Hi, and welcome. My name is Anthony Varela. And today we're going to solve rational equations representing work, rate, and time. So we'll begin by reviewing this relationship between work, rate, and time. And we'll develop rational equations then to solve for a work, rate, or a time.
So work is a product of rate and time. We multiply our rate by how time we spend. And that equals then the amount of work that we've completed. So the equation here is w equals r times t. w is the work, r is the rate, and t is the time.
We can divide this entire equation by t. And we have an equivalent equation, rate equals work divided by time. Now, thinking more about work, I'm going to call that a job. And so our rate then would be how much of the job we complete per hour. And time then would be measured in hours.
So thinking about rate equaling work over time, we can say that work is one job, and t is x number of hours. And 1/x is a rational expression. So we're going to be using rational equations to solve some problems here. Another thing that we'll see in our examples is that when people are working together, we can add their individual work to represent what they complete together.
So our first scenario is that Andrew can mow the backyard in 20 minutes by himself. If Amanda helps, they can finish the yard in 12 minutes. So how long would it take if Amanda mowed the lawn alone?
So we're going to organize our information in a rate, time, and work table. So I have a row for Andrew, a row for Amanda, and then I have columns for rate, time, and work.
Now let's fill in what we know about Andrew. We know that his rate is one backyard in 20 minutes. I'm going to call this one job over 20 minutes.
Now, the time that it takes for both Amanda and Andrew to complete this work together is 12 minutes. So I can say then that Andrew's portion of the work is 3/5 of the entire job. That's multiplying the rate and the time together. And we notice that our units of minutes cancel. So our work is 3/5 of the job.
Now, with Amanda, she can complete one job in x number of minutes. We don't know what that is. That's what we'd like to solve for.
We have 12 minutes in her column for time. That's the amount of time it takes for the two people to finish together. And so Amanda's portion then of the entire job can be expressed as 12/x. And our unit here is in jobs.
So now that we have our information organized in our table, we're going to create a rational equation. And we're going to sum Andrew's portion of the work and Amanda's portion of the work. So we have 3/5 plus 12/x. And what does that equal? Well, that equals one entire job. So here is the rational equation that we'd like to solve for.
Now, in order to solve this equation, I need to create common denominators. So looking at 3/5 and 12/x, how can I create a common denominator? Well, I'm going to multiply 3/5 by x/x. And then I'm going to multiply 12/x by 5/5. So now I have a common denominator of 5x in both of my fractions that we see on the left side of the equal sign.
Now I'm also going to create a common denominator on the other side of the equation. And we'll see why we do that in a minute. So I'm going to write 1 as 5x/5x.
So now that in all of these terms, I have a common denominator of 5x, I can focus on my numerators. So I have that 3x plus 60-- that's 12 times 5-- equals 5x. And this equation is a lot easier to solve.
So I'm going to take away 3x from both sides. So 60 equals 2x. So I know that x equals 30. And what does x represent? Well, that represents minutes. And that's how long it would take Amanda to complete this job all by herself.
Now, in our second scenario, we have Andrew and Amanda cleaning windows. Now, Andrew can clean 10 windows in 3 hours. Amanda can clean 10 windows in 2 hours. So how long will it take to clean 10 windows together?
So once again, we're going to organize our information in this table. Let's fill out what we know about Andrew. Well, his rate is 10 windows in 3 hours.
The time that it takes for them to complete this job together we don't know, so we're going to write in x hours. So multiplying the rate times the time, we get 10/3 times x. And our unit here is a job.
For Amanda, her rate is 10 windows in 2 hours. And once again, we don't know the time that it takes for them to complete this work together. So we have x hours in the middle column.
And multiplying rate and time then, we can describe Amanda's portion of the work as being 10/2 times x. The unit here is job.
So now let's create a rational equation by adding Andrew's work and Amanda's work. So we have 10x/3 plus 10x/2. Now, what does this equal when we add these two together? That equals one job, and one job is 10 windows. So this equals 10 windows. x is the amount of time it takes for Andrew and Amanda to clean 10 windows together.
So once again, I would like a common denominator. So what I'm going to do to 10x/3 is multiply that by 2/2. Now, to get a common denominator for 10x/2, I'm going to multiply that by 3/3. And once again, I want a denominator of 6 on the right of the equals sign. So I'm going to multiply 10 by 6/6.
So now I have 6 as a common denominator in all parts of my equation. So I'm going to focus on the numerators. 20x plus 30x equals 60. I'll combine our x terms, 50x equals 60. So now I know then that x equals 1.2. And remember, x is the number of hours that it takes Andrew and Amanda to complete this job together. So they can clean 10 windows in 1.2 hours.
Let's review our lesson on rational equations representing work, rate, and time. We have our equation work equals rate times time. And we used information about people's work, rate, and time to create a table. And by adding work together and multiplying rate and time to represent work, we solved rational equations by finding common denominators, and then focusing on the numerators in the equation.
So thanks for watching this tutorial on rational equations with work, rate, and time. I hope to see you next time.
