Draw a line beneath the dividend, and, by the multiplication table, find how many times the divisor is contained in the first figure of the dividend, or in the number expressed by the first two figures, or even in the number expressed by the first three figures, should the number given by the first, and even by the first two, be smaller than the divisor; and write the quotient under the line, taking care to observe what is over, as the divisor may be contained a certain number of times in the number expressed by the leading figure or figures, and leave something over. Proceed to the next figure of the dividend; regard what was over, if anything, to be prefixed to it; and find how many times the divisor is contained in the number you thus get; putting the quotient down, and, as before, carrying what is over to the next figure of the dividend, to which you must regard it as prefixed. And in this way figure after figure of the complete quotient is to be found, till all the figures of the dividend have been used. Should there be anything over at the end, this is called the remainder it is to be written beside the quotient figures, with the divisor placed under it, and a line of separation between them. 3)25602 8534 Suppose, for example, we have to divide 25602 by 3, then placing dividend and divisor (3) as in the margin, we proceed thus :-3 is contained in 2, no times; so that nothing is to be placed under the 2: 3 is contained in 25, 8 times and 1 over; 8 and carry 1: this 1, regarded as prefixed to the 6, gives the number 16: we therefore say: 3 in 16, 5 times and 1 over: 3 in 10, 3 times and 1 over: 3 in 12, 4 times. Therefore, the quotient is 8534; and this is the complete quotient, as there is no remainder. Again, suppose it were proposed to divide 7804623 by 5, we should say, 5 in 7, 1; and 2 over: 5 in 28, 5; and 3 over: 5 in 30, 6: 5 in 4, 0: 5 in 46, 9; and 1 over: 5 in 12, 2; As there is here a remainder 3, 5 under it, to the figures of the quotient, and call 15609243, the com over. plete quotient. and 2 over: 5 in 23, 4; and 3 we annex it, with the divisor 5)7804623 1560924% The principle upon which the foregoing operation depends is pretty evident: the leading figure in the dividend above is 7000000: the fifth part of this is 1000000 and 2000000 over; that is, with the local value of the next figure 8, 2800000; the fifth part of this is 500000, and 300000 besides; the fifth part of which is 60000: the fifth part of the 4000-the local value of the next figure-is 0 thousands, and 4000 over; this, with the local value of the 6, is 4600; of which the fifth part is 900, and 100 over; this, with the 20, is 120; the fifth part of which is 20, and 20 over; and lastly, the fifth part of the remaining 23 is 4, and 3 over; and, to imply that this 3 still remains to be divided, it is put down with the 5 underneath; because one number, placed in this way under another, is a form frequently used to denote that the upper number is to be divided by the lower. Hence the fifth part of the proposed number is 1560924, and the fifth part of 3 besides this quotient being made up of the several parts which arise from taking a fifth of each of the above-mentioned component portions of the number. The sign for division is÷, which stands for the words divided by: thus, 6 ÷ 2 =3 is a short way of stating that 6 divided by 2 is equal to 3. As noticed above, there is another way of indicating division, namely, by putting the dividend above and the divisor below, a short line separating the two: thus, 3 expresses the same thing as the notation above. The learner may exercise himself in the rule just explained by proving by it the truth of the following statements expressed in one or other of the forms of notation here adverted to: It is plain that division by 10 requires no work: the quotient is always the dividend itself, wanting the last figure, which is the remainder, and which, therefore, written as in this last example, with the divisor underneath, completes the quotient. In a similar way, to divide by 100 we have simply to cut off two figures from the dividend for remainder; to divide by 1000, to cut of three figures; and so on: thus, 78546 All this is obvious, because 7854678500 + 46 = 78000+ 546, &c. RULE.-Place the divisor to the left of the dividend as in the former case, and to the right mark off a place for the figures of the quotient. Find how many times the leading figure of the divisor is contained in that of the dividend, or in the number expressed by the first two figures, if the leading figure of the dividend be smaller than that of the divisor; and put the figure expressing the number of times in the quotient's place. Multiply the divisor by this first quotient-figure, and subtract the product from the number formed by the leading figures of the dividend, and to the remainder annex the next figure of the dividend. The number thus formed will be a new dividend, and the number of times it contains the divisor-to be found as before—will be the second quotientfigure, the product of which and the divisor, being subtracted from the new dividend, will give a second remainder, to which the next figure of the original dividend is to be joined, and the operation continued till all the figures of the dividend have been used. An example worked at length will explain the operation better than any verbal rule. Let it be required to divide 256438 by 346. Placing the divisor on the left of the dividend, and marking off a place for the quotient on the right, we look at the leading figure of the divisor and also at that of the dividend, with the view of seeing whether the latter contains the former, which it does not, 3 being greater than 2: we therefore commence with the number 25, formed by the first two figures of the dividend, and seeing that 3 is contained in 25, 8 times, we should put 8 for the first quotient figure; but bearing in mind that, when the whole divisor is multiplied by this 8, we must attend to the carryings, we perceive that 8 346)256438 (741 2422 1423 1384 398 346 52 is too great, we therefore try 7, and find 7 times 346 to be 2422, a number less than 2564 above it, so that we can obey the direction of the rule and subtract: the remainder is 142, which, when the next figure of the dividend is brought down, becomes 1423. We now take this as a dividend; and, looking only at leading figures in this new dividend and in the divisor, we see that the latter will go, as it is called, 4 times; we therefore put 4 for the second quotient-figure; and multiplying and subtracting, we get 39 for the second