MULTIPLICATION OF FRACTIONS. In order to multiply one fraction by another, multiply the two numerators together, and likewise the two denominators. EXAMPLE. Suppose the fraction is to be multiplied by which is thus written. I multiply 3 by 2, and 4 by 5; and the product is. To prove it, let us first take the simplest case, and suppose we have not, but simply 3 to multiply by 3, or 3 x; without the least notion of the process to be followed, any one will be apt to multiply first 3 by 2; but I observe this product to be too great; for it was not by 2 I had to multiply, but by 2 divided by 5, or by the 5th part of two units; this product of 3 by 2, is then 5 times too great; to reduce it to its proper value I must then divide it by 5, and 3X2 write 5 We shall here observe, that to multiply a whole number by a fraction, we must multiply that number by the numerator of the fraction, and divide the product by the denominator of said fraction. Let us now return to the question proposed; I say now, that it was not 3 I had to multiply by a fraction, but 3 divided by 4, or the 4th part of 3; therefore, in 3 x 2 the expression 5 the numerator is 4 times too great; I must then, in order to bring down this fraction to its true value, make its denominator 4 times greater, and then write 3X2 ; we must then 5X4 multiply numerator by numerator, and denominator by denominator. DIVISION OF FRACTIONS. The process in this operation, is quite inverse of the former, and is accounted for by the same explications. . EXAMPLE. Be this fraction to be divided by ; write itd. Here you have to multiply the numerator of the dividend, by the denominator of the divisor, for the numerator of the new fraction, and reciprocally for the denominator of the said new fraction; the quotient will then be 15 15. In order to prove it, let us take the simplest case, and suppose we have not, but only 3 to divide by, or 3 d. ; what naturally occurs, is, to divide 3 by 2, which gives : but I observe this fraction is too small, for its denominator is too great, since it was not by 2 I had to divide, but by 2 divided by 5, or by the 5th part of 2. I must then, to re-establish the value of this fraction, make it 5 times bigger by multiplying its numerator by 5. I will 3+5 then have 2 ; then the division of a whole number by a fraction, consists in dividing that number by the numerator of the fraction, and multiplying this. result by its denominator. But now again. I say; It was not (by the state of the question) 3 I had to divide, but 3 already divided by 4, or the 4th part of 3; therefore, in the expression. 3 X 5 2 the numerator is 4 times too great; I must then, to make it 4 times less, make the denominator 4 times larger, and writę 3 X 5 4X2 15 --which was to be demonstrated.. FRACTIONS OF FRACTIONS. Be it proposed to take the of; this is what is called fraction of a fraction, as likewise to take the of the of; or of a still greater number of fractions. These questions, at first sight, seem somewhat puzzling, yet in itself, nothing is more simple than the process to be followed in their solution. Multiply all the numerators together, and likewise the denominators; the result of the first question will then be and of the second. In order to prove it, let us stick to the 1st example. It is asked for the of; that is; we must take the of a quantity expressed by, or divide this last quantity into three equal parts, and take two of them. Now, to divide by 3, is to render this fraction 3 times smaller, and consequently multiply its denominator by 3, which gives 4 - 5×3 ; but we must take this fraction twice, the exact result will then be, 4x2 ; therefore the two 5X3 numerators must be multiplied by each other, and so also the denominators. For the 2d case, or any greater number of fractions, the demonstration would be the same; for first, in order to find out the of, we must multiply by, which gives and reduces the question to the taking the of, case similar to the former, the result of which is 4%, as we had advanced it. 2 REDUCTION OF FRACTIONS. It frequently happens that both the numerator and denominator of a fraction are large numbers, as it is the case in most divisions, where the quotient commonly contains a fraction; some of these fractions can be reduced to a very simple expression; there are two methods to accomplish it; the way of trying, and that of the greatest common divisor. Here is the process to be followed in the first case, which will be easily understood by means of an example. 1260 630 315 105 35 5 Whenever the numerator and denominator are both even numbers, they can both be divided by 2, or the half of them be taken, as we have done here, to effect the first reduction; when 2 can no longer be a common divisor, we have recourse to 3, and divide by 3 as often as possible; after 3 we try 5; It would be useless to try 4, because 2, which is an under multiple of 4, being no more contained in it, 4 cannot be; 5 being exhausted, it is not necessary to try 6, as it is a multiple of 3, we pass to 7, then 11, 13, &c. taking only the primary numbers, which can be divided but by itself or by 1. It would be tedious to carry your trials above 11, as the primary numbers above it, not being contained in the multiplication table, this reduction would require a complicated division. NOTE. To facilitate this kind of reduction, it is important to know, that a number is exactly divisible by 3, whenever the sum of its figures, added up together as simple units, is an exact multiple of 3; and that a number can be divided by 5, only when it is terminated by 0 or by 5. GREATEST COMMON DIVISOR. The above method, though only a trying, is preferable to the general rule, whenever the two numbers are not very large; if they be, take the following easy method. If Divide the denominator of the fraction by its numerator; if the quotient leaves no remainder, the numerator itself will be the greatest common divisor. there be a remainder, divide by this the numerator; if the quotient be exact, then this remainder will be the common divisor; should there be a new remainder, the former remainder must be divided by the second, and so on, 'till your division leaves no remainder; then the last divisor is the greatest common divisor, which will be used to divide by it both terms of the fraction, and the two quotients will be the two terms of the fraction reduced to its lowest expression. EXAMPLE. Reduce the fraction . I have divided the denominator by the numerator, then the numerator by the first remainder 884, then this by the 2d remainder 663, the 2d by the 3d 221, which is found to be the greatest common divisor, for the division leaves no more remainder. Now, dividing the numerator of the fraction by 221, we find 11 for exact quotient, and dividing the denominator by the same, we find 15, then is the value of the fraction reduced to its lowest terms. NOTE. In the above operation, we have written |