In the last two lessons, we focused on motion problems that involve rates. The rates were miles per hour, which is how fact an object was traveling. In this lesson, we're going to focus on a different kind of rate involving work. If it takes you two hours to complete a homework assignment, what fraction of the assignment can you complete in one hour? In this case, we're talking about the rate, about how fast you can work on an assignment. When you think you know the answer, enter that fraction here.
It turns out you can do one half of the job in one hour's time. We can get one half by drawing a diagram to help us out. We know that it takes two hours for us to do the entire assignment. So, if two hours passes, then we've finished our complete assignment; both of these bars would be full. So, what we can do is we can take our entire two hours and split it up into one hour increments. Each of these would represent one hour. And each of these on the corresponding side would represent the amount of work for the fraction being done in that hour. So if we split our time in half we could split our assignment in half. We'll know that we'll finish half the assignment in one hour. And then the other half in the next hour. So this is how we know that in one hour we only finish half of the assignment. Great work if you got that fraction.
So, we know you can complete half of the assignment in one hour. Well, you learn that Anthony, another student, can complete the same assignment in three hours. If that's true, what fraction of the work can he complete in one hour? Write that answer here.
Anthony can complete 1 3rd of the assignment in 1 hour's time. Great job if you found this fraction. For Anthony we know it takes him 3 hours time to complete one assignment. So let's take his 3 hours and break it up into three equally spaced intervals. So in 1 hour, Anthony can complete 1 3rd of his job. And the And the last hour, Anthony finishes his job, which would be 1 or 3 3rds. This is how we know is Anthony only is allowed to work for 1 hour, he completes a time.
But who's to say that you always have to work alone? What if you guys solved the problems together? The amount of work you do in 1 hour plus the amount of work Anthony does in 1 hour, would equal the total fraction of work you both can do in 1 hour. In other words, we knew that you could complete half of the job in 1 hour. And Anthony can complete a third of the job in 1 hour. So if we add these 2 fractions together, this would be the total sum of the fraction of the job that you both could do in 1 hour's time. Notice that the denominator here represents the time that it took for you do the entire job. And for Anthony, the 3 represents the amount of time it took for him to do the entire job. This means that x represents the total number of hours for you and Anthony to do that job together. So working together, how long would it actually take you and Anthony to complete the assignment together? Write the number of hours it would take them to complete the work together here. And then convert that answer to the number of minutes it would take them to complete the work here. Keep in mind I don't want you to write your answer as hours and minutes. I want you to write the fraction or the number of hours here and then convert that fraction to the toal number of minutes here.
It would take them one and one fifth of an hour or six fifth hours to complete that work together. And converting that to minutes, it would take them 72 minutes total time to complete the work or one hour and 12 minutes. Great work if you found these two numbers. We want to take our equation and solve for x. We know that this fraction represents the amount of work that they both can complete in one hours time. So x represents the amount of time it takes for them to do all of that work, or to complete the entire job. So we can solve this equation by multiplying through by the LCD, which is 6x. We simply multiply these denominators together, since the greatest common factor among them is one. We simplify this to 3x, this to 2x, and this fraction to the 6. We add like terms on the left to get 5x equals 6, and finally we divide both sides by 5 to get x equals 6 5th hours. We know 6 5th's hours is the same thing as 1 hour and 1 5th of an hour. And digging all the way back to fractions in our clock, we know 1 fifth of an hour is really 12 minutes. Now, if you didn't know this off the top of your head, you could have simply divided 60 minutes, which is 1 hour, by 5 to get 12 minutes. We know an hour is 60 minutes. So, all together, this would make 72 minutes of time. So this is how we know that the 2 of you can solve all of the problems in 72 minutes or 6 5ths hours.
In general if you have a person or thing that can do a job in a hours then 1 over a represents the amount of work done by that person or thing in one hour time. If you have a second person or thing that takes b hours to do that same job, then 1 divided by b represents the total amount of work done. By that thing in one hour. So if it takes them x hours to do the job together, then 1 over x is how much they have done together in that one hour. This would be the fraction of the job completed. Again, this x is the number that represents the total time it takes for the job to be completed by two people or objects working together.
