HOW T0 BECOME some. E | ie Bes LK F ARS | FOB) WR THE UNIVERSITY OF ILLINOIS LIBRARY The Frank Hall collection of arithmetics, presented by Professor H. L. Rietz of the Universit . hes, 70% Bef AY". \ peste Lee Xx SMe A EMATICS LIBRARY ay eS ARI | F e T +. HOW TO BECOME QUICK AT FIGURES. ——-COMPRISING — THE SHORTEST, QUICKEST, AND BEST METHODS OF BUSI- NESS CALCULATIONS. e/a ee GONE: MPitLOor, Sl00 FPOsTPATD. For Sale by all Book and Newsdealers. 1885. BOSTON, MASS: THE WOODBURY COMPANY, PUBLISHERS. at brarian’s Office the Li in Washing ton, 1883. Entered according to act of Congress ML aA SSe 2\ MEN WANTED. “It has been truly said that the great want of the age is men. Men of thought; men of action. Men who are not for sale. ‘Men who are honest to the heart’s core. Men who will condemn wrong in friend or foe —in themselves as well as others. Men whose consciences are as steady as the needle to the pole. Men who will stand for the right if the heavens totter and the earth reels. Men who can tell the truth and look the world and the devil right in the eye. Men who neither swagger nor flinch. Men who are Quick at Figures. Men who can have courage without whistling for it, and joy without shouting to bring it. Men through whom the current of everlasting life runs still, and deep and strong. Men too large for cer tain limits, and too strong for sectarian bands. Men who know their message and tell it. Men who know their duty and do it. Men who know their place and fill it. Men who mind their own business. Men who will not lie. Men who are not too lazy to work, nor too proud to be poor. When in office, the workshop, in the counting-room, in the bank, in every place of trust and responsibility, we can have such men as these, we shall have a christian civiliza- tion — the highest and best the world ever saw.” 464239 CONTENTS. PAGE Abbreviations in Prescr apsions, ; : 2 : : 107 Addition, ; , F . : TOT, Drill tables, . ; < : - i11—17 “e From left to right, ; r : A 17 as General rules, : ; ‘4 ri ° : oT $6 Grouping, : A : y 2 . 10 es Ledger columns, . : “ . 2 22 6 Lightning agers : . - ‘ : 19—22 66 Results only, ‘ : ; : . 8 “ the Easy method, 8 “ z 4 f 23—26 ‘s with periods, < ‘ ae ; 4 ; 26 “6 2 columns at once, . ; ‘ : is 15 : - 5 105 Carpenters’ Estimates, : ‘ : : : ; 115 Check, How to Endorse . : Sgt, ee: ‘ Ve Cisterns, round A : “ : : . - 104—105 Cisterns, square. : : : ; : 104 Clapboards, . - : : ‘ A 115 Coal, How to Estimate in Bulk . : . ; 100 Coins of of oclids Nations, . ; : F . T2—T74 Contents of S 2 ; - ; 4 : 112 Corn in Crib, . * ; : ‘ - p 108 Cubic Measure, . ; : 4 ; : 112 Day of Week, How to abies n : ‘ : : A 128 Decimals, . ; ; : Pe : 67—68 Division, Contractions : é : : ; - 53—54 Do Something, ; ° : : : : - 6 Drafts and Acceptances, 3 : : ‘ : : 76 Dry Measure, 4 ; ° : i : : 108 English to U. S. Money, . p : : 80 Fractions, . 4 : n ; , : : 55—66 és Addition, . ; ; ; : : ; 57 BE Contractions, . A : ‘ : : 60—66 a Division, . A ; ‘ < ‘ 58 cM Mixed N umbers, : A ‘ ‘ : 63 es Multiplication, : ‘ ‘ , A F 57 J Relation to . zs - A ; : 58 ie Subtraction, ; - : Z “ 57 #6 To a Common Denominator, ; : 4 56 ss To Lower ee: ‘ : 2 56 Freight, R.R. 2 mn ‘ ‘ 5 * 102 Great Britain’s Money, : : ; : - A 74 Hay, To Estimate . : - A : 4 . 96—98 Interest, 5 : - : ? 4 ‘ - 81—94 “ Bankers’ Method, : : - ; : 85 ee by Cancellation, . : : ; : : 87 CONTENTS — CONTINUED. Interest, Common Method, r for Days only, “More or Less than 60 Days, ES Partial Payments, . BS to find the Principal, . : ° < to find the Rate, : “ s&s to find the Time, é A . “6 Vermont Rule, < A A ie 6 per cent. Method, . ee $12 Rule, or Lightning Method, Land Measure, : . : i Laths, . : : Length of Nails, : Liquid Measure, - : ° Long Measure, - : - “ Long Ton Weight, Mariner’s Measure, ; Marking Goods, Masonry, . ° Measures, : Metric System, F Multiplication, . ; aS Aliquot Parts, . - Bo Contractions, 3 - as Cross Method, . és Sliding Method, se Squaring, 6c when the Tens are © Alike, 4 Multiples, Table of Nails, Calculating Rates, ‘ « Length “s Meaning, ‘at Penny, s¢ Number in a Pound, Notes, Description of A . . Paper, Names of various styles, Particular Branches, 4 : = Perches, A 66 How to Estimate “6 Short Method, Printers’ Table, . : 4 , Publishers’ Table, Z - 3 A Rates on Nails, how to calculate F Round Cisterns, ; ! : ; Shingles, to Estimate. J Shoemakers’ Measure, ; : Silver Coins, ; . Square Measure, . State Currency, A Surveyors’ Measure, . . The Day of the Week, - Troy Weight, . . : . U.S. Coins, : ; J ; U.S. Money, . ‘ : : Valuable Information, . : Ls eg ny echt 128—130 105 70 69 75 95 99 DO SOMETHING. Do not spend your precious time in wishing, and watch- ing, and waiting for something to turn up. If you do, you may wish and watch and wait forever. You can do it if you wish, but you must put forth the effort. Idleness and indifference never accomplished anything. It takes energy and push to make headway in the world, and an active, energetic, persevering man is sure to succeed. If he can not do one thing he will do something else. If he can not succeed in one direction he will in some other. He will do something. He will not waste his time in idleness. There is no lack of work, no lack of opportunities. Do what comes to your hand, and doit well. True progress is from the less tothe greater. You must begin low if you would build high. Work is ordinarily the measure of suc- cess. Quit resolving and re-resolving and go and do something. — School Supplement. ADDITION. The adding of one or more columns of figures should be done without mental labor, and may be acquired by anyone with a good deal of practice. The art of adding quickly is acquired by learning to vead a column of figures as you would a sentence composed of words, and those words composed of letters. By Practice we have become so familiar with letters that when we see them grouped together, it is unnecessary to separate them, or spell out the words, but we can tell at a glance what the word is. By Practice we may become so familiar with figures that when we see a group of them, we can tell ata glance what the sum of them reads, without spelling the figures at all. In practicing the reading of a column of figures in this way, we do not let the brain work at all, but simply pass the eye over the figures (see drill tables) as if you were reading a sentence, slowly at first, but increase the speed as proficiency is acquired. A few minutes daily practice will produce astonish- ing results in a very short time ; beginning with two figures, then three, four, and so on until finally we become able to write the Swm total of long columns. For example, when we see the figures 9, 8, 6, 4, we know at a glance that the sum is 27 without reading the figures themselves or spelling them out. Reading a column of figures as the reading of a sentence, is done by dividing a large group of figures ADDITION. into smaller ones and from group to group through the column, just as from word to word we read through a sentence. We give various methods, but commend the gvoup- ing method as the best and most practiced. Addition is more frequently used than all other operations combined. The most important qualities of an accountant are accuracy and speed. The most speedy calculators are usually the most correct. It is a deplorable fact that not one in one hundred of our graduates fresh from the High School can add a column of figures correctly without many trials. No labor should be regarded too great to master this, the key to all numerical as well as business transactions. RESULTS ONLY. Never spell your way through a column, thus: 6 and 8 are 14, and 9 are 23, etc. It is just as easy to name results only, and much more rapid. For the purpose of explaining a method, examples will be sometimes spelled out, but it is never recom- mended as a method to be adopted. 367 Begin at the bottom of the right hand 854 column, and name results only: 976 14, 23, 31, 37, 46, 52, 56, 63. 389 Then adding 6, the carrying figure, to the 736 second column, we have: j ; i 15, 17, 24, 28, 31, 39, 46, 51, 57. 726 Again carrying 5 to the next columr we 98 SY: 11, 18, 22, 27, 34, 37, 46, 54, 57, © 7 «3 which completes the operation. NotTre.—When 9 occurs in addition it is easier to add ten and sub- ract one mentally. thus: instead of 9+ 8=17, say, 10-+8=18—1=—17. ADDITION. GROUPING. SPELLING PROCESS. 145. 11+10+4+410+3-4 10 = 48 Perea tO 8 10 2 Tg 10 Demi be 5 10 1-110 11011-45104 Sen The sum of any two figures can never exceed a ten; therefore, the left hand figure of the sum of any two figures must always be one. It is, by this method, no more difficult to read the sum of two figures, than to read a single figure. In the process we pay no attention to the figures as they stand written, but only the sum of two of them. In the above example, instead of reading 11, 20, 25, 33, etc., we say: | 11, 21, 25, 35, 38, 48. 4, 14, 17, 27, 29, 39, 49, 51. 5, 15, 16, 26, 36, 37, 47, 50. It will be noticed that the left han figure increases by regular notation, 1, 2, 3, 4, 5, 6, etc. Practice will enable one to add much more rapidly than by the old method, with less liability to error, because it is systematic and simple, and requires less ‘mental labor. 10 GROUPING. By this method we mentally group the figures above 10 and under 20. To the first group add the tens of the next group, and to this sum add the units of the second group. In order to become proficient in grouping, famil- iarity of totals will be absolutely necessary. The sum of the figures must be read instead of the figures them- selves. By a proper understanding of this method, the mental labor is much less than by the old ones ; half the labor is simply counting by regular notation, because the left hand or tens figure to be added is always the same, and the right hand or units figure usually of a small denomination. By an analysis of the example under this head, we find the first col- umn in group to read :— 7 ; 10 To 11, the first group, add the tens of 3 the second group, we have 21, adding the aM 13 unit 4 of this group we have 25, adding the 5 ) tens of the third group we have 35, and add: 3 14 ing 8, the units figure of the group we have . at 11 238, adding the tens of the fourth group we 3 have 48 which completes the operation. SECOND COLUMN. 4 12 4 the carrying figure of the previous col- a ns umn, added to the first group we have :— 4+10+3+10+24-10+1012=51 4 3 12 By naming result only, we have :— : 13. 4,14, 17, 27, 29, 89, 49, 51 We recommend to name the carrying figure the first, and when a nine occurs call it 10, and subtract lin the operation. 1] ADDITION. DRILL TABLES. These Drill Tables must be thoroughly mastered. Read from left to right and right to left, the sum only, as rapidly as possible ; when you falter or make: an error, go back and start again. We would advise teachers to have a daily black- board exercise of these drill tables for addition. Divide the exercises. Write series that are sim- ple and easily comprehended. Ist. Read the sum of two figures only, in every: possible combination. 2d. Read the sum of three figures only. 3d. Read in two columns of two figures, then three, and also mixed numbers. It will make a pleasant, entertaining, as well as useful change from the regular routine. One month’s. daily exercise will result in surprising proficiency. The whole school should be engaged in these exer- cises, and combinations varied and made more diffi- cult, slowly. Hfint. One of the most accomplished accountants in the city of Boston, informed the writer that he owed his success as an accountant entirely to his being a confident adder, which he acquired largely by appropriating all figures in sight, and reading their sum only. Numbers on the streets, numbers of railroad cars, etc., etc., served as drill masters to him on many occasions. 12 } bo aj { ~1 ©o9 | Or In ce 1 oO cS Or i Kor! ioe) J oo 1 oc — 1 @ 1 on ©29 ¢© DRILL TABLE No, 1. oo 1 & 1 < oO Pees) tn 1 oc oo ADDITION. 8 4 | < | co 4 9 J} or 1 © pp a | {| or 4 9) co | < lees lor) 1 oo 1 “ oO {-< in 1 ~] { er i | © 1 on bo Le 1° 9 ease! On | — } <3 a © 1 G0 b> ao low cs loons 1 oon 1 cw o toner 1 Or Go | © 0& © [| own f on wv | Cc ~ © Iunrwon J} or oo CO loonn loons” ADDITION. DRILL TABLE No, 2. Ani Sins ae a Ag Sieh ie Oy ae aaa D SLA A Bat ea! oy Lene Quince gn SSA Gey Ose Ads Gt: 7 an MEV as OER Gin Aad be Suis cine 8 ie Ciba Ma Ger Sees ule Tenn () uh Said. 1 Ow 8 Sh Giicd ny Ging Ga Se ae Sins Gi Fee Gyaw Biel ft sind Wht hes ioc elon Tat Ox ae oes Subas Been Gamay Skt G Pity Gah Sali? Sie Gate Givi Qa ware SO AP ivS eG Asi aie eR ake Rs ae 7 en eee UL (ECT POO =toe os lrwwan | © CO fh {on { © Co yy { co ® bo Loe 1 rm ne 1» m® § oO Orc [orn © » CO O98 © oO oo 1 or ron co Lr oe © panos 33 loan 1 GC go © 1 3G GH | om © poo > 13 14 ADDITION, DRILL TABLE No. 3. ct~ | 15 TWO COLUMNS AT ONE OPERATION, FROM LEFT TO RIGHT If two or more columns can be added at one oper- ation, there must be some rule or method by which it is done. The following illustrates one of the most practical methods :— 2 5 SPELLING PROCESS. 63 27-+50=77-14— 81 78 81-30 = 1b oS 34 115+ 70 =185+8 — 193 54 193 +. 60 = 258-13 = 256 2 7 256-20 == 276.1 281 281 This process consists simply in adding the tens first and then the units. BEADING PROCESS. 27, 77, 81, 115, 185, 193, 253, 256, 276, 281. Practice will enable one to add two or more col- umns without much mental effort, because it is just as easy to say, 27 +50 — 77-+- 4 — 81, as it is to say, 5+2=—7, or 7+ 4=11. Proficiency in the above method will soon enable one to add without separating each number into tens and units, thus :— 27, 81, 115, 193, 256, 281. The all important thing is familiarity of totals, and we herewith append drill tables, applicable to this kind of addition, which must be read from each direction as advised before. 16 ADDITION. DRILL TABLE No. 4. 43 52 25 34 64 25 52 38. 63 89 94 98 38 68 46 89 93 38 36 48 97 39 84 98 78 63 87 36 85 95 98 79 63 Ya) 84 21 he ADDITION. DRILL TABLES. Read rapidly from left to right and right to left the sum only in the following combinations : — SEE NO, HOM. 2 Oi Gis Om LOL dat) Oy Vas Bete ihe 01 6S) Oy oA EOL COUN On Ban Onn alec d 24 35 72 94 39 36 28 32 46 82 38 48 59 39 Daim aoe km Octane SOONG Merete ah Gh eG Dear AO Oca” ih ) LOR! Oc OY ES PEDRO A Oe) Oe Oem On aie O Rd Oe ee OE) 1.25 12.50 9°50 29.40 45.90 84 3.42 21 3.50 1.50 5.62 40 4.21 6.30 56.89 Combinations should be varied and made more difficult as proficiency is acquired. The secret in adding rapidly consists in familiarity with totals of combinations. Counting is not adding, and spelling is not reading. 18 ADDITION, THREE COLUMNS AT ONE OPERATION, FROM RIGHT TO LEFT, Three or more columns mar be added at one op eration thus :— 223 425 384 256 1288 OPERATION.— 256-+-4—=260, 260-+-80=340, 340-+- 300—640, 640-+-5—645, 645+-20—665, 665+400— 1065, 1065-+-3—1068, 1068+4-20=1088, 1088-+-200 =1288, By naming results only, we have :— 260, 3840, 640, 645, 665, 1065, 1068, 1088, 1288. It will be noticed that beyond two, or at most. three, columns wide and a fewcolumns deep, this method requires more mental labor than the previ- ous ones, and is not considered very practical except for development of the mental faculties. EXAMPLE 2. 2.25 .385 8.46 PROCESS. -08 340, 350, 353, 1853, 1853, 1918, 1921, 15.60 2721, 2761, 2767, 2797, 2802, 2.13 3002, 3022, 3027. 1.40 19 ADDITION. THE LIGHTNING METHOD. AN OLD TRICK EXPOSED. This simple and wonderful combination of figures has deluded many into believing that adding from left to right such a large body of figures, instantane- ously and correctly, to be a Herculean task. Large sums have been paid by the unsuspecting and credulous for the possession of the wonderful secret. It is probably the simplest, as well as the most delusive combination of numbers, because if opera- ted in the hands of an expert it is almost impossible to be detected, unless by those who know the secret. EXAMPLE: ay PEGs BTg ST Mtoe mak fete bites ene: Seep 4 Gr roiaan oie be Lt Oi Grn ad SPAY Vite Oe: DONS a rata: ULES MIE Bibs Gay shia cp Geel venue Oa Lin O Fi hts at cin ts RUNS ehube tiie: 8 ta yee Che aK | PLING Oost 2 OG an Ott Ono The operation is as follows: A line of figures is given you, to the length of which you are wholly in- different, as in above example. ered 00 OF USN) Soke Gel,e 4s. (Bizet ting Now you claim the privilege of writing a line im. mediately underneath it. Your line will be in pairs of 9’s, and will read: 6 ly 4 3 1 1 0) 6 5 / (Second line.) Now «sk another line immediately under these ; any figures whatever, for example: 3 5 8 4 8 8 4 8 8 , Y (Third Line. 20 ADDITION. You write another to pair it into 9’s: 66 40512 868 46241455°.6 2o1%,.0) fourth Line) Another line of figures is given you: 78 6 (8) 90 AUiey Rist tiie This is called the key line, because the sum of the entire column is simply this line repeated, with 2 subtracted from the right, and 2 annexed to the left. In other words, the result being just like it, except the units’ place, which is as many less than the units in the key line as there are pairs of lines, and annex a similar number to the extreme left of the sum. The number of lines is necessarily odd. For the purpose of: explanation we will add the column without the key line. BF: Marts lm ones Whitt Fak NS colts Satie: Bolo OD ee Omak eas Oe OU paltn| SO Bae Sk Br Lh Oly Chane GSA sa aes ohne 00s Gemea aan | ane Bis Basie ae irs kiN wes Dn aes ee melee en (eget 2 yan UA Vs ODAC RE aed Pie TAU RR ie Subtracting two from the right hand figure of the key line is equivalent to adding two to the other col- umns, which would bring all the other figures ciphers as above. It matters not, therefore, what the key line is, so long as we call the right hand figure two less, and annex two to the left. A little practice will make this a ve’y interesting trick. 21 ADDITION. SECOND APPLICATION OF THE “LIGHTNING” METHOD. Take any three columns of figures as follows :— 63456 38429 25636 Pair the last two with 9’s and all the columns will stand thus :— 63456 38429 61570 25636 74363 263454 The top line repeated with 2 subtracted from the _ right hand figure and 2 annexed on the left, com- pletes the operation. THIRD APPLICATION OF THE LIGHTNING METHOD: 8987. Key Line. —$——$—$—<$— 48983 In this example we have four pairs of nines; therefore, subtract 4 from the right hand figure in the key line, and annex 4 on the left. 22 ADDITION. LEDGER COLUMNS. A great part of the work of an accountant consists in adding long ledger columns, like the following. Let the pupil find the sum of the numbers in each, being as careful to obtain a correct result as he would be if he were to receive or pay the several amounts. Combinations should be varied and made more difficult as proficiency is acquired. The secret in adding rapidly consists in familiarity with totals of combinations. Counting is not adding, and spelling is not reading. 