THE ROBERT E. COWAN COLLECTION I'RKSKXTED TO THE UNIVERSITY OF CHLIFORNIfl C. P. HUNTINGTON dUNE, 1897. flccession No./T? / 3 ^ Class No, HEXOERSOX & HAMI.TN'S LIGHTNING CALCULATOR; CONTAINING The Shortest, Siin/rfc-sf, and most Raintf Method of Computing Numbers, to uU kinds of Iitsiness^ find -wifhi.it the Coni/re- heusion of every one htirinf/ the sfif/htest knowledge of Figures. ENERGY IS THE PRICE OF SUCCESS. "The methods of calculation, by Prof. J. A. HEXDF.USOX. arc invaluable to business men, and will prove n light u all coming generations." A. -I. W\i;M-:i;, Pres. Elmira Commercial Cell' The above two methods are the finest known for liyhtnin^ niultipftcatiOJl.1'- I'rof. 1). I!. l-'olJD. I-'emale . Klmira. "I have examined Prof. J. A. HENDERSON'S new methods of calculation: They are renuu-kable for originality, and of great prtuttkul value. Ui.s methods of calculating Interest are peculiarly dear and comprehensive ill their adaptation to all possible cases." Rev. DR. O. P. FITZGERALD Address all orders for this Book to Prof. J. .A.. HENDERSON, SAN FRANCISCO, GAL. SAN FRANCISCO: A. L. BANCROFT & CO., PRINTERS AND LITHOGRAPHERS, 721 Market Si; 1872. i ing to Act of '.-irian at Washington. HENDERSON & HAMLWS LIGHTNING CALCULATOR; CONTAINING The Shortest, Simplest, and most Rapid Method of Computing Numbers, adapted to all kinds of Business, and within the Compre- hension of every one having the slightest knowledge of Figures. ENERGY IS THE PRICE OF SUCCESS, "The methods of calculation, by Prof. J. A. HENDERSON, are invaluable to business men, and -will prove a light in science to all coming generations." A. J. WARNER, Pres. Elmira Commercial College. "The above two methods are the finest known for lightning multiplication." Prof. D. R. FORD, Female College, Elmira. "I have examined Prof. J. A. HENDERSON'S new methods of calculation: They are remarkable for originality, and of great practical value. His methods of calculating Interest are peculiarly clear and comprehensive in then- adaptation to all possible cases." Rev. DB. O. P. FITZGERALD Address all orders for this Book to IProf. J. ^. HENDERSON, SAN FRANCISCO, CAL. SAN FRANCISCO: A. L. BANCKOFT & CO., PRINTERS AND LITHOGRAPHERS, 721 Market Street, San Francisco. 1872. AGE 7o /3J~ It is better to know everything about something, than some- thing about everything. Early ideas are not usually true ideas, but need to be revised and re-revised. Right means straight, and wrong means crooked. And knowing that thought kindles at the fire of thought, we do not hesitate or offer any apology for presenting to the Public some new seed-thoughts, and right methods of operation in busi- ness calculations. The practical utility of this book is found in the brevity and conciseness of its rules. Particular attention is invited to the grand improvements in the subjects of computing time, all possible cases in Interest, Squaring and Multiplying Numbers, Dividing and Multiplying Fractions, and an infinite num- ber of methods of Extracting Square and Cube Root. A? THK JNIVERSITT ADDITION. To be able to add two, three, or four columns of figures at once is deemed by many to be a Herculean task, and only to be accomplished by the gifted few; or, in other words, by mathematical prodigies. If we can succeed in dispelling this illusion, it will more than repay us ; and we feel very confident that we can, if the student will lay aside all prejudice, bearing steadily in mind that to be- come proficient in any new branch or principle, a little wholesome appli- cation is necessary. On the contrary, we cannot teach a student who takes no interest in the matter, one who will always be a drone in society. Such men have no need of this prin- ciple. If two, three, or more columns can be carried up at a time, there must be some law or rule by which it is done. We have two principles of Ad- dition ; one for adding short columns, and one for adding very long columns. They are much alike, differing only in detail. When one is thoroughly learned, it is very easy to learn the second. By a little attention to the following example, much time in future will be saved, ADDITION OF SHORT COLUMNS OF FIG- URES. Addition is the basis of all numerical operations, and is used in all depart- ments of business. To aid the business man in acquiring facility and accuracy in adding short columns of figures, the following method is presented as the best PROCESS. Commence at 274 the right hand column, add 346 thus : 16, 22, 32; then carry 134 the 3 tens to the second column; 342 then add thus: 7, 14, 25; carry 727 the 2 hundreds to the third col- 329 umn, and add the same way: 12, 16, 21. 2152. In this way you name the sum of two figures at once, which is quite as easy as it is to add one figure at a time,. Never permit yourself, for once, to add up a column in this manner: 9 and 7 are 16, and 2 are 18 and 4 are 22, and 6 are 28, and 4 are 32. It is just as easy to name the result of two figures at once, and four times as rapid. The following method is recom- mended for the ADDITION OF LONG COLUMNS OF FIGURES. In the addition of long columns of figures, which frequently occur in books of accounts, in order to add them with certainty, and, at the same time, with ease and expedition, study well the fol- lowing method, which practice will ren- der familiar, easy, rapid, and certain. THE EASY WAY TO ADD. EXAMPLE 2 EXPLANATION. Commence at 9 to add, and add as near 20 as possible, thus : 9+2+4+ 3=18, place the 8 to the right of the 3, as in example ; commence T at 6 to add 6+4+8=18; place 4 the 8 to the right of the 8, as in 6 example ; commence at 6 to add 3 6 6-f-4-j-7=17 ; place the 7 to the 9 right of the 7, as in example; 4 commence at 4 to add 4+9+3 7 7 =16 ; place the 6 to the right of 4 the 3, as in example ; commence 6 at 6 to add 6+4+7=17 ; place 8 8 the 7 to the right of the 7 as in 4 example; now, having arrived at 6 the top of the column, we add 3 8 and figures in the new column, thus: 7-f6-{-7+8-f-8=36; place 4 the right-hand figure of 36, 2 which is a 6, under the original 9 column, as in example, and add the left-hand figure, which is a 86 3, to the number of figures in the new column; there are 5 figures in the new column, therefore 3+5=8; prefix the 8 with the 6, under the original column, as in example; this makes 86, which is the sum of the column. Remark 1. If , upon arriving at the top of the column, there should be one, two or three figures whose sum will not equal 10, add them on to the sum of the figures of the new column, never placing an extra figure in the new column, unless it be an excess of units over ten. Remark 2. By this system of addition you can stop at any place in the column, where the sum of the figures will equal 10 or the excess of 10; but the addition will be more rapid by your adding as near 20 as possible, because you will save the forming of extra figures in your new column. EXAMPLE EXPLANATION. 2+6+7=15, drop 10, place the 5 to the right of the 7; 6+5+4=15, drop 10, place the 5 to the right of the 4, as in example; 8+3+7=18, drop 10, place the 8 to the right 4 of the 7, as in example; now we 7 8 have an extra figure, which is 4; 3 add this 4 to the top figure of the 8 new column, and this sum on the 4 s balance of the figures in the new 5 column, thus: 4+8+5+5=22; 6 olace the right-hand figure of 22 7 s under the original column, as in 6 example, and add the left-hand 2 Eigure of 22 to the number of fig ures in the new column, which 52 are three, thus: 2+3=5; prefix this 5 to the figure 2, under the orig- inal column; this makes 52, which is the sum of the column, RULE. For 'adding two or more col- umns, commence at the right-hand, or units' column; proceed in the same man- ner as in adding one column; after the sum of the first column is obtained, add all except the right-hand figure of this sum to the second column, adding the second column the same way you added the first; proceed in like manner with all the columns, always adding to each successive column the sum of the column in the next lower order, minus the right-hand figure, N. B. The small figures which we place to the right of the column when adding are called intergers The addition by intergers, or by forming a new column, as explained in the preceding examples, should be used only in adding very long columns of figures, say a long ledger column, where the footings of each column would be two or three hundred, in which case it is superior and much more easy than any other mode of addition; but in adding short columns it would be useless to form an extra column, where there is only, say six or eight figures to be added. In making short additions, the following sugges- tions will, we trust, be of use to the accountant who seeks for information on this