Changing to least similar fractions. A common denominator of two or more fractions is a number that contains all the denominators of the fractions an integral number of times; thus, 24 is a common denominator of 1, 1, and 1. The least common denominator (l. c. d) of two or more fractions is the least number that contains all the denominators of the fractions an integral number of times; thus, 12 is the 1. c. d. of 1, 1, and 1. The 1. c. d is the least common multiple of the denominators. Similar fractions are fractions that express the same unit value. They must therefore have a common denominator. By inspection : 1. Change,, and to similar fractions having the least common denominator. It is evident by inspection that 18 is the least common multiple of 2, 3, and 9. It is therefore the least common denominator of the given fractions. Changing the given fraction to 18ths, we find that ; and // = 18. Hence, the fractions,, and may be changed to the similar fractions, 1, and 18. Change to least similar fractions : = = By factoring the denominators: 1. Change,, and to similar fractions having the least common denominator. = 45 60 50 60 54 10 × 6 60 In finding the 1. c. m., use each factor as often as it occurs in any one number. 2 × 2 × 3 × 5 = 60 The least common multiple of the denominators is 60, which is the least common denominator of the given fractions. Changing the given fractions to 60ths, we find that = 15; =58; and 1% = 54. Size and kind of fractional units. 1. Why is it not possible to add 4 ft. to 5 oz. ? 2. What is the sum of and ? of and ? 3. What is the size of the fractional unit of and ? 4. What is the kind of fractional unit in each? 5. Are the fractional units of and alike in size and kind? 6. What is the size of the fractional unit in $ and ft.? What is the kind of unit in each? 7. Why can we add or subtract $2 and $3, or 4 ft. and ft.? 8. What kind of fractions can be added or subtracted? Before fractions can be added or subtracted they must be expressed in similar fractional units. Fractions having the same kind of units or having related units can be added; thus, } and 1, or } yd. and } ft. (= 1⁄2yd.) can be changed to similar fractions and then added. But fractions whose units are unrelated, as } yd. and ‡ oz., cannot be added, because they cannot be changed to similar fractions. The least common denominator is 36. Changing the given fractions to 36ths, we find that }=}; 1= }; =. The sum of these fractions ==211. 1. What is the first step in the work? 3. What is the third step? Since the numbers are mixed numbers, the integers and fractions are added separately, and their sums are united. The sum of, §, and is The sum of 3, 2, and 7 is 12. The sum 1 is 13. 7 or 1% of 12 and 30. A bicycler rode 8 miles the first hour, 78 miles the second hour, 6 miles the third hour, and 8 miles the fourth hour. How many miles did he ride in the four hours? 31. Find the distance around a field 80 rods long and 607 rods wide. 32. Find the sum of the improper fractions, 28, 41, 17. 33. What number is that from which if 327 is taken, the remainder is 23,99? Subtracting fractions. What kind of whole numbers can be added? what kind of fractions? In adding like fractions we find the sum of the numerators; in subtracting like fractions we find the difference of the numerators. Observe the three steps in subtraction of fractions: 1. Change the fractions, if necessary, to like fractions. 2. Take the difference of their numerators. 3. Change the difference to its simplest form. 14. A boy wishes to buy a pair of skates costing $%, but he has only $3. What part of a dollar is lacking? 15. From take 4. 16. What fraction added to will give ? 17. From of an acre of land, subtract ğ of an acre. 18. What part of a teacher's salary remained after he had spent, and of it? 3 16 |