Mathematics OF Accounting and Finance By SEYMOUR WALTON, A.B., C.P.A. Dean of the Walton School of Commerce, Chicago And H. A. FINNEY, Ph.B., C.P.A. Professor of Accounting, Northwestern University Chicago Third Printing NEW YORK THE RONALD PRESS COMPANY 1922 Copyright, 1921, by The Ronald Press Company All Rights Reserved PREFACE This book has been prepared as a manual of business calcu- lation, for use in connection with the problems which continually arise in the course of business as conducted today. In deciding what subjects to include the authors have been guided throughout by the desire to make the book directly useful in business offices, particularly to persons working in accounting and in the various lines of finance. There are many school texts on commercial arithmetic, but they are necessarily too rudi- mentary to be of much aid to readers of mature experience. No space has been given here to a consideration of the elementary and fundamental processes of mathematics. The effort has been, instead, to present material of more advanced nature which has not been generally available. This material falls into three general classes. The earlier chapters explain in considerable detail a number of short processes and practical suggestions that may be applied in routine computations of any sort. Particular attention has been given to the matter of adequate checks upon calculations. The central portion of the book treats of the special appli- cations of arithmetical principles and short methods to the problems of individual lines of business. In the final chapters an attempt has been made to explain, in simple terms, convenient ways of using logarithmic and ac- tuarial methods in the solution of business problems relating to compound interest, investments, annuities, bond discount and premium, effective bond rates, leaseholds and depreciation. During the last months of his life, Mr. Walton devoted most of his strength to this book, and it is a source of profound personal iv PREFACE regret that he did not live to see its publication. Few men were his equal as an accountant, a teacher and a writer. None surpassed him as a friend. H. A. Finney. EvANSTON, III. August 15, 1921 CONTENTS Chapter Page I Short Methods and Practical Suggestions ... 3 Short Methods of Computation Balancing an Account Combined Addition and Subtraction Adding a Part of a Column Deducting Several vSubtrahends from one Minuend Table with Net Decrease Complements Short Methods of Multiplication Multiplying by 1 1 Multiplying by 11 1 Tabulating Multiples of a Multiplier Tabulating Multiples of a Divisor Division by Use of Reciprocals II Fractions and Proof Figures 16 Addition and Subtraction of Fractions Cross Multiplication Equivalent Common and Decimal Fractions Proof Figures III Arithmetical Progression 22 Elements in Arithmetical Progression Analysis of Simple Progression Computing Total Simple Interest C. P. A. Problem IV Average 29 Utility of Average Simple Average Moving Average Progressive Average Periodic Average Weighted Average V Averaging Accounts 42 Settling an Account Calculating Interest Items of Varying Amounts Focal Date Rules Applied VI CONTENTS Chapter Page Reducing Days to Months Compound Average Example of Compound Average Another Illustration VI Percentage 50 Percentage Terms Used in Percentage Fundamental Processes Percentage of Increase and Decrease Some Applications of Percentage in Business Per Cent of Goods Sold — Various Manufacturers Monthly Sales Compared on a Percentage Basis Sales for the Week Ending December 18, 1920 Comparison of Sales by Departments Individual Sales Compared with Average Individual Sales Compared with Maximum Apportionment Division of Profits Distribution of Factory Overhead Gross Profit Method of Approximating Inventory Analysis of Statements Percentage Analysis to Determine Causes of Variation in Profits VII Equation in the Solution of Problems .... 66 Solving Equations Illustration I Illustration 2 Illustration 3 Illustration 4 VIII Trade and Cash Discount ......... 76 Trade Discount Cumulative Trade Discounts Methods of Finding Net Price Cash Discount Discount as a Protection Against Loss Cash Discount Regarded as an Expense IX Turnover 83 Indefinite Meaning of "Turnover" Normal Inventories Necessary Different Bases of Comparison Working Capital as Basis of Turnover De'in'tion of Working Capital Need of Exact Definitions CONTENTS vii Chapter Page X Partnerships 89 Division of Profits Liquidation of Partnerships Periodical Distributions Reducing Capitals to Profit and Loss Ratio XI The Clearing House 08 Principle of the Clearing House Debits and Credits with Clearing House Clearing House Transactions Manager's Sheet Economy of System Application of Principle Extended XII Building and Loan Association 103 General Characteristics of Building and Loan Associations Terminating Plan Practicabihty of Plan Serial Plan Distribution by Partnership Plan Distribution by Dexter's Rule Withdrawal of Shares Sources of Income Premiums Individual Plan Dayton or Ohio Plan XIII Good-Will and Consolidation 116 Purchasing a Business with Stock Allocation of Net Earnings Good-Will Appraising Good- Will — Year's Purchase Method Capitalizing Gross Income Issue of Two Classes of Stock XIV Foreign Exchange 123 Conversion of Foreign Coinage Reverse Conversion Dealing in Foreign Exchange Average Date of Current Account Conversion of Foreign Branch Accounts Reconciliation of Accounts XV Logarithms i^p Use Multiplication Division Vlll CONTENTS Chapter Page Calculating Powers Roots Nature of Logarithms The Characteristic and the Mantissa Tables of Logarithms Characteristics of Logarithms of Numbers Between i and lO Use of the Characteristic in Pointing off Results Logarithms of Numbers Smaller Than i Use of Negative Characteristics in Pointing oflF Answers Computing with Logarithms having Negative Characteristics Determining Mantissa by Interpolation Determining Numbers by Interpolation XVI Simple and Compound Interest 154 Simple Interest — Methods of Calculating Rates Other Than 6% 365-day Basis Partial Payments Compound Interest Symbols Amount of Principal Frequency of Compounding Determining the Amount Determining Interest Determining Present Worth Determining the Compound Discount Summary XVII Annuities 171 Definition of Annuities Symbols Amount of an Annuity Sinking Fund Contribution Present Worth of an Annuity Rent of an Annuity Equal Periodical Payments on Principal and Interest Annuities Due To Find the Amount of an Annuity Due Sinking Fund Required Annual Contribution Present Worth of an Annuity Due Rents XVIII Logarithms in Compound Interest and Annuity Com- putations Calculating Compound Interest and Annuities by Logarithms To Find the Compound Interest on i To Find the Principal To Find the Rate To Find the Time 190 CONTENTS IX Chapter Page To Find the Present Value of i To Find the Compound Discount on i To Find the Amount of an Annuity To Find the Amount of Sinking Fund Contributions To Find the Present Worth of an Annuit}' XIX Bond Discount and Premium 194 Bonds Purchased Below and Above Par Discount Scientific Method of Amortization Income Rates Bond Premium Computing the Premium and the Price First Method Second Method Computing the Discount and the Price First Method Purchases at Intermediate Date Serial Bonds XX Leaseholds 217 Commuted Rents XXI Depreciation Methods 221 Annual Depreciation Appendix A — Values of Foreign Coins 231 B — Logarithms of Numbers 233 C — Compound Interest and Other Computations . 254 Mathematics of Accounting and Finance CHAPTER I SHORT METHODS AND PRACTICAL SUGGESTIONS Short Methods of Computation Since the work of an accountant necessarily involves a great deal of computation, it is desirable that he be familiar with cer- tain labor-saving devices. Those which are presented in this chapter have been selected because of their simplicity and because an accountant has occasion to apply them so frequently as to make their use habitual. No attempt is made here to describe all the innumerable "short methods" which, however ingenious, are difficult to remember and rarely available. Balancing an Account In striking a balance in an account, the proper method for determining and inserting the balance is to add the larger column and enter its total in both columns; then to add the smaller column and insert each figure of the balance necessary to produce the total. The following example is given to illustrate the method. Debits Credits $1,846.22 $ 126.13 2,913.68 248.71 4,327-11 1,635.48 2,319.11 4,757-58 $9,087.01 $9,087.01 The balance of $4,757.58 in this account is found and entered in the following way. First, the larger side of the account, which happens to be the debit side, is added for a total of $9,087.01 and the amount is placed on both sides; then the items on the credit 3 4 MATHEMATICS OP ACCOUNTING AND FINANCE side are added and the figure in each digit column necessary to give the figure in the corresponding column of the total is in- serted in the place reserved for the balance. Thus, adding the first column, 3 + i + 8 + i, gives 13, to which 8 is added to make 21 and produce the i in the total. The 8 is inserted in the balance as shown, while the 2 of 21 is carried. The second column of digits and the 2 carried forward are added to 15, and to this a 5 is added to make 20 and give o in the total. The 5 is entered in the same column of the balance and the 2 of 20 is carried. In the same way the third column is added to 30, 7 is inserted in the balance to make 37, and the 3 of 37 is carried. The fourth column is added to 13, 5 is inserted to produce 18, and I is carried. The fifth column is added to 13, 7 is inserted to make 20, and 2 is carried. The last column requires a 4 to make 9 in the total, and the entire balance is found to be $4,757.58. Combined Addition and Subtraction It often happens that columns of figures are given in which are included both positive and negative values, that is, plus and minus numbers. It may be necessary, for example, to compare the figures of two years to show the increase or decrease of certain items and to determine the net increase or decrease for the second year. If the figures are written in ink, the minus quantities may be inserted in red. In a newspaper or book the minus numbers are usually printed in italics or indicated by a star. Two methods may be followed in adding a column of such figures. The first method is the ordinary way of adding the positive numbers and subtracting from their sum the sum of the negative numbers. The other method is to add the numbers continuously, the negative digits being in each case added nega- tively, that is subtracted. Thus, in adding the third column of the following table, which is a comparison of the manufacturing costs of 19 1 6 with those of 191 5, the procedure is as explained below: SHORT METHODS AND PRACTICAL SUGGESTIONS Account 1915 Material used $12,638.13 Direct labor 16,469.42 Indirect labor 5,827.59 Other manufacturing expense 7,962.28 Incre.\se or I9I6 Decrease $14,228.64 $1,590.51 19,672.18 3,202.76 4,713-92 1,113.67 8,242.37 280.09 $46,857.11 $3,959.69 ^2,897•42 The net increase of $3,959.69 for all four items is found by adding the increases and decreases in the following manner. The first column of digits on the right is added thus, 1+6 = 7, 7 — 7 = o, o + 9 = 9. The second column is added thus, 5 + 7 = 12, 12 — 6 = 6. In the third column 3 is subtracted from 2 to make — I, and the i is subtracted from 10 borrowed from the next column. The addition is therefore, 2 — 3 + 10 = 9. In the fourth column the borrowed 10 is represented by — i, and the column adds thus, —1+9 — 1+8 = 15. The i of 15 is carried, as in ordinary addition, to the next column, which added is, 1 + 5 + 2 — 1 + 2 = 9. The final column adds, i + 3 — I = 3- In practice, items of the same value but of opposite signs may be offset at once. Thus, a glance at the cents column in the fore- going example shows that i and 6 offset — 7, and the remaining 9 can, therefore, be put down at once in the result. Adding a Part of a Column This manner of addition is useful when it is desired to find the sum of all but the last few numbers of a long column whose total is known. For instance, a page may have an amount on each of 40 lines, and the addition of the entire column has been verified. It is now necessary to ascertain the total of the first 35 fines, as in the illustration below, in which these first 35 lines are omitted to save space. 6 MATHEMATICS OF ACCOUNTING AND FINANCE 36th line $ 2,382.46 37th " 1,896.28 38th " 1,237.61 39th " 2,068.42 40th " 1,923.16 Total of 40 amounts on page $82,642.57 The total of the items to and including the 35th may be found by adding the last five items on a memorandum paper and deduct- ing their sum from the total of the page. A much quicker and more workmanlike procedure, however, is to subtract them by addition as explained in the previous section. Adding the first column of digits, 6 + 8 + I + 2 + 6 gives 23. As 4 is required to make the 7 in the grand total, 4 must be the last digit in the footing at the 35th hne. Carrying 2 and proceeding in the same way with the second and the other columns, the required footing is found to be $73,134.64. The entire operation involves no more writing than the insertion of the figures under the 35th line. Deducting Several Subtrahends from One Minuend It is sometimes necessary to subtract several amounts from one amount, as, for example, to make various deductions from total sales. If there is room enough the various items may be listed in an inside column, their total inserted under gross sales, and the difference carried out. If only one column is possible, the subtraction can be made by addition, as shown in the follow- ing table, in which the items between the lines are deductions from the total sales, and the remainder is the amount of the net sales. Total sales $132,629.14 Less: Cash discount $ 1,436.27 Freight 2,389.16 Breakages 948.63 Sundry allowances 769.42 Net sales $1 27,085.66 SHORT METHODS AND PRACTICAL SUGGESTIONS 7 In performing the operation it should be remembered that the digits of the remainder are always those necessary in each column to make the digit of the same column in the total. Table with Net Decrease At times the amounts to be deducted are greater than the amount from which they are to be deducted. The table below is an example of this. It represents a comparison between two years to show the increaseor decrease of business in each of seven depart- ments and the total net result of the whole business. The table might have been arranged in four columns, one for the figures of 1920, one for the figures of 1921 , one for the increases, and one for the decreases; or the departments might have been listed in order, with the changes appearing in one column but the decreases indicated by red ink or italics. In the table as given the depart- ments showing an increase are listed first, and those showing a decrease are listed underneath. Dept. 1920 1 92 1 Increase or Decrease I $168,242.19 $174,629.46 Inc. $ 6,387.27 3 192,429.36 195,716.83 " 3,287.47 4 156,283.27 160,142.76 " 3,859-49 7 84,619.43 86,223.62 " 1,604.19 Total $601,574.25 $616,712.67 Tot. Inc. $15,138.42 2 $165,328.46 $158,693.82 Dec. $ 6,634.64 5 212,642.88 204,976.38 " 7,666.50 6 198,731.54 189,218.23 " 9.513-31 Total $576,702.88 $552,888.43 Tot. Dec. $23,814.45 Grand total $1,178,277.13 $1,169,601.10 Net Dec. $8,676.03 The net decrease in the foregoing table is the sum of the de- creases less the sum of the increases. The total of the incieases is, therefore, added negatively to the total of the decreases, that is, deducted digit by digit. 8 MATHEMATICS OF ACCOUNTING AND FINANCE Complements The arithmetical complement of a number is the difference between that number and the next higher power of lo. Thus the complement of 7 is 3, or the difference between 10 and 7, and the complement of 81 is 19, or the difference between 100 and 81. If instead of subtracting a number less than 10, its comple- ment is added, the result is 10 larger than the result of the sub- traction. Thus 8 subtracted from 15 gives 7; while the com- plement of 8, which is 2, added to 15, gives 17. Hence to subtract digits, in adding a column of numbers negatively, their complements may be added to the other digits and the sum reduced by ten times the number of complements added. The following table represents an addition of positive and negative quantities: 929 248 - 1,086 - 74 3,126 - 174 306 3,275 In the unit column of this table, 9 -]- 8 -f 4 (complement of 6) + 6 (complement of 4) + 6 + 6 (complement of 4) + 6 = 45, and 45 ~" 30 = 15; 30 being subtracted because three complements have been added. The 5 of the 15 is entered in the footing and the I is carried. In the tens column i (carried) + 2 -f 4 + 2 (complement of 8) +3 (complement of 7) + 2 -f 3 (complement of 7) = 17. The 7 is entered in the footing, and since three complements were added the i is dropped and — 2 carried to the hundreds column. In the hundreds column 8 (complement of the — 2 carried) + 9 + 2-f-i + 9 (complement of i) + 3 = 32, and 32 — 20 (there being only two complements) = 12. The 2 is entered and the i is carried. In the thousands column i + 9 SHORT METHODS AND PRACTICAL SUGGESTIONS 9 (complement of i) + 3 = 13, and 13 — 10 (there being only one complement) = 3, which is entered in the footing. The result of the addition is, therefore, 3,275. Short Methods of Multiplication To multiply numbers ending in zeros, both of which are integers, the significant figures of the numbers are multiplied and to the product are annexed as many zeros as there are final zeros in both the multiplicand and the multiplier. Thus, 3,400 times 1,200 equals 34 times 12, or 408, with four zeros annexed, or 4,080,000. When one of the two numbers multiplied is a decimal fraction the significant figures in the two numbers are multiplied and the decimal point in the product is moved as many places to the right as there are final zeros in the integer. This may necessitate annex- ing zeros. In the example .486X300, .486 is first multiplied by 3, to make 1.458. As there are two zeros in the multiplier, the deci- mal point in this product is moved two places to the right, making it 145.8. To multiply by 9, 99, or any number that contains no other figure than 9, as many zeros are annexed to the multiphcand as the multipher has 9's, and from the result the multiplicand is deducted. Thus, in multiplying 293 by 99, two zeros are annexed to 293, making 29,300, and from this 293 is deducted. The result is 29,007. Multiplying by 1 1 To multiply an amount by 1 1 it is not necessary to put down the figures, but only the result, which is secured by addition. The procedure is as follows : Beginning with the right-hand or unit figure in the multipli- cand, add to each digit the digit next to the right of it. Put the last digit of this sum down in the result, carrying the other digit of the sum if there is one. Continue this until each figure in the 10 MATHEMATICS OF ACCOUNTING AND FINANCE multiplicand has been used twice. The method is illustrated in the following example : Illustration Multiply 1,342 by 11 Solution: 1,342 Product 14,762 The first digit, 2, at the right of the multiplicand, has no figure to the right of it. Therefore nothing is added and it remains 2 The second digit, 4, is added to the first digit, 2, making 6 The third digit, 3, is added to the second digit, 4, making 7 The fourth digit, i, is added to the third digit, 3, making 4 Since the fourth digit has been used only once it is necessary to use it a second time, thus, i The product is, therefore, 14,762 An analysis of the ordinary method of multiplication shows that the procedure is exactly the same in both cases. Looking at the example when multiplied in the usual way, it is seen that, 1342 II 1342 1342 [4.762 As in the above explanation, the product is formed by putting down 2 alone, then adding 4 to 2, 3 to 4. i to 3 and repeating the i. It is, however, unnecessary to write out these figures, as the whole operation can be performed mentally and the product put down at once. SHORT METHODS AND PRACTICAL SUGGESTIONS 1 1 Multiplying by 1 1 1 To multiply by 1 1 1 the same procedure is followed as when the multiplier is ii, except that each figure in the multiplicand is added to the two figures to the right, and is, therefore, used three times, as shown in the following illustration: Illustration Multiply 1,342 by 11 1 Solution: 1,342 Product 148,962 The first digit in the multiplicand is put down alone, thus, 2 The second digit, 4, is added to the first digit, 2, making 6 The third digit, 3, is added to the second digit, 4, and the first digit, 2, making 9 The fourth digit, i, is added to the third digit, 3, and the second digit, 4, making 8 The fourth digit, i, is added to the third digit, 3, making 4 The fourth digit, i, is put down alone, thus, i 148,962 If the same method is applied in multiplying 1,796 by in, it will be found necessary to carry over i, 2 and i, in adding the digits. In multiplying by any number that consists of a series of i 's each digit in the multiplicand is used as many times in the way explained above as there are digits in the multiplier; twice when the multipher is 1 1 , three times when it is in, four times when it is 1,1 1 1, and so on. Tabulating Multiples of a Multiplier When a number containing several digits is to be used re- peatedly as a multiplier, it saves time and promotes accuracy to make a table of its multiples. The best way of compiling the 12 MATHEMATICS OF ACCOUNTING AND FINANCE table is to determine each multiple by adding the number to the preceding multiple. The addition is, of course, made on the table itself and not on a separate sheet of paper. The table is carried out to ten multiples in order to prove the work, as the tenth line must be the same as the first with one zero annexed. This is proof only in case the multiples are built up by addition. A table of this sort constructed for 683,947 as a multiplier is the following: ULTIPLIER Multiple I 683,947 2 1,367,894 3 2,051,841 4 2,735,788 5 3,419,73s 6 4,103,682 7 4,787,629 8 5,471,576 9 6,155,523 10 6,839,470 Supposing it is desired to multiply 683,947 by 3,469, the solu- tion is as follows : 683,947 multiplied by 9 = 6155523 " 6 = 4103682 " " 4 = 2735788 " 3 = 2051841 Product 237261 2143 Tabulating Multiples of a Divisor When a number containing several figures is to be used re- peatedly as a divisor, a table of multiples of the divisor may be prepared, to show at a glance how many times the divisor is contained in successive remainders and do away with the necessity of performing the multiplication. This method may also save considerable labor by reason of the fact that instead of writing SHORT METHODS AND PRACTICAL SUGGESTIONS 1 3 down the multiples to make each subtraction, they may be written on a card and the particular multiple to be deducted placed alongside the amount from which it is to be subtracted. The subtraction is made from the card across to the amount, and only the remainder is set down. The directions for the use of this method of division may be given as follows : Placing the card at the left of the dividend, find by inspection which multiple is to be used; sHde the card until that multiple is on a line with the dividend; deduct the multiple from the corre- sponding first figures of the dividend, putting the remainder under the dividend figures; then bring down the next figure and proceed in the same way with the new amount to be divided. The follow- ing example, in which 872,976,654 is divided by 683,947, illus- trates this method. Card OF Multiples Dividend Quotient (as slid for each division) I 683,947 872976654 I 2 1,367,894 1890296 2 7 4,787,629 5224025 7 6 4,103,682 4363964 6 3 2,051,841 2602820 3 8 5,471,576 5509790 38214 8 Quotient is 1,276.38, and the remainder 382.14. Division by Use of Reciprocals Two numbers are reciprocal when their product is i. For instance, 2 and .5 are reciprocal numbers, because 2 X .5 = i. It is evident from this that the reciprocal of a number is found by dividing i by the number. The quotient obtained by dividing by a divisor is the same as the product obtained by multiplying by the reciprocal of the divisor. For example, 12 -^ 2 and 12 X -5 are each equal to 6. 14 MATHEMATICS OF ACCOUNTING AND FINANCE Repeated division by the same divisor can be made easier, particularly if an adding machine is available, by determining the reciprocal of the number, tabulating the multiples of the recipro- cal, and proceeding as in multiplication. In dividing, for example, 872,976,654 by 683,947, the first step is to divide i by the divisor in order to obtain its reciprocal, which is .0000014621. The multiples of the reciprocal are then tabulated, as in the table below. The zeros in the reciprocal are not repeated in the table, as it is only necessary to remember the number of places to be pointed off in the product, ten in the present case. Having the multipHers, the procedure thenceforth is precisely as in multipli- cation, as shown in the second of the subjoined tables. ULTiPLEs OF Reciprocal I 14,621 14621 2 29,242 872976654 3 43,863 58484 4 58,484 73105 5 73,105 87726 6 87,726 87726 7 102,347 102347 8 116,968 131589 9 131,589 29242 10 146,210 102347 1 16968 12763791658134 Pointing off ten decimals, the result is 1,276.38 Since the reciprocal is approximate only, the last figures of the product must be ignored, as they do not show the exact remainder. The work may be shortened considerably by beginning at the left and ignoring all multiples and portions thereof beyond the first, second, or third place to the right of the last place desired in the product, as follows: SHORT METHODS AND PRACTICAL SUGGESTIONS 15 14621 872976654 8 I 16968 7 102347 2 29242 9 131589 7 10235 6 877 6 88 5 7 4 I 127637917 or 1,276.38 CHAPTER II FRACTIONS AND PROOF FIGURES Addition and Subtraction of Fractions Fractions having the same numerator may be added by multiplying the sum of the denominators by the common numera- tor to obtain the numerator of the result, and by multiplying together the denominators of the fractions to obtain the de- nominator of the result which is then reduced to its simplest form. To add 1/9 and 1/7 by this method the procedure is as follows: 9 + 7 = 16; 16 X I = 16, the numerator of sum 9X7 =63, the denominator of stun Sum = 16/63 The solution of 3/1 1 + 3/16 is as follows: II -|- 16 = 27; 27 X 3 = 81, the numerator of sum II X 16 = 176, the denominator of sum Sum = 81/176 To subtract fractions when the numerators are the same, the difference between the denominators is first multiplied by the common numerator. This gives the numerator of the result. Then the denominators are multipHed to obtain the denominator of the result, which is reduced to its simplest terms. This method is illustrated in the solution of the following subtraction, 3/1 1 — 3/16: 16— 11=515X3= 15, the numerator of the difference 16 X II =176, the denominator of the difference Difference = 15/176 Cross Multiplication Two fractions may be added by the method of cross multipli- cation. The numerator of each is multiplied by the denominator 16 FRACTIONS AND PROOF FIGURES l^ of the other, and the two products are added to form the numera- tor of the result. The denominators of the fractions are multi- pHed to form the denominator of the result, which is then reduced to its simplest terms. This method is illustrated in the following example, in which 5/8 and 7/9 are added. 5X9= 45 7X 8= j6 loi, numerator of the sum 8X9= 72, denominator of the sum Sum = 101/72 or I 29/72 Cross multiplication may also be used in subtracting fractions. The numerator of each fraction is multiplied by the denominator of the other, and the smaller product is subtracted from the larger to form the numerator of the result. The two denominators are multiplied to form the denominator of the result. Reduction to the simplest form then follows. In the following example 5/8 is subtracted from 7/9 by this method: 7 X 8 = 56 5 X 9 = 45 II, numerator of the difference 9X8= 72, denominator of the difference Difference = 11/72 The approximate product of two mixed numbers may be found by multiplying first the integers, then each integer by the other fraction to the nearest unit, and finally by adding the three products. The following illustration in which 1 7 1/5 is multiphed by 14 1/3 shows the application of the method : 14 X 17 = 238 1/5 of 14 = 3 (nearest unit) i/3 0fi7 = __6 Approximate product 247 The exact product of this multiplication is 246 8/15. If the multiplication of the fractions is carried out to at least one decimal place, the result is more exact, as the following shows: i8 MATHEMATICS OF ACCOUNTING AND FINANCE 14X 17 = 238 1/5 of 14 = 2A 1/3 of 17= 5.; Result 246.5 Equivalent Common and Decimal Fractions The following is a list of the most frequently used decimal fractions and their equivalent common fractions. Equivalent Decimal Common Fractions Fractions 5 y'^ ■33 1/3 1/3 662/3 2/3 25 1/4 75 3/4 16 2/3 1/6 ^3^/3 5/6 125 1/8 375 3/8 625 5/8 87s 7/8 081/3 1/12 41 2/3 5/12 581/3 7/12 91 2/3 11/ 1 2 0625 1/16 1875 3/16 3125 5/16 4375 7/16 5625 9/16 6875 11/16 8x25 13/16 9375 15/16 These figures can be used in a variety of ways. One of the most obvious is to substitute the common fraction for its equiva- lent decimal, as it is usually easier to multiply or divide by the FRACTIONS AND PROOF FIGURES I9 common fraction than by the decimal. Thus, if it is desired to find 12 1/2% of a number, it is far easier to divide the number by 8 than to multiply it by the decimal .125. This substitution of a common fraction for the equivalent decimal is simple enough when the numerator of the common fraction is i. When, however, the numerator is greater than i, it is necessary to split up the decimal into such parts as will per- mit the substitution of equivalent common fractions with a nu- merator of I . The number taken is multiplied by each of these parts and the products are added to find the result. The fewer the parts into which the decimal is split, the simpler is the opera- tion. The appHcation of the rule is illustrated in the following problem. Example How much will 3,648 yards of cloth cost at 43 3/4 cents per yard? Solution: The answer can be found in three ways: 1. By multiplying 3,648 by $.4375. 2. By substituting for $.4375 its equivalent, 7/16 of $1, and mul- tiplying 3,648 by the latter. 3. By spHtting up the common fraction of 7/16 into 1/4, 1/8, and 1/16, or the decimal of .4375 into .25, .12 1/2, and .06 1/4, and substituting the equivalent fractions, and then multiplying 3,648 by each of these fractions and adding the products, as shown below: At $1.00 per yard the price would be $3,648.00 At 1/4 or .25 per yard the price would be g 12.00 At 1/8 or .12 1/2 (1/2 of above) 456.00 At 1/16 or .06 1/4 " " " 228.00 At 7/16 or .43 3/4 the price is $1,596.00 The operation can be performed much more quickly if it is recognized that 7/16 is the same as 8/16 minus 1/16, that is, 1/2 minus 1/8 of 1/2, and the calculation is made as shown below: 20 MATHEMATICS OF ACCOUNTING AND FINANCE 1/2 of $3,648 = $1,824.00 1/8 of 1,824= 228.00 7/16 of $3,648 = $1,596.00 It is always advisable to prove all computations when it does not involve too much work. This is easily done in most cases where a fraction can be split up. Thus, in the above example it has been found that 7/16 of $3,648 is $1,596. If the figure for 1/16 is added to instead of subtracted from that for 8/16 the result is 9/16, and the calculation of $1,596 can be proved as follows : 8/16 minus 1/16 is 7/16 or 8/16 plus 1/16 is 9/16 or $1,596.00 2,052.00 16/16 or original amount $3,648.00 Proof Figures There are two systems of proof figures for testing the accuracy of mathematical computations. One is based on the casting out of 9's and the other on the casting out of ii's. Many bookkeepers use one or the other system to check all their work. Both systems are used in the illustration below to check an addition, the figure at the right of each amount being the excess over the 9's or ii's in the amount, or the remainder left after the digits are added and the largest multiple of 9 or 11 is subtracted. 365,592 286,548 64,320 94,094 810,554 Excess OVER 9's 3 6 6 8 _ Total 23, or an 5 excess of 5 Excess OVER ii's 365,592 7 286,548 9 64,320 3 94,094 ° Total 19, or an 810,554 8 excess of 8 To cast out 9's from a number, the digits are added ignoring 9's, zeros and combinations of digits which add to 9. From the FRACTIONS AND PROOF FIGURES 21 sum thus obtained the largest multiple of 9 contained therein is subtracted. Thus, the digits in the first number in the above example are added as follows, beginning at left: Ignore the 3 and 6 because they add to 9; 5 + 5 = 10; ignore the 9; 10 + 2 = 12. 12 — 9 = 3, th^ excess. To cast out 1 1 's begin with the first figure at the right and add to it the third, fifth and so on, and take the excess of their sum over ii's. Then add together the second, fourth, sixth figures, etc., and find the excess of their sum over i I's. Subtract this excess from the first excess and the result is the check num- ber. If the first excess is smaller than the second, add 11 to it before subtracting. Thus, in the first number of the foregoing example, 2 + 5 + 6 = 13, or an excess of 2 over 11. Similarly, 9 + 5 + 3 = 17, or an excess of 6 over 11. Adding 11 to the 2 and subtracting 6 gives 7 as the check number. Addition is proved if the excess over 9's or ii's in the sum of the individual check numbers is the same as the check number of the total of the numbers added, as shown in the preceding example. This is not an absolute proof, as an error of 9 or 1 1 or multiples thereof may be made. Multiplication is proved by multiplying the check figure of the multipHcand by that of the multiplier. The excess over 9's or ii's in the result should be the check figure of the product, as is seen from the following: Excess Excess Numbers over 9's overii's 4,621 4 I 3,274 7 7 15,129,154 28 or I 7 CHAPTER III ARITHMETICAL PROGRESSION Elements in an Arithmetical Progression An arithmetical progression is a series of numbers increasing or decreasing by a common difference. The numbers in the series are called the terms; the first and last terms are called the extremes, and the intermediate terms the means. An in- creasing or ascending series is formed by adding the common difference to each preceding term. For example, 7, 12, 17, 22, 27, is an ascending series with a common difference of 5. A decreasing or descending series is formed by subtracting the com- mon difference from each preceding term. Thus, 26, 23, 20, 17, 14, is a descending series with a common difference of 3. There are five elements in an arithmetical progression, which in the formulas to be presently derived are represented by the following symbols : First term / Last term / Common difference d Number of terms n Sum of series 5 When any three of these elements are known, the other two can be computed. Analysis of Simple Progression The following is an example of a short progression: ist term 3 2nd " 6 3d " 9 4th " 12 Sth " 15 6th or last term 18 r 22 ARITHMETICAL PROGRESSION 23 The common difference here is 3, the number of terms is 6, and the sum of the terms is 63. It will be seen that, although the number of terms is 6, the common difference is added only five times. This explains why in Cases I and 2 considered below, i is subtracted, and why in Case 3, I is added. It will also be seen that the series consists of a number of pairs, as follows: ist and last terms make a pair the sum of which is 21 2nd " 5th " " " " " " " " " 21 3d " 4th " " " " " " " " " 21 The sum of series is 63 It is evident from this that the sum of a series is the sum of the first and last terms multiplied by half of the number of terms. It is also evident that since the sum of each pair of terms is 21, the average single term is 10^, which multiplied by 6, or the number of terms, gives 63, or the sum of the series. If the number of terms is uneven, the number of pairs includes a half pair. Thus, if the series given above is extended to seven terms, the last term is 21, and the sum of the terms is 84, which is 3>2 times 24, the sum of the first and last terms. In the following four cases the derivation of the formulas for computing the various elements of an arithmetical progression is explained. Case I. Given the first term, common difference and the num- ber of terms, to find the last term. If the series is ascending, the common difference must be added as many times, less one, as there are terms in the series. Hence the formula is: / = /+(«- i)^ Example The first term is 9, the common difference is 3, and the number of terms is 5. Find the last term. 24 MATHEMATICS OF ACCOUNTING AND FINANCE Solution: /= 9+(5- 1)3 = 9+ 12 = 21 If the series is descending, the common difference must be deducted as many times, less one, as there are terms in the series. Hence the formula is : / = /- („- i)d Example The first term is 21, the common difference is 3, and the number of terms is 5. Find the last term. Solution: /= 21 - (5 - i) 3 = 21—12 = 9 Case 2. Given the extremes and the number of terms, to find the common difference. The number of common differences is one less than the num- ber of terms; and the sum of the common differences is the differ- ence between the extremes. Hence in an ascending series the formula for finding the common difference is : .= ^ w — I and in a descending series it is : Example The first term is 9, the last term is 21, and the number of terms is 5. Find the common difference. ARITHMETICAL PROGRESSION 25 Solution: 21 — Q d= s- I 12 4 = 3 Ca^e J. Given the extremes and the common difference, to find the number of the terms. The difference between the extremes is the sum of the common differences, and the number of the common differences is one less than the number of terms. Hence in an ascending series the formula for calculating the number of terms is: l-f d and in a descending series it is : f-l + I + I Example The first term is 9, the last term is 21, and the common difference is 3. Find the number of terms: Solution: 21 — 9 n = 12 = -+ I 3 = 4+1 = 5 Case 4. If the extremes and the number of terms are given, the sum of the terms is found by the following formula : s = X n 26 MATHEMATICS OP ACCOUNTING AND FINANCE or by the following: 5= (/+/)X- 2 Example The extremes are g and 21, and the number of terms is 5. Find the sum of the terms. Solution: 9+21 5 X 5 = 75 or 5 = (9 + 21) X - = 75 2 2 Computing Total Simple Interest The most important applications of arithmetical progression with which accountants are concerned fall under Case 4. They are made in computing the total simple interest on a principal which constantly increases or decreases from period to period by a common difference. For example, a $1 ,000,000 bond issue bearing interest at 5% is to be repaid in forty equal annual instalments, and it is necessary to compute the total interest to be paid during that period. Since 1/40 of the loan is to be paid annually, the principal during the last year will be 1/40 of $1,000,000 or $25,000. The interest for the first year will be 5% of $1,000,000, or $50,000, which is the first term of an arithmetical progression containing forty terms; the interest for the last year will be 5% of $25,000, or $1,250, which is the last term of the progression. Since 2 the total interest, which is s, will be $50,000 + $1,250 X 40, or $1,025,000 ARITHMETICAL PROGRESSION 27 C. P. A. Problem The application of the principle of arithmetical progression to the computation of interest on a given principal for an increas- ing or decreasing series of time periods, may be illustrated by a problem similar to one given in an Illinois C. P. A. examination. The problem is as follows : Problem Upon the death of a retired business man in June, 1910, a will is found conveying real and personal property aggregating $300,000 to the widow, who is his second wife, for her life, and upon her death to four children in equal shares. It is discovered after his death that his first wife had left to her two children, Henry and Emma, $20,000, consisting of securities for $10,000 bearing 6% interest, and uninvested cash of $10,000. The father had regularly collected the semi-annual interest on the investment, but there was no evidence as to his disposition of the cash portion of the be- quest. Exactly ten years elapsed between the death of his first wife and his own death, so that he had collected twenty items of interest, the last one just before he died. Henry and Emma were of age at the time of their father's death, and had never been informed of their legacy. Prepare a statement showing what would accrue to each of the four children at the death of the widow, who died immediately after her hus- band, including the amounts to which Henry and Emma would be entitled on account of their mother's estate. Exclude and do not consider any accrued income of the estate unexpended. In Illinois the legal rate of interest on undisclosed debts is 5%. Solution: The proceeds of the property that belonged to his first wife constitute a trust fund belonging to the two children, Henry and Emma; the total of this fund on June i, 1900, comprises: Open account for cash collected by decedent, June, 1890 $10,000.00 Interest thereon at 5% (legal rate on undisclosed debt) for ten years 5,000.00 Principal of securities 10,000.00 Interest collected on securities 6,000.00 Each collected coupon increased the undisclosed debt of the father as guardian. The first coupon was collected 28 MATHEMATICS OF ACCOUNTING AND FINANCE 91/2 years or 19 half years before the father's death. The twentieth coupon had just been collected. The computa- tion of the interest on these coupons can be accomplished thus: First $300 earns interest at 5% for 9 1/2 years. .$142.50 Last $300 " " " " " o " . . o Sum of extremes $142.50 Number of terms (coupons) 20 Applying the formula: s = X n 2 or 5= (/+/)X - 2 The total interest = 142.50 X 10 or 1,425.00 Total due Henry and Emma from mother's estate. . . $32,425.00 The division of the estate would be as follows: Total real and personal property $300,000.00 Amount due Henry and Emma from mother's estate 32,425.00 Balance divided among four children $267,575.00 Of which one-fourth is $ 66,893.75 The allowance of $1,425 for interest on the coupons collected must not be confused with compound interest. The item of $1,425 is interest on actual cash received by the father, for which he did not account. How he came into possession of the cash is immaterial. CHAPTER IV AVERAGE Utility of Average The principle of average may be used for determining proba- bilities, for comparing numbers with a standard or with each other, or for the purpose of simplifying calculations by using an average instead of a number of related values. Where it is pos- sible to collate statistics covering extensive and varied observa- tions, it is possible to determine an average which may be assumed, from the law of averages, to be standard. Reliance can then be placed on the probability that other similar cases, though individually at great variance from the average, will in the aggregate closely approximate the average. Mortality tables, for instance, are averages determined by exhaustive investigation. While they do not determine probabiHties for individuals, they do determine probabilities for large groups of individuals. When the principle of average is utilized, the basis of com- parison may be a simple average, a moving average, a progressive average, a periodic average, or a weighted average as the case may be. The choice of a base depends on the information desired and, in the case of weighted average, on the necessity imposed by the facts themselves. Simple Average The process of determining a simple average consists merely of adding the units to be averaged and dividing the sum by the number of units. If the sum of the units is known, the process requires division only. 29 30 MATHEMATICS OF ACCOUNTING AND FINANCE Suppose the daily sales for a week are as follows : Monday. $3,560.35; Tuesday, $3,115.95; Wednesday, $2,946.86; Thursday, $2,868.79; Friday, $3,269.87; Saturday, $3,896.43. Total sales, $19,658.25. The daily average of the sales for the week is found by divid- ing the total of $19,658.25 by the number of days taken, or 6, which gives a result of $3,276.37^. It must be remembered in figuring average that the divisor is the number of units in the dividend, and not necessarily the num- ber of items added to obtain the dividend. For example, suppose it is desired to determine the average contribution made by a number of persons to a benevolent fund, where 2 men each contributed ' I5OO, totaling : $1,000.00 4 " a (I 250. " 1,000.00 10 " u u 100, " 1,000.00 16 " a li 50, u 800.00 18 " U (( 25> " 450.00 so " " " a total of $4,250.00 The number of items added here to find the dividend of $4,250 is only 5, but the number of units to be used as the divisor is 50, making the average contribution equal to $4,250 divided by 50 or $85. Moving Average When it is desired to compare a series of numbers relating to units of time of uniform duration and of uninterrupted sequence, a moving average of a number of these units may be used as a basis of comparison. This kind of average is determined by tak- ing a simple average of the numbers to serve as the starting point ^or the moving series, and dropping after the lapse of each time unit the first number of the series and adding the number of the next time unit in order to form a new series and obtain a new average. AVERAGE 31 Assume, for example, the following conditions : 1. The values averaged are monthly sales approximated to the nearest $1,000. 2. The time units are months. 3. The number of units is 12, making up one year. 4. The first month of each series is dropped, as a new month is included in the next series. In an example of this sort each month's sales can be compared with any one of the twelve averages in the calculation of which the month's sales are included. For instance, the sales of Decem- ber, 1914, can be compared with the twelve monthly averages for the years ending with each month from December, 1914, and to November, 191 5, inclusive. Some of the possible comparisons in which the moving aver- age may be used are illustrated in the table below. Their value depends on the nature of the business whose figures are used. The most significant fact shown in the illustration is the almost uninterrupted increase in the moving average, although the monthly sales show wide variation due to the seasonal nature of the business. The decline in the moving average during the last months of 19 14 and the first months of 191 5, shows the effects of the war. Following is an explanation of the methods by which the figures in the various columns are determined : Moving average: Total sales for 1 2 months of 1913 335 Moving average including December, 1913, 335 -r- 12... 27. q 335 — 20 (January, 1913) + 53 (January, 1914) 368 Moving average including January, 1914, 368 -^ 12 ... . 30.66 Increase or decrease* in moving average — i month: Moving average including January, 1914 30.7 " " " December, 1913 27.9 Increase 2.8 32 MATHEMATICS OF ACCOUNTING AND FINANCE More or less* than annual average: December, 1913, sales 27 Moving average including December 27.9 Decrease .9* January, 1914, sales 53 Moving average including January 30.7 Increase 22.3 Increase in moving average — i year: Moving average including December, 1914 35.4 " " " December, 1913 27.9 Increase 7.5 Increase in moving average — 2 years: Moving average including January, 1916 48 " " " January, 1 914 30.7 Increase 1 7.3 Moving Average Sales Moving I9I3 (000 omitted) average Jan. 20 Feb. 17 Mar. 21 Apr. 17 May 26 June 40 July 28 Aug. 21 Sept. 36 Oct. 55 Nov. 27 Dec. 27 27.9 More or less* than annual average AVERAGE 33 Moving Average — Continued I9I4 Sales (000 omitted) Moving average Increase or decrease* in moving average — I month More or less* than annual average Increase in mov- ing aver- age — I year Jan. S3 30.7 2.8 22.3 Feb. 47 33-2 2-5 13-8 Mar. 47 35-3 2.1 II. 7 Apr. 45 37-7 2.4 7-3 May ?