1 f5 addition and subtraction of fractions

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1 f5 addition and subtraction of fractions

  1. 1. Addition and Subtraction of Fractions<br />Frank Ma © 2011<br />
  2. 2. Addition and Subtraction of Fractions<br />Suppose a pizza is cut into 4 equal slices<br />
  3. 3. Addition and Subtraction of Fractions<br />Suppose a pizza is cut into 4 equal slices<br />
  4. 4. Addition and Subtraction of Fractions<br />Suppose a pizza is cut into 4 equal slices and Joe takes one slice or ¼ of the pizza, <br />1<br />4<br />
  5. 5. Addition and Subtraction of Fractions<br />Suppose a pizza is cut into 4 equal slices and Joe takes one slice or ¼ of the pizza, Mary takes two slices or 2/4 of the pizza, <br />1<br />2<br />4<br />4<br />
  6. 6. Addition and Subtraction of Fractions<br />Suppose a pizza is cut into 4 equal slices and Joe takes one slice or ¼ of the pizza, Mary takes two slices or 2/4 of the pizza, altogether they take <br />1<br />2<br />+<br />4<br />4<br />1<br />2<br />4<br />4<br />
  7. 7. Addition and Subtraction of Fractions<br />Suppose a pizza is cut into 4 equal slices and Joe takes one slice or ¼ of the pizza, Mary takes two slices or 2/4 of the pizza, altogether they take <br />1<br />2<br />3<br />+<br />=<br />4<br />4<br />4<br />of the entire pizza. <br />1<br />2<br />4<br />4<br />
  8. 8. Addition and Subtraction of Fractions<br />Suppose a pizza is cut into 4 equal slices and Joe takes one slice or ¼ of the pizza, Mary takes two slices or 2/4 of the pizza, altogether they take <br />1<br />2<br />3<br />+<br />=<br />4<br />4<br />4<br />of the entire pizza. In picture: <br />=<br />+<br />1<br />2<br />3<br />4<br />4<br />4<br />
  9. 9. Addition and Subtraction of Fractions<br />Suppose a pizza is cut into 4 equal slices and Joe takes one slice or ¼ of the pizza, Mary takes two slices or 2/4 of the pizza, altogether they take <br />1<br />2<br />3<br />+<br />=<br />4<br />4<br />4<br />of the entire pizza. In picture: <br />=<br />+<br />1<br />2<br />3<br />4<br />4<br />4<br />Addition and Subtraction of Fractions With the Same Denominator<br />
  10. 10. Addition and Subtraction of Fractions<br />Suppose a pizza is cut into 4 equal slices and Joe takes one slice or ¼ of the pizza, Mary takes two slices or 2/4 of the pizza, altogether they take <br />1<br />2<br />3<br />+<br />=<br />4<br />4<br />4<br />of the entire pizza. In picture: <br />=<br />+<br />1<br />2<br />3<br />4<br />4<br />4<br />Addition and Subtraction of Fractions With the Same Denominator<br />To add or subtract fractions of the same denominator, keep the same denominator, add or subtract the numerators<br />
  11. 11. Addition and Subtraction of Fractions<br />Suppose a pizza is cut into 4 equal slices and Joe takes one slice or ¼ of the pizza, Mary takes two slices or 2/4 of the pizza, altogether they take <br />1<br />2<br />3<br />+<br />=<br />4<br />4<br />4<br />of the entire pizza. In picture: <br />=<br />+<br />1<br />2<br />3<br />4<br />4<br />4<br />Addition and Subtraction of Fractions With the Same Denominator<br />To add or subtract fractions of the same denominator, keep the same denominator, add or subtract the numerators<br />a<br />b<br />a ± b<br />±<br />=<br />d<br />d<br />d<br />
  12. 12. Addition and Subtraction of Fractions<br />Suppose a pizza is cut into 4 equal slices and Joe takes one slice or ¼ of the pizza, Mary takes two slices or 2/4 of the pizza, altogether they take <br />1<br />2<br />3<br />+<br />=<br />4<br />4<br />4<br />of the entire pizza. In picture: <br />=<br />+<br />1<br />2<br />3<br />4<br />4<br />4<br />Addition and Subtraction of Fractions With the Same Denominator<br />To add or subtract fractions of the same denominator, keep the same denominator, add or subtract the numerators<br />,then simplify the result.<br />a<br />b<br />a ± b<br />±<br />=<br />d<br />d<br />d<br />
  13. 13. Addition and Subtraction of Fractions<br />Example A:<br />a.<br />7<br />11<br />+<br />12<br />12<br />
  14. 14. Addition and Subtraction of Fractions<br />Example A:<br />a.<br />7<br />7 + 11<br />11<br />+<br />=<br />=<br />12<br />12<br />12<br />
  15. 15. Addition and Subtraction of Fractions<br />Example A:<br />a.<br />7<br />7 + 11<br />18<br />11<br />+<br />=<br />=<br />12<br />12<br />12<br />12<br />
  16. 16. Addition and Subtraction of Fractions<br />Example A:<br />a.<br />7<br />7 + 11<br />18<br />11<br />18/6<br />=<br />+<br />=<br />=<br />=<br />12<br />12<br />12<br />12/6<br />12<br />
  17. 17. Addition and Subtraction of Fractions<br />Example A:<br />a.<br />7<br />7 + 11<br />18<br />3<br />11<br />18/6<br />=<br />+<br />=<br />=<br />=<br />12<br />12<br />12<br />2<br />12/6<br />12<br />
  18. 18. Addition and Subtraction of Fractions<br />Example A:<br />a.<br />7<br />7 + 11<br />18<br />3<br />11<br />18/6<br />=<br />+<br />=<br />=<br />=<br />12<br />12<br />12<br />2<br />12/6<br />12<br />8<br />4<br />2<br />b.<br />+<br />=<br />–<br />15<br />15<br />15<br />
  19. 19. Addition and Subtraction of Fractions<br />Example A:<br />a.<br />7<br />7 + 11<br />18<br />3<br />11<br />18/6<br />=<br />+<br />=<br />=<br />=<br />12<br />12<br />12<br />2<br />12/6<br />12<br />8 + 4 – 2 <br />8<br />4<br />2<br />b.<br />+<br />=<br />=<br />–<br />15<br />15<br />15<br />15<br />
  20. 20. Addition and Subtraction of Fractions<br />Example A:<br />a.<br />7<br />7 + 11<br />18<br />3<br />11<br />18/6<br />=<br />+<br />=<br />=<br />=<br />12<br />12<br />12<br />2<br />12/6<br />12<br />8 + 4 – 2 <br />8<br />4<br />2<br />10<br />b.<br />+<br />=<br />=<br />–<br />15<br />15<br />15<br />15<br />15<br />