8.37 18 673.28 4.353 AT 597.84 7.62 ys) 3426.87 48 2.795 219.48 Od 1.20 8.37 2.50 4.37 167,84 6.19 8.29 5986.32 10.00 13.85 6749.31 4.28 2.00 4863.27 8.07 62 1542.35 4.37 20 2986.28 9.48 1.37 379.87 4.21 9.83 2.99 13.26 6.75 69.80 1.20 8.43 4060.75 ot %).48 309.71 3.08 6.00 124.87 4.96 1.00 8520.06 85 1.50 2493.28 4.00 (.69 48.75 THE EASY WAY TO ADD. This method of adding numbers is especially de- signed for those whose minds are constantly em- ployed with business affairs, the mind being relieved at intervals, and the mental labor of carrying over the sum of an entire column is obviated by the use of /ntegers or “catch figures.” 86 EXAMPLE. Process.—Begin at 9 to add as near 20 as you can, thus: 9-+ 2+ 4-+ 3= 18, reject the tens and place the 8 to the right of the 3, as in example ; begin at 6 and add 6+ 8-+-4= 18, reject the tens, as before, and place 8 to the right of 4, as in example; begin at 6 +- 7 4-4 = 17, reject tens, place 7 to the right of 4, as in example; then 9 + 4+ 3= 16, reject tens, place 6 to the right of 3; then 6 + 7+ 4 = 17, reject tens and place 7 to the right, as before ; having arrived at the top of the column, add the figures in the new column, thus: 8-+ 8 +7-+ 6-+ 7=36, or 3 tens and 6 units ; place the 6 units as the unit’s figure of the sum, hav- ing 3 tens to carry to.5 tens, the number of integers or catch figures already rejected. 3 + 5 = 8 tens, which prefixed with the 6 makes 86 the sum. 24 ADDITION. N. B.—The small figures we set to the right are called integers, ‘‘tally,” or catch figures. If upon arriving at the top of the column there should be one or more figures whose sum will not equal 10, add them to the sum of the figures in the new column ; never place an extra figure in the new column unless it is an excess over 10. 3 EXAMPLE.—Proceed as before ; begin at the bottom, 2-+ 7-+- 6 = 15, reject the tens, place 3s 5 to the right of 6; 6+ 4-+5=— 15, reject as 7 before, place 5 to the right of 5; say 8 + 7 +- 8 3-== 18, reject as before, etc. Now we have 5° three figures which do not add 10; add them ; to the new column and say 5 + 5-+- 8 + 4-4 gs 2-+3= 27; place the 7 under the original 7 column, add 2 to the number of tally figures, 2 which is 3, thus: 3 -+- 2 =5, the tens figure in oi the sum, and makes 57, the answer. Two or more columns can be added in the same manner. 6939 EXAMPLE. — Proceed as in adding a single - 65 column. The sum of the first column being 49, we write the 9 and carry the 4 to the next 49s column, thus: 4-- 2-8-3 == 17, reject 53 the tens as before, write the 7 to the left of the 765 3; then proceed as with a single column. REMARK.—To add very long columns, it is 734 better to add as near 30 as possible, instead of 82 20asin above examples. The reason for sug- 23 gesting this method is to decrease the number 659 of “tally” figures. It must be remembered, however, by adopting 380 as a standard, that two tens will be rejected instead of one, as in the former ex. amples, which will be observed in the following: 25 ADDITION. to EXAMPLE. Process.—Here we begin at the bottom as be- fore, adding over 20 and under 30, placing the excess or tally figure wherever it occurs in the line, thus: 4+5-+4+6+3-+442 = 28, reject the 2 tens, place the 8 to the right of 2, etc. © Ot OO # bO OS a By an inspection of the example it will be ; seen that8-+8-+9-+7-+ 8+ 2= 42, the sum of the tally figures ; place 2 as the unit’s figure of the sum ; carry 4 to twice the number of tally figures, since the twenties, not the tens, were rejected as tallies, so we say twice 6 are 12 and 4 to carry are 16; prefix this with the 2 units, and we have 162, the sum of the whole column. Notr. — The reason for adding numbers by tally figures must be clear, since it is nothing more than a condensing process, which can be briefly explained g thus: take for instance the tally figures in the pre- ceding example which are 8, 8, 9, 7, 8, 2, which. when fully written out, would make the column read, 28 But since the tens’ figures are all alike, it is 28 necessary to write the units only, and simply 29 bear in mind that for every unit’s figure written 27 out we have two tens, and thus abbreviate. 28 REMARK.—This mode of adding is espe- ae cially designed for those whose minds are con- startly employed on business affairs, and who are apt to be interrupted. A little practice will enable anyone to add rapidly and a/ways cor- rectly without any mental labor or fatigue. 162 But the young accountant, whose dusiness it is to add, and xothing else, should rely entirely upon the mind and adopt the preceding rules. eoOP DWE NWN WRAP ER DOWD LR WONe OSL DAIS ODP AO = 26 iT S| Co NS Oo RD OO RDI GO OTE BD ADDITION. Adding with Periods. Another condensed method of Addition is by periods, which is illustrated in the following example: Commence thus?)$. 454095273 2 = 18, reject ten, place a period to the right of 2, carry 8 to the next figure, thus: 8 3 4 2 == 17, place the period to the right of 2, reject ten, carry 7 to next figure,7 4 3 4 = 18 ; place period to the right of 4, reject ten, .carry $ to next figures (0). ae ee 19; reject ten, place 9 as the unit’s figure in the sum ; the number of periods, which are 4 in this case, will be the tens’ figure in the sum of the column and completes the addition. 27 General Rules for Addition. 1. Write numbers plainly and distinctly, so that 9’s may not be mistaken for 7’s, or 5’s for 3’s. 2. Write the numbers in vertical lines. _Ir- regularity in placing of figures is the cause of many errors. 3. Think of results, and not of the numbers them- selves. Thus do not ue 4 and 5 are 9, and 6 are 15, and 7 are 22, etc.; but‘9, 15, 22, etc. 4. Make Mere tinte of 10 or other numbers as often as possible, and add them as single numbers. Thus: Ce ee in adding 9 34 73 214 9 54 82 123, say 9 16 26 33 42 51 61 67, taking each group at a glance as a single number. When a figure is repeated several times, multiply instead of adding. 5. In adding horizontally begin at the left, since the eye is more accustomed to moving from left to right than from right to left. 6. In adding long columns, prove the work by adding each column separately in the opposite direction before adding the next column. We believe that addition should be drilled into every boy, every day, from the beginning to the end of his school life. It is ten times more important than measures, multiples, interest, percentages, stocks, etc., all put together, and the sooner teachers thoroughly understand this fact the more practical and beneficial will be the result ofour school system. Pupils should be trained to add figures when placed in a horizontal position as well as when placed in a vertical position. In the following exercises add both ways, and then prove the results by adding them. Ex.1. 4621 38946 4256 8432 1562 =— 22817 421 5000 7060 85 984 — **** 2012 12138 214 143 O75 = **** 1604 21038 1524 21388 4215 = **** 2385 6214 3121 1562 1428 = **** 11043 RR aio takok 3 |80]| 214 |20}| 14/20}]) 14 |27 2) 17 25 2 |60}| 181/10 1 |00}; 13/84 175 |16 1 |25}| 19/40}/ 125 |10}} 184/15 10 |80}} 13 )75}) 161)15 2 |00 6)17 —— |—__ ee ed i ee eee Ex. 2. ¢42 |50|/ $13 |40||9243/10]| $3 |o4|l$136 ees | 1 MULTIPLICATION. In the ordinary process of multiplication we ob- tain partial products and then add these together for the entire product. With a little wholesome prac- tice, however, we multiply by a number consisting of several figures, without writing out the partial products. There are those who can multiply by a number consisting of 10 to 12 or even more digits, writing the result under the given number with great readiness. This is a very unusual degree of profi- ciency ; but almost anyone can learn to do the same with a multiplier consisting of from 2 to 6 places. We indicate the method by the following problems and solutions. To make the operations easily un- derstood, we have selected small numbers at first, - and advance into higher and more difficult ones, step by step, and whoever studies the method thoroughly, and practices perseveringly, will be am- ply rewarded for the time devoted to the task. EXAMPLES. — PROCESS. 23 6 lst. 236, we write down the 6 for the ______ units figure in the product. 736 =. 2d. (2 2)-+-(83)=138, we write down the 3 for the ten’s figure in the product, and reserve 1 to carry. 3d. 32=—6-+1 (we carried) =7, which com- pletes the product. 29 MULTIPLICATION. EXAMPLE 2. 78 Ist. 7 X 856, we write down 6 as the unit’s _____ figure in the product, reserving 5 to carry. 2106 2d. (7 X7)4+(2X8)}+5 (to carry )—=70; We write down the 0 for the ten’s figure in the product, reserving 7 to carry. 3d. 2 X +-+-7=21, which completes the product. EXAMPLE 3. 126 Ist. 5 X 630, write down O as the unit’s _____ figure in the product, reserving 3 to carry. 4410 2d. (5X2)+(3X6)+3=31, write 1 for the ten’s figure in the product, reserving 38 to carry. 3d. (5X 1)+(8 X 2)+3=14, write 4 for the hun- dred’s place in the product, reserving 1 to carry. 4th. 3 X 1+1=—4, which completes the product. EXAMPLE 4. Ist. 6 X 424, write 4 as the unit’s figure in the product, reserve 2 to carry. 7 141264 2d. (3 X 4)-+(2 X 6)4+-2=26, write 6 as the ten’s figure in the product, reserving 2 to carry. 3d. (6X3)+(4 X 4)+(3 X 2) +-2=—42, write 2 as the hundred’s figure in the product, reserving 4 to carry. 4th. (8 X3)+(4 X 2)+-4—21, write 1 as the thov- sand’s figure in the product, reserving 2 to carry. 5th. 4 X 34+-2—14, which completes the product. 324 THE SLIDING METHOD MULTIPLICATION: Probably the easiest method to learn to multiply | large numbers in a single line is the sliding method as used by Peter M. Deshong, which is in reality nothing more than cross multiplication, as_ illus- trated ; but for the beginner it is the best that can be adopted. When familiar with the slide the stu- dent can proceed without it, and perform operations astonishing to himself and those who witness the operation, the largest numbers being readily multi- plied in a single line. This method can easily be understood by following the examples and solutions here given with paper and pencil. 31 MULTIPLICATION. EXAMPLE. — PROCESS, 324 : aay : 436 Write the multiplier on a slip of paper separate from that on which the multipli- 141264 cand is written, in an inverted order, thus: 634; place this slip directly over the multiplicand, so that the 4 will be directly over the 6, thus: 643 324 then say 6 X 424, write 4 as your unit’s figure in the product, reserving 2 to carry ; now slide the pa- per to the left so that 2 will come under 6, and 4 under 3, thus: 634 324 now (6X 2)-+-(4 X 3)+-2==26, write 6 as the ten’s figure, reserving 2 to carry ; again slide the paper to the left so that 3 falls under 6, 2 under 3, and 4 under 4, thus: 634 324 and you have (6 X 3)-+-(3 X 2) x (4 X 4)-+-2==42, write 2 as hundred’s figure in product, reserve 4 to carry; slide the paper again and the 3 will be under 3, and 2 un- der 4, thus: 634 324 and you have (3 X3)-+(4X 2)-++-4=21, write 1 as the thousand’s figure in the product, reserving 2 to car- ry ; now slide again, that 3 will be under 4, thus: 634 324 and you have 3X4-+2=—14, which completes the produs:, 141264. We have used the same figures in this, as in the preceeding example, and by ciose observation it can readily be seen that the work is all the same. The sliding method, however, saves the mental labor of 32 MULTIPLICATION. carrying over, in the mind, so many figures, which is quite wearisome to the unpractised mind. ‘These additions will soon be performed at a glance, as the producis are obvious without the formality of naming factors, which the student should never allow him- self to do in any operation; it is just as easy to name products only. To understand these directions thoroughly, factors must be placed upon slips of pa- per, and the directions strictly complied with, which will give an insight into the mode of operation, and the reason will be better understood in ten minutes, than in three hours without them. When once fam- iliar with the slide, the student may proceed without it. We will solve another example upon the same principle, naming products only, as it should be oper- ated. EXAMPLE. — PROCESS. 5768 324 On a separate slip of paper, as before, invert the multiplier, thus: 423 ; place the 1868832 multiplier so that 8 will be under 4, and you have 32; write 2 for units in product, reserve 3, carry 4; slide, say 24-++-16-+-3—43, write 3, carry 4; slide, 28--12-+-24--4—68, or thus: 28, 40, 64, 68, write 8, carry 6 as before, and slide again, 20, 34, 52, 58, write 8, carry 5; slide, 10, 31, 36, write 6, . carry 3; slide once more, 15, 18: you have the com- plete product. REMARK.—Proceed towards the left until the multipli- cand passes from under the multiplier, each time adding what you carry to the sever:l products that stand one over the other, and the result will be the product. 33 MULTIPLICATION. SLIDING METHOD. 5768 324 1,868,832 425 4658 2,901,934 326 4658 326 4658 326 4658 326 4658 326 4658 326 4658 EXAMPLE FIRST Reverse the multiplier, thus: 423. Reading products only. 32 9441-1643 me 43 98 112i 94 “Ws 2168 20-+14-+18+6 = 58 JO ton G28 86 Ub igt Geet 8 EXAMPLE SECOND. Reverse the multiplier, thus: 326. 24 16-+15 + 2 = 338 48+10+184+38= 79 30+124+124+7=61 36 +8-+ 6 = 50 24+5=—=29 CONTRACTIONS MULTIPETOCATION Contractions can often be advantageously em- ployed in business calculations ; but, like by-paths in a forest, they are convenient only to those who know . the whole ground. Strangers will do better to keep the highway. TO MULTIPLY BY ELEVEN. Ruiz. — Add the figures in the multiplicand, after the first, from right to left. APPLICATION 1.—45 X11. 4-++-5=9; place this sum between 4 and 5, thus 495. 35 X11. 3+ 5= 8; place this sum between 3 and 5, thus 385. / AD 11 == 462. 63 X 11 = 693. oh B78 Be eet 44 X 11 = 484, 95 "Aili "1045: AO XoUT e589: 35 CONTRACTIONS IN MULTIPLICATION. APPLICATION 2.—345 X11. Here we write 5; we say 4+5= 9. write 9; then4-+3—7; write 7; then write 3, thus 3795. 254 X11. Write 4 for the first figure in the pro- duct; 4+ 5= 9, write 9 for the second figure ; 5 + 2 = 7 which is the third figure, and write 2 for the last figure, and we have 2794. B20 x LP '3575:; 12 Xl Ba 189 353 X 11 = 8883. PMD ATG Dyker (LE Yap Nore.—If the sum of two figures is over 9, carry the one to the next figure. APPLICATION 8.—58 X 11 = 638 ; here we write 8 as the first figure, and say 5 + 8 = 13; write 3 for the middle figure, and carry the one to the next fig- ure 5, making the product 638. TALL o20. SDM O45. 885 X 11 = 4235. 5863 X 11 = 64493, TO MULTIPLY TWO FIGURES BY TWO FIGURES WHEN THE TENS ARE ALIKE. To multiply 87 by 82. Multiply units by units for the first figure of the product, the sum of the units by tens for the second figure, and tens by tens for the remaining figures, carrying when necessary. Wir weal AS carry” 1; Damo 2 he ON io oe anal ho 7134 8X 8 = 64, and 7 to carry = 71. E-xercises. 81 X 87 81 X 87 ints, ee ive 62 X 63 AT ae LOTS 54 X 55 56 X 52 107 X 105 ABIX AT 79 X 75 Wns gy, 27, X22 44 xX 43 113 X 114 36 MULTIPLICATION. CONTRACTIONS. TO SQUARE ANY NUMBER OF NINES. Rule, —Write from left to right as many nines, less one, as the given number contains, an 8, as many ciphers as nines, and 1. Thus the 9:9 9 Be rae TO I square 9999 99980001 of 9/1 9>929°9 99998000014 TO MULTIPLY BY ANY NUMBER OF NINES. Rule. — Annex as many ciphers to the right of the multiplicand as there are nines in the multiplier, and from this number subtract the multiplicand ; the remainder will be the product required. EXAMPLE. — 37645 & 9999. 8376412355 The reason is obvious. By annexing four ciphers, we multiply the given number 10000 times; and by subtracting the given number, we have the product one less than 10000, or 9999 times the number. TO MULTIPLY BY ANY NUMBER ENDING IN NINE, Rule. — Multiply by the next higher number, and subtract the multiplicand. EXAMPLE. — 42 X 389. 39+ 1= 40 42 « 40 — 1680 minus 42 = 1638.—Azs. 37 MULTIPLICATION. CONTRACTIONS. TO MULTIPLY BY ANY NUMBER FROM TWELVE TO TWENTY. Rule, — Multiply in regular succession the figures of the multiplicand by the unit’s figure of the mul- tiplier, and add to the product of each multiplication that figure in the multiplicand which stands next on the right of the one which you multiply ; add, also, the figure to carry, if any. EXAMPLE. — 3 6 4 3 5 13 473655 Here we say 8 X 5 = 15; write down 5, carry 1; say 3 X 3-++ 1+ 5, the figure which stands on the right of 3 = 15; write 5, carry 1; say3 X 4+1 + 3 = 16; write 6 and carry 1;3 xX 6+1-+4 = 23; write 8, carry 2; 3X 38+2+6=17; write 7, carry 1 to 38 = 4. TO MULTIPLY BY 21, 31, 41, 51, 61, 71, 81, 91. Rule. — Write down the units figure of the multi- plicand as the first figure of the product. Multiply in regular succession every figure in the multiplicand by the left hand figure of the multiplier, and to each product add the figure which stands next on the left of that which you multiply, and you have the requir- ed product. EXAMPLE: 3725 Here we write the 5 as the units figure 21. in the multiplier; then say 5 XK 2 +2 = 12; write 2, carry 1;say2 KX 2+1+ 78225 7 = 12; write 2, carryl1; say2 x 7+1 + 3 = 18: write 8, carry 1; then say ¢x 3-1 = 7, which completes the product. 38 MULTIPLICATION. CONTRACTIONS. TO MULTIPLY ANY NUMBERS OF TWO PLACES EACH, WHEN THE UNITS OR TENS ARE ALIKE. ule. — Multiply units by units ; then, if the units are alike, multiply the sum of the tens, and the tens by the tens. If the tens are alike, multiply the sum of the units by the tens, and the tens by tens; in all cases carrying as usual. EXAMPLE 1: 34 4X 4 = 16; write 6, carry 1. 54 548 x41 — 33; write 3, carry 3. —— 5X3+ 3= 18, which completes the 1836 product. EXAMPLE 2: 45 5 & 8 = 15; write 5, carry 1. 1 2 43 5+3 x 4+ 1 = 33; write 3, carry 3. 3 = 19, completes the product. 1938 This rule will apply to the square of any number. It is the most useful of all contractions, and should be carefully studied. TO MULTIPLY BY NUMBERS WHICH ARE FROM ONE TO TWELVE LESS THAN ONE HUNDRED, ONE THOUSAND, ETC. Rule.— Multiply the multiplicand by the differ- ence between the multiplier and 100, 1000, &c., and subtract the product from the product of the multi- plicand by 100, 1000, &c. EXAMPLE. — Multiply 35 by 98. 98 — 100 — 2. SD (270 35 & 100 = 3500, 3500 — 70 = 3430. — Ans. When from 1 to 12 more than 100, add the prod- uct of the multiplicand by the unit figure, after annexing the required number of ciphers, thus: EXAMPLE. — Multiply 325 by 102. 325 & 100 = 32500. 325 & 2 = 650 + 32500 = 38150. 