Manual / Guide
Contina's worked-examples booklet for the Curta: the four basic operations and the short-cut methods built on them, shown step by step with dial settings. Scanned copy; the text layer is OCR and carries errors.
i = a 4 ene —_ i, me 4 I i i iT iM a ey uta: ba 7 ; . Pies ry j | } \ ma) mo i 1 io fs i | | re i ‘ \ ti dah ¥ x all = te ss " Fe Te ae i By I: si | i ' i 1 pay. i : a if F i i rt 1. fy i TO ee “CONTINA. AG, ik VADL HN ae OTL ee f: PRINCIPALITY | OF des ve ICA ND CUSTOMS MS S UN ION WITH SWITZERLAND) EVs " Hs he 2) il Ba fi ee i a tie ss ne ire, fet! antlees f se ee * J is rani age at § Cina Paces taf a ES Be ee ee 3) tha, =e Ree: “ Ha = LE i. ue ant ne i a cS UA 1) ES ne est srr Lele it T ma il = " =| at Fal pat) Fh ad a hi 7 Ne ’ SES ly ain rash ens, a he aie eat ep aA ee rea Ca i ay! La Harst A Ms i ALA, Re ah af oe Ke fa pet aie : o 5 Pray ie? fet Ait ove ayia mi Biss “ate Ne igi =f fl bee i ey Ba use 13 ve iy eae I | fi. vty iL i MY : Ae LL i fasts LGN a LIMA agai We iy, bie : te A he Bs ee ie ee - : L, 7 4 ' ke . | ang ] * ‘be Fa - . Hue y * ie ay ik in . \ cul aif dee =f a Lae | be a at Co aan asp ig rel wl a J if ig) suiles Cig ‘ He) ee H he 2 Computing Examples for the CURTA Calculating Machine Ci CONTINA AG, VADUZ PRINCIPALITY OF LIECHTENSTEIN (ECONOMIC AND GUSTOMS UNION WITH SWITZERLAND) Reproduction of text and illustrations, in whole or in par ls subject to our written consent Gopyright by CONTINA AG, Vaduz, Liechtenstain Introduction These compufing examples complete the short instructions sheef “YOUR CURTA CALCULATOR” which accompanies every CURTA machine, and if Is assumed that the use of fhe machine for effecting the four arithmetical rules, as described therein, is undersfood, Every example in this collection relates fo both CURTA models, with the exception of those which are marked "only CURTA Model II". The two machines differ only in the number of figures which they can handle, These are: setting Counter Result Register Register Register for the CURTA Model | 8 6 11 for the CURTA Model Il 11 8 15 In every example the following abbrevia- tions are used: S.R. = Setting Register; C.R. = Counter Register; R.R. = Resulf Register. The expression "Machine ready" signifies thal 1. S.R., C.R. and R.R, have been cleared; 2. The operating handle is in its zero stop position; 3. The carriage is in position 1; 4, The reversing lever is in ifs normal (upper) position. The following list of contents will give the reader a survey of the subjects considered. CONTINA AG. Vaduz / Liechtenstein 3 Hal manaxs Wieve ‘nl jee she e ao I veo as “aa. sath er noteesrane mr Daas tiell dlisns doriescac ent, ey hg ‘oHized “i's ca 7 ae | pnitieves sft ib | ina protons (rece) | - nottoubot “her ets anoil A Oe ihe Rai shea io 2° fag | a i gi Soslniey edt e & fall priwollo} sill Et | ho Vevive 6 1sbes% aa elon pallugmos sz26rnT triajtauutent tore’ W. "AOTAIUDIAD: H bre. rnirnpes ATAUD soniisem oft te oun neti teoitionntltlns | (bosluebru, _ > oe ee Date et — oy rb cai i" ayiova | intact ats ricte g2arit wi ont, “It daboM: Ho yedmuin ant rt ete oven} jalbnarlt 4 a lobo ATRUD od sol FS pero! ata ent vol CONTENTS General Division by breaking down (subtractive method) Division by multiplying by a reciprocal The rule of three . The rule of three in a single cnloulation fonly, CURTA Medal II) Extended rule of three axbxXc Calculation of roots : Continued multiplcation aXbXeXd . Cubes without intermediate notes Commerce and Industry Checking of invoices and goods . Percentage calculations A) Percentage increase B) Percentage decrease C) Profit margin . D) Compound percentages E) Profit and loss. F) Capital and interest 10 11 12 13 16 nizd 22 24 24 25 26 27 ai Costing : Costing with simultaneous coniral only CURTA Model II) Calculations with nines transfer : Exchange calculations . . Calculations with English currency Siatistics : Simultaneous accumulation of a sum and a sum of squares (only CURTA Model I!) Computation of arithmetic mean and standard deviation Technical and Survey calculations Division info a negative number aie aa number) Calculation of co-ordinates . Determination of the amount of silver in an alloy! (only CURTA Model Hl) . Deiermination of the angles in an acufe-angled friangle, given three sides (only CURTA Model II) . . . , Determination of a side of an AN angled friangle (diver the: two othes sides and their included angle) Calculation of area from co-ordinates . : : Calculation of the distance between two pains given their Povordinates fusing Pythagoras’ theorem) Calculation of distance and azimuth nly CURTA Model 11) Linear interpolation 6 28 29 30 31 33 36 37 40 Ai 42 43 45 46 48 49 a1 General Division by breaking down (subtractive method) actb+e 9526.8 + 37.51 + 645.62 =i? d 3 (5 or 6 figures are required in the answer.) Machine ready, To begin with if is necessary to add fogether the three ferms in the numerator, In order fo accumulafe fhe sum as far fo the leff as possible in the R.R,, if is necessary fo move the carriage as far fo the right as possible. Bul care must be faken to move the car- riage only so far to the right that space is left to accommodate the most significant figures, Put the carriage in position 5. Set decimal points between positions 2 and 3 in 3.R., and positions 6 and 7 in R.R, Step 1. Set the first ferm of the sum on the right hand side of S.R. (00... 