How to Solve Algebra Problems of the Form "If One Person Can Paint One House In X Hours..."

A common problem in algebra is of the form, "If one person can point one house in X hours, and another person can paint the same house in Y hours, how long would it take them to paint the house if they worked together?" Such problems are sometimes referred to as work problems.

Most versions of these problems are actually very easy to solve, and involve little more than adding simple fractions and doing very basic arithmetic. However, there is one special tip that is essential to avoid confusion, and for whatever reason, both teachers and textbooks tend to not mention it.

This article shows you what that tip is, and the steps required to solve problems of this form. The article assumes that the reader has all the required prerequisite knowledge.

Difficulty: Moderate

Instructions

1. 
    

   Let's start out right way with the tip mentioned in the introduction. As we extract the information from the word problem, we're going to write it in fraction form. The trick to avoiding confusion is to label your numerators and denominators so that by the time you get to the end of the problem, and end up with a fraction such as 3/17, you'll know what each of those numbers actually represents. In particular, we're going to use our numerators for jobs, and our denominators for hours.

2. 
    

   Here is a typical problem: Rachael can paint one house in four hours. Giada can paint one house in six hours. How long will it take them to paint one house if they work together? Look at the equation at left to see how we set this up. In this case we're going to consider painting one house to be one job. If the task was to assemble 100 widgets, then doing so would be one job, assembling 200 would be two jobs, and so forth.
   
   Rachael's work is represented as 1/4, remembering that we're going to put jobs over hours. Giada's part is represented as 1/6. Since they are going to be working together, we must add those two fractions. It's just as simple as that. We get 5/12 after reducing.

3. 
    

   Here is where most students get very confused. We get an answer of 5/12 which is certainly the sum of those two fractions, but what does it mean? How does it help us answer the question? We always have to read the question carefully because sometimes it's about jobs, and sometimes it's about hours.
   
   In this case they want to know how long one job will take. The significance of our answer is that working together, they can do 5 jobs in 12 hours.
   
   It's good to know that, but it doesn't tell us how long it will take them to do just one job. We need for the numerator to be 1. To accomplish that, we must divide both top and bottom by 5, which is allowed. We end up with 1 over 12/5. That may look awful, but it's OK, and it's correct. Together they can do one job in 12/5 hours. That fraction can be converted into 2 and 2/5, or 2.4, or even 2 hours and 24 minutes, any of which is the right answer depending on the format requested.

4. 
    

   Let's take a fast break to review how to convert a decimal portion of an hour into minutes since it comes up quite a bit. One way is to multiply it by 60 since there are 60 minutes in an hour. If it's a clean number, you can also convert it into a fraction. For example, 0.4 is 2/5 after reducing. Look at the chart at left to see that 1/5 of an hour is 12 minutes, so 2/5 must be 24 minutes.

5. 
    

   It's important to understand that the problem could have also asked us what fraction of the house they would be finished together within one hour. We do the problem exactly the same way, and still end up with 5/12, but this time we need our denominator (hours) to be equal to 1. Divide both top and bottom by 12 which is allowed, giving us 5/12 over 1.
   
   Again, that may look scary, but it just means that together they can do 5/12 of a job in 1 hour. You could leave your answer like that, or if the problem told you to answer in decimal form you could convert 5/12 jobs to approximately 0.42 jobs (a bit less than half the house).

6. 
    

   Here is one more example to look at. Be certain to study the equation at left carefully. This is what we would do if we were told that three people were working on a task together. We just add the three fractions. Note that the equation implies that the second worker can do 2 jobs in 7 hours. That's fine. We won't always be given the information in terms of just one job. Also, the problem won't always ask for an answer in terms of one job or one hour. They could ask us for 2 jobs, or 3 hours, or similar. First get the answer for 1, then multiply as needed. In this example, we got our answer in terms of one job.

7. 
    

   This equation is for the exact same problem as above, but solving it terms of what fraction of the job can be done in one hour, instead of how many hours one full job takes.

8. 
    

   Sometimes you'll be given a problem like this, but instead of the combined work being unknown, you'll be given that, but you won't be given one of the other pieces of the puzzle. Set up the problem just as we did with these problems, but use X to represent any unknown value, and use basic algebra to solve for it.
   
   Do as many of these problems as you can until they become second nature. They will always follow the same pattern. Just remember to put jobs on top and hours on the bottom so that you'll remember what values represent what. ☺

Resources

• Math with Larry (Free Online Math Help)

[cherold598]

on 3/24/2009 Thanks for the how to solve algebra problems of if one person can paint one house in X hours. Those kinds of algebra problems are so scary.