39 MULTIPLICATION. CONTRACTIONS. WHEN THE SUM OF THE UNITS IS TEN, AND THE TENS ARE ALIKE. Method. — Multiply the units and write the result as the first two figures in the product. Then call the tens figure one more and write their product for the last two figures in the final product. EXAMPLE, 86 4xX6=2 8 4 8-+-1x< 8=7 7224 93 85 48 63 56 71 66 97 85 42 67 54 79 64 This contraction will only apply where the units equal ten and the tens are alike, as in above exam- ple. If the product of units does contain ten, as in 9 X 10, the place of tens must be supplied with a cipher. TO SQUARE ANY NUMBER ENDING IN FIVE. Method, — Multiply the figure preceding the units as they will stand by the next higher number, and ‘ annex 26 to the product. EXAMPLE. 795 T+1ixK7= 56, 75 Annex 26. 5625 WHAT IS THE SQUARE OF 257 S85? 45? 75? 85? 95? 105? 115? 125? 185? 145? 155? 165? 175? 185? 195? 205? 40 MULTIPLICATION. CONTRACTIONS. TO FIND THE PRODUCT OF ANY TWO NUMBERS WHOSE UNIT FIGURES ARE FIVE. * Method. — Take the product of the figures pre- ceding the 5 in each number, increase this by one- half of the sum of these figures, and prefix the result to 25. EXAMPLE, 25 4*%2+3=11. 45 1125 WHAT IS THE VALUE OF 25x45? 55X75? 7x95? 65> 95? 385 85? 85X45? 155 X 85: 165 X 45? 185 X65? 175% 65? 225 % 105? Notre. —If the sum of the figures preceding the 5 is odd, when we take one-half of it, the one-half or five-tenths which remains must be added to the figure 2 of the 25; or we may take one-half of the next smaller number, and use 75 as the suffix. TO MULTIPLY BY TWO FIGURES AT ONCE. Rule. — Multiply both figures in the multiplier by each figure in the multiplicand separately. Note. — When large numbers are to be multiplied, for the purpose of remembering which figure has been used, place a dot over each figure of the multiplicand as soon as multiplied. . EXAMPLE. 3265 Bx D4 ae 207, 24 6x 24+12=—156 (pha ss 2x24115=—63 78360 3x 2416=78 To Multiply by Aliquot Parts of 100, 1000, Ete. It is very important for an accountant to have a perfect knowledge of the table of Aliquot Parts of 100 and 1000. All goods sold at wholesale are bought and sold by these calculations, and those not ‘amiliar with the operation will often lose much val- uable time in obtaining a correct result. By this method they can arrive at the result in one-tenth of the time, and are not so apt to make mistakes. ALIQUOT PARTS. Ororo: 2=4 49> Ws ood St US) Of 100. 6z = 1s 16g = 4 50 == 4 Slow. 2 = 4 62 — gs 12,5 = 4 Boa 3 fi) mel? Mis} 8a 8m =k 18¢ = 3 3814 — +, Of 1000. Soh esp ieee bey fer ie 250 \—= 4 375 = 8 625 == 8 or 1-16 of 10,000. 8334 = 2 or 1-16 of 1,000. 875 = 42 ExaMPLeE 1.— Multiply 464 by 25 = 11600. 4)46400 11600 This is, in effect, the same as to multiply by 100 we divide by 4 because 25 is 4 of 100, which is th. same as multiplying by 25. In the same manner, annex two ciphers and divide by 2 multiplies by 50 ; annex two ciphers and divide by 8 multiplies by 124 ; or annex two ciphers and divide by 8 to multiply by 125, etc., etc. This same principle may be applied in any Aliquot Part of 10, 100, 1000, as shown in preceding table. RuLE.—Add ciphers to the multiplicand and divide by the number, as the multiplier is a part of 100 or 1000. When the multiplicand is a mixed number, reduce the fraction to a decimal and proceed as before. ExampPLeE 1.— Multiply 434 by 24. 4) 43400 10850 EXAMPLE 2.— Multiply 535 by 25. 4)53500 13375 EXAMPLE 3.— Multiply 5642 by 34. 3)56420 188062 EXAMPLE 4.— Multiply 4321 by 334. 3) 4382100 1440332 EXAMPLE 5.— Multiply 1254 by 13. 6)123840 20568 EXAMPLE EXAMPLE EXAMPLE EXAMPLE EXAMPLE EXAMPLE EXAMPLE EXAMPLE EXAMPLE 6.— Multiply 2245 by 13. 7)22450 32071 7.— Multiply 4456 by 14. 8) 44560 5570 8.— Multiply 5644 by 14. 9)56440 9.— Multiply 4324 by 64. 16) 432400 27025 10.— Multiply 5642 by 84. 12) 564200 470162 11.—Multiply 5648 by 124. 8) 564800 70600 12.— Multiply 6843 by 142. 7) 684300 97757} 13.— Multiply 7824 by 163. 6) 782400 130400 14.— Multiply 7846 by 834. 12)7846000 6538334 43 44 Examp.e 15.— Multiply 7896 by 125 $4of 1000. 8)7896000 987000 EXAMPLE 16.— Multiply 1246 by 16632. 4of 1000. 6)1246000 2076664 EXAMPLE 17.— Multiply 8453 by 250. $ of 100. 4)8453000 2113250 EXAMPLE 18.— Multiply 4642 by 625. qs of 10000. 16)46420000 or, 2901250 & of 1000. 8) 4642000 580250 alas isha niiee a2, 2901250 EXAMPLE 19.— Multiply 5642 by 8334, vz of 10000. 12)56420000 or, 47016662 2 of 1000. §)5642000 9403334 i A 47016663 EXAMPLE 20.— Multiply 1342 by 875. Z0f 1000. 8)1342000 167750 7 1174250 4D APPLICATION OF THE TABLE OF ALIQUT PARTS, In order to give an idea of rapid calculation in multiplication, a few examples will here be given, and from these others may be created without limit by anyone: 35 yards cloth @ $2.50. Add one cipher and divide by 4. Answer, 874 or $87.50. 216 yards cloth @ $2.25. Multiply $24 by sp setting down the amounts thus: “$486 48 yards cloth @ $2.124. Multiply by $24 in same manner as by $24. 55 yards cloth @ $1.95. Move decimal point 97.50 in price two places to the right, divide 195 9.75 by 2 and add +, thus: $107.25 162 yards cloth @ $1.80. Multiply by 2 and eet deduct ay thus : $291.60 36 yards cloth @ $1.75. Multiply by 2 and ta deduct 4, thus: $63 29 29 yards cloth @ $1.624. Add 4 and to 14.50 the whole, thus- 3.623 $47.12 114,50 1144 yards cloth @ $1.50. Add 4, thus: 57,25 $171.75 37.75 372? yards cloth @ $1.25. Add 4, thus: 9.44 $47.19 83 83 yards cloth @ $1.20. Add }, thus: 16.60 $99.60 46 APPLICATION OF THE TABLE OF ALIQUOT PARTS. 50 yards cloth @ $1.18. Find 4 of $118 = $59. 114 75 yards cloth @ $1.14. Deduct from $114 28.50 4 of that amount, thus: $85.50 24 yards cloth @ 95c. Deduct sh from $24, 74 yards c c. Deduct 5'5 from $ 1.20 thus: “$22.80 68.50 684 yards cloth @ 75c. Deduct 4, thus: 17.12 | $51.38 Eathetiy 23 46 yards cloth @ 55c. Find $ and add to 2 30 same ;,, thus: $25.30 : 16 32 yards cloth @ 45c. Find 4 and deduct 1.60 Zo, thus : $14.40 96 yards cloth @ 25c. Find 4 of $96 = $24. The reason for making the computations in this manner will at once be apparent, from the fact that when the price is either more or less than $1, the fractional part of a dollar is either taken from, or added to, the amount that sum would be if @ $1 per yard. It is sometimes more convenient to call the number of yards the price, and the price the number of yards, in order to make the computation, as in the example above ; 50 yards @ $1.18 would be the same as 118 yards @ 50c. ; or to say, if 100 yards @ $1.18 would be $118, 50 yards would be half of that amount, or $59. In the first example we say, if 35 yards @ $10 per yard would be $350, at $2.50 per yard it would be one-fourth of that amount, or $87.50. 47 It is of course expected that the student will per- form these operations in his mind. Nothing is more desirable for an Entry Clerk or Book-keeper than to have a thorough knowledge of the aliquot parts of 100 or 1000. Many remunerative situations have been obtained by those who thoroughly understood the practice, though their general education was very limited. Constant practice will enable anyone to give the products as fast as the questions are given, Another mode of multiplying, when the multiplier can be divided into factors, is an improvement on the common method ; but to multiply in a single line is still better. ‘The objection to this method is that when an error occurs in the first line it-will run into the second. EXAMPLE: — 1234 124 4936 14808 153016 Here we multiply through by 4; now, since 12 is 3 times 4, if we tuultiply the first line by 3 we have the product of 12 ina single line. Quite a variety of examples can be worked in this way. HOW TO PROVE MULTIPLICATION BY CASTING OUT THE NINES. RULE.— Find out the excess over nine in your multipli- cand and multiplier, and if the excess in the product of these excesses is the same as the excess in the product, the operation is correct. EXAMPLE :— oo 4 | Leap ae 8 a ere ~] “xX x = 153016 MULTIPLICATION ——— BY —— SOUARING NUMBERS. = 1 spans 9? 4 13? — 169 S7=. 9 142 — 196 4? .. 16 15? = 225 2 25 167 — 256 6? = 36 172 = 289 7? == 49 , 18? = 324 8? — 64 19? — 361 9? = 81 20? = 400 10? —100 212 = 441 11? —=171 222 — 484 Notrre.—The product of any two numbers is equal to the square of the mean, minus the square of half their differ- ence. The meax isa number as much greater than the less, as it is less than the gréater. RuLeE.—From the square of the mean subtract the square of the difference between either of the given numbers and the mean. EXAMPLE 1. 17 13 15* — 4 o= 221 221 REMARK.— 15 is as much greater than 13 as :t is less than 17, 15 is therefore the mean between 17 and 13. Half the difference is 2, the square of 2 is 4; or, we may say, the difference between the mean and either of the giv- en numbers is 2, square of which is 4. 49 EXAMPLE 2.---What cost 19 books at 13 cts. each ? SoLuTIon.—The mean of 13 and 19 is 16, the square of which is 256 —9, the square of a/f the difference of the given numbers = 245. EXAMPLE 3.—What cost 13 tons of hay at $14.00 a ton? Thus: 13? + 13 = 182 Where the difference of two numbers is a unit, we add the less number to its square, for 13 X 13 = 14 times 13. EXAMPLE 4.—What cost 16 ounces of gold dust at $17.00 an ounce? Thus : 167 +. 16 = 272. Table of Square Numbers continued. 26? — 676 277 = 729 237'== 529 287 = 784 247 — 576 29? = 841 257 == 625 30? = 900 Here let the student observe that the two right- hand figures in the square of any number, as much less than 25 as another is greater, are the same in one case asin the other. For example, in the squares of 23 and 27, the one as much less as the other is greater than 25, the two right-hand figures are the same. ‘This law holds true in all cases. EXAMPLE 1.—What cost 23 shad at 27 cts. each ? DA Ven A ie OTL EXAMPLE 2.—What cost 26 tons of hay at $27.00 a ton? 26? + 26 — 703 50 Table of Square Numbers continued. 317 = 961 197 == (361 CUE Lae 18? = 324 33° ==) 1089 LZ? c=) 1289 347 = 1156 16% 25256 $57 == 1225 157)== 1225 SO =F 12960 147° i396 377 == 1369 13? — 169 387 = 1444 12? = 144 Sones Lee 1H poet Brg 40? — 1600 10? == 7100 The first column is placed here to afford the stu- dent an opportunity of observing that the ‘wo right- hand figures in the square of any number which is as much less than 25 as the other is greater, are the same in the former case as in the latter; or, which is the same thing, the two right-hand figures in the square of any number as much és than 20 as anoth- er is greater than 80 are the same in one case as in the other. Table of Squares continued. 4}? — 1681, 9? 46? = 2116, 4? 42? — 1764, 8? 47? == 2209, 3? 43? — 1849, 7? 48? — 2304, 2? 442 — 1936, 6 49? — 2401, 1° 45? — 2025, 5? 50? = 2500, 0° The student need have no difficulty in remember- ing the squares in the above columns. Observe that the two right-hand figures in the square are in every instance the square of the difference between the — units figure of the root and 10; the two left-hand fig- ures in the square will also be easily remembered if we observe that the number is formed by adding 1 less than the right-hand or units figure of the root 61 to the square of the left-hand, or tens figure; thus, in the square of 47 the left-hand figures are 22, or 1 less than 7 added to the square of 4, thus: 47 == 16 + (7—1) =22. The right-hand fignres are obtained by subtracting 7 from 10 = 3; the square of 3 = 9 Note.— When the square of the difference between right- hand figures and ten is not over 9, prefix a cipher, as in above case. EXAMPLE 2.—437 = 10 — 3= 7 = 49, right- hand figures; 4 X 4 +- 2 = 18, left-hand figures ; therefore 43° — 1849. Table of Squares continued. Olt o01 567 — 3186 ra) 2704 WY feditomensts PY 4!) Dose 2509 58? == 3364 54? — 2916 TE bipeeegitey 23H) Wes OULD 60? == 3600 This square can also easily be remembered if we observe that the left-hand figures in every instance may be produced by adding the right-hand figure of the root to the square of the left-hand figure, and the number expressed by the two right-hana figures in the square, is the square of the right-hand figures in the root. . Example, square 56. Here we say 6? = 36, which are the right-hand figures in the square. Again, 5? + 6 = 31, which are the left hand figures in the square. Table of Squares continued. Beemer ete. te P10) OF ee AQ 00 4* — 576 ee Pte 4A ee O41 2 5* == GBD age OO ath = 1692) 72% O 184.) 967 == 676 Bae OG nL 4 1905 8 fos Doeo ae 27° => 729 Cie oa hye 229) 14> 47 G., 28) C84 52 16? — 256 66 —= 43856 7h" = D620). oo ee 677 — 4489. 177=- 289 | 767 5776 © 807 = 900 687 — 4624) 3/188 = 324 7 77 5929) ae 69274761) 197 Sols 006 eee 707 — "4900. © 207 == 4000 70? = 6247 25s 80? = 6400 In the above tables the columns of squares on the right are thus placed with their roots, in order that the student may associate them with the squares of the left-hand column. He will of course observe that two figures (units and tens ) are in regular suc- cession, the same in one column as in the other. In the following columns the same principle will be observed. 1 Gob Lalo 1361 Oi4=2"8281 2 9a0 sed S20 C7240 9 1 By 324 927 -2"S464 28ers So 0909 | 1d) ag 937 S649 aia 47 (0560-16 26 947 — 8836 67== 386 Fait 1 ya Sea dh meme 43) 957 == 19025 Wap are Bie (odOea Laci 962 02 [Ger eee Bie OUTU Mla RD O74 9409 ar ee Sore 71744 12h eas 95729604) 632. one (op eteess TAPMEL I OT dean NIB 9073/9 OO La cane 90781002 107 == 00) 1 OUt = 0000 ame The square of last column may be obtained by adding twice the right-hand figure of the root to 80. thus: 917—2X1—=2+80—82. EXAMPLE 2,— 99? = 80 + 18 = 98, left-hand fig- ures in the square. TO SQUARE ANY NUMBER ENDING IN 5. RuLE.— Multiply the part preceding the units by itself, increase by a unit and prefix the product to 25, thus: 65?== 6 X 7 == 42 ; prefix 25.== 4225. EXAMPLE 2,—72?== 7X 8= 56; 5620. prelxm2ome— DIVISION. ' The work in division can be abbreviated by not writing out the product figures, and finding the re- mainders as we pass along. RULE. — Subtract each product figure as it is formed (that is, the right-hand figure of the product ), and when it is greater than the figure from which you subtract, carry one more to the next product figure than you would other- wise carry. EXAMPLE :— 29)15341(529 8 2 6 Say5X9= 45. 5 from 13 = 8; then 5 X24 4+-1=—15. Our next quotient figure is 2; now say 2X9=—18; 8 from 14=—6; 2X2+1-+1=6, subtracted from 8 leaves 2, or 26, our next, etc. Vhe Italian method of dividing is to place the di- visor to the right of the dividend, and the quotient underneath it. The advantage in this is, you bring the operation closer together and can readily multi- ply the quotient by the divisor, to prove the work. EXAMPLE :— 04 CONTRACTIONS IN DIVISION. The table of Aliquot Parts on page 25 can be ap- plied to division. To divide by 25 multiply the dividend by 4 and point off two figures to the right-hand of the product as so many hundredths, or take one-fourth of the two right-hand figures of the product as so many twenty-fifths. ExaMPLE 1.— Divide 3757 by 25. De ok 4 15028 or, 1507, EXAMPLE 2.— Divide 437924 by 125. 437924 8 3503392 = 350349 This method will hold good through the entire ta- ble of Aliquot Parts. It is simply the reverse of multiplication. Another convenient mode of divid. ing is to reduce the divisor into factors. ExampLe.— Divide 34969 by 24. Dd rece Ok 6)34569 4) 576 1— 3 over. 1 44 0 — 1 over. 6 X 1+ 3 = 9 true remainder. or, 4) 34569 he aan 1440—2 2X 4 + 1 = 9 true remainder. FRACTIONS. A Proper Fraction is one whose value is less than a unit, as 4, #. An Jmproper Fraction is one whose value is equal to or more than a unit, as 3, 3. A Mixed Fraction consists of a whole number and a fraction, as 24, 44. 1.—Multiplying the numerator of a fraction by any number multiplies the value of the fraction by that number. | 2.—Dividing the numerator of a fraction by any number divides the value of the fraction by that number. 3.—Multiplying the denominator of a fraction by any number divides the value of the fraction by that number. 4.—Dividing the denominator of a fraction by any number multiplies the value of the fraction by that number. 5.—Multiplying both numerator and denominator of a fraction by any number does not change the value of the fraction. 6.—Dividing both numerator and denominator of a fraction by any number does not change the value of the fraction. 56 To Higher Terms How many fourths in $? SoLuTion. —In 1 there are }, and in 4 there are $ of 3, Ores ' How many sixths in ae 4? 8? §? How many eighths in Ae AP EP ee How many tenths in a? 4? 3? SP How many twelfths in 4? 42? 4? How many fourteenths in $? #? 4? #2? How many fifteenths in 3? $2? 4? 3? How many sixteenths in 4? #? 2? 3? How many eighteenths in 3? 4? #3? 8? How many twentieths in 4? 3? 73? #3? Te Lower Terms. How many thirds are equal to ¢? SoLUTION.—+# is equal to 2, therefore 4 of the number of sixths equals the number of thirds; 4 of 4 is 2. How many halvesin 2? $$? 8? 149? How many thirdsin #2? ¢? §? 48? How many fourths in $2? 8? 3%? le How many sixthsin 18? ? 8? 48%? How many eighths in 4? 42? 8? 32? How many fifths in go? 8h S22? 38? How many sevenths in +9? 8? 3%? 32? How many ninthsin 427 48? 35? 38? How many tenthsin 48? 21? 24? 25? To a Common Denominator. When fractions have the same denominator, they are said to have a Common Denominator. Ex.— Reduce % and # to a common denominator. SoLUTION.—A common eae ecnage for 3ds and 4ths is I2ths; in 1 there are 13s and in # there are # of +4, or 53 and in § there are % of 14, or 75. Reduce 4 and } to a common denominator. Reduce 4 and to a common denominator. Addition of Fractions. What is the sum of 3 and #? SoLuTIon.—# equals 38;, and $ equals 3;; 38 plus #5 are 1f, which equals 135. What is the sum Of 4 and }? Of 2 and 3? Of? and st Of 3 and 4? Of # and #? Of 34 and 44? Of 34 and 22? Of 43 and 5}? Of 73 and 83? Of 4, 4, and } Of 24 and 34? Of 2% and 13? Of 62 and 52? Of 61 and 54? Of 4, 4, and 2? ? Of 4, 4, and 4? Subtraction of Fractions. What is the difference between SOLUTION. — } is equal to 5%, and 3 2 # and 8? 2 3 minus ,5, equals 4. % from 4? # from 8? 4 from 3? epost 8 from §? 24 from 3}? Subtract 2from ¢? 3 from 2? 4 from }? 1 from 3? 2 from ¥? 4 from 2? 4 from 3? 2 from #? 24 from 34? 31 from 5}? 31 from 44? Multiplication of Fractions, How many are 4 times 2? SoLuTion.—4 times ~ are 12, which equals 3 or J, How many are 3 times How many are 4 times How many are 7 times How many are 4 times How many are 8 times How many are 8 times How many are 5 times 2? 3 times 2? §? 5 times +? 7? 38 times 2? tf? 6 times 7? St tunes yee 2? 8 times 3? sco: meses? 57 is equal to &; *% 58 Division of Fractions. How many times is % contained in 4? SoLuUTION.—1 contained in 4, 4 times; and if 1 is cone tained in 4, 4 times, 4 is contained in 4, 3 times 4 times, which are 12 times, and 2 thirds is contained in 4 4 of 12 times, or 6 times. How many times is 2 contained in 2? 4 How many times is ? contained in 8? 