2. One third of a mile is how many sixths of a mile? 3. One half of a dollar is how many fourths of a dollar? 4. Name some equivalent fractions for halves. Thirds. 5. Express in terms 3 times as great. 4 times as great. 6. The number of twelfths in a unit is how many times the number of halves? Of thirds? Fourths? 7. How is the fraction changed to tenths expressing the same value? To 15ths? To 20ths? To 40ths? 8. The denominators 4, 6, 8, and 10 are multiples of what number? 9. Name the first three multiples of 3, the denominator of . The first three of 4 in 2. 10. Change 2, 7, 8, and 11 to 24ths, to 48ths, and 96ths. 11. Multiply both terms of by 3, and show that the value of the fraction is not changed. ANALYSIS.-If both terms of are multiplied by 3, the resulting fraction is, which is equivalent to, since the fractional unit is as large, while the number of fractional units taken is 3 times as many (102, III). 12. Name three equivalent fractions for ; for ; for . ANALYSIS.-Since 1 third is equal to 4 twelfths, 8 twelfths are equal to as many thirds as 4 twelfths are contained times in 8 twelfths, which is 2 times. Hence there are in. 17. How many fourths of a rod are 9 twelfths of a rod? 18. Divide both terms of 15 by 5, and show that the value of the fraction is not changed. ANALYSIS.-If both terms of 15 are divided by 5, the resulting fraction is, which is equivalent to, since the fractional unit is 5 times as large, while the number taken is as many (102, III). 19. Change to a fraction having lower terms; 14; 1. as great; as great; 20. How many thirds in 18? Fifths in ? 21. Express the value of in terms as great; as great; as great. 22. What common divisors have the terms of 14? ? #8? 23. Is in its lowest terms? Why not? Name two factors of its terms. Why is equal to 1? 24. In what lower terms can be expressed? 25. Change to its lowest terms; 1, t; H. EXERCISES. WRITTEN 106. 1. Reduce to sixteenths. EXPLANATION.—Since the given denominator 8 is contained in the required denominator 2 times, multiply both terms of the fraction by 2. Hence, { = 18.* Reduce 2. to thirty-fifths. 3. to sixty-thirds. 4. to eighty-fourths. 8. Reduce to its lowest terms. EXPLANATION. - Dividing both by 4, terms of the given fraction the result is (102, III). Change 5. 6. 1 to 108ths. 7. to 256ths. 36+4 to seventy-fifths. or, +3 += 1; 36+12 11. 48+12 ૐ. = Again, dividing both terms of s by 3, the result is . Since the terms of are prime to each other, the lowest terms of 3 are 3. The same result may be obtained more directly by dividing the terms of the given fraction by their greatest common divisor 12. *Rules are omitted when the oral exercises and explanations are so full and explicit that the pupil can readily construct a rule for himself. 25. Express in its simplest form the quotient of 441 divided by 462. Of 189 divided by 273. EXERCISES. 4680 10600 5222 26. Express in the simplest form 1344 divided by 1536. 27. Which is the greater fraction, and why,, or 188? 28. Change 168÷252 to the form of a fraction in its lowest terms. 81 63. 160-400. 324÷612. 107. To change an integer or a mixed number to the form of an improper fraction. ORAL 1. How many thirds in 4? ANALYSIS. Since in any number there are 3 times as many thirds as whole ones, in 4 whole ones there are 3 times 4 thirds 12. Hence, etc. 2. In 4 bushels, how many eighths of a bushel? 3. In $1 how many fifths of a dollar? Sixths? Eighths? Tenths? Twelfths? 4. In 5 yards, how many fourths of a yard? ANALYSIS. Since in any number there are 4 times as many fourths as there are whole ones, in 5 yards there are 4 times 5 fourths or 2o of a yard, plus of a yard, equals 3 yards. Hence, etc. 5. How many sixths in $2? In 5 rods? In 7 acres? 6. In 6 cords of wood, how many fourths of a cord ? 7. Among how many boys can you distribute 5 quarts of chestnuts, if you give them † of a quart each? 8. How is an integer or a mixed number changed to the form of an improper fraction? WRITTEN 108. 1. Change 81 to a fraction having 24 for its denom inator. SOLUTION. 81 x 24 1944. Hence, 81 = EXERCISES. 7. 75. 8. 224. 9. 30718. 2. Change 49 to twelfths. SOLUTION. 4949+7; 49 x 13588; 588+7=525. = 3. Change 44 bushels to fourths of a bushel. 5. Change 294 weeks to sevenths of a week. = 10. 8617 11. 45018. 1.2. 261. 109. To change an improper fraction to the form of an integer, or a mixed number. ORAL EXERCISES. 13. 122713. 14. 2475. 15. 1307. 1. In 18, how many units or ones? ANALYSIS. Since 4 fourths equal 1, 18 fourths are as many times 1 as 4 fourths is contained times in 18 fourths, which is 4 times. Hence, 1818÷442. 2. How many times 1 are 28? 40? 78? 80 ? ff? 5. In of a foot, how many feet? In 188 of an acre, how many acres? In 182 of a ton, how many tons? 6. If a man buy 3 of a yard of cloth, how many yards does he buy? If ? If 108? 7. How is an improper fraction changed to the form of an integer, or a mixed number? WRITTEN EXERCISES. 110. 1. Change 141 to the form of a mixed number. 161. Hence, 147 — 16}. = SOLUTION.-147 = 147÷9 = Reduce to integers or mixed numbers, 10. 4407. 113 11. 4483. 5. 651. 6. 241. 7. 476. 8. 181. 13. 14. 15. 16. ORAL EXERCISES. 2184. 3872. 13007. 33 32460. 48 111. To reduce two or more fractions to equivalent fractions having a common denominator. In? In ? In ? In ? In ? 1. How many sixths in 1? 2. How many eighths in 1? 3. How can and be changed to 12ths? To 24ths? 4. Change and to fractions of the same denominator? 5. If the denominator is multiplied by any number, how is the value of the fraction preserved? Why? (102, I, 1.) 6. Change and to 15ths; to 30ths; to 45ths. 7. Change 18 and 18 to 5ths; to 10ths. 8. If the denominator be divided by any number, how is the value of the fraction preserved? Why? (102, I, 2.) A common multiple 9. What is a multiple of a number? of two or more numbers? (93, 94.) 10. Name a multiple of 2; of 4; of 6; of 8. 11. What is the common multiple of 3 and 4? Of 4 and 5? 12. What is the L. C. M. of 2, 5, and 10? Of 2, 3, and 4? 13. What is the L.C.M. of the denominators, †, and 1? Of †, §, and § ? Of †,†, and ? |
