Shared Work Word Problems
By Robert O
Rates of doing work differ from person to person. Even if they are machines, their rate of doing work is slightly different even if they look very identical. Think of taps filling a tank. We can’t assume that both can take the same time filling the tank. That is because we don’t know about the water pressure from the source.
In many cases, you will work with other people or have employees. You will always have a feeling that your colleagues are not performing as you or some of your employees are lazing around. Because of that, the shared work word problems always exist.
How do I solve shared work word problems?
Fortunately, there is a formula that makes the entire process of solving shared work problems easy. The formula is:
![\[\frac{T}{a} \pm \frac{T}{b}=1\]](data:image/svg+xml;nitro-empty-id=MjUwOjE2MTY=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgODUgMzciIHdpZHRoPSI4NSIgaGVpZ2h0PSIzNyIgeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzIwMDAvc3ZnIj48L3N2Zz4= "Rendered by QuickLaTeX.com")
Where: T is the total time when two or more people or machines are working together, and a and b are the times each takes when working alone.
To understand how this formula works, we will solve example problems to represent any other shared work problem that you will come across.
Example 1
Two painters can paint a house in 12 hours when working together. If one painter can take 18 hours painting the house alone, how long would it take the other painter to paint the same house when working alone?
Solution
The work is shared between two painters and both are contributing positively to finish the work. We, therefore, use the formula:
![\[\frac{T}{a}+\frac{T}{b}=1\]](data:image/svg+xml;nitro-empty-id=MjUwOjI4NDQ=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgODUgMzciIHdpZHRoPSI4NSIgaGVpZ2h0PSIzNyIgeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzIwMDAvc3ZnIj48L3N2Zz4= "Rendered by QuickLaTeX.com")
By substitution:
![\[\frac{12}{18}+\frac{12}{b}=1\]](data:image/svg+xml;nitro-empty-id=MjUwOjMzMTU=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgOTUgMzYiIHdpZHRoPSI5NSIgaGVpZ2h0PSIzNiIgeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzIwMDAvc3ZnIj48L3N2Zz4= "Rendered by QuickLaTeX.com")
We simplify the fraction by multiplying every term by the common denominator, which is 18b, and solve for b.
![\[18 b \times \frac{12}{18}+\frac{12}{b} \times 18 b=1 \times 18 b\]](data:image/svg+xml;nitro-empty-id=MjUwOjM5MTU=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMjM4IDM2IiB3aWR0aD0iMjM4IiBoZWlnaHQ9IjM2IiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
![\[12 b+216=18 b\]](data:image/svg+xml;nitro-empty-id=MjUwOjQzNDk=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMTIyIDE0IiB3aWR0aD0iMTIyIiBoZWlnaHQ9IjE0IiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
![\[b=36\]](data:image/svg+xml;nitro-empty-id=MjUwOjQ3NzM=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgNTAgMTIiIHdpZHRoPSI1MCIgaGVpZ2h0PSIxMiIgeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzIwMDAvc3ZnIj48L3N2Zz4= "Rendered by QuickLaTeX.com")
Answer the question:
The other painter would take 36 hours to paint the house when working alone.
Example 2
Faith, Mercy, and Mary can clean a room in 2hours, 5 hours, and 3hours, respectively. How long would it take to clean the room when the three girls are working together?