5 itow many times is ? contained in 2? How many times is 3 contained in 5? How many times is 2 contained in 4? How many times is § contained in 2? 8 How many times is ? contained in 3? SOLUTION.—? is equal to $$, and 3 is equal to 42; 439 is contained as many times in $2? as 10 is contained 12, which is 12 or £ times. How many times is ? contained in 2? How many times is ? contained in 3? How many times is 2 contained in }? How many times #is $3? 3? #2? How many times 2 is}? $? %? Relation of Fractions. What part of 2 is #? SoLuTION. — 1 is 4 of 2; and, if 1 is 4 of 2, 4 is 4 of 4, which is 4 of 2, and # is 8 times 3, or 2 of 2. What part of 3 is #? Of 2 is What part of 4 is 2? Of 5 is What part of 4 is ¢? Of 7is 2: What part of 9is 3? OF 5 is 4 of 2? sokn eco ICO wv wv In 3? In 5? In 4? In 7? In 5? In 4? What part of 6 is of 2? Of 7 is 2 of 8? What part of % is #? SoLuTIon,—} is } of 3, and 3, or ome, is 3 times 4, or 3 of 3. Since ove is $ of 3, 4 is t of $, which is }%) of %, and # is 4 times ;*;, which are +4, or § of 3. What. part of #'isi2?,. OL 2 ister), Of fis 22 What part of $ is $#? Of 2? is #? Of 3 is 23? kieducing to Fractions. What is 4 of 4? So.tuTion.—} of 1 is 3, and if } of Lis }, 4 of 4 is 4 times 3}, which are }. What is 4 of 5? 4 of 6? tof 7? 4 of 9? 4 of 5? Z4of 10? 4Fof122 ~4 of 20? py of 24? qs of 32? 2 of 6? # of 10? What is 3 of §? SoLuTIon. — + times 3, which are { or $. of § is 2. and if } of $18 2, ¢ of § are 2 What is 3 1 2 vi 3 6 2 4 1 % of 13? 2 of $? 8 of §: % of 38? 3 15 2 1 9 § of +3? Zof té? gof wy? What is 4 of 4? SoLuTion.—4 is one of the 4 equal parts into which a unit may be divided ; if we divide each fourth into 3 equal parts, each part is + of 4, and since there are 4 times 3, or 12 parts in all, each part is =); of a unit? ANOTHER SOLUTION.—}# equals #,,.and } of ,%, is +4. This is a simpler solution, but not so explanatory. What is 3 of 4? What is 4 of }? What is 4 of 3? What is } What is What is What is What is of 4? of 4? of Jy? of py? of #? (Xoo o> eras Uo 4times 4, which are ,4, and # of # are 2 times ,4, What is 4 of 3? What is 4 of 3? What is } What is 3 What is 2 4 of 4? 4 of f? 4 of 4? 4 of 3? 1 of 3? 4 of +? + of +? % of +? 1 of 3? 4 of 3? 4 of 3? lof p>? po Of pr? yy of ry? SoLurion. — 4 of | is 4, and if 4 of 4 is A, $ of § is or &. 4 of 3? } of 4? Zof 8? 40f 8? Zof%? sof 4? 4rnek # of 3? € of $? Bofe? 3% of 3? CONTRACTIONS Hal An@ le EG Nis To square any number containing 4, as 74, 84. RuLE.—Multiply the whole number by the next higher whole number, and annex to the product. Ex. 1. What is the square of 74? Ans. 564. We simply say, 7 times 8 are 56, to which we add 4. What will 94 lbs. beef cost at 94 cts. a lb. ? What will 125 yds. tape cost at 124 cts. a yd.? What will 53 lbs. nails cost at 54 cts. a Ib.? What will 114 yds. tape cost at 114 cts. a yd.? What will 195 bu. bran cost at 193 cts. a bu.? cat ner Reason.—We multiply the whole number by the next higher whole number, because half of any number taken twice and added to its square is the same as to multiply the given number by ONE more than it- self. The same principle will multiply any two “Ze numbers together, when the sum of the fractions is ONE, as 84 by 88, or 112 by 118, etc. It is obvious 61 CONTRACTIONS IN FRACTIONS. that to multiply any number by any two fractions whose sum is ONE, that the sum of the products must be the original number, and adding the number to its square is simply to multiply it by onE more than it- self ; for instance, to multiply 74 by 73, we simply say, 7 times 8 are 56, and then, to complete the multi- plication, we add, of course, the product of the frac- tions (} times 4 are ,3,), making 56%, the answer. Multiplication, WHERE THE SUM OF FRACTIONS IS ONE. To multiply any two like numbers together when the sum of the fractions is ONE. Ro.e.—Multiply the whole number by the next higher whole number ; after which, add the product of the fractions. Multiply 32 by 34 in a single line. Multiply 4 < #, which gives ~,andsetdown 3% the result ; then we multiply the 3 in the multi- 34 plicand, increased by unity, by the 3 in the multiplier, 3 X 4, which gives 12 and completes 12,5, the product. Multiply 72 by 72 in a single line. Multiply 2 X 2, which gives 5%, andset down 7 the result ; then we multiply the 7 in the multi- 73 plicand, increased by unity, by the 7 in the multiplier, 7 X 8, which gives 56 and completes 563%; the product. Multiply 114 by 112 in a single line. Multiply 3X 4, which gives 3, and set down 114 the result; then we multiply the 11 in the 113% multiplicand, increased by unity, by the 11 in —— the multiplier, 11 X12, which gives 132, and 1323 completes the product. 62 CONTRACTIONS IN FRACTIONS. To multiply any two numbers together, each of which involves the fraction 4, as 74 by 94, etc. RuLe.—To the product of the whole numbers add half their sum plus 4. 1, What will 34 doz. eggs cost at 74 cts. a doz.? Here the sum of 7 and 3 is 10, and half this 34 sum is 5, so we simply say, 7 times 3 are 21 73 and 5 are 26, to which we add 4. 264 N. B. 1f the sum be an odd number, call it one less to make it even, and in such cases the fraction must be 3. 2. What will 114 lbs. cheese cost at 94 cts. a lb.? 3. What will 84 yds. tape cost at 154 cts. a yd.? 4. What will 74 lbs. rice cost at 134 cts. a lb.? 5, What will 1U§ bu. coal cost at 124 cts. a bu.? ReEason.—In explaining the above rule, we add half their sum, because half of either number added to half the other would be half their sum, and we add 4, because 4 by 4 is 4. The same principle will multiply any two numbers together, each of which has the same fraction ; for instance, if the fraction was 4, we would add one-third their sum; if #, we . would add three-fourths their sum, etc. ; and then, to complete the multiplication, we would add, of course, _ the product of the fractions. MULTIPLYING ANY TWO NUMBERS TOGETHER, EACH INVOLVING THE SAME FRACTION. RuLe.-—To the product of the whole numbers, add the product of their sum by either fraction; after which, add the product of their fractions. 1. What will 113 lbs. rice cost at 92 cts. a lb.? Here the sum of 9 and 11 is 20, and three- 113 fourths of this sum is 15, so we simply say,9 93 times 11 are 99 and 15 are 114, to which we add the product of the fractions (9). 114,5, 63 CONTRACTIONS IN FRACTIONS, What will 72 doz. eggs cost at 8% cts. a doz. ? What will 62 bu. coal cost at 62 cts. a bu. ? What will 452 bu. seed cost at 32 dols. a bu.? What will 32 yds. cloth cost at 53 dols. a yd.? What will 172 ft. boards cost at 132 cts. a ft.? 7. What will 182 lbs. butter cost at 182 cts. a lb.? N.B. If the product of the sum by either fraction is a whole number with a fraction, it is better to re- serve the fraction until we are through with the whole numbers, and then add it to the Lieec of the frac- tions ; for instance, to multiply 34 by 74, we find the sum of 7 and 3, which is 10, and one-fourth of this sum is 2}; setting the 4 down in some waste spot, we simply say, 7 times 3 are 21 and 2 are 23; then, adding the 4 to the product of the fractions (9), gives 78, making 23,9, Ans. LEA ara ae MULTIPLYING ANY MIXED NUMBERS. Ru Le 1.—Multiply the whole numbers together. 2.—Multiply the upper digit by the lower fraction. 3.—Multiply the lower digit by the upper fraction. 4.—Multiply the fractions together. 5.—Add these four products together. 122 ExaMPLE.— Multiply 122% by 9%. 93 1.—We multiply the whole numbers = 108 2.—Multiply 12 by i — 9 3 —Multiply 9 by $ = 6 4.—Multiply # by $ = 0-8; 5.—Add these four aie together, 123.5 and we have the complete result. N.B. When the student has become familiar with the above process it is better to do the intermediate work mentally, and, instead of writing out the partial products, add them in the mind as you pass along, and thus proceed very rapidly. 64 CONTRACTIONS IN FRACTIONS. PRACTICAL BUSINESS METHOD FOR MULTIPLYING ALL MIXED NUMBERS. Business men generally. in multiplying the mixed numbers, only care about having the answer correct to the nearest cent; that is, they disregard the frac- tion. When it is a half cent or more, they call it another cent; if less than half a cent, they drop it. And the object of the following rule is to show the easiest and most rapid process of finding the product to the nearest unit of any two numbers, one or both of which involves a fraction. Multiply 81 by 104. Here we simply say 10 times 8 are 80 and} 8} of 8 is 2, making 82, and } of 10 is 2, which 104 makes 84; then 4 times } is 54, making 84,4 the answer. 84545 TO MULTIPLY ANY TWO NUMBERS TO THE NEAREST UNIT. GENERAL RULE 1.—Multiply the whole number in the multiplicand by the fraction in the multiplier to the near- - est unit. 2.—Maultiply the whole number in the multiplier by the fraction in the multiplicand to the nearest unit. 3.—Multiply the whole numbers together and add the three products in your mind as you proceed. N. B. In actual business the work can generally be done mentally; only easy fractions occur in business N. B. This rule is so simple and so true, according to all business usage, that every accountant should make him- self perfectly familiar with its application. There being no such thing as a fraction to add in, there is scarcely any liability to error or mistake. By no other arithmetical process can the result be obtained by so few figures. 65 CONTRACTIONS IN FRACTIONS. EXAMPLES FOR MENTAL OPERATION. EXAMPLE FIRST. Multiply 114 by 84 by business method. 114 Here 4 of 11 to the nearest unit is 8, and4 84 of 8 to the nearest unit is 3, making 6, so we simply say, 8 times 11 are 88 and 6 are 94, Ans. 94 Reason.—{ of 11 1s nearer 3 than 2, and } of 8 is nearer 3 than 2. Make the nearest whole number the quotient. EXAMPLE SECOND. Multiply 72 by 9% by business method. Here ? of 7 to the nearest unit is 3, and # 7? of 9 to the nearest unitis 7; then 3 plus 7is 92 10, so we simply say, 9 times 7 are 63 and 10 are 73, Ans. 73 EXAMPLE THIRD. Multiply 234 by 194 by business method. Here 4 of 28 to the nearest unit is 6, and4 234 of 19 to the nearest unit is 6; then 6 plus 6is 194 12, so we simply say, 19 times 23 are 437 and — 12 are 449, Ans. 449 EXAMPLE FOURTH. Multiply 128% by 25 by business method. 128% Here 2 of 25 to the nearest unit is 17,so 26 we simply say, 25 times 128 are 3200 and 17 - are 3217, the answer. $217 PRACTICAL EXAMPLES FOR BUSINESS MEN. 1 What is the cost of 174 lbs. sugar at 182 cts. per lb.? 66 CONTRACTIONS IN FRACTIONS. Here # of 17 to the nearest unit is 13, and = 174 4 of 18 is 9, 13 plus 9 is 22, so we simply 1832 say, 18 times 17 are 306 and 22 are 328, the ——— answer. Bo: 26 2. What is the cost of 11 lbs. 5 oz. of butter at 334 cts. per Ib. ? Here § of 11 to the nearest unitis 4,and 11 - 1g of 38 to the nearest unit is 10; then 4 334 plus 10 is 14, so we simply say, 33 times 11 are 363 and 14 are 377, Ans. 3. What is the cost of 17 doz. and 9 eggs at 124 cts. per doz.? Here 4 of 17 to the nearest unit is 9,and = 17,5 fz of 12 is 9; then nine plus 9 is 18,sowe 124 simply say, 12 times 17 are 204 and 18 are 222, the answer. 4, What will be the cost of 153 yds. calico at 124 cts. per yd.? Ans. $1.97. $3.77 , —s DECIMALS. To reduce a decimal to a common fraction. Ru.e.— Write the decimal as it stands, omitting the deci- mal point, for the numerator. For the denominator write 1 with as many ciphers annexed as there are decimal places in the numerator. EXAMPLES, Reduce .25 to an equivalent common fraction. Ans. 7°5, which reduced to its lowest terms = 4. Reduce .375 to acommon fraction. Ans. 73,4°5 = 2. Reduce .1875 to a common fraction. Ans. 4. Reduce .625 to a common fraction. Ans. %. To reduce common fractions to decimals. RuLEe.—Annex ciphers to the numerator, and divide by the denominator, prefixing a point to the quotient. There must be as many places in the quotient as there have been ciphers annexed; if not enough, prefix ciphers. 68 EXAMPLES. Reduce # to a decimal. 4)3.00 .75 = hy Ans. Reduce # to a decimal. Ans. .375, AI pa HS Ans. .8571-++. 66 4 73 be bs ce 3 66 6é MULTIPLICATION OF DECIMALS. RuLe.— Multiply as in whole numbers, and from the right of the product point off as many figures for decimals as there are decimal places in both mult¢plicand and mul- tiplier. If there are not figures enough in thef product, prefix ciphers. EXAMPLES. Multiply 4.25 by 6.5. 4.25 6.5 2125 2550 27.625 Ans. Multiply 84.5 by 4. pM! LASS SD 4 338.0 Multiply 6.425 by 4.25. 3 ibe Mg ES Ge * LOD UN Oe He 25. ** 6.0025 275 ** 3.0025 «18.625 « 5.25. What is the cost of 12% lbs. at 64 cts. per-lb? What is the cost of 72 yds. at 182 cts. per yd? NOTE.—It is sometimes more convenient to change common frac- tions to decimals before multiplying. 18} x 124 = 18.75 x 12.5. 69 UNITED STATES MONEY. 10 Mills(m)= 1 cent, ct. 10 Cents =1dime, d. 10 Dimes = 1 dollar, §. 10 Dollars = 1 eagle, E. The origin of the symbol $, or the United States dollar mark, has been ascribed to several sources. By some it is supposed to represent the ¢/ written upon the S, denoting U. S. (United States). Some think it is a modification of the figure 8, having refer- ence to 8 reals, or piece of Eight, as the dollar was formerly called; others, that it represents the “ Pil- lars of Hercules,”’ which were stamped on the Pillar Dollar ; and others, still, that it is a combination of the initials P. and S., from the Spanish Peso Duro, signifying Hard Dollar. As it is used in Portugal to note the thousands’ place, it is probable that it ori- ginated in that country: a Mil-reis, or thousand. reis, is written thus, 1$000. The term Dime is from the French dsme, mean- ing ten. The term Cent is from the Latin centum, a hundred. The term Mill is from the Latin mz//e, a thousand. 70 UNITED STATES COINS, The Gold Coins are the Double Eagle, $20.00 ; Eagle, $10.00 ; Half Eagle, $5.00; Quarter Eagle, $2.50 ; three dollar piece and dollar. The Fifty Dollar Piece is not a legal coin. The Half Copper Cent is no longer coined. The Mill is not a coin. Gold coin contains nine parts gold and one part copper and silver. The Silver Coins are Dollar, Half Dollar, Quarter Dollar, Dime, Half Dime and Three Cent Piece. Silver coins contain 9 parts silver and 1 part copper. The Nickel Coins are the Cent the new Three Cent Piece and new Five Cent Piece. The Nickel contains 88 parts copper and 12 parts nickel. The Copper Coins are the Cent and Two Cent Pieces. The Two Cent and Cent Pieces are made of nickel and copper. One Eagle (Gold) weighs 258 troy grains. One Dollar (Silver) weighs 412.5 troy grains. One Cent (Copper) weighs 168 troy grains. 23.2 grains of pure gold = $1.00. Gold Coins prior to 1834, like that of England = 88.8 per dwt. By an act of Congress of 1834 its value was made 94.8 cents per dwt. The old U.S. Eagle coined previous to 1834 is worth $10.66.8. By an act of Congress the payment of debt with coin is regulated as follows: All Gold Coins at their respective value for any amount. The Half Dollar, Quarter Dollar, Dime and Half Dime at their respective value for debt under $5.00. The Three Cent Piece for debts of any amount under thirty cents. The one cent pieces for debts of any amount under 10 cents. 71 STATE CURRENCY. The money of this country before the adoption of the decimal currency by Congress in 1786 was in the denominations of pounds, shillings, and pence. The Colonial notes which were then in circulation had depreciated in value; and the number of shillings equivalent to a dollar at that time are given in the following table :— New ENGLAND CURRENCY. New England States, Virginia, ' $i 6s 27, Kentucky, and Tennessee, Vso) Lbs.cts, ——— rs New YorK CURRENCY. New York, Ohio, Michigan, $1°—= 8s, == 962. and North Carolina, 15, ==) 124 cts. PENNSYLVANIA CURRENCY. Pennsylvania, New Jersey, at Sli 26.00. Us aware, and Maryland, Ise=13% cts; GEORGIA CURRENCY. Georgia and South Carolina, $1 = 4s. 8d. = 56d. ieee ueeCts. 72 The Coins of Foreign Nations, With their value in United States coins, as determined by the recognized standard at the Mint in Philadelphia. | Gm pn Pa BD BN a te 2 eS COUNTRY. $ AQBtCTiA wits wie oe wate -|Fourfold ducat ..........| 9 bs o\4,0/0:8:0'. sna son's (4 HOFIDS (NEW) vino x'ae wares 1 SeAEe Peeves lettis clots leer eis Duca tires statics sees atatste les Z Belgium .....-+-...... 25 TENCE sen ws sew siarseees 4 lite bales oR eae ics” BO aNtITets sss sce se'ele ase 10 Central America ..... FPSCRAGE) rd sabe ales o aterbio ee 3 CET UN Nie dels Toe, A Teal sited sae 6 ose eo Chili ....--...2------ 10 pesos (dollar) ........| 9 Columbia and South Value in U.S. money DENOMINATION. America generally..|/Old doubloon ..........- 15 Columbia ........-... 20 pesos, ‘‘ Bogota” ..... 18 ay pee e cree recs 20 pesos, ‘: Medellin”’....{19 , sccesecnnnee 20 pesos, ‘* Popayan” 18 Costa Rica. ..-+seces. 10) PesOSiesa te roca renee 8 Denmark © <<. ss e008 AY Crowne 4. 2 we ns ae ee 5 SET aR ous sctaatete Old ten thaler. Dn eS ort n 7 Egypt .----ccsccccees Bedidlik (100 piasters)...| 4 England . ...2seee.ee. Pound, or sovereign (new) 4 Fraticec ccc ss ss neateO ATA Csiia a enlete vis ete ene 3 German eas re Cae New 20 marks..........- 4 Greece .. -cese e120 drachMS o2-scceen-ese 3 India (British) in Gia tele Mohur. or 15 rupees...--.| 7 tal. ~ cee oly aie wares QO Tite: h alae choad one cen 3 JAPAN .--- es ee seceece, 20: VET ~ ceces svccereccwes 19 Mexito Vy oh sag es a neues Doublponaues ety sate ce 15 Netherlands.........- 10; Piidera Mun Wares aanlen 3 New Grenada ........ 10 p: sos (dollars) .....-.. 9 Norway --++--+++eee-- PU CVO WAS atts s ele, wt wtelae ecets 5 Perit tegeny aah ea ante AQ-GO1E8. uaa! 19 SEM ae wie vee eg esiae -|Coroa (crown) «-+++.-... 5 Russia -..-cccescoees [5 rubles. ..--sccccvcccess 3 SOE cases + 4's ones ot 100 Tealsrsiv Ah = to New florist. « wesepisssslee shee 0 SOMO alive's; ¢.0 «10,2 New Union dollar....... ) Belgium’ .-..+-.+,..... S4Tance +: eee tides eoee tO Bolivia ..... Shales ore coats New dollar ......-. wee Prasil .is\> o* tetrateare fot Wouble milreis .........-. 0 Seer aclaee oe sli) eno 120) CONtSec.cvee we coeneese t 0 Central America...... Dotlars set a ets la Bee. 0 WETS Hs ca aches be oa dicts Olo Gollarvows « ce sie tioies 0 66 SAP DOE Date Ee New dollar........- tO OTR ES SSeS ae -|Dollar (English mint) ...| 0 Ree Mr ctarevels siatsto\le «i? ie Oo tateacs Sot kactsaow 0 Be enmark ice ce s xSialse'ss 2 rigsdaler ..----.+--- +. 1 Egypt SICH BaAee Piaster (new) Morte vete Oisie es (@) England ....... ./Shilling (new) --+-e+---- 0 a tees -| Shilling Oa ek 0 66 Se MEL OLiiieecterele sake crete Selene nO TANCE: | «1s pet TaATiCS ce caretelee stelerii eleists 0) North German States. Thaler, before 1857 ...... 0 -|Thaler (new) biG AOR TORE ) Bath eet States.|Florin «-.. 