GMAT: Math 39

We're on problem 195 and they have drawn a little bit of a grid here. Let me try to draw it as well. 1, 2, 3. And then they do this in the other direction. 1, 2, 3. And they call this First Street. And there's actually 4 going that direction. Let me draw that one. So, there's another street like that. And they call this First Street, Second Street, Third Street, Fourth Street. And Avenue A, B, and C. And they say that this point right here-- I'll do that in a different color. That point right there is x, and that point right there is y. Pat will walk from intersection x to intersection y along a route that is confined to the square grid of 4 streets and 3 avenues shown on the map. How many routes from x to y can Pat take that have the minimum possible length? So essentially the minimum possible length so he doesn't waste time. So let's think about it this way. So he could go this way-- so we could think about how many different ways can you get to each point on this graph. So to get there, there's one way. One possible way. To get there-- this seems to be a very similar problem to what I saw on a Computer Science exam a long, long time ago. But there's one way to get there. That's obvious. So, how many ways are there to get here? Well, you can view it as-- well, to go from this path to this path and this path to this path, you can sum the two ways to get there. So it's 2. Which makes sense. You could go like that or you could go like that. How many ways are there to go here? Well, there's only one way to go here. Likewise only one way to go here. Now this is where it gets interesting. How many ways are there to go here? I can either come from there, and there's only one way to get there, so that's one path. Or I could come from here. But there's two ways to get here. So there's two ways to get here via this path, and there's one way to get here via this path. So there are three ways to get here. So essentially I just added the one and the two. You think of the same logic here. How many ways are there to get there? Well, I can come via this path, and that'll be-- there's only one way to get here so there's only one way to get from this direction. Or I could come from this direction, but there's two ways to get here. So there's two ways to come from this direction. So, there's three ways to get from that direction. If you use the same logic, there's one way to get here. You just go straight up. How many ways are there to get here? Well, one way to get there, three ways to get there, so there are four ways to get here because you can go through that one way or through these three ways. So how many ways are there to get here? You go three ways to there, three ways to there, and I can come from either here or here. So there are six ways. So, how many ways to get to y? Four ways to get here, six ways to get here, 6 plus 4. Ten ways to get here. I could either come from one of the six ways from this direction, or one of the four ways from this direction. So ten ways. And that is choice C. Next problem. 196. The ratio, by volume, of soap to alcohol to water in a certain solution is equal to 2:50:100. The solution will be altered so that the ratio of soap to alcohol is doubled, while the ratio of soap to water is halved. If the altered solution will contain 100 cubic centimeters of alcohol, how many cubic centimeters of water will it contain? So, let's think about what the new ratio is. The old ratio of soap to alcohol was 2:50. Now they want to double this ratio. So the new ratio of soap to alcohol is going to be 4:50. Fair enough. Now the old ratio of soap to water was 2:100. Now what's the new ratio of soap to water? It's going to be halved. So, that equals soap to water. We want to half this ratio, so now the new ratio of soap to water is 1:100. So let's see how we can think about it. Let's try this as a ratio of 4 to something. So that's the same. 1:100 is the same thing as 4:400. So, let's see if we can rewrite these ratios. So the ratio of soap to alcohol to water now becomes 4:50 and the ratio of soap to water is 4:400. So, they give us alcohol and they want to figure out water. So the ratio of alcohol to water is equal to 50:400. And that's another way of saying 1:8. Alcohol to water is equal to 1:8. And they're saying that we have 100 cubic centimeters of alcohol. So, how much water? So 100 over water is equal to 1:8. And you can even eyeball that. That's 100 over 800. So, water has to be equal to 800 cubic centimeters. And that's choice E. Next question. 197. If 75% of a class answered the first question on a test correctly-- so 75%, let's say 3/4, question one correctly. 55% answered the second question on the test correctly. Let me write it this way. Question one, 75% true. Question two, 55% answered the second question correctly, and 20% answered neither of the questions correctly. So three, 20%. So, that's not question three. 20% neither. What percent answered both correctly? So 3/4 answered question one correctly and 1/4 answered it incorrectly. Q1, they got it wrong. They're telling us that 20% got neither correct. So that means that they got Q2 wrong, as well. And this is 20% of the entire population. So this is 1/5. This is 1/5 of the entire population. So my question is, if this is 1/5 of the entire population, what fraction of this population was it? Well, of these people, the fraction that got Q2 incorrect, let's write that as x. So, x times 1/4 is equal to 1/5. Multiply both sides by a fourth, you get x is equal 4/5. So, what we know is since this is 1/5 of the entire population, that of the people who got question one incorrect, 4/5 also got question two incorrect. And if you want to know what percentage that is of the whole population, multiply 1/4 times 4/5. And you get 1/5 which is that data that they had given us. But this helps us. Because this tells us, of these people who got question one incorrect, what fraction got it right? Got question two right? Well, if 4/5 got it wrong, 1/5 got question two correct. So let me just take a step back. I want to make sure I'm not doing this the slowest possible way. They tell us that 75% got question one right and 55% got question two right. So let's think about it. So of this, some percentage got question two wrong. Actually, let's think of it this way. Of this, some percentage got question two right. Let's call this y. Let's call that y and let's call this-- I'm going to use x again. Let's call this x. I'll use x. This is a different x. So, first of all, what is the total proportion of people who got question one wrong and then question two right? Well, it's 1/4 times 1/5. So, it's 1/20 of the people got question one wrong and question two right. And that's equal to x. Now, what percentage of the people got question one right and question two wrong? Well, that's going to be y. Now if you think about-- and y is actually what we're solving for. They want to know how many students answered both correctly. So, if you think about it, y plus 1/20, that represents all of the people who answered question two correctly. All the people who answered question two correctly-- I know it's a little confusing-- are those people and those people. That's all the question twos answered correctly. We already know that this is 1/20 of the population. And we know that y plus 1/20 of the population is equal to 55% of the population. So, we can write that as 55 over 100. Or, we could divide both by 5 and you get 11 over 20. This is 11/20 of the population. So, what is y? Well, subtract 1/20 from both sides. y is equal to-- 11/20 minus 1/20 is equal to 10/20, which is equal to 1/2. So, 1/2 of the population got both the first and the second question right. Actually, we didn't even have to use-- well, we did have to use this 3/4 information to get this 1/4 here. But anyway. And let's see if that's one of the choices. Right. D. 50%. And I'm all out of time. See you in the next video.