Manual / Guide
A loose-leaf set of Curta methods issued free by the British agent, Automatic Business Machines Ltd of London: general techniques, commercial work in sterling, engineering examples, and mathematical procedures such as roots, polynomials and series, with numbered Curta tables. Scanned copy; the text layer is OCR and carries errors.
CURTA CALCULATING TECHNIQUES AUTOMATIC BUSINESS MACHINES LIMITED I5, CROMWELL ROAD, LONDON, S.W.7. Not for Sale - Free Distribution INDER Calculation with the Curta Preface. Abbreviated Notation 2 Handling the Curta 4 GENERAL TECHNIQUES. Successive Multiplication Al Transfer Multiplication A? nuccessive Division . A3 Complementary Division AA Accumulation of Quotients A5 COMMERCIAL, Sterling C1 Curta Tables Nos. 1, 2, 3 and 4. Wages G2 Percentages C5 Amortisation of Debts by Annuities C10 ENGINEERS. Gear Ratios Kl nides of a Triangle 2 Heat Insulation E3 MATHEMATICAL. Square Roots M1 Curta Table No.10l. Cube Roots M2 Curta Table No.102. Fractional Powers of Numbers M3 Quadratic Equations 4 Cubic Equations WM eummation of Squares M10 Accumulation of Products Mil Evaluation of a Polynomial or of a Power Series M18 Evaluation of Series th M19 Construction of an n ~ order Polynomial from n Differences M20 Interpolation We] Curta Tables 103 and 104. Cotton Cloth Calculations Tl Curta Table No.20. CALOULATION WitTtkH THE CURTA PREFACE This manual is primarily intended to supplement the Manufecturera'! little handbook ‘Instructions for the use of the Curta Calculating Machine! which showld first be read by the Curta user. It is hoped, however, that it will prove useful to calculating machine users renerally. The operation of the wachine itself, whether Model I or Model II, ig sO simple that many users may not at first appreciate the large variety of calculations which can be aolved with the Gurta. Basically, foe Curta only performs repeated addition, but there are few operations of numerical analysis and virtually no commercial calculations «hich cannot be reduced to 4 process of repeated addition. It is the purpose of this menual to show the most suitable method or 'programme' by which common commercial ard scientific calculations can best be adapted for solution by the Curta. Further instruction sheets will be meade available from time to time and it is hoped that readers will send us a note of any methods, not so far indicated, which they have found useful for special purposes. In this way, a really comprehensive ‘Library’ can be built up. Every six months we will circularise users with a copy of the latest Index so that they may apply for any new instruction sheets likely to be of interest to then. Anyone who has attempted to put instructions for calculating procedure on paper will appreciate the difficulty of explaining 4 technique clearly, yet concisely. Calewlations which take only a few seconds to perform are liable to take as many minutes to describe, and Examples are more useful thar detailed descriptions. For these reasons we have evolved an abbreviated netation for describing the process of solution which we hope will be readily understood. We strongly recommend that the exemples should be worked out on a Gurte when the instructions are first read. Generally, a Model I will be sufficient unless otherwise indicated. fhe various uses of the Gurta will be indexed by a letter shaving the general type of calculation and a serial number indicating the particular calculation described. -% - ABBREVIATED WOTATIONR The Curta has 3 Registers, or groups of dials, which will be indicated by their initials:- Setting Register: SR Counter Register: CR Preduct Hegister: PR A carriage position will be indicated thus:- Carriage to position 3: G3 The carriage position indicated is normally that for the lst step of the calculation. The position of the Reversing Lever, that is the Lever which reverses the gearing of CR, will be indicated by an R immediately after the Carriage Position when the reversing lever is dow, i.6. in the Reverse Position (to count negative turns). If the lever is up, the R will be omitted, The position of decimal markers will be indicated as follows:- Decimal place: * Decimal places where markers aré alao sét for other purposes: Marker used for other purpose: Fy Where two markers are required, ¢.g. to indicate a split GR: Ki In order to indicate the position of a setting or result on the corresponding register and to avoid printing unnecessary zeros, an oblique stroke / will be printed either to the