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one problem 129 129 if N is a positive integer is the value of B minus a at least okay so it's a so n is positive and positive integer and they're saying is a value of B minus a at least twice the value so that's greater than or equal to twice the value of 3 to the N minus 2 to the N and that's what we need to test statement one tells us a is equal to 2 to the n plus 1 and statement and B this is still statement 1 is equal to 3 to the n plus 1 so let's think about this a little bit let's think about this a little bit so let's substitute this into this equation so you get 3 to the n plus 1 3 to the n plus 1 minus 2 to the n plus 1 is greater than or equal to 1 let's multiply this out 2 times 3 to the n minus 2 times 2 to the N so what's 2 times 2 to the N well that's just 2 to the n plus 1 right that's just 2 to 2 times 2 to the N is to the n plus 1 there's a minus sign there so that's where I got it from so interesting just like that we have a minus to the n plus 1 on both sides so we can just subtract that from both side or add 2 to the n plus 1 from both sides to cancel them out so we get rid of that get rid of that and then our statement boils down to is 3 to the n plus 1 greater than or equal to 2 times 3 to the N well we could rewrite 3 to the n plus 1 as 3 times 3 to the N and we're testing whether that's greater than 2 times 3 to the N and they tell us that n is a positive number but it actually doesn't matter I mean because we have 3 to the N on both sides and there's no way you can even if you rose this to a negative exponent you would still have a positive number so you divide both sides by 3 to the N and you the statement rolls down to 3 is greater than 2 which is absolutely correct so statement 1 is enough information to mate to say whether this statement is true or not and actually proves that it's true let's look at statement number 2 statement number two tells us that n is equal to 3 n is equal to 3 let's see if we look at the original statement so if n is equal to 3 then we have to say this is this is what the statement reduces to it reduces to B minus a is greater than or equal to 2 times 3 to the third power so that's 27 minus 2 to the third power well 2/3 is 8 so it reduces to B minus a is greater than or equal to 27 minus 8 is 19 times 2 is 38 well I don't know we don't they don't tell us anything else about B or a or anything so this is a useless statement and actually going through this was a little bit of a waste of time you could have just said hey you're talking about n is I can come up with a number here but that still doesn't give me any information over what B or a is so statement 1 alone is sufficient to answer this question and 2 is is not do some let me go to sky blue problem 130 the inflation index for the year 1989 relative to the year 1970 was three point five six indicating that on average for each dollar spent in 1970 for goods 356 had to be spent for the same goods in 1989 let me write that down 1970 one dollar in 1986 that same was 1989 they're talking about 1989 in that same year in 1989 that same thing would cost $3.59 Pharaoh 3 dollars and 56 cents I mean my sixes and nines confused three dollars 56 had to be sent if the price of a model came mixer increased precisely according to the inflation index what was the price of the mixer in 1970 so we want to know the mixer K sir K in 1970 so statement number one the price of the model K mixer was 102 point four dollars more in 1989 than 1970 so let's let's call this K we're trying to figure out K 70 let's call that's what we're trying to figure out mixer K in 1970 so they say the price of the model K mixer was a hundred two dollars and 40 cents more in 1989 than 1970 so this essentially tells us that K mixer K in 1989 - mixer K in 1970 is equal to 102 dollars and 40 cents but we also have other information we know that mixer K it it went exactly with inflation so the increase in inflation price so we know that so mixer K in 1989 is equal to three point five six times mixer K in 1970 well how do we know that because they told us that mixer K the price of it increased directly with this inflation and anything that was $1 pretty much anything that went with inflation was worth three point five six times more in 1989 so mixer K in 89 is three point five six times more than it was in 1970 so now we have we have this linear equation which was given in the problem statement and we have this linear equation so we have two equations and two unknowns so we actually have enough information just with statement one to solve the problem and I'm actually not even gonna go into it could be a waste of time you should hopefully we're getting into the rhythm of this two linear equations two unknowns enough information if they're quadratic or something then you'd have to dig a little bit deeper statement number two or if they were somehow parallel equations but I wouldn't worry that about that too much on the GMAT statement number two the price of the model K mixer was one forty to forty in 1989 so they're telling us that K in 1989 was 142 40 well this is definitely enough information because we already have this equation that the 89 price which is 142 40 is equal to three point five six times the 1970 price so you just divide this by three point five six and you have your answer so each statement independently is enough to solve this problem problem 131 is 5 to the K less than a thousand so 5 to the K less than a thousand statement number one tells us that 5 to the k plus 1 is greater than 3000 I already know the census is going to help us because it's essentially setting a lower bound on 5 2 so let me just explain it so 5 to the k plus 1 that's the same thing as 5 to the K times 5 right so that's equal to 5 times 5 to the K right you could use this 5 to the first power and then you would add the exponents to get here and that they say is greater than 3000 you divide both sides by 5 and you get 5 to the K 5 to the K that's a K is greater than 600 that doesn't help us 5 to the K could be 6,000 well 5 to the K could be 700 in which case this would be true 5 to the K could be 7000 in which case this wouldn't be true so statement 1 by itself isn't enough information statement number two tells us 5 to the K minus 1 is equal to 5 to the K minus 500 this is interesting so let's see if what we can do here so this is the same thing let's get all the k's on one side so the same thing is 5 to the k minus 1 minus 5 to the k is equal to minus 5 is equal to minus 500 and let's just say this is well this is the same thing as 1/5 or 5 to the negative 1 let me write that 5 to the negative 1 times 5 to the K and then minus 5 to the K is equal to minus 500 let's factor 5 to the K out of this so you get 5 to the K times 5 to the negative 1 that's 1/5 minus 1 right we factored a 5 to the K out there is equal to minus 500 and let's see what we can do 5 to the K 1/5 minus 1 that's minus 4/5 right times minus 4/5 is equal to minus 500 and get rid of the minus sign on both sides multiply both sides by negative 1 and then we can multiply both sides by 5 for it so we get 5 to the K is equal to 500 times what 5 over 4 5 over 4 and that is what that's 125 that's 5 let's see 2500 divided by 4 125 well you don't see 2,500 divided by 4 I want to say it's 475 is that 5/4 no no so each 4th is 125 and you have another so it's equal to 625 even if I just got my math wrong that doesn't matter because they just want to know whether 5 to the K is less than a thousand and it definitely is even if my math 500 times 5/4 1/4 of 500 is 125 times 125 times 5 120 so we'll have one more of those all right 625 so statement 2 alone statement 2 alone is sufficient to answer the question statement 1 was kind of useless I've run out of time see you in the next video