subject. IVERSITY In the addition of several columns of figures, where there are only four or five deep, or when their respective sums will range from twenty-five to forty, the accountant should com- mence with the unit column, adding the sum of the first two figures to the sum of the next two, and so on, naming only the results, that is, the sum of every two figures, In the . present example, in 346 adding the unit column instead 235 of saying 8 and 4 are 12 and 5 724 are 17 and 6 are 23, it is better 598 to let the eye glide up the col- umn, reading only, 8, 12, 17, 23; and still better, instead of making . a, separate addition for each figure, group the figures thus: 12 and 11 are 23, and proceed in like manner with each column. For short columns this is a very expeditious way, and indeed to be preferred, but for long columns, the addition by integers is the most useful, as the mind is relieved at intervals, and the mental labor of retaining the whole amount, as you add, is avoided, which is very impor- tant to any person whose mind is con- stantly -employed in various commer- cial calculations In adding a long column, where the figures are of a medium size, that is, as many 8s and 9s as there are 2s and Hs, it is better to add about three figures at a time, because the eye will distinctly see that many at once, and the ingenious student will in a short time, if he adds by integers, be able to read bhe amount of three fig- ures at a glance, or as quick, we mi(^ht say, as he would read a single figure. Here we begin to add at the 5 26 8 bottom of the unit column and 67 add successively three figures 43 at a time, and place their re- 38 4 spective sums, minus 10, to the e 54 right of the last figure added ; 62 if the three figures do not 87* make 10, add on more figures; 5 65 if the three figures make 20 or 53 more, only add two of the fig- 44 4 ures. The little figures that 8 77 are placed to the right and left 33 of the column are called inte- 84 4 gers. The integers in the 3 56 present example, belonging to 14 the units' column, are 4, 4, 5, - 4, 6, which we add together 803 making 23; place down 3 and add 2 to the number of integers, which gives 7, which we add to the tens and proceed as before. KEASON. In the above example, every time we placed down an in- teger we discarded a ten, and when we set down the 3 in the answer we discarded two tens; hence, we add 2 on to the number of integers to ascer- tain how many tens were discarded; there being 5 integers, it made 7 tens, which we now add to the column of tens; on the same principle we might add between 20 and 30, always set- ting down a figure before we got to 30; then every integer set down would count for 2 tens, being dis- carded in the same way, it does in the present instance for one ten. When we add between 10 and 20, and in very long columns, it would be much better to go as near 30 as possible, and count 2 tens for every integer set down, in wnich case we would set down about one-half as many integers as when we write an in- teger for every ten we discard "When adding long columns in a ledger or day-book, and where the ac- countant wishes to avoid the writing of extra figures in the book, he can place a strip of paper alongside of the col- umn he wishes to add, and write the integers on the paper, and in this way the column can be added as conven- iently almost as if the integers were written in the book. Perhaps, too, this would be as proper a time as any other to urge the importance of another good habit; I mean that of making plain figures. Some persons accustom themselves to making mere scrawls, and impor- tant blunders are often the result. If letters be badly made, you may judge from such as are known; but if one figure be illegible, its value can- not be inferred from the others. The vexation of the man who wrote for 2 or 3 monkeys, and had 203 ^ent him, was of far less importance than errors and disappointments some- times resulting from this inexcusable practice. "We will now proceed to give some methods of proof. Many persons are fond of proving the correctness of work, and pupils are often instructed to do so, for the double purpose of giving them exercise in calculation and saving their teacher the trouble of re- viewing their work. There are special modes of proof of elementary operations, as by cast- ing out threes or nines, or by chang- ing the order of the operation, as in adding upward and then downward. In addition, some prefer reviewing the work by performing the Addition downward, rather than repeating the ordinary operation. This is better, for if a mistake be inadvertently made in any calculation, and the same routine be again followed, we are very liable to fall again into the same error. If, for instance, in running up a column of Addition you should say 84 and 8 are 93, you would be liable, in going over the same again, in the same way to slide insensibly into a similar error; but by begin- ning at a different point this is avoided. This fact is one of the strongest objections to the plan of cutting off the upper line and adding it to the sum of the rest, and hence some cut off the lower line by which the spell is broken. The most thoughtless can- not fail to see that adding a line to the sum of the rest is the same as add- ing it in with the rest. The mode of proof by casting out the nines and threes will be fully ex- plained in a following chapter. A very excellent mode of avoiding error in adding long columns is to set down the result of each column on some waste spot, observing to place the numbers successively a place further to the left each time, as in putting down the product fig- ures in multiplication ; and afterward add up the amount. In this way if the operator lose his count, he is not compelled to go back to units, but only to the foot of the column 011 which he is operating. It is also true that the brisk accountant, who thinks on what he is doing, is less liable to err than the dilatory one, who allows his mind to wander. Practice, too, will enable a person to read accounts without naming each figure : thus, instead of saying 8 and 6 are 14, and 7 are 21 and 5 are 26, it is better to let the eye glide up the col- umn, reading only 8, 14, 21, 26, etc. ; and, still further, it is quite practicable to accustom one's self 87 to group the figures in adding, 23 and thus proceed very rapidly. 45 Thus in adding the units' column, 62 instead of adding a figure at a 24 time, we see at a glance that 4 - and 2 are 6, and that 5 and 3 are 8; then 6 and 8 are 14; we may then, if expert, add constantly the sum of two or three figures at a time, and with practice this will be found highly advantageous in long columns of fig- ures; or two or three columns may be added at a time, as the prac- tised eye will see that 24 and 62 are 86 almost as readily as that 4 and 2 are 6, MULTIPLICATION. Multiplication, in its most general sense, is a series of additions of the same number; therefore, in mul- tiplication, a number is repeated a certain number of times, and the re- sult thus obtained is called the prod- uct. When the multiplicand and the multiplier are each composed of only two figures, to ascertain the product, we have the following RULE. Set down the smaller fac~ tor under the larger, units under units, tens under tens. Begin with the unit figure of the multiplier, mul- tiply by it, first the units of the multiplicand, setting the units of the product, and reserving the tens to ie added to the next product; now mul- tiply the tens of the multiplicand by the unit figure of the multiplier, and the units of the multiplicand by tens figure of the multiplier; add two products together, setting down the units of their sum, and reserving the tens to be 'added to the next prod- uct; now multiply the tens of the multiplicand by the tens' figure of the multiplier, and set down the whole amount. This will be the complete product. Remark. Always add in the tens that are reserved as soon as you form the first product. EXAMPLE 1. EXPLANATION. 1. Multiply the units of the 24 multiplicand by the unit fig- 31 ure of the multiplier, thus: 1 X4 is 4; set the 4 down as in 744 example. 2. Multiply the tens in the multiplicand by the unit figure in the multiplier, and the units in the multiplicand by the tens figure in the multiplier, thus: 1x2 is 2; 3x4 are 12, add these two products to- gether, 2 plus 12 are 14, set the 4 down as in example, and reserve the 1 to be added to the next product. 