9526.80), One plus furn of the operating handle. Step 2. Set 00... 37.51 in S.R. One plus turn. Step 3, Third term in S.R. One plus furn. sum in R.R. is now 10209.93. Clear only the C.R, and S.R, Now follows the division by 3. by the break- ing down method; Carriage in position 6, Set the divisor 3 in S.R. as far fo the leff as possible, but not so far that a minus furn would cause a ne- galive number to be indicated in R.R.; i. e. sel 3 in posifion 5. Place a decimal point between positions 4 and 5 of S.R. Place the position. reversing lever in its lower With the operating handle pulled upwards we continue to subiract the contents of S.R. from the contents of R.R. until the latter indicates a negative number, i.e. until a row of 9s appears (or possibly a row of 9's and an 8); thus turn the handle in its minus position and observe R.R. After 4 minus turns R.R. indicates 9... 8209930. Now make one plus furn; a row of 0's now appears in R.R., and the number in R.R, is in fact 0... 1209.930 ... 0. This number is the re- mainder after dividing the original dividend by the number now in C.R,, i.e. 3, which is the most significant figure in the quotient, Set the carriage in the next lower position, i.e, posifion 5. Again furn the handle in its minus position until a row of 9's appears in R.R., which occurs in this case after 5 turns. Make one plus turn. Carriage in position 4. Turn the handle in iis minus position until a row of 9's appears 3 in R.R, as described pre..cu_y, and again one plus turn, Carriage in posifion 3. 4 minus ijurns, 1 plus turn. Carriage in position 2, 4 minus turns, 1 plus turn, Carriage in position 1, Make one minus turn and zero appears in R.R., which indi- cafes that the division is at an end. (In the case of a division in which there is a re- mainder, the remainder appears in R.R.) Check that the number 3 (the divisor) is in S.R. From the rule concerning decimal points in a division”) we have 6 (R.R.) —4 (S.R.) = 2 decimal places in C.R, Thus we set a de- cimal point between. posifions 2 and 3 of C.R. and read the resulf (i.e. the quotient) as 3403.31. Return the reversing lever to its normal position. By using a CURTA *) Soe short Instructions sheet “Your CURTA calculator”. Model II (which has at its disposal 8 places in C.R.) a quotient of 8 significant figures may be obtained. Division by multiplying by a reciprocal (If we require a number of dividends {to be divided by the same divisor we can set the divisor as a constant number jin S.R. and build up the successive dividends in R.R., without resetting the divisor in S.R. and clearing C.R, and R.R. each time. The successive quotients appear in C.R. and can be noted down. See the costing example on p. 28.) If, however, a large number of dividends are to be divided by the same divisor, it is simplest to calculate the reciprocal of the divisor once and for all, and place this quantity as a consfant mulfiplier in S.R. The divisions are then transformed into multiplications, e. g, 16330 = 717 344.5 11.7 67.8 = 11.7 The division 1 + 11,7 to 6 or 7 figures gives us the reciprocal of the divisor 11.7, namely 0.0854701. We place this number in the right hand end of S.R., clear C.R, and R.R., and set a decimal point between 7 and 8 of S.R. The three divisions may now be carried out as 3 normal multiplications, namely, 139.572 0.0854701 x 1633 = 0,0854701 x 341.5 = 29.1880 0.0854701. xX 67.8 =. 5.79487 An experienced computor will dispense with the intermediate clearing of C.R, and R.R. affer each multiplication, and will build up the conients of C.R. to each successive multiplier by appropriafe plus and minus furns of the handle. The rule of three 180 X 46 axb . ness c 144 First: Method (subtractive division) Machine ready, First we form the product 180 xX 46 and then divide by 144, The multiplication 180 X 46 is performed in such a way thal the product appears as far fo the left as possible in R.R. Set the number 46 in S.R. by means of knobs 2 and 1, With the carriage in posi- tion 6 make 2 plus turns and in position 5 2 minus turns (shori-cut multiplication). The product 8280.000 is now in R.R. (with 3 de- cimal places in accordance with the decimal rule), 10 Now follows the division 8280 — 144 by the subtractive method (see the previous exam- ple), Clear only C.R. Set the number 144 in the right hand of S.R. Place the reversing lever in the lower position. The process of subtractive division now fakes place with the carriage starling in position 5, In this case the division produces no remainder, and