>2 38.2 •5 6.2* June 3,2 37-5 •7* 5.5* July 24 37-2 ■3* 13.2* Aug. 17 36.8 .4* 19.8* Sept. 39 37-1 •3 1.9 Oct. 42 36.0 I.I* 6.0 Nov. 20 35-4 .6* IS.4* Dec. 27 35-4 .0 8.4* 7-S Increase I9IS Sales (000 omited) Moving average Increase or decrease* in moving average — I month More or less* than aimual average or decrease* in moving average — I year Jan. 45 34-8 .6* 10.2 4.1 Feb. 43 34-4 .4* 8.6 1.2 Mar. 39 33.8 .6* 5-2 i-S* Apr. 38 33-2 .6* 4.8 4.5* May 32 33-2 .0 1.2* 5-0* June 25 32.6 .6* 7.6* 4.9* July 27 32.8 .2* 5.8* 4.4* Aug. 32 34-1 1-3 2.1* 2.7* Sept. 49 34-9 .8 14.1 2.2* Oct. 85 38.5 3-6 46.5 2-5 Nov. 67 42.4 3-9 24.6 7.0 Dec. 47 44.1 1-7 2.9 8.7 34 MATHEMATICS OF ACCOUNTING AND FINANCE Moving Average — Continued I9I6 Sales (000 omitted) Moving average Increase or decrease* in moving average— I month More or less* than annual average Increase in moving average — I year Increase in moving average —2 years Jan. 92 48.0 3-9 44.0 13.2 ^7-3 Feb. 97 52.5 4 5 44-5 18.I 19-3 Mar. 83 56.2 3 7 26.8 22.4 20.9 Apr. 44 56.7 5 12.7* 23-5 19.0 May 33 56.8 I 23.8* 23.6 18.6 June 42 58.2 I 4 16.2* 25.6 20.7 July 38 59-1 9 21. I* 26.3 21.9 Aug. 55 61.0 I 9 6.0* 26.9 24.2 Sept. 80 63.6 2 6 16.4 28.7 26.5 Oct. 97 64.6 I 32.4 26.1 28.6 Nov. 84 66.0 I 4 18.0 23.6 30.6 Dec. S8 66.9 9 8.9* 22.8 3^-5 Progressive Average Progressive average is cumulative, a new unit being added to form each successive dividend, and the divisor being constantly increased in an arithmetical progression of i . In the table given below the figures used are those of the preceding illustration. The first average is that of the first two months' sales; the second, of the first three months' sales, etc. The table shows the continuous growth of the business; but the differences between the successive progressive averages are not so significant as the differences between the successive mov- ing averages, because the former are borne down by the smaller sales of the first months and years. Moreover, an increase of sales in an early month increases the progressive average for that month to a greater extent than the same increase in sales in a later month will increase the progessive average of that month, because the total sales in the latter case are divided by a larger number of months. The last column of the table indicates the months in which the sales run above or below the average. It AVERAGE 35 will be noted that the current month is not included in the aver- age used as a base. If it were included, any increase or decrease in the month's sales would affect the standard as well as the month compared. Following is an explanation of the methods of determining the figures appearing in the various columns : Progressive average: February line (20 + 1 7) -J- 2 18.5 March line (20 + 17 + 21) -r- 3 1Q.3 Increase or decrease"^ in progressive average: Average for first three months 19.3 " " " two months 18.5 Increase .8 More or less* than progressive average: Sales of March, 1913 21 Progressive average for prior months 18.5 Increase 2.5 Progressive Average Increase or More or DECREASE* IN LESS* THAN Progressive PROGRESSIVE PROGRESSIVE I9I3 Sales AVERAGE AVERAGE AVERAGE January 20 February 17 18.5 March 21 19-3 .8 2-5 April 17 18.8 .5* 2.3* May 26 20.2 1.4 7.2 June 40 23-5 3-3 19.8 July 28 24.1 .6 4-5 August 21 23-8 •3* 3-1* September 36 25.1 1-3 12.2 October 55 28.1 30 29.9 November 27 28.0 .1* I.I* December 27 27.9 .1* I.O* 36 MATHEMATICS OF ACCOUNTING AND FINANCE Progressive Average — Continued Increase or More or decrease* in less* than Progressive progressive progressive I9I4 Sales average average average January S3 29.8 1.9 25-1 February 47 3I-I 1-3 17.2 March 47 32.1 I.O 15-9 April 45 32.9 .8 12.9 May 32 32.9 .0 .9* June 32 32.8 .1* .9* July 24 32.4 .4* 8.4* August 17 31.6 .8* 15-4* September 39 32.0 •4 7-4 October 42 32.4 •4 10. November 20 319 •5* 12.4* December 27 31-7 .2* 4.9* 1915 January 45 32-2 •S 133 February 43 32.7 •s 10.8 March 39 32.9 .2 6.3 April 38 33.0 .1 5-1 May 32 33-0 .0 I.O* June 25 32.7 .3* 8.0* July 27 32.5 .2* S-7* August 32 32.S .0 •S* September 49 330 •5 16.S October 85 34-6 1.6 52.0 November 67 35-5 •9 32.4 December 47 35.8 •3 II-5 1916 January 92 37-3 1-5 56.2 February 97 38.9 1.6 59-7 March 83 40.0 I.I 44.1 April 44 40.1 .1 4.0 May 33 40.0 .1* 7.1* June 42 40.0 .0 2.0 July 38 40.0 .0 2.0* AVERAGE 37 Progressive Average — Continued Increase or More or DECREASE* IN LESS* THAN Progressive PROGRESSIVE PROGRESSIVE 19 1 6 — Cont. Sales average AVERAGE AVERAGE August 55 40-3 •3 15.0 September 80 41.2 •9 39-7 October 97 42.4 1.2 SS-8 November 84 43-3 •9 41.6 December 58 43-6 •3 14.7 Periodic Average In order to show the variation in the volume of business be- tween seasons, periodic average may be utilized, as in the illus- tration below, in which a simple average is taken of the figures for the same month in the years 1913-1916 inclusive: Month 1913 1914 1915 1916 Total Average January 20 53 45 92 210 52.5 February 17 47 43 97 204 51.0 March 21 47 39 83 190 47-5 April 17 45 38 44 144 36.0 May 26 32 32 33 123 30.7s June 40 32 25 42 139 34.75 July 28 24 27 38 117 29.25 August 21 17 32 55 125 31-25 September 36 39 49 80 204 5I-0 October 55 42 85 97 279 69-75 November 27 20 67 84 198 49-5° December 27 27 47 58 159 39-75 The principle of progressive average may be utilized in con- nection with the principle of periodic average, as indicated in the following table : i^EAR January Progressive average More or LESS* than preceding PROGRESSIVE AVERAGE I913 20 I914 53 36.5 I915 45 39-3 8.5 I916 92 52.5 52.7 38 MATHEMATICS OF ACCOUNTING AND FINANCE Each progressive average in the foregoing illustration is also a periodic average. This organization of the numbers and aver- ages makes possible a comparison of the sales of each month with the average sales of the same month in all preceding years. The figures in the final column show the difference between the Janu- ary sales of the year and the progressive average of the January sales of the preceding years. Thus the figure 45 for 191 5 is 8.5 greater than the 36.5 progressive average for January, 191 3, and 1914. Weighted Average When the values entering into the computation of an average differ in two or more particulars, a simple average is impossible. Take the following case, for example : 3 men earn $5.00 per day S " " 6.00 " " 4 " " 7-00 " " In computing the average daily wage of these twelve men, the fact must be recognized that the wage payments differ in two particulars: 1 . The daily wage 2. The number of men receiving each wage The daily average is found by dividing the aggregate of their daily wages by their number, as shown below : Men Wage Product 3 $5 $15 5 6 30 4 7 28 12 $73 $73 -^ 12 = $6Vi2, the average daily wage This example serves to illustrate the principle that each value must be weighted by multiplying it by the number of units to AVERAGE $9 which the value is applicable. Additional illustrations will make the principle clearer. Illustration i What is the average rate of interest earned on the following investments made for one year? $100 at 7% $5,000 at 6% $15,000 at 5% Solution: Principal Rate Product $ lOO 7% $ 7.00 5,000 6% 300.00 15,000 5% 750.00 $20,100 $1,057.00 i)057 ^ 20,100 = 5.258%, the average rate of interest Illustration 2 What is the average life of a plant, the various fixed assets of which have the following costs and estimated lives? ( Class of Asset Cost Life A $ 3,000.00 5 years B 15,000.00 10 " C 35.000.00 20 " Solution: Depreclation Annual Class Cost Life Rate D EPRECIATIC A $ 3,000.00 5 years 20% $ 600.00 B 15,000 .00 10 " 10% 1,500.00 C 35,000, .00 -^0 •' 5% 1,750.00 $53,000. 00 $3,850.00 53,000-^ 3.850= 13.76+ ^o the average life 40 MATHEMATICS OF ACCOUNTING AND FINANCE Illustration 3 What is the average per cent of increase in the cost of manufacture under the following conditions: Where one pound of each item is used in the manufacture of each completed article, and the cost of each iiem has increased by the per cent shown in the last column of the first of the sub- joined tables? Material Cost per Pound Per cent Increase 1917 918 A $4 200 B 5 10 C 8 6X D IS 10 E 18 20 Solution i : No. Cost per Per cent Material Pounds Pound Increase I NCREASE .1917 1918 1918 A $ 4 200 $8.00 B 5 10 •50 C 8 ty^ •50 D IS 10 1.50 E 18 20 3.60 $14.10 14.10 -^ 50 = 28.2%, average increase Solution 2: Cost per Per cent Per cent Weighted Material Pound OF Total Increase Per cent 1917 1918 1918 A $4 8 200 16.0 B S 10 10 I.O C 8 16 6K I.O D IS 30 10 30 E 18 36 20 7.2 $50 28.2 AVERAGE 41 Illustration 4 The same conditions are taken here as in the preceding illustration, except that the items composing the finished article are each of a different weight, as indicated in the second column of the following two solutions: Solution i: No. % Cost per Weighted % Weighted Material Pounds Total Pound Product Increase % 1917 1917 1918 1918 A 2 10 $ 4 $ 40 200 80 B 8 40 5 200 10 20 C 4 20 8 160 (^Va 10 D 4 20 15 300 10 30 E 2 10 iS 180 20 36 20 100 fooo 176 -^ 880 = 20%, weighted average percent 176 Solution 2: No. Cost per Total % Material Pounds Pound Cost Increase Increase 1917 1917 1918 1918 A 2 $ 4 $ 8 200 $16.00 B 8 5 40 10 4.00 C 4 8 ^2 6>^ 2.00 D 4 15 60 10 6.00 E 2 18 36 20 7.20 $176 $35-20 35.20 -j- 176 = 20%, weighted average per cent CHAPTER V AVERAGING ACCOUNTS Settling an Account The object of averaging an account is to determine a single date, known as the average date, on which the account may be settled with fairness to both debtor and creditor. As a simple illustration of how an account is averaged, sup- pose B's account with A, to whom he is indebted, is as follows: March i — 60 days $1,000 March 31 — 60 days 1,000 The first item of this account is due on April 30 and the second on May 30. B may pay each amount at its maturity, or the entire $2,000 at the average maturity, which is May 15. He can make an equitable settlement on this average date because the time he gains in deferring payment of the first $1,000 for fifteen days is exactly offset by the time he loses in paying the second $1,000 the same number of days in advance of the due date. It may be desirable to compute the average date for the purpose of dating or determining the maturity of a note or other document given in settlement of an account. For instance, B might cover his account by giving A a single non-interest bearing note for $2,000, due May 15, instead of two non-interest bearing notes of $1,000 each, one due April 30, and the other due May 30. Calculating Interest The average date may also be desired for the purpose of com- puting interest on the balance instead of the individual items of an account. For instance, if A should settle the account on June 14 by a single payment of $2,000, equity would require that he pay interest, say, at 6% as follows: 42 AVERAGING ACCOUNTS 43 On $i,ooo from April 30 to June 14 — 45 days $ 7.50 On $1,000 from May 30 to June 14 — 15 days 2.50 Total $10.00 The interest might, however, be computed on the entire bal- ance of $2,000 for a period of thirty days from the average date, May 15 to June 14. Its amount in this case would also be $10. Assume that instead of paying cash, B, on April 14 gave A a note for $2,000 due in two months. The maturity of this note would be June 14, or thirty days after the average date. An equitable settlement would require the addition of $10 interest to the face of the note, which would make it $2,010. Items of Varying Amounts In the foregoing illustrations the amounts are the same and only the varying number of days has had to be considered in arriving at the average date. If, however, the amounts are not the same, they must also be considered. Suppose, for example, that B's account with A was as follows: March i — 60 days $2,000 " 31 — 60 days 1,000 The $2,000 item is due April 30 and the $1,000 item is due May 30. In this case the average date of maturity for the total of $3,000 would not be May 15 as in the previous example, since the payment of $2,000 fifteen days after maturity would not be off- set by the payment of $1,000 fifteen days before maturity. It would be May 10, because $2,000 paid ten days after maturity would be counterbalanced by $1,000 paid twenty days before maturity. It is evident from this that in averaging accounts due consideration must be given to amounts as well as to dates. The dates to be used in averaging an account are those at which the items may be assumed to have a cash value equivalent to the amount at which they appear in the accounts. Thus, sales on cash terms, being due on the day of sale, take their invoice 44 MATHEMATICS OF ACCOUNTING AND FINANCE dates in averaging; sales with credit terms take the dates on which the invoices are due; returns and allowances take the dates when the invoices to which they apply are due; an interest-bear- ing note takes the date of the note since the face is the cash value at that date; a non-interest bearing note is not worth its face until due and hence its maturity is used in the calculation of the average date. Focal Date The average dates in the preceding illustrations were deter- mined by inspection. When an account is not so simple the com- putation of its average date requires a method involving the principle of weighted average and the selection of a basic date for calculating the time of each item. This basic date is called the focal date and the one most advantageously employed is the last day of the month preceding the earliest date used in averaging. Take, for example, the following account: Date of Terms of Date Used Transaction Payment in Averaging Amount March i 60 days April 30 $2,000.00 " 31 60 " May 30 1,000.00 The first date to be used in averaging the account is April 30. Hence March 31, the last day of the preceding month, is selected for the focal date. The subsequent steps in the process of averag- ing the account may be enumerated as follows: 1. Assume that each item is paid on the focal date. 2. Determine the number of days each item would be pre- paid if it were paid on the focal date. 3. Multiply its amount by this number of days. 4. Add the products thus obtained. 5. Divide this sum by the total of the items; the quotient represents the number of days the focal date precedes the average date. AVERAGING ACCOUNTS 45 Rules Applied The application of these rules to the foregoing account is as follows : 1. The two items are assumed to be paid on March 31, the focal date taken. 2. The first item is therefore assumed to be prepaid 30 days, and the second 60 days. 3. Paying $2,000 30 days before it is due is equivalent to paying $1, 60,000 (2,000 X 30) days before it is due; and paying $1 ,000 60 days before it is due is equivalent to paying $1 , 60,000 (i ,000 X 60) days before it is due. 4. Hence the two assumed prepayments are equivalent to a prepayment of $1 by 120,000 days. 5. One dollar prepaid 120,000 days is equivalent to $3,000 prepaid 1/3 ,000 of 1 20,000 days, or 40 days. The aver- age date is therefore 40 days forward from March 31. Forty days are taken as the equivalent of i month and 10 days, making the average date May 10. The computation of the average date is shown in tabular form as follows : Date of Terms of Date in Time from Focal Transaction Payment Average to Maturity Date Amount Product March i 60 days April 30 30 days $2,000 $60,000 March 31 60 " May 30 60 " 1,000 60,000 $3,000 $120,000 Dividing the sum of the products by the sum of the amounts (120,000 -^ 3,000) gives 40, or the number of days the average date follows the focal date. Reducing Days to Months When the time between the focal date and the date of any of the items in the account is more than one month, the reduction of this time to days may be avoided by the method outlined below : 46 MATHEMATICS OF ACCOUNTING AND FINANCE 1 . Express the time in months and days. 2. Multiply the amount of each item by the number of months to obtain a product of months; and by the number of days to obtain a product of days. 3. Add the products of months; also the products of days. 4. Reduce the sum of the products of months to days by multiplying by 30. 5. To the product of days thus obtained, add the product of days previously obtained. 6. Divide this sum by the balance of the account to find the time in days between the focal and the average dates. 7. Reduce this time to months and days on the basis of thirty days to a month. This method is illustrated in the following example, in which the focal date taken is February 28, being the last day of the month preceding the earliest date in the average, which is March 3. Date of Terms of Date in Time Product Transaction Payment Average Mos 1. Days Amount Mos. Days March 3 cash March 3 3 $250 $ $ 750 March 18 I month April 18 I 18 500 500 9,000 April 10 30 days May 10 2 10 200 400 2,000 May 8 cash May 8 2 8 400 800 3,200 $1,350 $1,700 $14,950 30 X 1,700=51,000 $65,950 65,950 -T- 1,350 =48 115/135. or 49 days 49 days = I month and 19 days One month and nineteen days forward from February 28, the focal date, is April 19, which is the average date. This computation is based on the assumption that there are 30 days in each month and 360 days in a year. Strictly speaking, however, the item due May 10 runs for seventy-one days instead AVERAGING ACCOUNTS 47 of seventy, and the item due May 8 runs for sixty-nine days in- stead of sixty-eight. The result, however, would not be ma- terially modified if the exact number of days were taken. The other method is, therefore, sufficiently accurate for all ordinary commercial transactions. Compound Average The average date of an account is computed by simple aver- age, when the account contains either debit or credit items, but not both. When both debits and credits are included, the aver- age date is found by means of compound average, which involves the following steps: 1. Determine the products of months and days for the debits and credits. 2. Determine the diff'erence between the sum of the debit products and the sum of the credit products. 3. Divide this difference by the balance of the account. 4. If the difference of the products is on the same side of the account as its balance, the average date is forward from the focal date; but if the difference of the products and the balance of the account are on different sides, the average date is backward from the focal date. This latter condition rarely occurs if the focal date selected is prior to the dates of the items in the account. Example of Compound Average In illustrating compound averaging the following account is taken : Debit Credit June I $500 July 5 Note (2 mo. with- " 20, I mo 400 out int.) $500 July 10 600 " 10 Returns (Inv. August 5 500 June 20) 50 August I Cash 300 48 MATHEMATICS OF ACCOUNTING AND FINANCE As the July 5 th note on the credit side of the account is non- interest bearing, it does not have a cash value of $500 until its maturity on September 5. It therefore takes this date in the average. The July loth credit, being an offset to the debit of June 20, takes the same maturity date in the average as the debit item, or July 20th. As June i is the earliest date on which any of the items in the account has a cash value equal to the face of the item, the most convenient focal date is May 31. The sum of the products of months and days for the debit items of the account is found as follows : n Date of Terms of Date in Transaction Payment Average Mos. June I June i o " 20 I month July 20 i July 10 July 10 I August 5 August 5 2 Total debits $2,000 $2,000 $17,000 30 X 2,000 = 60,000 Sum of the debit products $77,000 The sum of the products of months and days for the credit items is found by the following computation. Date of Terms of Date in ^ime Products Transaction Payment Average Mos. Days Amount Mos. Days ^500 $1,500 $ 2,500 SO 50 1,000 300 600 300 T Products )ays Amount Mos. Days I $500 $ $ 500 20 400 400 8,000 10 600 600 6,000 5 500 1,000 2,500 July 5 2 months Sept. 5 3 5 July 10 offset June 20 Dr. July 20 I 20 August I August I 2 I Total credits 50 $2,150 $ 3,800 2,150 X 30 = 64,500 Sum of the credit products $68,300 The account has a debit balance of $1,150 and the sum of its debit products exceeds the sum of its credit products by 8,700. AVERAGING ACCOUNTS 49 The average date is forward from May 3 1 , the focal date, as many days as the number of times 1,150 is contained in 8,700, or ap- proximately 8. June 8 is, therefore, the focal date. The account could be averaged by deducting the $50 credit from the June 20th debit of $400 and dealing only with the net debit of $350. Another Illustration To illustrate the conditions under which the average date is backward from the focal date, assume that the July 5th non- interest bearing note was due in six months. All of the debit items are due on or before August 5, but since the creditor would have to wait until January 5 for the $500 payable on the note, which in the meantime would earn no interest, the balance of the account should carry an early average maturity. The sum of the debit products of days would be computed as in the preceding example and would total 77,000, while the sum of the credit pro- ducts of days would be computed as follows: Date of Terms of Date in Time Product Trans.\ction Payment Average Mos. Days Amount Mos. Days July 5 6 months January 57 5 $500 $3,500 $ 2,500 July 10 offset July 20 i 20 50 50 1,000 August I August 12 I 300 600 300 Total credits $850 $4,150 $ 3,800 30 X 4,150 = 124,500 Sum of the credit products $128,300 The balance of the account would still be a debit of $1,150, but the difference of the products of days would now be on the credit side, and would amount to 51,300. The average date would, therefore, be backward from June i, the focal date, by forty-five days, or the number of times $1,150 is contained in 51,300. Counting thirty days to the month, the average date would be one month and fifteen days backward from the focal date and would, therefore, be April 15. CHAPTER VI PERCENTAGE Percentage Percentage is a method of computing by hundredths. The symbol % means per cent or hundredths. A rate per cent is equivalent to a common fraction the numerator of which is ex- pressed and the denominator of which is indicated by the symbol % as being loo. Thus the same facts may be stated in the form of a common fraction, a decimal fraction or a per cent. The following are equivalent: Common Fractions Decim.\l Fractions % 17/100 .17 17 iX 1. 25 125 iH 7-So 750 12/4 3.00 300 Terms Used in Percentage Base. The number of which a given per cent is to be taken is called the base. Rate. The per cent of the base to be taken is called the rate. Percentage. The result obtained by taking a certain per cent of the base is called the percentage. Fundamental Processes All mathematical computations involving percentage may be grouped under three headings : I. To find a given per cent of a number; that is, to find the percentage. Rule: Multiply the base by the rate Example: $60 X 20% = $12 Base X Rate = Percentage 50 PERCENTAGE 5 1 2. To find what per cent one number is of another; that is, to find the rate. Rule: Divide the percentage by the base Example: $12 -r- $60 = 20% Percentage -H Base = Rate 3. To find a number when a certain per cent of it is known; that is, to find the base. Rule: Divide the percentage by the rate Example: $12 -7- 20% = $60 Percentage -^ Rate = Base Percentage of Increase and Decrease Percentage is frequently employed to compare numbers and to show how much larger or smaller one number is than the other. No new mathematical principles are involved in such computa- tions, as may be shown by the following illustrations: 1 . Percentage of Increase. The sales of a certain business in May, 1919, were $16,000 while the sales in May, 1920, were $18,000. The smaller number is taken as the base; the difference be- tween the two numbers is the percentage of increase. Then 2,000 -f- 16,000 = 12^% Percentage of increase -^ Base = Per cent of increase 2. Percentage of Decrease. The profits of a business for the year 1919 were $20,000 while the profits for 1920 were $17,000. The larger number is taken as the base; the difference be- tween the two numbers is the percentage of decrease. Then 3,000 -7- 20,000 = 15% Percentage of decrease 4- Base = Per cent of decrease Some Applications of Percentage in Business Comparisons Business statistics may be tabulated and compared on a per- centage basis to determine the relative effectiveness, desirabihty Number Per Cent Sold Sold 293 79.6 316 64.1 582 91. 1 416 79-5 52 MATHEMATICS OF ACCOUNTING AND FINANCE or productivity of similar factors. The illustrations given in this chapter are intended to be suggestive but not exhaustive. The following tabulation shows the number of units of a cer- tain commodity purchased from various manufacturers during a year, the number sold, and the per cent sold. A comparison of the per cents indicates the relative salability of the goods pur- chased from the various manufacturers. Per Cent of Goods Sold — Various Manufacturers Number Manufacturer Purchased Walker & Co 368 White & Dudley 493 Davis Mfg. Co 639 Barton-Walsh 523 The following tabulation compares the sales of various sales- men during a month. Each man's sales (as a percentage) divided by the total sales (as a base) produces a rate which measures his portion of the total. Monthly Sales Compared on a Percentage Basis Name Sales Per Cent of Total Arthur Bradley $ 1,264.90 7.8 J.B.Henderson 1,913.52 11. 8 Fred Bates 1,732.69 10.6 Arthur Dutton 2,213.72 13.6 J. L. Weston 1,963.45 12.1 Carter Doane 1,627.32 lo.o Walter S. Waite 1,692.18 10.4 Frank Chesley 2,138.45 13. i Harold Peters 1,728.46 10.6 Total $16,274.69 loo.o The statistics may be so arranged as to obtain two percentage analyses, as illustrated in the following tabulation which shows PERCENTAGE 53 what per cent of the sales of the week was made by each salesman, and what per cent was made each day. Sales for the Week Ending December i8, 1920 D\\^ Monday Tuesday Wednesday Thursday Friday Saturday Salesmen's totals Per cents Smith I362.50 415-75 396.21 472.96 387.29 493-89 $2,528.60 Brown J562.83 475-92 415-60 516.29 42936 562.64 32,962.64 Jones I862.94 732.83 769-42 640.20 721.32 816.25 $4,542.96 White 5126.39 143-62 129.38 145-17 96.27 163.92 504.75 Daily Totals $1,914.66 1,768.12 1,710.61 1,774-62 1,634-24 2,036.70 $10,838.95 Per Cents 17-6 16.3 15.8 16.4 15-1 The following tabulation illustrates the use of percentage of increase and decrease as a means of comparing the sales of each department of a store on the corresponding days of two years. Comparison of Sales by Departments Department Sales-Thursday December is. 1919 Sales-Thursday December 16, 1920 Increase Decrease* % Increase — Decrease* A $ 826.95 1.034-78 1.237.62 2,643.80 1,413.80 962.40 2,642.16 1.964-39 1,636.48 1,213.42 $ 914-32 1,231.64 1,196.14 2.843-27 1.376.29 1,235.96 2,927.92 2,129.80 1,596.27 1,723-96 $ 87-37 196.86 41.48* 199.47 37-51* 273-56 285-76 165.41 40.21* 510.54 10.57 19.02 3-35* 7-54 2.6s* 28.42 10.81 8.42 2.46* 42.07 B C D E f G H I J Total $15,575-80 S17. 175-57 Si, 599-77 10.27 The average of a number of quantities may be accepted as the basis of comparison, the relation of each quantity to the average 54 MATHEMATICS OF ACCOUNTING AND FINANCE being shown in terms of per cents. The figures in the preceding tabulation of "Monthly Sales Compared on a Percentage Basis" are used for the following illustration: Individual Sales Compared with Average Name Sales Per Cent of Average Arthur Bradley $1,264.90 69.95 J.B.Henderson .... 1,913.52 105.82 Fred Bates 1,732.69 95.82 Arthur Button 2,213.72 122.42 J.L.Weston 1,963.45 108.58 Carter Doane 1,627.32 89.99 Walter' S. Waite 1,692.18 93.58 Frank Chesley 2,138.45 118.26 Harold Peters 1,728.46 95-58 Average $1,808.30 100.00 Or the maximum may be accepted as the basis of comparison, the relation of all quantities to the maximum being shown in terms of per cents. Using the same statistics for an illustration: Individual Sales Compared with Maximum Name Sales Per Cent of Maximum Arthur Bradley $1,264.90 57-i4 J.B.Henderson 1,913.52 86.44 Fred Bates 1,732.69 78.27 Arthur Button 2,213.72 100.00 J. L. Weston 1,963.45 88.69 Carter Boane 1,627.32 73-5i Walter S. Waite 1,692.18 76.44 Frank Chesley 2 138.45 96.60 Harold Peters 1,728.46 78.08 The following tabulation is suggestive of the use which may be made of percentage in comparing quantities with two or more similar quantities and with the average thereof. In this case the average is a progressive one. PERCENTAGE Comparison of Sales of Successive Years 55 Year Sales Inc. or Dec* FROM Preceding Year Inc. or Dec* from First Year Progressive Average Prior Years Inc. or Dec* FROM Progressive Average Amount Amount rf /O Amount % 1917 1918 1919 1920 $200,000 238,000 190,000 285,000 $38,000 48,000* 95.000 19.00 20.17* 50.00 $38,000 10,000* 85,000 19.00 s-oo* 42.50 $219,000 209,333 $29,000* 75.667 13.24* 36.15 Apportionment When a quantity is to be divided or partitioned, the basis of the partition may be expressed in rates per cent. The partition is then accompHshed by applying to the base a number of rates, the total of which is 100%. The division of partnership profits is a familiar illustration. Division of Profits Partners P. & L. Ratio Profits A 20% $3,200.00 B 35% s,6oo.oo C 45% 7,200.00 Total 100% $16,000.00 Or the apportionment may be accomphshed by applying the same rate to a number of bases to obtain the desired percentages. In this case the rate is computed by dividing the total percentage by the total of the bases. The distribution of factory overhead is illustrative. Distribution of Factory Overhead (Direct Labor Cost Basis) Total direct labor, all departments $5,295.00 (base) Total factory overhead 3,460.00 (percentage) Then 3,460 -^ 5,295 = 65.34+ % (rate) 56 MATHEMATICS OF ACCOUNTING AND FINANCE Since the rate is approximate only, the distribution will not be exact; a remainder of 24 cents will be undistributed. Process or Proportion Department Direct Labor of Overhead I $2,140.00 $1,398.28 2 1,965-00 1,283.93 3 1,190.00 777-55 Total $5.295-00 $3,459.76 Gross Profit Method of Approximating Inventory The rate of gross profit of prior periods may be used to ap- proximate an inventory when it is impracticable or impossible to take a physical inventory. This is accompHshed by utilizing the elements involved in the computation of gross profits. In a bal- anced table, when all but one element is known, the unknown element is found as the amount necessary to balance the table. The gross profit on sales may be computed by setting up a mer- chandise account as follows : Merchandise Inventory, Jan. i . . .$100,000.00 Sales $350,000.00 Purchases 300,000.00 Inventory, Dec. 31 105,000.00 The gross profit would be $55,000.00, the amount necessary to bring the account into balance. Now if the inventory were not known, but the gross profit could be estimated at $55,000.00, the inventory could be determined thus: Merchandise Inventory, Jan. i $100,000.00 Sales $350,000.00 Purchases 300,000.00 Gross profit 55,000.00 The inventory would be $105,000, the amount necessary to bring the account into balance. Of course the gross profit could not be definitely ascertained without an inventory, but it could be approximated by using the PERCENTAGE 57 average rate of gross profit on sales of former years, if no radical variations have occurred in this rate and if there is no reason to believe that the rate of the current period has been radically different from the average rate of the past. To illustrate, let us assume that the sales and gross profits of the business whose merchandise account appears above, were as follows : Year S.vles Gross Profit Per Cent Third preceding .... $200,000.00 $31,400.00 15.7 Second " .... 310,000.00 48,050.00 15.5 First " .... 385,000.00 60,170.00 15.6 ,000.00 $139,620.00 15.6 The annual rates are computed to determine whether there has been any considerable variation in the rates of gross profit. Then, on the assumption that the rate of gross profit for the cur- rent period was the same as the average of the rates of the three last preceding years: 15.6% of $350,000.00 (sales) = $54,600.00, approximate gross profit Then: Inventory, January i $100,000.00 Add purchases 300,000.00 Total $400,000.00 Deduct cost of goods sold (approximate) : Sales $350,000.00 Less estimated gross profit 54,600.00 295,400.00 Inventory, Dec. 31 (approximate) $104,600.00 The inventory thus computed is $400 less than that shown by the merchandise account. The following problem from a C. P. A. examination will fur- ther illustrate the method, the principal uses of which are in 58 MATHEMATICS OF ACCOUNTING AND FINANCE approximating the value of merchandise destroyed by fire and in applying the gross profit test to the verification of an inventory. Problem The accountant is called on to confirm the inventory of a mercantile establishment. Investigation shows that inventories have been incor- rectly taken and are padded. It is mutually agreed that all inventories, except the first one, which is to be used as the basis, shall be entirely ignored. The accountant is to ascertain, on a fixed percentage of profit which it is decided shall be 33yj% of sales, what stock should be on hand December 31, 1914, with the following data obtained from the various books: Inventory referred to as a basis, January i, 191 1 $47,350.29 Gross purchases for the year ending December 31, 1911 . . . 76,320.15 Returned purchases 4,350.16 Freight and drayage on purchases 325.14 Gross sales 115,469.31 Returned sales 1,317.12 Gross purchases for the year ending December 31, 191 2 . . . $65,506.80 Returned purchases 3,715.16 Freight and drayage on purchases 41 7- 15 Gross sales 105,716.10 Returned sales 1,215.84 Gross purchases for the year ending December 31, 1913 . . . $62,517.10 Returned purchases 1,314.17 Freight and drayage on purchases 316.17 Gross sales 101,317.18 Returned sales 1,216.06 Gross purchases for the year ending December 31, 1914. . . . $58,715.16 Returned purchases 287.50 Freight and drayage on purchases 290.10 Gross sales 95,371.16 Returned sales 41 7- n Solution: Since the rate of gross profit was constant throughout the four years, and since only the final inventory is required by the problem, the data can be summarized and the four years' totals used in the inventory calculation. PERCENTAGE 59 Summary 1911-1914 Returned Returned Year Purchases Purchases Freight Sales Sales 1911 $76,320.15 $4,350.16 $325-14 $115,469.31 $1,317.12 1912 65,506.80 3,715-16 417-15 105,716.10 1,215.84 1913 62,517.10 1,314-17 316.17 101,317.18 1,216.06 1914 58,715-16 $263,059.21 287.50 290.10 95,371-16 417. II Total $9,666.99 $1,348.56 $417,873.75 $4,166.13 Statement of Approximation of Inventory At December 31, 1914 Inventory, January i, 19 11 $47,350.29 Add cost of goods purchased, 191 1-1914: Purchases $263,059.21 Z,e55 returned purchases 9,666.99 $253,392.22 Add freight. 1,348.56 254,740.78 $302,091.07 Deduct cost of goods sold, 1911-1914: Sales Less returned sales ^17,873-75 4,166.13 $413,707.62 Less gross profit (s^HVo of sales) 137,902.54 275,805.08 Inventory, December 31, 19 14 (estimated) $26,285.99 Analysis of Statements The following statements indicate the use which may be made of percentage in analyzing the financial statements of a business to show such facts as the ratio of cost of sales, expenses and profits to sales; the relative cost of the various elements of manufactured goods, and the variation in operating costs of different years. Problem From the following data obtained from the books of Johnson and Com- pany, construct a profit and loss statement showing cost of goods manufac- tured, and cost and gross profit of the goods sold. Also show percentage of each eli^ment based upon cost of manufacture and based upon sales. 6o MATHEMATICS OF ACCOUNTING AND FINANCE Raw material, January i, 1915 $42,000,00 " " December 31, 1915 45,000.00 " " purchases during 1915 130,000.00 Freight inward 4,218.00 Wages (productive) 70,000.00 Sundry manufacturing expenses 3,500.00 Sales 280,000.00 Finished goods, January i, 1915 18,000.00 " " December 31, 191 5 22,000.00 Selling expenses 22,000.00 Administrative expenses 20,000.00 Solution: Johnson and Company Profit and Loss Statement Year Ending December 31, 1915 Sales 1280,000.00 Deduct: Cost of goods sold: Raw material: Inventory, Jan. i, 1915 . . $ 42.000.00 Purchases, 1915 130,000.00 Total $172,000.00 Inventory, Dec. 31, 1915. 45,000.00 $127,000.00 Freight inward 4,218.00 Productive labor 70,000.00 Manufacturing expense. . .. 3,500.00 Cost of goods manufactured $204,718.00 Deduct: Inventory variation — finished goods: December 31, 1915 $22,000.00 January i, 1915 i8,ooo.f)0 4,000.00 Cost of goods sold 200,718.00 Gross profit on sales 179,282.00 Deduct selling expenses 22,000.00 Net profit on sales $57,282.00 Deduct administrative expenses. 20,000.00 Net profit on operations $37,282.00 % OF % OF Cost Sales 62.0 2.1 34-2 1.7 71-7 28.3 7.9 PERCENTAGE 6l This illustration, showing the per cent of net profit and gross profit on sales raises the question whether sales or cost of sales should be used as the base in the computation of rates of gross profit. In common parlance, when a statement is made that a sale has resulted in realizing a certain rate of profit, the rate is understood to have been applied to the cost. Thus, if it is said that an article costing $2 was sold at a 10% profit, one assumes that the profit was 20 cents and the selling price $2.20. But in percentage analyses of revenue statements it is much more con- venient to use the net sales as the base. Selling expenses nor- mally are proportionate to sales and the per cent of selling expense is computed on the basis of sales. By computing the cost of goods sold and afl other deductions from sales as percentages of the net sales, the statement begins with 100% as the base and continues on the same basis throughout. But if cost were taken as the base, the sales would be represented by a rate exceeding 100%, and the selling expenses, including such items as advertising, salesmen's commissions, freight out and store expense, would hsive to be rated on the illogical basis of cost. Therefore the percentage analysis of the revenue statement should properly be made on two bases : elements of manufacturing cost, including material, labor and factory overhead, should be considered as percentages of cost; and all deductions from sales, including the cost of goods sold, the selling expenses and the administrative expenses, should be rated as percentages of the net sales. This method was followed in the preceding illustration. As an illustration of the use of percentage in the comparison of successive revenue statements, the following condensed state- ments of the J. E. Smith Wire and Iron Company are presented, together with comparative percentage analyses thereof. The analytical statement shows that the increase in sales and the increase in cost of sales have not been proportionate, as the proportion of cost to sales has steadily increased, causing a corresponding decrease in the percentage of profit. 62 MATHEMATICS OF ACCOUNTING AND FINANCE H O o o o o o o o o o o a, o o o o o in o in o o H CO •* in u Vi J2 u > t: « 5 w o o o o o « z o o o o o H H o o o o o S2 a- 1/5 o in o m Z ^ N N ro M s w ^ Q < M (« z w o o o o o X o o o o o w o o o o o o o o O in z w» vO <^ CO t^ J >J (d W u u <; o o n o o o o O o o o o o o o o o o o o o o O vO m Tt- "5 w u c/) (d o o o o o o o o o o < o o o o o o o o lO o in in o 2: ^ ^ in in « < o. o M N n l>i o a a a o o M W rR ^ ^ « IIJOHJ 00 O M t~ O ct in tt n o ■4 t^ PI r-- d N M M W XldOHtJ xajvi $85,000 70,000 55,000 40,000 50,000 saivs do % (> vO in 0) in 't q Tt in in in rn asNadxa a.MivHXsiNiKav 00000 00000 q q q q q in d in 0' in « P) N ro w 6^ saivg NO xiJOHd xaN do % t^ 00 N m m t~ t- CO ci t.1 N ro W W M W M saivs NO XIdOH(J xaM 00000 00000 (» q d d 0" d in Oi 00 t^ sasNadxg ONmag HO % 0> 10 m M q in in q Tt in in ^ in sasNadxg ONmas 00000 00000 q q q q q d 0" 0" d in m -0 t~ 00 t^ xidoaj SSOHO -iio % t^ 00 in r-p N q ri t^ fO t^ 06 •^% gross profit (1916 rate) on $100,000.00 ad- ditional business done in 191 7 $1 2,500.00 Deduct excess of additional cost to manufacture over increase in selling prices: Additional cost of material $25,000.00 Additional cost of labor 80,000.00 Total $105,000.00 Less decrease in cost of manufacturing expense 2,500.00 Net excess in manufacturing cost $102,500.00 Less increase in selling prices 100,000.00 2,500.00 Excess of gross profit of 191 7 over 191 6 $10,000.00 Deduct addhional selling expenses of 191 7. . . 5,000.00 Additional net profit of 191 7 $5,000.00 CHAPTER VII EQUATIONS IN THE SOLUTION OF PROBLEMS Solving Equations Problems may often be solved by stating the conditions in the form of an equation and solving the equation by applying one or more of the following processes : 1. Multiplying both sides of the equation by the same number 2. Dividing both sides of the equation by the same number 3. Adding the same number to both sides of the equation 4. Subtracting the same number from both sides of the equa- tion Illustration i A manufacturer produced a certain commodity which he sold the first year at a certain price; he raised the price 25% the second year; increased that price 20% the third year; and in the fourth year he increased the third-year price by 1673%. The price the fourth year was $35. What was the price the first year? Solution: In order to obtain an equation it is necessary to represent some value by 100%. The value chosen to be represented by 100% will depend on the conditions of the problem; where convenient, 100% should represent the value required by the problem. In this case — Let 100% = the selling price the first year then 100% X 125% = 125% the selling price the second year and 125% X 120% =150% " " " " third " and 150% X ii6V3% = i75% " " " " fourth " Since the selling price the fourth year was $35 we obtain the equation: 175% = l3S 66 EQUATIONS IN THE SOLUTION OF PROBLEMS 67 But it is desired to determine 100%, which is accomplished by dividing both ?ides of the equation by 1.75: 100% = $20, the selling price the first year Illustration 2 A entered into partnership with B and was to act as manager of the business. Before dividing profits equally with B, A was to receive a special bonus of 25% of the net profit. Before calculating A's commission, the profits were shown by the revenue statement to be $5,000. How should the $5,000 be divided between A and B? Solution: This problem illustrates the difficulty which frequently arises in interpreting contracts which provide that commissions and bonuses shall be determined as percentages of the net profit. The difii- culty arises from the uncertainty as to what is the amount of the net profit; for, if the bonus is to be considered as an expense of the business, the net profits are less than $5,000; if the bonus is not to be considered as an expense but as part of the distribution of profits, the net profit is $5,000. From the statement of the problem, it is impossible to tell whether or not the commission is to be considered as an expense; hence it is necessary to give two solutions. Assuming the bonus is not an expense: Since the bonus is 25% of the net profit, the net profit must be 100%. Since the bonus is not an expense, the $5,000 is all to be considered net profit — Then, 100% = $5,000, the net profit and 25% = 1,250, the bonus and 75% = $3,750, the remaining profit to be divided equally. 37^2% = $1,875, share to A and B each Therefore, the division is as follows: A B Total Rate Bonus $1,250 $1,250 25% Remainder, K each 1,875 $1,875 3,75° 75% Total $3,125 $1,875 $5,000 100% 68 MATHEMATICS OF ACCOUNTING AND FINANCE Assuming that the bonus is an expense to be deducted from the $5,000 to obtain the net profit: 25% = bonus 100% = net profit 125% = bonus plus profit, or $5,000 Then 100% = $4,000, net profit and 25% = $1,000, bonus Proof 125% = $5,000 The division of the $5,000 on this assumption would be: A B Total Rate Gross profit $5,000 125% Deduct bonus $1,000 1,000 25% Net profit, y2 each 2,000 $2,000 $4,000 100% Total distribution $3,000 $2,000 The net difference of $125 in the distribution under the two interpretations emphasizes the necessity for care in drawing up contracts of this character. In such cases an accountant should always be consulted as to the wording of the contract, as any competent accountant would recognize the danger of there being two constructions placed on the contract, with a consequent dispute. Illustration 3 (From Ohio C. P. A. Examination, October, 1919) The American Manufacturing Company commenced business on January i, 1918, with a paid-up cash capital equal to the sales for the year 1918. The net profits for the year 1918 were $26,100. Of the total charges to manufacturing during the year, 40% was for materials, 30% for productive labor, and 30% for manufacturing ex- EQUATIONS IN THE SOLUTION OF PROBLEMS 69 penses (including 5% depreciation on plant and machinery, amounting to $3,000). The value of the materials used was 80% ot the amount purchased, and 90% of the amount purchased was paid during the year. The inventory value of finished goods on hand at December 31, 1918, was 10% of the cost of finished units delivered to the warehouse, and the work in process at that date was equal to 50% of the cost of units delivered to the warehouse. The selling and administrative expenses were equal to 20% of the sales; also to 40% of the cost of goods sold. Ninety per cent of these ex- penses were paid during the year 1918. Plant and machinery purchased during the year were paid for in cash. All labor and manufacturing expenses (exclusive of depreciation) were paid in full up to and including December 31, 1918. Of the total sales for the year, 80% was collected and 1% charged ofT as worthless. From the given data you are required to prepare a balance sheet and a profit and loss statement, showing cost of goods delivered to the warehouse, cost of goods sold, and net profit for the year. Solution: Let sales equal 100% Then (since selling and administrative expense is 20% of sales or 40% of cost of goods sold) the cost of goods sold is half of the sales or 50% and the gross profit is 50% (That is, if a is 20% of x and is also 40% of y, then y must be half of x.) Selling and administrative expenses are equal to 20% and bad debts equal 1% 21% Hence the net profit is 29% Then, 29% = $26,100 and 100% = $90,000, sales for the year, and the cash capital at the beginning of the year. Cost of goods sold = 50% of $90,000 $45,000 Gross profit 45,000 Selling and administrative expense 18,000 Bad debts 900 70 MATHEMATICS OF ACCOUNTING AND FINANCE Since the inventory of finished goods at December 31, i9i8,wasio%of the cost of finished units delivered to the warehouse, the cost of goods sold was 90% of the finished goods manufactured during the year. Then, $45,000-7-90% = $50,000, cost of finished goods manufactured And $50,000 — $45,000 = $5,000, inventory of finished goods at Decem- ber 31, 1918 Since the work in process at December 31, 191 8, was 50% of the cost of finished goods delivered to the warehouse, 50% of $50,000 = $25,000, Work in process inventory $50,000, Goods finished during 1918 25,000, Work in process at Dec. 31, 1918 $75,000, Total manufacturing cost of 1918 40% of $75,000.00 = $30,000, Cost of materials used 30% of 75,000.00 = 22,500 " " productive labor 30% of 75,000.00 = 22,500 " " manufacturing expense Of this manufacturing expense, $3,000 was depreciation on plant and machinery; hence the manufacturing expense paid in cash was $19,500. Since the rate of depreciation was 5%, the cost of plant and machinery was $60,000, all of which was paid for in cash. Since 80% of the material purchased was used in manufacturing, $30,000-^- 80% = $37,500, the cost of the material purchased; and $7,500 is the inventory of raw material at December 31, 191 8. Also 90% of $37,500, or $33,750, is the amount of cash paid for purchases; and $37,500 ~~ ^33-75° = $3,750- the accounts payable at December 31, 1918, for purchases. The selling and administrative expenses were $1 8,000. Of this amount, 90%, or $16,200, was paid in cash. The remainder, $1,800, is an addition to the accounts payable. 80% of the sales of $90,000 were collected. 1% was written off. Hence, Sales $90,000 Less: Cash collections $72,000 Bad debts 900 72,900 Balance of accounts receivable. . . . $17,100 EQUATIONS IN THE SOLUTION OF PROBLEMS 71 The cash summary is : Cash capital paid in $90,000 Collections on accounts receivable 72,000 $162,000 Deduct: Plant and machinery $60,000 Materials 33-75° Productive labor 22,500 Manufacturing expense IQ.500 Selling and administrative expense 16.200 Total disbursements 151,950 Balance $ 10,050 American Manufacturing Company Trial Balance December 31, 191S Capital stock $90,000 Plant and machinery $60,000 Reserve for depreciation, plant and machinery .... 3,000 Sales 90,000 Purchases 3 7,500 Productive labor 22,500 Manufacturing expense 19,500 Depreciation, plant and machinery 3,000 Selling and administrative expense 18,000 Bad debts 900 Accounts receivable 17,100 Accounts payable (3,750 + 1,800) 5, 550 Cash 10,050 $188,550 $188,550 Inventories: raw material, $7,500; goods in process, $25,000; finished goods, $5,000. 72 MATHEMATICS OF ACCOUNTING AND FINANCE American Manufacturing Company Profit and Loss Statement Year Ending December 31, 1918 Sales $90,000.00 Deduct: Cost of goods sold : Material: Purchases $37,500.00 Less inventory, Dec. 31, igi8 7,500.00 $30,000.00 Productive labor 22,500.00 Manufacturing expense 19,500.00 Depreciation — plant and machinery 3,000.00 Total manufacturing cost . $75,000.00 Deduct goods in process — Dec. 31, 1918 25,000.00 Cost of finished goods manu- factured $50,000.00 Deduct inventory finished goods — Dec. 31, 1918 5.000.00 45,000.00 Gross profit on sales $45,000.00 Deduct: Selling and administrative ex- $18,000.00 pense Bad debts 900.00 18,900.00 Net profit $26,100.00 EQUATIONS IN THE SOLUTION OF PROBLEMS 73 American Manufacturing Company Assets Plant and machinery . . Less depreciation . . . . Balance Shej December 31, i . 160,000.00 3,000.00 $ 57,000.00 ET 918 Liabilities Accounts payable . . . Raw material Goods in process 7,500.00 25,000.00 5.000.00 17,100.00 10,050.00 5,550.00 Accounts receivable . . . . Cash $121,650.00 $121,650.00 Illustration 4 As another illustration of the use of percentage in the solution of prob- lems, the following C. P. A. problem is given. The Orinoco Coal Company was incorporated under the laws of the state of Illinois, with an authorized capital of $8,000, divided into eighty shares of the par value of f 100 each, which were subscribed for as follows: Samuel Black 60 shares William Green 10 " John White 10 " Samuel Black was elected president; George Brown, vice-president and manager, and Charles Pinck, secretary and treasurer. Neither Brown nor Pinck held any stock, but were to receive in addition to their salaries a percentage of the profits after charging off all losses from whatever source — Brown 15% and Pinck 10%. These shares in the profits were not to be considered an expense deductible to obtain the basis of the bonuses. At the end of the year a meeting of the stockholders was held and the following balance sheet was presented: Assets Liabilities Cash $ 8,031.12 Accounts payable .... $ 740.22 Accounts receivable . . 817-32 Reserve for bad debts 376.05 Coal 6,644-15 Capital stock 8,000.00 Charles Pinck 2,264.14 Undivided profits .... 