Rational equations word problem: combined rates

Ian can rake a lawn and bag the leaves in 5 hours. Kyandre 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. It says Ian can rake a lawn and bag of the leaves in 5 hours. So for Ian, I for Ian. For 1 lawn, he can do 1 lawn in 5 hours. We could have also written this as 5 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 1/5 lawns per hour. Or 1/5 of a lawn per hour. That's the rate at which Ian can rake a lawn, at 1/5 of a lawn per hour. Now let's do the same thing for Kyandre. 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. Kyandre can rake the same lawn and bag the leaves in 3 hours. So for Kyandre, Kyandre can, for 1 lawn-- I'll assume it's a boy's name-- he can do it in 3 hours. Or if we were to write it as a rate, this is 1/3 of a lawn per hour. Now let's think about what the combined rate is. So let's say if we have Ian plus Kyandre. What's going to be their combined rate? Well, 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. So that's how long they would take together. And if we say t is how long they take together, then we could say that combined, they will do 1 lawn in every t hours, if we're assuming t is in hours. For every t hours. Or as a rate, their combined rate is going to be 1/t of a lawn per hour. So that's their combined rate. So here, we have the rate of Ian and the rate of Kyandre, and the combined rate. So the combined rate's just 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 units here, just because it gets redundant. 1/5 lawn per hour I could write over here. Plus 1/3 lawn per hour for Kyandre. That's going to be their total rate, which is 1/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 have a common denominator of 15. So this is the same thing as 3/15 plus 5/15 is going to be equal to 1/t. And then we have a common denominator now. So 3 plus 5 is 8. So this is going to be 8/15. I'll go over here now. So now we have 3 plus 5 is 8. Over 15 is equal to 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 15/8. And if we flip the right side, we get t/1, or just t. So it'll take them 15/8 hours. Or if we want that in kind of a way that we can think about it a little bit better, 15/8-- so t is equal to 15 over 8 hours. And I should say 15 over 8 hours per lawn. This whole time here we had lawns per hour. This was lawn per hour. And this was here as well, lawns per hour. When we flip it, it becomes hours per lawn. So that's exactly what we want. But 15/8 is the same thing as 1 and 7/8 hours. And 7/8 of an hour-- we can get our calculator out. If we have 60 minutes in an hour, times 7/8, we get 52.5 minutes. So this is equal to our answer. Combined, it will take them 1 hour and 52.5 minutes per lawn. Or to do this lawn right over here, the lawn in question. Hopefully, you found that useful.