each, begin-
ning at the decimal point and passing to the right.
:!. To find the cube root of a common fraction, take the cube root of
the numerator and the denominator separately, or reduce the fraction
to a decimal and then extract its cube root.
Find the cube root of :
1. 2744 7. 373,248
2. 4096 8. 941,192
3. 13,824 9. 1,860,867
4. 19,683 10. fife
5. 32,768 11. T VW,
e. 91,125 12. T y^
13.
.729
14.
15.625
15.
39.304
16.
2i^ti.981
17.
.003375
18.
3|
132 OPTIONAL SUBJECTS
Written Work
1. The volume of a cubical box is 5832 cu. in. What is
its edge ?
Note. — Since the volume of a cube is a number of cubic units equal
to the product of its three equal dimensions, the cube root of the volume
of a cube gives the length of its edge.
2. If 9261 cubic inches are built into one cube, what is
the area of its base ?
3. Find the depth of a cubical cistern that will contain
2197 cu. ft.
4. What is the edge of a cubical box that will contain 50
bushels of 2150.42 cu. in. each?
5. A box is 16 ft. long, 8 ft. wide, 4 ft. high. What is
the edge of a cubical box of the same volume ?
6. Find the edge of a cube whose volume is equal to the
volume of three cubes whose edges are respectively 3, 4, and
5 inches.
Geometry shows that :
The corresponding lines of similar solids are proportional
to the cube roots of their volumes.
7. If a ball 10 in. in diameter weighs 125 lb., what is the
diameter of a similar ball that weighs 216 lb. ?
8. Of two similar solids, one contains 8 times the volume
of the other. The diameter of the smaller is 8| feet; what is
the diameter of the other ?
Suggestion : </l : #8 = 8£ ft. : x.
9. The weights of two balls of metal are as 125 to 343.
What is the ratio of their diameters ?
Suggestion : D. of 1st : D. of 2d = ^125 : v^43.
10. If a stack of hay 12 ft. high contains 8 tons, how high
is a similar stack that contains 27 tons ?
11. A square grain bin which contains 1200 bushels of wheat
has a depth of only J of its width. Find its dimensions.
REFERENCE TABLES OF MEASURES
Liquid Measures
4 gills
2 pints
4 quarts
1 gal.
The gill is now seldom used.
The standard unit of liquid measure is the gallon.
= 1 pint
= 1 quart
= 1 gallon
= 4 qt. = 8 pt.
1 gallon = 231 cubic inches
1 cubic foot = nearly 7\ gallons
3H gallons = 1 barrel ] in measuring the capacity
63 gallons = 1 hogshead J of cisterns and vats
1 gallon of water weighs nearly S\ pounds
1 cubic foot of water weighs nearly 62 i pounds
Dry Measures
2 pints = 1 quart
8 quarts = 1 peck
4 pecks = 1 bushel
1 bu. =4 pk. = 32 qt. = 64 pt.
Our standard unit, the Win-
chester bushel, used for measuring
shelled grains, = 2150.42 cu. in., or
nearly \\ cubic feet. In form it
is a cylinder 1SJ- inches in diame-
ter and 8 inches deep.
The dry gallon = 268.8 cu. in.
The heaped bushel, used for
measuring corn in the ear, apples,
potatoes, etc., = 2747.71 cu. in., or
nearly 1§ cu. ft.
The standard English bushel =
2218.102 cu. in.
Measures of Length
12 inches = 1 foot
3 feet = 1 yard
5 1 yards j
16.i feet j
320 rods
5280 feet
1 rod
1 mile
1 mi. = 320 rd. = 1760 yd. = 5280
ft. = 63360 in.
The standard unit of length is
the yard.
A nautical mile (knot) = 6080.27
ft. or nearly 1.15 common miles.
A league = 3 nautical miles ; a
fathom, used in measuring the
depth of water, = 6 ft.; a hand,
used in measuring the height of
horses, = 4 in. A furlongs £ mi.
II \M. compl. \ l : I I 1 1 28
433
434 REFERENCE TABLES OF MEASURES
Measures of Surface
144 square inches = 1 square foot
'9 square feet = 1 square yard
30^ square yards = 1 square rod
160 square rods )
43560 squai'e feet j
640 acres = 1 square mile
1 mile square = 1 section
36 square miles = 1 township
The acre is not a square unit like the square foot, the square yard, etc.
When in the form of a square, it is nearly 209 feet on a side.
Surveyors' Measures
Surveyors and engineers formerly used the Gunter's Chain. It is 66 feet
long and divided into 100 links of 7.92 inches each. The tables are as
follows :
Length
7.92 inches = 1 link
100 links = 1 chain
80 chains = 1 mile
Surface
16 square rods = 1 square chain
10 square chains = 1 acre
They now generally use a steel tape 50 ft. to 100 ft. long divided into
feet and tenths of a foot; or a chain 50 ft. to 100 ft. long having links
each a foot in length, divided into tenths of a foot.
Land Measure is computed by dividing the number of square feet of
surface by 43560, the number of square feet in an acre, and changing the
decimal of an acre to square rods, etc.
Measures of Volume
1728 cubic inches = 1 cubic foot
27 cubic feet = 1 cubic yard
A cubic yard of earth is considered a load.
A cord of 4 foot wood is a pile of wood 8 feet long and 4 feet high, the
sticks averaging 4 feet in length, making 128 cubic feet in the pile.
A cord of short wood is a pile of wood 8 feet long and 4 feet high, the
number of cords in a pile being computed by multiplying the length of
the pile in feet by the height in feet, and dividing the product by 32.
REFERENCE TABLES OF MEASURES
435
Avoirdupois
10 ounces = 1 pound
100 pounds = 1 hundredweight
2000 pounds = 1 ton
•J-2 in pounds = 1 long ton
1 T. = 20 cwt. = 2000 lb. =
32000 oz.
The standard unit of weight is
the pound = 7000 grains.
1 Av. oz. = 437 £ grains.
The long ton is used in the United
wholesale transactions in coal and iron
Weight
* 60 pounds = 1 bu. of wheat or
potatoes
* 56 pounds =1 bu. of shelled
corn or rye
*32 pounds = 1 bu. of oats
196 pounds = 1 bbl. of flour
200 pounds = 1 bbl. of beef or pork
* In most states.
States custom houses and in the
. The long cwt. = 112 lb.
. Troy Weight
24 grains = 1 pennyweight
20 pennyweights = 1 ounce
12 ounces = 1 pound
1 pound = 12 oz. = 240 pwt. = 5760 gr.
The unit generally used for weighing diamonds, gems, etc., is the
carat, which is about 3.2 Troy grains. It is used also to express the
fineness of gold. 18 carats fine means || pure gold and 2 6 ? baser metal.
The Troy pound = 5760 grains
The Troy ounce = 480 grains
Apothecaries' Weight
This is used only in filling
medical prescriptions.
20 grains = 1 scruple — sc. or 3
:) scruples = 1 dram — dr. or 3
8 drams = 1 ounce — oz. or §
12 ounces =1 lb. or lb
Counting
12 things = 1 dozen (doz.)
12 dozen - 1 gross (gro.)
12 gross = 1 great gross
20 things = 1 score.
Apothecaries' Liquid Measures
This is used only in filling
medical prescriptions.
10 minims (m) = 1 fluid dram (f5)
8 fluid drains = 1 fluid ounce (f^)
16 fluid ounces = 1 pint (0)
Stationers' Measures
24 sheets = 1 quire
20 quires = 1 ream
Paper is frequently sold by the
pad or hulk of 100, 500, or 1000
sheets, or by the pound.
436
REFERENCE TABLES OF MEASURES
Measures of Time
60 seconds = 1 minute
60 minutes = 1 hour
24 hours = 1 day
7 days = 1 week
12 months "I
V = 1 common year
= 1 leap year
= 1 decade
= 1 century
Thirty days have September,
April, June, and November.
All the rest have thirty-one
Save February, which alone
Has 28, and one day more
We add to it one year in four.
365 days
366 days
10 years
100 years
The true solar year is 365 days, 5 hr., 48 min., 46 sec. The standard
unit of time is the day which is divided into 24 hours counting from
midnight to midnight. Tn business transactions 30 days are considered
a month, and 12 months are regarded as a year.
The centennial years divisible by 400, and all other years divisible by 4,
are leap years.
Measures of Angles and Arcs
60 seconds = 1 minute
60 minutes = 1 degree
360 degrees = 1 circumference
1 right angle = 90 degrees
United States Money
10 mills = 1 cent
10 cents = 1 dime
10 dimes = 1 dollar
10 dollars = 1 eagle
English Money
4 farthings = 1 penny = $.02025
12 pence = 1 shilling = §.243
20 shillings = 1 pound = 14.8665
The unit of English money is the pound.
The value in United States money of other foreign coins is as follows :
Ruble
Russia = $ .515
Yen
Japan = .498
Franc
France (Belgium) = .193
Mark
Germany = -238
Crown
Austria-Hungary = .203
Lira
Italy = .193
Peseta
Spain = .193
Peso
Chile = .365
Crown
Sweden = .268
INDEX
Abbreviations, 109.
Abstract number, 24.
Accident insurance, 262.
Accounts, 112-114, 170-174.
Acre, 192, 193, 356, 434.
Acute angle, 196.
Ad valorem duty. 273.
Addends, 14.
Addition, defined, 13.
of common fractions, 52-54.
of decimals, 96, 97.
of denominate numbers, 119,
1S4, 1S5.
of integers, 13-18.
sign of, 13.
Agent, 147, 255.
Agricultural Problems, 392-
400.
Aliquot parts, of$l, 108-111.
interest by, '276, 277.
Altitude. 129, 196, --"-"2. 368.
Amount, in adding. 14.
in interest, 151, 275.
in percentage, 23S.
Analysis, 84-89, 231-230, 345-
348.
Angle, defined, 194.
measures of, 195, 190, 436.
Annual interest, 286, 287.
Antecedent, 337.
Apothecaries' liquid measures,
435.
Apothecaries' weight, 435.
Approximate measurements,
224, 385, 3S9, 433.
Arabic notation and numera-
tion, 7-lo.
Arc measures, 195, 196, 436.
Are, 387.
Areas, 124, 129, 199-201, -'or,
211.
Arithmetic, defined, 7.
Assessed valuation, 271.
Assessment of stockholder,
325.
Assessors, 269.
Avoirdupois weight, 435.
Balance of account, 112, 173,
174.
Bank, 303.
Bank discount. 307-314, 421.
Bank draft. 317.
Banking. 808-314.
Barrel, 223, L"J4, 4.33.
Base in figures, 129. 190.
Base, in percentage, 145, 238.
Base line, 350.
Belgian money. 181-183.
Hill of exchange, 422.
of hiding, 821.
Billions, 8, 9.
Bills and accounts, 160-174.
Bins, 2i':;.
Blank indorsement, 294.
Board font, 135, 130, 217.
Bonds, 331-334.
Borrowing from banks, 307-
bl4.
Brickwork. 220, 221.
Broker. 'J.'"'.
Brokerage, 255-259, 325.
Bunch of shingles, 203.
Bundle of laths, 201.
Bushel, 115, '223, 224, 433.
Business applications of deci-
mals, 108-111.
Canadian money, 182.
Cancellation, 42, 43.
Capacity, common measures,
of, 224, 433.
metric measures of, 3SS, 3S9.
Capital. 324. 342.
i apital stock, 324.
i larat, 435.
Caret in division of decimals,
104.
Carpeting, 204. 205.
i ash discount, 264.
( lashier, 303.
Cent, 430.
Centare, 3S7.
Center of circle, 209.
Centigram, 3S9.
Centiliter. 889.
t 'entime, 1S1.
Centimeter. 884.
Central time, 355.
i lentury, 436.
Certificate of stock, 325.
Certified check, 317.
Chain. 434.
Check. 15-. 304.
Check book, 304.
Circle, 209.
t ircumference, 19 I. 210.
Cisterns, 223.
eting bills, 820-823.
Collector, 255, 269.
Commercial bills, 268. 269.
Commercial discount, 148,
264-269.
Commercial draft, 212.
437
Commission and brokers
147, 255-259.
Commission broker, 255.
Commission merchant, 255.
Common denominator, 51.
Common divisor. 40.
Common factor, 40.
Common fractions. 44-S'».
Common multiple, 41.
Complex decimal, 95, 107.
Complex fraction, 75.
Composite number, 88.
Compound denominate num-
ber, 115, 175.
Compound fraction, 00.
Compound interest. 288-292.
t lompound proportion, i25.
Concrete number. 24.
Concrete work, 220, 221.
Cones, 369-437 1 , 373-370.
Consequent, 337.
Consignor, 255.
Convex surface, 222,368.
Cud, of wood, 137,213,220.
Corporation, 324.
Cost. 250.
Counting table, 435.
Coupon bond, 332
Credit, 112, 173.
Creditor, 112, 109.
Crown, 436.
Cube, 212.
Cubes of numbers, 358, 426.
t !ube root, 426—432.
by factoring, 360.
Cubic common units, 132, 212,
434.
Cubic metric units, 3S7.
i 'ustomhouse. 272.
Customs, 272-271.
Cylinders. 221 223, 30S-373.
Date line, 354.
Dates, time between, 1S5.
Day, 115, 186.
Daybook, 173.
Daj s of grace, 295.
Debit, 112. 173.
Debtor, 112. 169.
1 (ecade, 436.
Decimal fractions,
addition of, '.'i', '.'-,
business applications of.
108-111.
compared with fractions, 94,
95.
defined, '.">.
438
INDEX
Decimal fractions,
division of, 101-106.
multiplication of, 98-101.
notation and numeration of,
91-94.
reduction of, 95.
subtraction of, 96-98.
Decimal point, 90.
Degree, of angles and arcs.
194, 196, 436.
of longitude, 350, 351.
Demand note, 294.
Denominate numbers, 115—
122, 175-191.
Denominator, 44.
Deposit slip, 303, 304.
Diagonal, 123, 207.
Diameter, 210.
Difference, in percentage, 238.
in subtraction, 19.
in time, 120.
Digits, 38.
Dime, 12, 382, 436.
Discount, bank, 307-314, 421.
commercial, 264—269.
on stocks, 324, 326, 328.
true, 421.
Discounting notes, 303, 308.
Dividend, in division, 29.
in insurance, 263.
in stocks, 325, 329.
Divisibility, tests of, 3S, 39.
Division, denned, 29.
of Common fractions, 6S-74.
of decimals, 101-106.
of denominate numbers, 121,
180, 187.
of integers, 29-32.
sign of, 29.
Divisor, 29.
Dollar, 12, 3S2, 436.
Domestic exchange, 315-323.
Dozen, 25, 435.
Drachma, 182.
Draft, 317-323, 422.
Dram, 435.
Drawee, 319.
Drawer, 319.
Dry measures, common, 134,
433.
metric, 388, 389.
Dutiei or customs, 272-274.
Eagle, 436.
Eastern time, 355.
Endowment policy, 262.
English money, 181-183, 436.
Equality, sign of, 13.
Equations, analvsis bv, 231-
236.
Equator. 349.
Equilateral triangle, 197-36".
Even number, 38.
Evolution, 360-36G, 426-432.
Exact divisor, 37.
Exact interest, 287.
Excavations, 132.
Exchange, 315, 323. 422-424.
Exponent, 37, 359.
Express money order, 315, 422.
Extracting roots, 860-866, 426-
432.
Extremes, 338.
Face, of note, 293.
of solid, 368.
Factor, defined, 37.
Factoring, 39, 40.
roots extracted by, 361.
Factors and divisors, 37—43.
Farthing, 181.
Fathom, 433.
Feeding stock, 392-394.
Fertilizers, 395, 396.
Figures, notation by, 7.
Fire insurance, 260.
Firm, 342.
Flooring, 202, 203.
Fluid dram and ounce, 435.
Foot, 433.
Feinting of account, 174.
