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GMAT: Data sufficiency 17

77-79, pg. 284. Created by Sal Khan.

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  • blobby green style avatar for user Frank Alfieri
    the original question asks is QR/PR=1; however the moderator does not finish the calculations to find the actual slope with the numbers plugged in. Why?
    (1 vote)
    Default Khan Academy avatar avatar for user
    • leaf blue style avatar for user abfoeva
      The question is whether the slope equals one, but this is a data sufficiency test which means you only have to find out whether you could find out the slope with the additional information provided to you. After you plug in the numbers the slope may or may not equal one. Your task is to select the option that sufficiently enables you to get an answer and not the answer itself and it so happens in this case that if you plug in the numbers it does equal one.
      (3 votes)

Video transcript

Problem 77. If Miss Smith's income was 20% more for 1991 than it was for 1990, how much was her income in 1991? Let's call it income 1991, is 20% more than 1990. So that means 1.2 times income in 1990. I hope you get that. If something is 20% more than something else, it's going to be 1.2 times that. 12 is 20% more than 10. And 12 is 1.2 times 10. Or you could view it as income plus 20% of income, which is 1.2 times income. Either way. So let's see what they tell us. Well, what do we have to figure out? OK, we were trying to figure out this, income in 1991. Statement number one. Miss Smith's income for the first six months of 1990 was $17,500, and the income for the last six months of 1990 was $20,000. So they're telling that she made $17,500 in the first six months of 1990, and her income for the last six months of 1990 was $20,000. Well, they're essentially telling us the total income for 1990. The first six months and the last six months. There are 12 months in a year. So her total income for 1990 was $37,500. That equals income for 1990. So clearly, if we know this is this, we just multiply that times 1.2, and we get the income for 1991. So this statement alone is sufficient. Let's see what they give us for statement number two. Miss Smith's income for 1991 was $7,500 greater than for 1990. So they say income of 1991 was $7,500, so it equals income for 1990 plus $7,500. Well, this alone does help us, because they've already given us this. So we have two linear equations, right? This is one linear equation and two unknowns. This is another linear equation and two unknowns. So we have two linear equations and two unknowns. We can solve this. Probably the easiest way is just to substitute, depending on what you want to solve for. But we've done that multiple times. You could substitute 1.2 times 1990 here, and then solve for it. Or you could do the other way. You could divide by 1.2 here and then substitute it there. But either way, this is trivial algebra, hopefully, by this point, to solve. But this and this is definitely enough information to solve the problem. So two equations with two unknowns. And you can do that in your spare time if you don't believe me. So both statements alone are sufficient for this one. 78. In the figure above-- so I think I have to draw. This is the y-axis, and that's the x-axis. And then they have a line. Let's see what I can do. The line looks something like that. And then they tell us-- what do they tell us? This, of course, is y. This is x. And then they say this is P. And this right here is Q. And then they draw this. They call this point right here R. And they draw-- that's like that. And this Q is at point c, d. And P is at point a, b. And then they say, in the figure above, segments PR and QR-- so let me draw that out a little bit better. --are each parallel to one of the rectangular coordinate axes. OK, fair enough. This is parallel to the y-axis, PR is parallel to the x-axis. Fair enough. Is the ratio of the length of QR:PR equal to 1? So is the ratio of QR:PR equal to 1? So they want to know QR/PR, is that equal to 1? And immediately this should trigger something from Algebra 1. They're asking you, essentially, is the slope of this line equal to 1? Right? Is the ratio of QR:PR, so rise over run, is the slope of this line equal to 1? So let's see what we can do. And slope is just change in y over change in x. So what's this point, first of all? You actually don't have to know anything about slope. I don't want to make you feel like you have to memorize some formulas. What's this point going to be? So it's going to be-- actually, let's do it even better. What is the length of QR going to be? I haven't looked at any of