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GMAT: Math 44

210-213, pg. 181. Created by Sal Khan.

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Video transcript

We're on problem 210. A television manufacturer produces 600 units of a certain model each month at a cost to the manufacturer of $90 per unit. And all of the produced units are sold each month. What is the minimum selling price per unit that will ensure that the monthly profit-- revenue from sales minus manufacturer's cost to produce-- on the sales of these units will be at least $42,000. OK. So, they want to know the minimum selling price. So, let's call that p. So how much are they going to make per unit? p minus whatever their cost was. p minus $90. That's how much they make per unit. And they do that for 600 units. And they say that this has to be at least $42,000. So, this has to be greater than or equal to 42,000. OK. So we get 600p minus-- what's 9 times 6? It's 54. And then we have one, two, three 0's. 54,000-- Right? 9 times 6. Yep. --is greater than or equal to 42,000. So then you get 600p. Let's add 54,000 to both sides. Has to be greater than or equal to 54 plus 42 is 96,000. And just to simplify things, let's divide both sides by 100. So, you get 6p. If you get two 0's here, two 0's here. 6p has to be greater than 960. And then p has to be greater than or equal to-- we don't have to worry about this because it's a positive number. 6 goes into 96, 16 times, so it goes into 960, 160 times. So the minimum selling price is $160 which is choice d. Problem 211. If 4x plus 3y is equal to minus 2, and 3x plus 6 is equal to 0, what is the value of y? Well this is a quick one. 3x would be equal to minus 6. So x is equal to minus 2. Substitute back here. 4 times minus 2, minus 8 plus 3y is equal to minus 2. Add 8 to both sides. 3y is equal to minus 2 plus 8. Or 3y is equal to 6. y is equal to 2. And so that is choice e. Problem 212. I'll do it in a different color. 212. The data sets one, two, and three above are ordered from greatest standard deviation to least standard deviation in which of the following? OK. So they want us to order them essentially. Greatest-- So, set one was 72, 73, 74, 75, 76. Set two is 74-- it says 74 a bunch of times-- 74, 74. And then set three has 62, and then 74, 74, 74, and then 89. So, standard deviation is how much, on average, do the numbers vary from the mean. And all of these cases, the mean is 74. I'm just eyeballing it. Let's see 89 is 15 more. Actually here the mean isn't 74. So, first of all, every number here is the mean. So, the standard deviation is actually 0. So this is going to be-- they want to know from greatest standard deviation to least standard deviation. So, 2 is definitely the least standard deviation. So that leaves us with choices either b or d. And now have to figure out which of these have a lower standard deviation. So here we can just eyeball and say the average is 74. If you average this and this you get 74, that and that you get 74. So, what's the difference from 74? Here it's 2. Here it's 1. Here it's 0. Here it's 1. Here it's 2. You just want the magnitude, how far away you are from 74 each time. And if you were to average these numbers, you get 2 plus 1, plus 1 plus 2. That's 6 divided by 5. So, you get 6 over 5. You know they square-- you could even do the variance if you really cared about it, you could square these numbers. But you just want to get some measure of, on average, how far is each of the numbers from the mean. So here's 6 over 5. Let's see 2 plus 1, 3. Yep. 6-- and there's five numbers. And 6 over 5 is the standard deviation there. What's the mean here? This is interesting. So, 74 to 89. Do I have those numbers right? Yeah. to go from 74 to 89 you have to go plus-- let me add these up. You get 62-- actually what's the average of 62 and 89? No, it's not 74. Let's see 2 plus 11 is-- well I could just take the average-- I just want to figure out the mean as fast as possible. So 89 and 62. 9 plus 2 is 11. 9 plus 6-- 151. You divide that by 2. The mean of these two numbers are 75.5. So, at least if you're trying to figure out the mean, that's going to be the same thing is the mean of-- so actually let's just do this. It's 151. Let's just take the average of everything. I tried to be cute, but it's better to just be brute force. So, 62 plus 89 is 151, plus 74 times 3, divided by 5. Let's see. 74 times 3. 4 times 3 is 12. 7 times 3 is 21. 22. And then 222 plus 151 is 373. And then we're going to divide that by 5. So, 5 goes into 373-- 7 times 5 is 35, 23, 5 goes into 23 4 times, 4 times 5 is 20. 3. 74.6. So the mean here is 74.6. This is going to have the largest standard deviation because this is 0.6 from the means, this is 0.6 from the mean, this is 0.6 from the mean. This number is 12.6 from the mean. And then this number is 14.4 from the mean. So, if you're to average these numbers, you're going to get a bigger number than 6/5, but let's just confirm it. So, 12.6 plus 14.4-- oh yeah, you're definitely going to get a much bigger number because when you summed all these up, you got a 6. When you just sum up the 12 and the 14.4 you get-- 15-- you get 27. And then it's going to get to 29. You divide that by 5. The standard deviation here is going to be 4. So, 3 has the largest standard deviation. 2 has the least. So, the correct answer is d. I should've been able to eyeball that. I shouldn't have had to go through all that computation. I took too much time. That wouldn't have been acceptable on the GMAT. Problem 213. Which of the following is the lowest positive integer that is the divisible by all of those numbers? So, it's 2, 3, 4, 5, 6, 7, 8, 8. So the lowest positive integer that is divisible by all of those. So, one way to think about it is let's just try to minimize our multiplication. If we're divisible bye 8, then we don't have to worry about 2 and 4. Being divisible by 8 is enough. Likewise if we're divisible by 9, we don't have to worry about being divisible by 3 because we're definitely going to be divisible by 3. We need to be divisible by 7. If we're divisible by 6-- well, if we're divisible by 8 and 9 then that means we're already divisible by 2 and 3. And if we're divisible by 2 and 3 then we don't have to worry about 6. So, we only have to worry about 5, 7, 8, and 9. So what are those numbers when you multiply by each other? So if we do 8 times 9 is 72. 8 times 9 is equal to 72, we got rid of those, times 7 is equal to 2 times 7 is 14. 7 times 7 is 49, plus 1 is 50. And that times 5. 4 times 5 is 20. 5 times 0 is 0. Bring down the 2. 5 times 5 is 25. So, it's 2,520. And that is choice c. And I'm out of time. See you in the next video.