GMAT: Math 9

We're on problem 49. And it says, what is the square root of 7 plus the square root of 7, and all of that squared. So this is really just a little bit of simplification. So if I have two of something, in this case, I have 2 square roots of 7, if I add them together, I get 2 of it, so I get 2 square roots of 7, and I'm squaring that. And 2 square roots of 7 squared, that's equal to 2 squared times the square root of 7 squared, right? That's just your exponent properties, so that is equal to 4 times-- what's the square root of 7 squared? Well, it's just 7, right? So it's 4 times 7, which is equal to 28, which is choice C. Next problem. 50. In a certain population, there are three times as many people aged 21 or under as there are people over 21. OK, so let's say that x is equal to under 21 and y is equal to over 21. So in a certain population, there are three times as many people aged 21 or under as there are people over 21. So this is the number of people under 21 and there are three times as many as there are over 21. The ratio of those 21 or under to the total population is? So they want to know the ratio of x, 21 or under, to the total population, not just to the over 21 population. So what's the total population? That's everyone, right? x plus y is equal to total. So what they want is the ratio of x to x plus y. And then we could do a little bit of substitution. We know x is equal to 3y, so let's substitute that into this. Remember, this is the ratio they're trying to figure out: the ratio of 21 or under to the total population. So x is equal to 3y over 3y plus y. And I just did that because, wow, if we do that then it looks like we could be able to cancel out all the y's. So we get 3y over what? What's 3y plus another y? Well, it's 4y. The y's cancel out. You get 3/4. And that is choice E, 3:4. Next question. 51. And actually, you know, you didn't even have to do it algebraically. If you just think about question 50, you could have just played with numbers. You can say, OK, there are three times as many people aged 21 or under as there are people over 21. So you just say, OK, if there's one person over 21, there'd be three people under 21, and there'd be a total of four people, so it would be 3:4. But anyway, either way it didn't take us too long. So they've drawn a figure here. Let me draw it as well. So they have two lines that intersect something like that. And they say that this angle right here is 3x degrees. This angle is y plus 30 degrees, and they say this angle is 2x degrees. And they say, in the figure above, the value of y is what? Well, there's a bunch of ways you can do it. The way this pops out to me is we could figure out x because we know that this angle plus this angle is equal to 180 degrees, because they are supplementary. They go all the way halfway around a circle. So we know that if we add 3x plus 2x, that should equal 180 degrees, or 5x is equal to 180. Or that x is equal to what? Let's see, 30 times 5 is 150, then you have 30 left over, so it's 36 degrees, right? 36 times 5 is, yep, 180. And now we know if x is 36 degrees, what is this angle? Well, this angle is going to be 72 degrees. It's 2 times x. 72 degrees. And then this angle and this angle are opposite angles. And you could do the angle game module on Khan Academy, if that doesn't ring a bell, but that should ring a bell from geometry. And there's another way you could do it. You could say that this angle plus this angle have to equal 180, either way. You could do it either way. But the easiest way is to say that this angle and this angle are equal, because they're opposite angles. So we know that y plus 30 has to equal this angle, equals 72. Subtract 30 from both sides, you get y is equal to 42 degrees, which is choice E. Next problem. Let me make it a different color now. 52. Kelly and Chris packed several boxes with books. If Chris packed 60% of the total number of boxes, what was the ratio of the number of boxes Kelly packed to the number of boxes Chris packed? So they want to know the ratio of Kelly to Chris, right? Well, if Chris packed 60% of the total, we could say, let C be the number that Chris packed, so that'll be equal to 0.6 of the total. Well, then that means that Kelly packed the rest of them, right? So, I'm assuming it's a she, Kelly would have packed the remainder, which is 40% of the total. So the ratio of the number that Chris-- what do they want to know? What was the ratio of the number of boxes Kelly packed to the number of boxes Chris packed. So the want to know the ratio of K to C. Well, K is 0.4T and C is 0.6T. T's cancel out. You get, essentially, 4/6, which is 2/3, right? If you multiply the top and the bottom by 10, it becomes 4/6, so you get 2:3, and that is choice E. Problem 53. Of the following, which is the closest approximation of 50.2 times 0.49 over 199.8? So I haven't looked at the choices yet, but if I had to approximate what this is, I would say, well, that's close to 50, right? 50.2 is really close to 50 times-- 0.49 is really close to 0.5, all of that over-- 199.8 is really close to 200, right? So you could view this as being equal to -- let's see, 25/200. And 25/200, you could divide the top and the bottom by 25. You get 1 over-- 25 goes into 100 four times, goes into 200-- eight times. And that is one of our choices, so I'll go with that. B. 1/8. Problem 54. The average of 10, 30, and 50. So the average is going to be 10 plus 30 plus 50. So the average of those three is that, divided by 3. First of all, what is that? Well, it looks like it should be 30, right? Because you get 60 plus 30/3, so that's 90. So the average of 10, 30, and 50 is 5 more than the average of 20, 40, and one of the choices, and x. So to write this, we say, OK, we figured out that the average-- oh, sorry, the average here isn't-- sorry, let me redo this for you. So they say the average of 10, 30, and 50, and we already figured out 10 plus 30 plus 50. If you average that, divided by 3, you get 90 divided by 3 is equal to 30. And you actually didn't even have to do the math. You could have said, oh, we'll have 30 in the middle. 50 is 20 more than 30. 10 is 20 less than 30. So the average is going to be 30. But this didn't take you much time. So the average is 30. And they say that that is 5 more than the average of 20, 40, and another number. So 30 is equal to 5 plus the average of those three numbers. So the average of those three numbers is 20 plus 40 plus x, over 3, right? So let's just simplify this. Let's subtract 5 from both sides. You get 25 is equal to 20 plus 40 plus x, over 3. So you multiply both sides by 3. You get 75 is equal to 60 plus x. And then you subtract 60 from both sides. You get x is equal to 15, and that is choice A. And I'm almost out of time, so I'll see you in the next video. See you soon.