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1 06642837
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BUSINESS MATHEMATICS
A TEXTBOOK
By
EDWARD I. EDGERTON, B.S.
Instructor of Mathematics in the Wm. L. Dickinson High
School. Jersey City. N. J., and Examiner in Mathematics for
the New York Sute Board of Regents
And
WALLACE E.^mRTHOLOMEW
State Specialist in Commercial Education, New York State
Board of Regents
Th^fd 'PHfitrn^ , *
NEW YORK
THE RONALD PRESS COMPANY
i
1922
\'
i
' "^' NKVV YORK
^^'JhLlC LIBRARY
--^''^"••. r.h:wox AND
Copyright, 192 1, by
The Ronald Press Company
AU Rights Reserved
• « • »
•
. « •
•• • ••
. • ,
« • • •
%% , • • • • •
' *:• •
PREFACE
The course in applied mathematics outlined in this book
is much more advanced and thorough than the usual course
in commercial arithmetic. The attempt has been made to
construct a practical course which will contain all the essen-
tial mathematical knowledge required in a business career,
either as employee, manager, or employer.
The fact that the field has been covered in this text both
more intensively and more comprehensively than it has yet
been covered in other texts, and the added fact that the
material gathered together has stood the test of six years'
experience in the teaching of large and varied classes, seem
sufficient warrant for its publication.
The work is adapted not only for use in the classroom but
also as a reference manual for those actively engaged in
business life. Thus it will be found a practical guide for
young employees who wish through private study to master
the fundamental mathematics involved in "running a busi-
ness." The tabulations, forms, illustrative examples, charts,
logarithmic applications, and simple rules, are all applicable
^ to the financial and other mathematical problems which
^ business presents. Lack of knowledge of this side of a busi-
- ness, or inability to work out its mathematics, often results
in haphazard guessing where accurate and careful calcula-
tions are required.
h The material has been submitted to the criticism of many
prominent business men and specialists in the commercial
field, from whom valuable suggestions and criticisms have
iii
• ■• j.'fii? 1:1 '"t '■t->:r. .::-.iTn
. L":' •■ "i • VloL-^c B>>
.. ^...'j :lt Preisenting F:il:?."
^fatistics.** and Kcis-er".-
.. •.. .:i::ig.' and such period:-
t \tw York Times, and
xi'i-wiedge the helpful sug-
'. :vi;'>. Ph.D., Principal of tht
« , ^ .. .iui Frank Tibbetts, Head
V ... i 'he same school, as well
I '.litriuseript by Hamlet P.
X . I'.is character is prepared,
.•..' o oreop in undiscovered.
1 1 .'i\ and glad to acknowl-
. . .xis.s :ho reader may care to
••.% v'i«» I. KOCKKTON
S \ WK v.. H.\KTH0L0MEW
CONTENTS
Chapter Page
I Sales and Profits Statistics 1
II Profits Based on Sales 13
III Pay.Roll Calculations 22
IV Interest 34
V Depreciation 50
VI Insurance 56
VII Exchange. 77
VIII Taxes 102
IX Interest on Bank Accounts 118
X Building and Loan Associations .... 127
XI Graphical Representation 135
XII Short Methods and Checks 172
XIII Averages, Simple and Weighted .... 191
XIV The Progressions 198
XV Logarithms 203
XVI Commercial Applications of Logarithms 223
XVII The Slide Rule 238
XVIII Denominate Numbers 250
XIX Practical Measurements 262
Appendix — Tables and Formulas 289
FORMS
Form Page
1. Depreciation Chart, showing Rate of Depreciation Computed
^ According to the Straight- Line Method 51
% Express Money Order 80
^. Sight Draft 83
4. Bank Draft 87
5. Bankers' Bill of Exchange 93
6. Letter of Credit 97
7. Travelers' Check 99
8. Circle Chart Showing Distribution of Income 139
9. Rectangle Chart 110
10. A Variation of the Straight-Line Graph 143
11. Curve Graph 146
12. Comparative Curves 151
13. Period Chart 152
14. Composite Chart Showing Relation Between Income and
Outgo 153
15. Chart Showing Component Parts 157
16. Correlative and Cumulative Curves 159
17. Map Chart 161
18. Frequency Curves Showing Changes in Costs 162
19. The SUde Rule 239
w\
BUSINESS MATHEMATICS
CHAPTER I
SALES AND PROFITS STATISTICS
1. Use of Comparative Records. — Every business is carried
on for the purpose of selling something at a profit. The things
sold may be the goods of the retailer, wholesaler, or manu-
facturer, or the services of an advertising concern, a bank, an
insurance company, or public utility corporation. No matter
what the kind of organization or enterprise, comparative
figures of its sales and profits play an important part in its
management.
The figures should be compiled regularly and tabulated to
show increases or decreases covering corresponding periods of
time. The tabulations may be made to show the trend of
sales or profits by departments, lines of goods, or salesmen, or
they may be worked out for the business as a whole. The
computations involved seldom require more than the use of
simple arithmetic and percentage calculations.
2. Comparative Percentage Figures. — In addition to show-
ing sales and profits by quantities, it is advantageous to
reduce these quantities to percentage figures, and show in-
creases or decreases in this way. In finding percentages the
proper adjustment should be made in the last decimal figure,
up or down as the case requires.
1
^ BUSINESS MATHEMATICS
Illustrative Example. The total sales for a given month are $12,896.37,
of which sales amounting to $1,596.32 are credited to salesman A. Find,
correct to two decimal places, the per cent of total sales made by A.
Solution i
$1,596.32 ^ $12,896.37 = .12378, or 12.378%
Adjusted to two decimals the last figure is 12.38%.
3. Tabulated Sales Records. — The tabulations most com-
monly used are those covering the daily, weekly, and monthly
sales by departments or by salesmen. Columns may be ruled
to show increases or decreases both by quantities and per-
centages or the cumulative figures to date, or the average
figures reduced to percentages. The possible combinations
are numerous and are determined by the kind of business.
4. Monthly Sales by Departments. — The figures in the
tabulation shown below represent the monthly department
sales totals for the year. The totals at the foot of each column
give the annual sales of all departments.
Supply the missing totals at the foot of each column and in
the last column.
Monthly Record op Sales by Departments
Months
Dbpt. 1
Dept. 2
Dept. 3
Dept. 4
Dept. 5
Total
Jan
Feb
$1,327.76
1.094.25
1.213.06
1,164.36
1,086.79
987.57
975.64
976.66
1.234.43
1.321.26
1.109.60
1.437.87
$2,976.47
3.462.45
3.126.87
3.879.65
2.580.56
2.784.66
2.564.43
2.376.65
3.107.52
3.245.63
2.895.64
3.256.76
$4,567.34
4.809.67
5.003.29
4.782.54
4.347.83
4.476.21
4.357.81
4.235.68
4.460.34
4.532.25
4.463.38
4.987.56
$3,091.23
2,890.67
2.901.56
2.875.69
2.784.35
2.783.52
2.569.58
2.467.92
2,984.62
3.012.56
2.982.29
3.248.90
$5,684.92
5.892.43
6.045.42
5.587.67
5.469.57
6.472.31
5.216.49
6.127.65
6.436.63
5.542.32
5.463.3F
5.873.24
Mar
Aor
•
May
June
July
Auff
Sept
Oct
Nov
bee
Total
SALES AND PROFITS STATISTICS
6. Daily Record of Sales by Departments. — The figures in
the following tabulation give the daily sales in each depart-
ment and are designed to show total daily and weekly sales
both by departments and for the business as a whole. In the
last line is shown the percentage of each day's sales to the
grand total of sales, and in the last column the percentage of
each department's weekly sales to the grand total of sales is
diown.
Supply the required totals, and compute tiie percentages in
each case as shown below.
Comparative Daily Record of Sales by Departments
Week beginnii
IK. . . .
'r*
Deft.
MON.
•
Tubs.
Wbd.
Thurs.
Pri.
Sat.
Total
%of
Grand
Total
I
II....
Ill
IV
V
VI
Sl.321.76
987.56
1,276.41
i. 107.63
987.64
3.217.78
$1,210.34
1.324.65
2.109.72
2.371.10
1.097.47
3.241.36
$1,040.30
2.134.67
1.967.73
1.986.78
897.74
3.269.91
$1,243.65
1.432.46
1.563.27
2.083.52
1.107.27
2.987.21
$1,121.09
1.253.54
2.136.76
2.376.82
985.71
3.009.21
$1,324.65
1.421.78
1.038.08
2.171.19
1,207.24
3.218.18
Total. .
%of
Grand
Total
6. Monthly Sales by Salesmen. — The tabulation below is
designed to furnish the monthly total sales by salesmen ; the
figures entered thereon are each salesman's weekly sales.
This gives a comparison of salesmen's totals.
Compute the weekly and monthly totals, and compute the
percentage of each salesman's sales to the total monthly
sales correct to the nearest tenth per cent.
BUSINESS MATHEMATICS
Comparative Monthly Record op Sales
Salesman's
Number
1. ..
2.. .
3...
4...
5...
6...
7.. .
8.. .
9.. .
10...
Total
First
Week
$350.65
456.76
387.57
341.25
324.43
678.93
426.47
276.34
576.27
264.64
Second
Week
$567.87
765.53
476.53
675.83
546.67
468.59
578.64
527.35
364.14
475.37
Third
Week
$436.65
876.65
587.76
436.54
478.35
359.48
658.47
621.34
713.26
718.88
Fourth
Week
$654.43
465.35
745.36
456.76
765.28
468.75
536.28
438.27
465.27
635.47
Total
% of
Grand
Total
7. Per Cent of Increase or Decrease. — The tabulation be-
low gives the daily departmental sales figures and is designed
to show the percentage of increase or decrease in each case.
Compute the percentages correct to one decimal place of
per cent and indicate a decrease by an asterisk (*), or by
red ink.
Comparative Sales for Corresponding Days of Two Years
Dept.
No.
Sales, Wep.
Dec. 4t 1919
Sales. Wed.
Dec. 3. 1920
Increase
Dkcreasf
%of
Increase
Decrcasb
1
2
3
4
5
6
7
8
9
10
$1,052.37
1.342.54
1.254.32
1.576.57
2.564.34
465.76
1.467.43
1.564.37
1.231.12
1 ..357.48
$1,781.65
1.254.46
1.324.56
1,456.53
2,657.62
612.35
1.647.25
1.652.48
1.2.3.').65
1,469.51
.
Total
SALES AND PROFITS STATISTICS 5
8. Per Cent of Average. — It is sometimes desirable to com-
pare the weekly or monthly sales of a clerk or department,
with the average weekly or monthly figures as the case may
be.
In the following table compute the total and average
sales and the per cent of the sales of each clerk to the
average figures.
Monthly Sales of a Number of Clerks
Clerk's Number
Sales
Per Cent of Average
350
$1,256.43
1.356.87
1.124.34
1.067.27
987.56
1.246.47
1.456.32
1.245.36
1.034.75
97586
1.326 52
1.137.63
1.364.37
351
352
353
354
355
356
357
358
359
360
361
362
Total
V
Average
9. Sales Returned — Per Cent. — In some lines of business
it is desirable to keep a close watch on goods returned. This
can be effectively done by means of the percentage figures
shown in the two following tables.
Compute the per cent of the departmental sales returned,
to total sales, and the per cent of all sales returned, to total
sales.
6
BUSINESS MATHEMATICS
Sales and Returned Goods by Departi^(ents
Year ending,
Dept.
Sales
Returned
Goods
Nft Sales
% OF Sales
Returned
1
$ 25,431.76
48.976.53
76.432.56
98,742.27
67.834.62
110.532.65
$ 768.63
876.52
1.097.57
1.210.78
895.68
873.45
2
3
4 '. .
5
6
Total
10. Average Net Sales per Check. — Compute the total
and net sales and the average net sales per check for each
clerk.
iNDivmuAL Daily Sales Sheet
Section 13, Dry Goods
Date, May 11, 19—
Clerk's
Number
Gross
Sales
Returned
Goods
Net Sales
Checks
Average Net
Sales Per Check
121
$312.67
413.36
215.23
318.56
456.78
235.67
102.46
189.67
213.53
346.76
S 6.75
11.23
8.79
1.78
2.34
3.21
9.56
121
136
97
118
124
79
81
104
115
121
122
123
124
12.')
126
127
128
i29
130
Total
1 —
•
SALES AND PROFITS STATISTICS 7
11. Tabulations for Other Comparative Purposes. — Tabu-
lations similar to those given may be used to compare sales
from month to month and from year to year, and also the
month of one year with the same month in preceding
years.
Show the monthly increase or decrease as the case may re-
quire and compute the percentage of increase or decrease of
sales during the present year.
Comparison of Sales by Corresponding Months
Salesman, John Doe
Month
Jan.
Feb.
Mar.
Apr.
May
June
July.
Aug.
Sept ,
Oct..
Noy.
Dec.,
Sales
Last Yfar
Salss
This Year
Increase
^ %OF
Increase
$356.76
$430.12
$73.36
20.6
456.87
515.60
345.65
356.45
450.10
390.50
287.65
324.36
231.90
245.87
a
450.65
467.54
346.75
356.52
436.47
420.75
567.35
580.67
478.56
487.64
564.32
545.53
Decrease
%OF
Decrease
12. Cumulative Record of Sales. — By means of this form
we may have not only a very complete record of each de-
partment's sales by any particular road or store salesman
from month to month, but also a comparative record of the
total sales for two or three or more months of any year, or
a comparison of these totals for any previous year since
this particular salesman has been connected with the
business.
8
BUSINESS MATHEMATICS
Compute the cumulative sales to the end of April and
show the average at the foot of each column.
Salesman's Cumulative Record of Sales by Departments
Salesman, H
. William
Dept.
January
February
Total
2 Months
March
Total
3 Months
April
Tot • l
4 Month]
1
S 546.56
768.56
876.45
1.134.76
1,056.87
S 348.76
756.46
983.24
1.234.58
1.121.09
$ 435.65
675.87
875.45
1 ,346.43
1.234.57
S 350.46
763.54
865.73
1.265.87
1,364.24
2
3
4
5
Total
AvcraKc. . .
13. Computation of Loss or Gain. — In figuring the profits
of a business, whether by departments or for the business as
a whole, the deduction of the cost of the goods sold, from the
goods sold or net sales, gives the gain for the period of time
cov(jred by the figures. To determine the cost of the goods
sold it is necessary to deduct the cost of the goods unsold at
the end of the month or year from the purchases made during
the same period. Assuming that there are no goods on hand,
i.e., no opening inventory at the beginning of the period, the
computation would be as follows:
Goods sold (sales) $70,000.00
Purchases S7:),0()0.00
Cost of goods unsold (inventory end of year). 10,000.00
Cost of Roods sold J)5,000.00
Qj^in yu/.J.O- $ 5,00 0.00
Gala % W,000 - $65,000 = 7.7%
SALES AND PROFITS STATISTICS 9
WRITTEN EXERCISES
1. Goods sold during the year $6r),743.87
Original cost of the good3 (K), 126.75
Cost of goods unsold (inventory end) 2,234.76
Find the gain and the gain per cent.
2. Goods sold during the year $75,000.00
Original cost of the goods 70,000.00
Cost of goods unsold 2,564.85
Find the gain and the gain per cent.
3. Sales $80,000.00
Purchases 90,000.00
Inventory at end of year 5,000.00
Find the loss and the loss per cent.
14. Accounting for Opening Inventories. — Assuming an
opening inventory, then the gain for the period may be com-
puted as shown below. Given the sales, inventories, and
the cost of the purchases, this plan may be followed by
any business to arrive at gain or loss.
Illustrative Example.
Sales $232,314.26
Opening inventory $ 36,756.65
Purchases of year 214,643.53
Total $251,400.18
Closing inventory 75,654.78
Difference 175,745.40
Gain $ 56,568.86
WRITTEN EXERCISES
1. A merchant had goods on hand January 1, 1920, $46,756.87.
IHffing that year he purchased goods to the amount of $314,567.89, and
sold goods to the amount of $298,654.65. His inventory at the end ol
30 1 tiiat year showed goods on hand $29,675.76.
"^ • Find his gain or loss for the year.
10
BUSINESS MATHEMATICS
2. Inventory at the beginning of the year $ 56,765.45
Inventory at the end of the year 44,643.44
Purchases during the year 347,124.96.
Sales during the year 357,649.85
Find the gain or loss.
3. Purchases during the year $250,669.25
Sales during the year 276,040.34
Beginning inventory 32,675.26
End inventory 31,575.45
Find the gain or loss.
Find the gain or loss per cent.
15. Comparative Records and Statements. — Increases or
decreases in profits of departments, salesmen, lines of good%
expenses, and sales, are sometimes compared as illustrated
below. Percentages usually give the fairer comparison.
WRITTEN EXERCISES
1. Complete the records to show the required totals and percentagM*
Halesman's Record of Comparative Sales by Departmento
\)ifpt.
\
u
■I.-
TuUl .
Salesman
I, A. John
First Year
Second Year
or 1
, %orj£
Sales
Profits
%
Sales
Profits
%
OR •
DBCRt4dt
11:1.145.56
U.217.4A
4.M52.:iO
1 j.imu.n?
$1,324.75
578.97
1.678.98
412.36
487.63
1.265.42
$12,345.64
3.567.56
14.126.34
4.215.76
4.978.34
14.357.26
$1,265.54
585.74
1.612.32
398.67
423.53
1.196.57
• 1
\
SALES AND PROFITS STATISTICS
2. Fmd the totals and percentages required in the following:
11
Comparative Statement of Sales, Earnings, and Expenses fqr
Three Years
Income:
Gross sales
Less: Cost of
goods sold
Ratio to sales
Gross profits on
sales
Ratio to sales
jRxppfwes;
Selling expenses ....
Ratio to sales ....
Administrative and
general expenses
Ratio to sales ....
Total expenses
Ratio to sales ....
Net profit on sales . . . .
Ratio to sales
Profit and loss charges.
. Ratio to sales
Net Profit
Ratio to sales
First Year
$896,437.65
569.568.26
--7
Mf-
t
$129,562.75
76.837.56
$ 16.283.16
Second Year
$1,062,792.80
703.610.15
$ 146.927.64
87.167.35
^P^^.^v,^^
$ 10.341.62
Third Year
$963,416.50
692.519.46
$119,816.75
72.963.10
>1
$ 7.189.42
Total
Average
3. How would you check Exercise 2? Check it.
The following is another form of profit and loss statement
in which each item is a certain percentage of sales, as shown
at the right.
Check : It will be noticed that the total of the figures in italics sub-
tracted from the total of the other figures should equal 100.
12 BUSINESS MATHEMATICS
Profit and Loss Statement
Sales— Less Returned Goods $1,080,416.70
Less Discount and Freight Allowance 75.215.63 6.96%
$1,005,201.07
Cost of Goods Sold $ 749.153.80 ^
Add Inventory. January 1 . .. 2,663.774.56
$3,412,928.36
L<?55 Inventory, December 31. 2,711,355.62 701.572.74 64.94%
$303.62&33
Add:
Purchase Cash Discount $ 6,864.07 -64%
Interest Received $ 101.44 '0^%
Miscellaneous Income 268.97 .02%
7,234.48
Gross Profit $310,862.81
Less:
Selling Expense S 153.215.58 14.18%
Administration Expense $ 31.171.73 2.89%
Taxes— State and Federal .. . 11,530.89 1.07%
Interest Paid 4.028.30 .37%
46.730.92
Reserve for Bad Debts 5.402.08 .50% *
Miscellaneous Expense 686.17 -06%
206,034.75
NetProfit $104,828.06 9.70%
100%
CHAPTER II
PROFITS BASED ON SALES
16. Methods of Markmg-up Goods. — In the tiiurking of
goods bought for resale the percentage of profit may bo added
to the cost price of the goods, which is their invoice price plus
ireight and cartage; or the percentage of profit may be com-
puted on the sales, which is their cost price plus the expense
of carrying on the business. The business man who adds a.
certam per cent of profit to the cost price of the goods rarely
inows how much profit he is actually making, because he
rarely knows how much the overhead expense of carrying on
the business is. If, for instance, his sales are $50,000 and his
expenses for the year are $10,000, it is apparent that each $1
worth of goods costs $.20 to sell. Therefore, to make 20%
clear profit on his goods he must first add 20% to the cost
price to give him their gross cost price, and then the 20%
profit required. Thus the sales represent the cost to buy
and sell.
n. Overhead Expenses. — Overhead expenses are those in-
curred in doing business — such as rent, taxes, salaries, Ught
and heat, insurance, telephone, advertising, postage, depre-
ciation, etc. These expenses must be taken into considera-
tion when marking-up goods. Overhead expenses usually
have a fairly constant ratio to gross sales, and from experi-
ence the merchant determines what this ratio is. This per-
centage plus the percentage of profit decided upon is deducted
fn)m 100% — representing the selling price — to determine the i
13 1
..:";.■ • t'lir? :o the
:: :•:»'. la r< dividec
.:. dollars. Tlii;
.:: i gain as a per
•-• fJo: ovcrhoad eha^^(
:ilo>;: freight is SI. Fin
-^. ..::\ii l»ri'T
:..: ciiariros and gain
. r. Sales = Gross Profit ' ^ o
Sale^
>a. ^.^■■.: ' ( = Cost of Sales ' ^
>^- fight = Cost
a fj '
■; = Selling Price
• ' I o t* o
v-\ what is the gain |ht (vnt u
'■• >v'Uing i)ri(.'o?
5-»0. What is the jxir cont (
V , s.'i on tho soiling price is 259i
\ Ki' i'i the gain jxt rent on th
•o I- V t' on the selling pric<
V . -v. \\ hioh cost him 12 ccntj
^\» '.hi* M'lhng j)rice?
.:.f.k il h» st'll for Sl.25, ho^
W ti.li |H'r ivnt is this on th
PROFITS BASED ON SALES
15
7. If an article cost $1, and you sell it for $1.50, what percentage of
profit do you make, minus overhead?
8. If overhead expense is 20% of sales, what will an article that <*()st
II and you sell for $1.50, figure as profit?
WRITTEN EXERCISES
1. Complete the following form.
NustBER
Cost
Marked Prick
% o^ Reduction to
Produce Cost
1
S 1.00
.20
.60
.03
.09
10.00
2.50
.40
50.00
3.50
$ 1.20
.25
.75
.05
.12
15.00
2.75
.50
75.00
7.00
^
2
3
4
5
6
7
8
9
10
2. Find the gain per cent on the selling price of the following;
No.
Cost
Selling Price
1
$ 30.00
$ 40.00
2
50.00
70.00
3
6.00
8.00
4
12f00
15.00
5
16.00
20.00
6
108.00
120.00
7
130.00
150.00
8
17
.20
9
15
.20
10
03
.05
8. A man sells goods for $15,000. . The overhead charges are 15% of
sales, and the profits 10% of sales. The freight is $125. Find the in-
voice price of the goods.
16 BUSINESS MATHEMATICS
4. An automobile is invoiced at $1,140. The freight charges ai-e $50.
If we allow 15,% for overhead and 15% for profit, what should be the
Belling pric^ of the automobile?
6. An invoice of Merchandise amounts to $r,575.25. If the overhead *
charges are 20%, the gair^ 10%, and the freight $125, find the selling
price.
6. The invoice cost of a lady's coat is $30, the overhead charges are
25%, the profit is 15%, and the freight is $2. Find the selling price.
7. A merchant marked goods at 20% above the cost. Owing to the
fact that these goods did not sell well he reduced them 20% and
claimed that he was selling them out at cost. Find the amount of his
error in per cent. If he had reduced them 25%, how much would he,
have lost?
8. If a man ouys some articles and marks them so as to gain 25%, and
then reduces them to coat in order to move them, what per cent must
he reducxj them on the marked price?
9. The invoice cast of an article was $3.50. The freight is $.25, the
overhead charges are 20%, and the profit is computed at 15%. Find
the selling price.
10. A retailer buys tables at $40, which he marks to sell at a profit '
of 40% on the cost. On account of slow business he decides to retail
them at 25% less on the marked price. At what price does he sell them?
Does he gain or lose and how much? What per cent is this on the selling
price.' ^
11. An automobile is invoiced at $3,000. The freight is $50. IS we
.allow 10% for overhead, aixd 20% for profit, what should be the scaling
price of the automobile?
12. ('Omplete the following form:
No. COST Gain ^<^^^^ ' Vc^^^
1 $ 5.00 $ 2.00 » -
2 .10 .04
3 .08 .02
4 24.00 8.00
5 500.00 250.00
18. Adding Per Cent of Cost.— The tables given on the
remaining pages of this chapter are "short-cuts" for quickly
calculating profits and selling prices.
k .
PROFITS BASED OX SALES
17
The following table shows the porwmtage of cost which
must be added to effect a given percentage of profit on sales.
. Add%
To Makk % Profit
Add %
To Makf % P::oKiT
TO Cost
ON Salfs
to Cost
ON Sali;s
1
.99
26
20.63
2
1.96
27
21.26
3"
2.91
28
21.88
4
3.85
29
22.48
5
4,76
30
23.08
6
5.66
31
23.66
7
6.54
32
24.24
8
7.41
33
24.81
9
8.27
335
25.(K)
10
9.09
34 •
25.37
1]
9.91
35
25.93
12
10.71
36
26.47
12}
11.11
37
27.01
13
11.50
37}
27.27
14
12.28 ^
38
27.54
15
13.04
39
28.06
16
13.79
40
28.57 •
165 .
14.29
41
29.08
• . 17
14.53
42
29.58
18
15.25
43
30.07
19
15.97
44
30.56
20^
21
16.67
45
31.03
17.36
46
31.51
22
18.03
47
31.97
23
18.70
48
32.43
24
19.35
49
32.88
25
20.00
50
33}
49. Computing Net Profits. — The following table shows the
per cent of the net profit when the per cent of expense to
sales and the per cent of mark-up are known.
If the cost of doing business figured on sales is represented
in the top line of the table beloyv, and the mark-up on the
goods is one of the percentages shown in the fii*st column to
the left, the percentage of net profit will be found at the junc-
tion of the line and column.
Is
Ul:<LNE{<6 >L\THEMATICS
X c c? X M t^ la
; O _ ^ s«|
o
o
.J
^ db CO X o
•-I «-! ?i
t I
n
2 M lO O "«t« Si I'-
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>^
X
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r?5
rj-
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1-H
t-H
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—•—•MM
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cv
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*♦- — n
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M f^ o e p m o
n f^ ^ m o t« 5
PROnXS BASED ON SALES
19
ntnstratiTe Exun|de. If your cost of doing buaincM w 15% of your
i;roea sales and you mark a line at 25% above cost, your net profit in .'i%
on Bales, as shown in the tabulation. If your cost of doinR l)U8in<% is
18% and you mark a line at 60% above coat, your net profit in 10)% on
20. Finding the Selling Price. — The tabic below is iiBcd to
find the selling price of an article after the desired tiot ]mt
cent of profit is added and when the cost to do busincmi ia
known.
To find the selHng price divide the cost (invoice price plus
the freight) by the percentage found at the junction of the
"desired net per cent profit" with the percentage of the
"cost to do busineea."
Table fob FnniiNa the Seluko Price or Ant Articu:
£.-1 FH I II
'
so
:
4
«
4-
4
'.
i
4
:
I
M
3!
3
niustratiTe Example. Article coat S60.00
Freight 1.20
J61.20
You desire to make a net profit of 5%
It coBt jou to do business 19%
20
BUSINESS MATHEMATICS
Solution: Take the figures in oolumn 5 on the line with 19, \;^
76.
$61.20 -^ .76 = $80.52, the selling iHioe
The percentage of cost of doing business and profit are figu
selling price
21. Per Cent of Profit on Sales.— The following
8how8 the per cents of profit made on the selling price
the jKT cents shown in the first column are added to th
of the goods sold.
Table for Computing Profit
5% a
iddc<
Ho
cost
=
^4 /O
profit
on
selling price
^2 /O
<(
=
7<^
• /o
tl n
10%
(>
=
9%
It It
12 J %
It
^
lli%
tt tl
15%
n
=
13%
It <l
16%
n
=
131%
n tt
17i%
It
=
15%
i» tt
20%
(<
=
161%
H ti
25%
il
=
20%
It tt
30%
u
=
23%
t. it
33}%
n
=
25%
it ti
35%
<<
=
26%
tt • 1
37 J %
11
=
27i%
<> It
40%
il
=
28}%
11 it
45%
n
=
31%
il tt
50%
it
=
33}%
tt tt
55%
li
=
35}%
tl tt
60%
li
^
37}%
tt ((
05%
<>
=
39}%
.1 i<
663%
<'
^
40%
tt .t
70';^
li
=
41 f^
It *i
75%
(.
=
^-^3 /O
«• ((
H0%
i :
=:
44 i%
tt t>
nr>'/„
I (
=
40%
ti tt
<K)%
• •
»•
=
47 J %
it tt
100%
(•
4 •
=
50%
tt t(
PROFITS BASED OX SALES 21
WKITTEN EXERCISES
1. A man buys an article for $125 and wishra to moke 20% profit on
sales. How much profit ahall ho rompiitc on (he post?
3. A merchant buys an arlirlc for a certain Rum of money and wiabot
to mark it to sell so that he can make 33 J % on the cost. What per cent
piofitiathiBequivalent tonhen computed on the fiel ling I>i^re'.*
3. A merchant marks an article to aell for S2(M], thereby making 33i^p
OQ the Mist. Find the cost. What is the e()uivalcnt |>cr t'Ciit on the
Belling price?
4. It your cost of doing business is 12% of gross sales and you mark
aline at 50% above cost, what is your net profit on sales?
G. If you mark a line 75% above cost and your cost of doing business
IS 16% of your gross sales, what is your net proRton sales?
t. If goods sold amount to St,OO0and your cost ol doing business is
I'i^o, and you have marked the line at 40^0 above t hi! cost, what is the
Mt profit OB the sales? What is the cost?
">■ If you buy an article for $12, and the freight is $.7r,, and you desire
to makes net profit of 10%, and it costs 18% to do busincRS, what is the
wUing price?
*■ If you desire to make a net profit of 8% on an article which costs
i^iUtd it costs 16% to do business, what should be the selling price of
Uutirtide?
9, If the selling price of on article is 8100, the net profit is 4%, cosi
oNoing business is 16%, find the cost of that article.
' 10. H a man sella an article for $100 and makes 26% on the selling
Pi^i "hat per ceni does he make on the cost?
i
CHAPTER III
PAY-ROLL CALCULATIONS
22. Methods of Wage Payment.— Every manufacturing
business and many mercantile concerns maintain a pay-
roll department. The duty of this department is to keep
the employees' time records or records of the quantity <rf
work done, and, at the end of the week or other period, to
compute the wages or salary earned by each employee.
Where work is paid for by the hour, day, or week, or accord-
ing to the number of pieces made, the computing of the pay-
roll may be a simple arithmetical problem of multiplication.
Where scales of wages vary with the eiBSciency of each
workman, and where good work or quick work is rewarded
by the payment of a premium or bonus in addition to the
regular hourly or piece-rate wage, the problem may involvie
intricate fractional and percentage calculations. The
method of computing the premium or bonus may be based
upon the number of pieces produced within a given time or
upon the amount of time saved in the performance of a
given operation.
23. Efficiency Pajrment Systems. — The more complicated
methods of wage payment are met with in the highly or-
ganized mechanical industries where the modern method of
management known as *' scientific management*' is in-
creasingly employed. Different types of industry often
adopt different methods; and the efficiency expert who has
22
PAY-ROLL CALCULATIONS 23
been responsible for the introduction of a special sj-stcni of
wage payment usually gives it his came to distinguish it
from others.
24. Day-Rate System. — Wages which are based on time
worked are usually computed at an hourly rate, with one and
one-half times the regular rate for overtime and lioliday
work, and sometimes double time for holiday work and
Sundays. The practice with respect to overtime wages
varies with different manufacturing concerns. The time
clock is generally used to record each employee's time, and
the time cards serve as a basis for computing the wages of
the employees.
WSITTEN EXERCISES
1. Id the following section of & pay-roll, the regular working dny ix
assumed to be S br. If a man works moro than 8 hr. on any Kinplc
<lsy, he ia paid time and a half for overtime, although on tuimc da>'K li:;
oay work less than the number of hours in the standard day.
Hake the required computations to show the wages due to each
Ho.^P..D.v
s
1
1
1
1
ii
M
II
NAin
M
w
T
P
S
13^
J=4.D™..,,
H.]o™,...
ct™,
*n. Oaborn
H-Orr
J-Wmer...
B-Pb.lp,...
6
S
s
8
S
s
s
9
g
s
s
6
7
8
Si
s
>
■to
s
40
36
IIS.fB
S3.l»l
(13.00
24 BUSINESS MATHEMATICS
2. Rule a pay-roll like the preceding model, enter the following data
and find the amount of wages due to each employee. A full day is 8
hr., and time and a half is paid for overtime.
No. Employee's Name Hourly Rate
1 Henry Jones $.34
2 Wm. Johnson 30
3 Chas. Bell .32
4 A. T. Wigham .35
5 G.R.Martin .34
During the week ending Oct. 13, time cards were turned in by the
foreman, showing the number of hours worked by each employee to be
as follows :
Monday 1, 8
*
2, 10; 3, 8; 4, 7; 5, 8
2, 8; 3, 8; 4, 9; 5, 6
2, 9; 3, 8; 4, 8; 5, 7
2, 8; 3, 9; 4, 8; 5, 8
2, 8; 3, 9; 4, 7; 5, 8
2, 8; 3, 8; 4, 8; 5, 6
Tuesday.. 1, 7
Wednesday 1,8
Thursday 1,8
Friday 1,8
Saturday 1, G
The cashier has already advanced the following sums: to No. 1, $3;
No. 3, $5; No. 5, $2.50.
* Workman No. 1, 8 hr.
25. Piecework System. — Instead of basing the wage rate
upon time, it may be based upon the quantity produced.
The principle of all straight piecework systems is that the
employee is likely to work harder and produce more if he is
paid in proportion to his production than he would if paid
by the day or hour. A pay-roll designed to record piece-
work wage payments is similar to the one used under the
hourly rate system, excepting that no provision need be
made for overtime, as overtime is generally paid for at the
same rate as regular time.
WRITTEN EXERCISES
1. Complete the following pay-roll by determining the amount due to
each employee.
PAY-ROLL CALCULATIONS 25
Piecework Pat-Roll for Week Endino December 27, 193 —
NUMBEB PaOOLX-H.
Opeiu.
riON No
1
S
If
u
T
W
T
p
=
ll
1 H,j!Uno...
214
IS
1A
1-1
17
IS
13
m
tifi
S13.3i>
n.oo
«10.3S
m
2.1
,i;i
3 W-R^erty,
315
42S
4S
41
43
4.1
34
4n
.07
snn
£ G.Willii.ms.
27 i
27
ilH
28
;!5
27
20
.10
26. The Differential Rate. — Under this plan a careful
eatimate is made of the number of pieces each employee can
produce in a day, and each employee is expected to produce
the standard number. If he produces less than the standard,
lie receives less per piece. If he produces more than the
Btandard, he receives more than the standard rate.
This system is based upon the idea that the expense of
heat, light, rent, power, etc., remains the same whether the
employees produce a small amount of product or a large
amount. By increasing the amount of the production, these
expenses are distributed over a larger quantity of manu-
factured goods, and the cost of making each article is thereby
decreased. The saving effected by increasing the output is
•divided between the owner of the factory and the workmen
who by their skill and industry increase the output. On
the other hand, the employees who by producing a small
quantity increase the cost per article, arc paid less.
27. Computing the Differential Rate. — The problem of
determining the standard number of pieces which shall con-
stitute a fair day's work is often a difficult matter. The
Wployer is naturally desirous of basing the rate on the
26
BUSINESS \L\THEMATICS
amount of pieces produced or work done by his most efficient
f employees. The less efficient employees or those who are
not temperamentally fast workers may be penaUzed if the
rate is placed at so high a figure that they earn less than the
Htandard rate. Usually a compromise is made by placing
the rate at such a figure that all industrious workmen can
efinily earn the standard rate and only the inefficient (»
lazy fail to earn it.
Illustrative Example. If a manufacturer found by experience that
ih^ avftragc workman in his factory could produce 10 articles per dayf
and that 35^ per article could be paid for the work, he might then outline
tfic following differential rate:
No. OF Pieces
Produced
8
9
10 (standard number and rate)
11
12
13
If Williams produces 8 pieces, he receives 8 X $.33 * $2.64.
If Hartman " 13 " " " 13 X $.38 = $4.94.
Rate per Piece
$.33
.34
.35
.36
.37
.38
WRITTEN EXERCISES
1. Work out a pay-roll blank showing each employee's production
and wages, using the daily production record and table of rates given
below.
No.
1
3
3
4
5
Namk
John Jones
Henry Edwards
ChM. Tync ....
J. Williams
W. Rmilh
Daily Production
17
17
16
19
18
w
T
F
16
17
14
18
18
19
17
18
17
21
22
21
19
18
19
15
17
17
20
IS
PAY-ROLL CALCULATIONS
No. OF Pieces
Rate
No. OF Pieces
Rate
No. OF Pieces
Rate
10
$.22
15
$.29
20
$37
11
.23
16
.31
21
..39
12
.24
17
.32
22
.41
13
.25
18 (standard)
.34
23
.43
14
.27
19
.35
24
25
.44
.45
28. Task and Bonus Plan. — This plan of wage payment
issometimes termed the " Gantt Bonus Plan. " In principle
it is based on an extra payment per hour in addition to
the regular hourly wage rate when the worker exceeds the
standard.
Illustrative Example. Assume that in an office where typewriter
work is paid for at piece rates the standard task at which the bonu^*.
payment begins is 150 or more sq. in. an hour, and that the bonus l^cgins
with an extra payment of $.012 per hr. Assume again that for every
2 in. above the standard task, the bonus increases by $.0012 up to 158
in.; by $.0016 from 160 to 168 in. ; and by $.002 for 170 in. and above.
The bonus payment per hr. in addition to the regular hourly rate
would then work out as shown in the following table:
Bonus Table
Sq. In. per
Hr. Bonus per Hr
150
$.0120
152
.0132
154
.0148
156
.0160
158
.0172
160
.0188
162
.0204
164
.0220
166
0236
168
.0252
170
.0272
172
.0292
174
.0312
28
BUSINESS MATHEMATICS
WRITTEN EXERCISES
1. By the aid of the above table, compute the total wages of the five
operators below.
Production and Solxthon of the Example
Operator
No.
Sq. In
Hours
1
2,160
4,150
3.510
3.968
4,975
51
41
43
351
46 i
2
3
4
5
Wage Rate
$.40
.36
.44
.50
.48
Bonus Per
Hour
Total Hourly
Rate
Total
Wages
2. Using the bonus table given above, compute the total earnings of
the following six typewriter operators.
Operatoii
No.
Sq. In.
Hours
Wage Rate
Bonus
Total Earnings
1
5.948
7.902
6.546
11.190
8.704
11.118
46
45 J
53 J
47
54
46 i
$.40
.42
.44
.46
.43
.45
2
3
4
;)
6
29. Halsey-Rowan Premium Rate. — Under this method
the workman is paid a premium which is generally from
one-half to one-third the value of the time saved. The time
saved is computed by setting a standard task to be done
within a certain time, and the difference between the stand-
ard time and the actual time (where the actual time is less
than the standard time) constitutes the time saved on the
task.
PAY-ROLL CALCULATIONS 29
IlhistratiTe Ezamide. If the standaixl time for curling a dozen ost rich
feathers is 3 hr., and a worker whose hourly rate is 50f( does the to^k in
2 hr., he saves 1 hr., or 53^. Therefore he receives $1 for the 2 hr. work
plus a premium of 25^ , this being half the value of the hour saved. He
still has one hour left in which he can do other work for which he should
leceive at least 50ff, so that his total payment for 3 hr. work should be:
Explanation:
2 hr. 1 doz. (actual time) at $.50 per hr $1.00
i hr. (premium) " .50 " " 25
Ihr.saved " .50 " " 50
Total $1.75
30. Emerson Efficiency Wage-System. — By this plan an
"efficiency" bonus is paid to those workers who maintain or
exceed a given rate of production. The bonus added is a
percentage of the wages earned at the regular hourly rate,
and the percentage is calculated by dividing the standard
time by the actual number of hours taken to do the work.
If the worker reaches 70% efficiency, he receives a bonus of
1^ on every dollar of wages; if 80%, 5^ on the dollar; if 90%
10^ on the dollar; if 100%, 25^ on the dollar; and so on
progressively.
The method of calculating efficiency is:
If A = actual time in hours
S = standard time in hours
E == eflficiency per cent
Then E = S/A
ninstratiTe Example.
Standard time is 25.5 pieces per hr.
" " 100 *' in 3.92+ hr.
Wage rate is 50(i
30
BUSINESS MATHEMATICS
If the worker makes 30 pieces, the standard time would be: 30 X
.0392 = 1.176 hr.
If the actual time on the 30 pieces were 1.7 hr., the "eflficiency per
cent*' would be: 1.176 -^ 1.7 = 69.2%.
Referring to the bonus table given below, an eflficiency of 69.2% corre-
sponds approximately to a bonus per cent of .7%.
The computation would then be as follows: 1.7 (hr.) X $.50 (wage
rate) X .7% (bonus factor) = $.00595 (bonus earned.)
Bonus Scale
Efficiency
Of
/o
Bonus
%
Efficiency
%
Bonus
%
Efficiency
%
Bonus
%
69
.7
81
5.2
93
13.
70
1.
82
5.6
94
14.
71
1.4
83
6.
95
15.
72
1.7
84
6.5
96
16.
73
2.2
85
7.
97
17.
74
2.6
86
7.5
98
18.
75
3.
87
8.
99
19.
76
3.3
88
8.5
100
25.
77
3.7
89
9.
101
26.
78
4.
90
10.
102
27.
79
4.4
91
11.
103
35.
80
5.
92
12.
104
45.
Allow for 5% increase in bonus for each increase of 1% in eflSciency
above 104.
The "efficiency per cent" corresponding to pieces per
hour and the bonus in cents is as follows:
Pieces per
Hour
Efft'-^ency
%
Bonus in Cents
50
70
1 per dollar of wage
60
80
e << << t( n
75
100
25 " " " "
WRITTEN EXERCISES
1. Complete the following tabulation from the following data;
Standard time is 100 pieces in 4 hr.
tt
u
tl
25
1
it
tt
per hr.
in .04 hr.
Wage rate is 40fi per hr.
PAY-ROLL CALCULATIONS
31
Name
J. B. Roads..
H. Waraer. . .
L. Smith
A. R. Jones. .
W. L. Brown
No. OF
PiBCBS
Time
Effi-
ciency
Made
Actual
SUndard
50
li
2
7i
2J
3
100
3
4
40
11
li
80
21
31
Amt.
Earned at
Reg. Rate
Bonus Earned
2. Tabulate the aboye-mentioned employees, with the same num-
ber of pieces of wo^ and the same actual time, from the figures
following:
Standard time is 100 pieces in 5 hr.
" " 20 " perhr.
" " 1 " in - hr.
Wage rate is 50f( per hr.
31. Salaries on a Commission Basis. — The salaries of
employees are usually fixed in their amount and no extra
pay is given for overtime. An exception to this custom is
sometiines made in the case of salesmen whose salaries are
frequently paid on what is known as a commission basis;
that is, a certain percentage of the sales of each employee is
paid to him instead of a regular salary or a small salary
may bo paid and supplemented with a commission on
sales made.
WRITTEN EXERCISE
!• The wages of a certain firm are determined on the following com-
mission basis:
On goods
sold
in
Department
1.
10%
(<
11
u
<(
ii
2,
12%
ti
tt
11
<<
11
3,
12%
tc
11
11
<<
n
4,
15%
BCSINESS MATHEMATICS
Salesman
Dept. 1
Derr. 2
Depi. 3
Dbpt. i
H Jon«
t360.24
634.45
879.87
$415.67
645. 4 J
t30Z.2i
587.65
W12 53
Plan a form for the abov« which will show:
(a) Each salesman's commission in each department.
(b) Total commissions paid to each salesman.
(c) Total commissions paid for each department.
(d) Total commissions paid to all s)
{e) Total sales in each department
(f) Total sales of each salesman.
(r) Total sales In all departments.
32. Pay-Roll Slips.— It 18 the practice of many concerns
to hand out the weekly wage in envelopes, each employee
receiving an envelope containing the exact amount of his
pay. In this case, a coin sheet is prepared as below.
WRITTEH EXERCISE
1. Find the total wiges and the total of bills and c
pay the wages shown below. Check your work.
Coin Sheet
W*c:i.s
na.77
;10,67
R„.L.
t.m
t.3r>
t.io
t.U5
,0.
no
ts
t2
St
—
_
H. JwhT
cnnu
J. Sullcit... .
T..t=i1
PAY-ROLL CALCULATIONS
3,3
33. Currency Memorandum. — This is used to take to the
bank to show the paying teller just what kinds of money are
required and how many of each kind are required to pay the
weekly pay-roll.
Chase National Bank
Currency Memorandum
New York, April 1, 192-
Depoeitor — ^John Doe
Dollars
Cenis
5 Billa $1
5
20
1
2
29
2
4 ** 5
10
20
50
*' 100
Coin : Pennies
6
5 ** Nickels
25
8 ** Dimes
80
6 " Quarters
4 " Halves
oO
" Gold
Total
01
WRITTEN EXERCISE
1. Make a currency memorandum for the exercise in § 3'/.
CHAPTZZ IT
EWTEHEST
ii. Natiuc ji Interest. — [.""r^r^* -r jn-jiii'.nr :.:r "he us»
. . .'u 'iLn: for v'..'':r. •.:.-': i--r.-. > jijijih*: i:* -ytfila;
...■ \ (■ v.-«.iir •:i'i:i.c'£^-M Ti.'z Trji-vTii- is "iie sua
'■'u uiiicipai i.::(i .r.'.f-r^r., i.:i-:C ".:«?^rher, an
I <, ...,,■■.■
'. ,..l: -.i.c :> "i.'s.-'i. T.'.i: 'rv'i.'rir.z :■: iz.*:.TrvSC at a
X.. ill- !iiiii'';*i \fj ^/y .:. i'.i i*:i*:ci* wirfa. th<
.;»i:iii Puko'^i; ^'^ / :r. Alirima. Alaska
' . .;.i. \loat.inu. 1'*:j;.:.. .^.rA Wyociine: I'J^^j
* . j.\-uo rally bi-i:*:f->:::i! to *hf' lender l'»^vau9i
.jvo a ivturr; f'.r "he nionoy earned b}
». 'M.si. It a:>«.' 'r> .r-i the Narrower, be
' ' .. *• v^i'i a lai>:ir rt-rurn from his work.
.'KAl EXERCISES
^ v*. AK» to viuiMc him :o nuimifaoturv a nen
.. ',si whiolv ho invist pay icr tho money?
INTEREST 35
2. If he pays $1,200 for the use of this money for 1 yr., what should
he pay for its use for 2 yr.? For 3 yr.7 For 3 yr. and 6 mo.?
3. Suppos3 at the end of 3 yr. and 6 mo. he sells out his business for
$100,000. If we deduct the sum borrowed and the interest on it, how
much will he have left?
4. If a man borrows $50,000 in one state at 6% and loans it in another
state for 8%, what will he make in 1 yr.? \ Sl *.> \
5. If a man pays you 3]% for the use of ^l^OOO of your money and
loans it at 6%, what will he make on it? 'uJT^ i
35. Interest Problems in Business. — It is a simple matter
to calculate interest for definite terms such as 1 yr., 6 mo.,
or 3 mo. In business, however, it is frequently necessary
to calculate the interest earned or due on a given sum of
money up to a certain date and this involves fractional cal-
culations. This has led to the devising of methods and
tables which simplify the calculation of interest for a given
number of days. These methods are explained in this
chapter.
36. Interest for Years and Months. — Such calculation
may easily be made by the following method :
1. Express the time in years only.
2. Find the interest at the given rate on the given
amount for 1 yr. and multiply this by the number
of years.
3. Parts of a year may easily be reduced from months
and days.
Illustrative Example. Find the interest on $450 for 3 yr. and 3 mo
at 5%.
Solutign: 3 yr. and 3 mo. = 3i yr.
$450 principal
.05 rate
^H-5^
$7ZA2\ or t73.I3 = iateresn ::r 3yr- acd 3 n>x at ofi
WRITTEN EXERCISES
F'ihrJ tiift iriV:T»-^t on:
1. $240 for 2 >T. ») mo. at 5^.
2. $375 for 4 yr. S mo. at 6^.
3. $325.16 for 9 mo. at 3^.
4. $456.76 for 3 >t. 6 mo. at 1%.
37. Short Methods of Computmg Interest. — The follow-
ing principU-s and methods of computing interest are short-
cut h for calculating interest when the rate is C^c-
To find the interest at 6^c ^or:
6 da. point off 3 places to the left in the princqiaL
A/\ i( ii It n «• n ti »< »» t» n
600
<(
<(
(<
1
fe •
*i
ii
ii
*i
ii
«
BfiOO
<<
<(
<.
no
ti
li
it
i»
»(
>(
«
WRITTEN EXERCISES
1. rind \\u' total amount of interest at 6% on the following:
$l,.'j7.'> for r»0 <l;i.
Find int. for 00 da. divide by 2 and add..
" i\ '' and multiply by 7.
" ()0 " divide by 3 and subtract-
" (M) " divide by 12 and addi-. ' •
1 ,o7.'>
" 00
Find
int
1,212
" 12
<<
<<
1 ,;<r>o
" 10
<(
<(
1,170
" (>:>
<<
. *<
I.(i7l
" 21
s7;{
•• so
1 .S2 1
'• 70
2. I'ind (he total amount of intereHt lit 6% on:
INTEREST
37
51,948 for 45 da.
2,648 " 15 '
3,642 " 25 '
2.600 " 21
3,400 " 33
3,600 " 55
Find int. for 6 da., divide by 2, and multiply by 7.
Find 30 da. and 3 da. int. and add.
Find 60 da. and 5 da. int. and subtract.
3. Find the total amount of interest at 6% on;
$1,673.00 for 72 da.
1,236.00 " 84
465.46 " 48
474.89 " 6
127.46 " 9
656.48 " 10
824.34 " 7
4. Find the total interest at 6% on:
I 4,568.47 for IS da. ,
356.35
2
6,000.00
1
1,245.60
3
7,454.00
2
9,000.00
8
5,648.00
4
1.242.00
5. Find the total amount of interest at 6% on: •
$ 456.84 ion 7 mo. 15 da. 7 mo. =3X2 mo. + 1 mo.
1,264.00 " 1 yr. 2 mo. 10 da. 60 da. = 2 mo.
1,952.00 " 6 mo. 20 da.
1,000.00 " 1 yr. 6 mo. 12 da.
1,275.87 " 8 mo. 8 da.
Rule : Pointing off two places in any principal gives the interest at
rates other than 6% as follows:
Atl% forlyr.
