Khan Academy Problem Solving

Khan Academy Problem Solving-33
So we have to pick an n that is essentially not divisible into 100. 1 is clearly an integer, so this can't be an integer, right? And now we can invert both sides or just set the denominators as equal to each other.

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If you're seeing this message, it means we're having trouble loading external resources on our website. And you might have a temptation to just invert both sides, but you can't do that just yet. And then 1/16, that's the same thing as 1/2 to the third power is 1/8, so it's to the fourth power. And then this is-- we'll just use what we know about exponent rules, and we've proven this in other Khan Academy videos. You can just multiply these exponents, so that's equal to 1/2 to the minus 4. They want to know how many red beads were removed, right? So all the beads, you multiply them together and you get 147,000.

If you're behind a web filter, please make sure that the domains *.and *.are unblocked. Because if you took the inverse of this, it's not like you just get x minus x plus 1. And then 1/2 to the 4 to the minus 1, you multiply them and that's equal to minus 4. So we've essentially factored out all the 7's that we can fit into 147,000.

Consecutive positive, so, you know, we could say that b would be equal to a plus 1, and c would be equal to a plus 2 or b plus 1. I don't know if that helps, I just felt like writing that down. Well, instead of going into some algebraic thing where I represent even and odd integers and all of that, let's just try it with two cases, one where a is odd and one where a is even. So the numbers would be 2-- sorry, would be 2 times 3 times 4, right? And so if you're divisible, and by definition, if one of these numbers are even, it's divisible by 2. I don't know, even if we were to take-- let's see, 3 plus 4 plus-- no, that works. Let's write it as-- so this is a plus a plus 1, right?

Multiply both sides by y and you get y is equal to 2. And you could've actually eyeballed it, because, say, 1 plus 1 is equal to 2, so this has to be 1. That equals 2, but sometimes it's easier just to mechanically go through the algebra, because sometimes, you can't eyeball it, and it's good to just keep moving forward. If a, b, and c are consecutive positive integers, and a is less than b, which is less than c, which of the following must be true? This has to be even because one of these numbers are going to be even, right?

So this left-hand side, the x's cancel out and you get 1 over-- well, we could just make that x squared plus x, right? The first thing I do when I see negative exponents, just so they don't confuse me, is that I just invert all of the numbers. This is the exact same thing as equal to-- this is the same thing as 2 to the third power, right? I'm going down the wrong path, because they have it in completely different format. OK, a number of beads are then removed from the container.

So now this simplifies to-- the left-hand side is 1 over x squared plus x, and the right-hand side is 1 over x plus 4. So if you take the inverse of both sides and you get-- and I'll just switch colors just randomly-- x squared plus x is equal to x plus 4. When you take something to a negative exponent, that's the same thing as the inverse of that thing to the positive exponent, so this is 2 to the third power. And so 2 to the third is 8 times 16 times 16, right? So actually, maybe I should look at the choices before I continue these problems. So they all have it as -- you know, 1/2 to some power, 1/8 to some power, so let me rewrite this. If the product of the point of values of the removed beads is 147,000, how many red beads were removed? So there's something that hopefully will jump out at you about these numbers. So if you look at any number, you can tell how many of a certain prime number there is in that number by just doing a prime factorization. And you'd be surprised how quickly a number can get smaller. So choice 3 will not produce an integer if it's selected for n, so we're at C. So they're saying it has the same area, so essentially they're saying that 9 times-- and they want to know its length. So the area of floor y, which is 9l, is equal to the area of floor x, which is equal to 18 soon. And I even multiply this out, because I don't know off the top of my head what 18 times 12 is, but we can just divide both sides by 9, and we get l is equal to-- let's divide the 18 by the 9. you A rainstorm increased the amount of water stored in state J reservoirs from 124 billion gallons to 138 billion gallons. So capacity is going to be equal to 138 divided by 0.82, right? So it's going to be this number-- this is total capacity-- minus what they were prior to the storm, 124. I don't know if they allow you to use a calculator, but I think we can do this on our own. And we could take this decimal point two over, so we can move this decimal point two to the right. Well, let's see how much they should have spent after four months. That's how much they should have spent after four months if they were on budget. If you're behind a web filter, please make sure that the domains *.and *.are unblocked. A project scheduled to be carried out over a single fiscal year has a budget of ,600. So if we divide it by 12 how much are we going to spend per month? At the end of the fourth month of that fiscal year, the total amount actually spent on the project was 4,580. So after four months they should have spent 1,050 times 4. Sure enough, you know, you could divide the top and the bottom by 3 and you get a plus 1. This is always going to evaluate to a plus 1, which is an integer. If you're seeing this message, it means we're having trouble loading external resources on our website.

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