22. sce. se0- oe 0 - German ppt se eeee 5 marks (new) -----«---- 0 Greece. ee ae re aI PCHIIS) we «dies ce a weir 0 Pendoetan: ca ee Rupee «.... eR etic cys -oi08 0 TEAL: cc ees esas hs vba BD hivecer assis wee ot Me usieca® 0 Japan aistaters elateisis erent aise i YEN weer cece cece eeeeee: O Care Te La oles bere SOTO att hilerns rath ORO aC oe 0) WMIGTICOT. fo ost orien ee Ticats $e fa a ts bo ole 0 Netherlands.....- 216 gilders .....--....-- 0 Norway. . --se++-+e/Specie daler..-- .-.+.-./ 1 New cease heredaue a TSCA ELLE LOD Cisie nvete sts aca’ c) s 0 Vd REE ie ae rN eer Old:dotlarce: sea e. cereeen 0 Portugal ......-.--+-- BOO POLS e dis nie ad alavale ete'e,s x be 0 ROUM ania. ds clench 2 lei (francs), TW widens ahs 0 Re TION T Agi i is ss ae crewed ere eae Oca ceeaelak 0 SOAIN (oe mews scces sees 5 pesetas (dollars)... sepean SWEET fac. stirs ceciene Riksdaleriwes ti sisiciers eleeie's 0 SRV IEZOTI AT) Gictisiclec\e« ene DoT AT CSohee henel atone ioretievel slatens 0 Mbt AS eet a dss ss 0 212"0'= B plasters -.-sseceesesees|Q Turkey oeoeereseeseeens 20 piasters ecseeceseescece 0) cts. 95 45 68 9] 91 95 Le 93 99 a1 99 09 03 03 21 20 42 HM OOOR ER WOHD + DOHNIBNWDOODAOCOR DON OH AAMAS UHH OPE 7 GREAT BRITAIN’S MONEY. 4 Farthings==1 Penny, d. 12 pence aE Ing, 5. 20 shillings ==1 Pound, 4f. The Gold coms are the sovereign, which repre- sents the pound, and the half-sovereign. The guinea, of 21 shillings, and its subdivisions, have not been coined since 1816. The Szlver coins are crowns of 5s., half-crowns, florins of 2s., shillings, the 6¢, the 4d. or groats, and 3d pieces. The Copper coins are the penny, half-penny, and farthing, coined at the rate of 24 pence per pound avoirdupois Bank of England Notes are a legal tender for any sum over £9; silver is not a legal tender over 40s. ; copper, for not more than 12d. in pennies or half- pennies ; or 6d., in farthings. ; % is acontraction of “brae, s. of solidi, d. of de- narii, and g. of guadrantes ; farthing is another word for fourthing. } The word sterling is supposed to be derived from the first coiners of English silver, who came into England from Germany in the reign of Richard L., and were called EZasterlings. It is used to distinguish the currency of Great Britain from that of the Colo- nies, and from some continental money bearing the same denominations. © 75 Valuable Information for Business Men. NOTES. Demand Notes are payable on presentation, without grace, and bear legal interest after a demand has been made, if not so written. An endorser on a de- mand note is holden only for a limited time, variable in different States. A Negotiable Note must be made payable either to bearer, or be properly endorsed by the person to whose order it is made. If the endorser wishes to avoid responsibility, he can endorse “ without re- course.” , A Foint Note is one signed by two or more persons, who can each become liable for the whole amount, Three Days’ Grace are allowed on all time notes, after the time for payment expires ; if not then paid, the endorser, if any, should be Jegally notified to be holden. Notes Falling Due Sunday, or on a legal holiday, must be paid the day previous. Notes Dated Sunday are void, Altering a Note in any manner, by the holder, makes it void. Notes Given by Minors are void. The Maker of a Note that is lost or stolen is not re- leased from payment if the amount and consideration can be proven. Notes Obtained by Fraud, or given by intoxicated persons, cannot be collected. An Endorser has a right of action against all whose names were previously on a note endorsed by him. 76 BILLS OF EXCHANGE, DRAFTS, ACCEPTANCES. A Lill of Exchange, or Draft is an order drawn by one person, or firm, upon another, payable either at sight or at a stated future time. Lt becomes an “ Acceptance” when the party upon whom it is drawn writes across the face ‘‘ Accepted,” and signs his name thereto; and is negotiable and bankable the same as a note, and is subject to the same laws. In many States both Sight and Time Drafts are en- titled to three days’ grace, the same as notes ; but if made in form of bank check, “ pay to,” without the words “‘at sight,” it is payable on presentation, with- out grace. 7 HOW TO ENDORSE A CHECK. Very few otherwise intelligent and educated peo- ple understand how to properly endorse a Bank check payable to their order, and few realize the in- convenience they cause, by placing their endorse- ment in an awkward position. An observance of the following rules will enable anyone to place their signature in the proper place. 1. Write across the back—not lengthwise. 2. The top of the dace is the 4ft end of the face. 3. To deposit a check, write ‘‘ For Deposit,” and below this your name. A clerk not having power of attorney to sign or indorse checks, can deposit his firm’s checks by writing on the top of the back ‘‘ For deposit only to credit of abet Bs etal and below this write his own name. 4. Simply writing your name on the back of a check signifies thatit has passed through your hands, and is payable to bearer. 5. Always indorse a check just as it appears on the face. For instance, if the check is payable to *¢ G, Read,” indorse “‘G. Read ;” if to “ Geo. Read,” indorse *‘Geo. Read ;” if to “George F. Read,” in- dorse “George F. Read.” If the spelling of the name on the face of the check is wrong, indorse first just as the face appears, and below, the proper way. For instance, the check is payable on face to “‘ George Reade ;” indorse “ George Reade,” and below this first indorsement write what it shouid have been, “George Read.” 6. If you wish to make the check payable to some particular person, write, “‘ Pay to_[E3&son’s NAmx] DROLET st OUR MAME) « Note. In England all checks are payable to bearer; but in this country all strangers presenting checks for payment must be identified by some one known to the bank. 78 CALCULATIONS USED IN PARTICULAR BRANCHES OF BUSINESS. To find the value of tous and hundred-weight with- out the use of fractions? Rue. — Multiply the number of hundred- weight by 5, and annex the product to the tons, as so many hundredths of tons; then multiply by the given price per ton, and point off two decimals. EXAMPLES. 1. What is the cost of 18 tons, 17 cwt. coal at $4 per ton? L2aA 100 18.85 X 4 = 75.40. Ans. $75.40. 2. What is the cost of 35 tons, 15 cwt. hay at $12 per ton? 3. What is the cost of 48 tons, 17 cwt. coal at $6.50 per ton? To find the value of shillings and pence in the decimals of a pound sterling. Rou.te.—Multiply the shillings by 5, and call the product hundredths. Multiply the pence by 4%. and call the product thou- sandths. The sum of these two values will be the decimal required. EXAMPLES, 1. Reduce 12s. 67. to the decimal of a pound. 12 iKCD re OU 6 X 44 = .025 625 79 SPECIAL CONTRACTIONS. 2. Reduce £187 138s. 37. to the decimal of a pound. Laks se) 3X41 — 0125 6625 Ans. £187.6625. To change aunes to yards. NOTE.—An aune is a French measure, equal to 114 yards. RuLr.—Annex a cipher and divide by 8. EXAMPLES. 1, In 484 aunes, how many yards? 4540-275 == 605-02 A ns: 2. In 3848 aunes, how many yards? In 1265? In 1847? NoTre.—14 = 3, or1,.. This rule can easily be applied to numerous other calculations. The contents of boards 14 inches thick, etc., may be computed in this manner; the selling price of goods in order to gain 25 per cent on the cost, and others. 3. What is the selling price of goods, which cost 64 cents per yard, to gain 25 per cent? 640 — 8 = 80 64 Lo 2o wf of 64. 80 To find how many gallons of linseed oil in a given number of pounds, at 74 lbs. per gallon. RuLe.—Add one-third of the number of pounds to itself, and point off one decimal. EXAMPLES. 1. How many gallons in 675 lbs. ? 675 225 = 4 of 675 90.0 Ans. 90 gals. 2. In 1846 lbs. how many gallons? In 675? In 338 lbs. ? 80 Another Method To reduce pounds, shillings and pence to the decimal of a pound. Rute. -- Write one-half of the greatest even number of shillings as tenths, and if there be an odd shilling write five hundredths; reduce the pence and farthings to farth- ings, and write their numberasthousandths. If the num- ber of farthings is between 12 and 36, add one to the thousandths;.1f between 36 and 48, add 2 to the thousandths. EXAMPLE :— £3 14s. 6 = £38.725. Put down the 43, then divide the 14 by 2 and put down 7, then multiply 6 by 4 and add 1 to get the 25. Again, £4 15s. 10d. = £4- 732 = £4.792. In this example one-half of 15 is 74; therefore we put it down in the decimal form .75, and 4 times 10 are 40, add 2 and we have 42, which added to .75 gives .792. EXERCISES. £2 '88.-6d." 6. 43S 8Ss20e Tea 2s ee 216468. Oda 0-7 Genes, Od wl 2g) os alos : £8 10s. 3d.y 8. h4. Tsi 4d. odd 2S Teed £9 4s. 5d. 9. £6 9s.5d. 14. £6 18s. 8d. . £1025. 8d. 10. $1 11s. 6d. 15. £5 16s. 73d. Cr yw OF DO = To reduce pounds to dollars. Rutle.-- Multiply the number of pounds by $4.84. EXAMPLE : — Reduce £3.725 to dollars a cents. 3.725 X 4.84 = 18.029. 81 LNG IO Ree) he Interest is the money which is paid for the use of money. The Principal is the sum for the use of which interest is paid. The Rute is the per cent. of the principal paid for an given time. ; Norte. — When no time is mentioned, ser annum, or by the year, is understood. The .fmouwnt is the sum of the principal and interest. Simple Interest is the interest on the sum loaned for the given time, at the given rate. Legal Interest is the interest according to a certain rate per annum, fixed by law. Nore 1. —A higher rate of interest than that prescribed by law is termed wsury, and is prohibited by law. Nore 2.— \When the rate per cent. is not named in notes, or other business documents, the legal rate must be taken. NoTE 3.—In most of the States, and on debts due in the United States, 6 per cent. is the legal rate, although a higher rate may be agreed upon by special contract. When no rate is mentioned, the legal rate is understood. Interest may be simple or compound. In simple interest the principal alone draws interest. Interest on interest remaining unpaid is considered illegal. In compound in- terest the entire amount due at regular intervals is con- verted into a new principal. 1t is compounded annually, semi-annually or quarterly, and sometimes monthly, ac- cording to agreement. To be an expert in computing interest it is necessary to be fami‘iar with all the methods, and apply the one that will give the correct result with the least labor. To com- pute interest for years, multiply the principal by the per cent., and you have the interest for one year. EXAMPLE 1. — What is the interest on $84 at 6 per cent. for 3 years? $84 X .06 = $5.04 X 3 = $15.12. 82 INTEREST. IF THE TIME CONSISTS OF MONTHS. Multiply the principal by the per cent, and you have the interest for 1 year. 1 month being ;; of a year, divide by 12 for the interest of 1 month; multiply this result by the number of months for the interest. EXAMPLE 2.—What is the interest on $120, at 8 per cent for 8 months? ComMMON METHOD. CANCELLATION. 120 12 ) 9.60 120 ‘ : 6.40 Nots.—If the time consists of years and months, reduce the years to months, adding the number of months, and proceed as above. IF THE TIME CONSISTS OF YEARS, MONTHS AND DAYS. Reduce years to months, adding the number of months, then place 4 of the number of days to the right of the months and proceed as before. REMARK. — Placing 4 of the days to the right, is reducing the days to the decimal of a month. The reason of this is obvious from the fact that we calcu- late 30 days for 1 month, then 1 day is J, of a month, and 3 days ;3,, or z5, or in decimal form .1; hence by taking 4 of the number of days we obtain tenths of a month. EXAMPLE 3.—What is the interest on $159, at 9 per cent, for i year, 4 months, and 12 days: 83 INTEREST. CoMMON METHOD. 150 .09 1 year = 12 months. 12\.5:50, Nh i 1125 12 idavs7— tu __ 16.4 16.4 months 18.4500 By CANCELLATION. 4 | 150 3X 15 = 45 X 41 = 18.45 12 g 3 164 41 41X38 X15 = 18.45 This operation comes under the head of contrac- tions, and can be multiplied mentally. ON ALL NOTES THAT BEAR $12 PER ANNUM, OR ANY ALIQUOT PART OR MULTIPLE OF 12, Any principal that bears $12 per year will bring $1 per month; hence, the time in months must be the interest. ILLUSTRATION. — If the interest for 1 month is $1, for 15.3 months it is 15.3 times $1, or $15.30; since the multiplication by the figure 1 is altogether super- fluous we can dispense with it, and at once say $15.3 or $15.80. Hence the propriety of the following: RULE. — Reduce years to months, add in the given months, place § the number of days to the right, and we have the interest in dimes. EXAMPLE 1.— Required the interest for $150, at 8 per cent, for 1 year, 6 months, and 11 days, 150 4 of 11 days = 38. 08° Years. Months. Days. 12.00 1 6 11 = 18.38, therefore $18.33 dimes, or $18.36. Ans. 84 INTEREST. A note that bears an aliquot part or multiple of 12; such as 6, 18, 24, 36, etc. EXAMPLE 2,—What is the interest on $300, at 6 per cent, for 3 years, 6 months, and 18 days? $300 4, 0f 13-days’—— 6, .06 Years. Months. Days. 18.00 3 6 Loa 2: $42.60 Interest at $12 a year. 21.30 i “$6, or 4 of $12, a year. 63.90 CANCELLATION. 300 12) hohe 213 x 300 = 63.90 ( Note. — To find the time any sum will double itself at simple interest, simply divide 100 by the rate per cent. ANOTHER METHOD TO COMPUTE INTEREST FOR DAYS. Rute. — Find the interest for 1 year and divide by 360 (the number of interest days ina year), and multiply by the number of days for the interest. ‘ EXAMPLE.—Required the interest on $720, for 60 days, at 8 per cent. $720 .08 CANCELLATION. 360 ) 57.60 ( .16 120 2 360 60 369 | 60 2160 9.60 | 8 X 60 X 2 = 9.60 2160 RemaARK. — In using 360 for a divisor the cipher may be rejected, because it avails nothing in dividing, and makes the divisor ten times as short ; your answer will be mills instead of cents, as before; cut off the tight hand figure, and you have the interest in cents. 4 85 INTEREST, For 10 per cent add 4. ve tt multiply by 2. ExAaMPLE.— What is the interest of $124, at 7 per cent, for 54 days? 124 4 | 31 an 3g | 124 1.116 | 54 6 .186 = }. 7X 6X 31 = 1.302. $1.302 Bankers’ Method. Banking business being nearly all transacted on the basis of 30, 60, and 90 days, the work can be very much abbreviated and the interest, sometimes, obtained without any calculation whatever. The following example will best illustrate this rule. ExAaMPLE.—What is the interest on $120, for 60 days, at 6,per cent? 120 3B | 60 10X 120 =1.200 Ans. 8 Observe that in this case we cancel the factors in the time and rate, and that the figures in the principal remain unchanged, therefore: Rue. — For any note at 6 per cent for 60 days remove the decimal point two places to the left, and you have the interest. EXAMPLE.—Required the interest on $350, at 6 per cent, for 60 days. SOLUTION. — Remove the point in the $350 two places to the left, thus: 3.50, and you have the result. 86 INTEREST. This is the shortest and best rule for days that can be adopted, because it may be applied in any per cent and any number of days. The great superiority of this rule consists in its simplicity, and when once understood is not readily forgotten. Some accountants have a different rule for every per cent, many of which are not only apt to be forgotten, except by those who apply them daily, but are actu- ally deduced from the above rule. In reckoning 360 days instead of 365 gives gz, or 7x, too much. But the difference is so small that in ordinary transactions it is not noticed. It is now universally adopted in all business transactions. To find the accurate interest divide by 365 instead of 360, 7 When the time is less than 1 month, the cents in the principal may be disregarded, because the inter- est on ¢hat sum for ¢ha¢ time would not amount to a cent; when less than 2 months, all under 50 cents, when less than 3 months, all under 33, and so on. To illustrate we will give the table of divisors for the different per cents. Any sum multiplied by the time in days, and divided by the number opposite the per cent, will give the interest at that per cent. At 5 % divide by 72 At 12% divide by 30 “6G yp ‘6 73 60 (a3 15 % be ‘é 94 6“ 7, % 66 66 52 6“ 20 We 66 66 18 73 8 ve 6 66 45 6c 24 70 6b 66 15 be “6 6 (74 6 6é 2H 40 A OA esate 66 10 % 66 66 36 It will be observed that these divisors are obtained by dividing 360 by the rate per cent ; and the student will have to retain in his memory a different divisor for every per cent when, by using 36, once for always 87 INTEREST, he need remember but one. The great advantage of using 386 must at once be admitted. Some authors use 12 and 30 which, of course, is the same thing ; we will now give a few solutions of problems as solved by the old method, and also by cancellation, that the student may perfectly understand them. EXAMPLE 1.—What is the interest of $80 for 1 year, ° months, and 12 days? 80 1 yr., 9 mo., 12 days = 522 days. .06 480 522 960 960 2400 36 360 ) 250560 ( $6.96 BY) 529 2160 87 X 80 = 6.960. 3456 3240 2160 2160 The student will here notice the vast amount of labor saved in the cancelling method. EXAMPLE 2.— What is the interest on $48 for 2 years, 3 months, and 6 days, at 8 per cent? 2 years, 3 months, 6 days = 816 days. 6 | Ap 8 BG 8 | 816 136 X 8 X 8 = 8.704 Another short rule for computing interest is called THE SIX PER CENT BASIS. TO FIND THE INTEREST FOR MONTHS AT 6 PER CENT. RuLe.—Multiply the principal by one half the number of months; when the principal is dollars only, point off two 88 INTEREST. places for cents in the product; when dollars and cent point off four places. EXAMPLE 1.— What is the interest on $153, at 6 per cent, for eight months? sof 8=4 153 2 1153 x 4 = $6.12 Ans. 4 12| 6 wards Chal SS an te $ 4 $6.12 Ans. Solving the above by cancellation will show why half the number of months, at 6 per cent, will bring the interest. TO FIND THE INTEREST FOR DAYS AT 6 PER CENT. RuLE.—Multiply by } of the number of days, and the product will be the interest in mills. ExaMPpLe.—What is the interest on $124, at 6 per cent, for 04 days? $ of i <2. | 124 X 9 = 1.116. a 6| 5A 9 | 3 St.i16)) “Ans. It will be observed, also, in this that it is an ab- breviation of cancellation. TO FIND THE INTEREST AT ANY GIVEN RATE. RuLE —Find the interest at 6 per cent as above; divide by 6 for 1 percent, then multip:y by the given rate; or, increase or diminish the result obtained by the rule for § per cent, in the same ratio that the rate is increased or diminished. For 4 per cent subtract 4. ~l pe) Qu. Qo. OH om obo 89 INTEREST. When the Time is more or less than 60 Days. Increase or diminish in the same ratio as the time is increased or diminished. For 90 days add 3 itself. eta) wee INCI PLY) Vo. wou les CL pvicesDye, 6é 15 “cc be Ae “ 45 “ subtract 4. Tey Ms eclwide byis. ce 10 oe 66 6. ica Aceon ft “10. SUB Oats “20, Norre.