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Fractions,[object Object],Frank Ma © 2011,[object Object]
Fractions,[object Object],p,[object Object],Fractions are numbers of the form       (or p/q) where ,[object Object],p,q  0 are whole numbers. ,[object Object],q,[object Object]
Fractions,[object Object],p,[object Object],Fractions are numbers of the form       (or p/q) where ,[object Object],p,q  0 are whole numbers. ,[object Object],q,[object Object],3,[object Object],6,[object Object]
Fractions,[object Object],p,[object Object],Fractions are numbers of the form       (or p/q) where ,[object Object],p,q  0 are whole numbers. Fractions are numbers that measure parts of whole items.,[object Object],q,[object Object],3,[object Object],6,[object Object]
Fractions,[object Object],p,[object Object],Fractions are numbers of the form       (or p/q) where ,[object Object],p,q  0 are whole numbers. Fractions are numbers that measure parts of whole items.,[object Object],Suppose a pizza is cut into 6 equal slices and we have 3 of,[object Object],them, the fraction that represents this quantity is      .  ,[object Object],q,[object Object],3,[object Object],6,[object Object],3,[object Object],6,[object Object]
Fractions,[object Object],p,[object Object],Fractions are numbers of the form       (or p/q) where ,[object Object],p,q  0 are whole numbers. Fractions are numbers that measure parts of whole items.,[object Object],Suppose a pizza is cut into 6 equal slices and we have 3 of,[object Object],them, the fraction that represents this quantity is      .  ,[object Object],q,[object Object],3,[object Object],6,[object Object],3,[object Object],6,[object Object]
Fractions,[object Object],p,[object Object],Fractions are numbers of the form       (or p/q) where ,[object Object],p,q  0 are whole numbers. Fractions are numbers that measure parts of whole items.,[object Object],Suppose a pizza is cut into 6 equal slices and we have 3 of,[object Object],them, the fraction that represents this quantity is      .  ,[object Object],q,[object Object],3,[object Object],6,[object Object],3,[object Object],6,[object Object],The bottom number is the number of equal parts in the division and it is called the denominator.,[object Object]
Fractions,[object Object],p,[object Object],Fractions are numbers of the form       (or p/q) where ,[object Object],p,q  0 are whole numbers. Fractions are numbers that measure parts of whole items.,[object Object],Suppose a pizza is cut into 6 equal slices and we have 3 of,[object Object],them, the fraction that represents this quantity is      .  ,[object Object],q,[object Object],3,[object Object],6,[object Object],3,[object Object],6,[object Object],The bottom number is the number of equal parts in the division and it is called the denominator.,[object Object]
Fractions,[object Object],p,[object Object],Fractions are numbers of the form       (or p/q) where ,[object Object],p,q  0 are whole numbers. Fractions are numbers that measure parts of whole items.,[object Object],Suppose a pizza is cut into 6 equal slices and we have 3 of,[object Object],them, the fraction that represents this quantity is      .  ,[object Object],q,[object Object],3,[object Object],6,[object Object],The top number “3” is the number of parts that we have and it is called the numerator.,[object Object],3,[object Object],6,[object Object],The bottom number is the number of equal parts in the division and it is called the denominator.,[object Object]
Fractions,[object Object],p,[object Object],Fractions are numbers of the form       (or p/q) where ,[object Object],p,q  0 are whole numbers. Fractions are numbers that measure parts of whole items.,[object Object],Suppose a pizza is cut into 6 equal slices and we have 3 of,[object Object],them, the fraction that represents this quantity is      .  ,[object Object],q,[object Object],3,[object Object],6,[object Object],The top number “3” is the number of parts that we have and it is called the numerator.,[object Object],3,[object Object],6,[object Object],The bottom number is the number of equal parts in the division and it is called the denominator.,[object Object],3/6 of a pizza,[object Object]
Fractions,[object Object],For larger denominators we can use a pan–pizza for pictures. For example,,[object Object],5,[object Object],8,[object Object]
Fractions,[object Object],For larger denominators we can use a pan–pizza for pictures. For example,,[object Object],5,[object Object],8,[object Object],How many slices should we cut the pizza into and how do we do this?,[object Object]
Fractions,[object Object],For larger denominators we can use a pan–pizza for pictures. For example,,[object Object],5,[object Object],8,[object Object],Cut the pizza into 8 pieces,,[object Object]
Fractions,[object Object],For larger denominators we can use a pan–pizza for pictures. For example,,[object Object],5,[object Object],8,[object Object],Cut the pizza into 8 pieces, take 5 of them. ,[object Object]