  21. 21. Addition and Subtraction of Fractions<br />Example A:<br />a.<br />7<br />7 + 11<br />18<br />3<br />11<br />18/6<br />=<br />+<br />=<br />=<br />=<br />12<br />12<br />12<br />2<br />12/6<br />12<br />8 + 4 – 2 <br />2<br />8<br />4<br />2<br />10<br />b.<br />+<br />=<br />=<br />=<br />–<br />3<br />15<br />15<br />15<br />15<br />15<br />
  22. 22. Addition and Subtraction of Fractions<br />Example A:<br />a.<br />7<br />7 + 11<br />18<br />3<br />11<br />18/6<br />=<br />+<br />=<br />=<br />=<br />12<br />12<br />12<br />2<br />12/6<br />12<br />8 + 4 – 2 <br />2<br />8<br />4<br />2<br />10<br />b.<br />+<br />=<br />=<br />=<br />–<br />3<br />15<br />15<br />15<br />15<br />15<br />Fractions with different denominators can’t be added directly since the “size” of the fractions don’t match. <br />
  23. 23. Addition and Subtraction of Fractions<br />Example A:<br />a.<br />7<br />7 + 11<br />18<br />3<br />11<br />18/6<br />=<br />+<br />=<br />=<br />=<br />12<br />12<br />12<br />2<br />12/6<br />12<br />8 + 4 – 2 <br />2<br />8<br />4<br />2<br />10<br />b.<br />+<br />=<br />=<br />=<br />–<br />3<br />15<br />15<br />15<br />15<br />15<br />Fractions with different denominators can’t be added directly since the “size” of the fractions don’t match. For example <br />1<br />2<br />
  24. 24. Addition and Subtraction of Fractions<br />Example A:<br />a.<br />7<br />7 + 11<br />18<br />3<br />11<br />18/6<br />=<br />+<br />=<br />=<br />=<br />12<br />12<br />12<br />2<br />12/6<br />12<br />8 + 4 – 2 <br />2<br />8<br />4<br />2<br />10<br />b.<br />+<br />=<br />=<br />=<br />–<br />3<br />15<br />15<br />15<br />15<br />15<br />Fractions with different denominators can’t be added directly since the “size” of the fractions don’t match. For example <br />+<br />1<br />1<br />3<br />2<br />
  25. 25. Addition and Subtraction of Fractions<br />Example A:<br />a.<br />7<br />7 + 11<br />18<br />3<br />11<br />18/6<br />=<br />+<br />=<br />=<br />=<br />12<br />12<br />12<br />2<br />12/6<br />12<br />8 + 4 – 2 <br />2<br />8<br />4<br />2<br />10<br />b.<br />+<br />=<br />=<br />=<br />–<br />3<br />15<br />15<br />15<br />15<br />15<br />Fractions with different denominators can’t be added directly since the “size” of the fractions don’t match. For example <br />=<br />+<br />?<br />1<br />1<br />3<br />2<br />?<br />
  26. 26. Addition and Subtraction of Fractions<br />Example A:<br />a.<br />7<br />7 + 11<br />18<br />3<br />11<br />18/6<br />=<br />+<br />=<br />=<br />=<br />12<br />12<br />12<br />2<br />12/6<br />12<br />8 + 4 – 2 <br />2<br />8<br />4<br />2<br />10<br />b.<br />+<br />=<br />=<br />=<br />–<br />3<br />15<br />15<br />15<br />15<br />15<br />Fractions with different denominators can’t be added directly since the “size” of the fractions don’t match. For example <br />=<br />+<br />?<br />1<br />1<br />3<br />2<br />?<br />To add them, first find the LCD of ½ and 1/3, which is 6.<br />
  27. 27. Addition and Subtraction of Fractions<br />Example A:<br />a.<br />7<br />7 + 11<br />18<br />3<br />11<br />18/6<br />=<br />+<br />=<br />=<br />=<br />12<br />12<br />12<br />2<br />12/6<br />12<br />8 + 4 – 2 <br />2<br />8<br />4<br />2<br />10<br />b.<br />+<br />=<br />=<br />=<br />–<br />3<br />15<br />15<br />15<br />15<br />15<br />Fractions with different denominators can’t be added directly since the “size” of the fractions don’t match. For example <br />=<br />+<br />?<br />1<br />1<br />3<br />2<br />?<br />To add them, first find the LCD of ½ and 1/3, which is 6. <br />We then cut each pizza into 6 slices. <br />
  28. 28. Addition and Subtraction of Fractions<br />Example A:<br />a.<br />7<br />7 + 11<br />18<br />3<br />11<br />18/6<br />=<br />+<br />=<br />=<br />=<br />12<br />12<br />12<br />2<br />12/6<br />12<br />8 + 4 – 2 <br />2<br />8<br />4<br />2<br />10<br />b.<br />+<br />=<br />=<br />=<br />–<br />3<br />15<br />15<br />15<br />15<br />15<br />Fractions with different denominators can’t be added directly since the “size” of the fractions don’t match. For example <br />=<br />+<br />?<br />1<br />1<br />3<br />2<br />?<br />To add them, first find the LCD of ½ and 1/3, which is 6. <br />We then cut each pizza into 6 slices. Both fractions may be converted to have the denominator 6. <br />
  29. 29. Addition and Subtraction of Fractions<br />Example A:<br />a.<br />7<br />7 + 11<br />18<br />3<br />11<br />18/6<br />=<br />+<br />=<br />=<br />=<br />12<br />12<br />12<br />2<br />12/6<br />12<br />8 + 4 – 2 <br />2<br />8<br />4<br />2<br />10<br />b.<br />+<br />=<br />=<br />=<br />–<br />3<br />15<br />15<br />15<br />15<br />15<br />Fractions with different denominators can’t be added directly since the “size” of the fractions don’t match. For example <br />=<br />+<br />?<br />1<br />1<br />3<br />2<br />?<br />To add them, first find the LCD of ½ and 1/3, which is 6. <br />We then cut each pizza into 6 slices. Both fractions may be converted to have the denominator 6. Specifically, <br />1<br />3<br />1<br />2<br />=<br />=<br />2<br />6<br />3<br />6<br />
