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1 f2 fractions

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1 f2 fractions

  1. 1. Fractions<br />Frank Ma © 2011<br />
  2. 2. Fractions<br />p<br />Fractions are numbers of the form (or p/q) where <br />p,q  0 are whole numbers. <br />q<br />
  3. 3. Fractions<br />p<br />Fractions are numbers of the form (or p/q) where <br />p,q  0 are whole numbers. <br />q<br />3<br />6<br />
  4. 4. Fractions<br />p<br />Fractions are numbers of the form (or p/q) where <br />p,q  0 are whole numbers. Fractions are numbers that measure parts of whole items.<br />q<br />3<br />6<br />
  5. 5. Fractions<br />p<br />Fractions are numbers of the form (or p/q) where <br />p,q  0 are whole numbers. Fractions are numbers that measure parts of whole items.<br />Suppose a pizza is cut into 6 equal slices and we have 3 of<br />them, the fraction that represents this quantity is . <br />q<br />3<br />6<br />3<br />6<br />
  6. 6. Fractions<br />p<br />Fractions are numbers of the form (or p/q) where <br />p,q  0 are whole numbers. Fractions are numbers that measure parts of whole items.<br />Suppose a pizza is cut into 6 equal slices and we have 3 of<br />them, the fraction that represents this quantity is . <br />q<br />3<br />6<br />3<br />6<br />
  7. 7. Fractions<br />p<br />Fractions are numbers of the form (or p/q) where <br />p,q  0 are whole numbers. Fractions are numbers that measure parts of whole items.<br />Suppose a pizza is cut into 6 equal slices and we have 3 of<br />them, the fraction that represents this quantity is . <br />q<br />3<br />6<br />3<br />6<br />The bottom number is the number of equal parts in the division and it is called the denominator.<br />
  8. 8. Fractions<br />p<br />Fractions are numbers of the form (or p/q) where <br />p,q  0 are whole numbers. Fractions are numbers that measure parts of whole items.<br />Suppose a pizza is cut into 6 equal slices and we have 3 of<br />them, the fraction that represents this quantity is . <br />q<br />3<br />6<br />3<br />6<br />The bottom number is the number of equal parts in the division and it is called the denominator.<br />
  9. 9. Fractions<br />p<br />Fractions are numbers of the form (or p/q) where <br />p,q  0 are whole numbers. Fractions are numbers that measure parts of whole items.<br />Suppose a pizza is cut into 6 equal slices and we have 3 of<br />them, the fraction that represents this quantity is . <br />q<br />3<br />6<br />The top number “3” is the number of parts that we have and it is called the numerator.<br />3<br />6<br />The bottom number is the number of equal parts in the division and it is called the denominator.<br />
  10. 10. Fractions<br />p<br />Fractions are numbers of the form (or p/q) where <br />p,q  0 are whole numbers. Fractions are numbers that measure parts of whole items.<br />Suppose a pizza is cut into 6 equal slices and we have 3 of<br />them, the fraction that represents this quantity is . <br />q<br />3<br />6<br />The top number “3” is the number of parts that we have and it is called the numerator.<br />3<br />6<br />The bottom number is the number of equal parts in the division and it is called the denominator.<br />3/6 of a pizza<br />
  11. 11. Fractions<br />For larger denominators we can use a pan–pizza for pictures. For example,<br />5<br />8<br />
  12. 12. Fractions<br />For larger denominators we can use a pan–pizza for pictures. For example,<br />5<br />8<br />How many slices should we cut the pizza into and how do we do this?<br />
  13. 13. Fractions<br />For larger denominators we can use a pan–pizza for pictures. For example,<br />5<br />8<br />Cut the pizza into 8 pieces,<br />
  14. 14. Fractions<br />For larger denominators we can use a pan–pizza for pictures. For example,<br />5<br />8<br />Cut the pizza into 8 pieces, take 5 of them. <br />
  15. 15. Fractions<br />For larger denominators we can use a pan–pizza for pictures. For example,<br />5<br />5/8 of a pizza<br />8<br />Cut the pizza into 8 pieces, take 5 of them. <br />