the result (quotient) In C.R, is 57.5. Return the reversing lever to its normal position. Second Method In numerous calculations involving the rule of three the calculation of the intermediate quantity — — is also required; e. g. If certain articles cost 180 francs/gross, how much does each article cost, and how much do 46 articles cost? Machine ready. First we must calculate the cost of one article (180 + 144) by additive division, so the number 144 is set in S.R. by means of knobs 3—1, and with the carriage in posi- tion 6, fhe dividend 180 is built up figure by figure in R.R. by positive turns of the operating handle. The quotient appearing in C.R, is 1.25 = cost of one arficle, Since we are about to multiply the cost of one article (1.25) by 46, and the number 1.25 is already in C.R., clear only R.R. Sel the number 46 in the right hand end of S.R. and place the reversing lever in the lower posifion. Starting with the carriage in position 4, reduce every figure in C.R. fo zero with plus turns of the handle, This indicates a mullti- plication by 1.25, and the reduction of the contents of C.R. fo zero is a check, (See the examples on pages 12, 34, 43.) The result in R.R. is 57,50 francs, namely the price of 46 articles. (Refurn the reversing lever to iis normal position.) The rule of three in a single calculation (only CURTA Model I!) In those cases in which the numbers in- volved are given only fo a small number of figures, the rule of three may be effected af one blow by using a CURTA Model Il, the quotient being obtained with 4 or 5 figures. 1764 X 375 144 Machine ready. The division 1764 + 144 is carried out al the left hand end of R.R. by the additive method. At the same time the quotient of this division is multiplied by 375 in the right hand end of R.R. S.R. 14400000375. Place the carriage in position 5 and by turn- ing the operating handle in the appropriate 11 manner build up the number 1764 figure by figure in R.R. R.R. now reads 176400004593750 and C.R. 00012250. 1764 is the dividend 12.25 is the quotient 1764 — 144 and 4593.75 is the final result (i.e. the product of 12.25 and 375). 19.45 X 87.2 34,4 Machine ready. S.R. 34.40000087.2. Carriage in position 4. (Since the first figure of the dividend is smaller than the first figure of the divisor room for one more digit [i.e. position 15] in R.R., must be Jeff free.) Build up the number 19.45 from the number 34.4, R.R. 19.44976 O 49.30[288). C.R. 000.5654. axbxe Extended rule ef three dsce ct 325 * 677 12* 119 12 Firstly the division 325 ~ 12 is carried out by the additive method to give 5 figures in the quotient; the division is commenced with the carriage in posifion 5. The quotient 27.083 appears in the first 5 figures of C.R. Clear only R.R. Sef 677 in the right hand end of S.R. Place fhe reversing lever fo ifs lower posifion. Starling with the carriage in position 1 reduce the number in C.R. to zero by plus turns of the handle (see p, 11). The product 18335.191 appears in R.R. Carriage in position 6. Set the new divisor 119 at the right hand end of S.R. Carry ouf the division by means of the subfractive method as far as position 1 of the carriage. R.R. now indicafes a remainder of 0.028, C.R. indicates the quotient 154.077. (Replace the reversing lever in its normal position.) By using this method if is possible that a small rounding off error can occur which, however, is insignificant in most practical cases, lf if is required to eliminate this error completely, if is recommended that first the factors in the denominator be multiplied together and noted down, and secondly the factors in the numerator be mulfiplied to- gether with the product being acuumulated as far to the leff as is possible in R.R. The divisor may then be sel in S.R. and the divi- sion carried out by the subtractive method. Calculation of roofs Square Roots (Hermann’s Method) In the method fo be described it is sup- posed, that by means of a slide rule or auxiliary fables or by judicious guessing, an approximate square root has been found. We wish to obtain a betfer approximation. Let N be the approximate value of R, the square root of R?, and denofe the error in the approximation by E, so that R = N + E. The method proceeds by setting N in S.R., multiplying by N (which appears in C.R.) fo produce N? in R.R. The quantity 2 N is then set in S.R. (without clearing R.R. or C.R.), and R? is built up from N? in R.R. Since R2 = N? + 2NE + E? it follows that (if we neglect E?) E is added fo C.R. Since C.R, already contained N, C.R. now reads N + E, the new approx- imation, | ¥ 150 =? Inifial approximation = 12.2. Machine ready. Carriage position 6, Sef 12.2 in S.R. with knobs 3 to 1. Multiply by 12.2 starting from posifion 6, Carriage position 6 — 1 plus turn. Carriage position 5 — 2 plus turns. Carriage position 4 — 2 plus turns. Set a decimal point between position 6 and 5S of R.R.: 148.84000. S.R. Sef twice the initial approximation (i. e. 24.4) in