Foreign exchange, 422.
Foreign money, 181-1S3.
Fractional parts, of fractions,
65, 66.
of integers, 60.
Fractional relations, 76, 77.
Fractional units, 44. 45.
Fractions, common, 44-83,
106. 107.
decimal, 90-111.
Franc, 181, 182,436.
Free list, 273.
French money, 181-183.
Full indorsement, 294.
Fundamental processes, 7-35.
Furlong, 433.
Gain and loss, 249-253.
Gallon, 115, 223, 224, 433.
German money, 182.
Gill, 433.
Globe, 369, 375, 376.
Government bond, 332.
Government land, 356, 357.
Government revenue, 272-
274.
Grace, days of, 295.
Grain, 435.
Gram, 383, 389.
Great gross, 435.
Greatest common divisor, 40,
41.
Greek money, 182.
Gross, 435.
Gross cost. 250.
Gunter's chain, 356.
Hand, 433.
Hectare, 387.
Hectoliter, 389.
Hexagon, 367.
Hurher terms, 47,48.
Hour, 436.
House, 342.
Hundreds, S.
Hundredths, 91.
Hundredweight, 435.
Hypotenuse, 365.
Improper fraction, 45.
Inch, 433.
Incomes, from stocks and
bonds, 329.
Incomplete decimal, H)7.
Index of root, 360.
Individual note. 295.
Indorsements, 15>. 293, 294.
Indorser of note, 293. 294.
Insurance, 259-263.
Integer, 37.
Interest, annual, 286, 287.
compound, 288-292.
exact, 287, 288.
simple, 151-154, 275-284.
table of, 291.
Interest term, 289.
Internal revenue, 272.
International date line, 354.
Inverting terms, of fractions,
72, 73.
Investments, 291, 329.
Involution, 35S, 359, 362.
Isosceles triangle, 197.
Italian money, 1S1-1S3.
Joint and several note, 295.
Kilogram or kilo, 389.
Kilometer, 384.
Knot, 433.
Land measures, 356, 357, 387,
434.
Lateral surface, 368.
Laths, 201.
Leap year, 436.
Least common denominator,
51.
Least common multiple, 41,
42.
Ledger, 173.
Ledger accounts, 173, 174.
Length, common measures of,
123, 192, 193, 433.
metric measures of, 384—386.
Letters, notation by, 892—
405.
Life insurance, 262.
Life policy, 262.
Like numbers, 14.
Linear measures, 433.
Lines, 194.
Link, 434.
Liquid measures, common,
433.
metric. 388. 3S9.
Lira, 181, 132, 436.
List price. 264.
Liter, 383, 389.
Load, 212.
Long division, 31, 32.
Long ton, 435.
Longitude and time, 349-355.
INDEX
139
Loss, 250.
Lowest terms, 48, 49.
Lumber, 134, 135, 217-220.
.Milker, of check, 305.
of imt.'. 298.
Making change, 17.
Murine Insurance, 260.
Mark. 182, 186.
Market value of stock, 324.
Maturity of note, 295, 808.
Means, in proportion, 388.
Measurements, practical, 193—
230.
Measures, tables of, 198, 196,
212, 224, 884-590, 433-
486.
Mensuration, 307-381.
Merchants' rule for partial
payments, 302.
Meridian, 349.
Meter. 382.
Metric system, 382-391.
Metric ton, 389.
Mile, 3S5, 434.
Mill, 12, 382, 436.
Millimeter, 384.
Millions, 8, 9.
Minim. 435.
Minuend, 19.
Minus, 19.
Minute. 436.
Mixed decimal, 92, 101, 102.
Mixed number, 45, 49, 50, 56,
57, 61-65.
Money order, 315, 316, 422.
Months, of year, 436.
Mortgage, 331, 332.
Mountain time, 355.
Multiple, 41.
Multiplicand, 24.
Multiplication, defined, 24.
of common fractions, 58-
68.
of decimals, 9S-101.
of denominate numbers,
186, 187.
of integers, 23-28.
sign of, 24.
Multiplier, 24.
National bank, 303.
Nautical mile, 433.
Negotiable note, 293.
Net price, 148, '-'04.
Net proceeds, 147, 250, 255.
Non-negotiable note, 293.
Notation and numeration, of
common fractions, 46.
of decimals, '.'1-94.
of integers, 7-1 o
Notes, promissory, 292-302.
Number. 7.
symbol, Jf, 169.
Numeration, 7-12. 46, '.'I B4,
392.
Numerator. II.
Obtuse angle, 196.
odd number, 88.
One dollar six per cent inter-
est method, 281, 2S2.
< Operation, signs of, 36.
Orders for goods, 167.
Orders of units, 8.
Ounce, 185.
Pacific time, 355.
Painting. 201.
Papering and carpeting, 204,
205.
Par value of stock, 324.
Parallel lines. 124, 194.
Parallelogram. 198, 207.
Parenthesis, 36.
Partial payments, 298-302.
Partition. 29.
Partitive proportion, H41-344.
Partnership, 342-344.
Payee, 293, 319.
Peck, 433.
Penny, 181.
Pennyweight, 435.
Pentagon, 367.
Per cent, 237.
Percentage, 141-146, 237-274,
335, 336.
Perch, 220.
Perfect square, 360, 361.
Perimeter, 193, 369.
Periods of figures, 8.
Perpendicular, 194.
Personal insurance, 262, 263.
Personal property, 269.
Peseta, 181, 182,436.
Peso, 436.
Pfennig, 182.
Pi (tt), 210, 372, 375.
Pint, 433.
Plane, 367.
Plastering and painting, 128,
201, 202.
Plus, 13.
Policy, 259.
Poll tax, 269.
Polygons, regular, 367, 868.
Postal money order, 315, 422.
Pound (sovereign 1 ). 181. 182,
436.
in weight, 115. 435.
symbol, $, 169.
Powers and roots, 358-366.
Practical measurements. 128-
140, 192-280.
Premium, on exchange, 259,
262.
on policy , 262.
on -toeks. 824, 320, 328.
Present worth, 421.
Price list, 266.
Prime factor, 39.
Prime meridian, 349.
Prime number, 37, 88.
Principal, in Interest, 151, 276.
consignor), 255.
Prism, :o;s-:f7:i.
Private bank, 808.
Problems in Interest, 288, 284.
Proceeds, net. 147, 250, 255.
of note. 808.
Product, •.'!.
Profit and loss, 249-268.
Promissory notes, 292-302.
Proper fraction. 45,
Property insurance, 260, 261.
Proportion, 888 844, 126.
Protesting a note. 294.
Protractor, 195.
Public land, 350, 357.
Pyramid, 369-371,373-375.
Quadrilaterals, L98, 199.
Quart, 1 88
Quintal, 889.
Quire, 435.
Quotations, of stocks and
bonds, 333.
Quotient, 29.
Radical sign, 360.
Radius, 210.
Range, 356.
Kate, of insurance, 259.
of interest, 151, 275.
of taxation. 269.
per cent, 145, 288.
Ratio, 337, 338, 425.
Real estate or real property,
269.
Ream, 435.
Receipts, 157, 166, 168, 169.
Reciprocal, 72.
Rectangles, 124, 198-201.
Rectangular solids, 131, 213,
214.
Reduction, of common frac-
tions, 47-52, 106, 107.
of decimals to common frac-
tions, 95.
of denominate numbers,
116-119, 175-184.
Registered bond, 832.
Regular polygons, 367, 368.
Remainder, in division. 29.
in subtraction, 19.
Revenue, government, 272-
274.
Reviews, 7S-s3, 107, 122, 138,
155, 156, 159-105. 1- 191,
225-280, 247-249, 253, 254,
284,285,885,886,881,401-
420.
Rhomboid, P.m.
Rhombus, 199.
Righl angle, 124,129,194, 196.
Right-angled triangle, 128, 206,
865, 306.
Rod, i
Roll of paper, 204.
Roman notation. 11.
Roofing and flooring, 202. 208.
R 9, 858 360, 426-482.
Ruble, 1 tfl
440
INDEX
Savings accounts, 289-291.
Savings bank, 303.
Scalene triangle, 197.
Score, 435.
Scruple, 435.
Second, 436.
Section, 192, 357.
Selling price, 250.
Share of stock, 324.
Shilling, 181, 436.
Shingles, 203.
Shipper, 255.
Short rates, 259.
Sight draft, 320, 321.
Similar figures, 376-378, 432.
Similar fractions, 51.
Similar solids, 37S, 379, 432.
Similar surfaces, 376-37S.
Simple denominate number,
115, 175.
Simple interest, 275-285.
Simple proportion, 338-341.
Simplification of complex frac-
tions, 75.
Six per cent interest method,
278-282.
Slxtv-day six per cent interest
"method, 278-280.
Slant height, 369.
Solar year, 436.
Solids, 212-216, 36.9-376, 37S,
379.
similar, 378, 379, 432.
Sovereign, 181, 182, 436.
Spanish money, 181-183.
Specific duty, 273.
Specific gravity, 379, 380.
Sphere, 369, 375, 376.
Spraying plants, 397-400.
Square, of numbers, 35S.
of roofing and flooring, 202.
(rectangle), 124, 19S, 367.
Square common units, 12S,
193, 206. 434.
Square metric units, 386.
Square root, 360-366.
Standard time, 354, 355.
State bank, 303.
Statements, on bills, 171.
Stationers' measures, 435.
Stere, 388.
Sterling money, 181, 182, 486.
Stock broker, 325.
Stock company, 324.
Stockholder, 325.
Stocks and bonds, 324-334.
Stonework, 220, 221.
Stub of check, 304, 305.
Subtraction, defined, 19.
of common fractions, 55-57.
of decimals, 96-98.
of denominate numbers, 120,
184, 185.
of integers, 19-22.
sign of, 19.
Subtrahend, 19.
Sum, in addition, 14.
in percentage, 238.
Surfaces, common measures
of, 124, 193, 194, 434.
metric measures of, 386, 387.
of solids, 213, 369-372.
similar, 376-378.
Surveyors' linear measures,
434.
Surveyors' square measures,
356, 434.
Swiss money, 181-183.
Tables, interest, 291.
of common measures, 126,
193, 196, 212, 223, 224, 356,
433-436.
of life insurance, 262.
of longitude, 350, 351.
of metric measures, 384-390.
of money, 181, 182, 436.
of specific gravitv, 380.
Tanks. 223.
Tare, 273.
Tariff, 272.
Taxes, 269-272.
Telegraphic money order, 316,
422.
Tens, 8.
Tenths, 91.
Term of discount, 308.
Term policy, 262.
Terms, of fraction, 44.
of ratio, 337.
Thousands, 8.
Thousandths, 91.
Time, and longitude, 349-355.
standard, 354, 355.
Time belts, 355.
Time discount, 264.
Time draft, 320-322.
Time measures, 436.
Time note, 294.
Ton, common, 224, 435.
metric, 3S9.
Township, 356.
Trade discount, 14S, 264.
Trapezium, 199, 209.
Trapezoid, 199, 208.
Triangle, 128, 196, 367.
Trillions, 8, 9.
Troy weight, 435.
True discount, 421.
Trust company, 303.
Unit, 7, 44.
Unit of measure, 175, 199.
United States money, 12, 382,
436.
United States rule for partial
payments, 298-302.
Unlike numbers, 14.
Vertex of triangle, 196.
of pyramid, 369.
Vinculum, 36.
Volume, 130, 214-216, 368, 372-
376, 378, 379.
common measures, 212, 434.
metric measures, 3S7, 388.
Water, weight of, 224, 390.
Week, 436.
Weight, common measures,
435.
metric measures, 389-391.
Winchester bushel, 433.
Wood measures, common, 137,
217-220, 434.
metric, 3S6.
Yard, 115, 384, 385, 433.
Tear, 436.
Yen, 436.
ANSWEES
Page 11. — 1. 43; 449; 1,000,502; L492. 2. 90; 1908; 1576; 1801.
3. XLI; LXIII; LXXXIV ; XCIX ; CVII ; CCXVIII ; DLXXII ;
DCCXXXV; CMXCVJ ; MCMVII ; MDLXIV ; MDCXVI ; M; CCLX.
Page 15. — 2. §3603.18. 3.2,595,237. 4.3,303,261. 5.3,996,337.
Page 16. — 7. 5182. 8. 5261. 9. 6684. 10. 5269. 11. 2799.
12. 83,302. 13. 82,276. 14. 76,385. 15. 84,503. 16. 62,872. 17. 71,809.
Page 18. — 1. 913 rai. 2. $11,276. 3: 3,805,074 sq. mi. 4. $ 2,376,-
000,000. 5. $1,254,060,661. 6. $748,152,215.
Page 20. — 7. 5997. 8. 22,962. 9. 27.4:!4. 10. 2600. 11. 7999.
Page 21. —3. 14.571. 4.36,906. 5.18,393. 6.30,996. 7.38,131.
8. 28,981. 9. $329.04. 10. $379.21. 11. $561.47. 12. $194.39.
13. $ 190.39. 14. $190.33. 15.300,884. 16.565,065. 17.295,404.
18. 186,418. 19. 433,983. 20. 225,545. 21. 3,472,273. 22. 2,698,987
23. 3,12-3,092. 24. 4,197,541. 25. 2,304,902. 26. 2.919.062.
Page 22.— 1. 2029. 2. 7290. 3. 966,779. 4. 144,752,500 bu.
5. A, 81375; B, $3075. 6. $9 gained. 7.8 4800. 8. 52,407,000 sq. mi.
9. James, 86750 ; Henry, $11,925 ; Frank. $5000.
Page 27. —2. 6000. 3. 3745. 4. 12.880. 5. 35,000. 6. 21,584. 7. 51,134.
8 a. 1,574,125; 8 6.2,210,200; 8-'. 3,623,700; 8./. 4,793,050; 8 e . "lr,. .700 ;
8 f. 6,174,425; % n. 5.583,325; 8 ft. 5,114,300; 8 i. 6,219,400; 8./. 5,769,650.
9a. 250,880; 9 b. 352,256; 9 <\ 577,536 ; 9d. 763,904 : 9^. 823,296 ; 9/. 984,
064 ; 9 g. 889,856 ; 9 ft. 815,104 ; 9 «. 991,232 : 9/. 919,552. 10.'. 2.130,4<iu ;
10 6.2.099,680; 10 c. 4,918,080 ; 10 d. 6,505,120; 10 e. 7,010,880 ; 10/8,379,-
920; 10 g. 7,577,680; 10 h. 6,941,120 ; 10 i. 8,440,960 ; 10 j. 7,830,560.
11 a. 2,364,740; 11 b. 3,320,288; 11 c. 5,433,728 ; 11 d. 7,200,392 ; 11 e. 7,760,
208 ; 11/. 9,275,572 ; 11 g. 8,387.588 ; 11 ft. 7,682,992 ; 11 i. 9,343,136 ;
11 j. 8,667,496. 12 a. 2,122,925 ; 12 6.2,980,760; 12 c. 4,887,060 ;
12 d. 6,464,090 ; 12 e. 6,966,660 ; 12 f. 8,327,065 : 12 g. 7,529,885 ; 12 ft. 6,897,-
340; 12 i. 8.387.720; 12 j. 7,781,170. 13. a. 1,934,030; 13 A. 2,715,536;
13 c. 4,452,216; 13 a". 5,888,924 ; 13 e. 6,346,776 ; 13/. 7,586,134 ;
13 g. 6.859.886 ; 13 ft. 6,283,624 ; 13 i. 7,641,392 ; 13 j. 708,812. 14 a. 2,073,-
925; 146. 2,911,960; 14 c. 4,774,060; 14-/. 6,31 i,890 ; 14*. 6,805,860 ;
14 f. 8,134,865 ; 14 g. 7,356,085 ; 14ft. 6,738,140 ; 14/. 8,194,120; 14./. 7,601,-
570. 15 a. 1,885,275; 15 6. 2,647,080 ; 15 c. 4,339,980 ; 15 d. 5,740,470 ;
15^.6,186,780; 15/7,394,895; 15 g. 6,686,955 ; 15 h. 6,125,220 ; 15 /. 7,448,-
760 ; 15 ). 6,910,110. 16 a. 2,064,125 ; 16 6. 2,898,200 ; 16 c. 4.751,700;
16 rt\ 6,285,050; 16 e. 6,773,700 ; 16/8,096,425; 16 0.7,321,826;
166. 6,706,300 ; 16 i. 8,155,400; 16j. 7,565,650. 17 a. 2,321,620 ; 176.8,269,-
744; 17 c. 5,344,464 ; 17 d. 7,069,096 ; 17 e. 7,618,704 ; 17/0,106,486;
17 g. 8,234,644 ; 17 ft. 7,542,896 ; 17 i. 9,172,768 ; 17 i 8,509,448.
id
442 ANSWERS
Page 28. — 18. 8155. 19. $7750. 20. $866.95. 21. $84.
