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GMAT: Math 8

42-48, pgs. 157-158. Created by Sal Khan.

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Video transcript

We're on problem 42. And they've drawn this little, looks like a pie graph. And they say, in the circular region with center o shown above, the two unshaded sections constitute 3/7 and 1/3 of the area of the circular region. Fair enough. The shaded section constitutes what fractional part of the area of the circular region? So the shaded section is just the whole area minus these two fractions, right? So if you said what fraction of the whole area is the whole area, you would say that's 1. And you would subtract out these two areas to get the shaded area. So 1 minus 3/7 minus 1/3 is equal to the fraction of the totally area that this shaded area is. And let's just add or subtract those fractions. The least common multiple is 21. 1 is the same thing as 21/21. Minus 3/7. Let's see, 7 goes into 21 3 times. So 3 times 3 is 9. So this is the same thing as minus 9 over 21. And minus 1/3 is the same thing as minus 7 over 21. So this is equal to 21 minus 16 over 21. And that's 5/21, which is choice D. Next question. 0.3. [SNEEZES] Excuse me. My apologies. 0.3 to the 5th over 0.3 to the 3rd. Well, anything to the 5th divided by anything to the 3rd, you could essentially say divide the top and the bottom by 0.3 squared. Well, actually, you could divide the top and the bottom by 0.3 cubed. You could say this is the same thing as 0.3 to the 5th times 0.3 to the minus third. That's just another way of doing this. And so if you're dividing these two numbers, you would subtract the exponents. But now we're multiplying. We're adding the exponents. But either way, it becomes 0.3 squared. And that is equal to 3 times 3, which is 9. And you're going to have two numbers behind the decimal points, right? 0.3 times 0.3. Two numbers behind the decimal points. 1, 2. So two numbers behind the decimal point. So 0.09. Or another way of saying it is 30% of 0.3 is 0.09. And that is choice C. 44. In a horticultural experiment-- this is sounding interesting already-- 200 seeds were planted in plot 1. So plot 1 got 200 seeds. And 300 were planted in plot 2. So plot 2 got 300 seeds. If 57% of the seeds in plot 1 germinated, and 42% of the seeds in plot 2 germinated, what percentage of the total number of planted seeds germinated? So the total number of planted, what percent? So how many total seeds germinated is going to be 200 times 0.57. That's how many in plot 1 germinated. Plus 300 times 42%, or 0.42. That's how many in plot 2 germinated. All of that divided by 500. Right? And how do I know 500? Because there were a total of 500 seeds. So just to simplify the math, we could just divide everything by 100, right from the get go. So if you divide the bottom by 100 and the top by 100, you have to do both terms by 100. So you get 2 times 0.57 plus 3 times 0.42 divided by 5. 2 times 0.57. That is what? Let's see, this is 1.14 plus 1.26. Is that right? 3 times 4 is 12. 3 times 2 is 6. Right? 1.26. All of that over 5. This becomes what? This is equal to 2.4 divided by 5. And so 5 goes into 2.4. Let's see, goes into 4, 4 times 5 is 20. 48. So 0.48. So the answer is 48% or 100 times 0.48. And that's choice C. Question 45. Let's switch to a more interesting color. 3 and 8 are the lengths of two sides of a triangular region. Which of the following can be the length of the third side? OK. So let's think about it a little bit. 3 and 8 are the lengths of two sides of a triangular region. Let me write their choices down. Choice one is 5. I can easily imagine a triangle that has sides 3, 8, and 5. That seems completely reasonable. I'm just experimenting. I don't know where this is going. 8. Well, sure, that's just an isosceles triangle. You can easily have a triangle that has 8, 8, and 3. Choice three. 11. Now, this is interesting. Let me ask you a question. Can I have a triangle that looks like this? 11 and then 3 and then 8. Is this possible? Well, no, because 3 plus 8 is 11. So the only way you're going to get 11 is if you push this side all the way flat. That's the only way you're going to get the length of this third side to be 11. In fact, 11 is the upper bound on what this third side could be. Because imagine this. Imagine if I made the triangle really flat, I made this angle right here really wide, as close to 180 as I can. And I make it really flat. Right? If this length plus this length, or this length plus this length is equal to 11, this length is going to be shorter than it. This length right here has to be shorter than this length plus this length, right? Because it's kind of a straight-line distance between this point and this point. So 11 is the upper bound, right? The only way to get 11 is if you completely flatten out this triangle, at which point that's not a triangle anymore. It'll be a line. So it can't be choice three. So the only possibility. They say, which of the following can be the length of the third side? So it's only choices one and two. And that is choice C. Next problem. 46. How many integers n are there such that 1 is less than 5n plus 5, which is less than 25? OK. So they say how many integers n are there so that 5n plus 5. So they didn't say positive integers, right? So that's an interesting thing to keep in mind. So let's just try to simplify this a little bit. Let's subtract 5 from all sides of this double inequality. So if you subtract 5 from everything you get minus 4 is less than 5n, which is less than 20. Right? So another way you could say it, is let's just divide everything by 5. So because 5 is positive you don't have to change the inequalities. So you get minus 4/5 is less than n, which is less than 20 divided by 5, is 4. So now the question becomes a lot simpler. How many integers n are there such that this? How many integers are there between minus 4/5 and 4? And it's not equal to any of those. So 0 is an integer. 1, 2, and 3. So there are 4 integers. So that is B. OK, next problem. 47. A car dealer sold x used cars and y new cars during May. So number used is equal to x. Number new is equal to y. During May. If the number of used cars sold was 10 greater than the number new cars, which of the following expresses this relationship? So the number of used cars, x, was 10 greater than the number of new cars. So it's 10 greater than y, so it equals y plus 10, right? This says that the number of used cars is 10 more than the number of new cars. So we just have to look for that. x is equal to y plus 10. That's choice D. I think we have time for one more. 48. If a 10% deposit that has been paid toward the purchase of a certain product is $110, how much more remains on the product? So essentially they're saying, 110 is 10% of what number? That's the first thing you have to say. So 110 is equal to 0.1 times what number? So that's the price. Let's call that the original price of the product. So the original price of the product is going to be what? It's going to be 110 divided by 0.1, which is just this times 10. Which is 1,100, right? Just add a 0. So that's original purchase price of the product. The deposit is $110 and they want to know what do you have left. So you're going to put $110 deposit on it. This was the original purchase price. You put $110 deposit. Let's see, 1,100 minus 100 would be 1,000, but then we have another 10. So it would be 990. So that is choice B. And you could do it the other way. You could just do a little bit of borrowing. Anyway, you get the idea. And you would get choice B, which is 990. And I'm out of time. See you in the next video.