3. Multiply the tens in the multipli- cand by the tens' figures in the mul- tiplier, and add in the tens that were reserved, thus; 3x2 are 6, and 6 plus 1 equal 7; now set down the whole amount, which is 7. EXAMPLE 1. EXPLANATION. Multiply first upper by units, 123 5x3 are 15, set down the 5, re- 45 serve the 1 to carry to the next product; now multiply second 5535 upper by units and first upper by tens, 5X2 are 10, plus 1 are 11, 4x3 are 12, add these products together; 11 plus 12 are 23, set clown the 3, re- serve the 2 to carry; now multiply third upper by units, and second up- per by tens, add these two products together, always adding on the re- 8 served figure to the first product; 5 Xl are 5, plus 2 are 7, 4X2 are 8, and 7 plus 8 are 15, set down the 5, re- serve the 1 ; now multiply third upper by tens, and set down the whole amount; 4x1 are 4 plus 1 are 5, set down the 5. This will give the com- plete product. Multiply 32 by 45 in a single line. Here we multiply 5X2 and set 32 down and carry as usual; tljen to 45 what you carry add 5X3 and 4X 2, which gives 24; set down 4 1440 and carry 2 to 4X3, which gives 14 and completes the product. Multiply 123 by 456 in a single line. Here the first and second 123 places are found as before; for 456 the third, add 6X1, 5x2, 4X - 3, with the 2 you had to carry, 56088 making 30; e?t down and carry 3; then drop the units 3 place and multiply the hundreds and tens crosswise, as you did the tens and units, and you find the thousand fig- ure; then, dropping both units and tens, multiply the 4X1, adding the 1 you carried, and you have 5, which completes the product. The same principle may be extended to any num- ber of places; but let each step be made perfectly familiar before advanc- ing to another. Begin with two places, then take three, then four, but always practising some time on each number, for any hesitation as you progress will confuse you. CURIOUS AND USEFUL CONTRACTIONS. To multiply any number, of two fig- ures, by 11. RULE. Write the sum of the figures be- tween them. I. Multiply 45 by 11. Ans. 495. Here 4 and 5 are 9, which write be* tween 4 and 5, 2. Multiply 34 by 11. Ans. 374. , N. B. When the sum of the two figures is over 9, increase the left-hand figure by the 1 to carry, 3. Multiply 87 by 11. Ans. 957. To square any number of 9s in- stantaneously, and without multiply- ing. RULE. Write down as many 9s less one as there are 9s in the given number, an 8, as many Os as 9s, and a 1. 4. What is the square of 9999? Ans. 99980001. EXPLANATION. We have four 9s in the given number, so we write down three 9s, then an 8, then three Os, and al. 5. Square 999999. Answer 999998- 000001. To square any number ending in 5. RULE. Omit the 5 and multiply the number as it will then stand by the next higher number, and annex 25 to the product, 6. What is the square of 75? Ans. 5625. EXPLANATION. We simply say, 7 times 8 are 56, to which we annex 25. 7. What is the square of 95? Ans. 9025. PRACTICAL BUSINESS METHOD For Multiplying all Mixed Numbers. Merchants, grocers, and business men generally, in multiplying the mixed numbers that arise in the prac- tical calculations of their business, only care about having the answer correct to the nearest cent; that is, they disregard the fraction. W T hen it is a half cent or more, they call it 9 another cent, if less than half a cent, they drop it. And the object of the following rule is to show the business man the easiest and most rapid pro- cess of finding the product to the nearest unit of any two numbers, one or both of which involves a fraction. GENEBAL RULE. To multiply any two numbers to the nearest unit. 1st. Multiply the whole number in the multiplicand by the fraction in the multiplier to the nearest unit, 2d. Multiply the whole nmmber in the multiplier by the fraction in the multipli- cand to the nearest unit. 3d. Multiply the whole numbers to- gether and add the three products in your mind as you proceed. N. B. In actual business the work can generally be done mentally, for 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 ap- plication. There being no such thing as a fraction to add in, there is scarcely any liability to error or mis- take. By no other arithmetical pro- cess can the result be obtained by so few figures. EXAMPLE FOR MENTAL OPERATION. Multiply 11J by 8J by business method. Here J of 11 to the nearest unit is 3, and J of 8 to the 11 J nearest unit is 3, making 6, so 8J we simply say, 8 times 11 are 88 and 6 are 94. Ans. 94 REASON. \ of 11 is nearer 3 than 2, and J of 8 is nearer 3 than 2, Make the nearest whole number the quotient. A VALUABLE HINT TO MERCHANTS AND ALL RETAIL DEALERS IN FOREIGN AND DOMES- TIC DRY GOODS. Retail merchants, in buying goods by wholesale, buy a great many ar- ticles by the dozen, such as boots and shoes, hats and caps, and notions of various kinds. Now, the mer- chant, in buying, for instance, a dozen hats, knows exactly what one of those hats will retail for in the mar- ket where he deals ; and, unless he is a good accountant, it will often take him some time to determine whether he can afford to purchase the dozen hats and make a living profit in selling them by the single hat; and in buying his goods by auction, as the merchant often does, he has not time to make the calculation be- fore the goods are cried off. He, therefore, loses the chance of making good bargains by being afraid to bid at random, or if he bids, and the goods are cried off, he may have made a poor bargain by bidding thus at a venture. It then becomes a useful and practical problem to de- termine instantly what per cent, he would gain if he retailed the hats at a certain price. RAPID PROCESS OF MARKING GOODS To tell what an article should retail for to make a profit of 20 per cent, is done by removing the decimal point one place to the left. For instance, if hats costs $17.50 per dozen, remove the decimal point one place to the left, making $1.75, what they should be sold for a piece to gain 20 per cent, on the cost. If they cost $31.00 per dozen, they should be sold for $3.10 apiece, etc. 10 We take 20 per cent as the basis, for the following reasons, namely: be- cause we can determine instantly, by simply removing the decimal point, without changing a figure; and, if the goods would not bring at least 20 per cent, profit in the home market, the merchant could not afford to pur- chase and would look for goods at lower figures. Now, as removing the decimal point one place to the left, on the cost of a dozen articles gives the selling price of a single one with 20 per cent, added to the cost, and, as tile cost of any article is 100 per cent. , it is ob- vious that the selling price would be 20 per cent, more, or 120 per cent; hence, to find 50 per cent, profit, which would make the selling price 150 per cent., we would first find 120 per cent., then add 30 per cent., by increasing it one-fourth itself; to make 40 per cent., add 20 per cent., by increasing it ore-sixth itself; for 35 per cent, increase it one-eighth itself, etc. Hence, to mark an article at any per cent, profit, we have the follow- ing GENERAL RULE. First find 20 per cent, profit by removing the decimal point one place to the left on the price the articles cost a dozen; then, as 20 per cent, profit is 120 percent., add to, or substr act from, this amount the fractional part that the re- quired per cent, added to 100 is more or less than 120. TABLE. For Marking all Articles bought by the Dozen. N. B. Most of these are used in business. To make 20 pr ct. remove the point one place to the left. To make 33 % p ct. remove point and add one-ninth itself. one-tenth ' 30 28 26 25 one- twelfth " one-fifteenth one-twentieth ' one-twer x "-fourth subtr ct one-sixteenth one-thirty-sixth one-ninety-sixth 80 60 50 44 40 37% 35 and add one-half itself, one-third one-fourth one-fifth one-sixth one-seventh one-eighth 18 ?i If I buy 1 doz. shirts for $28.00, what shall I retail them for to make 50 per ct.V Ans. $3.50. EXPLANATION. Remove the point one place to the left, and add on J it- self. Where the Multiplier is an Aliquot part of 100. Merchants in selling goods gener- ally make the price of an article some aliquot part of 100, as in selling sugar at 12 J cents a pound or 8 pounds for 1 dollar, or in selling calico, for 16 2-3 cents a yard or 6 yards for 1 dollar, etc. And to be- come familiar with all the aliquot parts of 100, so that you can apply them readily when occasion requires, is perhaps the most useful, and, at the same time, one of the easiest arrived at of all the computations the ac- countant must perform in the prac- tical calculations of the counting- room. TABLE. Of the Aliquot parts of 100 and 1000. N. B. Most of these are used in business. 12% is J$ part of 100. 8>j is 1-12 part of 100 25 is 2-8 or % of 100. 16 8 $ is 2-12 or 1-6 of 100 37% is 3-8 part of 100. 33^ is 4-12 or % of 100 50 is 4-8 or % of 100. 6G 2 s is 8-12 or % of 100 62% is % part of 100. 88^ is 10-12 or 5-6 of 100 75 is 6-8 or 3 of 100. 125 is X part of 1000 87% is % part of 100. 250 is 2-8 or % of 1000 6% is 1-16 part of 100. 