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CCSS.Math: , , ,

in the last video we made a visual argument as to why the this expression has to be less than one-third and this expression we already figured out is the fraction that are bears now we'll make an algebraic argument or I could call it an analytic argument and to make this argument I'm going to leave this expression we know this is the fraction that are bears and I'm going to write this one-third in a form that looks a lot like this and then based on the information have we can directly compare them so how can I write 1/3 maybe with the bee as a numerator well 1/3 is the same thing as B over 3 B which is the exact same thing as B over B plus B plus B so now this is looking pretty similar the only difference between this expression right over here B over C plus D plus B and B over B plus B plus B is that our denominators are different and the only difference in our denominators this denominator has a C plus D here well this has a B plus B over here now we have to ask ourselves a question what is larger is C plus D larger than B plus B and I encourage you to pause that and think about that for a second well yes we already see right over here it was given to us that C is greater than D that is greater than B so both C and D are greater than B so C plus D is definitely going to be greater than B plus B so this denominator right over here is greater so this has a larger denominator larger larger denominator or denominator this right over here has a smaller smaller denominator denominator denominator and since we know this has a larger denominator this has a smaller denominator they have the exact same numerator they both have B as a numerator we know that this whole thing we know that this whole thing must be a smaller quantity smaller quantity if you have the same numerator but one expression has a larger denominator it must be smaller lightweight so how does that work we just remember and we just imagine you have the same numerator what's going to be bigger a over seven or a over five well here you're dividing a by seven you're dividing into many more chunks and over here so this right over here is smaller this right over here is larger so this is the larger this right over here this right over here is smaller so it's same numerator the larger the denominator the smaller the quantity is going to be so going back to the original question this is the smaller quantity and this right over here one-third is the larger quantity