  16. 16. Fractions<br />For larger denominators we can use a pan–pizza for pictures. For example,<br />5<br />5/8 of a pizza<br />8<br />7<br />12<br />
  17. 17. Fractions<br />For larger denominators we can use a pan–pizza for pictures. For example,<br />5<br />5/8 of a pizza<br />8<br />7<br />12<br />Cut the pizza into 12 pieces, <br />
  18. 18. Fractions<br />For larger denominators we can use a pan–pizza for pictures. For example,<br />5<br />5/8 of a pizza<br />8<br />7<br />12<br />Cut the pizza into 12 pieces, <br />
  19. 19. Fractions<br />For larger denominators we can use a pan–pizza for pictures. For example,<br />5<br />5/8 of a pizza<br />8<br />7<br />12<br />Cut the pizza into 12 pieces, take 7 of them. <br />
  20. 20. Fractions<br />For larger denominators we can use a pan–pizza for pictures. For example,<br />5<br />5/8 of a pizza<br />8<br />7<br />or<br />12<br />Cut the pizza into 12 pieces, take 7 of them. <br />
  21. 21. Fractions<br />For larger denominators we can use a pan–pizza for pictures. For example,<br />5<br />5/8 of a pizza<br />8<br />7/12 of a pizza<br />7<br />or<br />12<br />Cut the pizza into 12 pieces, take 7 of them. <br />
  22. 22. Fractions<br />For larger denominators we can use a pan–pizza for pictures. For example,<br />5<br />5/8 of a pizza<br />8<br />7/12 of a pizza<br />7<br />or<br />12<br />8<br />12<br />Note that or is the same as 1.<br />8<br />12<br />
  23. 23. Fractions<br />For larger denominators we can use a pan–pizza for pictures. For example,<br />5<br />5/8 of a pizza<br />8<br />7/12 of a pizza<br />7<br />or<br />12<br />8<br />12<br />Note that or is the same as 1.<br />8<br />12<br />a<br />Fact:<br />= 1 (provided that a = 0.)<br />a<br />
  24. 24. Fractions<br />Whole numbers can be viewed as fractions with denominator 1. <br />
  25. 25. Fractions<br />Whole numbers can be viewed as fractions with denominator 1. <br />Thus 5 = and x = . <br />x<br />5<br />1<br />1<br />
  26. 26. Fractions<br />Whole numbers can be viewed as fractions with denominator 1. <br />Thus 5 = and x = . The fraction = 0, where x  0. <br />x<br />0<br />5<br />1<br />x<br />1<br />
  27. 27. Fractions<br />Whole numbers can be viewed as fractions with denominator 1. <br />Thus 5 = and x = . The fraction = 0, where x  0. <br />However, does not have any meaning, it is undefined. <br />x<br />0<br />5<br />1<br />x<br />1<br />x<br />0<br />
  28. 28. Fractions<br />Whole numbers can be viewed as fractions with denominator 1. <br />Thus 5 = and x = . The fraction = 0, where x  0. <br />However, does not have any meaning, it is undefined. <br />x<br />0<br />5<br />1<br />x<br />1<br />x<br />0<br />The Ultimate No-No of Mathematics:<br />
  29. 29. Fractions<br />Whole numbers can be viewed as fractions with denominator 1. <br />Thus 5 = and x = . The fraction = 0, where x  0. <br />However, does not have any meaning, it is undefined. <br />x<br />0<br />5<br />1<br />x<br />1<br />x<br />0<br />The Ultimate No-No of Mathematics:<br />The denominator (bottom) of a fraction can't be 0. <br />
  30. 30. Fractions<br />Whole numbers can be viewed as fractions with denominator 1. <br />Thus 5 = and x = . The fraction = 0, where x  0. <br />However, does not have any meaning, it is undefined. <br />x<br />0<br />5<br />1<br />x<br />1<br />x<br />0<br />The Ultimate No-No of Mathematics:<br />The denominator (bottom) of a fraction can't be 0. (It's undefined if the denominator is 0.)<br />
  31. 31. Fractions<br />Whole numbers can be viewed as fractions with denominator 1. <br />Thus 5 = and x = . The fraction = 0, where x  0. <br />However, does not have any meaning, it is undefined. <br />x<br />0<br />5<br />1<br />x<br />1<br />x<br />0<br />The Ultimate No-No of Mathematics:<br />The denominator (bottom) of a fraction can't be 0. (It's undefined if the denominator is 0.)<br />Fractions that represents the same quantity are called equivalent fractions. <br />
