Rational equations word problem: eliminating solutions

two different hoses are being used to fill a fish pond used together the two hoses take 12 minutes to fill the pond if used alone one hose is able to fill the pond 10 minutes faster than the other so 10 minutes faster than the other hose how long does each hose take to fill the pond by itself so let's think about each of the hoses we have a faster hose we have a faster hose and then we have a slower hose and let's say that the faster hose fills the pond let's say it takes him him F minutes F minutes per pond F minutes per pond now how long is it going to take the slower hose well the faster hose duck takes it in does it in 10 minutes less it's 10 minutes faster so the slower hose is going to take 10 minutes more so the slower hose is going to take F Plus 10 minutes minutes per per fish pond or per pond now this is in minutes per pond but if we want to be able to add rates together we should really think about it in terms of ponds per minute so let's rewrite each of these statements as ponds per minute you could write this as F minutes per one pond or F over 10 F plus 10 minutes per one pond and if you just take the inverse of each of these statements these these ratios are equivalent to saying one pond one pond per F minutes so it's really not saying anything else oh I'm just inverting the ratio or you could think of it as 1 over F ponds per minute 1 over F ponds per minute same logic right here we could we could essentially rewrite this ratio as 1 over F plus 10 ponds ponds per minute so now we have the rate of the faster hose we have the rate of the slower hose how many ponds per minute for the faster hose how many ponds per minute for the slower hose if we add these two rates will know the ponds per minute when they're acting together so if we have 1 over F ponds per minute plus one over F plus ten ponds per minute this is the faster hose is the slower hose this will tell us how many ponds per minute they can do together now we know that information they say the two hoses the two hoses take 12 minutes the two hoses take 12 minutes so let me write that over here so combined the combined take 12 minutes per pond twelve minutes per pond so what is their combined rate in terms of ponds per minute so you could view this 12 minutes per one pond you could take the inverse of this or take the ratio in terms of ponds per minute instead and you get one over 12 minutes sorry 1 over 12 pound ponds ponds per minute in one minute a combined they'll fill 1/12 of of a pond which makes complete sense because it takes them 12 minutes to fill the whole thing so in 1 minute they'll only do 1/12 of it so this is their combined rate in ponds per minute this is also their combined combined rate in ponds per minute so this is going to be equal to 1 over 12 and now we just have to solve for F and then F plus 10 is going to be what the how long it takes the slower the slower hose so let's multiply let's see what we could do we could multiply both sides of this equation times F and times F plus 10 so let's do that so I'm going to multiply both sides of this equation times F and F Plus 10 times both sides of this equation so F and F plus 10 scroll down a little bit let me scroll to the left so we have some real estate so let's distribute this F times F Plus 10 so if we multiply F times F Plus 10 times 1 over F that F and that F will cancel out and we're just going to be left with an F Plus 10 that's what when you multiply that term times the F times F plus 10 now when you multiply this term when you multiply 1 over F Plus 10 times F times F plus 10 this and this will cancel out and you're just left with an F so you have plus F is equal to and you have over 12 actually well in the second let me multiply all side this equation by 12 I'll do that next in a second so let's let's just say this is going to be equal to 1 over 12 times s squared times F squared F times F is f squared plus 10 F and now let's just multiply both sides of this equation by 12 I could have done it in the last step so that we don't have any so that we don't have any fractions here so we don't have any fractions and so the left hand side we get 12 times F we get 12 F plus 120 plus 12 F the right hand side that and that cancels out and you are left with F squared plus 10 F and now we have a quadratic we just have to get into a form that we know how to how to manipulate or deal with and before that we can simplify it we have a 12 F and a 12 F so this becomes 24 F plus 120 is equal to F squared plus 10 F and then let's get all let's get all of this stuff out of the left hand side but look at all on the right-hand side so from both sides of this equation let's subtract 24 F and a negative 120 or minus 120 so you have minus 24 F minus 120 left-hand side just becomes 0 that was the whole point right hand side is F squared 10 minus 24 F is negative 14 F minus minus 120 now we could factor this let's see if you do 20 times sit yeah that looks like it would work so 20 and negative 20 and 6 when you take their product give you negative hundred 20 and negative and negative 20 plus 6 is negative 14 so we could factor this right hand side is 0 is equal to F minus 20 times F plus 6 when you multiply negative 20 times 6 you get negative 120 negative 20 plus 6 is negative 14 and the only way that that's going to be equal to 0 is if F minus 20 if F minus 20 is equal to 0 or F plus six is equal to zero add 20 to both sides of this equation you get F is equal to 20 remember F is how many minutes does it take for the fast hose to fill the pond and then if you take this one you subtract 6 from both sides you get F is equal to negative 6 now when we're talking about how many minutes does it take for the fast hose to fill the pond it doesn't make any sense to say that it takes it negative 6 minutes to fill the pond so we can't use this answer we need a positive answer so this is how many minutes it takes the fast hose to fill the pond F is equal to 20 so the fought this right here the faster hose takes 20 minutes takes 20 minutes per pond so 20 I'll write it here 20 minutes per pond is the fast hose and then the slower hose is takes 10 minutes more its F Plus 10 so it takes 30 minutes per pond and we're done I want to confuse you with this stuff the faster hose takes 20 minutes slower two hose takes 30 minutes per pond if they were to do it together it would take 12 minutes which is a little bit more than half if you had to faster hoses it would take 10 minutes but this guy is a little bit slower so it's taking you a little bit more than 10 minutes so it makes sense it takes it takes you 12 minutes when they're working together