left or right of a number indicating that the dials on the left or right will read 0. To indieste that all the dials to the left of a number will read 9, a nine followed by an oblique stroke / will be placed on the left of the number, thus 9/. =-3- Examples: A setting of 4.5 38,/ indicates that 1.53) has been set on the four left hand dials of SR, 9/7,641:27 on PR of Model I indicates that the figures 999997,541:27 appear in the FR dials. One decimal marker has been set to indicate the ‘thousands’ and two decimal markers to indicate the decimal place. In shewing the steps of 4 calculation, SR will always be placed on the left, GR in the centre and PR on the right. An asterisk will be placed on the right of the result. Thus, if 1.534 ia to be multiplied by 37.2 and the result shown positively in the extreme left hand side of PR in Model I, the abbreviated description will be:= C6 (1253400 37.2/ 57. 06h8/* This means that we start with the carriage in position 6, the reversing lever up, set 1,534 in SH dials 6 to 3, multiply by 37.2 from the left and obtain the result 57.0648 in the left hand dials of FR, It is scmetimes desirable to set a complement into the machine or to convert a number, read from FR or OR, into its complement. This can very easily-be done mentally by deducting each digit from 9, except the right hand digit, which should be deducted from 10, Rrampte (4) 5390724 Set complement aa ; 9/4609276 9/4.00912 (41) Read complement as 599088 It will be found with 4 little practice that complements can be set or read almost 4s easily as the numbers themselves. There are several techniques in which the CR dials are not uaed and where it is not necessary to set the Reversing Lever down. In the examples, however, the figures appearing in the CR dials and the position of the Reversing Lever are generally given and the CR dials are cleared in order to show the turns of the handle which are made at each step. Of course full advantage should be taken of short cuts so that the number of turns actually made will be many less than the reading on the CR dials, =i HANDLING THE CURTA There are a few voints in handling the Curta which we de not think have been made sufficiently clear in the Makers' Handbook. tT. ae When holding the machine in the left hand, the third and little fingers should be clear of the setting levers to avoid moving these levers by mistake during operation. The carriage should not be gripped tightly by the thumb and fore- finger. When the carriage has been lifted by thumb and fore- finger, preasure of the forefinger should be inatinctively released ami the carriage rotated by the thumb only so that it must drop into the next position. During @ calculation the earriage should only be stepped one place at a time, even if no turns are to be made in a position. In this way, the carriage can be moved without looking at the machine and there will be no danger of moving two steps instead of one. The crank handle should be held lightly between the thumb and fore- finger of right hand. This will avoid the danger of passing over the zero position by mistake. Left handed operators will generally find it more convenient to hold the Gurta in their right hand, since they will Wish to hold their pencil in the left hand niilat Operating the machine. Normlly, the clearing ring should be left in the right hand position, i.e. between the first dial of PR and the last dial of CR. It is possible to clear a portion of FR or CR if desired and this can be very useful for certain calculations, If the left hand side of PR is to be cleared after each calculation, it will of ¢ourse be necessary to leave the clearing ring in the left hand position. The use of short cut methods of multiplication and division cannot be too atrongly satresased. The operator shauld get accustomed to short cutting when multiplying from the left or right and when using such special techniques as successive and transfer multi- plication describdd in Section A. Time gaved in rotating the handle is not the only advantage of short cutting in multiplication. It also minimises mistakes in counting turns cf the handle as the number of turns in any one position of the carriage (other than the left hand position) need seldom be more than five. When not in use, the Curta should be replaced in its rubber-sealed metal case, both to avoid accidental damage and the ccllection of dust and dirt on the setting levers, resulting in stiffness of these levers. If the levera should become at all atiff, the Curta should be returned to Automatic Business Machines Ltd., 1¢ Cromwell Road, London $W7. SBUCCESSIVE MULTIPLICATION Ad We give this name to multiplication of several factors, ¢.