  32. 32. Fractions<br />Whole numbers can be viewed as fractions with denominator 1. <br />Thus 5 = and x = . The fraction = 0, where x  0. <br />However, does not have any meaning, it is undefined. <br />x<br />0<br />5<br />1<br />x<br />1<br />x<br />0<br />The Ultimate No-No of Mathematics:<br />The denominator (bottom) of a fraction can't be 0. (It's undefined if the denominator is 0.)<br />Fractions that represents the same quantity are called equivalent fractions. <br />1<br />2<br />=<br />2<br />4<br />
  33. 33. Fractions<br />Whole numbers can be viewed as fractions with denominator 1. <br />Thus 5 = and x = . The fraction = 0, where x  0. <br />However, does not have any meaning, it is undefined. <br />x<br />0<br />5<br />1<br />x<br />1<br />x<br />0<br />The Ultimate No-No of Mathematics:<br />The denominator (bottom) of a fraction can't be 0. (It's undefined if the denominator is 0.)<br />Fractions that represents the same quantity are called equivalent fractions. <br />1<br />2<br />3<br />=<br />=<br />2<br />4<br />6<br />
  34. 34. Fractions<br />Whole numbers can be viewed as fractions with denominator 1. <br />Thus 5 = and x = . The fraction = 0, where x  0. <br />However, does not have any meaning, it is undefined. <br />x<br />0<br />5<br />1<br />x<br />1<br />x<br />0<br />The Ultimate No-No of Mathematics:<br />The denominator (bottom) of a fraction can't be 0. (It's undefined if the denominator is 0.)<br />Fractions that represents the same quantity are called equivalent fractions. <br /> … are equivalent fractions.<br />1<br />2<br />3<br />4<br />=<br />=<br />=<br />2<br />4<br />6<br />8<br />
  35. 35. Fractions<br />Whole numbers can be viewed as fractions with denominator 1. <br />Thus 5 = and x = . The fraction = 0, where x  0. <br />However, does not have any meaning, it is undefined. <br />x<br />0<br />5<br />1<br />x<br />1<br />x<br />0<br />The Ultimate No-No of Mathematics:<br />The denominator (bottom) of a fraction can't be 0. (It's undefined if the denominator is 0.)<br />Fractions that represents the same quantity are called equivalent fractions. <br /> … are equivalent fractions.<br />The fraction with the smallest denominator of all the equivalent fractions is called the reduced fraction. <br />1<br />2<br />3<br />4<br />=<br />=<br />=<br />2<br />4<br />6<br />8<br />
  36. 36. Fractions<br />Whole numbers can be viewed as fractions with denominator 1. <br />Thus 5 = and x = . The fraction = 0, where x  0. <br />However, does not have any meaning, it is undefined. <br />x<br />0<br />5<br />1<br />x<br />1<br />x<br />0<br />The Ultimate No-No of Mathematics:<br />The denominator (bottom) of a fraction can't be 0. (It's undefined if the denominator is 0.)<br />Fractions that represents the same quantity are called equivalent fractions. <br /> … are equivalent fractions.<br />The fraction with the smallest denominator of all the equivalent fractions is called the reduced fraction. <br />1<br />2<br />3<br />4<br />=<br />=<br />=<br />2<br />4<br />6<br />8<br />1<br /> is the reduced one in the above list.<br />2<br />
  37. 37. Fractions<br />Factor Cancellation Rule<br />Given a fraction , then<br />that is, if the numerator and denominator are divided by the same quantity c, the result will be an equivalent fraction.<br />a<br />a<br />a / c<br />=<br />b<br />b<br />b / c<br />
  38. 38. Fractions<br />Factor Cancellation Rule<br />Given a fraction , then<br />that is, if the numerator and denominator are divided by the same quantity c, the result will be an equivalent fraction.<br />In other words, a common factor of the numerator and the <br />denominator may be canceled as 1, <br />a<br />a<br />a / c<br />=<br />b<br />b<br />b / c<br />
