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on problem 55 55 at a certain picnic each of the guests was served either a single scoop or a double scoop of ice cream how many of the guests were served a double scoop of ice cream so let let's say that D is the number of guests that were served double scoop and that's what we're trying to figure out so what are their statements statement number one says at the picnic sixty percent of the guests were served a double scoop of ice cream so 60% of the total were served double scoop let's see if we can write that algebraically so 60 percent of the guests so 60 percent of the guests so the guests the total number of guests are the are the number of guests that so that were served double scoops plus the number of guests that were served single scoops then we have all of the guests right this is this is equal to the total number of guests number of guests and they tell us that 60 percent of them were served double scoop so the number that served were served double scoop is 60 percent of the total so that statement number one is just an algebraic equation and this alone though doesn't help me solve this equation maybe in the next statement they'll tell us how many total guests there are what this is and that'll help me but I don't know let's see statement number two statement number two they say a total of 120 scoops of ice cream were served to all the guests at the picnic so once again they're telling us that they're 120 scoops of ice cream they're not telling us that they're 120 guests so let's see what are the total number of scoops so there's the number of single scoops and each of those are the number of guests that got single scoops are asked and then the number of scoops is just one times that cause each got one scoop and then you have the number of guests that got double scoops but then the number of scoops they got is going to be two times that so this is an total number of scoops right the number of people who got single scoops types one plus the number that got double scoops times two and that is going to be equal to the total number of scoops and this alone isn't going to help us solve how many double scoops were served depends on the single scooped and all of that but if you look at both of these statements we have two linear equations in two unknowns so at this point you should immediately say unless somehow these are you know the same equation or they don't are somehow these are two parallel lines these should intersect and come up with and and allow you to solve this equation so the answer you should immediately if you especially if you don't want to waste time you should say that both statements together are sufficient and if you want me to prove this to you let me see if I can solve this I'll do it in a different color so let me do some let me do some substitution so we here we get s is equal to 120 minus 2d and we can substitute this back here so you get 0.6 times D plus s which is 120 minus 2 D is equal to D and so you get C 0.6 times 120 minus D is equal to D and what's 0.6 times 120 that's 72 right 6 times 12 is 72 fixed right so that's 72 minus 0.6 D is equal to D so you get 72 is equal to 1.6 D and then let's see 72 divided by one point six is equal to 45 so that we solved the problem D is equal to 45 45 double scoops were served and then we could substitute back into the equation to figure out how many single scoops or how many total guests over but this is all that we had to figure out and we were able to using both statements combined to the next problem 56 what is the value of XY okay statement number one they tell us that Y is equal to X plus one statement and that's gonna tell me what XY is I mean I could there's a bunch of X's and a bunch of different Y's depending on which what I choose to they tell us y is equal to x squared plus one once again this by itself it completely depends on which ex-wives I pick all right I can pick any combination and get different values you could try that out yourself but I have two equations in two unknowns it's a little tricky because one of them is nonlinear but let's see if we can figure it out with substitution if y equals this and y equals that we could set these two equal to each other and see if we could solve for X so we get X plus 1 that's the first one should equal this one because they're both equal to Y x squared plus 1 let's see we can subtract 1 from both sides and you get X is equal to x squared and let's see you can write you could you could say X well you can divide if we assume that X does not is if X does not equal 0 so there's two solutions here right X could be equal to 0 that's completely legitimate they never told us that X is an equal 0 that would satisfy that equation and then what other X would satisfy X could be equal 0 or X could be equal to 1 either of those are completely legitimate answers here right fair enough right if X is 0 this is true 0 equals 0 if X is 1 1 is equal to 1 squared X can't be negative 1 negative 1 is not equal to negative 1 square so these are the two solutions for X and now let's so right now it seems like oh well that's a little shady but remember we're not trying to solve for X or solve for y we're trying to solve for XY so maybe when we solve for y is something I don't know interesting could happen let's solve for y if you if you assume X is equal to 0 if X is equal to 0 what's Y well Y is equal to 0 plus 1 Y is equal to 1 in this case that's one solution and if you put X is equal to 0 here you also get Y is equal to 1 so this is one set of solutions and in this reality XY is going to be equal to what XY it's going to be equal to 0 right 0 times 1 is 0 now if X is equal to 1 what is y well if X is equal to 1 Y is equal to 1 plus 1 Y is equal to 2 well then X Y is equal to 2 so using both of the equipment both both of the statements I still don't know whether X Y is equal to 0 or 2 someone would have to tell me that X is not equal to 0 or X is equal to 0 and only then can I really solve this equation so the answer to this even though we have two equations and two unknowns because one of them was a quadratic and had two solutions we actually don't know what X Y is so the answer is each that the statements together are not sufficient or e that's interesting because they actually kind of make you do a little bit more work than just eyeballing it and saying oh I have a two equations two unknowns you got to be a little careful when the equations are nonlinear at least I think that's right alright question 57 they want to know what 101 over X plus 1 over Y so the police sirens outside I live in Palo Alto there's not normally a high crime scene here but you never know anyway so 1 over X plus 1 over Y so statement number one is X plus y is equal to 14 see that doesn't well actually let me let me try to simplify this a little bit if I were to try to add these two what could I do well when you add two fractions you have to find a common denominator and easy common denominator for x and y is just X Y so 1 over X is 1 over X Y what's Y over X y plus right this is the same thing as 1 over X plus and if I had X Y here what's 1 over Y well that's x over X Y so that can be re-written as y plus x over X Y and now when you rewrite this statement like this then statement1 looks a little interesting they gave us the numerator at least they haven't told us what XY is but they told us that Y plus X or X plus y is equal to 14 so statement 1 it's kind of part of the puzzle but it's not completely helping us let's see what statement 2 tells us I suspect if this is solvable the hospitalist with XY yeah telephone XY is it tell us that XY is equal to 24 so then we're done right this simplifies to this and they're telling us what this is X plus y is 14 so it equals 14 over 24 not that we have to really solve for it but that is equal to 7 over 12 but as soon as we got you say oh that's that that's that you're done you say statement both statements together are sufficient so see 58 58 if d denotes a decimal is D greater than greater than or equal to 0.5 so there's D greater than or equal to 0.5 if it is 0 here so you make sure you notice the decimal D greater than or equal to 0.5 statement number one tells us when D is rounded to the nearest tenth the result is 0.5 when it's rounded the nearest tenth the result is 0.5 so let's see it could be if we're this could be 0.5 to and when you round it to the nearest tenth it becomes 0.5 but on the other hand it could be point 4 6 and when you round it to the nearest tenth you also get point 5 so since statement one says it could be either one of these it really doesn't help us know whether we're greater than or equal to 0.5 this one is greater this one isn't so statement 1 by itself doesn't help statement 2 when D is rounded to the nearest integer the result is 1 now this is interesting if something is is rounded to the nearest integer it has to be well it has to be greater than 0.5 right I mean what are it the smallest number that when you round it to the nearest integer equal to 1 is 0.5 right so it has to be 0.5 or it has to be 0.5 1 because point 4 9 9 9 9 9 when you round it to the nearest integer is going to be equal to 0 these have to be right that's the only way you can get round into one if you're 0.5 or above so actually statement 2 alone is all you need to know that D is greater than or equal to 0.5 so that is choice B statement two alone is sufficient I've realized I've run out of time see you in the next video