# Full text of "Business Mathematics: A Textbook"

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About Google Book Search Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web at |http : //books . google . com/| 1 06642837 > t^- \ <J' "x^-C *• 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#^ f»-^ ^.r.. 'J -li-rt '-^an Tie- ill. C/imgaZistaa if Tiaes jv- x j5s il -n;i r arr. l.T tllIs ml ine iailar* ^ » j^ - - — " f ^ ^ » • • r X. X < - _* A / * • ^ X ^ 4 • ^ / ^ • ^ ^ • ^ s •«» X ■< ' ^ i» « ^^ • ^■^ ^ ^^ % '. %.'f. ^. V, ?. -//. \ : • » ' $: : :o: r..3i:..:i: flJti.JtiO fI.3iQL00 />'/ '/» /Jf. > A ;? '/^, '>. V jT. :i.»: uji. :ij 3CC.JC0 3.ooaoo //, i'. >//, ^ y. < '//. i.: ■.»', c.: -.ii: 43;. :*: JW.OW 4.3.xiLao 1', 1 'IV, <> '/, t '//, ^^ , X t-: >:•- ■SiX. >} -WlJ-XV «.twai» 'ATRITTfcN EXERCISES 1, »x:;.^;'i'» .'S^.^MX) $20,rXjo - $0,000 -h $600 112. How the Tax Rate Is Determined.— The assessed viiluation of iIm! proiMTty is found from the assessors' lists. TAXES 107 The amount of money needed, divided by the assessed [uation of the property, gives the tax rate to be levied, le following type solution shows the method and a form 'Ht finding the tax rate. ■Statb Tax: State budget S2.500.000 Assessed value of property in state $1,250,000,000 S2.500.000 -^ $1,250,000,000 .20% State rate County Tax: County budget Assessed value of property in county $40,000 -^ City Tax: City budget Assessed value of property in city $16,000 -^- $40,000 $5,000,000 $5,000,000 $16,000 $1,280,000 $1,280,000 .80% County rate 1.25% City rate 2.25% Total rate ORAL EXERCISE 1. State some reasons why the tax rate will vary in different cities. WRITTEN EXERCISES 1. Find the total tax rate on the following: State budget $ 2,500,000 State assessed valuation 1,500,000,000 County budget County assessed valuation. City budget City assessed valuation. School budget district Assessed valuation, school 28,000 4,500,000 15,000 1,500,000 35,000 1,500,000 state rate county rate city rate school rate total rate 108 atb^lIXJIi:^ MJOHEIIATICS %. Ih. s^ «ngR:MiL •BiErr t^ff^ tSKE iai» B :» Camugr *** .39 Cicir * 1j075 SdtodI '^.. l.« 3^ If pcQ^«viprQ<^:sas«R«i;a |<^E&»RalTalDe and Hortoa owns the SiCMno Sto*. 7,500 25,500 Find ld$ to<ai t;UL 4. Find the aoiMKmt o£ lii^ tax for the state. Tbe coanty. The city. 5. Wbatperopntof tlirt<i>taItaxin£sera&e2ai^pliestoeachdiv^^ lis. Inheritance Taxes. — An inheritance tax is a tax on the property of a deceased person. This is assessed in the greater number of the states but varies in different states, and a person interested in this subject should look up the law for each state. The following rates are for New Jersey and New York. New Jersey. To husband or wife, child, adopted child or its issue, or lineal descendant, the rates are: 1% from $ 5,000 to $ 50,000 1{% " 50,000 " 150,000 2% " 150,000 " 250,000 3% above 250,000 $5,000 exempt To parents, brother 'aw, and daughter-in- law, the rates are: TAXES 109 2% from S 5,000 to S 50,000 2J% " 50,000 " 150,000 3% " 150.000 " 250,000 4% above 250,000 All others 5% $5,000 exempt Preferred obligations: 1. Judgments 2. Funeral expenses 3. Medical expenses of last sickness New York. If inheritance is received by father, mother, usbandy wife or child, adopted child: Exemption to amount of $5,000 1% on amounts up to $25,000 2% on next $75,000 3% on next $100,000 4% upon all additional sums If inheritance is received by brother, sister, wife or widow f son, or husband of daughter: Exemption to the amount of $500 2% on amounts up to $25,000 3% on next $75,000 4% on the next $100,000 5% thereafter If inherit3,nce is received by any person or corporation ►ther than those above named : Exemption to the amount of $500 5% on amounts up to $25,000 6% on the next $75,000 7% on the next $100,000 8% thereafter 110 BUSINESS MATHEMATICS Preferred obligations: 1. Funeral and administration expenses 2. Debts preferred under United States