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we're on problem 110 scroll this up 110 whenever Martin has a restaurant bill with an amount between ten dollars and ninety nine dollars he calculates the dollar amount of the tip is two times the tens digit of the amount of his bill fair enough so essentially if there is a ten cent he just multiplies it by two if the amount of Martin's most recent restaurant bill was between ten and ninety nine fair enough was a tip calculated by Martin on his bill greater than 15% of the amount of the bill so was tip greater than 15% that is the question statement one says the amount of the bill was between 15 and 50 dollars fifteen and fifty dollars so one it will essentially is enough for this problem is sufficient if every for every bill based on his calculation where you double the tens digit it's going to be greater than fifteen and I suspect let's see if we take the lower end of this on this he'll pay two dollars right he'll pay $2 tip on 15 and what percentage is that 15 goes into 2.0 11550 15 goes into 50 three times three times 50 45 and it just keeps going so that's a 13% tip so if we're closer to well I guess it's never gonna well it's going to be even lower than that at$16 right maybe the bill wasn't fifteen the bill was $16 so the tip is I don't know at the lower end it's 13 12% and at the higher end if it's on if the bill is oh I don't know let's say the bill is$40 exactly if the bill is $40 exactly and he pays he would pay$8 on that right if the bill is 40 then bill would pay $8 which would be 20% so I can pick different numbers in this range and based on the way bill calculates his tip oh no not bill I don't know how I got that name oh they said the amount of the bill right depends how Martin calculates his bill he he can either pay less than 15% or more than Dean percent so statement one alone is not sufficient what a statement to tell us the tip calculated by Martin was$8 so the tip is equal to $8 well this is going to be 2 times the tens digit so that means that the bill was equal to 40 I don't know 40 something right so let's think about it if the bill so the worst case is if if the bill if you paid$8 on a $40 bill that's definitely more than 15% right that's 8 of 40 that's 20% that's what I just actually calculated but if you paid it let's see the worst case is on a$49 bill that the bill keeps going up and he just pays $8 so 8 is 8 bigger than 15% of 49 well yeah because 8 over 58 over 50 is equal to what it see that's equal to 16 percent right so if 8 over 50 is 16 percent 8 over 49 if we lower the denominator a little bit that's going to be greater than 16 percent so no matter what range in the 40s the bill was whether it's$40 or $49 or anything in between an$8 tip is going to be more than it's actually going to be more than 16 percent not not to speak of even 15% so statement number two alone is sufficient to answer this question and statement number one is fairly useless problem 111 the price per share of stock X increased by 10 percent over the same time period that the price per share of stock Y decreased by 10 percent the reduced price per share of stock Y was what percent of the original price per share of stock X fascinating so let's do initial and final so the price so X final is equal to it increased by 10 percent from the initial period okay so it equals one point one times X initial fair enough and then over the same time period y decreased by 10 percent so Y final is equal to 10% less than Y initial so that's 0.9 times y initial and what they want to know is the reduced price per share of stock Y so that's why F was what percent of the original price per share of stock X of X initial so this is what they want to figure out this is a percentage so let's see if the statements help us out at all the increased price per share of stock X was equal to the original price per share of stock Y so the increased price per share of stock X so that's XF that's the final that's the increased share price right an increase from the initial to the final so that increased price per share of stock X was equal to the original price per share of stock Y so that equals Y initial y initial so this is interesting I don't know if it gets us anywhere this deals with X final and X initial so I think we can so we could bright right if we could write all of it in terms of Y initial so Y final equals so let's rewrite this this is equal to Y final is point nine times y initial right that's just from this equation 0.9 times y initial and let's see if we could write the initial x in terms of in terms of initial Y so X final is equal to Y initial so that means that Y initial is equal to this so that equals one point one X initial and that means that we can divide both sides of this equality by one point one and we get X initial is equal to one over one point one times y initial right so then we have this one over one point one times y initial and then these two would cancel out and you would actually have your answer so statement one alone is sufficient to answer the question and it wasn't obvious to me at first but then you have to realize that the terminology is confusing but that you can actually write both of these in terms of y-initial given that information given the fact that X final is equal to Y initial let's see what statement 2 does for us the increase in the price per share of stock X was 10 xi the decrease in the price per share of stock Y let me think about that the increase in the price per share of stock X so that means that X final minus X initial right that's the increase that this was equal to 10 over 11 times the decrease in the price per share of stock Y so what was the decrease this was Y initial minus y final minus y final because this was a larger number and we wanted a positive number here cuz we're just saying the decrease we're not saying you know the negative increase so let's see if we can simplify this at all if we can simplify this at all so let's see you get well you get X final minus X initial is equal to 10 11 Y initial minus 10 11 y final and remember the whole time we just want to figure out what Y final over X initial our Y final over X initial so let's think about this so X final minus the next let's write X X final is equal to one point one times X initial right so we have one point one times X initial minus X initial is equal to actually I should have done that in the first step let's just skip this right now let's just write the 10 over 11 10 over 11 we want what do we want in the fund we want Y final so we just want to substitute for Y initial so Y initial sorry I'm confusing myself so Y initial is going to be equal to Y final divided by 0.9 so 1 over 0.9 Y final and it's a little confusing minus y final I just did a substitution for y initial and then here well this is actually we know that this is we can solve this problem although it gets quite hairy because here if you think about it you're gonna get some number you're gonna get well you're gonna get 0.1 X initial is equal to you know 1 divided by point you're gonna get some constant after you do all this math times y final y final and so you can easily figure out what Y final divided by X initial is just by dividing both sides by X initial and then dividing both sides by a and you would have solved the problem and I'm not gonna do that because it's actually kind of hairy and I don't have you know 1 divided by 0.9 and then multiplying it by 10 11 is a fairly convoluted way of doing it but hopefully you see that this is solvable and let me just review that again because I think I I did in my own head in a little secure dissuasive a statement itself said the change the gain in X X final minus X initial was 10 11 times the loss in Y so Y initial minus y final cuz Y final was a smaller 1x final we can rewrite in terms of X initial just using our initial the fact that it was 10% more so I just did that here right and Y initial we can rewrite as Y final Y initial you could say Y initial is equal to Y final divided by 0.9 and that's what we did there and then you can see here this will simplify to some constant times y final and you multiply that times 10 11 so you get some constant times y final and then you have point one times X initial right one point one minus one and then you can just do some simple algebra to figure out what Y final over X initial is so both statements independently are sufficient to solve this problem I think that was the hardest one we've done so far see in the next video