—Nearly all business paper is calculated on 30, 60, 90 days, or an aliquot part or multiple of a month. ExAMPLE 1 —What is the interest on $120. for 90 days, at 6 per cent? $1.20 interest for 60 days. OUD kee ‘¢ 4 of 60 or 80 days. $1.80 interest for 90 days. EXAMPLE 2,—What is the interest on $134.24, for 75 days, at 6 per cent? $1.3424 interest for 60 days. SOD Dine met Soli, S..,0r + of 60 days: $1.6780 The interest at any other rate can be obtained as in preceding rule, or by the following, which will show in what time, at the different rates, any number of dollars will give the interest in cents corresponding with the same figures in the principal. Thus the interest on $140 for 90 days, at 4 per cent, is $1.40. This rule is no shortér than the other, unless the time corresponds with the same figures in the table, or when it is an aliquot part of the time in the table. 90 INTEREST. RuLE. — When the time and rate correspond with the time and rate in the table, remove the decimai point twa places to the left, as in the preceding rule. 4 per cent for 90 8 per cent for 45 5 ce 6é bps 9 ce 66 40 6 66 “cc 60 10 ce 6< 36 7 " SoZ 12 %, oa 8) The cancelling system is very much preferred to this, because it very frequently takes advantage of both time and principal, as will be seen in the follow- ing solution: EXAMPLE. —Required the interest on $540, for 49 days, at 6 per cent? BANKER’S METHOD. $5.40 = Interest for 60 days. 2) 5.40 2) 2.70 = Interest for 30 days. Od Bes ee ede Bhat Oh) hua th eee as Ce eh autaes a beac nN ty id. LLY $4.41 CANCELLATION. g | #49 90 36 | . x 90 = 4410 N. B.—Where the time is not an aliquot part of the time in the table, but the principal is, reverse the operation and point off two places in your time for the interest, thus: a note that bears $1.17 in 60 days, at 6 per cent, on $117, is the same as $60 for 117 days. NoTE.—We have now conclusively proven that a/ the abbreviated processes of computing interest are based entirely upon the system of cancellation. 9t INTEREST. Indeed, there are many other methods that might abbreviate the work, ¢/ you have examples to sutt, But the canceling system will give you the advan- tage at 44, 6, 8,9, 12, and 15 per cents, and very frequently of the time and principal. And, to say the least, if the numbers are @// prime, and you can cancel zone, you have stated your problem in its simplest form to be solved by any other rule. We therefore recommend without hesitation the adoption of the system of Cancellation, as a general and ui- versal rule. The student, by close observation and considerable practice, may deduce rules from ¢/zs. How to find the PRINCIPAL, the rate, time and interest being given. RuLE.—Divide the given interest by the interest on one dollar for your time and rate. To find the RATE, when principal, time and interest are given, RuLE.—Divide the given interest by the interest on the principal at one per cent. To find the TIME, principal, rate and interest being given, RuLe.—Divide the given interest by the interest on the principal for one day, the quotient will be the required time in days. How to Compute Time. RvuLE. — Subtract as in compound numbers, reckoning 30 days to the month. EXAMPLE. — What is the time from January 30th, 1869, to March 13th, 1870? 92 INTEREST. 1870 3 13 1869 1 30 1 1 13 Year. Month. Days. SUGGESTION. — When you are obliged to borrow from the next higher number, in subtraction of com- pound numbers, subtract the number in the subtra- hend from the borrowed number first, then add the number to the minuend, thus, as in above, 30 from 30 = 0 + 13 = 13. Partial Payments. The manner of computing interest where partial payments have been made, has given rise to much litigation. The law in the different states on the subject very often does not clearly indicate the prin- ciple applicable in all cases. The aim of the law, of course, is to avoid usury and compound interest. The difficulty is in deciding whether the payment shall be applied to liquidate the interest or the prin- cipal. The U.S. rule involves compound interest, - as often as a payment is made greater than the interest then due. When more than a year intervenes, the U. S. rule is more favorable. The Vermont rule is more favorable, for there is no compound interest. All payments draw interest. We give an illustration of a problem under these rules, and the pupil can see the difference readily. The United States Rule. I.—The rule for casting interest when partial pay- ments have been made, is to apply the payment, in the first place, to the discharge of the interest then due. 93 INTEREST. II.—If the payment exceeds the interest, the sur- plus goes towards discharging the principal, and the subsequent interest is to be computed on the balance of the principal remaining due. I{I.—If the payment be less than the interest, the surplus of the interest must not be taken to augment the principal; but the interest continues on th? former principal until the period when the payments exceed the interest due, and hen the surplus is to be applied towards discharging the principal, and the interest is to be computed on the balance, as afore- said.—The above is the decision of Chancellor Kent ; Johnson’s Chancery Reports, Vol. 1, Page 17, and is adopted by the Supreme Court of the United States. EXAMPLE 1. PHILADELPHIA, May 1, 1842. For value received, I promise to pay to the order of J. THORNTON the sum of Three Hundred Dollars, with interest. THos. CLARK. The following endorsements were made on this note: 1842, Oct. 16, - - - $ 60.00 1843, March 4, - - - 17.50 LSier Auer sitncee 26:40 1844, Aprill, - - - 182.25. What was the balance due Sept. 19, 1844? PRRTOTITIT EGET EAI OLC). sieve ards ave rds diacele 910" 5 tes $300.00 Mrterest to. Oct. 16; 184250. ice) wee see ewes 8.25 PYRE ITIOU Cite tiers <9 sie ele 4 Oh Sac el ateis ok $308.25 First Payment. ...sseecevcscccsccccesves 60.00 New Principal.........+e0.. Mish «sts es Reto eo Interest from Oct. 16, 1842 to Mch. 21,1843 641 Second Amount Padwwisic melaisie ale aalscbve sine se B24: 00 94 INTEREST. Second Amount F Sanaa tal hia oie wt arateinl elas lete eterete as el eine Second Payment....+..... Rh ein Gelso ie ols c ie En aa New Principal.....- iat bastath fot Mists. efetieice dole os PAOIeLO Interest from Mch. 21, 1843, to Aug. 27,1843 6.17 Third SA morntse oniele ttoratersts cree shia is eel emiewie Pare Third Payment ..-.sescecessecsccovcccee 28.40 New Principal. ..ccecsccccccccecscss 2066 $214.93 Interest from Aug. 27, 1843, to Apr. 1, 1844 7.66 New Principal .... «sseccecceees cose cee $222.09 Fourth Payment ...eeseesseoees otelate.e ste ie es mia New Principal....+..se- Sess such dbl se Sep moUseEs Interest from Apr. 1, 1844, to Sept. 19, 1844 2.52 Balance due Sept. 19, 1844. ...+ceeee.ceesf 92.86 The following is called The Vermont Rule, And is generally applied when the time is less than a year. I.—Compute the interest on the whole debt from the time it was due until it is paid. . II.—Compute the interest on all payments, from the time of payment until the time of settlement. III.—Subtract the amount of all the payments, interest included, from the amount of debt, interest included ; the balance will be the amount due. KiXAMPLE 2. One year after date I promise to pay, to the oraer of D. B. JonEs, the sum of Three Hundred and Sixty Dollars, for value received wth use. SHELBY, O., Jan. 1, 1869. JOHN MILEs. N. B.— When no rate is mentioned, 6 per cent. is un- derstood. WEIGHTS AND MEASURES. AVOIRDUPOIS WEIGHT. PUPeS Urs) pet JEGUAl OUNCE) sesh) as) (OZ. TOIOUNC eS mee tee hl LLDOUD Al aniaanus ve LB. SI OOUIIS Etre ae 4 ce le Quarter. st aw aniat CT. 4 quarters ... “ 1 hundred weight, cwt. CUMUNULeC Were a sw eet tet at see Tae sii es The term Avoirdupois is derived from the French avoir du poids, signifyiug ‘to have weight.’”” The pound consists of 7000 Troy grains. This weight is used for weighing almost all articles except gold, silver, platina, and precious stones, which are weighed by Troy Weight. LONG TON WEIGHT. 28 Ibs. . . A . : “ - 5 1 quarter. 4 quarters, or 112 lbs. , ; : : 1 hundred weight. 20 cwt., or 2240 lbs. . : : s 1 ton. This measurement is nearly obsolete. It is allowed at the Custom House in estimating duties, and in the wholesale Coal and Iron trade. MISCELLANEOUS WEIGHTS. CEA YAS ie Sem yar teen W ASANTE TA9 Katey (8) 01} LOU uGaY ane nic viel os Celt in oat eS Cental: PEs Ne ASIN so) G.\s, 1 erp isap aul ate eT SAS De esr yiLiSib; ie te he ease LCouintal, TRE IALIS = st "ho fats tats Nett lent CCD PGs H OULS el cite einer” alien onie tak Darrel, MEM MGMULAL OTK |< coi 6 viilentel etiuke tated be Darrel. AMR EMEM ISTIC oihig canipritetip rratnaye via hc: GaSke PHOS Ali ate te hslvetecaie tee. barrel. 96 WEIGHTS AND MEASURES. THE STONE WEIGHT So often spoken of in English measures, is 14 lbs. when weighiug wool, feathers, hay, etc. ; but a stone of beef, fish, butter, cheese, etc., is only 8 pounds. HAY. In England, a truss, when new, is 60 lbs., or 56 Ibs. of old hay. A truss of straw, 40 lbs. A load of hay is 36 trusses. In this country, a load is just what it may happen to weigh; and a ton of hay is either 2,000 lbs. or 2,240 lbs., according to the custom of the locality. A bale of hay is generally considered about 300 lbs., but there is no regularity in the weight. There is no accurate mode of measuring hay but by weighing it. This, on account of its bulk and character, is very difficult, unless it is baled or other- wise compacted. This difficulty has led farmers to. estimate the weight by the bulk or cubic contents, a mode which, from the nature of the commodity, is. only approximately correct. Some kinds of hay are light, while others are heavy, their equal bulks vary- ing in weight. But for all ordinary farming purposes of estimating the amount of hay in meadows, mows, and stacks, the following rules will be found sufficient: As nearly as can be ascertained, 25 cubic yards of average meadow hay, in windrows, make a ton. When well settled in mows or stacks, 15 or 18 cubic yards make a ton. When taken out of mows or old stacks, and loaded on wagons, 20 or 25 cubic yards make a ton. Twenty or twenty-five cubic yards of clover, when dry, make a ton. 97 WEIGHTS AND MEASURES. TO FIND THE NUMBER OF TONS OF MEADOW HAY RAKED INTO WINDROWS. RuLeE. — Multiply the length of the windrow in yards by the width in yards, and that product by the height in yards, and divide by 25; the quotient will be the number of tons in the windrow. EXAMPLE. — How many tons of hay in a windrow 40 yards long by 2 wide and 2 high? SoLuTion.—40 X 2 X 2= 160 —- 25=— 62. Ans. TO FIND THE NUMBER OF TONS OF HAY IN A MOW. RuLE. — Multiply the length in yards by the height in yards, and that by the width in yards, and divide the pro- duct by 15; the quotient will be the number of tons. EXAMPLE.—How many tons of well-settled hay in a mow 10 yards long by 6 wide and 8 high? SOLUTION.—10 X 6 X 8 = 480 —— 15 = 32 tons. TO FIND THE NUMBER OF TONS OF HAY IN OLD STACKS. Rute.—Find the area of the base in square yards, in the table of areas of circles; then multiply the area of the base by half the altitude of the stack in yards, and divide the product by 15; the quotient will be the number of tons. EXAMPLE. — How many tons of hay in a circular stack, whose diameter at the base is 8 yards, and height 9 yards. | SoLUTION.—50.265, area of base in sq. yards, X 44, half the altitude, = 226.192 —- 15 = 15.079 tons, TO FIND THE NUMBER OF TONS IN LONG SQUARE STACKS. Rute. — Multiply the length in yards by the width in yards, and that by half the altitude in yards, and divide the product by 15; the quotient will be the number of tons. 98 WEIGHTS AND MEASURES. EXAMPLE. — How many tons of hay in a square stack 10 yards long, 5 wide, and 9 high? SoLuTION.—10 X 5 X 44 = 225 —- 15 = 15 tons. TO FIND THE NUMBER OF TONS OF HAY WHEN TAKEN OUT OF MOWS OR OLD STACKS. RULE. — Multiply the length of the load in yards by the width in yards, and that by the height in yards, and divide the product by 20; the quotient will be the number of tons. }iXAMPLE.—How many tons of hay can be taken from an old stack, in a load 6 yards long by 3 wide and 3 high? SOLUTION.—6 X 3X 3= 64 — 20 = 2,5 tons. These estimates are for medium sized mows or stacks. Ifthe hay is piled to a great height, as it often is where horse hay-forks are used, the mow will be much heavier per cubic yard. AN EASY MODE OF ASCERTAINING THE VALUE OF A GIVEN NUMBER OF LBS. OF HAY, AT A GIVEN PRICE PER TON OF 2000 LBS. RuLe.—Multiply the number of pounds of hay (coal, or anything else which is bought and sold by the ton) by one half the price per ton, pointing off three figures from the right hand; the remaining figures will be the price of the hay (or any article by the ton). ExaMPLE.— What will be the cost of 658 lbs. of hay, at $7.50 per ton? SOLUTION. — $7.50 divided by 2 equals $3.75, by which multiply the number of pounds, thus: 658 $3.75 $2.46||750. Ans. Nors.—38.75 is §; therefore } of 658 = 82, and # is 3 times 82 or $2.46. 99 WEIGHTS AND MEASURES. A BALE OF COTTON In Egypt is 90 lbs.; in America a commercial bale is 400 lbs. ; though put up to vary from 280 to 720, in different localities. A bale or bag of Sea Island cotton is 300 lbs. WOOL, In England wool is sold by the sack or boll, of 22 stones, which, at 14 lbs. the stone, is 308 lbs. A pack of wool is 17 stones and 2 lbs., which is rated as a pack load fora horse. It is 240 lbs. - 2240 = 54 tons. 2. How many pounds of bituminous coal in a car 30 ft. long and 7 ft. wide, the depth of the coal being 16 in.? 3 How many pounds of anthracite coal can be placed in a cart which measures 6 ft. in length, 44 ft. in width, and 16 in. in depth? 4, I wish to build a bin in my cellar to hold 8 tons of anthracite coal, 2240 lbs. to the ton ; I have made the length 12 ft., and the width 10 ft. ; what must be the height of the bin? 5. How many pounds of bituminous coal can be stored in a space 50 X 50 X 123 ft.? 6. How many tons of anthracite coal, 2000 Ibs. to the ton, can be stored in a yard which measures 60 ft. in length, and 380 ft. in width, the depth of the coal being 6 ft.? 7. A dealer purchases 1500 tons of anthracite coal, 2240 lbs. to the ton, which he wishes to store in an inclosure 100 ft. long, and 80 ft. wide ; what will be the depth of the coal? 101 WEIGHTS AND MEASURES. TABLE OF AVOIRDUPOIS POUNDS IN A BUSHEL. The following Table shows the weight of a bushel as prescribed by statute, in the several States named. 2 rae abe Si a wees 3 nN E wT AN| . 8 aa ES S/S ,| 3 <} ei Si. / 81S] 2] % 84 Seles COMMODITIES. F/R] |g 21S) STE Seis] P88 ei S iS] Ry ais SPSS TSE S| SPSL RT STS] ST si Sia gs] Si 818/81 815 PPE PEIN SB] SLRS Sis lglagiciaterecie stsisis ies SIS/ISIE/ SIS ISISIS/S ISIS [SIS FSIS IS {S]alRIN ARIA Barley, 5 : . |50|.. 148/48] 48]48 | 32]... | 46] 48] 48) 48] .. | 48] 48 | 48 1°48 | 46 | 47 | .- | 46 | 48 | 45 Beans, é 5 | eres terse GOI OOH GO aHO0 © | sere dee) Sore wesw] cote AOUL|: ore lee pmeti-etote| O-culae eit) cast lene | Mora ese emcee eete Bituminous Coal, Seen eee Zu O10 -livconll soo Borsetebetee’| eee |" GU|Se1eb [rors] dead |-eedl vers], erecta Oll| scart arom oteial ls Blue Grass Seed, See | ee ree Sd ah | DA als ae lees | a | Sater al Valeo celle col lster4eten| nee || tere. |eele dawre nines tlaete ti te’s Buckwheat, : . 1401451 40| 50/52) 52 | .. | .. | 46 | 42] 42152) .. | 50150) 43] .. | 42] 48] .. | 46 | 42 | 42 Castor Beans, . Reis cr eee 46a AO nhs eine [eich weet sere "| tern ater |) 4 Ons letemalieeren | ares Peres | ere [tie waa erei| evens eetemlne on fret Clover Seed, . . | ee |... |60|60]60)60 | ..]..|.- | 60] 60]60]..| .. | 64] 60] 60] 60|..|.- | -- | 60] 60 Dried Apples, . ae [etfs PAO Ge) OA | Aneel ea) — eels ck | CSc aon aed decree ew [cores Lpoean cies [52Onl Foren eee leewn| ao: |e Dried Peaches, . Fe eo er AB Se 1 Bat L we altos he ochre 1S SB°SS Pe ee seth oa Le | Teesk. wet Piva eed ame Flax Seed, r hice dose 1 DO ROGA DO | G6 cl wel Sel ee les ed-ee | OG ies lan 1 OO 100-1 BG Lise ced en |-een Dodie. Hemp Seed, ° eat es eed 4 ta | Ae oleae te Lae nae [eae oleee ne at | veer) ouellanven [ae em] ween] hematite tart kere Indian Corn, O . |52/56| 52) 56156156 | 56].. | 56 | 56] 56] 52] .. | 54 | 56 | 58] 56 | 56 | 56 | .. | 56 | 56 | 56 Indian Corn in ear, . ee ee 70 68 68 cael oe ee oo ee ee ee ara oe pee ee ee ee ee ee ee ee ee Indian Corn Meal, eae licey eal eee | ASI CVO lesrect fe. | uroyein)|| Os sO retour lucien [ate om | eran leer | athe Sen teever| Semel OO Wee oe eater |aere Cates . | 32 | 28] 32] 32] 35 | 334] 32 | 30 | 30 | 32 | 32 | 35] 30| .. | 30 | 32] 32] 34] 32] .. | 32/32) 36 | Onions, : 7 ET BTL Bil BT ABT = loon, hea OS Pact sa OF aettes lise bees Iueebicasei | OOM sr eaieeeeboumas Potatoes, . - |e. |60/60| 60/60/60 |..|/60|-.]|..]..|60] 60] .. | 60/60] .. | 60] .. | 60|60) 60/60 | Rye, . . . . 1541561541561 56156 | 82].. | 56) 56156) 56]..|.. | 56] 56 | 56 | 56 | 56] .. | 56 | 56 | 56 | Rye Meal, . y eee eS dee | eet Sea) LOO eed feet | tae ikea |) seat Se arke aoe heen bleh ae ere re Salt, . ; Se relies seen DOs DOA -DO. Worst gteclite ss |e ietl| Meret OO Mlnertal mates pict MOO ll worte leerattecemnrese tae ee Jee Timothy Seed, . Bee cee (Dea a cS | AB alae arte eull eee Meee |e ce AD dice cele ory mote eed lerorom aneteke Pevcaeal gerald On ase Wheat, oe i . |60]5C 160] 60] 60}60 | 60] .. | €0| 60] 60 | 60] .. | 60 | 60 | 60 | GO| GO | 6D] .. | 60 | 60 | 60 Wheat Bran, . pre hee atria’ | Oat ay ov tO OO | tas te ee ares’ £20 bee soliwinc Gacy = AMOR aeauey cok. OUTS. ea eee 102 RATLROAD FREIGHT, GROSS WEIGHTS. The articles named are billed at actual weights, if possible, but usually at the weights in the table below when it is not convenient to weigh them. Ale and Beer ......320 lbs. per bbl. | High wines.........3501bs. per bbl, Kf Pre a ae 170 ** } * | Hung’n Grass Seed 45 ‘“ bu. ss Oe acre eta LOO ie Aare Wy Shoat GLO caretaite wi tptetey 2 200 hee bp Apples, dried...... 24 ‘ bu. | Malt..... Gta dare S806) 2 ba. 66 DVOON | Asia's 6 SOO Jute ‘¢ | Millet..... eosseencs a res ‘f “ Shee taele be 150, “bbl. | Nails ..... Bs p.ule ein «108.7 FF Kee Barley......« se oy EO oe SU GRITS) | CORE Wart orale diareals ihe. 6 Aare FE Pee 0s Beans, white....... 60) 5,556 56 PUN CULES wie cieiatata Nal Jicisiatatata 400 * bbl a castor .....- 46 ost FF Onions. 3. con's aie Gattis Lenn bu. BOGE 6 66h obese veee.520.° **) ebb Peaches, dried... ses Ropdaad) VANE 2 INS dapeaiir Gop ooGor 6st 2.25] |10d Floor, and larger....... Py (3) Bereta ictces'c ose Se ins ei 1.50) | ———__$___—___—_——__ Ad Light. .a0) onsen ns tinas 1.50} | 6d Brad Head........ 1.50 4a Swedes, com........ vetls (90500), (0 8a. AM hats Aad cua 1°25 4a 66 Dar icetceces. 4) 4.00) 110d sé and larger...... 1.00 4d & 5d) Common, ....-.-. 15 f 6d & 74$ Fence and ‘ .50| | 8d Cooper & slate... ....: 2.00 8d & 9d ) Sheathing. oe eo 4d. és AE eres ieee a my ds 5d Hf J i yee Be 1.25 TOS Ae hae Wa eae, os .25| | 6d “< Ce Series ie Meal ea TAO oleae - Seth aati Pha eine 15 38d Fine Box.....