22. 1.254.400 1b. 23. $3729.38. 24. $26,020. 25. 48,720 rd. 26. 747 mi.
27. $13.65.
Page 30. —2. 114, rem. 2. 3. 127, rem. 1 ft. 4. 51, rem. 70. 5 a. $ 101.87
rem. 10; 5 6. $67.91, rem. 2? ; 5 c. $50.93, rem. 3? ; 5 d. $40.75; 5e.$33.95
rem. 5? ; 5/. $29.10, rem. 50 ; 5a. 8 25.40. rem. If; bh. $22.63, rem. 80
57. $20.37, rem. 50; 5j. $18.52. rem. 30. 6 a. $339.17; 66. $226.11
rem. 10; 6c. $169.58, rem. 2f ; 6d. $ 135. 66, rem. if ; 6 e. $ 113.05, rem
4 - : 6 f. s 96.90, rem. if; 6g. $84.79, rem. 2<i ; 6 h. $ 75.37, rem. 1 f
6*. $67.83, rem. 40; 6 j. $61.66, rem. 80. 7 </. $ 104.53, rem. 1 ; 76. $69.69
7 c. $52.26, rem. 30; 1 d. $41.81, rem. 20 ; 7 e. $34.84, rem. 30 ; 7/". $29.86
rem.50 ; 7 g. $26.13, rem. 3 * : 7 h. $23.23; 1i. $20.90, rem. 7 f ; 7j.$19.00
rem. If. 8a. $195.04; 8 b. § 130.02. rem. 2 f ; 8c. $97.52; 8c?. $78.01
rem. 30; 8 e. $65.01, rem. 20 ; 8 f. $ 55.72, rem. if ; 8 a. $48.76; 8ft.$43.34
rem. 2 0; 8 i. $ 39.00, rem. 80; 8j. $ 35.46, rem. 20. 9 a. $ 360.46, rem. 1
9 6. $240.31; 9 c. $ 180.23, rem. 1 ? ; 9(7. $ 144.18, rem. 3 ; 9 c. $120.15
rem. 30; 9 7*. $102.99; dg. $90.11, rem. 50; 9h. $80.10, rem. 30
9 7. $72.09, rem. 30 ; 9 }. $65.53, rem. 10 0. 10a. $189.69; 10 6.8126.46
10c. $94.84, rem.20; 10(7. $75.87, rem. 30 ; 10 e. $03.23; 10 f. $54.10
rem. 50; 10a. $ 47.42, rem. 20 ; 10/t. 842.15, rem. 30 ; 10?. $37.93
rem. 80; 10/'. $34.48, rem. 10 0. 11a. 8148.67; 116. $ 99.11, rem. 1
lie. ft 74.33, rem. 20 ; lid. $59.46. rem. 4 ; lie. $49.55, rem. 40
11 f. $42.47, rem. 50; 11a. $37.16, rem. 60; 11 h. $33.03. rem. 70
Hi. s 29.73, rem. 4 ; 11 j. $27.03, rem. 10. 12a. $213.92 ; 12 6. $142.61
rem. 10; 12 c. $106.96; 12 d. $85.56, rem. 40; 12c. $71.30, rem. 40
12/. $61.12; 12a. 853.48; 12ft. $47.53, rem.70; 12i. $42.78, rem. 40
12'/'. $38.89, rem. 5 0. 13 a. $459.03, rem. 10; 136. $306.02, rem. 10
13c. $229.51, rem. 30; 13d. $ 183.01, rem. 20 ; 13e. $ 153.01, rem. 10
13 r. $131.15, rem. 20; Ug. $ 114.75. rem. 7 ; IZh. $ 102.00, rem. 7
13V. $91.80, rem. 70; 13 j. $83.46, rem. 10. 14 a. $423.56; 14 6. $282.37
rem. 10; 14c. $211.78; 14 d. $ 169.42, rem. 2 ; 14c. $ 141.18, rem. 40
14/. $121.01, rem. 50; 14 a. $105.89; 14 h. $ 94.12, rem. 4 ; 14 7. $84.71,
rem. 20 ; 14 j. $77.01, rem. 10.
Page 31. — 5. 118, rem. 15; 11, rem. 175. 6. 1892, rem. 5; 756, rem. 45;
473, rem. 5 ; 75, rem. 345. 7. 2512 ; 1004, rem. 40 ; 628; 100, rem. 240.
8. 4510 ; 1804 ; 1127, rem. 40 ; 180, rem. 200. 9. 3703, rem. 19 ; 1481, rem.
29 ; 925, rem. 79 ; 148, rem. 79. 10. 20,490, rem. 5 ; 8196, rem. 6 ; 5122,
rem. 45 ; 819, rem. 305. 11. 39,504, rem. ; 15,801, rem. 36 ; 0876, rem. 6 ;
1580. rem. 86. 12. 19,503, rem. 15 ; 7801. rem. 25; 4875, rem. 75 ; 780,
rem. 75. 13. 49,250; 19,700; 12.312. rem. 40; 1970.
2. 12. 3. 25. 4. 32, rem. 20. 5. 20, rem. 22. 6. 11, rem. 4. 7. 105,
rem. 9. 8. 209, rem. 3. 9. 41. 10. 32. 11. 32. 12. 26. 13. 31. 14. 43.
15. 77, rem. 3. 16. 84, rem. 70.
Page 32. — 17. 62. 18. 45. 19. 41, rem. 6. 20. 43. 21.41. 22. 41.
23. 60, rem. 55. 24. 85, rem. 57. 25. 81, rem. 51. 26. 96, rem. 44.
27. 77. rem. 3. 28. 79, rem. 16. 29. 84, rem. 70. 30. 84, rem. 14. 31. 81,
rem. 78. 32. 91, rem. 34. 33. 90, rem. 78. 34. 87, rem. 77. 35. 108,
rem. 12. 36. 145, rem. 14. 37. 83, rem. 29. 38. 100, rem. 38. 39. 82,
rem. 37. 40. 109. rem. 47. 44 a. 24,120, rem. 181; 44 b. 18.260, rem. 301 ;
ANSWERS 443
44 c. 9562, rem. 429; 44 d. 8528, rem. 117 ; 44 e. 747:1, rem. 196; 44/. 6669,
rem. 445 ; 44 y. 8193, rem. 64 ; 44 h. 7206, rem. 569 ; 44 i. 12,700, rem. 41 ;
44 / 7263, rem. 271. 45 u. 29,278, rem. 256; 45/.. -!2,165, rem. 350;
45 C. 11,607, rem. 428; 45 d. 10,351, rem. 702; 45 e. 9071, rem. 345;
45/. 7974, rem. :J44; 45 y. 9945, rem. 155 ; 45 h. 8747, rem. 701 ; 45 i. 15,416,
rem. 16; 45/. 8816, rem. 520. 46(7. 21,882, rem. 172; 46 b. 16,566, rem.
184; 46 c 8075, rem. 248; 46-/. 7736, rem. 600; 46 e. 0779, rem. 713;
46 f. 5959, rem. 892 ; 46 g. 7432, rem. 700; 46 h. 0537, rem. 859 ; 46 i. 1 1.521,
rem. 359 ; 46 /. 0589, rem. 338. 47 a. 32,258, rem. 197 ; 47 6. 24,421, rem.
307 ; 47 c. 12.788. rem. 653; 47 d. 11,405, rem. 351 ; 47 e. 9994. rem. 531 ;
47 f. 8785, rem. 901 ; 47 g. 10,957, rem. 268 ; 47 h. 9638, rem. 55; 47 i. 10,984,
rem. 485; 47/ 9713, rem. 771. 48 a. 35,913. rem. 188; 48 6. 27,188, rem.
320; 48 c. 14,237, rem. 660; 48 d. 12,697, rem. 540; 48 e. 11,127, rem. 17;
48 f. 9781, rem. 368; 48 g. 12,198, rem. 650; 48 h. 10,730, rem. 62;
48 i. 18,909, rem. 191; 48/ 10,814, rem. 412. 49 a. 29.045, rem. 40;
49 b. 21,988, rem. 348; 49 c. 11,514, rem. 636; 49 d. 10,209, rem. 198;
49 c. 8998, rem. 830; 49 f. 7910. rem. 000; 49</. 9865, rem. 015; 49 h. 8077,
rem. 831; 49 i. 15.292, rem. 472; 49/. 8740, rem. 160. 50 a. 25,411, rem.
250 ; 50 b. 19,238, rem. 152 ; 50 c. 10.074, rem. 380 ; 50 d. 8984, rem. 532 ;
50 e. 7873, rem. 259; 50 f. 6921, rem. 140; 50 g. 8031. rem. 545; 50 h. 7592,
rem. 380; 50 i. 13,379, rem. 493; 50/. 7652, rem. 124. 51 a. 29,492, rem.
169 ; 51 b. 22.327. rem. 207 ; 51 c. 11,092, rem. 233; 51 d. 10,427, rem. 359 ;
51 e. 9137, rem. 520 ; 51 f. 8032, rem. 537 ; 51 y. 10.017, rem. 612 ;
51 h. 8811, rem. 558 ; 51 i. 15,528, rem. 273 ; 51 / 8880, rem. 825. 52 a. 18,160,
rem. 165; 52 b. 13,748, rem. 253; 52 c 7199, rem. 521; 52 d. 6420, rem.
685; 52 e. 5625, rem. 555; 52/ 4940. rem. 181; 52 y. 6168, rem. 493;
52 /(. 5425. rem. 820; 52 i. 9561, rem. 490 ; 52 /. 5468, rem. 525. 53 a. 12,-
068, rem. 343; 53 b. 9137, rem. 69; 53 c. 4784, rem. 583; 53 d. 4267, rem.
181; 53 e. 3739. rem. 332; 53 f. 3287, rem. 159; 53 y. 4099, rem. 456;
53 h. 3005, rem.882 ; 53 i. 6354,'rem. 381 ; 53 j. 3634, rem. 307.
Page 33.— 2. S199.92. 3. 40 hr. 4. 16 da. 5. §289.50. 6. §63.58+.
7. B, 84800; A, 85075.
Page 34. — 9. 8270. 10. §217.05. 11. 1710 T. 12. S 5.30, gain.
13. §1660, total sale; §.87 + , average cost. 14. §5247.45.
Page 35.-2. f 22.50. 3. So. 4. 837.50. 5. §2.40. 6. -825. 7. §10.
Page 36. —1. 44. 2. 37. 3. 144. 4. 2832. 5. 88,894. 6. 546. 7. 8.
8. L30. 9. 62. 10. 75. 11. 24. 12. 47. 13. 109. 14. 80. 15. 72.
16. 137.
Page 40. — 2. 5, 5, 5. 3. 2, 3, 5, 7. 4. 3, 3, 5, 5. 5. 2 4 , 5 2 . 6. 3 8 , 5, 7.
7. 2, 3*, 163. 8. 2 2 , 3, 5, 7, 11. 9. 2, 3, 7 2 , 13. 10. 11, 221. 11. 2 5 , 3 2 , 5 2 .
12. 2 s , 5 s , 7, 11. 13. 3*, 5 2 ,29. 14. 2 10 , 5 s . 15. 2', 5, 101. 16. 3,58, 7) 37.
17. 2 4 , 3, 5 3 . 13.
Page 41.— 2. 21. 3. 18. 4. 12. 5. 4. 6. 3. 7. 14. 8. 22. 9. 7.
10. 17. 11. 144.
Page 42— 3. 144. 4. 1260. 5. 480. 6. 1400. 7. 420. 8. 96.
9. 830. 10. 570. 11. 720. 12. 255. 13. 5400. 14. 882. 15. 1080.
16. 2200.
444 ANSWERS
Page 43.— 2. 108. 3. 7. 4. 10. 5. 8. 6. 12. 7- 37. 8. 40 1b.
9. 80 bu.
Page 48. -2. ff 3. ff 4. ff 5. ff. 6. flfe 7. ff. 8. flfe
9. Iff 10. Hi- 11. f*t- 12 - HI- 13. fff 14. |ff. 15- iff
Page 49.-3. |. 4. &. 5. f 6. |. 7. r V 8. f 9. { s . 10. f
11. f 12. 11. 13. |. 14. T \. 15. |. 16. §. 17. |f. 18. r 8 T . 19. ih
20. ff 21. T \. 22. T V r . 23. T V 24. if. 25. ^V 26. ff 27. f
28. If 29. f*. 30. ^ 31. ft 32. ff. 33. ft. 34. |f 35. £,
36. ff. 37. ff 38. ff. 39. &. 40. f. 41. ff. 42. f. 43. f 44. if.
45. ii. 46. ff 47. ff
Page 50. — 2. *£. 3. -\ 5 -. 4. J-fi. 5. *&> 6 - W- 7 - W-
8. iff*. 9. «jp> 10. tffft. ii. i|i.. 12. iff a. 13. Jfffi. 14. ifff*.
15. HV-*- 16- ^tM-- 17- fit*. 18. HV*- I 9 - ^f 3 - 20. a^M.
21. iff*. 22. ^A 23. ijfi. 24. i^ 4 A 25. ^W. 27. 2 T V
28. 47if 29. 136f 30. 0ff 31. 32f. 32. 15 2 ". ; . 33. 22ff 34. 22.
35. 18f 36. 25. 37. 24. 38. 15. 39. 21ff 40. 23 T V 41. 24.
42. 10ff 43. 5 T 9 ? . 44. 61f 45. 157J§. 46. 45ff. 47. IlOf 48. 92&.
49. 183f 50. 112.
p 31yo ci 9 4 3 5 Q 10 3 2 A 1 5 1 2 T K _4_ 9_ K
X-dge JJ.. <S. j 2 , jj, iJ. v>- 13) xJ) is- •*• igi iJj if- <•»• 24' 54' 24"
6 4 8 5_ t _8_ 12-7 ft 8 1_0 1 Q J_B 15 9 Irt 2 3 5
• T5' T5' 3?' •■ 40' 40' ¥5* °- 24' 24' 24' "• 40' 45' ?0" 1W - 1?' 1 ?' 1 J"
U12 14 1 lO 24 _6 19 1» 2 3 1 Id. 12 22 19 IB 27 25 21
• T >5' TO' 50' i#s# '36' 3?' 36' x "* 5' 6' 6* trx ' 58' 58"' 58" ±0 ' 45' l3' 43*
1R16 6 1 I? 85 18 2? 10 1615 34 1Q2 4 4J.9
1D - 58"' 58' 58". Xl - 45" 42' 45" i0 - 48' 48' 48"" ±C '' 50 ' 60' 67J"
On 3 2 35 1 Ol 4 4 8 4 5
^ U - 35' 35' 35" ,4± - 60' 60' 60"
P,„o KO O 10 16 15 20 q 32 15 2.8 12 A 42 36 32 27
rd 5 e •*"■ *■ 40' 40' 40' 45 - "• 48' 48» 48"' 4?' *" 60' 50' 50' 67"
B 80 96 105 100 fi 250 216 425 7 191 385 5<J.4 Q 24 57 2.5.
