SECTION IV. COMMON FRACTIONS. 134. A Fraction is a number of the equal parts of a unit. 135. Fractions are divided into two classes; commos tractions and decimal fractions. 136. A Common Fraction is one in which the unit is divided into any number of equal parts. 137. A Decimal Fraction is one in which the unit is divided into tenths, hundredths, etc. 138. A Common Fraction is expressed by two nombers, one written above the other, with a short line between them; thus, expresses 3 fourths. 139. The Denominator of a fraction denotes the num. ber of equal parts into which the unit is divided. 140. The Numerator of a fraction denotes the number of equal parts which are taken: 141. The Numerator and Denominator are called the Terms of the fraction. The numerator is written above the line, and the denominator below it. 142. Common Fractions consist of three principal classes ; namely, Simple, Compound, and Complex. 143. A Simple Fraction is a fraction having a single integral numerator and denominator; as j, k, etc. 144. A Proper Fraction is a simple fraction whose value is less than a unit; as , & 145. An Improper Fraction is a simple fraction whose value is equal to or greater than a unit; as 5, 7, 4, etc. 146. A Compound Fraction is a fraction of a fraction; 2 of 4, 6 of of }, etc. 147. A Complex Fraction is one whose namerator, or denominator, or both, are fractional; as 5 of 148. A Mixed Number consists of an integer and a fraction; as, 43, 5%, etc. 149. The Reciprocal of a number is a unit divided by that number; thus, the reciprocal of 3 is 150. The Number of Cases of common fractions is oight. They are as follows: 1. Reduction, 5. Division, 2. Addition, 6. Relation of Fractions, 3. Subtraction, 7. Greatest Common Dipisor, 4. Multiplication, 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 c. & number of things. Thus, i may be regarded as * of one or of three. 7. 54. 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 denominator. Name the kind and read the following: 8. 18. 9. off 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 the kind of fractional units. Write the following fractions:1. Two-thirds. 5. Seven-elevenths. 2. Five-sixths. 6. Eight-tenths. 3. Six-eighths. 7. Eleven-fifteenths. 4. Nine-tenths. 8. Twelve-twentieths. ANALYSIS OF FRACTIONS. 153. To Analyze a fraction is to explain what is ex. pressed by the fractional notation. 1. Analyze the fraction : SOLUTION.-In the fraction 3, 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 Analyze the following fractions : 5. to 8. 11 3. $. 6. i 9. H. 1. . 7. H. 10. 14. METHOD OF TREATMENT. 154. There are Two Methods of treating common fractions, which may be distinguished as the Inductive and Deductive Methods. 165. 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, cach of the same size as before, hence the value of the fraction is 5 times as great. 2. Dividing the numerator of a friction 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 re sulting, 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 he 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 imes 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 fraction 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 fraction 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 naultiplies 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 mul biplication or division produces a SIMILAR change in the value of the fraction, but such a change in the DENOMINATOR produces an OPPOBITE 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 : let 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 az cases. CASE I. 180. To reduce whole or mixed numbers to impropor fractions. MENTAL EXERCISES, 1. How many fifths in 4*? SOLUTION.-In one there are 5 Afths, and in 4 there are 4 times 5 fifths, or 20 fifths, which added to 3 Afths, equal 23 tisths; therefore 4= 4. 2. How many fourths in 7*? in 54? in 94? in 128? 6. Describe the operation we perform in reducing a mixed number to & fraction. WRITTEN EXERCISES. 1. Reduce 27} to fourths. OPERATION. SOLUTION.-In one there are 4 fourths, and in 27 there 27 are 27 times 4 fourths, or 108, which added to the , 명 equals 11. 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. Reduce to improper fractions, 2. 54. Ans. 1. 7. 516. Ans. ift. 3. 124 Ans. 68. 8. 3541. Ans. 43 4. 186 Ans. 44.9. 2477 ; 5178. Ans. 4; 470 6. 115. Ans. 184. 10. 8211; 100%. Ans. 114; 1882 6. 275. Ans. 221.1 11. 4914; 2354. Ans. 412; 4. CASX II. 161. To reduce improper fractions to whole or mixed numbers. MENTAL EXERCISES. 1. How many units in 4? SOLUTION.-In one there are 4, hence in 23 fourths there are as many ones as 4 is contained times in 23, which are 54. Therefore a 35 2. How many units in Y? in 48 ? in ? 3. How many units in ? in 4 ? in 4? 4. How many units in *? in ti? in #? 5. How many units in 47? in *? in f}? 6. Describe the process of reducing an improper fraction to a mixed aumber. |