£. Ax bxeG « weeeeee Tt is, of course, possible to re-set each partial product but this wastes time and re-setting or writing dom partial products may lead to errors. Having obtained the first partial product on PR, sét the next factor on SR WITH THE RIGHT HAND OPERATIVE DIGIT (digit other than zero) reduced by 1. Step the carriage so that this digit is below the left hand digit in PR and rotate the handle through the number of turns corresponding to that digit of PR. Step the carriage one turn to the left and turn the handle through the next digit and so on, What we are really doing ia to add ab{e - 1) to ab, gettdng abe, Example (i) 123 x 2345 x 567 cl {e345 /123 /u2.435" oS /566 /y2435 /24.060, 645" Anawer: 24,060,645 Example (ii) 24,060,645 x 192.1, where the first figure is a preduct already produced on FR. If we have more than the CR capacity set on PR, we turn the carriage to ite extreme left hand position, 6 with Model I Curta, and set the new factor with the last digit reduced by 1 immediately under the left hand digit of PR, thus, C6 /192, 000 24,0606 1622041. 2645" Answer: 4,622,041, 2645 Hote that we set .O under the deft hand digit of PR. Note also that we have not been able to step the carriage so as to multiply by the two right hand digits, 4 and 5, and the result will therefore not be exact. In fact, it will have an error up to 5 in the sixth place of the answer, If we want an exact answer we must either re-set the last factor on the right of GR and multiply by the remaining digits, having stepped the carriage to the left hand of these digits, or we must re-set the last partial product and mul tiply by the last factor in the nermal manner. Thus, with the former method, we have % G2 #1920 fus 4622049, 9045 Answer: 4,622,049, 90,5 Ad An objection may be made to this method of succeasive multiplication on the grounds that we cannot check the multiplier, unless we Write down the partial product and clear CR, A method of overcoming thia is to reduce, not the right hand operative digit, but the next sero on the right, by one. Thus, taking the same calculation as before, viz 123 x .345 x 567 C4 fe 34S /1283 /42 435" Clear OR C5 (566.9 /h2h35 (24060. 6450" Netice that as we rotate the handle through the number of terms corresponding to the digit of PR immediately above ',9', this digit ie reduced to zero and therefore there is no danger that we turn through the wrong number of turns. Notice alao that if we wish to use the decimal markers we must move the marker in FR one place to the left in the second stage 4s there is then one decimal place in SR. There are occasions when it is required to multiply a negative quantity on FR (appearing as a complement) by a positive quantity, producing the product as a complement on FPR; or by & megative quantity, producing a positive amount on PR; or to multiply a positive quantity on PR by a negative quantity, producing a negative ancunt on FR, A variation of the Successive Multiplication technique is used and is applicable te such problems as evaluation of Polynomials {see M18) where some of the terms are negative. Exemple (i) —123 x AT, where the complement, 9/877 ia set on FR. Set /46 on SR, Carriage to Ch so that right hand figure '6' on SR eomes below the right hand '9' on PR. Make one negative turn. Carriage to 63 and proceed as for normal successive multiplication, thus: - Ch. /h6 of 877 9/4219" The amount showing on PR is the complement of 123 x 47, i.e. of 5781, Example (ii) -123 x -47, where 9/877 is set on FR. Set /48 on SR and proceed as before but making one positive turn in position Ch and megative turna thereafter, thus:- ch oF /48 9/877 /5781* A Example (i1i). 