Let's see if we can use our concept of working together to solve this problem. Vanessa and Tim want to repaint a room. Tim's painted the room alone before, and done it in 4 hours. While Vanessa's painted the room alone, and done it in room? Try setting up an equation, and then solve and write your final answer here.
Working together, they can paint the room in 12 7ths hours time. Great thinking if you found this fraction. 12 7ths is equal to 1 hour and 5 7ths of an hour. If we do some conversion and convert this to minutes, it's about one hour and paint the room in 4 hours time. So he can complete one job in 4 hours. The other way of thinking about this is that he will complete 1 4th of the job in 1 hour. Vanessa's painted the room in 3 hours. Which means she can complete the full job in 3 hours time, job per hours. So, notice that we can think about this as a rate. We have Tim completing 1 job in 4 hours and Vanessa completing that same job in 3 hours. So if we add their two rates together, we would get them completing the job in x hours time, the time you want to find. And now here's the other way we were thinking about it. Tim can complete 1 4th of the job in 1 hours time. Vanessa can complete 1 3rd of the job in 1 hours time. So together they can complete 1 over x of the job in 1 hour's time. We start solving this equation by multiplying though by the LCD. The least common denominator is 4 times 3 times x. Or 12 x. We multiply each fraction by this number, and that gives us 3 x here, 4 x here, and 12 here. We add our like terms on the left to get 7 x equals 12. And finally we divide by 7 to get 1 x is equal to 12 7ths. This is the amount of time it takes them to paint the room together.
Let's try another problem Lucy and Javier can mow their lawn in 1.2 hours time if they work together. If it takes Javier one hour longer than Lucy, how long would it take them to mow the yard alone? Now, we want to use our same approach from before. We're going to use our 1 over a plus 1 over b equals 1 over x. Where x is the total amount of time it takes them to do a job together. We know that total time is 1.2 hours. So we can replace x with 1.2. Now we don't know how long it took Javier or how long it took Lucy. Those are both unknowns. So what do you think we could write here and here, in order to finish out our equation? You'll need to think about how Lucy's time to mow the yard and how Javier's time to mow the yard alone relate to one another. When you think you know what should go in this denominator, and what should go in this denominator, type them in here. As a hint, you'll have to use a variable, and for this one I want you to use the variable t.
The key to finishing this equation is that it takes Javier one hour longer than Lucy. So let's let Lucy's time equal t. We'll use that for our first fraction. This fraction would represent one job completed in t hours. This means that Javier can do the same job in t plus 1 hours. So his fraction would be one job divided by t plus 1 hours. His time is simply 1 hour more than Lucy's. So t plus 1.
Now that we have an equation to work with, let's try and solve for t. Once you solve for t, tell me what Lucy's time would be, and what Javier's time would be to mow the lawn alone.
It would take Lucy 2 hours, and Javier 3 hours. Great thinking if you found these numbers. We want to keep in mind that t represents the number of hours it takes for Lucy to mow the yard alone. So when we solve for t in this equation, we'll find Lucy's time. To begin solving this equation, we want to multiply through by the LCD, which is 1 and 2 10ths times t times t plus 1. It's the product of these 3 denominators since they're all different and the greatest common factor is 1. When we multiply through by the LCD, we'll see that the factors cancel. The t's will cancel. The quantity t plus 1 will reduce to 1. And then number 1.2 or 1 and 2 10ths reduces to 1 as well. This leaves us with times t here. We'll distribute the 1 and 2 10th and the t to the terms to get this equation. Then we'll combine the like terms on the left hand side to get 2 and 4 10th times t, plus 1 and 2 10th equals t squared plus 2. To get rid of these decimals, we'll multiply through by 10. So we'll have 24t plus 12 equals factor. We want to move all of these terms to the right hand side and then see if we can factor. So, subtracting 24t and 12, we'll get 10t squared, minus 14t, minus 12 on the right hand side of our equation. And we'll get 0 on the left. I notice that all of these terms are even. So I can divide my entire equation by quadratic equation, we use factoring by grouping to get the factors of 5t plus for the values of t. We find that t can equal positive 2, or t can equal negative 3 5ths. We know that t represents the time for Lucy to mow the yard so we don't want to use this negative value, it wouldn't make sense here. Instead, we'll use t equals 2. So, she takes 2 hours to mow the yard. If we remember, Javier's time was t plus 1, he took an hour longer than Lucy, so this would mean he would mow the yard in 3 hours. We just needed to add 1 on to Lucy's time.
