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let's say that someone's running a fairly unusual pet store that only has three types of animals in it it has cats and we represent the number of cats with the letter C it has dogs and we represent the number of dogs by the letter D and it has bears that's what makes it unusual and we're going to represent the number of bears with the letter B now this person also tells us that at this unusual pet store the number of cats is greater than the number of dogs which is greater than the number of bears now based just on this information they then ask us which expression is greater B divided by C plus D plus B or 1/3 and I encourage you to pause the video now and come up with your own rationale behind which of these is greater or maybe they're not maybe neither is greater maybe they're equal or maybe you just don't have enough information from what this pet store owner told us to actually figure out so pause the video now so let's reason through what this expression is actually representing this is B the number of bears over the number of cats plus the number of dogs plus the number of bears so there's the number of bears over the sum of all the animals so it's the number of bears over all of the animals so this really is the fraction that are bears the fraction that are bears that are bears now so this question really reduces to is the fraction that are bears greater than a third less than a third equal to a third or can we not tell there's a bunch of different ways to do this problem and I'll try to expose you to many ways in this video and the next so one let's do a visual one doing different cases so let's assume that the fraction that are bears let's do this an area where the fraction that our bears are greater are greater than third so let's visualize this so this is if this little diagram right over here this is let's see this is all of the pets in the pet store and I've divided into thirds right over here if the fraction that our bears is greater than one-third then it might look something like this the fraction of bears so this is a third under the dotted lines if I want to be greater than one-third I'll go a little bit more than greater than one-third so we go greater than one-third right over there but if the fraction of bears is greater than one-third well the fraction of dogs has to be even greater than that so it has to be at least as much as this blue area so it's going to be even greater than that and the fraction that our cats has to be even greater than that and I didn't even make this one that great so it could even be you know has to be more than this blue area and the fraction that our cats this to be even greater than that and you see you can't have three things that are all greater than one-third adding up adding up to a whole you can't have all three parts of it being greater than one-third so this scenario breaks down another way you could have thought about it is if B if this expression right over here is greater than 1/3 then C plus D over this has to be less than 2/3 but that means one of them C or D would have to be definitely less than 1/3 in order for that to work out so at least thinking about this way you know that this thing cannot be greater than third let's think about whether it can be equal to third let's imagine if this expression the fraction that are bears were equal to 1/3 so let's draw let's make our diagram again so in this situation the Bears are exactly 1/3 so I'll just color in this third right over here so they're exactly exactly 1/3 our Bears but we know that the number of dogs is greater than the number of bears so the fraction that our dogs has to be greater than the fraction that are bears so the fraction that our dogs would have to be greater than 1/3 greater than 1/3 but if the fraction our dogs are greater than 1/3 then all we have less is something less than 1/3 for the cats and the cats that's actually the largest fraction of the animals or cats so we know we know that this also is not a possible so what we're really left with is that this has to be less than one third and let's make sure that that's reasonable so let's let's paste another example right over here so if the blue or if the bears I should say are less than one third are less than one third you can you can imagine and I may make it a little bit more dramatic so that you can you can make it clearer so this is less than a third and let's say the blue let's say the the the or net that with the purple the dogs let's say they're roughly exactly one-third they're exactly one-third then you could make a scenario where you have a largest proportion of of cats so this would be the cat's right over here that's not the same colors that can't let me do the same colour is that that pink color so this right over here is our cats so this is completely believable c over c plus d plus b c over c plus D plus B does indeed look bigger than then D over C plus D plus B which does indeed look bigger than B over C plus D plus B so this is completely this is completely believable to be less than one-third so based on just this visual argument that we just made you could say hey look this right over here is the larger quantity in the next video I'll give what I would call more of an analytical argument where I won't draw graphs - or diagrams to show that this has to be less than one-third