375 is % part of 1000 1834 is 3-16 part of 100. 625 is 5 / 8 part of 1000 31% is 5-16 part of 100. 875 is % part of 1000 To multiply by an aliquot part of 100. RULE. Add two ciphers to the multi- plicand, then take such part of it as the multipliers is part of 100. N. B. If the multiplicand is a mixed number reduce the fraction to 11 "Cl B f. OP TJTK UNIVERSITY a decimal of two places before divid- ing. General Rules for Cancellation. RULE IST. Draw a perpendicular line ; observe this line represents the sign of equality. On the right-hand side of this line place dividends only ; on the left hand side place divisors only; having placed dividends on the right and divisors on the left as above directed, 2d. Notice whether there are ciphers both on the right and left of the line ; if so, erase an equal number from each side. 3d. Notice whether the same num- ber stands both on the right and left of the line ; if so, erase them both. 4th. Notice again if any number on either side of the line will divide any number on the opposite side without a remainder ; if so, divide and erase the two numbers, retaining the quotient figure on the side of the larger number. 5th. See if any two numbers, one on flach side, can be divided by any as- sumed number without a remainder ; if so, divide them by that number, and retain only their quotients. Proceed in the same manner as far as practicable, then, 6th. Multiply all the numbers re- maining on the right-hand side of the line for a dividend, and those remain- ing on the left for a divisor. 7th. Divide, and the quotient is the answer. SIMPLE INTEREST BY CANCELLATION. RULE. Place the principal, time and rate per cent, on the right-hand side of the line. If the time consists of years and months, reduce them to months, and place 12 (the number of months in a year) on the left-hand side of the line. Should sist of months and days, reduce them to days or decimal parts of a month. If reduced to days, place 36 on the left. If to decimal parts of a month, place 12 only, as before. Point off two decimal places when the time is in months, and three decimal places when the time is in days. NOTE. If the principal contains cents, point off four decimal places when the time is in months, and five decimal places when the time is in days, NOTE. We place 36 on the left because there are 360 interest days in a year. (Custom has made this law- ful.) LIGHTNING METHOD OF COMPUTING INTEREST. On all notes that bear $12 per an- num, or any aliquot part or multiple of $12. If a note bears $12 per annum, it will certainly ^"bear $1 per month : hence the time in months would be the interest in $ ; and the decimal parts of a month would be the in- terest in decimal parts of a $ ; there- fore when the note bears $12 per annum we have the following rule : RULE. Reduce the years to months, add in the given months, and place one-third of the days to the right of this number, and you have the interest in dimes. EXAMPLE 1. Required the interest of $200 for 3 years, 7 months, and 12 days, at 6 per cent. 200 y* of 12 days = 4. 6 -- Yr. Mo. Da. $12.00 == int. for 1 yr. 37 12 = 43. 4 mo. Hence 43.4 dimes, or $43.40cts., Ans We see by inspection that this note bears $12 interest a year; hence 12 the time reduced to months, with one-third of the days to the right, is the interest in dimes. If this note bore $6 a year, instead of $12, we would take one-half of the above in- terest; if it bore $18 instead of $12, we would add one-half; if it bore $24, instead of $12, we would multiply by 2, etc. EXAMPLE 2. Required the interest of $150 for two years, 5 months, and 13 days, at 8 per cent. % of 13 days =4% 150 8 Yr. Mo. Da. $12.00 = int. for 1 yr. 2 5 13 = 29. 4y 3 mos. Hence $29. 4% dimes, or $29.433%cts., Ans. We see by inspection that this note bears $12 interest a year; hence the time reduced to months, with one-third of the days placed to the right, gives the interest at once. EXAMPLE 3. Required the interest of $160 for 11 years, 11 months, and 11 days, at 7j- per cent. 160 %of 11 days = 3%- 7% Tr. Mo, Da. $12. 00 = int. for 1 yr. 11 11 11 = 143%mos. Hence 143.3% dimes, or $143.36%cts., Ans. When the interest is more or less than $12 a Tear. RULE. First find the interest for the given time on the base of $12 in- terest a year; then, if the interest on the note is only $6 a year, divide by 2; if $24 a year, multiply by 2; if $18 a year, add on one-half, etc. EXAMPLE 1. "What is the interest of $300 for 4 years, 7 months, and 18 days, at 6 per cent.? % of 18 days =6. 300 4yr. 7mo. 18da.=55.6mo. 6 $18.00 =int. for 1 yr. 2)55.6, int. -at $12 a yr. $18=1% times $12. 27.8. $83.4. Ans. If the interest was $12 a year, $55.60 would be the answer; because 55.6 is the time reduced to months; but it bears $18 a year, or 1J times 12; hence 1J times 55.6 gives the interest at once. EXAMPLE 2. Required the interest of $150 for three years, 9 months, and 27 days, at four per cent 150 4 $6.00 = int. for 1 yr. $6 = % times S12. % of 27 days =9. 3yr. 9mo. 27da = 45.9mo. 2)45.9. int. at $12 a year. $22.95, Ans. If the interest was $12 a year, $45.90 would be the answer; because $45.9 is the time reduced to months ; but it bears $6 a year, or \ times 12; hence \ times 45.9 gives the interest at once. RULES FOE DETERMINING THE WEIGHT OF LIVE CATTLE. Measure in inches the girth round the breast., just behind the shoulder- blade, and the length of the back from the tail to the fore part of the shoulder- blade. Multiply the girth by the length, and divide by 144. If the girth is less than three feet, multiply the quotient by 11 ; if between three feet and five feet, multiply by 16; if between five feet and seven feet, multiply by 23', if between seven and nine feet, multiply by 31. If the animal is lean, deduct l-20th from the result. Take the girth and length in feet, multiply the square of the girth by the length, and multiply the product by 3.36. The result will be the an- swer in pounds. The live weight, multiplied by 605, gives a near ap- proximation to the net weight. 13 ASTRONOMICAL CALCULATIONS. A scientific method of telling imme- diately what day of the week any date transpired or will transpire, 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 Ratio of September is.. . .1 Ratio of December- is 1 Ratio of April is 2 Ratio of July is 2 Ratio of January is 3 Ratio of October is 3 Ratio of May is 4 Ratio of August is 5 Ratio of March is 6 Ratio of February is G Ratio of November is 6 NOTE. On Leap Year the ratio of January is 2, and the ratio of Feb- ruary 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 : I fl 200, 900, 1800, 2200, 2600, 3000, ratio is 3 300, 1000, ratio is 6 | 400, 1100, 1900, 2300, 2700 ratio is 5 500, 1200, 1600, 2000, 2400, 2800, ratio is 4 600, 1300, ratio is 3 000, 700, 1400, 1700, 2100,2500, 2900 ratio is 2 100, 800, 1500 ratiois 1 NOTE. The figure opposite each century is its ratio ; thus the ratio for 200, 900, etc., is 0. To find the ra- tio of any century, first find the cen- tury 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, METHOD* OF OPERATION. RULE.* To the given year add its fourth part, rejecting the fractions; *When dividing the year by 4, always leave off the centuries. We divide by 4 to find the number <>f Leap Tears. 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, Thurs- day as the fifth, Friday as the Sixth, Saturday as the seventh ; the remainder for Saturday will be O or zero. EXAMPLE 1. Required the day of the week for the 4th of July, 1810. To the given year, which is 10 Add its fourth part, rejecting fractions 2 Now add the day of the month, which is 4 Now add the ratio of July, which is 2 Now add the ratio of 1800, which is. 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. Eule for finding the number of feet of boards which can be cut from any log whatever. From the diameter of the log, in inches, substfact 4 for the slabs and saw-calf. Then multiply the remainder by half itself and the product by the length of the log in feet, and divide the result by 8; the quotient will be the number of square feet. EXAMPLE 1. What is the number of feet of boards which can be cut from a log 24 inches in diameter and 12 feet long ? Diameter 24 inches For slabs and saw-call 4 Remainder 20 Half remainder 10 200 Length of log 12 300 the number of feet. HENDERSON'S LIGHTNING PROCESS, For Computing Time and Interest, Squaring and Multiplying Numbers, and a Fine Method for Dividing Fractions, and an infinite number of of ways of Extracting Square and Cube Root. The following Table gives the Interest on any amount at 7 per cent. , by simply removing the point to right or left, as the case may require : Number of Days. $100 $90 $80 $ro $60 $50 $40 $30 $30 1. .0192 .01726 .01534 .01342 .01151 .00950 .00767 .00575 .00384 2. .. .0384 .03452 .03058 .02685 .02301 .01918 .01534 .01151 .00767 3. .. .0575 .05178 .04603 .04027 .03452 .02877 .02301 .01726 .01151 4. . .0767 .06904 .06137 .05370 .04603 .02836 .03068 .02301 .01536 5. . .0959 .08630 .07671 .06712 .05753 .04795 .03836 .02877 .01918 6. . .1151 .10356 .09205 .08055 .06904 .05753 .04603 .03452 .02313 7. . .1342 .12082 . 