  39. 39. Fractions<br />Factor Cancellation Rule<br />Given a fraction , then<br />that is, if the numerator and denominator are divided by the same quantity c, the result will be an equivalent fraction.<br />In other words, a common factor of the numerator and the <br />denominator may be canceled as 1, i.e.<br />a<br />a<br />a / c<br />=<br />b<br />b<br />b / c<br />1<br />a*c<br />a*c<br />=<br />b*c <br />b*c <br />
  40. 40. Fractions<br />Factor Cancellation Rule<br />Given a fraction , then<br />that is, if the numerator and denominator are divided by the same quantity c, the result will be an equivalent fraction.<br />In other words, a common factor of the numerator and the <br />denominator may be canceled as 1, i.e.<br />a<br />a<br />a / c<br />=<br />b<br />b<br />b / c<br />1<br />a<br />a*c<br />a*c<br />=<br />=<br />b .<br />b*c <br />b*c <br />
  41. 41. Fractions<br />Factor Cancellation Rule<br />Given a fraction , then<br />that is, if the numerator and denominator are divided by the same quantity c, the result will be an equivalent fraction.<br />In other words, a common factor of the numerator and the <br />denominator may be canceled as 1, i.e.<br />a<br />a<br />a / c<br />=<br />b<br />b<br />b / c<br />1<br />a<br />a*c<br />a*c<br />=<br />=<br />b .<br />b*c <br />b*c <br />(Often we omit writing the 1’s after the cancellation.)<br />
  42. 42. Fractions<br />Factor Cancellation Rule<br />Given a fraction , then<br />that is, if the numerator and denominator are divided by the same quantity c, the result will be an equivalent fraction.<br />In other words, a common factor of the numerator and the <br />denominator may be canceled as 1, i.e.<br />a<br />a<br />a / c<br />=<br />b<br />b<br />b / c<br />1<br />a<br />a*c<br />a*c<br />=<br />=<br />b .<br />b*c <br />b*c <br />(Often we omit writing the 1’s after the cancellation.)<br />To reduce a fraction, we keep divide the top and bottom by common numbers until no more division is possible. <br />What's left is the reduced version.<br />
  43. 43. Fractions<br />Factor Cancellation Rule<br />Given a fraction , then<br />that is, if the numerator and denominator are divided by the same quantity c, the result will be an equivalent fraction.<br />In other words, a common factor of the numerator and the <br />denominator may be canceled as 1, i.e.<br />a<br />a<br />a / c<br />=<br />b<br />b<br />b / c<br />1<br />a<br />a*c<br />a*c<br />=<br />=<br />b .<br />b*c <br />b*c <br />(Often we omit writing the 1’s after the cancellation.)<br />To reduce a fraction, we keep divide the top and bottom by common numbers until no more division is possible. <br />What's left is the reduced version.<br />78<br />Example A: Reduce the fraction . <br />54<br />
  44. 44. Fractions<br />Factor Cancellation Rule<br />Given a fraction , then<br />that is, if the numerator and denominator are divided by the same quantity c, the result will be an equivalent fraction.<br />In other words, a common factor of the numerator and the <br />denominator may be canceled as 1, i.e.<br />a<br />a<br />a / c<br />=<br />b<br />b<br />b / c<br />1<br />a<br />a*c<br />a*c<br />=<br />=<br />b .<br />b*c <br />b*c <br />(Often we omit writing the 1’s after the cancellation.)<br />To reduce a fraction, we keep divide the top and bottom by common numbers until no more division is possible. <br />What's left is the reduced version.<br />78<br />Example A: Reduce the fraction . <br />54<br />78<br />=<br />54<br />
  45. 45. Fractions<br />Factor Cancellation Rule<br />Given a fraction , then<br />that is, if the numerator and denominator are divided by the same quantity c, the result will be an equivalent fraction.<br />In other words, a common factor of the numerator and the <br />denominator may be canceled as 1, i.e.<br />a<br />a<br />a / c<br />=<br />b<br />b<br />b / c<br />1<br />a<br />a*c<br />a*c<br />=<br />=<br />b .<br />b*c <br />b*c <br />(Often we omit writing the 1’s after the cancellation.)<br />To reduce a fraction, we keep divide the top and bottom by common numbers until no more division is possible. <br />What's left is the reduced version.<br />78<br />Example A: Reduce the fraction . <br />54<br />78<br />78/2<br />=<br />54<br />54/2<br />
