Find the least common multiple of: 3. 15, 20, 30 13. 16, 24, 32 4. 18, 24, 36 14. 20, 35, 42 5. 14, 21, 42 15. 27, 45, 63 6. 27, 54, 63 16. 28, 40, 56 7. 32, 48, 96 17. 36, 48, 64 8. 36, 54, 63 18. 32, 52, 65 9. 64, 72, 108 19. 50, 60, 70 10. 72, 84, 120 20. 55, 75, 88 11. 54, 81, 135 21. 60, 72, 96 12. 8, 12, 16, 20 22. 84, 108, 120 Find the g. c. d. and the l. c. m. of each of the following: 23. 45, 105, 180 27. 42, 70, 126 24. 36, 132, 192 28. 48, 80, 144 25. 54, 108, 198 29. 66, 134, 220 26. 39, 91, 130 30. 60, 135, 180 FRACTIONS 1. Divide 1 into 5 equal parts ; into 10 equal parts ; into 15 equal parts. 2. What is meant by }, id, or 15, of 1? 5. How many integral units are there in 5 ? in 10 ft. ? in 5 yd. ? in 4 mi. ? in $10 ? 6. If 1 ft. is divided into 12 equal parts, what is each part called ? How many twelfths of a foot does it take to equal a foot ? A fractional unit is one of the equal parts into which an integral unit has been divided; as, \, , }, it, ita, etc. A fraction is one or more fractional units; as, }, }, }, }, 16. 7. How many fractional units are there in 12? in is? in ? in ? in 1ā ? in 1? in 19? How many are there in 8? in ? in ? in ? in? 8. Observe that the sum of the fractional units of an integral unit equals the integral unit. Thus, 13 =1; s = 1; 1=1; b = 1; 18=1. 9. In the fraction , what is 3 called and what does it show ? 10. In the fraction , what is 4 called and what does it show? Since the denominator (p. 23) of a fraction shows into how many equal parts the integral unit has been divided, as, tenths, thirds, fourths, it shows the size of the fractional unit. Since the numerator (p. 23) of a fraction shows the number of fractional units taken, as, two thirds, ş, it names the number of the fractional units. 11. Observe that the unit of measure in 12 yd. is 1 yd.; thus, 12 yd. means 12 x1 yd; $8 means 8 x $1. What does 16 acres mean? 6 dozen ? 12. Name the different units of measure in problem 11. 13. Observe that the unit of measure in 19 in. is 12 in.; thus, 11 in. means 7 x 12 in.; * means 4 x*}. What does {acre mean? 12 dozen ? 14. What are the different units of measure in problem 13? 15. With what kind of a unit do we measure or compare whole numbers? 16. With what kind of a unit do we measure or compare fractions ? 17. What is a proper fraction ? (see p. 59). Explain why the fractional forms g, g, 19, 4, 4, are not proper fractions. 18. What is an improper fraction ? (see p. 59). A mixed number? (see p. 35). REDUCTION OF FRACTIONS Changing a mixed number to an improper fraction. 1. Change 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, each to the fractional unit 4ths ; to 5ths; to 6ths; to 7ths. Thus, 1=4, 2= , etc. 2. Change 13, 14, 2}, 35, 23, 816, 96, 107, each to the fractional units indicated by the fraction of the mixed numbers. Thus, 14 = 3. 3. What kind of fractions are j, 1, 12 ? Written Work 1. Change 111 to an improper fraction. 1173 15 165 Since 1 unit = 1), 11 units = 11 x if = 19; 480 + = 7 172 15 To change a mixed number to an improper fraction, multiply the whole number by the denominator. To this product add the numerator, and write the sum over the denominator. Change to improper fractions: 2. 15 12. 753 22. 8321 24. 283 26. 7541 28. 46% 31. 5031 Changing an improper fraction to a mixed number. 1. Name the integers and the fractions in 5, 1, 1 ft., 84 lb., $20, 4 yd., 10 oz., 42 in. 2. How much greater than 1 are 4 ? ? 8? 10 ? 16.? ? 18? 18? 3. In changing 1 ft. to integers and fractions, think first how many fourths it takes to make one whole unit. Then = how many whole units and if remaining ? HAM. INT. ARITH. - 10. Written Work 1. Change 145 to a mixed number. Since 1 = 8)145 8 eighths, in 145 eighths there are as many l's as 8 eighths are contained times in 145 eighths, or 18}. 18] Therefore 19= 183. Change to mixed numbers : 2. 4 8. 88 14. 11 15. 7 mi. 16. j rd. 17. Bol bu. 12. 12 hr. 18. He in. 7. The 13. 114 min. 19. 10 A. Changing the size of fractional units without changing the value of the fraction. 1. 1= š= 16 = 24 4. In changing 1 to å, we multiply both terms of the fraction by 3. Does this affect the value of j? Draw a figure to show that į and å are the same in value. 5. In changing i to j we divide both terms of the fraction by 3. Does this change its value? Illustrate by the figure drawn for example 4. Multiplying or dividing both terms of a fraction by the same number does not change its value. 6. Change }, }, , j, d, e, , fa, each to twenty-fourths. 7. Change 1, 12, 13, 14, 16, 34, 35, each to sixths. |