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1 06642837 



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*• ii*tt 



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'- 

O ..H -I r- 91 





?1 














>^ 


X 


CO 




r-t 


r?5 


rj- 


X 


(« 








1-H 


t-H 


M 


W 



X 



5^ -1* r* M «0 — c: 

—•—•MM 



















^ 






•»- 


■«• 


•«< 


cv 




e 


s 


»0 


X 


C5 


t^ 


M 


^ 






t-H 


t-« 


M 


CO 


^ 






•»- 


^ 


•«• 


«»• 




Ok 

M 


•-« 


o 


Ci 


t 


X 


CO 


— 








t-H 


1— < 


M 


CO 


vO 






















♦I- 


■Av 


•«• 


«»- 




00 


M 


I'. 


w 


•0 


a 


t 


M 


M 






1— < 


•-H 


1— < 


CI 


CO 



«!• a^n .4n •:>• 

CO Vl ^ O C "O CO 
— ' — • fl "TJ CO 





















«H >«p| 


•«n 


c»- 




•^ 


Ol 


CI 1* 


^4 


<o -t< 






f— 1 ^H 


CI 


CI CO 


• W 


1 




«r> a^n 


■4n 


«*• 


V4 


»j: 


^ 


CO X 


^1 

•  


I* 10 




•-H 


•— < 1— < 


CJ 


CI CO 




















♦♦. >«pi 


•«• 


Cl- 




o 


f-M 


f Ci 


CO 


cr ?c 




*"■ 


-^ — ' 


CI 


CI CO 


1 ^^ 


















«*> -«• 


••• 


•;»• 




!>. 


M 


lO c 


V4« 


c: r>. 


- 


^-i 


— M 


CI 


CJ CO 


«^ 
















«*> -«• 


m^% 


Tt- 




X 


^? 


« — 


»-0 


C X 






-< M 


M 


CO CO 








♦»- — • 


••«» 


et. 




Si 


'I' 


!>. M 


e 


— CJ 


1 • 




^-i 


— M 


M 


CO CO 








*♦- — n 


.«« 


■.•l» 


U 


^ 


1.-? 


X CO 


t>. 


CI o 


r"^ 


t-H 


« M 


M 


CO "^ 



'\ 



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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j5s il -n;i r arr. 



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• 




r 




X. 




X 




< 








- 




_* 




A 






/ 


* 


• 


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X 


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4 


• 


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• 


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• 


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s 


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' 


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'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 $ 



n 

tt 

tt 

({ 

n 

it 

(t 

it 

it 

tt 

It 

tt 

tt 

It 

tt 

tt 

tt 

tt 

tt 

tt 

tt 

tt 

tt 

tt 

It 

It 

It 

tt 

It 

It 

II 

tt 

tt 

tt 

tt 

t( 

li 

(( 



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 



^