the same positions as were used for 13 the inifial approximation before. By turning the handle in the appropriate manner in conseculive positions of the carriage, in- crease the number in R.R, to 150, thus Carriage position 4 — 1 plus turn. Carriage position 3 — 5 minus turns. Carriage position 2 — 3 minus turns. Carriage position 1 — 5 plus turns. R.R, now reads 149.999, C.R, now gives fo 6 correct figures the root 12.2475, This method determines as many additional correct figures as there were correct figures in the approximation. In this case the initial approximation is correct fo 3 figures, and the desired approximation fo 6 figures, From this rule if is unnecessary fo proceed further; furthermore we have in this particullar case come very near to the required number (150) with 149.999. lf one requires 8 correct figures in the root and a CURTA Model Il is available, one can 14 repeat fhe previous example using the initial approximation 12.25 (i.e. with 4 correct figures), One then obiains the root 12.247449 in CLR. Remark: The rule thaf one gains as many correct figures in the roof as one has fo start with is capable of a limifed number of exceptions. (See p, 49 same method by subtraction.) Cube roots CURTA users who have at their disposal our CURTA Tables may, by use of these tables, obfain a cube root correct to 5 figures by means of an addition and one subsequent mulfiplication. These fables will be for- warded free of charge on application fo our headquarters, Use of fables can be avoided by using an extension of the method for obtaining square roots described in the last section. Let N be the approximate value of R, the cube root of R?, and denofe the error in the approximation by E, so that R = N + E, and R? = N? + 3 N? E + ferms in E? and E’. The computation proceeds by arriving al a sifuation in which we have N? in R.R,, 3 N? in S.R., and N in C.R. We then build up the contents of R.R. to R3, i.e, from the above equation, adding approximately E to the contents of C.R. C.R. thus contains a quantity which approximates fo N + E. It will be appreciated that (and this remark applies equally well to the method for deriving square roots) if the derived ap- proximation is insufficiently accurate, if may be improved by repeating the process. a V 132.651 = ? Machine ready. Initial approximation = 5.0, Carriage in position 6: Sef 5 in S.R. with knob 1. Multiply by 5 in position 6, R.R.: 25. Set a decimal point in C.R. befween positions 5 and 6. Clear C.R. and R.R. Sef 25 in S.R. with knobs 2 and 1. Multiply by 5 in position 6. Replace 25 in S.R, by 75 (= 3 X 25). Increase the confents of R.R. fo 132.651, thus Carriage in position 5: 1 plus turn. Carriage in position 3: 2 plus turns. Carriage in position 1: 1 plus furn. R.R. now contains 132.65075, C.R. contains 5.10201, The cube root of 132.651 is 5.1. lf the process is repeated with the first approximation 5.10201, C.R. finally contains Sat) Further roots may sometimes be computed by application of the preceding methods, 6 eg. V14=% Firstly we compule the square root 3.74166 15 and then the cube root of this number, namely a é V 3.74166 = 1.5524 = /14 Continued multiplication axbxcxXdXxX... ele. Introductory Remark: The limit of a com- putation of this type is reached when the number of figures in the product is equal to the number of figures in the Result Register, i.e. 11 figures for the CURTA Model | and 15 figures for the CURTA Model Il, Before the computation is begun it is possible to estimate the number of figures in the product by adding together the number of figures in the different factors, One can, if the need arises, omit the last figures of the partial products in order to avoid an excessively long end product. | Obviously one can compute a function of the form aX bxXexX dX .,. efc. by mul- 16 tiplying the first two factors together, setfing the product in S.R., multiplying by the third factor, and so on. Use of this method, how- ever, means that each partial product must be sef at each stage in S.R. We now des- cribe two methods which may readily be applied in many cases, First Method 38 X 24 X 57 X 63.44 =? Machine reaay, |. S.R.a:t Set the number 38 with knobs 2 and 1. C.R.: Develop 24 (normal multiplication). R.R.: Resulf 912 = Partial producti |. Il. S.R.: Set the next factor diminished by a tenth, i. e. 56.9 = (57—1/10) with knobs 3 to 1. Carriage: Place the last figure on the right of S.R. immediately under the first figure on the left of the partial product in R.R., thus move the carriage to position 3, Handle: Continue fo turn the operating handle in its plus position until the number immediately above the 9 in S.R. goes to 0. One thus observes this number in R.R. and continues to turn the handle until it reads 0 (optical control). 