Fractions,[object Object],For larger denominators we can use a pan–pizza for pictures. For example,,[object Object],5,[object Object],5/8 of a pizza,[object Object],8,[object Object],Cut the pizza into 8 pieces, take 5 of them. ,[object Object]
Fractions,[object Object],For larger denominators we can use a pan–pizza for pictures. For example,,[object Object],5,[object Object],5/8 of a pizza,[object Object],8,[object Object],7,[object Object],12,[object Object]
Fractions,[object Object],For larger denominators we can use a pan–pizza for pictures. For example,,[object Object],5,[object Object],5/8 of a pizza,[object Object],8,[object Object],7,[object Object],12,[object Object],Cut the pizza into 12 pieces, ,[object Object]
Fractions,[object Object],For larger denominators we can use a pan–pizza for pictures. For example,,[object Object],5,[object Object],5/8 of a pizza,[object Object],8,[object Object],7,[object Object],12,[object Object],Cut the pizza into 12 pieces, ,[object Object]
Fractions,[object Object],For larger denominators we can use a pan–pizza for pictures. For example,,[object Object],5,[object Object],5/8 of a pizza,[object Object],8,[object Object],7,[object Object],12,[object Object],Cut the pizza into 12 pieces, take 7 of them. ,[object Object]
Fractions,[object Object],For larger denominators we can use a pan–pizza for pictures. For example,,[object Object],5,[object Object],5/8 of a pizza,[object Object],8,[object Object],7,[object Object],or,[object Object],12,[object Object],Cut the pizza into 12 pieces, take 7 of them. ,[object Object]
Fractions,[object Object],For larger denominators we can use a pan–pizza for pictures. For example,,[object Object],5,[object Object],5/8 of a pizza,[object Object],8,[object Object],7/12 of a pizza,[object Object],7,[object Object],or,[object Object],12,[object Object],Cut the pizza into 12 pieces, take 7 of them. ,[object Object]
Fractions,[object Object],For larger denominators we can use a pan–pizza for pictures. For example,,[object Object],5,[object Object],5/8 of a pizza,[object Object],8,[object Object],7/12 of a pizza,[object Object],7,[object Object],or,[object Object],12,[object Object],8,[object Object],12,[object Object],Note that         or          is the same as 1.,[object Object],8,[object Object],12,[object Object]
Fractions,[object Object],For larger denominators we can use a pan–pizza for pictures. For example,,[object Object],5,[object Object],5/8 of a pizza,[object Object],8,[object Object],7/12 of a pizza,[object Object],7,[object Object],or,[object Object],12,[object Object],8,[object Object],12,[object Object],Note that         or          is the same as 1.,[object Object],8,[object Object],12,[object Object],a,[object Object],Fact:,[object Object],= 1 (provided that a = 0.),[object Object],a,[object Object]
Fractions,[object Object],Whole numbers can be viewed as fractions with denominator 1. ,[object Object]
Fractions,[object Object],Whole numbers can be viewed as fractions with denominator 1.  ,[object Object],Thus 5 =       and x =     .  ,[object Object],x,[object Object],5,[object Object],1,[object Object],1,[object Object]
Fractions,[object Object],Whole numbers can be viewed as fractions with denominator 1.  ,[object Object],Thus 5 =       and x =     .  The fraction       = 0, where x  0. ,[object Object],x,[object Object],0,[object Object],5,[object Object],1,[object Object],x,[object Object],1,[object Object]
Fractions,[object Object],Whole numbers can be viewed as fractions with denominator 1.  ,[object Object],Thus 5 =       and x =     .  The fraction       = 0, where x  0.  ,[object Object],However,        does not have any meaning, it is undefined. ,[object Object],x,[object Object],0,[object Object],5,[object Object],1,[object Object],x,[object Object],1,[object Object],x,[object Object],0,[object Object]
Fractions,[object Object],Whole numbers can be viewed as fractions with denominator 1.  ,[object Object],Thus 5 =       and x =     .  The fraction       = 0, where x  0.  ,[object Object],However,        does not have any meaning, it is undefined. ,[object Object],x,[object Object],0,[object Object],5,[object Object],1,[object Object],x,[object Object],1,[object Object],x,[object Object],0,[object Object],The Ultimate No-No of Mathematics:,[object Object]
Fractions,[object Object],Whole numbers can be viewed as fractions with denominator 1.  ,[object Object],Thus 5 =       and x =     .  The fraction       = 0, where x  0.  ,[object Object],However,        does not have any meaning, it is undefined. ,[object Object],x,[object Object],0,[object Object],5,[object Object],1,[object Object],x,[object Object],1,[object Object],x,[object Object],0,[object Object],The Ultimate No-No of Mathematics:,[object Object],The denominator (bottom) of a fraction can't be 0. ,[object Object]