°- T55' T50"' T55' T20" "■ 437' 4o0' ¥30" •• 6 93' 553' 653" ° - 50' 60' 50'
34 Q 36 126 81 154 Ifl 140 504 12 8 1 5 11 3 5 54 J_6_0 39
55" «• 535' 535' 535' 552- ±u - 360' 35~5» 35"0' 367' *■*•• 5c To' 95' 55"
19 42 180 243 4.'.4 1 J» 25' 44 J 3 2 66 14. 42 25 27 5_S IK JUL
i4 - 756' T..6' 736' 736- 10 ' 72 ' T5' ?5~1 72' A *' 60' 65' 60' 60" 10 " 336'
1«9 1134 944 Ifi 240 408 36 357 17 10 5_ 112 3 7 4 10 2 00
3T5' :!35~' S"36" 10, 845' '8~T5' 5T5' 840" A ' " 126' 126' l55» T55' xo ' ^25'
126 780 1Q 54 64 3S 67
553- "I~l~5- 1 *'- T35' 135' 1T5' 135'
Page 54.-2. 2^. 3.14. 4. lfff 5. 1 T \. 6. 2^. 7. 8^-
8. Iff. 9. lfff 10. 2fa 11. 2^. 12. 2ff- 13- Iff i4 - HU-
15. 2-^- 16. 1 T 4 55- 18- 80ff. 19. 3111. 20. 35,V 21. 44J ff .
22. 92ff 23. 115if. 24. 39^. 25. 70fff. 26. 86fff. 27. 224^.
28. 22if|. 29. 194^2- 30. 3011 mi. 31. 2~83ird. 32. 7 T %. 33. 56^.
Page 55.-2. ^. 3. f 4. &. 5. T ^. 6. ff- 7. A- 8 - iV-
9. f 10. T %\. 11. fff 12. Jft. 13. f^. 14. § 2 V 15. A- I 6 - I-
17. 1A. 18. T 5 ?
r5-
Page 56. —19. ff. 20. $6.
2. 4i. 3. 7ff 4. lOff. 5. »ff 6. 20ff- ?• 88ff 8. 39^-
9. 51 fff 10. 54ff. 11. 68fff. 12. 58ff. 13. 48i|i. 14. 24|i.
15. 16ff. 16. 20 T «^. 17. 37ff. 18. 17 T %. 19. 5^. 20. |8f
21. 5af
ANSWERS 445
Page 57. — 22. fa. 23. Increased ft. 24. $41ft; $434J.
25. 32£$lb. ; 6£§lb. 26. ■ 5J mi. ; L|mi. 27. 23?. 28. lfthr.;
ftbr.; Ifbr. 29. 84}f|* 30. 17ft T.
Page 60. — 2. 2§. 3. 6J. 4. (if. 5. Of. 6. 6§. 7. 252. 8 8i
9. 105. 10. 36. 11. 54. 12. 41. 13. 9. 14. 126. 15. 81. 16. 4ft.
17. 2f 18. 3f 19. 2ft. 20. ISf. 21. 3£. 22. Iff. 23. 9. 24. 74.
25. 15f. 26. 27ft. 27. 11£. 28. 51.
Page 61. — 1. 552,000 T. 2. 23,005,284. 3. ('.55,200 T.
Page 62. — 2. $328.50. 3. $7.62. 4. 9192.19. 5. $90.57.
6. $68.43. 7. 8181.86. 8. $125.38. 9. $10.11. 10. $97.95.
11. $41.40. 12. $13.28. 13. $61.43. 14. 966| mi. 15. $85.73.
16. $468.31. 17. $2.66; $2.34; $3.44. 18. $74,131.20.
Page 63. —19. $10.12.
2. 344. 3. 530. 4. 805. 5. 1926. 6. 5832. 7. 3620. 8. 0274.
9. 4950. 10. 9373. 11. 32.202. 12. 0391. 13. 15,420. 14. 45534.
15. 12,798. 16. 5173. 17. 13,063^. 18. 16,6724,. 19. 15,449.
Page 64. — 20. 129,8081- 21. 29,7434. 22. 80,726|. 23. 24,970|.
24. 19,929f. 25. 33,823'. 26. 31,644. 27. 31,2211 28. 84,458f.
29. 35,502. 30. 18,836ft. 31. 22,810. 32. 39,201 f. 33. 361,150|.
34. 410,050|. 35. 198 ft. 36. $13564,. 37. $56. 38. $1.20. 39. $18.
40. 190 T. 41. $15.86. 42. $109.50. 43. $170,276. 44. 14.898 lb.
Page 65. — 45. 25| da. 46. $881.25. 47. $25,521.88. 48. 2,187,000
gal. 49. $105.
Page 67. -2. §. 3. |. 4. ft. 5. f. 6. ft. 7. ft. 8. |. 9. ft.
10. U- 11. ft- 12- h 13. ft. 14. ft. 15. U. 16. |i. 17. i.
18. 915. 19 5- 20. 10f. 21. 12ft. 22. Oft. 23. 3. 24. 6f. 25. 5ft,
26. 5f. 27. 20j£. 28. 45|. 29. "25f|. 30. 10§§. 32. 118$. 33. 64f.
34. 108J. 35. 21251. 36. 745(1. 37. 2624;. 38. 9269 ,»ft. 39. 12,222.
40. 75ft. 41. 86,060. 42. 3430}-3|. 43. 1485f.
Page 68. — 44. 450f£ mi. 45. 228 ff T. 46. 90f| mi. 47. $841.50.
Page 70. -2. ft. 3. ft 4. ft\.' 5. T | -. 6. ft. ?• ft. 8. ,4ft.
9. ft. 10. ft. 11. jft. 12. ft. 13. ft. 14. ft. 15. -,-' : . 16. ft.
17. ft. 18. rfft. 19. f | T . 20. ft. 21. ftV- 22. ,f,. 23. T § 2 . 24. ft^
25. ^. 26. T f 5 . 27. rife. 28. ftV 30. f 31. f. 32. ft. 33. ft.
34. ft. 35. 2f. 36. 11. 37. lft. 38. Iff 39. J. 40. ft. 41. |f.
42. ft. 43. ft. 44. ff|. 45. 1$. 46. f. "47. ft.
Page 71.— 49. 3668^. 50. 1643f|. 51. 816f|. 52. 1375|f
53. 3306|J. 54. 666^$. 55. 309ft%. 56. 2733ft. 57. 883$ jj. 58. 787^.
Page 73. —1. 21. 2. 15. 3. 284,. 4. 45$. 5. 57^. 6. 51!,. 7. L93£.
8. 293J. 9. 1050. 10. 2058. 11. 1242. 12. 2624. 13. If 14. §. 15. l\.
16. U. 17. lft. 18. ;. 19. 2ft. 20. §. 21. |f 22. 11. 23. ff
446
ANSWERS
24. lfc. 25.
32. 6|. 33.
40. 2$. 41.
48. 16. 49.
56. f.
If 26. 1||. 27. B- 28. IH-
lfi. 34. 16f. 35. Hf. 36. 21f.
1J. 42. 8f 43. 8*. 44. Jf.
ft. 50. H. 51. 3ft. 52. 3.
29. ft 7 . . 30. ljf 31. 5$.
37. lift. 38. 6£. 39. 14f
45. 7f£$. 46. 341 L 47. 65.
53. f. 54. 3 T *ft- 55. fff
Page 74.
6. 9ft. 7.
pupils. 12.
— 1. 45 da. 2. If. 3. i\.
9ff. 8. 25gfhr. 9. 6125 gal.
82.20.
4. -55000; if 5. 85.4:,.
10. 204^f min. 11. 3090
Page 75.
9. i, 10.
16. "l8|.
-2. ft. 3. ft. 4. f 5. i.
24ff. 11. 7i|. 12. ft. 13
fi 91 7 7? R 1 5
0. Zj. <. T75 . B. 1 T5 .
. ft. 14. 1ft. 15. 33tf.
Page 76.
10. ft. 11
-3. ^. 4. A. 5. T fo. 6.
. J. 12. ftfc. 13. f|. 14.
|. 7. ft. 8. \\. 9. ft.
i
T5-
Page 77.
-3. 96. 4. 539. 5. 4J.
6. 13*. 7. 1|. 8. 1ft.
9. |f. 10. 7. 11. 81300. 12. 314,400. 13. 10,824 books. 14. 64,725
men. 15. -34500.
Page 78. —1. 35. 2
6. 3 505. 7. 24. 8. 149
12. 7f£mi. 13. 5365,250.
Increased ft.
16-
9.
14. 7
19
19
¥0"'
. ff. 4. 25ft. 5. 364.76.
1*0. 14 da. 11. 160 sq. rd.
15. 812,800. 16. $382.50.
Page 79.
22. 354 mi.
1421 r d.
17.
8bu. 18. 82480. 19. 8 43.20. 20. 857. 21. 86408.
24. 16 da. 25. 104 r 6 ft mi. ; 20f££ mi. 26. 30£ rd. ;
27. 306.50. 28. 86.30. 29. 108 bu. 30.
23. |f.
300
3 25-
Page 80. —31. 812,500. 32. 168 trees. 33. 300. 34. Ji 35. 5200
gain. 36. 8 70. 37. 812, room; 520, board. 38. 8 16,000, son's ;
514,000, daughter's. 39. 8900. 40. 131. 41. 8| hr. 42. 6 doz.
Page 81. — 43. 53289.60
47. 33 ft. 48. 540.
52. 867.500. 53. 4 A
44.
Q7
45. 8214.50. 46.
!'s; 8 4
55. 120 A.
120 lemons.
49. l T i ¥ . 50. 5 16,000, B's ; 8 42,000. A's. 51. AJ
54.
8 1000 gain ; 60 A
Page 82. — 56. 56000. 57. 7$ da. 58. 37800.
60. 5 2400. 61. 3 5 per barrel. 62. $145i. 63. 860.
59. 51120.
64. 2621 lb.
Page 83. —65. 8151.05.
69. 76$ ft. ;
123£f| hr.
Page 84.
7. 510 lots.
601 ft.
59j ft.
72. 18 hr. ; 8 hr.
-1. 31.6 T. 2.
8. 84 sheep.
66. a . 81 f gain. 67. 81075 gain. 68. 35.43.
67| ft. ; 109 ft. 70. 55 JJ. 71. 3906ft mi. ;
8.75. 3. 8120.
9. 8100.
4. 10. 5. f 6. 90.
Page 85. — 10. 818.
15. 8250. 16. $350.
11. 81.56. 12. 87.80. 18. 821. 14. $1.86.
17. 5 lb. 3 oz. 18. 82.17. 20. 32.35.
Page 86. — 21. $70.50. 22. $11. 23. \, ft, \. 25. 85 pupils. 26. 3450.
27. 128 cu. ft. 28. $1690. 29. 462 cu. in. 30. 8 50. 31. 56^. 32. 224 bu.
33. $770.
ANSWERS 117
Page 87. — 34. 6 da. 35. $880f. 36. $600. 37. 2000 bu. 38. §1200.
39. 100 da. 40. 34 head. 41. $1.60. 42. $81. 44. 35 and 6.
Page 88. -45. $6000. 46. 600 pupils. 47. 300 A.; 200 A. 48. $ 1G0.
49. §720. 50. 30. 51. 117. 52. 32. 53. 160 A. 54. 16.
55. A, $35; H, 8 4:.. 56. $12. 57. 65. 58. §4000.
Page 89. — 59. *4800. 60. §180. 61. f. 62. $60. 63. 12 da.
64. $96. 65. $1050. 66. §18. 67. £. 68. !, 2, ,'-',, |. 69. §300.
70. §4800. 71. 96 gal. 72. *2.40.
Page 93. — 1. .:JI. 2. .0075. 3.10.000075. 4.400.045. 5.6006.0066.
6 81)005 7. .074(1. 8. 000.000084. 9. 5,000,009.0000400. 10. .005095.
11. 8.0017. 12. .000125. 13. 896.00:101. 14. 1000.001. 15. 18,051.957.
16 07 0003. 17. .009864. 18. 2135.000032. 19. 1.000001. 20. 1,000,000.1.
21. 90,000.071. 22. 1830.11684. 23. 429,000.0040. 24. 7035.97.
25. 07,375.00035. 26. .05815. 27. 375.069.
Page 94. — 28. 419,863.023456. 29. 81.00921. 30. 2986.0298643.
31. 3020.00302. 32. 70.07. 33. 8000.008. 34. 645,000,000.000009.
Page 95. -1. \. 2. ft. 3. 5 fo. 4. ft. 5. J. 6. |. 7. ,fo.
8. jfa. 11. f. 12. ft. 13. f 14. rife. 15. |. 16. ft. 17. fc. 18. ft.
19. f. 20. J. 21. ft. 22. ft. 23. |. 24. i. 25. ft. 26. |f
Page 96. — 5. 111.55. 6. 187.64. 7. 66.51. 8. 105.9. 9. 168.507.
10. 19.079. 11. 179.992. 12. 103.422. 13. 471.0. 14. 1.260. 15. 2.2249.
16. 210.80. 17. 16.2. 18. 230.4909. 19. 25.823. 20. 45.3407. 21. 35.755.
Page 97. — 22. 11.1751. 23. 354.32685. 24. 189.61752. 25. 343.9706.
26 2. 28-2506. 27. 984.37936. 28. 298.09398. 29. 266.1172. 30. 454.4365.
31. 858.3827. 32. 47.0625 lb. 33. 23.0875 A. 34. 230.7177.
35. § 7154.95. 36. 866.9375 sq. ft.
Page 98. — 37. .5675. 38. .0466. 39. 2.4194. 40. 9.993. 41. 19.724.
42. 33.58. 43. 38.068. 44. 71.125. 45. 113.8909. 46. .101. 47. 100.192.
48. 26.483. 49. .101. 50. 4.3935. 51. 8.2415. 52. .01. 53. 13.5.
54. 18.021. 55. 99.91. 56. 58.625 yr. 57. 161.6.
Page 100. — 3. .055. 4. .0476. 5. .00081. 6. 3.24. 7. 14.7.
8. 80.214. 9. 4.25. 10. .486. 11. .011055. 12. 911.2. 13. 339.
14. 2759.1. 15. 719.44. 16. .0003424. 17. .030335. 18. 125.472.
19. 42.504. 20. .1183952. 21. 2.0736. 22. 42.6725. 23. 1203.03208.
24. 1.925. 25. 1.296. 26. 6. 27. 7.930. 28. 31.290. 29. 489.00.
30. 73.8. 31. 2.352. 32. 37.468. 33. 10.7325. 34. 59.2144.
35. 3135.65. 36. 12.054. 37. .0428790. 38. .12. 39. 28.65.
40. .000225. 41. 7.762536. 42. 2.75804. 43. 328.252. 44. .12048.
45. .760125. 46. 404.02944. 47. .01548. 48. 14.8:556. 49. .04936.
50. .000608. 51. 1.006008. 52. .00025. 53. 30.04812. 54. §3888.75.
55. §1200; §3000; §4800. 56. §20,530.25. 57. 472.44 in. ; 13.123+ yd.
Page 102. — 3. 12.02. 4. 141.6. 5. 6.07. 6. 1.5. 7. 5.06. 8. .1026.
9. .03002. 10. 20.03. 11. .084. 12. .061. 13. .87. 14. .009. 15. .076.
16. .450. 17. .15. 18. 50.6. 19. 10.25. 20. 3.002. 21. 200.3. 22. .904.
23. .0906. 24. .704. 25. 9.2945+. 26. .0027. 27. .084. 28. .0704.
448
ANSWERS
Page 105. — 3. 15. 4. 1.8. 5.
9. 560. 10. 1.24. 11. 12.5. 12.
16. 218. 17. .018. 18. .0055.
