If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

# Rational equations word problem: combined rates

CCSS.Math:

## Video transcript

Ian can rake a lawn and bag the leaves in 5 hours keiondre can rake the same lawn and bag the leaves in 3 hours working together how long would it take them to rake the lawn and bag the leaves so let's think about it a little bit so this e in can rake along and bag the leaves in 5 hours so for E and I for Ian he can for 1 lon one lawn he can do one lawn in one lawn in five hours one lawn in five hours we could have also written this is five hours per lawn but we'll see that writing it this way is more useful because it's actually a rate because this is the same thing as one over five lawns lawns per hour or 1/5 of a lawn per hour that's the rate at which en can rake a lawn at 1/5 of a lawn per hour now let's do the same thing for kyandre keiondre and this actually probably shouldn't be plural 1/5 lawns 1/5 of a lawn so let me just erase that s right there 1/5 of a lawn per hour now let's do the same thing for kyandre keiondre can rake the same lawn and bag the leaves in 3 hours so for kyandre keiondre can for 1 lon I'll assume it's a boy's name he can do it in 3 hours in 3 hours or if we were to write it as a rate this is 1/3 1/3 of a lawn 1/3 of a lawn per hour now let's think about what the combined rate is so let's say if we have en Plus keiondre en + keiondre what's going to be there combined what's going to be their combined rate well they tell us they tell us that working together how long would it take them to rake and bag the leaves so let's let let's let T be how long it will take them together how long it will take together so that's how long they would take together and if we say T is how long take together then we could say that combined they will do one lawn they will do one lawn in every t hours of resuming T is in hours for every t hours or as a rate their combined rate is going to be one over T one teeth of a lawn one teeth of a lawn per hour one over T lawns per hour so that's their combined rate so here we have the rate of en and the rate of chiandra keiondre and the combined rate so the combined rate is going to be the sum of each of their rates if he can do 1/5 of a lawn per hour and he can do 1/3 of a lawn per hour their combined rate is going to be 1/5 of a lawn per hour plus 1/3 of an hour because in an hour he'll do 1/5 and he'll do 1/3 so you'll add those two together to figure out how much they can do in an hour so their combined rate is going to be 1/5 lawn per hour and I won't write the unit's here just because it gets redundant 1/5 lawn per hour could right over here plus 1/3 lawn per hour for keiondre plus 1/3 of a lawn per hour for Kandra that's going to be their total rate which is 1 over T lawns per hour which is 1 over T lawns per hour and now we just have to solve for T and we'll know the total number of hours it will take them so let's do that so to do that we just have to add 1/5 plus 1/3 well we can have a common denominator of 15 so this is the same thing as 3 over 15 plus 5 over 15 plus 5 over 15 is going to be equal to 1 over T 1 over T and then we have a common denominator now so three plus five is eight so this is going to be 8 over 15 I'll go over here now so now we have three plus five is eight over 15 is equal to one over T 1 over T if we want to solve for T we could take the reciprocal of both sides so if we flip the left side we get 8 over we get 15 over 8 15 over 8 and if we flip the right side we get T over 1 or just T so it'll take them 15 over 8 hours or if we want that in kind of a way that we can think about it a little bit better 15 over 8 so T is equal to 15 over eight hours 15 over eight and I should say 15 over eight hours per lawn this whole time here we had lawns per hour this was lawns per hour this was lawn per hour and this was here as well lawns per hour when we flip it and become hours it becomes hours per lawn so that's exactly what we want but 15 over 8 is the same thing as 1 and 7/8 hours and 7/8 of an hour we can get a calculator out get our calculator out if we have 60 minutes in an hour times 7 8 we get 50 2.5 minutes so this is equal to our answer combined it will take them 1 hour 1 hour and 52 point 5 minutes and 52 point 5 minutes per lawn or to do this lawn right over here the lawn in question hopefully found that useful