Fractions,[object Object],Whole numbers can be viewed as fractions with denominator 1.  ,[object Object],Thus 5 =       and x =     .  The fraction       = 0, where x  0.  ,[object Object],However,        does not have any meaning, it is undefined. ,[object Object],x,[object Object],0,[object Object],5,[object Object],1,[object Object],x,[object Object],1,[object Object],x,[object Object],0,[object Object],The Ultimate No-No of Mathematics:,[object Object],The denominator (bottom) of a fraction can't be 0. (It's undefined if the denominator is 0.),[object Object]
Fractions,[object Object],Whole numbers can be viewed as fractions with denominator 1.  ,[object Object],Thus 5 =       and x =     .  The fraction       = 0, where x  0.  ,[object Object],However,        does not have any meaning, it is undefined. ,[object Object],x,[object Object],0,[object Object],5,[object Object],1,[object Object],x,[object Object],1,[object Object],x,[object Object],0,[object Object],The Ultimate No-No of Mathematics:,[object Object],The denominator (bottom) of a fraction can't be 0. (It's undefined if the denominator is 0.),[object Object],Fractions that represents the same quantity are called equivalent fractions. ,[object Object]
Fractions,[object Object],Whole numbers can be viewed as fractions with denominator 1.  ,[object Object],Thus 5 =       and x =     .  The fraction       = 0, where x  0.  ,[object Object],However,        does not have any meaning, it is undefined. ,[object Object],x,[object Object],0,[object Object],5,[object Object],1,[object Object],x,[object Object],1,[object Object],x,[object Object],0,[object Object],The Ultimate No-No of Mathematics:,[object Object],The denominator (bottom) of a fraction can't be 0. (It's undefined if the denominator is 0.),[object Object],Fractions that represents the same quantity are called equivalent fractions. ,[object Object],1,[object Object],2,[object Object],=,[object Object],2,[object Object],4,[object Object]
Fractions,[object Object],Whole numbers can be viewed as fractions with denominator 1.  ,[object Object],Thus 5 =       and x =     .  The fraction       = 0, where x  0.  ,[object Object],However,        does not have any meaning, it is undefined. ,[object Object],x,[object Object],0,[object Object],5,[object Object],1,[object Object],x,[object Object],1,[object Object],x,[object Object],0,[object Object],The Ultimate No-No of Mathematics:,[object Object],The denominator (bottom) of a fraction can't be 0. (It's undefined if the denominator is 0.),[object Object],Fractions that represents the same quantity are called equivalent fractions. ,[object Object],1,[object Object],2,[object Object],3,[object Object],=,[object Object],=,[object Object],2,[object Object],4,[object Object],6,[object Object]
Fractions,[object Object],Whole numbers can be viewed as fractions with denominator 1.  ,[object Object],Thus 5 =       and x =     .  The fraction       = 0, where x  0.  ,[object Object],However,        does not have any meaning, it is undefined. ,[object Object],x,[object Object],0,[object Object],5,[object Object],1,[object Object],x,[object Object],1,[object Object],x,[object Object],0,[object Object],The Ultimate No-No of Mathematics:,[object Object],The denominator (bottom) of a fraction can't be 0. (It's undefined if the denominator is 0.),[object Object],Fractions that represents the same quantity are called equivalent fractions. ,[object Object],                             … are equivalent fractions.,[object Object],1,[object Object],2,[object Object],3,[object Object],4,[object Object],=,[object Object],=,[object Object],=,[object Object],2,[object Object],4,[object Object],6,[object Object],8,[object Object]
Fractions,[object Object],Whole numbers can be viewed as fractions with denominator 1.  ,[object Object],Thus 5 =       and x =     .  The fraction       = 0, where x  0.  ,[object Object],However,        does not have any meaning, it is undefined. ,[object Object],x,[object Object],0,[object Object],5,[object Object],1,[object Object],x,[object Object],1,[object Object],x,[object Object],0,[object Object],The Ultimate No-No of Mathematics:,[object Object],The denominator (bottom) of a fraction can't be 0. (It's undefined if the denominator is 0.),[object Object],Fractions that represents the same quantity are called equivalent fractions. ,[object Object],                             … are equivalent fractions.,[object Object],The fraction with the smallest denominator of all the equivalent fractions is called the reduced fraction. ,[object Object],1,[object Object],2,[object Object],3,[object Object],4,[object Object],=,[object Object],=,[object Object],=,[object Object],2,[object Object],4,[object Object],6,[object Object],8,[object Object]