At 2%
At 3%
At 4% " 3 " '' 90
At4i% " 80
'' 6 mo. or 180 da.
tl
<(
I
38
BUSINESS ^L\THE^L\TICS
ORAL
XH'iWkiX
1. At 5% for how many ds.?
2. At 7i%
8. At 8%
4. At 9%
6. At 10%
6. At 12%
7. At 15%
8. How is the number of days for the two-fdaoe point-off obtai
IC
u
tl
it
u
tt
«
«
«
tt
tt
tl
It
it
tt
tt
it
tt
WRITTEN EXERCISES
1. Find the total interest on the following:
$1,256.75 for 1 >t. at 1%
456.45 " 3 mo. "4%
876.48 " 6 " "2%
986.54 " 80 da. " 4J%
984.42 " 72 " " 5%
2. Find the total amount of interest on the following:
$ 800 for 40 da. at 9%
800
" 20
"9%
1,600
" 48
" 7i%
1,600
" 8
" 7h%
2,400
" 43
^
3,000
" 15
"8%
4,000
*' 60
"8%
Rule: To find the interest at:
6|%, add 1-2 to the interest at 6%.
7%
o»
it
6J%, " J " " " " 6%,
i
tt tt
tt tt
tt
tt
" 6%.
ORAL EXERCISES
1. 7J%, add ? to the interest at 6%.
2. 7J%, " ? " " '' " 6%.
3. S%. '^ ? *' " " " 6%.
4. 10%, divide 6% interest by 6, and multiply by what*
5. 12%, multiply 0% interest by?
INTEREST 39
Rule: To find the interest at;:
6j%, deduct h from the 6% interest
6}% " J " " 6% "
• 3%, divide 6% interest by 2.
Any rate, divide 6% interest by 6, and multiply by that rate.
ORAL EXERCISES
To find:
1. 5%, deduct ? from 6% interest.
2. 4t%, " ? " 6% "
3. 4i%. " ? " 6% "
4. 4%, " ? " 6% "
6. 2%, divide 6% interest by what ?
Illustrative Example. What is the ijntercst on $2,400 for 60 da. at
%?at4%?at7%?
Solution:
$24 = 60 da. int. at 6% $24 = 60 da. int. at 6%
4 = 60 " " " 1% 8 = 60 " " " 2%
$20 = 60 " " " 5% $16 = 60 '* " " 4%
$24 = 60 da. int. at 6%
4 = 60 " '' " V/o
$28 = 60 " " " 7%
WRITTEN EXERCISES
Find the tatal interest on the following:
$1,225.00 for
6 da.
at 7%
1,175.00
12
"5%
1,456.00
3
^ /o
1,524.00
15
it QCf
O/o
1,200.00
30
" !)%
1,800.00
60
"^%
3,624.00
60
" 7i%
1,464.76
24
"3%
4,468.74
36
" 1%
40 BUSINESS MATHEMATICS
Rule: Interchangiiig prmcipal and tims. To find the interes
$600 for 63 da. at 6%, change this to finding the interest on $63 foi
da., at 6%.
WRITTEN EXERCISE
1. Find the total interest on:
$ 600 for 25 da. at 6%
60
<(
17
((
"6%
200
ft
13
(«
"6%
1,500
ti
115
n
"6%
1,200
t<
67
tt
"6%
1,000
((
123
t
"6%
660
a
46
tl
"6%
Rule : To find accurate interest.
First : Find ordinary interest by above principles.
Second: Deduct 7*3 of the ordinary interest from itself.
Example. Find the accurate interest on $7,300 at 6% for 60 d
Solution:
$73 = 60 da. int. at 6% (ordinary interest)
1 = $73 -^ 73
$72 = 60 da. int. " 6% (accurate interest)
MISCELLANEOUS WRITTEN EXERCISES
1. Find the total interest on:
$600 for 80 da. at 45%
550
((
72
'' " 5%
780
u
45
" " 8%
800
tt
40
'' ." 9%
720
K
60
" *' 6i9
2. Find the total interest on:
$ 6^10 for 60 da. at 62%
1,000 '' 60 *' " 7i%
600 " 60 " " 10%
INTEREST 41
$1,440 for 60 da. at 5)%
560 " 60 *• •* r)l%
3. Find the total interest on :
$2,000 for 60 da. at 4t%
800 *• 60 *' "4i%
800 '* 60 " "2J%
800 " 60 " '*9%
4. If the ordinary interest on a certain sum of money for a certain
time is $250,098, at a given rate, what is the ac'curate interest for t he same
time and the same rate?
38. To Find the Time from Principal, Rate, and Interest.
—It is sometimes necessary to find the time at which a
certain sum of money at a given rate will produce a certain
amount of interest. This is computed l)y finding the in-
terest on the given amount of money for the given rate for
one year, and dividing the amount of interest required by
the amount for 1 yr. The result will be the number of
years and a decimal or fraction of a year.
Illustrative Example 1. If the interest on $(KK) is $108 iit ij^/c, annually,
find the time it was on interest.
Solution:
$600 at 6% for 1 yr. = $30
$10S 4- iiC) = ;{
• •> yr.
Illustrative Example 2. If $400 at 6^ ;, yields SS4 int.Tost, find the
time.
Solution :
$400 at 6% for 1 yr. = $24
$84 -^ $24 = V,
.'.15 yr. nio.
42 BUSINESS MATHEMATICS
WRITTEN EXERCISES
1. If $500 yields $120 at 6%, find the time.
2. In what time will it take $1,200 to produce $210 at 5%?
3. How long will it take $1,000 to produce $260 at 6%?
39. To Find Interest Between Certain Dates. — It is often
necessary to find the time between certain dates and then
find the interest for that amount of time. This is accom-
plished by:
(a) Subtracting the dates as illustrated below.
Illustrative Example. Find the interest on $400 at 6% from July
12, 1919, to Jan. 3, 1921.
Solution:
yr.
mo.
da.
1921
1
)3-
1919
7
12
1 5 21
Explanation: Subtract. In doing so, borrow 1 mo., and add its
equivalent 30 da. to the 3 da. in the minuend, then take 12 from 33 = 21,
Next take 7 from 0, but in order to do this, we must borrow 1 yr. and
add its equivalent 12 mo. to the 0, then take 7 from 12. Finally subtract
1919 from 1920. Then find the interest for the given time at the given
rate on $400.
(b) Finding the exact number of days from one date to
the next date.
Illustrative Example. Find the number of days from Mar. 15, 1919
to June 26, 1919.
Solution:
Left in March
16 da.
April has -
30 ''
May "
31 "
Time in June
26 *'
T^^al 103 "
INTEREST
43
WRITTEN EXERCISES
1. Fiad the time between June 17, 1919, and Oct. 14, 1919.
2. Find the time from Apr. 1, 1919, to May 6, 1923.
3. Find the time from Dec. 1, 1919, to March 18, 1920.
4. Find the time from Jan. 1, 1920, to March 1, 1920.
(c) The table method of finding a date in the future.
Illustrative Example 1. Find the date 5 mo. from May 6, 1920;
also 6 mo. from Oct. 20, 1920. Note that May is the 6th mo.,
adding 5, gives the 10th mo. (see the number on
the left) or October; therefore Oct. 6, 1920, is the
date. October is the 10th mo., 10 -f 6 = 16, and
the 16th n^o. (see the number on the right) is
April; therefore the date is Apr. 20, 1921.
Illustrative Example 2. Find the date 90 da.
from Dec. 7, 1920. 12 + 3 = 15th mo. if in 3
rao., or Mar. 7, 1921; but note in the table that
2 da. (1 da. -f- 1 da.) must be subtracted for De-
cember and January, and 2 da. added for Febru-
ary. Therefore the date is Mar. 7, 1921.
WRITTEN EXERCISES
!• Find the date 6 mo. after Jan. 2, 1921.
2. " " *' 60 da. " June 17, 1921.
8. " " " 90 " " Aug. 10, 1921.
^ " " " 45 " " Sept. 16, 1921.
1
Jan.
^"* 1
13
2
Feb.
+2
14
3
Mar.
— 1
15
4
Apr.
16
5
May
— 1
17 '
6
June
18
7
July
— 1
19
8
Aug.
— 1
20
9
Sept.
21
10
Oct.
— 1
22
11
Nov.
23
12
Dec.
^~ \
24
W. Compound Interest. — Compound interest is the in-
terest on the principal and on the unpaid interest after it
l^ecoines due. It is usually paid on the deposits in savings
banks. Premiums in life insurance are determined by it,
^^d it is also used in sinking funds.
The interest may be compounded quarterly, semi-
annually, annually i or at any regular interval of time. Its
collection is not permissible in some states.
44 BUSINESS MATHEMATICS
Illustrative Example. Find the compound interest on $1,000 for 3
yr. at 5%, compounded annually.
Solution:
5% of $1,000 = $50, 1st year's interest
$1,000 + $50 = $1,050, new principal 2nd yr.
5% of $1,050 = $52.50, 2nd year's interest
$1,050 + $52.50 = $1,102.50, new principal 3rd yr.
5% of $1,102.50 = $55.13, 3rd year's interest
$1,102.50 + $55.13 = $1,157.63, amount end of 3rd yr.
$1,157.63 - $1,000 = $157.63, compound interest
WRITTEN EXERCISES
1. Find the compound interest on $1,200 for 4 yr. at 6% compounded
annually.
2. Find the compound interest on $3,000 for 4 jrr. at 4% compounded
semiannually.
3. A boy has $1,000 deposited in a savings bank for him on his 14th
birthday. If the bank pays 4% compound interest, semiannually, what
amount will he have on his 21st birthday if there are no other deposits
or withdrawals?
4. Find the difference between the simple interest on $1,000 for 6 yr.
at 5%, and the compound interest on the same amount for the same
time at the same rate if the interest is compounded semiannually. If
compounded quarterly.
5. Find the compound interest on $5,000 for 2 yr. at 4% compounded
quarterly.
6. If $200 is deposited in a savings bank which pays 4% compound
interest compounded semiannually, on Jan. 1, 1921, what will it amount
to July 1, 1924?
7. What is the compound amount on $975 for 3 yr. at 5%, com-
pounded semiannually?
41. Compound Interest Table. — If a person has much
compound interest calculation work to do, he should resort
to the table. It is much easier, simpler, and quicker that
the method exphiined in § 40.
INTEREST
45
This table shows the amount of $1 conqiounded annually
at the different rates.
Years
3%
3*%
4%
4h%
sSi
t%
Tears
I
1.030000
1.035000
1.040000
1.045000
1.050000
1.060000
I
2
1.06O9OO
1.071225
1.081600
1.092025
1.102500
1.123600
a
3
1.092727
1.108718
1.124864
1.141166
1.157625
1.191016
3
4
1.125509
1.147523
1.169859
1.192519
1.215506
1.262477
4
5
1.159274
1.187686
1.216653
1.246182
1.276282
1.338226
5
6
1.194052
1.229255
1.265319
1.302260
1.340096
1.418519
6
7
1.229874
1.272279
1.315932
1.360862
1.407100
1.503630
I
8
1.266770
1.316809
1.368569
1.422101
1.477455
1.593848
9
1.304773
1.362897
1.423312
1.486095
1.551.328
1.689479
9
10
1.343916
1.410599
1.480244
1.552969
1.628895
1.790848
lO
II
1.384234
1.459970
1.539454
1.622853
1.710339
1.898299
II
12
1.425761
1.511069
1.601032
1.695881
1 .795856
2.012197
la
13
1.468534
1.563956
1.665074
1.772196
1.885649
2.132928
13
14
1.512590
1.618695
a. 73 1676
1.851945
1.979932
2.260904
14
IS
1.557967
1.675349
1.800944
1.935282
2.078928
2.396558
IS
i6
1.604706
1.733986
1.872981
2.022370
2.182875
2.540352
x6
17
1.652848
1.794676
1.947901
2.025417
2.113377
2.292018
2.692773
\l
i8
1.702433
1.857489
2.208479
2.406119
2.854339
19
1.753506
1.922501
2.106849
2.307860
2.526950
3.025600
19
20
1.806111
1.989789
2.191123
2.411714
2.653298
3.207136
ao
21
1.860295
2.059431
2.278768
2.520241
2.785963
3.399564
ai
22
1.916103
2.131512
2.369919
2.633652
2.925261
3.603537
aa
23
1.97.3587
2.206114
2.464716
2 752166
3.071524
3.819750
a3
24
2 032794
2.283328
2.563304
2.876014
3.225100
4.048935
34
25
2.093778
2.363245
2.665836
3.005434
3.386355
4.291871
25
Dlustrative Example. Find the compound interest on $4,000 for 5
yr- at 6%.
Solution: $1 compounded annually at G% for 5 yr. amounts to
11.338226, as shown by the above table.
4000. X $1.338226 = $5,352.90
$5,352.90 - $4,000 = $1,352.90. compound interest
Note: If the interest is compounded semiannually, take i the rate
for twice the time.
If the interest is compounded quarterly, take i the rate for 4 times the
time.
Illustrative Example. Find the compound interest on $4,000 for 5
yr. at 6%, interest compounded semiannually.
46 BUSINESS MATHEMATICS
Solution:
J of 6% = 3%.
2 times 5 yr. = 10 jrr.
The amount of $1 compounded at 3% for 10 jrr. is $1.343916.
$4,000 X $1.343916 = $5,375.66
$5,375.66 - $4,000 = $1,375.66, compound interest
WRITTEN EXERCISES
1. Find the compound interest on $3,500 for 6 yr. at 4%, compounded
annually.
2. To what sum will $2,000 amount in 9 yr. if invested at 6%, interest
compounded semiannually?
3. What is the compound interest on a loan of $500 at 12%, com-
pounded quarterly for 5 yr.?
4. What sum must be invested at 4% compound interest to amount
to $800 in 10 yr. if the interest is compounded annually?
5. What sum must be deposited on Jan. 1, 1921, so that on Jan. 1,
1931, with interest at 5% compounded annually, the amoimt will be
$1,000?
6. What is the value of a 10-year endowment hfe insurance premium
of $100.60, if placed at 6% compound interest, compounded semi-
annually, at the end of the 10 yr.?
7. If the average daily number of passengers carried on the Inter-
borough subways and elevated lines of New York was 1,011,053 in
1920, and the average increase per annum is 6%, how many passengers
must be provided for 25 yr. later?
42. Annual Deposits at Compound Interest. — The follow-
ing table is very useful when one has to find the amount of
$1 deposited annually at compound interest for any number
of years up to 25 inclusive. It is perf(»ctly obvious that such
an example would be an endless task without a table of this
nature. Its practical use will become very apparent with
the written exercises which follow it.
INTEREST
>
47
This table shows the amount of $1 deposited annually at
compound interest for any number of years to 26 inclusive.
Years
a%
3%
4%
4i%
5%
6%
X
1.02
1.03
1.04
1.045
1.06
1.06
2
2.0604
2.0909
2.1216
2.137025
2.1525
2.1836
3
3.121608
3.183627
3.246464
3.278191
3.310125
3.374616
4
4.204040
4.309136
4.416323
4.470710
4.525631
4.637093
5
5.308121
5.468410
5.632975
5.716892
5.801913
5.975319
6
6.434283
6.662462
6.898294
7.019152
7.142008
7.393838
7
7.582969
7.892336
8.214226
8.380014
8.549109
8.897468
3
8.754628
9.159106
9.582795
9.802114
10.026564
10.491316
9
9.949721
10.463879
11.006107
11.288209
11.577893
12.180795
10
11.168715
11.807796
12.48^351
12.841179
13.206787
13.971643
II
12.412090
13.192030
14.025805
14.464032
14.917127
15.869941
la
13.680332
14.617790
15.626838
16.159913
16.712983
17.882138
13
14.973938
16.086324
17.291911
17.932109
18.598632
20.015066
14
16 293417
17.598914
19.023588
19.784054
20.578564
22.275970
15
17.639285
19.156881
20.824531
21.719337
22.657492
24.672528
i6
19.012071
20.761588
22.697512
23.741707
24.840366
27.2128S0
17
20.412312
22.414435
24.645413
25.855084
27.132385
29.905653
i8
n. 840559
24.116868
26.671229
28.063562
29.539004
32.759992
19
23.297370
25.870374
28.778079
30.371423
32.065954
35.785591
20
24.783317
27.676486
30.969202
32.783137
34.719252
38.992727
31
26.298984
29.536780
33.247970
35.303378
37.505214
42.392290
as
27.844963
31.452884
35.617889
37.937030
40.430475
45.995828
33
29.421862
33.426470
38.082604
40.689196
43.501999
49.815577
24
31.030300
35 459264
40.645908
43.565210
46.727099
53.864512
25
32.670906
37.553042
43.311745
46.570645
50.113454
58.156383
Illustrative Example. Find the amount of $10 deposited annually
for 10 yr. in a savings bank paying 4% compound interest.
Solution: In the column headed 4%, and down opposite 10 yr., we
find that $1 under the stated conditions will amount to $12.486351;
then$10 wiU amount tolO X $12.486351, or $124.86351.
WRITTEN EXERCISES
1. A man 28 yr. of age has his life insured for $2,000 by taking out a
20 yr. endowment policy, for which he pays annually $49.95 per $1,000.
K at the expiration of the 20th yr. he receives the face value of the
policy, find the gain to the insurance company if money is worth 4%
compound interest to them. (See above table.)
2. If the insured in Exercise 1 had died at the age of 37, would the
insurance company have gained or lost, and how much?
48 BUSINESS MATHEMATICS
3. A young man starts a savings bank account on his 16th birthday
by depositing $30. If he deposits $30 every 6 mo. thereafter until he is
25 yr. of age, what amount will he have to his credit, if the bank pays
4% interest compounded semiannually?
4. What amount of money deposited in a savings bank paying 4}%
annually will amount to $1,000 in 20 yr.?
43. Sinking Funds. — A sinking fund is a sum of money
set aside at regular periods for the purpose of paying oflF an
existing or anticipated indebtedness, or of replacing a value
which will disappear by depreciation, exhaustion, or
cermi nation.
The payment of a public or a corporation debt and the
replacing of certain public, corporate, or private values due
to depreciation or other causes are often made easier by
regularly investing a certain sum in some form of security.
The interest and principal from these investments from
year to year form a sinking fund, which, it is planned, shall
accumulate to an amount needed to redeem the debt when
it falls due, or replace the value when it disappears.
Illustrative Example. A corporation sets aside annually out of
profits of the preceding year $25,000 for 20 yr. If this amount is in-
vested at 4J% compound interest, compounded annually, find the
amount at the end of the 20th yr.
Solution: Amount of $1 deposited annually for 20 yr. at 4J% =
$32.783137.
Amount of $25,000 deposited annually for 20 yr. = 25,000 X
$32.783137 = $819,578.40.
Refer to above table.
WRITTEN EXERCISES
1. At the beginning of each year for 10 yr. a certain company set aside
out of the profits of the previous year $25,000 as a sinking fund. If this
sum was invested at 4% comiK)und interest, compounded annually,
what did it amount to at the end of the 10th year?
. INTEREST 49
2. Jan. 1, 1910, a certain city borrowed $100,000 and aji^rood to pay
the same on Jan. 1, 1920. What sum should have been invested on
Jan. 1, 1910, and each succeeding year for 10 yr. in bonds iiaying 5%
compound interest, compounded annually, in order to pay the loan when
it became due?
3. What sum must a city set aside and invest annually to build a
school building costing $50,000 if it is to be paid for in 20 30*. and the city
receives 4J% on the money thus set aside?
4. What sum must a large printing company set aside to meet the
costof a printing press, through depreciation, in 15 yr., if it cost $5,000,
and the money is worth 4% compounded annually?
4
CHAPTER V
DEPRECIATION
44. Nature of Depreciation. — Depreciation is the loss a
expense incurred in business through decline in the value o
property. While repairs may be made to prolong the useful
ness of a building or a machine, sooner or later the tim
comes when the property is either worn out or it is
business economy to replace it.
A machine, for instance, costing $2,400 is worn out in 1
yr., at the end of which time, when the machine is replaced,
there will have 'been a loss of $2,300 due to depreciation.
Unless a portion of the depreciation is charged to profits
an annual expense, the entire $2,300 loss will be charged
against the profits of the last year. The practice in business
is to spread this loss over the life of the property by charg-
ing off part of the loss to the operations of each year. These
charges are called depreciation charges.
45. Methods of Computing Depreciation. — The following
methods are those most commonly used to compute the de-
preciation charges:
1. The straight-line method.
2. A fixed rate, computed each year on the original value
of the property.
3. A decreasing rate, computed on the original value of
the property.
4. A fixed rate, computed on a decreasing value.
DEPRECIATION
51
46. Straight-lane Method.— First, the probable life of
the machine and the scrap value at the end of its life are
determined. If it has been determined that 10 yr. is the
soo -^. \ I a' -
SM -^-|
w ^_3_
BM Syg
m SJ
« ^s|- ~
, . _ S,^_s
M - — ^
-. ^ J
IK— „ . _N
Depreciation Chart, showing Deprt'ciuted Viiluc Cotiiimtccl
According to the Straight- Line Method
life of the machine, then each year ^\ of the original cost of
"•e machine less its scrap value is charged to factory ex-
panse account, and the depreciation reserve account is
credited with the same amount. For example, if a machine
^ttl^oOO and its probable life is 10 yr., and its scrap value
iatlOO, we take jV (11,000 - $100) = $90, to be written oft
52 BUSINESS MATHEMATICS
each year. This is called the straight-line method or the
fixed proportion method, because if the remainder values are
plotted on the vertical lines (see Form 1) and the years on
the horizontal line, then the remainder value is shown by the
oblique straight line.
47. Fixed Rate Computed on Originel Value. — This is a
very simple method. The difference between the original
value and the probable scrap value is first obtained. This
difference is then divided by the number of years that the
machine is estimated to last, and this result is called the
depreciation per year. The depreciation per year is then
divided by the original value of the machine, which gives
the rate per cent of the original value to be charged off
each year.
Illustrative Example. A printing press is purcharfed at a cost of $8,000
and it is expected that this press can be used for 10 yr., when it will have
a value of $2,000. Therefore, during the 10 yr. of use, a depreciation of
$G,000 will occur. This is an annual depreciation of $600. $600 -j-
$8,000 = 7i%. Therefore75%of the original value is charged off each
year as an expense.
48. Decreasing Rate Computed on Original Value of
Property. — It is sometimes preferred to charge the largest
amount of depreciation the first year, gradually reducing it
each year thereafter. This is done because a greater depre-
ciation actually occurs during the first year than during any
later year. For example, an automobile is ''second-hand"
after only a few months' use, and the owner suffers a much
greater loss from its use during the first year than he does
during the second year. It will always depend upon the
article as to what amount must be deducted each year.
DEPRECIATION 53
49. Fixed Rate Computed on a Decreasing Value. — This
method in a somewhat similar way as in \\ 48 results in a de-
creasing annual charge for depreciation. That is, the depre-
ciation for the first year will amount to more than that for
the second year, and that of the second year will be more
than for the third year, etc.
Illustrative Example. Suppose the orifdnal value of the property is
11,200 and the rate of depreciation is 10% a year, then depreciation
under this method is computed as follows:
$1,200.00 original value
.10
$120.00 depreciation first year
$1,200.00
120.00
$1,080.00 decreased or carrying value, beginnihjx
of second year
.10
$108.00 depreciation second year
$1,080.00
108.00
$972.00 decreased value, beginning of third year
.10
$97.20 depreciation third year, etc.
The fixed rate is obtained by somewhat more complicated
calculations, which usually involve logarithms, and can
readily be understood by anyone having a working knowl-
edge of them. In general the fixed rate is found as shown
'Ji the following :
54 BUSINESS MATHEMATICS
Illustrative Example. A printing firm purchased a printing press
$5,000. It was estimated that it would last 10 yr. and have a sc
value of $200. Find the annual rate of depreciation.
Solution: The following equation is used:
where V = present value of the asset
R = residual value after n periods
n = number of periods
r = percentage of diminishing value to be deducted
annually, or rate of depreciation
If we substitute the values of the problem we have:
T"200
$5,000
04'^^
r = 1 - "^
= 1 - .7221 +
= .2779-
.*. 27.79 — % will be the rate to be used on the decreasing value €
year.
WRITTEN EXERCISES
1. Find the annual depreciation of a building worth $15,560, if
is charged olY each year.
2. How much is charged off annually for depreciation by a mi
facturer who owns property which depreciates at the following rates
Propkrty Value Depreciation Rate
Factory building $40,000 5 %
Machinery 4,800 7i %
Tools I,2o0 12i%
Patents 5,000 6i%
3. The owner of a building estimates the annual depreciation as
xif it s cost. The building cost $4,000. What is the amount of the am
depreciation?
'Vhv, building is rented at $40 per month. If the taxes, insurance, i
ot luT expenses amount to $80 per year, what net income does the ow
of this property receive on his investment after allowing for depreciati
►
It
DEPRECIATION 55
4. It is estimated that a machine costing $2,220 can be Kold at the
end of 8 yr. for $500. What per cent should be charged annually for
depreciation?
6. Machinery in a factory cost $24,000. Depreciation i« computed
as follows:
10% of the original value the Ist year
8% " " " " " 2d
6% " " " " " 3d
3% " " " " " 4th " and each year thereafter.
What was the amount of depreciation charged off each year for o yr.?
What was the inventory value of the machinery at the beginning of
each year?
What was the inventory value of the machinery at the end of 12 yr.?
6. A flour mill was equipped with machmery costing $60,000. Depre-
ciation was computed at 12 J % of its cost the Ist yr., 8% of its cost the
2d>T., 5% the 3d yr., 2J% the 4th yr., and 2% each year thereafter.
Find the amount of depreciation each year for 7 yr. Find the inven-
. tory value of the machinery at the beginning of each year.
f * 7. Depreciation on certain property costing $3,200, was computed at
8% of the decreased value for 4 yr. Find the annual depreciation and
the decreased value each year.
8. A manuf actm-er was engaged in business for 10 yr. His machin-
ery cost $14,500, and he charged 6% depreciation annually on decreased
values. Find the annual depreciation and the decreased value each year.
9. Machinery in a factory cost $7,460, and depreciation was computed
At 7% on decreased values. What was the depreciation during the 4th
yr., and the inventory value at the end of the 4th yr.?
10. The cost of machinery was $23,746, and depreciation was com-
puted at 12i% on decreasing annual values. What was the amount of
the depreciation during the 6th yr., and the reduced value at the end of
*Juit year? The depreciation during this 6th yr. was what per cent of
the original cost of the machinery? ^
^^ *The authors acknowledge their indebtedness to Finney and Brown.
Modem Business Arithmetic," for these problems and much of the
Hher material in this chapter.
1
CHAPTER VI
INSURANCE
50. Necessity of Insurance. — Every business must
the precaution of insuring its premises and stock-in-tn
against fire, its workmen against accident; and in mj
cases the life of a partner, a managing director, or an ii
portant officer, must also be insured to protect the busine
against the loss that his death might cause.
It is also considered wise for each employer or employ
to insure himself against death or accident.
51. Kinds of Insurance. — There are very many kinds d
insurance. The first four named below will be considered
quite in detail in this work. Some kinds of insurance are: ]
1. Fire 17. Transportation
2. Life 18. Keys (loss of)
3. Fraternal 19. Mail
4. Accident 20. Flood
5. Liability 21. Profit (loss of)
6. Inspection 22. Use and occupancy (loss oO
7. Burglary 23. War risk
8. Plate glass 24. Riot
9. Steam boiler 25. Damage
10. Automatic sprinkler and claim 26. Furniture
11. Casualty 27. Indemnity
12. Automobile 28. Musicians' fingers
13. Live stock 29. Earthquake
14. Marine - 30. Title to property
15. Hail 31. Express
16. Cyclone 32. Health
56
INSURANCE 57
i
52. Fire Insurance. — Fire insurance is guaranty of in-
lemnity for loss or damage to property by fire. Such con-.
:racts usually cover losses by lightning, and sometimes loss
caused by cyclones and tornadoes. Insurance companies
are Uable for loss or damage resulting from the use of water
or chemicals used in extinguishing the fire, and from smoke.
The fire insurance policies of all the companies in the
states of New York, New Jersey, Connecticut, and Penn-
sylvania, are uniform and conjbain the''' New York stand-
ard" (80%) clause. This in pattris as fpllows:
"This company shall not be liable for a larger proportion of
any loss or damage to the property described herein than the
sum hereby insured bears to 80% of the actual cash value of
said property at the time such loss shall happen." This is
easily understood with an example.
Per instance, if a piece of property is valued at $100,000
and is insured for $60,000, and a fire and water loss is
140,000, the amount paid by the insurance company would
be as follows: ,.
't '
80% of $100,000 = $80,000 ,
$60,000
3
$80,000 *
I of $40,000 = $30,000 (amount paid by the company on this loss)
It will be observed that the company pays much less than
the actual loss, owing to this clause in the policy. It has
been claimed that the companies use this clause to force
Daanufacturers and other large owners of property to insure
their property for what it is worth. " It can be easily seen
that a large plant composed of many detached buildings is
not as liable to burn up completely as a loft building situated
in the city; and consequently the owner of the latter is keen
/
58 BUSINESS MATHEMATICS
to insure his building for more nearly what it is actually
worth, while the owners of large manufacturing plants are
more liable to take a chance and not insure for what they
are actually worth. Consequently the insurance companies
by the aid of the 80% clause are able to penalize the large
plant. The insurance company would have had to pay a
much larger amount had the owners insured the property
for $100,000 at the beginning, as shown in the following
computation:
80% of $100,000 = $80,000
$100,000 , _
'^^^ = U, or 125%
$80,000 ' ^
li of $40,000 = $50,000
Of course, it is hardly conceivable that the insurance com-
pany would have insured the plant for $100,000 at the begin-
ning; but they might have insured it for $80,000 or $90,000,
and then the amount received for the loss would have been
much larger than it was.
These policies also contain a "waiver" clause which is:
'^ In case of loss, if the value of the property described herein
does not exceed $2,500 the 80% average clause shall be
waived." Some states have passed laws which require the
policy to state definitely the amount of loss for which the
company is liable. By this policy the company is compelled
to pay the actual loss not exceeding the face of the poUcy.
In some states the policy contains a coinsurance clause, which
specifies that only such a part of the loss will be paid as the
face of the policy bears to the value of the property insured.
If more than one company insures the same property, each
company pays only its pro rata share of any loss on the
property.
INSURANCE 59
63. Kinds of Policies. — A valued policy states the exact
amount that the company agrees to pay.
An open policy covers goods in storage and elsewhere.
The amount varies as the quantity of goods is increased or
decreased. When goods are received they are recorded.
The premium charged is based upon the annual rate. If the
goods are returned within 1 yr., the company returns the
unearned part of the premium.
If the company cancels a policy it will return to the in-
sured such a part of the premium as the unexpired time of
the policy is a part of the entire term of the policy. If the
insured cancels the policy, the company will return to him
only the amount by which the premium paid is more than
the premium calculated at the short rate, which is a higher
rate, as is explained in § 56. . A policy is sometimes issued
for 3 yr. at a premium of 23^ times the annual premium,
and for 5 yr. at 4 times the annual premium.
64. How to Find the Premium. — This is obtained by
finding a certain rate on a certain number of dollars, or
by finding a certain per cent of the amount of the policy,
or by finding a certain rate in cents on a certain number of
dollars with a possible discount on the latter in some cases.
Illustrative Example 1. If property is insured for $20,000 at 18^ per
$100 per annum, what is the annual premium?
Solution: $20,000 = 200 hundreds of dollars
200 X $.18 = $36, the annual premium
Illustrative Example 2. If property is insured for $12,000 at li%,
less 6%, what is the annual premium?
Solution: li% of $12,000 = $150
5% of $150 = $7.50
$150 - $7.50 = $142.50, premium
60
BUSINESS IVIATHEMATICS
WRITTEN EXERCISES
1. Find the premium on each of the following policies;
No.
1
2
3
4
5
Face of Policy
$15,300
17.500
9.500
23.500
65.000
Rate of Insurance
$.21 per $100
$.35 •• $100
i%
U% less 10%
$.45 per $100 less 10%
Amount of Premium
55. To Find the Amount Paid by the Insurer.
Illustratiye Example. If property valued at $50,000 is insured for
$30,000 at 1% per annum, and fire and water cause a loss of $24,000,
find the amount that would be paid by "the insurance company
(a) Under an ordinary policy
(b) Under a coinsurance clause policy
(c) Under the New York standard (80%) average clause policy
Solution:
(a)
(b)
(c)
$24,000
$30,000
$50,000
I of $24,000
80% of $50,000
$30.000
$40,000
J of $24,000
$14,400
$40,000
3
4
= $18,000
ORAL EXERCISES
1. What amount of the loss docs the company pay in (a)?
2. State the part of the loss paid in (b) as a fraction. Write the names
in the numerator and the denominator.
3. Same as Exercise 2 with (c) in the place of (b).
1
i
I
f
INSURANCE 61
lUustratiTe Example. A stock of merchandise is insured in Company
X for $10,000, in Company Y for $14,000, and in Company Z for $16,000.
If the damage is $10,000, how much should each company pay?
Solution:
$10,000 -I- $14,000 + $16,000 = $40,000 total amount of insurance
$10,000
— ^ = i; i of $10,000 = $2,500, paid by Co. X
$14,000
•—^ = 2V, ^\ of $10,000 = $3,500, paid by Co. Y
$16,000
$40000 = ^' * °^ S10,000 = $4,000, paid by Co. Z
Check: $2,500 + $3,500 + $4,000 = $10,000
WRITTEN EXERCISES
!• If a house is valued at $12,000 and is insured for § of its value at
I %. and its contents are valued at $5,000 and are insured for J of their
value at |%, and fire causes a total loss of the building and a loss of
$2,000 on the contents, find how much the insurance company will pay.
(a) Under an ordinary policy
(b) Under a coinsurance clause policy
(c) Under a New York standard 80% clause policy
2. A store and its contents are insured in Company A for $40,000 at
55^ per $100; in Company B for $48,000 at J%; and in Company C for
^.000 at 60^ per $100. This property is damaged by fire and water
to the amount of $20,000.
(a) What will each company pay?
(b) What is each company's net loss if they have held the insurance
for 8 yr. when money is worth 5%?
66. Standard Short-Rate Table. — This table is used for
the purpose of computing premiums for terms less than 1
yr-, or for the purpose of computing the amount of premium
to be returned by the insurer (the insurance company) when
the policy is canceled by the insured. It is used as follows:
I
62
BUSINESS MATHEMATICS
Take the percentage opposite the number of dajrs th^
risk is to run, on the premium for 1 jrr. at the given rate, aa
this result will be the premium to be charged in case of shoi:
risks, or earned in case of cancellation.
1
da. 2%
of
Euinual ]
jTemi
2
" 4%
It
tt
3
" 5%
<<
tt
4
" 6%
tt
tt
5
" 7%
11
tt
6
" 8%
It
tt
7
" 9%
u
ft
8
" 9%
n
tt
9
" 10%
(t
ft
10
" 10%
n
tt
11
" 11%
11
tt
12
" 12%
n
tt
13
" 13%
it
tt
14
" 13%
tt
tt
15
" 14%
il
tt
16
" 14%
tt
tt
17
" 15%
tt
tt
18
" 16%
It
tt
19
" 16%
tt
tt
20
" 17%
tt
tt
25
" 19%
tt
tt
30
" 20%
tt
tt
35
" 23%
tt
ft
40
" 26%
tt
it
45
" 27%
it
tt
50 da. 28% of annual premiux:
55
tt
29%
tt
tt
It
60
tt
30%
tt
tt
tt
65
tt
33%
tt
tt
tt
70
it
36%
tt
tt
«
75
tt
37%
tt
tt
U
80
tt
38%
tt
tt
tt
85
tt
• 39%
tt
tt
tt
90
tt
40%
tt
tt
tt
105
tt
45%
tt
tt
tt
120
tt
50%
tt
tt
tt
135
tt
55%
tt
tt
tt
150
tt
60%
tt
tt
tt
165
tt
65%
tt
tt
tt
180
tt
70%
tt
tt
tt
195
tt
73%
tt
tt
tt
210
tt
75%
tt
tt
tt
225
tt
78%
tt
It
tt
240
tt
80%
tt
It
tt
255
tt
83%
tt
tt
tt
270
tt
85%
tt
tt
tt
285
tt
88%
tt
tt
tt
300
tt
90%
tt
tt
tt
315
tt
93%
tt
tt
tt
330
tt
95%
tt
tt
tt
360
tt
100%
tt
tt
tt
Illustrative Example 1. Find the cost of insuring a stock of goods foi
$30,000 for 4 mo. if the annual rate is |%.
Solution:
J% or 1% of $30,000 = $300 6 ) $300
i of $300 = $50 60
$300 - $50 = $250, annual premium 2 ) $250
50% of $250 = $125, premium for 4 mo $125
INSURANCE 63
Explanation: 1% = J of the value of 1% less than the value of
1% of the amount.
Illustrati7e Example 2. Merchandise valued at $48,000 is insured for
i of its value for 1 yr. at 55f( per $100. How much of the premium should
be returned if the policy is canc^led at the expiration of 9 mo. (a) by
the insured? (b) By the insurance company?
Solution:
(a) I of $4^,000 = $40,000
400 X $.55 — $220, annual premium
15% of $220 = $3^ amount returned if insured cancels policy
(b) 3mo. : 12 mo. = $ oc :$220
3 X $220
12
= $55, amount returned if insurer cancels poUcy
^ WRITTEN EXERCISES
1. An insurance policy for $15,000 at }% per annum was dated Jan.,
1921. Six months later it was canceled by the insured. How much of
the premium was returned?
2. June 1, 1920, 1 took out a pohcy on my furniture for $1,800 at 45^
per $100 per annum. Feb. 10, 1921, 1 canceled the policy.
(a) How much of the premium should be returned to me?
(b) How much would have been returned to me had the company
canceled the policy on that date?
3. Goods valued at $10,000 are insured at 60^ per $100 per annum for
3 yr- The policy is canceled by the insurer at the end of 2 yr.
(a) How much premium should be returned to the insured?
(b) How much would have been returned in case the policy had been
canceled by the insured.
4« Find the cost of insuring a stock of goods for $14,000 for 7 mo. at
70^ per $100 per annum.
6. An open policy of insurance is issued on merchandise stored in a
warehouse, the premium on which is to be 75ff per $100 per year.
Goods which are withdrawn within 1 yr. are to be charged the short
rat^. Find the total premiums paid, and the total returns on the
following:
64
BUSINESS MATHEMATICS
Receipts
Withdrawals
Premium
Total Re-
Date
Amount
Date
Amount
Feb. 10. 1920
Purs
$15,000
Dec. 7. 1920
Furs
$14,500
Mar. 15. 1920
Silk
9.800
Aug. 25. 1920
Silk
9.000
May 5. 1920
Woolen
8.900
Nov. 5. 1920
Woolen
8.500
June 8. 1920
Hosiery
7.600
Dec. 4. 1920
Hosiery
7.600
Aug. 7. 1920
Gloves
9.500
Nov. 25. 1920
Gloves
9.500
67. Life Insurance. — Life insurance is a contract by whic
a company, in consideration of payments made at state
intervals by an individual (or by a company for the in
dividual), agrees to pay a certain sum of money to his heir
at his death, or to himself if he attains a certain age.
The contract is called a policy and the money paid b;
the individual a premium. Premiums are payable eithe
weekly, monthly, quarterly, semiannually, or annually.
Many policies now contain the permanent disability claus
which states that the company shall waive payment of all fut
ure premiums and pay 10% of the face of the policy annuall
during disability, or make some other similar provision.
58. Principal Kinds of Life Insurance Policies. — Th
principal kinds of policies are:
1. Ordinary life policy
2. Limited life policy
3. Endowment policy
4. Term policy
5. Life income policy
6. Joint life policy
7. Survivorship annuity
INSURANCE
They differ in three important ways:
1. The number of premiums paid by the insured
2. The amount of each premium
3. The time when payment is made by the company
65
59. Comparison of Different Kinds of Policies. —
Kind of Policy
NUMBZR OF Yb.\RS
Premiums Arb Paid
Time when Payment Is Made
DY THE Company
Ordinary Life
During life of insured.
This period may be
shortened in some com-
panies if the dividends
are allowed to accumu-
late.
At death of insured.
Limited Life
lO-payment life
2D. '•
10 yr.
20 "
At death of insured.
At death of insured.
Endowment Policy:
23-yr. endowment
lO-yr. endowment
20-payraent
30-yr. endowment
20 yr.
10 "
20 "
At death of insured payment made
to beneficiary ; or at expiration of
20 yr. payment made to insured
if still living.
At death of insured or at expiration
of 10 yr.
At death of insured or at expiration
of 30 yr.
Term Policy;
20-yr.
20 yr.
At death of insured if he dies with-
in 20 yr. If he lives beyond this
term no payment is made
Other periods of time may be obtained in the last three
above.
Life income policy provides an annual payment to the in-
sured after a stated date. j
66
BUSINESS MATHEMATICS
60. Premiums and Premium Rates. — The amount of the
premium is a certain amount on $1,000 worth of insurance,
and depends upon the age of the insured at the time of buy-
ing the poUcy, and on the kind of poUcy. The younger the
person, the cheaper the cost of the insurance.
The premium rates per thousand for different kinds of
participating policies at different ages are shown in the
following table.
Annual Premium on Different Kinds of Insurance per $1,000
Age
Ordinary
Life
20-Pay.
Life
10-Pay.
Life
10-Yr.
Endowment
20-Yr.
Endowment
15
$17.40
$27.34
$44.63
$100.60
$47.79
20
19.21
29.39
47.85
101.57
48.48
25
21.49
31.83
51.67
102.73
49.33
26
22.01
32.37
62.51
102.99
49.53
28
23.14
33.52
64.28
103.54
49.95
30
24.38
34.76
56.18
104.14
50.43
35
28.11
38.34
61.53
105.87
51.91
38
30.88
40.89
65.21
107.13
53.10
40
33.01
42.79
67.90
108.07
54.06
61. Computation of Premiums. —
WRITTEN EXERCISES
From the preceding table find the annual premium for the following:
1. An ordinary life policy for $3,000, taken by a man 25 yr. of age.
2. A 20-yr. endowment policy for $4,000 for a man 20 yr. of age.
3. A man 28 yr. of age took out a 10-yr. endowment policy for $4,060,
and died after making 6 payments. How much less would the combined
premiums have been en an ordinary life policy?
4. A man on his 26th birthday took out a 20-payment life policy for
$2,000, and 4 yr. later he took out an ordinary life policy for $4,000. He
died after making 12 payments on his first policy. How much more did
the beneficiary receive from the insurance company than the insured
had paid premiums? (Dividends not to be considered.)
INSURANCE 67
62. Dividend Payments on Policies. — Certain companies
write policies which provide that a portion of the profits of
the company shall be paid to the holders of the policies
These annual payments of the share of the profits are called
dividends. The profits of a company naturally vary from
year to year, so that no specific amount can be guaranteed
the policyholder. The policyholder generally receives no
dividend the first year.
Dividends are applied, at the option of the insured, as
follows:
1. Paid to him in cash.
2. x\pplied to the payment of his premium.
3. Left with the company and allowed to accumulate at
compound interest.
4. Left with the company to increase the amount of in-
surance carried.
5. Left with the company in order to decrease the num-
ber of payments of premiums.
WRITTEN EXERCISES
!• Find the amount paid in in premiums on each of the following
policies. Find the difference between the net cost of each policy and
the amount received by the insured. How much per year did the pro-
tection he gave his family cost him?
2. Find the net cost for each year and the total net cost.
3. If the holder of these policies had died Nov., 1896, which policy
would have netted the family the better returns?
4. How much more would the family have received than if the holder
had put the same amount of savings in a savings bank paying 4% com-
pound interest, compounded semiannually? (See § 42.)
6. Give reasons for a man's carrying life insurance.
The following table shows the actual dividends that were
applied upon two different kinds of policies.
68
BUSINESS MATHEMATICS
Annual Cash Dividends and Net Cost op Insurance on Pod
OF $1.000— Age 25
Year
20-Pay. Life Issued in 1893
Annual Premium $27.28
Net
Cost
•
20- Yr. Endowment Policy
Issued in 1893
Annual Premium $46.82
>
C
1894
$2.78
$24.50
$2.79
$4
1895
2.95
3.17
1896
3.14
3.56
1897
3.34
3.97
1898
3.53
4.40
1899
3.73
4.85
1900
3.93
5.31
1901
4.17
5.80
1902
4.15
5.0rl
1903
4.26
5.29
1904
4.41
5.58
1905
4.56
5.89
1906
4.71
6.21
1907
4.97
6.54
1908
5.04
6.88
1909
5.21
7.23
1910
6.18
•8.64
ion
6.36
8.99
1912
6.56
9.35
19i:i
7.09
9.72
63. Cash Surrender, Loan, and Paid-up Insurance. —
policies of most companies have certain privileges which
iiiBiired may take advantage of after the policies have 1:
in force for 2 or 3 yr.
These privileges are as follows:^
1. Borrowing money from the company.
2. Hurrendering the policy for a cash payment.
3. Receiving a *^ paid-up** policy which states a fi
amoimt of insurance during the remainder of
without further payment of premiums.
4. Being insured for the face of the poUcy for a fi
number of years and months.
9
INSURANCE 69
For example, after a 20-yr. endowment policy for $1,000,
^ taken at the age of 33 in a certain company, has oeen in force
for 10 yr., the insured can:
1. Borrow $402.41 from the -company on the security of
the policy.
2. Surrender the policy and receive $402.41 in cash.
3. Stop paying premiums and be insured for the re-
mainder of his life for $532.
4. Stop the payment of premimns and be insured for
$1,000 for 10 yr. or receive $460 in cash at the end
of 20 yr. from the date of the policy.
64. Methods of Settlement. — Upon proof of the death
of insured (or in the case of an endowment policy, at
the expiration of the endowment period) the policy is to
be paid by the company. The various ways of settle-
ment are:
!• Payment of cash to the beneficiary.
2. Annual payment of interest during the life of the
beneficiary, and payment of the face of the policy
at the death of the beneficiary.
3- Payment of equal annual instalments for the number
of years specified to the beneficiary.
^' Payment of equal annual instalments for a certain
period (usually 20 yr.), and for as many years
thereafter as the beneficiary shall live. The
amount of each annual payment depends upon
the age of the beneficiary at the death of the
insured.
5. Payment to the survivor of the face of the policy or of
an annuity.
70 BUSINESS MATHEMATICS
65. Lapses. — If the premium is not paid when due, tb
policy lapses. Most companies allow (and many state
make it a law that they shall allow) 30 da. of grace durinj
which time the premium may be paid plus the interest oi
that premium for the overdue time. If the insured wishe
to pay the premium after this 30 days' time, he must under
go another examination by a physician and if successful h
may pay it plus the interest on that premium for the tim
overdue.
66. Fraternal Insurance. — This is an insurance offered b:
different fraternal organizations. The following statement
are answers to a questionnaire sent to officers or thos
thoroughly acquainted with the financial obligations of eacl
of these organizations.
The arabic numbers in each group refer to the same nuni
ber of group I, for example, III (3) refers to the Jersey Cit^
Teacher's life insurance organization.
I. Names of the organizations.
1. Now and Then Association.
2. Name omitted.
3. Jersey City Teachers.
4. Name omitted.
6. United States Immigration Service Beneficial Association.
[J. Amount paid to dependents of members at the death of tb
member.
1. $.25 from each member of the association.
2. $100 (some claim $50).
3. $1 from each member of the association.
4. $1,000 or $2,000.
5. $500 to $3,000.
6. $1 from each member in good standing (average $500).
INSURANCE 71
III. How assessment to meet number II is paid.
1. Assessmants upon the death of member, 20 da. in which to
pay before second notice, with additional fee, called a tax,
is sent.
2. Weekly dues 13^.
3. Assessment of $1 due by the first of the following month for
each member who has died during the month.
4. At age 27 cost 99fi per month. Increase is large for older
men.
5. Age 27-29, 75^ a month; monthly, semiannually (2% off);
annually (4% off).
6. Assessment at death of member.
IV. To whom is money payable? When payable?
1. To one designated by deceased. Immediately u|)on proof of
death.
2. To wife, if living, and if not, then to the children if any sur-
vive; if no wife or children, then to the nearest of kin. At
once upon proof of death.
3. To the nearest of kin. Inmiediately upon proof of death.
4. To person designated.
5. To dependent (usually a relative) designated.
6. To anyone designated by the member. As soon as proof of
death is shown (by official certificate or personal view of
the remains by an officer of the society).
V- Is there a legal prior claim which can be put on this money?
1. None except for unpaid dues to the association.
2. None.
3. None.
4. None.
5. None.
6. None.
vl. Any other information you think advisable.
1. Dues in the association take care of insurance and other
expenses of the club house. When the sick benefit fund
reaches a stated amount an assessment is levied on
members.
2. No reply.
3. A good thing from the point of view that the money is paid
over immediately at a time when the dependents may
72 BUSINESS MATHEMATICS
need it very much, while in an *' old-line" company it
generally takes from 30 to 60 da. to prove death and get
the insurance.
4. ProoF of death is suflficient to get the money.
5. $.10 a month to pay supreme oflScers, etc. Company pays
$60 for funeral expenses.
6. It affords quick relief at small cost, now averages about $16
per year per $1,000. The only salaries paid are $50 per
annum for the secretary and treasurer.
VII. Amount of sick benefits. Are they uniform, or do they differ for
different illnesses or accidents, and may a member pay more
and receive more, or are they the same for all?
1. Sick benefits of $5 per week, not paid for 1 week's illness.
2. $4 per wk. for 13 wk.
d|>0 H i( II II n
d»2 << <* << << <<
%\ " " ^' '^ '^
Uniform.
3. None.
4. Local organization (lodge) $2.50 dues quarterly. Lodge
raises money for sick members.
5. $.50 once in 3 mo. for home expenses. Sick and accident
benefits cost $.50 a month, pay $6 per wk. for 12 wk-
Also pay $750 for loss of both eyes, or both legs or both
arms.
6. Amount of payments the same for all, depending upon the
number of members in good standing. No accident ^r
sick benefits. An annual ball or social affair is held, whicl^
usually nets about $200 which pays incidental expenses-
67. Accident Insurance. — Accident insurance is insurance
which covers loss by accidents. Accident and health insur-
ance is insurance which covers loss to the insured througl^
accidents, loss of health, or both.
Some important data for the layman to know follow.
68. What Constitutes the Occupation?— The profession,
business, trade, employment, or any vocation followed as »
INSURANCE 73
means of livelihood constitutes the occupation. Should a
person engage in work, for hire, in any other occupation,
such shall constitute his occupation.
69. Greatest Hazard Determines the Classification. —
If the applicant has more than one occupation, all occupa-
tions must be named in the application. The one involving
the greatest hazard determines the classification.
70. Age of Applicants. — Applications will not be accepted
from persons under 18 nor over 65, but those who insured
before the age of 65 are usually carried to the 66th birthday.
Disability or health insurance is issued only to male persons
between 18 and 59
71. Beneficiaries. — They must always be named in the
application blank, and must be persons having an insur-
able interest in the Ufe of the insured, as a wife, father,
mother, brother, sister, or other relative, a dependent or a
creditor. If the insured does not care to mention a bene-
ficiary, he can state ''my estate" or ''my executors, ad-
Diimstrators, or assigns," in which case the money goes
into the estate and has the same legal meaning as cash in
the bank.
