INSTRUCTOR: Welcome to a video on the applications of rational equations. Let's go ahead and get started. Martin can pour a concrete walkway in six hours working alone. Victor has more experience and can pour the same walkway in four hours working alone. How long will it take both people to pour the concrete walkway working together? 

So let's jot down some information here. If Martin can pour the concrete walkway in six hours, Martin's rate would be 1/6 of the job per hour. And if Victor can do it in four hours, his rate would be 1/4 of the job per hour. Since they'll be working together, we're going to be adding their rates together. So if we let t equal hours, then at any t, the fraction of the job that would be complete would be Martin's rate times the number of hours plus Victor's rate times the number of hours. 

Since we want to know when this job would be 100% complete, 100% as a fraction would be equal to 1. And this is the equation we need to solve in order to determine how long it will take if they both work together. Now, if we wanted to, we could rewrite this. This is t/1 and t/1. So we could write this equation as t/6 plus t/4 must equal 1. 

Let's go ahead and take this equation over to the next screen and then solve for t. I'm going to go ahead and rewrite 1 as 1/1. And what we'll do here is we'll clear the fractions by multiplying every term by the LCD. Well, the LCD would be 24. So we'll multiply each term by 24/1. 

Now, before we multiply, we'll simplify. This would change to a 1. This would be 4. So we'll have 4t. Here, we have a common factor of 4. That would change to 1. That would be a 6 plus 6t must equal 24. So now we'll solve this equation for t. This would be 10t equals 24. Dividing both sides by 10, t equals-- this would be 2.4 hours. If they work together, they can complete the job in 2.4 hours. 

Now, if we want to know specifically how many minutes, we could convert 0.4 hours into minutes. Let's go ahead and do that. 0.4 hours over 1. Multiply this by 60 minutes over 1 hour. Notice how the units of hours simplify out. 0.4 times 60 would be 24 minutes. So more specifically, the total time to complete the job if they work together would be 2 hours, 24 minutes. 

Now, that should make sense. Because if we go back to the original problem, if Martin can complete the job in six hours and Victor can complete it in four hours, of course, if they work together, it should be faster than the fastest person. So having an answer less than four hours doesn't make sense. 

Let's go and take a look at another problem. An inlet pipe can fill a tank in 12 hours. An outlet pipe can drain the tank in 20 hours. If both pipes are mistakenly left open, how long will it take to fill the tank. So what's happening here, we have water coming in and also water going out. Let's go ahead and determine the rates. 

If it takes 12 hours to fill the tank, the rate of the water going into the tank would be 1/12 of the tank per hour. And if it takes 20 hours to drain the tank, the rate of the water leaving the tank would be 1/20 of the tank per hour. So again, if you let t equal time in hours, we can write an equation to represent how much water would be in the tank at any time. 

In this case, they're working against each other. So this is the rate of the water going in. This is the rate of the water going out. So the equation that we would come up with would be 1/12 times the number of hours. And it would be minus 1/20 times t. Now, again, at any time t, this would represent what fraction of the tank would be full. But since we want to know when the tank would be full or 100% full as a fraction, that would be equal to 1 or 1/1. 

Let's go ahead and rewrite this one more time, and then we can solve it. Again, these t's could be written over 1. So the equation that we can set up and solve now would be t/12 minus t/20 must equal 1/1. Again, let's take this to the next slide, where we have more room. 

We're going to multiply every term by the LCD. Well, the least common denominator of 12 and 20 would be 60. So multiply each term by 1. This should clear our denominators. Simplifying here, we'd have a 1 and a 5. That would be 5t. Here, we're going to have a 1 and a 3. So we have minus 3t equals 60. 

Again, the result is a very straightforward equation to solve. We have 2t equals 60. Dividing both sides by 2, the time is 30 hours. Let's go back and take a look at the original problem. Notice that if the drain wasn't left open, it would have taken 12 hours. So it does make sense that it's going to take a lot longer to fill if we're draining the water at the same time. So having a total fill time of 30 hours in this situation makes a lot of sense. 

OK. We'll go ahead and take a look at a couple more questions in the next video. You'll probably find the next examples a little more challenging, but this is a good place to start. Thank you for watching.