Fractions,[object Object],Whole numbers can be viewed as fractions with denominator 1.  ,[object Object],Thus 5 =       and x =     .  The fraction       = 0, where x  0.  ,[object Object],However,        does not have any meaning, it is undefined. ,[object Object],x,[object Object],0,[object Object],5,[object Object],1,[object Object],x,[object Object],1,[object Object],x,[object Object],0,[object Object],The Ultimate No-No of Mathematics:,[object Object],The denominator (bottom) of a fraction can't be 0. (It's undefined if the denominator is 0.),[object Object],Fractions that represents the same quantity are called equivalent fractions. ,[object Object],                             … are equivalent fractions.,[object Object],The fraction with the smallest denominator of all the equivalent fractions is called the reduced fraction. ,[object Object],1,[object Object],2,[object Object],3,[object Object],4,[object Object],=,[object Object],=,[object Object],=,[object Object],2,[object Object],4,[object Object],6,[object Object],8,[object Object],1,[object Object], is the reduced one in the above list.,[object Object],2,[object Object]
Fractions,[object Object],Factor Cancellation Rule,[object Object],Given a fraction      , then,[object Object],that is, if the numerator and denominator are divided by the same quantity c, the result will be an equivalent fraction.,[object Object],a,[object Object],a,[object Object],a / c,[object Object],=,[object Object],b,[object Object],b,[object Object],b / c,[object Object]
Fractions,[object Object],Factor Cancellation Rule,[object Object],Given a fraction      , then,[object Object],that is, if the numerator and denominator are divided by the same quantity c, the result will be an equivalent fraction.,[object Object],In other words, a common factor of the numerator and the ,[object Object],denominator may be canceled as 1, ,[object Object],a,[object Object],a,[object Object],a / c,[object Object],=,[object Object],b,[object Object],b,[object Object],b / c,[object Object]
Fractions,[object Object],Factor Cancellation Rule,[object Object],Given a fraction      , then,[object Object],that is, if the numerator and denominator are divided by the same quantity c, the result will be an equivalent fraction.,[object Object],In other words, a common factor of the numerator and the ,[object Object],denominator may be canceled as 1, i.e.,[object Object],a,[object Object],a,[object Object],a / c,[object Object],=,[object Object],b,[object Object],b,[object Object],b / c,[object Object],1,[object Object],a*c,[object Object],a*c,[object Object],=,[object Object],b*c   ,[object Object],b*c   ,[object Object]
Fractions,[object Object],Factor Cancellation Rule,[object Object],Given a fraction      , then,[object Object],that is, if the numerator and denominator are divided by the same quantity c, the result will be an equivalent fraction.,[object Object],In other words, a common factor of the numerator and the ,[object Object],denominator may be canceled as 1, i.e.,[object Object],a,[object Object],a,[object Object],a / c,[object Object],=,[object Object],b,[object Object],b,[object Object],b / c,[object Object],1,[object Object],a,[object Object],a*c,[object Object],a*c,[object Object],=,[object Object],=,[object Object],b .,[object Object],b*c   ,[object Object],b*c   ,[object Object]
Fractions,[object Object],Factor Cancellation Rule,[object Object],Given a fraction      , then,[object Object],that is, if the numerator and denominator are divided by the same quantity c, the result will be an equivalent fraction.,[object Object],In other words, a common factor of the numerator and the ,[object Object],denominator may be canceled as 1, i.e.,[object Object],a,[object Object],a,[object Object],a / c,[object Object],=,[object Object],b,[object Object],b,[object Object],b / c,[object Object],1,[object Object],a,[object Object],a*c,[object Object],a*c,[object Object],=,[object Object],=,[object Object],b .,[object Object],b*c   ,[object Object],b*c   ,[object Object],(Often we omit writing the 1’s after the cancellation.),[object Object]
Fractions,[object Object],Factor Cancellation Rule,[object Object],Given a fraction      , then,[object Object],that is, if the numerator and denominator are divided by the same quantity c, the result will be an equivalent fraction.,[object Object],In other words, a common factor of the numerator and the ,[object Object],denominator may be canceled as 1, i.e.,[object Object],a,[object Object],a,[object Object],a / c,[object Object],=,[object Object],b,[object Object],b,[object Object],b / c,[object Object],1,[object Object],a,[object Object],a*c,[object Object],a*c,[object Object],=,[object Object],=,[object Object],b .,[object Object],b*c   ,[object Object],b*c   ,[object Object],(Often we omit writing the 1’s after the cancellation.),[object Object],To reduce a fraction, we keep divide the top and bottom by  common numbers until no more division is possible.  ,[object Object],What's left is the reduced version.,[object Object]
Fractions,[object Object],Factor Cancellation Rule,[object Object],Given a fraction      , then,[object Object],that is, if the numerator and denominator are divided by the same quantity c, the result will be an equivalent fraction.,[object Object],In other words, a common factor of the numerator and the ,[object Object],denominator may be canceled as 1, i.e.,[object Object],a,[object Object],a,[object Object],a / c,[object Object],=,[object Object],b,[object Object],b,[object Object],b / c,[object Object],1,[object Object],a,[object Object],a*c,[object Object],a*c,[object Object],=,[object Object],=,[object Object],b .,[object Object],b*c   ,[object Object],b*c   ,[object Object],(Often we omit writing the 1’s after the cancellation.),[object Object],To reduce a fraction, we keep divide the top and bottom by  common numbers until no more division is possible.  ,[object Object],What's left is the reduced version.,[object Object],78,[object Object],Example A:  Reduce the fraction       . ,[object Object],54,[object Object]