72. Scope of Policies. — The poUcies usually cover acci-
dents sustained while residing or traveling for business or
pleasure in any part of the civilized globe; while discharging
the usual duties pertaining to the occupation named in the
policy; while pursuing any ordinary form of pleasure or re-
creation; and while engaged in athletic exercises usually in-
dulge in by business and professional men.
74 BUSINESS MATHEMATICS
73. Limit of Risk. — The maximum amount of death
benefit and weekly indemnity most companies will carry on
any one risk (exclusive of the double clause) is stated op-
posite each classification, and cannot be increased except by
special authority from the home oflSce.
74. Prohibited Risks. — Persons are not insurable who are
blind, deaf, or compelled to use a crutch or cane; who are
insane, demented, feeble-minded, or subject to fits; who
have suffered from paralysis or are paralyzed; who are in-
temperate, reckless, or disreputable; who are suffering from
any bodily injury; or have any deformity, disease, or in-
firmity.
75. Cripples. — A person who has lost a hand, or a foot, or
the sight of an eye, but is otherwise an able-bodied and ac-
ceptable risk, and whose occupation is not classed as more
hazardous than ordinary, may be insured at an advanced
rate by applying to the home office; but one who has lost
a leg above the knee, or who is obliged to use a crutch or
cane, or who, having any of the aforesaid defects, is engaged
in an occupation more hazardous than ordinary, will not be
accepted.
76. Insurance of Women. — A woman will not be accepted
for weekly indemnity unless engaged in a stated business or
employment from which she derives a regular income on
which she is dependent for support. If in receipt of any
such income, she may be insured for death benefit* and
weekly indemnity (without doubling clause) at the rate
named for her occupation, but such policy will not be issued
for more than $3,000 death benefit and $15 weekly indem-
INSURANCE 75
nity. Housewives, housekeepers, boarding-house keepers,
and canvassers are not to be included in the above.
A woman who, by reason of her circumstances and posi-
tion in life, is not Uable to loss or suspension of income on
account of a disabling injury, may be insured under an acci-
dental death policy at regular rates, but in no case for more
than a maximum amount of $5,000.
DisabiUty or health policies are not issued to women.
77. Overinsurance. — Agents must guard against over-
insurance of an applicant. The weekly indemnity should
not equal the actual money value of the insured's time or of
Ws weekly salary.
78. Designations of Occupations. —
A
1
Select
B
2
Preferred
BS
2+
Extra preferred
C
3
Ordinary
D
4
Medium
DS
5
Special
E
6
Hazardous
F
7
Extra hazardous
X
Prohibited
79. Industrial Insurance. — This is the kind of life insur-
^lice in which small investments can be made. Premiums,
^tead of being paid in a large sum once, twice, or four
times a year in advance, are deposited in small sums, 3^,
¥> 10^, 15pf, 20^ a week, and so on, in exchange for which
the company gives a Ufe insurance policy payable at death
0^ the insured, or after a certain number of years. Indus-
Wal insurance furnishes a means of saving, a little at a time.
76 BUSINESS MATHEMATICS
week by week. Every time a 6^ premium is deposite-
something is saved. Every man who works for wages eg
secure the insurance protection which his means aflFori
without making a great strain on his income.
The companies usually send agents to the home ever
week to collect the -premiums. This is done for the coi
venience of the policyholder, but if for any reason the ager
should fail to call, the money may be sent to the home oflSc
of the company. One company was started in 1866, an
no^ has more than twenty million policies of this kind i
force. Any member of the family over the age of 1 yr. an
up to the age of 65 yr. next birthday, who is in good healtl:
can obtain one of these policies.
Illustrative Example. Suppose a young man of 25 pays 10^ a week t
the company. He secures a policy upon which he has to pay the week!
premium each week and the company agrees to pay in case of his deat
the sum of $180, provided he has paid all premiums for 6 mo. or more u
to the time of his death. If his death should occur any time within th
first 6 mo. and he has paid regularly up to that time, the company woul
pay one-half of the $180, even if he died immediately after the deliver
of the policy to him. Some companies do not require the payment c
further premium on any industrial policy after the insured reaches 7
yr. of age. Industrial endowment, cash values, paid-up insuranc(
paid-up endowment, and automatic extended insurance are common i
this kind of insurance.
CHAPTER VII
EXCHANGE
80. Domestic Exchange. — Exchange is the pajrment of a
debt for goods bought, or for some other purpose without
the sending of money. Domestic exchange is such pay-
ment between persons or corporations in the same country.
ORAL EXERCISES
!• State some objections to sending money through the mails even if
the letter is registered.
2. State some objections to sending it by express.
81. Methods of Exchange. — The paying of debts without
the transmission of real money is effected by:
1- Personal checks
2- Postal office money orders
3- Bank drafts
4- Express money orders
^' Telegraphic money orders
^' Conamercial drafts
If a merchant sends his check to a manufacturer in De-
troit, the latter will deposit it in his bank, and it will be
credited to his account. The check is then returned through
proper channels to the bank of the maker, and the merchant
IS charged with that amount on his account.
If A owes B, in another town, $25, and A does not have a
hank account, he may go to the post-office and buy a money
77
78 BUSINESS MATHEMATICS
order for the amount payable to B. This he sends by mail
to B, which pays the debt. Find out from your post-oflSce
who is liable in case the money order and letter are lost or
destroyed.
If A so wishes, he can go to a bank and pay the cash for a
bank draft, payable to B, for the amount of the debt and
then send it to B. B can take it to his bank and get the
cash or have that amount credited to his account.
Or A can go to the express office and pay cash and buy an
express money order payable to B, and send this on to B.
Or A can go to a telegraph office, pay cash and buy a
telegraphic money order payable to B, and can send this by
telegraphic communication to B in his city. B then can ge:^'
the money at the telegraph office in his city.
Or if A is a merchant to whom a debt is due from sonc^^
person who lives in B^s town, A may send B a commerci-®^
draft drawn on C for the amount that A owes B.
82. Postal Money Order. — This is a government order 7" ^^
a post-office in one place to a post-office in some other pla^^^^^
to pay a stated amount to a specified person. In order ^
obtain a postal money order a person must fill out an appd^^*^'
cation blank which states:
1. The name and address of the payee.
2. The amount to be paid.
3. The name and address of the one who buys the ord«
The rates charged for postal money orders are:
For $ 2.50 or less
$.03
From $30.01 to $ 40.00
^^.10
From 2.51 to $ 5.00
.05
u
40.01 '* 50.00
.18
5.01 ^' 10.00
.08
tl
50.01 *' 60.00
.20
" 10.01 •* 20.00
.10
It
60.01 " 75.00
.25
*' 20.01 " 30.00
.12
tt
75.01 " 100.00
.30
EXCHANGE 79
The largest amount for which a single postal money order
may be issued is $100. If larger sums are to be sent, one
must purchase additional money orders. These orders are
to be presented for payment at the post-oflSce on which they
are drawn, or at a bank. If they are not presented within
1 yr. or if they are lost, a duplicate may be obtained from
Washington upon proper presentation of evidence of such
WRITTEN EXERCISE
!• Find the cost of the following money orders:
(a) $ 17.25
(b) 4.50
(c) 35.00
(d) 64.13
(e) 125.00
83. Express Money Orders. — An express money order
(Form 2) is quite similar to a postal money order. It is a
written order by one express agent to another agent to pay
a stated sum to a specified person. The largest amount of an
express money order is $50. If one desires to send more he
must purchase additional orders. Express money orders
"^ay be indorsed and transferred in a manner similar to bank
^'rafts and checks. Rates are the same as for postal money
orders.
WRITTEN EXERCISES
1. Find the total fee for the transfer of $125 by express money order.
2. What is the best way to buy express money orders for $87.75, and
what would be the total cost of the same?
3. A man having a bank account prefers to send a check of $37.50 to
pay a bill he owes, rather than express money order or postal money
order. Why?
80
BUSINESS MATHEMATICS
^
EXCHANGE 81
84. Telegraphic Money Orders. — Sometimes it becomes
necessary to send money immediately for some special pur-
pose, say one of the following :
1. To banks to meet maturing obligations.
2. To fire and life insurance companies for premiums.
3. To travelers and traveling salesmen.
4* To students and pupils at schools, seminaries, colleges,
etc.
5. To guarantee purchases.
6. To accompany bids for contracts.
7. For payment of bills.
8. For the purchase of railroad, steamship, and theater
tickets.
9. For purchases of all kinds.
10- For hoUday gifts and other remembrances.
11. For memorial occasions and anniversaries.
12. For payment of taxes and assessments, and for all
other purposes requiring the quick remittance of money.
Illustrative Example. To any place where the 10-word telegram rate
^60^ one can send $50 and a 15-word message for $1.13.
WRITTEN EXERCISES
!• What is the total charge for sending $30 by telegraph to a place
^*iere the cost of a 10-word message is 60fi?
2* Find the cost of sending $250 by telegraph to the same place.
3' A man finds that he is in a strange place and needs money from his
^ at once. He telegraphs for $75. The charge for a ten-word message
^tween those places is $.72. Find the cost, including the charge for the
message.
4. A man pays a bill of $65.50 by postal money order, another bill of
123.65 by express money order, and another bill of $115 by telegraph.
If the 10-word rate is 42fi, find the total cost to him.
82
BUSINESS MATHEMATICS
In cases like these money may be sent by the use of a
telegraph money order. The rates for such orders are ob-
tained in a table like the one following.
Table op Charges for Telegraph Money Orders*
For a Transfer
OF
$ 25.00 or less
25.0
60.0
75.0
100.0
200.0
300.0
400.0
500.0
600.0
700.0
800.0
900.0
1.000.0
1.100.0
1.200.0
1.300.0
1.400.0
1,500.0
1.600.0
1.700.0
1,800.0
1.900.0
2.000.0
2,100.0
2.200.0
2.300.0
2,400.0
2,500.0
2,600.0
2,700.0
2.800.0
2.900.0
to $
to
to
to
to
to
to
to
to
to
to
to
to
to
to
to
to
to
to
to
to
to
to
to
to
to
to
to
to
to
to
to
50.00
75.00
100.00
200.00
300.00
400.00
500.00
600.00
700.00
800.00
900.00
1.000.00
1,100.00
1.200.00
1,300.00
1,400.00
1.500.00
1.600.00
1,700.00
1,800.00
1,900.00
2,000.00
2,100.00
2.200.00
2.300.00
2.400.00
2.500.00
2,600.00
2,700.00
2.800.00
2,900.00
3.000.00
To ANY Place Where the 10- Word Telegram Rate Is
f.30
$0.68
.78
1.03
1.28
1.53
1.78
2.03
2.28
2.53
2.78
3.03
3.28
3.53
3.78
4.03
4.28
4.53
4.78
5.03
5.28
5.53
5.78
6.03
6.28
6.53
6.78
7.03
7.28
7.53
7.78
8.03
8.28
8.53
f.36
$0.74
.84
1.09
1.34
1.59
1.84
2.09
2.34
2.59
2.84
3.09
3.34
3.59
3.84
4.09
4.34
4.59
4.84
5.09
5.34
5.59
5.84
6.09
6.34
6.59
6.84
7.09
7.34
7.59
7.84
8.09
8.34
8.59
$.42
$0.80
.90
1.15
1.40
1.65
1.90
2.15
2.40
2.65
2.90
3.15
3.40
3.65
3.90
4.15
4.40
4.65
4.90
5.15
5.40
5.65
5.90
6.15
6.40
6.65
6.90
7.15
7.40
7.65
7.90
8.15
8.40
8.60
$.48
$.60
$.72
$.90
$0.91
$1.03
$1.22
$1.45
1.01
1.13
1.32
1.55
1.26
1.38
1.57
1.80
1.51
1.63
1.82
2.05
1.76
1.88
2.07
2.30
2.01
2.13
2.32
2.55
2.26
2.38
2.57
2.80
2.51
2.63
2.82
3.05
2.76
2.88
3.07
3.30
3.01
3.13
3.32
3.55
3.26
3.38
3.57
3.80
3.51
3.63
3.82
4.05
3.76
3.88
4.07
4.30
4.01
4.13
4.32
4.55
4.26
4.38
4.57
4.80
4.51
4.63
4.82
5.05
4.76
4.88
5.07
5.30
5.01
5.13
5.32
5.55
5.26
5.38
5.57
5.80
5.51
5.63
5.82
6.05
5.76
5.88
6.07
6.30
6.01
6.13
6.32
6.55
6.26
6.38
6.57
6.80
6.51
6.63
6.82
7.05
B76
6.88
7.07
7.30
7.01
7.13
7.32
7.5^
7.26
7.38
7.57
7.80
7.51
7.63
7.82
8.05
7.76
7.88
8.07
8.30
8.01
8.13
8.32
8.55
8.26
8.38
8.57
8.80
8.51
8.63
8.82
9.05
8.76
8.88
9.07
9.30
$1.20
$1.88
1.98
2.23
2.48
2.73
2.98
3.23
3.48
3.73
3.98
4.23
4.48
4.73
4.98
5.23
5.48
5.73
5.98
6.23
6.48
6.73
6.98
7.23
7.48
7.73
7.98
8.23
8.48
8.73
8.98
9.23
9.48
9.73
$3,000.00 and up add $.20 for
thereof to the charges for $3,000.
each additional one hundred dollars or fractioi
Includ
ng 1 5- word message.
EXCHANGE 83
86. Commercial Drafts. — Commercial drafts arc used as
an effective means of collecting an account overdue. It is an
order drawn by the party to whom the money is due, asking
the debtor to pay a specified sum of money to the drawer or
to another person stated in the draft.
86. Kinds of Commercial Drafts. — 1. One in which the
debtor is asked to pay the money to the drawer. This kind
of a draft is collected through a bank.
2. One in which the debtor is asked to pay the money to a
third party.
87. Business Use of Commercial Drafts. — Company A of
Binghamton, N. Y., sells A. H. Jones of Albany a shipment
1500.00 Binghamton, N. Y., April 14. 1921
At sight pay to the order of
First National Bank of Binghamton
Five Hundred and OO/lOO Dollars
Value received and charge to the account of
To A. H. Jones, A Company
Albany, N. Y.
No. 589
Form 3. Sight Draft
of shoes worth $500. Company A wishes to get a quick
return of the cash, or a promise from A. H. Jones that the
latter will pay for them on a specified date. In order to do
this Company A uses what is known as a bill of lading to-
gether with the commercial draft. Company A receives
from the Binghamton freight office where these goods are
shipped, this bill of lading, which is a written statement
showing that these goods have been received there for ship- j
84 BUSINESS MATHEMATICS
ment. A. H. Jones must receive this bill of lading and pre-
sent it to the freight office in Albany before he can obtaiii
the goods. Company A takes this bill of lading to the First
National Bank in Bingham ton and deposits it along with a
draft similar to that shown in Form 3.
The First National Bank of Binghamton sends this billrf
lading and the draft to, say, the Albany Second National
Bank, which in turn takes the draft to A. H. Jones for pay-
ment. If the latter pays it (or accepts it, in case it is a time
draft), he then receives the bill of lading and in turn can se-
cure the shoes from the freight office. Should A. H. Jones
refuse to pay the draft, then the Albany Second National
Bank so informs the Binghamton bank. The latter thffl
notifies Company A, who must then sell the shoes in some
other manner.
In case it is a time draft and A. H. Jones accepts it, the
draft is then sent back to Company A, and the latter can
take it to the bank and have it discounted and the proceeds
credited to their account in the bank, or Company A can
ask the Albany bank to discount it and send them the pro-
ceeds. If a time draft is discounted, time is figured from the
day it is discounted.
88. Advantages of the Commercial Draft. — 1. By the
above method the purchaser must pay for the goods before
he receives them.
2. If an account is past due, and a draft is sent to be
collected by the purchaser's bank, it is often very effective
in securing the money.
3. Company A, if they have good credit in their hom€
bank, can immediately increase their funds because the
home bank will discount the draft at once for them.
EXCHANGE 85
4. A. H. Jones cannot send a check which is worthless
and thus cause Company A much loss in money and trouble.
5. The purchaser cannot claim that he has not received
the bill of the goods.
6. The purchaser cannot maintain that he has already
sent a remittance in some other way.
WRITTEN EXERCISES
1. If a commercial draft of $3,000 payable 90 da. after date were dis-
Bounted 15 da. after date, and the rate of discount were 6%, and the
cost of exchange were i %, find the net proceeds of this draft.
2. Suppose that R. D. Ford & Co. of New York were to sell A. H.
Williams of Chicago a bill of goods amounting to $1,500 on Apr. 14.
Suppose that they draw a 60-da. draft on A. H. Williams through the
First National Bank of New York. If a Chicago bank should buy thia
draft at 6% discount and exchange were i*o%i what are the proceeds for
R. D. Ford & Co.?
3. On Apr. 14, you purchased an invoice of goods of $100 from 11. R.
Brown & Co. of Albany, N. Y. Terms, 30-da. draft from date of sale,
less 2%. On Apr. 16, you received by mail the draft dated Apr. 14,
and due in 30 days. You accepted it and returned it to H. R. Brown &
Co. When will you have to pay this draft?
89. Terms Used in Domestic Exchange. — The maker or
drawer of a draft is the person who signs it.
The drawee is the one who is to pay it.
The payee is the one to whom the money is paid.
Par means that a draft is bought exactly for its face
value.
Premium means that the buyer of the draft must pay
Diore than its face value. Exchange is then said to be at a
premium. This occurs when, say, the banks of St. Louis
owe the banks of New York a large sum, because the banks
of St. Louis must either pay for the transportation of the
^oney to New York or pay interest upon it. If A lived in
86 BUSINESS MATHEMATICS
St. Louis at that time he would have to pay a premiu
cause his draft would increase the amount that tl
Louis banks owe New York banks.
Discount is a term used when the draft can be b<
slightly below face value. This might occur if D in
York were to send a draft to A, in St. Louis (given i
above paragraph) at the same time, because it would '.
the balance which New York had against St. Louis.
Exchange on checks is a small sum usually charged
bank for paying a check from another city, since the bj
put to some expense in sending the check back and c(
ing the money on it. The siuns vary from 10^ up, bu
ally are not more than 25^.
A sight draft is one which must be paid immediately
it is presented.
A time draft is payable after a stated time. It i
sen ted at once to the drawee; if he accepts it, he writ
word ** Accepted" across its face and signs it as well a
ing it. This shows that he promises to pay it, and
makes it a promissory note.
90. Bank Drafts. — A bank in a small town, for ins
will keep funds in some large city bank on which it car
checks just as an individual can draw a check on his
for some other person.
A bank draft (Form 4) is an order drawn by one
against its deposits in another bank.
91. Usefulness of Bank Drafts. — A bank draft is
by the bank while a check is given by a person or a
The cashing of the former is considered safer than that
latter.
EXCHANGE 87
There is less expense in collecting a bank draft than a
personal check because the former are drawn on banks in
large cities while the latter are very often drawn on small
banks in the country.
Explanation: The cashier of the Franklin National
Bank of Franklin is W. D. Ogden. This bank deposits its
funds with the Chase National Bank of New York. When
PsANKLiN National Bank
No. 1687
Franklin.
N.
Y.. May 1, 1921
Pay to the order of
> . • • • !£•
H.
Williams . . .
• • • a
$25 ,V\i
Twenty-five and i%
• • • •
DoUars
To Chasb National Bank
W. D.
Ogden,
Cashier
of New York
Form 4. Bank Draft
the latter bank pays the amount of this draft it will reduce
the balance of the account of the Franklin National Bank for
this amount.
E. H. Williams, who lives in Franklin, wants to pay this
amount to T. H. Morse of Albany. He accordingly buys
this draft to send to Morse. He could have had it made
payable to Morse, but if it should reach Morse with no
letter to explain it, Morse might not credit it to the right
^an. Therefore Williams has it made payable to himself
and, after receiving it, indorses it about as follows: ^'Pay
to the order of T. H. Morse," and then signs his (Williams)
name to it. This assures him of the proper credit by Morse
and also acts as a receipt if Morse afterwards receives it and
uses it.
Mr. Morse receives it and has it cashed at his bank, the
Albany National Bank. The Albany National Bank col-
Ie<^« it an rVyilowa: Thia 'Tank has x Yeir Y«xk: <Ziiy^
rlu^ Chemirai Xahomu Bunk. It '"hPTKore^ ^esufr the
f A ♦^he Ohi*mical yarioiiai Bank js a 'iepiHr. The C
Xattionai IWik -»pniia 'he imrr to The ^Tleami^ Qbiaae.
in t»im 'Viilef^trf :r. from 'he CTmse ^^anonaL Bank. The!
hftntc iip^n rjn^-in^ *he irafr '*hargfy The ^imounr <if rfae
fA the Praniciin Nairionai Bank.
9i, C&at (tf Bflufc Drafts. — The Fmnkim 3(atifiiiaL
wo»iW make a .^mail ^hars5& for die lirafr. This chac^
^^ilM ^x^hanjp*. The exi^hange woakt fac paid by the
f'hiwf^r ''Willianw . The usual dbarisB is t'i% with, a mim*
mum ^harjre of Ite,
The foil<>win<( rates are in cent? for f 1.000 faerweoi these
dtie«,
Chi«*aari->^'^wyork $.05 ifiscounc
Hfiri f rin/riscrr-X<*w York J!5 pRmiuziL
h-jftor. N'^TT York i)5 pi^minizL
H^. r^/'^.s-N'^w York j3) discount
tthi<itr»trrt Exam^e 1. Ar 20e pn»innxzii find the cast oc a SkOQOcbifl
ExFJ^AMATTo.v: ^',,(lflf) = 3 thousands
innstratfve Example 2. If 'oc/^rh^nt^f; k^i selling at 25e discount, find Uw
cr;»tof?i$.VJ^J^)'Jraft.
•SoLrmoN': 3/ $.25 -■ S.75, rli-oount on $3,000
%yf(Jfl) - $.75 - $2,(f(i(K2o
WRITTEN EXERCISES
Vmug, the aU»vf; nit^n find :
1. The cmt rif a draft nn N^w York at f 'hiraKf> for $13,000.
2. The c^jst of a $12,H(K) cJraft at San Franrisro on New York.
t. The cost of $6,500 draft on New York at liostoc.
EXCHANGE
89
93. To Find the Proceeds of a Draft. —
Illustrative Example 1. Find the proceeds of a sight draft of $3,500,
if the collection and exchange is i%.
Solution:
4% of $3,500 = $4,375
$3,500 - $4,375 = $3,495,625
Illustrative Example 2. Find the proceeds of a 60-da. commercial
draft of $4,000, if sold the day it was dated at J % discount, when money
is worth 6%.
Solution:
$40 = 60-da interest.
i% of $4,000 = $10
$40 + $10 = $50, total discount
$4,000 - $50 = $3,950, proceeds
WRITTEN EXERCISES
Find the proceeds of the following sight drafts :
1. $1,800 draft when collection and exchange is J%.
2. $8,600 draft when collection and exchange is i'o%.
3. Find the proceeds of a 90-da. draft of $1,250.75 if sold at J% dis-
count, when money is worth 6%.
4. Find the proceeds of a draft for $75,000 when collection and ex-
change is i%.
6. If collection and exchange is i% what are the proceeds of a draft
for $8,750?
6. What are the proceeds of a sight draft for $1,375, if collection and
exchange is \%.
94. Postal Money Orders, Bank Drafts and Trade Ac-
ceptances. — (a) The following dilTerences exist between a
postal money order and a bank draft:
1. A postal money order must be presented to the post-
office on which it is drawn, or to some bank which
can present it to that post-office.
2. A bank draft can be cashed at any bank.
90 BUSINESS MATHEMATICS
3. A postal money order is to be indorsed but once.
4. A bank draft may be indorsed any number of times.
5. A postal money order will not be cashed until th<
post master I'eceives a notice of such order from th<
office which wrote the order.
6. A bank draft can be cashed as soon as it is presented
(b) A trade acceptance is like an ordinary bill of exchangi
except that it has a written guarantee upon it that the in
debtedness has originated on an exchange of merchandise.
The advantage of the trade acceptance is shown by th
following example:
A firm in Boston buys from a firm in New York $1,000 worth of goock
Simultaneously with the shipment the seller draws on the buyer a draf
at 90 days from date or sight (according to the terms of sale) and mails i
to the latter with the invoice and the bill of lading. The usual form c
draft is used with this additional clause; "the obligation of the accepts
hereof arises out of the purchase of goods from the drawer."
The buyer accepts the draft by writing across the face of it, " Aceepte
Payable at Bank, Boston." He dsM
and signs this acceptance and returns the accepted draft to the seller i
New York. The document is now a ** trade acceptance," becomin
liue for payment in Boston 90 days from the date of the draft, if draw
"aft<»r date.. " but in 90 da. from date of acceptance, if drawn "90 day
mKlit."
1 f the seller requires the money represented by this acceptance 1
may take the accepted draft to his bank and the bank will purchase
juovidtHl the names appearing thereon seem satisfactory. The bank
I urn \\u\y rediscount the acceptance with the federal reserve bank, as t1
ol»hgalit)n arises out of the purchase of goods, and thus falls under tl
ju^iNinitins of the Federal Reserve Act. The market rate of discount
^ Iw^iitul by the hank for its courtesy.*
* "J v\ i\\ , NiU'bert, "Principlesof Foreign Trade." New York, Rona
k'u ... UMU.
EXCHANGE 91
Manufacturers, wholesalers and jobbers are urging the use
of trade acceptances, because they do away with much book-
ing and collections. RetaUers ai^e opposing it because they *
have to keep a close watch on their bank accounts to meet
the payment of these acceptances.
95. Foreign Exchange and Foreign Money. — Oiur trade
with foreign countries compels us to change United States
money into the value of different foreign money, jis well as
to change foreign money into our money. International
debts are settled by means of bills of exchange, postal
money orders, bankers' bills, commercial bills, and the
sending of actual money. Foreign exchange also deals with
travelers' checks, letters of credit, etc.
96. How to Find the Value of Foreign Coins. — Find the
value of the foreign coin in United States money. Multiply
this value by the number of coins.
Illustrative Example. Find the value of £1,000 English money in
United States money.
£1 = $4.8665
1,000 X $4.8665 = $4,866.50
■■. Multiply the number of coins by the value of one coin.
97. To Change United States Money into Foreign
Money. —
Illustrative Example. Find the value of $1,000 United States money
in pounds English money.
$1,000 ^ $4.8665 = £205.486
.*. Divide the number of dollars by the value of the foreign coin. i
92
BUSINESS MATHEMATICS
Values of Foreign Coins in United States Money
Country
Legal
Standard
Monetary
Unit
Value in Terms
OF U. S. Money
.q
Rfmarks
Argentine Republic
Austria-Hungary ..
Canada
gold
gold
gold
gold, silver
gold
gold
gold, silver
gold
peso
krone
dollar
franc
mark
pound sterling
lira
yen
$0.9648
.2026
1.00
.193
.2382
4.8665
.193
.4985
Greatly depreciated
France
Exchange value $.0001
Greatly depreciated
Exchange value $3 Jlj
Exchange value $'Olflj
Exchange value $.554
Germany
Great Britain
Italy
Taoan
98. Bills of Exchange. — Bills of exchange are drafts
of a person or bank in one country on a person or a
bank in another country. They are divided into three
classes :
1. Bankers' bills (Form 5).
2. Commercial bills drawn by one merchant on an-
other.
3. Documentary bills, which are drawn by one merchant
upon another and have a bill of lading attached,
together with an insurance policy covering the
goods en route.
Bills of exchange are usually issued in duplicate, called the
original and the dupHcate. They are sent by different mails
and the payment of one of them cancels the other. Some-
times the original is sent, and the duplicate is placed on file
and sent later if needed.
99. Par of Exchange. — This is the actual value of the
pure metal of the monetary unit of one country expressed in
terms of the monetary unit of another country.
f Uodn |([D<1 SlUSUIIOOQ J3AI[3Q
94 BUSINESS MATHEMATICS
Illustrative Example. One pound sterling ( £1 ) contains 113.0016 gr.
of fine gold. $1 contains 23.22 gr. of fine gold. 113.0016 -5- 23.22 =
4.8665. Therefore, £1 = $4.8665, which is called the par of exchange
between the United States and England.
100. Rate of Exchange. — This is the market value in one
country of a bill of exchange of another country.
The price paid for a bill of exchange is constantly fluctuat-
ing, due Uke other things to supply and demand. If the
United States should owe Great Britain the same amount
that Great Britain owes the United States, then exchange
would be at par. If the United States should owe Great
Britain more than Great Britain owes the United States,
then exchange in the United States would be at a premium
and in Great Britain it would be at a discoimt. If Americans
are exporting much more than they import, they will have
many bills of exchange in the form of documentary bills to
sell the American banker, and supply will exceed demand,
thus causing exchange to fall below par. On the other hand,
if Great Britain is exporting to the United States much
more than the United States is exporting to Great Britain,
then bills in the United States will be scarce and sell at a
premium.
101. Quotations of Rates of Exchange. — Exchange on
Great Britain is usually quoted at the number of dollars
to the pound sterling; 4.86 means that a pound bill on Lon-
don will cost $4.86.
Exchange on France, Belgium, etc., is quoted at the num-
ber of cents to the franc. Thus, exchange on France quoted
at 19.02 means that 1 franc costs 19.02^.
Exchange on Germany is quoted at the number of cents
to 1 mark. Thus, 8.5 means that SM will purchase 1 mark.
EXCHANGE 95
The following foreign exchange rates were quoted on the
dateiDdicated:
Normal Ratbs
ofExchangb Pbb. 1. 1921
$4.8665 London $3.83J
19.30^ Paris 7.04^
19.30^ Belgium 7.39ff
40.20^ Holland 33.85^
19.30ff Italy 3.64^
19.30^ Spain 13.94^
19.30^ Switzerland 15.95^
23.83ff Germany 1.58^
•
Qlastrative Example 1. How to find the cost of a banker's bill on
London:
What is the cost of a £500 draft on London bought at $3.83 J?
SoLunoN :
$3,835, cost of 1 pound
500, number of pounds
$1,917.50, cost of 500 pounds
Dlttstrative Example 2. How to find the cost of a draft on Paris:
What is the cost of a 200-franc draft on Paris, bought at 7.01?
Solution:
$.0704, cost of 1 franc
200, number of francs
$14.08, cost of 200 francs
Illustrative Example 3. How to find the cost of a draft on Berlin:
What i» the cost of a 2,000-mark draft bought at 1.58?
Solution:
$.0158, cost of 1 mark
2,000, number of marks
$31.60, cost of 2,000 marks
96 BUSINESS MATHEMATICS
WRITTEN EXERCISES
Using the foregoing quotations, find the cost of drafts of each of the
llowing amoi
quoted prices:
bllowing amounts, first at the normal rate of exchange, and second at the
1. £200 6. 1,800 marks
2. £1,500 6. 240 marks
3. 500 francs 7. 150 guilders (Holland)
4. 600 marks 8. 350 guilders
9. John Doe of New York owes T. H. Jones of London, £300 8s 5d.
He buys a foreign draft of this amount to pay his bill. Suppose that
exchange on London is $4.75, what is the cost of the draft?
102. Letter of Credit. — This is a circular letter (Form 6)
issued by some bank or banker, introducing the holder and
instructing the bank^s correspondents in stated places of the
world to pay the holder any amount up to the face of the
letter.
The holder deposits with the bank cash or securities to
the face amount of the letter of credit. The purchaser must
sign this letter at the purchasing bank in order that he may
be properly identified by their correspondents. He also
writes other copies of his signature which the bank forwards
to its correspondents.
When the holder requires money he presents the letter to
some one of the banks specified (as a correspondent) to-
gether with a draft drawn by the holder for the amount re-
quired. If the signatures agree he is paid the sum asked for,
and such sum is indorsed on the back of the letter by the
I)aying bank. The bank making the last payment retains
the letter of credit and returns it to the drawee. Banks
usually charge a connnission of 1% for issuing a letter of
credit.
98
BUSINESS MATHEMATICS
103. Travelers' Check. — This is a circular check (1
7) which is made payable for a stated amount in the
rency of the foreign countries nained on the face of the cl
They are usually in amounts of $10, $20, $50, $100,
$200. A conmiission of \% is the customary charge.
104. Postal Money Orders. — The following rates pr
for foreign postal money orders, if payable in Aug
Belgium, Bolivia, Cape Colony, Costa Rica, Denn
Egypt, Germany, Great Britain, Honduras, Hongli
Hungary, Italy, Japan, Liberia, Luxemburg, New S
Wales, New Zealand, Peru, Portugal, Queensland, Ru
Salvador, South Australia, Switzerland, Tasmania,
Transvaal, Uruguay, and Victoria.
For Orders
COJT
For Orders
From $ .01 to $ 2.50
$.10
From $30.01 to $ 40.00
2.51 " 5.00
.15
*' 40.01 '
' 50.00
5.01 *' 7.50
.20
" 50.01 *
* 60.00
7.51 " 10.00
.25
" 60.01 '
' 70.00
10.01 " 15.00
.30
" 70.01 '
' 80.00
15.01 " 20.00
.35
" 80.01 *
' 90.00
'' 20.01 " 30.00
.40
" 90.01 '
* 100.00
If payable in any
other for
eign country.
For Ord£rs
Cost
For Cri
jers
From $ .01 to $10.00
$.10
From $:0.01 t
0$ 60.00
10.01 " 20.00
.20
" 00.01 '
' 70.00
" 20.01 " 30.00
.30
" 70.01 '
* 80.00
" 30.01 " 40.00
.40
80.01 '
' 90.00
" 40.01 ** 50.00
.50
" 90.01 '
' 100.00
WRITTEN EXERCISES
1. Find tlio cost of a i)ostiil money order for $35 sent to a pen
Canada.
100 BUSINESS MATHEMATICS
2. What will it cost me to buy $250 worth of orders for a man in Paris?
3. How much would it cost me to buy the following orders made pay-
able to myself in the following countries :
Amount of Order
Payable in
$25
London
50
Paris
25
Constantinople
50
Calcutta
4. A sends 200 francs to B in Switzerland. If 1 franc costs 19f5, find
cost of postal order.
105. Use of Commercial Bills. — If A in England owes B,
a merchant in the United States, and B wishes to collect, he
may draw up a commercial bill and let his bank collect it in
a manner similar to the collection of sight or time drafts in
domestic exchange.
106. Immediate Payment by Bill of Lading. — Mr. Jones,
a merchant in the United States, sends a bill of goods to Mr.
Williams, who lives in London. Mr. Jones delivers the
goods to the transportation company and receives a bill of
lading. He also insures the goods against loss in transit and
receives the certificate of insurance from the insurance
company. Mr. Jones then draws a bill of exchange on Mr.
Williams, and attaches the bill of lading and the insurance
certificate to this bill of exchange. All these papers are
indorsed to the order of the bank which buys the draft. Mr.
Jones then receives pay for the goods he has shipped. U
the goods are lost the bank is reimbursed by the insurance
company. If the bill is uncollectible, the goods are taken
over by the bank. The United States bank indorses theses
papei's and sends them to a foreign bank, thereby receiving^
credit by the latter for the amount. The foreign bank then
collects the bill.
EXCHANGE 101
MISCELLANEOUS EXERCISES
1. At 25f5 a word and 1% of the amount, find the cost of a 25-word
;afc»le money order from New York to Paris for 20,000 francs, if exchange
s cjuoted at 7.04 (cents).
2. An exporter sold to a broker the following bills of exchange: £500
L-t 3.90; 1,250 francs at 7.18; and 12,400 marks at 1.58. Find the total
:iet proceeds, if the broker charged J% for collection.
3. What would be the cost of a London draft for £220, exchange being
quoted at 4.85?
4. A man sends his son $200 to London. How much English money
would the son receive if exchange is quoted at 4.8665?
6. A Frenchman presents to a New York bank a draft for 1,000 francs.
Wbat money should he receive if exchange is quoted at 7.21?
CHAPTER VIII
TAXES
107. Kinds of Tax. — Many businesses as well as ms^iy
persons are required to pay some form of tax. A tajc is
money levied upon a person or property for the payment ^^
the public expenses.
A direct tax is a tax levied on a person, his property, or b^
business. If it is upon his business it usually takes the fori^
of a license fee, and if upon his person it is called a poll ta^*
An indirect tax is a tax (called a duty) on imported goad^
or a tax (called an internal revenue) on tobacco products-
The latter tax need not be paid on goods exported.
* An income tax is a tax on the income of a person or a firro-
An excess profits tax is a tax upon the excess profits of ^
business.
The taxes are levied by officers called assessorSi or t>y
people called income tax collectors.
108. Purpose of Taxes. — The purpose of taxes is to me^^
the expenses of the government. These purposes may t>^
classified somewhat as follows :
1. National taxes are to pay the army and navy, ib-^
salaries of the officers and employees, pensions, sjcxd
for any other United States government expenses -
2. State taxes are to pay their officers and employees,
to support their schools, universities, asylums, add
to pay all state expenses.
102
TAXES 103
3. County taxes are to pay for the cost of the roads, the
salaries of employees, for charities, and for any
other county expenses.
4. City taxes are to pay for fire and police protection,
the salaries of employees, the support of the schools,
and for any other city expenses.
5. Town taxes are to pay for their schools, the salaries of
employees, and for any other town expenses.
109. Method of Assessing Taxes in a State. — The state
legislature determines the amount of money to be spent.
The amount of taxable property is usually determined by
local oflScers called assessors. The total amount to be col-
lected is then divided by the number of dollars of taxable
property. Therefore the tax is a certain per cent of the
property assessed.
The property is usually assessed at some part of its real i^
value. This part will vary in different localities or states.
In some states it is becoming the policy to assess for nearly
the full value.
110. How to Find Amount of a Tax. — The amount of the
tax is also determined by various methods. The following
examples will perhaps explain the different methods in the
clearest way.
Illustrative Example 1. When the rate is stated as a certain number
of mills on the dollar:
Warner's property is assessed at $4,000; the tax rate is 35 mills on the
dollar. Find his tax.
Solution: $4,000 assessed valuation
.035 tax rate
$140 his tax
i
104 BUSINESS MATHEMATICS
Illustrative Example 2. When the tax rato is stated as a certa^ ^ P^^
cent:
Baker's property is assessed at $5,000; the tax rate is 1.5%. Fi*:^^^ "^
tax.
Solution : $5,000 assessed valuation
.015 tax rate
$75 tax
Illustrative Example 3. When the tax rate is stated as a certain j::^^^'
ber of dollars on each hundred of dollars.
White's property is assessed at $3,000; the tax is $1.75 per $100. ^^i^^
the amount of his tax.
Solution : $3,000 = 30 hundreds of dollars
30 X $1.75 = $52.50 tax
Illustrative Example 4. Jones' property is assessed at $5,000. ^"^
tax rate is $25 per $1,000. Find the amount of his tax.
Solution: $5,000 = 5 thousands of dollars
5 X $25 = $125
ORAL EXERCISES
1. State each of the following tax rates in two other ways:
2.5% 32 mills $1.87 per $100
2. If a certain state assesses property at f of its real value, what is * «
?
assessed value of property worth $4,000? Of a building worth $12,CH^^*
Of a manufacturing plant worth $8,720?
3. If property valued at $10,000 is assessed at J of its value and the *"
rate is $2.30 per $100, what is the tax?
4. A man pays 2% tax on J of the real value of his house. What ^
cent of the real value docs he pay? i
5. Jones owns property worth $5,000 and is taxed $2 per hund** ,
dollars on | of the real value. Williams owns a house worth $5,000 ^^ ^
is taxed 19 mills on f of the real value. Which pays the larger t^^
How much does he pay?
TAXES
105
WRITTEN EXERCISES
1. Complete the following form:
Real Vallte
OF Property
S135.500
23O.00O
38.400
25,00O
Fraction* of
Value Assessed
Full value
•• «•
Rate of Tax
S .004 on $1
$1.64 iH-rJKK)
$6,845 per $1,000
2 mills on $1
Amount of Tax
In a certain city, a discount of 1% is allowed on all taxes paid before
Feb. 10; if paid on or after Feb. 10 and before Mar. 1, .i% discount is
allo"wed; if paid on or after Mar. 1 and before Apr. 1, no discount is al-
lowed; if paid on or after Apr. 1, J% is added (.n the first day of each
month for the remainder of the year. Find the amount of tax on each
of the following in that city at.$1.95 per $100:
2. A house assessed at $3,000, tax paid Feb. 10.
3. A store assessed at $10,000, tax paid Mar. 29.
4. A building assessed at $50,000, tax paid July 25.
6. An apartment house assessed at $75,(XX), tax paid Nov. 15.
In New York City one-half of the tax on real estate and all the tax on
personal prop)erty are due and payable on and after May 1. The other
half of the real estate tax is due and paj'able on and after Nov. 1. A
discount of 4% per annum is allowed on the second half of t he real estate
tax if paid before Nov. 1, provided the first half has been })iiid. Interest
at 7% per annum from May 1 is added to all payments of the first half
of the real estate tax and all personal taxes paid on and after June 1.
Interest at 7% per annum from Nov. 1 is added to all payments of the
second half of the real estate tax on and after Dec. 1. Find the tax due
and payable, according to these regulations, on the following:
6 7 8 9
Property Business building Railroad Apt. house Lot
Assessed valuation... $13,000,000 $17,800,000 $100,000 $3,000
Borough Manhattan Queens Brooklyn Richmond
Rate (1920) $2.53 $2.54 $2.54 $2.53
Date of Payment:
First half June 15, 1920 May 20. 1920 Sept. 5.1920 Feb. 10. 192C
Second half .. . Oct. 10, 1920 Nov. 15, 1920 Feb. 10.1921 Sept. 1.1920
Total Tax
.i#^
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-WlJ-XV
«.twai»
'ATRITTfcN EXERCISES
1, »x:;.^;'i'» .'S^.^MX) $20,rXjo - $0,000 -h $600
112. How the Tax Rate Is Determined.— The assessed
viiluation of iIm! proiMTty is found from the assessors' lists.
TAXES
107
The amount of money needed, divided by the assessed
[uation of the property, gives the tax rate to be levied,
le following type solution shows the method and a form
'Ht finding the tax rate.
■Statb Tax:
State budget S2.500.000
Assessed value of property in state $1,250,000,000
S2.500.000 -^ $1,250,000,000
.20% State rate
County Tax:
County budget
Assessed value of property in county
$40,000 -^
City Tax:
City budget
Assessed value of property in city
$16,000 -^-
$40,000
$5,000,000
$5,000,000
$16,000
$1,280,000
$1,280,000
.80% County rate
1.25% City rate
2.25% Total rate
ORAL EXERCISE
1. State some reasons why the tax rate will vary in different cities.
WRITTEN EXERCISES
1. Find the total tax rate on the following:
State budget $ 2,500,000
State assessed valuation 1,500,000,000
County budget
County assessed valuation.
City budget
City assessed valuation.
School budget district
Assessed valuation, school
28,000
4,500,000
15,000
1,500,000
35,000
1,500,000
state rate
county rate
city rate
school rate
total rate
108 atb^lIXJIi:^ MJOHEIIATICS
%. Ih. s^ «ngR:MiL •BiErr t^ff^ tSKE iai» B :»
Camugr *** .39
Cicir * 1j075
SdtodI '^.. l.«
3^ If pcQ^«viprQ<^:sas«R«i;a |<^E&»RalTalDe and Hortoa owns the
SiCMno
Sto*. 7,500
25,500
Find ld$ to<ai t;UL
4. Find the aoiMKmt o£ lii^ tax for the state. Tbe coanty. The city.
5. Wbatperopntof tlirt<i>taItaxin£sera&e2ai^pliestoeachdiv^^
lis. Inheritance Taxes. — An inheritance tax is a tax on
the property of a deceased person. This is assessed in the
greater number of the states but varies in different states,
and a person interested in this subject should look up the
law for each state. The following rates are for New Jersey
and New York.
New Jersey. To husband or wife, child, adopted child or
its issue, or lineal descendant, the rates are:
1% from $ 5,000 to $ 50,000
1{% " 50,000 " 150,000
2% " 150,000 " 250,000
3% above 250,000
$5,000 exempt
To parents, brother 'aw, and daughter-in-
law, the rates are:
TAXES 109
2% from S 5,000 to S 50,000
2J% " 50,000 " 150,000
3% " 150.000 " 250,000
4% above 250,000
All others 5%
$5,000 exempt
Preferred obligations:
1. Judgments
2. Funeral expenses
3. Medical expenses of last sickness
New York. If inheritance is received by father, mother,
usbandy wife or child, adopted child:
Exemption to amount of $5,000
1% on amounts up to $25,000
2% on next $75,000
3% on next $100,000
4% upon all additional sums
If inheritance is received by brother, sister, wife or widow
f son, or husband of daughter:
Exemption to the amount of $500
2% on amounts up to $25,000
3% on next $75,000
4% on the next $100,000
5% thereafter
If inherit3,nce is received by any person or corporation
►ther than those above named :
Exemption to the amount of $500
5% on amounts up to $25,000
6% on the next $75,000
7% on the next $100,000
8% thereafter
110 BUSINESS MATHEMATICS
Preferred obligations:
1. Funeral and administration expenses
2. Debts preferred under United States laws
3. Taxes
4. Judgments and decrees
114. Federal Inheritance Tax. — The federal tax is
posed on the estate as a whole, not on the shares oi
several legatees, irrespective of the beneficiaries to
decedent.
$50,000 of each estate is exempt from tax. The tslU
the excess are as follows:
Not exceeding
$ 50,000
1%
From $ 50,000 to
150,000
2%
150,000 "
250,000
3%
250,000 "
450,000
4%
450,000 "
750,000
6%
750,000 "
1,000,000
8%
" 1,000,000 "
1,500,000
10%
1,500,000 "
2,000,000
12%
" 2,000,000 "
3,000,000
14%
" 3,000,000 "
4,000,000
16%
" 4,000,000 "
5,000,000
18%
*' 5,000,000 "
8,000,000
20%
" 8,000,000 "
10,000,000
22%
Exceeding
10,000,000
25%
WRITTEN EXERCISES
1. Find the amount of inheritance tax to be paid to each, the st
New Jersey and the United States, on an estate of $600,00C* wil
follows: $200,000 to decedent's wife; $100,000 to each of three chi
balance divided eciiially among the mother, a brother, and two sist
2. What would the state and federal inheritance tax amount to ii
York?
3. Find the amount of inheritance tax to be deducted from each
following amounts willed by a decedent of New Jersey;
TAXES 111
$40,000 to the wife
25,000 '' a son
25,000 '' a sister
10,000 '' the father
6,000 " a brother
4. Apply Exercise 3 to New York.
116. Mortgage Tax Law of New York. — All mortgages
upon real estate in New York can be made tax-exempt by
paying the recording tax of ^ of 1%. This will permanently
exempt the mortgage from taxation.
WRITTEN EXERCISES
1* Find the recording tax on a mortgage of $56,000.
2. What is the interest for the first year and the recording tax on a
mortgage of $4,800?
116. Income Tax. — The present income tax law was en-
acted on account of the increased expenses of the world war.
The law is here given, with some of the features.
The tax shall begin at 4%, on incomes above the amount
exempted up to $4,000 normal tax, and 8% normal tax on
all incomes in excess of $4,000 with the exemption allowed.
Ill the case of a head of a family, married man or woman, an
exemption of $2,500 for incomes up to $5,000, and $2,000 for
those over $5,000, and $400 for each dependent child under
^8 is allowed. The exemption for a single man or woman
•
^8 $1,000. Exemptions are allowed state and municipal
paid employees. Only one deduction is allowed from the
^gregate income of both husband and wife living together.
The gross income includes gains, profits, and income from
^^y source whatever, except the interest on some United
.States government bonds or bonds of the political sub- ^
divisions of the United States.
112
BUSINESS MATHEMATICS
The net income is obtained by deducting from the gross
income all necessary expenses actually paid in carrying on
the business, such as interest paid on indebtedness; loss from
bad debts, if charged off; taxes; fire losses not covered by
insurance or otherwise; and a reasonable depreciation on
the value of the property. Personal, household, and living
expenses are not included.
Illustrative Example. A man living with his wife has a net income of
$5,000. How much income tax is he required to pay?
Solution: $5,000 net income
2,500 exemption
$2,500 taxable income
.04 normal rate
$ 100 income tax
117. Super-Tax or Surtax. — An additional tax is assessed
on large incomes in excess of a certain amount. The table
below shows the additional rates charged on the portions
of net income above certain stated amounts (not deduct-
ing the $1,000 or $2,500 or $2,000 exemption applying to
the normal tax.)
Cumulative
Rate
Amounts
Exceeds $ 6,000 but does not exceed $
8,000
1%
20
8,000
(( - (( (
10,000
1
40
10,000
(( 11 i
12,000
2
80
12,000
(I K (
14,000
3
140
14,000
(( (( (
16,000
4
220
16,000
(( (I i
18,000
5
320
18,000
It il (
20,000
6
440
20,000
t • ( ( (
22,000
8
600
22,000
(. (( (
24,000
9
780
24,000
(( a (
26,000
10
980
26,000
(( (( (
28,000
11
1,200
28,000
(( (( (
30,000
12
1,440
TAXES
113
Exceeds $
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30,000 but does not exceed $
32,000 "
34,000 "
36,000 '^
38,000 "
40,000 "
42,000 "
44,000 "
46,000 "
48,000 "
50,000 "
52,000 "
54,000 "
56,000 "
58,000 "
60,000 "
62,000 "
64,000 "
66,000 "
68,000 "
70,000 "
72,000 "
74,000 "
76,000 "
78,000 "
80,000 *'
82,000 "
84,000 "
86,000 "
88,000 *'
90,000 "
92,000 "
94,000 "
96,000 "
98,000 "
100,000 "
150,000 "
200,000 "
300,000 "
500,000 "
1,000,000 '*
32,000 13
34,000 15
36,000 15
38,000 16
40,000 17
42,000 18
44,000 19
46,000 20
48,000 21
50,000 22
52,000 23
54,000 24
56,000 25
58,000 26
60,000 27
62,000 28
64,000 29
66,000 30
68,000 31
70,000 32
72,000 33
74,000 34
76,000 35
78,000 36
80,000 37
82,000 38
84,000 39
86,000 ' 40
88,000 41
90,000 42
92,000 43
94,000 44
96,000 45
98,000 46
100,000 47
150,000 48
200,000 49
300,000 50
400,000 50
1,000,000 50
50
1,700
2,000
2,300
2,620
2,960
3,320
3,700
4,100
4,520
4,960
5,420
5,900
6,400
6,920
7,460
8,020
8,600
9,200
9,820
10,460
11,120
11,800
12,500
13,220
13,960
14,720
15,500
16,300
17,120
17,960
18,820
19,700
20,600
21,520
22,460
46,460
70,960
120,960
220,960
470,960
114 BUSINESS MATHEMATICS
Illustrative Example: Jones, a married man with no children, re-
ceives a salary of $12,000 a year but has no other income. Find his
total income tax.
Solution :
Normal Tax
$12,000 net income
2,500 exemption
$9,500 taxable income (normal tax)
$4,000 at 4% =$160
$5,500 " S% = 440
Surtax
From $ 6,000 to $10,000 $4,000 at 1% = $ 40
*' 10,000 " 12,000 2,000 " 2% = 40
$ 80
Total Tax
$160 + $140 + $80 = $jSO
Credits for determining normal tax :
1. Dividends from corporations which are subject to the
tax upon net income.
