Read and analyze the following fractions:10. 4: 7. 17. 45. 72. 48 84 456 9, 12, 38, 100 375 1009, 7863, 537 87. 95 48. 75 436. 150. 150072. 100001 13. 159; 48; 13385; 158873; 488881. 537 972 47956 475000 200002' Fractions are distinguished as Proper and Improper. 170. A Proper Fraction is one whose numerator is less than its denominator; its value is less than the unit 1. 171. An Improper Fraction is one whose numerator equals or exceeds its denominator; its value is never less than the unit 1. NOTES.-1. The value of a proper fraction, always being less than a unit, can only be expressed in a fractional form, hence, its name. 2. The value of an improper fraction, always being equal to, or greater than a unit, can always be expressed in some other form; hence its name. 172. A Mixed Number is a number expressed by an integer and a fraction. 173. Since fractions indicate division, (168, II), all changes in the terms of a fraction will affect the value of that fraction according to the laws of division; and we have only to modify the language of the General Principles of Division, by substituting the words numerator, denominator, and fraction, or value of the fraction, for the words dividend, divisor, and quotient, respectively, and we shall have the following GENERAL PRINCIPLES OF FRACTIONS. 174. PRIN. I. Multiplying the numerator multiplies the fraction, and dividing the numerator divides the fraction. PRIN. II. Multiplying the denominator divides the fraction, and dividing the denominator multiplies the fraction. PRIN. III. Multiplying or dividing both terms of the fraction by the same number, does not alter the value of the fraction. 175. These three principles may be embraced in one GENERAL LAW. A change in the NUMERATOR produces a LIKE change in the value of the fraction; but a change in the DENOMINATOR produces an OPPOSITE change in the value of the fraction. REDUCTION. 176. The Reduction of a fraction is the process of changing its terms, or its form, without altering its value. CASE I. 177. To reduce fractions to their lowest terms. A fraction is in its lowest terms when its numerator and denominator are prime to each other; that is, when both terms have no common divisor. the result, 1, by 3, and obtain 4 for the final result. As 4 and 7 are prime to each other, the lowest terms of 60 105 are 4. Instead of dividing by the factors 5 and 3 successively, we may divide by the greatest common divisor of the given terms, and reduce the fraction to its lowest terms at a single operation. Hence, the Cancel or reject all factors common to both numerator and denominator. Or, RULE. 15. Reduce 41369, and 47166 to their lowest terms. 62443) 46418 CASE II. 178. To reduce an improper fraction to a whole or mixed number. 1. Reduce 297 to a whole or mixed number. ANALYSIS. Since the value of a fraction in integral units is equal to the quotient of the numerator divided by the denominator, (168, 1,) we divide the given numerator, 297, by the given denominator, 12, and obtain for the value of the fraction, the mixed number 24,2=24}. Hence the RULE. Divide the numerator by the denominator. NOTES. 1. When the denominator is an exact divisor of the numerator, the result will be a whole number. 2. In all answers containing fractions, reduce the fractions to their lowest terms. Ans. $31 14. In 407 of a dollar. how many dollars? 15. If 1000 dollars be distributed equally among part of a dollar must each man receive in change? CASE III. 179. To reduce a whole number to a fraction having a given denominator. 1. Reduce 37 to an equivalent fraction whose denominator shall be 5. OPERATION. 37 x 5 185 = 37 185 Ans = 5 1 ANALYSIS. Since in each unit there are 5 fifths, in 37 units there must be 37 times 5 fifths, or 185 fifths = 185. The numerator, 185, is obtained in the operation by multiplying the whole number, 37, by the given denominator, 5. Hence the RULE. Multiply the whole number by the given denominator; take the product for a numerator, under which write the given denominator. NOTE. A whole number may be reduced to a fractional form by writing 1 under it for a denominator; thus, 9=2. EXAMPLES FOR PRACTICE. 1. Reduce 17 to an equivalent fraction whose denominator shall be 6. Ans. 102. 2. Change 375 to a fraction whose denominator shall be 8. 3. Change 478 to a fraction whose denominator shall be 24. 4. Reduce 36 pounds to ninths of a pound. 5. Reduce 359 days to sevenths of a day. 6. Reduce 763 feet to fourteenths of a foot. 7. Reduce 937 to a fractional form. CASE IV. Ans. 2513. Ans. 10682. 14 Ans. 937. 180. To reduce a mixed number to an improper frac tion. 1. In 12 how many sevenths? OPERATION. 125 7. 8,9 ANALYSIS. In the whole number 12, there are 12 x 7 sevenths 84 sevenths, (Case III), and 84 sevenths + 5 sevenths 89 sevenths, or 3. Hence the following RULE. Multiply the whole number by the denominator of the fraction; to the product add the numerator, and under the sum write the denominator. EXAMPLES FOR PRACTICE. 1. Reduce 15 to fifths. 9. In 135 how many twentieths? 10. In 433 bushels how many fourths of a bushel? 11. In 760 days how many tenths of a day? CASE V. 181. To reduce a fraction to a given denominator. We have seen that fractions may be reduced to lower terms by division. Conversely, I. Fractions may be reduced to higher terms by multiplication. II. All higher terms of a fraction must be multiples of its lowest terms. 1. Reduce to a fraction whose denominator is 40. OPERATION, 40 ÷ 8 = 5 3 x 5 = 40 15, Ans 8 x 5 ANALYSIS. We first divide 40, the required denominator, by 8, the denominator of the given fraction, to ascertain if it be a multiple of this term, 8. The division shows that it is a multiple, and that 5 is the factor which must be employed to produce it. We therefore multiply both terms of 3 by 5, (174, III), and obtain 1%, the re quired result. Hence the RULE Divide the required denominator by the denominator of the given fraction, and multiply both terms of the fraction by the quotient. |