GMAT: Math 21

Problem 110. If candy bars that regularly sell for $0.40 each are on sale at 2 for $0.75-- so they used to be $0.40, and now they're going 2 for $0.75-- what is the percent reduction in the price of 2 such candy bars purchased at the sale price? What is the price reduction? So before, if I was going to buy 2 candy bars, I'd be paying $0.80, and now for 2 candy bars I'm paying $0.75. So I'm saving $0.05 or I'm getting a $0.05 price reduction. So they want to know the percent price reduction. So it's essentially what percent is $0.05 of $0.80? 5 over 80 is equal to what percentage? So let's just figure it out. 80 goes into 5-- and actually, we could divide the top and the bottom, make it a little bit easier. 5.00, add some 0's there. So 80 does not go into 50. 80 goes into 500, so how many times does 8-- It goes into it six times, because 8 times 6 is 48. 6 times 80 is 480. 500 minus 480 is 200. 80 goes into 200 two times. 2 times 80 is 160, and then remainder of 400. 800 goes into 400 five times. So that's 0.0625, which is the same thing as 6.25%. And that is choice B, 6 1/4%. Same thing. Choice B. Problem 111. If s is greater than 0, and the square root of r over s is equal to s, what is r in terms of s? OK, so let's just square both sides of this equation, so you get r over s. The square root of r over s squared is just r over s. And that's going to be equal to s squared. We've got to square both sides of the equation. Now let's multiply both sides of the equation times s, and we get r is equal to s to the third. r is equal to s to the third, either way. And that is choice D. Next question, 112. They've drawn us a neat little diagram here. Let me see if I can draw what they have drawn. They drew it in a dark color, so I'll draw it-- That's kind of overly bright, but it gets the job done. And then they have two boxes inside of it. It looks like that. One box, something like that. One box, and then there's another box in there. It looks something like that, although they look like the same size in the picture. And they tell us that this distance is 6 feet. They tell us that this distance is 8 feet. The front of a 6 by 8 foot rectangular door has brass rectangular trim, as indicated by the shading in the figure above. OK, this green stuff is brass. If the trim is uniformly 1 foot wide-- so that's 1 foot-- what fraction of the door's front surface is covered by trim? So we need to figure out the area that is essentially green. Well, the easier thing might be to figure out the area that's black, and then subtract it from the total area. So what's the area that's black? So both of these-- well, they don't tell us. Well, it doesn't matter, we could figure it out. Well, there's a couple of ways to figure it out. Let's just figure out the area that's green. I'm going to do it in different color. So let's figure out this area right here. I hope you can see that. So that area is what? Let me actually draw all of the different boxes and then we can figure out the areas. Then we have that one. And then we have this one. And then we have this one right here. What's the area of this? Its height is 8, this is 8, and its width is 1. So the area here is 8, and so the area here is going to be 8 of this box right here, that whole box. What's the width of this? Well, it's not the whole 6, because you have 1 here and 1 here. So the width is going to be 4 by 1, so this is going to have an area of 4. This is the same thing, width of 4 by 1, and that's 4 and 4. So all of the green stuff has an area of-- 4 plus 4, plus 4-- that's 12-- plus 8, plus 8, plus 16, is equal to 28. They want to know what fraction of the door's front surface is covered by the trim. So 28 is the amount of square feet covered by the trim. And what's the total surface area of the front? The total area's going to be 6 times 8, which is equal to 48. So if you want to know the fraction that's covered by the trim, it'd be 28 over 48, which is equal to-- let's see, you divide the top and bottom by 4-- 7 over 12, 7/12. And that's choice D. Next question, 113. If a is equal to minus 0.3, which of the following is true? OK, so they have a bunch of statements that relate a-- I'm just trying to see the pattern in the-- So in all of these choices, they're relating a to a squared, to a to the third, and then they're putting it in order. They're saying which of these is the least, then next, and then the greatest. So let's just figure it out. a is minus 0.3. What's a squared? So it's going to be positive. Negative times a negative is a positive. And then 3 times 3, and we're going to have two digits behind the decimal, right? Because 0.3 times 0.3. So it's going to be positive 0.09. And then what's a to the third? Well, 0.09 times 0.3. 9 times 3 is 27. And we're going to have 1, 2, 3 numbers behind the decimal spot, so it'd be 0.027. And we have a negative times a positive, so it's going to be negative. So let's put these in order. Which of these is the smallest number? So think about it. The smallest number is going to be the one that's actually the most negative. So this right here is the most negative. Minus 0.3 is the most negative. We could even draw a number line here. You could have minus 0.3, which is a. Then you have minus 0.027, which is a to the third. And then you have 0 some place. And then you have 0.09, which is a squared. So the correct answer should be a is less than a to the third, which is less than a squared. And that is choice B. Problem 114. Let me switch colors. Which of the following is the product of two integers whose sum is 11. So they're telling us that x plus y is equal to 11. And which one is the product? So xy is equal to what? So let's think about it. Which of the following is a product of two integers whose sum is 11? So this is interesting. Maybe we just have to do some experimentation right now. We could we could be dealing with negative numbers here. So let's think about how we can rearrange this. I'm just experimenting. Let's see, if we have y is equal to 11 minus x, does that give us an intuition? Then xy is equal to 11 minus x, because y is 11 minus x. That equals 11x minus x squared, where x is really any integer. So if you put a 1 there, you could you could get a 10. They don't tell us that they're greater than 0. So let me think about this. So 32 is 8 and 4. 26 is 13. Let me look at all those choices. This one has me temporarily stumped, hopefully temporarily. So minus 42, what are the different ways I could get there? I can get 21 and 2. I could get 7 and 6, but one would have to be positive and one would have to be negative. I can get 14 and 3, 14 times 3. And actually if I have 14 times minus 3, what is that equal to? It's minus 42, and then 14 plus minus 3, and that equals 11. So it's answer A. I wish I could give you an analytical way of doing it instead of just experimenting with numbers. I was just trying to figure out what equals 42. Which of the following is a product of two integers whose sum is 11? I guess a more systematic way of doing it might have been to just say what are the factors of 42? It'd be 1 and 42, 2 and 21, 3 and 14, and then 6 and 7. And you say OK, if I-- and since this is a negative number, one of these has to be positive and one has to be negative-- so if I were to take a positive 1 of one of these, and a negative 1 of one of these, and add them up, can I get to 11? And I guess that's where your brain should say, oh, 14 minus 3, that is equal to 11. And luckily it was choice A, and you didn't have to go any further. But you would have to do this with every one from there. You might want to look at the solutions. Maybe they have more elegant solution to this. Anyway, I'll see you in the next video.