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We're on problem 65. They wrote 2, 4, 6, 8, n, 3, 5, 7, and 9. And they tell us, in the list above, if n is an integer between 1 and 10, inclusive-- so that means n could be 1, and it could be 10, or any number inbetween, and it's an integer-- then the median must be-- the median, right, not the mean, the median. So the median means the middle number, right? Let's put all the other numbers in order. So we have 2, 3, 4, 5, 6, 7, 8, 9. And they're essentially saying that n could be anywhere in this. It could be a 1 here, it could be right here, it could be another 2, it doesn't tell us. It could be any number, it could be 2, could be another 3, could be 10. We don't know. But before we stick n in here, let's figure out what the middle is. Right here we have eight numbers, so the middle of the range is actually-- there is no middle number-- but when we add n, there will be a middle number. So let's just think about it for a little bit. If n is 5 or less, so n goes into this bucket, n goes someplace here, then what happens? Then what becomes the middle number? If n is 5 or less, then we'll have n here. And n might be 5, n might go right here. But what becomes the middle number? Then we have 1, 2, 3, 4 on this side of the middle number, and we have 1, 2, 3, 4 on that side of the number. So then, the median, which is the middle number, becomes 5. That's if n is 5 or less. If n is less than or equal to 5, then we know that the median will be equal to 5. Now let's do the opposite thought experiment. What if n is 6 or more? What if n is greater than or equal to 6? So it goes someplace over here, so n will go someplace here. Could be 6, I might have to put it right there. But wherever I put it on the side, I'll have 4 to the right of 6, and I'll have 4 to the left of 6. n isn't on this side anymore, 4 to the left of 6. So then the median would be 6. And either of these cases have to be true for n. So the median is either 5 or 6, and that is choice B. Problem 66. I like this brownish color. Reminds me of when I was in middle school math competitions. 4u7, n23, 162. And you add them all together, you get one 1,222. And what they want to know is, if n and u represent single digits in the correctly worked computation above, what is the value of n plus u? So the best thing to do is to try to work these out, and see what happens. So 7 plus 3 is 10, plus 2 is 12. So you would write a 2 here, which they already wrote, and carry a 1. And now we have 1 plus u-- let me write this down-- 1 plus u plus 2 plus 6 is equal to, well, it's either going to be equal to 2, or some digit and a 2. So let's just think about it a little bit. 1 plus u plus 2 plus 6, that's equal to u plus 9. u plus 9 is going to have to equal something. So if u is 9, and it has to be something that ends with a 2. And remember, u can only be between 0 and 9. It's a digit, it has to be an integer, it has to be a digit. So let's think about it a little bit. It has to be something that ends in a 2. So the only next thing above 9 that ends as a 2 is 12. Because you can't get to 22. To get to 22, you would have to add 13, and you can't be 13. So this has to be a 12, so u has to be equal to 3. That's the only possibility that'll give you something that ends in a 2. So if we assume that u is 3, then we have 1 plus 3 plus 2 plus 6, which is 12, carry the 1, you get 1 plus 4 plus n plus 1 is equal to 12. So you get 1 plus 4 plus 1. That's 6, plus n is equal to 12, and n would be equal to 6. And they want to know what n plus u is. So n plus u, 6 plus 3. That's equal to 9. And that is choice B, 9. 67. Look at that, this is a dense looking one, but let me write down the little equation they wrote on top. r is equal to 400 times d plus s minus p, all of that over p. If stock is sold three months after it is purchased, the formula above relates p, d, s, and r, where p is the purchase price of the stock. That's the purchase price. d is the amount of any dividend received. Fair enough. s is the selling price of the stock, and r is the yield of the investment as a percent. Fair enough. If Rose purchased $400 worth of stock-- so p is equal to $400-- received a $5 dividend-- so d is equal to $5-- and sold the stock for $420, for three months after purchasing-- so this formula applies, because this is the formula for selling after three months-- what was the yield of her investment, according to the formula? Assuming she paid no commissions. We just substitute in. So let's see. The yield would be equal to 400 times 5 plus 420 minus 400, p is 400, all of that over 420. That is equal to 420 minus 400 is 20, plus 5 is 25. 400 times 25 over-- this shouldn't be 420, this is p. The price you paid was 400. 25 over 400. I just substituted these values into this equation. This cancels out, and I'm just left with a yield of 25, and it's probably in terms of percent. Yep, 25%. E. All right, problem 68. Let's switch colors. 68. The temperatures in degree Celsius recorded at 6:00 in the morning in various parts of a certain country were 10 degrees, I'm not going to write all of it, 5, minus 2, minus 1, minus 5, and 15. What is the median of these temperatures? The median just means the middle. Don't confuse that with the average, which is the mean. Or the mean, which is the average. Median means middle, so let's just put them in order and figure out the middle. So the smallest of these numbers is minus 5, cross it out. Then we have minus 2, cross it out. Then we have minus 1, cross it out. Then we have 5, cross it out, then we have 10, cross it out. Then we have 15. So this is interesting about median. If there is no true middle number, you have three on this side of negative 1, or negative 1 and less, and you have three on that side, so there's no middle number, because we have an even number of numbers. So what you want to do is, you take the two middle numbers, which are negative 1 and 5, and you average them. So it's negative 1 plus 5 over 2, which equals 4 over 2, which is equal to 2. 2 degrees Celsius, which is choice C. Problem 69. If y times three 3x minus 5 over 2 is equal to y, and y does not equal 0, then x is equal to what? Well the simplest thing, since y doesn't equal 0, we can divide both sides of this equation by y. You divide both sides by y. y divided by y is 1, y divided by y is 1. So then we're left with 3x minus 5 over 2 is equal to 1. Multiply both sides by 2, you get 3x minus 5 is equal to 2. Add 5 to both sides, 3x is equal to 7. And you get x is equal to 7 over 3, which is choice C. And I'm out of time, so I'll see you in the next video.