2. Interest from United States Liberty bonds.
3. Single persons have an exemption of $1,000; married
persons or heads of families have exemptions of $2,000, but
the total exemption of both husband and wife shall not
exceed $2,000.
4. An additional credit of $200 is allowed each head of a
family for each dependent under 18 years of age or incapable
of self-support because mentally or physically defective.
TAXES 115
Credits for determining surtaxes :
1. Income from a total of $5,000 par value invested in
Second, Third, and Fourth Liberty Loan bonds.
2. In addition to the above, the income from $30,000 par
value of Fourth Liberty Loan bonds will be exempt from
surtaxed until 2 yr. after the termination of the war. The
income from an amount of Second and Third Liberty Loan
bonds not exceeding one and one-half times the amount of
bonds of the Fourth Liberty Loan originally subscribed for
and still owned at the time of making return, but not to
exceed a total of $45,000 par value, will be exempt from sur-
taxes until 2 yr. after the war. And in addition, the income
from $30,000 par value of Fourth Liberty Loan bonds con-
verted from the First Liberty 3^'s will also be exempt from
surtaxes until 2 yr. after the war.
3. Income from Federal Farm Loan bonds will be tax-free.
lUustrative Example. To find the income tax on an income of $50,-
000 made up as follows:
Income to be Reported but Exempt from Tax
Income from municipal bonds $15,300
Interest on 3J% liberty's 1,050
$18,350
Income Which Must Be Reported
Salary $10,000
Interest from real estate, mortgages, and rents 15,000
Corporation dividends 1,000
Interest on Liberty 4's and 4}*s issued previous to the 4th
Loan. (The owner originally subscribed for $30,000 of the
4th Loan and still holds them) 1,250
Interest on railroad and utility bonds not tax-free 6,400
$33,650
116 BUSINESS MATHEMATICS
To Determine Normal Tax
Net income subject to tax $33,650
Less Credits:
1. Corporation dividends $1,000
2. Interest on Liberty bonds 1.250
3. Fixed exemption for married person 2,000
4. One dependent child 400 4,650
Income subject to normal tax $29,000
To Determine Surtax
Net income $33,650
Less Credits:
A and B income from Liberty bonds 1,250
Income subject to surtax $32,400
Accumulated surtax at various rates on $32,000 $l,70O
Surtax on $400 at 15% ftO
Total surtax $1,7^^^
Subject to 4% rate, $4,000 $ 160
Subject to 8% rate, $25,000 2,000
Normal Tax $2,160
Surtax 1,760
Total tax $3,920
The interest on $1,250 (A and B) might be considered i\»-^
following year.
WRITTEN EXERCISES
1. A single man has a salary of $8,000 a year and no other incom
Find his total income tax.
2. A married man with no children has a yearly income of $8,000 i
salary and no other income. Find his income tax.
3. A single man has a yearly salary of $20,000 and no other inconn
Find his total income tax.
TAXES 117
4. A married man has a yearly income of $20,000 salary. He has one
dependent child. What is his total income tax?
6. Find the net income and the total income tax for a married man
with no children, from the following data:
Salary $3,200; interest on money loaned by him in the form of a note,
$500; rent on buildings owned by him, $1,400.
He pays $1,200 interest on money which he has borrowed; repairs, in-
surance, and depreciation on his rented property $175; state and local
taxes $225.
6. Find the net income and the total tax for a married man from the
follo^ng information:
Cost of goods sold during the year $135,000. Gross sales $185,000.
Wages of employees, insurance, rent, and other business expenses
$12,000. Loss from bad debts, determined and charged off $895.
He had borrowed $4,000, on which he paid a year's interest at 6%.
His store building was worth $7,000, and he estimated the annual de-
preciation at 2% of the value of the building.
Net Sales = Gross Sales — Returned Sales
Gross Profit = Net Sales — Cost of Goods Sold
Cost of Goods Sold = Net Sales — Gross Profit
Net Profit = Gross Profit — Expenses
7. A married man with no children has an income of $80,000. Find
the income tax made up as follows:
Income to Be Reported but Exempt from Tax
Income from Municipal bonds $20,000
Interest on 3 J Liberty's 2,000
$22,000
Income Which Must Be Reported
Salary $20,000
Interest from real estate and rent 18,000
Interest on Liberty 4's and 4i's issued previous to the 4th
Lioan . (The owner originally subscribed for $30,000 of the
4th L#oan and still holds them) 2,000
Interest on raih-oad bonds not tax-free 18,000
$58,000
CHAPTER IX
INTEREST ON BANK ACCOUNTS
118. Bank Interest. — Every business banks its money
and if the amount on deposit is large or is not required for
current use, interest is usually earned thereon. The com- ■'
puting of the amount of interest earned is often a difficult
problem, the diflHiculty being in part due to the different
methods of paying interest used by different banks and in
part to the fact that the amount on deposit is often changing,
owing to the frequent deposit and withdrawal of cash.
119. Kinds of Banks. — Money may be deposited in a
savings bank, a postal savings bank, or an ordinary com-
mercial bank.
A savings bank is primarily for the purpose of accepting
deposits from persons who wish to put their savings in an
institution which shall guarantee them a certain rate pe^
cent upon their money. These banks are chartered by the
state and are under the supervision of the state banking
(U^partment. Common rates of interest paid by these bank^
are 3%, 3|%, and 4%. Money in these banks cannot be
tlrawn out, except by a check payable to the depositor him-
s<M. The law allows these banks the privilege of requiring
from 15 to 00 days' notice of withdrawal, but this is seldom
ovriiMsinl.
A postal savings bank is one conducted by the United
?^i.U^^ i;i>vtM*ninent, in which savings may be deposited where
\\w\ w \\\ draw u small rate of interest from the government
118
INTEREST ON BANK ACCOUNTS 119
A commercial bank is one which accepts deposits which
are subject to check at any time and to any person or firm.
These banks are under the control and supervision of the
state banking department, or, in the case of national
banks, the supervision of the United States Treasury
Department.
120. Savings Banks. — The interest term in these banks is
the period of time between the dates on which interest pay-
ments are due; i.e., if the interest payments are due Jan. 1
and July 1, the interest term is 6 mo. If due Jan. 1, Apr. 1,
July 1, and Oct. 1, the interest term is 3 mo., etc. In some
savings banks deposits begin to draw interest from the
first of each month. In most banks, however, only such
sums as have been on deposit for the full term may draw
interest.
Interest is computed on the number of dollars, and the
parts of a dollar are not considered. When the interest is
due it may be withdrawn, or left in the bank, in which case
it is added to the balance and draws interest as any deposit.
Savings banks, therefore, pay compound interest. In com-
puting savings bank interest it is important that the form
of the solution and the form of the work be carried along
together if one desires to make the work simple. The two
forms below will show the solution of the following example,
and the explanation at the end should also be read as the
computer works over the example.
Illustrative Example. The interest days of the Franklin Savings
Bank are Jan. 1, Apr. 1, July 1, and Oct. 1. On each of these interest
days, interest at the rate of 4% per annum is computed on the smallest
quarterly balance.
The account of J. B. Jackson follows:
120
BUSINESS
HEMATICS
J. B. jAcmsfKX
Datx
DZTOSTS l3CTEZ£SX WfTHDKAVaIjS BaLASKB'
Jan. 18.
Apr, 15.
Mar 2.
June 1 .
Jalr 1
July 16
Aug. 25.
Oct. 1.
Nor. 12.
Jan.
1920
1921
S200
250
50
150
100
S2.00
4.02
5 SI
SIQO
45000
to
asim
Second form acoompanying the above form:
First quarter
Second " .
Third ** .
Fourth " . .
smai.ixst
Balance
(k'akteklt
Inteksst
$ 0.00
$0.00
200.00
2.00
402.00
4.02
581.02
5.81
Explanation: The first interest term was from Jan. 1 to Apr. 1; but
since Mr. Jaclo^^n made no deposit until Jan. 18, the smallest balanoe
on depr^sit during the entire interest term was $0.00, and therefore no
interest is added Apr. 1.
The second interest term was from Apr. 1 to July 1. The smalksl
balance on deposit during the entire term was $200. The interest was
$2.
The third interest term was from July 1 to Oct. 1, the smallest balanoe
being $402. The interest was $4.02.
The fourth term was from Oct. 1 to Jan. 1, the smallest balanoe
$681.02. The interest was $5.81.
121. Other Methods of Computing Interest. — Some
savings banks compute the interest on monthly balances,
some on quarterly balances, and some on semiannual
INTEREST ON BANK ACCOUNTS
121
balances. Some banks add the interest quarterly, and some
add it semiannually.
The following illustrations will make these principles
clear. The explanation form should be carried along with
the other form.
Illustrative Example. Suppose that the above-mentioned bank com-
puted the interest on quarterly balances, but added tlic interest semi-
annually — we would then have:
Explanation Form:
Semiannual
Smallest Quarterly Dividend uf
Balance Interest Interest
First quarter $ 0.00
Second " 200.00 $2.00 $2.00
Third " 402.00 4.02
Fourth " 577.00 6.77 9.79
Then the account would read as follows:
J. B. Jackson
Date
Deposits
Interest
Withdrawals
Balance
1920
Tan 18
1
$200
250
50
150
100
$2.00
9.79
$100
75
$200.00
Anr 15
450.00
Vfa V 2
:i50.00
Hw^y *•
400.00
Tnlv 1
402.00
JUly ■»•
Tnl V lO
552.00
Alter 2S
G52.00
/vile* ^'*^
1^^» 12
577.00
1921
fo« 1
586.79
jsn* •*•
In case the bank computed the interest on the monthly
balances and added interest quarterly, the explanation
form of Jackson's account would appear thus:
122
BU.-^IN
B.
VfATRFTW
ATTTS^
llO'CTHLY 3AJLAaH:E
•t
Moiith: L^sm
EKEST
m r -mw ^^ -
.VfONTH
-Z£-7 Jivusaa
Oh MkU&r
>n..
P^-b
\f ar
200.1)0
:MO.i)0
201.32
.^Sl-12
401-12
404.49
.>>L49
654.40
I
.06
.as
1
1
1
&J&
Apr.
May
S .67
L17
L33
xir
A'j^r- . .
SL24:
Ld4r
2:ia
5.38
0-t .
.Vov
92.19
L94
OLOT
N'vrK Th*^ -/?nt^ in */«!*» principal are dropped, and the parts of cents in tbi
WRITTEN EXERCISES
1. Makft a/*/v>unt<? for E. C. Stewart to and inciudini^ Jan. 1, 1921-
(phc UfT f'fif'h (A the following mcXhcxU of interest at 4^:
(h) Interf st r^omputerl rjiiarterly and added quarterly.
(\t) Int/r^st r/>rripnt^d quarterly and added semiannually.
(v,) \uU:rf'.9,\, ry>rrif>i]tf!d monthly and added quarterly.
\)Ktv. Deposits Withdrawai^
Jan. 19 $400
Vch. \.\ .VK)
Mar. r, $ 50
Mar. 17 75
June 150
July 25 75
AuR. ;j 200
(M. 15 125
Nov.21 100
INTEREST ON BANK ACCOUNTS 123
2. Apply the methods of Exercise 1 to the following data and make
an account with J. H. Harris.
Date Deposits Withdrawals
1919
Apr. 1 $128
May 15 85
July 3 $30
Oct. 3 60
Dec. 22 75
1920
June 1 60
Aug. 12 100
1921
Mar.l5 175
Find balance July 1, 1921.
3. Using 4%, computed quarterly and added semiannually, find from
the following data the balance Jan. 1, 1921 :
Balance Jan. 1, 1919, $400.50; deposited Sept. 15, 1919, $250; with-
drew May 15, 1920, $100; deposited July 1, 1920, $75, withdrew Oct.
1, 1920, $60.
4. Copy and complete the following account, supplying the missing
amounts. Interest days, Jan. 1, Apr. 1, July 1, and Oct. 1. Rate 4%.
American Savings Bank
In Account with Mr. Joseph B. Oliver
Date
1920
Jan. 1
Mar. 8
Mar. 28
Apr. 15
July 1
Aug- 13
Sept. 15
Oct. 1
Dec. 5
Deposits
S250.00
350.00
125.50
76.00
60.25
150.00
Interest
Withdrawals
SlOO
125
50
Balance
r
124 BUSINESS MATHEMATICS
122. Postal Savings Banks. — No person can deposit
than $1 nor more tliau $100 a month in these banks, nor
he have a balance at any time of more than $500, excltu
of interest. Interest is allowed at the rate of 2% per yi
for each lull year that the money remains on depa
beginning with the first day of the month following thai
which it is deposited.
A depositor may exchange his certificates, under defii
conditions, in multiples of $20, for United States gove
ment registered or coupon bonds paying 2^%, but this ii
be done at the beginning of the year if such bonds are av
able at that time. These may be held in addition to '
S500 mentioned above.
Money may be withdrawn if one gives up to the pa
officer, where the deposit was made, the savings certifies
for the withdrawal amount.
Illustrative Example 1. How much interest would I receive in a pa
BQviaga bank, if I deposited 38 Jan. 1, 1G20, and withdrew it Jai
1921?
SoLtmoN: The answer is, none, because the money does not bi
to draw interest until Feb. 1, 1920.
Illustrative Example 2. How much interest will I receive if I dep
$15 on Jan. 1, 1920, and withdraw it on Juiy 1, 1921?
Solution: Since the money is in the bank I full year but not a
the 2nd year, I would receive only 1 year's interest or 2% of $15 = |
123. Computation of Depositors' Daily Balances. — 1
following method is one employed by some banks to co
puts a depositor's daily balance on a checking account,
will be noted that from Oct. 31 to Nov. 2 there were
da. upon which the depositor had 1 thousand dollars
deposit. This means the same in value as 2 thousa
dollars for 1 da. Again on Nov. 8, there had been 7 da.
INTEREST ON BANK ACCOUNTS
125
which the deposits had not fallen below 2 hundred dollars.
This is equivalent to 14 hundred or 1 thousand, 4 hundred
dollars for 1 da. The account may thus be carried along
for the month and at the end of the month the aggregate
for 1 da. may be found. The number of hundreds is then
dropped, and the interest on the aggregate at the given per
cent may be found as follows:
$50,000 at 2% = $1,000
$1,000 -^ 365 = $2.74, amount of interest to be credited to
the account.
H. N. Trust Company
Statement of Interest
Account of B. H. Jones
Days
Balance
Aggregate
Rate
• Date
Interest
Thous.
Hund.
Thous.
Ilund.
Oct. 31
2
5
1
Nov. 2
2
1
2
4
4
6
2
4
8
7
2
1
4
" 15
2
2
3
4
6
•• 17
13
3
30
" 30
50
14
2%
$2.74
WRITTEN EXERCISES
1. How much interest would I receive if I should deposit $25 in a
postal savings bank Jan. 1, and withdraw it 4 mo. later?
2. How much would I receive if I should withdraw it 1 yr. and 1 mo.
later?
3* X make the following deposits in a postal savings bank:
126 BUSINESS MATHEMATICS
June 1,1021 $10.00
July 15, 1021 8.00
Sept. 1,1021 15.00
Nov. 10, 1021 20.00
When would I be able to receive a full year's interest on each of these
deposits, and how much interest would I receive for all?
4. If a man has on deposit in a postal savings bank on Jan. 1, $125|
and he decides to withdraw $120 and purchase United States bonds ^
bearing 2|% interest, how much will all his money have earned in iaUx^^
est for him 1 yr. later?
CHAFTEK X
BDIUMBG iUn> UMX ikSBOCXlUCnS
loan
purpose is to
buying of then* homwi EmA
one or more shares of stock, aad zo z&j : :c ittisi i.' ih^ 735^2^
rate of 25yr a wirft or $1 ptr Eicc^i. : :c -ea.?^ *^^ir^> Tb^
money so obtained i$ koiKd ai ibe j*tx^ ThZr- <€ 'jLze^^^i 10
members wishii^ to binr or iwul ifco^.
The corporatiofi r e c e i yie g a< the prc*^i£ oc ihe as^onsiioc
the interest on loans, fines from :*5 nfeffnr«rr> who f^ :o
pay their does at the spmfied tinie. pr^ecaiusns on ]a:in<
(i.e., some association? reqoife their ID€lnt«e^^ to I4d for a
loan — the member mar make an oSer oi a bonus of say
$50 more than the legal rate of inieresi for ibe first year, to
gain the privilege of having the money loan€d to him ^t ih^t
time, while another member may bid only $25. and thus
fail to obtain the loan at that partictilar time \ and the
difference between the book value and the withdrawal
value of members who must leave the association.
The book vafaie is the actual value of the money paid in.
plus all acctunulated profits in which that money shall shart\
The ¥nthdrawal valtie is the actual value less a certain
per cent, which is determined by each association. It
generally nms about 90% of the actual book value. That is,
if a person withdraws he is unable to obtain as much as t he
actual book value of his shares at that time.
127
128 BUSINESS ^UTHEMATICS
The distribution of profits must naturally be computed in
order that each share shall have its pro rata share of these
profits added to the amount of money paid in by the owner
of each share, to obtain the actual booli value. This is al?o
necessary for the annual reports of the association, whicb
may be required by its members, or by the state banking:
department, or by both.
125. The Series Plan.— This plan is to sell whatever
number of shares the association shall deem necessary, say
on Jan. 1; then on Apr. 1, to sell another series of sharns
which shall mature 3 mo. later than the first series; then
another series July 1, and another Oct. 1. These may be
issued but twice or even once a year, if the association thinks
beat or if there is little call for money for building purposes.
Earnings are usually determined and divided semiannually.
There arc thr(!e types of problems which are of interest to
the average person in an association, or connected with it
as an employee or director.
126. To Find the Withdrawal Value.—
mustratii'e Erample. A man owns 10 shares in a buJIdiof: and lott"
association and hiis paid hia diica at the rate of SI per mo. for enC^
share for 5 jr., when he is crjmpelled to withdraw. If he is allowe*^
profits at .5% per annum, to what amount is he entitled?
Explanation ; On 10 slmrcs at SI per month for each Bhare, the du^^
would amount to S600 (5 yr. = 60 mo.). The first dues (»10) haV^
earned profita for 60 mo., the second dues ($10) have earned profits 1^*
SS mo., and so on until the last dues have earned profits for 1 mo. Th»^
gives an arithmetical series of numbers which has 60 for the first ten**
Mid 1 for the last term, and in which the number of terms is 60. Th^
algebraic sum of such a scries of numbers is equal to the sum of the firs ^
tmd last terms multiplied by one-half the number of terms, or iu thi^
«ainplo (1 + 60) X <00 -<■ 2) ■= 1,830. The general method of compui^
BUILDING AND LOAN ASSOCIATIONS 129
ing the interest is to calculate it on the total dues paid in for the average
time. The average time is obtained by dividing the total number of
months, 1,830, by 60, the number of months in which dues have been
paid, which equals 30J. The interest on $600 for BOj mo. at 5% is
$76.25, which added to the amount paid in in dues gives $676.25, the
withdrawal value.
The average time may be easily calculated by taking one-half the
number of months and adding i mo. to it.
Solution:
60 X $10 = $600, total amount paid in, in dues
$600 X -;^ X .05 = $76.25, profits
$600 + $76.25 = $676.25, withdrawal value
WRITTEN EXERCISES
1* A man has paid dues of $1 per mo. per share on 20 shares for 6
F. when he is compelled to withdraw from an association. If the profits
^ere 4% per annum, to what amount is he entitled?
2. State reasons why a person might be compelled to withdraw from
& building and loan association.
3. A man at the end of 7 jn*. finds that he must withdraw from an as-
sociation. He has carried 15 shares at $1 per mo. per share. He has
uiipaid fines against him of $4.50. If his profits are calculated at 4J%,
and all unpaid fines are deducted, find his withdrawal value.
^» A man holds 5 shares in an association that pays 5}%. He has
^n in for 8 yr. and has paid $1 per share per month. Find the amount
to which he is entitled if he withdraws.
6* If I own 12 shares in a building and loan association and must
y^thdraw at the end of 5 yr. and 6 mo., to how much will I be entitled
if the association allows 6%?
127. Conq)utatioii of Profits on Shares. — When the dues
^d profits combined amount to the par value of the stock
(usually $200), all shares are canceled if the borrower has
Duilt or purchased a house, or each member is paid in cash
^ he has not borrowed from the association. To be able to
faiow when each share shall amount to $200 with the profi.ta
130 BUSINESS MATHEMATICS
added to the amount of money paid in by the member, it ia
necessary to know how to compute the profits for each share
of each series. This is done in the following manner:
Illustrative Example. An association has issued 4 series of shares as
follows:
1st series of 500 shares, dated Jan. 1, 1919
2d " " 400 " " July 1, 1919
3d " " 300 " . " Jan. 1, 1920
4th " " 400 " " July 1, 1920
The dues in each series were $1 per share per month. If the entire
profits on Jan. 1, 1921 were $2,765, what would be the value of 1 share
of each series at that time?
Explanation; Dues of $1 a month have been paid on each of the
500 shares for 24 mo., in the first series (the average time is 12i), which
makes an average investment of $24 X 500Xl2i, or $150,000 for 1 mo.
In the same manner the average investments in the other series are found
and the total for 1 mo. is $250,200. The share of the profits belonging to
each series is in the same ratio as these average investments. The profit
for 1 share in each series is obtained by dividing the number of shares in
a series into the entire profit of each series. The value of 1 share in each
series is the sum of all dues paid in on that share and the profit for that
share.
Form of Solution:
$24 X 500 X 12^ = $150,000 1st series invest, for 1 mo.
18 X 400 X 9^ = 68,400, 2d " " " 1
12 X 300 X 6i = 23,400, 3d " " " 1
tt
It
6 X 400 X 3J = 8,400, 4th " " " 1
$250,200, total
It tt it 1 ((
155|2? of $2,765 = $1,657,673, share of 1st series
_ ^8.40 g f 2,765 = 755.899, " " 2d "
250.200 " '
23.400 r 2 765 = 258.597, " " 3d "
250.'200 "' '
8.400 . 2.765 = 92.831, " " 4th "
250.200 "' '
OB 1 J
Bkwrd lia
*- 1
*• **2d
^ 1
-u ^3^
^ I
-^ - *tJb
oflfli
fe0raf IfS «
*- 1
- - 3d
- 1
- --Sd
^ 1
- *- 4th
BUILDING AXD LQAX ASBDCIATIOXS 131
$1,657,673 ^ dOO ^ SLSla, pnife
755.899 ^ 400 = L,8»,
25S.597 ^ 300 ^ JdSV,
92.S31 ^ 400 = .222,
$24 + $3,315 = $27^5,
18 + 1.889 = 19.889L
12 + .861 = 12..861,
6+ -232 = 6.232,
One of the larigest bufldmg and loan associations in the
United States gives thdr method of computing profits as
follows:
niustratiYe lgy*mpU, An asBooatioa haTin^ 3 series of 100 sIultrs
each and profits of $100.
S=UE.s Shakes Yejlks Half Yeaiis
1 100X3 = 300XlJ=4oO
2 100X2 = 200X1 =300
3 100X1 = 100X1 =50
roo
Computation of the profits per share:
($100 X tU = $64,284) -- 100 = $.642 profit per share
( 100 X «3S = 28.572) ^ 100 = $.28.5 "
( 100 X V% = 7.144) H- 100 = .071
n
it 11
$100
Another association uses the following plan: The profits
of the association are divided and ascertained at the end of
each fiscal year on the partnership plan, in the following
manner: Each series investment, being the amount paid
in for dues, is multipUed by the average time invested and
the results added together for a sum of results. Each result
is multiplied by the total earnings of the association from its
institution to date, and this product divided by the sum of
the results, the quotient in each case will show each series
132 BUSINESS MATHEMATICS
share of the net earnings. Add the net earnings in each
series to the principal of that series investment and divide
the sum by the number of shares outstanding in such series,
and the i-esult will be the net result of each share in sucJi
Illustrative Example. First series 3 yr. old; second scries I ;
SUAHES MOKTHS IsVKST. TI^d^ RESliLIS
lat, 2,500 X 36 = $90,000 X 1| = 8135,000
2d, 1,000 X 12 = 12,000 X i = 6,000
Net Profits, S 13,000
($135,000 X 13.000 = 1,755,000,000) -r- 141,000 :
$12,446.80 iBt aeries profits
90,000.00
2,500 }$102,446.8O
[6,000 X $13,000 = 78.000,000) ■;- 141.000 =
553.20 2d series profits
12,000.00
1,000 )$12,545.20
WRITTEN EXERCISES
1. A buildiii(! and lonn aasotiiition issued a new series at the beginning
of each year. The 1st aeries Iiaa 300 shures, the 2d 500, the 3d 400, and
the 4th 500. If the dues are SI per month per share and the profits at
the end of the 5th year are $4,500, find the value of 1 share in each series
at the end of the 5th year.
2. Try this out with the second plan mentioned above a
your results.
BUILDING AND LOAN ASSOCIATIONS 133
8. Try the first plan on the second plan example, and compare results.
Which appears to be the better plan for the membera of the aasociation?
128. Distributton of Profits Statement — It is necessary
for an asfxiciation to publish such a statement at certain
intervals, either semiannually or annually. The following
plan, used by some associations, will give a member a com-
prehensive idea of the standing of the association.
niustrative Example, Suppose that a new scries is opened each 3
mo., with dues 25f per wk, per share. Series number 49 has been open
520 wk. and there are 1,225 Hhareein the scries Dec. 31, 1920. The total
subscriptions paid in on aeries number 49 is equal to 25ti per wk. for 520
wk., or SI30 per shore, and on 1,225 shares SI59,250. During 1920 the
aubacriptionsequal $15,925 (52 wk. at 25(ion each share of l,225share8).
Subtracting $15,925 from $159,250 gives $143,325, the total subscrip-
tions paid in, Dec. 31, 1919, and this amount earns profits for all of the
year 1920. Theprofitsfor 1920arc paid on one-half of thesubacriptions
paid in, in 1920, or on $7,963 .50, and this added to $143,325 gives $151,-
287.50, the total amount on which profits are allowed for 1920 in series
49. The per cent of profits isfound by dividing the total profit on all the
series by the total subscriptions in all the series sharing in profits, as
shown by the total of the column headed "Total Profit-Sharing Sub-
criptions, Dec. 31, 1920." Allow the same rate on all series.
^"
^
^.
=
^
"S
1
s.
IL
£3
I
u
1
I
s
s
s
1=
i
ji
1
is
s^l
.go
1
£
1
40
520
1.225
(15B.350
»15,92C
»143,325
S7.96a.30
1151.287.50
«
m.orr.ss
50
SOT
51
1,058
62
481
1,740
40H
64
4S5
1.202
SS
442
1,714
134 BUSINESS MATHEMATICS
WRITTEN EXERCISES
1. Using the same rate per cent of profits, complete the above form.
2. Given the 1st series 4 yr. old and having 3,000 shares, the 2nd
series 3 yr. old and having 1,200 shares, the 3rd series 2 yr. old and
having 1,500 shares, and the 4th ^ries 1 jn*. old and having 2,400 shares,
net assets $262,080, find :
(a) Net profits
(b) First series profits
(c) Value of 1st series shares
(d) Second series profits
(e) Value of 2nd series shares
CHAPTER XI
GRAPHICAL REPRESENTATION'
129. Why Graphs Are Used. — The object of presenting
statistical data in graphic form is to enable the reader to
interpret more readily the facts contained in the collected
data, and to draw proper conclusions from these facts as
presented. Graphs or diagrams do not add anything to the
meaning of statistics, but when drawn and studied intelU-
gently they bring to view more clearly the various parts of a
group of facts in relation to one another and to the whole
group, or show effectively the fluctuations or trend charac-
teristic of the data under consideration.
The construction of graphs is essentially mathematical
and for that reason has been emphasized in this book.
Moreover, some graphs are of material help in the work of
bookkeeping and accounting.
The graph is designed to show:
1. The true proportion of the component parts of a
group total.
2. The relation of one part of a group to other parts of
the same group.
« Teachers are advised to study the following books if they wish to
make a further study of graphical representation: U. S. Census Bureau,
United States Statistical Atlas, 13th Census, 1910, 1914; W. C. Brinton,
Graphic Methods for Presenting Facts, New York, The Engineering
Magazine Company, 1914; A. L. Bowley, Elementary Manual of Static
tics. New York, Charles Scribner's Sons, 1915.
135
136 BUSINESS MATHEMATICS
3. The fluctuations or general trend in a series of similar
magnitudes (or sizes), arranged date by date for a
given period of time.
130. Kinds of Graphs or Charts. — Graphs used to show
the true proportion of the component parts of a group total
are:
1. The circle.
2. The rectangle.
3. The straight line, representing the total, and divided
into segments (or sects) of proportionate lengths to
represent the quantities making up the total.
Graphs used to show the relation of one part of a group
to other parts of the same group are:
1 . Parallel lines or bars of the same width, drawn either
horizontally or vertically from a common base line.
2. Pictograms, or illustrations in perspective, showing
the relative values represented by the quantities
compared. This form is very unsatisfactory
because the relative values are not comprehended
easily by the reader.
3. Circles, squares, or any figures in which the relative
values are represented in more than one dimension,
the attempt being to show relative values by the
size of the different circles, etc. This form is un-
satisfactory because the eye cannot grasp the true
relation from these sizes.
Graphs used to show the fluctuations or general trend in
a series of similar magnitudes, o-rranged date by date for a
given period of time, are :
GRAPHICAL REPRESENTATION 137
1. The curve (Form 11), connecting points located at
distances to the right of a vertical axis, as deter-
mined by the time variable, and at a distance up-
ward from a horizontal axis, as determined by the
quantity variable.
2. Comparative curves (Form 12), with a common time
variable but with different quantity variable deter-
mining the location of the points in the respective
graphs.
131. Construction of Graphs. — Squared (or co-ordinate)
paper is necessary. Loose-leaf size is preferred for school
use, since the graphs prepared should be kept as a part of the
required notebook work. Paper ruled into inches and tenths
of an inch will usually be found most convenient, the lines
at the inches and half-inches being heavier than the rest.
All rulings should be of such a color as will bring the graph
into proper relief. If desired, paper with metric rulings
may be used instead of that on an inch scale. A ruler with
the same graduations as that of the paper can be used to
advantage. Select a scale which will work well on your
paper — 10, 100, etc., will naturally work well on paper ruled
in tenths or twentieths. The construction of curve graphs
will be facilitated if the steps are completed in the following
order:
1. Arrange the quantities given in the statistical table
in the order of the time units, the earliest date first, and use
round numbers only, e.g., use 22,000 for 22,345, etc.
2. Mark off on the horizontal axis the time points from
left to right, using the vertical axis as the earliest date line.
3. Mark on the vertical axis upward from the point of
intersection, the quantity scale, using the horizontal axis as
138 BUSINESS MATHEMATICS
the " O " (or zero) line. The quantity scale selected is deter-
mined by the largest quantity in the series; the time scale,
by the number of years or months.
4. Locate the points at the correct distance from the two
axes to represent the quantities given in the series with
reference to their respective dates and connect these points.
The graph will then be plotted (or drawn) in the first quad-
rant, that is, in the space to the right of the vertical axis and
above the horizontal axis.
5. Add legends to interpret the quantitative scale and
time points.
6. If two or more curves are plotted on the same chart
field, the procedure is the same except that a second scale,
if there is one, may be indicated on the right. The curves
should be distinguished by the character of the line or by
color and an explanatory legend should be given.
132. Facts Shown by the Component Parts of a Circle. —
This is a form of graph quite commonly used by business
men. The circle is divided into parts which will show how
much of the group total is represented by each of its parts.
In the following example the division of the circle is deter*
mined as follows: 15% of it is shown, representing the
amount that might be used for clothing. There are 360
degrees in the whole circle. 15% of 360 degrees = .15 X
300"^ = 54°. By the use of the protractor, which is ex-
plained in Chapter XIX, it is an easy matter to lay off
54°. The divisions should always be from the center of th^
circle.
Illustrative Example. Some good authorities claim that the follov^'
ing per cents are the largest per cents which should be expended out C^^
the family income for the various expenses of the home.
GRAPHICAL REPRESENTATION
Food
Rent
Fuel and light
Clothing
Carfare
Sundries
Doctor aod dentist 2t%
Insurance 2^%
Distribution of a $3,500 salary ac
cording to thig schedule (Form 8)
Fuel and light...
Clothing
Carfare
Sundries
Doctor and dentist...
375 00
250 00
2SO0O
62 50
Total
. $2 500 00
WRITTEN EXERCISES
1. The Adirondack park is classified by ownersliip by (he New York
State Conservation Commission as follows:
State 48%
Improved 6%
Private parka 15%
Lumber and pulp companies 23%
1 Private 6%
Mineral companies 2%
Sbuw ihm by the above method.
1. The causes of divorce a.s taken from the report of a judge of the
tTurt of domestic relations in a large city were:
Disease 12%
Alcoholic drink 46%
Immorality 1S%
Ill-temper and abuse 10%
Inlcrference of parents 7%
Miscellaneous 10%
r
140
^V Make a chart to show these facts.
^1 3. 'fhp payments out of the milk dollar as rciwrted hy the Bordrn's
^K Farm Products Co. Inc., in 1916 wore as follows:
BUSINESS M4TUEMATICS
To dairymen 45.87)^
To labor 25.41^
Torftilroada 9.03t^
To shareholders 3.25i
For materials, stipislios, aiid expenses of
bottles, boxes, etc 16.44^
Chart these fads,
4. If the total sales are 88,761, cost of the goods sold S0,645, Feliirri?
S1.50. ETOSS profit $2,IlG,-aellin(t expenses SSOO, and
general expenses S400, find Ihe net profit and firri|i)i
MATERIALS,
RENT,
TRAVELING,
ETC., 20%
ihei
facts i
133. Graphs by Component Parts of a
Rectangle. — This plan is to divide a given
roptangie up into ils proportionate pari?.
The length of the rectangle should first he
determined from the business facts, am!
then subdivide d in proportion to t he
amounts which shall represent the correcl.
parts of the total group. Paper ruietl 1<i
tenths is very useful for this purpose. Il is
well illustrated in the following:
Illustrative Example. The gross revenue of rhe
Bell Telephone .System tor 1 yr. was disposed of in
llie following manner:
Salaries and wages 50%
Interest 19%
Surplus a%
Taxes 5%
Materials, rent, traveling, ef e 20%
This is shown by Ihe component parts of a rectangle
(Form !)1.
GRAPHICAL REPRESENTATION 141
WRITTEN EXERCISES
1. Graph Exercise 1 under § 132 by this method.
2. The disposition of a 5ff carfare paid to a certain city railroad in a
year was as follows:
General expenses, including pensions ar d insurance 223if
Cost of power 422if
Wages and conducting transportation 1.34ji
Other transportation expenses 128ji
Maintenance of way 488ji
Maintenance of equipment 36if
Depreciation 069if
Damages and legal expenses 291if
Taxes 367^
Rentals, subways, and tunnels 171^
Interest 36oif
Rental, surface lines 486^
Dividends 277ff
Surplus 031^
Make a graph by the above method which will show this.
3. The utilization and accompanying waste of 1 year's coal supply for
locomotives on the railroads of the United States was as follows :
Millions Millions
OF Tons of
OF Coal Dollars
Consumed in starting fires, keeping engine hot
while standing, and left in fire box at the end of
run 18. 34.
Utilized by the boiler 42. 79.3
Lost in vaporizing moisture in coal 2.5 4.7
Lost through the company 75 1.4
Lost in gases discharged from the stack 9.25 17.4
Lost in the form of unconsumed fuel in cinders,
sparks 9.5 17.9
Lost through unconsumed fuel in the ash 3.5 6.6
Lost through radiation, leakage of steam, etc 4.5 8.7
Hint: Make 90 equal spaces on one side of the rectangle to represent
miUions of tons of coal, and 170 equal spaces on the other side to repre-
*ient millions of dollars and plot by the above method.
4. Graph Exercise 2 under § 132 by the above method.
142 BUSINESS MATHEMATICS
Sl Make a flttit of tbe foUowing:
Dc^rmiBmox of Raiuioad $100 Income
FOB A Cektaix Year
Labor $43.20
Fuel aDd ^1h>p supplies 8.12
M^iieiuJ 16.90
Dtimai^e 2.22
Tax 4.72
Divisko supeiintendent 5.00
Betierment 1.08
RentaR 3.97
Inteiest balance
6. Graph the foUowing data:
Source or Railroad $1 Income for One Year
Passengers, 22.2^
Product of mines 23.9fi
Manufacturer 15.1fi
Product of agriculture 11. 7f^
PRxiuot of iorest:> 7^
Pnxluot of anim^k 4.2f^
Merohandisi^ 4.2f^
Mail 1.9f^
Expres: 2.3^
Miscellaneous balance »
7. Graph Exercise 4 under § 132 by the above method.
8. The caiL^es of crime as reported by a noted detective are as followi
Poverty 40%
Gambling and debt 25%
Strict parents 13%
Having too much money 1%
Opium 1%
Easy going parents 12%
Drink 8%
Plot this by the above method or by that of § 132.
134. Simple Comparisons by Graphs. — The graph show
in Form 10 is often better than any other form for som
kinds of business facts.
GRAPHICAL REPRESENTATION
143
Illustrative Example. The repairs and renewals of locomotives per
ton tractive force for an average over 5 yr. as compared with the 4 yr.
preceding this period is shown by the following graph (Form 10).
% Decrease % Increase
2015 10 5 5 10 152021
S
D. L. & W.
EA
iTERN
ROADS
Penn. R. R.
1
ROADS
N. Y. Cent.
B. &0.
L. V.
Wabash
C. M. & St. P.
WE
iTERN
C. R. I. & P.
C. &N.W.
C. & A.
C. B. & Q.
A. T- & S. P.
»
(From "Railroad Operating Costs," New York. SuflEern and Son. 1911)
Form 10. A Variation of the Straigth-Line Graph
WRITTEN EXERCISES
1. The distance that different kinds of trucks can travel on $1 ex-
penditure has been reported as follows:
5 ton Horse 1.6 miles 2 ton Gas 2.6 miles
5 "Gas 1.8 " 3J •• Electric... 2.75 "
3i •' Horse 2.2 '' 2 " Horse 2.93 ''
5 " Electric 2.3 '' 2 " Electric... 3.3
3} " Gas 2.4 "
Graph this from a zero line toward the right.
?. Chart the following causes of death in a certain city for a certain
year.
Disease Number of Deaths
Organic heart 1,325
Tuberculosis 1,120
Pneumonia 940
144 BUSINESS MATHEMATICS
Brights disease
Stomach and bowel (under 2 yr.)
Apoplexy
Broncho-pneumonia
Cancer of stomach and liver
Diphtheria and croup
Alcoholism
Appendicitis
Scarlet fever
3. Make a chart of the following fluctuations in the pric^
the United States.
Per Cent of Average Per Month
1909 1910 1911 iS
Jan 20% Inc. 42% Inc. 41% Inc. 389^
Apr 20% Dec. 11% Dec. 31% Dec. 18%
July 12% •* 15% *^ 32% " 22%
Oct 8% " 5% Inc. 8% " 3??
4. The following table shows the amount of various foods that c^^
eaten to secure the same number of calories that are found in 100
of ordinary white bread. Make a graph to show this simple comp^^
No. OF Grams No. of C^
Equivalent Equiva^^
Food in Calories to Food in Calor. ^
100 Grams of 100 GraVJ
White Bread White B ^
White Bread 100 Butter 35
Wheat flour 70 Gruyere cheese 73
Tapioca 70 Smoked ham 74
Meat pie 70 Pork cutlets 90
Macaroni 70 Sliced mutton 90
Maize flour 70 Jellied fruits 100
Potato starch 75 Sirloin of beef 138
^* War "bread 70 Chicken eggs 170
Hulledrice 70 Chicken 175
Vermicelli 73 Veal loin 180
Dried beans 73 Unsalted herring 330
Split peas 70 Potatoes 370
Noodles 74 Milk 380
Barley flour 70 Apples 500
Lentils 74 Mussels 600
Rye Bread 100 Spinach 954
GRAPHICAL REPRESENTATION 145
6. Make a chart of the following, showing the causes of leavinf; posi-
tions.
Reasons Per Cent
Not enough money 9
Never started 6
Working conditions 20
Discharged 6
Laid off 1
Dissatisfied 2
Better job 12
Needed at home 8
Living conditions 2
Failed to report 28
Personal reasons 7
135. Curve Plotting. — This plan is considered one of the
best to present business facts in such a manner that they
^ay be easily grasped by the reader. Before we take this
subject up in detail, however, it will be well to master cer-
tafa well-defined rules which are very important if one is to
"^ successful in graph work. These rules follow.
^^1^8 for graphical representation of facts:
^' JMake the title of the chart very complete and clear.
^' ^ake the general arrangement of a chart read from left to right.
'^- The horizontal scale figures are placed at the bottom of the chart,
^^^^ures may also be used at the top if needed.
^' The figures for the vertical scale are to be placed at the left of the
Right-hand figures may be added if needed.
^- include with the chart the data from which it was made.
"• ^^lace the lettering and figures so that they may be read from the
t or from the right-hand side of the chart.
' • ^Earliest date should be shown at the left and later dates to the
tight.
^- CDharts should usually read from left to right and from the bottom
"top.
^' ^Ijreen may be used to express desirable features, and red to in-
Qicatci londesirable features.
^0- ^ero line should show on the chart whenever possible.
chart.
botto
to thfe
146
BUSINESS MATHEMATICS
11. Make the zero line much heavier than the squared paper lines,
12. The bottom line should be wavy if the zero line cannot be ahown
13. If the chart refers to percentages, the 100% line should be bro^
like the zero hue.
S 30
•s
1
1
:;:/''■"■
: :::::T:::
: ::::::::::
is 20
10
:::
w
- --i------
c
1111 =
lill.
III.MII
Second IfiDT rUrd lur
Form 11. Curve Graph
14. When the horizontal scoic beKms with zero, the vertical lin^
the left which represents zero should be broad.
15. If the horizontal scale denotes time, the left- and right-hand li 3
are wavy, aa the beginninR and ■md of time cannot be shown.
16. If curves arc to be printed, be careful not to show any more 1l3
of the co-ordinate paper than is oecessary. Lines one-fourth of an iK
apart are better.
GRAPHICAL REPRESENTATION
147
17. Make the curve lines much broader than the co-ordinate ruling
of the paper.
18. It is often advisable to show at the top of the chart the value of
each point plotted.
19. If figures are shown at the top for each point plotted, have these
figures added when possible so as to show monthly or yearly totals.
20. If a number of curves are to be shown on the same chart field,
use different-colored inks, or different kinds of lines.
The graph illustration in Form 11 shows the monthly net
earnings of a steel corporation over a period of 2 yr. Observe
that in this graph 1 space up on the vertical axis represents
1 million of dollars, and 2 spaces on the horizontal axis
represent 1 mo. of time. Note that the peak of the net
earnings was in March of the 3d yr., and that the ebb (or
'owest amount) of the net earnings was in October of the 1st
yr. Thus any business can be pictured over a term of years.
This can also be done for sales, profits, or anything that is
desired.
WRITTEN EXERCISES
1« Show the rise and fall of the United States Steel Corporation's
^^^lled orders from the following data:
l^^O 1921
*^^^ 7.75 millions of tons Jan 11.5 millions of tons
8
8.5
9.3
9.8
9.9
9.6
9.55
9.6
9.5
10
11
Feb 11.4
Mar 11.6
Apr 11.75
May 12
June 11.75
July 11.3
Aug 10.75
Sept 10.4
Oct 9 8
Nov 9
Dec 8.8
(<
((
It
ti
II
tc
n
CI
It
it
u
ti
ft
It
ft
It
II
((
II
II
11
II
148
BUSINESS ^L\THEMATICS
2. Make a curve to show the rise of the New York City budget for a
period of 20 yr. from the foUowing data:
Jst year
2d "
... 77
... 93
. . . 98.6
. . . 98.1
... 98.7
... 99
... 99
. . . 98.6
... 115
... 130
millions
11th:
12th
13th
14th
15th
16th
17th
18th
19th
20th
^ear
143 niillions
156 "
3d "
163
4th "
173
5th "
180
6th "
7th "
«••••••
192
196
8th " . ...
198
9th "
199
mh "
... 200
3. Make a curve to show the increase in the deposits of a certain na-
tional bank.
Deposits
Istyear $ 4,100,000
10,600,000
2d
3d
4th
5th
6th
14,200,000
17,500,000
21,100,000
30,200,000
4. The following annual report of a railroad is based on a certain year
and is for the 4 yr. following it. Make a curve for each of the items men-
tioned on one chart field whose vertical column represents per cent and
whose horizontal line represents years.
In Terms of Per Cent
BaseYr. IstYr. 2ndYr. 3rd Yr. 4th Yr.
1. Operating income:
Net operating revenue after
deducting taxes 39 125 113.5 147
2, Gross operating revenue 4.5 28.7 26 39
S, Operating expenses 2.3 11.1 10.4 17
Sl Make a curve to show the proper inflation pressure per square inch
OD diffcwant-siaed automobile tires. About 20 lb. is allowed to the square
iBdiiif action.
GRAPHICAL REPRESENTATION 149
Si7E IN Tire
Inflation Pressure
Cross Section
Lb. Per Sq. In.
3 sq. in.
60 lbs.
3J " "
70 "
4 ti u
80 "
4i " "
90 "
5 " "
95 "
5i " "
100 ''
6 " "
105 "
6. Using the vertical line to represent relative value of coals in dollars,
and the horizontal line to represent price of anthracite coal in dollars,
and using one space (of any chosen length) for $1 on the base line, and
one-half of that space for $1 on the vertical line, draw four lines to show
the comparative values.
Relative
Price Value
Anthracite coal 5 5
" ''
Illinois coal
" " 8 5
Coke
" 6 6.25
Pocohontas Coal
'' '' 8 9
7. Using one space (of chosen length) on the horizontal scale to repre-
sent years and one of chosen space on the vertical scale to represent $2,
show the increasing demand upon the New York City transit lines.
Year Per Capita Year Per Capita
1860 $ 0.00 1906 $14.40
1870 6.00 1907 15.20
1880 8.50 1908 15.20
1890 10.00 1909 15.20
1900 12.00 1910 16.00
1901 12.40 1911 16.40
1902 12.80 1912 16.80
1903 13.20 1913 17.60
1904 13.60 1914 17.20
1905 14.00 1915 17.00
1916 18.00
\
150 BUSINESS MATHEMATICS
136. Comparisons Involving Time. — It is quite necessary
at times to show a comparison from one year to another on
the same line or article, such as sales, wages, cost of food
product, etc. This is accomplished very simply by the use
of the graph shown in Form 12. Note that the same amount
is placed on the top side of the rectangle as on the bottom
side on corresponding lines.
WRITTEN EXERCISES
1. Show the changes after 10 yr. in costs of materials and in freight
rates with a graph similar to that shown in Form 12, using $100 as the
basis for each at the beginning of that time.
Labor From $100 to $120
Interest
Fuel
Railroad rates
Tracks per mile
Rails
Pine lumber
2. Show the rising wage scale of railroad labor from 1899 to 1911
to 1920 using each unit to represent $.25 and starting with $1 per da.
in both the left- and right-hand columns.
Trackmen
Station agent
Trainmen
Machinists
General office clerks
Conductors
Engine men
3. Obtain similar information concerning some railroad or large
manufacturing plant and make a graph to show the results of the in-
formation which you obtain.
137. Period Charts. — It is often advisable to arrange the
working hours of a number of employees in chart form.
100 " 125
100 " 130
100 '' 95
100 " 145
100 " 150
100 " 183
Wages in
Wages in
Wages in
1899
1911
1920
$1.00
$1.50
5po.oo
1.75
2.20
6.96
1.95
2.95
6.40
2.30
3.20
6.80
2.15
2.40
4.50
3.15
4.20
7.00
3.65
4.70
7.52
GRAPHICAL REPRESENTATION
Form 12. Comparative Curvea
152
BCStNISS MATHEMATICS
IthMV
This IB partimlariy trae vhea tbey work in shifts and t]
relieve one another at stated intenrals of time or at certain
bottrs. It is also applicable in the airaogemeiit of their
VBcatioos as will be shown in the following.
Assign 2 v^eks' vacatian to each id IG ntco
-e than 2 sliaJl be away at once, sad dxm graphically.
D.Smith
^1
. 1 , , ,
P.J.Haaar
1
^
N.L.R<«,
1 ^^ 1
1
J.R.TnffiUc
1 ^_
W.D. Little
^m
1
1
E-Oonrin
1
i_ 1
1
E-Dari.
1
/
B.P.CdOin.
1 1 1
K-E-lhrUii
^H
UD-Rhodi
_B
C.S. Wood^ud
W.N.Scnuler
H. B.Baker
Z1B~
F-.J.Il<:Markin
^H
F.A. Ttbb«tt8
^^
H.D.BurKhardt
^H
1
Form 13. Period Chart
WRITTEN EXERCISES
. Make a chart aimilar to the nhovc for 25 men,
. Make a chart for 60 mi'n in wtich uo more than 3 shall be away at
e time, bcgiuniDg May 17, 1920.
3. Three patrolmeo in a certain city take the work from 8 to 4, 4 lo
12, and 12 to 8 respectively. Each man gets 1 da. off in 27. Arrange
t, diart to show this. Plan one off, etc., sifter a certain number of days.
H.nt: Isf
S-4
<-12
12-3
i
GRAPHICAL REPRESENTATION
153
138. Comparison of Curves. — The officials of a company
frequently wish to be able to see a comparison of certain
items of the business. This may be easily accomplished by
drawing two or more curves on the same chart field. These
curves will immediately, if drawn correctly, show the compari-
sons desired. Form 14 is a good type of this form of graph:
Illustrative Example. The earnings and expenses of a certain railroad
by years are given below:
Earnings .
Expenses
1915
1916
1917
191S
1919
$400,000
260.000
$660,000
440.000
$680,000
460.000
$800,000
500.000
$960,000
640.000
1920
$1,000,000
680.000
Using a solid line for the earnings and a dotted line for the expenses,
these are shown on the same chart (Form 14).
Dollars
1,000,000
800,000
600,000
400,000
200,000
y
^artiVng^
y
,^^^-— -
/ *
Expenses
_ ^^ 1
^-^-
-*»^'
•
1915 .1916 1917 1918 1919 1920
Form 14. Composite Chart showing Relation between Income and Outgo
These curves are similar to the comparative curves shown in Form 3 2
154
BUSINESS MATHEMATICS
WRITTEN EXERCISES
1. The monthly earnings and expenses of a railroad are as
llmxra*
follows:
Jan.
Feb.
March
April
May
June
Earnings
Expenses
$ 83.000
57,000
$ 73.000
61.000
$87,000
63.000
$86,000
61.000
$95,000
70.000
$101,000
73.000
Expenses
Earnings .
Aug.
$119,000
87.000
Sept.
Oct.
Nov.
$98,000
84.000
$92,000
87.000
$85,000
70.000
Dec.
$ 80.000
65.000
Make two curves on the same chart to show the contrast.