513012 now appears in R.R. Carriage: Position 2. One observes the 2nd place in R.R.; affer one plus furn this in- dicafes a 0, R.R. now reads 518702, Carriage: Position 1. One observes the first figure of R.R.; this indicates 0 after 2 plus furns. R.R. then reads 519840. The partial product Il is 51984.0 (because 56.9 was sef instead of 57 which gives one decimal place in R.R.), The partial product Il was derived as follows: (56.9 X partial product!) + ‘/10 partial product! = 57 X partial product |. Ill. S.R.: Set the number 63.439 = (63.44-'/10) with knobs 5 fo 1. Carriage: Position 6, Turn the handle in its plus position and at the same time observe the 6th position of R.R. When this reaches 0 R.R, reads 31720019840, Carriage; Position 5. When the 5th number in R.R, has been reduced to zero, R.R. reads 32354409840. Proceed in this manner from position to position up to and including position 1 of the carriage. Finally R.R. in- dicates 32970649600, The final product, after bearing in mind the three decimal places of the number, is thus 3297864.96, Remark: If can occur that fhe number of figures in the partial product is greater than fhe number of positions through which the carriage may be moved. The next example shows nevertheless, that the same method can still be used, It is, however, advisable fo arrange the computation in such a manner, where possible, that fhe factor with the largest number of figures is. sef last in S.R. (see the first example), 63.44 < 38 X 24 X 57 =? 17 (the result is already known), The mullfipli- cation 63.44 X 38 = 2410.72 is nermal. Next se} 23.9 in S.R, with knobs 3 fo 1. Carriage: Position 6. By turning the handle in its plus position in positions 6, 5, 4, 3, 2 and 1 of the carriage reduce every digit in R.R. fo 0. R.R. now reads 57857280. The partial product is 57857.280, Continuation of the calculation with a CURTA Model I. Carriage: Position 6, S.R.: Set 56.9 with knobs 5 to 3 (knoks 1 and 2 to 0), By turning the handle in its plus posifion in positions 6 to 1 of the car- riage reduce each digit in turn to O (first the 8th digit, then the 7th and so on). The 2nd and 1st digits of R.R. are not re- duced fo zero. R.R. now reads 3297860.4080. We can identify this result with that pro- duced in the previous example, If com- plete accuracy is required for the last two 18 figures the computation may be conducted as follows: S.R.: Set 56.9 with knobs 3 to 1 (knobs 4 and 5 fo 0). Carriage: Posifion 2, By turning the handle in ifs plus posifion reduce the digits in po- sitions 8 to 2 to 0, Carriage: Position 1, This digit is already zero, the calculation is finished. R.R. reads 3297864.9600, Continuation of the calculation with a CURTA Model Il The carriage of a CURTA Model II may be moved through 8 positions, and in this ins- tance the calculation in hand may be com- pleted in the normal manner as described inf, However, as an exercise for the case in which the number of figures in the partial product exceeds the number of posifions through which the carriage may be moved, and in order fully to appreciate the principle described, it is advisable fo carry out the exercise given for the CURTA Model I. We therefore request you fo perform the pre- vious method on your CURTA Model Il. The end resulf in R.R. is the same as before, Second Method 38 X 24 X 57 X 63.44 = ? Machine ready. Proceed as for section | (page 16) of the previous example. This method differs from the previous one in that the next factor is reduced by a unit in the last figure instead of by a tenth. Thus sei 57 —1 = 56 in SLR. with knobs 2 and 1. Clear only C.R, Carriage: position 3. Handle: the arrow points to a 9 in R.R. Memorize this number and carry out 9 plus turns of the handle. Naturally a 9 appears in C.R, Carriage: Position 2. The arrow now points fo a 1 in R.R., thus one plus furn of the handle, Carriage: Position 1. The arrow points to a 2 in R.R., thus 2 plus furns of the handle. R.R. shows partial product Il, namely 51984, and C.R, indicates partial product |, namely 912. Set in S.R. there is the third factor minus a unit, namely 56. Partial product Il was arrived at as follows: (56 X Partial product |) + (1 X Partial product 1), (which was already in R.R.) = 57 X Partial product |. Clear only C.R., and set in the right hand end of S.R. the factor 63.44 with fhe last figure diminished by a unit, i.e. 63.43. Beginning with the carriage in position 5 turn the handle in its plus position so as jo build up in C.R. the number now in R.R. The final result indicated in R.R. is 3297864,96. The advantage of this second method is thal no extra zeros appear at the right hand end 19 of R.R., and thus the full capacity of the machine can be ulfilised, lt is, however necessary fo check the calculations by noting that the partial products appear in C.R., whereas in the first method this was aufomatically assured by the appearance of a 0 in the successive positions of R.R. Cubes without intermediate notes 3273 = ? We tirst obtain 3277, i.e, 327 X 327 by normal multiplication, The square 106929 is builf up in R.R. S.R. indicafing 327 and C.R. also indicating 327 are not