22. 115.8. 23. 16.4. 24. .0027.
28. 111,000. 29. 100. 30. .0002.
2.21. 6. 2.05.
.0207. 13. .8.
19. 1,000,000.
25. 17,500. 26.
31. 210. 32.
7. 1200. 8. .012
14. 745. 15. 9.7.
20. 2.5. 21. 2.44.
.033. 27. .000011.
.00137. 33. .00023.
34. 125. 35. 2(5.175. 36. 1700.82. 37. 14,096.4043.
Page 106. -
43. .009. 44.
--38. .001.
570.8108+.
39. 83 da.
45. 1.25.
40.
46.
1. .625. 2. .6. 3. .416|. 4. .3125.
8. .59375. 9. .4375. 10. .68.
§.405. 41.
18.5 sq. i - d.
5. .875.
88.5. 42. $18.25.
47. §20,000.
6. .44. 7. .5625.
Page 107. — 11. .6363+.
15. .6296+.
20. .7551+.
25. 1.5882+.
1. 19.97.
7. 96.309125.
12. 104.1635.
16. .4736+,
21. .5384+.
26. .75.
2. 1.4714. 3.
8. 16.2921.
13. 10. 5805 &.
12. .2903+.
17. 1.5714+.
22. .4545+.
13. .6164+.
18. 1.4705+.
23. .0166+.
14. .5471+.
19. 1.2962+.
24. .4615+.
214.065. 4. 4.9222.
9. 130.5808. 10
14. 105.4865. 15
5. 5.996+. 6. .02+.
2.31225. 11. 15.5225.
16. 245.494275.
9.7416|.
Page 109.— 2. §20. 3. $6.25. 4. §60. 5. §6.25. 6. §10.
8. §200. 9. §300. 10. §200. 11. §100. 12. $5.37*. 13. §1000.
7. §45.
14. § 16.
Page 110.
Page 111.
7. 150 yd.
12. 1600 lb.:
•15. §585. 16. §528. 17. $760.
— 2. 2560 1b. 3. 3000 1b.
8. 12,800 qt. 9. 800 lb.
750 bu. ; 2400 cakes. 13.
4. 60 1b. 5. 200 yd. 6. 260 yd.
10. 32 collars. 11. 60 bananas.
48 yd. ; 48 lb. 14. 260 lb.
Page 113. — 1. §5.02.
6. §0.20.
Page 114. — 7. §6.50.
11. §21.65.
Page 117. — 3. 248 oz. 4.
8. 1396 min. 9. 14,608 yd.
2. >4.22. 3. §1.75. 4. $.90. 5. §2.80-
8. $ 10.65.
9. $10.60.
10. $28.65.
13.
19.
23.
27.
30.
34.
14,500 lb.
3 mi. 244 id.
17 T. 7 cwt.
119 gal.
14 bu. .
$3.84.
pk.
1157 in. 5.
10. 27 in.
14. 508 pt. 15. 28 qt.
J yd. 20. 1 mi.l ft, 21.
24. 3 T. 95 lb. 25. 52 mi.
28. 221 rd. 4 yd. 1 ft. 6 in.
31. 2 mi. 44.5 rd. 32.
176 qt. 6. 56 pt. 7. 1825 lb.
11. 9600 lb. 12. 465 sec.
16. 43 pk. 18. 1 rd. 4 yd.
34 min. 3 sec. 22. 64 lb. 9 oz.
217 rd. 26. 346 bu. 3 pk. 1 qt.
29. 1 hr. 54 min. 35 sec.
min. 33.
16 hr. 4f
46 1 gal.
36. §26.88.
41. 3 mi. 40 rd.
7.
48
8.
3
IS'
Page 118. —35. §39.83.
39. 32 mi. 40. 44 mi. 80 rd
3. i. 4. T \- 5. f 6.
Page 119. — 9. f. 10. &. 11. f
15. .1. 16. .09. 17. .28. 18. §2.95.
2. 29 bu. 3 qt, 1 pt. 3. 40 T. 11 cwt. 4 lb
5. 16 wk. 1 da. 14 hr.
37. §41.60. 38. 1920 hr.
12.
13. \. 14. .7.
4. 21 yr. 239 da. 1 hr.
Page 120. —7. 22 mi. 194 rd
9. 8 hr. 43 min. 15 sec. 11
14. Lee, 58 yr. 2 mo. 20 da.;
3 mo. 8 da.
4 yd. 2 ft. 6 in.
(i yr. 9 mo. 27 da.
Grant, 42 yr. 11 mo.
8. 13 gal. 1 qt. 1 pt.
12. 15 yr. 16 da.
12 da.; dif. 15 yr.
ANSWERS ' I 19
Page 121. 2 10 gal. 3 qt. 1 pt. 3. 77 bu: 2 qt. 4. 152 T. 10 cwt.
7 lb. s oz. 5. 247 wk. 4 da. 6 hr. 6. 280 mi. 216 rd. 1 yd. 1 ft.
8. 3 wk. 5 da. it hr. 9. 26 gal. 3 qt. 1 pt. 10. 9 bu. 3 pk. 4 qt.
11. 5 yr. 7 mo. 18 da. 12. 14 T. 7 cwt. 24 lb. 13. $ 22. so. 14. 26 rd. 1 yd.
15. 10 packages.
Page 122. — 1. 6 min. 18 sec. 2. 32 bbl. 3. 55$jj bu. 4. 818.30.
5. 137 T. 12 cwt. 88 lb. 6. 50 ft. 8 in. 7. 110,340 gal. 8. 35.8605 T.
9. 3.451b.; $1.29+. 10. 54 mi. 200 rd. 11. 20,007 doz.
Page 126. —20. 272.25 sq. ft. 21. 2800 sq. rd. ; 17* A. 23. 24,300
sq.ft. 24. 7200 sq. in. 25. 10 A. 26. 260 sq. rd. ' 27. 26 r V r sq. rd.
28. 250,470 sq. ft. 29. 33| A.; $2025. 30. 8488 sq. ft. 31. $76.
32. $170.10. 33. 8 40,425.
Page 127. — 34. $6450. 35. 640 A. ; $54,400. 36. 17? bu. 37. 8 500;
$.16|. 38. $88.32. 39. 40 rd. 40. (3) 2]^ A. ' 41. (3) lf& A -
Page 128. — 1. §3. 2.8 8. 3. 140 sq. yd. 4. $29.55. 5.8 181.
Page 129. —5. 40 sq. in. 6. 36 sq. in. 7. 225 sq. in. 8. 432 sq. in.
9. 10 A.
Page 133. — 2. 028 sq. ft, 3. 120 ft. 4. $37.21. 5. 8 62.80.
6.72. 7. 288 cu. ft. 8. 166$ loads ; $ 41f . 9. 99 sq. f t. 10.810,000
cu. in.; 3506ft gal. 11. 8 462.
Page 134. — 13. f| ; 48 oz. 14. 153$ bu. 15. 4308*$ gal. 16.2800
sq. rd. larger. 17. 4 ft. 18. 04 sq. ft. ; 384 sq. ft.
Page 135. — 4. 24 bd. ft. 5. 9 bd. ft. 6. 45 bd. ft, 7. 48 bd. ft.
8. 100 bd. ft. 9. 360 bd. ft.
1. 8 200. 2. 812. 3. 8260.40. 4. 8400.
Page 136. —6. 868.75. 7. 87.17. 8. $13.44. 9. 82.52. 10. 88.06.
11. $67.20. 12.8 40.32. 13. $58.80. 14.8 26.88. 15.16 ft.
16. 2112 bd. ft. 17. 224 bd. ft. 18. $25.96.
Page 137. — 1. 20 cords. 2.8 440. 3.815. 4.8 44.80. 5. 12$
cords.
Page 138. —1. 540 tiles. 2. 289$ f t. 3. 277$ cu. yd. 4. $105.
5. 1,760,000 times. 6.8 24. 7. $327.68. 8. 718114 sal. 9.8108.
10. $16.33. 11. 50 rd. 12. 8 4.84. 13. 10 rd.
Page 139— 14. 1728 cakes. 15. 8116.07. 16. 792 bd. ft. : $23.76.
17. 15 A. 18. 2048 bu. 19. 4847^2 gal. 20.150 ft. 81. $96.80.
22. 40 ft. 23. 8 800.
Page 140. —25. 400 ft. ; 202 ft. : 292 ft. ; 26 ft. : 216 ft. 26. 1111$ sq.
yd. 87. 511$ sq. yd. 88. 4$ sq. yd. 29. 88| sq. yd. 80. $90£f
31. $137.78. 32.8.22. 33.8120.07. 34. ^ ft. 35. $23.11.
36. 2154f* gal. 37. $ 10.80. 38. 60 farms. 39. 3000 sods.
40. $62.40.
Page 144. — 2. 375. 3. 21.375. 4. 3.5. 5. 235.2. 6. 320.
7. 735. 8. $ 16.20.
Page 145 —9. 312. 10. $30.
HAM. COM PL. \UIT1I. — 29
450 * ANSWERS
Page 146— 12. ft 12. 13. 2 mo. 14. 18 da. 15. 1 A. 16. 5 A.
17. 32. 18. 28 horses. 19. 3 T \. 20. 9 1b. 21. ft If 22. 20.
23. ft 200. 24. §50. 25. 8 words. 26. ft .20. 27.280. 28. §3125.
29. ft 60. 30. 90 A. 31. §400. 32. $150. 33. 1209 people.
Page 147. — 2. §90. 3. §79.70. 4. §97.70. 5. §70.04. 6. §76.40.
7. ft 118.53. 8. §532.37. 9. § 135, commission ; §6615, net proceeds.
10. §1800. 11. §456.
Page 148. — 1. §12. 2. § 12. 3. § 3. 4. § 20. 5. §6. 6. $7.50.
7. § Hi. 8. §36. 9. §5.10. 10. §3. 11. §8. 12. §3.
1. §126.56. 2. §1145. 3. § 1248. 4. §174. 5. §1818. 6. §180.52.
7. §612.85. 8. §251.01. 9. §2283.36. 10. §6436.44. 11. $176.64.
12. §52.90.
Page 149. —14. §288. 15. §267.54. 16. §157.03. 17. $6.13.
18. §4.59. 19. 8 7.65. 20. ft 129.60. 21. §4080. 22. §1890.
23. §1353.75. 24. §53.55. 25. §64.60. 26. §291.60; §21.00 dif.
27. $641.25. 28. Both, §360.
Page 150.— 30. $6643. 31. §31.82. 32. §583.20 ; §648. 33. §203.74.
Page 152. — 2. §15. 3. §128. 4. §29.25. 5. §126. 6. §15.
7. §12. 8. §25.07. 9. §18. 10. §6.50. 11. §52.50. 12. §43.20.
13. §173.25. 14 §29.43. 15. §15.40.
Page 153. — 16. §3. 17. § 10. 18. §19.50. 19. $12.48. 20. $6.80.
21. §7. (19.
Page 154.— 2. §63; §363. 3. §40.83; §290.83. 4. §34; §194.
5. $4.17; 854.17. 6. §152; §952. 7. §3.81; §54.61. 8. §.64;
§16.64. 9. §3; §78. 10. §35; §455. 11. $2.63; §43.13. 12. §8.76;
§309.16. 13. §4.08; §104.08. 14. $27.50 ; §527.50. 15. 85; §1005.
16. §3.23; §63.83. 17. $69.44; $319.44. 18. $15.64; $91.44.
19. $251.25; §1751.25. 20. §24.85; §150.35. 21. §296.78; $1436.78.
22. §130.68; §1043.28. 23. §701.17; §3910.17. 24. §48.22; $682.72.
25. §297.
Page 155. — 1. §33. 2. §800. 3. $3. 4. 30 girls. 5. .0125;
.125; 1.375; .0625. 6. §12.50. 7. §64. 8. §22.50. 9. § .83. 10. §76.50.
11. 8.24. 12. §1.50.
Page 156. — 13. §1.32. 14. $45. 15. $2. 16. $420. 17. $20.
18. 20,UU0. 19. §325. 20. §3. 21. §9.
Page 158. — 7. §8.75. 4. §274.32.
Page 159. —1. 110,092. 2. 37. 3. 50.601. 4. 25.061; 125.5; 300.0002.
5. 4742. 6. .0119. 8. 8jL 10. 55.12. 11. $.30. 12. 2771 It.
13. 39 mi. 319 rd. 5 yd. 1 ft. 1 in.
Page 160. — 14. §132. 15. $2160. 16. §103.74. 17. 2f. 18. .1.
19. T ",V 20. §42. 21. 38104 mi. 22. 2| T. 23. §375.75. 24 §1080.
25. §"100. 26. §272.50. 27. 56 A.
Page 161. —28. §3892.50. 29. §16.80. 30. 1 da. 2 hr. 53 min. 20 sec.
31. §750. 32. §20.17. 33. -^-, -% 7 -, ^, §£. 34. §288. 35. 2||ff mi.
36. 048,045 cu. in. ; 375?} cu. ft, 37. 26$% gain.
ANSWERS J.".1
Page 162. — 38. 8; $1.60; $20.83$. 39. $5625. 40. $ 101.25.
41. $4.50. 42.2i)rd. 43. $ 707 J. 44. $55.08. 45. $2.
Page 163.- 47. 156f cu. yd. 48. 4 rd. 1 ft. <i in. 49. 28 rd. 52. §420.
53. 10%. 54. 62.5 da. 55. 24 rd. 56. 48 people.
Page 164. — 57. 999.00199+. 58. $16.88. 59. 177 rd. 4 yd. 10 in.
60. $8. 61. 2 1b. 8 oz. 62. $ 1900. 63. 1152 gal. 64. 8 30. 65.1}$.
66. John, 55 A.; James, 110 A. 67. ^ da. 68. .0072. 69. $46.20.
70. $.18.
Page 165. — 71. §193.33. 72. 50£$ A. 73. 1108.8 ft. 74. 81926.
75.814.40. 76. T %. 77.81800. 78. $111.30. 79- 320 rd. 80.2.5.
81. $247.50. 82. 840.50. 83. $20.97. 84. Latter by §75.
Page 169. — 1. 8564.35.
Page 170. — 2. 8257.53. 3. $365.80. 4. 8213.17. 5. $12,677.94.
6. 86.20.
Note. — In business, a half cent or more is usually counted as an addi-
tional cent.
Page 172. — 1. 8159.88. 2.8107.21. 3. $17.87.
Page 174. — 1. By Bal. 87.87. 2. By Bal. $97.68.
Page 178— 3. 920 qt. 4. 218 pt. 5. 247 hr. 6. 2640 rd.
7. 22, 160 min. 8. 33,585 lb. 9. 8765 T f£§ hr. 10. 924 in. 11. 423$ ft.
12. 31,451.35 ft. 13. 184 oz. 15. 28 oz. 16. 112 pt. 17. 174,960 in.
18. 280 rd. 19. 16 cwt. 80 lb. 20. 202 da. 18 hr. 40 min. 21. 128$ ft.
22. 5 da. 6 hr. 23. 255 min. 24. 9768 ft.
Page 179. — 2. 56 gal. 1 qt. 3. 20 bbl. 7 gal. 2 qt. 4. 365 bu. 3 pk.
5. 10 da. 10 hr. 6. 16 rd. 1 ft. 4 in. 7. 94 min. 35 sec. 8. 1 mi. 58 rd.
4 yd. 1 ft. 9. 2 leagues. 10. 4 T. 2020 lb. 11. 45 rd. 3 yd. 2 ft. 7 in.
13. .02|| mi. 14. .30025 T. 16. & ft. ; ^ yd. 17. f lb.
Page 180. — 18. f. 19. .87+. 20. 820.36. 21. $6. 22. Ill boxes.
23. 150 qt. 24. 81.08. 25. 113,796 ft. 26. 8263.01. 27. 191 bottles.
28. 9500 packages. 29. 81.65.