2. Draw two curves to represent the average rate paid and the aver-
age number of phones in use in a city, from the following data:
1907
1908
1909
1910
1911
1912
1913
Av. rate paid
Av. No. used
$0
$120
11,000
$113
20,000
$100
26.000
$85
36.000
$69
48,000
$6?r
64.000""
1914
1915
1916
1917
1918
1919
Av. rate paid
Av. No. used
$60
80,000
$56
94,000
$52
116,000
$46
140,000
$40
170.000
$39
196.000
3. Representing the per cents on the vertical column and the grades
on the horizontal column, make three curves to show the percentage
of pupils attaining the various grades as given by three teachers in the
same subject.
GRAPHICAL REPRESENTATION
155
1st teacher.
2nd
3rd
40-50
50-60
60-70
70-80
80-90
4%
5%
3%
7%
8%
10%
10%
12%
7%
25%
35%
20%
30%
25%
30%
90-100
24%
15%
30%
4. Represent with four curves the outward messages from 12 selected
stations of the American Telephone Company for the 12 mo. of the year.
Year
Jan.
Feb.
March
April
May
June
1917.. .
1918.. .
1919.. .
1920 . . .
13.450
14.400
15.550
16.350
13.700
14,800
15.950
16.400
13.600
15.150
15,450
15.600
13.550
14,750
15.800
16.100
13.750
14.850
15.800
16.100
13.400
14.600
15.650
16.150
Year
July
August
Sept.
October
November
December
1917.. .
1918.. .
1919.. .
1920 . - .
12.400
13.650
14.900
15.500
12.350
13.600
14.500
15,750
13.250
14.750
15.750
15.200
14.000
15.650
16.000
15.550
14.300
15.400
16.250
16.550
14.100
15.550
16.000
16.500
6. If the American Telephone Company wished to show how its busi-
ness was running, it might plot the data given in Exercise 4 and draw
one continuous curve for the 4-yr. period. The work can be simplified
somewhat by plotting only 1 yr., and connecting these points with
straight lines. Do this and note whether the hne tends upward or down-
w^ard. If it gradually rises what will it show about the business?
6. Construct 2 curves on the same chart to represent the debt and
the stock of gold coin (in dollars) in the United States from the follov^^ing
data. Use solid and broken lines or two colors.
Year
1860.
1865.
1870.
Net Debt of the
United States
50 millions
2,680
2,300
Stock of Gold
156 BUSINESS MATHEMATICS
Net Debt of the
Year United States Stock of Gold
1875 2,050 millions 110 millions
1880 1,900 " 340
1885 1,350 " 600 "
1890 900 " 700 "
1895 900 " 625 "
1900 1,100 " 1,000 "
1905 1,000 " 1,350 "
1910 1,050 " 1,600 "
1915 1,100 " 2,000 "
1917 1,150 " 3,100 "
7. The following facts show advances in rentals from 1900 to 1913.
They are based on a study of charges for the same properties in 48 cities
having a population of 10,000 or over. Make three curves to show these.
Use a solid hne for the stores in the first-class business districts, a dashed
line for second-class districts, and a dotted line for third-class districts,
or use colors.
Rents 1st class
" 2nd "
" 3rd "
1900
1901
1902
1903
1904
1905
100
110
120
130
135
140
100
120
130
145
155
175
100
107
115
120
125
130
i9oa
165
190
13r>
Rents 1st class
•' 2nd "
" 3rd "
1907
1908
1909
1910
1911
1912
190
220
240
265
295
330
215
240
255
275
297
325
150
155
170
180
190
200
191
21{>
139. Component Parts Shown by Curves. — The chart
illustrated in Form 15 shows the total cost plus the profit in
a manufacturing business. This is of especial value to the
executive, for he may see at a glance how the manufacturing
end of the business is working. It will also show if there are
GRAPHICAL REPRESENTATIOX 157
"regular, extra heavy expenses in some particular part of
t|>e manufacturing costs and allow him to take means to
.. L.-il-u...! ,1 1 1 1 ] 1
- - ^
2.500
c ^ ^^
2,fm
IGOO
lOOD
500
.
3. —
sb Mar Apr May Juno July Aug Sept. Oct. Nov Dec.
Form 15. Chart showing Component Parts
■emedy them. Note that each item of the expense is added
""i so that the top curve will show the total.
WRITTEN EXERCISES
I- Using the following data, construct cu
1 Form 15.
rvcssi
inilar to those shown
Jan.
Feb.
Mar.
APR,.
Mav
,™
SI, 000
1.200
300
500
$1,100
1,20J
500
350
»1,200
1.300
475
1 800
1.100
350
. 500
(1,500
1.300
300
1,250
Brvidon and detka
fd charges
3^
158 BUSINESS MATHEMATICS
3. Make a graph showing the percentage of distribution of the ex-
pensps of operating the railroads of the United Statea per year. Total
of all per cents should be 100.
™o
™.
15
31
i™
»i
,»s
».
»
H
31
Y
30
7
14
30
3
32
2
13
33
3
14
3D
^
Main lena nee of equipment
:o
CancJuctinH transportation
41
»
,«»
190O
■m
i«»
..»
,.„
IZ
15
13
37
13
16
S9
13
18
18
36
16
4i
140. Correlative and Cumulative Curves.— Such curves
are constructed so that they show a relation as well as the
total to date. For instance, in Form 16 the sales up to Apn'
are $22,000, while the total collections to April are $19,00O-
Similarly, the total sales to September are $44,000, and the
total collections fo September are $46,000. The collections
for March show against the sales for January, etc. If the
business conditions of the firm are- ideal, the two curves
should run nearly parallel to each other. A 60-da. lag
means that the sales are 60 da. ahead of the collections.
WRITTEN EXERCISES
1. Construct a graph similar to that showD in Form 16, to represei*
the following data on production in a [aanufacturing plant:
GRAPHICAL REPRESENTATION
159
Month
Number Planned
Per Week
Actual Output
1st Week
2d Week
3d Week
4th Week
m
40
40
50
60
60
75
75
30
35
40
50
60
70
70
40
45
55
65
55
70
80
45
40
60
60
65
65
80
50
40
50
65
60
80
85
eb
SAar
\pT
May
June
July.;
60,000
50.000
FOR SALES
Jan. Feb. Mar. Apr. May June July Auar. Sept. Oct. Nor. D«e.
40^000
s
20;00O
10,000
>
r^
»^^
^
^
.^
^
^^
^y
1>^
^
^A
■^^
^
f^
60 da
1 Lag:
Har. Apr. May June July Ausr. Sept. Oct. Nov. Dec. Jan. Feb.
FOR COLLECTIONS
Form 16. Correlative and Cumulative Curves
2. The following information represents the cost per car-mile of differ-
ent Weight trucks, as estimated by a manufacturer, as well as the actual
^t- Use a dotted line for the former and a solid line for the latter (or
Afferent colors), to show the two curves.
Estimated Cost
Per
Actual Cost Per
;ht of Truck
Car- Mile
Car-Mile
1,000 lbs.
.IH
2.63fi
2,000 "
2.13ff
H
3,000 "
3.38ff
4.12ff
4,000 "
H
ZH
160 BUSINESS MATHEMATICS
Estimated Cost Per
Actual Cost Per
Weight of Truck
Car-Mile
Car- Mile
5,000
it
S.5t
3.8fi
6,000
a
3.75fi
4fi
7,000
it
4.25fi
H
8,000
tt
4.5ff
6.5fJ
141. Map Representations. — Map graphs^ are very use-
ful in the executive's office. They show at a glance the loca-
tion of all the subsidiary offices or manufacturing plants.
Form 17 shows the location of the various cantonments for
the United States Army in 1918. Pins with colored heads
and letters or figures can well be used with such maps. The
letter could stand for the name of the city or town. A
tabulation showing the names of the places opposite its
corresponding number could accompany a chart.
WRITTEN EXERCISES
1. Make a map of the United States and on it locate the following
selling agencies of a prominent article :
Boston, Mass. St. Louis, Mo. Salt Lake, Utah
Albany, N. Y. New Orleans, La. Butte, Mont.
New York City Dallas, Texas Portland, Ore.
Harrisburg, Pa. Des Moines, Iowa San Francisco, Cal.
Detroit, Mich. Tulsa, Okla. Sacramento, Cal.
Chicago, 111. Denver, Colo. Milwaukee, Wis.
^ Two types of maps should be used in the teaching of map graphs',
the blackboard outline map for the class recitation, and the desk outline
map for the notebook work by the individual pupil. Maps of the world,
the United States, and your own state will be sufficient for this work.
The desk outline maps should be punched to fit the notebook cover, in
order that the maps prepared by the pupil may be made a part of his
notebook work.
The general use of the pin map in business offices justifies the author's
belief that they belong in this work. Any good wall maps mounted on
cork composition and framed, together with a box of map pins of assorted
colors and sizes will serve. Exercises in which pupils are required to
show on the map the location of important commercial cities, industrial
sections^ etc., should give added interest to this subject.
GRAPHICAL REPRESENTATION
161
2. Construct a map of New York and New Jersey and on it show the
elevation of the following points on the New York, Ontario, and Western
Railroad:
Weehawken
Cornwall
Middletown
Youngs Gap s .
Oneida
Cadosia 1,000 "
Apex 1,500
20 ft.
25 "
550 "
1,800 "
400 "
n
Walton 1,100ft.
Northfield 1,700 "
Oswego 250 "
Sidney 1,050 "
Summit 1,600 "
Norwich 950 "
Eaton 1,350 "
Form 17. Map Chart
142. Frequency Charts or Curves. — These curves are
intended to show how often or when certain things occur.
The curve given in Form 18 shows when the commodities
•
fise and fall in price. The peak of a curve will show when
^he particular thing will reach the greatest amount or the
greatest number of years, etc.
WRITTEN EXERCISES
!• Plot a curve to show the age at marriage of 439 ladies who were
allege graduates.
n
I
162
BUSINESS MATHEMATICS
First
240
220
200
180
160
140
120
100
Age %
20 6*
21 7
22 2.8
23 5.5
24 12.5
25 15.2
26 14.1
27 14.4
28 11
29 8.2
Age %
30 5.2
31 2.8
32 2
33 2
34 1.1
35 1.1
36 2
37 7
38
Ninth
YEARS
Tenth Eleventh
Twelfth
Thirteenth
240
220
^
ALL COMMODITIES
1. Food.
2. Clothing.
8. Cloths.
4. Drugs.
6. Farm Products.
6. Metal.
7. Metal Products.
8. Lumber.
9. Building Material.
10. House Furnishing.
11. Miscellaneous.
y
^
/
/^
/
/
j^
/
^
^
y^
200
180
160
140
120
100
Form 18. Frequency Curve showing Changes in Costs
2. Construct a curve to show the death rate changes of Americans at
various ages since 1880, as reported by an insurance company.
Age % Decrease % Increase
Under 20 17.9
20 to 30 11.8
30 to 40 2.3
40 to 50 13.2
50 to 60 29.2
60 and over 26.4
GRAPHICAL REPRESENTATION
163
• Using one space on the horisontal scale to denote a 2-wk. period,
one space on the vertical scale to denote 10 deaths per thousand,
itruct two curves on the same chart to show the comparison of deaths
lildren in similar parts of two successive years in New York City,
sported by the Sheffield Milk Company, who claim their milk was
to poor people in 15 Board of Health stations in the latter year at
V cost while these people were not accorded that privilege in the
er year.
2wk.
li u
u u
u u
if u
il u
II u
« <(
« ((
It <(
« ((
(( ((
K U
(( l(
« ((
« l(
a ((
« «
First Year
Deaths Per 1,000
Second Ybak
Deaths Per 1.000
140
98
123
96
118
110
127
120
110
130
130
120
115
132
138
131
130
124
110
135
112
108
120
118
148
155
200
145
240
86
200
145
170
165
165
170
140
140
MISCELLANEOUS WRITTEN EXERCISES
Construct on the same chart the weekly sales of four salesmen from
)nowing data:
MEN
1st Wk.
2d Wk.
3d Wk.
4th Wk.
5th Wk.
6th Wk.
7th Wk.
8th Wk.
$92
$ 30
$176
$150
$105
$110
$ 80
$ 88
' • • •
56
58
86
88
140
95
98
76
, , ^
60
116
120
150
245
210
76
220
76
380
230
290
310
270
240
325
164
BUSINESS MATHEMATICS
The above-mentioned sales were actually made by dififerent men selling
a house-to-house article on a commission basis. The curves will ina-
mediately show up the comparison of their sales.
2. The following data shows that an automobile should stop withii*
the given distances according to the miles per hour of speed, if the brake©
are in proper order. Construct a curve from this data.
At Speed of
10 miles per hr
15 "
20 "
25
30
35
40
50
({
t(
it
ii
It
A Car Should Stop in
9.2 ft.
20.8
37
58
83.3
104
148
231
3. The following information shows the maximum percentages
different family annual incomes which ought to be expended in no
times for the variously named parts of a family expense account.
cdJ
^
Various Items
Percentage of Annual Income
$2,000-$4.000
Food
Rent
Clothing
Miscellaneous operating. .
Higher living, books,
savings, insurance, re-
ligious, etc
Railroad
Street-car
Water rent
Gas
Electricity
Telephone
25
20
15
8
24
2
1
1
2
1
1
100
$1.000-$2,000
25
20
20
5
21
2
1
1
3
1
1
100
S800-$1.000
30
20
15
5
18
2
3
1
5
1
100
$500-$8
45
15
10
7
11
2
5
1
4
100
Construct a circle or a rectangle for each income and divide it up into
its component parts. Then place the circles or rectangles side by side
to show comparisons.
GRAPHICAL REPRESENTATION 165
4. The following facts taken from the New York Times Annalist show
the trend of bond prices over a certain period of time. Plot a curve
from the following data:
1917 Price 1918 Pricb
Jan 90 Jan 77
Feb 87 Feb 76|
Mar 86i Mar 76
Apr 85 Apr 761
May 83 May 77i
June 84 June 76 J
July , 82 i July 76$
Aug 83 Aug 76 1
Sept 80 Sept 76
Oct 79i Oct 77f
Nov. 77 Nov 82
Dec 74i Dec 80
5« Each dollar of cash income of the New York Life Insurance Com-
pany, was expended as follows according to a recent yearly report of the
company. Construct a graph to represent these facts.
Paid for death claims $ .21
Paid to living policyholders .38
Set aside for reserve and dividends .29
Paid to agents 06
For branch expenses, agency supervision, and
medical inspection .02
For administration and investment expenses .03
For insurance, debt, taxes, fees, and licenses .01
Total $1.00
^' Construct three curves on the same chart (preferably with different
^^ored inks) to show the changes in the population of New York, Chi-
^So, and Pittsburgh respectively, from the following data:
Population of Population of Population of
Year New York Chicago Pittsburgh
1850 480,000 100,000 20,000
1860 325,000 200,000 80,000
1870 425.000 210,000 90,000
1900 3,437,202 1,698,575 451,512
1910 4,766,883 2, 185,283 533,905
1920 5,620,048 2,701,705 588,193
166 BUSINESS MATHEMATICS
7. The United States life tables, Census 1910, report the following
death rates per 1,000 among the white males at the various ages. Con-
struct the curve from the data and compare the numbers of different ages.
Death Rate
PER 1.000
69
137
245
386
585
Age
Death Rate
PER 1.000
Age
12 . . . .
.... 2.4
70.
20
.... 4.5
80.
30
6.8
90.
40
.... 12
100.
50
.... 18
106.
60
36
8. Refer to §138 and construct a chart showing the following informa-
tion from averaged figures, taken from many trade organizations, the
Harvard Bureau of Business, and individual investigations.
Percentage of Total Sales
Cost of Doing
Kind of Business Net Profits Cash Discounts Business
Variety goods 6 3 19
Dry goods 5 4 23
Clothing 5.25 5.5 23.5
Furniture 7.75 3.25 23.75
Jewelrv 3 6.5 25
Drugs 5 3 25
Hardware 7.75 6.25 19.5
Shoes 4 3 25
Department stores 6 25
Implements and vehicles 2.25 6.5 17.5
Groceries 2 2 17
9. Refer to § 138 and construct a chart from the following information
of profits, costs, and discounts by lines (net profits for one turnover).
Percentage of Total Sales
Cost of Doing
Line
Net Profits
Cash Discounts
Business
Books . . .
2
3
22
Corsets . .
8
4
24
Furs
7.5
3.25
26
Gloves. . .
4
5
24
Hosiery. .
5
5.5
23.5
Handkerchiefs
4
3.5
24.5
Laces. . . .
9.5
3.25
23.25
Linens. . .
6
3
24
Millinery.
10
6
25
GRAPHICAL REPRESENTATION
167
Pbrcgntagb of Total Sales
Cost of Doing
Line Net Profits Cash Discounis Business
Pictures 13 4 25
Ribbons 3.5 4 24
SUks 4 5 23
Toys 10.75 3.25 22
Umbrellas 5 4 26
(Vaah Goods 7 3.5 20.5
10. Referring to the tables in S 5 18 and 21 , which show the correspond-
□g profit on cost represented by certain per cents profit on seUing price,
lOlve one of the above examples for profits on cost and note comparison
jf your answers with those enumerated above.
11. Make a comprehensive chart from the following fact?, somewhat
ifter the idea of £ 138, with these suggestions: Use a scale on the left for
i^apitalization up to 145 millions; one on the right up to 25 mtflions; but
the right-hand scale to be 4 times the left-hand scale, i.e., 20 milhons on
the left la the same line as 5 millions on the right. The lower right-hand
;cale is to represent earnings. In the upper right-half of the chart have
a scale for price range of the company's general and refunding 5% bonds.
Let 75 on this scale be on a line with 100 on the left, and 80 shall cor-
respond with 108 on the left, etc. Put in the price range curve with red.
I. ^
....
DOL.A.S
X
a
s
£
"fi
1
S,
1
S
1
Si
•5
3
I
l-S
'-'
^
w
o
Oio
1909
leio
s,
.ill
,0
.IS
.;
;;
K
1611
103
^s
HI
H.1
31
IK
$92
77
34.7
na
57
iei3
83
IBH
130
3,1
Id
7H
i:<
88
191S
132
,>K
1A
42
i«
75
90
me
17
IR
7S
Hi
im
Ul
JB
23
feu
ao
15
yo
SS
168 BUSINESS MATHEMATICS
12. From the following data, taken from Paul Nystrom's book, "Tlie
Economics of Retailing," make ten circles of equal size or ten rectangles
of equal length, showing the component parts for the apportionmea^ of
the rent in 10 stores of different numbers of floors. Study and compare
your results.
Percentage of Rent
123456789 ^10
Basement 35 25 15 10 10 15 12.5 15
Main floor
Second floor
.... 65
65
35
50
25
60
30
10
45
25
15
45
25
10
10
50
20
10
10
40
20
15
10
35
20
15
10
7.6
20
Third
lO
Fourth
lO
Fifth
5
Sixth
3
First
35
__ — —
13. Construct two curves to represent the time it takes two oper»."tx^^
to do various operations. Add the time for each operation to the to*'^
time spent on the preceding operations, thus forming a cumulative 'ti^'^^^
for the whole number of operations
Time in Seconds
First Second
Kinds of Operations Operator Operatof^
Reaches for label 2 3
Reaches for brush 3 3
Wipes brush on glue pot 6 9
Brings brush to label 2 3
Covers label with glue 5 9
Replaces brush 2 2
Puts label on package 3 4
Adjusts and smooths label 6 12
14. Construct three curves on the same chart field to show the j>^'"
centage of the pupils of each teacher's class which attained the vario^is
values (or percentages) in the same grade of work.
Values
Number of
Students
Per Cent of
Total Number in Class
First Teacher
40- 50
2
5
50- 60
2
5
60- 70
4
10
70- 80
20
50
80- 90
10
25
90-100
2
5
GRAPHICAL REPRESENTATION
169
VAttrfc^* P'* Cent of
^^ NuiiBEK OF Students Total Number in Class
-_ Second Teacher
^^02 6
5J:'3^0 2 6
^ ^O 15 37.5
^ ^O 10 25
^^100 11 27.5
Third Teacher
5^ 7o
ftT ^Q 20 50
^ ^ " 10 25
^^100 10 25
15. struct a chart to show the comparison of the expenditure of
%\ in 2 successive years, of the raikoads of the United States, from the
ioUowing ^^^.
In Cents
1st Yr. 2d Yr.
Operating expenses 68.1 62.55
"^axes 7.03 4.25
^cess of fixed charges over non-operat-
ing income 10.13 11.14
I^vidends 14.74 22.06
, • Construct from the following information three curves on each of
^ charts of the same height and scale, to show how a certain business
^ ^ning.
n 1st Chart
^^^ipts
Ji'^^chases
^^1 expense
p 2d Chart
^ ^1 expense
p ^tal salary expenses . .
^sonal
P» 3d Chart
^^'Oss profit
v^^Pense
^t profit
Jan.
Feb.
Mar.
Apr.
May
$1,375
$1,350
$1,450
$1,425
$1,400
925
850
1.100
1.000
950
600
525
520
550
500
625
550
525
500
550
440
420
450
420
410
340
320
310
320
340
38
43
31
38
35
35
25
25
26
25
3
18
6
12
10
June
$1,350
1,000
475
525
420
320
40
28
12
170
BUSINESS MATHEMATICS
17. Chart the following facts by the use of the same-sized circles
parts of a circle to show the dollar's buying power and its changes durin.^
a part of the World War.
% Buying Power
1st yr 100
2d " 90.6
3d " 84.28
4th " 58.8
18. Construct two rectangles of the same length and divide each infco
its component parts to show how rents vary in different lines. The^€
figures were compiled by the Bureau of Business Research of HarvaK~<l
University. Note how the rents of the two respective lines vary.
Ratio Between Rent and Net Sales
High Low Cobimon or Typical
Groceries 4.1% .3% 1.3%
Shoes 14.6% .8% 5%
19. Chart three curves on the same field to show the following facets
concerning the railroads of the United States for the year. These facts
are from the Bureau of Railroad Economics, Washington, D. C.
Revenues . . .
Expenses
Net revenues
Jan.
Feb.
Mar.
Apr.
May
$1,125
800
325
$1,150
800
350
$1,250
850
400
$1,225
825
400
$1,300
860
440
CE
JUN'
$1.3O0
850
450
Revenues . . . .
Expenses . . . .
Net revenues
July
$1,320
850
470
Aug.
$1,410
890
520
Sept.
$1,400
890
510
Oct.
$1,470
900
570
Nov,
$1,400
890
510
DEC
$1,350
900
450
20. A man walks to a place at the rate of 4 miles per hour, remain^
hr., then rides back at the rate of 10 miles per hour. He was absent-
hr. How far did he walk? ^
3 From H. E. Cobb, Elements of Applied Mathematics. Bosta^'
Ginn and Company, 1911.
GRAPHICAL REPRESENTATION 171
Hint: To graph, or solve graphicaUy, take 8 spaces from the inter-
ction of the vertical and horiiontal axes (called 0) on the horixontal.
Ekll it P. Let the horizontal scale represent hours, and the vertical
ale miles. Draw a line from through X the intersection of the lines
spresenting 1 hr. and 4 miles, to the opposite side of the chart. Do the
une from P toward the left 1 hr. and up 10 miles to a point Y. Draw
ae from P through Y until it crosses the former line. Find the points
1 the two lines on a horizontal, which are 1 hr. apart. Follow that line
o the left and you will find the number of miles.
21. A man rows at the rate of 6 miles per hr. down stream to a place,
knd 2 miles per hr. in returning. How many miles distant was the
3lace if he was absent 12 hr. and remained at the place 6 hr.?
22. The expenses of a firm were reported as follows for a certain year.
Graph these facts so that you could present them in good form to the
president of the company.
Postage S 1,000 Advertising $17,000
Telephone 2,000 Equipment 1,400
Selling 17,000 Managing 6,000
23. Try to find out the main branches of the Ford Motor Company,
their location, etc., and make a pin map of the branches that are in the
United States.
2^. Find out the branch offices of some large corporation or manu-
facturing plant in your locality and then construct a map, and on it
Daark these branch offices.
25. Apply Exercise 23 to Swift and Company, Chicago, 111.
26. Try to obtain similar information from the United Cigar Stores
Company of New York and make a pin map.
27. Obtain similar information concerning the Woolworth 5 and 10
cent stores and make a pin map.
28. Find out the location of the branches of the United States Steel
^rporation and apply Exercise 23.
25* Do the same for the Standard Oil Company.
CHAPTER XII
SHORT METHODS AND CHECKS
143. Value of Short-Cut and Checking Methods.— A
computer naturally should be the master of short-cuts and
methods for checking his work. The latter is perhaps of
more importance than the former, because one must be
able to show his employer that he knows that his work is
correct.
The executive should also know these methods in order
that he may be able to check the work of his employees.
He must also be sure of the facts presented as a basid for a
satisfactory and successful business.
It is the object of this chapter to give to any person who
will devote a small amount of time to it each day, such a
knowledge that he will be well equipped to do computation
work in the shortest as well as the best way, and at the same
time be able to make an accurate check on these computa-
tions.
144. Addition. — *^ Anything worth doing is worth doing
welV^ therefore we should not be satisfied with a piece of
work unless we check it in some way, if it is at all possible,
and with most work it is possible. The checks that are
quite common in addition are:
1. Adding the example from top to bottom, then adding
from bottom to top, or vice versa.
2. Adding by columns.
172
SHORT METHODS AND CHECKS 173
Illustrative Example.
(a)
(b)
1456
1456
7238
7238
3564 or 3564
18
11
14
11
11
14
11
18
12258 12258
Explanation: (a) Add each column separately, beginning at the
right, and set the result for each column down by itself. The result
for the tens column should be placed one place to the left of that for the
units column. Then add the totals. This method is valuable when
adding long columns, especially if one is interrupted while adding.
(b) Add in a similar manner beginning at the left column and moving
each result one-place to the right.
3. Excesses of 9's, often called ''casting out 9\s."
Illustrative Example. 1457 8
7238 2
3564
12259 y 1
x/
EIxplanation: Add the digits (or figures) in each number (as for
example in 1,457), 1+4 + 5 + 7 = 17; divide 17 by 9 and the amount
remaining, 8, is placed out at the right. Do the same for all the numbers.
Then add these excesses (or amounts left over) and the result is 10 in
the above example; find the excesses of 9 in this sum, giving 1 in the
ahove. Then find the excesses of 9 in the total, 12,259, which we also
find is 1.
This gives a fairly satisfactory check. This check, however, is not
always to be depended upon because one may make a mistake of 9 (or
some multiple of 9) or in the addition which will not show as an error in
the excesses of 9's in the total or a mistake in arranging the figures in a
number, Le., 4,175 instead 1457, above.
174 BUSINESS MATHEMATICS
WRITTEN EXERCISES
Try out these checks on the followiDg examples:
1. 4,567 2. 6,597 3. $1,467.75
3,784 5,836 5,693.83
2,543 6,784 7,649.75
6,738 5,987 6,573.81
6,774 5,967.48
5,965 6,796.54
Hint: In all addition try finding numbers that group and make
10, or 5, or something similar.
Example: 7 ^ ^
3 1 ^"
2
4
I
145. Subtraction. — The common check is the process
opposite to the usual one, i.e., add the remainder to the
subtrahend and it will produce the minuend if the work is
correct.
Illustrative Example.
567
198 Check: 369 + 198 = 567
369
WRITTEN EXERCISES
Kind the difference in the following examples and check:
1. 145.674 2. 7,564 3. 4,563
96.785 3,783 7,862
AiuMluM' check is the subtraction of the excesses of 9*s.
SHORT METHODS AND CHECKS 175
Illustrative Example.
196 7
157 4
39 ^3
/
WRITTEN EXERCISE
1. Add columns (a) and (c) and subtract column (b) from the result
for column (d)
(a) (b) (c) (d)
liALANCE
AT Beginning
Checks
Deposits Balance
$150
215
$ 75
205
$125
137
316
187
49
149
258
325
256
385
234
274
425
324
124
How would you check this problem? Check it.
Another check is to subtract by addition or to subtract a
sum of two or more numbers from a certain number.
niustrative Example. 14563
3754
2649
5365
2795
ExpliANATiON: Adding from the bottom up in the first column, 5, 14,
18, and what make 23, put the result, 5, down and carry 2; then the
second column 2, 8, 12, 17, and what make 26. Set down the 9 and carry
the 2 to the third colimin; 2,5,11,18, and what make 25, answer 7. Set
down the 7 and carry 2 to the last column ; 2, 7, 9, 12, and what make 14.
Set down the answer 2. Now check the work by adding up all the num-
bers but the top one, and the result should be the top number.
176 BrSIXE^ MATHEMATICS
Try :Lii liir: or. 'hr : -ILi-T-^i: r:tiinpics and find the baJaoccs in the
1. $3>4o ItikLuMe in ifae haiAk in the morning
243
L^ ' Checks eiven oat during the day
3ai^nce Csbck:? Last Balance
2. $ 5d7.So $124 ft5: $ 56.76: $7.5.61
1.256.65 546.73: 124.75; 75.75
li6. Multiplication. — Methods of checking multiplica-
tion are as follows:
1. Divide the product by the multiplier to obtain the
multiplicand, or by the multiplicand to obtain the multiplier.
2. Repeat the multiplication and if the result is as before
we mav assume that the work is correct.
3. Cast out 9's, as follows:
niustrative Example. 43 X 21
43 7 excess
'21 3 *'
903 21 .
3=3
ExPLAN ATioN ! Find the excesses of 9's in the multiplier and the multi-
plicand. Multiply these excesses together and find the excess of 9's in
this prfxluct.
WRITTEN EXERCISES
Multiply the following and check:
1. 25G 2. 345 3. 4oG 4. 375 6. 451
25 31 123 256 231
SHORT METHODS AND CHECKS 177
By the use of the excesses of 9's, without actually multiplying out,
state which of the following results are correct:
6. 456 7. 376 8. 415 9. 575 10. 56
25 75 16 125 22
11,400 28,200 6,440 71,875 1,232
Short methods prove to be very valuable. Thoroughly
understand each method as you go along and practice it
whenever possible.
147. To Multiply by Any Multiple of 10.— Move the deci-
mal point as many places to the right as there are zeros in
the multiplier. It is obviously necessary to annex zeros if
the multiplicand is a whole number as a decimal point is
understood at the end of each whole number.
Illustrative Example 1. .456 X 100
Solution: Move the decimal point two places to the right, giving
4">.6
Illustrative Example 2. 1.347 X 10,000
SoLimoN: Move the decimal point four places to the right, giving
13,470.
Illustrative Example 3. 15 X' 1,000
Solution: Since there is a decimal point understood at the right of
any whole number, therefore in this case we move it three places to the
right, giving 15,000.
ORAL EXERCISES
Multiply the following mentally:
6. 2.467 X 10,000
7. .00035 X 10,000
8. 4.00016 X 1,000
9. 546.1 X 1,000
10. 310 X 10,000
12
1.
456 X 10
2.
64.7 X 100
3.
456.25 X 1,000
4.
6.413 X 100
6.
.00005 X 1,000
178 BUSINESS MATHEMATICS
148. To Multiply Numbers Having Zeros as End
Multiply by the significant figures, then annex as
zeros as there are in both the multiplicand and the
plier.
mustrative Example 1. 430 X 400
Solution: Multiply 43 by 4, giving 172, then annex thrc
giving 172,000.
ORAL EXERCISES
Multiply the following
I'
1.
2.
3.
4.
6.
236 X 20
352 X 300
456 X 1,200
2,145 X 2,200
120 X 400 :
6. 1,350 X 900
7. 18,000 X 12
8. 140 X 2,500
9. 230 X 4,000
10. 1,600 X 1,200
Illustrative Example 2
. .256 X 20
Solution:
.256 X 10 =
2.56 X 2 =
or
.256 X 2 =
.512 X 10 =
2.56
5.12
.512
5.12
Multiply the following:
1. .53 X 300
2. 16.351 X 400
3. .056 X 120
4. .0027 X 9,000
6. 2.136 X 30,000
149. To Multiply by 9, 99, 999, etc.— To mul
number by 9, multiply the number by 10, and th(
tract the original number from the result of the
plication.
SHORT METHODS AND CHECKS 179
niustratiye Example. 456 X 9
Solution:
4560 = 10 X 456
456
4.104
To multiply a number by 99, multiply the number by
100, and subtract the number.
Wustrati?e Example. 345 X 99
Solution:
34500 = 100 X 345
345
34155
WRITTEN EXERCISES
^^Itiply each of the following numbers by 9, 99, and 999:
1. 238 4. 642
2. 426 6. 233
3. 175
^ ^6o, To Multiply by 25, 50, 75, etc.— Since 25 is equal to
* therefore to multiply a number by 25, multiply the
^^^ber by 100, and divide the result by 4.
^^strative Example 1. 348 X 25
4 ) 34,800
8,700
Since 75 = 100 - 25 = 100 - ( i of 100), therefore to
multiply a number by 75, multiply the number by 100, di- ^
vide by 4, and subtract.
180 BUSINESS MATHEMATICS
nhtstntive Example 2. 576 X 75
Solution:
4 ) 57,600
14,400
43,200
WRITTEN EXERCISES
1. Multiply 679 by 50. How would you do it?
2. How would you multiply by 125?
Find the results of the following by the above methods:
3. 32 X 25 7. 67 X 50 11. 1,248 X 125
4. 76 X 25 8. 88 X 75 12. 3,457 X 25
6. 84 X 25 9. 145 X 50 13. 1,256 X 750
6. 43 X 50 10. 726 X 250 14. 496 X 250
151. To 'Multiply Two Numbers, Each Ending in 5.-
lUustrative Example 1. 65 X 65 = 4,225
Explanation : When the same numbers are to be multiplied, writ>^
25 for the last two figures at the right. Then add 1 to the tens figur^
(6 + 1=7) and multiply by the other tens figure (7 X 6 = 42), anc/
write this result at the left of the 25 just written. The product i^
4,225.
Illustrative Example 2. When the sum of the tens figures is an even
number. 55 X 75 = 4,125.
Explanation: Set down the 25 as the first part of the product.
Then find i of the sum of the tens figures (J of 12 = 6), and add it to the
product of the tens figures. 5 X 7 = 35. 35 + 6 = 41. Place the 41
at the left of the 25 previously written.
Illustrative Example 3. When the sum of the tens figures is an odd
number. 35 X 25 = 875
Explanation: Set down 75 as the first part of the product. Then
find J of the sum of the tens figures (2 of 5 = 2 J). Drop the i, and add
the 2 to the product of the tens figures.
SHORT METHODS AND CHECKS 181
ORAL EXERCISES
Multiply the following:
1. 35 X 35 6. 65 X 15
2. 85 X 85 7. 35 X 45
3. 15 X 15 8. 45 X 55
4. 65 X 25 9. 125^
6. 85 X 45 10. 75^
152. To Square Any Number of Two Figures. — This is
based upon the principle that the square of the sum of two
quantities, like (a + 6)^, is equal to the square of the first
quantity, plus twice the first quantity times the second
quantity, plus the square of the second quantity.
Ulustrative Example.
(3 +4)^ =3^+24+4^
= 49
WRITTEN EXERCISES
Square the following numbers by the above method: '
1. 64 4. 52
2. 36 6. 124 (Call it 12 tens.)
3. 27
163. Sum of Two Quantities Times Their Difference. —
Certain number products come under the principle that the
sum of two quantities times the difference of those same two
products, as (a + b){a — b) equals the square of the first
quantity minus the square of the second quantity, or a' — b^.
Illustrative Example.
21 X 19 = (20 + 1) (20 - 1)
= 400-1
182 BUSINESS MATHEMATICS
WRITTEN EXERCISES
Find the product of the following numbers by the above method:
1. 22 X 18 4. 64 X 56
2. 36 X 44 6. 62 X 58
3. 53 X 47
164. To Multiply by 11, 22, or Any Multiple of 11.—
Illustrative Example 1. 264 X 11
Solution: 264
11
2,904
Explanation : Set down the units figure of the given number in the
product, 4. Add the units and tens figures (6 + 4 = 10). Set down the
zero and carry 1. Add the tens and hundreds figures (6 + 2 = 8) ; add
the 1 carried, making 9. Multiply the last figure (hundreds in this ex-
ample) by 1, set the result, 2, down. Result = 2,904.
Illustrative Example 2. 416 X 22
Solution: 416
22
9,152
Explanation: Two times the unit figure = 12. Set down the 2 and
carry 1. Add the units and tens figures (6+1 =7). Multiply the 7 by
2 = 14, and 1 carried makes 15. Set down the 5 and carry 1. Add the
tens and hundrnds figures (1 +4 = 5). Multiply the 5 by 2 and add the
1 carried, making 11. Set down the 1 and carry 1. Multiply the last
figure (4 in this example) by 2 and add the 1 carried, making 9. Result
= 9,152.
WRITTEN EXERCISES
Multiply the following:
1. 340 X 11 5. 1,246 X 22
2. 562 X 22 6. 322 X 44
3. 124 X 22 7. 1,416 X 88
4. 214 X 33 8. 3,514 X 77
SHORT METHODS AND CHECKS 183
155. To Multiply by a Number Composed of Factors. —
If certain figures of the multiplier are factors of the other
figures in the multiplier, the following method can be used:
Illustrative Example. 356 X 568
Solution: 356
568
2848 = 8 X 356
19936 = 7 X 2848
202,208
Hint: Be careful about placing the result of the second operation.
WRITTEN EXERCISES
Multiply the following according to the method illustrated above:
1. 624 X 426 3. 126 X 124
2. 358 X 244 4. 1,718 X 186
156. To Multiply Two Numbers between 10 and 19
Inclusive. —
lUustrative Example. 16 X 14
Solution : 200
24
224
Explanation: To either number add the units figure of the other
number and annex 0, then add the product of the units figures.
Multiply the following according to the method explained above :
1. 17
3. 18
6. 17
7. 19
18
13
15
13
2. 16
4. 19
6. 18
8. 14
15
17
16
IS
i>
- ■■■IliiJg -
*• • - -
^ * . // - / *^ . / ^v«< « *•— -• ',*■
/ -'
/' y ^,./.,. f ' .^ yy,/^r.'. s? *ji*- J^"i»-TF-
i^
//
/ ^^
*
■/-^
/ <-
4
/ /
/ 'y
*
/>'
/ ,
*
/' /^
/ •.
^1-
"t-
.sr^jcj in- 3
M#H/4W4WlWi rtlM<*« Vft - HK) - 2r, - 100 - rj of 100)
SHORT METHODS AND i^nx^v...
WRITTEN EXERCISES
>he total value of each of the following:
1.
2.
3.
25 yd. at 44^
25 lb. at 45^
12 i yd. at 72^
33i " "
36ff
37J " " 72ff
8i " '* 24ff
12i '* "
48^
311 " " 32ff
33J " " 14^
161 " "
60ff
16| " " 42ff
62J " " 24ff
4.
6.
6.
60 lb. at lOfff
120 yd. at 62Jff
16 articles at $12§
90 " " QU
300
" " 83Jff
12
" $11
48 " " 6J<f
32 articles at $25
28
" $125
56 " " S7U
996 bu. at $o2J
18
'' $1§
159. To Divide by 10, 100, 1,000, etc.—
Illustrative Example. 4,675 -^ 10 = 467.5
Explanation: Move the decimal point as many places to the lefi
as there are zeros in the divisor.
ORAL EXERCISES
Divide each of these numbers by 10, 100, 1,000:
1. 14,500 6. .124 9. 67.346
2. 675.64 6. .000643 10. 7,865.4
3. 567.52 7. 12 36 11. 649
4. 695.6 8. 724.74 12. 7,567.4
160, To Divide by 25, 50, 75, etc.—
lUustrative Example 1. 12,400 -^ 25 = 124 X 4 = 496
Explanation : Since 25 = -?^, therefore to divide by 25, divi(?
LOO, and multiply by 4.
lUustrative Example 2. 3,0Q0 ^ 75
3, 00 -ir 100 = 30
30 + (i of 30) = 40
U5 VUSSSES
ffVffioni «adbi of the kJkniv <
a. 4,0004-50 CllU3i-£-73
4. 12;W0 4-75 H - 3&S -s- 9
i. 2 JOO 4-50 ML iMi -3- S
161. To Dmde by 2|, 31, etc—
niiislraliirt ExMBple L 50 -s- 2}
HfpumoH: 2| = i of 10
50 -5- 10 » 5
4X5 »20
lUttstratiTe Enunpie 2. 80 -s- 6i
H/iLirno.v: 10 - Of =« 3|
3} ^ I of 6i
80 -^ 10 = 8
i of 8 = 4
8 + 4 = 12
WRITTEN EXERCISES
Perform tho following cJiviHions:
1. 120 -5- 21 4. 60 -r 6J 7. 360 -5- li
2. 150 4- 3i 6. .624 ^5 8. 540 ^ 7J
3. 240 -5- 7i 6. 31.6 -i- 2i
162. To Divide by a Number Composed of Factors.
niustrative Example. 2,352 -^ 42
Solution: 7 ) 2,352
6 )336
56
Explanation : 42 - 6 X 7
SHORT METHODS AND CHECKS 187
163. To Divide by Continued Subtraction. — This method
been used as a catch question.
Illustrative Example.
645 -^ 147
Solution:
645
(1) 147
498
(2) 147
351
(3) 147
204
(4) 147
Result = 4
57
= remainder
164. Checks in Division. — 1. Multiply the quotient by
the divisor, and add the remainder.
2. Cast out 9's, as follows:
mustrative Example. 7,564 -^ 143 = 52, with a remainder of 128.
Solution: Excesses of 9's in 7,564 = 4
« u u a J43 ^ g
li tt " " 52 = 7
a it li u J28 = 2
8 X 7 = 56
Excesses of 9's in 56 = 2
2+2=4
/. The excesses in the dividend = the excesses in the remainder + the
excesses in the product of the excesses of the divisor and theiquotient .
WRITTEN EXERCISES
Divide the following and check:
1. 7,563 -^ 246 4. 6,574 -5- 925
2. 13,674 -J- 345 6. 76.348 -^ 34.6
3. 5,692 -^ 74
188 BUSINESS MATHEMATICS
166. Addition of Numbers Containing
niustrative Example 1. i + | + | + f = ?
Solution: J 45
i 48 Write the least cominon denominator
I 40 only at the end of the work.
I 50
183
60
Illustrative Example 2. 15i + 14| + 65} = ?
Solution: 15 J 2
14? 8
652 9
94 + 15 =95/ff
Illustrative Example 3. How many yards in 4 pieces of cloth contain-
ing 21 ^ 36 S 43^ 56'?
Solution: 21^
36' (The small numbers at the right and
43^ above represent quarter yards.)
56'
157^
Illustrative Example 4. J + i = ? Ans. ih
Note: It will be noted that the numerator is the sum of the denom-
inators, and the denominator is the product of the denominators.
Illustrative Example 5. H + i = f § or H
Note : Observe that the numerator is 2 times what it was in Example
4, and the denominator the same.
Illustrative Example 6. To add fractions by cross-multiplying the
denominators and numerators. + 7=?
Solution: 5X7 = 35
3X6 = IS
35 + IS = 53, the new numerator
7X6 = 42, the new denominator
51! = result
SHORT METHODS AND CHECKS 189
166. Subtraction of Fractions. —
Illustrative Example. When the numerators are alike, i — i
Solution: 5 X 3 = 15
4 X 3 « 12
15 — 12 = 3, the new numerator
4 X 5 = 20. the new denominator
/o = result
167. Multiplication of Mixed Numbers. —
Illustrative Example. 14} X 8f
Solution: 14}
^1
8f
1 — 1 V I
4 = 8 X }
2
s
112 = 8 X 14
9§ = § X 14
1255 = result
168. Multiplication of Mixed Numbers Whose Fractions
Are |. —
Illustrative Example 1. 8} X 8} = ?
Solution:
8J
i = ix J
8 = i of 8 + i of the other 8, or J of (8 + 8)
64 =8 X8
72i = result
190 BUSINESS MATHEMATICS
or
8i
8i
i = i Xi
72 = (8 + 1) times the other 8
72i = result
Illustrative Example 2. 4.5 X 4.5 = ?
Solution: 4.5
4.5
20.25
169. Division of Numbers Containing Fractions. —
Illustrative Example 1. To divide a mixed number by a whole number.
64J -J- 9 = ?
Solution: 641 = H^
5
9 =¥
194 ^ 27 = 7/7
or
9)641
7 5
• 2 7
Explanation:
64 -J- 9 = 7, and 1 over
14-2 — ^
'' -i. Q — 5
3 -7- y — Af
Illustrative Example 2. To divide a whole number by a mixed number.
35 -^ 7h = ?
Solution:
35 = 70 halves
7i = 15 "
70 -h 15 = 4f result
CHAPTER XIII
AVERAGES, SIMPLE AND WEIGHTED
170. Kinds of Averages. — In commercial work as well as
in other fields, there often occurs a need for knowledge of
how to find the average. Unless one has a thorough under-
standing of this subject he is very liable to arrive at an
entirely erroneous conclusion owing to his inability to com-
prehend the meaning of weighted averages as well as simple
averages. For example, it is an easy matter to find the
simple average weight of four people, whose weights are
140, 160, 180, and 200 lb., by adding the weights and divid-
ing by four; but were we to find the average wages of four
men who earn $3 each a day, 6 men who earn $4 each a day,
and 5 men who earn $5 each a day, we would then have to
compute the average in a slightly different manner in order
to arrive at the correct result.
It is the purpose of this chapter to acquaint the student
with the various kinds of averages as well as the proper
methods to use in order to arrive at correct conclusions.
171. Simple Average. — This is perhaps best understood
by the use of some examples and their solutions. If one
will read these carefully and pay particular attention to
the solution as indicated he should be able to find the
simple average, if the problem is such as to come under
this method.
191
192 BUSINESS MATHEMATICS
Illustrative Example 1. What is the average weight of a dozen eggs
weighing 666 grams?
Solution:
666 grams -^ 12 = 55.5 grams
Illustrative Example 2. What is the average wage of the following men
if 10 men receive $4 per day; 12 men receive $3.75 per day; 8 men re-
ceive $4.50 per day; and 5 men receive $5.25 per day?
Explanation: It is first necessary to find the total wages earned
by each group of men, then divide the total of those totals by the total
number of men.
Solution:
Number of
Men
Day Wage
Total
10
$4.00
$40.00
12
3.75
45.00
8
4.50
36.00
5
5.25
26.25
35 $147.25
Average wage = $147.25 -J- 35 = $4,207
WRITTEN EXERCISES
1. If six boys weigh respectively 118, 104, 168, 156, 132, and 112 lb.,
what is their average weight?
2. If a merchant mixes 2 lb. of coffee worth 37ff a pound, 3 lb. worth
39^ a pound, and 1 lb. worth 42^ a pound, what is a pound of the mixture
worth?
3. A pay-roll shows 12 hands are employed at $2.75 per day, 14 hands
at $2.50 per day, 18 hands at $2.60 per day, and 6 hands at $5.75 per
day. Find the average daily wages.
4. The following clerks in a store made these sales in one day:
Clerk No. Sales
122 $256.75
123
137.80
124
243.60
125
260.80
Find the average sales of these clerks for that day. WJhich sold above
the average? Which below the average?
AVERAGES, SIMPLE AND WEIGHTED 193
6. If the expense of running the city of New York is $316,000,000 in
192Q, and the population of the city is approximately 6,000,000, what is
the per capita expense?
6. In a certain school of 3,100 pupils, 350 are 13 yr. of age; 800, 14
yr. of age; 750, 15 yr. of age; 600, 16 yr. of age; 550, 17 yr. of age. 40,
18 yr. of age; and 10, 19 yr. of age. Find the average age of the school.
7. Find the difference in cost for a trip from New York to Orange,
N. J., if a 10-trip ticket cost $2.48 and a 50-trip ticket costs $9.90.
172. Weighted Averages. — These take their name from
the fact that they must be weighted or reduced to a common
basis in order to obtain a correct average, and unless this is
done an entirely erroneous result will be found. This plan
is well illustrated in the following example.
Illustrative Example. John Smith has given to William Jones notes
as follows: $150 due May 14 ; $200 due June 29 ; $500 due July 20. He
wishes to pay them all at one time. When shall they be considered due?
Explanation: In order to arrive at a correct solution it is necessary
10 reduce each of these to a 1-da. basis, for if all the notes were paid
% 14 Smith would lose the use of $200 for 46 da., and of $500 for 67 da. ;
or reduced to a 1-da. basis we would have:
$200 X 46 = $ 9,200 for 1 da.
$500 X 67 = 33,500 " 1 "
Total = $42,700 " 1 "
"ttiith, therefore, would lose what is equivalent to $42,700 for 1 da.
^^ is entitled to keep the $150 + $200 + $500 = $850 as many days
after May 14 as are required for the use of $850 to equal the use of $42,-
7jX)for 1 da., or *|b^o^ = 50.2 da. Hence the equated time for paying
^ the notes is 50 da. after May 14 or July 3, or arranged as follows :
$150 X = $ 0,000
200 X 46 = 9,200
500 X 67 = 33,500
$850 ) $42,700
50.2
13
194 BUSINESS MATHEI^LITICS
WRITTEN EXERCISES
1. Find the equated time for the pa>iDent of $250 due in 3 mo., S^
due in () nio.. and $700 due in 8 mo.
2. Find the equated time for the payment of $300 due in 30 da., 1^
duo in i>0 da., and $200 due in 90 da.
3. F'ind tiie equated time for the payment of $275 due June 21, |1
duo July 10, $200 due Aug. 6. and $150 due Sept. 3.
4. A owes B $200 due in 10 mo. If he pays $120 in 4 mo., wl:
should ho pay the balance?
Soli'tion: By paying $120 in 4 mo. A loses the use of $120 fo
luo., which is equal to the use of $720 for 1 mo. Therefore, he is entit
to ki^p the balance, $80, Vo* mo* or (reducing the fraction) 9 no. al
its maturity.
5. A owes B $2,000 payable in 4 mo., but at the end of 1 mo. he p
him $500, at the end of 2 mo. $500, and at the end of 3 mo. $500.
lu>w many months is the balance due?
6. A man bought Feb. 11, a bill of goods amounting to $1,700, o
mo. credit ; but he paid Mar. 22, $400; Apr. 20, $220; and May 10, $3
W hen is the balance due?
7. Find the equated time of maturity of each of the following bi
aiul I he amount due at settlement including interest at 6%.
John Doe to William Price, Jr.
Vpr. 5 To mdse. on 4 mo. credit $120.50
Vpr. 15 To ♦' '' 3 " " 87.33
May 7 To '' '' 3 " " 218.17
MayJlTo " "4 " ** 317.00
$743.00
Paid Oct. 18
' t ' " I- iiul t \\v equated time for payment, reckoning from July
'. K i llu t .nlii^i ilute that any item becomes due, and find the in
' t.u 111. vNhnlf lull hum tluMMjuated time loOct. 18.
tt. 1 1 iwi. 1 1 HI I lu-^m business together with $6,000 borrowed car
■«« »h. , ill .1 u\\. ihi'ir net worth is ^18,000, what was their a
t IlllUi \\ u illl
\». ll \\\ Wilh III! . ^U^n^•>^ts on Apr. 1, in a bank that pays interest
V jl i-.i»m*il\ 1. 1 la I ivi.,. the Sinn of $2,000, and on Apr. 21 he redi
'» :. -.uui ill "nil ,r»(V, *k u luiuu* his average balance for April.
