18 x 24 x 16 x 15 x 14 x 9 = 130603680, which is 2592 times greater than the least common multiple, and would greatly increase the labour and chance of error in any calculation into which it enters. One of the neatest applications of the least common multiple is the reducing of fractions to a common denominator, in order to add them; and the least common multiple is, of course, the least common denominator. Calling the numerator n, the denominator d, and the least where we have a multiplication and a division to perform, and know that we can divide one of the factors without remainder, we abridge our labour very considerably by performing the division first, and thus converting the quotient into a multiplier. Now as the least common multiple is necessarily divisible by all the denominators, we can divide it and obtain a multiplier for each numerator. The formula will thus become n x m d the division of m by d being performed, and the quotient used as a multiplier. Let us illustrate this by an example: Uncle Nathan, who by great skill in calculating the arbitration of exchanges, and various little other arts which are well known on the Stock Exchange in the city of London, at which his mind had been so constantly, so silently, and so cautiously at work for half a century, that he got the name of "the calçulating clock with the dead beat 'scapement," was in the fulness of time, and the abundance of his accumulations, gathered to his fathers. He left a goodly fortune; but as part of it was in the hands of half the kings of the world, he could not tell its amount, and therefore could not bequeath it to his six nephews in specific sums; therefore he devised it fractionally as follows: Grippy, and 50 to Gad, who, being something of a wandering blade, stood lowest in the favour of Uncle Nathan. Furthermore, he willed that Goosewing, who had been his trusty and well-beloved scout and scribe for many years, should arrange the whole matter, transfer to each of the six nephews his legacy, and take 2500l. for his trouble. It is required to find the amount of Uncle Nathan's savings, and the portion which came to each of the nephews. It is clear that the fortune, whatever it is, is 1, and may be represented by any fraction of which the numerator and denominator are exactly equal; and it is also clear that the fortune may likewise be represented by all the fractions which the nephews are to receive, together with Goosewing's 25007.; therefore, 13 6 11 5 7 3 + + + + + +25001. 1. 50 25 64 32 64 50 We shall, in the mean time, call the 25007. a; and our first business will be to colleet all the fractions into one sum, which +a will give us the fortune, or at least the proportion which a bears to the other shares, and then from that we can easily get the shares themselves. In order to do this let us first find the least common multiple of the denominators, 50, 25, 64, 32, 64, 50. Here we see at once that 50 × 64 can be divided by all these numbers; therefore, 64 × 50=3200=least common multiple. This least common multiple is the common denominator of the fractions; and to find the numerator of each in terms of this denominator we have merely to divide it by each denominator, and multiply the respective numerator by the quotient. For this we have We have next to multiply each one's share by his multiplier, and we get the proportional shares in terms of the denominator Thus the shares of the six nephews amount altogether to L The value of a is evidently the difference between the numerator and denominator of this fraction. If the numerator were the greater it would be and if the terms were equal it would be 0; but the numerator is less than the denominator, therefore Goosewing's fee is 8 3200 ; or, disregarding the denominator, as it is the same in all, his share is 8, and the money value of it, according to the will, is 25007. If we divide 2500 by 8 we get 3127. 10s. as the value of every unit in the numbers; and this, multiplied by the proportional numbers, will give the shares, the sum of which, together with the scribe's fee, will be the whole fortune. 3127. 10s. is, changing the 10s. to a decimal, 312·5; wherefore, We have given this example chiefly on account of its simplicity, and the consequent ease with which a reader not much conversant with figures can understand it; and having done so, we shall proceed, in the next section, very shortly to examine those principles in decimals which are most generally useful. 147 SECTION VIII. SOME PROPERTIES OF DECIMALS. IN a former part of this work we pointed out the general nature of decimals: such as, that they are a continuation of the scale of numbers below the last place of integers, or the place of units, and that the several places in a decimal number bear exactly the same relation to each other as the places in an integer number—that is to say, that 1 in any place is equal to 10 in the place immediately to the right of it, to 100 two places to the right, and so on; that decimal numbers are added, subtracted, multiplied, and divided, exactly in the same manner as integer numbers; that the removing of the decimal point any number of places to the right is equivalent to the multiplying of the number as often successively by 10 as the number of places which the point is so removed; and that the removing of the decimal point in any number of places toward the left is equivalent to the dividing of the number as often by 10 as the number of places that the point is removed. It will also appear evident, from what has been said respecting fractions, that the denominator of a decimal number must consist of 1 with as many Os annexed as there are figures in the decimal; and as the number of Os is always equal to the number of times that 10 is a factor, if the decimal consists of n figures, the expression for the denominator will be 10". The ready use of the decimals enables us, however, so much to simplify all the common applications of arithmetic to the business of life, and is so indispensable whenever the relations of magnitude enter into those calculations, that it is necessary for every one to understand their nature more thoroughly than it can be understood, without a knowledge not only of arith |