remainder; and, by bringing down another figure, 398 for a new dividend: the divisor goes into this once; so that the quotient is 741, and the final remainder 52: this remainder, as in the former case, must be annexed, with the divisor underneath, to the quotientfigures; so that the complete quotient is 741, which is the 346th part of 256438. Of the truth of this you may convince yourself by observing that 256438 has been cut up into portions, and the 346th part of each portion found; for the work above is nothing else but that here annexed, with useless repetitions suppressed. According to this arrangement it is at once seen that 700 is the 346th part of 242200, that 40 is the 346th part of 13840, and that 1 is the 346th part of 346, and that of 52, the 346th part, still remains to be taken. Now, 242200+ 13840 + 346 + 52 = 256438; consequently 741, together with, is the 346th part of the number proposed. It must be noticed that if any dividend, formed by a remainder and a figure brought down, should be less than the divisor, that the divisor will go no times in that dividend; so that a 0 will be the corresponding quotient-figure; and that then a second figure must be brought down, as in the operation here annexed; where the complete quotient is 1022 There is another thing also to be attended to. Sometimes the divisor ends with zeros or noughts: when such is the case, the best way is to cut the ciphers off, and entirely to disregard them in the division, cutting off, however, at the same time, as many figures from the end of the dividend, which latter figures help to form the final remainder you will see by operating on the same example first with the ciphers retained, and then with the ciphers dismissed, that nothing is omitted but useless ciphers: the complete quotient being, by either way, 61,12. In thus completing the quotient, by means of the final remainder, you must, of course, take care to restore the ciphers that were temporarily cut off from the divisor: in some books on arithmetic this has been forgotten. The following examples are subjoined for practice : 346)256438(700 242200 346) 14238(40 13840 346) 398 (1 346 52 472) 48165(102 472 965 944 21 2700)164826(61 16200 2826 2700 Rem. 126 27,00)1648,26(61 162 28 27 Rem. 126 To prove whether the quotient, in any case, is correct, you have only to multiply it 472 102 and the divisor together: the product will be the dividend, if the operation is correct, as is obvious; for the object of division is to find a number such, that that number of times the divisor shall make the dividend. In thus proving division, if there be a remainder, you add this remainder to the product of the divisor and quotient-figures. For example: to prove whether the work at page 16 is correct, we multiply and add as in the margin; and as the result is the same as the dividend, we may be sure that the quotient is right. 472 944 21 Rem. 48165 III. When the Divisor is composed of Factors, none of which exceeds 12. The operation just explained is called long division, to distinguish it from the shorter process of the preceding case, where the divisor is not greater than 12. If any divisor be found to be the product of factors, each of which does not exceed this limit, the division may be performed by successive applications of the shorter rule: for you may divide first by one factor, then the quotient by another factor, the new quotient by a third factor; and so on, till all the factors have been used. : 7) 38214 9) 5459...1 It is possible that the division by the first factor may leave a remainder; if so, it must, of course, be preserved: the division by the second factor may also leave a remainder; if so, you must multiply it by the first divisor, and add in the former remainder; the result will be the complete remainder as far as the operation has been carried if there be a third division, and a third remainder, you must multiply it by both the first and second divisors, adding in the former complete remainder and so on, till all the divisions are completed. For example, suppose we have to divide 38214 by 63: then since 63=7×9, we may operate as in the margin the final quotient being 60639. In like manner, if we have to divide 24611 by 126; then, since 126=3×6×7, the operation, by short division, is that here annexed: the remainder from the first division is 2; that from the second is 1; and this 1, multiplied by the first divisor 3, and the former remainder being taken in, gives 5 for the complete second remainder: the remainder from the third division is 2, which multiplied by 3, and by 6, both the former divisors, that is by 18, gives 36; which, with the preceding remainder 5, makes 41;-the final complete remainder : hence the complete quotient is 1954 606...36rem. 3) 24611 6) 8203...2 7) 1367...5 195...41rem. The method here described of obtaining the final remainder, and thence completing the final quotient, cannot be clearly explained till some knowledge of fractions is acquired: parts of a whole, as one-half, one-third, two-fifths, &c., and which are denoted by,,, &c., are called fractions: when you are a little acquainted with the management of these, you will plainly see the reason of the foregoing directions. Upon the principles now delivered depend all the operations of arithmetic. In what has preceded, they have been applied only to what are called abstract numbers, without any reference to particular objects or articles. It remains to show the application of the same principles to concrete quantities; that is, to real commodities, or things—as to money, weights, measures, &c. And, in order to this, a few Tables connected with these matters must first be given. 18 TABLES OF MONEY, TIME, WEIGHTS, AND MEASURES. I. TABLES OF MONEY, TIME, WEIGHTS, AND MEASURES. MONEY. A farthing, that is, one-fourth of a penny,-is represented thus, d.; a halfpenny, thus, d.; and three farthings, thus, d. To express a fraction of a farthing, the letter fis put against the fraction: thus, f. means half a farthing; f, three-fifths of a farthing, &c. 60 seconds 60 minutes 24 hours II.-TIME. 7 days 1 week. 52 weeks 1 day, or 4 quarters (gr.), or 112lb. 20 hundredweight (cwt.) 1 hundredwt. 366 days 1 leap year. The mark dwt. stands for pennyweights, and gr. for grains: see the table of Troy weight. The learner can scarcely require to be informed that £ stands for pounds, s. for shillings, and d. for pence. The coin guinea has been long abolished, but the name is still retained for 21s. The name pound is given to 20s., because the quantity of silver in this sum originally weighed a pound troy. |