AVERAGES, SIMPLE AND WEIGHTED 195
oolution:
20 X $2,000 -h 10 X $1,600 $56,000
30 "30
= $1,866.67
10. Find the average daily balance at the end of the month of a man's
bank account from the following information:
Deposits Checked Out
$300 Jan. 1 $50 Jan. 5
50 " 10
50 " 15
50 " 20
50 " 25
11. On the organization of a partnership, A invests $12,000. but with-
draws $2,000 at the end of 8 mo.; B invests $6,000, withdrawing $3,000
after 6 mo. How should the Ist year's gains of $4,750 be apportioned?
Solution:
A's investment of $12,000 for 8 mo. = $ 96,000 for 1 mo.
A's " " 10,000 '' 4 " = 40,000 " 1 "
A's average investment = $136,000 " 1 mo.
B's investment of $ 6,000 for 6 mo. = $ 36,000 for 1 mo.
B's " " 3,000 " 6 " = 18,000 " 1 '*
B's average investment
= $ 54,000 '' 1 mo.
A's average + B's average
= $190,000 " 1 •'
A's share = ISo = 8S
•D>« tt 2 7
r> s =95
Using 6%:
^'5 of profit 9i $4,750 = $50
Jl = 68 X $50 = $3,400 = A's profit
iJ = 27 X $50 = $1,350 = B's '*
Interest on $12,000 for 8 mo. = $480
" " 10,000 " 4 '' = 200
A's invested earning power = $680
Interest on $6,000 for 6 mo. = $180
3,000 '' 6 " = 90
<( ((
B's invested earning power = $270
196
BUSINESS MATHEMATICS
Total earning power, $680 + $270 = $950, of which A's mvestment
represents Ji and B's H-
12. X and Y form a partnership. X invests $12,000 for 9 mo., aii&
^hcn adds $4,000. Y invests $28,000, but withdraws $8,000 at theeiid(f
4 mo. At the end of a year their accounts stand as follows:
Goods of Dept. A
Cost $12,480.00 Sales $10,537.00
Onhand 7,300.00
Goods of Dept. B
Cost $4,264.00 Sales $7,172.80
Onhand 2,250.00
Goods of Dept. C
Cost $11,384.00 Sales $14,436.40
Onhand 1,930.00
General Expense
Cost $ 2,592.36
Apportion the profits according to the average investment.
13. The following is a memorandum of flour stored by B. G. Jackson
with the Heights Storage Co. at 4^ per bbl. per term of 30 da., average
storage. What was the amount of the bill?
Solution :
Date
Receipts
Deliveries
Balance
Time in
Storage
In Storage
FOR 1 Da.
Feb 6
200 bbl.
150 "
400 "
200 "
100 bbl.
150 "
300 "
400 "
200 bbl.
100 "
250 "
650 ••
500 "
200 *•
400 "
"
6 da.
9 ••
15 "
1 "
12 '•
8 "
6 "
1.200 bbl.
•• 12
900 "
•• 21
3.750 "
xf ar 8
650 "
9
6,000 ••
•• 21
1.600 "
" 29
r,400 •'
A r»r 4
16.500 bbh
Apr. ^
The storage items arc equivalent to the storage of 1 bbl. for 16,500 da.
16 500 da. = ^^^^^ terms of 30 da. each. At 4ff per term, the storage =»
550 X 4fi = S22.
AVERAGES, SIMPLE AND WEIGHTED 197
14. The Dobeon Storage Co. received and delivered on account of
W. T. Johnson sundry barrels of flour as follows: received Nov. 10, 2,000
bbl., Nov. 20, 1,200 bbl., Dec. 15, 800 bbl., Jan. 20, 2,000 bbl. ; delivered
Dec. 2, 1,200 bbl., Dec. 28, 1,400 bbl., Jan. 24, 500 bbl., Feb. 4, 800 bbl.,
Mar. 30, 300 bbl. If the charges were 4>^^ per bbl. per term of 30 da.
average storage, what was the amount of the bill?
CHAPTER XIV
THE PROGRESSIONS
173. Arithmetic Progression. — This name is given to a
certain series of numbers, each term of which is formed by
adding a constant quantity, called the difference, to the
preceding term; for example, 1, 4, 7, 10, etc. Therefore, to
find the difference, subtract any term from the following
term. It is useful at times in order to find the total of a
given series of numbers or to find any particular term of that
series. For example, if a car, going down an inclined plane,
travels in successive seconds, 2 ft., 6 ft., 10 ft., 14 ft., etc.,
how far will it go in 30 seconds? The long method would be
to set down all the 30 numbers and add them; but we shall
see that it can be done in a much quicker and easier way.
174. Quantities and Symbols. — There are certaiti well-
established symbols which are used in the consideration of
an arithmetic series of numbers, which are:
a — the first term
d — the common difference
/ = the last term
n — the number of terms
5 = the sum of the terms
175. General Form of an Arithmetic Progression. — This
general form is a, a + d, a + 2d, a + 3d. ; . . Therefore,
coefficient of d in each term is 1 less than the number of the
term.
1^^
THE PROGRESSIONS 199
[ lUttstrative Example 1.
7th term » a + W
12th " » a + llrf
nth " =- a + (n - l)d
Hence / = a + (« - 1) <^
Also:
. 5 = a 4- (a + rf) + (a + M) + . . . . + (/ - rf) + /
(1) or writinR this in
*=/ + (/- rf) + (/ - 2</) +....+ (a + rf) + a
(2) the reverse ordet
2* « (« + /) + (a + /) + (a + /) + ... + (a + + (fl
,v By adding (1) and
' ^ '^ (2).
'»2j««(a + /)
f » il/2(a + 1)
II
» « 11/2 [2a + (ii - 1) <^
III Py substitut-
ing I in II
niustratiTe Example 2. Find the 12th term
and the sum of this
series of numbers 5. 3, 1, - 1 ... .
SonmoN: / = o + (n - l)d
a = 5
= 5 + (12 - l)(-2)
rf = -2
= 5 + (-22)
n = 12
« -17
Sum = jro + I)
o = 5
= ¥(5 - 17)
rf = -2
= 6(-12)
n = 12
= -72
WRITTEN EXERCISES
1- Find the 8th tenn of the series 3, 7, 11
^' Find the sum of the first 30 odd numbers.
3- If a man saves $100 in his 20th yr., $150 the next year, $200 the
^ext, and so on through his 50th yr., how much will he save in all?
*• If a body falls 16.1 ft. in 1 second, three times as far in the next
*cond, and 80.5 ft. in the 3d second, and so on, how far will it fall in 6
*cond8? In 20 seconds?
"• If a clerk received $900 for his 1st year's salary, and a regular
^^y increase of $50 for the next 10 yr., find his salary for the 11th yr.
^^ the total salary for the 11 yr.
'• Find the sum of Iff, 2^, 3ff, etc., to $1 inclusive.
'• Find the sum of the first 30 even numbers. Of the first 100 num-
bers.
*• Find the sum of Iff, Iff, liff, etc., to $1 inclusive.
200 BUSINESS MATHEMATICS
9. Compare your answer in Exercise 8 with that in Exercise 6, and
find the gain if lottery tickets are sold as by Exercise 8, rather than by
Exercise 6.
10. A man invests his savings in the shares of a building and loan
association, depositing $1,000 the 1st yr. At the beginning of the 2d yr.
he is credited with $60 interest on the amount deposited the Istyr.,
and pays in only $940, making his total credit $2,C00. At the beginning
of the 3d yr. he is credited with $120 interest, and pays in $880 cash,
etc. What is his credit at the end of 10 yr., and how much cash has he
paid in?
176. Geometrical Progression. — This is a series of num-
bers, each term of which is formed by multiplying the pre-
ceding term by a constant called the ratio.
The series 1, 3, 9, 27, etc., is one in which the ratio is 3.
The same symbols except d are used as in the arithmetic
progression with the addition of r for the ratio.
The type series is a, ar, ar', ar^y ar^ . . . . Hence the
exponent of r in each term is 1 less than the number of the
term.
Illustrative Example 1.
10th term = ar^
15th " = ari4
nth '* = ar""- ^
In deriving an equation for the sum, we know, also:
s = a -\- ar -V ar^ -\- ar^ -\- ar'^ -V -f ar^~ ^ (1) Now multiply (1) by;
rs = ar 4- ar^ -\- ar"^ -\- ar^ ■\- + ar"" » + ar^ (2)
rs — 5 = ar^ — a Subtracting (1) from (2)
six — 1) = ar^—a
ar^-a
s = — - n
r — I
Multiplying l = ar^~^ by r, we get rl = ar^
Substituting rl for ar^ in (II).
rl-a m
8 =
THE PROGRESSIONS 201
Ultistnthre Ezunple 2. Find the 8th tenn and the sum of the S tenns
of the g^metric progreflsioD, 1, 3, 9, 27
Soujtion: / = ar^~~ ' a — \
= 1(3)7 r = 3
= 2,187 11 = 8
rl-a,
3 X 2,187 - 1
3-1
6,561 - 1
2
= 3.280
niustrative Example 3. Find the 10th term and the sum of 10 tenns
of the geometical progression, 4, —2, 1, —J
Solution: I — ar^~ ' a = 4
= 4(-5h)
= —lis
(-J)(-ri.)-4
8 =
_ 1 _ 1
_ 341
— 12B
WRITTEN EXERCISES
!• Find r, then find the 6th term in the series 2, 6, 18
2. Find the 9th term in the series 2, 2|/2, 4
3. A ship was built at a cost of $70,000. Her owners at the end of
each year deducted 10% from her value as estimated at the beginning
of the year. What is her estimated value at the end of 10 yr.?
4' The population of a city increases in 4 yr. from 10,000 to 14,641.
"hat is the rate of increase if
n-iil
5« The population of the United States in the year 1900 was 76,300,-
^' If this should increase 50% every 25 3T., what would the popula-
tion be in the year 2000?
S* If the annual depreciation of a building is estimated at 4% of its
value at the beginning of each year and the building cost $25,000, wha^
is its estimated value at the end of 20 y r. ?
202 BUSINESS MATHEMATICS
7. A machine costing $9,000 depreciates 7% of its value at the be-
ginning of each year. Find its estimated value at the end of 8 yr.
8. In making an inventory at the close of each year, a manufacture^
deducted 10% from the valus of his machinery at the previous inventory,
because of deterioration. The machine cost $20,000. What was the
value at the end of the 5th yr.?
Hint: Value at end of 1st yr. = .90 X $20,000
" " " " 2d " = .90^ X $20,000
What was the common ratio?
9. A boy puts $100 in a savings bank, which pays 3% compound inter-
est , compounded annually. What does it amount to at the end of 6 jt.?
Hint: Value end of 1st yr. = 1.03 X $100
10. The population of a city is 100,000, and increases 10% each year
for 10 yr. Find the population in 10 3rr.
11. A person has two parents, each of his parents has two parents,
and so on. How many ancestors has a person, going back ten genera-
lions, counting his grand parents as the first generation and assuming
that each ancestor is an ancestor in only one line of descent?
CHAPTER XV
LOGARITHMS
177. Logarithms. — Many kinds of commercial work, as
well as academic or technical work, require computation by
the use of logarithm tables. Such forms are multiphcation,
division, raising numbers to required powers, and extracting
roots of numbers. This work is made much easier as well as
much quicker at times by the use of these tables or calcula-
ting devices. The tables are not difficult to understand if one
will study thoroughly the explanation of their use.
178. Calculations Made Through the Use of Exponents. —
Whenever numbers which are the powers of one number, as
for instance, of 2, are to be multiplied, divided, raised to
powers, or have their roots extracted, these operations can
be performed very quickly if a table containing the various
powers of the numbers has been prepared beforehand.
Table
Power of 2
Number
2^
2
22
4
23
8
24
16
25
32
20
64
T
128
2fi
256
29
512
2^0 =
1024
1''
/-
1
; i.
?03
■■..'\
204 BUSINESS MATHEMATICS
Table
Poi^^x OF :
2
Number
2"
:=
2048
212
:=
4096
2^3
^
8192
2^4
^
16384
2^5
^
32768
216
^
65536
217
=:
131072
2i8
=
262144
2^9
^
524288
220
:=
1048576
WRITTEN EXERCISES
1. Calculate 16 X 64 by the use of the table.
Solution: 16 = 24
64 = 2^
16 X 64 = 24 X 2^
= 2^0 Same as .t4 xP = x^^
= 1,024 (From the table)
2. Calculate 8 X 128 X IG by the use of the table.
3. Calculate 16, 384 -J- 250 by the use of the table.
Solution: 16,384 = 2^4
256 = 28
16,384 -^ 256 = 2^4 -^ 2«
= 2^' Same as x^^ -^ x^ = sfi
= 64 (From the table)
4. Apply the table to:
(a) 1,024 -^ 16
(b) 512 -r 64
(c) 32,708 -T- 1,024
5. Square 32 by the use of the table.
Solution: 32^ = (2S)2
= 2^0 Same as {x^y = x^^
= 1,024 (From the table)
LOGARITHMS 205
6. Apply the table to the following:
(a) 323
(b) 642
(c) 324
7. Find the square root of 256 by the use of the table.
Solution: 1/256 = 1/28
= 24 Same as y/x^ = x^
= 16
Explanation : Divide the root (2) into the power (8) and it gives 4.
:)f 16,384. 1'" If ^
" 32,768. r^ 5^ z<^l
8. Apply the table to find the square root of
9. " " " " " " cube "
10. '' " " *' " " fourth '' '• 4,096.
11. ' " " " '' " fifth " " 1,024.
^^ ^ _ 64X256X16. . ^ .^
12. Solve ; by the table.
13. Calculate 1,024 X 16 " *'
14. Calculate 512 X 64 " "
179. Logarithms Are Exponents. — The logaritkm of a
given number is the exponent of the power to which a base
number must be raised to produce this number.
In logarithms three different numbers are always involved:
1. A number.
2. Its logarithm.
3. The base used. /j . ^
i
Illustrative Example. In 3^ = 9, we can say 2 is the logarithm of 9
to the base 3. Similarly, since 3^ = 81, we can say 4 is the logarithm of
81 to the base 3. Therefore, B^ = A^, we can say L is the logarithm of
N to the base B.
180. A System of Logarithms. — This is a set of numbers
with their logarithms all taken to the same base. Notice
that the logarithm of 1 in any system is 0, since a^ = 1.
206
BUSINESS MATHEMATICS
System of Logarithms with Base 2
Number
Logarithm
Reason
Number
Logarithm
1
20 =1
2
1
2' =2
i
- 1
3
1.585
2»-s8s = 3'
4
2
2» =4
i
- 2
5
2.3223
22.3323= 5
8
3
23 =8
&
- 3
Reason
2-»
2-2
= i
= 2^ =i
2-3 =
(2)3
= i
The student need not know exactly how decimal loga-
rithms like 1.585 arc found. Originally they were found by
a long process of extracting roots. Since logarithms are
exponents, they may be interpreted as such. Thus in the
equation 2roo2S = 3, we see that the 15,850th power of
the 10,000th root of 2 equals 3, and if these operations were
actually performed on 2, the result would be 3.
181. Notation and Terms. — To avoid writing long expo-
nents, such an equation as 2^^^^° = 3 is changed into
log 23 = 1.5850, and is read ''logarithm of 3 to the base 2
equals 1 .5850.^' The subscript indicating the base is usually
omitted when. 10 is the base.
The integral part of the logarithm is called its character
istic and the decimal part the mantissa.
WRITTEN EXERCISES
Express the following in the language of logarithms:
1. 24 = 16
Solution: Log 2 16 = 4. Read, logarithm of 16 to base 2 is 4.
2. 33=27 6. 3-2 = J
3. 103= 1,000 7. 102= 100
4. 42= 16 8. 10-1= ^
6. 25= 32 9. 10-4= .0001
LOGARITHMS 207
Express in the language of exponents, and find the value of each :
= X
= X
= X
— X
= X
= X
— X
= X
= X
= X
= X
= X
29. What is the logarithm of 9 in a system whose base is 3? Of 81?
Of 27? Of 3? Of i?
30. What is the logarithm of 256 in a system whose base is 16? Of
16? Of 4? Of 8? Of 64?
31. What is the logarithm of 100 in a system whose base is 10? Of
1,000? Of 100,000? Of A? Of .01? Of 1? Of .001?
32. What is the logarithm of 81 in a system whose base is 27? Of
3? Of 9? Of 243? Of A? Of J? Of «S?
182. Briggsian or Common System of Logarithms. — This
system uses 10 for the base.
10. log,8 = 3
17. Iog327
11. log464 = 3
18. Iog464
12. logs25 = 2
19. log, A
13. log84 = 1
20. log3{
14. log,o.01 = -2
21. log3A
16. log927 = 2
22. log,oA
16. log39 = X
23. log.oOl
Find the value of x
24. log.o.OOl
Solution: 3* = 9
26. log232
3* = 3»
28. logos'*
.-. X = 2
27. log48
23. logs 16
Illustrative Example.
Since 104 = iq^qOO then log 10,000 = 4
103 = 1,000 " log
1,000 = 3
102 = 100 " log
100 = 2
10^ = IO4 " log
10 = 1
10° = 1 " log
1 =
10-^ = .1 " log
.1 = -1
10-2 ^ 01 " log
.01 = -2
10-3 = ^001 " log
.001 = -3
to r.^l
183. Positive Characteristics. — You will observe that
^ny number between 1,000 and 10,000 has for its logarithm,
^ + a decimal. Any number between 100 and 1,000 has
N\^
208 BUSINESS MATHEMATICS
2 + a decimal; and between 10 and 100, 1 + a decimal; and
a number between 1 and 10 has + a decimal. Hence, in
general, the characteristic of the logarithm of any number
greater than 1, in the Briggsian or common system of loga-
rithms, is 1 less than the number of places at the left of the
decimal point. Thus the characteristic of 729.4 is 2; of
7,460 is 3; of 3.96 is 0.
184. No Change in Mantissa When Decimal Point is
Moved. — In the common system, in which the base is 10,
the mantissas do not change when the decimal point is
moved. To understand why this is true, we take 10 '°; '^
= 1.27, and multiply or divide both members of this equa-
tion by 10' = 100, or by 10' = 10. Recalling that when
X is multipUed by o^ we obtain x"^^ or x^, and when
X is divided by a; we obtain x or x , then by the same
process of reasoning we have:
102.1038 = 127 or log 127 = 2.1038
101.1038 = 12.7 '' log 12.7 = 1.1038
IO.1038--1 = 1^7^ <^ log 127 = 1.1038 or 9.1038-10
10.1038-2 ^ 012/ " log .0127 = 2.1038 or 8.1038-10
The minus sign over the characteristic at the right be-
longs to the characteristic only. Thus, by regarding the
characteristic only as changing in signs, mantissas stay the
same no matter where the decimal point in the number is
changed to, and mantissas are always positive.
185. Negative Characteristics. — Any number betwe^
.001 and .01, having two ciphers (or zeros) before the fi^*^^
significant (i.e., first figure other than 0) figure, hSls 3 for i^
characteristic, since its logarithm lies between 3 and 2 a^^"
LOGARITHMS 209
the mantissa added is positive. Any number between .01
and .1 has 2 for its characteristic, since its logarithm hes
between 2 and 1, and the mantissa added is positive; also,
any number between .1 and 1, there being no cipher before its
first significant figure, has T for its characteristic, since its
logarithm lies between 1 and and the mantissa added is
positive. Hence, in general, the characteristic of any num-
ber less than 1 is one more than the number of ciphers be-
tween the decimal point and the first significant figure, and
is negative. Thus, the characteristic of the logarithm of
.00468 is 3; of .7396 is T; of .000076 is 5.
186. Explanation of a Logarithm Table. — In the logarithm
table the left-hand column is a column of ordinary numbers.
The first two figures of the given number whose mantissa
is sought are found in this column. In the top row are the
figures from to 9. The third number is found there.
Hence, to obtain the mantissa of 364, we take 36 in the
first column and look along the row beginning with 36 until
we come to the column headed 4. The mantissa thus ob-
tained is 5611. To find the mantissa of 2,710 we find the
mantissa of 271, and the mantissa of 7 is the same as that of
70 or 700.
>
187. To Find the Mantissa of a Number Containing More
than Three Figures (Interpolation). — Find the mantissa for
the first three figures and add a correction for the remaining
figures. This correction is computed on the assumption
that the differences in logarithms are proportional to the
differences in the numbers to which they belong. Though
this prop^tion is not strictly accurate, it is suflBciently
accurate for practical purposes.
14
210 BUSINESS MATHEMATICS
Illustrative Example : Find the mantissa for 1,581 .47.
mantissa for 159 = .2014 mantissa for 158 = .1987
'' 158 = .1987 .0027 X .147 = .0004
difference for 1 = .0027 mantissa for 1,581.47 = .1991
The difference between the mantissas of two successive
numbers is called the tabular difference. Hence, to find
from the table the mantissa for a number containing more
than three figures: Obtain from the table the mantissa for
the first three figures, and also that for the next higher
number, and subtract. Multiply the difference betwreen
the two mantissas by the remaining figures with a decimal
point at their left, and add the result to the mantissa for
the first three figures.
188. To Find the Logarithm of a Given Number.— Deter-
mine the characteristic. Neglect the decimal point (in the
given number) and obtain from the table the mantissa for
the given figure.
Illustrative Example 1. Find the logarithm of 3.6257.
Solution: The characteristic of 3.6257 is 0, since 1 (the number oi
places at the left of the decimal point) — 1=0.
mantissa of 363 = .5599 log of 3.62 = 0.5587
'' 362 = .5587 .0012 X .57 = .0007
difference for 1 = .0012 log of 3.6257 = 0.5594
Illustrative Example 2. Find the logarithm of .078546.
Solution: The characteristic of .078546 is 2, since 1 (the numt>^*
of ciphers at the right of the decimal point) +1=2.
mantissa of 786 = .8954 log of .0785 = 2.8949
" ^^ 785 = .8949 = 8.8949-10
difference for 1 = .0005 .0005 X .46= .0002
log of .078546 = 8.8951-10
Explanation: Instead of using the 2.8949, we change the — ^
g - 10, which it equals.
LOGARITHMS 211
WRITTEN EXERCISES
Find the logarithms of the following numbers whose mantissas are
found directly in the table. Common fractions and mixed numbers must
first be reduced to decimals.
1. 400
10. 471
18. 70,000
2. 4
^^
11. 699
19. 25.4
3. .0372
y
12. A
20. 3.56
4 i
13. 4J
21. 356
5. 37
14. 1
. 22. 3,560
6. 40
15. 12J
23. 5,670
7. 7,000
16. 3,680
24. .00046
8. .000029
17. .000451
25. .0000564
9. .06i
189. How to Use Tables of Proportionate Parts. — These
tables are time-savers in finding the mantissas of given num-
bers. They are used as follows :
Illustrative Example. If the number is 36.54 and we desire its loga-
rithm, we note that its characteristic is 1.
Then we find the difference between the mantissa of 365 and that of
366, which is 12. Looking down the extra-digit column headed 12 until
We find 4 (the fourth figure in the number), and following across to the
right, we find 4.8. We add this difference to the last figure of the man-
tissa of 365, which gives 5,628 (as 4.8 gives 5 with the .8 correction made
to the next figure at the left).
log 36.54 = 1.5628
WRITTEN EXERCISES
To find the logarithms of numbers whose mantissas are not in the
table.
Find the logarithm of the following:
1. 921.5
6.
.6757
11. 31.393
2. 3.1416
7.
.09496
12. 48387.
3. V2~= 1.414.
8.
4.288
13. 7.3165
4. V3
9.
.0000 j023
14. .019698
6. 1079
10.
.0002625
16. 810.39
/*"
•
\
no
212 BUSINESS MATHEMATICS
190. How to Find Antilogarithms. — Since the character-
istic depends only on the position of the decimal point and
not on the figures forming the given number, the character-
istic is neglected at the outset of the process of finding the
antilogarithm.
1. If the given mantissa can be found in the table: Take
from the table the figures corresponding to the mantissa of
the given logarithm; use the characteristic of the given
logarithm to fix the decimal point in the number obtained
from the table.
Illustrative Example. Find the number whose logarithm is 1.4425.
Solution: The figures corresponding to the mantissa .4425 are 277.
Since the characteristic is 1, there are two figures at the left of the deci-
mal point.
Therefore, if log x = 1.4425
X = 27.7
2. If the given mantissa does not occur in the table:
Obtain from the table the next lower mantissa with the cor-
responding three figures of the antilogarithm. Subtract the
tabular mantissa from the given mantissa; divide the latter
difference by the difference between the next lower and the
next. higher mantissa in the table; annex this quotient to the
three figures of the antilogarithm already obtained from
the table. Use the characteristic to place the decimal point
in the result.
Illustrative Example 1. Find the number (or antilogarithm) whose
logarithm is 2.4237.
Solittion: .4237 is not in the table; the next lower is .4232. The
difTcronoo between them is .0005. If a difference of 17 in the last two
figures of the mantissa makes a difference of 1 in the third figure of the
number (antilog), a difference of 5 in the last figure of the mantissa will
make a diflorence of A of 1, or .294, with respect to the third figure of
(lie number.
./
LOGARITHMS 213
Hence if log x = 2.4237
X = 265.294
Illustrative Example 2. If log x = 7.2661 - 10, find x.
Solution: The nearest less mantissa is .2660, of which the number
is 1845.
The tabular difference = 2
l-h2 = .5 ^ ^i^'
:. X = ^0018455 - ). ' , 1,1 cir,,
191. To Find the Number by Use of Proportionate Parts.
— If the logarithm is given the number may be found by
use of proportionate parts. The method is outhned in the
following:
t»
V
Illustrative Example. Find the number of which the logarithm is
1.8678.
Solution: Find the mantissa just below this, and put down the
corresponding first three figures of the number, or 737.
Find the difference between the mantissa just below the given mantissa
and just above the given mantissa, which in this case is 6.
Find the difference between the given mantissa and the next lower
mantissa, which in this case is 3..
Find the proportionate parts column headed 6, follow down it until
you come to the figure 3 (or the figure nearest to it in value), then follow
across to the left-hand column headed extra digit, which in this case
is 5.
Annex this figure to the first three figures (737 in this case), making
7,375.
Use the characteristic to point this off, giving the number whose
logarithm is 1.8678 as 73.75.
192. Table of Logarithms. — The following is a table of
logarithms of numbers, such as may be applied to the
problems in this book. The table of proportionate parts ig
at the right.
7)7
J. X
BUSINESS MATHEMATICS
»..
1. 1 I 4
. . , . >
'
.,.,.
«.
10
3010 3032 3Qj4 3075 3086
3118^139 3:B0 3181 3-'01
■a
122i 32M 3263 J M 3J04
3J24 334o 3J63 J3S5 3404
I
■1
3424 3144 ^4fl4 ■i4S3 350_^
DiBuan
IS
3802 3820 l^l JH56 3874
3979 39'> 4014 4031 4048
3892 3909 m-1 3045 3962
406 4082 409B 4116 4133
1
■a
*lfiO 4166 4ia3 4''00 4216
42414249 4385 4281 4298
43dT^409 44234440 44 j6
11
n
4314 4310 4J48 4162 437S
3,3
.11
4WS 4 64 45Sf 4 94 4600
a
4i
19
4624 46J9 s.6j4 4669 4683
4BB8 4 13 47ffl 4742 47u7
«i
Vi
30
4771 4 -ia 41 n 4SI4 4S''3
tS43 4Sj7 43 1 4886 4900
i
ito
lo-
31
4r8s 4017 son M 4 ,03s
6
133
u'i
31
lU
33
ISO 51(8 0211 62 4r
"* Tl
1
ili
lis
SS
3«
jai 5j75 55S7 S59B S61
St
i(
M
1.9
3
%
33
D 1 MID
a
txi
41
CI "^
1
lis
4
n.
iS
14.0
1.1
8
ifl.n
9
is.fl
IT.
it)
ta
" Jm
\'ii
",
2
A
i
u
3
e.£
so
6^100 69eS 1 6
5
6X,
g.
7J76 7081 7( 1 -
6
o.s
10.
■1
2.6
11.
4,4
9
ij.
BS
, 4 4
''l
U
!*.
»
11 74
f
6A
1
h 1 sn 1 ' ^46
H
T
aa
7 4 7in 7M1 (4 -1
7
9.e
11.B
'do
8
'?■?
"
1(6 Boeq 107^ 808 BOS) mi91 si 1 Hi. SI
14.4
13-S
''I
LOGARITHMS
/""'.».»
1 t I «
84S1 8457 g4S3 S4T0 S476
'■ a 8525 8531 86:!7
9 S5S5SJ91 SJa7
19 8645 8S51 8657
« 8704 8710 8716
8782 g76a S774
8820 88ZS 8831
S876 Ras2 SSH7
■I 8756
_. 8 8814
8865 8871
8921
0031 9036
r?«I43
. II 9196
U243 924S
9294 9209
9345 9350
9395 9400
9494 0490
9542 9547
. 9590 9595
I 9633 9643
9685 9689
9777 0782
" "3 9827
8921 8927 8932 8938 8943
9U42 9047 9053
9096 9101 91IJ6
9801 02(» 92'ia
9263 9268 9263
9304 93D9to315
___J 0415
9460 94fio
9405 9410 0415
9455 9460 946
0504 95D0 051
0-152 or..
7 9021 9926 9031
0000 0004 0000 0013 0017
3 0048
1 1 0128 0133
0052 0058 0060
0137 0141 0145
8494 fl50a 8506 '
85,i5 8581 8.587
83751 86R1 8688
8949 89.54 SOOO 8965 8971
9269 9274
9320 9325
n.l70 9;i75
0420 942;
D27D 9
7 9232 92;iK
9335 9340
9130 94:!5 9«0
0479 94H4 9.
9528 9033 0.
9.581 9.5S6
9628 9633
967J 967.1 9680
9773
.9 9474 9479 94H4 9
[)614 9619 9624
984.1 (
9809 9;
9890 0SII4 9899 (
9934
9978 9983
0022 0026
0065 OOT"
0107 01
0191 0195
0069 0073 0077 01
4 9118
3 990K
9087 9901 OoiiO
00.10 01
0110 0120 0124
0199 0204 02Ut{
.7 0241 0345 0L._
8 02K2 0286 029(1
S 032a 0326 D.'ii'O
a 0362 0366 0370
8 0402 0406 0410
BUSINESS MATHEMATICS
Dili 041S wiz D4ze 0430
l>'33 (H57 0461 OiOa 046Q
U4»:j (Hue O^UO 0504 0508
la'Al 0535 05:18 -0543 0548
0569 US73 0577 05H0 05S4
OODT OeU 061G 0B18 0622
0645 own 0652 0650 06BU
UaS2 068(1 06Se 1X03 060T
0719 07:^2 0726 0730 0734
0755 0750 0763 0786 0770
3 0970 09S0 0983
1106 100
133S 1330
.343 l:
i7,'i2 1733
ir-iS 17
7fl7
0626 0630 0633 0637 0641
7 0741 0746 0748 0752
144B
1386
1449
421 l'
!S
it??
5!4 '.
5B1)
laoa
605 6
1775
ffi?
l=Rf
i8oa
1S9S 1
H
LOGARITHMS
3 1906 looa lais l:
1 2154 2156 215»
2271 2274 2276
2297 2299 2)U2
2322 2325 2327
2348 33SU 23G3
237.1 237G 237H
2398 Z4U0 24U3
218
BUSINESS AL\THEAL\TICS
WRITTEll EXERCISES
Find the numbers (or antilogarithms) which correspond to the fol
lowing logarithms:
1. 0.8189
2. 7.60640-10
8. 1.87670
4. 2.67600
5. 3.98260
6. 8.79540-10
7. 6.59930-10
8. 9.94370-10
9. 0.77810
10. 5.45710-10
U. 1.30190
12. 4.25270-10
13. 2.01590
14. 3.72640-10
15. 4.49290
16. 1.81418
17. 1.41863
18. 0.98349
19. 9.22321-10
20. 5.00400
21. 2.34578
22. 1.63350
23. 0.57750
24. 3.92430
26. 9.79730-10
26. 7.70070-10
27. 1.49000
28. 1.89040
29. 2.45270
30. 9.64020-10
193. Computation by the Use of Logarithms.— It
shown in algebra that,
X^X^ = X5
and that,
a'^ay = a* + y
It is shown that,
(3-2)4 = x^
and that
(ax)P=a^x
It can also be shown that:
LOGARITHMS 219
1. log (mn) = log m -\- log n
For if m = 10* Then log m = x
And if n = 10^ '' log n = y
mn = 10* "♦■y or log wn = x + 2/
= log w + log n (by substitution)
2. log — = log m — log n
n
m 10* ^ _ , ^ ,
— = -— = 10* y or log — = X — y = log m — log n
n 10^ n
3. log m^ = p log w
m^ = (10*)^ = 10^* or log m^ = px = plogm
Hence y/m = 10^ or log ^w = - = log-
V V
p
m
4. log\ w = log —
^ p
I. To multiply numbers.
Add their logarithms and find the antilogarithm
of the sum. This will be the product of the
numbers.
II. To divide one number by another number.
Subtract the logarithm of the divisor from the
logarithm of the dividend and obtain the anti-
logarithm of the difference. This will be the
quotient.
III. To raise a number to a required power.
Multiply the logarithm of the number by the
index of the required power, and find the anti-
logarithm of the product.
IV. To extract the required root of a number.
Divide the logarithm of the number by the re-
(juired root and find the antilogarithm of the
quotient.
220 BTSINiib ilATHEiLkTKS
" .75 H OS^S?
biff J]Cia?»> = >.,jIt57-W
kjffPmdu«!t: = Ij!K«i:
HbHtntiTe Ilxample 2. 46.72 ^ .0998
S^crmo^: kn? 46.72 = LfitiSG
loff iWQs = S-9WI — H>
kj)E QufjCieac = L.67l>4
Qu>>aent =46i^j2
IDostnitiTe Example 3. Find the <kh pt^wer of .7929. cr .?Ji9^-
SoLcnox: , log .7929 = 9.S992-10
6
I.jg jr = .'^9.3tr/2— 60
= 9.30 j2- 10 tf^' .j-^i ^
X = .24n4
Illustrative Example 4. Find the cube root of '»o2.76S or {^ 33- 76i
Sol'JTIon: log.732.76S = 2.7-26o
J log 532.768 = 0.90SS
X = 8.1060
WRITTEN EXERCISES
1. Multiply 763 by 298 by the use of logarithms, and check the res
by actual multiplication.
2. Multiply 3.245 by 63.29.
5 Divide 19.65 by 2.843, and check by actual division.
4 Raise 1764 to fourth power or find value of (17.64)4.
- Calculate the 5th root of 29.34 or find value of 1/29.34.
LOGARITHMS 221
Find value of the following:
6. 26.45 X .02687 X 3.194 11. 862 X 48.75
7. 336^1984 7.862X6.827
8. 527 X .083 12. .0734 raised to the fourth power
9. 42.316 13. .6374
.06214 14. 89.76 X 98.54 X 26.6 3
10. 1.78 X 19 .005862 X 8271
23.7 15. .076
16. Extract the fifth root of .0329.
17. Extract the fourth root of .0072.
Find the value of the following:
18. 47.1 X 3.56 X .0079
19. 4.77 X ( — .71) Hint: Work the same as if the sign were +,
.83 and put in correct sign at end.
20. -523 X 249 28. 79 X 470 X .982
767 X 396 ^^' ^'^^^ ^ ^^^'^
30. - .643 X 7095
21. (1.032)15
22. (.795)^ ,, ^7 X 9 X - .462
23 CS 57)4 '^* (^'388)5
24* V 05429 ^2- (1014)^5
QK* J/7vT^ V IV 7W^ 33. .0325 X .6425 X 5.26
25. V.005 X l7 .0765 ^. ^, xu 4. r nnono
26 -^'Tol seventh root of .00898
27. 529
'■4
36. 1/15
36. 4.26
^^ ^ ^^^ 7.42 X .058
37. Find the circumference of a circle whose diameter is 17.63 inches,
from the equation Co = '^d. (x = 3.1416)
38. Find the area of an ellipse whose semi-axis a is 22.18 in. and
shorter axis h is 16.88 in. from the equation, A ellipse = "^ «&•
39. Find the area of a triangle whose sides are a = 16.35, h = 18.97,
and c = 24.77 in. respectively, using the equation:
' fl "4~ h "4~ c
A^ = y/s (s — a) (s — b) (s — c) where s = •
£1
40. The diameter of a spherical balloon which is to lift a given weight
is caJpuJated by the equation:
»=f-
where
5236 {A - G)
i22: ir:?SI?^33i3^ 3ll£rSE3Sl.^TEjf
.V -
tf ixiIfviH n
O - * ^^^ n ~tB!M]ttIcilIIL
^Hit // I' ? - jw^n : = .mm. ir = ijsa Ji.
Z'' - v^su^^sTtr.-jr* TiJr=Enmer if "an 71^5
/' - Tvjwr^t Ji. TijR tiniimiiE
••J
a. If ;r - 'jr., \\u'\ X.
40, If 10'- t,iUi,UwU.
47. IfJ*- <H, firi^J X.
CHAPTER XVI
COMMERCIAL APPLICATIONS OF LOGARITHMS
194. Calculation of Compound Interest. —
1. To find the amount when the principal, rate, and time
are given. The amount at the end of one year is p + pr or
p(l + r), since p is the principal and pr is 1 year's interest.
Thus to get the amount at the end of 1 yr. always multiply
the principal by 1 + the rate.
Now, in compound interest the principal at the beginning
of the second year is p (1 + r). Then the amount at the
end of the second year is p(l + r) (1 + r) or p(l + r) ;
and so on for n years. Hence
niustratiye Example. Find the amount of $725.15 at 5% compound
interest compounded annually at the end of 6 yr. Also find compound
interest.
Solution:
A = p(l +r)«
= 725.15 (1 + .05)6 log 1 05 = .0212
log A = log 725.15 + 6 log 1.05 6 log 1.05 = .1272
= 2.8604 + .1272
= 2.987Q
A = $971.80
a = $971.80 - $725.15 = $246.65
2. To find the cost or "present worth'' of a sum payable
a years hence, supposing interest to be compounded. By
solving the above equation for p we get,
A
V =
(1 + r)«
223
224 BUSINESS MATHEMATICS
Illustrative Example. Find the cost or present worth which will
amount to $923 at 4% compound interest in 12 yr.
Solution A
P =
(1 + r)^
= 923
log p = log 923 - 12 log 1.04
3. To find the amount when interest is compounded q
times a year, use the formula: ^
WRITTEN EXERCISES
1. Find the amount of $933 at 5% compounded annually for
7yr.
2. Find the principal which amounts to $775.20 in 15 yr. at 5%
compounded annually.
3. Find the cost or present worth of $918 to be paid in 10 yr. allowing
5% interest compounded annually.
4. Find the amount of $700 which ran for 12 yr., interest being com-
pounded semiannually at 6%.
A = p (1 + -2)^« .
6. Find the amount of $1 at 6% compound interest for 20 yr., com-
pounded annually. Also find the compound interest.
6. Find the amount of $1,250 for 12 yr. at 6% compounded annu-
ally.
7. Find the amount of $25 for 500 yr. compounded annually at
5%.
8. Find the amount of $300 for 50 yr. at 6% compounded annu-
ally.
9. Find the amount of $300 for 50 yr. at 6% compounded semi-
annually.
10. Find the amount of $300 for 50 yr. at 6% compounded quarterly.
11. In how many years will $1 double itself at 3% interest com-
pounded annually?
COMMERCIAL APPLICATIONS OF LOGARITHMS 225
Solution: 1.03*= 2
xlog LOS = log 2
log 2
X =
log L03
_ .30103 - 'tXiyit<^
" .01284 Cf, ,^ tf-y ^ /
= 23.5 years
12. In how many years will $1 double itself at 5% compounded
annually? At 4%? At 6%?
13. In how many years will $4,000 amount to $7,360.80 at 5% in-
terest compounded annually?
14. In how many years will $12 double itself at 4% interest com-
pounded semiannually?
A '
Solution: A= p {I + V)''*
24 = 12 (1 + .02) 2«
2 = 1.022"
2nlog 1.02 = log 2
log 2
2n =
log L02 . ^
.30103 ,; 'i-^i^;
.00860
.= 35
n = 17.5 yr.
16. In how many years will $100 double itself at 6% interest com-
pounded semiannually?
16. In how many years will $100 double itself at 6% interest com-
pounded quarterly?
17. What sum of money will amount to $400 in 10 yr. if placed at
interest at 4%, if compounded annually?
196. Sinking Fund Calculations. — (See also § 43). The
following explains the application of logarithms to sinking
fund calculations.
If the sum set apart at the end of each year to be put at
compound interest is represented by /S, and
IS
BUSIXES MATHEMATICS
P
r = in t g Tc g* on %\ for I jr.
R — *\ -i- r* = amooDt ot %\ lor 1 yr.
then the sum at the end of the
1st yr. - 5
M - -S
-i- 5J?
3d - -5
-r 5« ^ 5«*
mth" ^S
-1- i« -r 5R'^ .
— SR*^^
at is yt » 5
-j- 5« - 5J?*-r .
. — 5it« »
n)
/. .4 J? - SR
^ SR*^ SR^-i- .
. ^ SR*
«2f « X (1)
.AR— A = SR^
— S
i3» (2) - (1)
AiR— \) =5 <R^- If
(4j Factor left memba
Jf '''** ~
1)
A •
-1
(5) ^hjR-l
J"Jt*-
-ll
16) /? - 1 = ?
(Note that (1) above is a geometric pn^ression.)
niastrative Example 1. If $10,000 be set apart annually, and put at
6^ compound interest for 10 \t., what will be the amount?
Solution :
t
_ 10,000 (1.06^^ - 1)
.06
= $131,808 + (with a 6-place table)
Ohistratiye Example 2. A county owes $60,000. What sum must ^
set apart annually, as a sinking fund, to cancel the debt in 10 5^'
provided money is worth 6%? Find total cost each year.
Ar
Solution:
5 =
^-1
$60,000 (.06)
(1.06)^0- 1
3,600
1.79085 - 1
COMMERCIAL APPLICATIONS OF LOGARITHMS 227
= $4,552. + (6-place table)
Yearly Interest = 6% of $60,000
= $3,600
Total cost = $3,600 + $4,552
= $8,152
WRITTEN EXERCISES
1. Find the amount of $20,000 set apart annually and put at 5%
compound interest' for 20 yr.
2. A city is bonded for $50,000. What sum must be set aside annually
as a sinking fund, to cancel the debt in 20 yr., provided money is worth
5%?
3. If an annual Ufe insurance premium is $150 and money is worth
4%, what is the value of- the sum of the premiums at the end of 20 yr.?
196. Annuities. — An annuity is a sum of money that is
payable yearly, or in parts at fixed periods in the year.
197. Finding the Amount of an Unpaid Annuity. — To find
this amount when the interest, time, and rate per cent are
given, we let the sum due at the end of the
1st yr. = ^
2d " = etc., as in § 194
Illustrative Example. An annuity of $1,200 was unpaid for 6 yr.
Wliat was the amount due if interest is reckoned at 6%?
Solution: , ^ (/2« - 1)
A —
r
$1,200 (1.06^ - 1)
.06
= $8,360 +
' Unless otherwise stated, interest is compounded annually.
228 BUSINESS MATHEMATICS
WRITTEN EXERCISES
1. An annual pension of $600 was unpaid for 5 yr. Find the amount
due if interest is computed at 5%.
2. A widow receives a pension of $400 annually from the United States
goverimient and back-pay for 8 yr. What should she receive in back-
pay if interest is at 3% per year?
198. Finding the Present Value of an Annuity. — To find
the present value of an annuity, when the time it is to con-
tinue and the rate per cent are given, we use the following
formula:
P = present value
A = amount of P for n years, or the amount of the annuity
for n years
But the amount of P for n years
= P (1 + r)»
= PR""
S (/?" — 1)
And A = — — = amount of the unpaid annuity for n
R — 1
years
Hence PR"" = -^- — ^ Since A = PR»
R — 1
. J. _ S(R- -1)
R^(R-l)
S R^-1
P = X
/2« - 1
If the annuity is perpetual, the fraction — — — approaches 1 as its
R^
limit.
.'. P = - (when annuity is i)erpetual)
Illustrative Example 1. Find the present value of an annual pension
of $1,000 for 5 yr., at 4% interest.
COMMERCIAL APPLICATIONS OF LOGARITHMS 229
Solution: S R^ — I
p — — \^
i2« R -I
_ 1,000 (L04S - 1)
~ 1.04S ^ 1.04 - 1
_ 1,000 1.21661 - 1
~ 1.21661 ^ ^04
216.61
.0486644
= $4,451 +
Illustratiye Example 2. Find the present value of a perpetual scholar-
ship that pays $300 annually, at 6% interest.
Solution: S
P = -
r
_ 300
.06
= $5,000
WRITTEN EXERCISES
1. Find the present value of an annual pension of $1,200 to continue
12 yr., at 4% interest.
2. A man is retired by a railroad on a yearly pension of $900. He
lives 9 yr. and 6 mo. What is the value of such pension if money is
worth 4i%?
3. Find the present value of a perpetual scholarship of $450 per year
at 5%.
4. Find the present value of a property purchased on a basis of $500
paid annually for 15 yr., if money is worth 4%.
199. Finding Present Value of Annuities. — The present
value of a perpetual annuity which shall begin in a given
number of years, when the time it is to continue and the
rate per cent are given, mayj3e found by the following
formula:
230 BUSINESS MATHEMATICS
RP (R - 1)
(where p — number of years before annuity begins)
niustratiye Example 1. Find the present value of a perpetual annuity
of $1,000, to begin in 3 yr., at 4% interest.
Solution:
P =
RP (R - 1)
_ $1,000
"" (1.04)3 X .04
= $22,225
niustratiye Example 2. Find the present value of a term annuity of
$5,000, to begin in 6 yr., and to continue 12 yr. at 6%.
P =
Solution:
S J^- 1
RP ^ g J? — 1 (where q = number of years that
$5,000 (1.06) '2 _ 1 annuity is to continue)
= $29,550
WRITTEN EXERCISES
1. Find the present value of a perpetual annuity of $500, to begin in
8 yr. at 4% interest.
2. Find the present value of an annuity of $1,200, to begin in 10 yr.
and continue for 15 yr. at 5%.
3. A man is 55 yr. of age. He is to be retired at 70 on an annual
pension of $900. Suppose that he lives until he is 85. Find the present
value of such a j ension at 4% interest.
200. Finding the Annuity. — To find this when the present
value, the time, and the rate per cent are given, the follow-
ing formula may be applied :
„ S(R»-l)
R"{R - 1)
COMMERCIAL APPLICATIONS OF LOGARITHMS 231
PR^'iR - 1)
:.s =
= Pr X
i2«
ijj«- 1
Illustratiye Example. What annuity for 5 yr. will $4,675 give when
interest is reckoned at 4%?
Solution:
i2«
S = Pr X
i2«- 1
1.045
= $4,675 X .04 X ^^ ^^,_ J
= $1,050
WRITTEN EXERCISES
1. What annuity will $6,000 buy for 10 yr. if interest is reckoned at
3i%?
2. What annual pension will $10,000 buy for 20 yr. if money is worth
4%?
3. $4,000 will buy what annuity for 10 yr. at 4%?
201. Life Insurance. — In order that a certain sum may
be secured, to be payable at the death of a person, he
pays yearly a fixed premium, according to the following
formulas:
P = premium to be paid for n years
A = amount to be paid immediately after the last premium
P(ijJ«- 1)
A =
.-. P =
P =
R - 1
A{R-l)
/2«- 1
Ar
If A is to be paid I yr. after the last premium then
232 BUSINESS MATHEMATICS
p
n
AiR -
1)
" R(R'' -
At
- 1)
1
•
" R{po -
• I)
To find the number of years the premium should be paid,
in order that the company shall sustain no loss, the follow-
ing formula may be used :
/J« = 1 + — orn =
log (l + f )
S logR
In the calculation of life insurance it is necessary to em-
ploy tables which shall show for any age the probable dura-
tion of life.
The following table gives the number of survivors at the
different ages out of 100,000 persons alive at the age of 10.
A.GE
Survivors
Age
Survivors
10
100,000
55
64,563
15
96,285
60
57,917
20
92,637
65
49,341
25
89,032
70
38,569
30
85,441
75
26,237
35
81,822
80
14,474
40
78,106
85
5,485
45
74,173
90
847
50
69,804
95
3
Illustrative Example. Taking the figures of the above table, calculate
what the chance is that a person 15 yr. of age will live to the age of 35?
Solution:
81,822
96,285
r COMMERCIAL APPLICATIONS OF LOGARITHMS 233
ORAL AND WRITTEN EXERCISES
1. What is the chance thut a person 40 yr. old will live to bp SO yr. old?
That a person 70 yr. old will live to be 90 yr. old?
2. If a peraon now ie 20 yr. of age, what are Ihp chunres that he iviil
live to be 45? To be 507 To be 657 To be 80?
3. What annual premium should be charged for a jwlicy worth J1,000
a,t the end of 20 yr. it money ia worth 4%?
4. If the annual premium i^ $50, the amount of the ixilicy in t2,0(X],
and money ia worth 4%, for how many years must the premium be paid
that no josstihali be sustained by the company.'
6. What annual premium should bo cliarKfii for a polic'y worth $2,0(HJ
at the end of 10 yr,, if money ia worth 4%?
I' Use logitrilhms in aolving the following:
' 1. To what willS3,750amount in 2U yr. at^% compoundedannually?
3. To what does 81,000 amount in 10 yr. if left at n% compounded:
a) AnniiallyT (b) SemianmiallyT (c) Quarterly?
5. A sum of money left at 4^% compounded annually for 30 yr.
aniourtts to $30,000. What is the sum?
4. AtwhatpercentintercstmustS15,000beleft in order to amount to
300,000 in 32 yr. compounded annually?
6. At what per cent must $3,333 be left so that in 24 yr. it will amount
tn $10,000 compounded annually?
6. In how many years will a sum double itself it left at 6% interest
compounded annually?
7. Find the amount of $2,500 in 18 yr. ut 4% compounded annually,
find also the compound interest.
8. What sum should be paid tor an annual pension of $1,000 payable
fannually for 20 yr., money being worth 3% per anniun compound iu-
MISCELLANEODS WRITTEN EXERCISES
-7('
(1 +rr-
^ 9. What sum will amount to $l,2,''t0 if ])Ut at eoiiipoiiiid interest at
klO. if $1,600 ia placed at 35 % interest semi jnnua,lly for 13 yr., to how
b will it amount in that time?
234
BUSINESS MATHEMATICS
11. A person borrows $600. How much must he pay amiually that
the whole debt may be paid in 35 yr., allowing interest at 4% com-
pounded annually?
12. Find the amount of $100 in 25 yr., at 5% per annum, compounded
annually.
13. What is the present worth of $1,000 payable at the end of 100 yr.,
interest being at the rate of 5% per annum and compoimded annually?
