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Rational Equations Representing Work and Rate

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Today, we're going to talk about rational equations, representing work, rate, and time. So we're going to start by reviewing the relationship between work, rate, and time. And then, we'll do some real-world examples with work, rate, and time in rational equations.

So let's start by reviewing the relationship between work, rate, and time. We have a formula. w equals r times t, where w is the amount of work being done, rate is the speed that it's being completed at, and t is the time it takes to complete that work.

So we can rearrange this equation and write an equation for the rate. We would do that by isolating the r variable by dividing by t on both sides. And we would have that the rate is equal to work over time.

So if we want to find the rate to complete 1 job, then our value for work would just be 1. We're completing 1 job. And we can say that x is our time. So if x is the amount of time that it takes to complete a job, then the rate is going to be equal to 1 over x.

And finally, if we want to note that when people are working together, the total amount of work is going to be equal to the sum of the individual amount of work of each person.

All right, so now let's do a real-world example involving work, rate, and time. Let's say we have Anna and Becky who are working on tiling bathroom floors. Anna can tile a bathroom floor in five hours. Together, Anna and Becky can tile a bathroom floor in two hours. So we want to know how long it would take just Becky to tile the bathroom floor by herself.

So we can make a chart relating work, rate, and time for our two workers, Anna and Becky. So we have Anna. And we have Becky. And we're looking at their rate that they're working at, the time that they're working for, and then the amount of work that they're able to complete.

So we know-- it's given-- that Anna can complete 1 bathroom floor in 5 hours by herself. So she can do 1 job in 5 hours. Becky, on the other hand-- we don't know how long it takes her to do 1 job on her own, so we'll say she can do 1 job in x hours. x is what we're trying to find. We know that, together, they can complete the work in two hours.

And so, for Anna, we can write an expression for the amount of work that she completes. We know that work is equal to rate times time. So her work is going to be equal to 1 job over 5 hours, times 2 hours, which will just be equal to 2/5 of a job.

For Becky, we can write a similar expression. Again, we will multiply our rate times the time, so 1 job over x hours times the 2 hours is going to give us 2 over x jobs. Now, we know that Anna and Becky are going to be working together to complete 1 job. They're going to tile 1 bathroom floor together.

And we know that we can sum their individual amounts of work to equal the total amount of work being done-- 1 bathroom floor. So we can add both of their individual works. So 2/5 of a job plus 2 over x jobs should equal 1 job.

When I'm solving this equation, I see that it's rational because we have a ratio of algebraic expressions. So we are going to solve this by, first, finding a common denominator between our individual terms. And our common denominator is going to be 5x. So to make this 5 into 5x, I'm going to have to multiply by x in the denominator. And if I do that in the denominator, I'll also have to multiply by x in the numerator.

To make this denominator x to be 5x, I'll have to multiply by 5, again, in the denominator and the numerator. And to make this fraction have a denominator of 5x, I'm going to write it, first, as a fraction of 1 over 1. And then, I'll multiply both the denominator and the numerator by 5x.

So now, rewriting this equation, I have 2x over 5x plus-- 5 times 2 is-- 10 over 5x is equal to 5x over 5x. Solving this equation, I can see, since my denominators are all the same, I can pretty much ignore them and just write an equation using my numerators. So then, my equation becomes 2x plus 10 is equal to 5x.

I'm going to bring my x variables just on one side of the equation. So I'll cancel this one out by subtracting it. So that leaves me with 10 is equal to 3x. And then I'll divide by 3 to isolate x. And I find that x is equal to 10/3 or approximately 3.3. So we found that Becky can tile a kitchen floor by herself in 3.3-- and this will be-- hours.

Let's do another similar example. Let's say Anna and Becky are not painting cabinets. Anna can paint a cabinet in 20 minutes by herself. Together, Anna and Becky can paint a cabinet in only 15 minutes. Well, we want to know how long it would take Becky to paint a cabinet by herself.

So we can set up another work, rate, and time chart. We have Anna and Becky. And we're looking at their rate, time, and work completed. So we know that Anna can paint 1 cabinet-- she can do 1 job, and it takes her 20 minutes. We don't know how long it takes Becky.

So we'll say that Becky can do 1 job in x minutes. We know that together, it's going to take them 15 minutes to paint a cabinet. And so we can use the rate and the time to come up with expressions for work for both Anna and Becky.

So work is rate times time. So for Anna, her work is going to be equal to 1 job over 20 minutes times 15 minutes, which is the same as 15 over 20 jobs, which is 3/4 of a job. 15 over 20 reduces to 3/4. For Becky, her expression for work will be 1 job over x minutes multiplied by the 15 minutes is going to be equal to 15 over x jobs. So she can complete 15 over x jobs.

So if we want to find-- we can write an equation for the amount of work that they do individually. And that will add up to 1 job. So we can say that-- we know that 3/4 of a job for Anna plus the 15 over x of a job for Becky has to equal 1 job. Now again, when we're solving this equation, we want to first have a common denominator for our terms. And that common denominator is going to be 4x.

So to get a common denominator in my first fraction, I'm going to multiply by x in the denominator and the numerator. My second fraction-- I multiply it by 4. And for my last term 1, I'm going to, again, make that 1 over 1-- so it's a fraction-- and then multiply by 4x in the denominator and in the numerator. So now, my equation becomes 3x over 4x plus 60 over 4x is equal to 4x over 4x.

And I can solve this equation, now, simply by ignoring the denominators and writing an equation with just the numerators. And that's, again, because my denominators are all the same. So my equation is 3x plus 60 is equal to 4x.

To solve this, I'll subtract 3x from both sides. And this gives me 60 is equal to 1x, which is the same as just x is equal to 60 or 60 minutes. So I found that it will take Becky 60 minutes to paint a cabinet by herself.

So let's go over our key points from today. Work is equal to the product of rate and time. When people work together, the total work is the sum of individual work. And the rate an individual person works for can be expressed as a rational expression with a variable for time in the denominator.

So I hope that these key points and examples helped you understand a little bit more about work, rate, and time, and rational equations. Keep using your notes And keep on practicing. And soon, you'll be a pro. Thanks for watching.