21. What will be the result if both terms of be divided by 3? If both terms of be multiplied by 3? 22. Fourths are how many times as large as twelfths? How many times as many twelfths as fourths must be taken to give the same part of a unit? 23. How will the value of the fraction be affected if the numerator of be divided by 3, and the denominator of the quotient be divided by 3? 24. Show that if both terms of a fraction are divided by the same number, the value of the fraction is not changed. 148. General Principles of Fractions. I. The value of a fraction is multiplied 1. By multiplying its numerator; or, II. The value of a fraction is divided III. The value of a fraction is not changed 1. By multiplying both terms by the same number; or, 2. By dividing both terms by the same number. Exercises. Multiply the numerator of by 2, and tell how the value of the fraction is affected. 149. Model. If the numerator of is multiplied by 2, the result is, and the value of the fraction is made twice as great; because, while the size of each part remains the same, the number of the parts is made twice as great. Multiply both terms of by 3, and tell how the value of the fraction is affected. 150. Model. If both terms of are multiplied by 3, the result is, and the value of the fraction is not changed, because, while the size of each part is made 3 times as small, the number of parts is made 3 times as great. 1. Multiply each of the following fractions, and tell how the value of the fraction is affected: 2. Divide each of the following fractions, and tell how the value of 151. Reduction of Fractions is the process of changing their form without changing their value. 1. To Change a Fraction to Lower Terms. 1. How many thirds of an inch in 1 inch? In 6 sixths of an inch? In 2 sixths of an inch? In 4 sixths? 2. How many fourths in 6 eighths? 152. Analysis. — Since one fourth equals 2 eighths, 6 eighths equal as many fourths as the number of times 2 eighths are contained in 6 eighths, which is 3 times. Hence, in 6 eighths there are 3 fourths. 3. How many fifths in 2 tenths? In 4 tenths? How many fifths in? In? In 1? In 29? 4. Six ninths of a yard are how many thirds of a yard? Express in terms one third as large. 5. Express the value of as halves; fourths; eighths. 6. What factor is common to the terms of 19? Divide both terms by their common factor. 7. What common divisor have the terms of the fraction 14? What does the fraction become if both terms are divided by 7? 8. What fraction is produced by dividing both terms of the fraction by their greatest common divisor? Is the value of the fraction changed? Why? 153. A fraction is changed to lower terms when it is changed to a fraction of equal value having a smaller numerator and denominator. 154. A fraction is changed to its lowest terms when its numerator and denominator are made prime to each other. 18 24 9 Thus, expressed in lower terms equals but changed to its lowest terms equals since 3 and 4 are prime to each other. 9. Express the value of in terms as great; as great; as great;;2. 1 10. Is 18 in its lowest terms? Why not? What is 10 expressed in its lowest terms? Why is equal to 10? 11. By what number must both numerator and denominator of be divided to change them to fifths? To tenths? 12. Express the value of 14 in its lowest terms. By what common factor can both terms be divided? 75 13. Frank has g of a dollar, and Harry has of a dollar. Which has the greater sum of money? 14. From one tree a gardener took of a bushel of pears, from another, and from another 3. Which yielded most? 15. Divide both terms of by 9, their greatest common divisor, and show that the value of the fraction is not changed. 16. Show that the reduction of a fraction to its lowest terms does not change the value of the fraction. Tell by what process a fraction is reduced to its lowest terms. 155. Principle. Dividing both terms of a fraction by the same number does not change the value of the fraction. Exercises. Change to its lowest terms. 156. Model. -- Since dividing both terms of a fraction by the same number does not change the value of the fraction, 18 is changed to its lowest terms by dividing the numerator and the denominator by 6, their greatest common divisor, giving, the fraction required. Change, or reduce, to their lowest terms 2. To Change a Fraction to Higher Terms or to a Given Denominator. 1. How many eighths of a yard in 1 yard? In 1 fourth of a yard? In 2 fourths? 3 fourths? 2. In 2 thirds of a yard how many sixths? 157. Analysis. Since 1 is equal to 6 sixths, 1 third is equal to one third of 6 sixths, which is 2 sixths, and 2 thirds are 2 times 2 sixths, or 4 sixths. Hence, etc. 3. How many twelfths in 1 half? In 2 thirds? How many in? In &? 5 4. Express the value of as tenths; as fifteenths. In terms 4 times as great. In terms 5 times as great. 5. Name the first three multiples of 4, the denominator of . Express as eighths; as twelfths; sixteenths. 6. The number of twelfths in a unit is how many times the number of thirds? How many times the fourths? Sixths? 7. Express, 4, 3, g, and as 24ths and as 48ths. 3 7 8. What fraction is produced by multiplying both of the terms of the fraction by 3? Is the value of the fraction changed? Why? 9. Express the value of in terms 2 times as large; 3 times as large; 4 times; 5 times; 6 times. 10. What is expressed in terms 4 times as large? How are fifths changed to 20ths? Why is equal to 18? 11. By what number must both numerator and denominator of be multiplied to change it to 14ths? To 21sts? 12. Express the value of in sixteenths. By what factor must both terms be multiplied? 13. Change of a foot and 15 of a yard to fractions. having terms 3 times as great; 4 times; and 5 times. 14. Edward has of a dollar, and William has 17 of a dollar. Which has the greater sum of money? 15. In one lot are å of an acre, and in another of an Which is the larger lot? Prove it. acre. 16. How is a fraction changed to higher terms? Show that the reduction does not change the value of the fraction. 158. A fraction is changed to higher terms when it is changed to a fraction of equal value having a larger numerator and denominator. 3 5 Thus, & may be expressed as $, $$, etc. 12 20 |