-..+.- | hy 4.00) sro SPAT = SEs OX sacs siete aicleia.s wiels\als's.4/ S200} Lin HOON S F-ce).rs sere 3.00 4d o6 ce cece eereeeeeseer- ee 2.0 a att a ee ee oe ee oe ee, —— BOih ae eaten. got me's Pt setatsto Leite eS teCLIN@is ce e)s. == ecccece 1.00 SPSg APES ) MRSA Wal tea Camere 7 BOTS OU Met Pete tien ain'en -(5| | Lin. Clinch ...-.......... 4.00 10d Box, and larger. rere 5U| | yin. « Rear nt eres the 3 50 os LEAS Vi ec Gis Sehr can ta ae 3.00 4d Fine Finishing..... -.. 2.25} 14% I aie ewe ne vee 2.50 5d ‘6 one 2.00 2& 2M, in} Clinchiesaee-ee: 2.00 6d es 1 Perea 1.75] | 2144 & 2%, in. * Palsaleinewa | tle do 8d Fe as tee 2.0788 1.50 3& 3% in. aa -o- eeee 1.50 10d “6 TU Aweee |) 1.25) | Half Casks}‘additionaly 2% | \).25 TRATES AIVO Ls ater ce he.c ceeds sae 2.50| | Galvanizing.. aisela poleiotets 2.50 1i,zin. ‘6 2.2.00 wake eice's ZOO|p| SLINNING eign cts ctelwiere's o's SeSuIEGAO LENGTH OF NAILS. od 3d 4d 5d 6d 7d 8d 9d 10d 12d 20d 30d 40d 50d 60d lin. 14 1% 1% 2 2% 2% 2% 3 3% 4 4% 5 5% 6 NUMBER OF NAILS IN A POUND. 94 4d 5d 6d Td 8d 10d 12d 20d 30d 40d 50d 557 353 232 167 141 101 68 54 34 16 12 10 120 HOW TO MEASURE LAND. Land can be measured with satisfactory accuracy for many purposes, by pacing. Five paces are equal to one lineal rod. A man having long legs usually measures more than a rod at five paces, while a short legged man will be obliged to step unnaturally long to measure a rod at five paces. The correct way is to measure 164 feet on level ground, then practice gauging the steps until one can measure one rod at every five steps, then one hundred steps or paces will be equal to twenty rods. Ifa plat of land be two hundred paces long and fifty paces wide, call every five paces a rod, multiply the rods in length by the rods in width, and divide the product by 160, the square rodsinanacre. Thus: 100 paces = 20 rods, and o0)*paces —=10erods3 10 920 —— 200G square rods, which, divided by 160, gives 14 acres. A square acre is about 208 feet 84 inches on ev- ery side. In order to lay out one acre of land four times as long as the width, the length must be 417 feet 5 inches, and the width 104 feet 4 inches. Twenty feet front and 2,187 feet deep is one acre. Twenty-five feet front and 1,7424 feet deep is one acre. Thirty-three feet front and 1,320 feet deep is one acre. Forty feet front and 1,089 feet deep is one acre. Fifty feet front and 8764 feet deep is one acre. One hundred feet front and 4354 feet deep is one acre. In one square acre there are 48,560 superficial fect, 121 TRANSACTIONS WITH BANES. Make your deposits in the bank as early in the day as you conveniently can, and never without your bank- book. For your own security, it is well to have ONE PAR- TICULAR PERSON to do your business at the bank, who shall be competent to take charge of the money and papers you intrust to his care, and sufficiently in- telligent to understand and properly deliver the mes- sages and explanations you may have occasion to make ; also, that you write or stamp OVER YOUR IN- DORSEMENT, upon all checks which you send to be deposited to your credit in the bank, the words “ For DEPOSIT TO OUR CREDIT,” which will prevent their being used for any other purpose. Always use the deposit tickets furnished by the bank, and examine the date and indorsement of ev- ery check. When checks are deposited the banks require them to be indorsed by the depositor, whether drawn to his order or not. Keep your check-book, when not in use, under your own lock and key. Make it a rule to give checks only out of YOUR OWN CHECK-BOOK. Draw as few checks as possible. When you have several sums to pay, draw ONE CHECK for the whole, and take notes of such denominations as will enable you to distribute the amount among those you intend it for. Do not allow your bank-book to run too long with- out being balanced, and when returned by the bank compare it with your own account, and examine your cancelled checks without delay. If you wish to pre- serve your cancelled checks, deface or destroy the signature as soon as returned, in a manner that will 122 prevent their being copied, and place the checks out of the reach of others. In filling up checks, do not leave space in which the amount may be increased. It has been decided that when a check is so carelessly drawn that an al- teration may be easily made, the loss arising from the alteration, if any, must be borne by the drawer. Write your signature with your usual freedom, and never vary the style of it. Offer notes for discount or collection in good sea- son. Do not put off the offering of notes for dis- count until the last day of your need. When notes are discounted or collected for you, hand your bank- book to the clerk, that they may be entered in it to your credit. BROKERS’ TECHNICALITIES. A Butt is one who operates to depress the oe of stocks, that he may buy for a rise. : A Bear 1s one who sells stocks for future delivery, which he does not own at the time of sale. A CorRNER is when the Bears cannot buy or borrow the stock to deliver in fulfillment of their contracts. OVERLOADED is when the Bulls cannot take and pay for the stock they have purchased. SHorT is when a person or party sells stocks when they have none, and expect to buy or borrow in time to deliver. LonG is when a person or party has a plentiful supply of stocks. A Poot or RING is a coinbination formed to control the price of stocks. A broker is said to Carry stock for a customer when he has bought and is holding it for his account. A Wash is a pretended sale by special agreement be- tween buyer and seller, for the purpose of getting a quota- ‘ion reported. A Pur AND CALL is when a person gives so much per cent. for the option of buying or selling so much stock on a certain fixed day, at a price fixed the day the option is given. MARKING GOODS. In buymg goods the merchant is often at a loss: to know whether the price of the article suits his market or not; ard if he is not a good accountant it often takes him some time to determine. Those who buy largely can best appreciate the value of a short method of calculating the percentage desired. If you wish to calculate the per cent. on a single article, the following is considered the best method. If you wish to sell an article at any of the following per cents., say the article cost 70 cents, and you wish to make 10% Divide by 10, Multiply by 1177. 20 « “10, “19-84, 25 ‘* Multiply by 10, Divide “ 887%. 30 * Divide by.10, Multiply “© 18=91, 834 Add 4 of itself. = 934. 334% Divide by 3, Multiply by 4=-933. 50 “ Add 4 of itself. ==1.05. 124 Another method of marking 25 per cent. profit is to cut off the right-hand figure, and you have the price in shillings and pence: thus, if you buy an article for 60 cents, and wish to gain 25 per cent., cut off the right-hand figure and you have 6 shillings of 124 cents each, or 75 cents, the cost with 25 per cent added. If the figure you cut off is not a cipher, add 4, thus: 3 cents, add #; 5 cents, 14; 6 cents, 14, etc. EXAMPLE.—Suppose an article cost 74 cents, and you wish to make it 25 per cent. advance, cut off the right-hand figure and you have 7s.4d., 4=$=—1, added to 4 = 5, 7s. = 874 +5 = 925 cents. How to mark an article bought by the dozen, to taake 20 per cent. Remove the decimal point one place to the left. EXAMPLE.—Suppose a lot of hats cost $2.50 per dozen, by removing the decimal point one place to the left we have 20 per cent. and cost, or .25 apiece for the hats. To ascertain any other per cent., we take the basis at 20 per cent. and add or subtract, as the case may be. To make 25 per cent., remove the point one place | to the left, and add ,}. To make 30 % add 7, itself. 66 334 6 6 1 66 66 85 6 8s 4 66 66 384 66 6G 4 rT 66 40 “ «& 3 66 66 44 6 6 1 66 66 50 “é 66 4 66 fs OO Se es ‘ 80 « « 3 re These additions must be made after removing the point as above directed, and this sum will always be the selling price of a single article. 125 The above table contains all the per cents. gener- ally used in business, and can easily be applied. This rule is very valuable to the merchant in buy- ing goods; suppose he buys his goods at auction, he does not have sufficient time to make extensive cal- culations before the goods are cried off. But by knowing that at 20 per cent. profit, he need not change a figure, he can tell instantly whether he can afford to buy those goods or not. MARKING GOODS. It is customary for merchants to have a private mark, denoting the cost and often the selling price. These marks are sometimes made up of peculiar characters, but mostly letters of the alphabet that represent the nine digits. For example: BLACK HORSE 1234567890 Suppose an article cost $2.25, and you wish to sell it for $5.00, the mark would be thus: yi Usually they have what is called a repeater, that is to be used where a letter is repeated, as above. suppose G to be the repeater; then instead of using the letters L and E twice, we insert the repeat- er, thus: j&%. It sometimes happens that there are but two letters in the cost price and three in the selling price; to avoid this, place the letter repre senting 0 as the first letter in the cost, thus: write eok 75 cents cost. 1.00 selling price : fee. HOW TO TELL» THE DAY OF THE WEEK, THE DAY OF THE MONTH, THE MONTH IN THE YEAR, THE AGE IN YEARS, WITHOUT ASKING A SINGLE QUESTION, PROCESS. Ask the person you wish to figure out the above facts for, to write down first, the day of the week on which he or she was born; if this is not known, ascertain by preceding method ; next, the day of the month, next, the month in the year, then multiply the whole by 2, add 5, multiply by 50, add age, sub- tract 3865, add 115, the result will be ; the first figure will be the day of the week ; the next, the day of the. month ; the next the month in the year, and the last the age in years. 127 ExXaMPLE.—I was born Wednesday, May 31, 1843. Process.—Ist. Write 4 as the first figure, because Wednesday is the 4th day of the week. 2d. Write 5 as the second figure, because May is the 5th month. 3d. Write 31 as the 3d and 4th figures, because this is the day on which I was born. ‘The figures therefore read, ety rast 2 multiply. 5 0 multiply. 453350 4 1 add age. 4533891 3 6 5 subtract. 453026 1 1 5 add 4,6,3 1.4 1 Beis 27 Sf as & < B a ExAMPLE 2d.—A friend was born April 6th. Take number of month and day of month. 5 6 2 multiply. 5 added. hy J 5 0 multiply. add age. 90 0 3 6 5 subtract. 5 5 add. HOW 10 TELL THE DAT OF THE WEEE, A scientific method of telling immediately what day of the week any date transpired or will tran- spire, from the commencement of the Christian Era for the term of three thousand years. MONTHLY TABLE. The ratio to add for each month will be found in the following table : Ratio of June is....-..... 0 | Ratio of October is......- 3 Ratio of September is-.-.1 | Ratio of May is..-..-. oafald, Ratio of December is-..-. 1 , Ratio of August is.-...... 5 Ratio of April is......... 2 | Ratio of March is........ 6 Ratio of July is ..... ....2 | Ratio of February is......6 Ratio of January is....... 3 | Ratio of November is.---6 Notre.—On Leap Year the Ratio of January is 2, and the ratio of February is 5. The ratio of the other ten months do not change on Leap Years. CENTENNIAL TABLE. The ratio to add for each century will be found in~ the following table : , 200, 900, 1800, 2200, 2600, 3000, ratio is. ....-.-- 0 E 800, 1000, oes 5 iesss | seus ieee TALIO IS tensemee sO & 400, 1100, 1900, 2800, 2700, ---. ratio is. ........ 5 = 500, 1200, 1600, 2000, 2400, 2800 ratio is. ........ 4 EF (600; 1300), ss" Weeds hk Wee agtetien Hobie yee tienes 3 000, 700. 1400, 1700 2100 2500 2900 ratio is. ....-.-- 2 1002800) )2500,) sins) Ae ie ALTO aletam eee area d Notr.—The figure opposite each century is its ratio; thus the ratio for 200, 900, etc., is 0. To find the ratio of any century, first find the century in the above table then run the eye along the line until you arrive at the end; the small figure at the end is its ratio. 129 METHOD OF OPEKATION. Ru e.*—To the given year add its fourth part, rejecting the fractions; to this sum add the day of the month; then add the ratio of the month and the ratio of the century. Divide this sum by 7; the remainder is the day of the week, counting Sunday as the first, Monday as the second, Tuesday as the third, Wednesday as the fourth, Thursday as the fifth, Friday as the sixth, Saturday as the seventh; the remainder for Saturday will be 0 or zero. EXAMPLE 1.—Required the day of the week for the 4th of July, 1810. To the given year, which is.........s.. eo sevesecess 10 Add its fourth part, rejecting fractions....... ern ete oie 2 Now add the day of the month, which is.........-- - 4 Now add the ratio of July. which is....-....++e2e--- 2 Now add the ratio of 1800, which is..... Bt cietsete ol) eis 0 Divide the whole sum by 7. 7 | 18—4 2 We have 4 for a remainder which signifies the fourth day of the week, or Wednesday. Norz.—In finding the day of the week for the present century, no attention need be paid to the centennial ratio, as it is 0. EXAMPLE 2.—Required the day of the week for the 2d of June, 1805. To the given year, which is.......... iat acaak ese 5 Add its fourth part, rejecting fractions..........- +. 1 Now add the day of the month. which is........... ~ 2 Now add the ratio of June, which is.++--++s.ee++ eee. 0 Divide the whole sum by 7. 7| 8—1 1 We have 1 for a remainder, which signifies the first day of the week, or Sunday. The Declaration of American Independence was signed July 4, 1776. Required the day of the week. “* When dividing the year by 4, always leave off the centuries. We a.vide by 4 to find the number of Leap Years. i 130 To the given year, which is.......esecesecceceereee IO Add its fourth part, rejecting fractions...- sese++---19 Now add the day of the month, which is ....e+-e+++ 4 Now add the ratio of July, which is...... S wenn ta daw Now add the ratio of 1700, which is.....--.eee+eeees 2 Divide the whole sum by 7, 7 | 1083—5 14 We have 5 for a remainder, which signifies the fifth day of the week, or Thursday. The Pilgrim Fathers landed on Plymouth Rock Dec. 20, 1620. Required the day of the week. To the given year, which is.......- a ejnlete's Raima eee 20 Add its fourth part, rejecting fractions .---..-.....-- 5 Now add the day of the month, which is.........--.20 Now add the ratio of December, which is ..--.-.ee«- 1 Now add the ratio of 1600, which is.......-cseee ey: | Divide the whole by 7, 7 | 50—1 7 We have 1 for a remainder, which signifies the first day of the week, or Sundav. AMUSING ARITHMETIC. Under the head of Amusing Arithmetic we give a collection of problems particularly adapted to the social circle, or the fireside, of a winter evening. The most of those problems are in the form of puz- zles, and some of them particularly amusing, The majority of them are very old, their parentage being entirely unknown, so that no credit can be given to their authors. This is believed to be the largest collection ever published. 1. Think of a number of 3 or more figures, divide by 9, and name the remainder; erase one figure of the number, divide by 9, and tell me the remainder, and I will tell you what figure you erased. \TETHOD. --If the second remainder is less than -the first, the figure erased is the difference between the remainders; but if the second remainder is greater than the first, the figure erased equals 9, minus the difference of the re- mainders. 2, Think of a number, multiply it by 8, and mul- tiply it also by 4, take the sum of the squares of the products, extract the square root of this sum, divide by the first number, and I will name the quotient. MertTHop.—The quotient will always be 5. The same will be also true if we have them multiply and divide by the same multiples of 3, 4, and 5, as 6, 8, 10, &c. If we have them divide by 5, it will give the number they com- menced with. 3. Think of a number, multiply it by 5, also by 12 ; square each product, take their sum, extract the square root, divide by the number commenced with, and I will name the quotient. MetuHop.—The quotient is always 13. To give variety it is well to use multiples of 5,12; as 10, 24, &c., and then the quotient is 26, &c. 132 AMUSING ARITHMETIC. 4, Think of a number composed of two unequal digits, invert the digits, take the difference between this and the original number, name one of the digits and I will name the other. MeTHop.—The sum of the digits in the difference is al- ways 9; hence when one is named, the other equals 9 minus the one named. 5. Take any number consisting of three consecutive digits and permutate them, making 6 numbers, and take the sum of these numbers, divide by 6, and tell me the result, and I will tell you the digits of the number taken. METHOD.—The quotient consists of three equal digits; the digits of the number taken are, Ist, one of these equal digits; 2d, this digit increased by a unit; 3d, this digit diminished by a unit. The same principle holds when the digits of the number taken differ by 2, 3, or 4. It is a very pretty problem to prove that the sum is always divis- ible by 9, and 18. 6. Think of a number greater than 3, multiply it by 3; if even, divide it by 2; if odd, add 1, and then divide by two. Multiply the quotient by 3; if even, divide by 2; if odd, add 1, and then divide by 2. Now divide by 9 and tell the quotient, without the remainder, and I will tell you the number thought of, MeETHuHoD. — If evex both times, multiply the quotient by 4; if even 2d, and odd lst, multiply by 4, and add 1; if even Ist, and odd 2d, multiply by 4, and add 2; if odd beth times, multiply by 4, and add 3. 7. Take any number, divide it by 9 and name the remainder. Multiply the number by some number which I name, and divide this product by 9, and I will name the remainder. MeETHOD.—To tell the remainder, I multiply the first remainder by the number by which I told them to multi- ° ply the given number, and divide this product by 9. ‘The remainder is the second number that they obtained. 133 AMUSING ARITHMETIC. 8. A and Bhave an 8 gallon cask full of wine, which they wish to divide into two equal parts, and the only measures they have are a 5 gallon cask and a 3 gallon cask. How shall they make the division with these two vessels? MetTuop.—Fill the 3 and pour it into the 5, then fill it again, and from it fill up the 5, which will leave one gal- lon in the 3 gallon keg; empty the 5 in to the 8, and pour the one from the 3 into the 5; fill the 8 again and empty into the 5; then there are four gallons in the 5 gallon keg, and the same left in the 8. 