120 by the quotient, because it is only by such multiplication that the original numerator and denominator come to be both multiplied by the same number. Before giving you a general rule for finding the L. C. M., it is, perhaps, better that I should show you, by an example, what appears to me to be the most convenient and obvious method of proceeding. Suppose the numbers 24, 10, 9, 32, 6, 45, and 25, were proposed, we should have to discover the lowest number that would give a quotient free from fractions, when divided by any one of these seven numbers. Let us first divide the 24 by the 10, we shall have 1% = 2*; and as the fraction is in its lowest terms, with 5 for denominator, it is plain that we must take the dividend 5 times, at least, in order to render the quotient free from a fraction ; so that 24 x 5, or 120, is the least number which is exactly divisible by both 24 and 10. Again ; i 13}, so that we must multiply the 120 by 3, at least, to make the quotient by 9 a whole number: hence, 360 is the least number exactly divisible by 24, 10, and 9. Again ; $60 = 11% so that 360 x 4 = 1440, is the least number divisible by 24, 10, 9, and 32: we need not attend to the 6, because whatever is divisible by 24 is also divisible by 6:* taking, therefore, the next number, we have 1440 = 32, so that 1440 is the least number divisible by 24, 10, 9, 32, 6, and 45. Lastly, 15° = 53%; consequently, 1440 x 5 = 7200, is the L. c. M. of all the numbers. From this example you will be prepared for the rule I propose to give: it is as follows : RULE. Take any two of the numbers for dividend and divisor. If the quotient have a fraction, reduce it to its lowest terms, and multiply the dividend by its denominator. Take the product for a new dividend, and another of the numbers for divisor: if the quotient have a fraction, reduce it to its lowest terms, and, as before, multiply the dividend by the denominator: take the product for a new dividend, and another of the numbers for a divisor; and so on, till all the proposed numbers have been used, omitting those which are obviously contained in any of the others, or in a dividend already found : the last product will be the L. C. M. * As 45 is one of the given numbers, we might, for a like reason, have neglected the 9; since whatever is divisible by 45 must be divisible by 9 : or we might bave neglected the 45, since wbatever is divisible by 10, and 9, must obviously be divisible by 45. NOTE 1. If any quotient occur without a fraction, the same dividend is to be used with the next divisor. 2. When in going over the row of numbers you come to one which is a prime number (page 20), you should see, as you may readily do, whether this prime number be contained exactly in any of the preceding : if it be, you are to omit it, as the rule directs; but if it be not, then you will know that the quotient, arising from dividing your last dividend by it, must have a fraction with that prime number for denominator: you need not, therefore, be at the trouble of performing the division ; you will merely have to multiply the dividend by the prime number, and then to pass on to the next number; but a still better way will be, to reserve these prime numbers, after having selected them from the entire row, and to apply the rule only to the composite numbers into which the primes do not enter; and when you have got the L. C. M. of the composite numbers, to multiply it by the primes, one after another; of course, you are to take no account at all of such primes as are contained in any of the composite numbers. Exercises. Find the least common multiple of 1. 8, 12, 18, 20 4. 24, 16, 20, 30, 25 2. 3, 9, 27, 81 5. 27, 24, 15, 126 3. 2, 3, 4, 5, 6, 7, 8, 9 6. 242, 748, 21, 427. It may be as well to remind you here, that as 6 and 8, in Ex. 3, contain 2, 3, and 4, these latter numbers may be omitted ; and as 5 and 7 are primes, not contained in any of the other numbers, they may be reserved (NOTE 2); so that you need apply the rule only to 6, 8, 9, of which the L. C. M. is 72: therefore, 72 x 5x7 2520, the L. C. M. of the proposed numbers. The following example is also one in which you may, with advantage, avail yourself of NOTE 2. 7. Find the L. C. M. of 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21. Here there are four prime numbers, which you may reserve till you have applied the rule to the others. From what has now been said about the L. C. M. of a set of numbers, you can never be at a loss how to reduce a row of fractions to others equal to them, and having the least possible common denominator: as observed at the commencement of Art. 65, when you have found the L. C. M. of the deno minators, by aid of the above directions, you will only have to divide it by the denominator of the first fraction, to get the number by which the numerator must be multiplied, in order that you may have the proper numerator of the corresponding changed fraction ; and so of all the following fractions. In other books on Arithmetic, you will see rules for finding the L. C. M. different from that above: some of these may look shorter, but I think the one here given will be found as expeditious as any, while it is much more easy of proof, and of being borne in the memory. PRACTICE. (66.) PRACTICE