Page 181. — 30. 12,300 ft. 31. 429.14 lb. 32. 13 mi. 252 rd. 2 ft.
33. 84.
Page 183. — 1. $182.49. 2. 8130.27. 3. 8111.75. 4. 8170.33.
5. 813(1.85. 6. $73.34. 7. 8151.85. 8. 829.20. 10. 850. 11. 8131.62.
12. $481.64. 13. $23.94. 14. $425.89. 15. 8148.42. 16. $20.22.
17.8143.79. 19. £ 160 7s. 7.72d. 20. 500 marks. 21. 144 lire.
Page 184. — 3. 133 bu. 2 pk. 7 qt. 4. 97 hr. 52 min. 39 sec.
5. 5 hr. 46 min. 6. 11 yr. 9 mo. 3 \vk. 5 da.
Page 185. — 8. 57 yr. 2 mo. 8 da. 9. 84 vr. 9 mo. 10 da.
10. 42 vr. 11 nm. 12 da'. 11. £2006 18s. 9d. 12. £ 1 14 10s. Id. ;
£62 4.s\ lOd. ; £ 192 9s. id. 13. 32 hr. 40 min. 14. 5 T. 4 cwt. 15 lb.
Page 186. —3. 52 bu. 8 baskets. 4. 8 yd. 251 in. 5. 10,800 ft.
6. 42 gal. 2 qt. 1 pt.
452 ANSWERS
Page 187. — "I. Widow, $9733.65; children. sr,840.19. 8. $8.40.
9. 5940 turns. 10. $13.89. 11. 96 mi. 284 id. 4 yd. 12. 165 T. 13. $9.30.
14. 240 ft. 11 in. ; 8 lb. 10 oz. ; 27 bu. 2 pk. 5 qt. 15. $115.07. 16. $74.31.
Page 188. — 1. $73.88. 2. $305.50. 3. f*. 4. 10 bbl. 5. $166.88.
6. 10,000. 7. 2| doz. 8. 04.435. 9. 5~4.41^ mi. 10. 401.44 mi.
11. $.0311+.
Page 189. — 12. 55.07+ mi. per hour. 13. 191.75 1b. 14. 522.708^.
15. 1071f bu. 16. If. 17. $675. 18. $47.50. 19. .977 5 | 7 min.
20. 84.5375 mi. 21. 28.75 rd. 22. 1876 1b. 23. 7.
Page 190. — 24. $3.85. 25. 100 bbl. 26. $1000. 27. 3.74624625.
28. 4131. 29. .138. 30. $8.50. 31. 136 books. 32. $60,000.
33. $2018.75.
Page 191. — 34. $350. 35. $24,000. 36. $9.10. 37. $6000.
38. 2111.101. 39. 849.404. 40. 6270.835. 41. 101,260.269.
42. 1034.25478.
Page 193. — 1. 480 sq. in. 2. 075 sq. rd. 3. .78125 A. 4. 60 sq. rd.
Page 194. — 5. 174,240 sq. ft. 6. 108 sq. in. 7. 176,418 sq. in.
8. 256,000 sq. rd. 9. 11 A. 40 sq. rd. 10. 11 sq. ft, 11. 294,030 sq. ft.
12. 94.31625 A.; $29,473.83.
Page 200.— 2. 246 sq. ft. 3. 144.9 sq. ft. 4. 2916 sq. ft.
5. 218.5 sq. ft. 6. 517 sq. ft. 72 sq. in. 7. 045 sq. ft. 8. 27 sq. ft.
9. 54 sq. yd. 10. 180 sq. yd. 11. 2400 sq. yd. 12. First 16 times as large
as second ; first \ as large as second. 13. $30.94.
Page 201. — 14. bj% A. 15. 30 rd. 16. 3 lots.
2. $40. 3. $7.89. 4. 89 bundles.
Page 202. — 5. $54.15. 6. $305.21. 7. $231.22.
1. $134.40. 2. 12,960 tiles.
Page 203. — 3. 40 sq. in. 4. 360 slates. 5. 240 slates ; 450 1b.
6. 5760 slates; $88. 7. 157 bundles. 8. $172.38. 9. $105.
10. 30 squares.
Page 204. — 2. 15 rolls. 3. $18.
Page 205. — 5. 55 T ^ yd. ; 73^ yd. 6. 5, 5, 6, 6, 6. 7. 7, 6, 7, 8, 8.
8. $14.67. 9. $45.84.
Page 207. — 2. 9 sq. ft. 3. 140 sq. ft. 4. 60 sq. ft. 5. 150 sq. ft.
6. 450 sq. ft.
Page 208.— 1. 52* A. 2. 28.4062 A. 3. 75. 4. 25 rd. 5. 7 rd.
6. 80 rd.
2. 7 sq. ft. 42 sq. in.
Page 209. — 3. 10 rd. 4. 30 rd.
2. 440 sq. ft. 3. 308 sq. ft.
Page 210. — 1. 47.124 ft. 2. »78.54 ft. 3. 188.496 ft. 4. 33.5104 ft.
5. 39.27 ft. 6. 78.54 yd. 7. 126.7112 ft. 8. 7.854 ft. 9. 82.7282 ft.
10. 4 ft, 11. 40 yd.
ANSWERS 453
Page 211. —1. 78.54 sq.rd. 2. 314.16 sq. rd. 3. 254.4696 sq. rd.
4. 1017.8784 sq. ft. 5. 314.16 sq. in. 6. 1256.64 sq. in. 7. 5026.56 sq. rd.
8. 490.875 sq. rd. 9. 6026.56 sq. ft. 10. .78 sq. ft. 11. 3.14 sq. rd.
12. 706.86 sq. in. 13. 962.11 sq. yd. 14. 314. Hi sq. ft. 15. 78.54 sq. yd.
16. 49,087.5 sq. ft. 17. 11,309.70 sq. yd. 18. 1134.11 sq. yd. 19. 21.46
sq. ft. 20. $50.27.
Page 213. — 1. 432 sq. ft. 2. 68 sq. ft. 8 sq. in. 3. 118 sq. ft.
4. 412 sq. ft. 5. 170 sq. ft, 6. 334 sq. ft. 7. 706 sq. ft. 8. 96 sq. in.
9. 6 sq. ft. 10. 24 sq. ft. 11. 6 sq. ft. 73$ sq. in. 12. 8 sq. ft. 24
sq. in. 13. 88 J sq. yd. 14. 8 sq. ft. 109| sq. in.
Page 214. —2. 250 cu. ft. 3. 684 cu. ft. 4. 800 en. ft. 5. 9 cu. ft.
6. 4 cu. ft. 1248 cu. in. 7. 5 cu. yd.
Page 215.— 8. 213$ loads. 9. 1728 cakes. 10. 448 en. ft. 11. 5 C.
12. 22£C. 13. 64 boxes. 14. 22} C. ; S41.63. 15. 23 $ C. ; $ 42.78.
16. 28f C. ; 8 52.03. 17. 18| C. ; $34.69. 18. 106$ loads.
Page 216. — 19. 56 \ C. 20. J 7 . 21. 1417$ cu. in. 22. 111.475 cu. in.
23. 126.6 cu. in. 24. 1621| cu. in. 25. 10,936f cu. in. 26. 1595| cu. in.
27. 189 cu. in. 28. First; 1440 cu. in. 29. 124|| cu. in.; 451 T 9 g cu. in.
Page 218.— 2. 15 bd. ft. 3. 24 bd. ft. 4. 32 bd. ft. 5. 180 bd. ft.
6. 96 bd. ft. 7. 36 bd. ft. 8. 150 bd. ft. 9. 144 bd. ft. 10. 432 bd. ft.
11. 360bd. ft. 12. 2000 bd. ft. 13. 720 bd. ft. 14. 1200 bd. ft.
15. 240 bd. ft. 16. $21.60.
Page 219. —1. 8419.06. 2. 828. 3. 847.25. 4. $26.25. 5. 839.38.
6. 8 10.50. 7. 960 bd. ft. 8. 12,800 bd. ft. 9. 3500 bd. ft.
10. 4800 bd. ft. 11. 1600 bd. ft. 12. 1500 bd. ft. 13. 8000 bd. ft.
14. 21.000 bd. ft. 15. 8750 bd. ft. 16. 1200 bd. ft, 17. 16,666f bd. ft.
18. 18,200 bd. ft. 19. 24,000 bd. ft. 20. 7000 bd. ft. 21. 3600 bd. ft.
22. 10.000 bd. ft.
Page 220. — 23. 8837. 24. 81488. 25. 8118.56.
1. 448 cords. 2. 805f| cu. yd.
Page 221. — 3. lOOff loads; 503if loads; 201U loads. 4. 86429.63.
5. 488|cu. yd. 6. 37,842 bricks. 7. 41,328 bricks. 8. 8364.80.
Page 222.— 1. 753.984 sq. in. 2. 1413.72 sq. in. 3. 62.832 sq. ft.
4. 20.944 sq. ft. 5. 8.3776 sq. ft. 6. 282.744 sq. ft. 7. 678.5856 sq. ft.
8. 155.5092 sq. ft.
Page 223. — 10. 3180.87 cu. ft. 11. 14,726.25 cu. ft. 12. 4021.248
cu. ft. 13. 4712.4 cu. ft. 14. 100.5312 cu. ft. 15. 6031.872 cu. ft.
16. 12.5664 cu. ft. 17. 235.62 cu. ft.
1. 803.56+ bu. 2. 520.71+ bu. 3. 448.83 gal. 4. 119.36+ bbl.
5. 4.94+ bbl.
Page 224.
3. 29,687|lb,
— 1. 7500 lb.
4. 53331
; 120 cu. 1
bbl. 5.
Et.
072 1
2.
JU.
l|f§ T. hard
; i
i 9 T.
soft.
Page 225.
10. 81|fT.;
1. 50. 2
sq. in. ; 5832 i
— 6. 5
38? T.
. 15 ft.
3U. in.
3.
6.
7. 471.24
3280 tiles.
918 boxes.
^gal.
4.
7.
8. 84.823.2 1b.
WO sq. in.; 80 sq.
216 cakes.
9.
in.
1958.4 bu.
5. 1944
454 ANSWERS
Page 226. — 8. 264 sq. in. 9. $57,600. 10. $800. 11. $7168.
12. $398.47. 13. 63,910 bricks. 14. $553.96.
Page 227. — 15. 1.32+ ft. 16. 294f i T. 17. 72 yd. 18. $113.47.
19. $24.70. .20. Grading $798; $52,800. 21. 60°. 22. 78°.
Page 228. — 23. 75°; 30°. 24. 76£° ; 90°. . 25. 80|°. 26. A, 20.75
acres ; B, 26.93+ acres ; R. R. 2.32- acres. 27. 100 revolutions.
28. 26.6455 knots per hour. 29. $35.47. 30. 21 double rolls.
31. 345 sq. ft. 32. $100.45.
Page 229. — 33. $424.96. 34. 15 ft. 35. $96.36. 36. $80.
37 S666.67. 38. $ 468. 39. $826.20. 40. 2714.34 cu. ft. 41.960
blocks ; 55.2 T. 42. 16.755 bbl. 43. 3 T. 44. $72.50.
Page 230. — 45. $663.51. 46. $67.50. 47. $322.22.
48. $167.68. 49. 2025 1b. 50. $166.59. 51. 4 ft. 52. $712.80.
53. 1200 ft.
Page 232. — 2. $8. 3. 30 sheep. 4. 60 yr. 5. 300. 6. 24.
7. 100.
Page 233. — 11. $11. 12. $37.50. 13. 16 yr. 14. 30 books.
15.80?. 16. $100. 17. $9000. 18. $7.50. 19. 60 bu. 20. $96.
Page 234. — 21. $80,000. 22. $480. 23. $2400. 24. $1000.
25. $24. 26. $1.50. 27. 750 girls ; 450 boys. 28. 12 yr. 29. $24.
30. $300. 31. 30^. 32. $5000. 33. 30 girls. 34. $8000.
Page 235— 35. $75. 36. $3200. 37. 18 mi. first day. 38. $20.
39. 18; 54. 40. $150. 41. $3000. 42. $45,000. 43. $540.
44. $5000; $3000. 45. £, J.
Page 236. — 46. T %. 47. 2| da. 48. 12 da. 49. 6 da. 50. 4 da.
51. ft 48. 52. Harness, $20; sleigh, $40; horse, $160. 53. $2.
54. $2700. 55. $ da. 56. Second, $4200; third, $3000.
Page 240. — 2. 13.12. 3. 11.34. 4. 28.93. 5. 57. 6. 39.72.
7. 20.45. 8. 236.16. 9. 5.4. 10. 25. 11. ±. 12. 15. 13. f.
14. 147. 15. 112. 16. 180 girls ; 220 boys. 17. $7.50. 18. 25.2 T.
19. $2422.50. 20. $16.50. 21. The same. 22. $73.95*. 23. $1750.
24. $292,702.50.
Page 241. — 25. $2.60; $1.73. 26. $220, John; $198, James; $132,
Henry. 27. 240 A., wheat ; 108 A., corn ; 168 A., oats and grass ; 84 A., rem.
Page 242.— 2. 40%. 3. 374%. 4. 60%. 5. 75%. 6. 10%.
7. 62i%. 8. 15%. 9. 60%." 10. 6±%. 11. 16|%. 12. 4£%-
13. 14|%. 14. 200%. 15. 250%. 16. 360%. 17. 6%. 18. 90%.
19. 15%. 20. 21%. 21. 4%. 22. 42%. 23. 56J%. 24. 280%.
25. 7%. 26. 7i%- 27. 6%.
Page 244. — 3. $30. 4. $30. 5. $728. 6. $72. 7. $127.50.
8. $1200. 9. $11.25. 10. $8. 11. $56.80. 12. $32.32. 13. $4128.
14. $56. 15. 1)20 men. 16. $1800. 17. $4700. 18. $50. 19. 600 pupils.
ANSWERS 455
Page 245. — 2. 300. 3. 200. 4. 400. 5. 200. 6. 40. 7. 30. 8. 900.
9. 300. 10. $3. 11. $2. 12. $3500. 13. $1000. 14. *4700.
15. 10 sq. mi.
Page 246. — 2. 100. 3. 300. 4. 72. 5. 45. 6. 400. 7. 400. 8. 500.
9. 5. 10. $40. 11. -sC.O. 12. $384. 13. $2000; $800 ; $10,000.
14. 549 pupils. 15. $20. 16. J,j. 17. 1000. 18. $100 loss. 19. 16,000.
Page 247. — 1. 100. 2. 20. 3. 1. 4. $10; 12'%. 5. $854.40.
6. 50%; 10. 7. 300%; 331% g. 50%. 9. 10fi%. 10." 340. 11. $4000.
12. $500. 13. if.
Page 248—14. 85%. 15. $14,175. 16. 120. 17. $5.
18. 37', %; 2J%; 87.', ";. 19. $ 200, horse ;$ 100, buggy. 20. $12,000.
21. 2 t. 22. 5%. 23. $0000; $7200. 24.' $ 144 ; $150; $180.
25. 121%. 26. $1732.50. 27. 60^.
Page 249. — 28. $1620. 29. $9250. 30. $ 2760, increase. 31. 74f£%.
Page 250. — 2. $15. 3. $32. 4. $17.25. 5. $43.13. 6. $74.76.
7. $37.10. 8. $24.28. 9. $ 110.10. 10. $260. 11. $2.40. 12. $540.
Page 251. — 2. 5% gain. 3. 20% gain. 4. 4% loss. 5. 35% loss.
6. 30% gain. 7. 33£%. 8. 20%. 9. 22?%. ' 10. J % gain.
Page 252.— 2. $500. 3. $300. 4. $500. 5. *2450. 6. $8400.
7. $2500. 8. $35. 9. $225. 11. .s200. 12. S600. 13. $880.
14. $960. 15. $1020. 16. $600. 17. $11,600. 18. $1250. 19. $1.25.
