divisor by the last remainder, until nothing remains. The last divisor is the G. C. D. sought. To find the G. C. D. of more than two numbers, find the G. C. D. of any two of them, and then of that divisor and a third number, and so on till all the numbers are taken. 105. EXAMPLES. Find the G. C. D. of Ans. 99. 12. 229 and 954. Ans. 6. 13. 9. 198 and 297. 10. 222 and 564. 392, 1008, and 224. Ans. 23. 15. What is the width of the widest carpeting that will exactly fit either of two halls, 45 feet and 33 feet wide respectively? Ans. 3 ft. 16. A has a piece of ground 90 feet long and 42 feet wide; what is the length of the longest rails that will exactly suit its length and its width? Ans. 6 ft. 17. A lady has one flower bed measuring 10 feet around, and another measuring 18 feet. If she borders the beds with pinks, what is the greatest distance she can set her pink roots apart, and have them equally distant in the two beds? Ans. 2 ft. 18. A man has 90 bushels Kidney potatoes, CO bushels Jackson Whites, and 105 bushels Red Rileys. If he puts them all into the largest bins of equal size that will exactly measure either lot, how many bushels will each of his bins contain? 19. What is the length of the longest stepping-stones that will exactly fit 3 streets, 72, 51, and 87 feet wide, respectively? 20. What is the length of the longest curb-stones that will exactly fit 4 strips of sidewalk, the first being 273 feet long, the econd 294, the third 567, and the fourth 651? For Dictation Exercises, see Key. FRACTIONS. 106. A Fraction is one or more of the equal parts of a unit; thus,, read three fourths, shows that a unit has been divided into four equal parts, and that three of those parts are taken. 107. The number which shows into how many equal parts unit is divided, is called the Denominator of the fraction, because it denominates or names the parts; thus, 4 is the denominator of 2. 108. The number which shows how many parts are taken, is called the Numerator; thus, 3 is the numerator of 109. The numerator and denominator are called the Terms of a fraction. 110. A Common or Vulgar Fraction is a fraction whose denominator and numerator are both expressed, the numerator being written above, and the denominator below, a dividing line; as,,, 25. 9 111. A Decimal Fraction is one whose denominator is 10, or some integral power of 10. The denominator is not generally expressed. written .2, and 36 written .36, are decimal fractions. 112. A Mixed Number is a whole number and a fraction expressed together, as 7§, 213. 113. Common fractions may be either Proper, Improper, Compound, or Complex. 114. A Proper Fraction is one whose numerator is less than its denominator, as 2. 115. An Improper Fraction is one whose numerator equals or exceeds its denominator, as 35, 2. 116. A Compound Fraction is a fraction of a fraction, as of %. 117. A Complex Fraction is one which contains a fraction in either or both of its terms, as 2§, 8 7 7 QUESTIONS. What does the denominator of a fraction show? What does the numerator show? What is meant by the expression? Ans. 5 of the 8 equal parts into which a unit is divided. cu.. What is meant by the expression ? ? 18? 1? Fig. 1. 118. If we compare common fractions with the last expression for division in Art. 55, we shall see that their forms are alike. A fraction implies division, the numerator being the dividend, and the denominator the divisor. Thus, may be considered either three fourths of 1 or one fourth of 3. The following diagram will show that these are equivalent expressions, the one line in figure 1 being equal to of the three lines in figure 2. of of 12. 1 of 3 = 1. $$$ What does denote? Ans. It denotes either 5 of the seven equal parts into which 1 is divided, or one seventh of 5. What does show? ?1? 119. GENERAL PRINCIPLES. NOTE.-The following propositions should be copiously illustrated by the teacher, and frequently referred to, until they are fully comprehended by the pupil. PROPOSITION I. As the denominator of a fraction shows the number of parts into which a unit is divided, and the numerator shows how many parts are taken, it follows that if we multiply the numerator of a fraction by a whole number, we multiply the number of parts, and thus increase the value of the fraction; but if we multiply the denominator of a fraction, we multiply the number of parts into which a unit is divided, and thus diminish the size of the parts, and consequently decrease the value of the fraction. ILLUSTRA 2X2 TION. 5 2 5 X 2 ILLUSTRATION. 6 ILLUSTRATION. 3 X 2 }} ILLUSTRATION. 4÷2 2÷2 Fraction diminished. PROPOSITION II. If we divide the numerator of a fraction by a whole number, we divide the number of parts and thus diminish the value of the fraction; but if we divide the denominator of a fraction, we divide the number which shows into how many parts the unit is divided, and thus increase the size of the parts, and consequently increase the value of the fraction. mo 2 62 210 3 45 2 10 6 Fraction increased. PROPOSITION III. If we multiply the numerator and denom, inator, each by the same number, we increase the number of parts of the fraction, but diminish their size in the same proportion; consequently the value of the fraction is not altered. ++ Fraction diminis.ed. + PROPOSITION IV: If we divide the numerator and denominator, each by the same number, we diminish the number of parts in the same proportion as we increase their size, consequently the value of the fraction is not altered. 4 Fraction not altered Fraction not altered in value. + QUESTIONS. How does multiplying the numerator of a fraction affect the value of the fraction? Why? How does multiplying the denominator affect the value of the fraction? Why? How does dividing the numerator of a fraction affect the value OPERATION. 8 4X2 4 10 5X2 5' of the fraction? Why? How does dividing the denominator affect the value of the fraction? Why? If, then, you multiply the numerator and denominator each by the same number, what is the effect upon the fraction? Why? If you divide the numerator and denominator each by the same number, what is the effect upon the fraction? Why? REDUCTION OF FRACTIONS TO LOWEST TERMS. 120. A Fraction is expressed in its lowest terms when the numerator and denominator are prime to each other. 121. ILL. EX. Reduce to its lowest terms. 8=4X2; 10 = 5 × 2. Dividing the numerator and denominator each by striking out the common factor 2, the value of the fraction will not be altered (Art. 119, Prop. IV.), and will be expressed in its lowest terms. Hence the Ans. RULE. To reduce a fraction to its lowest terms: Remove from the numerator and denominator all their common factors. 122. EXAMPLES. Reduce to their lowest terms, 1. 18. Ans. . 4. 39 271 10. 1 11. 758. 12. 용골동. 2. 2. Ans. §. 5. 3. 18. Ans. 7. 6. 급분 18 25. 13. Reduce to its lowest terms. Ans. NOTE I. 1, being a factor of all numbers, will remain when all other factors are struck out, as in the numerator of example 13. 16. 17. 8. Ans. 11. 99 10. 562 252.8 50 7. 208. NOTE II. In case the factors of the numerator and denominator cannot readily be ascertained, find the G. C. D. of the two terms, and divide each of them by it. Reduce to their lowest terms, 14. 700. | 15. 81% 7290. For Dictation Exercises, see Key. 18. 19. 9. 송음. 2391 10361 737 6′58′5. 20. 21. |