  30. 30. Addition and Subtraction of Fractions<br />Example A:<br />a.<br />7<br />7 + 11<br />18<br />3<br />11<br />18/6<br />=<br />+<br />=<br />=<br />=<br />12<br />12<br />12<br />2<br />12/6<br />12<br />8 + 4 – 2 <br />2<br />8<br />4<br />2<br />10<br />b.<br />+<br />=<br />=<br />=<br />–<br />3<br />15<br />15<br />15<br />15<br />15<br />Fractions with different denominators can’t be added directly since the “size” of the fractions don’t match. For example <br />=<br />+<br />?<br />1<br />1<br />3<br />2<br />?<br />To add them, first find the LCD of ½ and 1/3, which is 6. <br />We then cut each pizza into 6 slices. Both fractions may be converted to have the denominator 6. Specifically, <br />1<br />3<br />1<br />2<br />=<br />=<br />2<br />6<br />3<br />6<br />
  31. 31. Addition and Subtraction of Fractions<br />Example A:<br />a.<br />7<br />7 + 11<br />18<br />3<br />11<br />18/6<br />=<br />+<br />=<br />=<br />=<br />12<br />12<br />12<br />2<br />12/6<br />12<br />8 + 4 – 2 <br />2<br />8<br />4<br />2<br />10<br />b.<br />+<br />=<br />=<br />=<br />–<br />3<br />15<br />15<br />15<br />15<br />15<br />Fractions with different denominators can’t be added directly since the “size” of the fractions don’t match. For example <br />=<br />+<br />?<br />1<br />3<br />3<br />6<br />?<br />To add them, first find the LCD of ½ and 1/3, which is 6. <br />We then cut each pizza into 6 slices. Both fractions may be converted to have the denominator 6. Specifically, <br />1<br />3<br />1<br />2<br />=<br />=<br />2<br />6<br />3<br />6<br />
  32. 32. Addition and Subtraction of Fractions<br />Example A:<br />a.<br />7<br />7 + 11<br />18<br />3<br />11<br />18/6<br />=<br />+<br />=<br />=<br />=<br />12<br />12<br />12<br />2<br />12/6<br />12<br />8 + 4 – 2 <br />2<br />8<br />4<br />2<br />10<br />b.<br />+<br />=<br />=<br />=<br />–<br />3<br />15<br />15<br />15<br />15<br />15<br />Fractions with different denominators can’t be added directly since the “size” of the fractions don’t match. For example <br />=<br />+<br />?<br />1<br />3<br />3<br />6<br />?<br />To add them, first find the LCD of ½ and 1/3, which is 6. <br />We then cut each pizza into 6 slices. Both fractions may be converted to have the denominator 6. Specifically, <br />1<br />3<br />1<br />2<br />=<br />=<br />2<br />6<br />3<br />6<br />
  33. 33. Addition and Subtraction of Fractions<br />Example A:<br />a.<br />7<br />7 + 11<br />18<br />3<br />11<br />18/6<br />=<br />+<br />=<br />=<br />=<br />12<br />12<br />12<br />2<br />12/6<br />12<br />8 + 4 – 2 <br />2<br />8<br />4<br />2<br />10<br />b.<br />+<br />=<br />=<br />=<br />–<br />3<br />15<br />15<br />15<br />15<br />15<br />Fractions with different denominators can’t be added directly since the “size” of the fractions don’t match. For example <br />=<br />+<br />?<br />2<br />3<br />6<br />6<br />?<br />To add them, first find the LCD of ½ and 1/3, which is 6. <br />We then cut each pizza into 6 slices. Both fractions may be converted to have the denominator 6. Specifically, <br />1<br />3<br />1<br />2<br />=<br />=<br />2<br />6<br />3<br />6<br />
  34. 34. Addition and Subtraction of Fractions<br />Example A:<br />a.<br />7<br />7 + 11<br />18<br />3<br />11<br />18/6<br />=<br />+<br />=<br />=<br />=<br />12<br />12<br />12<br />2<br />12/6<br />12<br />8 + 4 – 2 <br />2<br />8<br />4<br />2<br />10<br />b.<br />+<br />=<br />=<br />=<br />–<br />3<br />15<br />15<br />15<br />15<br />15<br />Fractions with different denominators can’t be added directly since the “size” of the fractions don’t match. For example <br />=<br />+<br />5<br />2<br />3<br />6<br />6<br />6<br />To add them, first find the LCD of ½ and 1/3, which is 6. <br />We then cut each pizza into 6 slices. Both fractions may be converted to have the denominator 6. Specifically, <br />1<br />3<br />1<br />2<br />1<br />1<br />3<br />2<br />5<br />=<br />=<br />Hence, <br />+<br />=<br />+<br />=<br />2<br />6<br />3<br />6<br />2<br />3<br />6<br />6<br />6<br />
  35. 35. Addition and Subtraction of Fractions<br />We need to convert fractions of different denominators to <br />a common denominator in order to add or subtract them.<br />
  36. 36. Addition and Subtraction of Fractions<br />We need to convert fractions of different denominators to <br />a common denominator in order to add or subtract them. <br />The easiest common denominator to use is the LCD, the least common denominator. <br />
  37. 37. Addition and Subtraction of Fractions<br />We need to convert fractions of different denominators to <br />a common denominator in order to add or subtract them. <br />The easiest common denominator to use is the LCD, the least common denominator. We list the steps below.<br />
  38. 38. Addition and Subtraction of Fractions<br />We need to convert fractions of different denominators to <br />a common denominator in order to add or subtract them. <br />The easiest common denominator to use is the LCD, the least common denominator. We list the steps below.<br />Addition and Subtraction of Fractions With the Different Denominator<br />
  39. 39. Addition and Subtraction of Fractions<br />We need to convert fractions of different denominators to <br />a common denominator in order to add or subtract them. <br />The easiest common denominator to use is the LCD, the least common denominator. We list the steps below.<br />Addition and Subtraction of Fractions With the Different Denominator<br />1. Find their LCD<br />
  40. 40. Addition and Subtraction of Fractions<br />We need to convert fractions of different denominators to <br />a common denominator in order to add or subtract them. <br />The easiest common denominator to use is the LCD, the least common denominator. We list the steps below.<br />Addition and Subtraction of Fractions With the Different Denominator<br />1. Find their LCD<br />2. Convert all the different-denominator-fractions to the have <br />the LCD as the denominator.<br />