8,640.46 $17,756.73 $17,756.73 74 MATHEMATICS OP ACCOUNTING AND FINANCE It was announced that Treasurer Pinck, who was not financially responsible and was not bonded, had disappeared and his account was uncollectible, but would be reduced by crediting the account with his share of the net profits. What amounts should Pinck and Brown receive as salary addition? What is the amount of Pinck's defalcation? If the remaining profit is divided among the stockholders, what divi- dend should each receive? Solution: Since the bonuses allowable to Pinck and Brown are to be calculated on the net profit after deducting all losses, the defalcation must be deducted from the undivided profits of $8,640.46 to obtain the net profit. Let 100% = the net profit, then $8,640.46 — defalcation = 100%. But the amount of the defalcation is not known, except that it is the amount of Pinck's debit balance, $2,264.14, minus his 10% of the net profit ; hence — $2,264.14 — 10% = defalcation Substituting the first term of this equation for "defalcation" in the first equation, we obtain — $8,640.46 — ($2,264.14 — 10%) = 100% Removing the parentheses and changing signs — $8,640.46 — $2,264.14+ 10% = 100% Subtracting 10% from both sides of the equation — $8,640.46 — $2,264.14 = 90% or $6,376.32 = 90% Dividing both sides by 90% — $7,084.80 = 100%, the net profit Then 708.48 = 10%, Pinck's bonus and 1,062.72 = 15%, Brown's bonus $2,264.14, Pinck's debit balance 708.48 " bonus credited '1)555-66 " defalcation EQUATIONS IN THE SOLUTION OF PROBLEMS 75 $8,640.46 undivided profit per balance sheet 1,555.66 Pinck's defalcation $7,084.80 net profit — basis of Brown's and Pinck's bonus $708.48 Pinck's bonus 1,062.72 1,771.20 Brown's bonus and total $5,313.60 profits remaining for dividends Vs of $5,313.60 = $3,985.20, Black's dividend yg " " = 664.20, Green's " 1/8 " " = 664.20, White's " i/g" " = $5,313.60, total (as above) CHAPTER VIII TRADE AND CASH DISCOUNT Trade Discount There are two kinds of discount affecting the amount received for goods sold or the amount paid for goods bought. The first of these is trade discount, which is a device for varying prices without interfering with basic or *'hst" prices, often called retail prices. The convenience of its use arises from the fact that an expensive catalogue can be made permanent by recording only the list prices. The real or trade prices are determined by the trade discounts offered by the seller, which are usually contained in confidential letters or circulars sent to customers. A great saving of expense is effected by not having to issue a new catalogue whenever market prices are modified. Moreover the printing and circulating of large catalogues would take con- siderable time so that it would be impossible to give effect to price revisions until weeks after the necessity for them had arisen. But a circular altering the trade discount and thus raising or lowering the actual prices, can be prepared on a reproducing ma- chine and sent out to customers in one or two days. The way in which a trade discount operates is as follows: A manufacturer or wholesaler sells to a retailer at 75 cents an article the list or gross price of which is $1.25. He bills the article to the buyer at the gross price but then deducts the discount of 40%, thus: 16 dozen of article I15.00 $240.00 Less 40% 96.00 $144.00 In this way the actual or net price of the article isestabhshed at 75 cents. This actual price is the amount entered in the books 76 TRADE AND CASH DISCOUNT 77 of both seller and purchaser, neither of whom makes any record of the list price or the trade discount. Cumulative Trade Discounts In order to provide for fluctuations in price the device of cumulative trade discounts has been adopted. Thus the dis- counts quoted may be 30, 20, 10, and 5. This does not mean a total discount of 65, because each successive rate is calculated on the amount left after deducting the discount at the preceding rate. Thus, if the list price is $240, and the discount is 30, 20, 10, and 5, the computation is as follows: List price $240.00 Less 30% of $240.00 72.00 $72.00 $i6S.oo Less 20% of $168.00 3360 33.60 $134.40 Less 10% of $134.40 13.44 13-44 $120.96 Less 5% of $1 20.96 6.05 6.05 $114-91 $125.09 The total discount of $125.09 is almost exactly 52 3^ % of the gross price of $240. The net or real price is $1 14.91 . If the wholesaler wishes to advance the price, he notifies the trade that the last discount of 5% is discontinued, which will make the amount $120.96, or that the last two discounts of 10 and 5 are replaced by 5 alone, raising the price to $127.68 ($134.40 - $6.72). If the price is to be lowered, it is done by adding a further dis- count. If another 5 is added the net price becomes $109.16 ($114.91 - $5.75). Methods of Finding Net Price In order to avoid the necessity of making a separate computa- tion for each discount, it is well to know how to find one rate that 78 MATHEMATICS OF ACCOUNTING AND FINANCE will give the same result as that reached by the successive steps of the combination rate. The rule for this is : Add the first two discounts; multiply the two discounts; sub- tract the second result from the first. With this result as one dis- count combine the third in the same manner; and take up in turn each of the other discounts. Thus in the illustration — The sum of .30 and .20 is .50 The product of .30 and .20 is .06 .44 The sum of .44 and .10 is .54 The product of .44 and .10 is .044 .496 The sum of .496 and .05 is .546 The product of .496 and .05 is .0248 .5212 Therefore the combination rate is .5212 It is not necessary to remember any rule, because this same result can be reached by computing the discount on $100 thus: List price $100.00 Less 30% 30.00 .30 $70.00 Less 20% 14.00 .14 $56.00 Less 10% 5.60 .056 $50.40 Less s% 2-52 .0252 Net price and total discount rate. . $47.88 .5212 This latter method has the further advantage of being sus- ceptible of easy proof, since the net price and the discount must add to 100. A still better way of reaching the same result is to compute the net amount at once, instead of finding the discount and deduct- TRADE AND CASH DISCOUNT 79 ing it. Thus, instead of ascertaining that 30% of $240 is $72 and deducting it to find $168, it is shorter to multiply 240 by 70%, or rather by .7, to get the same result, and then to multiply 168 by .8 to get 134.40, and so on. In other words, mul- tiply each successive amount by the complement of its discount rate, and the final result will be the net price, with all discounts deducted. In order to find a single rate for the net amount, multiply all the complements of the discount rates. Thus, the product of .7 X .8 X .9 X .95 is .4788, the single rate for the net price when the discounts are 30, 20, 10, and 5. Applying this single rate to $240 will give $114.91, the same result as by the other method. It will save a great deal of labor, as well as insure greater ac- curacy, if tables are made of the net amounts resulting from the application of the discounts usually given by the concern. These tables should be made for amounts of $1 to $100 and should be built up by successive additions and not multiplications. They will prove at every tenth amount, since if the net for $1 is $.4788, it will be $4.7880 for $10, thus proving every intermediate amount, whereas there is no proof if each amount is found by independent multiplication. It is not necessary to carry the tables beyond $100 if all the decimals are used. It is necessary only to be careful to move the decimal point enough spaces to the right to represent the higher numbers. Thus, if the list price is $4,836 and the net rate .4788, the table will show on line 36 the amount of 17.2368 and on line 48 the amount of 22.9824. Therefore: The net of $ 36.00 is $ 17.2368 " " " 4,800.00 " 2,298.24 " " " $4,836.00 " $2,315.48 The same result could be reached with a table of only ten lines, from $1 to $10, but a table of 100 lines shortens the computa- tions and is still easily contained on a comparatively small card. 8o MATHEMATICS OP ACCOUNTING AND FINANCE Cash Discount In order to induce prompt payment of accounts merchants frequently offer to deduct a certain per cent from bills if they are paid within a fixed number of days. This deduction is called a cash discount and is always applied to the net or trade price reached after all trade discounts have been deducted. The discount terms are expressed by the figure of the per cent followed by the number of days in which the discount is allowed and then by the total number of days that may elapse before the bill becomes due, thus: 2/ 10 ; 1/30 ; N/60 which means that if payment is made in 10 days 2% may be de- ducted; if paid in 30 days 1%; and that the bill is due in 60 days net, that is, without any discount. The advantages of giving cash discount are usually said to be that they decrease : 1. Loss from bad debts 2. Cost of collecting accounts 3. Amount of capital tied up in outstanding accounts As only those who are comparatively strong financially are able to avail themselves of discounts offered, the first advantage would not appear to be very often realized. The advantage to the purchaser is that he makes much more than normal interest by taking his discount. Thus, in the case of 2/10; 1/30; N/60, if the 2% is taken the purchaser gains 2% for 50 days' use of the money, which is at the rate of 14.6 per cent per annum, while if only 1% is taken he gains 1% for 30 days, or 1 2% per annum. If he has sufficient bank credit, he can well afford to borrow at 6 or 7% in order to take his discounts. Discount as a Protection against Loss The term "cash discount" is usually understood to refer to de- ductions that amount to a rather heavy interest. In some cases, TRADE AND CASH DISCOUNT 8l however, the deduction allowed is far more than this. For in- stance, one concern manufacturing electrical apparatus sells on terms of 40% discount if paid within 30 days. If not paid within the time the price is at list. So large a discount is not usually considered to come within the definition of a cash discount, al- though it is such, strictly speaking, since it is dependent upon the payment of cash. It would perhaps be better to call it a trade discount with a time limit. A discount of this kind is adopted as a protection against loss in case of the bankruptcy of a customer. If a discount is given purely as a trade discount, a price is established that remains the same whether a bill is paid at maturity or not. If the customer becomes bankrupt, the claim filed with the receiver must be the net with all trade discounts deducted. If the list price is $1,000 and the unconditional trade discount is 40%, the claim must be filed for $600. If the final settlement is for 50%, the creditor loses $300. But if the discount has a time limit of 30 or 60 days, the time will have expired and a claim can be filed for the list price of $1 ,000, on which the dividend will be $500. The creditor will lose only $100, instead of $300. Cash Discount Regarded as an Expense A view of cash discount not very generally accepted is that the net price is the real price and that if the bill is not paid in time the discount is added as a penalty. This, of course, reverses the usual understanding of the subject, which is that discount taken is a profit. If this view is adopted, cash discount taken is eliminated entirely, and discount not taken becomes an ex- pense. To illustrate, if the trade price is $1,000 with an option of a cash discount of 2%, the entries would be as follows: Purchases $980 Cash discount 20 Creditor $1,000 82 MATHEMATICS OF ACCOUNTING AND FINANCE Then if discount is taken, Creditor $i,ooo Cash $980 Cash discount 20 In this case the discount disappears entirely. If the discount is not taken the entry would be : Creditor $1 ,000 Cash $1 ,000 This leaves the charge of $20 in the cash discount account as an expense. CHAPTER IX TURNOVER Indefinite Meaning of " Turnover" The principal difficulty in discussing turnover is to obtain a clear idea of what is meant by the term. In spite of the efiforts of several committees of the American Association of Public Accountants, we are no nearer an authori- tative standard of accounting terminology than we were ten years ago. The principal difficulty in arriving at correct definitions is that few authors on accounting subjects attempt any definitions at all. They seem to take for granted that everyone else attaches the same meaning to a term as they do themselves and that a definition of it is, therefore, unnecessary. There does not seem to be any formal definition of "turnover" in any standard work on accounting, and the word does not appear to be used with any clearcut meaning. R. H. Montgomery comes nearest to defining the word, but even he only suggests a definition. After stating that authorities differ greatly as to what the term means, he says: "Uniformity is desirable in accounting terminology, so the author suggests this definition: The turnover of a merchant or manufacturer repre- sents the number of times his capital in the form of stock-in- trade is re-invested in stock-in-trade during a given period."^ This is the generally accepted definition of turnover; that is, the number of times the merchandise is turned over. Mr. Mont- gomery further says: "To ascertain the turnover, take the start- ing inventory, add the purchases or cost of manufactured goods, and deduct the inventory at the end; divide the total by the start- ' R. H. Montgomery, Auditing Theory and Practice, 1919, p. 455. 83 84 MATHEMATICS OF ACCOUNTING AND FINANCE ing inventory. The result will be the number of times the capital invested in stock-in-trade has been turned over during the period. The calculations are based upon a normal inventory."^ Normal Inventories Necessary The application of this rule to two successive years of a business will exhibit the importance of the qualification that the inventories must be normal. It will also show the difficulty of determining in the case of the majority of concerns how many times the stock-in-trade is turned over. If the starting inventory of the first year is $25,000, the pur- chases $200,000, and the inventory at the end is $50,000, the turnover during the year is 7, since $25,000 -j- $200,000 — $50,000, or $1 75,000, is seven times the first inventory. If in the following year the starting inventory is $50,000, the purchases $175,000, and the ending inventory $25,000, the turnover is 4, since $50,000 + $175,000 — $25,000, or $200,000, is four times the first inven- tory. Thus by the mere accident of a difference in the amount of the inventories at the beginning of the two years, the second year, which did the larger business, shows only a little more than half of the turnover of the first year. As the number of turnovers in the year is supposed to be a measure of the prosperity of the busi- ness, this method of determining it is evidently unsatisfactory. The difficulty is greatly reduced if it is possible to determine the average, normal quantity of stock-in-trade carried. This normal quantity may or may not be the same as the inventory at the beginning of the year as there is no necessary connection between the two. Usually, however, we only know that there was a total turnover of $175,000 or $200,000 during the year. Not knowing the amount of the concern's normal inventory from any figures contained in either its revenue statement or balance sheet, we have no means of finding how many times the stock has been ^ R. H. Montgomery "Auditing Theory and Practice," 1919. P- 455- TURNOVER 85 turned over. This we can ascertain only it inventories are taken monthly or at other intervals throughout the year, the average of which may be regarded as the normal inventory. In any event it is necessary to deline the word "normal" as it may mean the average inventory that should be carried or the inventory that is customarily carried. Different Bases of Comparison Even if this difficulty is met, another presents itself when choosing a basis for comparing the turnovers of two concerns. Mr. Montgomery probably represents at least the majority of American accountants in considering turnover to be the number of times the normal stock-in-trade is reinvested in the goods sold. British accountants, on the other hand, following Lisle, say that it should be the relation between the inventory and the sales. It can readily be seen that a serious misunderstanding results in the comparison of the turnovers of two concerns if one is calculated on the basis of cost and the other on the basis of sales, especially if high selling expenses make the cost of the goods a compara- tively small part of the selling price. This difference of opinion as to the proper basis for comparing turnovers has reference to trading businesses. In the case of manufacturing businesses the disagreement is even greater, as three bases of comparison may be used, raw material, raw ma- terial plus labor, and the total cost of the goods sold. Working Capital as Basis of Turnover Owing to this wide divergence of opinion, it has been sug- gested, and it would seem justifiably, that a better basis for cal- culating the turnover is the working capital of the business. If this method is adopted, most of the difficulties encountered in computing the turnover disappear, because the amount of work- ing capital is indicated in the balance sheet, being the excess of current assets over current liabilities. Except as affected by the 86 MATHEMATICS OF ACCOUNTING AND FINANCE slight increase due to undistributed profits, it remains the same throughout the entire period, and, therefore, affords a stable basis of comparison. Every business, whether it is engaged in trading or manufac- turing, requires capital for two purposes: first, to provide the necessary fixed assets to carry on the operations; and second, to furnish sufficient funds to carry the stock-in-trade and accounts receivable until cash is realized on them and more goods or mate- rial for manufacture are bought. The point that interests the proprietor is how many times a year he can invest his available floating capital by repeating the process of buying the goods, selling them, and collecting the proceeds. If one person with a working capital of $40,000 is able to sell in a year goods costing $200,000, while another with the same amount of working capital is able to sell goods costing only $160,000, the first person has turned over his capital five times to the other person's four times. Some exception is taken to this definition of turnover. It is said that working capital is used for many purposes, including advertising campaigns and similar items of expense that do not affect the cost of manufacture. It is also objected that accounts receivable vary to a great extent between different concerns, as some sell on short time and others on long time, and some keep a big stock of raw materials, while others do not consider it neces- sary, or cannot afford to do so. The question of what constitutes the proper basis for reckon- ing the turnover seems to turn on the object for which the turn- over is used. If it is merely for the purpose of showing that one manager can handle more goods than another with the same average amount of stock on hand, the proper test is the relation between the normal inventory and the cost of the goods sold. But a manager could establish a record on this basis by following the foohsh policy of buying in small quantities at retail prices and paying the high expressage instead of the low freight rates. His turnover would be large, but would lead to disaster. TURNOVER 87 If a profitable business is the object sought for, the use the manager makes of the working capital would seem to be one of the best measures of his success. If too much of the working capital is diverted from the production or purchase of goods for sale to carrying on an extensive advertising campaign, or if it is tempora- rily locked up in long-time accounts receivable, the quantity of goods sold is apt to decrease. This will show itself in a lessened turnover of working capital. On the other hand, if either the beginning or the normal inventory is made the measure, the turnover will remain the same even though the business done is smaller, because the inventory will decline with the volume of business. An inventory of $25,000 and cost of sales totaling $100,000 will show the same turnover on this basis as $50,000 in- ventory and cost of sales totaling $200,000. When the yardstick varies in length, comparative measurements are of little value. The use of working capital as the basis of turnover is logical, first, because the capital is put in the business for the purpose of being turned over as rapidly as possible; second, because it is virtually constant; and third, because it presents all the elements concerned in the turnover, not only the stock-in-trade, but also the accounts and notes receivable, by means of which the turnover is effected. The turnover of working capital also furnishes a better criterion of the excellence of the management. With the inventory as the only standard a manager can make an apparently good record by starving his stock-in-trade. If, however, he uses working capital as the standard, he makes his best record by dihgence in collecting outstanding accounts, and increasing the supply of cash for the development and handling of a more extensive business. Definition of Working Capital Even if this is granted, the difficulty does not come to an end, because there is no authoritative detinition of working capital. H. R. Hatfield says: "Working capital has long had a specific 88 MATHEMATICS OF ACCOUNTING AND FINANCE meaning as a collective term for what are often called quick assets, e. g., cash, accounts receivable, perhaps merchandise, etc." ^ H. C. Bentley says: "Working capital is the excess of quick assets over quick liabilities."'* Each of these authorities has his followers, as was shown in a recent discussion of the subject in the Journal of Accountancy. The definitions differ because in one the notes and accounts pay- able are considered to be borrowed capital, while in the other only the amount contributed by the proprietor is treated as capi- tal. The first is the economic view of what constitutes capital, while the second is the business and general accounting view. If the ordinary business man is asked how much capital he has in his business, he will always state the amount of his proprietary interest. Unlike Micawber, he does not think he has added to his capital whenever he issues a note payable. Need of Exact Definitions The final lesson to be learned from the consideration of this subject is that in preparing comparative statements of turnover, accountants should first define the terms used and should plainly state the basis of the calculations. Otherwise the conclusions reached will be entirely misleading to persons whose conception of the subject is dift'erent. It is to be hoped that the Institute of Accountants will eventually end the present ambiguity by formu- lating an authoritative definition of turnover. Having settled upon the basis to be used, the accountant de- termines the turnover by ascertaining how many times the normal inventory will go into the cost of the goods sold, or into the sales, according to which view is adopted, or how many times the work- ing capital has been reinvested in the purchase or production of goods sold. The greater the quotient in each case, the more pros- perous and better managed the business is supposed to be. 3 H. R. Hatfield, Modern Accounting, 1909, p. 179. 4 H. C. Bentley, Science of Accounts, 191 1. CHAPTER X PARTNERSHIPS Division of Profits Profits may be divided by partners in any proportions to which they agree; if they malce no express agreement the law im- plies an agreement to divide the profits equally, regardless of the capital or services contributed. The customary methods of dividing profits are: 1. In the ratio of the capital balances at the beginning of the period. 2. In the ratio of the average capitals for the period. 3. In an arbitrary ratio, usually expressed in terms of frac- tions or per cents. 4 . In an arbitrary ratio after allowing interest on the capitals. To illustrate these methods, assume the following facts: Illustration A and B are in partnership and the profits for division at the end of the year are $12,000. A's capital account during the year undergoes the following changes: Credit balance, January i $50,000 Investment, March i 2,000 " November i 3,000 Total credits $55,000 Withdrawal, May i $500 " December i 1,000 Total debits 1,500 Credit balance, December 31 $53,500 90 MATHEMATICS OF ACCOUNTING AND FINANCE B's capital account changes as follows: Credit balance, January i $25,000 Investment, February i 10,000 " June 1 5,000 Total credits Withdrawal, October i Credit balance, December 31 j.0,000 1,000 59, 000 Solution i: Division of profits in the ratio of the capital balances at the beginning of the year: This ratio is A, 50, and B, 25; or 2 to i. A, therefore, is credited with % of $12,000, or |8,ooo; and B with }4 of $12,000, or $4,000. Solution 2: Division of profits in the ratio of the average capital for the period: This ratio may be computed in either of two ways. The first is as follows: Multiply each capital account credit by the number of months or days from the date of the credit until the end of the period, and find the sum of these products. Multiply each capital account debit by the number of months or days from the date of the debit until the end of the period, and find the sum of these products. Find the difference between the credit products and the debit products. Do this with each capital account and determine what fraction each difference is of the sum of the differences. The difference between the credit and debit products of A's capital account is found thus: Credits Date Amount Time January i $50,000 12 mo. March i 2,000 10 " November i 3,000 2 " Debits May I $ 500 8 mo. December i i ,000 r " Difference Product $600,000 20,000 6,000 $626,000 $ 4,000 1,000 5,000 $621,000 PARTNERSHIPS 91 The difference between the credit and debit products of B's capital account is found thus: Credits Date Amount Time January i $25,000 12 mo. February i 10,000 11 " June 1 5.000 7 " Debits October i $1,000 3 mo. Difference Product $300,000 110,000 35,000 $445, 00c 3,000 |.2,000 The partnership profits of $12,000 are accordingly divided in the follow- ing ratios and amounts: Ratios Expressed in Fractions A 621/1063 B 442/1063 Division of Profits $ 7,010.35 4,989.65 $12,000.00 The second method by which profits may be divided in the ratio of the average capitals for the period is as follows: Multiply the opening balance of each account by the number of months or days it remained unchanged. Multiply each new balance resulting from investments or withdrawals by the number of months or daj's it remained unchanged. Find the sum of these products for each capital account. Find the ratio of each sum to the total for all. The products for A's capital account are computed in the following way: 92 MATHEMATICS OF ACCOUNTING AND FINANCE Balance From To Time Product $50,000 January March 2 mos. $100,000 52,000 March May 2 " 104,000 51,500 May November 6 " 309,000 54,500 November December I mo. 54,500 53,500 December December 31 I " 12 mos. 53,500 $621,000 The products for B's capital account are figured as follows: Balance From To Time Product $25,000 January i February I I mo. $ 25,000 35'000 February i June I 4 mos. 140,000 40,000 June I October I 4 " 160,000 39,000 October i December 31 3 " 117,000 This method results in the same ratios as the first method, and it has the advantage that the final balances shown in the computation ($53,500 and $39,000) are the same as the balances of the accounts; and that the time numbers used as multipliers add to a full year. Checks on the accuracy of the computations are thus provided. It must be understood that these computations determine the average capital ratios but not the average capitals. To compute the average capitals it would be necessary to divide by 12, thus: $621,000 -^ 12 = $51,750.00, A's average capital $442,000 H- 12 = $36,833.33, B's " Since only the ratio between the average capitals is required, the division by 12, or by 365 if the numbers of days have been used as multipliers, is unnecessary. Solution 3 : Division of profits in an arbitrary ratio. No limit can be placed on the variety of arbitrary ratios which can be agreed upon and illustrations are unnecessary. Solution 4: Division of profits in an arbitrary ratio after allowing interest on capital. No interest can be allowed unless there is a specific agreement to do so, and in that event the rate should be agreed upon. PARTNERSHIPS 93 Assuming a rate of 6% on the opening balances of the capital accounts, the distribution, if the remaining profits are divided equally, would be as follows : A B Total 6% of $50,000 $3,000 6% of 25,000 $1,500 Total interest $4,500 Balance equally 3,750 3,750 7,500 $6,750 $5,250 $12,000 If the agreement provides for interest on partners' capitals, it must be credited to them even though it exceeds the total profits. In that event the resulting debit balance in the profit and loss account is charged to the partners in the agreed ratio. Assuming that the profits were only $4,000, the division would be as follows: A B T0T.VL Credits for interest (as above) $3,000 $1,500 $4,500 Debits for excess of interest over profits 250 250 500 Net credits $2,750 $1,250 $4,000 If interest is provided for in the agreement, it must be credited to the partners even though a loss has been incurred instead of a profit earned. Assuming a loss of $1,000, the division would be as follows: A B Total Debits for sum of loss of $1 ,000 and in- terest of $4,500 $2,750 $2,750 $5,500 Credits for interest 3,000 1,500 4,500 Net credit $250 Net debit $1,2^0 $1,000 The result is that B bears all of the loss from operations as well as the $250 net credit to A. 94 MATHEMATICS OF ACCOUNTING AND FINANCE Liquidation of Partnerships When a partnership is terminated, the procedure to be fol- lowed in realizing the assets, liquidating the liabilities and dis- tributing the partners' capitals, depends on whether all losses on realization have been ascertained before payments are made to the partners. If they have been, the losses are deducted from the partners' capitals in the profit and loss ratio, and the remaining assets, after paying the outside creditors, are distributed to the partners in amounts sufficient to pay off the capitals. To illustrate, assume that all partnership debts have been paid, and that the capital accounts are: A $10,000 B $8,000 There must be assets of $18,000. These are sold for $15,000. With losses divided equally, the division of cash proceeds is as follows : A B Total Capitals $10,000 $8,000 $18,000 Losses on realization i,Soo 1,500 3,000 Balances paid in cash $8,500 $6,500 $15,000 Periodical Distributions If periodical distributions to the partners are made before all assets are realized and all losses ascertained, they should be made, if possible, in such a way as to reduce the balances of the capital accounts to the profit and loss ratio existing between the partners, so that if all remaining assets are lost each partner's capital account will be exactly sufficient to cover his share of the loss. To illustrate, assume that all liabilities are paid and the part- ners' capitals are as follows : A $15,000 B 20,000 C 25,000 PARTNERSHIPS 95 The assets total $60,000. In realizing on $i8,oco worth of assets a loss of $3,000 is incurred, so that there is $15,000 in cash to divide. The division of cash should be made as follows, as- suming that the partners share profits and losses equally. ABC Total Capitals $15,000 $20,000 $25,000 $60,000 Loss 1,000 1,000 1,000 3,000 Balance before dividing cash $14,000 $19,000 $24,000 $57,000 Cash 5.000 10,000 15,000 Balances left in P. & L. ratio $14,000 $14,000 $14,000 $42,000 It should be noted that the cash is not distributed in the capi- tal ratio, the profit and loss ratio, nor any other ratio, but in arbitrary amounts sufficient to reduce the capitals to the profit and loss ratio. Reducing Capitals to Profit and Loss Ratio It is not always possible to bring the capital account balances to the profit and loss ratio at the first distribution of cash. This is the case if the capital of any partner after charging off all ascer- tained losses is less than his profit and loss ratio of the assets which will remain after making the proposed distribution. To illustrate, assume the capitals to be as follows: A $10,000 B 20,000 C 30,000 Losses are to be shared equally. The assets total $60,000 and all Habilities are paid. Assets carried on the books at $30,000 are sold for $24,000, the loss of $6,000 being divided as follows: ABC Total Capitals before dividing loss Loss Capitals before distribution of cash $10,000 $20,000 $30,000 2,000 2,000 2,000 $60,000 6,000 $8,000 $18,000 $28,000 $54,000 96 MATHEMATICS OP ACCOUNTING AND FINANCE After the $24,000 is divided there will remain a total capital of $30,000. If possible the $24,000 cash should be divided in such a way as to leave each partner with a balance of $10,000; but this is clearly impossible since A's balance is already reduced to $8,000. Since A's capital is not sufficient to bear his share of the total possible loss of the $30,000 of assets which remain after the distribution, nothing should be paid to A. In the event that a total loss of $30,000 is incurred the charge of $10,000 to A would leave his account with a debit balance of $2,000. If he could not pay in the $2,000 it would have to be charged against B and C in their profit and loss ratios, which in this case happen to be equal. Therefore, B and C may possibly lose the following amounts : B C One-third each of total possible $30,000 loss $10,000 $10,000 Excess of A's share of possible loss over his capital 1 ,000 1 ,000 Total possible loss $11,000 $11,000 The $24,000 cash should be divided between the two partners in such a way as to reduce their balances to these amounts, as follows : ABC Total Capitals before distribu- tion of cash $8,000 $r8,ooo $28,000 $54,000 Cash distributed 7,000 17,000 24,000 Balances $8,000 $11,000 $11,000 $30,000 In the next realization, assets carried at $15,000 are realized at $12,000, the loss being $3,000. After dividing the loss the cash can be distributed in amounts which will reduce the capital balances to the profit and loss ratio of equality. PARTNERSHIPS ABC Total Balances (as above) $8,000 $11,000 $11,000 $30,000 °^^ i-ooo 1,000 1,000 3,000 Balances before distribut- ^"S cash $7,000 $10,000 $10,000 $27,000 ^^ 2,000 5,000 5,000 12,000 Balances (reduced to P. & ^- ^^^^°) $5,000 $ 5,000 $5,000 $15,000 The remaining assets are sold for $9,000, the di- vision of loss and cash being as follows: ^°^^ 2,000 2,000 2,000 6,000 Balances paid in cash ... . $3,000 $3,000 $ 3,000 $ 9,000 97 CHAPTER XI THE CLEARING HOUSE Principle of the Clearing House The general principle on which a clearing house operates is that of offsetting debits and credits and dealing only with net differences. Its simplest form is illustrated when two persons buy from and sell to each other. If instead of each paying his bill to the other in full, the one who owes the larger sum deducts the other's debt to him and pays the difference, he has to that extent adopted the clearing house principle. In the fullest application of the principle, an outside party is introduced to act as settling or clearing agent. Each unit in the combination that forms the clearing house has dealings with every other unit, but instead of settling its dealings with each of the other units individually it charges the total of all its debits and credits the total of all its credits to the clearing agent, with which it settles the net difference of the totals. In this way a single settlement takes the place of a large number of settlements. An example will make this clear. Debits and Credits with Clearing House Suppose there are six banks in a city. Every day they re- ceive on deposit and otherwise, checks on each other. If there is no clearing house, each bank presents to each of the others the checks drawn on it and collects or pays the difference due to or from it, according as the amount of the checks it pre- sents is greater or smaller than the amount of those presented to it. If the banks form a clearing house, each provides itself with a blank form, in which it inserts the amount of checks it holds 98 THE CLEARING HOUSE 99 against each of the others. The total it charges to clearing house checks on the teller's blotter. This form and the checks themselves are sent to the clearing house. The checks are de- livered to the representatives of the banks on which they are drawn, with a memorandum of the amount for each bank. As each bank receives the checks drawn on it, it enters the amounts in the second column of the form and the total becomes the credit to clearing house checks. If a bank's credits to the clearing house are greater than its debits, it pays the difference to the clearing house. The amounts thus paid in are afterwards paid to those banks whose debits to the clearing house are greater than their credits. Clearing House Transactions The following are assumed to be the transactions on a certain day of the six banks in the clearing house, it being remembered that the first column of figures on each blank is made up before the checks leave the bank and the second column at the clearing house. First National Bank Atlas Nat ^27,819.32 Merchants St 12,948.24 Traders St 22,687.19 State Tr. Co 18,729.63 Union Nat 14,963.48 Atlas National Bank $16,248.76 First Nat $16,248.76 14,629.83 Merchants St 11,283.42 26,963.42 Traders St 12,732.63 21,246.38 State Tr. Co 15,376.28 IS. 782. 16 Union Nat 9.659.87 C. H. owes . 197,147-86 $94,870.55 $ 2,277-31 Owes C. H. $27,819.32 8,664.19 10,497.26 11.958.47 7,326.58 $65,300.96 $66,265.82 $ 964.86 Merchants State Bank i^irst Nat . . $14,629.83 $12,948.24 Atlas Nat 8.664.19 11,283.42 Traders St 9.586.28 10,392.16 State Tr. Co . . . 11,473.47 9.627.59 Union Nat 10,827.42 11.432.27 $55,181.19 $55,683.68 Owes C. H $ 502. 49 Traders State Bank First Nat $26,963.42 Atlas Nat 10,497.26 Merchants St 10,392.16 State Tr. Co 11,788.39 Union Nat 9.673.57 C. H. $22,687.19 12,732.63 9,586.28 13.249.36 8,865.71 $69,314-80 $67,121.17 $ 2,193.63 100 MATHEMATICS OF ACCOUNTING AND FINANCE s First Nat . TATE Trust Compan Y $18,729.63 15,376.28 11.473-47 11,788.39 8,842.74 Union First Nat Atlas Nat National Ban . . $15,782-16 7,326.58 11,432.27 8,865.71 8,842.74 K Atlas Nat . St... 11,958.47 9.627.59 13,249.36 7.982.25 9,659.87 Merchants Traders St. Union Nat . Merchants St. . . Traders St State Tr. Co Owes C. H 10,827-42 9.673.57 7,982.2s Owes C. H. $64,064.05 $ 2,146.46 $66,210.51 $52,249.46 . . $ 857-13 $53,106.59 Debit Credit Debits Credits Balances Balances $ 94.870.SS $ 97.147.86 $ $2,277-31 66,265.82 65.300.96 964.86 55.683-68 SS,l8l-I9 502.49 67.121. 17 69.314-80 2,193.63 66,210.51 64.064.05 2,146.46 S3. 106.59 52,249-46 857.13 $403,258.32 $403,258.32 $4,470.94 $4,470-94 Manager's Sheet The representatives of the different banks are allowed a cer- tain number of minutes in which to finish the preparation of their respective sheets. They then in turn call out their figures which are registered by the manager of the clearing house on his own sheet in the following manner: Banks First National Bank Atlas National Bank Merchants State Bank Traders State Bank State Trust Company Union National Bank If the manager's sheet balances, he rings his bell to signify that the session is over. If it does not balance he announces the fact. A certain number of minutes is allowed for finding the error. If found within that time the one who made the error is fined a certain sum. The fine increases progressively with any added time taken to reach a balance. Later, the banks with debit balances make their payments to the clearing house manager, who shortly after pays off the banks with credit balances. The total figures are kept by the manager for statistical pur- poses. The sum of the checks presented ($403,258.32) is the clearing house movement for the day, and the sum of the balances ($4,470.94) is the amount of the balances reported in the financial THE CLEARING HOUSE lOI news. The corresponding figures for the week, month and year are published periodically in the financial columns of the news- papers, and are considered a barometer of business activity. Economy of System The convenience and the economy of the clearing house sys- tem may be demonstrated by an analysis of the transactions of the Merchants State Bank. This bank has collected checks aggregating $55,181.19 and has paid checks totaling $55,683.68, or a total settlement of $1 10,864.87, by a payment of only $502.49, or less than one half of 1%. Without the clearing house, it would have been compelled, even by offsetting with each of the other banks, to make the following payments : Atlas National Bank $2,619.23 Traders State Bank 805.88 Union National Bank 604.85 $4,029.96 It would have received the following amounts: First National Bank $1,681.59 State Trust Company 1,845.88 3,527.47 Total cash movement in the bank would have been. . . $7,557.43 One payment of $502.49 takes the place of five transactions aggregating fifteen times as much. Each bank must settle on the basis of the clearing house re- turns. The corrections of any errors in addition, the listing of checks, either for wrong amounts or on the wrong bank, etc., and the settlement for checks returned for lack of funds or on account of missing endorsements, are all matters left to the individual banks to adjust with each other. Application of Principle Extended The clearing house principle is applied to the settlement of accounts whenever several parties have reciprocal relations, in- 102 MATHEMATICS OF ACCOUNTING AND FINANCE volving the transfer of value in the form of money, securities, or merchandise. It is applied to the settlement of accounts between brokers on boards of trade and other exchanges, and also to the regulation of sales of merchandise when an agreement exists among concerns in the same line of business restricting the output of each to an agreed percentage. At present there is very little of the latter use of the method, owing to the danger of violating the law which prohibits agreements in restraint of trade. CHAPTER XII BUILDING AND LOAN ASSOCIATIONS General Characteristics of Building and Loan Associations A building and loan association is founded on the general principle of co-operation among a large number of small investors and a smaller number of intending borrowers, who wish to get advances for the purpose of erecting buildings on land they own, or of paying off existing loans on real estate that has been partially paid for. There are several different plans on which such associa- tions are organized, but the idea of co-operation is present in them all. In some foreign countries, notably Italy, this principle of co- operation has been extended to cover loans to workmen for the purchase of tools or machinery and even to the furnishing of large sums to working contractors for building railroads and other public works. The usual characteristics of a building and loan association are as follows: Each member subscribes to a definite number of shares, on which he pays monthly instalments of one half of i% of the par of the stock. If, for example, the par is fixed at $ioo, the monthly payment on each share is 50 cents. When these instalments together with the profits earned by each share amount to $100, the stock is said to have matured and is paid off. Terminating Plan The terminating plan of organization, the first to be adopted, is the simplest of all. It involves the subscription by a limited num- ber of persons to a certain number of shares which are to be paid in instalments, and the dissolution of the association upon the 103 I04 MATHEMATICS OF ACCOUNTING AND FINANCE maturity of the shares. As money is paid by subscribers it is loaned from time to time. As the interest collected is also loaned, the association is in theory receiving compound interest. For instance, if 2,000 shares are taken and 50 cents per share is paid monthly, a loan of $1,000, say at 6%, can be made at the end of the first month. At the close of the second month there is another $1,000 to lend, together with $5 interest collected on the first loan, and this is repeated from month to month. The com- pound interest theory, however, does not hold strictly here, as it is difficult to lend odd amounts, unless other security than real estate is taken. The objection to this plan is that it is impossible to employ the money profitably in real estate loans, because during the latter part of the association's life loans have to be made for so short a time that great difficulty is experienced in making them; and if the later loans should be made for the usual term of five years the dues and profits on the stock could not be paid when they matured. It can, however, be arranged that after the stock reaches par no further payments should be made on the shares and no shares should be withdrawn, but the money should be kept invested and cash dividends paid out of the interest collected. Practicability of Plan The terminating plan can be used to best advantage when a large number of persons wish to pay in a given time a large sum of money with interest, each person paying a small amount monthly for the given time. This plan, for example, is extremely useful in paying off a church debt. If a church without any rich members owes $10,000 on which it is paying 6% interest, it might seem impossible to raise so large an amount. If a building and loan association is formed with 100 shares of $100 each, to run for five years, on which the monthly payments are fixed at $1.93 per month, it will usually not be found difficult to persuade the con BUILDING AND LOAN ASSOCIATIONS I05 gregation to take the whole 100 shares. This will not only pay off the entire debt in five years, but will also pay the interest on the diminishing amount of the note, as the holder of the note will be willing to accept partial payments of even hundreds of dollars. The plan works out in this way — Each holder of a share agrees to pay $100 and interest in monthly instalments of $1.93. If he makes his payments regularly, his account for the first two months runs as follows : Original agreement $100.00 One month's interest at 6% .50 $100.50 First month's payment 1.93 $ 98.57 One month's interest at 6% .49 $ 99.06 Second month's payment 1.93 $ 07-13 The treasurer credits the interest account each month with the interest paid that month, and the principal with the balance of the payments, applying the money to the payment of the inter- est and principal of the note. The amount of the monthly payments theoretically necessary is found from the compound interest table headed, "Amount per annum necessary to pay a debt of $1 now due and the interest thereon." In order to reduce it to monthly payments it is neces- sary to use the table of ^ % for the number of months in the life of the association. In our illustration we find the amount necessary, or $1.93, in the Yi % column on the line of the 60th period. For six years or 72 months the amount is $1.65^. As all the amounts will not be paid promptly, it is better to make the monthly payments $2 and $1.70 respectively. io6 MATHEMATICS OF ACCOUNTING AND FINANCE Serial Plan The serial plan is virtually a succession of terminating plans combined in one association. A series is started at regular inter- vals of three, six or twelve months, with whatever number of shares may be subscribed. In some states the par of a share is $200 and the monthly payment per share is $1 ; but in most of the states the par is $100 and the payment is 50 cents per month, sometimes 25 cents per week. After a new series is begun no more shares can be issued in any previous series, although the holder of stock in any series can sell it or otherwise transfer it. A serial association is a partnership with limited liability, the subscribers to the different series being the partners. The profits are divided among the series in proportion to their capitals which are the amounts paid in plus the accumulated profits to date. A simple illustration will make the process clear. Suppose that an association is started on January i, 191 7, and that on September 30, 1918, the shares outstanding are as fol- lows: Series Date of Issue nu.mber of Shares Paid per Share Profit per Share V\\LUE PER Share I 2 3 January, 1917 April, July, October, January, 1018 April, July, 800 600 700 500 400 600 500 Jio.so 9.00 7. so 6.00 450 3 00 1.50 $2 I I 29 71 25 88 56 27 02 $12.79 10.71 8.7S 6.88 s 6 7 5.06 3-27 1-52 During the cjuarter beginning October i, 1918, the eighth series of 700 shares is sold. The net profits for the quarter from interest, fines, etc., less the secretary's salary and other expenses, are $489.92. The undivided profits on September 30, 1918, are BUILDING AND LOAN ASSOCIATIONS 107 $4,571.28. There are two plans by which the profits may be divided, the partnership plan and the Dexter rule. Distribution by Partnership Plan In the so-called partnership plan all previous profits are ig- nored and the capital of each series is taken as the money actually paid in multiplied by the equated number of months during which it has been left in the business. As the dues are payable on the first of each month, the equated time is found by adding i to the number of months the series has run and dividing the sum by 2. Thus, on December 31, 1918, the first series has run 24 months. Adding i and dividing by 2 gives 12^^ months as the equated time. As there are 800 shares in that series and as each share has paid in $12 the capital is $12 multiplied by 800 or $9,600 and it has been in an equated time of 12^ months. Multiplying $9,600 by 12^ gives $120,000 as the equated capital of Series i. Pursuing the same method the equated capitals of the several series are calculated as follows: Series I $12.00 X i2>^ mos. X 800 shares $120,000 2 10.50 X II "X 600 " 69,300 3 9.00 X 9/2 " X 700 " 59,850 4 7-5° X 8 "X 500 " 30,000 5 6.00 X 6>2 " X 400 " 15,600 6 4-5° X 5 " X 600 " 13,500 7 3-00 X 3>2 " X 500 " 5,250 8 1-50 X 2 "X 700 " 2,100 Total of equated capitals $315,600 The total profits to date, amounting to $5,061.20, are divided among the series in the proportion that each one's equated capital bears to the total equated capital. Reducing to the least com- mon denominator of 2,104, we have the following division of the profits among the several series. io8 MATHEMATICS OF ACCOUNTING AND FINANCE Series I . 