Fractions,[object Object],Factor Cancellation Rule,[object Object],Given a fraction      , then,[object Object],that is, if the numerator and denominator are divided by the same quantity c, the result will be an equivalent fraction.,[object Object],In other words, a common factor of the numerator and the ,[object Object],denominator may be canceled as 1, i.e.,[object Object],a,[object Object],a,[object Object],a / c,[object Object],=,[object Object],b,[object Object],b,[object Object],b / c,[object Object],1,[object Object],a,[object Object],a*c,[object Object],a*c,[object Object],=,[object Object],=,[object Object],b .,[object Object],b*c   ,[object Object],b*c   ,[object Object],(Often we omit writing the 1’s after the cancellation.),[object Object],To reduce a fraction, we keep divide the top and bottom by  common numbers until no more division is possible.  ,[object Object],What's left is the reduced version.,[object Object],78,[object Object],Example A:  Reduce the fraction       . ,[object Object],54,[object Object],78,[object Object],=,[object Object],54,[object Object]
Fractions,[object Object],Factor Cancellation Rule,[object Object],Given a fraction      , then,[object Object],that is, if the numerator and denominator are divided by the same quantity c, the result will be an equivalent fraction.,[object Object],In other words, a common factor of the numerator and the ,[object Object],denominator may be canceled as 1, i.e.,[object Object],a,[object Object],a,[object Object],a / c,[object Object],=,[object Object],b,[object Object],b,[object Object],b / c,[object Object],1,[object Object],a,[object Object],a*c,[object Object],a*c,[object Object],=,[object Object],=,[object Object],b .,[object Object],b*c   ,[object Object],b*c   ,[object Object],(Often we omit writing the 1’s after the cancellation.),[object Object],To reduce a fraction, we keep divide the top and bottom by  common numbers until no more division is possible.  ,[object Object],What's left is the reduced version.,[object Object],78,[object Object],Example A:  Reduce the fraction       . ,[object Object],54,[object Object],78,[object Object],78/2,[object Object],=,[object Object],54,[object Object],54/2,[object Object]
Fractions,[object Object],Factor Cancellation Rule,[object Object],Given a fraction      , then,[object Object],that is, if the numerator and denominator are divided by the same quantity c, the result will be an equivalent fraction.,[object Object],In other words, a common factor of the numerator and the ,[object Object],denominator may be canceled as 1, i.e.,[object Object],a,[object Object],a,[object Object],a / c,[object Object],=,[object Object],b,[object Object],b,[object Object],b / c,[object Object],1,[object Object],a,[object Object],a*c,[object Object],a*c,[object Object],=,[object Object],=,[object Object],b .,[object Object],b*c   ,[object Object],b*c   ,[object Object],(Often we omit writing the 1’s after the cancellation.),[object Object],To reduce a fraction, we keep divide the top and bottom by  common numbers until no more division is possible.  ,[object Object],What's left is the reduced version.,[object Object],78,[object Object],Example A:  Reduce the fraction       . ,[object Object],54,[object Object],39,[object Object],78,[object Object],78/2,[object Object],=,[object Object],=,[object Object],54,[object Object],54/2,[object Object],27,[object Object]
Fractions,[object Object],Factor Cancellation Rule,[object Object],Given a fraction      , then,[object Object],that is, if the numerator and denominator are divided by the same quantity c, the result will be an equivalent fraction.,[object Object],In other words, a common factor of the numerator and the ,[object Object],denominator may be canceled as 1, i.e.,[object Object],a,[object Object],a,[object Object],a / c,[object Object],=,[object Object],b,[object Object],b,[object Object],b / c,[object Object],1,[object Object],a,[object Object],a*c,[object Object],a*c,[object Object],=,[object Object],=,[object Object],b .,[object Object],b*c   ,[object Object],b*c   ,[object Object],(Often we omit writing the 1’s after the cancellation.),[object Object],To reduce a fraction, we keep divide the top and bottom by  common numbers until no more division is possible.  ,[object Object],What's left is the reduced version.,[object Object],78,[object Object],Example A:  Reduce the fraction       . ,[object Object],54,[object Object],39,[object Object],78,[object Object],78/2,[object Object],39/3,[object Object],=,[object Object],=,[object Object],54,[object Object],54/2,[object Object],27/3,[object Object],27,[object Object]
