GMAT: Data sufficiency 26

We're on problem 110. Let me scroll this up. Whenever Martin has a restaurant bill with an amount between $10 and $99 he calculates the dollar amount of the tip as two times the tens digit of the amount of his bill. Fair enough. So essentially, if there is a ten digit, he just multiplies it by 2. If the amount of Martin's most recent restaurant bill was between $10 and $99, was the tip calculated by Martin on his bill greater than 15% of the amount of the bill? That is the question. Statement 1 says, the amount of the bill was between $15 and $50. So 1 essentially is enough for this problem, is sufficient, if for every bill, based on his calculation where you double the tens digit, it's going to be greater than 15. And I suspect, let's see if we take the lower end of this, on this he'll pay $2. He'll pay a $2 tip on $15. And what percentage is that? 15 goes into 2.00. 1, 15, 50. 15 goes into 50 3 times. 3 times 50, 45 and it just keeps going. So that's a 13% tip. Well, it's going to be even lower than that at $16. Maybe the bill wasn't $15, the bill was $16. So the tip is, at the lower end it's 13%, 12%. And at the higher end, if the bill is $40 exactly and he pays $8 on that. If the bill is $40, then Martin would pay $8, which would be 20%. So I can pick different numbers in this range, and based on the way Martin calculates his tip, he can either pay less than 15% or more than 15%. So statement 1 alone is not sufficient. What does statement 2 tell us? The tip calculated by Martin was $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. So let's think about it. So the worst case is if the bill-- if he paid $8 an a $40 bill, that's definitely more than 15%. That's 20%. That's what I just actually calculated. Let's see the worst case is on a $49 bill. That the bill keeps going up and he just pays $8. So is 8 bigger than 15% of 49? Well yeah. Because 8/50 is equal to what? That's equal to 16%. So if 8/50 is 16%, 8/49, if we lower the denominator a little bit, that's going to be greater than 16%. So no matter what range in the forties 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%. Not to speak of even 15%. So statement number 2 alone is sufficient to answer this question. And statement number 1 is fairly useless. Problem 111. The price per share of stock x increased by 10% over the same time period that the price per share of stock y decreased by 10%. 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 x final is equal to an increase by 10% from the initial period. OK, so it equals 1.1 times x initial. Fair enough. And then, over the same time period y decreased by 10%. 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 y 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 as 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 x f. That's the final. That's the increased share price. It increased from the initial to the final. So the increased price per share of stock x was equal to the original price per share of stock y. So that equals 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 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 0.9 times y initial. 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 the initial y. So x final is equal to y initial. So that means that y initial is equal to this. So that equals 1.1 x initial. And that means that we can divide both sides of this equality by 1.1. And we get x initial is equal to 1 over 1.1 times y initial. So then we have this. 1 over 1.1 times y initial. And then these two would cancel out. And you would actually have your answer. So statement 1 alone is sufficient to answer the question. 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 10/11 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-- that's the increase-- that this was equal to 10/11 times the decrease in the price per share of stock y. So what was the decrease? This was y initial minus y final. Because this was a larger number and we wanted a positive number here. Because we're just saying the decrease. We're not saying the negative increase. So let's see if we can simplify this at all. So let's see, 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 is. So let's think about this. Let's write x final is equal to 1.1 times x initial. Right So we have 1.1 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. What do we want? We want y final. So we just want to substitute for y initial. I am confusing myself. So y initial is going to be equal to y final divided by 0.9. So 1 over 0.9 y final. I know this is a little confusing. Minus y final. I just did a substitution for y initial. And then here, well this is actually, we know that we can solve this problem. Although it gets quite hairy. Because here, if you think about it, you're going to get some number, well you're going to get 0.1 x initial is equal to-- you're going to get some constant after you do all this math-- times 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 going to do that because it's actually kind of hairy and I don't have-- 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 did it in my own head. So the 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. Because y final is the smaller one. x 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. And y initial, we can rewrite as y final. You could say y initial is equal to y final divided by 0.9. And that's what we did there. And then you could see here, this will simplify to some constant times y final. And then you multiply that times 10/11. So you get some constant times y final. And then you have 0.1 times x initial. 1.1 minus 1. 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 you in the next video.