Modeling with rational functions
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Working together Anya and Bill stained a large porch deck in 8 hours. Last year Anya stained the deck by herself. The year before Bill painted it by himself, but took twice as long as Anya did. How long did Anya and Bill take when each was painting the deck alone? So let's define some variables here. Let's define A. Let's define A as number of hours for Anya to paint a deck. And so we could say that Anya paints, or has A hours for one deck. Or we can invert this and say that she can do 1/A decks per hour. Now, let's do another variable for Bill just like that. Let's define B as the number of hours for Bill to paint a deck. And so for Bill, he can paint-- it takes him B hours per deck, per one deck. We could put it over 1 there if we want. We don't have to. And this is the same thing as saying that he can do 1 deck per every B hours. Or another way to think of it, is he can do 1/B deck per hour. Now, when they work together, Anya and Bill stained a large porch deck in 8 hours. So let's write this down. Anya plus Bill-- I'll do it in orange. Anya and Bill. This information right here. Anya and Bill stained a large porch deck in 8 hours. So we could say 8 hours per deck. Or 8 hours per 1 deck, which is the same thing as saying 1 deck per 8 hours. And this is going to be the combination of each of their rates. So this 1 deck of per 8 hours, this is going to be equal to Anya's rate, 1/A decks per hour, plus Bill's rate. Plus-- do that same color. Plus 1/B decks per hour. So we have one equation set up. Let me scroll down a little bit. And I won't write the units any more. We have 1/8 is equal to 1/A plus 1/B. Now, we have two unknowns, so we need another equation if we're going to solve these. Let's see. It tells us, it tell this right here the year before Bill painted it by himself, but took twice as long as Anya did. So the number of hours it takes Bill to paint the deck is twice as long as the number of hours it takes Anya to paint a deck. So B is equal to 2A. The number of hours Bill takes is twice as the number of hours Anya takes per deck. So B is equal to 2A. So we can rewrite this equation as-- I'll stick to the colors for now. We could say 1/8 is equal to 1 over Anya. Instead of writing 1 over Bill we would write, so plus 1 over Bill is 2 times Anya. The number of hours Bill takes is two times the number of hours Anya takes. So 2 times Anya. And now we have one equation and one unknown. And we can solve for A. And the easiest way to solve for A right here is if we just multiply both sides of the equation by 2A. So let's multiply both sides of this by 2A. Multiply the left side by 2A. Actually, let's multiply both sides by 8A, so we get rid of this 1/8 as well. And then multiply the right-hand side by 8A as well. The left-hand side, 8A divided by 8 is just A. A is equal to-- 1/A times 8A is just going to be 8. 1/2A times 8A. 8A divided by 2A is 4. So A, the number of hours it takes Anya to paint a deck-- and I made it lowercase, which I shouldn't have. Well, these should all be capital A's. This is A is equal to 8A over A is 8. 8A over 2A is 4. So the number of hours it takes Anya to paint a deck, or A, is 12 hours. Now what are they asking over here? They're asking, how long did Anya and Bill take when each was painting alone? So we figured out Anya. It takes her 8 hours. And then we know that it takes Bill twice as long as Anya. Bill is two times A. So bill is going to be 2 times 12. So Bill is equal to 2 times Anya, which is equal to 2 times 12 hours. Which is equal to 24 hours. So when they're alone it would take Anya 12 hours, it would take Bill 24 hours. When they do it combined, it takes 8 hours. Which makes sense. Because if Bill took 12 hours by himself, combined they would take 6 hours, they would take half as long. But Bill isn't that efficient. He's not as efficient as Anya, so it takes them a little bit longer. Takes them 8 hours. But it makes sense that combined they're going to take less time than individually.