Fractions,[object Object],Factor Cancellation Rule,[object Object],Given a fraction      , then,[object Object],that is, if the numerator and denominator are divided by the same quantity c, the result will be an equivalent fraction.,[object Object],In other words, a common factor of the numerator and the ,[object Object],denominator may be canceled as 1, i.e.,[object Object],a,[object Object],a,[object Object],a / c,[object Object],=,[object Object],b,[object Object],b,[object Object],b / c,[object Object],1,[object Object],a,[object Object],a*c,[object Object],a*c,[object Object],=,[object Object],=,[object Object],b .,[object Object],b*c   ,[object Object],b*c   ,[object Object],(Often we omit writing the 1’s after the cancellation.),[object Object],To reduce a fraction, we keep divide the top and bottom by  common numbers until no more division is possible.  ,[object Object],What's left is the reduced version.,[object Object],78,[object Object],Example A:  Reduce the fraction       . ,[object Object],54,[object Object],39,[object Object],78,[object Object],78/2,[object Object],39/3,[object Object],13,[object Object],=,[object Object],=,[object Object],=,[object Object],54,[object Object],54/2,[object Object],27/3,[object Object],9 .,[object Object],27,[object Object]
Fractions,[object Object],Factor Cancellation Rule,[object Object],Given a fraction      , then,[object Object],that is, if the numerator and denominator are divided by the same quantity c, the result will be an equivalent fraction.,[object Object],In other words, a common factor of the numerator and the ,[object Object],denominator may be canceled as 1, i.e.,[object Object],a,[object Object],a,[object Object],a / c,[object Object],=,[object Object],b,[object Object],b,[object Object],b / c,[object Object],1,[object Object],a,[object Object],a*c,[object Object],a*c,[object Object],=,[object Object],=,[object Object],b .,[object Object],b*c   ,[object Object],b*c   ,[object Object],(Often we omit writing the 1’s after the cancellation.),[object Object],To reduce a fraction, we keep divide the top and bottom by  common numbers until no more division is possible.  ,[object Object],What's left is the reduced version.,[object Object],78,[object Object],Example A:  Reduce the fraction       . ,[object Object],54,[object Object],39,[object Object],78,[object Object],78/2,[object Object],39/3,[object Object],13,[object Object],=,[object Object],=,[object Object],=,[object Object],54,[object Object],54/2,[object Object],27/3,[object Object],9 .,[object Object],27,[object Object],or divide both by 6 in one step.  ,[object Object]
Fractions,[object Object],One common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator.,[object Object]
Fractions,[object Object],One common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator.,[object Object],A participant in a sum or a difference is called a term. ,[object Object]
Fractions,[object Object],One common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator.,[object Object],A participant in a sum or a difference is called a term. ,[object Object],The “2” in the expression “2 + 3” is a term (of the expression).,[object Object]
Fractions,[object Object],One common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator.,[object Object],A participant in a sum or a difference is called a term. ,[object Object],The “2” in the expression “2 + 3” is a term (of the expression). ,[object Object],The “2” is in the expression “2 * 3” is called a factor.,[object Object]
Fractions,[object Object],One common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator.,[object Object],A participant in a sum or a difference is called a term. ,[object Object],The “2” in the expression “2 + 3” is a term (of the expression). ,[object Object],The “2” is in the expression “2 * 3” is called a factor.  ,[object Object],Terms may not be cancelled. Only factors may be canceled. ,[object Object]
Fractions,[object Object],One common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator.,[object Object],A participant in a sum or a difference is called a term. ,[object Object],The “2” in the expression “2 + 3” is a term (of the expression). ,[object Object],The “2” is in the expression “2 * 3” is called a factor.  ,[object Object],Terms may not be cancelled. Only factors may be canceled. ,[object Object],2 + 1,[object Object],3,[object Object],= ,[object Object],2 + 3,[object Object],5,[object Object]
Fractions,[object Object],One common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator.,[object Object],A participant in a sum or a difference is called a term. ,[object Object],The “2” in the expression “2 + 3” is a term (of the expression). ,[object Object],The “2” is in the expression “2 * 3” is called a factor.  ,[object Object],Terms may not be cancelled. Only factors may be canceled. ,[object Object],2 + 1,[object Object],3,[object Object],= ,[object Object],2 + 3,[object Object],5,[object Object],This is addition. Can’t cancel!,[object Object]
