This is the first part of the Solving Working Problems Series. In this post, we are going to discuss in details the basics of work problems.
Work problems usually involve the time for two or more persons or machines to complete the same job given the rate that they can work. For discussion purposes, let us have the following example.
Ariel can paint a house in 5 days and Ben can do the same job in 6 days. In how many days can they complete the job if they work together?
Discussion and Scratch Work
If Ariel can finish the job in 5 days, then if he were to work one day, he would have completed 1/5 of the job. If he works for two days, then he would have completed 2/5 of the job. Similarly, if Ben can finish the same job in 6 days, if he were to work for one day, then he would have completed 1/6 of the job. If he works for 2 days, he would have completed 2/6 of the job (or 1/3 of the job if reduced to lowest terms).
Suppose Ariel and Ben work together starting on a Monday. Then their progress can be described as shown in the table below.
Now, since Ariel can finish the job in 5 days, with Ben working with him even at a slower rate, the job can be finished less than 5 days (working together makes the completion shorter!). Looking at their progress in the table below, it is quite clear that the job can be finished in less than 3 days (can you see why?). Examine the table and see why this is so.
The Geometric Representation of Working Together
The rates of work of Ariel and Ben can be represented geometrically by comparing them to a length of a rectangle. The length of the red rectangle (Ariel’s rate per day) is 1/6 the length the white rectangle, while the length of the blue rectangle (Ben’s rate per day) is 1/5 the length of the white rectangle. Since we used the white rectangle as our unit of measure, its length is equal to 1. In this problem, 1 represents the completed job.
By the end of Monday (see next figure), the combined work done is shown as the length of the two colored rectangles. So, we can say that the number these pairs that can fit horizontally into the white rectangle is the number of days that the job will be completed.
By the end Tuesday, both of them have worked 1/5(2) + 1/6(2) as shown in the figure below, more than half of the job.
Since the pair of blue and red rectangle represents one day, this means that the job can be completed in less than 3 days if they both work together.
The Tabular Representation of Working Together
In the first part of this post, I have mentioned that from the table, it can be seen that the job can be completed in less than 3 days. Why? Because if you can see, 3/6 is already half of the job (50%), while 3/5 is more than half (60%). This means that on the third day, they would have completed 110% of the job. Therefore, the job will take less than 3 days.
The 110% is confirmed by adding the fractions. By the end of the Wednesday, they would have completed 33/30 which is equal to 110%.
Notice that both the tabular and geometric interpretation only gave us an approximation. This is why we need Algebra to solve this type of problem.
So, how many days will it take to finish the job if they work together?
The Algebraic Representation of Working Together
We can form the equation using the table above. If they both work for one day, then they have worked . If they both worked for two days, they have finished of the job. Since we are looking for the number of days that they have worked together, which we represent with days, we can form the expression . And since we are looking for the complete job, we will equate the expression with 1. That is,
Finalizing the Solution
Now, we have an equation with fractions. As we have learned in solving equations with fractions, we need to eliminate the denominators. To eliminate the denominators of the fractions, we multiply everything with the least common denominator of 1/5 and 1/6 which is 30. That is,
Again this confirms the tabular and geometric representations that the job can be completed in less than 3 days.
In the next post, we will have more examples.