3 25 12. Change to their lowest, or smallest, terms ; ; 28. 32. 36. 56. 65. 72 29; 33; 30; 48; 92; 95; 33. 60 72, 78 84. 226. A fraction is reduced to lower terms when it is changed to a fraction of equal value having a smaller numerator and denominator. 12 Thus, expressed in lower terms, equals or . " 18 227. A fraction is reduced to its lowest terms when its numerator and denominator are made prime to each other. Thus, is in its lowest terms, because no integer greater than 1 will exactly divide both 8 and 9. 228. Principle. Dividing both terms of a fraction by the same number does not change the value of the fraction. Written Exercises. 229. Example. Reduce to its lowest terms. SOLUTION. 225 25 9 250 25 10 225)250(1 225 25)225(9 225 EXPLANATION.-The greatest common divisor of the terms of the fraction, 225 and 250, is 25. Since dividing both terms of a fraction by the same number does not change the value of the fraction, divide the numerator and the denominator by 25, their greatest common factor or divisor, giving, which is in its lowest terms, since its terms, 9 and 10, are prime to each other. 230. Rule to Change Fractions to their Lowest Terms. Divide the terms of the fraction by their greatest common divisor. The resulting fraction will be in its lowest terms. Note. A fraction may be reduced to lower terms by rejecting any factor common to the terms of the given fraction. 52 RA132 7.13; 153; 788; 12,2; 23,73; 34133. 8. 3 of a ton; 105 9. $5,750; 23,2756; 45,672 tons; 56,375 gallons. 2520 10. Express in their lowest terms the value of the fractions 65 121 264 2700 1050 2226, and 968 1049 33, 2008, 243, 10752, 19133, 11. What are the lowest terms in which 3336, 4624 5004 10 592, 7625, 11311, 31031, and 1811 can be expressed? 22459 12. Find the lowest terms in which 36 2000 171 7180 1008, 1129 $125,956, 4444798, and 2001998 can be expressed. 10 5899 13. Which is the greater part of a cake, 5% of it, or of it? 14. Which is the greater sum of money, $15, or $2,300? 15. A farm was so divided that one of two persons re 20 160 232 1250 ceived of it, and the other of it. Which had the larger portion? CASE II. To Change a Fraction to Higher Terms or to a Given Denominator. 1. If an orange be cut into 4 equal pieces, what part of the orange is each piece? 2. If 1 fourth of an orange is cut into two equal pieces, what part of the orange is each piece? 3. In 3 fourths of an orange are how many eighths? 231. Analysis. — Since 1 is equal to 8 eighths, 1 fourth is onefourth of 8 eighths, which is 2 eighths; and 3 fourths are 3 times 2 eighths, which are 6 eighths. Hence, & of an orange=§ of an orange. 4. To how many sixths are 2 thirds equal? To how many ninths? How many twelfths? Fifteenths? 5. Express the value of in eighths; in twelfths. In terms 4 times as great; 5 times as great. 6. Which expresses the greater value, or ? or? or?or? 7. The number of eighths in a unit is how many times the number of halves? How many times the fourths? changed to sixths expressing the To twelfths? 8. How is the fraction same value? To ninths? 9. Name the first three multiples of 4, the denominator of. The first three of 5 in. 10. Name three fractions of equal value to which can 11. What fraction is produced by multiplying each term of by 2? By 3? By 4? By 5? 2 3 12. Change 3, 4, 5, 7, and 1 to 24ths, to 48ths, and 96ths. 232. A fraction is changed to higher terms when it is changed to a fraction of equal value having a larger numerator and denominator. Thus, $may be expressed in higher terms, as $ 2 4 5 4 ད 8 10 9 etc. 233. A Given Denominator is a denominator named to which another is to be changed. Thus, if is to be changed to eighths, the given denominator is eighths, and the equivalent fraction is 8 234. Principle. Multiplying both the terms of a fraction by the same number does not change the value of the fraction. Written Exercises. 235. Example.-Change to 108ths. SOLUTION. 1089 = 12 4 X 12 9 X 12 48 108 EXPLANATION.-Since 108, the given denominator, is a multiple of the denominator 9, the other factor required to produce it is the quotient of 108 divided by 9, which is 12. Since the denominator of must be multiplied by 12, the numerator must also be multiplied by 12, that the value of the fraction may not be changed. Hence, equals, the fraction required. 236. Rule to Change a Fraction to Higher Terms or to a Given Denominator. Divide the given denominator by the denominator of the fraction, and multiply both terms of the fraction by the quotient. The resulting fraction will be in the higher terms required. Reduce, or change Problems. 1. and to fifteenths. 2.,, to twelfths. 6 4 9 5. 8, 2, 17 to 36ths; 72nds. 3.,, to twentieths. 6. §, 1, 3 to 48ths; 96ths. 7. 10 and to fractions each having 35 as the denominator. 8. and to fractions whose denominator is 75. 11 25 13 9. 1 to 100ths; 17 to 108ths; to 150ths; to 300ths. 25 36 10. to 32nds, 48ths, 64ths, 80ths, 96ths, and 112ths. 7 16 11. Change, 6, 76, 2, 30, and to sixtieths. 11 12. Express 7, 54, 12, 20, 15, 20, 25, and 4 as fractions. 31, having 84 as their denominator; having 336. 13. Express the value of $3 in hundredths of a dollar. 14. Change of a foot, and of a yard to fractions having terms 3 times, 5 times, and 10 times as great. 9 36 15. Reduce 4, 1, 2, 7, 43, and 112 to 112ths of a ton. CASE III. To Change an Integer or a Mixed Number to an Improper Fraction. 1. How many halves of an apple are in 1 apple? How many thirds of an apple? How many fourths? 2. In 2 pears, how many halves of a pear? 237. Analysis. - Since in 1 pear there are 2 halves of a pear, in 2 pears there are 2 times 2 halves, or 4 halves of a pear. 3. In 3 pears, how many halves of a pear? How many thirds? How many fourths? Fifths? 4. How many fifths of a dollar are in $1? How many in $2? $3? $6? $9? $10? $15? 5. If 4 pints of berries are divided among some boys, to how many could a half-pint each be given? 6. How many fourths of a cake are 2 and 3 fourths cakes? 238. Analysis. Since in 1 cake there are 4 fourths, in 2 cakes there are 2 times 4 fourths, or 8 fourths; 8 fourths and 3 fourths are 11 fourths. Hence, in 2 cakes are of a cake. 7. How many sixths are in 2 apples? In 2 apples? In 3 yards? In 5 yards? 9 yards? 8. How many ninths of a dollar are in $7? In 7? Tenths of a foot in 8 feet? 8,3 feet? 10 9. How many tenths of an acre in 7% acres? How many quarter-pounds of candy in 123 pounds? 10. Change 5, 7, 6, and 10 to eighths, and to tenths; and change 51, 73, 95, to improper fractions. Written Exercises. 239. Example 1.-Change 7 to ninths. EXPLANATION. Since in 1 there are 9 ninths, in 7 there are 7 times 9 ninths, which are 63 ninths. Hence, 7 equals 3. |