14. Find the present value of an annuity of $100 to be paid for 30
yr., reckoning interest at 4% compounded annually.
16. Find the amount of $1 in 100 yr. at 5% compound interest com-
pounded annually.
16. Find the amount of $500 in 10 yr. at 4% compounded semi-
annually.
17. What is the present value of $1,000 which is to be paid at the end
of 15 yr., reckoning interest at 3% compounded annually?
18. What is the present value of an annuity of $500 that ceases at the
end of 25 yr., interest reckoned at 6%?
19. If the population of a state increases in 10 yr. from 2,009,000 to
2,487,000, find the average yearly rate of increase if
R"" =
Average rate = /2 — 1, and
population at end
population at beginning
ovR
20. If the population of a state now is 1,918,600 and the yearly rate
of increase is 2.38%, find the population after 10 yr. hence if
Population at end = Pi (yearly rate + 1)"
Pb — population at beginning
21. A man borrows a sum of money at 3J% interest annually, and
lends the same at 5% quarterly. If his annual gain is $441, find the sum
borrowed.
22. If the annual premium is $150, the amount of the policy is $5,000,
and money is worth 4%, for how many years must the premium be paid
that no loss shall bo sustained by the company?
23. If a city wishes to take up $2,500,000 worth of bonds at the end
of 4 yr., how much must it set aside each year, if the rate of interest is
5% and
S\ (1 +r)«- 11 ,
(J = _iJ^ ! — i !. where
C = number of dollars in debt
n = number of years
S = sum set aside annually
r = rate of interest
COMMERCIAL APPLICATIONS OF LOGARITHMS 235
24. Find the amount of $5,000 at the end of 10 yr., interest at 8%
compounded annually.
26. A sum of money is left 22 yr. at 4% compounded annually and
amoimts to $17,000. Find the principal which was originally put at
interest.
26. Find the amount and the compound interest on $1,000 at 4% for
10 yr. compounded annually; then find the amount and compound
interest on the same for 4 yr. at 10% con^pounded annually; then find
the difference between the two result's. Which is the greater?
27. What sum should be paid for an annuity of $1,200 a year to be
paid for 30 yr., money being worth 4% compounded annually?
28. A premium of $120 is paid each year for 10 yr. Find the value
of the sum of these premiums at the end of the 10th yr., with interest
at 4% compounded annually, if
Value = Premium X P
(5^)
29. A man invests $200 a year in a savings bank which pays 31 % per
annum on all deposits. What will be the total amount due him at the
end of 25 yr.?
30. Twenty annual payments of $500 each are deposited with an
assurance company for the benefit of a person to whom, beginning with
the 20th yr., the entire amount paid in, together with accruing interest,
is to be returned in 40 equal annual payments. Reckoning interest at
5%, what should be the amount of each payment?
31. The sum of $100 was deposited in a bank at compound interest on
Jan. 2 every year for 10 yr. At the beginning of the 1 1th yr. and on each
succeeding Jan. 2 during 10 yr., $100 was withdrawn. Interest being
reckoned at 5%, what amount remained on deposit Jan. 1 at the end of
the 10th JO", of withdrawals?
32. If the average death rate per annum in a city be 1t^% and the
average birth rate be 2t%, and if there be no increase or decrease in the
population by migration, in how many years will the population be
doubled?
33. A man borrows $6,000 to build a house, agreeing to pay $50
monthly until the principal, together with interest at 6% is paid. Find
the nmnber of full payments required.
34. If each payment in Exercise 33 is at once loaned at 6%, com-
pounded annually, what will they all amount to by the time the final
payment of $50 is made?
36. From Exercises 33 and 34 determine the totaii interest received
by the money lender up to the time of the last payment. What per cent
on the original $6,000 is this?
236 BUSINESS MATHEMATICS
202. Bonds. — To find what interest on his investment a
purchaser will receive, the following formulas may be used:
P — price of a bond that has n years to run
r = per cent it bears
S = face of bond (usually $100 or $1,000)
q — current rate ot interest
Let X = rate of interest on the investment
Then, P (l -\- x)^ = value of purchase money at the end of n years
Sr{\ -\- g)»— * -I- .S> (1 -f ^)« - ' -I- . .. i-Sr -\-S = amount of money received on
bond if interest on bond is
put immediately at com-
pound interest at q/c
But. Sr(l -|-(7)«-' + 6>(H-fl) «-^+.. -|-5r+5 =5 +:
Q
5rf(l+<z)«-l|
5r[(l +<?)"- II
F (1 + x)« = 5 +
l+JT
'\P^ ^/'
" \ Pa h
n'ustrative Examp'.e 1. What is the rate of interest on a 4^ bond at
114, that has 26 yr. to run, if money is worth 85%?
/ 3.5 + 4 (1.035)26 _ 4\ J
1 + X = [ ) 26
\ 114 X .035 /
1 -\- X = 1.033
X = .033
.*. Purchaser receives 3.30*^2.
Illustrative Example 2. At what price must 7% bonds be bought,
runnmg 12 yr., with the interest payable semiannually, in order that the
purchaser may receive on his investment 5% interest semiannually'
9 q = .025 (interest semiannual-
r/(l+T)« r = .035
_ 2 5 + 3.5(1 .025)^-^ -3.5 n = 24
.025 (1.025)24 X = .025
= 118
COMMERCIAL APPLICATIONS OF LOGARITHMS 237
m WRITTEN EXERCISES
ll. If $126 is paid for bonds due in V2 jr. and yielding 3J% semi-
annuiiUy, what per cent is realised on Iho investment, provided money is
worth 2% semiannually?
2. When money is worth 2% Bemiannuully, if bonds having 12 yr. tu
run and bearing semiannual coupons of 3i% each are bought at 1141,
what per cent is realized on the investment?
3. What may be paid for bonds due in 10 yr., and bearing semiannual
coupons (it 4% each, in order to realize 3% semiannually, if money is
worth 3^0 semiannually, when
r (I
-gl"-
?(1
z)"
4. If 4i% Liberty bondg maturing in 30 yr. are bought at 96.76 and
money ia worth 4%, what is the yield?
6. When ii% United States Third maturing in 10 yr. are bought at
97.16 and money ia worth 6%, what JH the yield?
6. What may be paid fur bonds due in 2.'jyr., and bearing semiannual
(riujioas of 4% each, in order to riailize 41% semiannually if money is
wurth 3% semiannually?
7. What is the yield on New York City 4i 'b due in 45 yr. ct 102^ if
money ia worth 4%7
CHAPTER XVII
THE SLIDE RULE
203. History and Use. — So far as has been determined,
the slide rule as an instrument having one piece arranged to
slide along another, was invented, according to Cajori, by
William Oughtred between 1620 and 1630. The present
arrangement of scales (see Form 19) was devised by Lieuten-
ant Mannheim of the French army about 1850.
It was originated undoubtedly because of the fact that it
is a time and labor saver. It should be borne in mind that in
nearly all practical calculations only an approximately
(correct answer is necessary, and the skill of the operator is
often best shown by his ability to approximate to the right
degree of accuracy. If the result is as accurate as the data
em])loyed to obtain it, or as accurate as our answer is re-
quired to be, then we have accomplished an economy of time
and labor.
It is entirely possible, after having attained proficiency
in handling the slide rule, to obtain results which shall not
have more than ^^ of 1% of error. This is perfectly satis-
factory for many of the problems of the business world.
The slide rule is used in the office, either to check figures,
or for original calculation. It computes mensuration, pay-
i.)!l, interest, percentage rates, discount, profit and loss,
foreign exchange, freight, prorating, compound interest, and
has many other applications of the kind in the business
field.
238
THE SLIDE RIXE
239
Eie of this chapter is to explain tlic uso of lliJs
ing stick so that you Ciin
apply it to j^oi
work.
204. Description of the Slide Rule.—
Since we found in the study of lueari I hnis
that we could multiply numbt'is by adding
their logarithms, we also have found out
that we can add lengths or logarithms on a
nile and oftpntimes simplify our work.
This is done by the use of two rulers which
slide along each other. The rulers are
marked to show logarithms of nitinber.s,
and by adding these logarithms we can
easily find the logarithm of their product,
and then the product. We can also use
the slide rule to divide, to find the roots
and powers of numbere. Each number
printed on the slide rule stands in the
position indicated by its logarithm.
In Form 19, BC is the slide, graduated
on the upper and lower edges, These
graduations were made as follows : CC waa
divided into 1,000 equal parta; log 2 = .301,
therefore 2 was placed at the 301st gradua-
tion; log 3 = .477, therefore 3 was placed
at the 477th graduation; and so on for all
the integers from 1 to 1,000.
In order to read the nundjcrs from 1 to
1,000, we go over the rule froru left to
right. We read fii-st 1, 2 ... 10; then
ill
11). TheSUde
Rule
240 BUSINESS MATHEMATICS
bt^DninR nt 1 again and calling it 10, read it 10, 20 . .
100; tlicn beginning at 1 again read it 100.200 . . . 1,000
This is allowable because the mantiesa for 10 is the same a.'^
that for 100, 1,000, etc. It will be noted that there is a do-
crease in the lengths of the spaces from left to right. Tliesi-
decreaspa in lengths correspond exactly to the differeneei
itetween the logarithms from 1 to 10. We can also putio
marks to show the mantissas for the logarithms of 1.5, 2.5,
etc. Noiv since log 1.5 is 0,176, thisisnot half the diflerenre
betwcpn log 1 and log 2, therefore the mark does not ex-
actly bisect (he line from I to 2
205. How to Read the SUde Rule.— On BB it will be
noled that the distance from I to 2 is divided into what we
shall call 10 large divisions and they will be read from 1 at
the left (toward the riglil) as follows: 11, 12 ... 19. 2
(read as telephone numbei-s one-one, one-two, etc., or under-
stowl as 1.1, etc.). It will also bo noted that each of these
large divisions is again divided into 5 parts, each of wliich
denotes .02, so that the second division after the f
mark of the large division would be read 1.14, and the foui
small division after the mark denoting the fifth large diw
sion would be read 1.58, etc. It will be noted that the nm
ber of large divisions from 2 to 3 is also 10, but that e
large division is subdivided again into only 2 small divisionSi*
f^o t hat the small mark after the first large division between 2
and 3 would be read 2. 15, etc. Thesamescheme works from 3
to 5. From 5 to 6 it will be observed that there are but 2 iai^
divisions and 5 small ones in each of the large divisions.
iirst large mark there would be read 5.5, and the i
small mark following this large mark would bo read 5.tf
The divisions are the same from to 10. If the little runi
THE SLIDE Rl'LE 241
having a hair line on it , which you find on Ihe ruler, should be
used and (he hair Hue should fall half way between the first
small line after 5 and the second small line after 5, it would
be read 5.15, etc. The same plan of ruhnR will be found on
the right-half of the ruler commcneing with the second 1
Bbarked on the loiler.
" 206. Operstions with the Slide Rule.— It is not difficult
to learn to use the slide rule if the student will use small
numbers at first. If in doubt how to do an operation, try it
first with small numbers which you can easily check mentally.
1. MuUipUmlion. Multiply 2 by 4. Move the slide
(the part of the rule in the middle which slides) so as to set
the 1 of the B scale directly under 2 of the A scale, and read
Ihe answer 8 on the A scale directly above the 4 of the B scale ;
or set the 1 of the C scale directly above the 2 of the D scale
uad read the answer 8 on the D scale directly below the 4
of the C scale. Hence, to find the product of two nurabcrs,
set the 1 of the C scale on one of the numbers on the D scale,
and under the other number ou the C scale read the product
on Ihe D scale.
Sometimes in multiplying we will have to use the 1 at the
n|gfat-hand end of the C scale. For example, multiply Sli by
Pw. Set 1 at the right-hand end of the C scale on 86 of D, and
iuider2of C read the product 172 on D. We simply use the
1 at the left end or the 1 at the right end of C, according an
it brings the other number over scale D. It will be observed
in the above exam])le that if wo had used the 1 on the lof(,
end of C, it would have brought the 2 of C off the scale I).
Place your decimal point by inspection. Thua to muiti-
VlO-o by 1.8, set 1 C on 18 D, and under 105 C rend tho
r 189 on D. Then make an approximate multipUc
242 BUSINESS MATHEMATICS
tion mentally, 10 X 2 = 20; hence we know that there are
two integral figures in the product, giving 18.9 as the result.
The decimal point will have to be placed by making an ap-
proximate calculation mentally.
2. Division. Divide 6 by 2. Set 2 C over 6 D, and read
the result directly under 1 C on D. Therefore to divide one
number by another, set the divisor on scale C over the divi-
dend on scale D, and under 1 C read the quotient on scale
D. Here again the decimal point is placed by inspection.
Thus to divide 2.85 by 15, set 15 C over 285 D, and under
1 C read the quotient 19 on D; but we can observe that 3
-r- 15 is about i'*b, or J or .2; hence our quotient is .19.
3. Combined Multiplication and Division, Find the
26 X 4
value of — - — . Set 8 C over 26 D, and under 4 C read the
result 13 on D. First the division of 26 by 8 is made by set-
ting 8 C over 26 D, and under 1 C we might read the quotient;
but we want to multiply this quotient by 4. As 1 C is al-
ready on this quotient we have only to read the product 13
on scale D under 4 C. By the use of this scheme we can find
the fourth term of a proportion. For example, in the pro-
26 X 4
portion 8:26 = 4: a?, a: = — r . Therefore to find the
o
fourth term of a proportion, set the first term over the sec-
ond, and under the third read the fourth term.
4. Continued Multiplication and Division, In this work
use the little glass (or celluloid) runner which has the hair
line on it.
Illustrative Example 1. Find the value of 4 X 6 X 3.
Solution : Set 1 C at the right over 4 D, set the runner on 6 C, set 1
C at the right on the runner (shding glass), under 3 C read 72 on D.
.-. 4 X 6 X 3 = 72.
THE SLIDE RULE 243
72
Illustrative Example 2. Find the value of .
4X9
Solution : Set 4 C over 72 D, runner on 1 C, set 9 C on runner, under
1 C read the result 2 on D.
16 X 36
Illustrative Example 3. Find the value of .
12 X 8
Solution: Set 12 C over 16 D, set runner on 36, set 8 C on runner,
and under 1 C read the result 6.
4X3X8
Illustrative Example 4. Find the value of .
16
Solution: Set 16 C over 4 D, set runner on 1 C, set 1 C at the right
end of the slide on the runner, set runner on 3 C, set 1 C on runner, under
8 C read the result 6 on D. We can work any continued multiplications
and divisions in a similar maimer.
5. Squares and Square Root Note that the graduations
on the upper scale A are the squares of the numbers directly
below on scale D. For example, the square of 2 is 4, and
above 3 is 9, above 6 is 36, above 15 is 225. The first 4 on A
is either 4 or 400, the square of either 2 or 20 respectively on
scale D. The second 4 on A is either 40 or 4000, the square
of 6.32 or 63.2 of D. Hence, to square any number, find
the number on scale D and read its square directly above it
on scale A. To find the square root of any number, find
the number on scale A and read its square root directly be-
low it on scale D.
Scale A will be found very useful when dividing or multi-
plying by square roots, finding area of circles, etc.
Illustrative Example 1. Find the value of 5 \/2.
Solution: Set 1 C at left end of scale on 2 A, under 5 C read the
result 7.07 on D.
6
Illustrative Example 2. l^lnd the value of —i= .
Solution: 5 _ 5 \/2
V2^ 2
Set 2 C on 2 A, and under 5 C read the result 3.53 + on D,
244
BUSINESS MATHEMATICS
niustntive Example 3. Find the value of
Vex vn
V7 •
Solution : Set 7 B on 6 A, and under 14 B read the result 3.47 on I
niustntive Example 4. Find the area of a circle whose radius is
inches.
Solution: Set 1 C on 3 D, and above x on B read the area, 28.2
sq. in. on A.
Note : If the student can do his work on the slide rule so that it i
correct to the first decimal, this will be satisfactory for most computf
tions.
EXERCISES
Find the value of the following:
1. 65 X 4
2. 4.6 X 3.5
3. 7.2 X 5.54
4. 10.5 X 22.8
6. .08 X 2.6
6. .28 X .004
7. .54 X 1.8
8. 2.6 X 18.5
9. 4.4 X 18.4
10. .54 X .92
11. 4.84 X .005
12. .128 X 64
13. 45.2 -T- 25
14. 144 -^ 24
16. 8.84 ^ .75
16. 128 -^ 4.4
,_ 26.8
11.
4.6
18.
4.28
.65
19.
17.28
1.2
20.
625
25
91
6.25
6.25
2.6
23. 38 4-18
26. 4-^8
26. 55 ^ 27
27. 31.25 -^ 25
3 X 4 X 12
2 X6
5 X 7 X 56
6X 14
35 X 64 X 8
7 X 16 X 4
49 X 54 X 9
28.
29.
30.
31.
18 X 27 X 7
32. 10 X 35 X 65
«« 3 X 8.4 X 6.6
33.
4 X4.6 X 2.6
^^ 16.4 X 12 X 4.2
34.
2.6 X 8.4
36. 3 Vo
36. 7 -^ Ve _
37.VW10
V's
.25
THE SLIDE RULE 245
38. Find the area of a rectangle whose length is 4.6 in. and whose
idth is 2.8 in.
39. Find the area of a circle whose radius is (a) 4 in.; (b) 2.8 in.; (c)
.2 ft.; (d) 9.6 yd.
40. Find the circumference of a circle whose radius is 6.4 ft. if the
Jcumference equals twice the radius times x.
41. Find the area of a lateral cylinder whose radius is 3 in. and whose
eight is 6.4 in.
42. If the wages of 6 men for 1 da. are $28.50, what are the wages of
2 men at the same rate?
43. A department store offered a sale of 7 articles for 47^, find the
ost of 13 articles at the same rate.
44. 8 is what per cent of 24?
Hint: 8 -^ 24 = what decimal = what %?
45. A = what %?
46. 2^^ = what %?
47. 1 ? = what %?
48. e** = what decimal?
49. Y*5 = what decimal?
50. Find the interest on $800 at 5% for 27 da.
Hint:
Principal X Rate X Time in days
Interest =
360
61. Compute the interest on $650 at 4% for 65 da.
62. What is the interest on $500 for 75 da. at 4^%?
63. Find the principal which will produce $120 in 5 yr. at 6%.
Hint:
Interest
Principal =
Rate X Time (in years)
64. Find the principal which will yield $2,400 in 8 yr. at 5i%.
55. Find the square of each of the following:
(a)
4.6
(f) 2.56
(b)
6.8
(g) 25.6
(c)
12
(h) .15
(d)
9
(i) 1.5
(e)
10.6
(i) 15
246 BUSINESS MATHEMATICS
06. Find the square root of each of the following:
(a) &4 (e) 1.44
(b) 49 (f) 14.4
(c) 9 (g) 2
(d) 144 (h) 3
Note: Check (g) and (h; by extracting their square roots, ther
memorize the result correct to 3 deciuLils.
67. Find the value of each of the following:
(a) t: X 12 (d) r, x (5.4)^
(b) T X 6 (e) t: x (4.6) ^
(c) t: X 4^ (f) ^ X (2 Ay
68. Find the radius of a circle whose area is 144 sq. in.
69. Find the radius of a circle whose circumference is 31.416.
60. Find the circumference of a circle whose area is 78.54 sq. in.
207. To Find Cubes and Cube Roots.— To find the cube
of 4, work as follows: Set 1 of B over 4 of D and read the
answer 64 on A directly over 4 of B.
To find the cube root, reverse the process. For example,
to find the cube root of 64, move the slide back and forth
until the number on B directly under 64 (on A) is the same
as that under 1 C', on D. If the left-hand 1 does not work,
use the right-hand 1 on C.
EXERCISES
1. Find the cube of the following numbers:
(a) 2
(f)
6.4
(b) 3
(g)
1.2
(c) 5
(h)
1.25
(d) 15
(i)
.04
(o) 4.2
(J)
.0015
2. Find the approximate cube roots of the following:
(a) 8 (e) OS
(b) 27 (f) 425
(c) 125 (g) 16.8
(d) 216 (h) 2.64
THE SLIDE RULE 247
S. 6 times the cube root of 16 — 7
4. 4 times the cube of 1.6 = T
5. Divide 18 by the cube root of 6.
6. Divide the cube root of 6 by 4.
MISCELLANEOUS EXERCISES
1. To find the 4th term of a proportion.
a c h
Hint: If r = *; » t^^n d = c X - .
ha a
Set the first term on C to the second term on D, run the rider to the
third term on C, and under the rider find the fourth term on D.
2. Find the fourth term of the proportion G : 18 = 7 : x.
3. To find the mean proportional between two given numbers.
a X ,
Hint: The proportion is - = - , or x = y/ac.
X c
Set index of scale B to a on scale A, and place the rider opposite c on
scale B, then under the rider on D scale find the mean proportional
required.
4. Find the mean proportional between 27 and 13.
5. To reduce to the decimal of a given quantity.
Express 4 oz. 10 dr. as a decimal of 1 ton.
74
Our fraction is 4 oz. 10 dr. = 74 dr.
16 X 16 X 2,000
Hint: Work out the denominator first, then the resulting fraction.
6. To find the interest on a sum of money.
Find the interest on $500 at 4% for 4.5 yr.
Pnr
Hint: / =
100
Railway apportionment of fares for different roads.
7. Suppose a fare of $10 has been paid and the traveler goes over
three different roads, making 250, 140, and 110 miles respectively on the
different roads, find the amounts that should be apportioned to the
different roads.
Hint: Hi = J for the first road.
W? = 60 == /s for the second road. What %?
W = iJ f or the third road. What %?
248 BUSINESS MATHEMATICS
Reductions and conversions.
8. Reduce 24 ft. to meters. (24 -^ 3.28 = ?)
9. Change 9 oz. to the decimal of a lb. (9 -f- 16 = ?)
10. Change 36 ft.-lb to ft.-tons. (36 -^ 2,000 = ?)
11. Convert 8 cubic ft. of water to lb. per sq. in. (8 X .4333 = ?)
12. Reduce 25 miles per hr. to knots. (25 X .8684 = ?)
13. Change 12 H. P. hours to kilowatt-hours. (12 X 1.34 = ?)
14. Find the value of 215 X \/2T^.
Hint: Set the rider to 24.2 on the right-hand end of A, bring 1 of C
to the rider and move the rider to 215 on C, when mider it we find the
answer on D.
"^o find the wages due.
16. If we wish to find the wages due for N hr. at $48 per week for 44
N
hr., we have the proportion 48= —
16. Find the wages due for 26 hr. if a man receives $42 for a 48-hr.
week.
17. Find the rate of interest on 2|% consols at 112f, neglecting
brokerage.
Hint: 112J _ 2i
"ioo ~ ?
18. An article costing $20 is sold at $50 less 25%. Find the per cent
of gain on the cost. What is the per cent of gain on the selling price?
19. Given:
Sales $500
Cost of goods sold 260
Selling expenses 110
General expenses 45
Profit
What per cent of the sales is each item?
20. Find the amount due an employee if he has worked 44 J hr. at
$20 per 48 hr. wk.
21. Find the selling price of an article bought for $4 on which a 24%
profit (on cost) is to be made.
22. If an article costs $2.35 and is to be sold so as to make 20% on the
selling price find the selling price.
Hint: Cost will be 80% of selling price, so set .8 of C scale to 2.25 of
D scale and under 1 of C scale read the answer.
THE SLIDE RLXE
'tS. Work tlic futluwiiig with the slide rul>^:
(;o,sT
To Make on
Price
W S 3.00
22%
(b) 5.00
34%
(c) iii.as
18%
(d) 9.00
40%
(e) 10.00
45%
34. Whut amounl.
a due nn
employee
n employee for 41 hr. of overtime work at
time and ii half, when the regular 44 iir. weekly wuge rate i±> $'25?
26. What is the value in £ of 875, when the exchange rate ia ;i.!)l?
Hikt: 75 -V 3.31
VSe. If exchange is 3.94, find tlie value in dnllur^ uf £40
Hint: S.?t 3.M of C to 40 of D, :iiid undrr 1 of C rp;,d the answer.
27. Find the value in franis of $75, when exchanRc is 6.97.
29. Find the value of l.OIW francs if e.tchange is 6.91,
39. Fitid the value in lira of }S5 when eychange is 4,09.
30. Find (he value of 2,500 lira in dollars if e.ii^hange is 3.95,
31. The liat price is $15, Ip^ 10% and 5%. Find the net cost.
32. Find the interest on a 47^ Liberty Bond of »50 from Jan. 1, 1921,
to Mar. 16, 1921.
.04 X 50 X 74
:(ti5
_iules for Characteristic,
Multiplication. If the slide projects to the left, the characteristic
equals the sum of the characteristics of the factors — if to the right, it
equals the sum + 1.
Division. If the slide project.? to the left, it equals the characteristic
of the dividend, minus the characteristic of the divisor — 1 — if to the
right it equals the difierence.
CHAPTER XVIII
DENOMINATE NUMBERS
208. Denominate Numbers. — A denominate number is a
number with a specific name, such as $5, 4 yd., 6 lb., 8 meters,
etc. Sooner or later any person is apt to have occasion to
use denominate numbers. Accordingly it is thought best to
introduce a short chapter of these numbers in this book both
as text for the student and as reference matter for the busi-
ness man, who has probably studied this subject at some
time in his school career and then forgotten most, if not all,
of it.
There are several tables of these numbers, including both
the English and the Metric systems. A few simple exer-
cises in connection with them have been introduced, to-
gether with methods of solution, in order that the reader
may recall them.
209. Tables.—
Long Measure
12 in.
= 1ft.
3 ft.
= 1yd.
5| yd. or IG^ ft.
= Ird.
320 rd.
= 1 mi.
17fc yd.
= 1 mi.
5280 ft.
= 1 ini.
?50
iSi2 Bl :CXES^ MATHEMATICS
Tto.*r Vd^aET ukc h: migJu ng-gBold, etc.)
:i!i> }^!fui}'wsifsr» = 1 az.
i:? uc = 1 Hq.
X^Joc. = 1 It
:*(Mj lo. = 1 XM
Tl¥) 1*^. = 1 Wu? '-at ustc m coal. «c.. tranBactimiB— ^lofea*
Ai»«jTKfiCAKiE6 Weight
2^J rx- =1 Btmple
3 wTuples = 1 dram
K dnuikf == 1 uL.
i:> '/z. = 1 lb.
O^Mj'AJKATivE Weight?
1 lb. XT'jv *)T a;yxL*-'-jLrie^* = o760 pr.
1 <jz. • •• • = 4S0 •*
1 io. avoird jr^i^ = 7CICO **
1 oz. • = 4371 ••
1 bM. of fiouT = l^^;Jb.
1 " " u^-f = :^^j() *•
1 cu. ft. of w^v.r wei;ilis OJi lb. (about 7| gal)
1 bu. wlicul = Wi lb.
1 bii. oal.s = '.M "
1 bu. i>otaUXfS = <)0 ''
1 bu. apples = /» '*
Liquid Me.vsure
4 gills
= 1 pt.
2 pt.
= 1 qt.
4 qt.
= 1 gal. = 231 cu. in.
:nj »ii.
= 1 bbl.
m gal.
= 1 iKjgshead
DENOMINATE NHJMBERS 253
Dbt M£ASUBE
2 pt. = 1 qt.
8qt. = Ipk.
4 1*. = 1 bu- = 2150.42 cu. in.
MsASUKEs OF Time
60 sec.
= 1 min. 12 mo. = 1 3t.
60 min.
= 1 hr. 360 da. =1 oonunerdal 3t
24 hr.
= 1 da. 365 da. = 1 oommon yr.
7 da.
= 1 wk. 366 da. = 1 leap yr.
30 da.
— 1 commercial mo. 100 3Tr. — 1 century
Centennial years divisible by 400, and other ^^ears divisible by 4 are
leap years.
Measures of Value
United States Money English Money
10 mills = 1 cent 4 farthings = 1 penny (d)
10 cents = 1 dime 12 pence = 1 shilling (s)
10 dimes = 1 dollar 20 shillings = 1 pound sterUng
10 dollars = 1 eagle = $4.8665
1 far. = 1$ cents; 1 d = 2A cents; 1 s = 24J cents.
French Money German Money
100 centimes = 1 franc = $.193 100 pfennigs = 1 mark = $.238
Miscellaneous Measures
12 things = 1 doz.
12 doz. = 1 gross
12 gross = 1 great gross
24 sheets = 1 quire
20 quires = 1 ream
= 480 sheets
210. Reducing to Lower Denominations. — It is sometimes
necessary to reduce a given quantity to a lower denomina-
tion, or to reduce quantities of different denominations to
the same denomination.
2oi BUSINESS MATHEMATICS
mostnitnre Ezamile 1. How many gills in 5 gal. 3 qt. 1 pt.?
SoLcnox: 5 gal. = 20 qt.
20 qt. -f 3 qt. = 23 qt.
2:3 qt. = 46 pt.
46 pt. -f 1 pt. = 47 pt.
47 pt. = 188 gills
fflostrmthre Example 2. Reduce .626 mi. to lower denominations.
SoLCTiox: 1 mi. = 320 rd.
.626 mi. = .626 X 320 = 200.32 rd.
1 rd. = 5.5 yd.
.32 rd. = .32 X 5.5 = 1.76 yd.
1 yd. = 3 ft.
.76 yd. = .76 X 3 = 2.28 ft.
1 ft. = 12 in.
.28 ft. = .28 X 12 = 3.36 in.
Therefore .626 mi. = 200 rd. 1 yd. 2 ft. 3.36 in.
WRITTEN EXERCISES
Reduce:
1. ; mi. to rd. 4. .50 bu. to qt.
2. .7.) gal. tj pt. 5. .otU degrees to min. and sec.
3. .874 mi. 6. .374 chains to lower denominations.
211. Reducing to Higher Denominations. — It is some
times necossary or convenient to change a given quantit)
to a higher denomination.
Illustrative Example 1. Change 1,268 hr. to higher denominations.
Soldtion:
24 hr. = 1 da.
1,268 hr. = 1,268 -I- 24 = 52 da., and 20 hr. remaining
7 da. =1 wk.
52 da. = 52 -r 7 = 7 wk. and 3 da. remaining
4 wk. = 1 mo.
7 wk. =7-7-4 = 1 mo. and 3 wk. remaining
Therefore 1,268 hr. = 1 mo. 3 wk. 3 da. 20 hr.
DENOMINATE NUMBERS 255
Illustrative Example 2. Reduce 2 qt. 1 pt. to the decimal part of
a gal.
Solution: 1 pt. = 1 -s- 2 = .5 qt.
2.5 qt. = 2.5 -5- 4 = .625 gal.
Therefore 2 qt. 1 pt. = .625 gal.
WRITTEN EXERCISES
Reduce to higher denominations.
1. 128 pt. (hquid).
2. 3 J qt. to decimal part of a bu.
3. 8 oz. to decimal part of a lb. troy.
4. Ij ft. to the decimal part of a rd.
6. 75 ft. to the decimal part of a mi.
6. 59 min. to the decimal part of a wk.
212. How to Add Denominate Numbers. — Denominate
numbers may be added as follows:
Illustrative Example: Add
5 gal. 3 qt. 1 pt.
6 " 2 " 1 "
'7 a 1 " 1 "
Solution: 18 gal. 6 qt. 3 pt.
19 " 3 " 1 "
WRITTEN EXERCISES
1. Add 5 ft. 8 in.
6 " 6 •'
2. Add 6 sq. rd. 4 sq. yd. 3 sq. ft. 12 sq. in.
o u << g ti <* g u << g tc <<
3. Find the length of wire necessary to wire a rectangular field 8 rd.
5 yd. 1 ft. 9 in. by 6 rd. 4 yd. 2 ft. 8 in. with four strands of wire.
213. How to Subtract Denominate Numbers. — Denor
inate numbers may be subtracted as follows:
256 BT^NESS MATHEMATICS
m r y^— fi> Soblnct 5 ft. 10 in. firm 12 ft. 9
Soumas: 12 ft. 9 in. = 11 ft. 21 in.
5 " 10 ^
6ft. 11 in.
'Ai'J-m^^K
Sobtnetthe
1. From 14 lb. 8 os.
7 lb. 7 OS.
S. From 36nL 4yd. 2 ft. Sin.
26 rd. 2 yd. 1ft. 9 in.
3. A nnn sold three lots each containing 8) sq. rd. from a field con-
taining 21 acres. How moch had he left?
214. Molt^lication Usiiig One Denommate Number.--
Denominate numbers may be multiplied as in the follow-
ing:
IHnstrative Example. Multiply 6 gal. 3 qt. 1 pt. by 6
Solution:
6 gal. 3 qt. 1 pt.
6
36 gal. 18 qt. 6 pt.
41 " 1 ''
WRITTEN EXEPXISES
Multiply the following:
1, 4 yd. 2 ft. 8 in. by 7.
%, 12 lb. 4 oz. by 15.
S, What is the weight of 6 J cu. ft. of cast iron if cast iron is 7i times as
liCAVV a$ water and watsr weighs 02. 5 lb. to the cu. ft.?
il6. Division Using One Denominate Number. — De-
tKMWittx^to numbers may be divided as in the following:
|||«;ftitti\« Bxwnple 1. Divide 356 gills by 4.
DENOMINATE NUMBERS 257
Solution: 356 gills -r- 4 = 89 gills
89 gills = 22 pt. 1 gill
22 pt. = 11 qt.
356 gills ^ 4 = 11 qt. 1 giU
Illustrative Example 2. Divide 46 yd. 2 ft. 8 in. by 12.
Solution: Reduce to inches and then proceed as in Example 1.
WRITTEN EXERCISES
Divide the following:
1. 37 yd. 2 ft. 8 in. by 8.
2. 16 lb. 7 oz. by 6.
3. If a man can walk 16 mi. in 6 hr., what is his rate of travel?
4. If an automobile makes 238 mi. in 8 hr., what is the rate per hr.?
216. The Metric System. — This is the system of weights
and measures in use in France. It is also used quite ex-
tensively in the United States and other countries at the
present time. Its great advantage is the fact that all the
tables use a scale of 10.
217. Terms. — The meter is the unit of length and is
approximately 39.37 in.
The liter is the unit of capacity and is equal in volume to
1 cu. decimeter.
The gram is the unit of weight, and is the weight of 1 cu.
centimeter of distilled water in a vacuum, at its greatest
density (39.2°) Fahrenheit. It weighs 15.432 + grains, Eng-
lish measure.
218. Prefixes, — The three Latin prefixes denote parts of
the unit;
»7
258 BUSINESS MATHEMATICS
milli- means one one-thousandth
centi- " ** one-hundredth
deci- " ** one-tenth
The four Greek prefixes denote multiples of the unit:
deka- means ten
heeto- " one hundred
kilo- " one thousand
myria- " ten thousand
219. Tables.—
Linear Measure
(The unit is the meter)
10 miUimeters (mm.) = 1 centimeter (cm.)
10 centimeters = 1 decimeter (dm.)
10 decimeters = 1 meter (m.)
10 meters = 1 dekameter (Dm.)
10 dekameters = 1 hectometer (Hm.)
10 hectometers = 1 kilometer (Km.)
10 kilometers = 1 myriamcter (Mm.)
WRITTEN EXERCISES
1. Change 356 m. to Dm. ; to dm. ; to Km. ; to mm. ^
2. Reduce 2642 cm. to m.; to Dm.; to Km.
3. A rectangle is 352.6 cm. long. How many meters long is it?
4. Change 5 Km. 3 Hm. 2 Dm. 4 cm. to m.
6. Reduce .25 m. to Mm,
Square Measure
100 sq. mm. = 1 sq. cm.
100 sq. cm. = 1 sq. dm.
100 sq. dm. = 1 sq. m.
100 sq. m. =1 sq. Dm.
100 sq. Dm. = 1 sq. Hm.
100 sq. Hm. = 1 sq. Km.
DENOMINATE NUMBERS - 259
Land Measure
(llie unit is the are)
100 centares (ca.) = 1 are (a) or 100 sq. m. (about 33 ft. square).
100 ares = 1 hectare (Ha.) or 10,000 sq. m. (about 2i acres).
WRITTEN EXERCISES
1. Reduce 565 sq. m. to sq. Hm.
2. Reduce .0674 sq. Km. to sq. m.
3. Reduce 1 sq. m. to sq. in. (correct to three decimals). (See com-
parative table below).
4. Reduce 1 sq. ft. to sq. m. (correct to three decimals).
Cubic Measure
1,000 cu. mm. = 1 cu. cm.
1,000 cu. cm. = 1 cu. dm.
1,000 cu. dm. = 1 cu. m. or stere.
etc.
Measure of Capacity
(The unit is the liter)
10 milliliters (ml.) = 1 centiliter (cl.)
10 cl. = 1 deciliter (dl.)
10 dl. =1 liter 0)
etc.
Measure of Weight
(The unit is the gram)
10 milligrams (mg.) = 1 centigram (eg.)
10 eg. =1 decigram (dg.)
10 dg. = 1 gram (g.)
10 g. =1 decagram (Dg.)
etc.
Comparative Table of Metric Values vs. English Values
1 in. = 2.54 cm.
1 ft. = .3048 of 1 m.
1 yd. = .9144 of 1 m.
Ird. = 5.029 m.
Imi. = 1.6093 Km.
1 sq. in. = 6.452 sq. cm.
aiii ]
Bl5INt:>£
» ^LVrHK\f\TICS
1 ari ft.
= >»29
sq. m.
1 arj. yd.
« -S361
sq. m.
1 flci, rd.
« 25.29;i
sq. m.
1 ari, mi.
= 2..S9
sq. Km.
1 cu, in.
= 1S..%7
CU- cm.
1 Cli. ft.
= 28.317
cu. dm.
1 cu. yd.
= .7W6
cu. m.
1 liquid qt
« .^163
L
.
1 dry qt.
= 1.101
1.
ir>k.
» 8.809
1.
1 bu.
= J5524
H.
Iff-.
= .0648
«-
1 oz. rtroy)
= 31.103-r
g.
1 oz. ravoirdupoiif)
= 28.35
g-
1 lb. ftroy;
= .3732
Kg.
1 lb. (ayoMupoiii)
= .4536
Kg.
1 cu. dm. of water
= 1
L of water and weighs 1 Kg.
or 2.2046 IK
1 cm.
= .3937
in.
1 m.
= 39.37
in.
1 Km.
= .6214
mi.
1 H(\. m.
= 1.190
sq. yd.
1 cu, m.
= 1 .308
cu. yd.
11.
= 1 .Or)07
liquid qt.
11.
= .1K)8
dry qt.
1 K.
= 15.432
U,r.
= .0321.^
I oz. troy
c= .03527
' oz. avoirdupois
IKg.
= 2.2010
lb. avoirdupois
1 metric toil
= 2,204,() lb. avoirdupois
WRITTEN EXERCISES
1. Reduce 25.55 K^. to lb. avoir.
2. Reduce 2 ft. 5 in. to m.
3. Change GO sq. m. to nq. ft.
4. How many in. in 30 mm.?
6. Reduce 2 gal. 3 qt. 1 pt. to 1.
6. Change 8,678 Kg. to tons and lower denominations.
7. If cast-iron weighs 7. 1 1 3 g. per cu. cm., how many lb. does a cu. ft.
m^gh?
8. Kind the cost of 25 yd. of cloth at $1.26 per m.
9. How many Km. in 25 mi. (to the nearest thousandth)?
DENOMINATE NUMBERS 261
10. What is the time of traveling \ mi. at the rate of 100 m. in 16 sec.?
11. If a stream of water 5 ft. wide and 9 in. deep is flowing at the rate
of 1 yd. per sec, find the weight of water in metric tons, supphed in
12 hr., if a cu. ft. of water weighs 1,000 oz.
12. Find the weight in lb. and in Kg. of 31.17 gal. of the best alcohol,
specific gravity .792.
13. If the pressure of the air is about 1 Kg. per sq. cm. how many lb.
is that to the sq. ft.?
14. What is the difference in yd. between 5 mi. and 8 Km.?
15. A bar of iron (specific gravity 7.8) is 6 ft. by 3 in. by 4 in. Find
its weight in Kg.
CHAPTER XIX
PRACTICAL MEASUREMENTS
220. Practical Measurements. — Such measurements are
what the term signifies; Le., measurements which are of
practical use to any person or any business at any time.
These include the measurements of or appertaining to differ-
ent kinds of angles; surfaces; polygons, including the paral-
lelogram, the rectangle, the square, and the triangle; circles,
including the diameter, the radius, the circumference, and
the area; problems involving square root; area of irregular-
shaped figures; solids, such as the cube, the cylinder, the
cone, the prismatoid, and the sphere.
221. The Angle. — An angle is the amount of opening be-
tween two straight lines which meet at a point.
The sides of the angle are the Unes whose
intersection forms the angle. The vertex of an
angle is the point in which the sides inter-
seci/.
222. Reading an Angle. —
1. The best way to read an angle is to place a small
letter or figure like a or 1 as in the following
figures, and call it angle a, or angle 1.
2. Another way is to use three letters, as angle ABC
in the following figure, putting the vertex letter
in the middle.
262
PRACTICAL MEASUREMENTS
263
3. Another plan is to use a capital letter, as angle C
in the following figure.
A
B c c
223. Unit Angle. — The unit of the angle is the degree.
If we divide the circle into 360 parts, and the ends of one of
these parts are joined with the center by two straight Hnes,
the angle formed at the center is 1°.
224. The Protractor. — A protractor is a convenient in-
strument for measuring angles. It is a half circle with its
rim divided into 180 equal parts, called degrees of the arc.
The center is also denoted at B,
To measure an angle ABC, place the protractor over the
angle so that the center of the protractor is directly over the
vertex of the angle and the ^ — ^
A
zero mark on the scale is over ^ B
one side ot the angle as CB. The point where the other side,
AB, of the angle ABC crosses the scale indicates the number
of degress in the angle, as 45 in this illustration.
2U
BUSINESS MATHEMATICS
\D
225. The Straight Angle. — This is an angle whose
lie in the same straight Une and extend in opposite direc-
tions from the vertex; as the angle ABC, in the accompany-
ing figure.
226. The Right Angle. — This is one of two equal angles
made by one straight line meeting another straight line.
Thus if the fine CD meets the hne AB
so as to make the angle DC A equal to
the angle DCB, each of these angles is a
right angle.
What part of a straight angle is a
^ ^ right angle? How many degrees in a
right angle?
227. Perpendicular Lines. — A line is said to be perpendi-
cular to another line when it meets it so as to form two
('(luul iingles.
What kind of angles do the lines form?
228. The Kinds of Angles. — The acute angle is an angle
Ir.sM than a right angle. An obtuse yc
\\\\^\v is an angle greater than a right
mi^lr, but less than a straight angle. ^ b ^
\ /i( ' iH an acute angle. CBD is an obtuse angle.
ViaW. Surfaces.— A surface is that which has length and
UuM^llli but no thickness.
\ \\\m\p Niuface is a level surface such as the surface of
. wU w \\\'\\ A si might edge will fit on it in any position.
\ |v|fiu« tlU"*"^^ ^^ ^^ figure all of whose points lie in the same
PRACTICAL MEASUREMENTS
265
230. Polygons. — A polygon is a portion of a plane
bounded by straight lines, as the following figure. The
perimeter of a polygon is the sum of all its sides. A diagonal
is a straight line joining two non-adjacent vertices as if, in
the figure below, a line should be drawn from any one
comer to an opposite corner.
231. Quadrilaterals. — A quadrilateral is a plane surface
bounded by four straight lines.
232. Parallelograms. — A parallelogram is a quadrilateral
having its opposite sides parallel.
Parallelogram
Rectangrle
233. Rectangles. — A rectangle is a
parallelogram all of whose angles are
right angles.
234. The Square. — A square is a rectangle
having four equal sides.
235. The Triangle. — A triangle is a plane figure bounded
by three sides and having three angles.
236. The Right Triangle. — A right triangle is a triangle
that has one right angle. No triangle has more than one
right angle.
The sum of the three angles of any triangle equals two
right angles or 180°.
266
BUSINESS MATHEMATICS
237. The Hypotenuse. — The hypotenuse of a right tri-
angle is the side opposite the right angle.
Triansrie
'««
2C8. The Equilateral Triangle. — An equilateral triangle
is a triangle having all its sides equal and all its angles
equal.
239. The Isosceles Triangle. — An isosceles triangle is a
triangle having two sides equal and two angles equal.
240. A 30-60 triangle is a triangle, one of
whose angles is 30°, another of whose angles
is 60° and the third angle obviously 90 .
The hypotenuse is twice the length of the
shorter arm BC.
241. The base of any plane figure is the side on which it is
supposed to stand, as AC in §240.
242. The altitude
of any plane figure is
the perpendicular
distance from the op-
posite point highest ^ ^ b a
from the base to the base or to the base extended as CD.
PRACTICAL MEASUREMENTS 267
243. A circle is a plane surface bounded by a curved line^
called the circumference, every point of which is equally
distant from the center of the circle.
244. The diameter of a circle is a
straight line drawn through the center
and terminated by the circumference.
248. The radius of a circle is a straight
line drawn from the center to any point
on the circumference.
246. An arc of a circle is any part of the circumference.
247. The perimeter of a circle is the length of the cir-
cumference.
248. The area of any plane figure is the number of square
units within its bounding line. A square whose side is one*
Mnit is said to have an area of one square imit. A square'
vhose side is 1 foot is said to have an area of 1 square foot.
WRITTEN EXERCISES
Draw a rectangle 8 iu. lung and 4 in. wido, mid divide it into
inch squarea by drawing Lnea parallel Ui the sides. Obtain the area
of this rectangle by counting the number of small aquarea thus formed
in the figure. Can you state any shorter way of obtaining the area of
■Oiis rectangle?
2. Complete the following equation where Ao means the area of t-
.tectangle, b = base, and a = altitude: Aa =
8. Write the equation of Exerpiso 2, and then substitute the proj
dues. Keep all equality signs under each other and find the
whose length is 6 in. and whose breadth (or width) is 4
268
BUSINESS MATHEMATICS
4. Do the same if the length (f) = i in. and the height or width,
or altitude (a) is \ in.
6. Find the area of a rectangle whose base (&) is .25 of an in. and whose
altitude (a) is .125 in.
6. A tennis court is 78 ft. long and 36 ft. wide. How many square
feet does it contain? What part of an acre is it?
7. Find the perimeter and the area of a rectangle 15 yd. by 12 yd.
8. How many paving blocks 1 ft. long and 5 in. wide will be required
to pave a street 2 mi. long and 35 ft. wide?
9. A rectangular field is 40 rd. long and 20 rd. wide. Find the cost of
fencing it at $2.25 a rod.
10. Find the cost of painting the four side walls of a room 12 ft. long,
10 ft. 6 in. wide, and 9 ft. high at 12 cents per sq. yd., no allowance being
made for openings.
11. The length of a rectangular piece of iron is 85.24 in. and the width
is 34.75 in. Find its area and perimeter.
12. If 1 sq. ft. of the above mentioned iron weighs 5.1 lb. what is the
weight of the entire piece if of same thickness throughout?
13. The plan of a slide valve is 10.5 in. by 7.75 in. and the pressure
back of it is 85 lb. per sq. in. Find the total force pressing the valve.
14. Find the area of a channel iron from the dimensions in the accom-
panying figure. I
//
5.23
.54"
/
T
-2.76-
^54
//
15. Find the area of the shaded part in the accompanying hollo^
square.
PRACTICAL MEASUREMENTS
269
16. How many pieces of sod will it take to sod a lawn 24 ft. wide and
28 ft. long if the pieces are 12 in. by 14 in.?
17. The area of a rectangle is 180 sq. in., and its base is 4 yd. Find
the altitude.
18. Find the area of a floor from the dimensions in the accompanying
figure. 20'
30
10'
12'
2S
40'
249. To Find the Area of a Parallelogram. — The parallelo-
gram ABCD may be shown equal in area to the rectangle
BE c F ^£/^D by cutting off the triangle 45^
and placing it on the triangle CDF.
This shows that the equation for the
area of a parallelogram is then the same
as that for the rectangle. What is that equation?
WRITTEN EXERCISES
1. Find the area of a parallelogram whose base (h) is 8 in. and whose
altitude (a, or ^^ in above figure) is 6 in.
2. Complete the following form for parallelograms whose dimensions
are:
Base
12 in.
6.5 in.
9J ft.
10.2 in.
Altitude
8 in.
3.25 in.
4f ft.
7.45 in.
Area of Parallelogram
3. A piece of metal in the form of a parallelogram has an area of 127.89
sq. in. and the base is 6.3 in. Find the altitude.
270 BUSINESS MATHEMATICS
260. To Find the Area of a Triangle.— If we draw iln
diagonal AC in the rectangle ABCD, then cut through fhr
diaKonal, wci shall find that the ttianEli'
ABC will exactly fit on the triangle ACD.
A triangle may therefore be shown to iw
■^ ^ equal in area to one-half of the area of a
rectangle with the same base and the aami; altitude.
State then the equation for the area of a triangle wba=<'
base is b and whoise aUiliide is a. Call the left nieiiiber of the
equation A^-
WRITTEN EXERCISES
1. Find the area of a trianglfl whosu base ('i) is 12 in. and whosp
altitude In) is Sin,
3. What is the area of a triangle whose base is S..3 ft. and wlios:; alli-
ludfi is 4.5 ft.?
3. Find the altitude of a Irianjile whose area (A) ia 144 an- >"■ uid
whoae faaae (b) ia 48 in.
251. To Find the Circumference of a Circle.— Find the
length of the ciriHunfercneo of a circle by taking a cardboard
circle, marking
some point on f \ (^ ''
i
it, as P, where
circle touches
level aa A and "* ^
roll it along on a level surface until P again touches the
level surface — say B. The distance AB will then represent
the circumference of this circle. It will be found also that
this length divided by the diameter of this circle will give
approximately 3.1410, called x (pi). Therefore, in the 60*
compan>4ng figure C -i- /) = tc, or C
e.
I
PRACTICAL MEASUREMENTS 271
WRITTEN EXERCISES
^p,t Find the circumference of a circlp whnse diameter is 8 It.
S. A circle ia 12.3 in. in diurncttr. What in il8 circumference?
3. Slate finother name tor the circumference of a circle.
4. A bicyclist travels 8S0 ft. per miD. The bicycle wlieclw are 28 in.
jidiameter. Find how many times each wheel revolves per min.
^U. A flywbeet 10.5 ft. in diameter revolves UO times in 1 min. Find
^^Bdistani^e that a point on the rim travels in 1 min.
HPr, The wire from a signal tower to the signal Ls 450 yd. long, and the
Ipide pulleys on the posla arc 1} in. in diameter. Assuming that the
wire must be pulled 12 in. to cause the signal to drop, how many revolu-
tions must a pulley make?
262. To Find the Area of a Circle. — Imagine the circle
divided tip into many Kiiiall parts as shown in the figure.
It will be oliSLTved tljiit wo liuve prucli-
cally nuraeroua small triuiigies whose
altitude is the radius of the circle and
wljoae base is an arc of the circle. We
then note that the sum of these arcs is
the circiunference of the circle. We
then have the area of the circle,
j1 O — ~~. but we have already seen that
C = TC B, or C = 2 •:: r, (r = radius).