Fractions,[object Object],One common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator.,[object Object],A participant in a sum or a difference is called a term. ,[object Object],The “2” in the expression “2 + 3” is a term (of the expression). ,[object Object],The “2” is in the expression “2 * 3” is called a factor.  ,[object Object],Terms may not be cancelled. Only factors may be canceled. ,[object Object],2 + 1,[object Object],3,[object Object],2 + 1 ,[object Object],= ,[object Object],= ,[object Object],2 + 3,[object Object],2 + 3,[object Object],5,[object Object],This is addition. Can’t cancel!,[object Object]
Fractions,[object Object],One common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator.,[object Object],A participant in a sum or a difference is called a term. ,[object Object],The “2” in the expression “2 + 3” is a term (of the expression). ,[object Object],The “2” is in the expression “2 * 3” is called a factor.  ,[object Object],Terms may not be cancelled. Only factors may be canceled. ,[object Object],!?,[object Object],2 + 1,[object Object],1,[object Object],3,[object Object],2 + 1 ,[object Object],= ,[object Object],= ,[object Object],= ,[object Object],2 + 3,[object Object],2 + 3,[object Object],3,[object Object],5,[object Object],This is addition. Can’t cancel!,[object Object]
Fractions,[object Object],One common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator.,[object Object],A participant in a sum or a difference is called a term. ,[object Object],The “2” in the expression “2 + 3” is a term (of the expression). ,[object Object],The “2” is in the expression “2 * 3” is called a factor.  ,[object Object],Terms may not be cancelled. Only factors may be canceled. ,[object Object],!?,[object Object],2 * 1,[object Object],1,[object Object],2 + 1,[object Object],1,[object Object],3,[object Object],2 + 1 ,[object Object],= ,[object Object],= ,[object Object],= ,[object Object],= ,[object Object],2 * 3,[object Object],3,[object Object],2 + 3,[object Object],2 + 3,[object Object],3,[object Object],5,[object Object],Yes,[object Object],This is addition. Can’t cancel!,[object Object]
Fractions,[object Object],One common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator.,[object Object],A participant in a sum or a difference is called a term. ,[object Object],The “2” in the expression “2 + 3” is a term (of the expression). ,[object Object],The “2” is in the expression “2 * 3” is called a factor.  ,[object Object],Terms may not be cancelled. Only factors may be canceled. ,[object Object],!?,[object Object],2 * 1,[object Object],1,[object Object],2 + 1,[object Object],1,[object Object],3,[object Object],2 + 1 ,[object Object],= ,[object Object],= ,[object Object],= ,[object Object],= ,[object Object],2 * 3,[object Object],3,[object Object],2 + 3,[object Object],2 + 3,[object Object],3,[object Object],5,[object Object],Yes,[object Object],This is addition. Can’t cancel!,[object Object],Improper Fractions and Mixed Numbers,[object Object]
Fractions,[object Object],One common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator.,[object Object],A participant in a sum or a difference is called a term. ,[object Object],The “2” in the expression “2 + 3” is a term (of the expression). ,[object Object],The “2” is in the expression “2 * 3” is called a factor.  ,[object Object],Terms may not be cancelled. Only factors may be canceled. ,[object Object],!?,[object Object],2 * 1,[object Object],1,[object Object],2 + 1,[object Object],1,[object Object],3,[object Object],2 + 1 ,[object Object],= ,[object Object],= ,[object Object],= ,[object Object],= ,[object Object],2 * 3,[object Object],3,[object Object],2 + 3,[object Object],2 + 3,[object Object],3,[object Object],5,[object Object],Yes,[object Object],This is addition. Can’t cancel!,[object Object],Improper Fractions and Mixed Numbers,[object Object],A fraction whose numerator is the same or more than its ,[object Object],denominator (e.g.) is said to be improper.,[object Object],3 ,[object Object],2,[object Object]
Fractions,[object Object],One common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator.,[object Object],A participant in a sum or a difference is called a term. ,[object Object],The “2” in the expression “2 + 3” is a term (of the expression). ,[object Object],The “2” is in the expression “2 * 3” is called a factor.  ,[object Object],Terms may not be cancelled. Only factors may be canceled. ,[object Object],!?,[object Object],2 * 1,[object Object],1,[object Object],2 + 1,[object Object],1,[object Object],3,[object Object],2 + 1 ,[object Object],= ,[object Object],= ,[object Object],= ,[object Object],= ,[object Object],2 * 3,[object Object],3,[object Object],2 + 3,[object Object],2 + 3,[object Object],3,[object Object],5,[object Object],Yes,[object Object],This is addition. Can’t cancel!,[object Object],Improper Fractions and Mixed Numbers,[object Object],A fraction whose numerator is the same or more than its ,[object Object],denominator (e.g.) is said to be improper.,[object Object],We may put an improper fraction into mixed form by division.,[object Object],3 ,[object Object],2,[object Object]