laws 3. Taxes 4. Judgments and decrees 114. Federal Inheritance Tax. — The federal tax is posed on the estate as a whole, not on the shares oi several legatees, irrespective of the beneficiaries to decedent. $50,000 of each estate is exempt from tax. The tslU the excess are as follows: Not exceeding $ 50,000 1% From $ 50,000 to 150,000 2% 150,000 " 250,000 3% 250,000 " 450,000 4% 450,000 " 750,000 6% 750,000 " 1,000,000 8% " 1,000,000 " 1,500,000 10% 1,500,000 " 2,000,000 12% " 2,000,000 " 3,000,000 14% " 3,000,000 " 4,000,000 16% " 4,000,000 " 5,000,000 18% *' 5,000,000 " 8,000,000 20% " 8,000,000 " 10,000,000 22% Exceeding 10,000,000 25% WRITTEN EXERCISES 1. Find the amount of inheritance tax to be paid to each, the st New Jersey and the United States, on an estate of $600,00C* wil follows: $200,000 to decedent's wife; $100,000 to each of three chi balance divided eciiially among the mother, a brother, and two sist 2. What would the state and federal inheritance tax amount to ii York? 3. Find the amount of inheritance tax to be deducted from each following amounts willed by a decedent of New Jersey; TAXES 111 $40,000 to the wife 25,000 '' a son 25,000 '' a sister 10,000 '' the father 6,000 " a brother 4. Apply Exercise 3 to New York. 116. Mortgage Tax Law of New York. — All mortgages upon real estate in New York can be made tax-exempt by paying the recording tax of ^ of 1%. This will permanently exempt the mortgage from taxation. WRITTEN EXERCISES 1* Find the recording tax on a mortgage of $56,000. 2. What is the interest for the first year and the recording tax on a mortgage of $4,800? 116. Income Tax. — The present income tax law was en- acted on account of the increased expenses of the world war. The law is here given, with some of the features. The tax shall begin at 4%, on incomes above the amount exempted up to $4,000 normal tax, and 8% normal tax on all incomes in excess of $4,000 with the exemption allowed. Ill the case of a head of a family, married man or woman, an exemption of $2,500 for incomes up to $5,000, and $2,000 for those over $5,000, and $400 for each dependent child under ^8 is allowed. The exemption for a single man or woman • ^8 $1,000. Exemptions are allowed state and municipal paid employees. Only one deduction is allowed from the ^gregate income of both husband and wife living together. The gross income includes gains, profits, and income from ^^y source whatever, except the interest on some United .States government bonds or bonds of the political sub- ^ divisions of the United States. 112 BUSINESS MATHEMATICS The net income is obtained by deducting from the gross income all necessary expenses actually paid in carrying on the business, such as interest paid on indebtedness; loss from bad debts, if charged off; taxes; fire losses not covered by insurance or otherwise; and a reasonable depreciation on the value of the property. Personal, household, and living expenses are not included. Illustrative Example. A man living with his wife has a net income of $5,000. How much income tax is he required to pay? Solution: $5,000 net income 2,500 exemption $2,500 taxable income .04 normal rate $ 100 income tax 117. Super-Tax or Surtax. — An additional tax is assessed on large incomes in excess of a certain amount. The table below shows the additional rates charged on the portions of net income above certain stated amounts (not deduct- ing the $1,000 or $2,500 or $2,000 exemption applying to the normal tax.) Cumulative Rate Amounts Exceeds $ 6,000 but does not exceed $ 8,000 1% 20 8,000 (( - (( ( 10,000 1 40 10,000 (( 11 i 12,000 2 80 12,000 (I K ( 14,000 3 140 14,000 (( (( ( 16,000 4 220 16,000 (( (I i 18,000 5 320 18,000 It il ( 20,000 6 440 20,000 t • ( ( ( 22,000 8 600 22,000 (. (( ( 24,000 9 780 24,000 (( a ( 26,000 10 980 26,000 (( (( ( 28,000 11 1,200 28,000 (( (( ( 30,000 12 1,440 TAXES 113 Exceeds $ 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 ^