2. 3- soo/ . ... 462/ • • ■ 399/ 4- 200/ 5- 6. 7- 8. ■ ■ ■ 104/ 90/ 35/ . . . 14/ 800/2104 of $5,061.20 equals $1,924.40 ,924.40 $2.41 Der share ,111-34 1.85 959-79 1-37 481.10 .96 250.17 .62 216.50 -36 84.19 -17 33-68 •OS 2,104 ;,o6i.i7 Of course it is impossible to balance exactly the total profits with the sum of the profits assigned to each series. Distribution by Dexter's Rule The first step in applying the Dexter rule is to calculate the capitals of the series. The capital of each series on September 30 is the value per share as shown in the table on page 106 [of this chapter] multipHed by the number of shares outstanding. To find the earning capital on December 31, it is necessary to add the contribution of 50 cents per share for each of the three months of the current quarter. This is an average of $1 per share, cal- culated thus : 50 cents paid October i was in 3 months or $1.50 for i month 50 " " November i " "2 " " i.oo " i 50 " " December i " " i month " .50 " i " Total $3.00 This is an average of $1 for three months. One dollar must, therefore, be added to the value per share shown in the first table. Bearing this in mind, the earning capital of the different series is found as follows: BUILDING AND LOAN ASSOCIATIONS 109 Series I 800 shares at $13.79 P^r share " II. 71 600 700 500 400 600 500 700 9-75 7.88 6.06 4.27 2.52 1. 00 )II,032 7,026 6,825 3,940 2,424 2,562 1,260 700 Total capital $35,769 This rule leaves undisturbed the previous distributions of profits and adds to them the profits of the current quarter on the basis of the present earning capitals. The current profits of $489.92 are 1.37 per cent of the capital of $35,769. Hence the profit per share for each series is calculated as follows : Series Capital Current Profit Previous Profit Total Profit Shares Profit per Share $11,032 7,026 6,825 3.940 2,424 2,562 1,260 700 $151.14 96.26 93-50 53-98 33-21 35-10 17.26 9-59 $1,832 1,026 875 440 224 162 10 $1,983.14 1,122.26 968.50 493-98 257.21 197-10 27.26 9-59 800 600 700 500 400 600 500 700 $2.48 1.87 1.38 -99 .64 6 ■ 33 • 05 8 .01 $35,769 $490.04 $4,569 $5,059-04 A comparison of the profits per share shows that the partner- ship plan gives larger profits to the later at the expense of the earlier series. This is a great injustice, as the earlier series are made to share the profits they have accumulated, with the later series, which had no part in producing those profits. This is not done in an ordinary commercial business when new partners are admitted. Dexter's rule is the true partnership plan and is the one usually adopted. no MATHEMATICS OF ACCOUNTING AND FINANCE Withdrawal of Shares In case any shares are withdrawn during the quarter the holders are given the amount they have paid in and interest for the equated time at 3 or 4 %. Thus, if fifty shares in the third series are withdrawn on December i, 191 8, the amount paid in is $8.50 per share or $425, the interest at 3% for the equated time of nine months is $9.56, and the total amount paid is $434.56. The book value of this stock is $9.75 per share or $487.50. There is a book profit of $5 2. 94 on the transaction. No entry is made for this profit, but it is added to the cash profits of $489.92, which makes the divisible profits for the quarter $542.86. Sources of Income The profits of building and loan associations arise from inter- est paid by the borrowers and also from fees, fines and premiums. The fees are membership fees, usually 25 cents per share, paid when the stock is originally subscribed. Sometimes these fees are capitalized in the series in which they are paid, making the value per share 25 cents more. Otherwise they are treated as a general profit, as in our example. In some associations a transfer fee is paid when stock changes hands. The fines are levied on stockholders who are delinquent in their monthly payments, usually 5 cents per share on stock of non-borrowers (investment stock), and 10 cents per share on stock of borrowers, for each month the delinquency continues. Premiums The premiums are either payable monthly or are deducted from the loan when made. They arise from the bidding for loans by those desiring to borrow. When the association is in posses- sion of loanable funds, bids are invited. The person offering to pay the highest premium for the money receives the loan, if his BUILDING AND LOAN ASSOCIATIONS III real estate security is found to be adequate. On the cash pre- mium plan, 1% of the premium is paid monthly in cash. On the deducted premium plan, the whole premium is deducted from the loan in advance and only the net amount is paid. If the loanable funds are $2,000 and the cash premium bid is 25%, the borrower makes his note for $2,000 and his monthly payments are: Dues $10, interest at 6% $10, and premium amounting to 1% of $500 premium or $5, making a total cash payment of $25. For the same amount of money on the deducted premium plan, the bid may be for $2,500 at a premium of 20%. The borrower receives $2,000 cash, but his note is made for $2,500 and his monthly payments are: Dues $12.50 and interest $12.50, a total cash payment of $25. The premium of $500 is credited to unearned premiums, and at the end of each quarter profit and loss is credited and unearned premiums charged with $15. Each borrower is obliged to subscribe for stock the par value bf which is equal to the face of his loan. When the dues paid on this stock and the profits credited to it cause it to be worth par, it is paid off by cancelling the loan. No payments are ever applied to the loan, which remains in full force until cancelled by the matured stock. It will be noted that in both the serial plans the treatment is entirely on the basis of the several series being fully paid to date. If stock in any series is delinquent, it still receives its share of the profit, the only offset being the small fine, which in the older series is much less than the earnings for the quarter. Individual Plan To meet this objection to the serial plan, some associations have adopted the individual plan, which really makes each stock- holder a series by himself. The capital is then the actual amount paid in plus accumulated profits. The procedure as to earning capital and profits is otherwise the same as in the serial plan. 112 MATHEMATICS OF ACCOUNTING AND FINANCE Dayton, or Ohio Plan One great objection to the plans hitherto considered, or to any modification of them, is that they place the borrowers in an uncertain position with regard to the time at which their loans will be cancelled, and consequently with regard to the amount of principal and interest they will be called upon to pay. Their loans are not cancelled until their stock matures. If the associ- ation proves to be very prosperous, the profits will be large and the stock will mature in a comparatively short time. On the other hand, in a poorly managed association, it may take a long time to mature the stock, and in the meantime the borrowers will have to continue the payment of their instalments and interest, making their loans much more expensive than they have anticipated. To meet this objection various forms of the "Dayton Plan" have been adopted, all of which agree in this — that the borrower is given a definite contract. His loan is to be repaid in a fixed number of months by the regular payment of a definite amount, which will cover both principal and interest. The calculation of the amount to be paid is based on the same principle as the one already shown for liquidating a church debt. The amount applicable to the principal is credited to the loan each month by the association, as the borrower does not have to own any stock. The loanable funds are obtained from the instalments paid by investing stockholders, with whom the accounts are kept on the individual plan. This system is really that of a co-operative savings bank, and is the fairest method of all. When properly managed, it is very successful. The two following problems illustrate principles dealt with in this chapter. Problem i A building and loan association published the following statement on December 31, iqi8: BUILDING AND LOAN ASSOCIATIONS Age and Condition of Shares 113 Series Date of Issue Number of Shares Paid per Share Profit per Share Value per Share I January, 191 7 April, July, October, " January, 1918 April, July. October, " 800 600 700 Soo 400 600 500 600 $12.00 10.50 9.00 7.50 6.00 4-50 3.00 I. so $2.56 1.97 ISI 1. 14 .82 • 53 .26 .02 $14.56 12.47 10.51 3 4--- s 8.64 6.82 6 3.26 8 The balance sheet showed that the profits to date were $5,647.82. On March 31, 1919, the net profits from interest, premiums, fines and fees, less salary and expenses, were found to be $483.16. There had been with- drawn during the quarter 50 shares of the 4th series on which $10.75 ^^- terest had been paid, not included in the expenses previously mentioned. The Qth series was opened and 400 shares were subscribed and paid for the quarter. Distribute the profits March 31, 1919, on the partnership plan. Solution: First find the equated time; which is ascertained by adding i to the number of months in the age of each series and dividing by 2. Next multi- ply the equated time by the amount paid and then by the number of shares outstanding in each series, as follows: Series I . . •• 27 months old. equated 14 mos. X 3 13.50 X 800 si 1. $151,200 2. . .. 24 I2>2 ' X 12.00 X 600 90,000 3-- 21 II ' X 10.50 X 700 80,850 4-- .. 18 9>4 ' X 9.00 X 450 38,475 S-- • 15 8 ' X 7.50 X 400 24,000 6.. 12 6>^ ' X 6.00 X 600 23,400 7- 9 5 " X 4.50 X 500 11,250 8.. 6 s'A " X 3.00 X 600 6,300 9.. •• 3 2 ' X 1.50 X 400 1,200 $426,675 114 MATHEMATICS OF ACCOUNTING AND FINANCE The profits to be divided are found as follows: Undivided profits, December 31, 1918 Profits of quarter, net Less interest paid on withdrawal Profits to be divided The profits are divided in the following manner: Series ;,647. 82 483-16 ),i30.98 IO-75 3,120.23 2oi6/5689*of|6,i2o.23equals$2,i68.8i,orpershare $ 1200 1078 513 320 312 150 84 16 5,689 ,290.96 ' ,159-71 ' 551-89' 344.26 ' 335-65 ' u u 161.37 ' u u 90.37 ' 17.22 ' 2.71 2-15 1.66 1.23 .86 •56 •32 •15 .04 ),I20.24 *i5i, 200/426,675 = 2016/5689 Problem 2 Using the same figures as in the previous problem, distribute the profits in accordance with Dexter's rule. Solution: First find the capitals of the series by multiplying the value per share plus$i by the number of shares outstanding, as follows: Series 800 shares by $15.56 equals $12,448 600 700 450 400 600 500 600 400 13-47 8,082 II. 51 8,057 9-64 4,338 7.82 3,128 6.03 3,618 4.26 2,130 2.52 ' 1. 512 1. 00 * 400 $43,713 BUILDING AND LOAN ASSOCIATIONS 115 The profits for the quarter are $483.16 Plus profit on withdrawn shares 46.25 Total to be divided $529.41 The withdrawal profits are ascertained thus: Book profit for 4th series is $1.14 per share, and for 50 shares. Actually paid Retained profit on withdrawal >S7-oc 10.75 '■25 The divisible profit of $529.41 is a trifle more than 1.21% of the capital of $43,713. Calculated for each series it is as follows: Series Shares Old Profit PER Share Old Profit OF Series New Profit OF Series Total Profit per Share New Profit PER Share I 800 J2.S6 $2,048.00 J150.62 $2,198.62 $2.75 2 600 1-97 1,182.00 97 79 1,279.79 2.13 3 700 l-Sl 1,057-00 97 49 1. 154-49 1-65 4 450 1. 14 51300 52 49 565-49 1.26 S 400 .82 328.00 37 85 365-85 -91 6 600 .53 318.00 43 78 361.78 .60 7 500 .26 130.00 25 77 155-77 • 31 8 600 .02 12.00 18 30 30-30 ■OS 9 400 4 84 4.84 .02 $5. 588. 00 S528 93 $6,116.93 The last series is usually given 2 cents as i cent is so very small. With the profit per share known the statement of age and condition of shares can now be made up if asked for. CHAPTER XIII GOOD-WILL AND CONSOLIDATIONS Purchasing a Business with Stock When two or more concerns are merged, or when a business is taken over by a holding company, or when a business is bought by a corporation already in existence, the payment for the busi- ness taken over is frequently made in the stock of the corporation acquiring the business. If the amount of stock to be given for the business is to be determined on an equitable basis, two elements in the business must be taken into consideration. These are the fair value of the net assets of the business as a going concern and the earning ca- pacity of the business. The fair value of the net assets may be reached by mutual agreement between the parties, or by an appraisal of the fixed assets by an appraisal company, and an estimate of the value of the active assets by an accountant. When the value is determined, stock of a corresponding amount in face value is allotted as the price paid for the net assets. Allocation of Net Earnings To ascertain the earning power for which further stock should be issued, it is first necessary to establish a normal, or standard rate of earnings. This can be done only by agreement between the parties concerned. When the rate is agreed upon, it is ap- plied to the stock already allotted for net assets. The result is the amount of net earnings to be devoted annually as dividends on the stock given in payment of the net assets. The net earnings remaining after deducting dividends at the agreed rate on the stock issued for net assets, is the basis for Ii6 GOOD-WILL AND CONSOLIDATIONS II7 calculating the amount of stock to be allotted for the excess earning capacity of the business acquired. The theory on which stock is allotted for excess earnings is based on the fact that one of the principal reasons for investing in a stock is that it will produce an income that is considered adequate return on the money invested. What the rate of return should be is determined by the conditions of the enterprise. An investment in a business whose conditions are stable and which may be depended upon to maintain a steady earning power, or even to increase it, yields a much lower rate of return than one in a speculative and unreliable business. Good-Will The name given to the valuecreatedby the excess earnings of a business is "good- will." Consideration of good- will is involved in the discussion of a stock distribution on the basis of earnings. The stock allotment is partly based on the value of the net assets acquired and partly on the good-will, or excess earning power. Since good-will is the measure of earning capacity, it follows that only an established business can possess it. It resides usually in the reputation the business has built up in consequence of the excellence of its product, the practice of fair dealing or any other characteristics of its management giving it an advantage over its competitors. The existence of good-will cannot be proved except by a more or less prolonged experience. There- fore, when a new business is started it is false accounting to issue any stock for good-will, because its promoters expect it to earn more than the normal rate of profit. A new company, however, which has taken over an old business with a developed good-will may carry good-will among its assets and issue stock therefor. Appraising Good- Will — Years' Purchase Method In allotting stock the good-will may be valued in several different ways. It may be expressed as a given number of years' Il8 MATHEMATICS OF ACCOUNTING AND FINANCE purchase of the total profits. This means that the purchaser is wilHng to forego profits from the business for the time agreed upon. Each condition of the terms must be agreed upon between the buyer and seller. Thus, if it is agreed that the basis shall be two and one half years' purchase of the average profits for the last ten years, it will be necessary to ascertain the total profits for ten years. One tenth of this sum will be the average yearly profit, and two and one half times the average profit will be two and one half years' purchase. If the total profit for lo years is $256,232.00 the average annual profit is $ 25,623.20 and 23^ times the average is $ 64,058.00 which is the value placed upon the good-will on the basis of two and one half years' purchase. It is better to ascertain the average of a number of years than to take the figures for the last two and one half years, as the latter may not be normal. This method is not as logical as allowing a certain number of years' purchase of the profits in excess of a rate agreed upon as normal. The latter calculation is the same as that given above, except that the normal profits are first deducted. Thus, if the invested capital is $200,000 and the normal rate agreed upon is 10%, the value of the good-will is figured as follows: Average profits (as shown above) .... $25,623.20 Normal profit 20,000.00 Excess profit $ 5,623.20 8-years' purchase $44,985.60 Capitalizing Gross Income A third method is to capitalize the gross income. If the capital is known and the per cent of income is also known, the income is found by multiplying the capital by the rate per cent. Thus, if the capital is $200,000 and the rate of profit is GOOD-WILL AND CONSOLIDATIONS 119 15%, the income is $30,000. On the other hand, if it is desired to ascertain what capital will produce $30,000 income if the rate per cent is 15, it is necessary to divide the income by the per cent expressed as a decimal, thus, $30,000 divided by .15 equals $200,000. Therefore, to capitalize a given income at a given rate per cent, the income is divided by the decimal expressing the rate. This method, like the first, is objectionable because it does not take into consideration the fact that part of the income must be used to furnish a return on the stock already allotted for net assets. That amount of the income is in fact duplicated, and the duplication reduces the amount of the return to the concern with the largest rate of profit and increases the return to the concern with the smallest rate. Suppose the following companies wish to combine on the basis of net assets and good-will representing gross income capitalized at 15%. ABC Net assets and capital $200,000 $300,000 $500,000 Average annual income 40,000 45,000 65,000 Rate of profit 20% 15^^ 13% If it is proposed to form a new company with a capital of $2,000,000, the distribution of the stock is as follows: For net assets $200,000 $300,000 $500,000 For income capitalized at 15%. . 266,667 300,000 43i^3,.^3 Total $466,667 $600,000 $933,333 The total income being $150,000, the rate of profit is 7>^ %, which will have to be distributed as follows: A receives y}4 % of $466,667 $35,000 B " " " " 600,000 45,000 C " " " " 933^333 70.000 Total $2,000,000 $150,000 120 MATHEMATICS OF ACCOUNTING AND FINANCE A would thus receive $5,000 less per year than when operating alone and will certainly object to the plan. B would not be affected because he is already receiving 15%, while C would gain the $5,000 lost by A. Therefore, a fourth plan should be adopted by which the duplication of profits will be avoided. After allowing for a fixed rate of return on the capital issued for net assets to each of the merging concerns, the amount of profits necessary to provide the dividend at that rate is deducted from the total profits previously earned by the concern and the capitalization of the remaining profits is made by employing the same rate. This amounts to fixing the rate desired on the new stock and issuing stock to each concern sufficient to earn the profits previously enjoyed. If 8% is fixed upon as the rate desired, the first allotment of stock for net assets is the same as before — A $200,000, B $300,000, and C $500,000. The remaining profits are then calculated as follows : A B C Total $ 40,000 16,000 $ 45,000 24,000 $ 65,000 40,000 $ 150,000 80,000 $ 24,000 $ 21,000 $ 25,000 $ 70,000 Capitalizing at 8% $300,000 200,000 $262,500 300,000 $312,500 500,000 $ ] 875,000 ,000,000 $500,000 $562,500 $812,500 $ ,875,000 Income at 8% $ 40,000 $ 45,000 $ 65,000 $ 150,000 In this way each concern will receive the same earnings after the merger as before, and none will suffer. Issue of Two Classes of Stock We have thus far considered the issue of only one kind of stock. This necessitates the use of only one rate of return on the stock GOOD-WILL AND CONSOLIDATIONS 121 issued. A more usual procedure when a combination of this kind is effected is the issuance of two classes of stock, preferred and common. It is an almost universal practice to issue preferred stock for the net assets and common stock for the good-will, or excess earnings. The objection to calculating the stock given for good-will on the basis of the total earnings is the same as in the third plan. If the preferred stock is issued at 6% and the good-will is capital- ized on the total profits at 20%, the common stock issued for good-will is as follows: A $200,000 B 225,000 C 3 25^000 Total $750,000 As it takes $60,000 to pay the preferred dividend, there will be left $90,000 for the common stock, which will allow a 12% divi- dend. The profits will therefore be divided as follows: A $200,000 preferred at 6% $12,000 200,000 common at 12% 24,000 $36,000 B 300,000 preferred at 6% 225,000 common at 12% C 500,000 preferred at 6% 325,000 common at 12% Total $18,000 27,000 45,000 $30,000 30,000 6g,ooo $: [50,000 In this way A makes a loss and C a gain of $4,000. The proper method is to issue the common by capitalization of the excess profits after the preferred dividends have been provided for, thus: A Original profits $40,000 Preferred dividends 6% 12,000 Excess remaining $28,000 $27,000 $35,000 Capitalized at 20% $140,000 $135,000 $175,000 B C $45,000 $65,000 18,000 30,000 122 MATHEMATICS OF ACCOUNTING AND FINANCE It is evident that dividends of 6% on the preferred and 20% on the common stock will give the same return as formerly. After the preferred dividends are deducted it does not make any difference what basis is adopted for capitalizing, because whatever rate is used the relative proportion remains the same. Thus, if the excess is capitalized at four years' purchase the result- ant common stock is as follows: A $112,000, B $108,000, and C $140,000, a total of $360,000, on which the $90,000 remaining profits will provide a dividend rate of 25%, and the distribution will be thus: A for preferred $12,000 " common, 25% of $112,000 28,000 $40,000 B " preferred $18,000 " common, 25% of $108,000 27,000 45,000 C " preferred $30,000 " common, 25% of $140,000 35, 000 65,000 The practice of starting a new business with a capital stock of a par value greater than the total of the net assets is often in- dulged in. In such a case a debit must be made to some account, in order to balance the books. The account debited is often called good-will. This is altogether wrong, since, as we have seen, good- will is a matter of growth and cannot be possessed by a business which has had no time to develop it. CHAPTER XIV FOREIGN EXCHANGE Conversion of Foreign Coinage In order to provide a basis for the comparative values of United States and foreign coinage, the Director of the United States Mint periodically estimates the values of foreign coins, which are proclaimed by the Secretary of the Treasury. The values thus fixed are called the mint pars of exchange, and repre- sent the intrinsic or bullion values of foreign coins in terms of United States money. A table of values recently proclaimed appears in the Appendix. To reduce a value in foreign coinage to United States money, multiply the value of the coin in United States money by the number of coins. Illustration What will a draft for £410 'i9/'6 cost in dollars at the mint par of exchange? Solution: There are two ways to calculate this: 1. Reduce shillings and pence to decimals of the pound, thus: £410. ig shillings are 19/20 of a pound .95 6 pence are 6/240 or 1/40 of a pound .025 £410.975 £410.975 multiplied by $4.8665 is $2,000.01. 2. Reduce pence, shillings and pounds to value in dollars, thus: 6 pence are 1/40 of $4.8665 $ .12 19 shillings are 19/20 of $4.8665 4.62 410 pounds at $4.8665 1,995.27 £410/19/6 at $4.8665 $2,000.01 123 124 MATHEMATICS OF ACCOUNTING AND FINANCE Reverse Conversion To reduce a value in United States coinage to a foreign money value, divide the value in United States money by the value of the foreign unit. Illustration What is the value in pounds of $2,000 at the mint par of exchange? Solution: There are again two ways to calculate this, which are as follows : 1. 2,000 -^ 4.8665 = 410.973, number of pounds .973 of 20 (shillings in the pound) = 19.46 .46 of 12 (pence in the shilling) = 5.52 Hence $2,000 = £410/19/6. 2. To find the number of pounds: 4.8665 1 2,000.00 I 4io£ 1,946.60 53-400 48.665 Remainder 4-735° Reduce to shillings 20 48665 947000 1 19 s 48665 460350 437985 Remainder 22365 Reduce to pence 12 48665I 268380 1 5.52 d 243325 250550 243325 72250 Result £ 4 1 0/19/6. In this particular case there is a much shorter method, which is as follows : FOREIGN EXCHANGE 1 25 £410/19/6 is the same as £411 minus 6d, or £411 minus 1/40 of £1. We can, therefore, obtain the result by performing the following sub- traction : £4ii/oo'o X $4.8665 $2,000.13 — 6 or 1/40 of $4.8665 —.12 £410/19/6 $2,000.01 Current rates of exchange vary from the unit par rate, because the supply and demand for foreign drafts are afifected by the balance of trade between countries. Transactions involving foreign exchange are made at current instead of mint par rates. Exchange is quoted at the current value in cents and con- versions are made as illustrated in the conversions at mint rates. Illustration What is the cost of a 300 franc draft on Paris, purchased at 8.49? Solution: $.0849 X 300 = $25.47 A United States merchant sends $5,000 to Paris when the rate is 8.49. What is the value in francs? Solution: 5,000 -^ .0849= 58,892.8, the number of francs. Dealing in Foreign Exchange There are two methods of recording transactions in foreign exchange. When the first method is used, the charges and credits are entered in two values, domestic and foreign, the conversion of each item being made at the rate applying to the transaction When the account is closed the balance in foreign money is valued by converting either at cost or current rate, and the profit or loss on exchange is ascertained. The other method attempts to ascertain the profit or loss on each item, by computing the difference between the conversion value at par and current rates. Both methods are illustrated below : 126 MATHEMATICS OF ACCOUNTING AND FINANCE Illustration A banking concern deals in foreign exchange and the following are the transactions with a London correspondent for one month: Debits Sept. 1 Remittance, 30-day bill £400 at $4.86 10 " sight bill £100/10 " 4-87 "15 " " " £200/0/6 " 4-86K Credits Sept. 2 Draft, sight £300 at $4.87^ 12 £200/12/5 " 4-87 20 Cable £100 " 4.88 1. Ascertain the profit or loss in the account for the month. 2. State the balance of the account at the end of the month in foreign and domestic currency, the current rate of sterling exchange for cable transfers being $4.89. Solution i : Although the problem states that the current rate of exchange for cable transfers is $4.89, it seems incorrect to inventory the balance of the account at this rate, since the rate is not only higher than cost but higher than any of the prices at which sales have been made. On the theory that the first items purchased are the first sold we find that the first two remittances are ofTset by the first two credits. Applying this theory, the balance of the account must have cost $4.86^, the rate apply- ing to the remittance of September 15. Setting up the account and valu- ing the balance at $4.86^, we have the following: Account with London Correspondent Debits Foreign Rate Domestic Sept. I, Remittance, 30-day bill £400/00/0 $4.86 $1,944.00 " xo, " sight bill loo/io/o 4.87 489.44 " 15, " " " 200/00/6 4.86K 973.62 " 30, Profit 6.74 £700/10/6 $3,413.80 Oct. I, Balance — Inventory £99/18/1 $4.86^ $486.28 FOREIGN EXCHANGE 127 Credits Sept. 2, Draft, sight £300/00/0 $4,871^ $1,462.50 " 12, " " 200/12/5 4.87 977-02 " 20, Cable 100/ 00/0 4.88 488.00 " 30, Balance 99/18/1 4-86^ 486.28 £700/10/6 $3,413.80 While $6.74 is shown as profit it must be remembered that the rate of 4.86^ is used because the current rate for checks is unknown, that interest has been disregarded in the calculation, and that the profit as shown is nominal only and may be increased or decreased when realized by a sale. If the balance were inventoried at $4.89, its domestic value would be $488.53, and the profit would be $8.gg. It would be correct to value the balance at the current buying rate on September 30, if that were known. This is the usual practice. In calculating the dollar value of the balance, the shortest way is to take £99/18/1 as I s 11 d less than £100/00/0, as follows: I shilling is 1/20 of $4.8675 or $0.24337 II pence is 1/12 less than i shilling, that is, $.24337 — .02027 .22310 5.47 £100 $486.75 - i/ii --47 £ 99/18/1 Solution 2: The above is the usual method of keeping a foreign ex- change account in this country. In Canada each transaction as recorded in the buying and selling registers is compared with the par of exchange and an entry made debiting or crediting Exchange, as the case may be. The par of sterling is $4.8665 or $4.86?/^. The registers, in the example taken, would show the following: Bought Date Time Sterling R.ate Paid Par Dr. Ex. Cr. Ex. Sept. 1 30 d £40000/0 $4.86 $1,944.00 51,946.67 S $2.67 10 St. loo/io/o 4.87 489.44 489.10 .34 IS " 200/00/6 4-8634 973.62 973-45 -'7 £700/10/6 J3,409-22 $.51 $2.67 128 MATHEMATICS OF ACCOUNTING AND FINANCE Sold Sept. 2 .. . . 12 ... . 20.. . . 30 St. . . . . ca. .... balance £300/00/0 200/12/5 loo/oo/o 99/18/1 4.87 4.88 $1,462.50 977-02 488.00 $1,460.00 976.35 486.67 486.20 £700/10/6 $3,409.22 *2.S0 .67 1.33 $4-50 The ledger would contain an account with the London correspondent charged and credited with the entries in the Par columns, and an account with Exchange, showing a debit of $0-51 and credits of $2.67 and $4.50. The sterling column is a memorandum inventory account only. The account with Exchange would indicate a profit of $6.66, the diflference from the profit of $6.74 shown by the United States method being caused by inventorying the balance at par instead of at cost. Of course, the postings would be made in daily totals in a bank in which a daily trial balance is taken off. The balance may be entered at cost of $4.86K, or $486.28 in the sold column and 8 cents in the credit exchange column, which will make the profit $6.74, the same as by the other method. Average Date of Current Account The principles already explained for determining the average date of an account apply when the account is kept in a foreign coinage. The following is an illustration. Illustration A of London, in current account with B of New York, engages an account- ant to prepare a statement, to be mailed to B, based on the following data: 1914 Debits: May 12 £750 May 30 117 June 12 340 July 1 150 Total debits £1,357 FOREIGN EXCHANGE 129 Credits: June 10 ^Soo June 30 300 Total credits 800 Balance i 557 Find the average due date of the account and the interest at 5% to July I, taking 365 days to the year. Solution: Focal date, April 30 — Debits May 12 £ 750 12 days £9,000 " 30 117 30 " 3,510 June 12 340 43 " 14,620 July 1 150 62 " 9,300 £1,357 £36,430 Credits June 10 £ 500 41 days £20,500 " 30 300 61 " 18,300 £ 800 38,800 Balances £ 557 Debit £2,370 Credit The amount, 2,370 divided by 557 equals 4- But since the balance of the account is a debit while the balance of the products is a credit, the average date is four days backward from April 30, or April 26. From April 26 to July i is 66 days. Interest on £557 for 66 days at 5% may be calculated in either of the following ways: I. £557 X .05 X ^ = £5-57 X - = ^5-036 £1 X .036= 2osX.o36= .72s isX -72= 12 dX .72 = 8.64 d Total interest £5/0/9 130 MATHEMATICS OF ACCOUNTING AND FINANCE 2. At 6% •fS-S? is int. for 6o days on 360 days basis " 6% .557 £6.127 " " " 66 " " 1% 1.021 n u u a II u ii u " 5% £5.106 " " Reducing to 365 day basis .07 Subtract 1/73 a a ic II (I £5.036 which is reduced to £5/0/9 as above The decimal .036 can also be reduced as follows: £1 = 240 d £ .036 = 240 X .036, or 9 d Conversion of Foreign Branch Accounts The following example shows how foreign branch account balances are converted to domestic values, and how the home office and foreign branch accounts are consolidated in the period- ical statements. Illustration A New York corporation builds a plant and establishes a branch in Liverpool, England. At the expiration of its fiscal period, a trial balance is forwarded to the New York ofhce, as follows: Plant £250,000 Accounts receivable 187,500 Expenses 25,000 Inventory (end of fiscal period) 50,000 Remittance account 150,000 Cash 12,500 Accounts payable £ 87,500 Income from sales 250,000 New York office 33 7,500 £675,000 £675,000 FOREIGN EXCHANGE 131 A trial balance of the New York books on the same date is as follows: Capital stock $2,500,000.00 Patents $1,500,000.00 Liverpool account 1,640,250.00 Remittance account 729,281.25 Expenses at New York 25,000.00 Cash 64,031.25 $3,229,281.25 $3,229,281.25 The remittance account is composed of four 60 -day drafts on Liverpool for £37,500 each, which were sold in New York at $4.85^^, $4.86, $4.86^^ and $4.86^ respectively. Prepare a balance sheet of the New York books after closing and a statement of assets and liabilities of the Liverpool branch consolidated with the New York books. Close the books at the rate of exchange on the last day of the fiscal period, which is $4.87 1<4, conversion of remittances to be made at the average rate for the four bills. Solution: Since the fixed assets are not subject to revaluation at current exchange rates, the plant cost must be taken out of the New York office account, which is subdivided to show indebtedness to the main office for fixed assets and for current assets. Since the cost of the fixed assets is not given, we must assume that it is at the same rate as the rest of the New York account. Dividing the Liverpool account balance of $1,640,250 by the New York office account balance of £337,500 we obtain the ex- change rate of 4.86, at which rate the plant has a value in United States money of $1,215,000. The reciprocal accounts are then subdivided as follows : On Liverpool books: New York office — plant £250,000 " " " — current account 87,500 On New York books: Liverpool account — plant $1,215,000.00 " current account 425,250.00 The next step is to close the Liverpool books into the New York office current account as follows: 132 MATHEMATICS OF ACCOUNTING AND FINANCE New York Office Current Account Expenses £ 25,000 Balance £ 87,500 Remittance account . . . 150,000 Income from sales 250,000 Balance 162,500 £337,500 £337,500 Balance. . . .• £162,500 The balance sheet of the Liverpool branch is now as follows: (Plant valued at cost, $4.86; current assets and liabilities valued at current ex- change rate of $4.87}^). Assets Plant $4-86 Accounts receivable 4-^7/4 Inventory 4-^7/^ Cash 4.87M' £250,000 $1,215,000.00 187,500 913,503-75 50,000 243,625.00 12,500 60,906.25 £500,000 $2,433,125.00 Liabilities Accounts payable $4-87K £87,500 $ 426,343.75 New York office — plant account .... 4.86 250,000 1,215,000.00 New York office — current account — A-^7% 162,500 791,781.25 £500,000 $2,433,125.00 On the New York books, the Liverpool current account is credited with the remittances, and a charge is made to the account (offset by a credit to profit and loss) sufficient to leave the account with a balance of $791,781.25, the balance of the net current assets in Liverpool at $4.87>^. It is not necessary to find the average rate of the remittance account since the whole account is used. It would be found, by dividing $729,- 281.25 by 150,000, to be $4.86 3/16. The figures to be used are then found by multiplying £150,000 by $4.86 3/16, which gives $729,281.25, an amount which we already have. FOREIGN EXCHANGE 133 Liverpool Current Account Balance $ 425,250.00 Remittances... $ 729,281.25 Profit 1,095,812.50 Balance 791,781.25 M, 521,062. 50 51,521,062.50 Balance $ 791,781.25 The New York profit and loss account will appear as follows: Profit and Loss New York expenses.. .$ 25,000.00 Liverpool Branch $1,095,812.50 Net profit to surplus. . 1,070,812.50 $1,095,812.50 $1,095,812.50 This profit is subject to a further charge for reduction of the patents account. The New York office trial balance is as follows: Capital stock Surplus Patents Liverpool plant account . . Liverpool current account Cash $2,500,000.00 1,070,812.50 $1,500,000.00 1,215.000.00 791,781.25 64,031-25 $3,570,812.50 $3,570,812.50 The balance sheet is as follows: Assets Fixed assets: Patents Liverpool Plant Current assets: Accounts receivable, Liverpool Inventory Cash — Liverpool $60,906.25 Cash — New York 64,031.25 51,500,000.00 1,215,000.00 5 013.593-75 243,625.00 52.715,000.00 124,937-50 1.282,156.25 $3,997,156.25 134 MATHEMATICS OF ACCOUNTING AND FINANCE Liabilities Capital stock $2,500,000.00 Surplus 1,070,812.50 $3,570,812.50 Accounts payable, Liverpool 426,343.75 $3,997,156.25 The Liverpool branch profit of $1,095,912.50 is made up as follows: Income from sales £250,000 Less expenses 25,000 Net profit on sales £225,000, at $4.87/,^ $1,096,312.50 Increase of current exchange rate ($4. 8714^) over original valuation ($4.86) of opening balance of current account: £87,500 at $4.87^ $426,343.75 £8/, 500 at $4.86 425,250.00 1,093.75 $1,097,406.25 Less exchange loss on remittances: £150,000 at $4.87^ (current rate) .... $730,875.00 £150,000 at $4.861875 (average con- version rate) 729,281.25 i, 593-75 Net profit $1,095,812.50 Even in this explanation it is not necessary to use the average con- version rate. CHAPTER XV LOGARITHMS Use Logarithms are a special compilation of figures, used to reduce the labor of multiplication, division, computation of powers and extraction of roots. After explaining the method of employing logarithms, their theory will be briefly discussed. Every number has its corresponding logarithm — but a knowl- edge of the method of determining the logarithms of various numbers is not essential, as they may be obtained from prepared tables. The logarithms of certain numbers are stated below for use in illustrations. For a more complete table, see Appendix. Table of Logarithms of Selected Numbers Number Logarithm 2 •30103 3 .47712 4 .60206 5 .69897 6 •7781S 7 .84510 8 .90309 9 •95424 lO 1. 00000 12 1. 07918 15 I.I 7609 16 1. 20412 30 1.47712 100 2.00000 160 2.20412 300 2.47712 1,000 3.00000 1,600 3.20412 3,000 3-47712 135 136 MATHEMATICS OF ACCOUNTING AND FINANCE Multiplication Multiplication is accomplished by the addition of logarithms, as follows: 1. In the table, find the logarithms of the numbers to be multipHed. 2. Add the logarithms; the sum is the logarithm of the desired product. 3. In the table, find the number corresponding to the logarithm obtained in 2. The following examples, though more easily performed with- out logarithms, will illustrate the steps of the process; Example 2X3=? Solution: 1. Table shows 2. Sum of logarithms Number Logarithm 2 .30103 3 -47712 .77815 log of product 3. Table shows .77815 is log of 6, the product Example 2X3X2=? Solution: 1. Table shows <( a 2. Sum of logarithms Number Logarithm 2 -30103 3 -47712 2 -30103 1.07918 log of product 3. Table shows 1.07918 is log of 12, the product LOGARITHMS 137 Without going beyond the selected table, the following multipli- cations may be performed by using logarithms: 5X6=? 4X3=? 3X2X5=? Division Division is accomplished by the subtraction of logarithms as follows: 1. In the table find the logarithms of the dividend and the divisor. 2. Subtract the logarithm of the divisor from the logarithm of the dividend; the difference is the logarithm of the desired quotient. 3. In the table, find the number corresponding to the loga- rithm obtained in 2. Example 6-^2=? Solution: Number Logarithm 1. Table shows: Dividend 6 -77815 Divisor 2 -30103 2. Difference of logarithms -47712 log of quotient 3. Table shows .47712 is log of 3, the quotient The following divisions may be performed by logarithms with the aid of the selected table: 8-i-2 10 4-5 16-J-4 300 -T- 30 Calculating Powers A power of a number i3 the product obtained by using that number repeatedly as a factor. The square of a. number is the 138 MATHEMATICS OF ACCOUNTING AND FINANCE second power; the cube, the third power, etc. The exponent of the power indicates the number of times the factor is to be used. For instance 3^ means the second power of 3; or 3 squared; the number used as a factor is 3. The superscript number "" is the exponent of the power, showing that 3 is to be used twice as a factor. Thus, 3' = 3 X 3 = 9 A number is raised to a power by logarithms as follows: 1 . In the table find the logarithm of the number to be raised to a power. 2. Multiply the logarithm by the exponent of the power; the product is the logarithm of the power. 3. In the table find the number corresponding to the loga- rithm obtained in 2. Example i 3'= ? Solution: Number Logarithm 1. Table shows 3 -47712 2. Multiply by exponent of power 2 .95424 log of power 3. Table shows .95424 is log of 9, the second power of 3. Example 2 2" = ? Solution: Number Logarithm 1 . Table shows 2 -30103 2. Multiply by exponent of power 4 1. 204 1 2 log of power 3. Table shows 1.204x2 is log of 16, the fourth power of 2. LOGARITHMS ^39 Roots A root is a number which, used repeatedly as a factor, will produce a given power. The square root of 9 is 3, because 3 used twice as a factor produces 9. , • r ^f^^ The fourth root of 16 is 2, because 2 used 4 times as a factor ^'""ThTroot of a number is expressed by writing the number under the radical sign; the index figure indicates the root to be extracted. The square root of 9 is expressed thus: The superscript ^ is the index of the root. Any root of a number may be extracted by logarithms as follows: I. In the table find the logarithm of the number, the root of which is to be extracted. 2 Divide the logarithm by the index of the root; the quo- tient is the logarithm of the desired root. 3. In the table find the number corresponding to the loga- rithm obtained in 2 . Example i Solution: I. Table shows log of 9 to be .95424 2 Divide by index of root: .954^4 - ^ = .47712, log of root ;. Table shows .47712 is log of 3, the square root of 9 Example 2 2>?) "I" to carry. This negative remainder causes confusion in the division of the positive mantissa. Hence the procedure below is followed: Add and subtract 20* Divide by 3 Number Logarithm 000729 4.86273 or 6.86273 — 10 20. — 20 3 ) 26.86273 - 30 8.95424 — 10 )r 2.95424 .95424 is the mantissa of 9 Characteristic 2; one zero; hence -y^ .000729 = .09 *2o is used to obtain the —30, which is divisible by 3 Examples Compute the following: I. .031 X .3 2 .0045 X .00021 3 .0036 X 2.2 4 .000011 X 56000 5 .0023 X 42000 6 .006 X 6.5 7 .038 X .0003 8 504 -f- 24 9 .368 -r 23 10 .00351 -^ .027 II. 390 H- •15 12. 13- .0994 - 28^ - 71 14. 1.3^ 15- 16. 17- .9^ .003 <^ 18. V.0625 19. V. 000343 20. 4^ X \/.ooi6 150 MATHEMATICS OF ACCOUNTING AND FINANCE Determining Mantissas by Interpolation Tables of logarithms are limited in scope and do not contain the mantissas of all possible numbers. For numbers beyond the scope of the table, the mantissa may be approximated by interpolation. Illustration What is the logarithm of 18565? Solution: Number Mantissa The table shows 18570 .26S81 " 18560 .26858 Difference 10 .00023 If a difference of 10 in the numbers causes a change of .00023 in the mantissa, a difference of 5 will cause a change of approximately V.o of .00023 in the mantissa. Now, .5 of .00023 = .000115 Then .26858, mantissa of 18560 plus .000115 sum .268695, mantissa of 18565 (approximate) Logarithm, 4.268695 Examples By interpolation determine the logarithms of the following numbers: 3642 1209 47629 758263 25.69 3.479 .004682 32.0046 Determining Numbers by Interpolation When a computation results in a mantissa not to be found in the table, the number corresponding to the mantissa may be determined approximately by interpolation. Solution: LOGARITHMS 151 Illustration 15^= ? Number Logarithm IS I. 17600 Multiply by exponent of power 4 Logarithm of 1 5 "* 4. 70436 The mantissas larger and smaller than .70436 shown in the table are: Number Mantissa 5063 5062 I .70441 •70432 Differences .... .00009 The mantissa oils'* is " 5,062 is ... . .70436 •70432 Difference .00004 If an increase of .00009 in the mantissa represents an increase of i in the number, an increase of .00004 in the mantissa represents an increase of approximately 4/9 of i, or .444 + in the number. Hence .70432 Mantissa of 5062 add interpolation for .00004 444 + Result .70436 mantissa of 5062444 + 4.70436 is the logarithm of 50624.44 or 15'' Exact answer is 50,625. Examples What are the approximate numbers corresponding to the following logarithms? 2.76354 4^94965 3.42345 2.4166572 1.82629 3.57927 152 MATHEMATICS OF ACCOUNTING AND FINANCE Illustration What is the value of 1856^? Solution: The logarithm of 1,856, is 3.26858 Multiply by exponent of power 2 Logarithm of 1856^ 6.53716 Table shows: Number Larger mantissa 3445 Smaller " 3444 Differences i M.-VNTISSA 53719 53706 Mantissa of. 1856^ 3444 Difference. 00013 53716 53706 10/13 of I = .769 approximately Hence 3444 Plus 760 •53706 .00010 Sum 3444769 .53716 Then 3444769 is the number represented by 6.537148 1856^ is accurately 3444736 Approximated by logarithms 3444769 Error 3^ If the problem had been 18.56'' The accurate power would be 344.4736 Approximate power (by logs) 344.4769 Error . ■0033 This example shows that fairly accurate results may be obtained by interpolation, but absolutely accurate work requires tables extended to show the mantissas of all numbers required by the problem. LOGARITHMS 153 Examples What are the values of: 1. 24 X 18 2. 360 X 800 3. 32.6 X 3 4. .27 X .71 5. 864 X .00321 6. 2462 X 3278 7. .00964 X 3425 8. 34400 -^ 43 21. ^%? Solution: The interest at 6% has already been computed being $16.41248. Since 4>2% is yi of 6%, the interest at 4^% is found as follows : $16.41248 interest at 6 % 4.. 03 1 2 " " iK%(>^% for 119 days? Solution: The interest at ^Yi^o on a 360 day basis has already been computed, being $12.30936 Deduct 7^3 of $12.30936 .16862 Exact interest on 365 day basis $12.14074 Partial Payments There are two methods of calculating interest on a debt on which partial payments are made. The one in common use among business men and known as the Merchants' Rule, consists in calculating the interest on the principal sum from its date to the date of settlement and a similiar calculation on the partial pay- ments, the difference between the total interest on the respective sides being the net interest due. On small sums and for short periods this is accurate enough for all practical purposes. The other, called the United States Rule, gives precedence to the interest due at the time of each payment, and requires that each payment shall be first applied to the liquidation of the inter- est then due, only the remainder after the interest is deducted being applicable to the reduction of the principal. If the pay- ment is not equal to the interest then due, it is applied in reducing the interest, but the excess interest is not added to the principal, 158 MATHEMATICS OF ACCOUNTING AND FINANCE as this would be compounding it, but it is carried down and added to the interest to be deducted from the next payment. This method is made legal by statute in nearly every state. An example of the two methods will show the difference between them, 30 days to the month being used for convenience. Illustration Jan. I. Original amount $600 March i, payment $200 May I, " 200 June I, " 100 Interest is to be charged at 6%. Required — amount due July i Solution: By the Merchants' Rule : Interest on $600 for 6 months is $18.00 " " 200 "4 " " $4.00 " " 200 "2 " " 2.00 " " 100 " I " " 50 6.50 Interest due July i $11.50 Unpaid principal 100.00 Total due July i $111.50 By the United Stales Rule: Original debt $600.00 Payment March i $200.00 Less interest on $600 for 2 months 6.00 194.00 $406.00 Payment May i $200.00 Less interest on $406 for 2 months 4.06 195.94 $210.06 Payment June i $100.00 Less interest on $210.06 for i month. . . 1.05 98.95 $111. II Interest on $111. 11 for i month .56 Total due July i $111.67 SIMPLE AND COMPOUND INTEREST 159 While the difference of 1 7 cents in the results obtained by the two methods is negligible, it is otherwise when the amounts are large and the time is long. The following example is taken from the accounts of an estate that was not settled for thirty years. It was virtually as follows : Illustration An amount of $36,000 was due one of the heirs of an estate, who was to receive 6% interest until paid. January i, 1886, legacy $36,000 " I, 1 89 1, payment 12,000 " I, 1896, " 12,000 " I, 1906, " 12,000 " I, 1911, " 19,000 The administrator, using the commercial rule, agreed to settle the account as of January i, 1916, as follows: Original amount $ 36,000 Interest for 30 years at 6% 64,800 $100,800 Less total cash paid $55,000 Interest on $12,000 for 25 years 18,000 Interest on $12,000 for 20 years 14,400 Interest on $12,000 for 10 years 7,200 Interest of $19,000 for 5 years 5, 700 100,300 Amount due January i, 1916 $500 The heir refused the settlement and made his claim under the law of the state (Illinois) as follows: Original amount $36,000.00 January i, 1891, payment $12,000.00 Interest 5 years on $36,000.00. . . 10,800.00 1,200.00 $34,800.00 January i, 1896, payment $12,000.00 Interest 5 years on $34,800.00. . . 10,440.00 1,560.00 $33,240.00 I60 MATHEMATICS OF ACCOUNTING AND FINANCE January i, 1906, payment $12,000.00 Interest 10 years on $33,240.00.. 