cleared. C.R. will now be built up to 106929 starting with the carriage in posifion 6 in order that the least significant figures in R.R, will not be modified in the ensuing multiplications. Carriage: Position 6, The arrow points to a 1 in R.R, In the corresponding 6th posi- tion of C.R. there is a 0; thus one plus turn. Carriage: Position 5, The arrow points to 20 a 0. In position 5 of C.R, there is similarly a 0. Thus this position can be skipped over, Carriage: Position 4, The arrow points to a 6 in R.R., thus turn the handle in its plus position until C.R, also indicates 6 in this position, Carriage: Position 3, The arrow points fo a 9 in R.R. Thus turn the handle in ifs plus position until the 3rd position in C.R. also indicates 9. — Carriage: Position 1. In this position both R.R. and C.R. indicate a 2. Thus this posifion can be skipped over, Carriage: Position 1, In this position R.R. indicates.a 9, and C.R. a 7; thus 2 plus furns. Now the basic number 327 is in S.R., in CLR, the square 106929, and in R.R, the 3rd power 34965783. Higher Powers Clearly the capacity of the machine used limifs the order of the power fo be comp- uted, In the previous case for example the fourth power may be computed with a Model | by setting the cube and multiplying by 327. With the result 11433811041 the limit of the capacity of R.R. is reached. Higher powers may be computed only by discarding the last 3 or 4 figures in S.R. Using the Model Il the 4th power also may be computed without clearing S.R. and C.R. by developing the appropriafe number in C.R, in a manner similar fo that which has been described for the 3rd power, the cube being finally contained in C.R. For higher powers if is suggested that some convenient number (square, 3rd power, etc.) be sef in S.R., in which case, when larger numbers are being dealt with, the necessity for curt- ailing the number is indicated in advance. 21 Commerce and Industry Checking of Invoices and goods a) Calculation of ifems and fotal amount in an invoice (Only with CURTA Model II) 13 Articles af 1.48 = 19.24 25 Articles at. 4.45 = 111.25 39 Articles at 7.25 = 282.75 31 Articles at 11.55 = 358.05 Total =e l29 Machine ready. sef 13000000013 in S.R. Develop 1.48 in C.R, with the handle, Set decimal points as follows: between posi- tions 2 and 3 of C.R., befween the 11th and 12th, and 2nd and 3rd positions of R.R. R.R. indicates in both its left and right hand ends fhe product 19.24, i.e. the first ifem. Clear C.R. and finally only the left hand 22 end of R.R. so that the number 19.24 remains in the right hand end of R.R, S.R. 25000000025. Develop 4.45 in C.R. with the handle, The leff hand end of R.R. now contains the second item: 111.25. In the right hand end of R.R. the second item has been accumulated with the first: 130.49. Clear only C.R. and the left hand end of R.R. S.R, 39000000039, Develop 7.25 in CLR. with the handle, !n the left hand end of R.R. there appears the third item 282.75 and in the right hand end the sum of the first three items 413.24. Clear only C.R. and the left hand end of R.R. 5.R. 31000000031. Develop 11.55 in CLR. with the handle, At the leff hand of R.R. appears the fourth item 358.05 and at the right hand end the final total 771.29. (Remark; For eventual percentage reduction or increase of the invoice we refer fo the exercises on percenfage calculations on page 24.) b) Check on the final fotal of an invoice lf we wish to calculate the final fotal of an invoice, without determining the separate items, then if is unnecessary fo set two numbers in S.R. and the calculation may be performed on a CURTA Model I: 38 Articles af 14.30 = 543.40 52 Articles af 23.75 = 1235,— Gross fotal = 1778.40 deducted returns: 17 Articles at 12.80 = 217.60 Nef fofal = 1560.80 Machine ready. Sef 14.30 at the right hand end of S.R. Set decimal points in S.R, between positions 2 and 3, and in R.R. between positions 2 and 3. Develop 38 in C.R. with the handle. Clear only C.R. Sef 23.75 at the right hand end of S.R, and develop 52 in C.R. with the handle, R.R. indicafes the gross total 1778.40. Clear only C.R. Reversing lever in the lower position. Set 12.80 at the right hand end of S.R. and turning the handle in its minus position develop the number 17 in positions 1 and 2 of the carriage, (Negative mulli- plication.) R.R. contains the net total 1560.80. (Return the reversing lever to ifs normal position.) c} Checking the tofal and number of artictes (Only with CURTA Model II) 74 Articles af 32.25 = 2386.50 38 Articles at 19.40 = 737.20 Tofal 112 Articles = 3123.70 deducted returns: 13 Articles af 26.35 = 342,55 net 99 Articles; net tot, = 2781.15 Machine ready, 23 Set ai at the extreme left (i.e. with knob 11 of S.R., at the extreme right the number 32.25, thus S.R.: 100000032.25. Set decimal points in S.R, between knobs 2 and 3, and in R.R, between positions 2 and 3. Develop the number 74 with the handle. The multi- plier 74 appears in C.R. as a check for R.R. Af the leff hand end of R.R. appears the number of arficles 74 and af the left hand end the item 2386.50, Clear only C.R. 5.R.:; 100000019.40. Develop 38 in CLR. with the handle, At the left hand of R.R. the number of articles (112) has been accum- ulated and at the right hand end the sum of the two ifems viz. 3123.70. Clear only C.R. Move the reversing lever to its lower position. S.R. 100000026.35, By turning the handle in ifs minus position develop 13 in C.R. The resulfs displayed in R.R. are at the leff hand end 99 arficles nef, and at the right hand end 2781.15 = net fotal, (Return the revers- ing lever fo its normal position.) 