:.Ao = XT- X r
Ao = Ti'orAo — T. times the Kquare of the radius.
WRITTEN EXERCISES
It. Find the area of a circle whose radius is 4 in.
t. Find the area of a circle whose radius is 8 in. X = 3.1416 or ^
3. (Jomparc the results of Exercises 1 and 2.
4. The radius of a high pressure cyUndcr of a marine engine is 13 in.,
and the cITcctive steam pressure at a certain instant is 45 lb. per sq. in.
Find the force working down on this piston if such force is equal to iMfi I
product of tbe area uf the cross section of the cylinder in sq. in. by JT
tHcctive pressure in lb. per sq. ii
272
BUSINESS MATHEMATICS
i. The diameter of a lever safety valve is 3 in., and the steam Ucms
off at 95 lb. per sq. in. Determine the upward pressure oo the vahe.
6. An Ef^yptian obtained the area of a circle by subtracting from the
diameter one ninth of its length and squaring the remainder. Try this
plan on a circle whose diameter is 9 in., and then aohre it by the above
method and obtain the amount of difference of areas.
7. The boiler of an engine has 300 tubes, each 3 in. in diameter, for
conducting the heat through the water. Find the total ctqgs sectional
area.
8. A piece of land is circular and 20 ft. in diameter. A circular walk
5 ft. wide is laid around it. What is the cost of this walk at $1.75 per
sq. ft.
263. The Trapezoid and its Area.— The
trapezoid is a quadrilateral having only
two sides parallel . The area of a trapezoid
is the product of one-half of the altitude
by the sum of its bases b and b'.
A trapezoid — fy \b ~T- b )
WRITTEN EXERCISES
1. Find the area of a trapezoid
whoso bases are 12 in. and 10 in.
and whose altitude is 4 in.
2. The bases of a trapezoid are
4 ft. 6 in. and 2 ft. 8 in., and the
altitude is 1 ft. B in. Find the
area in sq. in.
3. Find the area of the accom-
panying figure.
4. The area of a trapezoid is 66,
/; = 14, 6' = 8. Find the altitude.
i
T
■3 '4:
■8-
254. Extracting Square Root. — The square of a number
is the result o})taincd by multiplying some number by itself,
as 5^ = 5 X 5 = 25, and we say that the square of 5 is 25.
PRACTICAL MEASUREMENTS
273
The square root of a number is one of the two equal
factors of that number. From the statement above it is
obvious that the square root of 25 is 5. This may be rep-
resented by the following ways. V25 or 25* = 5.
WRITTEN EXERCISE
1. Complete the following form:
Number
9
16
36
49
64
125
1000
10000
Its Square Root
It will be observed from the above form that the square
root of any number between 1 and 100 is between 1 and 10;
of a number between 100 and 10000 is between 10 and 100.
The square of 25 may be found as follows, or as shown in
the accompanying figure.
25
20 + 5
20 + 5
(20 X 5) + 5=»
20^+ (20 X 5)
202+ 2 (20 X 5) + 52
20*
20x5
5*
25
This may be stated as follows:
The square of any number of two figures is equal to the
square of the tens plus twice the product of the tens by the
units plus the square of the units.
By applying this principle the square root of any nv
ber may be obtained.
r
The square of any number will contain twice as many
f figures or one less than twice as many figures as the niimber.
Therefore separate the number into groups of two 6giires
each beginning at the decimal point and working each way
from it. There will be as many figures in the square root aa
there are groups in the number.
lUustiative Example. Find the square root of 623, or 2_5
V^ = ? 6'25.
SoLnnoN: Begin at the decimal point and separate Ihe j^_.
number into groups of two figures each. The largest '"''L?''
equiire in 6 is 4, and the square root of 4 is 2. Obtain the -B ::^
remainder 2 and annex the next group <2o), giving 225.
Having taken the square of the t^na from the number, therefore the re-
mainder (225) must contuin twice the product of the ten^ by the umS
plus the square of the units. Two times 2 tens or 20 = 40. 40 is con-
tained 5 times in 225. 5 then ig the units figure of our root. Two lines
the tens, times the units, plus the square
sum o[ two times the tens, and the u:iits, times the units. Henrei ^
add 5 units to the 40 and multiply the sum by 5, obtaining 22.i. There-
fore the square root of 625 is 2,j.
Principle; To obtain the square root of a number:
1. Beginning at the decimal point, separate the number into
groups of two figures each.
2. Take the square root of the greatest perfect square con^
tained in the left-hand group for the first root figure;
subtract its square from the left-hand group,
the remainder bring down the next group.
3. Divide the number thus obtained, exclusive of its units^'
by twice the root figure already found for a second rootiil
figure; place this figure at the right of the root figure-
already found and also add this figure to the trial divisor
just used, Mulliply this sum by the last root figure..
Subtract and proceed in u. similar manner until tba
root is obtained.
274 BUSINESS MATHEMATICS
PRACTICAL MEASUREMENTS
275
Hints :
1. If the divisor is greater than the remainder, place a zero
in the root and also at the right of the divisor, bring
down another group and proceed as before,
2. If the root of a mixed decimal is required, form groups
each way from the decimal point. The last group at
the right must have two figures, even though a zero
must be annexed to form it.
3. To find the root of a common fraction first reduce the
fraction to a decimal, then obtain the root of that
decimal.
4. Obtain all roots to three places (at least) of decimals for
accuracy.
WRITTEN EXERCISES
Find the square root of each of the following:
2. 576.
3. 1225.
1. 42436.
9. .000624
13. i
10. 482.
14. J
11. 25.8
IB. i
12. 3.
16. A
266. Proportions of the
Right Triangle. — It is
shown in the figure and it
is proved in geometry that ,
the square on the hypote-
nuse of a right triangle is
equal to the sum of the
squares formed on each of
the legs of the right triangle.
Therefore the hypotenuse
equals the square root of
the sums of the squares of
-\>Cjx
Vt:
3
276
BUSINESS MATHER LVTICS
the two legs of the right triangle, or stated as an equatioD
It may also be proved that a = y/H^ — fr^ or that 6 ='
Illustrative Example. Find H \ia — \2,h = 16.
Solution: H — Va^ -f- 6^
= Vl44 -h 256
= Vioo
= 20.
WRITTEN EXERCISES
1. Find H, and the area of a right triangle if a = 20 in., & — 25 in.
2. The distance from home to first base on a baseball diamond is 90 ft.,
and the distance from first to second base is 90 ft., find the distance from
se(;ond base to home in a straight line.
3. If a park is rectangular in shape and 890 yd. long by 150 yd.
wide, how inuc;h is saved by walking from one corner to the opix)site
corner along a diagonal walk instead of walking along the sides?
4. If a body is at B^ h ft. above
the surface of the earth the num-
ber of miles 7«, at which it can be
seen, is limited by the curvature
of the earth. This distance is ob-
tained by the equation m = ^ 3A
T'se this in the following:
The light of a certain lighthouse is 200 ft. above the sea level. How
many miles distant can it he seen?
6. The bac k stay of a suspension bridge is 65 ft. long, and the distance
of the anchoring point from the foot of one of the piers is 54 ft. Find the
height (a) of the pier.
PRACTICAL MEASUREMENTS
277
6. A railway incline is 1 ft. in 150 ft.
zontal length (6)?
150'
What is the projected or hori-
7. Find the total area of the accompanying figure by dividing it up
into parts, as indicated, then finding the areas of each and then the total.
10"
*>. The hypotenuse of a right-triangle is 30 in. The altitude is 18 in.
(a) Find the base.
(b) What is the side of a square whose area is equal to area of this
triangle?
(c) Find perimeter of triangle.
256. Proportions of the 30°-60
Triangle. — The hypotenuse is twice
the shorter arm, a. The angle op-
posite the hypotenuse is 90°, therefore ^^^
the principles of the preceding section apply.
WRITTEN EXERCISES
1. The side opposite the 30° angle is 8, what is the hypotenuse? Find
the base by applying the principle in § 255.
2. The base of a 30° - 60° triangle is 9 in., the hypotenuse is 10.392 in.
Find the other leg of the triangle, and the area of the triangle. Find the
perimeter of the triangle.
278
BUSINESS MATHEMATICS
3. If the pitch of a 60° thread is 1 in., find the depth (d).
4. Find the altitude AH. of the rhombus depicted here if AC
AB = S in., &nd<ABD = 60°. Find the area.
A C
= BD»'
// D
6. Find the number of square inches in the surface of the pi^c®
sheet iron, shown in accompanying figure.
ot
/
>
\^
^0°
W^
« — "-^
2.841
12'
6''
90°
12"
IF::!-
-r-
1
!58
257. To Find Area of a Triangle, Given Three Side^
The area of a triangle, given the three sides without an a
>
ii-
tude, is sometimes required. It is C^^
tained by taking the square root of t -^
^
product of one-half the sum of the sid
MS
by one-half the sum of the sides minu ^
one of the sides by one-half the sum of the sides minus ttr
second side by one-half the sum of the sides minus the thir^
^)ide, or, expressed as a working equation,
, , /n + b -\- c'
Aa = \s{s - a) (s - b) (s - c) wliere 6- = I —
)
PRACTICAL MEASUREMENTS
279
WRITTEN EXERCISES
1. Find the area of a triangular piece of land whose sides are 8 rd.
12 rd. and 16 rd.
2. Find the area of a piece of tin whose sides are 4 in., 6 in., and 8 in.
3. The sides of an army camp were 7, 5, and 4 mi. What was the area
of the camp in sq. mi.?
4. A piece of metal has the form of a quadrilateral. The diagonal one
way is 7 in. Two of the sides on one side of this diagonal are 5 in. and
4 in. The two sides on the opposite side of this diagonal are 6 in. and 3
in. Find the area of the whole piece of metal.
258. To Find Areas of Surfaces. — The required area of
the figure may be found approximately by joining the ex-
tremities of the offsets by
straight lines, and then
finding the sum of the
areas of the trapezoids thus
contained.
TheTrapezoidal Rule :
To half the sum of the first and last offsets add the sum of
all intermediate offsets, and multiply this
result by the common distance between
them.
Another Method for Finding the
Area of an Irregular Piece: Find
the distance of each vertex as A, F, etc.,
in the figure from a given base line as XY.
These distances are called offsets and
are the bases of trapezoids whose alti-
tudes are A' B' B' ,& C\d\ etc. The area of ABCDEF may
now be found by the proper additions and subtrac-
2SD
BUSKESS MATHEMATICS
SiMPSOx's RtXE: Add together the &rst ordinate, (the
perpendicular) the last ordinate, twice the sum of the other
odd ordinates, and four times the
sum of the even ordinates. Multi-
ply this sura by the extreme lengtl
of the diagram and divide the
result by three times the numbet
of parts into which the diagram is divided.
WRITTEN EXBRCI^S
1. State Smpson's rule ss an equation, using tbe following notstkiu:
L = length or dia|;rain.
y, = first ordinate.
yl = last ordinate.
tf.. Vi, etc. = the other ordinates,
fi = number of parts into which the diagram is divided.
2. In the accompanying figure:
Find tiie area.
3. Trace the accompanying figt
the lines, and find tbf
PRACTICAL MEASUREMENTS 281
4. The diagram shows the cross
section of a gun metal oil ring. Find
its area by the above method, then
check by finding the area in some
other manner.
259. Solids. — A rectangular solid is bounded by six
rectangular surfaces. It is called a prism. The bases of a
jjrism are parallel and equal.
Principle: The voliune of a rectangular solid is equal
(in cubic units) to the product of the number of like units in
its three dimensions.
WRITTEN EXERCISES
1. Find the volume of a piece of metal 6 in. by 4 in. by i in.
2. What is the volume of a tank 25 ft. by 15 ft. by 8 ft.?
3. A cube has an edge of 9.54 in. Determine its volume in cu. in.
and cu. ft.^ and its surface in sq. in. and sq. ft.
4. A cubical tank is I full of water. An edge of this cube is 9 in. How
many cu. in. of water are in the tank? If 1 cu. ft. of this water weighs
62.5 lb. what is the weight of the water in the tank?
6. A piece of steel of f in. square section is chosen to make a lathe tool.
Determine the weight, if its length is 7.25". 1 cu. in. of steel weighs .28
lb.
6. A bar of steel 2J in. square and 2 ft. long is molded into a square
bar 12 ft. long. Find the dimensions of the bar after it is molded.
7. A rectangular tank measures on the inside 11 J in. by 13 J in. by
9 in. Compute the number of gallons which the tank contains when
£lled within an inch and a half of the top if 231 cu. in. holds 1 gal.
8. Allowing 30 cu. ft. of air per minute for each person in the class-
room, how much air must be driven into the room and how many times
must the air be changed during the recitation period to insure good
\rentilation?
«
9. If 38 cu. ft. of coal weigh a ton, how many tons can be put in a bin
12 ft. long, 8 ft. wide, and 6 ft. deep?
10. If 1 cu. ft. of lead weighs 700 lb., what will be the weight of
3. rectangular mass of lead 3 ft. 3 in. long, 2 ft. 4 in. wide, and 3 ft.
biah?
J- as:; business mathematics
Wk T© Rnd the Volume of a Prism. — The volume of ^
•ay prwui IS wjuiil tu tlie product of its base and ii:
WRITTEN EXERCISES
It VWI tbr vukuw til a Uianitul^ prism whose base Is an ccguilat
MWiWIJ>i *Ww sntr i^ Hi in., mhI tvhiist'' hci|i;bt ia 12 in.
ktkXV; ^'teJ Uk' khHiklr at Uu- irbiiKto. or apply S 2o7.
of a steel rod whose bu"
i in., if the length of tin-
t. FimI the roll
H :i h^xitipiu wiih
n<d » 10 ft.
llivt: A bex»^n is composed of G eqiiibti
(. yWt < W wilXBW tJ rtfri iwl whose base is the form of ji rhombij~
«iiAkiM»ikMwi9'tiit..iMMlb>dWi^Mi«ieiW, if itwrodis 10 ft. loug.
I. In ttll steel coast rut' lion work, such us
tke ntttslruHiua nf modem buildings, tbe
wv<4cht ut the steel is computed.
lu » vrrttua buihling the specifications
tiiliuiv 100 tlvfl. beams, a right cross scc-
titiii \i( which » ehuwn. Find ihe weight
i.4 ll»' Wnnvi Mii\ t\if cost at 9t |>er lb. when
)>iit ill itbtce. (Steel weighs 490 lb. per
cu. fl.)
PRACTICAL MEASUREMENTS 2S3
6. In another piece of construction work, T-beams 20 ft. 1on!:aDdU-
f^haped beams IQ ft Iod^ verc used, the n)cht sections of which are shown
in tlie diagram Find the weight of caih.
261. To Find the Volume of a CyUnder— The volume of
any cylinder is equal to the product of the base by the al-
titude.
WRITTEN EXERCISES
1. The diameter of the base of ii right circuLtr Mliiider is 24,5 in.
Find the volume if its height is 36,4 in.
2. A circular cast iron plate I
in. thick and having 26 holes,
i in. in diameter ia 4 ft. 8 in. in
diameter. Find the weight if 1
■eigha .26 lb.
S. How high must a tomato can that is to hold 1 qt. be matie, if its
Hint: 231 cu. in. holds 1 ga!.
4. The eyhnder of a steam pump, used for pumping city water, is 2 ft.
in diameter and 3 ft. long. It ia filled and emptied twice at each revolu-
tion of the piston. Find the number <if gallons delivered by the pump in
P a minute, if the piston makes 24 revolutions a minute.
5. If a gasoline tank on a motor car is a cylinder 35 in. long and 15 in.
& diameter, how many gallons of gasoline will it hold?
J J
284
BUSINESS ^L\THE^L\TICS
6. A hollow cylindrical can is partly filled with water. An irregular
piecf; of iron ore is placed in the water and causes it to rise 2.5 in. in the.
c>'linder. If the diameter of the cylinder is 8 in., what is the volume of
the ore?
7. A cylinder is 20 ft. long and its volume is 1,72S eu. in. What is the
diameter of the cylinder?
8. An iron pipe is a cylindrical sheU 2 in. in thickness. If the pipe is
10 ft. long and its inner diameter is 12 in., and 1 cu. ft. of iron weighs
4 ST) lb., find the weight of the pipe.
262. Pyramids.—
(a) The lateral area of any regular p>Tamid is equal to
the product of one half of the slant height (oH) by the peri-
meter of the base.
(b) The lateral area of a frustum (lower
part) of a regular p>Taniid is equal to one
half of the sum of the perimeters of the
bases multiplied by the slant height, (hH).
((') The volume of any regular pj-ramid
is equal to one third of the product of the
area of the base by its altitude (oO).
WRITTEN EXERCISES
1. The slant height of a regular pyramid is 24 ft., and the base is a
triangle each side of which is 8 ft. Find the lateral area.
2. The (Jn^at Pyramid of Egypt, when completed, was 481 ft. high,
and each side of its s(|uare base was 764 ft. long. How many sq. ft. in the
surface of the sides?
3. The figure shows the plan of a square
roof in the form of a frustum of a pyramid,
the upper base being a flat deck. CD is
IS ft., AB is 6 ft., and the height of the
roof, or altitude AO of the frustum, is 8
ft. Find the lengths that the rafters AC
and AE must be cut.
M>M
^ — ;=in«- TJ
IS if n
5i
-
IH^-IiTiJ
"Tli. li^-rt. '" . "^ "^.r '^ '
H
/
/
7r^#~T("- -r --.- i:— - --.. -.v
A"
^^^-
— — ' •■ - '- — "■ - «.-»»■• ■«■.«.■..
of the aldtude.
WMTTEN EXERCI^FS
1. The slant heigjit of a right omniljir ^vno ^^ S u\ v '^>^^^ ^^^* ^ ^^'^^^ '
the base is 6 in. Find the latonU arx\* , Tho \ y^\ fO .-on^-^
2. How many sq. yd. of canvas will Iv »V\\\uh^»l i»^ \\\^\^^' ^ »»m\\» \\
tent whose altitude is 12 ft. and dian\f>tor wi \\w Im»««* I M«
Hint: Find slant height by § 2^.\
I
286 BUSINESS MATHEMATICS
' 1. 9tai:e bow you would find the Uleial aiva of the fmslun) of a n°!it
nfcukir cone-
4. Find ihe amount ot aheet metal required to make a lot of 500 pai.'j.
each 10 in. deep, Hia. id diameter at the bottom, and -II in. iadi^metiT
ftl the lop, not allowing for seams nor waste in cutting.
6. The acFompsnjing diagram i^n? tbe
elevation of a conical friction dutch used in
automobiles. Find the contact area in si|.
in., i.e. the area of the curved surface.
S. The base of a marble column is a frustum of a cone. The hei^l i^
1 ft. 6 in., and the diametejs of the bases ti ft. and 4 ft. 6 in., respccti^^v.
If 1 cu.ft. weighs 170 lb. find its weight. VjI. = l{altO(e + 6 -f -^Bbi.
7. The lateral area of a cone of revolution is 240 sq. in., and the radius
of the base 6 in. Find (a; the slant height (bl the altitude (ci ibe
volume.
264. The Prismatoid. — A prismatoid is a polyhedron'
bounded by two polygons in parallel planes, called theb;
and by lateral faces which are either
triangles, parallelograms, or trap-
ezoids. If one base is a rectangle
and the other base a line parallel
to one side of this rectangle the
prismatoid is called a wedge.
The altitude is the perpendicular
liistance between the bases of the prismatoiii.
The volume of any prismatoid is equal to the sum of il
bases and four times the area of its mid-section (M), mult
plied by one sixth of the altitude or V" = i a(B + fc + 4JS
WRITTEH EXERCISES
1> Find the weight of a steel wedge whose base measures Sin. by 5i
Uie hf^ht ot the wedge losing 6 in., if one cu. in. of steel weighs 4.63 C
PRACTICAL MEASUREMENTS
287
if D
Z. liow mu[^h will it cost lo dig a diti-h 100 rd. long, 6 f1. wide at thft
lop, 2^ ft. wide at the bottom, and i ft. deep, at UOf per cu. yd.?
3. The volume of any truncated triangular prism is
equal to one third of the sum of the three lal^nil edges,
multiphed by the area of the right aectioa (b).
Find the volume of a truncated triangular prism
whose base is an equilatonU triangle forming & right
KCtion with the lateral edges. The aide of ihe tri-
BQgie being S in., and the edges being S, 10, and 6 in.
respectively.
4. In order to find the conteats of large excavations, the surface of the
grotmd is laid off into small equal rectangles. Stakes are driven nt the
comers of all of the rectangles, such as J, I, etc., and then the surveyor
finds (he depth of cut to be made at each of these
comers. If the rectangles are taken small their
surface may be considered plane for practical pur-
poses. The whole excavation is, therefore, divided
into a number of partial volumes, each in the form
of a truncated quadrangular prism. By computing
the volume of each of these, and adding, we are able
to obtain the whole excavation. The depth of the cut at any c
May be used 1, 2, or 3 times in the computation, depending upon the '
number of rectangles adjoining it.
The volume of the whole is obtained as follows:
Take each comer height as many times aa there are partial ai
joining it, add them all together, and multiply by one fourth of the oreK'
of a single rectangle.
In the figure here shown each rectangle is a square whose side ii
35 ft. The depths of the cuts at the various comers are as follows: A,
12 ft.; C, a ft.; D, 8 ft.; E. 8 ft.; F, 6 ft.; G, 10 ft.; H, 2 ft.;
/,6 ft.; J, 6 ft.; A, 4 ft.; L, 8 ft.; A/, 6 ft.; JV, 10 ft.; O, 8 ft.; P, fi
Q, 10 ft.; R, 12, ft. Find the number of cubic feet in the excavation-
How many cubic yards is this?
266. The Sphere.—
(a) The area of the surface of a sphere is equal to the a
of four great circles of the sphere, or A Bph«e =
288 BUSINESS MATHEMATICS
(b) The volume of a sphere is equal to the product of the
area of its surface by one third of the radius, or
(c) The volume of a spherical shell (a hollow sphere) is
equal to the volume of the outside sphere minus the volume
of the inside sphere whose radius is r or
y ipherical ■hall ^ »^A — i'Zr = JTT \R "~ r )•
WRITTEN EXERCISES
1. The surface of a tiled dome, in the form of a hemispherical surface
whose diameter is 24 ft. is made of colored tiles 1 in. sq. How many tiks
are required to make it?
2. If a cubic foot of ivory weighs 114 lb., what is the weight of an
ivory billiard ball 2 in. in diameter?
3. A hollow spherical steel shell is 1 in. thick, and its inner diameter
is 8 in. How much does it weigh if there are 490 lb. to the cu. ft.?
4. If a boiler is in the form of 4 ft. cylinder 2 ft. in diameter, with
hemispheriral ends, how many gallons will it hold?
6. There are two spheres whose diameters are 4 in. and 8 in., respec-
tively. Find (1) The area of each sphere. (2) The volume of each
sphere. (3) The relation between the areas or the volumes of these two
spheres.
6. The diameter of an arc lamp is 16 in. How many square inches C-
surface has the lamp, assuming it to be a sphere?
APPENDIX
TABLES AND FORMULAS
i. A Portion of a Bond Table and How It is Used. —
20 Year Interest Payable Semiannually
Bonds Bearing Interest at the Rate of:
Net Per
Annum
4
4.10
4i
4.20
4.25
4.30
4i
4.40
41
4.60
41
4.70
4f
4.80
41
4.90
6
5.10
5i
6.20
5i
7%
6%
6%
4J%
4%
3i%
141.03
127.36
113.68
106.84
100.00
93.16
139.32
125.76
112.20
105.42
* 98.64
91.86
138.90
125.37
111.84
105.07
98.31
91.54
137.63
124.19
110.75
104.03
97.31
90.59
136.80
123.42
110.04
103.35
96.65
89.96
135.98
122.65
109.33
102.66
96.00
89.34
134.75
121.51
108.27
101.65
95.04
88.42
134.35
121.14
107.93
101.32
94.72
88.11
132.74
119.65
106.55
100.00
93.45
86.90
131.16
118.18
105.19
98.70
92.21
85.72
130.77
117.82
104.86
98.38
91.90
85.42
129.61
116.74
103.86
97.43
90.99
84.55
128.84
116.02
103.20
96.80
90.39
83.98
128.08
115.32
102.55
96.17
89.79
83.40
126.95
114.27
101.59
95.24
88.90
82.56
126.58
113.92
101.27
94.94
88.61
82.28
125.10
112.55
100.00
93.72
87.45
81.17
123.65
111.20
98.76
92.53
86.31
80.09
123.29
110.87
98.45
92.24
86.03
79.82
122.22
109.87
97.53
91.36
85.19
79.02
121.51
109.22
96.93
90.78
84.64
78.49
3%
86.32
85.09
84.78
83.87
83.27
82.68
81.80
81.51
80.35
79.22
78.94
78.11
77.57
77.02
76.22
75.95
74.90
73.86
73.61
72.85
72.34
Illustrative Example. 1. If I wish a 5% investment, what price can I
afiford to pay for a 4§% bond maturing in 20 years?
Solution: Look in the left hand column for 5% and follow to the
right until the column headed by 4J%. The number is 93.72, therefore
I can pay as high as $93.72.
Illustrative Example 2. If a 6% bond maturing in 20 years costs
114.27, how much will this net the buyer?
19 289
290 APPENDIX
Solution: Look in the 6% column and follow down to 114^, then
follow across to the left until the left hand column is found, and you wiU
find 4i, therefore it will net 4}%.
The tables as used by the bond houses are much more extensive as to
the number of years and per cents.
WRITTEN EXERCISES
1. From the above table find the rate on the investment on bonds
maturing in 20 years if bought as follows:
(a) 5% bonds bought at 103.20
(b) 6% • " " " 109.87
(c) 3J% " " " 85.42
2. Find the price at which bonds maturing in 20 years can be pur-
chased to produce the following:
(a) 4i% bond to yield 5i%
(b) 7% " " " ^%
(c) 3i% " " " 5%
3. What is the rate on the investment on bonds which mature in 20
yr., if bought under the following conditions:
(a) 7% bonds bought at 134.35
(b)3% " " " 73.86
(c) 5% " " " 110.04
{d)^% " * '* 94.94
4. Find the price at which bonds maturing in 20 yr. can be bought,
to produce the following:
(a) 5% bond to yield 4.20%
(b) SWo "
(<
" H%
(c) 7% "
i(
" 4i%
(d) 6% "
H
" 4.9%
(e) 5% "
i<
" 4.1%
(f) 4% "
«
" 4i%
n
TABLES AND FORMULAS
291 1
2.
Table Showing Powers and Roots of Some Numbers.—
1
1
SOUABE
CUIIK
^,.....
CU»H
1
N
•'-
Sqvare
CUBI.
No.
Sqi;akk
CUBB
Rooi
"7
^
,
,
I
11
2601
32651
7 141
3 7084
L259()
3:7325
9
27
4NS77
3,7563
16
2.
l'--,l>U
S«
2B16
57464
7.3485
125
2.2361
1.710U
66375
7.416
38
BS
3.8259
49
2!645S
i:ui2U
324B
185193
7:549s
M
sia
S8
195112
7.6 1, iH
720
3.
S9
3:8930
10
100
3.1623
2'.]544
3800
218000
7:7480
121
1331
3.3166
2.2240
«1
3731
226981
7.8102
3.9363
238328
3.9579
13
3!6056
2:3513
3969
250047
7:9373
196
2744
3.7417
2,41m
4096
3375
»
274825
B:D623
4:0207
m
g
256
4096
4
2,5198
M
4356
38749H
8.1240
4.0412
1
■a8B
4!l231
300763
^
4.Z426
2:6207
314432
4:0s 17
1
1
4.3580
2.6684
69
4761
328509
83066
4.1016
4U0
4900
SI
441
9261
4.6826
2,7589
71
5041
357911
8,4261
4.1408
484
1064M
4.seu4
6184
373248
M.4S63
539
12187
4,7958
2:8439
Tl
389017
8,5440
4!l793
13824
4,8990
5478
405224
8.6023
4-1BSa
M
625
15625
S(
676
17576
S.09B0
2.9625
TE
5776
438976
8.717S
4,2358
19683
5929
456533
784
21951
474552
8:8318
4:2727
*'.
5;3852
3:0723
T9
6241
493039
8,8882
4.2908
30
2700(
e.4772
3.1072
60
961
S3144I
9,
4.3267
9!
32768
S:6fi89
3:i748
83
6724
551368
9,0554
4,3445
1089
35937
5.7446
3,2075
>S
571789
39304
7056
911652
4:3795
St
1225
42875
s!9iei
3:2711
7225
61412,1
9,2195
4.396B
ai
298
46651
6.
66
7396
638056
9,2736
6.0828
75B9
658503
9.3274
SI
5487:
6.1644
3:3620
68
7744
881472
S,38U8
4:448U
K
521
BO
704969
64b0(
6^3246
3:4200
ta
8100
739000
9:4868
414814
M
1681
58921
6.4031
3, 4S2
8281
753571
9.5394
4.4078
«t
1764
ta
795o;
e'.Soli
n
8649
804357
*t
1936
8518;
6.6332
3: 303
94
8836
830584
0:8054
4:3468
u
2025
91
857376
M
97336
6.7823
IB
2209
10383;
6.8557
3,6088
97
9409
9:S489
4:5947
41
2304
BB
, 96U4
941192
9,8995
A9
99
B 70299
M
2500
125000
7:0711
9^8
1,7725
4:6416
1.4648
IIk. .^^^
292
APPENDIX
3. Table of Decimal Eqtiivalents of Some of the Fractions
of 1 Inch. —
Fraction of
One Inch
Decimal
Equivalent
Fraction of
One Inch
Decimal
Equivalent
6\
.01563
.03125
.0625
i
.125
.25
4. Table of Wages by the Day — 8 Hours to a Day.—
Hours
$2
$2.25
$2.50
$2.75
$3.00
$3.25
$3.50
$4.00
$4.50
$5.00
1
.25
.28
.31
.34
.38
.41
.44
.50
.56
.63
2
.50
.56
.63
.69
.iO
.81
.88
1.00
1.13
1.25
3
.75
.84
.94
1.03
1.13
1.22
1.31
1.50
1.69
1.88
4
1.00
1.13
1.25
1.38
1.50
1.63
1.75
2.00
2.25
2.50
5
1.25
1.41
1.56
1.72
1.88
2.03
2.19
2.50
2.81
3.13
6
1.50
1.69
1.88
2.06
2.25
2.44
2.63
3.00
3.38
3.75
1.75
1.97
2.19
2.41
2.63
2.84
3.06
3.50
3.94
4.38
8
2.00
2.25
2.50
2.75
3.00
3.25
3.50
4.00
4.50
5.00
Explanation: At the rate of S3.25 per day, 5 hours' wages will be
S2.03.
Similar tables may be constructed for any commercial house at their
prevaihng wages.
5. Table of Formulas for Use in Commercial Work. —
1.
2.
3.
4.
= Prt.
_ I_
rt
Pt (1%)
Pr
= a + (« — 1) d
principle; / = interest; r
rate ; t = time
last term of arith. ser/if
1st term
number of terms
common difference
TABLES AND FOR!^IULAS
293
Co
a)
T
A
2 12 a + (» — 1) <fl
a (r»— 1) rl~ a
or
r— 1
r— 1
^O « ""fl*
ylA = V^ (5 - a) (5 - b) (5 — c)
\r
\ .5236 (.4 — G)
4 9.000
P (1 + r)"
5 (/2** - 1)
P. y. =
S_ R^— I
i?« ^ i? - 1
P. V. =
/^~ 1
/?^ + <7 R—i
L8. »
log (i + 4^)
log/?
19. P. V.
-( - ^ \
sum of arith. series
» ratio in a geom. series
s sum of a geom. series
= area
s semi-major axis of an ellipse
» semi-minor axis of an ellipse
a. b, and c are the sides of
a -\-b -\-c
triangle, 5 «
.
c « circumference
d = diameter of the circle
r = radius of the circle
D ■» diameter of a balloon
.4 ■» weight in lb. of 1 cu. ft. of air
G = weight in lb. of 1 cu. ft. of gas
in balloon
\V = weight to be raised, including
balloon
T = number of seconds required
for a bomb to fall from an
aeroplane
= height in feet
= amount; P = principal; r «
rate; n = no. of years
= amount of the principal for r
years
= amount of $1 for 1 yr. at the
given rate
= interest on $1 for 1 yr.
» sum to be set aside annually
« present value of an annual
pension
R, and n as in Formula 15
■« number of yr. before pension
begins
» number of yr. it is to be paid
R, and P. V. as in Formula 16
= no. of yr. premium should be
paid in order that Life Ins.
Co. shall sustain no loss
- 1 +r
■» amount to be paid immediately
after last premium
5 " amount of premium paid
annually
P. V. — present value
A — amount of annual x>ension
r « rate
H
A
A
R
r
S
'P.V.
15.
Q
S,
n
R
A
ihH
:hV
,?\.
aa.
'>7
a:u
34.
36.
3vS.
30.
APPENDIX
I
i»«i
>
>!vl +r^"- l|
r
rR
aA. x
I
Pb
R
C
H
s
r
V
P
R
P
S
Q
X
b
a
population at end
population at beginning
1 + r ■" rate oi increase <rf
population
number of dollars in debt
number of years
sum set aside annually
rate of interest
total value
premium paid each year
1 +r
price of a bond that has n jt.
to run
rate % it bears
face of bond (usually $100 or
$1,000)
current rate of interest
rate of interest yield
base
altitude
^1
6«i
a or - 1,6 + 6 )
2 2
X yiTVf^\x\Ar fi^wrv^ - sum of .4's of triangles, trapezoids, etc.
// = hypotenuse of a right triang'®
//
m
H
-Vu^+6^
r\e
3A
2a
32. r ^rcv^unjiuUr soHd^
V vpyramiiO
uN'
- \ Ba
»» altitude of a riijht triaoS'
when b is base
i A » no. of ft. object is above surf^'®
~ 'I of earth
\m » no. of mi. object can be secf-
H » hypotenuse of a 30-60. ri^^
triangle
«i » arm opposite the 30^ angle
la » 1st edge. 6 »= 2d edge.
\c = 3d edge, V-volimie
B » area of base and a = altiti^*- '
of pyramid
Lateral artra vP>*TanTiid) — \ slant height X P^ P/» = perimeter of base
V ^oone^
L. A . I, cone ^
37. V <.prismatoid)
- i cMfi + ft + 4 A/) '
.4 (sphere)
V (sphere)
40. V (spherical shell)
\ area k>{ base X altitude
i slant height X circumference of base
a =alt.;B =areaoflo^
base
b = area of upper base
M = area of mid-section
4 T /? ^ /? = radius ot the sphere
iT Ri
R = radius of outside <*
sheU
[ r « radius of inside of sh^
^ ir{Ri-ri)
TABLES AND FORMULAS
295
6. Abbreviations Used in Commercial Transactions.
acre
3ct.ora c. . .account
?t agent
nt amount
IS answer
pr April
^s account sales
ug August
V average
al balance
g bag, bags
bl. or bl barrel
dl bundle; bundles
k bank; book
I bale; bales
kt basket
/L bill of lading
/O back order
Dt bought
u bushel
K box; boxes
c, b cash; cash book
ish cashier
ct cent
I card ; cord
ir carton
% centigram
1 chain; chains; chest
ig charge
i. f carriage and insur-
ance free
c: check
XI centimeter
xii commercial
1 can
o company ; county
. O. D collect on delivery
>11 collection
^m commission
consg't coQsignmeDt
<r crate: credit; credi-
tor
cs case: cases
csk cask
cu. f t cubic feel
cu. in cubic inch
cu. vd cubic vard
cwt hundredweight
d pence
da dav
Dec I>ecember
dep't department
dft dnift
disc discoimt
do ditto
doz.; dz dozen
Dr debtor; debit; doc-
tor
E East
ea each
e. g for example
e. o. e errors and omis-
sions excepted
etc and so forth
ex express
exch exchange
exp expense
far farthing
Feb February
f . o. b free on board
frt freight
f , fr franc
ft foot
gal gallon
gi gill
gr grain
gro -r. .gross
guar ^v3Ltxxvw\\sifc
296
APPENDIX
hf half
hf . cht half chest
hhd hogshead
hr hour
i. c that is
in inch; inches
ins insurance
inst instant; the present
month
int interest
1.; inv invoice
inv't inventory
Jan January
kg keg; kegs
1 link; links
lb pound ; pounds
1. p list price
Mar March
mdjic merchandise
Messrs Gentlemen; Sirs
mi mile; miles
min minute; minutes
mo month; months
Mr Mister
Mrs Mistress
X .North
no number
Nov November
Oct October
o (I on demand
O. K correct
oz ounce; ounces
P page
pay't payment
pi' piece; pieces
|hI paid
|H'i by the; by
»K I \ A'Ut iK»r centum; by the
hundred
pfd prefened
pk peck; peck
pkg package
PP pages
pr pair; pairs
pt pint; pints
pwt pennyweight
qr quire
qt quart
rd rod
rec'd received
rm ream
Rm. (or M.) . .Reichsmark; Mark
s shilling; shillings
S South
sec second
Sept September
set settlement
ship shipment
shipt shipped
sig signature; signed
sq. ch square chain
sq. ft square foot
sq. mi square mile
sq. rd square rod
sq. yd square yard
T ton
tb tub
Tp township
tr transfer
treas treasurer
ult last month
via by way of
viz namely; to wit
vol volume
wk w^eek
wt weight
yd yard; yards
yr year; years
TABLES AND FORMULAS
297
e of Symbols.
. . . account M thousand
. . .account sales ** inch; inches; sec-
. . .addition ends
. . . aggregation > greater than
. . .and < less than
. . . and so on X multipUcation
. . .at; to ii number if written
...care of before a figure;
. . .cent; cents pounds if written
. . . check mark after a figure
. . .degree 1 ' one and one fourth
. . . division % per cent
. . .dollar; dollars £ pounds sterling
. . .equal; equals ".' since
. . . .foot; feet; minutes — subtraction
hundred .* therefore
r
INDEX ^^^^^H
IIS, taUe of. 29S
Banks, ^H
1.90
den»d.IlS ^H
|Bia««.72
iDtenstonaccotints. liS-136 ^^H
Iteiestom. llS-126
postal savings. 124 ^^M
saviDES. 118 ^H
ite ntimbere. 255
interest, 118-126 ^^H
bods. 172
B:^ plaae. de&wd, 286 ^^M
Sned,2Be
Between dates, interest calcula- ]
tiona. 42 1
Bills of exchange. 93
Bais of lading. 100 1
p»
Bonds,
interest to matiinty. 236
tables, 289 '
Bonus wage systems, 27, 29
fcbylDgaiiUiiiis,227-
Building and loan associations,
r
127-134
■hie, 228-230
^s,212
distribution of profits, 133
B weights, 2.j2
shares, 129
de defined, 267
I
shares,
series plan, 128 i
n-ithdrawal value, 128
b:!^
'
r
Cash surrender policies, 68 1
Checking methods, 172-190 i
279
Circle, 1
,272
arc, 267 l
470.278
are;, 271
progressions.
lilS-
circumference o£ 270
defined, 267 J
diameter, 267 M
of ascertaining,
191-
perimeter. 267 ^^^^^^M
Wntage, 5
Circumference circle. 270 ^^^^^^^H
1, table of weifihts,
Commercial bills. 100 ^^^^^^H
Commissions, ^^^^^^^^^1
1
^^^^^^^M
300
INDEX
Commodities, table of weights and
measures, 252
Compound interest, calculation by
logarithms, 223
Cone, 285
Conversion tables, currency, 91-
92. 250-253
Cost of sdUling, 13-21
Cube roots,
computed by slide rule, 246
table showing powers and roots,
291
Cubic measures,
Enghsh, 251
metric, 259
Currency, table of values, 91-92,
253
Cylinder, volume of, 283
Daily balances, interest, 124
Day-rate wage system, 23
Decimals, table of equivalents,
292
Denominate numbers, 250-261
addition, 255
division, 256
multiplication, 256
reducing to higher, 254
reducing to lower, 253
subtraction, 255
table showing powers and roots,
291
Deposits, banks, 46
Depreciation, 50-55
computation, 50
methods,
decreasing rate on original
value, 52
fixed rate on decreasing value,
53
fixed rate on original value,
52
straight line, 51
Differential rate wage system, 25
Division,
computed by slide rule, 242
denominate numbers, 256
life insurance, 67
short methods, 187
Dozen measures, 253
Drafts, 83-90
Dry measures, table, 253
Efficiency wage system, 29
Emerson wage system, 29
English weights and measures,
250-253
compared to metric, 259, 261
Exchange,
domestic, 77
acceptances, 90
bills of exchange, 93
bills of lading, 100
drafts, 83-90
express money orders, 79
methods of payment, 77
postal money orders, 78, 89
telegraph money orders, 81
terms, 85, 86
foreign, 85, 86
bills of exchange, 93
currency values, 91
commercial bills, 100
conversion tables, 91-92,250-
253
bankers' bills, 93
bills of lading, 100
letters of credit, 96
par of exchange, 92
postal money orders, 98
quotations, 94
rates of, 94
travelers* checks, 98
Expenses, business problem, 11
Expenses, selling, 13-17
Exponents, logarithms, 203-205
Express money orders, 79
Fire insurance, 57-64
Formulas,
table of, 292-294
Foreign exchange, 91-100 (^
also " Exchange, foreign")
Fractions,
short methods, 189
table of decimal equivalents,
292
Fraternal insurance, 70-72
^H
INDEX 301 ^
G
Insurance— Ciwih'B ued
liie— Continued
^metric progressions, 200-202
loans on policies, 68
Geometry, 262-28S
occupation of insured, 72, 75
iwds (See also "Sales")
paid-up policies, 68
marking up, 13
payments to beneficiaries, 69
iraphic presenlatioii, 135-171
policies. 64
Forms, 136
policies lapsed, 70
drcles, 138
policies, scope of, 73
comparisoDS,
premiums. 66
curves, 153
risks, 74
involving time, 150
simple, 142
occupational hazards, 73
construction of, 137
overinsurance, 75
curves, 145. 156-160
Interest. 34-^9
frequency charts, 161
accounts. 118-128
maps, 160
bank, 118-126
objects of, 135
bond tables, 289
rectangles, 14U
bonds. 236
compound. 43-19
H
annual deposits, 4%
computation by loEarithms
^alsey-Rowan wage system, 28
223
azards, occupational, 73
sinking funds, 48
igher number, reducing t
,254
table, 44
lypotenuse defined, 2tH5
I
dailv bank balances, 124
defined, 34
for months, 35
for years, 35
acometas, 111-117
postal savings bank accounling,
iheritance taxes, lOS
124
federal, 110
savings bank accounts, 118-136
state, 108-110
short methods of computing, SB
asurance, 56-76
simple,
between dates, 42
^57-64
by time, 35, 142
fceomputation of premiu
m, 59
Interpolation, 209
■iKilicies, 59
Inventories, 9
rshort rate table, 61
L
fraternal, 70-72
industrial, 75
Land measures.
kinds of, 56
English, 250, 251
life, C4-76
metric. 259
^ we of insured, 73
Letters of credit. 96
^Jumuitieij, calcula tion, 227-235
Life insurance, 64-76
annuities calculation, 227-235
^R^ surrender policies
68
Life tables, 233 _^
^BMnputation of premiums, 66
Ljne, perpendicular of angle, 26^* 1
EngUsh,250
HT^nies. 67
metric, 258-261
302
INDEX
Liquid measures,
English, 252
metric, 259
Loan associations (See ''Building
and loan associations")
Loans, on insurance policies, 68
Logarithms,
annuities, calculated by, 227-235
antilogarithms, 212
applications, 223-237
bond interest to maturity, 236
Briggsian or common system,
207
characteristic defined, 206
compound interest calculations,
223
exponents, 203-205
interpolation, 209
mantissa defined, 206
notation, 206
sinking fund calculations, 225
systems, 205-209
systems with same base, 205
tables, 214-217
explanation, 209
proportionate parts, 211
terms, 206
Loss or gain (See "Profit and loss
statements")
Lower number, reducing to, 253
M
Measures — Continued
metric, 257, 258
of time, 253
quantity, 252-253
sailors, 251
square,
English, 251
metric, 258
surveyors long or land,
English, 251
metric, 259
surveyors square, 251
tables,
EngUsh, 250-253
metric, 258-261
Metric system, 257, 258
Money (See "Currency")
Money orders, 78^2
Months, interest calculations for,
35
Mortgage tax. 111
Multiplication,
computed by slide rule, 241
denominate niunbers, 256
short methods, 177-186
N
Notation logarithms, 206
Numbers (See "Denominate num-
bers")
Mantissa, defined, 206
Marine measures, 251
Marking-up goods, 13
Pleasures,
angular, 251
capacity,
English, 252
metric, 259
cubic,
English, 251
metric, 259
currency, 253
dry, 253
linear,
English, 25('
metric, 258
liquid,
English, 252
metric, 259
Occupations,
hazards of, 73
of insured, 72, 75
Overhead, computing sales cost,
13
Parallelogram,
area of, 269
defined, 265
Pay-roll slips, 32
Percentage,
av'erage sales, 5
increase and decrease of sales, 4
profit and loss, 1
profit on sales, 20
INDEX
303
Perimeter of a circle defined,
267
Piecework wage system, 24
Plane measurement,
altitude, 266
angles, 262-264
arc of a circle, 267
area, 267
base plane, 266
circles, 267
definitions, 262-267
diameter of circle, 267
quadrilaterals, 265
parallelograms, 265
perimeter of circle, 267
polygons, 265
radius of circle, 267
rectangles, 265
square, 265
surfaces, 264
triangle, 265, 266
Policies,
fire insurance, 59
life insurance, 64
Polygons, defined, 265
Postal money orders, 78, 89, 98
Postal savings banks, 124
Powers, table showing, 291
Practical measurements, 262-287
(See also "Plane measure-
ments; solids")
Premivmi rate wage system, 28
Premiums,
fire insurance, 59
life insurance, 66
Price (See "Selling price")
Prism, 281
volume of, 282
Prismatoid, 286-287
Profit,
based on sale, 13-21
building and loan associations,
133
calculation for goods at resale
prices, 13-21
computation, building and loan
associations, 128
distribution, 133
net, 17
per cent on sales, 20
shares, building and loan asso-
ciations, 128
Profit and loss statements, 1-12
comparative percentages, 3
comparative records, 10-11
computation, 8
inventories, opening of, 9
Progressions,
arithmetic, 198-200
geometric, 200-202
Property,
tax computations, 103-108
taxation of, 103-108
Proportions of 30^-60° triangle,
277
Protractor, defined, 263
Pyramids, 284-285
Quadrilaterals, defined, 265
Quantity measures, 251-253
Quotations, foreign exchange, 94
Radius of circle, defined, 267
Records,
profit and loss, 10-1 1
returned goods, 5
sales, 2-7
Rectangle defined, 265
Returned goods record, 5
Right angle defined, 264
Right triangle,
defined, 265
proportions of, 275-277
Roots,
cube, 246
square, 243
table showing, 291
Sales,
average percentage, 5
cost of selling, 13-21
cumulative record, 7
daily records, 3
expenses, 13-17
marking-up goods, 13
monthly records, 3
3(A
INDEX
overhead, expense, 13
per cent of pro&t on, 20
percentage of increase and de-
crease, 4
profit and loss statements, 1-12
returned goods record, 5
selling price, 13
tabulated records, 2
Savings banks, 1 18
interest on accounts, 1 18-126
postal, 124
S^ng price, calculation of, 13-21
Shares,
profit on, building and loan
associations, 128
seriss plan, building and loan
associations, 128
withdrawal value, building and
loan associations, 128
Short methods, 172-190
arldition, 172
division, 187
fractions, 189
interest, 36
multiplication, 177-186
subtraction, 174-177
Sinking funds,
caU:ulation.s by logarithms, 225
calculation by compound inter-
est, 4H
Slide rul(!, 238-249
(!ul>(;s and cube roots by, 246
description, 239
division by, 242
history and use, 238
multiplication by, 241
reading' of, 240
scpian^ root by, 243
Solids, US I 1>S8
cone, liSf) 1»S()
cylinder, L»S3
pHstUMtoid, 2S()-2S7
pristus, JSl L>Sl>
pv tain ids, 2S1 12S5
sphere. 2S7 L\S8
^;phet•t^ :!sv l»ss
S(|M.»t(\ delinetl, '2(\')
Square root, 272-275
oompated by slide ride, 243
table showing powers and roots.
291
Subtraction,
short methods, 174-177
denominate numbers, 255
Surfaces, -
area of, 279
de6ned,26i
Surveyors long measure,
Enghsh, 251
metric, 258
Surveyors square measure, 251
Symbols, table of, 297
I '
.Min.ne t\ieasufes,
l'nr.'»'i». -d
meh\e, 1\*»S
snt'\ evois, l.\")l
Tables,
abbreviations, 295-296
angular measures, 251
Apothecaries weights, 252
avoirdupois weights, 252
bond, 289
comparative weights, 252
computing net profit, 18
computing profit, 20
cubic measures,
English, 251
metric, 259
currency values, 253
decimal equivalents of some
of the fractions of 1 inch,
292
dry measures, 253
finding selling price, 19
fire insurance short rate, 62
formulas, 292-294
interest, 45, 47
life, 233
linear measures,
English, 250
metric, 258
liquid measures,
English, 252
metric, 259
logarithms, 214-217
measures, English, 25Q-253
metric, 258-261
metric and English values com-
pared, 259-261
^ — ^ -^
INDEX 305
Tables — Cont inued
powers and roots. 291
proportions of nght, ZTo-Jii
sailoTS measures, 251
right. 265
square measures.
Troy wdght, 252
English, 251
metric. 258
surveyors long or land measure.
Valuation (See "Depredation")
English. 250
Value, shares, building and loan
metric, 2o»
associations, 128
surveyors square measure, 251
Volume of cone, 285
symbols, 297
Volume of cylinder, 283
tax tables, 106
Volume of prismatoid, 286-287
time measures. 253
Volume of pyramid, 284-285
Troy weight, 252
wages for S-hour day. 292
Volume of s^ere, 288
weights, 250-253
W
English, 250-253
metric, 258-261
Wage payments.
of commodities. 252
bonus, 29
Task and bonus v.-age system, 27
Taxes. 102-117 (See also "Income
currency memorandum, 33
tax," 'Tiiheritance tax,"
day-rate system, 23
differential rate system, 25
'■Mortgage tax")
assessments, sUte methods.
Emerson efficiency system, 29
103
Halsey-Rowan premiimi ratei
defined. 102
28
proper^', computation of. 103-
pay-roll sUps, 32
108
piecework system, 24
purpose of, 102
table for 8-hour day, 292
Tdesraphic money orders. 81
task and bonus system, 27
Time,
Weights,
interest calculation for, 35. 42
apothecaries, 252
table of measures. 253
avoirdupois, 252
Trapezoid, area of, 2/2
commodities, 252
Travelers' check, 98
comparative, 252
Triangle,
EngHsh, 2,50-253
area of, 270, 278
metric, 258-261
defined, 265
tables, 2,50-253
equilateral, 266
Troy, 252
hypotenuse. 266
isoceles, 266
T
aO'-eO', 266
proportions of. 277
Years, interest caleulatioti for, 35
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