123 x -47, where /123 is set on PR. Set /48 on SR, carriage to C3 and proceed as in normal successive multiplication but making subtractive, instead of additive, turna. cs /ho /123 9/4219 We can thus formulate the following rules. (14) When multiplying by a negative quantity, add '1' to last operative figure set on SR, instead of subtracting '1', and rotate the handle in subtractive position through the number of turns corresponding te the digits on FR, (2) If a complement is set on PR, step the carriage ao that the right hand figure on SR comes below the right hand 9/ on PE, turning the handle in the opposite direction to that used for the other digits of FR. TRANSFER MULTIPLICATION We give this name to 4a methed of multiplying «a figure produced in CR by a figure set on SR, without re-setting the CR figure. This method can be very useful in series of multiplications and divisions, such as axbxe dxe This celeulation is made in atages - firat ax b = f om PR, then f/d = g on GR, then g xe = h on PR, finally h/e = i on CR. Example: 123 x 345 x 567 23h x 450 C6 {35 123/ /421,35000°" Cleer CR Cc R 234 181346 {36 Clear PR cl Rk 567 G00000 /No028234182" Co OUR 456 225489" /198 Note that in the 3rd stage, when we have to multiply a figure ch GR, 161546 ty 567, we merely set /567 on SR and using positive turns, reduce each digit of CR, in turn, to zero, This is very quickly done, with virtually no risk of mistake, by positive turns. Since the Reversing Lever is dom, the respective figures in CR will be reduced by the number of turns made, At the same time the figure set in SR is multiplied by the number of turns made. We can, theoretically, carry on like this indefinitely, alternately multiplying and dividing, After the first stage the Reversing Lever can be left dow, Generally it is not worth while setting decimal markers, unless 4 number of very similar calculations have to be made, and the decimal markers can be left in position for al] stages, If the number of factors in the numerator exceed those in the denominator by more than one, we oan use successive multiplication, per method A?, at one or more stages, If the factors are identical in number, we can start with a division. If the factors in the denominator exceed those in the numerator, we must either multiply some of these together or re-set the quotient at some stage, or use the methad of successive division described in A5. SUCCESSIVE DIVISION AS Suppose we have a figure in PH at some stage of celculation, and we want to divide it by some divisor and obtain the quotient in PR, perhaps because we want to divide the quotient by another figure, We use a method which we have called "Suecessive Division", because of the analogy to successive multiplication. The method is sometimes called ‘Complementary Division'. Example: Divide 567 (already produced on PR as, say, 56700000) by 4666 Set the complement of 456 on SR, preceded by a figure %, thus cé / Dds, 17a Th2 /7126342 (0048) Anemer: /, 124342 Note that although extra figures may be preduced in PR, the quotient can only be obtained accurately to the same number of figures ag the capacity of CR. Note too that the quotient is £132 praluced in GR and this can he used at a check that the calculation has been performed accurately. Successive division ean bé very useful but takes a little practice. The operator snould carry the two left hand figures of the divisor jn his Mind during the operation, Sinoe the quotient remains in the PR, it can be divided by a further divisor, either by aubtractive division or by successive division, COMPLEMENTARY DIVISTON Ak In a division calculation, instead of setting the dividend on FR and proceeding by Subtractive Division, there may, in some cases, be an advantage in setting the complement of the dividend on PR and building wp PR te sero with the divisor set in SH. This is particularly the case with such calculations as ax bh, carried c out in one operation, when we wish to obtain the raximum capacity of which the machine is capable. Example! 123. x poeise the answer required to & places ead} of decimals. Using a Model II, cé R “56789 JI C0000 9/'54321 100000 Clear CR cé O1230235,567 /4194737 25952650997 3879 Answer = 2359527 We use the right hand side cf SR, 234567 to build up the right hand side of PR to as hear zero as possible. Meam-hile, the left. hand side of SR, 0123, is multiplied automatically by the quotient of 456789/234567 and the product shown in the left hand side of PR. Note: The positioning of the Reversing Lever and the clearing cf CR are actually unnecessary since the CK dia] is not used, but they are specified here to shor; that the handle has been given one hegative turn in the 6th position of the carriage for the lst operation and the quotient is shown on CH in the 2nd operation. Note also that if we had made the calevlation ty setting 01250234567 in Sk and building up to 456785 in PR, we should heve had the wrong answer, 239572, because the left hand portion of the multiplicand, 4.56769, would have overlapped the ccrrect answer, A506 ACCUMULATION OF QUOTIENTS Quotientes can be accumulated on CR ny build-up divigion, the reversing lever being pushed up for positive terms and down for negative terme. To set the decimal marker in PH, add the maximum number of decimals in the denominators to the number of decimal places required in the answer, plua 1. Ticimal markers ahould always be set when accumulating quotients. 