smaller term, and, as was shewn before, without altering its value. This is done by finding a figure or number that will divide both the numerator and denominator evenly ; that is, a number that will divide them without leaving a remainder: and for this purpose we have the following rule. Divide the numerator into the divisor, and if there should be a remainder you divide it into your last divisor, and so continue until nothing remain, and your last divisor will divide both the denominator and numerator of your fraction. 476, In the above exam- 1144) 2764 2 ple we divide the nu 2288 merator 1144 into the 476) 1144 2 denominator 2764, 952 and find there is a remainder of 192) 476 2 which remainder be 384 comes our next divi 92) 292 2 sor, and 1144, our 184 former divisor, becomes our new di 8)92 1 vidend; and these 88 numbers, when divid 4) 8 2 ed into one another, 8 leave a remainder of 192, which likewise we divide into our last divisor 476, and find our remainder is 92 ; this remainder divided into the former divisor 192, leaves a remainder of 8 ; this 8, when divided into 92, leaves a remainder of 4, and our remainder being contained evenly in our last divisor, is our greatest common measure, or the highest figure that will divide both terms of our fraction evenly, thus : 2 86 691 You will now recollect that it is the difference between the numerator and denominator that determines the value of the fraction ; therefore, to prove that the above division has not altered the value of the fraction, we will subtract the numerators from the denominators, and you will find the sums of their difference equal, that is to say, two hundred and eighty-six bears the same proportion to six 2764 691 1144 286 hundred and ninety-one that one thousand one hundred and forty-four 1620 405 bears to two thousand seven hundred and sixty-four; and also that if you multiply the remainder of the smaller fraction by the common measure, four, it will produce the remainder of the larger fraction. By studying the example already given, and working it on your slate as you read it over, you will be able to find a common measure for any fraction that can be reduced, that is, that can be expressed in shorter terms without altering their value. Reduce the following numbers to the lowest terms : (1.) 4% (2.) 48 (3.) (4.) 764 (5.) 276.48. (6.) 14.8 Any number may be written fractionally by taking one as the denominator, thus, i; but such fractions having their numerator greater than their denominator, are called improper fractions. These improper fractions often occur in the working of fractions. When there are mixed numbers it is ne cessary to reduce them to improper fractions, and this is performed by taking the denominator of your fraction for a multiplier and your whole number for a multiplicand, and adding the numerator; under the product write the figure you multiplied by for your denominator. Thus reduce the mixed number 698 to an improper fraction. The denominator of the fraction being 4, I take it for the multiplier, and beginning at 693 4 the units place in the whole number, I say 4 times 9 are 36, and 3, the numerator, 279 make 39; 4 times 6 are 24 and 3 make 27 ; thus I have 279 for a new numerator, and 4 under this I write 4, the multiplier, for a denominator. Reduce the following mixed numbers to their equivalent improper fractions. Ex. (1.) 174 (3.) 164 (5.) 7943 (2.) 1970 (4.) 2041 (6.) 816,7 An improper fraction can be changed into a whole or mixed number by dividing the denominator into the numerator, and under the remainder write the divisor for a denominator. Taking the example given for reducing a mixed number to an improper fraction, the mixed number was 693, and we found the improper fraction to be 279; now taking the denominator 4 as our divisor, and the numerator 279 as our 4) 279 dividend, we say 4 into 27, six times 4 are 24 from 27 and 3; and 4 into 39, nine 693 times 4 are thirty-six and 2 over, which three I take for a numerator, and write the four under it for a denominator, and the 69% is reproduced, proving that the work is correct, and that the rule we employ for doing it is also correct. Reduce the following improper fractions into mixed numbers. Ex. (1.) (2.) 1862. (3.) 116 (4.) 3275. (5.) 3179. (6.) 9192 These being the improper fractions produced by the reduction of the mixed numbers given in the six sums set down for practice, the answers to these two modes of reduction are not given. A compound fraction can be reduced to a simple one by multiplying the whole of the numerators for a new numerator, and the whole of the denominators for a new denominator. Thus, reduce i of of to a simple fraction. Beginning from the right we say, 5 times 2 are 10, and 3 关圣号 = times 10 are 30, which we write down for a numerator. We then commence multiplying the denominators, saying, 7 times 7 are 49, and 6 times 49 are which we write under the numerator 30, and the work is complete. Work for practice the following sums : (1.) Reduce the compound fraction of of} to its equivalent simple one. Ans. 15 (2.) Reduce # of of Á to a simple fraction. Ans. 105 352 (3.) Reduce 1 of 1 of to a simple fraction. Ans. 858 (4.) Reduce of % of 14 to a simple fraction. Ans. 378 30 294 2640 (5.) Reduce of 1 of 1% to a simple fraction. Ans. 273. (6.) Reduce of % of 5 to a simple fraction. Ans. 2. * To reduce fractions of different denominators to the same denominator, or finding a common denominator. We know that to multiply the numerator and denominator of a fraction by the same number, does not alter the value of the fraction; and knowing this, our work is simple. We have but to multiply the whole of the denominators together for a new denominator; and to preserve the value of the fractions unaltered, we multiply the numerators of each fraction by the denominators of the others, for new numerators. To shew that the value remains unaltered, let us take two fractions, say į and ž. In the first fraction, ], the denominator twice as much as the numerator, while the denominator of the second is three times as many as its numerator. Now if we multiply these two fractions in the manner before laid down, and find the proportion between their numerators and denominators remains unaltered, it will convince us that the mode of proceeding is correct. By multiplying the denominators i, j; , *. two and three, we find the common denominator six. Now we multiply the numerator of the } by the denominator of the }, and the product is three ; we then multiply the numerator of the į by the denominator of the į, the product is two, or two-sixths. Thus we find the denominator of the first fraction, or), is just twice as much as the numerator, and that the denominator of the second is three times as much ; which was just the difference between the denominators and numerators of these fractions before we began to reduce them. * The mixed number is reduced to an improper fraction, and then the numerators and denominators are multiplied as before. D |