Improper Fractions and Mixed Numbers,[object Object],23 ,[object Object],Example B.  Put        into mixed form. ,[object Object],4,[object Object]
Improper Fractions and Mixed Numbers,[object Object],23 ,[object Object],Example B.  Put        into mixed form. ,[object Object],4,[object Object],·,[object Object],23 ,[object Object],4 = 5 with remainder 3.  ,[object Object],·,[object Object]
Improper Fractions and Mixed Numbers,[object Object],23 ,[object Object],Example B.  Put        into mixed form. ,[object Object],4,[object Object],23 ,[object Object],3 ,[object Object],·,[object Object],23 ,[object Object],4 = 5 with remainder 3.  Hence, ,[object Object],= 5 +  ,[object Object],·,[object Object],4,[object Object],4  ,[object Object]
Improper Fractions and Mixed Numbers,[object Object],23 ,[object Object],Example B.  Put        into mixed form. ,[object Object],4,[object Object],23 ,[object Object],3 ,[object Object],3 ,[object Object],·,[object Object],23 ,[object Object],4 = 5 with remainder 3.  Hence, ,[object Object],5,[object Object],= ,[object Object],= 5 +  ,[object Object],·,[object Object],4,[object Object],4  .,[object Object],4  ,[object Object]
Improper Fractions and Mixed Numbers,[object Object],23 ,[object Object],Example B.  Put        into mixed form. ,[object Object],4,[object Object],23 ,[object Object],3 ,[object Object],3 ,[object Object],·,[object Object],23 ,[object Object],4 = 5 with remainder 3.  Hence, ,[object Object],5,[object Object],= ,[object Object],= 5 +  ,[object Object],·,[object Object],4,[object Object],4  .,[object Object],4  ,[object Object],We may put a mixed number into improper fraction by doing the reverse via multiplication.,[object Object]
Improper Fractions and Mixed Numbers,[object Object],23 ,[object Object],Example B.  Put        into mixed form. ,[object Object],4,[object Object],23 ,[object Object],3 ,[object Object],3 ,[object Object],·,[object Object],23 ,[object Object],4 = 5 with remainder 3.  Hence, ,[object Object],5,[object Object],= ,[object Object],= 5 +  ,[object Object],·,[object Object],4,[object Object],4  .,[object Object],4  ,[object Object],We may put a mixed number into improper fraction by doing the reverse via multiplication.,[object Object],3 ,[object Object],Example C:  Put           into improper form. ,[object Object],5,[object Object],4 ,[object Object]
Improper Fractions and Mixed Numbers,[object Object],23 ,[object Object],Example B.  Put        into mixed form. ,[object Object],4,[object Object],23 ,[object Object],3 ,[object Object],3 ,[object Object],·,[object Object],23 ,[object Object],4 = 5 with remainder 3.  Hence, ,[object Object],5,[object Object],= ,[object Object],= 5 +  ,[object Object],·,[object Object],4,[object Object],4  .,[object Object],4  ,[object Object],We may put a mixed number into improper fraction by doing the reverse via multiplication.,[object Object],3 ,[object Object],Example C:  Put           into improper form. ,[object Object],5,[object Object],4 ,[object Object],3 ,[object Object],4*5 + 3,[object Object],5,[object Object],= ,[object Object],4  ,[object Object],4,[object Object]
Improper Fractions and Mixed Numbers,[object Object],23 ,[object Object],Example B.  Put        into mixed form. ,[object Object],4,[object Object],23 ,[object Object],3 ,[object Object],3 ,[object Object],·,[object Object],23 ,[object Object],4 = 5 with remainder 3.  Hence, ,[object Object],5,[object Object],= ,[object Object],= 5 +  ,[object Object],·,[object Object],4,[object Object],4  .,[object Object],4  ,[object Object],We may put a mixed number into improper fraction by doing the reverse via multiplication.,[object Object],3 ,[object Object],Example C:  Put           into improper form. ,[object Object],5,[object Object],4 ,[object Object],3 ,[object Object],4*5 + 3,[object Object],23 ,[object Object],5,[object Object],= ,[object Object],= ,[object Object],4  ,[object Object],4,[object Object],4,[object Object]
Improper Fractions and Mixed Numbers,[object Object],23 ,[object Object],Example B.  Put        into mixed form. ,[object Object],4,[object Object],23 ,[object Object],3 ,[object Object],3 ,[object Object],·,[object Object],23 ,[object Object],4 = 5 with remainder 3.  Hence, ,[object Object],5,[object Object],= ,[object Object],= 5 +  ,[object Object],·,[object Object],4,[object Object],4  .,[object Object],4  ,[object Object],We may put a mixed number into improper fraction by doing the reverse via multiplication.,[object Object],3 ,[object Object],Example C:  Put           into improper form. ,[object Object],5,[object Object],4 ,[object Object],3 ,[object Object],4*5 + 3,[object Object],23 ,[object Object],5,[object Object],= ,[object Object],= ,[object Object],4  ,[object Object],4,[object Object],4,[object Object]

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