1.39 7-465 Lie The answer is required to three places of decimals. Decimal markers are set before positions 3, 7 and 4 in SR, PR and GR. Cé, /1.390 /23.3453" /32.4499670 Clear PR cé [1-465 /32.6956* —/69.7999895 "oo" cé R 11.700 /24.0341* /101.3395500 Anewer 24,034 When the numerators are the gum or differences of other numbers the accumulation ean still be carried out in the above way, but using subtractive division, provided care is taken to correct CR as in the example below, In calculating the Position for the decimal marker in PR, we must add to the maximum number of decimal places in denominators or numeratora, the number of decimal places required in the answer, plus 1. Example (2) 13.475 - 5.75 , 17.24 + 3.92 _ 18.715 6.29 7-56 9.5 The answer is required to three decimal places. Decimal markers are set before positions 3, 7 and 4 of SR, FPR and CR. A.5 Ze G5: -R /13.475 99,/ /13.4750000* R= /5..750 0 /7.T250000* coc; Rr /6,290 /1.2281* /0.0002510 Clear PR ce} Rr /17.240 /.2281 /17. 2400000 R /3.920 9/2261 /21.1 600000 Now clear SR and make two negative turns to correct OR. G5 t oO 1.2281 /21.1600000 ec} ‘R /7.860 /'3.9202 /oo.o000940 Clear PR The last quotient can be obtained by building-up division, but we keep reversing lever down because it has to be subtracted. C5 R /9.500 /1.9502 /18.7150000 Answer 1.950 STERLING C1 (a) Decimaliving oteriing calculations are normally carried out by converting shillings and pence into decimals of £1, or pence into decimals of oné shilling. The latter is used mainly for small amounts. Curta Tables 1 and 2 show the decinal equivalents of pence and fractions of pence expressed az decimals of £1 and one shilling respectively. Curta Table § shows shillings and pence as decimals of £1; shillings can, however, easily be expressed as decimals of £1 by dividing by #. Thus i7/- = .85 of £1 Example [1 Find the cost of 455 articles @ 3/644. each, From Table 2, 64 is equivalent to .54167 of 1/- We therefore get 3.54767 on SR and aultiply by 455 in the normal Manner. cl /3. 54467 {455 /1,614.45985" Mentally divide 1,611 shillings by 20 obtaining £80, 1i1/- .. Locking up the nearest figure to .45985 on Table 2 we obtain 4583 or Sed., so that the answer ig:- £80, 11,54. The pence may alternatively be found ty mentally multiplying the decimal by 1? or by memorising the first two décimal places, if much ef this work is being done, Example (ii) Find the cogt of 215 shares @ £1. 3. Tad. each. Sei. the pounds on SR foljowed by the shillings divided by 2 with, gay, § decimal places, thus 1.175000 From Table 1, Yad. is equivalent to .03125 of £1. We therefore add this to the previous setting, so that 1.18125 is set on SR, er /1AB125 (245 (253, 96875, The answer is thus £253.96875 or £253. 19. dad. The shillings can be found by reducing the first bro decimal figures to a multiple of 5, in this case 95, and multi- plying by 2. The remainder, .O18/5, can be looked up in Table 1, Biving Rad, or we can regard the 2nd and 3rd places of decimals, in this case 18, as farthings, mentally dividing by 4& te obtain ibd, If the pence figure thus found comes to 6d. or over we deduct a farthing. Thus, for .037, which lies between 8% and 94, we obtain Sid. less Jd,, equals 4. Gl Example (iii) Find the cost of 5,150 erticles at £3. 17. 7d. From Table 5, we obtain .&791667 for 17/74. cr 3.8791 667 [54150 19,977. 708508 Answer = £19,977. 14. 2d. To decide the number of decimal places +o be set on SR, at is advisable to add the number of digits in the cultiplier to 3, if decimals of fi are to be set, or 2 if decimals of 1/- are set, This will enable the answer to be given to the nearest farthing, Thus, in the above example, we hed & figures in the multiplier sa that we set 7 decimal places of £1. (b) Pence Transfer Tne disadvantages of decimalising sterling are that a table is required, unless a skilled operatcr is employed who cen hemorise the decimal equivalents, and, if the multiplier is large, 4 considerable number of decimal places have te be set. An alternative method, whick is especially usefy] when the zum, only, of a number ef sterling products ia required, is to set the shillings and pence, separated by several seres, After multiplication, the PR shows the answer in shillings and pence. The surplus pence (in excess of lld,) ean be converted into shillings by an adaptation of the method of Successive Division given in A 3. Example {i} Check the following Invoice in total. 