9. Two men have 24 ounces of fluid, which they wish to divide between them equally. How shall they effect the division, provided they have only three vessels , one containing 95 ozs., the other 11 ozs., and the third 13 ozs. 10. Two men, stopping at an oyster saloon, laid a wager as to which could eat the most oysters. One eat ninety-nine, and the other eat a hundred and won. How many did both eat? REMARK.—The ‘‘catch” is in ‘‘a hundred and won.” When this is repeated it sounds as if it meant ‘‘one eat 99 and the other eat 101;” hence the result usually given is 200. Thecorrect result, of course, is 199. 11. Six ears of corn are in a hollow stump. How long will it take a squirrel to carry them all out, if he takes out three ears a day? ReMARK.—The ‘‘catch” is in the word ears. He car- ries out two ears on his head, and one ear of corn each day; hence it will take him 6 days. 12. A and B went to market with 30 pigs each, A sold his at 2 for $1, and B at the rate of 3 for $1, and they, together, received $25. The next day A went to market alone with sixty pigs, and, wishing 134 AMUSING ARITHMETIC. to sell at the same rate, sold them at 5 for $2, and received only $24. Why should he not receive as much as when B owned half of the pigs ? MetHop—The insinuation that the first lot were sold at the rate of 5 for $2. being only true in part. They com- mence selling at that rate, but after making 10 sales, A’s pigs are exhausted, and they have received $20.; B still has 10, which he sells at ‘‘2 for a dollar,” and of course receives $5; whereas, had he sold them at the rate of 5 for $2, he would have received but $4. 13, In the bottom of a well, 45 feet deep, there was a frog which commenced travelling towards the top. In his journey he ascended 3 feet every day, but fell back two feet every night. In how many days did he get out of the well? 14. A man having a fox, a goose, and some corn, came to a river which it was necessary to cross. He could, however, take only ove across at a time, and if he left the goose and corn, while he took the fox over, the goose would eat the corn; but if he left the fox and goose, the fox would kill the Boos How shall he get them all safely over? Mrtuop.—Let him first take over the goose, leaving the fox and the corn, then let him take over the fox, and bring back the goose, then take over the corn, and lastly, take over the goose again. 15. A man went to a store and purchased a pair of boots worth $5, and hands out a $50 bill to pay for them ; the merchant, not being able to make the change, passes over the street to a broker and gets the bill changed, and then returns and gives the man who bought the boots his change. After the purchaser of the boots has been gone a few hours, the broker, finding the bill to be a counterfeit, re- turns and demands $50 of good money from the 135 AMUSING ARITHMETIC. merchant. How much did the merchant lose by the operation. REMARK.— At first glance some say $45 and the boots; some, $50 and the boots; some, $95 and the boots; and others, $100 and the boots. Which is correct? 16. What relation to me is my mother’s brother- in-law’s brother, provided he has but one brother ? 17. Three men, travelling with their wives, came to a river which they wished to cross. There was but one boat, and but two could cross at one time; and, since the husbands were jealous, no woman could be with a man unless her own husband was present. In what manner did they get across the river. Metuop.—Let A and wife go over, let A return, let B’s and C's wives go over, A’s wife returns, B and C go over, B and wife return, A and B go over, C’s wife returns, and A’s and B’s wives go over, then C comes back for his wife. Simple as this question may appear, it is found in the works of Alcuin who flourished a thousand years ago; hundreds of years before the art of printing was invented. —Parke. 18. Suppose it were possible for a man, in Cincin- nati, to start on Sunday noon, when the sun is in the meridian, and travel westward with the sun, so that it might be in his meridian all the time. He would arrive at Cincinnati next day at noon. Now, it was Sunday noon when he started, it has been noon with him all the way around, and is Monday noon when he returns. The question is at what point did it change from Sunday noon to Monday noon? 19. Suppose a hare is 10 rods before a hound, and that the hound runs 10 rods while the hare runs 1 rod. Now when the hound has run the 10 rods, the hare has run 1 rod; hence they are now one rod apart, and when the hound has run that 1 rod, the 136 AMUSING ARITHMETIC. hare has run 4, of a rod; hence they are now 7 of a rod apart, and when the hound has run the 7; of a rod they are z§, of a rod apart; and in the same way it may be shown the hare is always +5 of the previous distance ahead of the hound; hence the hound can never catch the hare. How is the con- trary shown mathematically. 20. Think of any three numbers less than 10. Multiply the first by 2, and add 5 to the product. Multiply this sum by 5, and add the second number to the product. Multiply this last result by 10, and add the third number to the product; then subtract 250. Name the remainder, and J will name the numbers thought of, and in the order in which they were thought of. Mertruop.—The three digits composing this remainder will be the numbers thought of; and the order in which they were thought of will be the order of hundreds, tens, and units. 21. Write 24 with three equal figures, neither of them being 8. METHOD.—22 + 2 = 24, or 3? —3 = 24: 22. Put down four marks, and then require a person to put down five more marks, and make ten. Metuop.—The four marks are as represented in the margin; the five more, making ten, are | | | | placed as in the margin. TEN 23. Which is the greater, and how much, six dozen dozen, or one-half a dozen dozen, or is there no difference between them? 24. Show what is wrong in the following reason- ing:—8—8 equals 2— 2; dividing both these equals by 2— 2 and the result must be equal ; 8—-8 divided by 2 —2 = 4, and 2—2 divided by 2—2 137 AMUSING ARITHMETIC. = 1; therefore, since the quotients of equals divid- ed by equals, must be equal, 4 must be equal to 1. 25. A man has a triangular lot of land, the largest side being 156 rods, and each of the other sides 68 rods ; required the value of the grass on it, at the rate of $10 an acre. ReMARrK.—The ‘‘ catch ’ in this is, that the sides given will form no triangle. 26. Says A to B, ‘Give me four weights, and I can weigh any number of pounds not exceeding 40.” Required the weights and method of weighing. ANSWER.—The weights are J, 3,9, and 27 pounds. In weighing, we must put one or more in both scales, or some in one scale and some in the other; thus, 7 lbs. = 9 lbs. +1 lb. —3 Ibs. 27. Mr. Frantz planted 13 trees in his garden, in such a manner that there were 12 rows. and only 3 trees‘ in each row. In what manner were they planted ? ANSWER.—They were in the form of a regular hexagon, having a tree in the centre, and one at the middle and ex- tremity of each side. 28. A and B raised 749 bushels of potatoes on shares ; A was to have #, and B # of them. Before they were divided, however, since A had used 49 bushels, B took 28 bushels from the heap, and then divided the remainder according to the above agreement. Was this division fair? if not, show how it should have been. 29. Two-thirds of six is nine, one-half of twelve is Ae SEVEN, The half of five is four, and six is half of eleven. So.ution.--Two thirds of §]X is |X; the upper half of X|| is Vii; the half of FEWE is [Y; and the upper half of X] is VI. 138 AMUSING ARITHMETIC. 30. Does the top of a carriage wheel move faster than the bottom? METHOD. —This seems absurd, but it is strictly true, as any one may Satisfy himself in a moment, by setting up a stake by the side of a wheel, and move the wheel forward a few inches. 31. Supposing there are more persons in the world than anyone has hairs on his head, there must be, at least, two persons who have the same number of hairs on the head, to a hair. Show how this is. 32. Place 17 little sticks — matches, for instance — making 6 | equal squares, as in the margin. Then remove 5 sticks, and leave 3 | | perfect squares of the same size. == 33. Three persons own 51 quarts of rice, and have only two measures; oné a 4 quart and the other a7 quart measure. How shall they divide it into three equal parts? MeEtTHOD.—Perhaps the easiest way is to give each one- 17 quarts, which may be obtained thus: fill the 7 quart measure, empty this into the 4 quart measure, and there will be 3 quarts in the 7 quart measure, which added ta two 7 quart measures, equals 17 quarts. 34, What four United States coins will amount to fifty-one cents? ANSWER.—Two 25 ct. pieces and two half-cents. 35. How may the nine digits be arranged in a rectangular form, so | 4 | 9 | 2 that the sum of any row, whether | | horizontal, vertical, or diagonal, 5? shall equal 10: A en oe AnsweErR.—As in the margin. 139 AMUSING ARITHMETIC. 36. How may the first 16 | 1 | 16/111] 6 digits be arranged, so that the sum of the vertical, and hor- | i138 | 4 | 7 | 10 izontal, and the two oblique rows, may equal 34? ANSWER.—As in the margin. 19°05 oN ay 37. In what manner may the-first 20 digits, bey) eis eb ie arranged, sothatthe sum | g | 31 | 90 | 22 | 3 of each row of five fig- |———|——_|——-|_|__ ures may be 65? 1s Rl i - R om | |) ANSWER.—“*As in the mar- gin. PREP OUT RS ORS Ce adh (eit REMARK.—The above are | 95 | 9 Sn TAL Org called Magic Squares. They are very interesting, and have engaged the attention of some of our greatest mathematicians, among whom we may mention Leibnitz, Stifels, &c. The methods of ar- rangement given above are by no means the only ones that may be used. For the second problem, Frenicle, a French mathematician, has shown that there may be 878 different arrangements. 38. Take 10 pieces of money, lay them in a row, and require some one to put them together in heaps 2 in each, by passing each piece over 2 others. MetTuHop.—Let the pieces be represented by the numbers bets ob4n 0507 44°) 95) LUs. bace 7,on 10,) Don 2. Sy Ons; 1 on 4, and 9 on 6. 39. An old Jew took a diamond cross to a jeweller, to have the diamonds reset; and fearing that the jeweller might be dishonest, he counted the dia- monds, and found that they numbered 7, in three different ways. Now the jeweller stole two dia- monds, but arranged the remainder so that they counted 7 each way. as before. How was it done? 140 AMUSING ARITHMETIC, MrtTHop.—The form of the cross when yy.y. wg. left is represented by Fig. 1, and when re- 7 7 turned by Fig. 2. It will be seen by the fig- wenee fet ures how the diamonds were counted by the 4 4 3 os old Jew, and how they were arranged by the 2 2 i i jeweller, who ‘‘jewed ” the Jew. 40. Let a person select a number greater than 1 and not exceeding 10; I will add to it a number not exceeding 10, alternately with himself ; and, although he has the advantage in selecting the number to start with, I will reach the even hundred first. METHOD.—I make my additions so that the sums are, respectively, 12, 23, 34, 45, &c., to 89, when it is evident Ican reach the hundred first. With one who does not mistrust the method, I need not run through the entire series, but merely aim for 89, or, when the secret of this is seen, for 78, then 67, &c. 41, Let a person think of any number on the dial- face of a watch ; I will then point to various num- bers, and at each he will silently add ove to the num- ber selected until he arrives at ¢wenty, which he will announce aloud, and my pointer will be upon the number he selected. MetHop.—I point promiscuously about the face of the watch until the eighth point, which should be upon “ 12;” and then pass regularly around, towards ‘‘1,” pointing at 611,” $10,” °9,” &c., until ‘‘twenty is called, when, as may be easily shown, my pointer will be over the number selected. 42. Is there any difference between the results of the two following problems, and if so, what is it? If the half of 6 be 3, what will the fourth of 20 be? If 3 be the half of 6, what will be the fourth of 20? 43. A vessel with a crew of 30 men, half of whom were black, became short of provisions ; and, fearing that unless half the crew were thrown overboard, all would perish, the captain proposed to the sailors ta 141 AMUSING ARITHMETIC, stand upon deck in a row, and every ninth man be thrown overboard, until half the crew were destroyed. It so happened that the whites were saved. Re- quired the order of arrangement. ANSWER.—W. W. W. W. B. B. B. B. B. W. W. B. W. W. W. B. W. B. B. W. W. B. B. B. W. B. B. W. W. B. This can easily be found by trial, using letters or figures to represent the men. 44, Think of a number, multiply it by 6, divide this product by 2, multiply by 4, divide by 3, add 40, divide by 4, subtract the number thought of, divide by 2, and the quotient is 5. Show why this is so. 45, If through passenger trains, running to and from Philadelphia and San Francisco daily, start at the same hour from each place (difference of longi- tude not being considered ) and take the same time, six days, for the trip, how many through trains will the Pacific Express that leaves the San Francisco depot at 9 p. M., Sunday, have met when it reaches the Philadelphia depot? 46. A switch siding to a single track railroad is just long enough to clear a train of eight cars and a locomotive. How can two trains of sixteen cars and a locomotive each, going in opposite directions, pass each other at this siding, and each locomotive remain with and have the same relative position to its own train after as before passing? 47. Two hunters killed a deer, and sold it by the pound in the woods. ‘They had no proper means of weighing it, but knowing their own weights—one 130 pounds and the other 190 pounds they placed a rail across a fence so that it balanced with one on each end. They then exchanged places, and the lighter man taking the deer in his lap, the rail again balanced. What was the weight of the deer ? 142 AMUSING ARITHMETIC. VARIATIONS. The doctrine of variations and combinations forms the basis of many forms of Lotteries, and of other calculations used in practical life. We shall com- mence with the simplest form of variations in which all the articles are taken at once and which is called PERMUTATION. To determine the number of permutations, com- mence with unity and multiply by the successive terms of the natural series 1, 2, 8, &c., until the highest multiplier shall express the number of indi- vidual things. The last product will indicate the number of possible changes. Example 1. How many changes can be made in the arrangement of 5 grains of corn, all of differ- ent colors, laid in a row? SOLUTION. —1X2X38X4X5=120, Axs. This may seem improbable, the number being so great, but if there were but a single grain more, the possible changes would be 720; and another would extend the limit to 5010; and so onward in a constantly increasing ratio. The reason, however, will be obvious on a little scrutiny. If there weré but one thing, as a, it would ad- mit of but one position; but if two, as a 6, it would admit of two positions, ad, da. If three things, as adc, then they will admit of 1X2X3=6 changes, for tne last two will admit of two variations, as adc, acé, and each of the three may successively be placed first, and two changes made to each of the others, so that 83X2=6, the number of possible changes. In the same way we may show that if there be four individual things, each one will be first in each of the six changes which the other three will under- go, and consequently, there will be 24 changes in all. In this way we might show that when there are 5 individual things, there will be 5 times as many changes as when there were but 4; and when 6, there will be 6 times as 1438 AMUSING ARITHMETIC, many changes as when there are only 5; and so on ad infinitum, according to the same law. Example, 2. In how many ways may a family of 10 persons seat themselves differently at dinner? ANSWER.—3628800. When we consider that this would require a period of 99353, years, the mind is lost in astonishment. The story of the man who bought a horse at a farthing for the first nail in his shoe, a penny for the second, &c., is thrown into the shade; and we incline to doubt whether there is not some mistake; and yet on just such chances as one to all these do gamblers constantly risk their money! Example 3. 1 have written the letters contained in the word NIMROD on 6 cards, being one letter on each, and having thrown them confusedly ‘into a hat, I am offered $10 to draw the cards successively, so as to spell the name correctly. What is my chance of success worth? ANSWER. —1,% cents. 48. Sold a horse for $56, and gained as much per cent. as the horse cost me. Required the cost. Roize.—‘‘ Multiply the selling price by 190, and add 2500 to the product; of the sum extract the square root, and from the root subtract 50. The remainder will be the prime cost.” Horse sold for $56 Proof— 100 Cost $40 a Percent. 40 5600 — +2500 Gain 16.00 Cost 40 ¥8100—90 — —50 Sold for $56 Leaves $40 cost. Human ingenuity would perhaps fail to find a reason for the above rule, by the aid of common arithmetic mere- ly, or to explain the steps satisfactorily to a learner. It seems to be without reason, and yet it will solve all ques- tions involving a similar principle. 144 AMUSING ARITHMETIC. 49. Suppose 2000 soldiers had been supplied with bread sufficient to last them 12 weeks, allowing each man 14 ozs. per day; but on examination they find 105 barrels, containing 200 lbs. each, wholly spoiled ; what must be the allowance to each man, that the remainder may last them the contemplated time? ANALYSIS, as follows :—I1st. If one man ate 14 ozs. in a day, he would eat 7 times 1498 ozs. in a week; and if he ate 98 ozs. in a week, he would eat 12X98=1176 ozs. in 12 weeks; and 2000 men would eat 2000X1176 ozs.=2352000 ozs. in 12 weeks. 105 barrels of 200 lbs. each=21000 lbs. destroyed; and 21000X16=836000 ozs., which deducted from the whole quantity 2352000 ozs. leaves 2016000 ozs. to be consumed. Then if 2000 men consume 2016000 ozs. in 12 weeks, 1 man will consume 2918°9°%==1008, ozs.; and if 1 man consume 1008 ozs. in 12 weeks, he will consume 199884 ozs. in 1 week, and 8412 ozs. inl day. Azs. 50. How may 100 be expressed with four nines? ANSWER. — 9932. 51. What three figures, multiplied by 4, will make precisely 5? | ANSWER. — 11, or 1.25. 52. Required to subtract 45 from 45 and leave 45 as a remainder. SOLUTION. — 9-+8--7-++-6-+-5+4+3121 145 11243441516174819—45 8-+6+4-+-1+9-+7-+5+3-2—=45 53. From 6 take 9; from 9 take 10; From 40 take 50, and 6 will remain! SoLu110N.— SIX IX XL IX ,¢ 1 S I X The majority of these puzzles and problems being founded upon principles quite easily comprehended, itis not thought necessary to explain the principles of the puzzles nor solve the problems. it is hoped that they may prove a source of pleasure and profit. fem 3 0112 017102598