  41. 41. Addition and Subtraction of Fractions<br />We need to convert fractions of different denominators to <br />a common denominator in order to add or subtract them. <br />The easiest common denominator to use is the LCD, the least common denominator. We list the steps below.<br />Addition and Subtraction of Fractions With the Different Denominator<br />1. Find their LCD<br />2. Convert all the different-denominator-fractions to the have <br />the LCD as the denominator.<br />3. Add and subtract the adjusted fractions then simplify the result.<br />
  42. 42. Addition and Subtraction of Fractions<br />We need to convert fractions of different denominators to <br />a common denominator in order to add or subtract them. <br />The easiest common denominator to use is the LCD, the least common denominator. We list the steps below.<br />Addition and Subtraction of Fractions With the Different Denominator<br />1. Find their LCD<br />2. Convert all the different-denominator-fractions to the have <br />the LCD as the denominator.<br />3. Add and subtract the adjusted fractions then simplify the result.<br />5<br />3<br />a. <br />Example B: <br />+<br />6<br />8 <br />
  43. 43. Addition and Subtraction of Fractions<br />We need to convert fractions of different denominators to <br />a common denominator in order to add or subtract them. <br />The easiest common denominator to use is the LCD, the least common denominator. We list the steps below.<br />Addition and Subtraction of Fractions With the Different Denominator<br />1. Find their LCD<br />2. Convert all the different-denominator-fractions to the have <br />the LCD as the denominator.<br />3. Add and subtract the adjusted fractions then simplify the result.<br />5<br />3<br />a. <br />Example B: <br />+<br />6<br />8 <br />Step 1: To find the LCD, list the multiples of 8 <br />
  44. 44. Addition and Subtraction of Fractions<br />We need to convert fractions of different denominators to <br />a common denominator in order to add or subtract them. <br />The easiest common denominator to use is the LCD, the least common denominator. We list the steps below.<br />Addition and Subtraction of Fractions With the Different Denominator<br />1. Find their LCD<br />2. Convert all the different-denominator-fractions to the have <br />the LCD as the denominator.<br />3. Add and subtract the adjusted fractions then simplify the result.<br />5<br />3<br />a. <br />Example B: <br />+<br />6<br />8 <br />Step 1: To find the LCD, list the multiples of 8 which are <br />8, 16, 24, ..<br />
  45. 45. Addition and Subtraction of Fractions<br />We need to convert fractions of different denominators to <br />a common denominator in order to add or subtract them. <br />The easiest common denominator to use is the LCD, the least common denominator. We list the steps below.<br />Addition and Subtraction of Fractions With the Different Denominator<br />1. Find their LCD<br />2. Convert all the different-denominator-fractions to the have <br />the LCD as the denominator.<br />3. Add and subtract the adjusted fractions then simplify the result.<br />5<br />3<br />a. <br />Example B: <br />+<br />6<br />8 <br />Step 1: To find the LCD, list the multiples of 8 which are <br />8, 16, 24, .. we see that the LCD is 24. <br />
  46. 46. Addition and Subtraction of Fractions<br />Step 2: Convert each fraction to have 24 as the denominator. <br />
  47. 47. Addition and Subtraction of Fractions<br />Step 2: Convert each fraction to have 24 as the denominator. <br />5<br />5<br />For , the new numerator is 24 * = 20, <br />6<br />6<br />
  48. 48. Addition and Subtraction of Fractions<br />Step 2: Convert each fraction to have 24 as the denominator. <br />5<br />5<br />5<br />20<br />For , the new numerator is 24 * = 20, hence <br />=<br />6<br />6<br />6<br />24<br />
  49. 49. Addition and Subtraction of Fractions<br />Step 2: Convert each fraction to have 24 as the denominator. <br />5<br />5<br />5<br />20<br />For , the new numerator is 24 * = 20, hence <br />=<br />6<br />6<br />6<br />24<br />3<br />3<br />For , the new numerator is 24 * = 9,<br />8<br />8<br />
  50. 50. Addition and Subtraction of Fractions<br />Step 2: Convert each fraction to have 24 as the denominator. <br />5<br />5<br />5<br />20<br />For , the new numerator is 24 * = 20, hence <br />=<br />6<br />6<br />6<br />24<br />3<br />3<br />3<br />9<br />For , the new numerator is 24 * = 9, hence <br />=<br />8<br />8<br />8<br />24<br />
  51. 51. Addition and Subtraction of Fractions<br />Step 2: Convert each fraction to have 24 as the denominator. <br />5<br />5<br />5<br />20<br />For , the new numerator is 24 * = 20, hence <br />=<br />6<br />6<br />6<br />24<br />3<br />3<br />3<br />9<br />For , the new numerator is 24 * = 9, hence <br />=<br />8<br />8<br />8<br />24<br />Step 3: Add the converted fractions. <br />
  52. 52. Addition and Subtraction of Fractions<br />Step 2: Convert each fraction to have 24 as the denominator. <br />5<br />5<br />5<br />20<br />For , the new numerator is 24 * = 20, hence <br />=<br />6<br />6<br />6<br />24<br />3<br />3<br />3<br />9<br />For , the new numerator is 24 * = 9, hence <br />=<br />8<br />8<br />8<br />24<br />Step 3: Add the converted fractions. <br />5<br />3<br />20<br />9<br />+<br />=<br />+<br />6<br />8 <br />24<br />24<br />