19,944.00 Interest carried forward $ 7,944.00 January i, 191 1, payment $19,000.00 Interest 5 years on $33,240.00. . . $9,972.00 Interest brought forward 7,944.00 17,916.00 1,084.00 $32,156.00 January i, 1916, interest 5 years on $32,156.00 9,646.80 Amount due January i, 1916.. $41,802.80 It can readily be seen that a little knowledge of correct principles was a valuable asset to the heir. Compound Interest The principles of interest are involved in the computations which an accountant may be required to make in connection with such matters as bond premium and discoimt, leasehold premiums, depreciation and sinking funds. These calculations involve not only compound interest and the amount of a given principal at compound interest, but also the more complex problems of annuities. All scientific computations of interest on an indebtedness or an investment extending over more than one period of time must be based on compound interest. This is because the indebtedness or investment increases with the lapse of time. If only simple interest is charged, interest is earned only on the original invest- ment instead of on the investment for each period. For instance, if $100 is invested at 6%, the investment at the beginning of the first year is $100. It increases $6 during the first year and amounts at the end of the first year to $106. Since the investment has now increased to $106, interest SIMPLE AND COMPOUND INTEREST l6l should be earned thereon. If simple interest only is charged, $6 will be earned during the second year on an investment of $io6. This is at the rate of only 5.66 +%. Scientific computations of interest are based on the sup- position that the increase arising from interest is re-invested. This is equitable in theory, for if the interest of one period in- creases the investment at the close of that period, the investor should, during the next period, earn interest on the increased investment. To maintain the agreed interest rate during a series of periods, computations must be made on the basis of periodical compounding. Symbols The following standard symbols by Sprague and Perrine, will be used throughout this book: I = $1, £1, or any other unit of value i = the rate of interest for a single period « = an indefinite number of periods a = the amount of $1 for a given time at a given rate / = the compound interest on $1 for a given time at a given rate p = the present worth of $1 for a given time at a given rate D = the discount on $1 for a given time at a given rate r = {i -\- i), the periodic ratio of increase' Amount of Principal In finding the amount of a given principal at compound inter- est for a given number of years at a given rate per year com- pounded annually, it is customary to compute the amount of $1 for the given time at the given rate, and multiply this result by the number of dollars in the principal. When i = 6% or .06, the accumulation of the amount may be computed by either of the following methods: -C. E. Sprague and L. L. Perrine, The Accountancy of Investment, 1914. l62 MATHEMATICS OF ACCOUNTING AND FINANCE First Method Second Method Dollars End of Dollars Symbols Principal i.oo i.oo I Interest on Ji. 00 .06 Multiply by ... . 1.06 (i + i) 1.06 I year 1.06 (i + «') " " 1.06 0636 " " .... 1.06 (i + »') 1. 1236 2 " 1. 1236 (i + «')' " " 1. 1236 067416 " " .... 1.06 (i + ») 1.191016 3 " 1.191016 (i +»)^ 1.191016 071461 " " .... 1.06 (i + ») 1.262477 4 " 1.262477 (i +')* The ratio of increase is, in symbols, (i+i), and in figures 1.06. The investment at the beginning of each year must, therefore, be multiplied by (i+i) to obtain the investment, or amount, at the end of that year. Hence i invested at the rate i becomes at the end of 1 year (i + O 2 " d + O' 3 " (i-hi)' 4 " d+O' n " (i + O" Thus the formula is obtained a= (1+ i)" If it is desired to find the amount of i at 5% for 20 years, compounded annually, the formula becomes Frequency of Compounding In the preceding illustration the interest was compounded annually. But the period of compounding may be shorter than one year, with the result that the compounding occurs with greater frequency than once a year. In such cases the formula stated above still applies. Although compounding may occur SIMPLE AND COMPOUND INTEREST 163 semiannually, quarterly, monthly or even daily, the rate is usually stated as a certain per cent per year, but i in the formula is the annual rate divided by the number representing the periods in a year. For instance, if the rate is 6% per year, iunder various conditions would be as follows: Frequency of Compounding Value of i Annually .06 Semiannually (.06 -^ 2 ) .03 Quarterly (.06 -^ 4 ) .015 Monthly (.06 -^ 12 ) .005 Daily (.06 -^ 365) -^ The number of periods, represented in the formula by n, will be the number of years times the number of periods per year. For instance n, under various conditions, is as follows: Frequency of Value of n Compounding One Year Two Years Three Years Annually i 2 3 Semiannually. . . 2 4 6 Quarterly 4 8 12 Monthly 12 24 36 Daily 365 730 1095 Illustration What is the amount of $1 at 6% interest for 20 years, compounded quarterly? Solution: i = .06 -4- 4 = .015, the rate for one period n = 20 X 4 = 80, the number of periods a = (i + i)** or (1.0x5) *°, which amounts to 3.29066279 Determining the Amount Interest tables show the amount of $1 at various rates for various periods. Such a table appears in the Appendix of this 1 64 MATHEMATICS OF ACCOUNTING AND FINANCE book. When a table is not available, the amount may be computed with a table of logarithms. When neither an inter- est table nor a table cf logarithms is available, it is necessary to compute the amount by repeated multiplication, thus: 1.03 1-03 1. 0609 1.03 1.092727 1.03 a = (1.03)'= ? = (1.03)^ = (1.03)^ 1.12550881 = (1.03)" At this point the number of decimal places becomes so large as to make the computation too laborious. The number may be reduced to six places, by approximation, thus: 1.12550881 becomes 1. 125509 The multiplication continued is as follows: 1.125509 = (i.03)'* 1.03 159274 03 (1-03) ■ 194052 = (1.03)' 03 229874 = (1.03)7 03 266770 = (1.03) This last amount is the amount of $1 at 6% interest for 4 years, compounded semiannually SIMPLE AND COMPOUND INTEREST 165 But when the number of periods is large, as will be the case when the interest must be calculated over a long term of years, these repeated multiplications become irksome. It is possible, however, to cut down on the number of operations necessary. A short method may be utilized which should be readily understood if it is remembered that (14-^)" means that (i +i) is used n times as a factor. If we know the value of (i +i) ^ we have the amoimt obtained by using (i+i) twice as a factor. By squaring this amount we get (1+^)'', which is the product resulting from the use of (1+/) four times as a factor. Squaring this amount in turn gives us (i +i) ^, the product obtained by using (i +/) eight times as a factor. By the application of this principle, a very material reduction may be effected in the number of multiplications, as is made clear in the example given below : 1.03 1.03 1.0609 1.0609 = (i + i) = (i + i) 1. 125509 1. 125509 = (1 + i)' 1.266770 = (i + i) The quantity 1.03'° can be determined in the following manner : 1.26677 (i + «)^ 1.0609 (i "f" ^)^ 1. 34391 6 (i + i)^" The general application of this principle may be made clearer by another illustration. I66 MATHEMATICS OF ACCOUNTING AND FINANCE Illustration Required — the value of (i + i)^°: Solution: (I + i) (I + i) : + i)' + iy + i)' + i)' + i)' + iV + ■yU + i)' + iy + iy + iV + iy Thus, only seven multiplications are required to obtain the 30th power. The 80th power could be obtained thus : (i + 7) '^ (obtained by four multiplications) (1+ i)" (1+ i)'' (1 + i)'' u + iy'' SIMPLE AND COMPOUND INTEREST 167 The same principle may be utilized in the use of compound interest tables which do not extend to the desired number of periods. Illustration Given a table of 40 periods, required — the value of (i + i) '^. Solution: ( I + /) '^ " shown by table (I + /V " " " (i + i)" Determining Interest The amount of $1 at compound interest is composed of two elements: the original investment of $1 and the accumulated interest. Hence, to find the compound interest, apply the follow- ing formula: / = c — I or / = (i + J-)" - I If $1.266770 is the amount of $1 at 6% interest, compounded semiannually for four years, $1 .000000 is the original investment, and $.266770 is the compound interest. Determining Present Worth The present worth of a sum due at a fixed future date is a smaller sum which with interest will amount to the future sum. For instance, the present value of $1 due in one year at 6% is a sum, smaller than $1, which with interest at 6% for one year will amount to $1. Representing the present value of this $1 by p p X 1.06 = $1 or in symbols, pX (i + i) = i when the dollar is due in one year. I68 MATHEMATICS OF ACCOUNTING AND FINANCE Since pX {i + i) = i, it follows that i -r- (i + t) = /» (Since p X 1.06 = i, it follows that I-^ 1.06 = .943396.) If the dollar is due two periods hence, PX (i-i- i)'= I and it follows that i -^ (i + f)^ = /> If the dollar is due three periods hence pX (i-\- i)'= I and it follows that i -^ {i -\- i)^ = p If the dollar is due ?i periods hence px {i-\- ir = I and it follows that i -^ (i -\- i)" = p But since (i + i)" = a the formula p = i -^ (i + i)" may also be stated p = i -^ a In determining the present value at 6% of $1 due in four years, interest compounded semiannually, several methods are available. 1 . It may be possible to refer to a table of present values, similar to the one in the Appendix. 2. If a table of present values is not available, the present value may be computed by using the formula p = I -T- a It will first be necessary to obtain the value of a. When in- terest at 6% per annum is compounded semiannually, 7 = .03 ; with semiannual compounding for four years, « = 8. Then, a = (1.03)^. The value of a may be found by any of the methods previously explained; it is 1.266770. Then $1 -V- 1.266770 = $.789409, the present value SIMPLE AND COMPOUND INTEREST 169 3. Instead of dividing $1 by (1.03) ^ i.e., by 1.266770, the same result can be obtained by dividing by 1.03 eight times, using I as the first dividend, each succeeding dividend being the quo- tient resulting from the preceding division, thus $1.000000 - - 1.03 = $. .970874 - - 1.03 = . .942596 - - 1.03 = . .915142 - - 103 = . .888487 - - I-03 = . .862609 - - 1.03 = . .837484 - - 1.03 = . .813092 - - 1.03 = . 970874, present value of $ 942596, 915142, 862609, 837484, 813092, 789409, due in i period 2 periods 3 4 5 6 7 Determining the Compound Discount As shown in the preceding section, one would, in exchange for a promise to pay $1 a given number of periods hence, loan the present value of $1. Present values for various numbers of periods, and the discount earned, are shown below (discounted at 3% per period) : Due Periods Hence Amount Loaned Present Value I $.970874 2 942596 3 915142 4 888487 8 .789409 Discount Earned (i-p) $.029126 .057404 .084858 •111513 .210591 The discount for two or more periods is called compound discount. Compound discount may be computed by the formula D = I - p It may also be computed by the formula D= I -^ a 170 MATHEMATICS OF ACCOUNTING AND FINANCE This formula requires explanation. Compound discount is really compound interest, deducted in advance; but it is the compound interest on the money actually loaned. For instance, if $.78940915 loaned on a promise to pay$i eight periods hence at 3%, the discount $.210591 is 3% compound interest on $.789409, the principal actually loaned. This fact can be demonstrated thus : The amount of $1 at 3% compound interest for eight periods is $1.266770. Hence the compound interest on $1 is $.266770 and the compound interest on $.789409 is the result in the fol- lowing multiplication: .789409 p Multiplied by .266770 i Product .210591 d Since the compound discount {D) is really the compound interest (/) on the actual loan {p) , D= IX p And since /> = i -^ a, we can substitute (i H-a) for p, and the formula becomes D= IX I or D = I -h a This formula may also be explained thus : The present value of $1 due in eight periods at 3% is $.789409, or $1 -^(1.03)^, or$i-^a. If the loan were i, the compound interest would be /, or $.266770; but since the loan is i-i-a, the compound discount is / -7-a, or $.266770^ 1. 266770=$. 210590. Summary In this chapter the following formulas have been derived: a = (i + ir I = a — I p = 1 -r- a; ori-T-(i + t)'' D = 1 — p, or / -7- a CHAPTER XVII ANNUITIES Definition of Annuities A series of equal payments, due at regular intervals, is an an- nuity. Although the word "annuity" suggests annum and year, the interval may be any period, as a month, quarter or half-year. Symbols In the discussion of annuities the following symbols will be employed : A = the amount of an annuity of $i for a given time at a given rate P = the present value of an annuity of |i for a given time at a given rate Amount of an Annuity Let us assume that a contract provides for payments as in the following : Example January i, 191 5 $100 " I, 1916 100 " I, 1917 100 " I, 1918 100 Total $400 Required — the accumulated value of this annuity at January i, 1918, interest at 5%. Solution: It is assumed that each payment is put at interest, in which case the amount of each payment computed separately, is as follows: 171 172 MATHEMATICS OP ACCOUNTING AND FINANCE Payment Made January i, 19 is I, 1916 I, 1917 " I, 1918 Amount of the annuity Periods at Interest Payment $100 100 100 100 Amount Symbol (I +i)' (I +.)' (I + 115.7625 110.2500 105.0000 100.0000 431.0125 Although the amount of an annuity may be computed by determining the amount of each payment, it is unnecessary to resort to this labor, as the following short method may be used: To find the amount of an annuity of $i for a given number of periods at a given rate, divide the compound interest on $i for the number of periods at the given rate, by the interest rate. Or, in symbols, A = I -^ i Applying this formula to the illustration above, /, the compound interest on $i for four periods at 5%, is shown by an interest table to be $.215506; then $ .215506 -J- .05 = $4.31012, amount of an annuity of $1 $4.31012 X 100 = $431,012, amount of an annuity of $100 This formula requires explanation. Let us suppose that $1 is loaned on January i, 1914, the contract requiring payment of simple interest annually at 5% as follows: D.\TE Interest January i, igi5 $.05 I, 1916 .05 I, 1917 05 1,1918 05 These payments, being equal in amount and made at regular intervals, constitute an annuity of $.05 per year. If put at 5% compound interest, the amount of each payment computed separately would be as follows: ANNUITIES 173 Payment Made January i, ipis I, 1916 I. 1917 I, 1918 Totals Periods at Interest Payment $.05 • OS • OS • OS f.20 Amount Symbols (l + i)i (I +i}' (I + i) .057881 •055125 .052500 .050000 .215506 The total of the annuity payments is $.20, the simple interest on $1 for four years ; and the amount of the annuity, $.215 506, is the compound interest on $1 for four years, or /. Hence, compound interest is merely the amount of an annuity. If the compound interest $.215506 is the amount at 5% of four annual payments of 5 cents, $.215506 -^ 5 = $.0431012, the amount at 5% of four annual pay- ments of $.01 or $.215506 -^ .05 = or in symbols I..31012, the amount at 5% of four annual payments of $1. 1 ^ i= A Illustration Required the amount of an annuity of $25 per month for five years at 6% per annum, interest compounded monthly. Solution: 12 (payments per year) X 5 = 60, number of terms 6% -^ 12 = K% or -ooS) the rate Now, I = a — 1 And a = (1.005)^°= 1.348849 Then \ = I -^ i = .348849 ^ .005 = 69.7698, amount of an annuity of $1 $69.7698 X 25 = $1,744.25, amount of an annuity of $25 174 MATHEMATICS OF ACCOUNTING AND FINANCE Sinking Fund Contribution The preceding section developed a method of determining the amount which results from the accumulation of a known annuity. It is the purpose of this section to discuss the converse problem, of finding an unknown annuity which will produce a required amount. This problem finds application among accountants in com- puting the periodical contribution necessary to accumulate a required sinking fund. Let us assume that a sinking fund of $100,000 is to be accumulated in five years by equal instalments made at the end of each year; what is the required contribution, assuming 4% interest compounded annually? The annual contributions constitute an annuity and the accumulated fund is the amount of the annuity. We shall first find what fund would be accumulated by a contribution of $1, applying the formula A = I ^ i To find I: 1.04 Multiplied by 1.04 1.0816 " 1.0816 = (1 + i)' 1.16Q85856 " " 1.04 = (1+ /)^ 1. 2166529024 Deduct I . = (i + f)5 .2166529024 = /, compound interest on $1 for five periods at 4% To find A : .2166529024 -^ .04 = 5.41632256, amount of an annuity of $1 ANNUITIES 175 Since annual contributions of $1 will produce a fund of $5.41632256, to find the contributions necessary to produce a fund of $100,000: $100,000 -r- 5.41632256 = $18,462.71, required contribution The computation was performed without using an interest table. If tables are available, the work may be materially decreased. For instance, a table of amounts of $1 per annum shows the amount of $1 for live periods at 4% to be $5.416323. Only one computation is necessary : $100,000 -j- 5.416323 = $18,462.71 If only a compound interest table is available, it will show the amount of $1 in five periods at 4% to be $1.216653. The following computations will be necessary: $.216653 -^ -04 = $5.416325, amount of annuity of $1 $100,000 -^ 5.416325 = $18,462.71 The following tabulation of sinking fund accumulations shows the accumulation of the fund from the two elements, annual contributions and interest: End of Ye.-vr Contribution Interest Total Fund $18,462.71 18,462.71 18,462.71 18,462.71 18,462.71 $ 738.51 1,506.56 2,305.33 3.136.05 $18,462.71 37.663.93 57.633.20 78,401.24 Totals $92,313.55 $7,686.45 Present Worth of an Annuity Let us assume that a contract, made on January i, 1914, provides for the following payments : 176 MATHEMATICS OP ACCOUNTING AND FINANCE January i, 1915 $100 " I, 1916 100 " I, 1917 100 " I, 1918 100 Total $400 We are to find the present value of this series of payments, or annuity, on January i, 1914, discounted at 5%. The present value of each payment, computed separately, is as follows: Payment Due Periods Hence Payment Present Value Symbol $ I 2 3 4 lioo 100 100 100 I -=- Cr 4- i) 95.2381 90.7029 86.3838 82.2702 I, 1916 . I - I - - (I +i)' - (i +i)^ - ri 4- .-14 I, 1917 I, 1918 Totals J400 3S4.S9S0 The present value of each payment, and the present value of the annuity, may be computed thus: 5100 95.2381 90.7029 86.3838 -T- 1.05 = $ 95.2381 present value of $100 due in i year 1.05 = 90.7029, 1.05 = 86.3838, 1.05 = 82.2702, $100 $100 $100 2 years 3 4 $354.5950, present value of the annuity. When interest tables are not available, this method is satis- factory; but w^ith interest tables available a short method may be used which is similar to the short method of computing the amount of an annuity. To find the present value of an annuity of $1 for a given ANNUITIES 177 number of periods at a given rate, divide the compound discount on $1 for the given number of periods by the interest rate. Or in symbols, P = D -^ t Applying this formula to the illustration above, the present value of $1 due four periods hence at 5%, is shown by an interest table to be $.822702; D, the compound discount, is $.177298. $.177298 -4- .05 = $3.54596, the present value of an annuity of $1 $3-54595 X 100 = $354,596, " " " " " " " $100 This formula also requires explanation. Let us suppose that a loan of $1 is made on January i, 1914, the contract requiring the payment of simple interest annually at 5% as follows : Date Interest January i, 191 5 $.05 " I, 1916 05 I, 1917 05 " I, 1918 05 These payments, being equal in amount, and made at regular intervals, constitute an annuity of five cents per year. If dis- counted at 5%, the present value of each payment on January I, 1 9 14, computed separately, is as follows: Payment Due Periods Henxe Payment Present Value Symbols $ January i, 1915 I, 1916 I 2 3 4 S.os • 05 • OS ■ 05 I - I - I - - (1 + .) - (i +.)' - (i +.-)3 - Ct -I- ,U .0476191 •0453514 .0431919 .0411351 Present value of the annuity .... •177297s 178 MATHEMATICS OF ACCOUNTING AND FINANCE Now, instead of paying five cents interest each year, and pay- ing the $1 at maturity, the present value ($.1772975) of the four interest payments might be paid, or deducted, in advance, thus: $1.0000000 payment to be made at end of four years .1772975 present value of interest payments $ .8227025 present value of $1 In other words, the simple interest on $1 at 5% for four years is an annuity of five cents; and $. 1772975, the present value of these in- terest payments, is the present value at 5% of an annuity of four five cents payments; and$.i 772975 is also the compound discount on $1 due four periods hence at 5%. Therefore the compound discount on $1 , due in four periods at 5%, is the present value of an annuity of five cents in four periods at 5%. If the compound discount $.1772975 is the present value at 5% of four annual payments of 5 cents, then $.1772975 H- 5 = $.0354595, the present value at 5% of four annual payments of i cent or $.1772975 -4- .05 = $3.54595, the present value at 5% of four annual payments of $1 or in symbols D -^ i = P Illustration Required the present value of an annuity of $25 per month for five years at 6% per annum, compounded monthly. Solution: 12 (payments per year) X 5 = 60, number of periods 6% -i- 12 = K% or .005, the rate D = 1 - p and /> = I -^ (1.005)^"= .741372 Then D = i — .741372 = .258628, the compound discount on $1 due in sixty periods at >2 % ANNUITIES 179 and D ^ i= P $ .258628 -^ .005 = $ 51.7256, present value of an annuity of $1 $51.7256 X 25 = $1,293.14, the present value of an an- nuity of $25 Rent of an Annuity The preceding section, containing an explanation of the method of determining the present worth of a known annuity, developed the formula, P = D-^i. Each periodical instalment of an annuity is known as the rent of an annuity. This section will deal with the question of determining what rent will be produced by a known present worth. It has been shown that the present value of an annuity of $1 for four periods at 5% is $3.54595, Since a present value of $3.54595 will produce four annual rents of $1 at 5%, a present value of $1 will produce four annual rents of of $1, or $.282012. 3-54595 Therefore, to find the rent of an annuity with a present value of $1 for a given number of periods at a given rate: 1. Find the present value of an annuity of $1 for the given time and rate 2. Divide $1 by this present value of an annuity of $1 Or expressed in symbols, R= I -^ P Illustration A man invests $3,000 in an annuity to be repaid to him in five annual instalments, interest computed at 6% annually. What annual rent will be produced? l8o MATHEMATICS OF ACCOUNTING AND FINANCE Solution: Formula — R = i -^ P The present value of $i due five periods hence at 6% may be com- puted thus: I -=- (i.o6)s or it may be determined from interest tables. The compound interest table shov/s 1. 06^ = 1.338226 then I -^ 1.338226 = .747258, the present value of $1 due in five periods I — .747258 = .252742, the compound discount on $1 due in five periods .252742 -T- .06 = 4.21236, the present value of an annuity of $1 for five periods at 6% then R = 1 -7- 4.21236 = .237396, rent produced by present worth of $1 $.237396 X 3,000 = $712.19, rent produced by present worth of $3,000 The following schedule shows the reduction of the investment due to the excess of the periodical payments over the interest earned on the decreasing investment. Original investment $3,000.00 Interest ist year, 6% of $3,000 180.00 $3,180.00 Deduct ist rent 712.19 Balance, end of ist year $2,467.81 Interest 2d year, 6% of $2,467.81 148.07 $2,615.88 Deduct 2d rent 712.19 Balance, end of 2d year $1,903.69 Interest 3d year, 6% of $1,903.69 114.22 $2,OI7.Qi Deduct 3d rent 712.19 ANNUITIES l8l Balance, end of 3d year $1,305.72 Interest 4th year, 6% of $1,305.72 78.34 $1,384.06 Deduct 4th rent 712.19 Balance, end of 4th year $671.87 Interest, 5th year, 6% of $671.87 40.31 $712.18 Deduct 5th rent 712.18 $ .00 The schedule could also be arranged as follows, showing that each payment is composed of two parts: 1 . Interest on the diminishing principal 2. Repayment of the principal Original investment $3,000.00 First rent: Interest on $3,000 at 6% $180.00 Repayment of principal 532.19 532.19 $712.19 $2,467.81 Second rent: Interest on $2,467.81 at 6% $148.07 Repayment of principal 564.12 564.12 $712.19 $1,903.69 Third rent: Interest on $1,903.69 at 6% $114.22 Repayment of principal 597-97 597-97 $712.19 $1,305.72 Fourth rent: Interest on $1,305.72 at 6% $ 78.34 Repayment of principal 633.85 633.85 $712.19 $ 671.87 Fifth rent: Interest on $671.87 at 6% $ 40.31 Repayment of principal 671.87 671.87 $712.18 $ .00 l82 MATHEMATICS OF ACCOUNTING AND FINANCE Or the schedule may be shown thus: Payment Rent I $ 712.19 2 712.19 3 712.19 4 712.19 5 712.18 Totals $3,560.94 [ntere-st Reduction of Investment Diminishing Investment $3,000.00 $180.00 $ 532.19 2,467.81 148.07 564.12 1,903.69 114.22 78.34 40.31 $3 597-97 633-85 671.87 ,000.00 1,305-72 671.87 $560.94 Equal Periodical Payments on Principal and Interest When a debt together with the interest thereon is to be paid in equal periodical instalments, the principal of the debt is the pres- ent value of an annuity, and the periodical payments are rents, to be computed by the method already explained. To illustrate, if $3,000 bearing 6% interest compounded annually, is to be paid in five annual instalments, each to include the accrued interest and a portion of the principal, $3,000 is the present value of five unknown rents. These rents were computed in the preceding illustration , being $712.19. The reduction of the debt may be tabulated thus : Original principal $3,000.00 First payment $712.19 Less I year's interest on $3,000 180.00 Payment on principal 532.19 Balance of principal $2,467.81 Second payment $712.19 Less I year's interest on $2,467.81 .... 148.07 Payment on principal 564.12 Balance of principal $1,903.69 Third payment $712.19 Less I year's interest on $1,903.69 114.22 Payment on principal 597-97 ANNUITIES 183 Balance of principal $1,305.72 Fourth payment $712.19 Less I year's interest on $1,305.72 78.34 Payment on principal 633,85 Balance of principal $671.87 Fifth payment $712.18 Less I year's interest on $671.87 40.31 Payment on principal 671.87 Balance of principal $ .00 Annuities Due In the foregoing discussion of annuities, rents, and sinking fund contributions, the formulas and methods described apply to the ordinary form of annuities, in which the payments are made at the end of the periods. When the payments are made at the beginning of the periods, the annuity is called an annuity due. Changing the payment from the end to the beginning of the period affects the compound interest, thus changing both the amount and the present value of the annuity. To Find the Amount of an Annuity Due To arrive at a method of computing the amount of an annuity due, let us compare the amounts of: 1. An ordinary annuity (A) of six periods (due at end of period) 2. An annuity due (B) of live periods (due at beginning of period) In each annuity, the interest is compoimded annually at 4%. A requires the payment of $1 on December 31 for each of six years, beginning December 31, 191 1. 1 84 MATHEMATICS OF ACCOUNTING AND FINANCE B requires the payment of $i on January i for each of five years, beginning January i, 191 2. It will be noted that the first five payments of annuities A and B are made on practically identical dates; but annuity A has one more payment than annuity B ; hence the amount of annuity A will exceed the amount of annuity B by one payment of $1. This may be more clearly shown by the following table which shows the amount of each annuity payment and the amount of the annuity. Annuity A Annuity B Date Payment Amount at Dec. 31. 1916 Date Payment Amount at Dec. 31. 1916 Dec. 31, 1911 •' 31. 1912 " 31. 1913 " 31. 1914 " 31. 191S " 31, 1916 Ji.oo 1. 00 1. 00 1. 00 1. 00 1. 00 Si. 216653 1. 169859 1. 124864 1. 081600 1.040000 1. 00 Jan. I, 1912 " I. 1913 " I. 1914 " I. 191S " I, 1916 $1.00 1. 00 1. 00 1. 00 r.oo Si. 216653 1. 169859 1. 1 24864 1. 081600 1.040000 $6.632976 55-632976 Annuity B is an annuity due of five rents; annuity A is an or- dinary annuity of six rents; the periodical payments and interest rate are the same in each case. The difference between the two amounts is $1, or one periodical payment. Hence we can find the amount of annuity B (annuity due) for five periods by ascertaining the amount of an ordinary annuity (A) of the same payment and rate for six periods, and deducting $1 . The amount of annuity A (ordinary) may be computed thus: $1.265319, amount of $1 for six periods 1. 00 $ .265319, compound interest for six periods $.265319 -^ .04 = $6.632975, amount of an ordinary annuity of six periods ANNUITIES 185 To compute the amount of B, the annuity due: $6.632975, amount of ordinary annuity of six periods Deduct 1. 000000, one rent $5.632975, amount of annuity due of five periods Sinking Fund Contributions to the sinking fund of a bond issue are ordin- arily made at the end of the period. If a bond issue is to run twenty years and provision is to be made for it by twenty con- tributions to a sinking fund, the first contribution is usually made at the end of the first year, thus allowing time in which to acquire cash from profits. Such contributions to a fund constitute an ordinary annuity. When contributions to a fund are made at the beginning of the period, the payments become an annuity due. To find the periodical contribution necessary to accumulate the required fund by such payments : 1. Find the amount of an ordinary annuity of $1 for one more than the number of periods 2. Deduct $1 in order to find the amount of an annuity due of $1 for the given number of periods 3. Divide the required fund by the amount of an annuity due of $1. Required Annual Contribution In the discussion of sinking fund contributions in the early part of this chapter it was required to find the sinking fund contribution necessary to invest at 4% at the end of each of five years to pay an obligation of $100,000. This was found to be $18,462.71. We shall now calculate the required annual contri- bution if made at the beginning of each year. These payments constitute an annuity due of five periods. The calculation will proceed by the following steps: 1 86 MATHEMATICS OF ACCOUNTING AND FINANCE I. Find the amount of an ordinary annuity of six periods: !i. 265319, amount of $1 at compound interest for six periods at 4% .265319, compound interest on $1 for six periods at 4% .265319 -r- .04 = $6.632975, amount of ordinary annuity of six periods 2. Find the amount of an annuity due of five periods: $6.632975 .632975, amount of annuity due of five periods 3. Divide total fund by amount of annuity of $1 : $100,000 -H 5.632975 = $17,752.61, required annual contribution The following sinking fund table shows the accumulation of the fund due to the two elements of contributions and compound interest. Ye.\r Contribution Fund Beginning Year Interest Fund End of Ye.\r I 2 3 4 5 $17,752.61 17,752.61 17,752.61 17,752.61 17,752.61 517.752-61 36,215.32 55.416.54 75,385.81 96,153.85 $ 710.10 1,448.61 2.216.66 3.015.43 3.846.15 $ 18,462.71 37,663.93 57,633.20 78,401.24 100,000.00 $88,763.05 Six, 236. 95 Present Worth of an Annuity Due When the annuity payments are due at the beginning of the period, the present value of the annuity is composed of two elements : 1. A rent, or payment, now due (a) 2. The present value of an ordinary annuity of one less than the given number of periods (b) ANNUITIES 187 For instance, $5.451822 deposited at 4% interest compounded annually permits the immediate withdrawal of: (a) $1 rent now due, and leaves a balance of (b) $4.451822, which is the present value of live annual pay- ments of $1, the first of which is due one year hence. Therefore, to find the present value of an annuity due, proceed as in the following illustration. Illustration What is the present value at 4% per annum of an annuity of six pay- ments of $1 , the first of which is due? Solution: i. Find the present value of an ordinary annuity for one less than the given number of periods: $1.000000 .821927, present value of $1 due in five years at 4% $ .178073, compound discount on $1 due in five years at 4% $ .178073 -^ .04 = $4.451825, present value of an ordinary annuity of five payments 2. To this present value add one rent: $4.451825, present value of ordinary annuity of five payments 1. 00 $5.451825, present value of an annuity due of six payments The reduction of this present value, due to the accumulation of interest and the payment of rents is, tabulated below. Present value of annuity due of six rents $5.451825 First rent i . Balance $4.451825 Interest earned ist period .178073 Balance end of " " $4.629898 Second rent i . I88 MATHEMATICS OF ACCOUNTING AND FINANCE Balance beginning 2d period $3.629898 Interest earned " " .145196 Balance end of " " $3.775094 Third rent i. Balance beginning 3d period $2.775094 Interest earned " " .111004 Balance end of " " $2.886098 Fourth rent i. Balance beginning 4th period $1. Interest earned " " .075444 Balance end of " " $1.961542 Fifth rent i . Balance beginning 5th period $ .961542 Interest earned " " .038461 Balance end of " " $1.000003 Sixth rent i . Since the sixth rent is withdrawn at the beginning of the sixth period, there is no balance at the beginning of, nor interest earned during, the sixth period. Rents The converse of the above problem is to determine what periodical payment or rent a known present value will produce, when the first payment is due immediately. For instance, what rent will an investment of $1,000 produce, the first of six annual rents being due at once and the interest being 4%? 1. Find what present value will produce a rent of $1 This was found above to be $5.451825 2. Divide the known present values by the present value of an annuity of $1 $1000.00 -j- 5.451825 = $183.42 ANNUITIES 189 The reduction of this present value may be tabulated thus: Period Rent Balance Interest Balance $1,000.00 I 3183.42 $816.58 I32.66 849.24 2 183.42 665.82 26.63 692.4s 3 183.42 sog.03 20.36 529.39 4 183.42 345.97 13.84 3S9.8I S 183.42 176.39 7.06 183.4s 6 183.4s 0. Si. 100.55 $100.55 CHAPTER XVIIi LOGARITHMS IN COMPOUND INTEREST AND ANNUITY COMPUTATIONS Calculating Compound Interest and Annuities by Logarithms The following examples illustrate the methods of utilizing logarithms in compound interest and annuity computations. Examples I. To find the amount of i. Required — the amount of $4,500 at 4% per annum for twenty years, compounded semiannually Solution: Amount = $4,500 X 1.02''° Log 1.02 = .00860 Multiply by 40 Log 1.02''" = .34400, the log of 2.208 Hence 1.02''''= 2.208 Amount = $4500 X 2.208 = $9936 Amount computed with aid of interest table = $9936.18 Logarithms may be used more extensively in this solution, as follows: Log 4,500 = 365321 Log 1.02 = .00860 Multiply by 40 Log 1.02''" = .34400 Log 4,500+ log 1.02'"'= 3.99721, log of 9,936 Hence $4,500 X 1.02'"'= $9,936 2. To find the compound interest on i. Solution: Compute the amount as above and deduct the principal Amount $9,936. Principal 4.500. Compound interest $5,436. iqo LOGARITHMS AND ANNUITY COMPUTATIONS 191 3. To find the principal. What sum invested at 4% compounded quarterly will amount to $5000 in eight years? Solution: Principal = 5000 -7- (i.oi)-*^ Log 5000 = 3.69897 Log 1. 01 = .00432 Multiply by 32 Log i.oi^^ = .13824 Log of principal = 3.56073, the log of 3636.92 Principal = $3636.92 Principal computed by interest table = $3636.52. 4. To find the rate. If $1,125 is to be returned at the expiration of seven years for a loan of $800, what rate of interest compounded an- nually is earned? Solution: Since 1,125 = 800 X (i + tV (i + i)^= 1,125 -J- 800 Log 1,125 = 305115 Log 800 Log (i +1)7 Log I + i Hence \ -\- i i 90309 14806 14806 -7- 7 = .021151, the log of 1.0499 0499 0499, or nearly 5% 5. To find the time. For how many years should $2,000 be placed at 5% interest compounded annually to produce $5,054? Solution: Since $5,054 = $2,000 X 1.05" 5,054 -^ 2,000 = 1.05" Hence 3.70364, which is log 5054 Minus 3.30103, " " " 2,000 Equals .40261 '' " " 1.05" Since .40261 = log 1.05 X n n = .40261 -T- log 1.05 Since log 1.05 = .02119 n = .40261 -^ .02119 — 19) the number of years 192 MATHEMATICS OF ACCOUNTING AND FINANCE 6. To find the present value of i. What is the present value of $15,000 due in five years at 4%, interest compounded quarterly? Solution: Present value = $15,000 -^ (i.oi)^" Log 15,000 = 4.17609 Log 1. 01 = .00432 Multiply by 20 Log i.oi^" = .08640 Log of present value = 4.08969 Present value = $12,293.89 Present value by compound interest table = $12,293.17 7. To find the compound discount on i. Solution: Compute the present value as in the preceding example: deduct this present value from the amount due at maturity. Amount due at maturity $15,000.00 Present value 12,293.89 Compound discount $ 2,706.11 8. To find the amount of an annuity. What is the amount of an ordinary annuity of $25 for twenty periods at 4% per period? Solution: 1.04'"- I Amount of an annuity of $1 = .04 Log 1.04 = .01703 Multiply by 20 Log 1.04^" = .34060, the log of 2.1908 1.04^" = 2.1908 Deduct 1. 0000 Compound interest = 1.1908 Amount of annuity of $ i = $ 1.1908 ^ .04 = $ 29.77 " " " " $25 = 29.77 X 25 = $744-25 " " " " " by interest tables = $744-45 Logarithms may be used more extensively in these computations, but it is desired to make the solutions as simple as possible. 9. To find the amount of sinkiui^ fund contributions. What annual contribution must be made at the end of each of twenty years to amount to $50,000 at 4K%? LOGARITHMS AND ANNUITY COMPUTATIONS I93 Solution: $50,000 -r- amount of annuity of $1 = contribution Log 1.045 . = .01912 Multiply by 20 Log 1.045^" = .38240, the log of 2.41 21 1 1 Hence 1.045^° = 2.41 21 11 Deduct 1. 000000 1.412111 Compound interest Amount of annuity of $1 = 1.412111 -^ .045 = 31.38 $50,000-=- 31.38 =$1593-37 10. To find the present worth of an annuity. What is the present value of an annuity of $50 per year for ten years at 6%? Solution: The present value of the annuity = 50 X I> D =.- ^ 1.06 Log 1.06 = .02531 Multiply by 10 Log of 1. 06'° = •-?53io, the k-gof amount uf i .00 Determine the present value as follows: Log I. = 0.00000 or 10.00000 — 10 Log 1.06'" = .25310 Hence log—— = g. 74690 - 10, the log of .5583375 1.06 1. 0000000 Minus .5583375 Equals .4416625, the compound discount .4416625 -^ .06 = 7.36104, present value of annuity of $ i $7.36104 X 50 = $368,052, " " " " " $50 CHAPTER XIX BOND DISCOUNT AND PREMIUM Bonds Purchased below and above Par Bonds are frequently purchased at prices either below or above par; that is, at a discount or a premium. When a bond purchased at a discount is held until maturity and paid at par, the owner makes a profit amounting to the difference between the cost and par; that is, to the discount. If the bond is purchased at a premium and repaid at par, the owner incurs a loss amount- ing to the premium. Conversely, one who issues a bond below par and repays it at par loses the discount, while one who issues a bond at a premium and repays it at par gains the premium. The question arises as to when such profit shall be credited, or such loss be charged, to profit and loss. The following discussion considers the subject from the point of view of the investor; the same principles apply to writing off premium or discount on the books of the business issuing the bonds. Discount Let us assume that a $ioo bond, due in two years and bearing 4% interest payable semiannually, is bought for $96.28. When paid at par there will be a gain of $3.72, the amount of the dis- count. The discount may be taken up by the following methods, which are, however, unscientific: I. By an immediatecredit to income of $3.72, thus raising the investment account on the books to the par of $100 and taking credit at once for the discount. This method is clearly wrong because it takes up at the time of purchase an income which is earned gradually as the bond approaches maturity. 194 BOND DISCOUNT AND PREMIUM 195 2. By carrying the investment at its cost of $96.28 until it is paid, at which time the difference between the cash received and the cost of the bond is credited to income. This method also is erroneous, although the error is not so apparent. In the first place, while the earning is made gradually as the bond approaches maturity, it would appear from the above treatment that none of the discount was earned until the very day of maturity; and in the second place the bond increases in value as the day of matur- ity approaches, when it is paid at par. 3. By considering the $3.72 as extra interest earned, divid- ing the amount by 4, the number of semiannual interest periods, charging }i of $3.72 each six months to the bond account and crediting it to interest. This has the effect of raising the invest- ment account gradually to par, while spreading the earning over the four periods in which the bond is held. The effect of this treatment is shown by the following schedule: Period Bond account beginning of 6 months period Portion of discou.nt TAKEN UP during ONE PERIOD Int. collected Total credit to interest I 2 3 4 Maturity $ 96.28 97-21 98.14 99.07 100.00 % .93 .93 .93 ■93 $2.00 2.00 2.00 2.00 J 2.93 2.93 2.93 2.93 Total I3.72 $8.00 $1 1.72 It may appear that this method answers all requirements, since it raises the investment account gradually to par, taking up the discount periodically. But it is subject to the criticism that, while the investment is gradually increasing, the interest remains constant. This means that the rate of interest is gradu- ally decreasing, as shown in the following table: 196 MATHEMATICS OF ACCOUNTING AND FINANCE Period Asset value in- bond ACCOUNT Credit to Income Semiannual rate of income I 2 3 4 $96.28 97-21 98.14 99-07 f2.93 2.93 2.93 2.93 3.043 +% 3-014 + % 2.98s + % 2.957 + % Since the investment or asset value is increasing as the bond approaches maturity, the credits to income should also increase, so that the same rate of income will be maintained each period. When the amounts are small, the error arising from this method is immaterial and the discount may be written off in equal amounts, as above. But when the amounts are large, this method may result in serious unfairness to some parties in interest. Assuming, for instance, that $96,280 is invested in the purchase of $100,000 of bonds for a trust and that the bene- ficiary changes at the expiration of the second six months' period, the first beneficiary would receive over 3% income semiannually on the assets of the trust, while the second beneficiary would receive less than 3%. It is not sufficient that they each receive the same number of dollars. The property, or investment, held for the second beneficiary is of greater value than that held for the first, and hence the second should receive more income per period, since each is entitled to the same rate of income on the investment of the trust. Scientific Method of Amortization Any method of writing off discount, to be scientific, must take the following into consideration : 1. The investment increases in value as the bond approaches maturity, and at each interest date the investment value should be increased on the books, thus raising it gradually to par. 2. The purchase of a bond at a discount results in the rate of interest earned on the investment being higher than the nominal BOND DISCOUNT AND PREMIUM 197 rate paid on the par of the bond. Hence, the amount added to the asset value of the bond at each interest date should also be credited to interest, together with the cash collected. 3. The total credit to interest (cash collected plus portion of discount) at the end of each period should be an increasing amount but always the same per cent of the carrying or asset value of the bond at the beginning of the period. When a bond is bought at a price other than par, there are two interest rates: 1 . The nominal or cash rate paid on the par of the bond 2. The effective, basic, or income rate actually earned on the investment When the bond is bought at a discount, the income rate is greater than the nominal rate for two reasons : 1. The investment is less than par, although it gradually increases to par. 2. The income is more than the coupons collected, since the income is composed of two parts : (a) The coupons collected (b) The periodical portion of the discount When the discount is scientifically amortized or written off the method is as follows: 1 . Determine the effective rate earned on the investment, as explained later. 2. At each interest date multiply the carrying value of the investment by the effective rate, to determine the amount of revenue earned during the period. 3. Make the following entry: Debit cash for coupon collected Debit investment for portion of discount Credit interest for total income computed in (2) above. 198 MATHEMATICS OF ACCOUNTING AND FINANCE The amount charged to investment will be the difference between the total income and the cash collected. To illustrate, the price of $96.28 in the example above, was chosen because it was known that a 4% bond due in two years, with semiannual interest, would yield an effective rate of income of 6% per year or 3% semiannually if purchased at this price. The following shows the scientific amortization of the discount : Cost of bond $96.28 First period: Income, 3% of $96.28 $2.89 Coupon, 2% of $100.00 2.00 Balance — portion of discount charged to investment .89 Second period: Carrying value of bond $97- 1 7 Income, 3% of $97.17 $2.92 Coupon 2.00 Balance — portion of discount charged to in- vestment -92 Third period: Carrying value of bond $98.09 Income, 3% of $98.09 $2.94 Coupon 2.00 Balance — portion of discount charged to investment -94 Fourth period: Carrying value of bond $99-03 Income, 3% of $99-03 $2.97 Coupon 2.00 Balance — portion of discount charged to in- vestment -97 Par of bond $100.00 BOND DISCOUNT AND PREMIUM 199 It will be noted that this process conforms to the three require- ments of the scientific method as outlined above. The investment is increased gradually to par by writing off a portion of the discount at each interest period, thus: Beginning of Period Investment I $96.28 2 97.17 3 98.09 4 Q9-03 Par 100.00 The amount added to the investment value at each interest period is also credited to interest, as follows: Credits to Interest End of Period For C.\sh For Portion' of Discount TOT.\L I 2 3 4 $2.00 2.00 2.00 2.00 $ .89 .92 •94 • 97 $2.89 2.92 2.94 2.97 The total credit to interest increases each period and is always the same per cent of the increased investment at the beginning of each period, as shown below: Period Investment, beginning of period 'Income for period R.\TE I 3 3 4 $96.28 97.17 98.09 99-03 $2.89 2.92 2.94 2.97 •i /o 3% 3% 3% 200 MATHEMATICS OF ACCOUNTING AND FINANCE While, as stated before, the difference is immaterial when the investment is small, the following table comparing results by the equal instalment method and the scientific amortization method will serve to show the injustice which might be caused by the improper method. Investment Value Income Equal Scientific Differ- Equal Scientific Differ- INSTAL. PLAN AMORTZN. PLAN ence INSTAL. PLAN AMORTZN. plan ence I $96.28 I96.28 i.oo 52. 93 $2.89 S+-04 2 97-21 97-17 .04 2.93 2.92 +.01 3 98.14 98.09 -05 2-93 2.94 — -01 4 99-07 99-03 -04 2.93 2.97 -.04 Assuming that the bond was held as an investment for a trust and that the beneficiary changed at the end of the second period, the column of income differences shows that, by the equal instal- ment method, the first beneficiary would receive 5 cents too much income, while the second beneficiary would receive 5 cents less than was rightfully his. Assuming that the bond was held for an estate for the benefit of a life tenant and a remainderman, that the life tenant died at the end of the second half year, and that the estate reverting to the remainderman, the equal instalment method would work detriment to the remainderman. Let us assume that the estate originally consisted of $100. By the equal instalment method the estate reverting to the remainderman would be less than $100. The income represented by the portion of discount written off would be paid to the life tenant in cash and would be added to the carrying value of the bond. The following would be a statement of the assets of the estate: BOND DISCOUNT AND PREMIUM 20I Equal Instalment Method 1 Scientific Amortiza- tion Method Cash Bond true value Total Cash Bond TRUE value Total $100.00 96.28 $ 3.72 2.00 J96.28 97-17 98.09 $100.00 100.00 9996 99-95 $100.00 96.28 J96.28 97.17 98.09 First period: Cash collected ... $ 3-72 2.00 $100.00 $ 5-72 2.93 $ 5.72 2.89 $ 2.79 2.00 S 2.83 2.00 100.00 Second period: Cash collected . . S 4-79 2.93 $ 4-83 2.92 S 1.86 S 1-91 100.00 It is seen that if the equal instalment method was followed, the assets of the estate would be impaired 5 cents by paying the life tenant in cash 5 cents erroneously computed as income. The result would be that the remainderman would be defrauded, the injustice being concealed by turning the bond back to him at a value of $98.14 when it really had a value of only $98.09. Income Rates Bond dealers frequently offer bonds at prices which net the investor a rate other than the cash or nominal rate. In the illustration above, where a 4% bond was purchased to yield an income rate of 6%, the quotation might be stated, "4% bond paying 6%," "4% bond yielding 6%," "4% bond to net 6%," or ' ' 4% bond on a 6% basis. ' ' To avoid the necessity of computing the price at which each sale must be made, bond tables have 202 MATHEMATICS OF ACCOUNTING AND FINANCE This price varies been prepared showing the price to be paid, with 1 . The nominal rate 2. The income rate 3. The number of periods mi til maturity Since bonds usually bear semiannual interest there are two periods per year, and bond tables usually show a price based on the assumption that the interest is paid semiannually. The following portion of a page in a bond table shows various prices to be paid for a 2-year bond, depending on the nominal and effective rates. The rates at the head of the columns are nominal rates; those at the side are the effective rates. For instance, in the 4% nominal rate column on the 6% effective rate line is found the price of $96.28 used in the preceding illustrations of a 2-year 4% bond netting 6%. 