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# Rational equations word problem: combined rates (example 2)

CCSS.Math:

## Video transcript

working together onion bill stained a large porch deck in eight hours last year Anya stain the deck by herself the year before bill painted it by himself but did 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 of hours for Anya to paint a deck to paint a deck and so we could say that Anya Anya paints or has eight hours eight hours for one deck or we can invert this and say that she can do one over a decks one over 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 paint deck and so for Bill he can paint it takes him B hours per deck for one deck we could put it over one there if we want we don't have to and this is the same thing as saying that it takes he can do one deck for every be hours be hours or another way to think of it is he can do 1 over B deck per hour now when they work together Anya and Bill stained a large porch deck in eight hours so let's write this down on your plus bill on your plus bill do an orange Anya and bill this information right here Anya and built stained a large porch deck in eight hours so we could say eight hours per deck or eight or eight hours per one deck which is the same thing as saying one deck one deck per eight hours and this is going to be the combination of each of their rates so this one deck of eight for eight hours this is going to be equal to Anya's rate one over a decks per hour decks per hour plus bills rate Plus do that same color plus 1 over B decks per hour decks per hour so we have one equation setup let me scroll down a little bit and I won't write the unit's anymore we have 1 8 is equal to 1 over a 1 over a plus 1 over B plus 1 over B now we have 2 unknowns so we need another equation if we're going to solve these let's see it tells us tells us right here the year before bill painted 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 on you to paint a deck so B is equal to 2a the number of hours bills 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 on you instead of writing 1 over bill we would write so plus 1 over bill is 2 times on the number of hours bill takes is 2 times the number of hours Anya takes so 2 times Anya and now we have one equation in one unknown and we can solve for a 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 8 a so we get rid of this 1/8 as well 8a and then multiply the right-hand side by 8 as well the left-hand side 8a divided by 8 divided by 8 is just a a is equal to 1 over a times 8a is just going to be 8 1 / 2 8 times 8 8 8 a divided by 2 a is 4 right 8a over 2 a is 4 so a the number of on the out number of hours it takes on you to paint a deck and I made it lowercase which I shouldn't have well these should all be capital A's you should all be capital A's so you have eight this is a is equal to 8 a over 8 over a is 8 8 8 over 2 a is 4 so the number of hours it takes on you 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 on you it takes her 8 hours and then we know that it takes bill twice as long as Anya bill is 2 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 as efficient design it so it takes them a little bit longer it takes some 8 hours but it makes sense that combined they're going to take less time than individually