ing 5 under it, thus, . The whole quotient of 13 divided by 5, is 23. From these remarks it appears, that in the fraetion, the upper number is to be divided, or is the dividend, being an undivided part of the dividend 13. The lower number is the divisor. The upper number, 3, is also called the numerator, and the lower number the denominator. The propriety of these terms will appear when we take another view of a fractional expression. Let an apple be divided into five equal parts, one part would be expressed thus, ; two parts thus, ; three parts thus, ; four parts thus, . From these expressions it is shown, that the figure representing the number of parts of the unit, taken in any example, is always made the numerator in a fractional expression. Thus three parts of the apple are represented in this manner, . This expression shows that three parts of the apple are taken. The numerator, then, numbers the parts of the unit in any fraction; and the meaning of the word is, one that numbers, or counts. From the same expressions appears that the denominator shows into how many parts the unit is divided. When an apple is divided into five equal parts, the numeral 5 is made the denominator. And universally, the numeral expressing the number of parts into which the unit is divided, is made the denominator. This numeral names the fraction; and the meaning of denominator is one that gives a name, or namer. it In reading fractional expressions, the denominator is considered an ordinal, and not a numeral. Thus, is read, one fifth, and not, one five; is read, two fifths, and not, two fives. And universally, in reading vulgar fractions, the ordinal of the denominating numeral is employed. Thus, is read, one third; is read, one fourth; is read, one sixth; is read, one seventh. The learner should now attend carefully to the following REMARKS. 1. Fractions are parts of a unit. 2. When the numerator of a fraction is less than the denominator, thus,, the fraction is called a proper fraction; when the numerator is not less than the denominator, thus, , or, the expression is called an improper fraction. 3. When a whole number and a fraction are written, thus, 24, the expression is called a mixed number. 4. A simple fraction has one numerator and one denominator in the same expression; thus, 2, . 5. A compound fraction is a fraction of a fraction, and may be known by the word of placed between the two simple fractions; thus, of. I. TO REDUCE A FRACTION TO ITS LOWEST TERMS. Divide the numerator and denominator by any number that will divide them both without a remainder, and that quotient by any number that will divide it without a remainder, and so on, till no number, exceeding unity, will divide the fraction without a remainder. NOTE.-The value of a fraction is not altered, unless the ratio of the numerator to the denominator is changed. But in the above example, the ratio which the numerator has to the denominator, is not changed; therefore the value of the fraction is not altered, The numerator, 15, is the same multiple of 3, that the denominator, 175, is of 35. Therefore, as have the same ratio as, the values of the two expressions are the same. The process of finding the lowest terms of a fraction may be varied by finding the greatest common divisor of that fraction. 1. Reduce to its lowest terms. OPERATION. 96 544|5 480 64|96|1 By this process, it is found that 32 are the greatest common divisor of 96 and 544. Therefore the fraction, reduced to its lowest terms, is 32, Ans. 96 544 32 6412 NOTE. By inspecting the above process, it will be seen that 32 are the measure of 64, the first remainder, and 96. Therefore they will be the measure of 96×5+64. As 32 are the measure of 96, they will be the measure of 96, multiplied by any number, and that product increased by 32 taken any number of times. Thus, 32 are the measure of 96×2+32. And universally, any number that is a measure of the remainder and divisor will be a measure of the dividend; and also of the divisor multiplied by any number, and that product increased by the remainder taken any number of times. 2. Reduce to its lowest terms. 3. Reduce to its lowest terms. 4. Reduce to its lowest terms. 268 1896 Ans. 4. Ans. Ans. §. II. TO REDUCE A MIXED NUMBER TO AN IMPROPER FRACTION. Multiply the whole number by the denominator, and to the product add the numerator; the sum will be the numerator of the improper fraction, and the denominator will be the same as before. 1. Reduce 131⁄2 to an improper fraction. OPERATION. 13g 6 83=93, Ans. 2. Reduce 197 to an improper fraction. 3. Reduce 62 to an improper fraction. III. Ans. 182. TO REDUCE AN IMPROPER FRACTION TO A WHOLE, OR MIXED NUMBER. Divide the numerator by the denominator, and the quotient will be the whole number. If there be a remainder, write it over the divisor, at the right hand of the quotient. 1. Reduce 189 to a whole, or mixed number. OPERATION. 4|189 47, Ans. In this example, the unit is divided into fourths. Every fourths make a unit. Therefore, as many times as 4 are contained in 189, so many units there will be. I find that 4 are contained in 189, 47 times, and there is I remainder. There are, then, in 189, 47 units and † of a unit. 2. Reduce 324 to a mixed number. 3. Reduce 1148 to a mixed number. 4. Reduce to a mixed number. 100 Ans. 1247 Ans. 1814 Ans. 99 5. Reduce 999 to a mixed number. 6. Reduce 100g to a whole number. Ans. 1000000. This process consists simply in ascertaining the number of sixths in 13 and . In one unit there are six sixths, and in 13 units there are thirteen times as many sixths as in 1 unit. Hence, 13×6, or 78, express the number of sixths in 13. To 78 add 5, because in there are 5 sixths, and the amount, 83, expresses the number of sixths in 135. IV. TO REDUCE A COMPOUND FRACTION TO A SIMPLE ONE. Multiply the numerators together for a new numerator, and the denominators for a new denominator. If there be mixed numbers, reduce them to improper fractions, and proceed as the rule directs. 1. Reduce of of to a simple fraction. OPERATION. 1X7X 963 2X8X10-160 The propriety of this process may be seen from the following analysis. One eighth of is nine eightieths; 7 of are seven times as much as one eighth of; and seven times are 88. of § is 3, Ans. 6 2. Reduce of of 7 of 11 to a simple fraction. Ans. 8. 3. Reduce & of 1 of of to a simple fraction. Ans. 25 4. Reduce of of 265 to a mixed number. Ans. 84. 5. Reduce 7 of 34 of 8041 to a whole number. |