2 Years Interest Payable Semiannually Per Cent Per Annum 3% 3 1/2% 4% 4 1/2% 5% 6% 7% 4.80 96.61 97.55 98.49 99.43 100.38 102.26 104.IS 4 7/8 96.47 97.41 98.3s 99.29 100.24 102.12 104.00 4.90 96.42 97.36 98-31 99-25 100.19 102.07 103.95 S 96.24 97.18 98.12 99 06 100.00 101.88 103.76 5-10 96.05 96.99 97-93 98.87 99.81 101.69 103.57 S 1/8 96.01 96.95 97.89 98.83 99.77 101.64 103.52 S-20 95.87 96.81 97-75 98-69 99.62 101.50 103.38 5 1/4 95.78 96.72 97-66 98.59 99-53 101.41 103.28 5.30 95.69 96.63 97-56 98.50 99-44 101.31 103.19 5 3/8 95-55 96.49 97.43 98.36 99-30 101.17 103.04 S-40 95.51 96.44 97.38 98.32 99-25 I0I.I2 103.00 SI/2 95-33 96.26 97-20 98.13 99-07 100.93 102.80 5 5/8 95-10 96.03 96.97 97.90 98.83 100.70 102-57 53/4 94-87 95.81 96.74 97-67 98.60 100.47 102.33 5 7/8 94-65 95-58 96.51 97-44 98.37 100.23 102.09 6 94.42 95-35 96.28 97-21 98.14 100.00 101.86 BOND DISCOUNT AND PREMIUM 203 Bond Premium When a bond is bought at a price above par, the effective rate is less than the nominal rate, for two reasons: 1 . The investment is more than the par to which the nominal rate applies. 2. The coupons collected are not all income; since only par will be repaid, a portion of the cash received at each interest date must be considered as a return of the premium (which is principal) and only the balance as income. Bond premium should be scientifically written off as follows: 1 . Determine the effective rate. 2. Multiply the gradually diminishing investment by the effective rate to compute the income earned during the period. 3. Make the following entry: Debit cash for coupon collected Credit income for amount computed in (2) above ' ' investment for difference between cash and income. To illustrate, the bond table shows that a 2-year $100 6% bond to net 4% should be purchased for $103.81 . The following shows the scientific amortization of the premium: Cost of bond $103.81 First period: Coupon, 3% of $100.00 $3.00 Income, 2% of $103.81 2.08 Balance, premium written off .92 Second period: Carrying value of bond $102.89 Coupon $3.00 Income, 2% of $102.89 2.06 Balance, premium written off .94 204 MATHEMATICS OP ACCOUNTING AND FINANCE Third period: Carrying value of bond $101.95 Coupon $3.00 Income, 2% of $101.95 2.04 Balance, premium written off .96 Fourth period: Carrying value of bond $100.99 Coupon $3.00 Income, 2% of $100.99 2.02 Balance, premium written off .98 Par of bond $100.01 The error of i cent arises from repeated approximations of interest to the nearest cent. The following table shows the periodical entries : Period Debit Cash Credit Interest Credit Investment (or Premium) Investment Cost $ 3.00 300 300 3- 00 $2.08 2.06 2.04 2.01* 1 .92 .94 .96 .99* $103.81 I 2 3 4 102.89 101.95 101.99 100.00 Total $12.00 18. 19 $3.81 ' Adjustment of i cent to correct discrepancy. The second column shows the total cash collected; the third, the amount of cash taken as income; the fourth, the portion of the coupon applied to repayment of the premium ($3.81) which, with the final repayment of the bond, completely realizes the invest- ment. Since a portion of the investment is realized periodically, the investment gradually diminishes in value. Hence the credits to interest decrease with each successive period. BOND DISCOUNT AND PREMIUM 205 Computing the Premium and the Price While the price above par to be paid for a bond to yield an effective rate less than the nominal rate may be found in a bond table, such tables are not always available. If a table of com- pound interest, present values or annuities is at hand, it will be of assistance, but the price may be computed without tables of any kind. Two methods are explained, as follows: First Method An interest bearing bond comprises two promises, as follows: 1. To pay the par at maturity — in the last preceding illustration, $ioo at the expiration of four six-months periods. 2. To pay a stipulated amount of interest periodically — in the illustration, $3 at the end of each of four six-months periods. The price to be paid for the bond is the present value, discounted at the effective rate, of all cash payments promised. In the illustration, the price is the sum of 1. The present value of $100 due four periods hence, dis- counted at 2% per period. 2. The present value of an annuity of $3 for four periods, discounted at 2% per period. These present values may be found in the interest table; thus the table shows that: Present value of $1 due four periods hence at 2% = $.923845 $.923845 X 100 = $92.3845, P. V. of principal Present value of an annuity of $1 for four periods at 2% = $3.807729 $3.807729 X 3 = 11.4232, P. V. of coupons Total $103.8077 or $103.81, price 206 MATHEMATICS OF ACCOUNTING AND FINANCE If the table does not show present values but does show com- pound interest, the computation may be made as follows: The table shows the amount of $i at compound interest for four periods at 2% to be $1.082432. Then, $1.00 -J- 1.082432= $ .923845, P. V.of$i $.923845 X 100 = $ 92.3845, P. V. of principal Also, $1.00 — .923845 = $ .076155, compound discount, four periods $.076155 -^ .02 = $ 3.8077, P. V. of annuity of $1 for four periods $3.8077 X3 = 1 1. 423 1, P. V. of coupons Total $103.8076, price When no interest table of any kind is available, recourse may be had to the methods explained in the chapter on compound interest and annuities. Perhaps the easiest method would be to compute the amount of $1 for four periods at 2% thus: 1.02 1.02 1.0404 1.0404 1.082432 The procedure following would be as shown above. Second Method This method determines only the premium to be paid to reduce the income from the nominal rate to the effective rate. The premium so determined, added to the par of the bond, com- prises the total price. The method is based on the following reasoning : Let us assume that a 4% annual, or 2% semiannual, income is required. If the bond bore 4% interest, the price would be par. Hence the payment of $100, or par, entitles the holder of the bond to receive: BOND DISCOUNT AND PREMIUM 20/ 1. Principal, at maturity 2. Interest, $2 at the end of each six months But if, as in the preceding illustration, the bond pays 3% semi- annually, each semiannual coupon collected will be $3. Of this, the payment of par entitles the holder to receive $2. And the payment of a premium entitles him to receive the remain- ing $1. Hence, the premium is the sum which must be paid to entitle the holder to collect that portion of the periodical coupon which is in excess of the product obtained by multiplying the par of the bond by the effective periodical rate. This excess interest is an annuity. In the case of a 2-year 6% bond bought to net 4%, the annuity is $1 for four periods. The premium is the present value of this annuity, discounted at the effective rate, and may be computed as follows: $100 X 3% (semiannual cash rate) = $3.00 coupon $100 X 2% ( " effective rate) = 2.00 effective income on par Excess $1.00 The present value at 2% of an annuity of $1 for four periods is shown by an interest table to be $3.807729, to which is added the par, $100, the total being the price of the bond or $103.81. When a table showing the present value of an annuity is not at hand, one of the methods already explained may be used to find the present value. Perhaps the easiest method is by successive divisions, as follows: Example What is the premium to be paid on a 3-year 7% bond, interest payable semiannually, bought to net 5%; par $100,000? Solution: $100,000 X 3K% (cash rate per period) =$3,500 $100,000 X 2>^% (effective rate per period) = 2,500 Excess $1,000 208 MATHEMATICS OF ACCOUNTING AND FINANCE The required premium, therefore, is the present value at 2^% of an annuity of $1,000 for six periods. 975.610 P. V. of $1000 due I period hence Now, $1,000.00 - - 1.025 = $ 975.610 975.610- - 1.025 = 951.814 951.814 - - 1.025 = 928.599 928.599 - - 1-025 = 905-950 905-950 - - 1.025 = 883.854 883.854 - - 1-025 = 862.297 $1000 $1000 $1000 $1000 $1000 2 periods 3 " 4 " 5 6 $ 5,508.124, premium 100,000.000, par $105,508.12, price The premium could also be computed as follows: Find amount of $1 for six periods at 2^%, as follows: 1.025 1.025 1.050625 amount of $1 for 2 periods 1.050625 1.103813 " " I " 4 " 1.050625 1.159693 " " I " 6 " Find compound discount on $1 due six periods hence at 2^ % thus: $1 -^ 1. 159693 = $.862297 P. V. of $1 due six periods hence $1 — $ .862297 = .137703 compound discount or $.159693 (comp. int.) -^ 1. 159693 (amt.) = $. 137703 compound discount Find present value of annuity of $1 ,000 for six periods at 2% %, thus: % -137703 -^ -025 = $5.50812 P. V. of annuity of $1 $5.50812 X 1000 = $5,508.12 P. V. of annuity of $1,000 BOND DISCOUNT AND PREMIUM 209 The following table shows the periodical entries and the diminishing balance of the investment. Period Debit Cash Credit Interest 2j^% OF Investment Credit Investment Investment Cost $ 3,500 3.500 3.500 3.500 3,500 3,500 $ 2,637.70 2,616.15 2,59405 2,571-40 2,548-19 2,524-38 $ 862.30 883.85 905.95 928.60 951-81 975-62 •?I05,508.I3 104.645.83 103,761.98 102,856.03 101.927.43 100,975-62 100,000.00 6 Total |2I,000 115,491.87 55,508.13 Computing the Discount and the Price When bond tables cannot be consulted to determine the price to pay for a bond to net an income rate higher than the nominal rate, the price may be computed by methods similar to those described for determining a premium. First Method. The first method consists of the following three steps: (a) Compute the present value of the par, discounted at the effective rate. (b) Compute the present value at the effective rate of all coupons to be collected. (c) Add the foregoing two items, the sum being the price to be paid. In the case of the $100 2 -year 4% bond bought to net 6%, at a price of $96.28, the first item (a) is the present value of $100 due four periods hence at 3% (the effective rate) per period. The second item (b) is the present value of an annuity of $2 for four periods discounted at 3%. These present values may be found in a book of tables, thus : 210 MATHEMATICS OP ACCOUNTING AND FINANCE Present value of par: P. V. of $1 due in four periods at 3% is $ .888487 $.888487 X 100 = $88.8487 Present value of coupons: P. V. of annuity of $1 for four periods at 3% is $3.717098 $3.717098X2= 7-4342 Total price $96.2829 or as shown by the bond table $96.28 If an interest table is not available, the methods already described for determining present values may be used. 2. Second Method. This method determines the discount to be deducted from par. Since the cash rate does not produce the required income, the seller permits the deduction of an amount which, invested at the effective rate, will produce the extra periodical income required. The discount is the present value of an annuity of the extra income. For instance, in the case of the $100 2-year 4% bond bought to net 6%, at a price of $96.28, the 3% effective rate is equivalent to $3 per period, while the 2% coupon produces only $2 per period, the required excess being $1 per period for four periods. The present value of 3% (the effective rate) of an annuity of $1 for four periods is shown by an interest table to be $3.717098. Then $100.00 par Less 3.72 discount $96.28 price When neither a bond table nor an interest table can be used, one must resort to the previously described method of comput- ing the present value of the annuity. Illustration What are the discount and the purchase price of a 3-year 5% bond, interest payable semiannually, bought to net 6%; par $100,000? BOND DISCOUNT AND PREMIUM 211 Solution: $100,000 X 3% (effective rate per period) $100,000 X 23^% (cash " " " ) )3,ooo 2,500 5 500 Deficient interest per period The required discount, therefore, is the present value at 3% of an annuity of $500 per period. -- $485,437 P. V. of $500 due I period Now, $500 - I. Hence, 485-437 " - i.( 47I.2Q8- - i.< 457-571- - I. 444.244- - I. 431.305 - - I. Discount 03 = 471 03 = 457 03 = 444 03 = 431 03 = 418 298 571 244 305 742 500 500 500 500 500 2 periods 3 4 " 5 6 $2,708,597 Then par Less discount Price $100,000.00 2,708.60 $ 97,291.40 The discount can also be computed as follows: Find the amount of $1 for six periods at 3%, thus: 1.03 1.03 1.0609 amount of $1 for 2 periods 1.0609 1. 125509 " •• I " 4 1.0609 1. 194052 " " I " 6 Find the compound discount on$i due six periods hence at 3%, thus: $1.00 -^ 1. 194052 = $.837484 P. V. of $1 due six periods hence $1.00— .837484= $.162516 compound discount Find the present value of an annuity of $500 for six periods at 3% thus: $.162516 (comp. dis.) -J- .03 = $5.4172 P. V. of annuity of $1 $5.4172 X 500 = $2708.60 present value of annuity of $500, or discount 212 MATHEMATICS OF ACCOUNTING AND FINANCE The following table, or schedule of amortization, shows the periodical entries and the remaining balance of the investment. Period Credit Income 3% OF Investment Debit Cash Debit Investment Investment I 2 3 4 5 6 $ 2,918.74 2,931-31 2,944-24 2,957-57 2,971-30 2,985-44 1 2,500 2,500 2,500 2,500 2.500 2,500 $ 418.74 431-31 444-24 457-57 471-30 485-44 $ 97.291-40 97.710.14 98,141.45 98,585.69 99.043.26 99.514-56 100,000.00 $17,708.60 $15,000.00 $2,708.60 Purchases at Intermediate Date In the preceding explanations of methods for computing prices for bonds either at a premium or at a discount, it has been assumed that the purchase occurred on an interest date. When this is not the case, the customary method of determining the price is as follows : 1. Compute the price as if the purchase had been made at the next preceding interest date; also the price as if the purchase were to be made at the next succeeding interest date. The difference is the portion of premium or discount to be amortized during the period. 2. Such a proportion of this premium or discount is amortized as the elapsed time between the preceding interest date and the date of purchase bears to the total interest period. 3. To the amortized value thus obtained add the accrued interest at the nominal rate. For instance, referring to the table on page 209 let us assume that the interest dates are January i and July i ; and that $105,508.13 is the value on Jan. i, 1918, on a 5% basis 104,645-83 " " " " July i> 1918, " " " " $ 862.30 " " premium to be amortized during the period BOND DISCOUNT AND PREMIUM 213 If the purchase is made on February i, 19 18, one-sixth of the interest period has elapsed, hence one-sixth of $862.30 should be amortized, thus: Value at January i, 1918 $105,508.13 Deduct Ye of $862.30 i43-72 $105,364.41 Add accrued interest: ^/g of $3500 583.33 Price $105,947.74 The price may also be computed by adding to the price on January i the accrued interest at the effective rate, thus: Price on January i, 1918 $105,508.13 Effective interest for six months: 2H% of $105,508.13 = $2,637.70 Ve of $2,637.70 439-6i Price on February i, 1918 $105,947.74 Of this amount $105,364.41 is charged to investment and $583.33 to accrued interest. When the coupon is collected on July i, it is applied as follows: Cash $3500.00 Accrued interest $ 583.33 Interest earned 2,198.09 Investment amortization ($862.30 — $143.72). .. 718.58 The investment of $105,364.41, reduced by the amortization of $718.58, is novi^ carried at the true value (seepage 209) on July i, $104,645.83. While this is the customary method of computing a flat price, it is unfair to the buyer, since he advances to the seller $583.33 accrued interest five months before it is due. When the bond is to be purchased at a discount, the propor- tion of discount to be amortized for the fractional period should be added. Referring to page 212, let us assume that interest is payable on January i and July i, that $97,291.40 is the value at 214 MATHEMATICS OF ACCOUNTING AND FINANCE January i, 1918, on a 6% basis, and that the transfer is to be made on May i, 1918. Then, $97,710.14 is the value on July i, igi8 97,291.40 " " " " January i, 1918 $ 418.74 " " discount to be amortized during the period % of $418.74 = $ 279.16 the discount to be amortized during two-thirds of the period $97,291.40 value at Jan. i, 1918 279.16 two-thirds of discount amortized in six months 1,666.67 accrued interest — two-thirds of $2,500 $99,237.23 price on May i, 1918 Or, value at January i, 1918 $97,291.40 Add two-thirds of 3% (effective rate) on $97,291.40 1,945.83 $99,237.23 Of this amount $97,570.56 is charged to investment and $1,666.67 to accrued interest. On July i, the investment is adjusted to its true value (see table on page 212) by the following entries : Cash $2500.00 Investment (discount $418.74 — $279.16) 139-58 Accrued interest $1666.67 Interest 972.91 All entries after July i, 1918, are as indicated in the schedule on page 212. Serial Bonds Instead of providing a sinking fund for the eventual redemption of a bond issue, the bonds may be retired gradually by serial redemption. In computing the price at which the entire issue may be purchased to net an effective rate other than the cash rate, the bonds maturing at each redemption date must be considered separately, the several values so obtained being added to fmd the total price. BOND DISCOUNT AND PREMIUM 215 As a simple illustration assume that five bonds of $100 each, bearing 6% interest payable semiannually, are to be retired in amounts of $100 at the end of each of five years. Required — the price to net 5%. The following values are taken from a bond table, though they could be computed by the methods already explained. Maturities Value Bond due in i year $100.96 " " " 2 years 101.88 " "3 " 102.75 4 103.59 5 " 104-38 (( u Total value of issue $513.56 The following schedule shows the reduction of the premium and the serial redemption of the bonds: Cost $513-56 First period: Coupons 3% of $500 $15.00 Income 2 1/2% of $513.56 12.84 2.16 Second period: Carrying value $511.40 Coupons 3% of $500 $15.00 Income 2 1/2% of $511. 40 12.78 2.22 .i« First redemption 100.00 Third period: Carrying value $409.18 Coupons 3% of $400 $1 2.00 Income 2 1/2% of $409.18 10.23 1.77 Fourth period: Carrying value $407.41 Coupons 3% of $400 $12.00 Income 2 1/2% of $407.41 10.18 1.82 $405.59 216 MATHEMATICS OF ACCOUNTING AND FINANCE Second redemption loo.oo Fifth period: Carrying value $305.59 Coupons 3% of $300 $ Q.oo Income 2 1/2% of $305.59 7.64 1.36 Sixth period: Carrying value $304. 23 Coupons 3% of $300 $ 9.00 Income 2 1/2% of $304.23 7.61 1.39 $302.84 Third redemption 100.00 Seventh period: Carrying value $202.84 Coupons 3% of $200 $ 6.00 Income 2 1/2% of $202.84 5.07 .93 Eighth period: Carrying value $201.91 Coupons 3% of $200 $ 6.00 Income 2 1/2% of $201.91 5.05 .95 $200.96 Fourth redemption 100.00 Ninth period: Carrying value $100.96 Coupons 3% of $100 $ 3.00 Income 2 1/2% of $100.96 2.53 ^ .47 Tenth period: Carrying value $100.49 Coupons 3% of $100 $ 3.00 Income 2 1/2% of $100.49 2.51 .49 $100.00 Fifth redemption 100.00 o CHAPTER XX LEASEHOLDS Commuted Rents The problem of determining the present value of an annuity arises when real estate is leased and an advance payment is made covering, or applying on, a series of rents which would otherwise be paid at regular intervals in the future. Let us assume that it is proposed to lease certain property on January i, 1918, for five years at an annual rental of $1,000, payable on January i of each year. This contract would require the following payments : Date Rent January i I I I I 1 91 8 $1,000 1919 1,000 1920 1,000 1 92 1 1,000 1922 1,000 The lessee desires to make one payment on January i, 1918, covering the entire rental; and it is agreed between the parties to discount at 5% the payments which would otherwise be made in 1919, 1920, 192 1 and 1922. The single payment to be made (i.e., the present value of the annuity) may be computed thus: Date Due Rent Symbols Present Value January r, 1918 I. 1919 " I, 1920 1, 1921 I, 1922 Now 1 year hence 2 years 3 " 4 " li.ooo 1,000 1,000 1,000 1,000 J - - (i + <•) - (r + <•)' - ( r + i)i - (i + i)^ Si. 000. 000 952.381 907.029 S63.838 822.702 Single payment Jj. 545.950 217 2l8 MATHEMATICS OF ACCOUNTING AND FINANCE The single payment can also be computed thus : Present value of first payment Present value of future payments: Present value of an annuity of $i,ooo for four periods at 5% Present value of $1 due in four years at 5% $.822702 Compound discount $.177298 Present value annuity of $1 = $.177298 ^ .05 $3-54596 $3.54596 X 1,000 Single payment 3,545-96 54,545-96 Although the single payment of $4,545.96 pays the rent for five years, the annual entries in the accounts must show the following : 1. Annual rental of $1,000 2. Annual interest earning on the advance payment 3 . Application of advance payment to annual rent 4. Reduction of value of advance payment The following schedule shows the figures used in the annual journal entries. Date Debit Rent Credit Interest Credit Leasehold Debit balance of Leasehold Account li.ooo 1,000 1,000 1,000 1,000 I177.30 136.16 92.97 47.62 $1,000.00 822.70 863.84 907.03 952.38 $4. 545. 95 3,545.95 2,723.2s 1,859.41 952.38 I, 1920 Totals S.5,000 1454-05 $4,545.95 It is manifest that the value of a leasehold depends largely on the rate of interest used. If the above lease had been com- LEASEHOLDS 219 muted on a basis of 4% its value would have been $4,629.90, or $83.95 more than it is worth at 5%. On a very short lease the difiference is not very great, but on a 99-year lease it amounts to a very large sum. Sublease A similar problem arises when property is subleased. Let us assume that A leases certain property on January i, 191 2, for a period of ten years at an annual rental of $3,000, and that he occupies the property until December 31, 191 7, at which time he assigns the lease to B. Due to a rise in land values, B is willing to assume the lease at an annual rental of $5,000. B may, therefore, make the following payments: Date Payment to Owner of Fee Payment to A $3,000 3.000 3.000 3.000 3.000 $2,000 2,000 I. 1919 Or B may, with A's consent, pay A a lump sum on January i, 1918, instead of annual payments. If the future payments of this annuity of $2,000 are discounted at 5%, the lump sum is com- puted as follows : Present value of first payment Present value of an annuity of $2,000 for four periods at 5%: Present value of an annuity of $1,000 (as computed above) $3,545.95 Then $3,545-95 X 2 = Single payment i2,000.00 7,091.90 ,091.90 220 MATHEMATICS OF ACCOUNTING AND FINANCE The following schedule shows the journal and cash entries to be made annually by B. Date Debit Credit Credit Credit Cash Balance of Rent Interest Leasehold (paid owner) Leasehold Account Cost of leasehold . . . $9,091-90 January i, 1918 .... $ 5.000 $2,000.00 $ 3,000.00 7,091.90 I. 1919 5.000 1354-60 1,645.40 3,000.00 5.446.50 I, 1920 .... 5.000 272.33 1.727.67 3,000.00 3.718.83 I, 1921 .... 5.000 185.94 1,814.06 3,000.00 1,904.77 I, 1922 .... 5.000 95-24 1,904.76 3,000.00 .01 Total $25,000 J908.11 $9,091-89 $15,000.00 CHAPTER XXI DEPRECIATION METHODS Annual Depreciation There are six methods in more or less general use for the de- termination of the amount of depreciation which business en- terprises should provide annually on their fixed assets. Writing of^ an arbitrary amount at any convenient time is not included, because it is too haphazard and unscientific to be called a method. The six recognized methods are as follows: 1. Straight line method 2. Diminishing value, using a fixed per cent 3. Diminishing value method based on the sum of the years' digits 4. Annuity method 5. Sinking fund method 6. Production method It is not within the province of this book to examine the merits of the different methods. The object is only to indicate how the amount to be written off annually is ascertained when the managers of the business have decided upon the method to be used. The illustration to be used throughout is that of an asset costing $5 ,000 which is estimated to have a life of six years and a residual or scrap value of $200, if any. I . The straight line method has the advantage of simplicity, as the annual amount is determined by the simple process of dividing the total depreciation by the number of years the asset is expected to remain effective, as shown below. 222 MATHEMATICS OF ACCOUNTING AND FINANCE Year Annual Depreciation Carrying Value I $ 8oo 800 8oo 800 800 8oo Is, 000 4,200 3,400 2 4 s 6 1,000 200 Total S4,8oo 2. The diminishing value method, when a fixed per cent is applied, is a desirable one to use, but it has the disadvantage that the rate on the diminishing value is difficult to compute. The process requires the extraction of a root the index of which is the same as the number of years covered by the depreciation. The formula for this method is In this formula r represents the desired rate; w the number of periods ; 5 the scrap value ; and c the cost. The formula is applied to the illustration as follows : 6' 200 \ 5,ooo 6/ = I — V -04 Log .04 is 8.60206 — 10 Add 50. ~ 50 58.60206 — 60 Divide by 6 9.76701 — 10 log of .5848 r= I- .5848 = .4152 = 41.52% The following table shows the depreciation charges and the diminishing value in the example taken. DEPRECIATION METHODS 223 Year Depreciation Diminishing value I 2 3 4 5 6 41.52% of Is, 000. 00 " 2,924.00 " 1,709.96 99998 584-79 341-99 12,076.00 1,214.04 709-98 415-19 242-80 141.99 $5,000.00 2,924.00 1,709.96 999-98 584-79 341-99 200-00 Total $4,800.00 The fixed per cent of diminishing value is very frequently used, but because of the difficulty or ignorance of the method of computing it, the rate is guessed at, or a purely arbitrary one is used, which, however, is usually too small. Thus, it might seem to many factory managers that 25% would be too large a rate, in the preceding illustration but it really is not, as the fol- lowing shows : Year 25% OF Diminishing value Diminishing value I 2 3 4 , S 6 $1,250-00 937-50 703-13 527-34 395-51 296.63 $5,000.00 3.750.00 2,812.50 2,109.37 1,582.03 1,186.52 889.89 Total $4,110.1 1 3. The diminishing value method based on the sum of the years' digits is one which is often used to avoid the difficulties of the percentage method described above. To find the sum of the years' digits the figures representing the successive years are added together to form a denominator, in our illustration, 21 224 MATHEMATICS OF ACCOUNTING AND FINANCE (1 + 2+3+4+5 + 6). Each year a numerator is used represent- ing the successive number of years the asset is expected to Hve, and the fraction thus obtained is apphed to the estimated total depreciation to determine the amount to be charged off that year. The result of applying this method in our illustration is as follows: Year Fraction Annual Charge Carrying Value $5. 000. 00 I 6/21 $1,371.43 3.628.57 2 S/21 1. 142. 86 2,485.71 3 4/21 914.29 1,571.42 4 3/21 685.71 885.71 5 2/21 457-14 428.57 6 1/21 228.57 200.00 Total . . . 21/21 $4,800.00 As is readily seen, this method spreads the depreciation with less difference between the early and the late years than does the second method. 4. The annuity method, to quote Hatfield . . . rests upon the assumption that the cost of production includes not only repairs and the depreciation of machinery, but as well interest on the amount of capital invested in the machine. Depreciation, on this theory, should be a sum figured as a constant annual charge, sufficient not only to write off the decline in value but also to write off annual interest charges on its diminishing value. ' In other words, this method treats the cost of machinery as an investment earning interest. Hence the cost of machinery is dealt with as the present value of an annuity; the depreciation to be written off periodically is an equal amount, and the credit to interest decreases periodically because of the diminishing value of the asset. ' H. R. Hatfield, Modern Accounting, 191 1. P- 131. DEPRECIATION METHODS 225 If there were no scrap value, the computation of the annual depreciation charge would be as follows: The cost of the machine is the present value of an annuity of unknown rents, or depreci- ation charges; this cost is divided by the present value of an annuity with rents of $1. If it is assumed that the asset in our illustration will have no scrap value and that the annuity is to be based on an interest rate of 6%, the computation is as follows: Present value of $1 due six periods hence. ... $ .70406054 Compound discount ($1 — $.70496054) .29503946 Present value of annuity of $1 ($.29503946 ■^ .06) 4.917324 Annual amount required ($5,000 -J- 4.917324). 1,016.81 The following is a table of depreciation for our example, based on the annuity method, assuming the asset has no scrap value. Year Debit Depreciation Credit Interest Credit Depreciation Reserve Balance-carrying value Basis of interest I 2 3 4 5 6 I1.016.81 i,oi6.8r 1. 016. 81 1. 016. 81 r.oi6.8i 1,016.81 $ 300.00 256.99 21 r.40 163.08 111.8s 57-56 $ 716.81 759.82 80s. 41 853-73 904.96 959-25 Is. 000.00 4.283.19 3,52337 2,717.96 1,864.23 959.27 .02 Total 16,100.86 Si, 100. 88 14,999-98 When the asset has a scrap value, the conditions are more complicated, because the cost of the asset consists of two elements : (a) The present worth of an annuity of the depreciation charges, and (b) The present worth of the scrap value. 226 MATHEMATICS OF ACCOUNTING AND FINANCE Referring to our illustration, if the asset has a scrap value of $200, the $5,000 invested in it is the sum of the following two items : (a) The present value at 6% of $200 remaining after six years (b) The present value on the basis of 6% of an annuity of six rents of unknown amounts The rents, or depreciation charges, are computed thus: Cost of asset $5,000.00 Deduct present value of $200 due in six years (.70496054 X 200) 140.99 Present value of six depreciation charges $4,859.01 In the preceding illustration the present value of an annuity of $1 for six periods at 6% was found to be $4.917324. Then $4,859.01 -r- 4.917324 = $988.14 annual depreciation The following is the table of depreciation based on the an- nuity method, when the asset has a scrap value of $200. Debit Credit Credit Year Depreciation Interest Depreciation Reserve Balance Is, 000. 00 I 1 988.14 J 300.00 $ 688.14 4. 311. 86 2 988.14 258.71 729-43 3,582.43 3 988.14 214.9s 773.19 2,809.24 4 988.14 168.55 819.59 1,989.6s S 988.14 119.38 868.76 1,120.89 6 988.14 67.25 920.89 200.00 Total JS.928.84 Ji, 128.84 14,800.00 It is not pertinent here to enter into a discussion of the propriety of including interest on fixed assets among the manu- facturing expenses. DEPRECIATION METHODS 227 5. The sinking fund method is based on the assumption that a fund is set aside to accumulate at compound interest with which to acquire a new asset when the old one is discarded. It assumes that the funds for the purchase of the asset will be pro- vided from two sources : (a) The scrap value of the old asset (b) The sinking fund Since the fund accumulates at compound interest, it is a sinking fund in the mathematical sense, but not in the accounting sense, which limits the term " sinking fund " to a fund accumu- lated to pay a definite hability. It would be preferable to call this fund a replacement fund accumulated on the sinking fund principle. At the end of the life of the asset the fund should equal the amount of the depreciation. The annual contributions to the fund will be the rents of an annuity which will produce this total depreciation fund. In our illustration the total depreciation is $4,800, or cost minus scrap value (c—s). Hence the annual contribution to the fund {SFC) will be $4,800 divided by the amount of an annuity of $1 for the given time and at the given rate. This is found by dividing the compound interest on $1 for the given time by the given rate of interest (I-^i). The formula is: SFC = {c- s)^r t or SFC = (c- s)X- Assuming that a fund is to be accumulated on a 4% basis, the compound interest (/) on $1 at 4% for six years being $0.265319, the formula is applied thus: SFC = ($5,000 - $200) X '^"^ .265319 22« MATHEMATICS OF ACCOUNTING AND FINANCE = $4,800 X = $192 .04 •265319 .265319 ^723-66 If a fund is established, the entries therefor will be a debit to fund and a credit to cash each year for $723.66. When the interest is collected, the cash goes into the fund by an entry debiting fund and crediting interest. In addition there must be entries for depreciation, debiting depreciation and crediting reserve for depreciation with an amount equal to the sum of the cash contributed and the interest earned each year. The following table shows the operation of the sinking fund method : End of Credit Credit Debit Total Fund Carrying Year Cash Interest Fund and Reserve Value $5,000.00 I 1 723-66 $ 723-66 J 723-66 4.276.34 3 723-66 1 28.9s 752-61 1,476.27 3.523.73 3 723-66 59-05 782.71 2,258.98 2,741.02 4 72366 90.36 814.02 3.073-00 1,927.00 5 723.66 122.92 846.58 3.919-58 1,080.42 6 72366 156.78 880.44 4,800.02 199-98 Total ... $4,341-96 I458.06 I4.800.02 The annual charges to depreciation are the amounts in the column headed "Debit Fund." Thus the reserve for depreci- ation is always equal to the fund. The charge to operations on account of depreciation increases annually, but this increase is offset by the credit to interest, making the net expense the same each year. 6. Concerning the production method ^lontgomery says: A method of making depreciation allowances which has its advantages under certain conditions is that of charging an established rate per unit of DEPRECIATION METHODS 229 output. This is especially applicable in the case, say, of a blast furnace where the frequency with which the linings will need to be renewed de- pends on the extent to which the furnace is being used. If it is being run at full capacity night and day, the wear on the linings is obviously much greater than if the furnace had not been in continual use during the entire fiscal period. ^ Mr. Montgomery applies this method with perfect justice to the lessening in value of a wasting asset, such as timber, or the coal or ore in a mine, but this lessening in value is not caused by depreciation, but by an actual consumption, or removal and conversion of the asset. In fact, this is the only possible method to be applied to those assets which diminish in exact ratio to the amount used. There can be no rules formulated for the determination of the amount to be written ofif against each unit of production. That is a matter that must be left to the judgment of the factory managers, guided by experience. The following is a comparative table of depreciation charges, made according to the first five methods : Per Cent of Sum of Straight Diminishing Years' Sinking Year Line Value Digits Annuity Fund I $ 800 $2,076.00 $1,371-43 1 988.14 $ 723.66 2 800 1,214.04 1,142.86 988 14 752.61 3 800 709-98 914.29 988 14 782.71 4 800 415-19 685.71 988 14 814.02 S 800 242.80 4S7.I4 988 14 846.58 6 800 141.99 228. S7 988 14 880.44 Totals I4.800 $4,800.00 $4,800.00 $5,928.84 $4,800.02 Credit to interest $1,128.84 Net ' R. H. Montgomery, Auditing Theory and Practice, 1919, p. 550. 230 MATHEMATICS OF ACCOUNTING AND FINANCE The comparative carrying valuts in the same example under the five methods are as follows: End of Year Straight Line Per Cent of Diminishing Value Sum of Years' Digits Annuity Sinking Fund $S,ooo I5.000.00 $5,000.00 $5,000.00 $5,000.00 I 4,200 2,924.00 3.628.57 4.3 1 1 86 4.276.34 2 3.400 1,709.96 2,485-71 3.582.43 3.523-73 3 2,600 999.98 1.571-42 2,809.24 2.74102 4 1,800 584-79 885-71 1,989.65 ■ 1,927-00 S 1. 000 341-99 428.57 1,120.89 1,080.42 6 200 200.00 200.00 200.00 199.98 APPENDIX A VALUES OF FOREIGN COINS Following is a list of foreign monetary units and their values, representing the pars of exchange, as estimated by the United States Director of the Mint. Value in COUNTRY Legal Standard Monetary Unit Terms of U. S. Money Argentine Republic Gold Peso S0.9648 Gold .1930 .3893 Bolivia Gold Boliviano Brazil Gold Milreis • 5462 British Colonies in Austral- asia and Africa Gold Pound sterling 4.866s Gold Dollar Central American States: Costa Rica Gold Colon .4653 British Honduras Gold Dollar 1. 0000 Gold 1. 0000 Guatemala \ Honduras / Silver Peso .4403 Gold Gold Peso Peso .5000 Chile ■3650 Amoy .7219 Canton ■ 7197 Cheefoo .6904 Chin Kiang .7052 Fuchau .6678 Haikwan • 7345 (customs) Hankow ■6754 Tael ■ Kiaochow Nankin Niuchwang Ningpo Peking .6995 .7143 .6770 .6940 ■ 7037 China Silver J Shanghai Swatow Takau ^Tientsin Yuan ■ 6594 .6668 .7264 .6995 •4730 Dollar Hongkong British Mexican .4748 .4748 .4783 231 232 VALUES OF FOREIGN COINS Country Legal Standrd Colombia Cuba Denmark Ecuador Egypt Finland France Germany Great Britain Greece Haiti India [British] Indo-China Italy Japan Liberia Mexico Netherlands Newfoundland Norway Panama Paraguay Persia Peru Philippine Islands . . Portugal Roumania Russia Santo Domingo . . . . Serbia Siam Spain Straits Settlements . Sweden Switzerland Turkey Uruguay Venezuela Gold Gold Gold Gold Gold Gold Gold and silver Gold Gold Gold and silver Gold Gold Silver Gold Gold Gold Gold Gold Gold Gold Gold Gold (Gold \ Silver Gold Gold Gold Gold Gold Gold Gold Gold Gold and silver Gold Gold Gold Gold Gold Gold Monetary Unit (Dollar) Peso Dollar Krone Sucre Pound (lOO piasters) Finmark Franc Mark Pound sterling Drachma Gourde Rupee Piaster Lira Yen Dollar Peso Guilder (Florin) Dollar Krone Dollar Peso (Argentine) Ashrafi Kran Libra Peso Escudo Leu Ruble Dollar Dinar Tical Peseta Dollar Krona Franc Turkish Pound Peso Bolivar Value in Terms of U. S. Money 1. 0000 .12680 .4867 4-94 J I • igjo .1930 .2382 4.866s • 1930 .2500 •3244 • 4755 .1930 .4985 1. 0000 ■ 4985 .4020 1. 0000 .2680 1. 0000 .9648 ■ 0959 .0811 4.866s .5000 1.080S .1930 .5146 r.oooo .1930 • 3709 .1930 .5678 .2680 .1930 .0440 10342 .1930 APPENDIX B LOGARITHMS OF NUMBERS^ No. I 2 3 4 5 6 7 8 9 lOO 00 000 00 043 00 087 00 130 00 173 00 217 00 260 00 303 00 346 00 389 lOI 00 432 00475 00 518 00 561 00 604 00 647 00 689 00 732 00 775 00 817 102 00 860 00903 00 945 00 988 01 030 01 072 01 IIS 01 IS7 01 199 01 242 103 01 284 01 326 01 368 01 410 01 452 or 494 01 536 01 578 01 620 01 662 104 01 703 01 745 01 787 01 828 01 870 01 912 01 953 01 995 02 036 02 078 105 02 119 02 160 02 202 02 243 02 284 02 32s 02 366 02 407 02 449 02 490 106 02 S3 I 02 572 02 612 02 653 02 694 02 735 02 776 02 816 02 857 02 898 107 02 938 02 979 03 019 03 060 03 100 03 141 03 181 03 222 03 262 03 302 108 03 342 03 383 03 423 03 463 03 5 03 03 543 03 583 03 623 03 663 03 703 109 03 743 03 782 03 822 03 862 03 902 03 941 03 981 04 021 04 060 04 100 IIO 04 139 04 179 04 218 04 258 04 297 04 336 04 376 04 4IS 04 454 04 493 III 04 S32 04 S7I 04 610 04 650 04 689 04 727 04 766 04 805 04 844 04 883 112 04 922 04 961 04 999 05 038 OS 077 OS 115 OS 154 05 192 05 231 OS 269 113 OS 308 OS 346 05 385 05 423 OS 461 OS 500 OS 538 OS 576 OS 614 OS 652 114 05 690 05 729 OS 767 05 80s OS 843 OS 881 05 918 OS 956 OS 994 06 032 IIS 06 070 06 108 06 14s 06 183 06 221 06 258 06 296 06 333 06 371 06 408 116 06 446 06 483 06 521 06 SS8 06 595 06 633 06 670 06 707 06 744 06 781 117 06 819 06 856 06 893 06 930 06 967 07 004 07 041 07 078 07 lis 07 isi 118 07 188 07 225 07 262 07 298 07 335 07 372 07 408 07 445 07 482 07 S18 119 07 555 07 591 07 628 07 664 07 700 07 737 07 773 07 809 07 846 07 882 120 07 918 07 954 07 990 08 027 08 063 08 099 08 US 08 171 08 207 08 243 121 08 279 08 314 08 350 08 386 08 422 08 458 08 493 08 529 08 56s 08 600 122 08 636 08 672 08 707 08 743 08 778 08 814 08 849 08 884 08 920 08 9SS 123 08 991 09 026 09 061 09 096 09 132 09 167 09 202 09 237 09 272 09 307 124 09 342 09 377 09 412 09 447 09 482 09 517 09 552 09 587 09 621 09 656 125 09 691 09 726 09 760 09 795 09 830 09 864 09 899 09 934 09 968 10 003 126 10 037 10 072 10 106 10 140 10 175 10 209 10 243 10 278 10 312 10 346 127 10 380 10 41S 10 449 10 483 10 S17 10 551 10 585 10 619 10 653 10 687 128 10 721 10 755 10 789 10 823 10 857 10 890 10 924 10 958 10 992 II 02s 129 II 059 II 093 II 126 II 160 II 193 II 227 II 261 II 294 II 327 II 361 130 ir 394 II 428 II 461 II 494 II 528 II 561 II 594 II 628 II 661 II 694 131 II 727 1 1 760 II 793 II 826 II 860 II 893 1 1 926 II 959 II 992 12 024 132 12 057 12 090 12 123 12 156 12 189 12 222 12 254 12 287 12 320 12 352 133 12 385 12 418 12 450 12 483 12 S16 12 548 12 S8l 12 613 12 646 12 678 134 12 710 12 743 12 775 12 808 12 840 12 872 12 905 12 937 12 969 13 001 ' E. H Barker, Computing Tables and Formulas. 1913, pages 22-39. 233 234 APPENDIX No. I 2 3 4 5 6 7 8 9 135 13 033 13 066 13 098 13 130 13 162 13 194 13 226 13 258 13 290 13 322 136 13 354 13 386 13 418 13 450 13 481 13 SI3 13 S4S 13 577 13 609 13 640 137 13 672 13 704 13 73S 13 767 13 799 13 830 13 862 13 893 13 92s 13 956 138 13 988 14 019 14 051 14 082 14 114 14 14s 14 176 14 208 14 239 14 270 139 14 301 14 333 14 364 14 395 14 426 14 457 14 489 14 520 14 551 14 582 140 14 613 14 644 14 675 14 706 14 737 14 768 14 799 14 829 14 860 14 891 141 14 922 14 953 14 983 IS 014 15 045 IS 076 IS 106 IS 137 IS 168 IS 198 142 15 229 IS 259 IS 290 IS 320 IS 3SI IS 381 IS 412 IS 442 IS 473 IS S03 143 IS 534 IS 564 IS 594 IS 62s 15 6s5 IS 685 IS 715 IS 746 IS 776 IS 806 144 IS 836 IS 866 IS 897 IS 927 15 957 IS 987 16 017 16 047 16 077 16 107 14s 16 137 16 167 16 197 16 227 16 256 16 286 16 316 16 346 16 376 16 406 146 16 435 16 46s 16 495 16 524 16 554 16 584 16 613 16 643 16 673 16 702 147 16 732 16 761 16 791 16 820 16 850 16 879 16 909 16 938 16 967 16 997 148 17 026 17 056 17 08s 17 114 17 143 17 173 17 202 17 231 17 260 17 289 149 17 319 17 348 17 377 17 406 17 435 17 464 17 493 17 522 17 551 17 580 ISO 17 609 17 638 17 667 17 696 17 72s 17 754 17 782 17 811 17 840 17 869 151 17 898 17 926 17 955 17 984 18 013 18 041 18 070 18 099 18 127 18 156 152 18 184 18 213 18 241 18 270 18 298 18 327 18 355 18 384 18 412 18 441 153 18 469 18 498 18 526 18 554 18 583 18 611 18 639 18 667 18 696 18 724 154 18 752 18 780 18 808 18 837 18 865 18 893 18 921 18 949 18 977 19 005 155 19 033 19 061 19 089 19 1 17 19 14s 19 173 19 201 19 229 19 257 19 285 156 19 312 19 340 19 368 19 396 19 424 19 451 19 479 19 S07 19 535 19 562 157 19 590 19 618 19 645 19 673 19 700 19 728 19 756 19 783 19 811 19 838 158 19 866 19 893 19 921 19 948 19 976 20 003 20 030 20 058 20 085 20 112 159 20 140 20 167 20 194 20 222 20 249 20 276 20 303 20330 20 3S8 2038s 160 20 412 20 439 20 466 20 493 20 520 20 548 20 575 20 602 20 629 20 656 161 20 683 20 710 20 737 20 763 20 790 20 817 20 844 20 871 20 898 2092s 162 20952 20 978 21 005 21 032 21 059 21 085 21 112 21 139 21 165 21 192 163 21 219 21 24s 21 272 21 299 21 32s 21 352 21 378 21 405 21 431 21 4S8 164 21 484 21 511 21 537 21 564 21 590 21 617 21 643 21 669 2 1 696 21 722 165 21 748 21 775 21 801 21 827 21 854 21 880 21 906 21 932 21 9S8 21 985 166 22 on 22 037 22 063 22 089 22 IIS 22 141 22 167 22 194 22 220 22 246 167 22 272 22 298 22 324 22 350 22 376 22 401 22 427 22 453 22 479 22 SOS 168 22 531 22 557 22 583 22 608 22 634 22 660 22 686 22 712 22 737 22 763 169 22 789 22 814 22 840 22 866 22 891 22 917 22 943 22 968 22 994 23 019 170 23 045 23 070 23 096 23 121 23 147 23 172 23 198 23 223 23 249 23 274 171 23 300 23 32s 23 350 23 376 23 401 23 426 23 452 23 477 23 S02 23 528 172 23 553 23 578 23 603 23 629 23 654 23 679 23 704 23 729 23 754 23 779 173 23 80s 23 830 23 855 23 880 23 90s 23 930 23 9S5 23 980 24 005 24030 174 24 055 24 080 24 105 24 130 24 155 24 180 24 204 24 229 24 254 24 279 175 24 304 24 329 24 353 24 378 24 403 24 428 24 452 24477 24 502 24 527 176 24 551 24 S76 24 601 24 62s 24 650 24 674 24699 24 724 24 748 24 773 177 24 797 24 822 24 846 24 871 24 895 24 920 24 944 24 969 24 993 25 018 178 25 042 2S 066 25 091 25 IIS 25 139 25 164 25 188 25 212 25 237 25 261 179 25 28s f 25 310 25 334 25 358 25 382 25 406 2S 431 25 455 25 479 25 S03 LOGARITHMS OF NUMBERS 235 No. I 2 3 4 5 6 7 8 9 180 25 527 25 551 25 575 25 600 25624 25 648 25 672 25 696 25 720 25 744 181 25 768 25 792 25 816 25 840 25 864 25 888 25 912 25 935 25 959 25 983 182 26 007 26 031 26 05s 26 079 26 102 26 126 26 150 26 174 26 198 26 221 183 26 24s 26 269 26 293 26 316 26 340 26 364 26 387 26 411 26 435 26 458 184 26 482 26 50s 26 529 26 553 26 576 26 600 26 623 26 647 26 670 26 694 I8S 26 717 26 741 26 764 26 788 26 811 26 834 26 858 26 881 26 905 26 928 186 26 951 26 975 26 998 27 021 27 04s 27 068 27 091 27 114 27 138 27 161 187 27 184 27 207 27 231 27 254 27 277 27 300 27 323 27 346 27 370 27 393 188 27 416 27 439 27 462 27 48s 27 SO8 27 531 27 554 27 577 27 600 27623 189 27 646 27 669 27 692 27 715 27 738 27 761 27 784 27 807 27 830 27 852 190 27 87s 27 898 27 921 27 944 27 967 27 989 28 012 28 035 28 058 28 081 191 28 103 28 126 28 149 28 171 28 194 28 217 28 240 28 262 28 285 28 307 192 28 330 28 353 2837s 28 398 28 421 28 443 28 466 28 488 28 SIX 28 533 193 28 5S6 28 578 28 601 28 623 28 646 28 668 28 691 28 713 28 735 28 758 194 28 780 28 803 28 82s 28 847 28 870 28 892 28 914 28 937 28 959 28 981 195 29 003 29 026 29 048 29 070 29 092 29 lis 29 137 29 159 29 i8r 29 203 196 29 226 29 248 29 270 29 292 29 314 29 336 29358 29 380 29403 29 425 197 29447 29 469 29491 29 513 29 535 29 557 29 579 29 601 29 623 29 645 198 29 667 29 688 29 710 29 732 29 754 29 776 29 798 29 820 29 842 29 863 199 29 88s 29 907 29 929 29 951 29 973 29 994 30 016 30 038 30 060 30 081 200 30 103 30 125 30 146 30 168 30 190 30 211 30 233 30 255 30 276 30 298 201 30 320 30341 30363 30384 30 406 30 428 30 449 30 471 30492 30 S14 202 30 535 30 557 30578 30 600 30 621 30 643 30 664 30 685 30 707 30 728 203 30 750 30 771 30 792 30 814 30 835 30 856 30 878 30 899 30 920 30 942 204 30 963 30 984 31 006 31 027 31 048 31 069 31 09X 31 112 31 133 31 154 20s 31 175 31 197 31 218 31 239 31 260 31 281 31 302 31 323 31 345 31366 206 31 387 31 408 31 429 31 450 31 471 31 492 31 S13 31 534 31 555 31 576 207 31 597 31 618 31 639 31 660 31 681 31 702 31 723 31 744 31 765 31 78s 208 31 806 31 827 31 848 31 869 31 890 31 911 31 931 31 952 31 973 31 994 209 32 015 32 035 32 056 32 077 32 098 32 118 32 139 32 160 32 181 32 201 210 32 222 32 243 32 263 32 284 32 305 32 325 32 346 32 366 32 387 32 408 211 32 428 32 449 32 469 32 490 32 510 32 531 32 552 32 572 32 593 32 613 212 32 634 32 654 i2 67s 32 695 32 715 32 736 32 756 32 777 32 797 32 818 213 32 838 32 858 32 879 32 899 32 919 32 940 32 960 32 980 33 001 33 021 214 a 041 a 062 3i 082 33 102 33 122 33 143 33 163 33 183 33 203 33 224 215 a 244 a 264 33 284 33 304 33 325 33 345 33 36s 33 385 33 405 33 42s 216 a 445 3i 465 33 486 33 S06 33 526 33 546 33 566 33 586 33 606 33 626 217 3i 646 a 666 33 686 33 706 33 726 33 746 33 766 33 786 33 806 33 826 218 a 846 a 866 33 88s 33 905 33 9-25 33 945 33 965 33 985 34 005 34 025 219 34 044 34 064 34 084 34 104 34 124 34 143 34 163 34 1S3 34 203 34 223 220 34 242 34 262 34 282 34 301 34 321 34 341 34 361 34 380 34 400 34 420 221 34439 34 459 34 479 34 498 34518 34 537 34 557 34 577 34 596 34616 222 34 63s 34 655 34674 34694 34 713 34 733 34 753 34 772 34 792 34 811 223 34 830 34 850 34 869 34 889 34 908 34 928 34 947 34 967 34986 35 OOS 224 35 025 35 044 35 064 35 083 35 102 35 122 35 141 35 i6o 35 180 35 199 236 APPENDIX No. I 2 3 4 5 6 7 8 9 225 3S 218 35 238 35 257 35 276 35 295 35 31S 35 334 35 353 35 372 35 392 226 35 411 35 430 35 449 35 468 35 488 35 507 35 526 35 545 35 564 35 583 227 35 603 35 622 35 641 35 660 35 679 35 698 35 717 35 736 35 755 35 774 228 35 793 35 813 35 832 35 8SI 35 870 35 889 35 908 35 927 35 946 35 96s 229 35 984 36 003 36 021 36 040 36 059 36 078 36 097 36 1x6 36 135 36 154 330 36 173 36 192 36 211 36 229 36 248 36 267 36 286 36 305 36 324 36 342 231 36 361 36 380 36 399 36 418 36 436 36 455 36 474 36 493 36 Sii 36 530 232 36 549 36 568 36 586 36 605 36 624 36 642 36 661 36 680 36 698 36 717 233 36 736 36 754 36 773 36 791 36 810 36 829 36 847 36 866 36 884 36 903 234 36 922 36 940 36959 36 977 36996 37 014 37 033 37 051 37 070 37 088 235 37 107 37 125 37 144 37 162 37 181 37 199 37 218 37 236 37 254 37 273 236 37 291 37 310 37 328 37 346 37 365 37 383 37 401 37 420 37 438 37 457 237 37 475 37 493 37 511 37 530 37 548 37 S66 37 585 37 603 37 621 37 639 238 37 658 37 676 37 694 37 712 37 731 37 749 37 767 37 785 37 803 37 822 239 37 840 37 858 37 876 37 894 37 912 37 931 37 949 37 967 37 98s 38 003 240 38 021 38 039 38 057 38 07S 38 093 38 112 38 130 38 148 38 166 38 184 241 38 202 38 220 38 238 38 256 38 274 38 292 38310 38328 38346 38364 242 38 382 38 399 38 417 38 435 38 453 38 471 38489 38 S07 38 525 38 543 243 38 S6i 38 578 38 596 38614 38 632 38 650 38 668 38686 38 703 38 721 244 38 739 38 757 38 775 38 792 38 810 38 828 38 846 38 863 38 881 38 899 24s 38 917 38 934 38 952 38 970 38 987 39 005 39 023 39 041 39 058 39 076 246 39 094 39 III 39 129 39 146 39 164 39 182 39 199 39 217 39 235 39 252 247 39 270 39 287 39 305 39 322 39 340 39 358 39 375 39 393 39 410 39428 248 39 445 39 463 39 480 39498 39 SIS 39 533 39 SSO 39 568 39 585 39602 249 39 620 39637 39 655 39672 39 690 39 707 39 724 39 742 39 759 39 777 250 39 794 39 811 39 829 39 846 39 863 39 881 39 898 39915 39 933 39 950 2SI 39 967 39 985 40 002 40 019 40 037 40 054 40 071 40 088 40 106 40 123 2S2 40 140 40 157 40 175 40 192 40 209 40 226 40 243 40 261 40 278 40 295 253 40 312 40 329 40 346 40 364 40 381 40 398 40 41S 40 432 40 449 40 466 254 40 483 40 500 40 S18 40 535 40 552 40 569 40 586 40 603 40 620 40 637 255 40 654 40 671 40 688 40 70s 40 722 40 739 40 756 40 773 40 790 40 807 256 40 824 40 841 40 858 40 875 40 892 40 909 40 926 40943 40 960 40976 257 40 993 41 010 41 027 41 044 41 061 41 078 41 095 41 II I 41 128 41 145 2S8 41 162 41 179 41 196 41 212 41 229 41 246 41 263 41 280 41 296 41 313 259 41 330 41 347 41 363 41 380 41 397 41 414 41 430 41 447 