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GMAT: Math 35

177-180, pg. 176. Created by Sal Khan.

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Video transcript

My wife is looking at me kind of strange right now. So forgive me if I have a problem doing some of these problems. But anyway, problem 177. She said she wanted to observe me making videos. Let's see how this goes. Do you want to say hi? No, no, OK. No, I'm not going to record it. A company accountant estimates that airfares next year for business trips of 1,000 miles or less will increase by 20%. So less than 1,000 miles. Airfares next year going are going to go up 20%. And for all other business trips, it will increase by 10%. So essentially greater than or equal to 1,000 miles, airfare is going to increase by 10%. This year total airfares for business trips of 1,000 miles or less were $9,900. So that's the total for 1,000 miles or less. And airfares for all other business trips were 13,000 miles. According to the accountant's estimate, if the same number of business trips will be made next year as this year, how much will be spent for airfares next year? I have a problem with what they say, because they say the same number of business trips. But we really, I'm thinking, have to assume that not only are the same number of business trips made, but the allocation between less than 1,000 and greater than 1,000 is also going to have to be the same. But anyway, let's just do what I think they're assuming. So essentially you just have to take the $9,900 and increase it by 20%. 9,900 times 1.2. And then add that to the 13,000 increased by 10%. So 1.1 times 13,000. That's a 13, not a B. So let's see. Let's just write 9,900 times 1.2. 2 times 0 is 0. 2 times 0 is 0. 2 times 9 is 18. 2 times 9, 18, plus 1 is 19. Add a 0 and then you have 9,900. 0, 0, 9, 9-- scroll down-- 0, 0, 8, 9 plus 9 is 18, 2 plus 9 is 11. And we have one number behind the decimal. So that's 11,880. And then 1.1 times 13,000. So it's 13,000 times 1.1. 1 times 13,000 is 13,000, 0, 1 times 13,000 is 13,000. 0, 0, 0, 3, 4, 1. And we have one number behind the decimal. So it's 14,300. And if we were to add these two together. Let's see, if I add 11,880 to this, I get 0, 8. 3 plus 8 is 11. So, 26,180. And that's choice B. Next problem. Problem 178. If x star y is equal to xy minus 2 times x plus y, for all integers x and y, then what does 2 star minus 3 equal? So we essentially just have to do pattern matching. Everywhere we see an x here, we substitute with 2. And everywhere we see a y, we substitute with minus 3. And just to be clear, this isn't an equal, that's a minus right there. So it's 2 times minus 3, minus 2 times 2, plus minus 3. 2 times minus 3 is minus 6. Minus 2 times 2 plus minus 3 is minus 1. That's the same thing as 2 minus 3. And let's see, you have a minus and a minus, so we can make those both plus. So you have minus 6 plus 2. When you multiply it by 1, that's the same thing. So minus 6 plus 2 is equal to minus 4. And that is choice C. Problem 179. I'll switch colors to ease the monotony. 179, the table above shows the number of students in the 3 clubs at [? McCullough ?] School. So let me write the table. So they have the club. And they say the number of students. And they have chess, drama, and math clubs. The chess club has 40 students, drama has 30, and math has 25. The table above shows the number of students in 3 clubs at [? McCullough ?] School. Although no student is in all 3 clubs, 10 students are in both chess and drama. I feel a Venn diagram coming on. 10 students are in both chess and drama. 5 students are in both chess and math. And 6 students are in both drama and math. How many different students are there in the 3 clubs? Yeah, I think we need to do a Venn diagram. So let's draw the chess students first in yellow. Actually I'm not going to do circles, because I think I have to draw it a little bit different. Because there's no students in all 3 clubs. So let's say that this is the chess club. There's 40 people in that. And let's do drama next. There are 30 students in the drama club. And then how many people are in both? Well they told us that there are 10 people in chess and drama. So this is right here is 10. And then let's draw math. So math will intersect with both of these. It looks like that. Math has a total of 25 students. And their overlap with math and drama is 6. And the overlap with math and chest is 5. So the main question we have to ask is, if we were to add up all of these-- if we were to add up the 40 plus the 30 plus the 25-- how many times are we counting each of these sections? So let's just try that, because we just want to know the total number of students in 3 clubs. So let's take the 40. We're already counting this 10 once, right, because they're in that 40. So let's add the 30. So 40 plus 30. If we add the 40 plus the 30, we're double counting this 10 that's in both clubs. So if we were to subtract one of them, this number right here is the number of students in the chess and drama clubs, right? So that is-- what is that-- that is 40 plus 30 minus 20. That's 60 students. So 60 students in chess and drama. And now if we add 25 students-- so we're adding the math club to that-- who are we double counting? Now we're double counting these 6 that we've already counted right? Because we added them in the 30 from the drama club. And we're also double counting these 5, which we added when we first took into account the 40 from the chess club, right? So since we've already counted those, we need to subtract those from the 25. So minus 5 and minus 6. This problem is all about not double counting. So let's see, 60 plus 25 is 85, minus 5 is 80, minus 6 is 74. So 74 total students. So that's choice C. Problem 180. In a nationwide poll, N people were interviewed. So question number one. That's in the question, question number 1. So it said for question number 1, 1/4 said yes. And of those, 1/3 answered yes to question 2. Which of the following expressions represents the number of people interviewed who did not answer yes to both questions? OK, so first of all, you have the people who answered no to question 1. So that was 3/4 of the people, right? No on question 1. And who cares what they answered on question 2, right? Because they definitely said no to one of the questions. Because they want to say who did not answer yes to both questions? So these guys fall into that category. They said no to one of the questions. And then you want to say, what percentage of the total people said yes on the first question and then no on the second? That combination would be if 1/3 of the people who said yes on the first question, said yes on the second, then 2/3 said no. And what percentage, or what fraction is this of the entire population? Well it's 2/3 of 1/4. So that is 2/3 times 1/4 is equal to, 1, 2. That's equal to 1/6. 1/6 of the entire population said yes on the first and no on the second. So if we were to add these two up, these are people who said no in either or both questions, you get 1/6 plus 3/4 is equal to-- let's give a common denominator of 12-- so that's 2/6 plus 3/4, that is 9/12. So that is 11/12. Oh, there were N people interviewed. So 11/12 of the people did not answer yes to both questions. And so if they want to know the number of people, you would say 11/12 times the total population, which is N. So 11N over 12. And that's choice E. And my wife is still staring at me. See you in the next video.