  46. 46. Fractions<br />Factor Cancellation Rule<br />Given a fraction , then<br />that is, if the numerator and denominator are divided by the same quantity c, the result will be an equivalent fraction.<br />In other words, a common factor of the numerator and the <br />denominator may be canceled as 1, i.e.<br />a<br />a<br />a / c<br />=<br />b<br />b<br />b / c<br />1<br />a<br />a*c<br />a*c<br />=<br />=<br />b .<br />b*c <br />b*c <br />(Often we omit writing the 1’s after the cancellation.)<br />To reduce a fraction, we keep divide the top and bottom by common numbers until no more division is possible. <br />What's left is the reduced version.<br />78<br />Example A: Reduce the fraction . <br />54<br />39<br />78<br />78/2<br />=<br />=<br />54<br />54/2<br />27<br />
  47. 47. Fractions<br />Factor Cancellation Rule<br />Given a fraction , then<br />that is, if the numerator and denominator are divided by the same quantity c, the result will be an equivalent fraction.<br />In other words, a common factor of the numerator and the <br />denominator may be canceled as 1, i.e.<br />a<br />a<br />a / c<br />=<br />b<br />b<br />b / c<br />1<br />a<br />a*c<br />a*c<br />=<br />=<br />b .<br />b*c <br />b*c <br />(Often we omit writing the 1’s after the cancellation.)<br />To reduce a fraction, we keep divide the top and bottom by common numbers until no more division is possible. <br />What's left is the reduced version.<br />78<br />Example A: Reduce the fraction . <br />54<br />39<br />78<br />78/2<br />39/3<br />=<br />=<br />54<br />54/2<br />27/3<br />27<br />
  48. 48. Fractions<br />Factor Cancellation Rule<br />Given a fraction , then<br />that is, if the numerator and denominator are divided by the same quantity c, the result will be an equivalent fraction.<br />In other words, a common factor of the numerator and the <br />denominator may be canceled as 1, i.e.<br />a<br />a<br />a / c<br />=<br />b<br />b<br />b / c<br />1<br />a<br />a*c<br />a*c<br />=<br />=<br />b .<br />b*c <br />b*c <br />(Often we omit writing the 1’s after the cancellation.)<br />To reduce a fraction, we keep divide the top and bottom by common numbers until no more division is possible. <br />What's left is the reduced version.<br />78<br />Example A: Reduce the fraction . <br />54<br />39<br />78<br />78/2<br />39/3<br />13<br />=<br />=<br />=<br />54<br />54/2<br />27/3<br />9 .<br />27<br />
  49. 49. Fractions<br />Factor Cancellation Rule<br />Given a fraction , then<br />that is, if the numerator and denominator are divided by the same quantity c, the result will be an equivalent fraction.<br />In other words, a common factor of the numerator and the <br />denominator may be canceled as 1, i.e.<br />a<br />a<br />a / c<br />=<br />b<br />b<br />b / c<br />1<br />a<br />a*c<br />a*c<br />=<br />=<br />b .<br />b*c <br />b*c <br />(Often we omit writing the 1’s after the cancellation.)<br />To reduce a fraction, we keep divide the top and bottom by common numbers until no more division is possible. <br />What's left is the reduced version.<br />78<br />Example A: Reduce the fraction . <br />54<br />39<br />78<br />78/2<br />39/3<br />13<br />=<br />=<br />=<br />54<br />54/2<br />27/3<br />9 .<br />27<br />or divide both by 6 in one step. <br />
  50. 50. Fractions<br />One common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator.<br />
  51. 51. Fractions<br />One common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator.<br />A participant in a sum or a difference is called a term. <br />
  52. 52. Fractions<br />One common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator.<br />A participant in a sum or a difference is called a term. <br />The “2” in the expression “2 + 3” is a term (of the expression).<br />