10740 .09897 .08055 .06712 .05370 .04027 .02685 8. . .1532 .13808 .12274 .10740 .09205 .07671 .06137 .04603 .03068 9. . . .1726 .15534 .13808 . 12089 1.0356 .08630 .06904 .05178 .03452 90. 1.7260 1.5342 1.38082 1.20822 1.03562 .86301 .69041 .51781 .34521 93. . . 1.7836 1.60521 1.42685 1.24849 1.07014 .89178 .71342 .53508 .35671 100... . 1.9178 1.82603 1.53425 1.24247 1.15065 .95890 .76712 .57534 .48356 For 10-7 of a year remove the decimal point one place to the left; 1-7, or 52 days, two places to the left. Increase or diminish the results to suit the time, When the Eate is 6 per cent. For 5-3 of a year, or 20 months, remove the point one place to the left; 60 days, two places, and 6 days three places to the left. $5. 7. 8. 9. 00 $ 94|7.50 50.25 3458.50 36.50 943.20 47.75 Is the interest at 7 per 64|9.3t) cent, for 52 days, or 1-7 of a year. At 6 per cent, for 20 months; for 60 days draw the line two places to the left of the decimal point; and for 6 days three places, etc. When the rate is 5 per cent. For two years remove the point one place to the left, and 73 days two places to the left. When the rate is 7J per cent. For 4-3 of a year or 16 months remove the point one place ; for 48 days two places, the result modifying to suit f he time given. 15 When the rate is 8 per cent. For 15 months, remove the point one place ; for 1-8 of a year, or 45 days, two places to the left. To MAKE a rule for all rates, divide 100 by the rate and the quotient is the time, when the principal equals the interest and the point remains the same ; divide 10 by the rate, and the quotient indicates the time or base you work from, when you remove the point 1 place to the left ; divide unity by the rate, and the result is the part of a year and the number of days, when the point is to be removed two places to the left. To FIND THE INTEREST by the table, for any given time and any number of dollars, look on the Time Table for the time, and on the Interest Table for the interest of twenty, thirty and forty dollars, etc. Modify by removing the point right or left to suit the example given. You can find the interest very conveniently by taking the number of months and J of the days, and multiply that by % of the principal, and you have the interest at 6 per cent, in cents. RULE. Eemove the point one place to the left, because one-tenth of the principal equals the interest. Remove the point two places, for one-hundredth of the principal equals the interest. Remove the point three places, because one-thousandth of the principal equals the interest. These methods give the interest of all finite sums of money, for the time and rate mentioned in each rule. To reach all other time, increase or diminish the results to suit the time given. Thus: $500 for 1-7 of a year "at 7 per cent, is five dollars ; for one half of that time $2.50 ; for one fourth, $1.25, &c.; for one year it is seven times $5.00, $35. $400, for 1-6 of a year, and rate 6, is $4 ; for one-half of that time, $2. For one year $24 ; for 1-60 of a year, or 6 days, remove the point three places and the interest is 40 cents ; for one half of that time it is 20 cents. The rule may thus be expressed : The reciprocal of the rate is the time when the point can be removed two places to the left in all cases ; ten times that time remove it one place to the left, one tenth of the same time three places to the left : Increase or diminish the results to suit the time -given. TO MULTIPLY NUMBERS, FIRST KNOW HOW TO SQUARE THEM. (99)=9801 Take the comple- (101)2=10201 When above the base, add the (11)2=121 ment of 99 from it, call (102)2=10404 supplement, call it hundreds, and (12)2=144 it hundreds, and add the (103)2=10609 increase it by the square of the (13)2=169 square of the comple- &c., &c. supplement, &c., &c. ment: Then n=99 and c=l n+ c=100 n _ c= 98 n2-c2=9800 16 Now add c 2 to both members of the equation, and we have the square of the number. In the same manner, let n equal the number and s the supple- ment, and the reason of the rule becomes evident. For same reason: (98)2=9604 (97)2=9409 Take any number that is easy to multiply by for the (96)2=9216 base 10, 20, 40, 50, &c. (95)2=9025 &c., &c. The product of any two numbers is the Square of the Mean diminished by the Square of Half the Difference. . 39 X41=(40)2 12=1599 38X42=(40)2 22=1596 37 X43=(40)2 32=1591 &c., &c. 79X81=6399 78X82=6396 23X27=821 22X26=616 24X26=624 &c., &c. From the square of the mean subtract the right hand digit of the greater number ; because it in- dicates half of the difference of the two numbers. 8J 33 Multiply both dividend and divisor = by the least common multiple of 6J 26 the denominators of the fraction- al parts. . Increase 2 by 1, and multiply by the other tens digit, and an- nex the product of unit's digits. Add 1, because the sum of the units digits is =10. 8 1-2X8 1-2=72 1-4 8 1-3X8 2-3=72 2-9 8 2-5X8 3-5=72 6-25 c^ for all similar ex- amples. HENDEKSON'S METHOD OF EXTRACTING- CUBE BOOT. 10000 30000 6000 400 36400 6000 800 43200 1800 25 45025 1953125(100+20+5 1000000 953125 728000 225125 225125 Add to each true divisor, as they occur, twice the surface of one side of the small cube, and one of each of the three parallelopipedons, for a trial divisor; because that will make three sides of the complete cube. By observation the reason is evident and the conclusion just, for making trial and true divisors by this method. We have an infinite number of ways of finding the square root, cube root, &c. a Presume the root to be divided into a certain number of parts. Square the parts in square root; cube them in cube root to find the divisor. Thus let a-\-a represent the square root of any number. The square of a-\-a is 4 a 2 : hence divide any number by 4 and extract the square root of the quotient, and we have half of the root. Divide any number by the square of 3, and extract the square root of the quotient, and we have one-third of the root, &c., for all numbers. In the cube root we cube the number representing the parts the root is divided into, for a divisor 17 To find the Day-of the- Week from the Day -of the-Month. Cast the sevens out of the day of the month, the ratio of the month, the ratio of the year, and the year. One of a remainder will be the first day of the week ; two second, &c., the last day of the week. The ratio of the month is found above its name. The ratio of every month except January and February is one more in Leap Years, Jan'y 3 Feb'y 6 March 6 April 2 May June July 2 August. Sept. 5 -1 October 3 Novem. 6 Decem. j 1 1 1 32 1 60 1 91 1 121 1 152 1 182 1 213 ! 1 244 1 274 1 305 1 335 2 2 2 33 2 61 2 92 2 122 2 153 2 183 2 214! 2 245 2 275 2 306 2 336 3 3 3 34 3 62 3 93 3 123 3 154 3 184 3 215 3 246 3 276 3 307 3 337 4 4 4 35 4 63 4 94 4 124 4 155 4 185 4 216 4 247 4 277 4 308 4 338 5 5 5 36 5 64 5 95 5 125 5 156 5 186 5- 217 5 248 5 278 5 309 5 339 6 6 6 37 6 65 6 96 6 126 6 157 6 187 6 218 6 249 6 279 6 310 6 340 7 7 7 38 7 66 7 97 7 127 7 158 7 188 7 219 7 250 7 280 7 311 7 341 8 8 8 39 8 67 8 98 8 128 8 159 8 189 8 220 8 251 8 281 8 312 8 342 9 9 9 40 9 68 9 99 9 129 9 160 9 190 9 221 9 252 9 282 9 313 9 343 10 10 10 41 10 69 10 '100 10 130 10 161 10 191 10 222 10 253 10 283 10 314 10 344 11 11 11 42 11 70 11 101 11 131 11 162 11 192 11 223 11 254 11 284 11 315 11 345 12 12 12 43 12 71 12 102 12 132 12 163 12 193 12 224 12 255 12 285 12 316 12 346 13 13 13 44 13 72 13 103 13 133 13 164 13 194 13 225 13 256 13 286 13 317 13 347 14 14 14 45 14 73 14 104 14 134 14 165 14 195 14 226 14 257 14 287 14 318 14 348 15 15 15 46 15 74 15 105 15 135 15 166 15 196 15 227 15 258 15 288 15 319 15 349 16 16 16 47 16 75 16 106 16 136 16 167 16 197 16 228 16 259 16 289 16 320 16 350 17 17 17 48 17 76 17 107 17 137 17 168 17 198 17 229 17 260 17 290 17 321 17 351 18 18 18 49 18 77 18 108 18 138 18 169 18 199 18 230 18 261 18 291 18 322 18 352 19 19 19 50 19 78 19 109 19 13* 19 170 19 200 19 231 19 262 19 292 19 323 19 353 20 20 20 51 20 79 20 110 20 140 20 171 20 201 20 232 20 263 20 293 20 324 20 354 21 21 21 52 21 80 21 111 21 141 21 172 21 202 21 233 21 264 21 294 21 325 21 355 22 22 22 53 22 81 22 112 22 142 22 173 22 203 22 234 22 265 22 295 22 326 22 356 23 23 23 54 23 82 23 113 23 143 23 174 23 204 23 235 23 266 23 296 23 327 23 357 24 24 24 55 24 83 24 114 24 144 24 175 24 205 24 236 24 267 24 297 24 328 24 358 25 25 25 56 25 84 25 115 25 145 25 176 25 206 25 237 25 268 25 298 25 329 25 359 26 26 '26 57 26 85 26 116 26 146 26 177 26 207 26 238 26 269 26 299 26 330 26 360 27 27 27 58 27 86 27 117 27 147 27 178 27 208 27 239 27 270 27 300 27 331 27 361 28 28 28 59 28 87 28 118 28 148 28 179 28 209 28 240 28 271 28 301 28 332 28 362 29 29 29 88 29 119 29 149 29 180 29 210 29 241 29 272 29 302 29 333 29 363 30 30 30 89 30 120 30 150 30 181 30 211 30 242 30 273 30 303 30 334 30 864 31 31 31 90 31 151 31 212 31 243 31 304 31 365 SUGGESTIONS ON TEACHING ARITHMETIC. Qualifications. The chief qualifica- tions requisite in teaching Arithmetic, as well as other branches, are the fol- lowing: A thorough knowledge of the subject; a love for th which gives 6 J 6J; this multiplied by the 6 in the 6J multiplier, 6 X 6J gives 39, to which we add the product of the 39.