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GMAT: Math 37

185-189, pgs. 177-178. Created by Sal Khan.

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Video transcript

We're on problem 185. Car x and car y traveled the same 80 mile route. If car x took 2 hours-- so, x went 80 miles in 2 hours, and that's 40 miles per hour-- and car y travelled at an average speed that was 50% percent faster than the average speed of car x. So, 50% faster. So y went 50% faster than x. 50% faster times 40 miles per hour. Well that's 40 plus 20. That's equal to 60 miles per hour. How many hours did it take car y to travel the route? So, distance is equal to rate times time. 80 miles is the distance. The rate is 60 miles an hour. So the time is equal to 80/60, which is equal to 1 and 20/60. Which is equal to 1 and 1/3 hours. And that is what they have there. They have choice c. I thought they might have 1 hour and 20 minutes written. Next problem. 186. If the average of the four numbers k, 2k plus 3, 3k minus 5, and 5k plus 1 is 63, what is the value of k? So let's just average the numbers. k, plus 2k plus 3, plus 3k minus 5, plus 5k plus 1. And that was four numbers. So, divide by 4. That is equal to 63. Let's add up all the k's. I have 1k plus 2k is 3k. So, 6k. 8k. So, I get 8k. And then I get 3 minus 5 is minus 2 plus 1 is minus 1. And if you multiply the 4 times the 63, if we just multiply both sides of this equation by 4 you get 4 times 63 is 252. Let me think about that. 4 times 60 is 240 plus 12. Yeah, 252. And then 8k would be-- add 1 to both sides-- 253. And so k is equal to 253 divided by 8. Let's see what that is. 8 goes into 253-- 8 goes into 25 three times. 24. Bring down-- equals one time. 1 times 8 is 8. You have a remainder of 5. So, it's 31 and 5/8 is k. And that is not one of the choices. Let me see where I made a mistake. So the average of k. 2k plus 3, 3k minus 5, and 5k plus 1. That's four numbers, right? What is the value of k? All right, so I add up the four numbers, and I divide by 4 and they should be equal to 63. So, maybe I just made a mistake in the simplification. So, I have k plus 2k is 3k, 3k plus 3k is 6k. Oh, 6k plus 5k is 11k. I don't know what I was-- that's 11k. And then you have 3 minus 5 which is minus 2 plus 1. Minus 1. So, 11k minus 1. So, this is 11k. So, k is equal to 253 divided by 11, which is-- 11 goes into 253 two times. 22, 33. Oh, that's nicer. It goes in exactly 23 times and that's choice D. Sorry for the error. Next question. 187. If p is an integer, p is even. They're telling us q is odd. Which is the following must be an odd integer? OK, so they give us a bunch of things. So, an even times an odd. Well, by definition, that's going to be even. Why is that? Well, you can view it as if you multiply p times q, if p is even that means that p is divisible by 2, right? So, if 2 is one of the factors of p, it's definitely going to be one of the factors of p times q. So, this one will not be odd. Let me think about it. If I add an even plus an odd, that gives me an odd. So, let's see if any of the choices are that. Let's see, p divided by q. An even divided by an odd does not necessarily-- well, this is an interesting one. 2p plus q. That's choice c. Now if p is even-- well, it doesn't matter whether p is even or not, but 2p is actually definitely going to be even. This is going to be even. And q, we already know, is odd. So, this is an even plus an odd. So, it's going to be odd. So, the choice is c. And we could even prove it a little bit. If you say that p is equal to 2 times some number k. Right? We don't know what that is. We say that q is equal to 2 times some number, I don't know, m plus 1, then what does this simplify to? This simplifies to 2 times p, which is 2 times some number k-- we don't know what k is-- plus q which is 2 times some number m plus 1. Some integer m. So, this becomes 4 times some number k, plus 2 times some number m, plus 1. Well, 4 times some number k, that's the same thing as 2 times some other number n. You know it's 2 times some other integer where n is twice what k is, right? Plus 2m plus 1. And this is the same thing as 2 times n plus m-- and remember, n, m k, these are all just integers-- plus 1. And so, if n and m are integers, then we could say, well, that's just equal to 2 times some other integer-- let's call it l-- plus 1. And this would be an odd number. Because this is even. And then you're adding 1 to an even number. But you didn't have to do that. You just had to recognize that an odd plus an even is equal to an odd. Choice c. Next problem. 188. Drum x is 1/2 full of oil. So x is 1/2 full. And drum y, which has twice the capacity of drum x, is 2/3 full. So y is 2/3 full and has two times x capacity. If all of the oil in drum x is poured into drum y, then drum y will be filled to what fraction of its capacity? OK. So if x is 1/2 full. So x is equal to 1/2 of its-- well, let me think of the best way to write this algebraically. x is 1/2 full and y is twice as big. So, if you poured all of this 1/2 into y, it would contribute to 1/4 of y's capacity. I guess another way, we could say capacity of x is equal to 1/2 the capacity of y. So, if we were to say 1/2 the capacity of x, if we were to multiply both sides of this by 1/2, that's equal to 1/4 the capacity of y. So, this 1/2, that's equal to 1/2 the capacity of x which is equal to 1/4 the capacity of y. So, if we add 1/4 of the capacity of y to 2/3 of the capacity of y-- that's 2/3 full, that's another way of saying 2/3 times the capacity of y-- what do we get? We get 1/4 plus 2/3 which is equal to-- get a common denominator 12-- 3/4 plus 8/12. 3/12 plus 8/12, that's 2/3. You get 11/12. So, y would be 11/12 full of its capacity. So, that is choice C. Problem 189. If x is greater than 0, x/50 plus x/25 is what percentage of x? So, you get a common denominator. This is equal to over 50. x/50 is x/50. x/25 is 2x over 50. So, that is equal to 3x over 50 which is the same thing as 3/50 x. And that's the same thing as-- since we want it as a fraction of 100, since we want to do a percentage-- that's the same thing as if you multiply the numerator and the denominator by 2, you get 6 over 100 x which is equal to .06x which is equal to 6% of x. And that's choice A. Anyway. See you in the next video.