  53. 53. Fractions<br />One common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator.<br />A participant in a sum or a difference is called a term. <br />The “2” in the expression “2 + 3” is a term (of the expression). <br />The “2” is in the expression “2 * 3” is called a factor.<br />
  54. 54. Fractions<br />One common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator.<br />A participant in a sum or a difference is called a term. <br />The “2” in the expression “2 + 3” is a term (of the expression). <br />The “2” is in the expression “2 * 3” is called a factor. <br />Terms may not be cancelled. Only factors may be canceled. <br />
  55. 55. Fractions<br />One common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator.<br />A participant in a sum or a difference is called a term. <br />The “2” in the expression “2 + 3” is a term (of the expression). <br />The “2” is in the expression “2 * 3” is called a factor. <br />Terms may not be cancelled. Only factors may be canceled. <br />2 + 1<br />3<br />= <br />2 + 3<br />5<br />
  56. 56. Fractions<br />One common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator.<br />A participant in a sum or a difference is called a term. <br />The “2” in the expression “2 + 3” is a term (of the expression). <br />The “2” is in the expression “2 * 3” is called a factor. <br />Terms may not be cancelled. Only factors may be canceled. <br />2 + 1<br />3<br />= <br />2 + 3<br />5<br />This is addition. Can’t cancel!<br />
  57. 57. Fractions<br />One common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator.<br />A participant in a sum or a difference is called a term. <br />The “2” in the expression “2 + 3” is a term (of the expression). <br />The “2” is in the expression “2 * 3” is called a factor. <br />Terms may not be cancelled. Only factors may be canceled. <br />2 + 1<br />3<br />2 + 1 <br />= <br />= <br />2 + 3<br />2 + 3<br />5<br />This is addition. Can’t cancel!<br />
  58. 58. Fractions<br />One common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator.<br />A participant in a sum or a difference is called a term. <br />The “2” in the expression “2 + 3” is a term (of the expression). <br />The “2” is in the expression “2 * 3” is called a factor. <br />Terms may not be cancelled. Only factors may be canceled. <br />!?<br />2 + 1<br />1<br />3<br />2 + 1 <br />= <br />= <br />= <br />2 + 3<br />2 + 3<br />3<br />5<br />This is addition. Can’t cancel!<br />
  59. 59. Fractions<br />One common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator.<br />A participant in a sum or a difference is called a term. <br />The “2” in the expression “2 + 3” is a term (of the expression). <br />The “2” is in the expression “2 * 3” is called a factor. <br />Terms may not be cancelled. Only factors may be canceled. <br />!?<br />2 * 1<br />1<br />2 + 1<br />1<br />3<br />2 + 1 <br />= <br />= <br />= <br />= <br />2 * 3<br />3<br />2 + 3<br />2 + 3<br />3<br />5<br />Yes<br />This is addition. Can’t cancel!<br />
  60. 60. Fractions<br />One common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator.<br />A participant in a sum or a difference is called a term. <br />The “2” in the expression “2 + 3” is a term (of the expression). <br />The “2” is in the expression “2 * 3” is called a factor. <br />Terms may not be cancelled. Only factors may be canceled. <br />!?<br />2 * 1<br />1<br />2 + 1<br />1<br />3<br />2 + 1 <br />= <br />= <br />= <br />= <br />2 * 3<br />3<br />2 + 3<br />2 + 3<br />3<br />5<br />Yes<br />This is addition. Can’t cancel!<br />Improper Fractions and Mixed Numbers<br />
  61. 61. Fractions<br />One common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator.<br />A participant in a sum or a difference is called a term. <br />The “2” in the expression “2 + 3” is a term (of the expression). <br />The “2” is in the expression “2 * 3” is called a factor. <br />Terms may not be cancelled. Only factors may be canceled. <br />!?<br />2 * 1<br />1<br />2 + 1<br />1<br />3<br />2 + 1 <br />= <br />= <br />= <br />= <br />2 * 3<br />3<br />2 + 3<br />2 + 3<br />3<br />5<br />Yes<br />This is addition. Can’t cancel!<br />Improper Fractions and Mixed Numbers<br />A fraction whose numerator is the same or more than its <br />denominator (e.g.) is said to be improper.<br />3 <br />2<br />