  53. 53. Addition and Subtraction of Fractions<br />Step 2: Convert each fraction to have 24 as the denominator. <br />5<br />5<br />5<br />20<br />For , the new numerator is 24 * = 20, hence <br />=<br />6<br />6<br />6<br />24<br />3<br />3<br />3<br />9<br />For , the new numerator is 24 * = 9, hence <br />=<br />8<br />8<br />8<br />24<br />Step 3: Add the converted fractions. <br />5<br />3<br />20<br />9<br />29<br />+<br />=<br />+<br />=<br />6<br />8 <br />24<br />24<br />24<br />
  54. 54. Addition and Subtraction of Fractions<br />Step 2: Convert each fraction to have 24 as the denominator. <br />5<br />5<br />5<br />20<br />For , the new numerator is 24 * = 20, hence <br />=<br />6<br />6<br />6<br />24<br />3<br />3<br />3<br />9<br />For , the new numerator is 24 * = 9, hence <br />=<br />8<br />8<br />8<br />24<br />Step 3: Add the converted fractions. <br />5<br />3<br />20<br />9<br />29<br />+<br />=<br />+<br />=<br />6<br />8 <br />24<br />24<br />24<br />7<br />5<br />9<br />b.<br />+<br />– <br />12<br />8 <br />16<br />
  55. 55. Addition and Subtraction of Fractions<br />Step 2: Convert each fraction to have 24 as the denominator. <br />5<br />5<br />5<br />20<br />For , the new numerator is 24 * = 20, hence <br />=<br />6<br />6<br />6<br />24<br />3<br />3<br />3<br />9<br />For , the new numerator is 24 * = 9, hence <br />=<br />8<br />8<br />8<br />24<br />Step 3: Add the converted fractions. <br />5<br />3<br />20<br />9<br />29<br />+<br />=<br />+<br />=<br />6<br />8 <br />24<br />24<br />24<br />7<br />5<br />9<br />b.<br />+<br />– <br />The LCD is 48.<br />12<br />8 <br />16<br />
  56. 56. Addition and Subtraction of Fractions<br />Step 2: Convert each fraction to have 24 as the denominator. <br />5<br />5<br />5<br />20<br />For , the new numerator is 24 * = 20, hence <br />=<br />6<br />6<br />6<br />24<br />3<br />3<br />3<br />9<br />For , the new numerator is 24 * = 9, hence <br />=<br />8<br />8<br />8<br />24<br />Step 3: Add the converted fractions. <br />5<br />3<br />20<br />9<br />29<br />+<br />=<br />+<br />=<br />6<br />8 <br />24<br />24<br />24<br />7<br />5<br />9<br />b.<br />+<br />– <br />The LCD is 48.<br />Convert: <br />12<br />8 <br />16<br />
  57. 57. Addition and Subtraction of Fractions<br />Step 2: Convert each fraction to have 24 as the denominator. <br />5<br />5<br />5<br />20<br />For , the new numerator is 24 * = 20, hence <br />=<br />6<br />6<br />6<br />24<br />3<br />3<br />3<br />9<br />For , the new numerator is 24 * = 9, hence <br />=<br />8<br />8<br />8<br />24<br />Step 3: Add the converted fractions. <br />5<br />3<br />20<br />9<br />29<br />+<br />=<br />+<br />=<br />6<br />8 <br />24<br />24<br />24<br />7<br />5<br />9<br />b.<br />+<br />– <br />The LCD is 48.<br />Convert: <br />12<br />8 <br />16<br />7<br />= 28<br />48 *<br />12<br />
  58. 58. Addition and Subtraction of Fractions<br />Step 2: Convert each fraction to have 24 as the denominator. <br />5<br />5<br />5<br />20<br />For , the new numerator is 24 * = 20, hence <br />=<br />6<br />6<br />6<br />24<br />3<br />3<br />3<br />9<br />For , the new numerator is 24 * = 9, hence <br />=<br />8<br />8<br />8<br />24<br />Step 3: Add the converted fractions. <br />5<br />3<br />20<br />9<br />29<br />+<br />=<br />+<br />=<br />6<br />8 <br />24<br />24<br />24<br />7<br />5<br />9<br />b.<br />+<br />– <br />The LCD is 48.<br />Convert: <br />12<br />8 <br />16<br />7<br />7<br />28<br />= 28<br />48 *<br />so<br />= <br />12<br />12<br />48<br />
  59. 59. Addition and Subtraction of Fractions<br />Step 2: Convert each fraction to have 24 as the denominator. <br />5<br />5<br />5<br />20<br />For , the new numerator is 24 * = 20, hence <br />=<br />6<br />6<br />6<br />24<br />3<br />3<br />3<br />9<br />For , the new numerator is 24 * = 9, hence <br />=<br />8<br />8<br />8<br />24<br />Step 3: Add the converted fractions. <br />5<br />3<br />20<br />9<br />29<br />+<br />=<br />+<br />=<br />6<br />8 <br />24<br />24<br />24<br />7<br />5<br />9<br />b.<br />+<br />– <br />The LCD is 48.<br />Convert: <br />12<br />8 <br />16<br />7<br />7<br />28<br />= 28<br />48 *<br />so<br />= <br />12<br />12<br />48<br />5<br />= 30<br />48 *<br />8<br />
  60. 60. Addition and Subtraction of Fractions<br />Step 2: Convert each fraction to have 24 as the denominator. <br />5<br />5<br />5<br />20<br />For , the new numerator is 24 * = 20, hence <br />=<br />6<br />6<br />6<br />24<br />3<br />3<br />3<br />9<br />For , the new numerator is 24 * = 9, hence <br />=<br />8<br />8<br />8<br />24<br />Step 3: Add the converted fractions. <br />5<br />3<br />20<br />9<br />29<br />+<br />=<br />+<br />=<br />6<br />8 <br />24<br />24<br />24<br />7<br />5<br />9<br />b.<br />+<br />– <br />The LCD is 48.<br />Convert: <br />12<br />8 <br />16<br />7<br />7<br />28<br />= 28<br />48 *<br />so<br />= <br />12<br />12<br />48<br />5<br />5<br />30<br />= 30<br />48 *<br />so<br />= <br />8<br />8<br />48<br />
  61. 61. Addition and Subtraction of Fractions<br />Step 2: Convert each fraction to have 24 as the denominator. <br />5<br />5<br />5<br />20<br />For , the new numerator is 24 * = 20, hence <br />=<br />6<br />6<br />6<br />24<br />3<br />3<br />3<br />9<br />For , the new numerator is 24 * = 9, hence <br />=<br />8<br />8<br />8<br />24<br />Step 3: Add the converted fractions. <br />5<br />3<br />20<br />9<br />29<br />+<br />=<br />+<br />=<br />6<br />8 <br />24<br />24<br />24<br />7<br />5<br />9<br />b.<br />+<br />– <br />The LCD is 48.<br />Convert: <br />12<br />8 <br />16<br />7<br />7<br />28<br />= 28<br />48 *<br />so<br />= <br />12<br />12<br />48<br />5<br />5<br />30<br />= 30<br />48 *<br />so<br />= <br />8<br />8<br />48<br />9<br />= 27<br />48 *<br />16<br />