^ fractions; thus JXj gives * B , added to 39 completes the pro- duct. EXAMPLE SECOND. Multiply 11J by llf in a single line. Here we would add HJ-f f , 11J which gives 12 ; this multiplied by llf the 11 in the multiplier gives 132, to which we add the product of the 132 j 3 6 fractions; thus fXj gives p e , which added to 132 completes the product. EXAMPLE THIRD. Multiply 12J by 12 J in a single line. Here we add 12-j-f , which gives 12J 13J; this multiplied by the 12 in 12} the multiplier, 12X13J, gives 159, - to which add the product of the 159$ fractions; thus Jxi gives , which added to 159 completes the product. WHERE THE SUM OF THE FRACTIONS IS ONE. To multiply any two like numbers to- gether when the sum of the fractions is one, RULE. Multiply the whole number by the next higher whole number, after which add the product of the fractions. N. B. In the following examples the product of the fractions are obtained first, for convenience : PRACTICAL EXAMPLES FOR BUSINESS MEN. Multiply 3| by 3} in a single line. Here we multiply JXf , which 3} gives 4 3 ff , and set down the result; 3J then we multiply the 3 in the mul- - tiplicand, increased by unity, by 12^ the 3 in the multiplier, 3X4, which gives 12 and completes the product. Multiply 7| by 7 3 in a single line. Here we multiply f Xf , which 7$ gives \ 6 , and set down the result : 7f then we multiply the 7 in the mul -- tiplicand, increased by unity, by the 7 in the multiplier, 7x8, which gives 56, and completes the product. Multiply 11J by llf in a single line. Here we multiply f X J, which 11 J gives f, and set down the result; llf then we multiply the 11 in the -- multiplicand, increased by unity, by 132f the 11 in the multiplier, 11X12, which gives 132, and completes the product. EXAMPLE FOURTH. Multiply 16f by 16J in a single line. 42 Here we multiply J X > which 16 j gives f , and set down the result; 16 J then we multiply the 16 in the mul tiplicand, increased by unity, by 272f the 16 in the multiplier, 16X17, which gives 272, and completes the product. EXAMPLE FIFTH. Multiply 29J by 29J in a single line. Here we multiply JXj, which 29 J gives J, and set down the result; 29 J then we multiply the 29 in the mul- tiplican, increased by unity, by 870J the 29 in the multiplier, 29x30, which gives 870, and completes the product. NOTE. The system of multiplication introduced in the preceding examples applies to all numbers. Where the sum of the fractions is one, and the whole numbers are alike, or differ by one, the learner is requested to study well these useful properties of numbers. WHERE THE FRACTIONS HAVE A LIKE DE- NOMINATOR. To multiply any two like numbers together, each of which has a fraction with a like denominator, as 4-JX4J, or lliXllf,orlOfXi,etc. RULE. Add to the multiplicand the fraction of the multiplier, and multiply this sum by the whole number, after which add the product of the fractions. PRACTICAL EXAMPLES FOR BUSINESS MEN. N, B. In the following example the sum of the fractions is one: 1. "What will 9f Ibs. of beef cost at 9J cts. a lb.? The sum of 9| and J is 10, so we 9f simply say 9 times 10 are 90; then 9J we add the product of the fractions, J times f are Sg. 90 A 3 6 N. B. In the following example the sum of the fractions is less than one: 2. What will 8J yds. tape cost at 8f cts. a yd.? The sum of 8 J and | is 8|, so we 8J simply say 8 times 8| are 70; then 8J we add the product of the fractions, - | times J are ^g , or J. 70J N. B. In the following example the sum of the fraction is greater than one: 3. What will 4 yds. cloth cost at $ a yd.? The sum of 4f and J is 5 J, so we 4f simply say 4 times 5J are 21; then 4 we add the product of the fractions, - times f are gi. 21i N. B. Where the fractions have dif- ferent denominators reduce them to a common denominator. RAPID PROCESS FOR MULTIPLYING- MIXED NUMBEBS A valuable and useful rule for the ac- countant in the practical calculations of the counting room. To multiply any two numbers to- gether, each of which involves the frac- tion \ as 7-JX9, etc. RULE. To the product of the whole numbers add half their sum, plus J. EXAMPLES FOR MENTAL OPERATIONS. 1 What will 3 J dozen eggs cost at 7J cts. a doz.? Here the sum of 7 and 3 is 10, 3J and half this sum is 5, so we simply 7^ say 7 times 3 are 21 and 5 are 26, - to which we add J. 26J N. B. If the sum be an odd number call it on less, to make it even, and in such cases the fraction must be - . 2. What will 11J Ibs. cheese cost at 9J cts. a lb.? * 43 3. What will 8J yds. tape cost at cts. a yd.? 4. What will 7J Ibs. rice cost at 18 J cts. a lb.? 5 What will lOJbu. coal cost at 12J cts. a bu.? REASON. 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 J- because 4X4 is J. The same principle will multiply any two numbers together, each of which has the same fraction for instance, if the fraction was J 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. 6. Multiply 4g by 4%. Ans. 21||. The sum of 4 and J- is 5J, and 4 ames6Jis21;add|Xj=J|. 21f|Ans. To multiply any two numbers together, each of which involves the fraction J. RULE. To the product of the whole numbers add half their sum, plus% 1. Multiply 34X7-J. Ans. 26J: Solution. The sum of 3 and 7 are 10, and one-half this sum is 5, so we say, 7 times 3 are 21 and 5 are 26, to which we annex J. 26J Ans. 8. What will 7J Ibs. cheese cost at 134 cts. a lb.? Ans. $1.01J. REMAKE. If the sum be an odd num- ber call it one less, to make it even; in which case the fraction must be f . 9. What will 8J Ibs, of sugar cost at 15J cts. alb.? Ans. $1.31 j. Here, 8-|-19fc=23, being an odd num- ber, we make it one less, 22, one-half of which is 11. Then 8 times 15 are 120, and 11 are 131, to which We add f. The same principle will multiply any two numbers together, each of which has the same fraction. For instance, if the fraction was i, we would add one- fifth their sum; if , we would add three-fourths their sum; if , add two- thirds their sum, etc., after which, of course, add the product of their frac- tions. 10. Multiply 8f X7f. Ans. 66|. The sum of 8 and 7 are 15, two-thirds of which is 10. We then say 8 times 7 are 56 and 10 makes 66, and add Xf =* INTEREST Is a sum paid for the use of money. Principal is a sum for the use of which interest is paid. Amount is the sum of the principal and interest, Rate per cent., commonly expressed decimally as hundredths, is the sum per cent, paid for the use of one dollar an- nually. Simple Interest is the sum paid for the use of the principal only during the whole time of the loan. Legal Interest is the rate per cent, es- tablished by law- Usury is illegal interest, or a greater per cent, than the legal rate. It is contended by many statesmen that the rate of interest should rot be established by statute, but that money is only a commodity that, like every other article of traffic, should be gov- erned by the law of supply and demand. If money is scarce the rate would be high; if plenty, then low. But as banks and other great moneyed institu- tions have the power, to a great extent, of controling the quantity of money in the market, thereby oppressing the great majority of the people, and taking ad- vantage of the times of scarcity, pub- lic opinion, at least, has established the law of usury. 44 To find the interest if the time con- sists of years. RULE. Multiply the principal by the rate per cent., and that product by the number of years. EXAMPLE 1. What is the interest of $150 for 3 years, at 8 per cent.? $150 .08 12.00 3 $36.00 Ans. The decimal for 8 per cent, is .08. There being two places of decimals in the multiplier we point off two places in the product. To find the interest when the time consists of years and months. KULE. Reduce the time to months. Multiply the principal by the rate per cent. , divide the product % 12, and the quotient multiplied by the number of months will be the interest required. OR BY CANCELLATION. Place the prin- cipal, rate and time in months on the right of the line, and 12 on the left, then cancel. 