41 464 41 481 260 41 497 41 514 41 531 41 547 41 564 41 S8l 41 597 41 614 41 631 41 647 261 41 664 41 681 41 697 41 714 41 731 41 747 41 764 41 780 41 797 41 814 262 41 830 41 847 41 863 41 880 41 896 41 913 41 929 41 946 41 963 41 979 263 41 996 42 012 42 029 42 045 42 062 42 078 42 095 42 III 42 127 42 144 264 42 160 42 177 42 193 42 210 42 226 42 243 42 259 42 275 42 292 42 308 265 42 325 42 341 42 357 42 374 42 390 42 406 42 423 42 439 42 455 42 472 266 42 488 42 SO4 42 521 42 537 42 553 42 570 42 586 42 602 42 619 42 635 267 42 651 42 667 42 684 42 700 42 716 42 732 42 749 42 765 42 781 42 797 268 42 813 42 830 42 846 42 862 42 878 42 894 42 911 42 927 42 943 42 959 269 42 97S 42 991 43 008 43 024 43 040 43 056 43 072 43 088 43 104 43 120 LOGARITHMS OF NUMBERS 237 No. I 2 3 4 5 6 7 8 9 270 43 136 43 152 43 169 43 i8s 43 201 43 217 43 233 43 249 43 26s 43 281 271 43 297 43 313 43 329 43 345 43 361 43 377 43 393 43 409 43 425 43 441 272 43 457 43 473 43 489 43 505 43 521 43 537 43 553 43 569 43 584 43 600 273 43 616 43 632 43648 43 664 43 680 43 696 43 712 43 727 43 743 43 759 274 43 775 43 791 43 807 43 823 43 838 43 854 43 870 43 886 43 902 43 917 27s 43 933 43 949 43 965 43 981 43 996 44 012 44 028 44 044 44 059 44 075 276 44 091 44 107 44 122 44 138 44 154 44 170 44 i8s 44 201 44 217 44 232 277 44 248 44 264 44 279 44 295 44 311 44 326 44 342 44 358 44 373 44389 278 44 404 44 420 44 436 44 451 44 467 44 483 44 498 44 514 44 529 44 545 279 44 560 44 576 44 592 44 607 44623 44638 44654 44 669 4468s 44 700 380 44 716 44 731 44 747 44 762 44 778 44 793 44 809 44 824 44 840 44 855 281 44871 44 886 44 902 44 917 44 932 44 948 44 963 44 979 44 994 45 010 282 45 025 45 040 45 056 45 071 45 086 45 102 45 117 45 133 45 148 45 163 283 45 179 45 194 45 209 45 22s 45 240 45 255 45 271 45 286 45 301 45 317 284 45 332 45 347 45 362 45 378 45 393 45 408 45 423 45 439 45 454 45 469 28S 45 484 45 500 45 515 45 530 45 545 45 561 45 576 45 591 45 606 45 621 286 45 637 45 652 45 667 45 682 45 697 45 712 45 728 45 743 45 758 45 773 287 45 788 45 803 45 818 45 834 45 849 45 864 45 879 45 894 45 909 45 924 288 45 939 45 954 45 969 45 984 46 000 46 015 46 030 46 045 46 060 46 075 289 46 090 46 105 46 120 46 135 46 150 46 165 46 180 46 195 46 210 46 22s 290 46 240 46 255 46 270 46 28s 46 300 46 315 46 330 46 345 46 359 46374 291 46 389 46 404 46 419 46 434 46 449 46 464 46 479 46 494 46 S09 46 523 292 46 538 46 553 46 568 46583 46 598 46 613 46 627 46 642 46 657 46 672 293 46 687 46 702 46 716 46 731 46 746 46 761 46 776 46 790 46 805 46 820 294 46 835 46 850 46 864 46 879 46 894 46 909 46 923 46 938 46 953 46 967 295 46 982 46 997 47 012 47 026 47 041 47 056 47 070 47 085 47 100 47 114 296 47 129 47 144 47 159 47 173 47 188 47 202 47 217 47 232 47 246 47 261 297 47 276 47 290 47 305 47 319 47 334 47 349 47363 47 378 47 392 47 407 298 47 422 47 436 47 451 47 465 47 480 47 494 47 509 47 524 47 538 47 553 299 47 567 47 582 47 596 47 611 47 62s 47 640 47 6S4 47 669 47 683 47 698 300 47 712 47 727 47 741 47 7S6 47 770 47 784 47 799 47 813 47 828 47 842 301 47 857 47 871 47 88s 47 900 47 914 47 929 47 943 47 958 47 972 47 986 302 48 001 48 015 48 029 48 044 48 058 48 073 48 087 48 lOI 48 116 48 130 303 48 144 48 159 48 173 48 187 48 202 48 216 48 230 48 244 48 259 48 273 304 48 287 48 302 48316 48 330 48 344 48 359 48 373 48 387 48 401 48 416 30s 48 430 48 444 48 4S8 48 473 48487 48 SOI 48 S15 48 530 48 544 48 5S8 306 48 572 48 586 48 601 48 61S 48 629 48 643 48 6S7 48671 48 686 48 700 307 48 714 48 728 48 742 48 756 48 770 48 785 48 799 48 813 48 827 48 841 308 48 85s 48 869 48 883 48 897 48 911 48 926 48 940 48 954 48 968 48 982 309 48 996 49 010 49 024 49 038 49 052 49 066 49 080 49 094 49 loS 49 122 310 49 136 49 ISO 49 164 49 178 49 192 49 206 49 220 49 234 49 248 49 262 311 49 276 49 290 49304 49 318 49332 49 346 49 360 49 374 49388 49 402 312 4941S 49429 49 443 49 457 49 47 r 4948s 49 499 49 513 49 527 49 541 313 49 554 49 568 49 582 49 596 49 610 49 624 49638 49 651 49 665 49679 314 49O93 49 707 49 721 49 734 49 748 49 762 49 776 49 790 49 803 49 817 238 APPENDIX No. , 2 3 4 5 6 7 8 9 315 49 831 49 84s 49 859 49 872 49 886 49 900 49 914 49 927 49 941 49 9SS 316 49 969 49 982 49 996 50 010 SO 024 50037 50 051 50 065 50 079 SO 092 317 50 106 50 120 SO 133 SO 147 50 161 so 174 SO 188 SO 202 50 215 50 229 318 SO 243 50 256 SO 270 50 284 SO 297 50311 SO 325 SO 338 50352 5036s 319 50379 50 393 50 406 50 420 50 433 50 447 SO 461 50474 50 488 50 SOI 320 50 515 50 529 50 542 50 556 50 569 50 583 50 596 50 610 50623 SO 637 321 50651 50664 50678 50691 SO 705 50 718 50 732 50 745 so 759 50 772 322 50 786 50 799 SO 813 so 826 50 840 50 853 SO 866 SO 880 SO 893 50907 323 50 920 50934 SO 947 SO 961 SO 974 so 987 SI 001 51 014 51 028 51 041 324 51 055 51 068 51 081 51 095 SI 108 SI 121 51 135 51 148 51 162 SI 175 325 51 188 51 202 SI 215 SI 228 SI 242 51 255 SI 268 SI 282 51 295 51 308 326 51 322 51 335 51 348 51 362 51 375 SI 388 51 402 SI 41S SI 428 SI 441 327 51 455 51 468 SI 481 51 495 51 S08 51 521 51 534 SI 548 51 561 51 574 328 51 587 51 601 51 614 51 627 SI 640 51 6S4 SI 667 51 680 51 693 51 706 329 SI 720 51 733 51 746 51 759 51 772 51 786 51 799 51 812 SI 825 SI 838 330 51 851 SI 865 51 878 SI 891 SI 904 51 917 51 930 51 943 51 957 SI 970 331 51 983 SI 996 5 2 009 52 022 52 035 52 048 52 061 52 075 52 088 52 lOI 332 52 114 52 127 52 140 52 153 52 166 52 179 52 192 52 205 52 218 52 231 333 52 244 52 257 52 270 52 284 52 297 52 310 52 323 52 336 52 349 52 362 334 52 375 52 388 52 401 52 414 52 427 52 440 52 453 52 466 52 479 52 492 335 5 2 504 52 517 52 530 52 543 52 556 52 569 52 582 52 595 52 608 52 621 336 52 634 52 647 52 600 52 673 52 686 52 699 52 711 52 724 52 737 52 750 337 52 763 52 776 52 789 52 802 52 81S 52 827 52 840 52 853 52 866 52 879 338 52 892 52 905 52 917 52 930 52 943 52 956 52 969 52 982 52 994 53 007 339 53 020 53 033 53 046 S3 058 S3 071 53 084 53 097 53 IIO 53 122 53 135 340 53 148 53 161 53 173 53 186 53 199 53 212 53 224 S3 237 53 250 53 263 341 53 275 53 288 53 301 53 314 53 326 53 339 53 352 53364 S3 377 53 390 342 53 403 53 41S 53 428 53 441 53 453 53 466 S3 479 53 491 53 504 53 517 343 53 529 53 542 53 555 53 567 53 580 53 593 53 605 53618 53 631 53 643 344 53 656 53 668 53681 S3 694 53 706 53 719 53 732 53 744 53 757 53 769 345 53 782 53 794 53 807 53 820 53 832 53 84s 53 857 53 870 53 882 53 895 346 53 908 53 920 53 933 S3 945 53 958 53 970 53 983 53 995 54 008 54 020 347 54 033 54 045 54 058 54 070 54 083 54 095 54 108 54 120 54 133 54 145 348 54 158 54 170 54 183 54 195 54 208 54 220 54 233 54 245 54 258 54 270 349 54 283 54 295 54 307 54 320 54 332 54 345 54 357 54 370 54382 54 394 350 54 407 54 419 54 432 54 444 54 456 54 469 54 481 54 494 54 S06 54 518 351 54 531 54 543 54 555 54 568 54 580 54 593 54 60s 54 617 54 630 54 642 352 54 654 54667 54679 54 691 54 704 54 716 54 728 54 741 54 753 54 76s 353 54 777 54 790 54 802 54 814 54 827 54 839 54 851 54 864 54 876 54 888 354 54 900 54 913 54925 54 937 54 949 54 962 54 974 54 986 54 998 55 on 355 55 023 55 035 55 047 55 060 55 072 55 084 55 096 55 108 55 121 55 133 356 55 145 55 157 55 169 55 182 55 194 55 206 55 218 55 230 55 242 55 255 357 55 267 55 279 55 291 55 303 55 315 55 328 55 340 55 352 55 364 55 376 358 55 388 55 400 55 413 55 42s 55 437 55 449 55 461 55 473 55 485 55 497 359 55 509 55 522 55 534 55 546 55 SS8 55 570 55 582 55 594 55 606 55 618 LOGARITHMS OP NUMBERS 239 No. I 2 3 4 5 6 7 8 9 360 55 630 55 O42 55 654 55 666 55678 55 691 55 703 55 715 55 727 55 739 361 55 751 55 763 55 775 55 787 55 799 55 811 55 823 55 83s 55 847 55 859 362 55 871 55 883 55 895 55 907 ss 919 SS 931 55 943 55 955 55 967 55 979 363 55 991 56 003 56 015 56 027 S6 038 S6 050 56 062 56 074 56 086 56 098 364 56 no 56 122 S6 134 56 146 56 158 56 170 56 182 56 194 S6 205 56 217 36s S6 229 S6 241 56 253 56 265 56 277 S6 289 56 301 56 312 56 324 S6 336 366 56 348 56 360 56 372 56384 S6 396 56 407 S6 419 56431 S6 443 5645s 367 S6 467 56478 56 490 56 502 56 514 56 526 56 538 S6 549 S6 561 S6 573 368 56 585 56 597 56 608 56 620 56 632 S6 644 56656 56667 56 679 56 691 369 S6 703 56 714 56 726 S6 738 S6 750 56 761 56 773 56 78s 56 797 56 808 370 56 820 56 832 56 844 S6 855 56 867 56 879 56 891 S6 902 56 914 56 926 371 56 937 S6 949 56 90 1 S6 972 56984 56 996 57 008 57 019 57 031 57 043 372 57 054 57 066 57 078 57 089 57 loi 57 113 57 124 57 136 57 148 57 159 373 57 171 57 183 57 194 57 206 57 217 57 229 57 241 57 252 57 264 57 276 374 57 287 57 299 57 310 57 322 57 334 57 345 57 357 57 368 57 380 57 392 375 57 403 57 41S 57 426 57 438 5 7 449 5 7 461 57 473 57 484 57 496 57 507 376 57 SI9 57 530 57 542 57 553 57 56s 57 576 57 588 5 7 600 57 611 57 623 377 57634 57 646 57 657 57 669 57 680 57 692 57 703 57 715 57 726 57 738 378 57 749 57 761 57 772 57 784 S7 795 57 807 57 818 57 830 57 841 57 852 379 57 864 57 875 57 887 57 898 57 910 57 921 57 933 57 944 57 955 57 967 380 57 978 57 990 58 001 58 013 58 024 58 035 S8 047 58 058 58 070 58 081 381 58 092 58 104 58 IIS 58 127 S8 138 58 149 58 161 58 172 58 184 58 195 382 S8 206 S8 218 58 229 S8 240 58 252 58 263 S8 274 S8 286 58 297 58 309 383 58 320 58331 58 343 58 354 58 36s 58377 58 388 58 399 58 410 58422 384 58 433 58 444 58 456 58 467 58 478 58 490 58 501 58 512 58 524 58 535 385 58 546 S8 557 58 569 S8 580 S8 591 S8 602 S8 614 58 62s 58 636 58 647 386 58 659 S8 670 S8 681 S8 692 58 704 58 715 58 726 58 737 58 749 58 760 387 58 771 58 782 58 794 58 805 58 816 S8 827 58 838 58 850 58 861 S8 872 388 58 883 S8894 58 906 58 917 58 928 58 939 S8 950 58 961 58973 58 984 389 58 995 59 006 59 017 59 028 59 040 59 051 59 062 59 073 59 084 59 09s 390 59 106 59 118 59 129 59 140 59 151 59 162 59 173 59 184 59 I9S 59 207 391 59 218 59 229 59 240 59 251 59 262 59 273 59 284 59 295 59 306 59318 392 59 329 59 340 59 351 59 362 59 373 59 384 59 395 59 406 59417 59 428 393 59 439 59 450 59 461 59472 59483 59 494 59 506 59 517 59 528 59 539 394 59 550 59 S6i 59 572 59 583 59 S94 59 605 59 616 59 627 59638 59 649 395 59 660 59 671 59 682 59 693 59 704 59 715 59 726 59 737 59 748 59 759 396 59 770 59 780 59 791 59 802 59 813 59 824 59 83s 59 846 59 857 59 868 397 59 879 59 890 59 901 59 912 59 923 59 934 59 945 59 950 59 966 59 977 398 59 988 59 999 60 010 60 02 1 60 032 60 043 60 054 60 06s 60 076 60 086 399 60 097 60 108 60 119 60 130 60 141 60 152 60 163 60 173 60 184 60 19s 400 60 206 60 217 60 228 60 239 60 249 60 260 60 271 60 282 60 293 60 304 401 60 314 6032s 60 336 60347 60358 60 369 60 379 60 390 60 401 60 412 402 60 423 60 433 60444 60 455 60 466 60477 60 487 60 498 60 S09 60 520 403 60 531 60 541 60 552 60 563 60 574 60 584 60 595 60 606 60 617 60 627 404 60 638 60 649 60 660 60 670 60 681 60 692 60 703 60 713 60 724 60 735 240 APPENDIX No. I 2 3 4 5 6 7 8 9 4CS 60 746 60 756 60 767 60 778 60 788 60 799 60 810 60 821 60 831 60 842 406 60 853 60 863 60 874 60 885 60 89s 60 906 60 917 60 927 60938 60 949 407 60 959 60 970 60 981 60 991 61 002 61 013 61 023 61 034 61 04s 61 OSS 408 61 066 61 077 61 087 61 098 61 109 61 119 61 130 61 140 61 151 61 162 409 61 172 61 183 61 194 61 204 61 215 61 225 61 236 61 247 61 257 61 268 410 61 278 61 289 61 300 61 310 61 321 61 331 61 342 61 352 61 363 61 374 411 61 384 61 395 61 405 61 416 61 426 61 437 61 448 61 458 61 469 61 479 412 61 490 61 500 61 511 61 521 61 532 61 542 61 553 61 563 61 574 61 584 413 61 595 61 606 61 616 61 627 61 637 61 648 61 638 6r 669 61 679 61 690 414 61 700 61 711 61 721 61 731 61 742 61 752 61 763 61 773 61 784 61 794 415 61 80s 61 815 61 826 61 836 61 847 61 8S7 61 868 61 878 61 888 61 899 416 61 909 61 920 61 930 61 941 61 951 61 962 61 972 61 982 61 993 62 003 417 62 014 62 024 62 034 62 04s 62 055 62 066 62 076 62 086 62 097 62 107 418 62 118 62 128 62 138 62 149 62 159 62 170 62 180 62 190 62 201 62 211 419 62 221 62 232 62 242 62 252 62 263 62 273 62 284 62 294 62 304 62 315 420 62 325 62 335 62 346 62 356 62 366 62 377 62 387 62 397 62 408 62 418 421 62 428 62 439 62 449 62 459 62 469 62 480 62 490 62 SCO 62 511 62 521 422 62 531 62 542 62 552 62 562 62 572 62 583 62 593 62 603 62 613 62 624 423 62 634 62 644 62 655 62 665 62 675 62 685 62 696 62 706 62 716 62 726 424 62 737 62 747 62 757 62 767 62 778 62 788 62 798 62 808 62 818 62 829 42s 62 839 62 849 62 859 62 870 62 880 62 890 62 900 62 910 62 921 62 931 426 62 941 62 951 62 961 62 972 62 982 62 992 63 002 63 012 63 022 63 033 427 63 043 63 053 63 063 63 073 63 083 63 094 63 104 63 114 63 124 63 134 428 63 144 63 155 63 165 63 175 63 18S 63 195 63 205 63 215 63 225 63 236 429 63 246 63 256 63 266 63 276 63 286 63 296 63 306 63 317 63 327 63 337 430 63 347 63 357 63 367 63 377 63 387 63 397 63 407 63 417 63 428 63 438 431 63 448 63 4S8 63 468 63 478 63 488 63 498 63 508 63 518 63 528 63 538 432 63 548 63 558 63 568 63 579 63 589 63 599 63 609 63 619 63 629 63 639 433 63 649 63 659 63 669 63 679 63 689 63 699 63 709 63 719 63 729 63 739 434 63 749 63 759 63 769 63 779 63 789 63 799 63 809 63 819 63 829 63 839 435 63 849 63 859 63 869 63 879 63 889 63 899 63 909 63 919 63 929 63 939 436 63 949 63 959 63 969 63 979 63 988 63 998 64 008 64 018 64 028 64 038 437 64 048 64 058 64 068 64 078 64 088 64 098 64 108 64 118 64 128 64 137 438 64 147 64 157 64 167 64 177 64 187 64 197 64 207 64 217 64 227 64 237 439 64 246 64 256 64 266 64 276 64 286 64 296 64 306 64 316 64 326 64 335 440 64 345 64 355 6436s 64 375 64 385 64 395 64 404 64 414 64 424 64434 441 64 444 64 454 64 464 64473 64 483 64 493 64 503 64 S13 64 523 64 532 442 64 542 64 552 64 562 64 572 64 582 64 591 64 601 64 61 1 64 621 64 631 443 O4 640 64 650 64 660 64 670 64 680 64689 64 699 64 709 64 719 64 729 444 64 738 64 748 64 7S8 64 768 64 777 64 787 64 797 64 807 64 816 64 826 445 64 836 64 846 64 856 64 865 64 875 64 88s 64 89s 64 904 64 914 64 924 446 64 933 64 943 64 953 64 963 64 972 04 982 04 992 65 002 65 on 6s 021 447 6S 031 65 040 65 050 65 060 65 070 6S 079 6S 089 65 099 65 108 6s n8 448 65 128 6S 137 65 147 65 157 6s 167 65 176 65 186 65 iq6 65 205 65 215 449 65 225 65 234 65 244 65 254 6s 263 6S 273 65 283 6s 292 6s 302 65 312 LOGARITHMS OF NUMBERS 241 No. I 2 3 4 5 6 7 8 9 450 65 321 65 331 65 341 65 350 65 360 65 369 65 379 65 389 65 398 65 408 45 1 65 418 65 427 65 437 65 447 65 456 6s 466 65 475 65 485 65 495 65 S04 452 65 S14 65 523 6s 533 65 543 6s 552 6s 562 6S 571 6s S8i 65 591 65 600 453 6s 610 65 619 65 629 65 639 65 648 65 658 65 667 6s 677 6s 686 65 696 454 6s 706 65 71S 65 725 6S 734 65 744 6s 753 6S 763 65 772 65 782 6s 792 455 6s 801 6s 811 6s 820 65 830 6s 839 65 849 6s 858 6s 868 65 877 65 887 456 6s 896 6s 906 65 916 65 92s 65 935 65 944 65 954 6s 963 6S 973 6s 982 457 65 992 66 001 66 on 66 020 66 030 66 039 66 049 66 058 66 068 66 077 458 66 087 66 096 66 106 66 IIS 66 124 66 134 66 143 66 153 66 162 66 172 459 66 181 66 191 66 200 66 210 66 219 66 229 66 238 66 247 66 257 66 266 460 66 276 66 28s 66 295 66 304 66 314 66 323 66 332 66 342 66 351 66 361 461 66 370 66 380 66 389 66 398 66 408 66 417 66 427 66 436 66 445 66 455 462 66 464 66 474 66483 66 492 66 502 66 511 66 521 66 530 66 539 66 549 463 66 558 66567 66 577 66 586 66 596 66 605 66614 66 624 66633 66642 464 66 652 66661 66 671 66 680 66 689 66 699 66 708 66 717 66 727 66 736 46s 66 745 66 755 66 764 66 773 66 783 66 792 66 801 66 811 66 820 66 829 466 66 839 66 848 66 857 66 867 66 876 66 88s 66 894 66 904 66 913 66 922 467 66 932 66 941 66 950 66 960 66 969 66 978 66 987 66 997 67 006 67 ors 468 67 025 67 034 67 043 67 052 67 062 67 071 67 080 67 089 67 099 67 108 469 67 117 67 127 67 136 67 14s 67 154 67 164 67 173 67 182 67 191 67 201 470 67 210 67 219 67 228 67 237 67 247 67 256 67 265 67 274 67 284 67 293 471 67 302 67 311 67 321 67 330 67 339 67 348 67 357 67367 67 376 67 385 472 67 394 67 403 67 413 67 422 67 431 67 440 67 449 67 459 67 468 67 477 473 67 486 67 495 67 504 67 514 67 523 67 532 67 541 67 550 67 560 67 569 474 67 578 67 587 67 596 67 60s 67 614 67 624 67633 67 642 67 651 67 660 475 67 669 67 679 67 688 67 697 67 706 67 71S 67 724 67 733 67 742 67 752 476 67 761 67 770 67 779 67 788 67 797 67 806 67 815 67 825 67 834 67 843 477 67852 67 861 67 870 67 879 67 888 67 897 67 906 67 916 67 92s 67 934 478 67 943 67 952 67 961 67 970 67 979 67 988 67 997 68 006 68 CIS 68 024 479 68 034 68 043 68 052 68 061 68 070 68 079 68 088 68 097 68 106 68 115 480 68 124 68 133 68 142 68 isi 68 160 68 169 68 178 68 187 68 196 68 205 481 68 215 68 224 68 233 68 242 68 251 68 260 68 269 68 278 68 287 68 296 482 68 305 68 314 68 323 68 332 68 341 68 350 68 359 68 368 68377 68 386 483 68 395 68 404 68 413 68 422 68 431 68 440 68 449 68 458 68 467 68 476 484 68 485 68 494 68 S02 68 sii 68 520 68 520 68 538 68 547 68 556 68 s6s 48s 68 574 68 583 68 592 68 601 68 610 68 619 68 628 68 637 68 646 68 655 486 68 664 68673 68 681 68 690 68 699 68 708 68 717 68 726 68 73S 68 744 487 68 753 68 762 68 771 68 780 68 789 68 797 68 806 68 815 68 824 68 833 488 68 842 68 851 68 860 68 869 68 878 68 886 68 895 68 904 68 913 68 922 489 68 931 68 940 68 949 68 958 68 966 68 975 68 984 68 993 69 002 69 on 490 69 020 69 028 60 037 69 046 69 055 69 064 69 073 69 082 69 090 69 099 491 69 108 69 117 69 126 69 135 69 144 69 152 69 161 69 170 69 179 69 188 492 69 197 69 205 69 214 69 223 69 232 69 241 69 249 69 258 69 267 69 276 493 69 285 69 294 69 302 69 311 69 320 69 329 69338 69 346 69 355 69364 494 69 373 69 381 69 390 69399 69 408 69 417 1 69 42s 69 434 69 443 69 452 242 APPENDIX No. I 2 3 4 5 6 7 8 9 495 69 461 69 469 69 478 69 487 69 496 69 504 69 S13 69 522 69 531 69 539 496 69 548 69 557 69 566 69 574 69 583 69 592 69 601 69 609 69 618 69 627 497 69 636 69 644 69 653 69 662 69 671 69 679 69 688 69 697 69 705 69 714 498 69 723 69 732 69 740 69 749 69 758 69 767 69 775 69 784 69 793 69 801 499 69 810 69 819 69 827 69 836 69 84s 69 854 69 862 69 871 69 880 69 888 500 69 897 69 906 69 914 69923 69 932 69 940 69 949 69 958 69 966 69975 SOI 69 984 69 992 70 001 70 010 70 018 70 027 70 036 70044 70 053 70 062 502 70 070 70 079 70 088 70 096 70 105 70 114 70 122 70 131 70 140 70 148 S03 70 157 70 165 70 174 70 183 70 191 70 200 70 209 70 217 70 226 70 234 5 04 70 243 70 252 70 260 70 269 70 278 70 286 70 295 70303 70312 70 321 S05 70 329 70338 70 346 70 355 70 364 70372 70 381 70 389 70 398 70 406 S06 70415 70424 70 432 70441 70 449 70 458 70 467 70 475 70 484 70492 S07 70 501 70 509 70 518 70 526 70 535 70 544 70 552 70 561 70 569 70578 508 70 s86 70 595 70 603 70 612 70 621 70 629 70638 70 646 70 655 70 663 509 70 672 70 680 70 689 70 697 70 706 70 714 70 723 70 731 70 740 70 749 510 70 757 70 766 70 774 70 783 70 791 70 800 70 808 70 817 70 82s 70 834 511 70 842 70 851 70 859 70 868 70 876 70 885 70 893 70 902 70 910 70 919 S12 70 927 70 935 70 944 70 952 70 961 70 969 70 978 70 986 70 995 71 003 513 71 012 71 020 71 029 71 037 71 046 71 054 71 063 71 071 71 079 71 088 514 71 096 "I 105 71 113 71 122 71 130 71 139 71 147 71 155 71 164 71 172 515 71 181 71 189 71 198 71 206 71 214 71 223 71 231 71 240 71 248 71 257 S16 71 26s 71 273 71 282 71 290 71 299 71 307 71 31S 71 324 71 332 71 341 S17 71 349 71 357 71 366 71 374 71 383 71 391 71 399 71 408 71 416 71 42s 518 71 433 71 441 71 450 71 458 71 466 71 475 71 483 71 492 71 500 71 508 519 71 517 71 525 71 533 71 542 71 550 71 559 71 567 71 575 71 584 71 592 520 71 600 71 609 71 617 71 625 71 634 71 642 71 650 71 659 71 667 71 675 S2I 71 684 71 692 71 700 71 709 71 717 71 725 71 734 7t 742 71 750 71 759 522 71 767 71 775 71 784 71 792 71 800 71 809 71 817 71 82s 71 834 71 842 523 71 850 71 858 71 867 71 87s 71 883 71 892 71 900 7 r 908 71 917 7 t 92s 524 71 933 71 941 71 950 71 958 71 966 71 975 71 983 71 991 71 999 72 008 525 72 016 72 024 72 032 72 041 72 049 72 057 72 066 72 074 72 082 72 090 526 72 099 72 107 72 IIS 72 123 72 132 72 140 72 148 72 156 72 165 72 173 527 72 181 72 189 72 198 72 206 72 214 72 222 72 230 72 239 72 247 72 255 528 72 263 72 272 72 280 72 288 72 296 72 304 72 313 72 321 72 329 72 337 529 72 346 72 354 72 362 72 370 72 378 72 387 72 395 72 403 72 411 72 419 530 72 428 72 436 72 444 72 452 72 460 72 469 72 477 72 48s 72 493 72 SOI 531 72 509 72 518 72 526 72 534 72 542 72 550 72 558 72 567 72 575 72 583 532 72 591 72 599 72 607 72 616 72 624 72 632 72 640 72 648 72 656 72 66s 533 72 673 72 681 72 689 72 697 72 70s 72 713 72 722 72 730 72 738 72 746 534 72 754 72 762 72 770 72 779 72 787 72 795 72 803 72 811 72 819 72 827 535 72 835 72 843 72 852 72 860 72 868 72 876 72 884 72 892 72 900 72 908 536 72 916 72 925 72 933 72 941 72 949 72 957 72 96s 72 973 72 981 72 989 537 72 997 73 006 73 014 73 022 73 030 73 038 73 046 73 054 73 062 73 070 538 73 078 73 086 73 094 73 102 73 III 73 119 73 127 73 135 73 143 73 151 539 73 159 73 167 73 175 73 183 73 191 73 199 73 207 73 215 73 223 73 231 LOGARITHMS OF NUMBERS 243 No. I 2 3 4 5 6 7 8 9 540 73 239 73 247 73 255 73 263 73 272 73 280 73 288 73 296 73 304 73 312 541 73 320 73 328 73 336 73 344 73 352 73 360 73 368 73 376 73 384 73 392 542 73 400 73 408 73 416 73 424 73 432 73 440 73 448 73 456 73 464 73 472 543 73 480 73 488 73 496 73 504 73 512 73 520 73 528 73 536 73 544 73 552 544 73 560 73 568 73 576 73 584 73 592 73 600 73 608 73 616 73 624 73 632 545 73 640 73 648 73656 73 664 73 672 73 679 73 687 73 695 73 703 73 711 546 73 719 73 727 73 735 73 743 73 751 73 759 73 767 73 775 73 783 73 791 547 73 799 73 807 73 815 73 823 73 830 73 838 73 846 73 854 73 862 73 870 548 73 878 73 886 73 894 73 902 73 910 73 918 73 926 73 933 73 941 73 949 549 73 957 73 96s 73 973 73 981 73 989 73 997 74 005 74 013 74 020 74 028 550 74 036 74 044 74 052 74 060 74 068 74 076 74 084 74 092 74 099 74 107 551 74 IIS 74 123 74 131 74 139 74 147 74 155 74 162 74 170 74 178 74 186 552 74 194 74 202 74 210 74 218 74 22s 74 233 74 241 74 249 74 257 74 265 553 74 273 74 280 74 288 74 296 74 304 74 312 74 320 74 327 74 335 74 343 554 74 351 74 359 74 367 74 374 74 382 74 390 74 398 74 406 74 414 74 421 555 74 429 74 437 74 445 74 453 74 461 74 468 74 476 74 484 74 492 74 500 556 74 507 74 51S 74 523 74 531 74 539 74 547 74 554 74 562 74 570 74 578 557 74 586 74 593 74 601 74 609 74 617 74 624 74 632 74 640 74648 74 656 558 74663 74 671 74 679 74687 74 69s 74 702 74 710 74 718 74 726 74 733 559 74 741 74 749 74 757 74 764 74 772 74 780 74 788 74 796 74 803 74 811 560 74 819 74 827 74 834 74 842 74 850 74858 74 865 74 873 74 881 74 889 561 74 896 74 904 74 912 74 920 74 927 74 935 74 943 74 950 74 958 74 966 562 74 974 74 981 74 989 74 997 75 005 75 012 75 020 75 028 75 035 75 043 563 75 051 75 059 75 066 75 074 75 082 75 089 75 097 75 105 75 113 75 120 564 75 128 75 136 75 143 75 iSi 75 159 75 166 75 174 75 182 75 189 75 197 S6S 75 20s 75 213 75 220 75 228 75 236 75 243 75 251 75 259 75 266 75 274 566 75 282 75 289 75 297 75 305 75 312 75 320 75 328 75 335 75 343 75 351 S67 75 3S8 75 366 75 374 75 381 75 389 75 397 75 404 75 412 75 420 75 427 568 75 435 75 442 75 4SO 75 458 75 465 75 473 75 481 75 488 75 496 75 504 569 75 511 75 519 75 526 7S 534 75 542 75 549 75 557 75 565 75 572 75 S8o 570 75 587 75 595 75 603 75 610 75 618 75 626 75 633 75 641 75 648 75 656 571 75 664 75 671 75 679 75 686 75 694 75 702 75 709 75 717 75 724 75 732 572 75 740 75 747 75 755 75 762 75 770 75 778 75 785 75 793 75 800 75 808 573 75 815 75 823 75 831 75 838 75 846 75 853 75 861 75 868 75 876 75884 574 75 891 75 899 75 906 75 914 75 921 75 929 75 937 75 944 75 952 75 959 575 75 967 75 974 75 982 75 989 75 997 76 005 76 012 76 020 76 027 76 035 576 76 042 76 050 76 OS 7 76 065 76 072 76 080 76 087 76 095 76 103 76 no 577 76 118 76 125 76 133 76 140 76 148 76 155 76 163 76 170 76 178 76 i8s 578 76 193 76 200 76 208 76 215 76 223 76 230 76 238 76 24s 76 253 76 260 5 79 76 268 76 275 76 283 76 290 76 298 76 305 76 313 76 320 76 328 76 335 580 76 343 76 350 76 358 76 365 76373 76 380 76 388 76 395 76 403 76 410 581 76 418 76 42s 76 433 76 440 76448 76 455 76 462 76 470 76 477 76485 582 76 492 76 500 76 507 76 515 76 522 76 530 76 537 76 545 76 552 76 559 583 76 567 76 574 76 S82 76 589 76 597 76 604 76 612 76 619 76 626 76 634 584 76 641 76 649 76 656 76 664 76671 76 678 76 686 76 693 76 701 76 708 244 APPENDIX No. I 2 3 4 5 6 7 8 9 S8S 76 716 76 723 76 730 76 738 76 745 76 753 76 760 76 768 76 775 76 782 S86 76 790 76 797 76 805 76 812 76 819 76 827 76834 76 842 76 849 76 856 S87 76 864 76 871 76 879 76 886 76 893 76 901 76 908 76 916 76 923 76 930 S88 76 938 76 945 76 953 76 960 76 967 76 975 76 982 76 989 76 997 77 004 S89 77 012 77 019 77 026 77 034 77 041 77 048 77 056 77 063 77 070 77 078 590 77 085 77 093 77 100 77 107 77 IIS 77 122 77 129 77 137 77 144 77 151 591 77 159 77 166 77 173 77 I8i 77 188 77 195 77 203 77 210 77 217 77 22s 592 77 232 77 240 77 247 77 254 77 262 77 269 77 276 77 283 77 291 77 298 593 77 30s 77 313 77 320 77 327 77 335 77 342 77 349 77357 77 364 77 371 594 77 379 77 386 77 393 77 401 77 408 77 415 77 422 77 430 77 437 77 444 595 77 452 77 459 77 466 77 474 77 481 77 488 77 495 77 503 77 510 77 517 596 77 525 77 532 77 539 77 546 77 554 77 S6i 77 568 77 576 77 583 77 590 597 77 597 77 60s 77 612 77 619 77 627 77 634 77 641 77 648 77 656 77 663 598 77 670 77 677 77 68s 77 692 77 699 77 706 77 714 77 721 77 728 77 735 599 77 743 77 750 77 757 77 764 77 772 77 779 77 786 77 793 77 801 77 808 600 77 815 77 822 77 830 77 837 77 844 77 851 77 859 77 866 77 873 77 880 6or 77 887 77 895 77 902 77 909 77 916 77 924 77 931 77 938 77 945 77 952 602 77 960 77 967 77 974 77 981 77 988 77 996 78 003 78 oro 78 017 78 025 603 78 032 78 039 78 046 78 053 78 061 78 068 78 075 78 082 78 089 78 097 604 78 104 78 III 78 118 78 125 78 132 78 140 78 147 78 IS4 78 161 78 168 605 78 176 78 183 78 190 78 197 78 204 78 211 78 219 78 226 78 233 78 240 606 78 247 78 254 78 262 78 269 78 276 78 283 78 290 78 297 78 30s 78 312 607 78 319 78 326 78 333 78 340 78 347 78355 78362 78 369 78 376 78 383 608 78 390 78 398 78 40s 78 412 78 419 78 426 78 433 78 440 78 447 78 455 609 78 462 78469 78 476 78 483 78 490 78 497 78 S04 78 512 78 S19 78 526 610 78 533 78 540 78 547 78 554 78 S6i 78 569 78 576 78 583 78 590 78 597 611 78 604 78 6ri 78 618 78 62s 78 633 78 640 78 647 78 6S4 78 661 78668 612 78 675 78 682 78 689 78 696 78 704 78 711 78 718 78 72s 78 732 78 739 613 78 746 78 753 78 760 78 767 78 774 78 781 78 789 78 796 78 803 78 810 614 78 817 78 824 78 831 78 838 78 84s 78 852 78 8S9 78 866 78 873 78 880 615 78 888 78 89s 78 902 78 909 78 916 78 923 78 930 78 937 78 944 78 951 616 78 958 78 965 78 972 78 979 78 986 78 993 79 000 79 007 79 014 79 021 617 79 029 79 036 79 043 79 O.SO 79 OS 7 79 064 79 071 79 078 79 08s 79 092 618 79 099 79 106 79 113 79 120 79 127 79 134 79 141 79 148 79 155 79 162 619 79 169 79 176 79 183 79 190 79 197 79 204 79 211 79 218 79 225 79 232 620 79 239 79 246 79 253 79 260 79 267 79 274 79 281 79 288 79 295 79 302 621 79 309 79 316 79 323 79 330 79 337 79 344 79 351 79 358 79 36s 79372 622 79 379 79386 79 393 79 400 79 407 79 414 79 421 79 428 79 435 79 442 623 79 449 79 456 79 463 79 470 79 477 79 484 79 491 79 498 79 50s 79 511 624 79 518 79525 79 532 79 539 79 546 79 553 79 S6o 79 567 79 574 79 581 62s 79 588 79 595 79 602 79 609 79 616 79 623 79 630 79 637 79 644 79 650 626 79657 79 664 79 671 79678 7968s 79 692 79 699 79 706 79 713 79 720 627 79 727 79 734 79 741 79 748 79 754 79 761 79 768 79 775 79 782 79 789 628 79 796 79 803 79 810 79 817 79 824 79 831 79 837 79 844 79 851 79 858 629 79 86s 79 872 79 879 79 886 79 893 79 900 79 906 79 913 79 920 79 927 LOGARITHMS OF NUMBERS 245 No. I 2 3 4 5 6 7 8 9 630 79 934 79 941 79 948 79 955 79 962 79 969 79 975 79 982 79 989 79 996 631 80 003 80 010 80 017 80 024 80 030 80 037 80 044 80 051 80 058 80 06s 632 80 072 80 079 80 08s 80 092 80 099 80 106 80 113 80 120 80 127 80 134 633 80 140 80 147 80 154 80 161 80 168 80 175 80 182 80 188 80 195 80 202 634 80 209 80 216 80 223 80 229 80 236 80 243 80 250 80 257 80 264 80 271 63 s 80 277 80 284 80 291 80 298 80 30s 80 312 80 318 80325 80332 80339 636 80 346 80353 80359 80 366 80373 80 380 80 387 80 393 80 400 80 407 637 80 414 80 421 80 428 80 434 80 441 80 448 80 455 80 462 80 468 80 475 638 80 482 80 489 80 496 80 502 80 509 80 516 80 523 80 530 80 536 80 543 639 80 550 80 557 80 564 80 570 80577 80 584 80 591 80 598 80 604 80 611 640 80 618 80 625 80 632 80 638 80 64s 80 652 80 659 80 66s 80 672 80 679 641 80 686 80 693 80 699 80 706 80 713 80 720 80 726 80 733 80 740 80 747 642 80 754 80 760 80 767 80 774 80 781 80 787 80 794 80 801 80 808 80 814 643 80 821 80 828 80 835 80 841 80 848 80 85s 80 862 80 868 80 875 80 882 644 80 889 80 895 80 902 80 909 80 916 80 922 80 929 80 936 80943 80 949 64s 80 956 80 963 80 969 80 976 80 983 80 990 80 996 81 003 81 010 81 017 646 81 023 81 030 81 037 81 043 81 050 81 057 81 064 81 070 81 077 81 084 647 8r 090 81 097 81 104 81 III 81 117 81 124 81 131 81 137 81 144 81 151 648 81 158 81 164 81 171 81 178 81 184 81 191 81 198 81 204 81 211 81 218 649 81 224 81 231 81 238 81 245 81 251 81 258 81 265 81 271 81 278 81 28s 650 81 291 81 298 81 305 81 311 81 318 81 325 81 331 81 338 81 345 81 351 6s I 81 3S8 81 365 81 371 81 378 81 385 81 391 81 398 81 405 81 411 81 418 652 81 42s 81 431 81 438 81 445 81 451 81 458 81 465 81 471 81 478 81 485 653 81 491 81 498 81 505 81 511 81 518 81 525 81 531 81 538 81 544 81 551 654 8r 558 81 564 81 571 81 578 81 584 81 591 81 598 81 604 81 611 81 617 655 81 624 81 631 81 637 81 644 81 651 81 657 81 664 81 671 81 677 81 684 656 81 690 8i 697 81 704 81 710 81 717 81 723 81 730 81 737 81 743 81 7SO 6S7 81 757 81 763 81 770 81 776 81 783 81 790 81 796 81 803 81 809 81 816 658 81 823 81 829 81 836 81 842 81 849 81 856 81 862 8i 869 81 875 81 882 659 81 889 81 895 81 902 81 908 81 91S 81 921 81 928 81 935 81 941 81 948 660 8r 954 81 961 81 968 81 974 81 981 81 987 81 994 82 000 82 007 82 014 661 82 020 82 027 82 033 82 040 82 046 82 053 82 060 82 066 82 073 82 079 662 82 086 82 092 82 099 82 105 82 112 82 119 82 125 82 132 82 138 82 145 663 82 151 82 158 82 164 82 171 82 178 82 184 82 191 82 197 82 204 82 210 664 82 217 82 223 82 230 82 236 82 243 82 249 82 256 82 263 82 269 82 276 66s 82 282 82 289 82 295 82 302 82 308 82 31S 82 321 82 328 82 334 82 341 666 82 347 82 354 83 360 82 367 82 373 82 380 82 387 82 393 82 400 82 406 667 82 413 82 419 82 426 82 432 82 439 82 445 82 452 82 458 82 465 82 471 668 82 478 82 484 82 491 82 497 82 504 82 510 82 517 82 52J 82 530 82 536 669 82 543 82 549 82 556 82 562 82 569 82 575 82 582 82 s88 82 595 82 6or 670 82 607 82 614 82 620 82 627 82 633 82 640 82 646 82 6s3 82 659 82 666 671 82 672 82 679 82 685 82 692 82 698 82 70s 82 711 82 718 82 724 82 730 672 82 737 82 743 82 750 82 7S6 82 763 82 769 82 776 82 782 82 789 82 795 673 82 802 82 808 82 814 82 821 82 827 82 834 82 840 82 847 82 8S3 82 860 674 82 866 82 872 82 879 82 88s 82 892 82 898 82 90s 82 911 82 918 82 924 246 APPENDIX No. I 2 3 4 5 6 7 8 9 675 82 930 82 937 82 943 82 950 82 956 82 963 82 969 82 975 82 982 82 988 676 82 995 83 001 83 008 83 014 83 020 83 027 83 033 83 040 83 046 83 052 677 83 0S9 83 06s 83 072 83 078 83 085 83 091 83 097 83 104 83 no 83 117 678 83 123 83 129 83 136 83 142 83 149 83 155 83 161 83 168 83 174 83 181 679 83 187 83 193 83 200 83 206 83 213 83 219 83 22s 83 232 83 238 83 24s 680 83 251 83 257 83 264 83 270 83 276 83 283 83 289 83 296 83 302 83 308 681 83 31S 83 321 83 327 83 334 83 340 83 347 83 353 83 359 83 366 83 372 682 83378 83 385 83 391 83 398 83 404 83 410 83 417 83 423 83 429 83 436 683 83 442 83 448 83 455 83 461 83 467 83 474 83 480 83 487 83 493 83 499 684 83 506 83 512 83 518 83 52s 83 531 83 537 83 544 83 550 83 5S6 83 563 68s 83 569 83 575 83 582 83 588 83 594 83 601 83 607 83 613 83 620 83 626 686 83 632 83639 83 645 83651 83 6s8 83 664 83 670 83 677 83 683 83 689 687 83 696 83 702 83 708 83 715 83 721 83 727 83 734 83 740 83 746 83 753 688 83 759 83 765 83 771 83 778 83 784 83 790 83 797 83 803 83 809 83 816 689 83 822 83 828 83 835 83 841 83 847 83 853 83 860 83 866 83 872 83 879 690 83 885 83 891 83 897 83 904 83 910 83 916 83 923 83 929 83 935 83 942 691 83 948 83 954 83 960 83 967 83 973 83 979 83 98s 83 992 83 998 84 004 692 84 on 84 017 84 023 84 029 84 036 84 042 84 048 84 OSS 84 061 84 067 693 84 073 84 080 84 086 84 092 84 098 84 105 84 III 84 117 84 123 84 130 694 84 136 84 142 84 148 84 155 84 161 84 167 84 173 84 180 84 186 84 192 69s 84 198 84 20s 84 211 84 217 84 223 84 230 84 236 84 242 84 248 84 2SS 696 84 261 84 267 84 273 84 280 84 286 84 292 84 298 84 30s 84 311 84 317 697 84323 84 330 84336 84342 84348 84 354 84361 84367 84 373 84 379 698 84 386 84 392 84398 84 404 84 410 84417 84423 84 429 84435 84442 699 84 448 84454 84 460 84 466 84 473 84 479 84 48s 84 491 84497 84 504 700 84 sio 84 S16 84 522 84 528 84 535 84 541 84 547 84 553 84 559 84 566 701 84 572 84 578 84 584 84 590 84 597 84 603 84 609 84 61S 84 621 84 628 702 84 634 84 640 84 646 84 652 84 658 8466s 84 671 84677 84 683 84 689 703 84 696 84 702 84 708 84 714 84 720 84 726 84 733 84 739 84 745 84 751 704 84 757 84 763 84 770 84 776 84 782 84 788 84 794 84 800 84 807 84 813 70s 84 819 84 82s 84831 84 837 84 844 84 850 84 8s6 84 862 84 868 84 874 706 84 880 84 887 84 893 84 899 84 905 84 911 84 917 84 924 84 930 84936 707 84 942 84948 84954 84 960 84 967 84 973 84 979 84 98s 84 991 84 997 708 8S 003 85 009 8s 016 8s 022 8s 028 85 034 8s 040 8s 046 85 OS2 85 0S8 709 85 06s 85 071 85 077 85 083 8s 089 85 095 85 lOI 8s 107 8s 114 8s 120 710 85 126 85 132 85 138 85 144 8S 150 8S 156 85 163 8s 169 85 175 8s i8l 711 85 187 85 193 85 199 85 205 85 211 85 217 85 224 8S 230 8s 236 85 242 712 8s 248 85 254 8s 260 8s 266 85 272 85 278 85 28s 85 291 85 297 85 303 713 8S 309 8S 315 8s 321 85 327 85 333 85 339 8s 345 85 352 85 358 8S 364 714 85 370 85 376 8s 382 85 388 85 394 85 400 85 406 85 412 8s 418 8S 42s 71S 8S 431 85 437 85 443 85 449 85 455 8s 461 85 467 85 473 85 479 85485 716 85 491 85 497 85 503 85 509 85 516 85 522 85 528 85 534 85 540 8S 546 717 85 55 2 85 558 85 564 85 570 85 576 85 582 85 588 85 594 85 600 85 606 718 85 612 8s 618 85625 85 631 85 637 85 643 85 649 85 6SS 8s 661 8s 667 719 8S 673 8S 679 85 68s 85 691 85 697 8S 703 85 709 85 715 85 721 85 727 LOGARITHMS? OF NUMBERS 247 No. I 2 3 4 5 6 7 8 9 7J0 85 733 85 739 85 745 85 751 8s 757 85 763 8s 769 85 775 85 78r 8s 788 721 85 794 8s 800 8s 806 85 812 8s 818 85 824 85 830 8s 836 85 842 85 848 722 85 8S4 8s 860 85 866 8s 872 85 878 8s 884 8s 890 8s 896 85 902 85 908 723 85 914 85 920 85 926 85 932 85 938 85 944 85 950 85 956 8s 962 8s 968 724 8S 974 8s 980 8s 986 8s 992 8s 998 86 004 86 010 86 016 86 022 86 028 72s 86 034 86 040 86 046 86 052 86 058 86 064 86 070 86 076 86 082 86088 726 86 094 86 100 86 106 86 112 86 118 86 124 86 130 86 136 86 141 86 147 727 86 IS3 86 IS9 86 165 86 171 86 177 86 183 86 189 86 195 86 201 86 207 728 86 213 86 219 86 225 86 231 86 237 86 243 86 249 86 255 86 261 86 267 729 86 273 86 279 86 28s 86 291 86 297 86 303 86 308 86314 86 320 86326 730 86 332 86 338 86344 86 350 86356 86362 86368 86 374 86 380 86386 731 86 392 86 398 86 404 86 410 86 415 86 421 86 427 86 433 86 439 8644s 732 86451 86457 86 463 86 469 86475 86 481 86 487 86 493 86 499 86 504 733 86 sio 86 516 86 522 86 528 86 534 86 540 86 546 86 552 86558 86 564 734 86 570 86 576 86 581 86 587 86 593 86 599 86 60s 86 611 86 617 86623 73S 86 629 86635 86 641 86 646 86652 86658 86 664 86 670 86676 86682 736 86 688 86 694 86 700 86 70s 86 711 86717 86 723 86 729 86 735 86 741 737 86 747 86 753 86 759 86 764 86 770 86 776 86 782 86 788 86 794 86 800 738 86 806 86 812 86 817 86 823 86 829 86 83s 86 841 86 847 86 853 86 859 739 86 864 86 870 86 876 86 882 86 888 86 894 86 900 86 906 86 911 86 917 740 86 923 86 929 86935 86 941 86 947 86953 86 958 86 964 86 970 86 976 741 86 982 86 988 86 994 86 999 87 005 87 on 87 017 87 023 87 029 87 03S 742 87 040 87 046 87 052 87 058 87 064 87 070 87 07s 87 081 87 087 87 093 743 87 099 87 105 87 III 87 116 87 122 87 128 87 134 87 140 87 146 87 151 744 87 157 87 163 87 169 87 175 87 181 87 186 87 192 87 198 87 204 87 210 745 87 216 87 221 87 227 87 233 87 239 87 24s 87 251 87 2S6 87 262 87 268 746 87 274 87 280 87 286 87 291 87 297 87 303 87 309 87315 87320 87 326 747 87 332 87 338 87 344 87 349 87 355 87 361 87367 87 373 87 379 87 384 748 87 390 87 396 87 402 87 408 87 413 87 419 87 42s 87 431 87 437 87 442 749 87 448 87 454 87 460 87 466 87 471 87 477 87 483 87 489 87 495 87 soo 750 87 506 87 512 87 S18 87 523 87 529 87 535 87 541 87 547 87 552 87 5S8 751 87 564 87 570 87 576 87 S8i 87 587 87 593 87 599 87 604 87 610 87 616 752 87 622 87 628 87633 87 639 8764s 87 651 87 656 87 662 87 668 87 674 753 87 679 87 685 87 691 87 697 87 703 87 708 87 714 87 720 87 726 87 731 754 87 737 87 743 87 749 87 754 87 760 87 766 87 772 87 777 87 783 87 789 755 87 795 87 800 87 806 87 812 87 818 87 823 87 829 87 83s 87 841 87 846 756 87 852 87 858 87 864 87 869 87 875 87 881 87 887 87 892 87 898 87 904 757 87 910 87 915 87 921 87 927 87 933 87 938 87 944 87 950 87 955 87 961 7S8 87 967 87 973 87 978 87 984 87 990 87 996 88 001 88 007 88 013 88 018 759 88 024 88 030 88 036 88 041 88 047 88 053 88 058 88 064 88 070 88 076 760 88 081 88 087 88 093 88 098 88 104 88 no 88 116 88 121 88 127 88 133 761 88 138 88 144 88 150 88 is6 88 161 88 167 88 173 88 178 88 184 88 190 762 88 195 88 201 88 207 88 213 88 218 88 224 88 230 88235 88 241 88 247 763 88 252 88 258 88 264 88 270 88 275 88 281 88 287 88 292 88 298 88 304 764 88 309 88315 88 321 88 326 88 3i2 88 338 88 343 88 349 88355 88 3O0 248 APPENDIX No. I 2 3 4 5 6 7 8 9 76s 88 366 88 372 88 377 88383 88 389 88 395 88 400 88 406 88 412 88 417 766 88 423 88 429 88434 88 440 88 446 88 451 88 457 88 463 88 468 88 474 767 88 480 88 48s 88 491 88 497 88 502 88 508 88 S13 88 519 88 525 88 530 768 88 536 88 542 88 S47 88 553 88 559 88 564 88 570 88 576 88 581 88 587 769 88 593 88 598 88 604 88 610 88 61S 88 621 88 627 88 632 88 638 88 643 770 88 649 88 655 88 660 88 666 88 672 88 677 88 683 88 689 88 694 88 700 771 88 705 88 711 88 717 88 722 88 728 88 734 88 739 88 745 88 750 88 756 772 88 762 88 767 88773 88 779 88 784 88 790 88 795 88 801 88 807 88 812 773 88 818 88 824 88 829 88 835 88 840 88 846 88 852 88 857 88 863 88 868 774 88 874 88 880 88 88s 88 891 88 897 88 902 88 908 88 913 88 919 88 925 775 88 930 88 936 88 941 88 947 88 953 88 958 88 964 88 969 88 975 38 981 776 88 986 88 992 88 997 89 003 89 009 89 014 89 020 89 02s 89 031 89 037 Tn 89 042 89 048 89 053 89 059 89 064 89 070 89 076 89 081 89 087 89 092 778 89 098 89 104 89 109 89 115 89 120 89 126 89 131 89 137 89 143 89 148 779 89 154 89 159 89 165 89 170 89 176 89 182 89 187 89 193 89 198 89 204 780 89 209 89 215 89 221 89 226 89 232 89 237 89 243 89 248 89 254 89 260 781 89 265 89 271 89 276 89 282 89 287 89 293 89 298 89 304 89 310 89 315 782 89 321 89326 89332 89 337 89 343 89348 89 354 89 360 89365 89 371 783 89376 89382 89387 89393 89 398 89 404 89 409 89 41S 89 421 89 426 784 89 432 89437 89443 89 448 89 454 89459 89 46s 89 470 89476 89 481 78s 89487 89 492 89 498 89 S04 89 509 89 515 89 520 89 526 89 531 89 S37 786 89 542 89548 89 553 89 559 89 564 89 570 89 575 89 581 89 586 89 592 787 89 597 89 603 89 609 89 614 89 620 89625 89 631 89 636 89 642 89647 788 89653 89 658 89 664 89 669 89675 89 680 89 686 89 691 89 697 89 702 789 89 708 89 713 89 719 89 724 89 730 89 735 89 741 89 746 89 752 89 757 790 89 763 89 768 89 774 89 779 89 785 89 790 89 706 89 8or 89 807 89 812 791 89 818 89 823 89 829 89 834 89 840 89 845 89 851 89 856 89 862 89 867 792 89873 89 878 89 883 89 889 89 894 89 900 89 905 89 911 89 916 89 922 793 89 927 89 933 89 938 89 944 89 949 89 955 89 960 89 966 89 971 89 977 794 89 982 89 988 89 993 89 998 90 004 90 009 90 ors 90 020 90 026 90 031 795 90 037 90 042 90 048 90 053 90 059 90 064 90 069 90 075 90 080 90 086 796 90 091 90097 90 102 90 108 90 113 90 1 19 90 124 90 129 90 135 90 140 797 90 146 90 151 90 157 90 162 90 168 90 173 90 179 90 184 90 189 90 195 798 90 200 90 206 90 211 90 217 90 222 90 227 90 233 90 238 90 244 90 249 709 90 255 90 260 90 266 90 271 90 276 90 282 90 287 90 293 90 298 90 304 800 90 309 90 314 90 320 90 325 90 a I 90336 90 342 90 347 90 352 90 3S8 801 90 363 90369 90374 90 380 90 385 90 390 90 396 90 401 90 407 90 412 802 90 417 90423 90 428 90 434 90 439 9044s 90 450 90 455 90 461 90 466 803 90 472 90477 90 482 90488 90493 90 499 90 504 90 509 90 515 90 520 804 90 526 90 531 90 536 90 542 90 547 90 553 90 558 90 563 90 S69 90S74 80s 90 580 90 58s 90 590 90 596 90 601 90 607 90 612 90 617 90 623 90628 806 90634 90 639 9C644 90650 90655 90 660 90 666 90 671 90 677 90 682 807 90 687 90 693 90 698 90 703 90 709 90 714 90 720 90 725 90 730 90 736 808 90 741 90 747 90 752 90 757 90 763 90 768 90 773 90 779 90 784 90 789 809 90 795 90 800 90 806 90 8 n 90 816 90 822 90 827 90 832 90 838 90 843 LOGARITHMS OF NUMBERS 249 No. I 2 3 4 5 6 7 8 9 810 90 849 90 854 90 859 90 86s 90 870 90 875 90 881 90 886 90 891 90 897 811 90 902 90907 90 913 90 918 90 924 90 929 90 934 90 940 90 945 90950 812 90 956 90 961 90 966 90 972 90 977 90 982 90 988 90993 90 998 91 004 813 91 009 91 014 91 020 91 025 91 030 91 036 91 041 91 046 91 052 91 057 814 91 062 91 068 91 073 91 078 91 084 91 089 91 094 91 100 91 105 91 no 81S 91 116 91 121 91 126 91 132 91 137 91 142 91 148 91 153 91 158 91 164 816 91 169 91 174 91 180 91 185 91 190 91 196 91 201 91 206 91 212 91 217 817 9t 222 91 228 91 233 91 238 91 243 91 249 91 254 91 259 91 265 91 270 818 91 275 91 281 91 286 91 291 91 297 91 302 91 307 91 312 91 318 91 323 819 91 328 91 334 91 339 91 344 91 350 91 355 91 360 91 365 91 371 91 376 820 91 381 91 387 91 392 91 397 91 403 91 408 91 413 91 418 91 424 91 429 821 91 434 91 440 91 445 91 450 91 455 91 461 91 466 91 471 91 477 91 482 822 91 487 91 492 91 498 91 S03 91 508 91 514 91 519 91 524 91 529 91 535 823 91 540 91 545 91 551 91 5S6 91 561 91 566 91 572 91 577 91 582 91 587 824 91 593 91 598 91 603 91 609 91 614 91 619 91 624 91 630 91 635 91 640 82s 91 64s 91 651 91 656 91 661 91 666 91 672 91 677 91 682 91 687 91 693 826 91 698 91 703 91 709 91 714 91 719 91 724 91 730 91 735 91 740 91 745 827 91 751 91 756 91 761 91 766 91 772 91 777 91 782 91 787 91 793 91 798 828 91 803 91 808 91 814 91 819 91 824 91 829 91 834 91 840 91 84s 91 850 829 91 8S5 91 861 91 866 91 871 91 876 91 882 91 887 91 892 91 897 91 903 830 91 908 91 913 91 918 91 924 91 929 91 934 91 939 91 944 91 950 91 955 831 91 960 91 965 91 971 91 976 91 981 91 986 91 991 91 997 92 002 92 007 832 92 012 92 018 92 023 92 028 92 033 92 038 92 044 92 049 92 054 92 059 833 92 06s 92 070 92 075 92 080 92 08s 92 091 92 096 92 lOI 92 106 92 III 834 92 117 92 122 92 127 92 132 92 137 92 143 92 148 92 153 92 158 92 163 83 s 92 169 92 174 92 179 92 184 92 189 92 195 92 200 92 205 92 210 92 21S 836 92 221 92 226 92 231 92 236 92 241 92 247 92 252 92 257 92 262 92 267 837 92 273 92 278 92 283 92 288 92 293 92 298 92 304 92 309 92 314 92319 838 92 324 92 330 92 335 92 340 92 345 92 350 92 355 92 361 92 366 92 371 839 92 376 92 381 92 387 92 392 92 397 92 402 92 407 92 412 92 418 92 423 840 92 428 92 433 92 438 92 443 92 449 92 454 92 459 92 464 92 469 92 474 841 92 480 92 485 92 490 92 495 92 500 92 505 92 511 92 516 92 521 92 526 842 92 531 92 536 92 542 92 547 92 552 92 557 92 562 92 567 92 572 92 578 843 92 583 92 588 92 593 92 598 92 603 92 609 92 614 92 619 92 624 92 629 844 92 634 92 639 92 645 92 650 92 655 92 660 92 665 92 670 92 67s 92 681 84s 92 686 92 691 92 696 92 701 92 706 92 711 92 716 92 722 92 727 92 732 846 92 737 92 742 92 747 92 752 92 758 92 763 92 768 92 773 92 778 92 783 847 92 788 92 793 92 799 92 804 92 809 92 814 92 819 92 824 92 829 92 834 848 92 840 92 845 92 850 92 8S5 92 860 92 86s 92 870 92 87s 92 881 92 886 849 92 891 92 896 92 901 92 906 92 911 92 916 92 921 92 927 92 932 92 937 850 92 942 92 947 92 952 92 957 92 962 92 967 92 973 92 978 92 983 92 988 851 92 993 92 998 93 003 93 008 93 013 93 018 93 024 93 029 93 034 93 039 852 93 044 93 049 93 054 93 059 93 064 93 069 93 075 93 080 93 08s 93 090 853 93 095 93 100 93 105 93 no 93 115 93 120 93 125 93 131 93 136 93 141 854 93 146 93 151 93 156 93 161 93 166 93 171 93 176 93 181 93 186 93 192 250 APPENDIX No. I 2 3 4 5 6 7 8 9 85S 93 197 93 202 93 207 93 212 93 217 93 222 93 227 93 232 93 237 93 242 8s6 93 247 93 252 93 258 93 263 93 268 93 273 93 278 93 283 93 288 93 293 857 93 298 93 303 93 308 93 313 93 318 93 323 93 328 93 334 93 339 93 344 858 93 349 93 354 93 359 93 364 93 369 93 374 93 379 93 384 93 389 93 394 859 93 399 93 404 93 409 93 414 93 420 93 425 93 430 93 435 93 440 93 445 860 93 450 93 455 93 460 93 465 93 470 93 475 93 480 93 485 93 490 93 495 861 93 500 93 SOS 93 510 93 515 93 520 93 526 93 S3 I 93 536 93 541 93 546 862 93 551 93 556 93 561 93 566 93 571 93 576 93 581 93 586 93 591 93 596 863 93 601 93 606 93 611 93 616 93 621 93 626 93 631 93 636 93 641 93 646 864 93 651 93 656 93 661 93 666 93 671 93 676 93 682 93 687 93 692 93 697 86s 93 702 93 707 93 712 93 717 93 722 93 727 93 732 93 737 93 742 93 747 866 93 752 93 757 93 762 93 767 93 772 93 777 93 782 93 787 93 792 93 797 867 93 802 93 807 93 812 93 817 93 822 93 827 93 832 93 837 93 842 93 847 868 93 852 93 8s 7 93 862 93 867 93 872 93 877 93 882 93 887 93 892 93 897 869 93 902 93 907 93 912 93 917 93 922 93 927 93 932 93 937 93 942 93 947 870 93 952 93 957 93 962 93 967 93 972 93 977 93 982 93 987 93 992 93 997 871 94 002 94 007 94 012 94017 94 022 94 027 94 032 94 037 94 042 94 047 872 94 052 94 057 94 062 94 067 94 072 94 077 94 082 94 086 94 091 94 096 873 94 lOl 94 106 94 III 94 116 94 121 94 126 94 131 94 136 94 141 94 146 874 94 151 94 156 94 161 94 166 94 171 94 176 94 181 94 186 94 191 94 196 875 94 201 94 206 94 211 94 216 94 221 94 226 94 231 94 236 94 240 94 24s 876 94 250 94 255 94 260 94 26s 94 270 94 275 94 280 94 28s 94 290 94 295 877 94 300 94 30s 94 310 9431S 94 320 94 32s 94 330 94 335 94 340 94 345 878 94 349 94 354 94 3S9 94 364 94 369 94 374 94 379 94 384 94 389 94 394 879 94 399 94 404 94 409 94 414 94 419 94424 94 429 94 433 94 438 94 443 880 94 448 94 453 94 458 94 463 94 468 94 473 94 478 94 483 94488 94 493 881 94 498 94 503 94 507 94 512 94 S17 94 522 94 527 94 532 94 537 94 542 882 94 547 94 552 94 557 94 562 94 567 94571 94 576 94 S8i 94 586 94 591 883 94 596 94 601 94 606 94 611 94 6i6 94 621 94 626 94 630 94 635 94 640 884 94 645 94 650 94 655 94 660 94 665 94 670 94 675 94 680 94 685 94 689 88s 94 694 94 699 94 704 94 709 94 714 94 719 94 724 94 729 94 734 94 738 886 94 743 94 748 94 753 94 758 94 763 94 768 94 773 94 778 94 783 94 787 887 94 792 94 797 94 802 94 807 94 812 94 817 94 822 94 827 94 832 94 836 888 94 841 94 846 94 851 94 856 94 861 94 866 94 871 94 876 94 880 94 885 889 94 890 94 895 94 900 94 905 94 910 94 91S 94 919 94 924 94 929 94 934 890 94 939 94 944 94 949 94 954 94 959 94 963 94 968 94 973 94978 94 983 891 94 988 94 993 94 998 95 002 95 007 95 012 95 017 95 022 95 027 95 032 892 95 036 95 041 95 046 95 OS I 95 056 95 061 95 066 95 071 95 075 95 080 893 95 085 95 090 95 095 95 100 95 105 95 109 95 114 95 119 95 124 95 129 894 95 134 95 139 95 143 95 148 95 153 95 158 95 163 95 168 95 173 95 177 89s 95 182 95 187 95 192 95 197 95 202 95 207 95 211 95 216 95 221 95 226 896 95 231 95 236 95 240 95 245 95 250 95 255 95 260 95 265 95 270 95 274 897 95 279 95 284 95 289 95 294 95 299 95 303 95 308 95 313 95 318 95 323 898 95 328 95 332 95 337 95 342 95 347 95 352 95 357 95 361 95 366 95 371 899 95 376 95 381 95 386 95 390 95 395 95 400 95 405 95 410 95 415 95 419 LOGARITHMS OF NUMBERS 251 No. I 2 3 4 5 6 7 8 9 900 95 424 95 429 95 434 95 439 95 444 95 448 95 453 95 458 95 463 95 468 901 95 472 95 477 95 482 95 487 95 492 95 497 95 501 95 506 95 511 95 516 902 95 521 95 52s 95 530 95 535 95 540 95 545 95 550 95 554 95 559 95 564 903 95 569 95 574 95 578 95 583 95 588 95 593 95 598 95 602 95 607 95 612 904 95617 95 622 95 626 95 631 95 636 95 641 95 646 95 650 95 6s5 95 660 90s 95665 95 670 95 674 95 679 95 684 95 689 95 694 95 698 95 703 95 708 906 95 713 95 718 95 722 95 727 95 732 95 737 95 742 95 746 95 751 95 756 907 95 761 95 766 95 770 95 775 95 780 95 785 95 789 95 794 95 799 95 804 908 95 809 95 813 95 818 95 823 95 828 95 832 95 837 95 842 95 847 95 852 909 95 856 9S 861 95 866 95 871 95 875 95 880 95 885 95 890 95 895 95 899 910 95 904 95 909 95 914 95 918 95 923 95 928 95 933 95 938 95 942 95 947 911 95 952 95 957 95 961 95 966 95 971 95 976 95 980 95 985 95 990 95 995 912 95 999 96 004 96 009 96 014 96 019 96 023 96 028 96 033 96 038 96 042 913 96 047 96 052 96 057 96 061 96 066 96 071 96 076 96 080 96 085 96 090 914 96 095 96 099 96 104 96 109 96 114 96 118 96 123 96 128 96 133 96 137 91S 96 142 96 147 96 152 96 156 96 161 96 166 96 171 96 175 96 180 96 185 916 96 190 96 194 96 199 96 204 96 209 96 213 96 218 96 223 96 227 96 232 917 96 237 96 242 96 246 96 25 I 96 256 96 261 96 265 96 270 96 275 96 280 918 96 284 96 289 96 294 96 298 96 303 96 308 96 313 96 317 96 322 96 327 919 96 332 96 336 96 341 96 346 96 350 96 355 96 360 96 365 96 369 96374 920 96 379 96 384 96 388 96 393 96 398 96 402 96 407 96 412 96 417 96 421 921 96 426 96 431 96 435 96 440 96 445 96 450 96 454 96 459 96 464 96 468 922 96 473 96 478 96 483 96 487 96 492 96 497 96 501 96 506 96 511 96 515 923 96 520 96 525 96 530 96 534 96 539 96 544 96 548 96 553 96 558 96 562 924 96 567 96 S72 96 577 96 S8l 96 586 96 591 96 595 96 600 96 605 96 609 92s 96 614 96 619 96 624 96 628 96 633 96 638 96 642 96 647 96 652 96 656 926 96 661 96 666 96 670 96 675 96 680 96 685 96 689 96 694 96 699 96 703 927 96 708 96 713 96 717 96 722 96 727 96 731 96 736 96 741 96 745 96 750 928 96 755 96 759 96 764 96 769 96 774 96 778 96 783 96 788 96 792 96 797 929 96 802 96 806 96 811 96 816 96 820 96 825 96 830 96 834 96 839 96 844 930 96 848 96 853 96 858 96 862 96 867 96 872 96 876 96 881 96 886 96 890 931 96 895 96 900 96 904 96 909 96 914 96 918 96 923 96 928 96 932 96 937 932 96 942 96 946 96 951 96 956 96 960 96 965 96 970 96 974 96 979 96 984 933 96 988 96 993 96 997 97 002 97 007 97 on 97 016 97 021 97 025 97 030 934 97 035 97 039 97 044 97 049 97 053 97 058 97 063 97 067 97 072 97 077 935 97 081 97 086 97 090 97 095 97 100 97 104 97 109 97 114 97 118 97 123 936 97 128 97 132 97 137 97 142 97 146 97 151 97 155 97 160 97 165 97 169 937 97 174 97 179 97 183 97 188 97 192 97 197 97 202 97 206 97 211 97 216 938 97 220 97 225 97 230 97 234 97 239 97 243 97 248 97 253 97 257 97 262 939 97 267 97 271 97 276 97 280 97 285 97 290 97 294 97 299 97 304 97 308 940 97313 97 317 97 322 97 327 97 331 97 336 97 340 97 345 97 350 97 354 941 97 3S9 97 364 97 368 97 373 97 377 97 382 97387 97 391 97 396 97 400 942 97 405 97 410 97 414 97 419 97 424 97 428 97 433 97 437 97 442 97 447 943 97 45 1 97 456 97 460 97 46s 97 470 97 474 97 479 97 483 97 488 97 493 944 97 497 97 502 97 506 97 511 97 516 97 520 97 525 97 529 97 534 97 539 252 APPENDIX No. I 2 3 4 5 6 7 8 9 945 97 543 97 548 97 552 97 557 97 562 97 566 97 571 97 575 97 580 97 585 946 97 589 97 594 97 598 97 603 97 607 97 612 97 617 97 621 97 626 97 630 947 97 635 97 640 97 644 97 649 97 653 97 658 97 663 97 667 97 672 97 676 948 97 681 97 685 97 690 97 695 97 699 97 704 97 708 97 713 97 717 97 722 949 97 727 97 731 97 736 97 740 97 745 97 749 97 754 97 759 97 763 9776C 950 97 772 97 777 97 782 97 786 97 791 97 795 97 800 97 804 97 809 97 813 951 97 818 97 823 97 827 97 832 97 836 97 841 97 845 97 850 97 855 97 859 952 97 864 97 868 97 873 97 877 97 882 97 886 97 891 97 896 97 900 97 905 953 97 909 97 914 97 918 97 923 97 928 97 932 97 937 97 941 97 946 97 950 9S4 97 955 97 959 97 964 97 968 97 973 97 978 97 982 97 987 97 991 97 996 955 98 000 98 005 98 009 98 014 98 019 98 023 98 028 98 032 98 037 98 041 956 98 046 98 050 98 055 98 059 98 064 98 068 98 073 98 078 98 082 98 087 957 98 091 98 096 98 100 98 105 98 109 98 114 98 118 98 123 98 127 98 132 958 98 137 98 141 98 146 98 150 98 155 98 159 98 164 98 168 98 173 98 177 959 98 182 98 186 98 191 98 195 98 200 98 204 98 209 98 214 98 218 98 223 960 98 227 98 232 98 236 98 241 98 245 98 250 98 254 98 259 98 263 98 268 961 98 272 08 277 98 281 98 286 98 290 98 295 98 299 98 304 98 308 98 313 962 98 318 98 322 98 327 98 331 98 336 98 340 98 345 98 349 98354 98358 963 98 363 98367 98 372 98 376 98 381 98 38s 98 390 98 394 98 399 98 403 964 98 408 98 412 98 417 98 421 98 426 98 430 98 435 98 439 98 444 98 448 96s 98 453 98 457 98 462 98 466 98 471 98 475 98 480 98 484 98 489 98 493 966 98 498 98 502 98 507 98 511 98 516 98 520 98 525 98 529 98 534 98 538 967 98 543 98 547 98 552 98 556 98 S6i 98 56s 98 570 98 574 98 579 98 583 968 98 S88 98 592 98 597 98 601 98 605 98 610 98 614 98 619 98 623 98 62S 969 98 632 98 637 98 641 98 646 98 650 98 655 98 659 98 664 98 668 98673 970 98 677 98 682 98 686 98 691 98 695 98 700 98 704 98 709 98 713 98 717 971 98 722 98 726 98 731 98 735 98 740 98 744 98 749 98 753 98 758 98 762 972 98 767 98 771 98 776 98 780 98 784 98 789 98 793 98 798 98 802 98 807 973 98 811 98 816 98 820 98 82s 98 829 98 834 98 838 98 843 98 847 98 851 974 98 856 98 860 98 865 98 869 98 874 98 878 98 883 98 887 98 892 98 896 975 98 900 98 90s 98 909 98 914 98 918 98 923 98 927 98 932 98 936 98 941 976 98 945 98 949 98 954 98 958 98 963 98 967 98 972 98 976 98 981 98985 977 98 989 98 994 98 998 99 003 99 007 99 012 99 016 99 021 99 025 99 029 978 99 034 99 038 99 043 99 047 99 052 99 056 99 061 99 065 99 069 99 074 979 99 078 99 083 99 087 99 092 99 096 99 100 99 105 99 109 99 114 99 118 980 99 123 09 127 99 131 99 136 99 140 99 145 99 149 99 154 99 158 99 162 981 99 167 99 171 99 176 99 180 99 185 99 189 99 193 99 198 99 202 99 207 982 99 211 99 216 99 2 20 99 224 99 229 99 233 99 238 99 242 99 247 99 251 983 99 255 99 260 99 264 99 269 99 273 99 277 99 282 99 286 99 291 99 295 984 99 300 99 304 99308 99 313 99 317 99 322 99 326 99 330 99 335 99 339 98s 99 344 99 348 99 352 99 357 99361 99 366 99 37c 99 374 99 379 99383 986 99388 99 392 99 396 99 401 99 405 99410 99 414 99 419 99 423 99427 987 99432 99 436 99 441 99 445 99 449 99 454 99 458 99463 99 467 99 471 988 99 476 99 480 99 484 99 489 99 493 99 498 99 502 99 506 99 5 I I 99 SIS 989 99 5 20 99 524 99 528 99 533 99 537 99 542 99 546 99 550 99 555 99 SS9 LOGARITHMS OP NUMBERS 253 No. I 2 3 4 5 6 7 8 9 990 99 564 99 568 99 572 99 577 99 581 99 585 99 590 99 594 99 599 99 603 991 99 607 99 612 99 616 99 621 9962s 99629 99 634 99638 99 642 99 647 992 99651 99 656 99 660 99664 99 669 99 673 99 677 99 682 99 686 99 691 993 99 695 99 699 99 704 99 708 99 712 99 717 99 721 99 726 99 730 99 734 994 99 739 99 743 99 747 99 752 99 756 99 760 99 76s 99 769 99 774 99 778 995 99 782 99 787 99 791 99 795 99 800 99 804 99 808 99 813 99 817 99 822 996 99 826 99 830 99 83s 99 839 99 843 99 848 99 852 99 856 99 861 99 865 997 99 870 99 874 99 878 99883 99 887 99 891 99 896 99 900 99 904 99 909 998 99 913 99 917 99922 99 926 99 930 99 935 99 939 99 944 99 948 99 952 999 99 957 99 961 99965 99 970 99 974 99 978 99 983 99987 99 991 99 996 1000 00 000 00 004 00 009 00 013 00 017 00 022 00 026 00 030 00 035 00 039 254 APPENDIX O HH H < H t) p^ ^ O UJ U H e^ »— 1 u w Q X H O KH O -< ■p Q t—l o U H w < M < 0-i hJ ^^ cu H < b < W H O P^ W D H ;z: < HH p ^ P o PLH 1^ o u fo » o r^ t vO t r^ o m m o r^ ^-^ lO o r^ a CO 00 •0 o •& eo on t o 00 00 o t 00 >o r^ O fn m CO ■* 00 M ■^ •^ on r» o. ro r^ o O M M M M -■ " -^ " "■ M rri ro •* •<1- f " M O M ~ t^ oo >fl f Oi Tt t •* fO o Oi ro o -c t vO r^ O O IN r^ CO ^ t <»> ■o Tt o M 00 •* m a t 00 •^ o M c« o ro >ri o fO m o. fo o a ■O ro o M p< r^ o a o ■o o 1-0 00 r^ r« ro fn o ►1 •>t 'O 00 o M t 00 I^ o f-l ro VO in lo o ■o 00 00 t^ p» fo o -t t O r^ m M m >o m Ci 1^ o f^ M 00 t Oi r^ -r> ■o r^ o> r< -n o 00 m (^ t ^ o. M ■^ O a. r< M- r^ o fO -O Oi r4 o fi -o o ro ■O o N M hI o q - «■ "■ "■ " "■ fO fn ro ro •>t ■^ " M -q H ■q o o r^ on t^ IN o. m vO t rs m o Tf r^ o nn M- O r^ •»*• fO r~ vO ro r^ M- ri M ^ vO on N o M m Oi M ■* t •o on on »o r^ ■o O o t^ ?