  62. 62. Addition and Subtraction of Fractions<br />Step 2: Convert each fraction to have 24 as the denominator. <br />5<br />5<br />5<br />20<br />For , the new numerator is 24 * = 20, hence <br />=<br />6<br />6<br />6<br />24<br />3<br />3<br />3<br />9<br />For , the new numerator is 24 * = 9, hence <br />=<br />8<br />8<br />8<br />24<br />Step 3: Add the converted fractions. <br />5<br />3<br />20<br />9<br />29<br />+<br />=<br />+<br />=<br />6<br />8 <br />24<br />24<br />24<br />7<br />5<br />9<br />b.<br />+<br />– <br />The LCD is 48.<br />Convert: <br />12<br />8 <br />16<br />7<br />7<br />28<br />= 28<br />48 *<br />so<br />= <br />12<br />12<br />48<br />5<br />5<br />30<br />= 30<br />48 *<br />so<br />= <br />8<br />8<br />48<br />9<br />9<br />27<br />= 27<br />48 *<br />so<br />= <br />16<br />16<br />48<br />
  63. 63. Addition and Subtraction of Fractions<br />Step 2: Convert each fraction to have 24 as the denominator. <br />5<br />5<br />5<br />20<br />For , the new numerator is 24 * = 20, hence <br />=<br />6<br />6<br />6<br />24<br />3<br />3<br />3<br />9<br />For , the new numerator is 24 * = 9, hence <br />=<br />8<br />8<br />8<br />24<br />Step 3: Add the converted fractions. <br />5<br />3<br />20<br />9<br />29<br />+<br />=<br />+<br />=<br />6<br />8 <br />24<br />24<br />24<br />7<br />5<br />9<br />b.<br />+<br />– <br />The LCD is 48.<br />Convert: <br />12<br />8 <br />16<br />7<br />7<br />28<br />30<br />27<br />28<br />= 28<br />=<br />48 *<br />so<br />= <br />+<br />– <br />12<br />12<br />48<br />48<br />48<br />48<br />5<br />5<br />30<br />= 30<br />48 *<br />so<br />= <br />8<br />8<br />48<br />9<br />9<br />27<br />= 27<br />48 *<br />so<br />= <br />16<br />16<br />48<br />
  64. 64. Addition and Subtraction of Fractions<br />Step 2: Convert each fraction to have 24 as the denominator. <br />5<br />5<br />5<br />20<br />For , the new numerator is 24 * = 20, hence <br />=<br />6<br />6<br />6<br />24<br />3<br />3<br />3<br />9<br />For , the new numerator is 24 * = 9, hence <br />=<br />8<br />8<br />8<br />24<br />Step 3: Add the converted fractions. <br />5<br />3<br />20<br />9<br />29<br />+<br />=<br />+<br />=<br />6<br />8 <br />24<br />24<br />24<br />7<br />5<br />9<br />b.<br />+<br />– <br />The LCD is 48.<br />Convert: <br />12<br />8 <br />16<br />7<br />7<br />28<br />30<br />27<br />28<br />= 28<br />=<br />48 *<br />so<br />= <br />+<br />– <br />12<br />12<br />48<br />48<br />48<br />48<br />5<br />5<br />28 + 30 – 27 <br />30<br />= 30<br />48 *<br />so<br />= <br />=<br />48<br />8<br />8<br />48<br />9<br />9<br />27<br />= 27<br />48 *<br />so<br />= <br />16<br />16<br />48<br />
  65. 65. Addition and Subtraction of Fractions<br />Step 2: Convert each fraction to have 24 as the denominator. <br />5<br />5<br />5<br />20<br />For , the new numerator is 24 * = 20, hence <br />=<br />6<br />6<br />6<br />24<br />3<br />3<br />3<br />9<br />For , the new numerator is 24 * = 9, hence <br />=<br />8<br />8<br />8<br />24<br />Step 3: Add the converted fractions. <br />5<br />3<br />20<br />9<br />29<br />+<br />=<br />+<br />=<br />6<br />8 <br />24<br />24<br />24<br />7<br />5<br />9<br />b.<br />+<br />– <br />The LCD is 48.<br />Convert: <br />12<br />8 <br />16<br />7<br />7<br />28<br />30<br />27<br />28<br />= 28<br />=<br />48 *<br />so<br />= <br />+<br />– <br />12<br />12<br />48<br />48<br />48<br />48<br />5<br />5<br />28 + 30 – 27 <br />30<br />= 30<br />48 *<br />so<br />= <br />=<br />48<br />8<br />8<br />48<br />31<br />9<br />9<br />27<br />=<br />= 27<br />48 *<br />so<br />= <br />48<br />16<br />16<br />48<br />
  66. 66. Addition and Subtraction of Fractions<br />Step 2: Convert each fraction to have 24 as the denominator. <br />5<br />5<br />5<br />20<br />For , the new numerator is 24 * = 20, hence <br />=<br />6<br />6<br />6<br />24<br />3<br />3<br />3<br />9<br />For , the new numerator is 24 * = 9, hence <br />=<br />8<br />8<br />8<br />24<br />Step 3: Add the converted fractions. <br />5<br />3<br />20<br />9<br />29<br />+<br />=<br />+<br />=<br />6<br />8 <br />24<br />24<br />24<br />7<br />5<br />9<br />b.<br />+<br />– <br />The LCD is 48.<br />Convert: <br />12<br />8 <br />16<br />7<br />7<br />28<br />30<br />27<br />28<br />= 28<br />=<br />48 *<br />so<br />= <br />+<br />– <br />12<br />12<br />48<br />48<br />48<br />48<br />5<br />5<br />28 + 30 – 27 <br />30<br />= 30<br />48 *<br />so<br />= <br />=<br />48<br />8<br />8<br />48<br />31<br />9<br />9<br />27<br />=<br />= 27<br />48 *<br />so<br />= <br />48<br />16<br />16<br />48<br />
  67. 67. Addition and Subtraction of Fractions<br />We introduce the following Multiplier-Method to add or subtract fractions to reduce the amount of repetitive copying. <br />
  68. 68. Addition and Subtraction of Fractions<br />We introduce the following Multiplier-Method to add or subtract fractions to reduce the amount of repetitive copying. This method is based on the fact that if we multiply a quantity x by a, then divide by a, we get back x. <br />
  69. 69. Addition and Subtraction of Fractions<br />We introduce the following Multiplier-Method to add or subtract fractions to reduce the amount of repetitive copying. This method is based on the fact that if we multiply a quantity x by a, then divide by a, we get back x. <br />For example, 2* 5 / 5 <br />