20. $1.25.
Page 253. — 22. $360. 23. 60'.'. 24. 2h%.
1. $4143.75. 2. 56}%. 3. 20% gain. 4. $88. 5. $3000. 6. $10,000.
7. $575. 8. $800.
Page 254. — 9. 25%,. 10. $30. 11. $7500. 12. 4% gain. 13. 12*%.
14. $2.70. 15. $153. 16. $8000. 17. $150.
Page 256. — 2. $68.75. 3. $78.18. 4. $365.75. 5. $5484.38.
6. 1%. -7. 3%. 8. 2%. 9 2£%.
Page 257. — 10. $09,040.80. 11. $152.50. 12. S624. 13. $616.
14. $1250. 15. $2268. 16. $3280. 17. $966.67. 18. $3795.
19. $19,750. 20. $1220. 21. $520 gained. 22. 2%.
1. $16,200. 2. S 7787.50.
Page 258. — 3. $1010; $1753. 4. $6085.25. 5. $306.
6. $ 1595.80 agent's commission ; 10^ % gain. 7. $210.75.
1. Com., $54; net pro.. $609.76. 2. Rate, 4°/ ; net pro., $524.
3. Gross sales, $ 1000 ; c .,$200. 4. Rate, 10% ; net pro., $504.
5. Gross sales, $600; rate, 8%. 6. Cora., $40; net pro., $960.
7. Grosssales, $6700; expenses, none. 8. Rate, 2%; net pro., $2404.
Page 259. — 9. $1851.35. 10. 360 A. 11. $131.60; $6580. 12. $1680.
13. $23,294.12.
Page 260 — 2. |%;$24. 3. $58.50. 4. 2J %;$ 2.50 per $ 100.
5. |%. 6. 23^. 7. $1900. 8. 1|%.
456 ANSWERS
Page 261. — 9. $53.25. 10. « 12,000. 11. 1%. 12. \\% 13. $42.
14. Farmer's loss, $210; company's loss, $3290. 15. Company's loss,
$17,415 ; owner's loss, $0585. 16. $4800. 17. First, $ 18,000 ; 'second,
$21,600 ; third, $32,400 ; fourth, $36,000.
Page 263. — 1. $119.80. 2. 33^|f%. 3. 48 5 2 5 %. 4. $821.60.
5. $268.45. 6. 45 yr. 7. $800.40. 8. $450.55.
Page 264. —1. $3; trade. 2. Time and cash. 3. 10%; trade. 4. $125.
Page 265. — 5. 30%; trade. 6. Trade. 7. Time, $3.75 ; cash, $7.50.
9. $15. 10. $30. 12. $10. 13. $1350. 14. 10% cash.
Page 266. — 4. $56.70. 5. $72. 6. $144. 7. $20.60. 8. $8.
9. $238.03. 10. $83.79. 11. $223.13. 12. $273.60. 13. $3249.
Page 267. — 14. $175. 15. $231.60. 16. 24£ if %. 17. $1.
18. First, $2.50. 19. No difference. 20. Third is $208 better than first
and $ 135 better than second. 21. $70.13. 22. $1525.95. 23. 66.8%.
Page 268. — 2. $547.20.
Page 269. — 3. $564.48. 4. $39.34.
Page 271. —2. Taxes, $38 ; poll tax, $3. 3. Taxes, $236 ; poll tax,
$4. 4. 3£ mills, rate. 5. Est. val., $937.50 ; assessed val., $ 750. 6. Est.
val., $222,900; assessed val., $148,600. 7. 18 mills ; $225. 8. $19.75.
9. $8125. 10. 8 22,460; $561.50.
Page 272. — 11. $74.70. 12. $65.60. 13. $4680. 14. $21,173.64.
Page 273. — 1. $198. 2. $2500. 3. 8 6000. 4. $144.
Page 274. — 5. $800. 6. $728. 7. 8 2687.50. 8. 3860 1b.
9. $2665.60. 10. $4.3045 per dozen. 11. 8 11,629.75. 12. $3289.49.
13. $2673. 14. 1600 yd.
Page 276 — 2. $37.50. 3. $198.24. 4. $45.50. 5. $40.
6. $19.50. 7. 888. 8. 8129. 9. $11. 10. $41.80. 11. $56.55.
Page 277. — 2. $1675.80. 3. 8 1226.55. 4. $448. 5. $573.80.
6. 83585. 7. 81439.3(1. 8. $215.12. 9. 82287.81. 10. $2952.50.
11. $225.50. 12. 81342.33. 13. $122.66. 14. $4056.50. 15. $210.43.
16. 8342.91. 17. $233.14. 18. $1087.75. 19. $167.67. 20. $656.43.
21. $58.68. 22. $314.61. 23. $110.89. 24. $53.79. 25. $56.28.
Page 278. — 2. $2.75. 3. $1.43. 4. $2.58. 5. $2.97. 6. $7.13.
7. $4.20. 9. *2.50. 10. $3.80. 11. $13.02. 12. $2.50. 13. $10.06.
14. $5.31. 15. $2.50. 16. $2.10.
Page 279. — 17. $18. 18. $15.75. 19. $42.75. 20. $60.50.
21. $57.60. 22. s 13.05. 23. $57.96. 24. $12.64. 25. ft 44.96.
26. $13.40. 27. $9.99. 28. ft 30. 29. $34.88. 30. $31.10. 31. $59.50.
32. $31.36. 33. $25.34. 34. $33.80. 35. $8.85. 36. $2.25.
37. $13. 38. $2.67. 39 $3.01. 41. $16.25. 42. ft 10.83. 43. $9.60.
44. 8 7.13. 45. $30.75. 46 $2.90. 47. 811. 48. $.51.
49. $3.07. 50. ft 7.05. 51. $13.24. 52.86.71. 53. $11.23. 54. $80.03.
55. $171.81. 56. $9.90.
ANSWERS 4. r >7
Page 280. — 57. $1500.75. 58. $121.60. 59. $207.80. 61. 827.54.
62. $74.30. 63. $74.80. 64. $160.05. 65. $306.10. 66. $144.15.
67. $202.0(1. 68. $20.50.
Page 281 . — 2. $115.48. 3. $91. 4. $65.33. 5. $101.28. 6. $4.58.
7. $9. 8. $19.85. 9. $114.49. 10. $73.02. 11. $77.25. 12. $169.30.
13. $337.40. 14. $219.75. 15. $247.50. 16. $2.40. 17. $4.77.
18. $6.18. 19. $10.50. 20. $47.60. 21. $60.50. 22. $38.25.
23. $145.13. 24. $10.66. 25. $19.99.
Page 282. — 26. $31.93. 27. $36.56. 28. $1312.19. 29. $966.60.
30. s 151.59. 31. $896.13. 32. $415.28. 33. $38.48. 34. $207.81.
35. $2246.44. 36. $7.16. 37. $18.27. 38. $12.33. 39. $22.25.
40. $77.19. 41. $9.61. 42. $1423.15. 43. $705.53. 44. $4044. 45. $134.
Page 283. — 2. $1000. 3. $680. 4. $1200. 5. $1375.
2. 0°/ .
Page 284. -3. 5%. 4. 5%. 5. 6%.
2. 3 yr. 4 mo. 3. 20 yr. ; 16f yr. ; 12} yr. 4. 40 yr. 5. Oct. 13, 1902.
1. $31.44. 2. 2 yr. 6 mo. 3. 3 yr. 6 mo.
Page 285. — 4. 41%. 5. $434. 6. $180. 7. $275. 8.6%.
9 In 3 vr. 4 urn. 10. $354.31. 11. $ 88.20. 12. Jan. 19, 1906. 13. $600.
14. July 1, 1906. 15. $86,700. 16. $3025.
Page 286. — 2. $116.19. 3. $8292.48.
Page 287.-4. $4954.20; $495.42.
2. $10.26. 3. $35.29. 4. $60.49. 5. $41.34. 6. $11.10. 7. $20.68.
'8. $299.18. 9. $116.99. 10. $88.70.
Page 288. — 11. $396.40. 12. $-655,912.33.
Page 289. —2. $103.81. 3. $10.45. 4. $27.41.
Page 290. — 2. $1.52. 3. $157.99. 4. $80.52. 5. $1056.07.
Page 291. — 6. $206.04. 7. $160.60. 8. $909.93. 9. $1233.35.
Page 292. — 1. $1526.74. 2. $14,233.12. 3. $5642.40. 4. $1324.90.
Page 298. — 1. $103. 2. $255.44. 3. $630.42. 4. $367.44.
5. $129.13. 6. $1219.50. 7. $316.60.
Page 301. — 2. $669.20. 3. $295.65. 4. $110.17.
Page 302. — 5. $355.64. 6. $9.57.
1. $ 439.70. 2. $2380.37.
Page 306. —1. $663.15. 2. $399.70, balance.
Page 309. — 2 *203. 3. $200. 4. 30;6<*.
1. Aug. 1. 2. Aug. 22. 3. Nov. 13. 4. Oct. 10. 5. April 2.
.May 17. 7. June^lO. 8. Sept. 5.
tj
458 ANSWERS
Page 310. —9. April 30. 10. July 10. 11. Nov. 10. 12. June 23.
13. Oct. 14. 14. April 2. 15. July 23. 16. Dec. 7. 17. Sept. 3.
18. Oct. 24.
3. Nov. 8; $2.25; $2.28. 4. Aug. 12 ; $ 5.67 ; $5.76. 5. Oct. 2 ;$ 11.50;
$11.63. 6. May 8; $15.63; $15.76. 7. April 24 ;$ 4.75; $4.85.
8. July 17 ; $4.53; $4.59. 9. Oct, 1 ;$ 8.50 ;$ 8.78. 10. March 11;
$6.46; $6.53. 11. May 16 ;$ 15.75 ;$ 15.97.
Page 312. —1. $2487.92 ; $2487.50. 2. $ 1568.89 ;$ 1568.63.
3. $ 1545.70 ;$ 1545.44. 4. $ 4521.82 ; 14521.06.
Page 313. — 5. $ 4455 ; $ 4454.25. 6. $ 3468.50 ; $ 3467.92. 7. $ 3537.87 ;
$3537.27. 8. $1000. 9. $ 1207.04 ;$ 1206.83. 10. $200; $200.
11. $209. 12. $ 205.62 ;$ 205.59. 14. $224.33. 15. $ 259.21 ;$ 259.10.
Page 314. — 16. 9%. 17. 12%. 18. $ 158.76 ;$ 158.73. 19. $1828.04;
$1827.74. 20. $1217.22; $1217.01. 21. $ 2467.50 ;$ 2467.08. 22.$6375.81;
$6374.98. 23. $1000. 24. $2800.
Page 319. — 1. $ 550.80. 2. $ 1575, face ; $ 2.63, exch. 3. $ 1.30.
4. 15?. 5. $ 1078.92.
Page 320. —7. Face, $989.01 ; exch., $.99. 8. Exch., $ 12.75; com.,
$275; face, $12,747.75. 9. $791.98. 10. $311.85.
Page 322. — 1. 60 da.; 61 da. 2. $1183 ; $1182.77.
Page 323. — 4. $ 11,016.75 ; $11,014.59. 5. $591.80 ; $591.70.
6. $3809.50. 7. $2456.25; $2455.83.
Page 326.-2. $6067.50. 3. $8987. 4. $12,142. 5. $11,355.
Page 327. —6. $19,936. 7. $8656.25. 8. $20,680.63. 9. $66,187.50.
10. $37,250. 12. 240 shares. 13. 8 shares. 14. 100 shares. 15. 40
shares. 16. 44 shares ; surplus $ 14.50. 17. 61+ shares ; surplus $54,625.
Page 328.-3. $825. 4. $1892.63. 5. $2153.13. 6. $22,137.50.
7. $7453.13. 8. $26,784. 9. $13,785. 10. $23,490.
Page 329. —12. 120 shares. 13. 112 shares. 14. 40 shares. 15. 168
shares. 16. 200 shares.
1. $60. 2. $80. 5.7%. 6.5%. 7. $2000. 8. $16,000;
$800.
Page 330.-10. 4%. 11.6%. 12. 4J%. 18.6$%. 14.6}%.
15.8%. 16. Less. 18. $1185. 19. Former, T \% better. 20. 12J %.
Page 331. — 21. $800,000. 22. $288. 24. $24,025. 25. $7288.75.
26. $36,556.88. 27. $9446.63.
Page 333. — 1. $15,581.25. 2.50. 3. $30,000. 4. $2000.
5. $35,437.50. 6. 12%.
Page 334. - 7. 6\ %. 8. 6 %. 9. $ 1487.50. 10. 6£ % ; 5| % ;
5%; 4J%; 4%. 11. $4000. 12. $1584.48. 13. $400. 14. 500%.
16. 5 r y/ .
Page 335. —1. $00.48. 2 $635.04. 3. $8870. 4. 30|%.
5. §520.52; $520.42. 6. $36.40. 7. $4000. p458
ANSWERS 459
Page 336.-8. 19$%. 9.23+%. 10. $212.50. 11. 6|%. 12. 16^%.
13. $585. 14. $1282.50; commission, $67.50. 15. $480. 16. $7100.
Page 338. — 1. 5. 2. .25. 3. 3. 4. &. 5. 8. 6. .4. 7. fa. 8. \.
9. 100. 10. 3. 11. 0. 12. 320.
Page 339.-3. 18f. 4. 15. 5. 10. 6. 5. 7. 5. 8. 10. 9. I. 10. .5.
11. 2.5. 12. 90. 14. 30 da. 15. $ 18. 16. 50 da.
Page 340. — 17. 8. 18. $1021.88. 19. $4.17. 20. 4 oz.
21. 41 j mi. 22. $40. 23. $70.55. 24. 4 da. 25. 37 Z sec. 26. 3875
letters. 27. 180 ft. 28. 31,250 bricks.
Page 341. —29. 36|| mi. 30. 30 men. 31. 24 men. 32. $60.30.
33. 38 cars. 34. $236.25.
Page 342. — 2. $81, man; $ 54, first boy ; $27, second boy. 3. Expenses,
$5200; net savings, $10,400. 4. $672,000; $28,000; $42,000; $03,000.
5. Lake, $1000; railroad, $ 1800. 6. $60,000; $230,000.
Page 343.-2. A's, $2520 ; B's, $5700 ; C's, $4320. 3. M's, $2520;
N's, $3024 ; K's. $2010. 4. E's, $1000 ; F\s, $857i ; G"s, $2142f
5. $1329 r 6 r ; $920 T \. 6. Smith's, $3225; Jones's, $4300; Brown's, $2150.
Page 344.-8. $3348 to N ; $40">8 to M. 9. R's, $1053.75 ;
S's, $1496.25. 10. A's, if ; B's, |§.
Page 345. —1. $600; $400. 3. $3200; $4000. 4. $15,000; $12,000.
5. 50%; 25%; 150%. 6. Frank, $ 18 ; Henry, $15.
Page 346. —7. Walter's, $1.20; Philip's, $1.50. 8.40%. 9. $5000.
10. Brown's, $1050; Long's, $1200. 11. $375. 12. $450.
13. Moore's, $1200; Silven's, $1800; Rogers's, $1500. 14. $45 per month.
15. 77 da. 16. To A, $2700 ; to B, $2400.
Page 347. — 17. $48. 18. 6 da. 19. 9 lots. 20. 48 persons.
21. 100 A. 22. 600 per dozen. 23. 24 da.- 24. $3200 ; $3000.
25. $1250. 26. $500; $000.
Page 348.-27. F, $1440; E, $900. 28. $500; 6%. 29. 500, 5000.
30. 15 da. 31. $32. 32. $320. 33. $1000. 34. 20,000 bu. 35. $5.
