ciples. 1. "Any number that measures each of two others, must also measure their sum and difference. 2. Any number that measures another, must also measure its double, treble, or any other multiplier of it." Let it be required to find the greatest common measure of 325 and 455. Here, since the number we are in search of must measure each of the numbers 325 and 455, it must, according to principle, 1st, measure their difference, 130; now, by principle, 2d, it must measure twice 130, or 260, which is the nearest multiple of 130 to 325. Again, since it measures both 325 and 260, it must measure their difference, 65. The greatest common measure then of the two given numbers must measure both 65 and 130. Now, 65 itself does this; hence 65 is the greatest common measure sought. It is obvious that it can be no number greater than 65, for no greater number would fulfil the requisite condition of dividing 65 without a remainder. From the preceding investigation, it appears that the process of finding the greatest common measure, consists in the successive decomposition of the two proposed numbers, and may be exhibited as follows: 325)455(1 130)325(2 65)130(2 Hence the general rule may be expressed thus: Divide the greater by the less, then divide the preceding divisor by the remainder, and so on continually until 0 remains; the last divisor is the greatest common measure sought. Hence the greatest common measure of three or more numbers may be found. For we have merely to find the greatest common measure of the first two numbers, then the greatest common measure of their common measure and the third number, and so on to the last of them. If, by the rule as now enunciated, we find the greatest common measure of the numerator and denominator of a fraction, then the fraction will be reduced to its lowest terms at once, by dividing both terms by it. Thus, if the terms of the fraction be both divided by 65, their greatest common measure, then this same fractional value will be expressed, in its 325 lowest terms, by . When, by the preceding process, we discover that the terms of a fraction have no common measure greater than unity, then the fraction is already in its lowest terms. 45. A whole number may be represented as a fraction having any given denominator. Ex. Reduce 7 to a fraction whose denominator shall be 5. Here 7 are equal to 7 units, or; this value will not be altered (No. 42,) if both numerator and denominator are multiplied by the same number; let them both be multiplied by 5, and we shall obtain 7 X 5 35 or for the fraction sought. 5 From this it is obvious that, to reduce a whole number to a fraction having a given denominator, we have merely to multiply the given number by the given denominator, and under their product write the denominator. 46. Hence it follows that a mixed number may be reduced to an improper fraction, and conversely. 72 Thus, 8 = and = and that is to 8 X 9 9 79 A little attention to the preceding steps renders the formal statement of any rule quite unnecessary. 47. A compound fraction may be reduced to a simple one, by multiplying all the numerators together for the numerator of the simple fraction, and all the denominators together for its denominator. The reason of this will appear from an example. Let it be required to reduce of to a simple fraction. The meaning of this expression is, that the third part of is to be taken twice. Now, the third part of is equal to 8 15 twice is equal to×2=5, (No. 41,) a simple fraction, whose 15 15 numerator is the product of the numerators, and denominator that of the denominators. If the compound fraction consist of three or more fractions, the two first may be reduced to a simple one, and then this and the third to another, and so on to the last. An expression of this kind may often be simplified by expunging those numbers which are common to the numerator and denominator. Thus, in the preceding example, the number 5 may be cancelled, since it appears both in the numerator and denominator, for this is nothing else than the division of both terms by the same number. 17 8 48. Having given a fraction expressed in one denomination, it is sometimes convenient to express it in another. Thus, for example, let it be required to reduce of a guinea to the fraction of L. 1. We operate on this fraction exactly in the same way as we would on a whole number, to reduce it from guineas to pounds; that is, we multiply it by 21 to reduce it to shillings, and then divide by 20 to reduce the shillings to pounds, observing that the general rule for multiplying a fraction is to multiply its numerator; and for dividing it, to multiply its denominator. (Nos. 41 and 42.) Hence the operation will be as follows, of a guinea 5 X 21 105 21 3 of a pound. 7x20-140-28-4 Ex. 2. Reduce of half a guinea to the fraction of a crown. Here the fraction must evidently be multiplied by 21, and divided by 10, because 21 sixpences makes half a guinea, and ten make a crown. 49. A quantity may be reduced to a fraction of any given denomination belonging to it, by reducing the given quantity to the lowest name it contains for the numerator, and the proposed integer to the same name for the denominator, and then reducing the fraction to its lowest terms. Ex. 1. Red. 21d. to the fraction of a shilling, 24—5 half pence; and 1 sh. 24 half-pence. Now, it is obvious, that 1 half-penny-of a shilling; and hence, that 5 half-pence, or 24d.sh. Ex. 2. Red. 17s. 6d. to the fraction of a £. 17s. 6d.=35 sixpences; and £1=40 sixpences; hence I sixpence is of a ; and, of course, 35 sixpences are or of a £. Ex. 3. Red. 3s. 41d. to the fraction of a guinea, 3s. 4d. =81 half-pence; and 1 guinea=504 half-pence; hence, 50. The converse of this problem is also easily managed, viz. given a fraction to find its value. Thus, let it be required to find the value of of a guinea. of a guinea is the same as the eight part of 5 guineas, (No. 35.) Now, 5 guineas are equal to 105 shillings; and hence, the eight part of 5 guineas is= = 13s. 1 d. 105 51. In order that fractions may be compared or added together, it is necessary that they be of the same kind, that is, that they have the same denominator. Fractions of different denominators may be reduced to equivalent ones, having the same denominator, by multiplying each numerator into all the denominators except its own, for the numerators of the equivalent fractions, and all the denominators together for their common denominator. The principle of this rule is, that if the numerator and denominator be both multiplied by the same number, the value of the fraction is not altered. That both terms are multiplied by the same number, by applying the preceding rule, will readily appear from an example. Let the fractions, and be reduced to equivalent fractions, having the same denominator. Fractions admit of being reduced to a common denominator, which is less than the product of their denominators, when the least common multiple of the denominators is less than their product. Thus, in the example now given, it is obvious that the least common multiple of the denominators is 24, that is, 24 is the least number that will contain them all without a remainder. Now, 24 contains 4, 6 times; 6, 4 times, and 8, 3 times. Hence, if the terms of the fraction be multiplied 6, those of by 4, and those of by 3, we obtain 18, and for the given fractions reduced to their least common denominator. This method of reducing fractions to their least common denominator is often exceedingly convenient; the student would therefore do well to make himself thoroughly acquainted with it. ADDITION OF FRACTIONS. 52. If the fractions to be added have the same denominator, their addition will be effected by writing the sum of their numerators above this denominator. vious to every one, since 2, 3, and 5 quantities of the same kind, are equal to 10 of that kind. If the fractions have different denominators, they must be reduced to others having the same denominator before they can be added. Ex. 5. Required the sum of £, of a guinea, and 3 of a shilling. The preceding example might be solved, by reducing the |