  70. 70. Addition and Subtraction of Fractions<br />We introduce the following Multiplier-Method to add or subtract fractions to reduce the amount of repetitive copying. This method is based on the fact that if we multiply a quantity x by a, then divide by a, we get back x. <br />For example, 2* 5 / 5 = 10/5 = 2, <br />
  71. 71. Addition and Subtraction of Fractions<br />We introduce the following Multiplier-Method to add or subtract fractions to reduce the amount of repetitive copying. This method is based on the fact that if we multiply a quantity x by a, then divide by a, we get back x. <br />For example, 2* 5 / 5 = 10/5 = 2, 3* 8 / 8 = <br />
  72. 72. Addition and Subtraction of Fractions<br />We introduce the following Multiplier-Method to add or subtract fractions to reduce the amount of repetitive copying. This method is based on the fact that if we multiply a quantity x by a, then divide by a, we get back x. <br />For example, 2* 5 / 5 = 10/5 = 2, 3* 8 / 8 = 24/8 = 3.<br />
  73. 73. Addition and Subtraction of Fractions<br />We introduce the following Multiplier-Method to add or subtract fractions to reduce the amount of repetitive copying. This method is based on the fact that if we multiply a quantity x by a, then divide by a, we get back x. <br />For example, 2* 5 / 5 = 10/5 = 2, 3* 8 / 8 = 24/8 = 3.<br />Multiplier Method(for adding and subtracting fractions)<br />
  74. 74. Addition and Subtraction of Fractions<br />We introduce the following Multiplier-Method to add or subtract fractions to reduce the amount of repetitive copying. This method is based on the fact that if we multiply a quantity x by a, then divide by a, we get back x. <br />For example, 2* 5 / 5 = 10/5 = 2, 3* 8 / 8 = 24/8 = 3.<br />Multiplier Method(for adding and subtracting fractions)<br />To add or subtract fractions, multiply the problem by the LCD (expand it distributive using law), then divide by the LCD. <br />
  75. 75. Addition and Subtraction of Fractions<br />We introduce the following Multiplier-Method to add or subtract fractions to reduce the amount of repetitive copying. This method is based on the fact that if we multiply a quantity x by a, then divide by a, we get back x. <br />For example, 2* 5 / 5 = 10/5 = 2, 3* 8 / 8 = 24/8 = 3.<br />Multiplier Method(for adding and subtracting fractions)<br />To add or subtract fractions, multiply the problem by the LCD (expand it distributive using law), then divide by the LCD. <br />5<br />3<br />Example C: a.<br />+<br />6<br />8 <br />
  76. 76. Addition and Subtraction of Fractions<br />We introduce the following Multiplier-Method to add or subtract fractions to reduce the amount of repetitive copying. This method is based on the fact that if we multiply a quantity x by a, then divide by a, we get back x. <br />For example, 2* 5 / 5 = 10/5 = 2, 3* 8 / 8 = 24/8 = 3.<br />Multiplier Method(for adding and subtracting fractions)<br />To add or subtract fractions, multiply the problem by the LCD (expand it distributive using law), then divide by the LCD. <br />5<br />3<br />Example C: a. <br />The LCD is 24. <br />+<br />6<br />8 <br />
  77. 77. Addition and Subtraction of Fractions<br />We introduce the following Multiplier-Method to add or subtract fractions to reduce the amount of repetitive copying. This method is based on the fact that if we multiply a quantity x by a, then divide by a, we get back x. <br />For example, 2* 5 / 5 = 10/5 = 2, 3* 8 / 8 = 24/8 = 3.<br />Multiplier Method(for adding and subtracting fractions)<br />To add or subtract fractions, multiply the problem by the LCD (expand it distributive using law), then divide by the LCD. <br />5<br />3<br />Example C: a. <br />The LCD is 24. Multiply the problem by 24, then divide by 24.<br />+<br />6<br />8 <br />
  78. 78. Addition and Subtraction of Fractions<br />We introduce the following Multiplier-Method to add or subtract fractions to reduce the amount of repetitive copying. This method is based on the fact that if we multiply a quantity x by a, then divide by a, we get back x. <br />For example, 2* 5 / 5 = 10/5 = 2, 3* 8 / 8 = 24/8 = 3.<br />Multiplier Method(for adding and subtracting fractions)<br />To add or subtract fractions, multiply the problem by the LCD (expand it distributive using law), then divide by the LCD. <br />5<br />3<br />Example C: a. <br />The LCD is 24. Multiply the problem by 24, then divide by 24.<br />+<br />6<br />8 <br />5<br />3<br />24 / 24 <br />(<br />)<br />+<br />*<br />6<br />8 <br />
  79. 79. Addition and Subtraction of Fractions<br />We introduce the following Multiplier-Method to add or subtract fractions to reduce the amount of repetitive copying. This method is based on the fact that if we multiply a quantity x by a, then divide by a, we get back x. <br />For example, 2* 5 / 5 = 10/5 = 2, 3* 8 / 8 = 24/8 = 3.<br />Multiplier Method(for adding and subtracting fractions)<br />To add or subtract fractions, multiply the problem by the LCD (expand it distributive using law), then divide by the LCD. <br />5<br />3<br />Example C: a. <br />The LCD is 24. Multiply the problem by 24, then divide by 24.<br />+<br />6<br />8 <br />4<br />5<br />3<br />24 / 24 <br />(<br />)<br />+<br />*<br />6<br />8 <br />