  62. 62. Fractions<br />One common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator.<br />A participant in a sum or a difference is called a term. <br />The “2” in the expression “2 + 3” is a term (of the expression). <br />The “2” is in the expression “2 * 3” is called a factor. <br />Terms may not be cancelled. Only factors may be canceled. <br />!?<br />2 * 1<br />1<br />2 + 1<br />1<br />3<br />2 + 1 <br />= <br />= <br />= <br />= <br />2 * 3<br />3<br />2 + 3<br />2 + 3<br />3<br />5<br />Yes<br />This is addition. Can’t cancel!<br />Improper Fractions and Mixed Numbers<br />A fraction whose numerator is the same or more than its <br />denominator (e.g.) is said to be improper.<br />We may put an improper fraction into mixed form by division.<br />3 <br />2<br />
  63. 63. Improper Fractions and Mixed Numbers<br />23 <br />Example B. Put into mixed form. <br />4<br />
  64. 64. Improper Fractions and Mixed Numbers<br />23 <br />Example B. Put into mixed form. <br />4<br />·<br />23 <br />4 = 5 with remainder 3. <br />·<br />
  65. 65. Improper Fractions and Mixed Numbers<br />23 <br />Example B. Put into mixed form. <br />4<br />23 <br />3 <br />·<br />23 <br />4 = 5 with remainder 3. Hence, <br />= 5 + <br />·<br />4<br />4 <br />
  66. 66. Improper Fractions and Mixed Numbers<br />23 <br />Example B. Put into mixed form. <br />4<br />23 <br />3 <br />3 <br />·<br />23 <br />4 = 5 with remainder 3. Hence, <br />5<br />= <br />= 5 + <br />·<br />4<br />4 .<br />4 <br />
  67. 67. Improper Fractions and Mixed Numbers<br />23 <br />Example B. Put into mixed form. <br />4<br />23 <br />3 <br />3 <br />·<br />23 <br />4 = 5 with remainder 3. Hence, <br />5<br />= <br />= 5 + <br />·<br />4<br />4 .<br />4 <br />We may put a mixed number into improper fraction by doing the reverse via multiplication.<br />
  68. 68. Improper Fractions and Mixed Numbers<br />23 <br />Example B. Put into mixed form. <br />4<br />23 <br />3 <br />3 <br />·<br />23 <br />4 = 5 with remainder 3. Hence, <br />5<br />= <br />= 5 + <br />·<br />4<br />4 .<br />4 <br />We may put a mixed number into improper fraction by doing the reverse via multiplication.<br />3 <br />Example C: Put into improper form. <br />5<br />4 <br />
  69. 69. Improper Fractions and Mixed Numbers<br />23 <br />Example B. Put into mixed form. <br />4<br />23 <br />3 <br />3 <br />·<br />23 <br />4 = 5 with remainder 3. Hence, <br />5<br />= <br />= 5 + <br />·<br />4<br />4 .<br />4 <br />We may put a mixed number into improper fraction by doing the reverse via multiplication.<br />3 <br />Example C: Put into improper form. <br />5<br />4 <br />3 <br />4*5 + 3<br />5<br />= <br />4 <br />4<br />
  70. 70. Improper Fractions and Mixed Numbers<br />23 <br />Example B. Put into mixed form. <br />4<br />23 <br />3 <br />3 <br />·<br />23 <br />4 = 5 with remainder 3. Hence, <br />5<br />= <br />= 5 + <br />·<br />4<br />4 .<br />4 <br />We may put a mixed number into improper fraction by doing the reverse via multiplication.<br />3 <br />Example C: Put into improper form. <br />5<br />4 <br />3 <br />4*5 + 3<br />23 <br />5<br />= <br />= <br />4 <br />4<br />4<br />
  71. 71. Improper Fractions and Mixed Numbers<br />23 <br />Example B. Put into mixed form. <br />4<br />23 <br />3 <br />3 <br />·<br />23 <br />4 = 5 with remainder 3. Hence, <br />5<br />= <br />= 5 + <br />·<br />4<br />4 .<br />4 <br />We may put a mixed number into improper fraction by doing the reverse via multiplication.<br />3 <br />Example C: Put into improper form. <br />5<br />4 <br />3 <br />4*5 + 3<br />23 <br />5<br />= <br />= <br />4 <br />4<br />4<br />

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