Let's try a third problem on working together. It takes Melissa 2 hours longer to edit a short film than it takes Carlos, who is a professional editor. If it takes them 2 and 4 10th hours to edit a short film as a team. How long would it take them to do it alone? Now, before we saw for each person's time to edit the film alone, let's find an equation. Enter the equation in here and you want to make sure that you use x as your variable. Good luck.
We'll start with our general case. We'll have one person completing one job in a hours and another person completing that same job in b hours. This means that their total time together would be x hours. And 1 over x would be the fraction of the job done together in one hour. We know x in this case would be 2 and 4 film together. Now, we need expressions for these two. If we let x equal the time it takes Carlos to edit, we can use x in one of the denominators. We know Melissa takes two hours longer than Carlos, so we can just add two to x to get x plus 2 for Melissa's time. Great thinking if you found this equation. I know it's not easy.
Now that we have an equation, I want you to solve for x, and then tell me what's Melissa's time to edit the film, and what is Carlos' time. Write them here.
Melissa would need six hours to edit the film and Carlos would only need four hours. Great thinking if you found these two answers. We'll start solving this equation by multiplying through by the LCD two and 4 tenths times x times x plus 2. The product of these three denominators. When we multiply through by the LCD, we can simplify each of the terms. The first term reduces to two and four tenths times x plus two. The second term reduces to two and four tenths times x And the last term simply reduces to x times x plus 2. We distribute 2 and 4 10ths to get these two terms, and we distribute x to get these two terms. Next we combine 2 and 4 10ths x, and 2 and 4 10ths x to get 4 and 8 10ths times x. Now to clear these decimals, I'm going to go ahead and multiply through by a factor of 10. When we multiply through by 10, we'll get 48x48 8, 10x squared, and 20x. Now we're ready to move these two terms, to the right hand side of our equation, so we can get 0 on the left side, and a quadratic expression on the right hand side. Now, we're ready to divide this equation by 2 since all these coefficients are even. When we divide each of these terms by 2, we'll get this new equation. Now we can factor. When we use factoring by grouping, we get the factors 5x plus 6 and x minus 4. We set each factor equal to 0 and then we can solve for the values of X. X can equal positive 4 or X can equal negative 6 fifths. We know X to the amount of time it takes for Carlos to edit the film. So, it would take him 4 hours we don't use this choice since it's negative. That wouldn't make sense with our concept of time. So, since it takes Carlos 4 hours, it must take Melissa 6 hours. We know this is true since x was 4, which is the amount of time it takes Carlos. And Melissa was X plus 2, two more hours than Carlos. If you are ever unsure, go back to the original problem. We knew Carlos time was represented by X, so he would take four hours to edit the film. Melissa takes two hours longer than Carlos so that means that she needs six hours in order to edit the film. I know this one was tough, so great work if you got it right.
Here's our fourth problem on working together. It takes Jamal and Danielle 4 hours to carpet a room together. If each of them worked alone, one could complete the work in 6 hours less time than the other. How long would it take the faster one to do the job? Before we get started to solve, let's see if you can set up the correct equation. Write your equation here and be sure to use x for the variable. Good luck.