2. Find the interest of $240 for 2 years and 7 months, at 7 per cent. Principal, $240 Bate, .07 12)16.80 1.40 , 2yrs,-}-7inos. 31 1.40 4.20 $43.40 Ans. BY CANCELLATION. $240 20 12 20X7X" 7 31 *1=$43.40. Ans. BANKERS' METHOD OF COMPUTING INTEREST AT 6 PER CENT. FOR ANY NUMBER OF DAYS. RULE. Draw a perpendicular line, cutting off the two right-hand figures of the $, and you have the interest for 60 days at 6 per cent. NOTE. The figures on the left of the line are dollars, and those on the right are decimals of dollars. EXAMPLE 1. What is the interest of $423 for 60 days, at 6 per cent.? $423= the principal. $4 | 23 cts.=rinterest for 60 days. NOTE. When the time is more or less than 60 days, first get the interest for 60 days, and from that to the time re- quired. EXAMPLE 2. What is the interest of $124 for 15 days at 6 per -cent.? Days. Days. 15=J of 60 $ 1 24=principal . 4) 1 | 24 cts.=interest for 60 days. | 31 cts.=interest for 15 days. EXAMPLE 3. What is the interest -of $123.40 for 90 days, at 6 per cent.? Days. Days. Days. 90 = 60 30 $123.40=principal. 2)1 2340=interest for 60 days. 6170=interest for 30 days. Ans $ | 851=interest for 90 days. EXAMPLE 4. What is the interest of $324 for 75 days, at 6 per cent.? Daj^s. Days. Days. 75 = 60 15 $324 principal. 4)3 24 cts. interest for 60 days. 81 cts. interest for 15 days. Ans. $4 | 05 cts. interest for 75 days. REMARK. This system of computing interest is very easy and simple, espec- ially when the days are aliquot parts 45 of 60, and one simple division will suf- fice. It is used extensively by a large majority of our most prominent bank- ers; and, indeed, is taught by most all commercial colleges as the shortest sys- tem of computing interest. METHOD OF CALCULATING AT DIFFERENT PER CENTS. This principle is not confined alone to 6 per cent. , as many suppose who teach and use it. It is their custom first to find the interest at 6 per cent. , and from that to other per cents. ; but it is equally applicable for all per cents. , from 1 to 15, inclusive. The following table shows the differ- ent per cents., with the time that a given number of $ will amount to the same number of cents when placed at interest : RULE. Draw a perpendicular line, cut- ting off the two right-hand figures of $, and you have the interest at the following per cents. : Interest at 4 per cent, for 90 days. Interest at 5 per cent, for 72 days. Interest at 6 per cent, for 60 days. Interest at 7 per cent, for 52 days. Interest at 8 per cent, for 45 days. Interest at 9 per cent, for 40 days. Interest at 10 per cent, for 36 days. Interest at 12 per cent, for 30 days. Interest at 7-30 per cent, for 50 days. Interest at 5-20 percent, for 70 days. Interest at 10-40 percent, for 35 days. Interest at 7^ per cent, for 48 days. Interest at 4J per cent, for 80 days. NOTE. The figures on the left of the perpendicular line are dollars, and on the right decimals of dollars. If the dollars are less than 10 prefix a cipher. EXAMPLE 1. What is the interest of $120 for 15 days at 4 per cent.? Days Cays. $120=principal. 15=4 of 90 6)1 20 cts. interest for 90 days. 20 cts. interest for 15 days. EXAMPLE 2. What is the interest of $132 for 13 days, at 7 per cent.? Days. Days. $132 principal. 13=J of 52. 4)1 32 cts. interest for 52 days. 33 cts. interest for 13 days. EXAMPLE 3. What is the interest of $520 for 9 days at 8 per cent.? Days. 9 Days. of 45 $520 principal. 5)5 20 cts. interest for 45 days. $1 04 cts. interest for 9 days. EXAMPLE 4. What is the interest of $462 for 64 days, at 7| per cent.? Days. Days. Days. $462=principal. 6448+16 3)4 $1 62 cts. interest for 48 days. 54 cts. interest for 16 days. $6 | 16 cts.^interest for 64 days, REMARK. We have now illustrated several examples by the different per cents., and if the student will study carefully the solution to the above ex- amples, he will in a short time be very rapid in this mode of computing inter- est. NOTE. The preceding mode of com- puting interest is derived and deduced from the cancelling system, as the in- genious student will readily see. It is a short and easy way of finding interest for days, when the days are even or ali- quot parts; but when they are not multiples, and three or four divisions are necessary, the cancelling system is much more simple and easy. We will here illustrate an example to show the difference. Required, the interest of $420 for 49 days, at 6 per cent. : BANKERS' METHOD. CANCELLING METHOD. 5)1 3) 42070 6 49 70 21 cts.=int. for 3 days. 7 cts.^int. for 1 day. $3.430 Ana. 2)420 cts.=int. for 60 days. 636 2)2 10 cts.=int. for 30 days. 05 cts.=int. for 15 days. $3 | 43 cts.=int. for 49 days. 46 The cancelling method is much more brief; we simply cancel 6 in 36, and the quotient -6 into 420; there is no devisor left; hence, 70X49 gives the interest at once. If the time had been 15 or 20 days, the bankers' method would have been equally as short, because 15 and 20 are aliquot parts of 60. The superiority of the cancelling system above all others is this, it takes advantage of the principal as well as the time. For the benefit of the student, and for the convenience of business men, we will investigate this system to its full extent, and explain how to take advan- tage of the principal when no advantage* can be taken of the days. This is one of the most important characteristics of interest, and very often saves much labor. It should be used when the days are not even or aliquot parts. The following table shows the dif- ferent sums of money (at the different per cents.) that bear one cent interest a day; hence, the time in days is always the interest in cents; therefore, to find the interest on any of the following notes, at the per cent, attached to it in the table, we have the following RULE. Draw a perpendicular line, cutting off the two right-hand figures of the days for cents, and you have the in- terest for the given time. Interest of $90 at 4 per cent, for 1 day is 1 cent. Interest of $72 at 5 per cent, for 1 day is 1 cent. Interest of $60 at 6 per cent, for 1 day is 1 cent. Interest of $52 at 7 per cent, for 1 day is 1 cent. Interest of $45 at 8 per cent, for 1 day is 1 cent. Interest of $40 at 9 per cent, for 1 day is 1 cent. Interest of $36 at 10 per cent, for 1 day is 1 cent. Interest of $30 at 12 per cent, for 1 day is 1 cent. Interest of $50 at 7.30 per cent, for 1 day is 1 cent. Interest of $70 at 520 per cent, for 1 day is 1 cent. Interest of $35 at 10.40 per cent, for 1 day is 1 cent. Interest of $48 at 7J per cent, for 1 day is 1 cent. Interest of $80 at 4J per cent, for 1 day is 1 cent. Interest of $24 at 15 per cent, for 1 day is 1 cent. NOTE. The 7-30 Government Bonds are calculated on the base of 365 days to the year, and the 5-20s and 10-40s on the base of 364 days to the year. PROBLEMS SOLVED BY BOTH METHODS. We will now solve some examples by both methods to further illustrate this system, and for the purpose of teaching the pupil how to use his judgment. He will then have learned a rule more val- uable than all others. EXAMPLE 5. What is the interest on $180 for 75 days, at 5 per cent. ? Operation by taking advantage of the dollar. 75=the days. $60X3 =$180. $0 75 cts.=the int. of $60 for 75 days. 3 Multiply by 3. Ans. $2 | 25cts.=the int. of $180for75 days. Operation by the Bankers' method. $180=the principal. 60 da. -f 15da.= 75 da. 4)$1 80 cts.=the int. for 60 days. 45 cts.=the int. for 15 days. Ans. $2 | 25 cts=the int. for 75 days. By the first method we multiplied by 3, because 3X$60=$180. By the sec- ond method we added on J, because 60 da.- da.=75 da. 47 N. B. .When advantage can be taken of both time and principal, if the stu- dent wishes to prove his work he can first work it by the Bankers 5 method, and then by taking advantage of the principal, or vice versa. And as the two operations are entirely different, if the same result is obtained by each, he may fairly conclude that the work is correct. PAKTIAL PAYMENTS ON NOTES, BONDS AND MORTGAGES. To compute interest on notes, bonds and mortgages, on which partial pay- ments have been made, two or three rules are given. The following is called the common rule, and applies to cases where the time is short, and payments made within a year of each other. This rule is sanctioned by custom and com- mon law ; it is true to the principles of simple interest, and requires no special enactment. The other rules are rules of law, made to suit such cases as re- quire (either expressed or implied) annual interest to be paid, and, of course, apply to no business transac- tions closed within a year. RULE. Compute the interest of the principal sum for the whole time to the day of settlement, and find the amount. Compute the interest on the several pay- ments from the time each was paid to the day of settlement; add the several pay- ments and the interest on each together and call the sum the amount of the pay- ments; subtracting the amount of the pay- ments from the amount of the principal will leave the sum due. EXAMPLE. A gave his note to B for $10,000 ; at the end of 4 months A paid $6,0#0, and at the expiration of another 4 months he paid an additional sum of $3,000; how much did he owe B at the close of the year ? Principal $10,000 Interest for the whole time . . 