125 dozen @ 3/7d. per dozen 23. Fa ay 65 articles @ 13/11d. each 45, he Fae 75 lbs, @ £1. 3, 5a, per 1b. 87.46. 34 ct /3,000007 /125 /375, 000875" Clear CB ot 13,00C014 /65 /1220,001550" Clear CR ct 23, 000005 /i5 /2545, 001965" Clear CR Now follow: the transfer of the surplus pence C3 / 999988 fi63 /3108 000009" Answer = 3108 shillings and ninepence = £155. &, od As the operation is one of checking only, it is not really necessary to Clear OR, The number of zeros to be left between shillings and pence depends on the quantity of items and number of entries on the invoice, Here we could have reduced the number of geros by 2, If, at some stage of the calculation, the pence threaten to flow over tothe shillings in the PR, we can always make a pence transfer and then continue with the other items on the invoice. In making the pence tranafer, we set BS under the right hand pence figures and 98 up to the shillings figure. We can deal with fractions of pence by decimalising theo, as this is easy, Example (ii) 38@ 2/780. 5. O. 640, 450 @ 413/ola, 99, 1. 34, 1h @ 5/1114. 42, 15 8, £145. 16, Sha, ct /2,0007. 75 (38 / 76,0294, 50* C1 43,0002,50 188 /2026 0669, 5c" cH /5,0011.25 (332 /27h6 , 2289, 50 * C3 / 9988.00 /522 / 2936 ,0009, 50 * Answer = £146. 16. 9d, This method of checking invoices is very speedy and can be carried out by operators with little training. It pives a more independent check than checking each item. 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Example Assuming a standard working week of 44 hours, caloulate the wage earned in the following cases, Employee Standard Wage Equivalent Nunber Wage of Hours Worked Earned, A. Smith 6& 15. 0. Lis Ts Se 9s w. Jones n Lg fe 2On Se H. Brown a 5as 9% 2. Ta R. Fobingson te Be! Be 375 & OO; 8. & Green A 59 % lil. IL. Tt is best to use 2a Curta II, The standard hours are set on the right of SR and not altered. The standard wage is set on the left of SR and only altered when the standard wage group alters. We build up the standard hours to the equivalent number of hours worked, on the right of PR and the wage earned is show on the left of PR. We only ¢lear the CR and PR dials when the standard wage changes. ch /6. 750000044 /10795 —/7.. 2866250, ,47.4980" G3 f14136 #7. 5168000, ,48, 998," oh /413352 «= 9, 0426000, 58. 7,88" Clear cR & PR ch060 6 /7.1250000), 866 /6.0320250, ,37.2 50K.” Ch (13409 95559125, 58,9996" The anawers, as given on the left of PR, are written dom on the wages sheet, as above, usually rounded off to the nearest penny above the figure calewlated, PERCENTAGES C5 Percentages are used in a great variety of ways in Commerce and Industry, A few of these are dealt with in "Instructions for Use of the Curta". The object of these notes is to summarise the various types of percentage calculations and to show the quickest way of obtaining the anawers on the Curta, i axtb (2) Add abtob = beaxbdb 100 (3) Rebate or Discount of a®onb = b-axb 100 (4) a expressed as Sage of b= 4 “4 109 a, b and ¢ expressed as es o elir am, a+b+e (5) b and c exp d as Sages of thei b 2 a x 700 ‘ 7 Abe cg? HS (6) Selling price to give a percentage profit of a % on sale for , b a cost price bo = = 1 = 706 nerease or rease expressed as a percentage o é (7) a Dec xpressed p tage of th larger or smaller figure, If b is greater than a (pb - a) x 100 a (b> --a) x 100 b % increase or decrease on a % increase or decreese on b (8) Successive Percentage corrections, 4 + ah + b%- ch cote, A percentage ia normally required to a certain standard number of decimal places, so it ia advisable to set the decimal markers, once and dor all, before the set of caleulations ig commenced, Thia Will reduce the work to the minimum, C5 . £5 (1) a®onb = 00 We merely set b, multiply by a and mark off 2 additional decimal places on FR, Example (i) - . Calenlate a dividend of 43% on £1,115. 10. 6. stock ; # ct /'44,4154525 fneS /50. 198625 Anawer = £50. 3. i124. Note that there are 2 more decimal places in PR than in SR + CR, Unless the percentage js more than 100, if is normally never necessary to set more than 3 decimal places in SR for sterling amon tS. (2) Add atob = braxt 100 In such cases, as in the extension of invoices, it is often desired to know both the percentage eddition and the total. If we only have one such calculation to make we can set b and multiply first by = and then by 1, without clearing PR, If we heve & humber of such caloulations to perform with the same percentage, we can set rece oh the right of SR, 1 + ris on the left and multiply by b, obtaining the percentage on the right of PR and the total on the left. Example (ii) Add 174% to an invoice for £13, 14. 