36. Claim, $1000 ; loss, $400. 37. & greater.
Page 352. —2. 15° ; 1 hr. ; 60° W. has earlier time. 3. 45° ; 3 hr. ;
120° W. 4. 60° ; 4 hr. ; 15° W. 5. 30° ; 2 hr. ; 30° E. 6. 105° ; 7 hr.;
75° W. 7. 150°; 10 hr. ; 120° W. 8. 120°; 8 hr. ; 90° W. 9. 105°; 7 hr. ;
30° E. 10. 105° ; 7 hr. ; 45° \V. 11. 30° ; 2 hr. ; 15° E.
Page 353. — 12. 10 hr. 8 min.20 sec. a.m. 13. 11 hr. 17 min. 45£ sec. a.m.
14. 8 hr. 58 min. 29£ sec. a.m. 15. 5 hr. 17 min. 33 sec. p.m. 16. 5 hr.
58 min. \§ sec. p.m. 17. 6 hr. 36 min. 49f sec. a.m. 18. hr. 1 min.
46^ sec. p.m. 19. 12 min. 11| sec. p.m. 20. 11 hr. 48 min. 4 sec. a.m.
21. 5 hr. 47 min. 7^ sec. 22. N.Y., 3 hr. 36 min. 71 sec. p.m. ; Cape
Town. 9 hr. 40 min. 3 sec. p.m. 23. 2 hr. 39 min. 35/. sec.
460 . ANSWERS
Page 354. — 26 Honolulu, 6 hr. 28 min. 37| sec. a.m. ; Berlin, 5 hr.
53 min. 34j| sec. p.m. ; San Francisco, 8 hr. 60 min. 17£ sec. a.m. ; Lon-
don, 4 hr. 59 min. 3G| sec. p.m.
1. 17 da. 2. 21 da.
Page 355. —1.3 hr.
Page 357.-5. 160 A. 6. 80 A. 7. 40 A. 8. 320 rd. 9. 480 rd.
Page 359.-8. 225. 9. 256. 10. 324. 11. 484. 12. 625. 13. 42.25.
14. .5625. 15. 2J$. 16. 272£. 17. 225 sq. in. 18. 625 sq. ft.
19. 256 sq. yd. 20. 72£ sq. ft. 21. 25 sq. in. 22. 72.25 sq. in.
23. 100 sq. yd. 24. 32^ sq. yd. 25. 36 sq. in. 26. 512 cu. in. 27. 8 cu. ft.
28. 34§£ cu. ft. 29. 42| cu. ft. 30. 56# 6 cu. ft. 31. 1876 J y cu. ft.
Page 361. —2. 15. 3. 24. 4. 21. 5. 14. 6. 40. 7. 28. 8. 36.
9. 16. 10. 48. 11. 35. 12. 42. 13. 24. 14. 56. 15. 18. 16. 20.
17. 72. 18. 25. 19. 32. 20. 64. 21. 27.
Page 364.-5. 22. 6. 24. 7. 26. 8. 31. 9. 33. 10. 43. 11. 51.
12. 63. 13. |. 14. if. 15. |f. 16. .5. 17. .15. 18. 2.1. 19. 2.5.
20. .25. 21. J7.74+. 22. 22.91+. 23. 34. 24. 66. 25. 14.5. 26. 13.35+.
27. 122. 28. 163. 29. 369. 30. 4.56. 31. 9.84. 32. 13|. 33. 23.25.
34. 56.5. 35. 75.12+. 36. .92. 37. .114. 38. .02+. 39. .66+. 40. 335.
Page 366.-2. 32.45+ in. 3. 3 ft. 4. 32 ft. 5. 25 yd. 6. 44.72+ in.
7. 8.94+ ft. 8. 1300 sq. rd. 9. 22.36+ ft. 10. 42.42 mi. 11. 21.21+ rd.
12. 1.41+ ft. 13. 45 ft.
Page 368. — 1. 370.8 sq. ft. 2. 28.28 rd. 3. 97.425 sq. in.
Page 370. — 1. 250 sq. ft. 2. 17^ sq. ft. 3. 108.3852 sq. ft.
4. 613.3974 sq. in.
Page 371. — 1. 169.6464 sq. ft. 2. 768 sq. ft, 3. 161.9 sq. ft.
4. 700 sq.ft. 5. 854.17. 6. *58.81.
Page 372. —1. 452.3904 sq. in. 2. 28.2744 sq. in. 3. 530.9304 sq. in.
4. 50.2656 sq. in. 5. •$17.45.
Page 373. — 1. 128 cu. in. 2. 2 cu. ft, 3. 185.4 T. 4. 190.8522 cu. ft.
Page 374. —1. 201.0624 cu. in. 2. 9 times. 3. 1440 cu. in.
4. 3456 cu. in. 5. 256 cu. ft. 6. 600 cu. ft. 7. 284+ bu.
Page 375. — 8. 4071.5136 cu. ft. 9. 2ff T. 10. 24 in. 11. 10 ft.
1. 904.7808 cu. in. 2. 268.0832 cu. in. 3. 288.696+ cu. in.
Page 377. — 1. 75 ft. 2. 80 rd. 3. 12 ft. 4. 2^. 5. 8 ft.
Page 378.-6. $8.67. 7. 21ft.
1. b
Page 379. - 2. T \. 3. ^f ,, 4. fr 6. 20 ft, ; 13$ ft, ; 8 ft, 7. 16 ft.
9. 200 cu. in. 10. 179^ bu.
ANSWERS 461
Page 380. —4. 172.51b. 5. C'.o.js lb. 6. 1501b. 7 1046.20+ lb.
8. 650.25 lb. 9. 64.375 lb. 10. -",7..") lb. 11. 556.25 lb. 12. 168.75 lb.
13.849.3751b. 14. 1206.251b. 15. 151b. 16. 556.25 1b. 17. 700.25 11..
18. 181.251b. 19.487.51b. 20.455.6251b. 21.489.3751b. 22. 10K.75 lb.
23. 114.375 lb.
Page 381. —1. 27.08 bbl. 2. 1000 sq. in. 3. 10.39+ in. in diam.
4. 3.4724 ini. 5. 23 A. 6. 0257 i bu. 7. 672.3024 cu. in. 8. 4.18+ sq.ft.
9. 18.1:5- gal. 10. 1 to 2|.
Page 385— 5. 1.363 in. ; 13.00025 m. 6. 1 Km. 088 m. 7. 177.102 mi.
Page 386.-8. 37.5 in. 9. 67.5906 mi. 10. 6 Mm. ; 5 Dm.; 1 m.
11. (1400 Km.
Page 388.-2. s 72.00. 3. 87.50. 4. 24 sq.m. 5. 24 steres.
Page 390. — 1. 937.51. 2. 188,4961.; 188.496 M. T. 3. $159.23.
4. 3,000,000 1. 5. 7.2 111. 6. 9000 bottles.
Page 391. — 7. 838(50. 8. 154.56 pf. ; $.3678. 9. 1000 1.
10. 89 T l T Kg. 11. 1.3591+ cu. m. 12. 3941.4 Kg. 13. 169.164 m.
14. 863.75. 15. 811, 812.50, cost of land; 81100, cost of fence.
16. 000,000 tiles. 17. 140.400 M. T. 18. 4080 cu. m. 19. 8 1.20.
Page 392. — 1. 86.10. 2. 80.33. 3. 84.04. 4. 82.31. 5. 83.30.
6. 811.74. 7. 811.93. 8. 812.06. 9. 8 13.48. 10.828.10. 11. s4.21.
12. 812.01. 13. 81.93. 14. 8 3.80.
Page 393. — 16. 1 to 7.0+. 17. 1 to 3.8+. 18. 1 to 5.27+.
19. 1 to 5.05*. 20. 1 to 5.8+. 21. 1 to 8.7+. 22. 1 to 13.7+.
Page 394. — 23. 1 to 5.08+. 24. 1 to 7.69+. 25. 1 to 8.8+. 26. Too
wide: lto7.1+. 27 Too wide ; 1 to 10.09+. 28. Too wide; 1 to 8.2+.
29. Too wide ; 1 to 10.3+. 30. 1 to 6.04+. 31. 1 to 4.06+.
Page 396. — 1. 234 lb. ; 01.2 lb. ; 52.25 lb. ; 184 lb. 2. .75 lb. ; 1.14 lb. ;
10.875 1b. 3. 39. tons. 4. 285 tons. 5.812.25. 6.8 23.16. 7.824.33.
8. 36.4 tons.
Page 397. — 1. 300 1b. of each. 2.82.60. 3. $62. 4.8274.
5. § 1 140.
Page 398—6. 8131.80. 7. $1. 8. $1.14. 9. 814.14.
10. Materials: .000+ per gal. ; 8 .115 per tree. Labor: 8 .01+ per gal. ; 8.13
per tree.
Page 399—11. 8.00. 12. 86.25; 81.28. 13. 830.19. 14. 300 gal. ;
7* lb.
Page 400. — 15. $9.80. 16. $.072- ; 84.28. 17. $58.33. 18. 22$%.
Page 401. — 1. $1007.17. 2. 3.1416 sq. rd. 3. 25.42+ in.; 42.42+ in.
4. $18. 5. 324 cu. ft. ; 10.8 min. 6. $200.72.
Page 402. — 7. times. 8. $300. 9. 8 2.88. 10. f 22.62.
11. 7.0086 cu. ft, ; 21.2058 cu. ft. ; 14.1372 cu. ft. 12. :! mills.
13 104.4 mi. 14. 00 lb. 15. $2000.
462 ANSWERS
Page 403. — 16. Cow, $50; horse, §140. 17. 150.7968 in.
18. 4ii rd. ; 160 rd. 19. BJ %. 20. 33$%. 21. 17.32 in. 22. 22.36 in.
23. Width, i» ft.; height, 6 it. 24. 1 to 3f. 25. 8 834.40.
26. 813,453.
Page 404. —27. 8 1080. 28. 82000. 29. 8181.43; 8181.40.
30. $ 175. 31. 20 A. 32. A's, 320 A. ; B's, 480 A. ; C's, 600 A.
33. 4.31+%. 34. 8630.40; 8630.29; 8639.86; 8639.75. 35. 81119.57;
81118.87.
Page 405.— 36. 852.17. 37. 85$ ft. 38. 848.72. 39. 8 570.
40. 73.24+ ft. 41. 25. 42. 887.14. 43. 5 02+%.
Page 406. —1. 60%. 2. 28?; $2.80. 3. 830,000. 4. 100%.
5. 6|hr. 6. 6i%. 7. $1.65. 8. 31ft %•
Page 407. — 10. M's, 8900; N's, 8800. 11. 240 A. 12. 20%.
13. 10 ft. 15. 300 crates. 16. 8120. 17. 40.15%. 18. 836. 19. 1081%.
20. 6i%.
Page 408. — 21. 64 cu. in. 22. 4%. 23. A's, 8450 ; B's, 8600 ; C's,
$1050. 24. $1400; 8 1500. 25. 84500. 26. 8 25,000. 27. a. 37$; b. f.
28.31%. 29. 5456 yd. ; 8342.4 yd. 30.8393.67. 31.8 2510. 32.32,768.
33. a. 4* ; b. 10^.
Page 409. —34. * 3600. 35. 288 cakes. 36.11,011,011.000011.
37. 3f. 38. 1121.112. 39. 8250,000. 40. 40 A. 41. .4, .375, .28,
.5625, .75, .15625. 42. 8 419.85. 43. 24, 15, f, 2. 44. 84800.
45. 84.32. 46. 12$ mills.
Page 410. —47. 60 da., 8986.67 ; 61 da., 8986.45. 48. 20^ sections ;
8 2880. 49. $429.84. 50. 8 505. 51. 86.75. 52. July 1, 1906. 53. 8 450.
54. 8 705. 55. A, T 2 3 ; B, £.&. 56. 8353. 57. 8168.48.
Page 411. —58. 60 shares. 59. 20 vr. ; 16f yr. ; 121 yr. 60. 8396.
61. 8 313.04. 62. 8 .921 per yard. 63. 8 336. 64. 14011b. 65. 819.20.
66. 8 2949.60. 67. 21 mi.
Page 412.— 68. 16.568 rd. 69. 75%. 70. 83$%. 71. 6283.2 sq. ft.
72. $39.90. 73. 13 J*. 74. 2 yr. 4 mo. 75. 1000 cu. ft. 76. $39.25.
77. 8887.50.
Page 413. — 78. Second, $15. 79. 8120; 20%. 80. 6%. 81. 81000.
82. s 38.40. 83. First, $13.50. 84. $6. 85. A, 1^ 5 ; B, f. 86.82125.60.
87. 143%.
Page 414. — 88. 827,984. 89. 1020.021. 90. 413.875 1b. 91. 8500 loss.
92. 8 4500 ; 81275. 93. Latter by § %. 94.48.55. 95.8 800. 96. 15 mills.
97. 630 Kl. ; 630,000 Kg.
Page 415. — 98. 10,000 A. 99. Nitrogen, 1890; oxygen. 7110.
100. $140.14 101. 200 men. 102 4V&. 103. .0125. 104. 8 643.40.
105. $345.60. 106. 50 rd. 107. 8 6000.
ANSWERS 463
Page 416. — 108. 836,720. 109. Ninety-nine hundred-thousandths.
110. Hi ft. 111. $4. 112. $2842.50. 113. $144. 114. $601.50.
115. 4 yr. 9 mo. 22 da. 116. $540. 117. 9720 shingles.
Page 417. — 118. 34%. 119. §640. 120. $120. 121. $6000.
122. 162.5 1b. 123. 79.5. 124. 47!'|'/ . 125. §300. 126. §320.-."..
127. $ 3554.40 ;$ 3655. 128. 500 ft.
Page 418. — 129. $ 1979.74 ;$ 1979.35. 130. $513.44. 131. $404.22.
132. $2937; $2936.33. 133. $1(370.14. 134. §2(57.48.
Page 419. — 135. 11 hr. 33 min. 57 sec. a.m. 136. $384. 137. $1550,
gain. 138. $558.33. 139. 57.7269 Kl. 140. 39 Kg. 141. $3213.42.
142. Dec. 7, 9 hr. 51 min. sec. a.m. 143. 1 mi. 22:! rd. (i in.
Page 420. — 144. 120. 145. 8 2:171.85. 146. 70. 147. $2298.75.
148. §1032.82. 149. $83,144.91. 150. §125,000. 151. §2090.
Page 422. — 2. $285. 3. $450. 4. $350. 5. $441.51. 6. $100.
7. $ 11.25. 8. The former is § 5 Letter.
Page 424.-4. §120.25. 5. $64.45+. 6. $ 1247. GO. 7. $1228.96.
8. §148.95. 9. $76.43+. 10. $1948. 11. 12,921 fr. 87.5 c. 12. $ 2GG.48.
13. $2697.50. 14. §457.90. 15. 6489 M. 80 pf. 16. £309 lis. lid. ' .
17. §1478. G4; 1260 francs.
Page 425. — 2. $90. 3. §140. 4. 54 rd.
Page 431. — 1. 14. 2. 16. 3. 24. 4. 27. 5. 32. 6.45. 7. 72.
8. 98. 9. 123. 10. 1. 11. I'i. 12. r \. 13. .9. 14. 2.5. 15. 3.4.
16. 6.1, 17. .15. 18. 'lh
Page 432. — 1. 18 in. 2. 441 sq. in. 3. 13 ft. 4. 47.55+ in. 5. 8 ft.
6. 6 in. 7. 12 in. 8. 17 ft. 9. 5 to 7. 10. 18 ft. high. 11. The base
of the bin is 172.8+ in. square ; the height of the bin is 86.4+ in.
= =• ASSESSED FOR FAii- pENA UTY
X%0 E OK S ON -H -J| N ? S U ON ThVeO.HTH
™1_l -ncrease to jo ceN the gEvENTH daY
oA Y AND TO
OVERDUE.
LD 21-100m-8,'34
F
YL 22560
918 !r^>
THE UNIVERSITY OF CALIFORNIA LIBRARY
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