24 Percentage calculations A. Percentage increase The amount 378.65 is fo be increased by 4.5 Yo, ne 378.65 X Foo. Set ihe number 378.65 at the extreme right of S.R. and develop 4.5 in C.R. with the handle, Set a decimal point befween positions 5 and 6 of R.R. The increase 17.03925 (rounded to 17.04) can now be read in R.R. Without clearing or resetting any number complete the number in C.R. to 104.5 (100 %o + 4.5%), The final amount 395.68925 now appears in R.R. rounded to 395,689, B. Percentage decrease 12% is jo be deducted from the amount 735.—. 45 735 X 400. The multiplication is quite normal. sef a decimal point between positions 2 and 3 of R.R. The 12% decrease can be read in R.R.: 88.20. Without clearing or resetting any number develop the number 88 (i.e, 100% —12%o = 88%) in C.R, by turning the handle in the required manner, R.R. now contains the nef amount 646.80, Calculations of this type in particular can be completed in one computation using a CURTA Model II. In the last example the numbers were relatively small and a CURTA|! might have been used in this way. Set 88 at the exfreme right and 12 af the extreme left of S.R. Sef decimal points between positions 6 and 7, 8 and 9, and 2 and 3 of RR. Develop 735 in C.R. with the handle. The results appear in R.R., namely on fhe leff the decrease 88.20 and on the right the net amount 646.80, With larger numbers there is always the danger that the results will overlap. We therefore recommend that when setting two numbers in S.R. a CURTA Model II which can accommodate 15 figures in R.R., be used, C. Profit margin Cost Price 1257.—; Selling Price 3840.—; Profit? Profit as a percentage of fhe Selling Price? Carriage in position 6, Set 1257 at the right hand end of S.R. One minus furn. Set 3840 at the right hand end of S.R. One plus turn. R.R. now indicates the profit 2583.—., Carriage in posifion 5. With the reversing lever in its lower position perform the sub- tractive division 2583 — 38.40 (= 1% of the Selling price). The result in C.R. is 67.266 %o and expresses the profif as a percentage of the Selling price. Return the reversing lever fo its normal position. If, in the same ex- ample we wish fo express the profit as a percentage of the cost price, the calculation is as follows: Carriage in position 6. Sef 3840 in S.R. One plus turn. Set 1257 in S.R. One minus furn. 25 Moving the reversing lever to its lower position the subtractive division 2583 — 12.57 (= 1% of the cost price) is finally carried out, The resulf in C.R. is 205.489 %o (Refurn the reversing lever to ifs normal position), D. Compound percentages lf we wish to calculate the same percentages of, or deductions from, a series of numbers (for example cost prices) i! is advantageous fo regard the percentage expression as ratios, and conduct the calculations as a series of multiplications. For example suppose that on a number of cost prices we wish fo calculate 57 %o profit and finally fo deduct 10% rebate and 2% discount. The appropriafe ratios are as follows: Profit = 57 %o; Cost price + profit = 157%; Rebafe 18% = 157 X18 100 < 100 = 28.26 %%o. 26 Net (deducting rebate) = 157 X (100—18) = 428.74 %o, 100 X 100 pee ane 128.74 < 2 ‘ 0 : rns = 0 Discount 2% 100 X 100 2.575 Vo, Net (deducting discount) 128.74 X (100—2) 700 & 100 126.165 2/5. We apply these to a cost price of 3755.—: 3755 is set in S.R. and the 6 ratios (in bold type) are developed in C.R. by turning the handle appropriately, the results being read one affer the other as follows: Cost price (in S.R.) 3755,— Profit’S7'%/or 7 0S 2140.35 Cost price + profit 5895.35 Rebafe 18% . . . . 1061.16 Nef (deducting rebate) 4834.19 Discounts? (ou lites hese ane 96.69 Net (deducting discount) 4737.50 E. Profit and loss 1. Financial year A: Profit 35676.— Financial year B: Profit 43217.— By how many percent is the profit in year B greater than that in year A? We compute 43 217 X 100 35 676 The result of the division 4321700 — 35676 is 121.14 "Yo 35676 = 100 %; the increase in profif is thus the difference 21.44%. The net amount of the increase in profit 7541.— may be de- termined in the same calculation as the difference 43217.— (R.R.) — 35676.— (S.R.), and occurs in the subtractive division after the first minus turn of the handle, and after the additive division it may be derived by carrying out one minus turn of the handle in the hundreds figure of the quotient in CLR. 2, Financial year A: Profit 17863,.