=,§ O ro 00 O 00 o. Oi t^ Tl- o r^ O 00 M- •t M • t r^ o m on •rl- r^ o q q " " - "■ " " " "■ ro rO ro •* ■* ^ •* in " in vq fo o in 00 o. r^ o n ■O o o o o o »^ Tl- r^ ro 00 on r- rt r^ o ^ C^ vO n (H r^ Tf o O o O r^ O 00 m 00 -o 00 fi o o N 00 00 Tt ''I 00 ro t ro N vO -t n Tt 00 vO 00 Oi t ■^ o. a a o M- r^ m ro t 1-0 r^ \r> f-1 o Oi o. 00 00 O. o ■rf r~ Oi ro O O ■f o Tt O vO f-1 U1 t^ o. o M- vO 00 f^ iri r^ o •^ •O O M ro ■O O. M o O q q q - " M " "■ " M " "■ ro ro M ro ro 't •<*■ •* •* in O) ui o vO Ov 00 fn ■^ r^ Tt ro m ro ^ M ro o lO 65 r-1 o a o 00 o 'I- o m ro -o o on r^ M 00 00 f^ M- r^ ■* N a o CO 00 m m O o O ro t^ (S r^ o 00 ■<+ 1- » 00 •* Tf m m ro 00 M- W5 -C r^ o •o fo •* 00 ■* fO o O r^ M o o ro O CO O ro «- O 1/1 r^ fO » •O m 1^ ro O 00 00 r^ <5 >o r^ r^ 00 o "^ t ^ r^ ^ o •* >o r* o. ro >o 00 O N M- O 00 O q q q "■ "■ "■ " ro ro ro ro »-i ro lO o ■* m en o 00 rO N N 00 r^ O ro O 00 Oi m ■* 0^ 1^ <1 r^ o o vO o in M fO o o f-^ o t M- m -J- o o 00 O. r^ ri 00 t^ 1/1 ■^ 00 en 00 Oi O ro O ro ■O on w a o o fO on •tt M t r^ O. 00 00 in O r^ r^ o ■* o t 00 vO O f> 1- o m o vO r< 00 O Ht ■o r^ o o w m f « r^ on o ro in O on O H o M M q q o q q q M « *-* "■ - - "■ "■ "■ r< " ro ro ro *-* m m 00 vO ■rt ro on O o o r^ M f^ o Oi ■o t ■* O o Oi O ■O o o ■O on m fO -t 00 00 ■n r-- on o £5: o o f) « N O Ov t^ o o •^ o o. o> O ro ro ■o o >n O ■O O 00 o t o ro ro r^ o O o M n m t m O 00 o. o M t vO 00 o 1- t~ O « ro T^ «n ■^ t^ 00 Oi o n t o r^ o. o ro ■^ m O o M M o M q q q " M " " M M r^ PI n ri o O " J^ t m -o r^ 00 ex o ro ■<»■ m o r^ 00 o o « ro t- 14 «^ w M « •"I r" ^^* •I <-" (1 n <1 « ^ COMPOUND INTEREST AND OTHER COMPUTATIONS 255 « „ 0. 00 r^ t vO fO 0. . on t on 00 00 fn ■0 PI PO m vO >o PO PI n <-1 m in r^ f«i ir r^ a 00 "1 r^ 1- ? on ■* 00 m f^ 00 -T 00 f-: on t i^ r^ r^ -0 m m 00 (N ir m 00 -^ r^ 0. en PO Ov 00 r^ 1^ t vO t 0. o> m f»i -t 1^ -g 0. o> m a on 00 m >o -0 ■* ■ m 00 00 -^ PI r^ r^ Ov PI 00 -0 'O ■o t^ 06 m m f^ m Oi m IT PI 1^ 00 0. r* Ov •0 00 PO ■* •n f^ in 00 m N 0. Nr> a t^ PO 00 Cl r>. fO r- m •^ *-" ro i»> ■* ■* in -0 1^ 00 00 » ►I in 1- 00 m Ov PO PI PI ■ 00 in ro 00 t> t rn Tt t N a Oi 0. r^ M rf Tf m o. PO •* T t^ 00 •^ PO 00 PO 'O tn •t fO M- m 0. fn >n N N 00 m f ■o 10 pn c> -0 t^ t m •q- m 0. ■o ■* 00 •+ 0. a ■<> rt o t 0. 00 00 a m o PO ■* PI Tf ^ Ov t fO ui Tt ro r^ on o> 00 1^ f> ■o m r^ 0. t^ •o •^ 0. r^ en r^ on fn 00 Tf -n 00 t r- t ^t t "0 t~ t^ r» 00 00 Oi '-' M M '^ in 'J- ^ m >o r^ l^ 00 a •t 00 PI r- PO Ov ^ p< PI M M M M M " " N N N M w CI ^ M M N M N M N M PI PI fO PO PO f) • t^ on t a M- o> m ■^ M PI i-i 00 fO 00 m Ov 00 PO 0. -0 00 rr -0 00 ^ m t^ 00 00 on r> t ro ■^ 00 m m ■n- r^ PI PO m fn PO t^ PO Ov • o. o •-- M (N M rn ro •^ •* m m ■o vO Ov N vO Ov Tf 00 PO Ov m PI M >-t M M " ►-» " " " " " M fj N N n M PI N N P) N PI PI PI N P) ro fo PO ■ 00 r^ t^ 0. r^ Ov Ov rr 00 PI Ov t- o> fO M- O r^ on m >o m r^ 1^ 00 00 r^ ^O 00 t^ SO 00 tn M Ifl T 00 00 m M fO fO r^ r^ t T 00 PI 00 PO 00 n Oi rf ■- r^ -r m t 0, t-f 00 t^ in 00 ^ p< in "1 "I- PO r» M 0, 10 n m m n 00 m r> Ov 00 00 PO vJO Ov ■«♦• >o » 10 00 M »»■ t^ fn m •0 f-1 r^ -t 00 N vO (?. PO 00 Ov PI 00 o NO l^ r* r- 00 00 00 0\ 0. on ►-I -t ro ifl Ifl t fn m -0 t vO PI ■T -I- Ov (^ f^ PO m vO ifl t Oi 0. 00 00 00 0. 'O m PO 0. 00 00 PI f 00 00 r^ 0. 00 r^ r^ Oi t r^ ^n I^ t^ P5 t •* •* 0. ■o M- Ov r^ 00 M a rn ■* o> 00 t^ r^ 00 00 fn N m PI M- Ov PI ^ m ■0 Ov 't- o> fo t 00 fn C^ -t r^ ■^ 00 -o tT rn PO PO ■* in PO PI m Tt t 00 «t tn 0. fO 00 ro 00 "1 -0 00 •* Oi m on ■* t^ -o •^ PO PO m Ov t • 1^ 00 t^ vO r^ IH •^ r^ Ov r^ m 00 M- Ov c> rt f^ on en 00 rt 1^ r^ on M Oi on t^ a Ov 0. n tn N vO fi m a I- -0 on t r^ n •^ vO PI rt Ov Tf <:> m l») vO m ^ a on -0 vO Tt M- m •* on 00 >-i Ov 0. a r^ r^ a t rn Ov ro o> 00 o> PO 00 t^ t <0 on 00 •>»• on a Ol vO t fn m rn m m t Tt r^ on 00 ►H r^ on » 1^ CO N -t on ■^ ■o on M tT 1^ 0. PO -0 00 ro '? •o •0 ■o " t^ M r- 00 M 00 M 00 Ov PI fi PO in pi 00 PO PO 00 00 M 00 0. vn 00 ■I- w m t^ 0. o> r^ •^ 00 t Ov rn 00 00 vO Ov •* IT, t^ fn 00 00 "1- PO 00 S so >o 00 n 00 ■o 00 •0 t^ N rn r^ 00 PO ■0 00 M- -0 PO on n t TT r^ r^ N -0 00 r^ >o N t m ■O ■o PI an t 00 rn Ov vO I^ 00 r^ I^ fn 00 rt PO vO so VO Ifl 00 •» r^ •t 00 ifl O t 00 r^ 00 m 0. T 00 •<}■ » ■o Ov VO 0, VO a 00 ") •* r^ 00 fO M- r^ 00 m t 00 0. PI t '? '? H rO '? fO •* ■<1- •* Tt •* ■^ M " " M 1/5 - 00 Ov ci PO pi pi in PI PI 10 >o t« 00 Ot ro t 10 00 ©. <*) t vO 00 0. m C9 « n « « to n ro ro n n fj n fO ro ■<1- ■» •f t t •t T T •«■ 10 ■o f- r* 90 00 On Ov 256 APPENDIX 6? O O 0\ 00 0\ O ^ M <5 M- O O O O lO Oi O 00 00 i-i r>. Ov ■^ Ov O f^ O lO t^ ro 0\ •O r^ Ov O PO 00 00 PO '^ Oi Oi O* lO fO t^ p^ ^ O 00 lO u^ ro r^ Oi ro O rO Tf r^ Ov f-O 0> oo P< (^OOOO MiOOO r^ 00 -^ 00 00 0\ OO PO PO O 'O f^^^p^^^^^ lo vO -<• (S Tt- N ro ■o t O 00 o> p) o M o> PI o t -t a PI t o o •i- t N <5 vO t^ r^ Oi Oi o vO lo -o lo o lO 00 Tf o> >o a on • M- PI r^ Oi o vO M a f^ >/) •o lO o •+ ro 00 r^ 00 o O Oi on o o Oi p) Oi 1/1 o PI vO -+ t^ lo o o r^ 00 O lO lo a 00 p) vO vO ro m lO vn lo PI o vO >0 lO M ^ o 1^ lO N Oi 00 r^ on Oi PI •O 00 r^ PI 00 ui m M M M M " p; f; M M <> r^ r^ 00 Oi o p< tH pi PJ pi pi Ov pi O pi rr> py po pi pi pi 4 ro o t « r^ 1- P) ~i o •^ K-l r^ M p« VO ■^ o o a » vO o. 00 vO •* o 'I- ir> ^ 00 1^ rO o o PO 00 rr o o> ■5J- ^^ on O o Oi vO o ■o •* p) vO ■* o. a Oi o ^ 00 f^ r^ vO ■* vO ro r^ o Oi PO M o 00 o a 0fl f^ o ^ 00 00 t o o '-' *-* c^ ro rO ■* ■* tn o vO t^ 00 a o *-f p) ro TJ- IT) o r- 00 o •-• (N ^ m M M PI " p< PJ P< P) P) P) PI ro P5 P5 PO ro o o N 00 M 00 -o O 00 ■^ r^ a ■^ -o PO 00 00 i/> O t^ 00 N o o t^ PO t 00 Oi Oi M- M- rO VO PI o ■o a fO 00 00 ■* vO o> 00 PI o w "> vO rt PO I^ >* 00 o t vO o ro o O o IT) &x ^o 00 00 o t^ o. ro 1- o o >(■> o. o> o 00 00 HH o. r- r-) 00 t PI vO t o> ■o 1/) PO o Oi VO o r^ m o 00 Oi rr ro lil fO on CO Tl- T^ 00 vO vO M o 00 fO o o ro o 1^ ■+ o f> r^ vC^ vO vO vO r* 00 Ov o o » "■ "■ "■ ro t lO "° <5 t^ 00 00 Oi o Pi Pi Pi pi pi "? PI pi 00 O. pi o M r^ VO 't lo o rn r^ o f^ n v£> 1/1 O. <^ t^ 00 lO o a. m O PI PI 00 00 00 t^ r- Ov cj o f^ ro t^ O -^ 00 PJ N ro PO O MOO M PO'+l'^O M r-t^O OirjoOOioM ro 00 O^OOO "^OOO lOOv'-' 0\00 "oOi-^oo t^oo o w^0\0 Ov-C '^CN'O "^lO O Ov"-" POOO u-irO'^r^M ■■■-- - POO""*" 0000 OiOOO wsOOO 10'^'^0l»0*-' 0\0> MTfOOTj-OOt^-'t- Tj-Tfioinoo t-i^ O 00 PO Ki M PI -* lO vO r^ r^ 't IT) '■) a 00 0\ Oi vO PO PO 1/1 t-i o 00 in PO o 00 vO t fO <> o PI P< PO Tf lO vO r^ « PI PI PI PI PI PI PI M PI O r^ 00 00 M Tt PO O »0 M o t- >o ^- 00 o tH fO 00 00 r^ r^ ■+ ■* -o o O PI XT, -t 1/1 H^ Ov c>4 o vO 1/1 fO o ro vO fO r^ o I^ -o 1^ ro vO t P-) -o •cf r^ tH Ov r^ rO O rO vO o M- ro o Oi o 00 Oi a 00 PI o 00 o. Ov 00 M- r^ Ov o vO t P5 •+ K1 00 Tt PO vO o vO p-1 PO vO r^ o vO o •T 00 vD o O 1/1 o VI PO a 00 PI ro rO PO rf T 1/1 t/1 o -O 1^ 00 00 o< Oi o o ^-« PI PO « « " " " " " " " - ■^ " -" " " " " " " " " " ~ " pi PI pi pi PI PI o t -o Ov lO PO M- on PO fO -o Tf lO -•o o fO 00 o PI i/> 1/0 PI vO PO ro t 1/1 o o ^ PO Ov Tf Ov 0\ t o 00 r^ r^ o V V. vO Ov •H ro 00 Ov o vO ro ^ on Ov o. PI 00 o PO PI vO o 1/1 vO ^ on I^ vO 00 t on o Ov Ov O vO 00 PI -t o vfl •i- o PO vO m Ov Cv r^ PO ro vO ro 1/1 M- t ^ lO O Ov O r^ PO m Ov tf o o o o •+ o vO t 1- t vO O o VO 00 PO o o o PI PI ro PO ■i- -t 1/) m vO vO 00 00 Ov Ov o o " " " "" " " " " " " •H " " " " PI PI PI PI (H P0'^»^0 r^oO a> M PO'l-lOO t--oo 0\ « fO tT lO O n fi OI ri M COMPOUND INTEREST AND OTHER COMPUTATIONS 257 ■ 00 00 00 OsONii-lO ^Ml^r^Mrv^oOrofO'^ ^^^)^^2l^?^7}% r* »r OD 00 00 »-A W i:^ "^ <*/! -^]- ■— i~ '-/ " ■ v -V ■ ' ■ ' ■ ' • ~ ,^ -. T Z. _*. z*. 0\ ^^ O « r^ N Oi O M li^ ) CO li^ TfMf-OlOOONO ^ooK^ttooo-^^^o oo'oiciiooo'vdo«^c>'t^ 5.^^'^j;riC^fi'£..'i iAsdosdt;.r^««dd 22::2:::?2"i?^^ "S^T^^SSo^^^s oo0^t^'^^»^^^'^O^l/^ i-iino^ooO> 5§5!«|"'~°^'^"- ??^ir5°E:«Lt:'.2':2 2°^g:C^^,'2J:?^ 0**^00 O MOO -^ ir; \0 fO O 00 't -^ r^ 0\ "^ sO -^ "^ "^ o 00 Tf 'V) p/^ sO O O »0 "-I tJ- Tj- Tf U^ 10 10 t^l^r^000000C>O\OO ^-^00'^OcOOn'*^0^ MTfO'-«irioo'l--tN_ ._ «„ oor*»-tOt-'t^ooOO*0\ Ti-o»'ioo^ ^ f^ 1/1 >o V> r^ Ov n Oi t 0. ^ i^ •^ r^ nn 00 M- f^ fO -1- 00 ►-' 1J1 t r^ t-i r^ 1-0 M n 10 •H N (N ro ^ 10 vO lOO TtOvii^vO »00^ (N UTj -O r^ '^ -rf -f fS p*5 "O O 0> f^ 0\ 10 00 rt 0\ ''I- ro 10 V-. SLJ uu w^ "»---- - « ^ -, ro M 10 O^ O PO CO OOt'-'^i'l^-'OOOOiO'^ sO'^Of^O'^jr-'OfJfO Jo^So?'?o^coooo ?^'^2'o:i'+'-2.^ ^^S^::!^^^:?? M ro 10 -O t^ Oi •-; ro ro (^ PO fO fO "^ t^ i>^ O O O »-« (sj 10 06 ro 00 'I- in ^ 00 o> Tt 00 00 t ^ •0 00 0. ^O If) o> c* 30 ->■ f t^ 'O ^ o> o> r-1 MJ t M- r^ 00 i-H n ro ^ -o 00 Oi N r^ ifl f~* (XJ (•^ L/J 1'/ •"■ *■■ ' ' ^ *^ N ro PO fO ro ro r^ r^ ro ro •q- ■* f ■* rr t m 10 vO 1^ o> " fO 10 00 " Pi n a Ov 1^ -t <. ro 00 T^ M o> re ^ t^ -0 VO ^ r^ 00 '^ a nv 00 t^ ro 10 a t 0. -f ^ 00 a M 0\ t o -0 vn ■<■> 00 Oi -i- 10 1/3 t^ 00 00 ro • PO <^ vO 00 •0 ro 00 .n r- -1- ^ ts N 00 N t •*• on f-> I-l O 00 Tt OS 10 a CN Ov T 'O vO 00 r^ 00 NO 00 Ov 00 00 00 1-0 ^ 0\ ^ O. Oi l») 00 r^ 00 Oi on 00 r^ M ro rn ^ in in vO r^ 00 00 a w n N rr) •* lo vO r^ 00 ^ vO *"• •^ N N (S N N r* N N ^ " " n fo r^ n m r«5 ro r^ r<5 fO ■>!• -0 t^ 00 " i*> w rr t -0 r^ 00 Oi \r> fO f*3 PO r*5 ro m PO 1^ t •a- "^ ■* •^ •* ■* ■0 P< 00 C< Irt ro rf- 'O lO O 00 w Oi m in ^ \0 'rt t^ (N r^ (-• O 00 l/l Oi 00 U^'HOOiy^i^M'-' OO O«OiO*Oi0000000000 Ooo OiOCOO •^'C *^ ^ 0\ -rf IT) Oi ^N^C^* 0^0 ooo i/^i^f^ t-f ooo o o -^ t^ t^ 00 O 0> ri ro 00 O 00 fO wOOOOOOOOO oooooooooo oooooooooo ^ -tM i/^M r^*o »/)•-* M ^ i»> "1 ro n "* f<1 so lO •t r^ ro t a 00 Ov 1^ ^ vO m Oi -t -t 00 r^ o t^ a IT) r^ vO lo •t -t Ifl o 00 r^ m fO O. r^ m f^ M o. OS Oi o. 00 00 00 00 00 ooooooooo roOioo tor-00 O cs ro't' ►-- Oi-rfO roMOO mOOO ~ " . 6666666666 I^OO t^^oo O\t^t^oo f*i ^mt'StN'O f*3O^'-<00 W 00 r>.0»'1-N f^r^PO'^io 0*0 (N 0»nO <^Ooo»0 't 0*00 i/jioioiow^toto odoodddddo r^ r^ O Oi lo . o o o « . . ^ ro'-t<^oor^O\io O •-< N ly^ in t^ O roo ooo Tj-N Ooo t^ioro OvO^OiOiOiOOOOOOOO ddddddddd O ■^ 00 re N ■^ O Ov ro r* rooi tOoOO-^"-! '-'^ O -^00 ror^'^oO rt-OO CN'^ O-t^oO •-< r*iy^u^QO ro •N 0> O 't OOOOOOOOOO ^ Tt -t 00 t>. i/^ Or^iOOONMrowM O OvoOio^H -^-^M Tf 0000 04 O^O^t-iO -^^ M 100*^0 >-< lOO*/) OOO -^rO"-^ OOOt-iO Oi0>0>0>0\0o00000 '^666666666 rt Tj- O O Ov -^ rjO i/jOiO r^'-'oo r^-O* t^P* O O rOOOO lOt^M OO NOO '^Ot-.'^'-i 0\ TfM (-1 O\oo t^lrtTff^M 000000 r»t^r>-r-t-*t^i^ 6666666666 00 O>00 h-0<^0004l^ looo ri M OfOt^t^^Ov ■^r^Oooooo Oi'ttooo MO "N r^roQ»TtO»<^ "^t ooo i^O»"i"0 0>0\oo0 loio-^ O O\00t^iO'^(*5M iH O r^OOOOOOOOO d d d o d d d d d d r^ioO\roroOiO*« ^ N^ MwO-^ONOwM oor^to-^M M Ooor- O>OlO^O^O^O^Ot0000 ddddddddd 00 O 't O On t^ M UD O fO O 0\ 00 O 'I" ro OOOOOOOOOO t>. 0» 00 ro Tt- O t "^ O O m ' r*5 (N Tf lo r^ m t^ N 00 O to On 00 O On O ro r^r>.t>-r-000000 d d d d d d d d d d ^ f^O roooooo roii^Ot roO fOCN Ooo On-^O rt ^ CC Ttt^rfiooO O lOO •-" Nr^t^"-" OvM O T}-rfior-i-< r>.r<)M r^iii^pO"-^ Ovooo io*t 00 t>-0 vnroN t-i oo» O»O»O>OiO»O»O\O»00 ddddddddd M (N »O00 '^Oit-OO r- r^ ^ i-^OOOvOiOnOvOn 00r-O»/^*t.TfTt-0 fOOO 00 00 r^ Tt O r* O r- O r*5 00 -^ O On O (N -^ M ro fO »/^ ' OOl^O»OTfro'N •-• O 0» 6666666666 h^ lO -^ O. ■^ u^ "-I (^ O M o * irj lO 00 O Tf •-« , 'Tf O t^ '^ r^ 000«ii<-"NMro^ Onoo i^O in'^ro<^ t-" OlC^ONOlO^C^ONO^O^ ddddddddd O f^^r^ MOv'-'t-r-Oi 00'-'MTtor- O 0*00 t^ ^ ^ u-jrfroN O\00000O000000Q00000 dddddddddd •-• o> PO tt o Tt" 0» On *0 »-i 'too i^Ooo r^roioN tfooi^oo to i^t O '^t 00 ro (N 't O Os r- O »« ■t r^ rr VT) t^ -O lO t^ On 00 00 00 r- dddddddddd o « N ro^*oot^oo o» -^ "t lo O t^ 00 0\ O •-< M fO t lO O N 0» C* 0* N CI COMPOUND INTEREST AND OTHER COMPUTATIONS 259 ? ir: 00 t r^ >« o. r^ vO c -c ■^ so 0. r^ 00 n Os n r^ >o DO I/) 0\ ~5 r^ a fo 00 00 f^ Tf r^ 0. t^ m m ^ SO r^ n •» n ^ •* • 00 00 00 o> 1- r^ Os n 00 -t n m 00 00 t 00 Oi t^ vO m ■^ fo 00 r^ in in rf m o> so n 00 -0 in 10 Tf -1- •:»■ T •t t t t TJ- T <~o rn "5 n in n m m n p-i •-• «-" ►"" *-■ d d d d d d 6 d d d d 6 d d d d d d d d d d d '^ d d d d d d d Ov r^ fO r^ n Oi fo ^ vO t 00 m r^ Os fn M r^ 00 C\ ~) r^ vO Oi t t^ 00 r^ fn ■* 00 so >o t^ N ■t Os 10 f^ 00 00 ro 00 ^ 00 n 00 r^ 00 t^ t M r^ 0. •* Os N 1^ 'f "5 00 t o> so o> m n in 00 m t^ so Os fn « c Oi 00 rn r^ f m Oi m m l^ -* r^ « o -0 0. m t^ tsi tn 00 00 « sO ■* p< » •* 00 N fn N m fn r^ t m 00 N » m t^ 00 Os t t 00 Oi « t^ N fn vO 0. ■o i^ t^ so Oi N Os Os Os Tf 10 •* vO 0. r^ 00 ■* N ro 00 00 Ov N Os m r- r^ 0. 00 r^ fo 00 t^ ■* o> 10 fO t^ n a Os -0 n 00 00 SO o> vO n -1- vO ■o t^ 0. o> OS m m r^ Os sO sO vO 00 o> r^ ro Ov Os ■^ Os so 00 r^ Os 00 fo «t t^ Tf 00 00 rn fn -0 N rn vO 00 r^ t^ » r^ so ■* •* in 00 00 00 00 a « r^ •* f^ sO 00 N 00 sO t^ N n in 00 f*5 Ov ■0 Tf t -T 1^ rr 0. t r^ r^ r^ 00 n 0. Os 00 00 N 00 fO in t^ 0. n vO 00 "5 00 Tj- r^ t t^ ^ •:t n t^ sO t^ 00 00 00 t^ i« t ro fo 00 r^ sO 'J- n r^ •* ►-t Os t^ t « 00 <5 >o ^O vO ^O •0 in in m m in m m m in ■* ■<)■ •<»■ m n in ro m Ci d d d 6 d d d d d d 6 d d d d d d d d d d d d d d d d d d d d N ♦-» r^ •* r^ t^ t N N -0 t 00 N so Os t t Os Ov "5 c> o> vO 0. 0. TT OS vO "3- 00 -0 00 t Os t^ t r^ r: T •* ro "^ r-> 00 Os m T^ rn Oi 00 00 a n • ■a- m vO ■O m •* fn a 0^ n -T 00 t t^ "* »:}■ r* fn m r^ ■^ r^ 00 m QO >n 00 m o> t 00 00 n 00 t 0. 00 QO o> ■>t fO o> 1^ ^o m m -T ~! fn t^ 0. M 00 (^ r^ r^ 1^ r- t^ -0 -0 <5 vO sO vO •0 so so in in m T •r «t t •* rn m d d d d d d d d d d 6 d d d d d d d 6 d d d d d d d d 6 d d d r^ t 10 vO 1^ 00 Ov m t m -0 r- 00 Os m in m (O n f^ n n fO to ro n fO t t ■* rr •o •o Ol rO 00 O 1^ N 00 TJ- t^ Ol Ol O sO M- ro ro Tj- lO t^ OO 00 o t^ O SO &5 a o Ol l/> VO to M a Ol 00 lO ro O sO •* so ^ o 1/5 O Ol t^ o so ro m ro a vO O Ol o •* 00 1-5 r^ Ol 00 ro so ro ro m 00 t^ o Ol 00 ro sO « -o IT) o Ol t^ ■* •o t^ 00 sO sO t^ ro O o t t^ so Ol t^ lO Tt ^ Ol •* o o Ol Ol sO t OS 1^ 1/5 ro o. r^ so •f ro o OS 00 a 00 00 1^ t^ so SO lO lO "7 t M- M- TJ- 00 ro ro ro ro N N 0) o< M M M d d d d d d d d d d 6 d d d d d d d d d d d d d d d d d d 00 o r^ t^ o in sO in Ol •o o Ol i« -O 00 so r^ *-t w r^ oo Tl- N Oi T^ lO ^ ro ro Ol ■* 00 Ol sO SO Ol M- oo ro Ol r^ t^ ro >0 ro o a t^ vO Ol 00 ro Ol t^ -O O ro Ol Ol M t^ O 00 ro so € 00 M r^ o 00 f^ O t^ ro so Ol ro 00 ^ ^ 1^ lO O ■* ^ Ol rl- ro o 00 vO 00 so Ol so 00 ro o o 1^ 00 a 00 O ro 00 O O i^ ro vO o sO f^ Tf sO O 1/5 00 sO -O 00 O m r^ m Ol •r> o O 00 •* ■* 00 00 o 00 ro Ol r^ m T^ Ol 00 so in "t Oi Ol 00 00 t^ r^ r^ so SO sO to If) lO lO ^ M- rf ^ ro 00 ro ro 00 00 M M M d d d d d d d d d d d 6 d d d d d d d d d d d d d d d d d o m o t -i- -o fo fo 00 -t so -* -o Tf 01 o r^ Ol >o 00 a 00 t^ o 00 t^ Ol Ol 00 Ol vO ro o t^ t ■o 00 tn •* 00 00 o Ol lO O r^ O o 00 O O lO fO Ol vO Ol 00 O Ol sO t^ t^ sO O O t 00 O lO o ro O Ci t^ M Ol o. 1^ M t 00 Ol Ol sO 01 Ol r^ T M 00 ro so r^ t^ ro t^ t^ Tf sO to O o vO 00 t^ t ro ro so Ol t Ol sO -t ro ro ■* so Ol ro t^ 00 ■i- " Ol r^ rO o vO fo O t^ -t 00 so Ol r^ ro M Ol l^ sO M- OO O Ol l^ ■^ o> Ol 00 00 oo l^ t^ t^ SO sO SO »0) lO >/5 -t ■* Tj- ^ t ro 00 ro ro ro 00 ro Ol 01 w d d d d d d d d d d d d d d d d d d d d d d d d d d d d d vO vO Ol ro rf l^ rO •o Ol 00 o 00 i/> 01 f< o Ol 00 t^ o ro t^ r* M ■o a 00 Ol o so 1/5 lO t o. ro so so ro 00 Ol r- SO 00 It so ■^f 00 -t O Ol o 01 »/) Ol -t o »o ro ro 00 ^ so 00 m o m o so ro O ^0 00 in Ol i~0 O t^ Tl- 01 o t^ 00 Ol r- lO ro 01 o Ol t-~ so -* ro 01 Ol Ol 00 00 00 t^ t^ t^ 1^ so sO sO so IJ^ no ^ T)- n- ^ ^ M vO lO 00 so f^ ro t-i 00 00 Tf 1J1 ■* 00 00 o >o "* r^ ro so 1« sO t^ Ol o o t^ Os 00 ro 00 OS Ov M- sO o t^ t^ m t^ o 1^ 00 ^ 1^ 00 t Ol sO 00 Ol t^ so -o 't o Ol ro m •* Ol so so t^ Ol ■* 00 o 00 Ol O Ol in so so Ol 00 t^ •* Ol Ol ro o a 00 t^ ■* 00 t o ^ O Tf Tf O tT 00 Ol 00 o ro so 00 SO OS so so O ro o ro o so rf O oo m o 00 SO i-t t^ 00 O t^ Tl- -f 00 vO ^ 00 -o -^ O 00 o Tf o OO ro o Ol (^ so 00 M Ol Ol Ol 00 00 00 00 r^ t^ t^ 1^ so so SO sO so in in in to 1/5 •* ■rt ■* -* ■>r ■* « d d d d d d d d d d d d d d d d d 6 6 6 d d d 6 6 d d d d f*^ Ol ro o ID O O o 00 00 t ^ 01 so t^ 00 in Tf Ol so >o t^ 00 o ro o a f*: 00 Ol Tf Ol Ol Ol r- t^ O Ol Ol so ^ o 00 t^ m rf T tM so r^ ro •I- t^ t^ o t Tf 00 ^ o ro ro 00 00 tH t^ 00 01 o ro 00 1-0 -o lO 00 ^ o O OS Ol ro o Ol Ol t- ro ro Ol 00 in ro rt 00 t^ O Ol fO 00 Ol 00 Ol sO 00 so N sO 00 ^ m Ol 00 00 ro (^ f^ Ol r^ ■^ 00 fO O ro r^ O oo o i~ m r^ •rt M Ol t O 00 sO tT 04 o oo -O t ro Ol 00 sO in 00 Ol O o 00 so m M S o Ol 00 00 00 CO 00 1^ r^ t^ so o so sO -O in in m in m m VO •51- •^ t ■* M d d d d d d d d d d d d d d d d d d 6 6 6 6 6 6 d d d d d ■r. a o o -> r. rO •* V, ■o t^ 00 Ol o i-» r. fO •* ui -o r^ 00 Ol o » OJ ro M 01 lO so t^ 00 Ol w Oh COMPOUND INTEREST AND OTHER COMPUTATIONS 261 fO -t fO r~ 00 10 a 00 r^ M vO vO •i- t ifl ro ro 00 t 10 1^ 00 rf -0 00 ro "i- vO ro Tt ro ■o ro 1- -0 ro 00 N Ov t^ Ov Ov t^ OC 00 't t^ Ov ro 00 t t^ ID 00 t Ov ro 10 M ro Cv ^ 00 -0 t vO t^ "t t Ov a t^ t^ r^ ■0 vO VO Ov ro Ov -t Ov Ov -* •t t vO r^ 10 a ro 1^ M CO -+ t^ t vO n Ov r^ 10 ro r^ 10 ^ ro ro M Ov Ov 00 t-^ t^ vO vO -0 ^ ro p) *~' *-< "-• ^ *-* >-< M •-* *-t q q 6 6 d d 6 d d d d 6 d d d d d 0" ^ d d d d d d d d d d d d d d tn t^ t^ t Ov ro vO t^ 00 ro 00 ro 00 r^ 00 Ov w t^ Ov t •* CO ^ vO ro Ov -0 vO -0 t ro ir, vO Ov t^ •* ^ 10 Ov a "t Tf r^ ^ r^ lO 10 t^ Ov ^ •0 vO a ro ro vO vO vO vO Ov vO in Tf r^ ifl t^ 0. 10 ro •* ■* ro vO Ov Ov -t M- >o ro 'I- vO 1/5 1^ 00 fO m 00 00 ro -0 ^ vO 00 t^ Ov Ov m ro Ov r^ 00 ro i^ vO a o\ Tf vO Ov 10 00 vO vO 00 ro to Ov t^ N Ov Ov 00 i^ « Tf ro « !i; Ov C3 00 ro q d 6 d d d d d d d d d d d d d d d d d d d 6 d d d d d d d d d Ov M- r^ •* Ov 1^ >o n- -0 t^ M 00 r^ •+ r^ Ov 00 ro r^ Ov -0 '^ a ■* ■* -t t^ tvi ro t^ •0 vO Ov 1- Tf Ov vO Oi 'I- Ov 10 00 N M- 00 t^ 0. Ov Ov ^ vO Ov ^ ro vO Oi r^ Ov r^ Ov vO p) r^ ■* vO 1^ vO r^ ro Ov Ov ro 00 ro 10 T Cv 00 1^ vO Ov rt vO a ro 00 vO 00 Ov 00 r^ 10 "* ro ro 'I- vO 1^ Ov ■* r^ •* 1^ vO Ov ro Ov m 111 ^ ro M pj 00 :^ 1^ vO M M H-l ^ ro ro i-i « !^ 00 1^ S ro q 6 6 d d d d d d d d d d d d d d d d d d d d d d d d d d d d d t^ -0 t t^ Ov r^ VI ro •* ro ro 10 vO -0 -0 -0 Ov 00 M 00 t^ r- vO -* Ov 00 Ov •* ro 00 00 00 ro t tn Tf -0 vO •o >o ro t^ ■o t^ ro vO ro 00 mD t ro ro ro t vO 00 in 00 r^ 00 vO 00 -I- ro 0. -t Ov a 00 t~~ -0 "5 t ro Ov 00 t^ yr, i« ^ ^ Ov r^ vO 't ro n p» r* M C< (S M M IH t-i q 6 d d d d d d d d d d d d d d d 6 6 d d d 6 d d d d d d d d d 10 t ^ t^ 00 t 1^ rf r^ t n 00 Ov 10 -? Ov ro 00 "t ro ro Ov 00 CO Ov 00 Ov vO (s a Ov 1^ Ov ro 00 •* vO ro i^ t^ r^ 00 "t ro ro vO r^ ro >o 10 10 ro t^ 00 VO Ov t^ ■0 r-) -t Ov 10 ^ 10 00 vO ^ 00 ro t^ Ov 00 Ov t^ t^ Tt Ov Ov t t^ 00 Ov vO vO Ov 10 ro ro 00 10 "+ ro N Ov 00 00 t^ vO 1- ro Ov 0. 00 t^ in 00 vO m -a- ro ro fO '? ro ro r^ rv» (v< fv* 01 r^ N rv* N C) rv| M w q q d d d d d d d d d d d d d d d d d d d d d d d d d d d 6 d d d fO ro -t -t t 00 ro vO Ov 00 r^ ^ vO t^ Ov N Tf Ov 1- t Ov 00 Ov vO 00 CS ro N 00 r^ 10 m •t 10 ITi vO 00 ■* -0 Ov ■* 00 vO a vO vO ro Ov a 00 1^ vO t ro l-l a 00 00 l^ m tT ^ ro Ov vO T)- Ov VO vO •^ ir> ro ro ro ro ro ro ro ro ro fv) w rs r* rv( (N CJ M (N M w 6 d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d ro cc t vO 0. vO in 00 irt -t ro N ro M Ov 00 ■* Ov rr M i> 00 Ov vD Ov t l^ •a- N 00 w M ^ -0 t n ro ro I^ 00 Ov ro Ov Ov •3- Ov Ov i-t t^ -0 r^ r^ vO tT 10 ro Tt- ■I- 00 ■:)- t^ ro 0. 00 ^ ^ Ov ro vO ri- vO 00 'I- M r^ 10 Ov •* vO 00 00 't 0. t Ov vO Ov ro T t^ t-^ vO Ov ro ro Ov t^ sO sO r^ i^ 00 ro •rt r^ Ov Tf 't vO Ov Tt Ov t^ in vO ■o o vO Pi •o >* O 00 1^ t^ rf t^ P5 00 o o ^ ro rO 00 vO rf Pi P5 rf o t^ p: vO O o o Ov o fO t^ ^ PI ■* ^ o. o> Pi t^ PI o vO 00 Tj- o t^ 1^ o 00 P) vO t^ t^ vO t r^ PI O 00 o -t vO VO ■* PI t Ov i^ Ov vO ' -^ (S r^ 1/1 >fl 00 rt fO O 00 o> rr P5 00 VO 00 ^ ■i- 00 vO 00 ^ in i-i t-4 m m ""* o q M N ro lO r^ Ov Pi ■+ i^ >-l m Oi P5 00 PI Ov in •-^ 00 in pp *-« o q 00 oo M ri (T) ^ >o t^ 00 o, M " pi lA •d " d. d Pi pi Pi " r^ 00 d P5 pi p> 4 PI vd P5 d ~ Ov O •& o o Tt- r^ t^ o 00 O ^ "7 vO PI p> o 7" 00 vO 00 o 00 1^ rO r^ rf 00 O 00 i^ t~~ oo p: m m vO o 00 o PI Ov Ov r^ r^ o vO o r- 1^ I^ 00 Ov Ov o Ov Ov P5 Ov vO m vO £5 o ro Ov vO Oi o r^ o Ov Ov vO ■* ^ r^ o VO o p« -O 1/1 o O t^ sO o> t^ O. p) P) o 1^ P5 m O VO ^ P5 •o ro i-O 't Ov I^ o o ■+ o o PI Ov 00 r^ vO Ov vO 1^ M o PI 00 •^ O o ro "t >o 00 q PI vq a Pi -q o M PI pi 4 vd t^ d d h4 pi 4 vd CO 6. ^ pi 4 vd 00 d ~ Pi Pi Pi Pi PO p> PI P3 P5 ■* o O 00 lO r^ o T fo T Pi ~ >o 00 PI o Ov ^ r^ N Pi P5 Ov ro o •t o M- 1/5 ►H Ov Pi Ov P5 vO 00 Pi Tt t^ t^ 00 P5 in 00 o O ro a 00 lo ol >o o 00 Ov t^ P5 PI a P5 O '^ O -* O r^ t^ o "1 Cv -t Pi o P) in P5 P5 Cv o. 00 Pi Ov P5 Pi o ■+ CO ^ Pi ^ O. OC O ^ P5 Cv o r^ PI 00 1- o o -* vO o o ro •^ O Ov PI 1- Ov 00 Ov ^ pn t^ Tt Ov O o M r^ ^ 1/1 r^ a •>»■ o o. Pi vO o ^ 00 Pi t-- Pi 00 ■* q vO Pp q t^ M P* rO 4 v> O 00 d 6 " pi - " " 00 d " s 4 in M 00 P< d Pi P5 pi P7 in PO 00 p> o o, « ui ^ lO -+ P5 PO lO vO 00 00 Ov 00 o o t-* m Ov 00 t^ t^ ro f^ Oi ro -* ^ o. o in P5 vO p< o. P5 vO PO vO -o O Oi 00 o a ro P5 ro Ov t^ vO Ov VO t vO Ov Ov Pi Ov o ^ O O ro o> •* O M- PI -t (^ o P5 o 00 Pi m PI PI t^ o. lO 00 o t t p> 00 ui 1- vO vO vO vO P5 o vO O ^ ■q- M- 00 _^ O CO 00 00 I^ o -t I') O o> ^ "^ M vO m o r^ O O 1^ PI >r, o O t^ O o o t o m r^ •* PI vO fn o O »-* "7 lO vO 00 q Pi ^ t^ Ov Pi vq Ov PJ t^ *-< in q m q in t-- PI ro 4 o 00 dv 6 pi pi 4 in •6 00 M d. d Pi CI pi S" vd 00 d p: pi PO 4 ui PI P) 00 ■^ fo o\ o ,^ a P5 o in 00 ^ Ov Pi >o o H-l t t o vO o r^ ^ a 0\ o o •* ^ vO o r^ 00 P5 r-5 Ov vO 00 vO Ov vO 00 vO o tT a a r^ Ov m vO t^ Ov P> o pn 00 oo' 00 o ^^ o so in a rn ro •o 00 P5 VO t^ >o Pi r^ ^ >o 1^ i^ o Oi a PO t^ 00 PI p^ PI p: 1^ vO in o Ov vC t l^ •N lO o o> o> PI ro o o Ov vO p: o i^ m m PO p: vC 00 -t Oi (N fO o o 't ~5 00 pn o 00 Ov t^ P5 PO vO 00 00 Ov '"' o O o f-H ro M- in t^ 00 O 't vq Ov Pi 't t^ M ^ 00 r^ vq o m q ■^ Ov « (S fO ■j- bi -d 00 d\ 6 " pi 4 - vd dv d Pi PI pi Pi 4 N ^ 00 Pi d P5 ^ pi P) 4 p: p? ~ •o a Tf so VO a vO t^ M t^ vC PO Pi t^ 00 I^ lO -o vO o O 00 PO vO O. PI Pi p: O a O o O ■* vO 00 P5 PO -t M o o 00 00 ro PO o vO vO -O vO Pi r^ 00 in o. o Ov m fo Oi O O 00 -o o 00 Ov •* Ov vO TJ- ■* in 00 pn o. 00 t- Oi M s q o t-< i-O •* m l^ 00 o ►H p? >n 00 o oc vr t 00 vq M CN ro 4 vd 00 d 6 " 4 " vd t^ od d •^ PI pi ^ vd 'cT d d P5 pi P3 4 P5 vO "7 o t^ Tl- t^ Tj- Pi Tf Ov ■* t^ •* Ov P5 00 PI m o o M Ov vO o o N P) ui -o o o PI m ■* m o Ov o Ov 00 00 in 00 vO vO m o r^ ■^ 00 m 't p) t^ P5 ^ in ■* Ov M -O r^ o o ro m o Ov vO ■* Tt a o Ov 00 o vO Ov PO 00 O. oo t? t o o vO 00 P5 1- 00 00 t 00 o PI ':^ vO 00 O PI o o ro 00 p< o a vO o ^ o Ov a vO PI p> o Ov o ^ o o 00 vO VO o o -t Ov r^ t^ -t o o q »-t M r4 "5 •* vO 00 q> O Pi ■* vO 00 q P4 ■* 1^ Ov Pi in CO M TJ- w pi ro 4 Pi 4 Pi 'c? vd Pi d Pi d P5 P) o „ p« n m vO ,, 00 O. o M M PI ^ in vO PI ,, oo o. COMPOUND INTEREST AND OTHER COMPUTATIONS 263 -0 •* 1/5 •* ro ■* Tf Tl- r^ co vO ro ro Ol vO 10 ^ 10 m m 00 00 -f t ro ■^ vO 0. >o m PI 00 Tf ro Tf Ov r^ ro Ol vO m Ov 1^ t~- >* r^ r^ >o Ov ro t^ ro PI Tf Tf r^ ro ro f r» r^ ro 00 ►-« ■:)• Ov t^ in ro p-> 00 00 r- Tf 00 ro Ov vO m m r^ 00 rf PI 0. m >o 00 CO ro 00 10 ro Ov 10 ro vO w •* 00 M t^ vO t r^ Tf 00 Tf 00 PI 00 f Tf CO ro P) 00 5 ro >o in vO in PI 00 in Ov 00 00 00 10 vO -f Oi ^ ro >o 0. PI r^ PJ r^ ■^ 00 vO in in in vO 00 •-* Tf in Ov v-i PI 00 ro PI »-" vq in f^ ■6 00 d ri 4 t^ d pi 4 r- d pi in 00 M 4 t-^ d 4 t^ in in d in 4 00 vd d 1^ pi ■51- M- ■* 10 10 10 10 w •0 vO vO t^ t^ t^ r^ 00 00 00 Ov Ov Ov » -0 00 PI Tf PI 00 ro ro ro Tf P) -0 vO 00 ro 00 t vO i/» •* Tt PI ro f vO vO ro r^ CO vO t^ 10 t^ ro tC N •0 -0 ro ro PI Ov vO Tf ro vO vO Ov 10 ro 1^ m m t^ ■* t^ (S 10 W 10 00 10 t^ ro VO 10 VO PI 00 Tf Tf Tf f^ m CO 00 m >o t^ ro t^ r^ r^ vO Ti- Tf vO r^ 00 ro Ov P5 ro N ro N Oi 00 vO Tt ro 10 ro ro ro -0 00 vO 00 t -0 10 Ov vO Ifl rl- t^ -0 Ov Ov t^ in Tl- t^ PI Tf r^ Ov PI vO » Tf ro 1^ ifl Ov vO Ov 1^ 00 PI »-( m vO PI Tf ri c^ 00 m vO Cv| ro I-, PI 0. t -r M t-t »-" M ro m r^ Tt t^ PI 1^ PI CO in PI 00 r^ t^ vO t rn m rn •-< v-v 1^ 10 00 « 4 vd 00 d P4 4 vd d ^ ro vd 00 t^ pi vd d ri 4 l-l d vd 4 4 •6 ^ d d 4 pi vd •. r^ vO r^ vO .0 ~ PI Tf vO r^ T 00 (N 1^ vO vO vO -t 00 ro r^ Tf 1^ vO 'f vO Tf Tf t^ 00 Ov ro o> t^ t^ rt Ov 00 ro PI ro r^ r- Tf 0. Ov ro Ov « vO r^ •* t^ S t^ vO ro ro 00 PI vO vO r^ c ro p^ 1^ m -0 ■* 00 t ro P) Ov a PI t vO 1^ 10 Tf Ov Ov vO Ov ro CO » t^ ro t •* Ti- -i- t^ Ov t^ ro Ov vO V-l vO t^ -0 vO t r- N ro Ov Ov ro ro vO 10 0. P5 -f 00 t^ Ov t^ t 00 P5 10 '? M M Ov Ov v-t Tf vq 00 '? 00 P^ q. 10 in q w Ov l^ •-< q d fj 4 -d 00 dv M 4 vd 00 d ri 4 ti d M 4 vd d M 4 06 4 H^ d d pi d r^ 00 pi ■3- Tt •* ^ 10 10 10 vO vO vO t^ ~ 00 Ov - ro t " Ov PI Tf p| P5 Ci T ~ m t^ Ov Ov ifl ~ vO ■* Tt 00 r^ vO 00 t^ 00 00 Ov 00 vO 'f vO 00 in Ov ^ N t r^ vO Ov PI 10 r- vO 00 ro n Ov vO 00 PI PI Ov 00 Ov ro Ov Ov ro ro vO PI ro ro Ov vO Tf PI t 10 ro 00 vO Tf ol r^ Tf tn >r> o> M M ro ro P) vO ro ro PI 00 ro 00 00 PI vO v-t P) 00 vO o> •* 00 vO 0. t^ ro ■0 t^ PI vO 1^ PI vO t^ o> N "* r^ ro ro ■* m Tf- ro 00 M PI 00 ro a Tf Ov N ro -0 ro ro PT) ro t^ Tt 0. 00 vO vO ro r^ ro ro Ov ro vO •f vO •* M Ov r^ f PJ PI n PJ Tt t-* Ov N vq P^ PJ t-» t^ 00 t^ Oi 6 ri 4 tfi t^ d M ro 10 t^ d w pi in t-^ d PI 4 vd 00 t-i 4 d 10 pi M PI IT, d vd f^ T ■«)■ M- ^ Ti- •o Ov ro Ov Ov Ov Ov 0. vO Tf in ro Tf ro 00 r^ 0. 00 ro ro PI 00 t^ ro Ov t^ 00 00 m ro 00 -0 00 a 00 -0 00 00 Ov Ti- vO Tf Ov PI Ov 10 t^ in 00 Tf f <5 t^ -0 Ov 00 -0 00 Ov Ov 00 Tf Ov vO 00 -0 t^ 00 0. 00 M 00 00 fO m Ov r^ ro 00 t 00 00 PI Ov Tf ro PI vO P5 ro 00 0. ro 1^ 00 00 vO 00 PI Ov 00 vO m 00 vO t^ vO N Ov in r^ Ov r^ f PI q. t^ vO vO 10 in vq vO m PI 00 ro t^ vO q vq 00 tC d d 04 ro 10 1-^ 00 d pi 4 vd t-^ d k-i ro 10 t^ d M* -0 4 vd 00 pi vd pi d 1^ r^ 06 r^ fO t t •* Ti- ■* t 10 m in in vO vO vO vO vO 00 Ov « ro 10 vO 00 Oi t ro P) ro „ 00 t ~ t^ 00 00 -f Ov PI vO -0 vO Tf vO PI 00 •o N ro vO irt ro Tl- PI 00 00 vO PI t^ Tf vO 00 ro a Ov Ov t^ vO 00 vO m 00 ro Ov t^ -0 ro nO 00 -t r^ vO Tt ^ Ti- ro ro Tf 00 Tf 00 in PI Ov 00 10 ro f ^ -T i/> ro 00 m t ro ro ro vO vO Tf ro -f m Ov Tf ro Oi 00 ir> Ov vO Ov ro t^ 00 f in 00 ro t t^ P5 00 00 vO Ci 00 10 ro t r^ (-1 t^ vO vO 00 ro 0. 00 00 vO ro Tf 00 1^ r- vO 00 IH 10 tf> "? v-i PI 00 •* »-< t^ Tl- V-* vO Tf PI 00 Tf in PI 00 •-< 0. i^ ro o t •* "§ a 1- m r^ ro 00 t- 10 00 -f ro 00 m vO Tf ro Tf Ov vO 00 t^ vO 00 vO r^ PI ro 0. 00 ro o ir> 00 10 vO Tf ro Ov vO -0 00 P^ ro 00 n 0. vO ■o M 00 r^ 0. ro Tf ro vO 10 vO ro r^ vO m 00 t^ M 00 PI -0 w-j 0. 1- 00 ro 00 Pp Ov Tf vO PI 00 Tf 00 vq Ov -0 vC 00 "7 f It- vd t^ 00 d M ro 4 •0 t^ 00 d M pi 4 vd 00 d M PI 4 pi K^ d d d »^ pi 4 t^ d fO ro ro ro ■q- ■* Tt- tT T Tf Tl- 10 vO vO vO t^ 00 Ov " ro Tf " « ro •* V) 00 p< ro t in vO t^ 00 Ov f) n ro ro ro ro ro ro ro ro ■* ■* Tl- t Tf t ■* ■» Tf Tf vO vO i^ 00 00 Ov Ov 264 APPENDIX n *^ vO -t lO 00 Tf -t o (^ fn 00 00 vO m o o 00 CO Ov m o rv] 00 M ri Oi vO Oi o. 0\ vO vO Ov 00 o m t^ -o VO fe o 1^ o vO re o N 00 r^ r^ in ■* f- m Ov 00 Ov vO o t-* t^ n le w 00 a ro vO Oi M- Tf ro vO vO r^ in a Ov Ov M r^ •-< 00 VO M Ov >o vO o ro 00 •I- ^ 1^ >o o. o Ov 00 vO Ov m f^ 00 re t^ I-. r^ fe? ro Tf i^ n t^ o Ov m Ov m M in -t vO 00 Ov o oo ro t^ Ov Ov Ov r^ vO 00 f^ r^ o 00 a Ov Ov vO o M re vO q ro O o. '? 00 •^ •-« q. 00 «5 q N vq M Ov r^ t^ Ov "? q 00 00 >-« t^ in vq ►-f (N ro 4 lO vd 00 dv h^ fO 4 >d 00 t^' IM q t^ m in vq 00 q r^ in •^ in r^ i-» vO ■* ^7 ro 4 o 00 d - ^i 4 - r^ d "' " 00 d ■n in 00 ^ 4 •? ■^ 4 in 00 (-J m vO f^ ro o c^ M ro t^ Ov -+ 1^ r- Ov fn Ov O ■o (^ o in vO ^ in o M r^ vO 1^ vO r^ ■o ^-5 i^ X 00 t^ -t t^ 00 Ov Ov re vO vO ir> Oi ro 't Ov 00 ~5 O. t vO o m vO Ov vO o 1- re re re ^ O O o> O rn o ~5 o 00 vO m Ov -f p) re re o t^ 00 O o Ov o ^n o ^ ►H ~> o vO re re O w t^ 00 o o Ov O CO t O. rf Ov fO l-l to ~5 Ov o re re -f fo t^ r^ 00 O 03 •i- vO 00 rj- o t^ 00 O fe 00 vO r^ Ov ri) ^ q (N 'J- t^ O ro 00 01 00 ■* M q t^ r^ r^ 00 O 1^ t^ f*) q vq in t^ q ■* t ro 4 tn vd 00 d d " ro lO " 00 d n 4 vd d r<5 r<5 vd 00 ■* 4 r^ -* d re 1-^ m m o xO 00 vO „ M ~ vO OO Ov •* ■* Ov 00 o 00 ^ a 00 ^ Ov N o 00 O 't ro "i- 't -O vO M i~5 00 -t m r- 1^ in vO ■* Ov re -1- •^ -o r^ 1^ M M Ov 00 Ov 00 Tf 00 ^ -t N -O >o r^ o. o. O o fn 00 ro N r^ o vO CO o o ^ 00 00 ■o t f-5 o rn 00 oo o. in r^ M O o. 00 vO Ov t^ kO ■o O rr vO vO vO ~5 ^ r^ 00 Ov t^ t^ 0) t t^ vO O- -t •t ro o. 00 o 00 o O. ^ t^ t^ vO -t 00 rr 00 vO vO t q ►H ^ vO 00 evi in o •t q vO o 00 •q vO vq t^ q M vq o vq re q Ov O. f^l 4 O d d - ^ " vd 00 d " "i in " d rr> 4 vd d ^e 5- 4 rr d !N 00 CO 00 r^ o T -t o o oo ~ o o oo ~ ~ Ov 00 t Tt- O 00 00 Ov vO ~3 ro o re re ^ lO lO vO '^i O vO o Ov o o re 00 vO o t^ Ov ■T o lO o 00 Ov Ov Ov vO r> Ov 00 t^ o o vO M Ov Ov t -T vO ^ ro Ov Ov Ov vO O o o vO •* Ov •^ 00 O vO t^ »o o ^ o Ov 00 'e o Ov r^ Ov Ov O o Ov re Ov o o (^ o vO vO fO -r o t^ Ov t^ o Ov m o vO vO •+ Ov " q N in t^ o ro o ^ -q Ov t fV) "? ■^ vO q re r- 1 q r<5 ro 4 Ifl o d d " ro 4 vd " d d ri 4 vd 00 d rn re 4 vd -e 00 re -t re vd 00 00 00 ~ ro Ov -o in -o o. O •* ~ "* o. tv» o o N M m tvi N o 00 CO o ro vO -^ rO 1^ ro TT ^ I^ J^ t^ fe t^ tr> o vO vO Ov m Ov o ^ n HH t- 00 •* o o ^ rs re rv) O r^ o ■o ro O 1^ a Ov 00 00 vO t^ 00 00 vO ^ re N in 0\ o t -r ro 00 o ro Ov 00 "1- 00 ro 't 00 -t o vO Ov 00 O r^ S 00 o. ro t^ vO 00 >o ^ vO o >o vO vO Ov re Ov O 00 ^ Oi 00 O o -o Ov m vO o Ov 00 Ov vO rn O re " o q ro -* -q 00 ■t 00 vO q lO ►H ^ *H 00 vq f vf ^ in l^ q N ro ■4 o t^ 00 d " " 4 - " 00 d (S ir> J? vd CO d re 4 fe vd re 00 re d rj in ~ o O o <5 tn o M ~ Ov r^ Ov N 00 N r^ o o ro in ~ a o M ^ N Tf 00 O M- t^ 00 00 00 o rn •T Ov vO m Ov Ov a VO ■* N O I^ o 00 CO 00 1- vO O Ov 00 t ^ m vO Ov 00 t in M -t o t o >o •-• re M 00 00 OO ^ Ov vO (VI -r o t- o ifl O o a 00 t Ov Ov -f T o f^ 00 m Ov 00 Ov ^ t^ ~5 CO -t ro VO 1- vO Ov n m ro r^ in oo vO o O 00 00 -O 00 o rO 't Ov Ov vO vO o o a 00 o vO 00 00 vO Oi q q ■i- •o 00 o iri o q "? ^ 00 in ^ q vO in <-} PI N (V) to rt N M N f^ 4 ui o l^ 00 d " « r^ in vd 00 d ^ S 4 vd " d re fe in re 1^ re re 71 Q O fO ^ ■o 00 Ov o rt ■n f -o 1~- 00 Ov o re 'I- in >c 00 Ov 5 " " ■^ M " •^ •^ " M " IN M D "^ N tv) o -0 i^ ■O 00 t^ vO 00 m m m 00 t^ vO t^ ro ^ vO ■0 r^ 00 ■* t Cv 00 10 00 ro t r^ ro ■* 00 •* vO Ov Ov r^ Ov 00 00 r^ 1^ vO W t^ vO vO 00 -f r^ ro vO vO Ov 'T 00 t vO (^ t^ 00 1- Ov m vO m •* Ov 00 vO t^ 00 Oi ro "* T r^ t^ 00 ro 00 00 ■^ 00 m 00 ts m tT 00 ^ 00 rj m vO -t m 10 t Ov vO ro t^ 00 ro •q- CO i^ vO 00 00 '? •-I 't •-I o rt- fO vO ro "t r^ t Ov r^ 00 m Ov ro vO Tl- ro vO 00 m t-- -0 t^ q •* q •-I 'I- q 00 q M q 00 vq in 00 in q -r 00 m vq vq 00 t^ 4 00 ri t^ M vd kH* vd M t^ N 00 in ^ 00 vd ro ^ dv 00 t^ di vd ^ ^ dv 4 in 04 d vO r^ 00 CO Ov Ov " " rj ro ro ^ vO vD ^ 00 vO ro 00 m t^ Ov ■* ro Cv ~ J^ !-4 in vO ^ ro 00 0. vO rr n ro -t Ov vO Ov in vO n in m 1- m ^ t^ VO •^ 00 00 rf l^ ro t^ Ov i^ ^ vO Ov ^ in 1- m "3- vO t^ in t^ 00 tT ro vO vO Ov vO t^ vD r^ vO ^ ro 00 vO vO ro ro 00 1- 1-^ fO MD tT ro t ro 00 t^ Ov Ov ro 00 00 00 10 vO t^ ro 'T a ro •^• a ro N JO 10 00 ro I--. vO ro tT ro ro vO •* 00 0. t^ 00 N Ov in vO Ov ro ro 00 Tl- ro tT ro 00 Ov t^ C<1 n t^ ^ vO ro vO in Ov vO Ov ro t Ov 00 00 n 00 vq in t^ qv M- q 00 00 Tf q 00 q 00 vq q 0. N -q r^ q t^ vO vd di N vd d. n t^ M in d in dv •4 d in ^ vd pj dv >n N M t^ 4 4 00 i_) vd 1-^ P^ t^ 10 10 vO vO vO t~ t^ 00 00 Ov Ov Ov " " " r< ro ro r)- m Ov ro Ov vO ro "3- m m vO 00 ro 00 00 " •O -r 00 N 1- m "7" 00 00 -0 00 a ^ T ro t^ t 0. ^ in 00 00 •T Ov t^ Ov ^ vO i^ Ov vO Ov vO ^ vO ■rt t^ vO t~ t^ vO Ov r^ ro ro t^ ro m 1- 00 vO Ov ■o ■* 1/1 vO 00 00 Ov ro vO ro vO Ov 00 Ov 00 00 CO t^ ro 00 vO 0. -3- tt ro -t t^ l^ 00 T Ov t^ 00 -^ 00 vO vO vO (^ N ro Tt t^ in ^ ro 00 00 in 00 a ^ vO ro ro vO 00 *-* ■* '? ro •* -q ■* t^ in in vq 00 n t^ M- ro ro vq 0. Ov in q in ro ro N t^ •0 ,4 4 t^ d f* t^ 00 't tT ro Ov Ov Ov vO Ov T -0 ro ro t vO t^ Ov T M 00 T 00 rr ro vO ■O ^ t^ 00 Ov tn t^ •^ M ro ro CO Ov vO 00 vO ro ■^ ro vO 00 M r^ t^ r«5 vO t^ ro vO 00 •q- Ov ■* Ov t^ in ro Ov ro vO •>t 00 "? t^ 't CN ^ M ^ vq ^ q f^ m ro ^ r^ q ro in ro 00 ro M t^ d ci i/i rC d rr> vd dv r< in 00 rj in dv rj vd d M- 00 r< vd ro 4 d N J, t^ fO dv r^ ■a- " in u-, in VC vO " 00 00 CO Ov Ov ro vO Ov ro ro ro ■ ro Ov 0. TT 00 Ov 't ro 00 '^ m m M- 00 t^ t^ ro 00 vO 00 vO 00 00 1- 00 ro t^ ro Ov 00 c t M -* •T •0 't s rr Ov Tf Ov I^ r- 00 ^ t^ 00 m m vO "T Tj- o> m ro t^ 00 00 t in 1- ^ r^ ro Cv 00 ro TT Oi ■0 Ov 00 00 vO vO Ov Ov r^ m ro Ov (^ >o a *-* vq •-• CO in ro N w N ro vq Cv t^ ro 00 ro 00 t^ m 00 r< ^ •* PI l^ li^ r^ d N in r^ d ts in 00 HH 4 r^ d ro vd d ro r^ IH •4 in 06 in vd ^ ri 00 ^ c^ ^ ■« 10 m 10 vO vO -0 ■O t^ t^ t^ 00 00 00 Ov Ov Ov h! t •^ M ro 00 ro rr ■. r- O r^ O f^ ^r^^fGO 1^01 'tr^'O O 00 M 1^ o o ■o o> a fO o fo 00 ^ ro o lo r^ 00 /I 00 a 00 •O N 0> o fo t-- fO a IT) N rf 00 rf 00 f>1 O ui vO ^ 00 1^ lO t Ov f -t 1- ^ 00 o ro 00 « ►"• r^ f-> 00 t 00 a o o fO ro "^ "^ toOO ^-r*0000 0^0^0 Pu W 6^ ^ ir> r^ H4 f^ o vO ^ r4 00 1/1 00 vO lO Oi vO o o NO ■^ vO a •O 0\ M- in i^ ■* 00 O o> en -1- ^ n r^ r^ •rf a 't a •* o I^ IT! r^u-)0\'-''0'00oo t^ ro -O 00 t^ »0 d w w ro-^iosd l>-o6 O O "^ ro 00 O 1^ Ov ^ ro r^ 4 MpgLooofOOOMOoiTj- i-H K- 00 ■^Tft^'T'O O t^^ lOOO rof^ CO Tj- lO O ro O t O (N lO M ro '^ >J^ O t^ 00 O r^t^oooo 0\0 O •Jt on '-I a r^ o t- r^ o i^ M N m o f> on •^ m vO vO t •^ o o •a rO Oi a on o 0\ 00 ■^ r^ on 1^ t^ fo 1-1 ■S- •o >o Oi o •* r^ 1^ o o o ■o ts -o a 0\ f- (I vO ■^ Oi M- Ov -1- c> (T -r o CO o on vO 00 t -o -t on no (N o nn O ^^ o\ cc a o O ■* M -t vO o on t O 00 o 00 O r^ t^ r^ a VO o 00 -+ o t^ o. r^ o ■o -o 1^ o o 00 -o fn Oi fO ^ -t fn vO o fn m Cn o t rn o VO Tt r^ 00 Ov Oi o> 00 t^ o T t> ^ o 00 m (^ ■t 00 •* t^ "^ Oi f_, (N in o t^ OO o a o HH (\l rO 'O -t in vO o r^ on 00 Oi O o M t-i r>i " M N -i r^ vn 0\ ii-> in f-i o ■o 00 on ^ O o vO r* o M i-^ on o 01 M r^ r»1 m >H M vO 00 O o o o o> 00 1^ O >o a O on o o m -t >i- m o -o o m n M o. -o f^ Cv o> O 00 t^ «o "* l~0 Ci o o. r^ in fO >-* a o •t 1-1 0\ ■o r^ o r^ ro o r^ ro O " fo f . o "^ 00 Ci o o " - " 1- rf » o " r^ 00 M o O " ^ " fO n ■n Ov 00 r^ r^ vO in M ,^ c m Cv fO in „ •t ^ in Oi in 00 -o O a Ov rO a -o Ov Ov in o. a ^ IT) vO a Ov m -t 00 00 vO -t r^ ■i- 00 Ov r^ r^ fN m O 1- O OO t O ^ O Ov vO r^ f; vO in a r> O in ir> OO ro vf) M o vO 00 Ov o o o o o o. Ov O. Ov o -t o O M- vO vO vO t M r^ rO t^ o f-5 on M 00 Ov o O o* Ov CO 00 t^ o lO ■* rO N Ov r^ 't Vvj lO r) 00 VO rn m ts o " N ro t ir, >o t^ 00 Ov O 1-1 " " t 1/5 vO vO " 00 Ov Ov O ^ M ^ 1-0 •t s? OOMmMI-iOI- Q\ iTi \ri \n t-i \0 -^t^ On O 00 O ro I- Oi I-- o ro o o. -t ■^- O O O M ro lO I 0» r^ 'I- O lO o» O\0\OvO>00 r>-r--0 "p 6 M M f^TtiONO »~*od O^ r^ CO rn <-> MO -I- lOO O lO-^f^iflONO ^lOO lO'-tM Olio*-" 00 00 o» o M M ro -^ »0 M M M M COMPOUND INTEREST AND OTHER COMPUTATIONS 267 t 00 ■* t~ ^ fO m rn Oi N in t~ bo 00 Ov m 00 Tf vO 00 Ov vO •0 •«• N 00 ri r^ t^ 00 f^ ^ m N vO Oi rn Ov tv) r<5 Ov « a vO r^ VO t>. Tf 00 Oi t^ 00 00 10 m ^ M ^ t vO 00 vO »-t in r^ •^ M vO ro 4 4 in in in -d vd vd t^ t-^ r^ 00 00 d d t^ N -o 4 ■n in vd vd « " n N « " " " M c< " c. " " " " 0, " « "* " ro r^ fO ro CO CO CO CO CO ir> ^O a t^ 00 t^ a 00 00 -o ^ vO Ov Ov r^ Ov ,^ 00 •n m N ■n Ov m in CO 00 I^ t^ r^ ":)- 1- r^ 0. w -0 in in m t'i 00 a (^ m 1- o> ^ 0. 10 00 vO ro rO Ov Ov 1- r>* vO 't N t -* vO VO Ov N (N o> -0 ro ^ tT vO -r Ov m t 00 00 (^ c rf f^ -0 00 t^ vO vO 0. 00 m 0. t^ 1- 00 m -0 ro vO Cv ^ 00 t^ 00 in t^ m r^ m -t ro fo in t^ in -t in in ^ » t^ Ov 00 1- t •^ fo ■^• 0. '^ vO 00 a 0. 00 1^ •* t^ vO fO t^ -+ 00 00 vO M- CO ^ t >-* vq »-< 10 >* a rn t^ i-" in Ov f^ r- i-i ■* 00 >-" m 00 fO t^ q q t^ ■* vq M N rj ro fr> 4 4 4 in in vd ■d t^ t^ 00 00 00 d d d w pi rn in -d >d t^ 00 d d P) N « M N M 0. f^ Ov m Ov 00 r^ vO Cv ^ CO Ov Ov lO T -t 00 00 a 00 r^ 00 00 vO 00 ■<1- r^ '^ 1- ■* t r^ Ov r^ vO 1^ ^ vO in Ov CO t^ 00 00 00 M o\ ■^ Ov VO rn ro "* 00 t^ ^ vO 00 Oi IT) 00 Oi a vO M- a rO vO Ov 0. 00 t^ m N t^ vO Cv Ov 1^ ■*■ 00 Ov ro a •f Oi •* Ov ^ q rf t> fo t^ N vq ^ 00 vO •* M t^ ^ -0 1^ I^ CO c r^ r^ 00 00 d d d d d M M rn 4 vd t^ 00 d d l_^ pi CO (M " c r, "* "" "* " c. " « " M ^ c. " f<3 ro r) f<5 fn rn ro fO en CO ^ •* ^ ■^ t t^ C-. -, ^ ,^ r^ m Oi 0. rn vO 7 00 r- ^ Ov t^ in -i- "5 10 0. 10 00 m' m 00 m 00 M vO ^ t^ vO t^ t^ -o vO Ov vO Oi i^ -t 00 o -1- Ov in vO P) t^ -T 1- CO t t^ ■O i^ 1^ vO -1- *1- 00 Ov t^ en CO 00 00 Ov r; 00 IN •3- Ov 00 M -I- Ov Cv ^ Cv -0 Cv Tf Oi ro r^ 1- ^ in 00 vO >o 00 ir> in "7 r^ 1^ 0. t^ 00 vO -t ^o t ~s 2" 0. 00 -0 rf vO Ov M 'S- fo vO ^ i^ 00 r^ vO in Tf i •-' r^ ro Oi "^ 10 ir> in in in Ov Tt 00 N t^ i-i Ov vq M m 00 ro ro 4 4 1^ vd IC iC 06 06 d d d d d M t^ pi (vi rn in vd 00 d ^ C) 4 in vd ,^ " " c " " M " " " c CM !-> ro rn ro ro rn ro m ro ro 1^ tT M- 't M- t ■^ ^ .. -t ., c\ 00 M ro 00 a t r^ m 00 ^ Cv t^ Ov 00 m -t 00 -t 00 (^ •* t rn T 00 Ov VO i~0 00 00 00 'O 'I- 00 00 00 •* ^ t^ r^ Ov m ~5 vO 00 vO CO tT "5 ■^ ro rs a 00 00 ro m a 00 00 vO vO 00 rs 00 (-■ Cv P) vO rO ^ >o vO ^ t^ 00 vO CO t^ 10 -0 t>. 00 I^ "I- 00 t vO IT) ^ Cv t^ ^ Cv -r t ^o t^ 00 rn -0 1- f Ov I^ 00 ro f vO vO -q M 00 'T •q 00 r^ o> -t a in in in in Ov N fO rn M 00 Tt- 00 1- vO 'i- 4 lA u^ •6 r^ tC 00 06 d d d d h-i N rj fO "i 4 4 4 t-^ d ^ -i ^ rj rO ro 4 4 in in vd t-1 d 01* 4 vd 00 d ci CO in vd " " n n " " m ~: fo r^ ro m m m ro ro 1-0 fo ro IT) ^ "^ ^ tl- m m 0, Oi •+ 00 -t -f 00 Ov Cv m vO r^ •t 00 r^ -t 00 Ov m C\ 10 •* T^ m •* t^ ro 00 a o> ro -0 0. GC 00 fo -0 Ov a 00 00 1^ ■* -0 vO 00 00 Ov vO 00 ro 00 00 in fo Ov m r^ Ov ro (^ 00 00 -0 10 >c -0 t vO Ov vO OC Ov 00 0\ a N 00 c- Ov ■^ m •* Ov 00 m -^■ m m 00 vO t^ m vO r^ 00 00 ^ ■o CO Ov 00 rO 0. Cv m r^ 00 Ov -5- m vO 00 r^ ^ 00 10 N 0. t^ ^ '-' t^ ^ -1- •-< 00 ■T t^ q in q >q m 00 q q u^ -d 1^ t^ 06 d 6 d •-^ rj ri ro 4 4 in vd vd t-^ t-l 00 d PI 4 r^ d p) 4 t^ d ^ CO " N N ro f^ ro ro f^ n f^ m 1^ ro rO t-5 ro n fo •3- ^ ■^ m •0 vO ro •^ r^ 00 rO ^ vO 1^ 00 Ov m in rO fj n to ro l«5 f<5 fO O ro ■* ^ ■f t ^ ^ ■a- -t •^ ^ m >o vO t^ 00 CO Ov Ov 268 APPENDIX ir> r^ Ov on •* t^ W) -t -o on Ov r^ Ov O M 01 00 vO Ov 00 01 N vO Ov vO r^ on ^ M O Ov Ov a c> VO •!f "^ O r^ Ov m vO lO rO ■!)■ on t^ t f) CN on 00 O m >o vO 00 i^ vn VO >:r M- Ov Q\ M vO N 00 a Ov 00 r^ •* 00 00 M- Ov O M r^ 00 r^ m m vO on vO 01 ro m o r*) on on r^ vO o on vO Ov on v(l Ov O m on oo oo 1^ r^ ro m in t^ Ov o vn on t r^ r^ 00 Ov •+ O o vO ^ ro i^ vO on o o vO 00 en Ov O vO VO o O M Ov 0\ 00 o ■* W a m M 00 on 00 r^ 00 N t^ M ^ 00 t-l •* t^ O r^ m o 01 •^ m o w N ro •^ t in vO VO r^ t^ 00 00 Ov Ov O O O " M " 01 01 M M on on ^ 00 Ifl ro fO O vO o vO 00 on N t Ov -t ^ vO m O vO t on 00 Ov r^ o 01 vO 00 a ^ o VO o 1- I^ VO Ov vO Ov Ov O Ov 00 r^ r^ m 00 r^ m OI vn m o O 00 O -O o 00 m O r^ M- ^ 00 on r^ S^ m ^ Ov -t vD on N 00 1^ ^ OJ in VO vO r^ O in 00 01 o m vO Ov O >H (N O, rn Ov m <<^ on r^ vO on on 00 Ov r^ ^ Ov 1^ en m O-V 00 l/l m t i^ oc vO o O vO Ov o. f^ 1^ on 00 -O o on Ov Ov 1^ 1- Ov ^ Ov GO t^ lO 00 o i^ 't »-< c^ on 00 m 00 rn 00 (V) vO o ■* 00 Tf t^ o on vO 00 tH o M " ro 't . m vO '^ ^ 00 CO Ov a o O tH " " 2 0) H M on Tf -t t ^ " n O t on ^ l^ O 00 o» on o o vO o 0< ■* ^ 00 00 oi on 00 ro o o r^ r^ o 00 Ov 00 M M- Ov Ov Ov on 01 r^ r^ o O 1^ 00 Oi ^ Ov Ov 00 f^ 00 f^ t^ o. 00 o Ov Ov l^ ^ ■* -o o) 00 00 o CO r^ Ov r^ N 00 N 00 00 Ov M- r^ Ov 00 t^ M- 't m 00 vO 01 M 1/1 r-^ -i- 00 00 in Ov a vO Ki CO OI r^ on O Ov o o on 't Ov t in M- oi •* Oi 00 t^ lO OO '-' IXJ 1/) rj Ov m v-i vO rj t^ (N t^ v-i m O ■+ r^ *-• ^ 00 v-v 't 1^ o " O) ro •* m m vO r^ t^ 00 a Ov o O " M 2 00 r^ 00 T^ M ^ m IH M* VO tH o r^ rO r^ M-) r^ r^ 00 vO m m m m r^ o -t n 1^ n- M- 00 m 0. on ■I- vO O IN ro on vO on o r^ r^ f^ 00 Ov ^ vO 00 Ov ■* Ov on vO Ov r^ vO ^ HH >o M- ^ vO 1^ OI i^ m en vO Ov VO <>v m on a Ov r^ ■+ Oi Ov a (N m m ■+ fn Ov t^ •* M on Ov vO Ov ^ vO VO 00 -n o o on cn o 1^ 00 -t o •a M vO M Ov on Ov o r^ 1^ o lo Ov N N o o in m 00 o. on O O vO vO o. on r^ n CO t o on r^ vO 00 00 vO vO on Ov 'i- OI 00 vf) Oi 00 1^ VO -t Tf w t^ <^ Ov m tH vO VO m O •i- 00 01 vO Ov 0:1 vO o ^ vn ns m on M- m vO r^ o t^ -t Ov Ov n r^ o en O vn Oi o oi r^ o o vD o 0\ O Ov vO OI 00 Ov 00 vO OO) ro o. 00 vO in on t-i CO VO m o vO r-^ Ov m o vO f^ (V| vO vO o Tl- 00 01 vO o M M r^ Tf in o -o *^ CO Ov Ov O o " l-i M on 1- ■* m m VO vO M VO M t^ - 00 o, c O Ov Tt VO o -r Ov 1^ •+ Ov ^ r^ 00 VO •!t •of Ov 01 Ov t^ 01 Ov ro "T t Ov Ov CO Ov 1^ O O ^ o 00 vO vO t ^ m fO Ov 00 r^ 00 ■1- r^ m on m 00 m o. t •t vO 00 01 00 •* vfl c^ O Ov 00 Ov o o O m m o Ov •* •t t OI m &^ on -t v(> o •o VO o o, O Ov t~i m -t O Ov vO l-( 00 tH ■* o en Ov o f^ VO o O) ^ •I- vO r^ vO on on l^ vO on in 07 vO •Of 00 o O -t Ov Ov 01 m -T -t •t m "I- on VO O O O O ■o Ov ■i- r^ on 01 Ov 00 m o Ov Ov *t t^ o -t vO -t 00 vO 1^ on Ov o vO ro ^ OO vO 00 Ov -t cn o o Ov in O on ^ t Ov 't o O o 00 m o> Ov 00 r~ -o Tf O 00 vO on 00 -t 00 t o -o 04 on 00 ■* Ov 't 00 01 00 O M P) 00 ^ in -o *^ l^ CO Ov o O " ►H M M fO t T^ h? M^ vO VO t^ " 00 00 Ov tH Ov Q O M O n ^f in ■o t^ 00 Ov o M N r') -t in -o r^ on o. O m t m vo t^ 00 Oi « "- M " t-i M "" M IH OI 01 01 01 01 N 01 M 01 M Pi COMPOUND INTEREST AND OTHER COMPUTATIONS 269 w o> 0. t PO vO 00 r- 't Ov 't r^ PI r^ 00 t -r PO Oi PO vo PO -0 -0 00 PO Ov Ov vO Ov r^ t^ t^ VO ^ t lAj fo a. •0 I^ tT PO p< Ov 00 vO pt P) PO 00 vO VO "7 00 tf ■Pj- ^ 00 00 t^ V r^ 00 PO vO r^ VO M- PI t M- PO 't PO ■t 00 t^ 00 PO 10 00 •o -t 00 Ov vO Ov 10 •^ Oi -t 00 vC vO -f vO PO -t Ov r- Ov t 10 Ov 00 00 •0 00 PO PO T 1- -p PO 00 10 » VO Ov vO 00 00 >o -t r^ t^ q PO -r vq t^ 00 q q pj PO PO Tf 10 vq t^ r^ q P-" PI PO t >o >o 10 vO VO r^ M ^ Tt 4 •t 't •t •i ^ 1h 1h »H Ih IC 10 10 ^ 10 10 " " vd vd vd vd vd vd vd vd vd "> On 't r^ VO 00 00 7 M 7 PO vO vO 7 PO 7 10 00 PO •pt Ov ui i^ PO PO o\ Ov PO 00 t^ t^ •t Ov 10 PO vO a rr r^ 10 0. ■* t t^ vO r^ r^ PO Ov vO r^ tn Ov vO Ov vO 10 r^ •!r 0\ 00 ^ ■O vO VO r^ 00 t^ t^ vO vO vO Ov •* 00 I/: Ov 05 00 PO Ov r^ t^ Ov t vO Ov -r 00 00 Ov w t -0 r^ l^ o> -t PO -t r^ 00 lO PO Ov vO PO >o t^ r^ Oi t> r^ •^ vO o> vt vO r^ 00 00 r^ vO lO PO vO Tj- 00 Ov 00 10 ■* CO W 00 PO 1^ 00 q HH "S- "? vq i^ 00 q q P) vq qv "-" PO t lO vO t^ 00 00 10 M IT) •6 ■d -d vd ■d r^ r^ 1^ r^ t^ ri r>- r^ 00 00 00 00 00 dv d dv d dv dv dv dv t Ifl vO t^ MD 00 a PO o> 0. o 00 ro 00 00 t^ lO PI PO PO PO t^ vO p) Ov vO PO 00 00 00 ^ 00 ■O o> vO PO Tf 00 vO 00 t 00 00 PO Ov Ov 00 •^ 00 -t ■ 00 "+ t Ov vO p« 00 P. 00 t^ Ov 00 t 10 vO Ov Ov fO 10 i/> r^ a CO 00 00 Ov P) t^ 00 vO Ov Ov Ov -0 00 ro ir> t^ PO T^ PO -0 't PO vO -l- 0. t 00 r^ 10 0. po o> ■pl- O. 00 -0 00 1^ •0 0. t PO PO i^ 00 00 Ov PO 1^ Ov -t •0 00 Tf r^ vO r^ OO vO Ov Tf vO M- vO •^ PO Ov N 00 po i^ t 00 t^ rf PO 00 -+ P) 00 PO vO t ir> t^ r^ ^ Oi 00 1^ ■* vO -t vO 00 a 00 t^ 1- 00 -t Ov •Pi- 00 Tf Ov 00 vO 0. N 00 M ■* vO q PO 10 Ov PO 1^ 00 q PO •pj- "-< vq q PO vq Ov k-< p< PO to t^ r^ t^ 00 00 00 00 d d 6. d> dv d p) d d d d ^ " ^ S ^ ' PO PI PO PI PO PO PI 4 PI 4 PI 4 PI 4 PI vO t^ 00 PO ~ vO r~ -t 7 vO PO 10 PO P, vO -t 00 t^ PO •t "7" ■* 1^ -t 00 t 00 r^ 00 t^ 00 00 vO vO vO vO •& ifl -T PO r^ o> 00 00 t^ Tf ^ PO Tr Ov vO 00 PO t m t t^ 00 vO Ov cc Ov r^ 00 PO •T vO to PO -t Ov 00 r^ 00 vO so Tf -t vO T)- Ov t p< vO t^ 00 PO vO r- 00 PO •0 ^ M 00 Ov t Ov -0 10 't ^ vO 00 vO Ov PO 10 o> 'O Ov r- Ov PO vO a 0. Ov t^ 10 VO •p(- Tf PO r^ 00 >o '? 1^ pr> 10 00 •H PO >o 00 P) t t^ 00 q p< rl- « Ov 10 t t^ q PI f •0 06 00 & 6< o. d 6 d d t-^ t^ M tj pi ri pi PI pi PO PO PO 4 4 10 vd vd vd r^ r-^ t^ [-!. N f>i P« p) pj Pi pj pj P) PI p< PI PI PI PI ^ tr, o> po 00 00 r^ •t PO t^ PO 7 -t PO -t PO t i^ 7 10 00 10 PO -t p, ■s- fO T)- t^ vO 10 T 10 a 0. Ov 10 00 vO Ov vO vO 10 PO ^ Cv PO 00 ir> vO Ov Ov PO Ov vO vO ■i- r^ PO vO PO ■* •0 a PO 10 PO l-l r- Ov t^ t-i 1- vO PJ vO Ov PI vO 10 10 Tt ^ t> i^ 00 Tf PO PO Ov -t vO 1^ •t 00 Tf 00 r^ Tt VO Ov 00 r^ t^ 00 ^ 1- 00 10 1- vO Ov 'I- PO 00 00 -0 PO 00 PO vO 00 vO PO PI 0. PO r^ -1- ^ 00 p^ ^ 1^ Ov P) 10 r^ q 10 t^ 1^ vq ^ t-< r^ PI vO q PO d d 6 6 »-l H4 t-4 ri pj PI PO PO PO PO 4 4 4 10 'O 10 UO vd r^ 00 dv d d d i-i w ^ M M w w N M PV) p) c< P4 p) M PI PI PO PO PO PO PO po N -c PO 00 t ■0 0. PO 0. ■»■ t 00 Ov 10 r^ 00 Ov Ov "5 PO vO vO -0 00 0. Ov r^ t lO Ov 00 I^ PO vO PO r^ 1^ 00 PO p< Ov 00 VO uo ^ vO 00 PO t 1- VO t^ i^ t^ Ov PO Ov vO vO vO VO r^ >o 'I- ^ 10 t ro po 00 tT PO -t Ov Ov PO ■* ^ Ov Ov ■0 •pf PO Ov Ov •i- 10 o> 00 vO PO Oi -0 00 vO vO -0 •pf Ov r^ 00 Ov 0. Ov ■* 00 0. PI o> t PO ^ PO t^ PO Ov 00 PO Ov •0 N M 10 Oi C4 vO q PO t^ q -t r^ PO vq Ov P< ■I- t^ Ov PI p-" Ov vq PI vq 6 6 )J M M pi p) ■0 PO PO 4 *f 4 •o 10 ui ui vd vd vd vd 00 dv d d ^ pi PI PO PO PO M .< " N c N " " p. P. P, P. p. p, N « "" p. p. p< PI PI PO PO ro PO PO PO PO PO PO •T « r^ 00 c> p< PO ^ vO 1^ oe 10