600 Amount $10,600 1st payment $6,000. Interest, 8 mos. . . . 240. 2d payment 3,000 Interest, 4 mos 60 Amount $9,300 9,300 Pue $1,300 PROBLEMS IN INTEREST. There are four parts or quantities con- nected with each operation in interest; these are the Principal, Rate per cent. , Time, Interest or Amount. If any three of them are given the other may be found. Principal, interest and time given to and the rate per cent. EXAMPLE 1. At what rate per cent, must $500 be put on interest to gain $120 in 4 years ? OPERATION. $500 .01 6.00 4 20.00)120.00(6 per cent. An&. 120.00 BY ANALYSIS. The interest of $1 for the given time at 1 per cent, is 4 cts. $500 will be 500 times as much=500X.04=$20. Then if $20 give 1 per ct., $120 will give yy> =6 per cent. RULE. Divide the given interest by tlie interest of the given sum at one per cent, for the given time, and the quotient will be the rate per cent, required- 48 Principal, interest and rate per cent, given to find the time. EXAMPLE 2. How long must $500 be on interest at 6 per cent, to gain $120? OPERATION. $500 .06 i-- 30 . 00) 120.00 (4 years. Ans. 120.00 ANALYSIS We find the interest of $1 at the given rate for one ye^r is six cents. $500 will be, therefore, 500 times as much = 500 X- 06 = $30.00. Now, if it take one year to gain $30, it will require 3 2 o to gain $120=4 years. Ans, EQUATION OF PAYMENTS. Equation of payments is the process of finding the equalized or average time for the payment of several sums due at different times, without loss to either party. To find the average or mean time of payment when the several sums have the same date. KULE. Multiply each payment by the time thai must elapse before it becomes due; then divide the sum of these products by the. sum of thepayments, and the quo- tient will be the average time required. NOTE. When a payment is to be made down it has no product, but it must be added with the other payments in finding the average time. EXAMPLE. I purchased goods to the amount of $1,200; $300 of which I am to pay in 4 months, $400 in 5 months, and $500 in 8 months. How long a credit ought I to receive if I pay the whole sum at once? Ans. 6 months. Mo. Mo. ( A credit on $300 for 4 months 4X300=1200 j is the same as the credit on *1 ( for 1200 months. ( A credit on $AOO for 5 months 5X400 = 2000 1 is the same as the credit on $1 ( for 2000 months. ( A credit on $.100 foi 8 months 8 X 500 = 4000 j is the same as the credit on $1 ( for 4000 months. Therefore, L should have the 1200) 7200 (6 nio. same credit as the credit on $1 7200 for 7200 months; and on $1200, the whole sum, one twelve hun- dredth part of 7200 months, which is 6 months. This rule is the one usually adopted by merchants, although not strictly correct; still, it is sufficiently accurate for all practical purposes. To find the average or mean time of payment when the several sums have different dates. EXAMPLE. Purchased of James Brown; at sundry times and on various terms of credit, as by the statement an- nexed. When is the medium time of payment? Jan. 1, a bill amounting to $360, on 3 months' credit. Jan. 15, a bill amounting to $186, on 4 months' credit. March 1, a bill amounting to $450, on 4 months' credit. May 15, a bill amounting to $300, on 3 months' credit. June 20, a bill amounting to $500, on 5 months' credit. Ans. July 24, or in 115 days. Due, April 1, $360. " May 15, 186 X 44 = 8184 " July 1, 450 X 91 = 40950 " Aug. 15, 300X136 = 40800 " Nov. 20, 500X233 = 116500 1796-j-into)206434(114|JJ dys We first find the time when each of the bills will become due. Then, since it will shorten the operation and bring the same result, we take the time when the first bill becomes due, instead of its date, for the period from which to com- pute the average time. Now, since 49 April 1 is the period from which the average time is computed, no time will be reckoned on the first bill, but the time for the payment of the second bill ex- tends 44 days beyond April 1, and we multiply it by 44. Proceeding in the same manner with the remaining bills, we find the average time of payment to be 114 days and a fraction from April 1, or on the 24th of July. RULE. Find the time ivhen each of the sums become due, and multiply each sum by the number of days from the time of the earliest payment to the payment of each sum respectively; then proceed as in the last rule, and the quotient will be the average time required, in days, from the earliest payment. NOTE. Nearly the same result may be obtained by reckoning the time in months. In mercantile transactions it is cus- tomary to give a credit of from 3 to 9 months on bills of sale. Merchants, in settling such accounts as consist of various items of debit and credit for different times, generally employ the following RULE. Place on the debtor or credit side such a sum (which may be called MERCHANDISE BALANCED as will balance the account. Multiply the number of dollars in each entry by the number of days from the time the entry was made to the time of settlement, and the merchandise balance by the number of days for which credit was given. Then multiply the dif- ference between the sum of the debit and the sum of the credit products by the in- terest o/$l for one day ; this product will be the INTEREST BALANCE. When the sum of the debit products ex- ceeds the sum of the credit products the interest balance is in favor of the debit side; but when the sum of the credit pro- ducts exceeds the sum of the debit pro- ducts it is in favor of the credit side. Now, to the merchandise balance add the interest balance, or substract it as the case may require, and you obtain the CASH BALANCE. A has with B the following account. 1849. Dr. Jan. 2. To merchandise, $200 April 20. " " 400 1849. Cr. Feb. 29. By merchandise, $100 May 10. " " 300 If interest is estimated at 7 per cent., and a credit of 60 days is allowed on the different sums, what is the cash bal- lance August 20, 1849? Ans. $206.54.' EXPLANATION. Without interest the cash balance would be $200. The object of these changes is to give the learner an accurate and complete knowledge of numbers and of division, and the result is not the only object sought for, as many young learners sup- pose. How many times is 75 contained in 575 ? or divide 575 by 75. Ans. 7f . Divide 800 by 12J. Quotient, 64. Divide 27 by 16f. Quotient, l^o, or 1ft. * A person spent $6 for oranges, at 6J cents a piece; how many did he pur- chase? Ans. 96. When two or more numbers are to be multiplied together, and one or more of them have a cipher on the right, as 24 by 20, we may take the cipher from one number and annex it to the other without affecting the product: thus 24X20 is the same as 240x2; 286X 1300=28600X13; and 350x70x40= 35X7X4X1000, etc. Every fact of this kind, though ex- tremely simple, will be very useful to those who wish to be skillful in operation. 50 h to the close of the opera- nght hand either of the multiplier or tion, when they must be annexed to the multiplicand, or of both, they may be! product TABLE FOB BANKING AND EQUATION. Showing the nuvtier of Days from any date in one Month to the same date in an,, other Month. Example: How many days from the 2d of February to the 2d of August ? Look for February at the left hand, and August at the top- in the angle is 181. In leap year, add 1 day if February be included From To 1 1 ft 1 :! i 1 P ha &b ^ 43 I | : ^ 1 i January 365 31 p;q Ofi i on -I (T-l February March 334 306 365 337 28 3fi^ i yu 59 q-i l^U 89 fii 151 120 QO 181 150 212 181 243 212 ! 2/3 | 242 304 273 334 303 April 275 3f)fi 004 iqftpr 01 on v2 1 122 153 184 214 245 275 May 245 97 fi Of) A ODD 33*; oU o^fr bl Q1 91 122 153 183 214 244 June 214 94. K 970 Of\A ODD OOA ol bl 92 123 153 184 214 July 184 2ia 94.3 OUtt 97 A. oo4 on/1 obo Oocr 30 61 92 122 163 183 August 153 184' A'to 219 <)AO oUtt 970 ooo Of\A ob5 QQ A 31 62 92 123 153 September 122 153 91 9 At o 9J.9 oU4 970 664: Of\O obo OO A 31 61 92 122 October 92 123 1^1 1 00 91 9 ZrO 9/< Q oUo O7O 664: 365 30 61 91 November 61 92 120 21 ^B -I Q-l z4o 91 9 216 f\A O 604: 335 365 31 61 December 31 62| 90 121 J.O1 151 ALA 182 24:2 212 2/o 9,43 304 9,74 334 304 3bb 335 30 ?>65 FNIVERSITY OF CALIFORNIA LIBRARY THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW ' 1915 I9J6 OCT 9 1916 11 IC22 30m-l,'15