is the name given to a set of operations in commercial arithmetic, by which the prices of commodities may often be calculated in an expeditious manner. From the examples of these operations which I shall here give, you will easily see what the peculiar method called Practice really is; and you will, at the same time, observe, that though it does not enable you to do anything you cannot do already by means of the rules for compound quantities before taught, yet, that in many cases, both time and figures are saved by replacing these rules by PRACTICE; and such a saving is matter of consideration in actual business. When you are familiar with the method, it will be for you to determine, in any example you may have to work, whether the former methods, or the method of Practice, or a combination of these two, be the most likely to save time and trouble. Ex. 1. What is the value of 48 tons, at £6 10s. a ton ? 48 Here, instead of reducing the money to 6 shillings, and then multiplying by 48, we should regard the £6 10s. as 6, working 288 the example by multiplying the 48 by 6, 10s. 24 and adding the half of 48 to the product: the sum is, of course, the whole product of £312 48 by 62, and the result comes out at once in pounds, without any reduction, as in the margin. The operation so conducted and so arranged is an operation of PRACTICE. 2. Suppose the price per ton had been 48 £6 158. : then, observing that 10s. is the 6 half of a pound, and 58. the half of 10s., the operation would have been as here annexed. 288 Here you see, that 48 times the £6 is 10s. 24 £288; 48 times the £, or 10s., is £24; 58. 12 and as 48 times 58. must be the half of this, we get the additional £12: so that the £324 whole sum is £324. You thus see, that by working examples of this kind by Practice, you avoid the trouble of reducing the compound quantity concerned to the lowest denomination, which appears in it, and then of reducing the result back again. In Practice the highest denomination is preserved, and the lower denominations considered as ALIQUOT Parts, that is, exact measures of the higher; so that the chief thing you have to do, is to consider how these lower denominations may be most conveniently cut up into aliquot parts of the higher : thus, in the last example, the 158. was cut up into 108., the half of the highest unit, a £, and into 58., the half of the 108., or the fourth of the £. 3. Suppose the price had been £6 178. 6d. per ton : then the aliquot parts would have been į of a 48 £ for 10s., į of 108. for 58., and į of 58. 6 for the remaining 28. 6d. : the work would therefore have stood as in the margin. 288 The fractions are put against the several 108. 24 sums into which the shillings and pence 12 are cut up, to show what aliquot parts 2s. 6d. 6 they are; but you must avoid the error of saying you divide by }, \, &c., for £330 you divide by 2, 4, &c., in order to get the half, the fourth part, &c. It is to prevent your falling into this error, that 48 each fraction is placed, not beside the 6 dividend, but beside the quotient. 4. Suppose, lastly, the price were 288 £6 78. 102d. Here you may cut up 58. 12 the shillings and pence as follows: 58. 28. 6d. 6 is £; 28. 6d. is į of 58.; 3d. is zo 3d. . 0 12 of 2s. 6d., or 30d. ; and, lastly, 1 d. 1 d. 0 6 is of 3d. Hence the work is as in the margin. £306 18 5s. 5. If the price had been £6 88. 10 d., the work might have been conducted as here annexed, from which you will see that 48 there is sometimes need for a little 6 reflection as to the most convenient set of aliquot parts. 288 When you come to the bottom of 68. 8d. 16 this page you will see how, by a little 1s. 4d. 3 4 contrivance, the work here given in 8d. 1 12 the margin may be abridged. I give 2d. 8 you the operation in this form at first, id. 3 2 as also that of the next example, in order that you may become acquainted £309 6 with the management of aliquot parts. 6. Find the value of 76 at 138. 11 d. each. In this example, the highest deno 76 mination being shillings, the first 13 aliquot part taken is that of a shilling; for the portion of the price 228 equal to one shilling, the 76 articles 76 would be 768.; therefore, for half that portion, that is, for the 6d., the 988 sum is 38s. The total amount of all 6d. 38 the component portions of the entire 3d. . 19 value is 10628. 5d. : therefore, di- 2d. of 6d. 12 8 viding the shillings by 20, we have d.of 3d. 4 9 £53 28. 5d.* Sometimes you may abridge 1062 5 the work by increasing the price of the single article, and then 48 at £6 10s. correcting the result, by sub 6 tracting what is due to the increase : for instance, when the 288 price is 178. 6d., preceded by 108. 1 24 pounds or not, you may increase it by 2s. 6d., and then allow for £312 the overplus, by subtracting the * As noticed above, the operation in the margin merely shows what the steps suggested by the rule of Practice really are; but the more completely the principles of arithmetic are taught, the more independent of rules does the learner become. He who knows these principles well, need never ransack his memory for rules. The value of the 76 articles above, at 148. each is 10648. ; this diminished by 76 farthings, or 19 pence, that is 18.7d., leaves 10628. 5d., as above. |