Alright, we're on problem 59. 59 on page 282. If a real estate agent received a commission of 6% of the selling price of a certain house, what was the selling price of the house? So the price of the house. And they just told us that the real estate agent received 6% percent but that alone doesn't tell us much. So statement number one, they tell us the selling price minus the real estate agent's commission was $84,600. So let's see if we can write that algebraically. The selling price, let's say p for price, minus the real estate agent's commission-- well, the they told us at the beginning the real estate agent received 6% of the selling price. So that's 0.06, 6% of the price. So they're telling us that that is equal to$84,600. Well, we're done! That's a linear equation with one unknown. This is algebra one. You can solve for this. Let's see, you could say this is, I mean if you had to, 0.94p is equal to 84,600 and then you'd have p is equal to 84,600 divided by 0.94, whatever that is. But we don't care. We just have to know that we could solve it. Well, since we got so close, let's just solve it. 84,600 divided by point-- this is a bad habit when you're taking the GMAT, you want to just know that you could solve it-- so the selling price of the house is $90,000. We didn't have to solve that. That would be a waste of time on the real GMAT but I just wanted to show you how easy it was to solve. Statement number two, let's see if this is independently useful. The selling price was 250% of the original purchase price of$36,000. We chose p as the selling price, right? The selling price minus-- right, everything we talked about before was the selling price. And now they introduce this thing called the purchase price. So the price selling is equal to-- and that's what we're going to figure out-- is equal to 250%. So that's 2.5 times the purchase price of $36,000. So times$36,000. Actually we didn't have to write this. So the price is 2.5 times 36,000 which is, I'm guessing, let's see, 72, yep, it's \$90,000 again. We didn't have to do that. But once again, this is just a very-- this is actually not even algebra. You just have to multiply 2.5 times 36,000 and your get the answer. So each of these, independently, are enough to solve this problem. So that's D. Next problem. 60. This yellow's a little bit over the top. Let me do more muted color. So they write, if the square root of x/y is equal to-- what does that say-- d? Right, is that what they wrote? Is equal to d? No, that's equal to n I think. Alright, is equal to n, what is the value of x? So statement number one, they tell us that yn is equal to 10. Well, this is pretty useful because this first equation they give us, we would just multiply both sides by y, we get square root of x is equal to yn, right? y times both sides gets us this. And if yn is equal to 10 then we know that the square root of x is equal to 10. So x is equal to 100. So statement one alone is enough. Now what does statement two do for us? They tell us y is equal to 40 and n is equal to 1/4. So once again, they substituted everything else, so we just have to solve for x. So both of these individually are enough so the answer is D. But we could solve for it just for fun. We get square root of x/40 is equal to 1/4 and then you get, essentially solving this, you get square root of x is equal to 10. And you get the same thing, x is equal to 100 and you are all done, right? And you might say, oh, but isn't there a plus or minus? No, because we're saying the square root of something is equal to 10. We're not saying the square root of 100 is what? If you said what is the square root of a 100 you'd say it's plus or minus 10. But if we said the square root of something is 10 that something has to be 100. Anyway, next problem. Don't want to dilly-dally. 61. I want to imbue you with a sense of urgency for the GMAT. How many integers are there between, but not including, r and s. Fair enough. So essentially we have to figure out what r and s are or how far part they are. So statement number one-- so integers between r and s, but not including, remember, not including. So statement number one tells us that r minus s is equal to 10. Now this is an interesting question because if we knew that r and s were integers then we could just pick. r minus s is, maybe r is equal to 11 and s is equal to 1. And then if you actually just want me to write it out you could say, well, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. How many integers are between the two and not equaling the two? 1, 2, 3, 4, 5, 6, 7, 8, 9, right? But that's if we assume that r and s are integers and this would work for any two sets of integers that are 10 apart. But what happens if we do it a little bit differently? What happens if we say that this is 1.1, 2.1, 3.1-- oh no, sorry. What if it's between 1.1 and 11.1? So if it's 1.1 is r-- oh no, let me write it this way. What if r was 11.1 and s was equal to 1.1? Then what are the integers between them but not equal? So if you start at 1.1 you would have 2, 3, 4, 5, 6, 7, 8, 9, 10-- and now 11 would be included because 11 would still be less than r-- 10, 11. So now you have 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. So statement number one is actually not enough because they didn't tell us that r and s are integers. If r and s were integers then the answer would be 9 in this case. But if r and s are not integers then the answer could be 10. So statement number one alone is not enough, the answer could be 9 or 10. OK, what does statement number two do for us. Let me do it in a different color. Statement number two. This is an interesting problem. There are 9 integers between, but not including, r plus 1 and s plus 1. OK. So this is interesting. So they're telling us that there are 9 integers between r plus 1 and s plus 1. So, I mean, actually just looking at it, inspection, you could say, oh, well then there are 9 integers between r and s. And they say but not including. So this is enough. This alone is sufficient and you don't need statement number one. Statement number one actually doesn't get you anywhere. So let me prove to you that if there are 9 integers between r plus 1 and s plus 1-- Well, think of it this way: if there are 9 integers between r plus 1 an s plus 1, if you subtract 1-- so s is one of the one's that are between-- now I want to make sure I do this right-- Let me think of a good way for me to prove it to you. Well, I'll just do it with an example. So let's say that the example is 11 and 1, right? This isn't a proof but I just want to give you the intuition, right? So if r is 11 and s is 1, so then you would have 12-- well, I don't want to do it that way. The easiest way to think about it is, it doesn't matter how much you are adding-- if you add the same amount to both the bottom of the range and the top of the range it should not change the total number of integers that are between them. So there would also be 9 integers between r minus 1 and s minus 1. You're just shifting the range along the number line but you're not going to actually change the number of integers in between them. So if they're 9 integers between r plus 1 and s plus 1, there's going to be 9 integers between r and s. So two is all you need. Hopefully that's a satisfactory explanation. I didn't want to go into something rigorous when we're trying to imbue you with a sense of urgency. See if I have time for the next problem. OK, 62. What is the number of members of club x who are at least 35 years of age. So number who's age is at least 35, so it's greater than or equal to 35. The number who's age is true. OK. They tell us statement number one, exactly 3/4 of the members of club x are under 35 years of age. So 3/4 are less than 35. That is fair enough but that doesn't tell us the number that are at least 35. This tells us the percentage. This tells us that 1/4 above or we say 1/4 greater than or equal to 35. 1/4 of total members greater than or equal to 35. So it tells us proportion but it doesn't tell us the total amount. And I just got the 1/4 from 3/4 are less than 35, 1/4 are going to be greater than or equal to 35 years of age. So one by itself doesn't help us. Let's see what two does for us. The 64 women-- am I reading the same problem? OK, yeah-- the 64 women in club x constitute 40% of the club's membership. OK. So 64 is equal to 40% of the total membership. Now we're set, right? Because we can use this equation to figure out the total membership. Call it t for total. t is going to be equal to 64 divided by 0.4. They just told us that essentially 64 is 40% percent of the total. They said it's the women and all that. But they could've just told us that 64 is 40% of the total. So then you can solve for the total. And then you substitute that here and you say 1/4 of the total are 35 five years or older and you get what we were trying to solve for. And so the answer is 1/4 times 64/0.4, and whatever that is, you can put it in your calculator. But all we had to care is that when you use both of these equations combined, you're able to solve for the number that are greater than or equal to 35 years old. And each statement alone isn't sufficient, so you need both statements. And that is C, both statements together are sufficient. See you in the next video.