7. c1 {13.729 (765 / 2.402575" oy. /117.5 /16,131575* Answer = Net amount of invoice £73, 14. Fe Add 174% 2, 8 of, fi6. 2, Fh. C5 de (345 Add 123% to a number of invoices, Using a Curta II, First invoice for £18. 4. 3 eg 1.1250000,125 /18.237 20,51662522. 279625" Answer = Net amount of invoice B18. kk. Fe Add 123% 2, 5a Ta £20, 19, 4. This method is obviously impractical if the amount of the invoice is large or there sre too many significant figures in the percentage. The setting will be left for the next calculation. (3) Rebate or Discount of afionb = b-axb 700 Precisely the same considerations apply as for (2) Example (an) Deduct a discount of 23% from £136. 15. 4. ct /136.766 (2.5 (3.419150 " Ch /97.5 /133.346850 " Answer = Grogs amount of invoice £135, 15. he Lesa Pay discount % 6B Oe £1353. 6. 11%. (4) ‘a expressed as a Sage of b = a x 100 b This is simply a division sum and is best carried out by build up division. Example (v) A mixture containing 1,137 litres includes 39 centilitres of alechol. What is the percentage of alcohol to 2 places of decimals, Set 3 places of decimals in OR {for the answer), There are 3 decimal places in SR, so there will be 3+ 3+ 2 (for Sage) = & in FR. Set 1.137 and build up to .39 in FR, cs 10137 f5n.501" 39000237 Answer = 34, 501% C4 (5) a, D.and c expressed as percentages of a+b+ece We set decimal markers on SH, CR and PR, adding # for FR as we require the answers as percentages. We set a + b + ¢ on SR and build up in turn to a, b and c in PR, without clearing any of the dials. Example (vi } The cost of a batch of articles 15 given as Tfollowrs:- Direct Wages E159. Th On Materials 57, Ws. Gs £310, 6, 6, ee es ee me Compute the percentage of each on the total cost, to 2 places of decimals. Ch. /310,.025 /4.5.06* /139,6972650 /18.65" /57.8196625 (36.29% /4142. 5080725 fAnewer = Direct Wages LS, O85 Materials 18.65% Overheads 36.29% 100, 00% at The carriage position is simply chosen by eye to enable the dividend to te built up correctly. In this case 4 happens to be the best position te start the build up in each case. None of the dials are cleared during the operation. The total of the pereentages should, of course, equal 100% vith .01% either way. If there is a large number of constituent parts, ¢.f. more than 5, if is worth while computing the reciprocal of the sums and multiplying each part by this reciprocal] in turn, C5 Thug, in this example we might compute 100/340, 025 by build up division to 100, obtaining .32255. Set this figure on SR and multiply in turn by the three constituent parts, 139.7, 57.825 and 112.5. (6) Selling price to give a percentage profit a on a cost price b = b ae {00 Example (vii) A number of articles costing 22/9, 25/6, 34/3, etc. have to be written up to their selling prices to the nearest penny to give a profit of 223% on the selling price. 100 If there are a large number of articles, we compute 700 - a! a 400 suai PT set the quotient on SR and multiply by each cost price in turn. It is better to work in decimals as if we require 2 places of decimals in the answer to give the nearest penny, and as there are at mogt 2 digits in the shillings, it is sufficient to have & decimal places in the quotient. We therefore set the carriage in position 5 for evalueting oe C5 7705 /1,2903* /.9999825 Clear CR & PR cl /1,2903 - /22.75 /29,354325* do ce (25.50 /32.902650" de C3 (31.25 /80, 321875* Anawer = 29/4, 32/14, 40/4, eter (7) Increase or Decrease expressed as a percentage of the larger figure, b, or the smaller figure, a. b= 700) b % increase or decrease on a 2 = ie - 1)100 4 b-ax 100 _ b Il i rh 1 lo Sama 5 & increase or decrease on b C5 We have merely to divide b by a or a by b, either by subtractive division or by build up division. We oan make use of the Reversing Lever, either to subtract one from the gquetient in the a firat case, or to give the complement of } on CR in the second cese. Example [viii] The turnover of a business increases from £567,450 to £786,58). Find the Sage increase to tre places of decimals, We require b- ax 100, Using subtractive division we set a 786584 and make. one positive turn with Reversing Lever down, 05 OR /786584 