148. A Mixed 'Number consists of an integer and a fraction; as, 4, 5, etc. 149. The Reciprocal of a number is a unit divided by that number; thus, the reciprocal of 3 is 3. 150. The Number of Cases of common fractions is eight. They are as follows: 1. Reduction, 2. Addition, 3. Subtraction, 4. Multiplication, 5. Division, 6. Relation of Fractions, 7. Greatest Common Divisor, 8. Least Common Multiple. NOTES.-1. Each fractional part is used as a single thing and is therefore a unit; hence, we have Units and fractional units. 2. The primary conception of a fraction is that it is a number of equal parts of a unit. It may, however, be regarded as a number of parts of one thing, or as one part of a number of things. Thus, may be regarded as of one or of three. NUMERATION AND NOTATION. 151. Numeration of Fractions is the art of reading a fraction when expressed by figures. Rule. Read the number of fractional units expressed by the numerator, and give them the name indicated by the de nominator. Name the kind and read the following: 152. Notation of Fractions is the art of expressing fractions by means of figures. Rule. Write the number of fractional units, draw a line beneath, under which write the number which indicates he kind of fractional units. Write the following fractions:-- 1. Two-thirds. 2. Five-sixths. 8. Six-eighths. 4. Nine-tenths. 5. Seven-elevenths. 6. Eight-tenths. 7. Eleven-fifteenths. 8. Twelve-twentieths ANALYSIS OF FRACTIONS. 153. To Analyze a fraction is to explain what is expressed by the fractional notation. 1. Analyze the fraction . SOLUTION.-In the fraction, the denominator 5 indicates that the unit is divided into 5 equal parts, and the numerator 4 denotes that 4 of these parts are taken. 154. There are Two Methods of treating common fractions, which may be distinguished as the Inductive and Deductive Methods. 155. By the Inductive Method we solve all the different cases by analysis, and derive the rules or methods of operation from these analyses by inference or induction. 156. By the Deductive Method we first establish a few general principles, and then derive all the rules or methods of operation from these general principles. NOTE. The Inductive Method will be used with the mental exercises; with the written exercises the method which is thought to be the simplest is used. PRINCIPLES OF FRACTIONS. 1. Multiplying the numerator of a fraction by any number multiplies the value of the fraction by that number. If we multiply the numerator of a fraction by any number, as 5, the resulting fraction will express 5 times as many fractional units, each of the same size as before, hence the value of the fraction is 5 times as great. 2. Dividing the numerator of a fraction by any number divides the value of the fraction by that number. If we divide the numerator of a fraction by any number, as 4, the resulting fraction will express as many fractional units, each of the same size as before, hence the value of the fraction is divided by 4. 3. Multiplying the denominator of a fraction by any num her divides the value of the fraction by that number Since the denominator denotes the number of equal parts into which the unit is divided, if we multiply the denominator of a fraction by any number, as 5, the unit will be divided into 5 times as many equal parts, hence each fractional unit will be as large as before, and the same number of fractional units being taken, the value of the fraction is as great. 4. Dividing the denominator of a fraction by any number multiplies the value of the fraction by that number. Since the denominator denotes the number of equal parts into which the unit is divided, if we divide it by any number, as 4, the unit will be divided into as many equal parts, hence each fractional unit will be 4 times as large as before, and the same number of fractional units being taken, the value of the fraction will be 4 times as great. 5. Multiplying both numerator and denominator of a frac tion by the same number does not change the value of the fraction. Since multiplying the numerator multiplies the value of the fraction, and multiplying the denominator divides the value of the fraction, multiplying both numerator and denominator both multiplies and divides the value of the fraction by the same number, and hence does not change its value. 6. Dividing both numerator and denominator of a frac tion by the same number does not change its value. Since dividing the numerator divides the value of the fraction, and dividing the denominator multiplies the value, dividing both numerator and denominator both divides and multiplies the value of the fraction, and hence does not change its value. 157. These principles may be embodied in one general law as follows: General Principle.—A change in the NUMERATOR by multiplication or division produces a SIMILAR change in the value of the fraction, but such a change in the DENOMINATOR produces an OPPOSITE change in the value of the fraction. REDUCTION OF FRACTIONS. 158. The Reduction of Fractions is the process of changing their form without altering their value. 159. There are Six Cases of reduction : 1st. Numbers to fractions. | 4th. To lower terms. 2d. Fractions to numbers. 5th. Compound to simple. 3d. To higher terms. 6th. Complex to simple. NOTE. -Reducing to a Common Denominator is included in these six eases CASE I. 160. To reduce whole or mixed numbers to improper fractions. 1. Reduce 27 to fourths. SOLUTION.-In one there are 4 fourths, and in 27 there are 27 times 4 fourths, or 103, which added to the equals 1. Therefore, etc. Rule.-Multiply the whole number by the denominator of the fraction, add the numerator to the product, and write the denominator under the sum. 5. 115. 6. 27ğ. 12. 593; 78413; 18,440. 30 Ans. 104. 10. 8211; 100%. Ans. 1244; 10889. CASE II. 161. To reduce improper fractions to whole or mixed numbers. 1. How many units in 2? SOLUTION. Since dividing both terms of a fraction by the same number does not change its value (Prin. 6), by dividing both terms by 5, we have 153, 1 or 153. OPERATION. 7-15%. Rule.-Divide the numerator by the denominator, and the quotient will be the whole or mixed number. Reduce to whole or mixed numbers, CASE III. 162. To reduce fractions to higher terms. 163. Reducing a Fraction to higher terms is the process of reducing it to an equivalent fraction, having a greater numerator and denominator. 1. How many twentieths in ? SOLUTION. Since multiplying both terms of a fraction by the same number does not change its value (Prin. 5), we multiply both terms by the number which will give the required denominator, which we see is 4; hence, }=}}. OPERATION. 3X4 3X4 Rule.-Multiply both numerator and denominator by the number which will give the required denominator. 165. Reducing a Fraction to lower terms is the process of reducing it to an equivalent fraction having a smaller numerator and denominator. Principle. A fraction is in its lowest terms when the numerator and denominator are prime to each other. 1. Reduce to fifths. OPERATION. SOLUTION. Since dividing both terms of a fraction by the same number does not change its value (Prin. 6), we may reduce to lower terms by dividing both numerator and denominator by 6; dividing, we have equal to ; and since the terms 4 and 5 are prime to each other, the fraction is in its lowest terms. Therefore, etc. Rule I.-Divide both terms successively by their common factors. Rule II-Divide both terms by their greatest comm divisor. |