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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.