  80. 80. Addition and Subtraction of Fractions<br />We introduce the following Multiplier-Method to add or subtract fractions to reduce the amount of repetitive copying. This method is based on the fact that if we multiply a quantity x by a, then divide by a, we get back x. <br />For example, 2* 5 / 5 = 10/5 = 2, 3* 8 / 8 = 24/8 = 3.<br />Multiplier Method(for adding and subtracting fractions)<br />To add or subtract fractions, multiply the problem by the LCD (expand it distributive using law), then divide by the LCD. <br />5<br />3<br />Example C: a. <br />The LCD is 24. Multiply the problem by 24, then divide by 24.<br />+<br />6<br />8 <br />4<br />3<br />5<br />3<br />24 / 24 <br />(<br />)<br />+<br />*<br />6<br />8 <br />
  81. 81. Addition and Subtraction of Fractions<br />We introduce the following Multiplier-Method to add or subtract fractions to reduce the amount of repetitive copying. This method is based on the fact that if we multiply a quantity x by a, then divide by a, we get back x. <br />For example, 2* 5 / 5 = 10/5 = 2, 3* 8 / 8 = 24/8 = 3.<br />Multiplier Method(for adding and subtracting fractions)<br />To add or subtract fractions, multiply the problem by the LCD (expand it distributive using law), then divide by the LCD. <br />5<br />3<br />Example C: a. <br />The LCD is 24. Multiply the problem by 24, then divide by 24.<br />+<br />6<br />8 <br />4<br />3<br />5<br />3<br />24 / 24 = (4*5 + 3*3) / 24 <br />(<br />)<br />+<br />*<br />6<br />8 <br />
  82. 82. Addition and Subtraction of Fractions<br />We introduce the following Multiplier-Method to add or subtract fractions to reduce the amount of repetitive copying. This method is based on the fact that if we multiply a quantity x by a, then divide by a, we get back x. <br />For example, 2* 5 / 5 = 10/5 = 2, 3* 8 / 8 = 24/8 = 3.<br />Multiplier Method(for adding and subtracting fractions)<br />To add or subtract fractions, multiply the problem by the LCD (expand it distributive using law), then divide by the LCD. <br />5<br />3<br />Example C: a. <br />The LCD is 24. Multiply the problem by 24, then divide by 24.<br />+<br />6<br />8 <br />29<br />4<br />3<br />5<br />3<br />24 / 24 = (4*5 + 3*3) / 24 = 29/24 = <br />(<br />)<br />+<br />*<br />24 <br />6<br />8 <br />
  83. 83. Addition and Subtraction of Fractions<br />7<br />5<br />9<br />b.<br />+<br />– <br />12<br />8 <br />16<br />
  84. 84. Addition and Subtraction of Fractions<br />7<br />5<br />9<br />b.<br />+<br />– <br />12<br />8 <br />16<br />The LCD is 48. Multiply the problem by 48, expand the multiplication, divide the result by 48.<br />
  85. 85. Addition and Subtraction of Fractions<br />7<br />5<br />9<br />b.<br />+<br />– <br />12<br />8 <br />16<br />The LCD is 48. Multiply the problem by 48, expand the multiplication, divide the result by 48.<br />5<br />9<br />7<br />+<br />– <br />(<br />)<br />* 48 / 48<br />12<br />8 <br />16<br />
  86. 86. Addition and Subtraction of Fractions<br />7<br />5<br />9<br />b.<br />+<br />– <br />12<br />8 <br />16<br />The LCD is 48. Multiply the problem by 48, expand the multiplication, divide the result by 48.<br />4<br />5<br />9<br />7<br />+<br />– <br />(<br />)<br />* 48 / 48<br />12<br />8 <br />16<br />
  87. 87. Addition and Subtraction of Fractions<br />7<br />5<br />9<br />b.<br />+<br />– <br />12<br />8 <br />16<br />The LCD is 48. Multiply the problem by 48, expand the multiplication, divide the result by 48.<br />6<br />4<br />5<br />9<br />7<br />+<br />– <br />(<br />)<br />* 48 / 48<br />12<br />8 <br />16<br />
  88. 88. Addition and Subtraction of Fractions<br />7<br />5<br />9<br />b.<br />+<br />– <br />12<br />8 <br />16<br />The LCD is 48. Multiply the problem by 48, expand the multiplication, divide the result by 48.<br />6<br />3<br />4<br />5<br />9<br />7<br />+<br />– <br />(<br />)<br />* 48 / 48<br />12<br />8 <br />16<br />
  89. 89. Addition and Subtraction of Fractions<br />7<br />5<br />9<br />b.<br />+<br />– <br />12<br />8 <br />16<br />The LCD is 48. Multiply the problem by 48, expand the multiplication, divide the result by 48.<br />6<br />3<br />4<br />5<br />9<br />7<br />+<br />– <br />(<br />)<br />* 48 / 48<br />12<br />8 <br />16<br />= (4*7 + 6*5 –3*9) / 48<br />
  90. 90. Addition and Subtraction of Fractions<br />7<br />5<br />9<br />b.<br />+<br />– <br />12<br />8 <br />16<br />The LCD is 48. Multiply the problem by 48, expand the multiplication, divide the result by 48.<br />6<br />3<br />4<br />5<br />9<br />7<br />+<br />– <br />(<br />)<br />* 48 / 48<br />12<br />8 <br />16<br />= (4*7 + 6*5 –3*9) / 48 <br />= (28 + 30 – 27) / 48<br />
  91. 91. Addition and Subtraction of Fractions<br />7<br />5<br />9<br />b.<br />+<br />– <br />12<br />8 <br />16<br />The LCD is 48. Multiply the problem by 48, expand the multiplication, divide the result by 48.<br />6<br />3<br />4<br />5<br />9<br />7<br />+<br />– <br />(<br />)<br />* 48 / 48<br />12<br />8 <br />16<br />= (4*7 + 6*5 –3*9) / 48 <br />= (28 + 30 – 27) / 48<br />= <br />31<br />48<br />
  92. 92. Addition and Subtraction of Fractions<br />7<br />5<br />9<br />b.<br />+<br />– <br />12<br />8 <br />16<br />The LCD is 48. Multiply the problem by 48, expand the multiplication, divide the result by 48.<br />6<br />3<br />4<br />5<br />9<br />7<br />+<br />– <br />(<br />)<br />* 48 / 48<br />12<br />8 <br />16<br />= (4*7 + 6*5 –3*9) / 48 <br />= (28 + 30 – 27) / 48<br />= <br />31<br />48<br />We will learn the cross–multiplication method to + or – <br />two fractions shortly. Together with the above multiplier method, these two methods offer the most efficient ways to handle problems containing + or – of fractions. These two methods extend to operations of the rational (fractional) formulas and will use these two methods extensively.<br />

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