99,/ O7B65E4, To net clear dials C5 R /5E7A5O /38.62* 9/ 848100 Anawer = 48.62% The above operation can also be appreciated in this way. After 567450 has been set and one subtractive turn has been made, we ‘are left with O on CR and b - aocn PR. We proceed to divide this by b =a on OR, aso that we obtain e (ix The turnover of a buginegs decreases from £786,584 to £672,410. Find the fege decresse to two places of decimals. We require b- ax 100, Using build up division, we set b 786584 and build up to a with reversing lever down, C5 / 786584, 9/14..52* 6723720032 dnswer = 14.52% The rule is therefore: Set the higher of the two figures, with the reversing lever down, and place carriage in position to give required decimal places in answer, If the change is an increase, make one positive turn, set the lewer figure and reduce FR to zero, If the change is a decrease, build up to the lower figure, C5 (8) Suceersive Percentage Corrections! d + af + b= c% etc, Such calculations are comron where prices are fluctuating too repidly for catalogues to he amended. Example (xc) £5. 1h. 6 + 5% + 122% - oom We can set 5.725 and multiply successively by 1.05, 1.4725 and ,3O75, Using Curta IT and method given in At. C1 /54725 1,05 f6.01125" Clear CR o6 1s 12k /6.01125 /6. 76265625" Clear CR Og #5. 740 (6. 7626562 /6,99358979505" Answer = £6. 11. 105. Mote: It ig not necessary to clear CR nor to write anything. down except the answer. Nor ig it necessary to proceed with the last step beyond 4 places on CR in this particular case. If such successive percentages have to be applied to a mmber of prices, it is better to work out the equivalent percentage once and for all. Thus, if we multiply 1.05 « 1.125 x -975 we obtain 1.45174875 so that + 5% 4+ 122% - 228 is equivalent to +75.172%. If we add this percentage to £5. ih. 6. we get 6.593597 or £6, 11, 102, as before, AWMORTISATION OF DEBTS EY ANNUITIES C410 Loans may be repayable in a number of ways but a common method is by equal payments at regular intervals. Such payments may be calculated to repay the loan over a fixed perscd or the annuities may be a fixed proportion of the loan, (a) Repayable over a fixed period, Part of the annuity represents interest and part repayment of capital. <A achedule is required showing the anounts outstanding at the end of each period and the interest and capital portions for the next period, The amount of the annuity can be found from an annuity table. Example: A loan of £750,000 on 1.1.54, carrying interest at 34% is to be repaid by 10 equal payments at half-yearly intervals. The amount is the same as if a 12% loan were repayable at yearly intervals, The amortisation factor found from the table is .1008753442. Multiplying this on the Curta by 75, we obtain the half yearly payment £82,106.51. We shall leave the figures in decimals. We enter the half-yearly payment on PR and set the rate of interest on SR. We multiply this negatively by the amount of the loan with the Reversing Lever down, so that we subtract the interest for the lat half year from the half-yearly payment, iéaving the capital portion in PR, whilat CR shows the amount of the loan, The former is noted down and used to miltiply the interest rate, positively, and added to PR. In this way the PR shows the capital value of the annuity for each half year and the Cr shows the outstanding amount of the loan at the beginning of the half-year, The operations on the Curta II are as follows, FPR and CR being scheduled at each step. The setting remains constant at the interest rate. C410 C1 (1098753442 {75 /82,406.508150* Clear CR and PR This gives us the + yearly payment, £82,406,54 C1 /hehe.510000 4 8205, 510000 Clear CR Date of Outatanding > Yearly Interest Payment SR Loan (CR) Amortisation (obtained ; (PR) later) 1.7.54. Ch R -/4.75.. 750,000.00 69,281.51 13,125,00 1.7055. CF R 610, 224.56 71,727.58 10,678,93 1.1.56. Ci R 5.38, 496, 98 72, 982,84 9,423, 70 4.7056. CP R 465,514.17 Fi, 260, 01 8,146.50 1.4.57, CH R 391, 25L.16 75,559.56 6,846.95 4.7.57. G7 R 515,694.60 76, 861.85 5,524.66 1.1.58, °C R 238,812. 75 78,227.29 L179. 22 1.7.55: C7 R 160,585.46 79,596.26 2,810.25 4.1.59 Cf 2&8 BO, 989, 20 60.989. 20 1,447.31 £750,000,00 £74,065.10 After the first 9 repayments the = yearly amortisation figure, £80,989,20, should equal the outstanding amount of the loan thus proving that the calculations have been made correctly and that the * yearly amortisation has been written down correctly. There may be a small discrepancy due to rounding