GMAT: Math 26

Problem 133. A certain country had a total annual expenditure of 1.2 times 10 to the 1/12 power last year. If the population of the country was 240 million last year, what was the per capita expenditure? 240 million, what is that? A million is 10 to the sixth power. So this is 240 times 10 to the sixth. And that's the same thing as 24 times 10 to the seventh. I just took 10 from here and multiply 10 there. And that is equal to 2.4 times 10 to the eighth power. Divide this by 10 and multiply this by 10. You get the same number. So 2.4 times 10 to the eighth. So the per capita income is going to be equal to the total annual expenditure, so 1.2 times 10 to the 12th, divided by the population, 2.4 times 10 to the eighth. 1.2 divided by 2.4, that's equal to 0.5, that's exactly half of that, times 10 to the 12th divided by 10 to the eighth. You just subtract the exponents, so that's 10 to the fourth. And that's not one of the answers, so we can multiply this 10 and divide this by 10. So this becomes 5 times 10 to the third. Ten to the third is 1,000, so 5 times 1,000 is equal to $5,000 in annual expenditure per capita. And that's choice E. Problem 134. A certain rectangular window is twice as long as it is wide. So if this is the width, its length is 2 times the width. If its perimeter is 10 feet, and its dimensions are, so its perimeter going to be w plus w plus the 2 lengths. So its perimeter is going to be 2w plus 2w plus w plus w. And that's going to be equal to 10. If we add up all the w's, that's 2, 4, 5, 6. So we get 6w is equal to 10. w is equal to 10/6, which is equal to 5/3 feet. And so its dimensions are 5/3 and then the 2w. So the length is 2 times that, which is 10/3. So it's 10/3 by 5/3. And that's choice B. They say 5/3 by 10/3, but same thing. So choice B. Problem 135. This is interesting. Let's see if I can draw this diagram. So to start, this is x, then it goes there and has a diamond like that, and it goes like that. And we have another diamond. I don't know what this is all for yet, but it looks interesting. It goes like that, then goes like that. There's a diamond there, looks like there, looks like there. There's a line that goes like that, goes like that, like that, and like that. And then finally, we're at y. The diagram above shows of various paths along which a mouse can travel from point x where does it lead to point y, where there's a reward with a food pellet. How many different paths from x to y can the mouse take if it goes directly from x to y without retracing any point along the path? There's only one way. It has to go here. So we're only at 1 path right now. And it can either go this way to get here, or it could go this way. So to get here, there's how many different paths? Like that, or like that. So at this point there's 2 ways to get right there. There's 1 way to get here, 2 ways to get here. Now, if there's 2 ways to get here, that means that there's 2 ways to get here. Now how many ways are there to get here? There's each of the 2 ways to get there, and now we each of those in combination with each of these 2 ways, so there's 4 ways to get here. You could think of it like this. You could go to the top 1 and the top, that's 1. Top and the bottom, 2. Bottom and the bottom, 3. Bottom and the top, 4. So you could just multiply how many ways it takes to get here, and then how many more ways to get from here to here. From here to here, there's 2 possible ways, and there's already 2 ways to get here, so you could multiply the 2. So there's 4 ways to get here, 4 ways to get here. To get from here to here, there are 3 ways, 1, 2, and 3. But since there are already 4 ways to get here, with each of those 4 ways you can go in 3 different ways. So we multiply by 4 times 3, so there's 12 ways to get right here. If there's 12 ways to get here, there's 12 ways to get to y. So the answer is C, 12. Problem 136. If the operation circle with a dot in it is defined by x circle with a dot y, is equal to the square root of xy, for all positive numbers x and y. So they're saying x and y are greater than 0. That's good, because that ensures we have a positive under the square root sign. Then they want to know what 5 circle with a dot 45 circle with a dot 60 is equal to. 5 circle 45 five, that's equal to the square root of 5 times 45. That's that, and whatever that is we're circling with that. So what you do when you have x circle y, you multiply the 2 and then you take the square root. So we would multiply 60 times this and take the square root of the whole thing. Let's just simplify this. This right here is equal to the square root of 5 times 45, which doesn't look simplifiable at face value. But if you think about, 45 is 9 times 5. You just want to take out any perfect squares you have there. So that's equal to 5 times 9 times 5. And that's equal to square root of 25 times 9. And that is equal to the square root of 25 times the square root of 9, and that's equal to 5 times 3, which is equal to 15. So this simplifies to 15. And 15 dot circle, whatever you want to call it, 60, is equal to the square root of 15 times 60. And let's see if we can somehow turn this into some type of a perfect square. Let's rewrite this as-- 60 is the same thing as 15 times 4. So this is equal to the square root of 15 times 15 times the square root of 4. The square root of 15 times 15 is going to be 15 times 2, which is equal to 30. And that's choice A. Problem 137. I'll do it in yellow. A bar over a sequence of digits in a decimal indicates that the sequence repeats indefinitely. What is the value of 10 to the fourth minus 10 to the 12 times 0.0012, and then this 12 will just keep repeating over and over again. So let's see what we can do. What's 10 to the fourth? Let's think about it this way. Let's distribute this. So what's 10 to the fourth? That's 10,000. So what's 10,000 times this number, 0.0012? This 12 keeps repeating, so we'll think about that in a second. So what's 12 times 10,000? It's 120,000. It's 12 with four 0's, 1, 2, 3, 4. And we have 4 numbers behind the decimal point, so it gets us there. You might say OK, we're done. It's 12. But it's actually not 12, because it's actually 12121212. The 12's just keep going. It would actually be 121212, and then 12's keep repeating on like that. I think you understand why, right? This would be if we just had 1 12 times 10,000, but if the 12's kept going, then these wouldn't be 0's, these would be 12's. Hopefully that make sense to you, maybe there's an easy way to think about it. So what's 100? We'll use the same logic. What's 100 times 0.0012? Actually, I probably could have done it a simpler way, but let's just continue this way. So 12 times 100 1,200, and we have 4 numbers behind the decimal point, 1, 2, 3, 4. And of course, it's not just 0.12, it'll just keep going. 0.1212 will just keep going. So we have this times this, which is this minus this times this. I'm just distributing it. So if I just subtract this from that, minus 0.12-- I wrote this right. This was 12, because we said 12 times 10,000 is 120,000. We had 1, 2, 3, 4 behind the decimal. So it's 12. But since we had multiple 12's behind it, it becomes 12.12. I don't know what I was doing with this right here. So let me write that again. Sorry I'm doing this a little bit messy. So 0.0012 repeating times 10 to the fourth is going to be equal to 12.12 repeating. And then we're going to subtract 0.0012 repeating times 10 squared, and that is equal to 0.12 repeating. So if I subtract out the 0.12 repeating from 12.12 repeating, I'm just going to be left with 12. And hopefully that's one of the answers. It is. Choice E. See you in the next video.