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problem 51 actually let me scroll all the way up problem 51 they drew a little triangle here which I assume I need so I'll draw it as well that side side is flat and the side goes down like that and they draw some angles if this is being angle X this is angle Y this is angle Z and they say what is the value of Z in the triangle above fair enough statement number one is X plus y X plus y is equal to 139 well as soon as you see a triangle you have to figure out one side you should always remember that the sum of the angles of a triangle add up to 180 so we know that X plus y plus Z is equal to 180 statement one just told us that X plus y is equal to 139 so we know from statement one that this is 139 so that would just make this equation 139 plus Z is equal to 180 which is very easy to solve subtract 39 from both sides would you get C's equal to 41 or something like that it doesn't matter you don't have to get the answer you just don't know that you can get the answer so statement 1 alone is enough now what do they give us in statement number 2 statement 2y plus Z is equal to 108 this statement alone is useless I mean I we don't know what Y is so we can't figure it out maybe one thing is true if you if you had both of these statements then you could figure out all of the angles of the triangle because you know you know that Z is 41 and you could figure out why then you could what you know Y then you can figure out X by using but that's top they're not asking us that they're asking we need to figure out what angle Z is and this statement alone will not tell us what angle Z is because we don't know what angle Y is but statement 1 alone works so the answer is a next problem turn the page 52:52 if X Y & Z are non zero so X Y or Z did not equal zero you know they could have written x times y times Z does not equal is not does not equal zero because that's the same thing because if any of them or zero would have been equal zero so I could have written X Y Z does not equal zero because if you multiply three nonzero numbers it's not going to equal zero but anyway they're asking is X Z is equal to is it equal to 12 X Z equal to 12 is the question statement one tells us and I'll do it in a different color ease the monotony statement one tells us that x squared times y Z is equal to 12 X Y so let's see what we can do what if we were to divide both sides of this equation by X Y so if you get let's divide both sides by X first so if you divide both sides by X and we can do that because we know that X does not equal zero so then you get divide both sides by X you get X Y Z is equal to twelve Y right divided by X on both sides now let's divide both sides by Y you get X Z is equal to twelve and we're done statement alone is enough and remember the only reason why we were able to divide both sides by y is because we knew it didn't equal zero now what a statement to tell us statement two tells us that Z over four is equal to 3 over X and here we can cross multiply x times Z is equal to three times four and if that didn't make sense to you could do it one step at a time multiply both sides by four you get Z is equal to 12 over X just multiplied both sides by four now multiply both sides by X and you get X Z is equal to 12 so each statement independently is enough to solve the equation so that's that's D now let's have to keep looking back and looking at which letter say I should memorize that you should at least you have to take the GMAT all right 53 53 a certain company currently has how many employees so we want to know the number of MPs how many employees okay so statement number one if three additional employees are hired by the company and all of the present employees remain there will be at least there will be at least 20 employees in the company so let's say e is the number of employees today so e plus 3 so the current number of employees and three additional are hired and all of them remain there will be at least 20 employees in the country so at least tells us greater than or equal to 20 employees that's what statement one is and that by itself doesn't tell me how many employees are at the company it tells us that there are at least 17 employees at the company but it doesn't tell us anything else I mean you can subtract 3 from both sides and you say e is greater than or equal to 17 but it could be a hundred I don't know okay statement two if no additional employees are hired by the company and three of the present employees resign so E minus three there will be fewer than 15 employees at the company so fewer fewer than 15 employees so not 15 or lower they say 15 fewer than 15 so less than 15 right here they said at least 20 so that's greater than or 20 greater than or equal to here they said fewer then they didn't say 15 or fewer they said fewer than 15 so if you simplify this you get add 3 to both sides you say e is less than 18 so let me ask you a question well first of all this by itself isn't helpful because this just tells us e we could have only 5 employees at the company or we could have 10 this doesn't tell us by itself but if we use both of these in conjunction if we know that E is less than so if we say 17 is less than or equal to e I just wrote this in a different order all right 17 is less than or equal to e which is less than 18 I'm using both statements what is he well it has to be greater than or equal to 17 so it could be 17 can it be 18 well no because that's to be less than 18 so E has to be 17 so both statements together are needed to solve this problem and that is C both statements together are sufficient next problem 54 54 let me erase this little 5 I drew here 54 all right what is the value of N in the equation minus 25 plus 19 plus N is equal to as well just to simplify it I'm we could add up my is 25 plus 19 you get what you get you get minus 6 so you get n minus 6 is equal to s all right that's right actually you don't have to do this remember your whole problem the whole point of this is to figure out whether you can solve the problem you don't always have to solve it but I'm solving them here just to prove it to you to give you the intuition of when you can solve it so if you add 6 to both sides anyway this is just a simplification let's read the statements I haven't read them yet maybe I didn't have to do any of that all right what problem was aisle 54 ok s is equal to 2 well clearly if we know what s is equal to 2 we can solve for n n would be equal to 8 this statement alone is sufficient now problem number statement number two tells us n over s is equal to four well once again as long as we have some relation with N and s and we could solve for n for s and then substitute back in we can solve for n and to prove it let me do it we have n is equal to four s and is equal to 4 s and then if we know that n is equal to 4 we're ordering trying to solve for n so we could also divide we could write s is equal to n divided by 4 and then you can substitute back into that equation so then you would get n is equal to s which is n over 4 plus 6 and then you could just solve for n and if you want to do it let's see multiply both sides by 4 you get 4 and is equal to n plus 24 that's an equal sign not not equal for n I just want to apply it both sides by 4 for N is equal to n plus 24 subtract 10 from both sides you get 3 n is equal to 24 and then you get n is equal to 8 again so each statement alone is sufficient and that is I should really memorize this that's d each statement alone is sufficient ok let's see do I have time for 55 let me wait for that for the next video see you soon