the data points right now. What is length of QR? Well, it's going to be this height. So it's this y, which is d, minus this y. This y is going to be b, right? Because all of this is y is equal to b, right here. So QR is going to be equal to d minus b. And PR is the length on the x-axis. It's going to be this x. What is this x? Well, this x is right here, c. x equal to c. It's going to be this x minus this x. Well, here, x is equal to a. And so the ratio is equal to d minus b, over c minus a. Which is, if you remember, the formula for the slope of a line. You just take the y1 minus y2, over x1 minus x2. But we didn't have to memorize that. It's intuition. See the x-coordinate is c and the y-coordinate is b. So hopefully that gives you intuition. Now let's look at the statements. You wouldn't have to do that on the real GMAT. That would all be a waste of time. Statement number one tells us, c is equal to 3 and d is equal to 4. So that by itself, that just gives us the first part of this. That doesn't help us figure out this entire ratio. So this by itself isn't that useful. Maybe in conjunction with what else they give us. Statement two. a is equal to minus 2, and b is equal to minus 1. Well, if you used both of these statements together, then we have everything here. We have d, we have b, we have c, and we have a. So we can solve it. So both statements together are sufficient for knowing whether the ratio of QR:PR is 1. Or essentially, is the slope of this line equal to 1? Next problem, 79. While on a straight road, car x and car y are travelling at different constant rates. If car x is now 1 mile ahead of car y, how many minutes from now will car x be 2 miles ahead of car y? So x is here. y is here. And they're going at constant rates. 1 mile. So they've been traveling for some amount of time and x is 1 mile ahead. And they're saying how long is it going to be before x is-- how many minutes before x is 2 miles ahead? And they're at constant rates. So if they started off-- and let's just think about it-- if they started off at the same point and it took 10 minutes for x to get 1 mile ahead, it would take another 10 minutes for it to get 2 miles. Well, that's how I'm thinking about. Let's see what they give us for the statements. Statement number one. Car x is traveling at 50 miles per hour, and car y is travelling at 40 miles per hour. Well, that seems to be pretty good information. So essentially, car y is moving away from car x at what? It's moving away 50 minus 40 miles per hour. So from car y's point of view, car x is always pulling away at 10 miles per hour. Right? Does that make sense? If car x was going at 40 miles per hour, you wouldn't be pulling away at all. If it was going at 41 miles per hour, it'd be pulling away at a increment of 1 mile per hour. And as long as we're not approaching the speed of light, we can assume Newtonian classical physics, and we could just take the difference between the two. So how long does it take for it to pull away another mile? Well, if you're going 10 miles per hour, relative to something else, how many minutes does it take to go a mile? Well, one, you know you can figure that out. But let me figure that out for you. So you know distance is equal to rate times time. So if your distance you want to know is 1 mile, and your rate is equal to 10 miles per hour, times time, what's the time going to be equal to? Time is going to be equal to 1/10 of an hour, or 6 minutes. So that's the answer to number one. One alone is sufficient. Or the answer to number 79, one alone is sufficient. Let's see what they give us for number two. Statement two. 3 minutes ago, car x was 1/2 mile ahead of car y. OK. So 3 minutes ago, the state of affairs was this: y was here, x was here. And it was a 1/2 mile difference. So what does that tell us? That's actually pretty good information too. 3 minutes ago, car x was 1/2 mile ahead of car y. Now car x is 1 mile ahead. So in 3 minutes, x pulls away by 1/2 a mile, right? And they're going at constant velocities. So the relative velocities between the two don't change. So if it takes 3 minutes for x to pull away by 1/2 a mile, it would take 6 minutes for x to pull away by a mile, right? You just multiply them by 2. They're all going at the same constant velocity. So in 6 minutes, x pulls away by 1 mile. And that's actually what they're asking. Because they say how many more minutes does it take x to pull away by another mile? They've probably been traveling for 6 minutes already, and then in another 6 minutes x would pull away by another mile. So two alone is also sufficient. So each of them independently are good enough to answer this question. See you in the next video.