— Financial year B: Profit 14937.— We wish to express the decrease in profits from year A to year B as a percentage. We must carry out the division 1493700 — 17863. The resulf in C.R, rounded to two decimal places is 83.62%. The reduction is thus 100 %o—83.62°%o = 16.38 °/o. This resulf can be obtained directly in C.R. by positioning the reversing lever appropri- ately at the beginning of each division: in the lower position for additive division and in the upper position for subtractive division, These positionings of the reversing lever cause the quofienf in C.R. fo appear as a complement, and it thus may occur that the most significant positions of C.R, are filled with 9's which are not taken into account when reading off the result (e. g, 9/1638). F, Capital and interest 1, The yearly interest on a capilal invest- ment of 67855.00 is 3912.00, What is the rate of interest? 27 We divide the interest 3912.00 by 1% of the capital. The division 3912 + 678.55 is carried out quife normally, and by inserting the appropriate decimal point in C.R, the result is 5.765 %o. 2. The yearly interest on a capital investment is 7593.00 from an interest rate of 4.75%. How much capital has been invested? We divide the interest 7953.00 by 4.75, and multiply the result by 100. Resulf 167432.00 = Capital investment. 3, The calculation of interest by using the known formulae Capital X rate of interest X days | 360 X 100 or Investment index X rate of interest 360 | can be performed by using various Interest tables and one of the examples on pages 10—12. 28 Costing The yearly expenditure in three departments are as follows 3545.00 in department A 6893.00 in deparfmenf B 2360.00 in department C in all: 12798,00, What is the expenditure of each depart- ment expressed as a percentage of the fotal expendifure? We proceed by dividing every expendifure by the total 12798. Machine ready, Set 12798 at the right hand end of S.R. The first division 3545 — 12798 is carried out by the additive’ method «starting with the carriage in position 5. After the subsequent division R.R. indicates 003545.04600 and C.R, 27.70% = Proportion A. Do not clear. Starting with the carriage in position 5 the dividend 3545 is replaced by the new dividend 6893 by plus and minus turns of the handle. Thus with the carriage in Position 5; 2 plus turns Position 4: 6 plus turns Position 3: 2 plus turns Position 2: 4 minus furns R.R. now contains 006893.00280, C.R.: 53.86 % = proportion B. Do nof clear, Starting with the carriage in position 5 the dividend 6893 is replaced by the new dividend 2360 by: turning the handle in the appropriate manner, Thus with the carriage in Position 5: 4 minus furns Position 4: 5 plus turns Position 3: 4 minus turns Position 2: 2 minus turns R.R, now contains 002359.95120, C.R.: 18.44 Yo = proportion C. It is easy to check the calculation by adding the percentages fogether, The sum must, within the prescribed accuracy, be 100%. By using the CURTA Model Il it is possible to incorporate this check automatically as will be demonstrated in the following cal- culation (see the next example). Remark: When a large number of items occur in the calculation, if is most econo- mical fo compute the reciprocal of the divisor and set this reciprocal in S.R, as a consianf multiplier, and fo develop the various ifems in C.R,. (See Division by Multiplying by a Reciprocal on page 9). Costing with simultaneous control (Only with CURTA Model II) The yearly expenditure in three departments are as follows: 3545.00 in department A 6893.00 in department B 2360.00 in department C in all: 12798.00. 29 What is the expenditure of each department expressed as a percentage of the fotal ex- pendifure? Machine ready. S.R.; 10000012798. Carriage in position 5. Divide into 3545 by the additive method. Thus with the carriage in Position 5: 3 plus turns Position 4: 2 minus furns Position 3: 3 minus turns The further passage to positions 2 and 1 is unnecessary in this case. The percentage 27.70%. = proportional ex- pendifure A appears at the left hand end of RR, The same number appears in C.R. In order fo check the calculation this number is nof cleared so that the perceniage may be accumulated in C.R, and be seen to toflal about 100%, Clear R.R. only. Carriage in position 5. 30 The division 6893 — 12798 is again carried oul by the addilive method, At the lefi hand end of R.R. appears the percentage 53.86 %o = proportional expenditure B. In C.R. the two percenfages have been ac- cumulated: 81.56 %o, Clear R.R. only. Commencing with the carriage in position 5 build up the next dividend 2360. The quantity 2359.95120 appears in R.R, as an approximation fo 2360. At the right hand end of R.R. appears the percenlage 18.44% = proportional expen- difure C. As a check the sum of the three percentages appears in C.R.: 100 %/o, Calcula