Here's the equation we want to solve. Great work if you found this one. I know this one was tough. The key idea to get started is to assign the variable. I think it's best to let x equal the time it takes for the slower person. If we let x equal the number of hours it takes the slower person to carpet the room, then we can use 1 over x for the first fraction. This would be one job completed in x hours. The faster person works in 6 hours less time. So this means the second person could complete that same job, one job in x minus 6 hours. Now we don't know who's faster Jamal or Danielle, we just know that the faster person does in in x minus 6 hours and the slower person does it in x hours. If we add the slower person's rate to the faster person's rate, we'll get 1 4th. One job completed in 4 hours. This is the amount it takes Jamal and Danielle to carpet the room together.
Now that we have an equation to solve, I want you to figure out the time it takes for the slower person, and the time it takes for the faster person. Keep in mind that the slower person would be x and the faster person would be x minus 6. Good luck.
Now we don't know who the faster or the slower person is. But we know the slower person takes 12 hours, and the faster person takes only 6 hours. Great algebra skills for getting that one correct. We'll start solving this equation by multiplying through by the lowest common denominator. Which is 4x times x minus 6, the product of these denominators. If we multiply through by the LCD, we'll find that he x cancels in the first term, the x minus 6 cancels in the second term. And then, on the right hand side of the equation, the 4s' simplify to 1. This leaves us with 4 times x minus 6, here. 4x for a second term and an x times x minus 6 on the right hand side. Next, we'll distribute this positive right, to get 8x minus 24 equals x squared minus 6x. Now we're almost to a quadratic expression that we can factor. We just need to subtract 8x and add 24 to both sides of the equation. This gives us 0 on the left side, x squared minus 14x, plus 24 on the right side. Now we want to find factors of positive set each of these factors equal to 0, and then we solve for x. So, we find the values of x equals 2 or x equals 12. And this might be confusing. Which answer should we take? Before we've always had at least one of the answers be negative. So we could eliminate it since a negative time didn't make sense. The key is that one person is 6 hours slower than the other. So if someone was 6 hours slower than someone who could work 2 hours, we'd wind up with a negative time. So this one would be out. This means that the slower person must take 12 hours, and the faster person could do it in 6.
When working together, we use the general formula 1 over a plus 1 over b equals hours equals 1 divided by x. The total time it takes for two things, or people to complete that job together. But not everything has to work together. Sometimes things can work against each other. So, what new equation do you think we could write here and use to help us problem solve?
Well if something is working against another thing, we want to subtract its rate from the original. So we'll have 1 over a minus 1 over b equals 1 over x. Good thinking if you found this equation.
Let's see if we can use this equation to help us solve a different type of problem. Our problem solving method will still be the same. We'll just want to use a minus sign instead of a plus sign. For this last problem we'll solve it using this general equation. Here, we'll have a pool that has a pipe system that can fill it with water. An inlet pipe can fill the pool in 10 hours time. While an outlet pipe can drain it in 12 hours. If both pipes are left often by mistake. How long would it take the pool to fill? Think carefully about where each of these numbers should go in your equation, and then solve for the number of hours.
It would take 60 hours or about two and a half days for the pool to fill. Great work if you found this. This is the equation that we'll use to help us solve. We can fill the pool in 10 hours, and we can drain one pool in 12 hours. We know that when you subtract the 1 12th here since it takes longer to drain the pool than it does to fill it. In other words this rate is it slower than this rate? And this should make sense, since we know 1 12th is smaller than 1 10th. So, the pool will fill up ever so slightly, and we can figure out the total time that it takes to fill the pool, this is x. We start solving this equation by multiplying through by the LCD 60x. The common factors of 10 and 12 are 2. The other factor of 10 is 5, and the other factor of 12 is 6. We multiply these factors together, which give us 60, and then we have to multiply by x, the other factor in our denominators. We multiply by x since x is not a common factor of 10 or 12. So 60x will be our LCD. This first term will reduce to 6x, since 60 divided by 10 equals 6. The second term will reduce to 5x, since 60 divided by 12 equals 5. This leaves us with the right hand side of just 60. Combining our like terms on the left, we'll have 1 x equals 60, so x equals 60 hours. The time it takes to fill the pool with the drain open.