Solution
The work involves three people who are both contributing positively to complete the task. So, the formula becomes:
![\[\frac{T}{a}+\frac{T}{b}+\frac{T}{c}=1\]](data:image/svg+xml;nitro-empty-id=MjUwOjU3Nzc=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMTIzIDM3IiB3aWR0aD0iMTIzIiBoZWlnaHQ9IjM3IiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
By substitution:
![\[\frac{T}{2}+\frac{T}{5}+\frac{T}{3}=1\]](data:image/svg+xml;nitro-empty-id=MjUwOjYyNTg=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMTIzIDM3IiB3aWR0aD0iMTIzIiBoZWlnaHQ9IjM3IiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
Find the LCM of the denominators and multiply each term by it. The LCM for 2, 5, and 3 is 30. The result is:
![\[30 \times \frac{T}{2}+30 \times \frac{T}{5}+30 \times \frac{T}{3}=1 \times 30\]](data:image/svg+xml;nitro-empty-id=MjUwOjY4NzE=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMjg0IDM3IiB3aWR0aD0iMjg0IiBoZWlnaHQ9IjM3IiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
![\[15 T+6 T+10 T=30\]](data:image/svg+xml;nitro-empty-id=MjUwOjczMDg=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMTY3IDE1IiB3aWR0aD0iMTY3IiBoZWlnaHQ9IjE1IiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
![\[31 T=30\]](data:image/svg+xml;nitro-empty-id=MjUwOjc3MzU=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgNzIgMTIiIHdpZHRoPSI3MiIgaGVpZ2h0PSIxMiIgeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzIwMDAvc3ZnIj48L3N2Zz4= "Rendered by QuickLaTeX.com")
![\[T=\frac{30}{31}\]](data:image/svg+xml;nitro-empty-id=MjUwOjgxNzA=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgNTYgMzYiIHdpZHRoPSI1NiIgaGVpZ2h0PSIzNiIgeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzIwMDAvc3ZnIj48L3N2Zz4= "Rendered by QuickLaTeX.com")
Answer the question:
It takes 30/31 hours to clean the room when all the three girls are working together.
Example 3
Tap A takes 5 hours to fill an overhead water tank and tap B takes 8 hours to empty the same tank. If both taps are accidentally opened at the same time, find the time it would take the two taps to fill the tank?
Solution
This problem is not as direct as the previous two. Here, we have two taps working together. Tap A is contributing positively while tap B is contributing negatively. From this statement, the formula becomes:
![\[\frac{T}{a}-\frac{T}{b}=1\]](data:image/svg+xml;nitro-empty-id=MjUwOjkzMDU=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgODUgMzciIHdpZHRoPSI4NSIgaGVpZ2h0PSIzNyIgeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzIwMDAvc3ZnIj48L3N2Zz4= "Rendered by QuickLaTeX.com")
By substitution:
![\[\frac{T}{5}-\frac{T}{8}=1\]](data:image/svg+xml;nitro-empty-id=MjUwOjk3NzM=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgODUgMzciIHdpZHRoPSI4NSIgaGVpZ2h0PSIzNyIgeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzIwMDAvc3ZnIj48L3N2Zz4= "Rendered by QuickLaTeX.com")
![\[40 \times \frac{T}{5}-\frac{T}{8} \times 40=1 \times 40\]](data:image/svg+xml;nitro-empty-id=MjUwOjEwMjQ5-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMjA2IDM3IiB3aWR0aD0iMjA2IiBoZWlnaHQ9IjM3IiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
![\[8 T-5 T=40\]](data:image/svg+xml;nitro-empty-id=MjUwOjEwNjgw-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMTA2IDEzIiB3aWR0aD0iMTA2IiBoZWlnaHQ9IjEzIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
![\[T=\frac{40}{3}=13 \frac{1}{3}\]](data:image/svg+xml;nitro-empty-id=MjUwOjExMTMw-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMTEwIDM2IiB3aWR0aD0iMTEwIiBoZWlnaHQ9IjM2IiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
Answer the question:
It takes both taps
![\[13 \frac{1}{3}\]](data:image/svg+xml;nitro-empty-id=MjUwOjExNjEw-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMjcgMzYiIHdpZHRoPSIyNyIgaGVpZ2h0PSIzNiIgeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzIwMDAvc3ZnIj48L3N2Zz4= "Rendered by QuickLaTeX.com")
Remarks
Regardless of the nature of a shared work problem, you can always find the solution by applying the right form of the formula. It also does not matter how many people or machines are sharing the work. Just extend the formula to accommodate every contributor.
About the Author
This lesson was prepared by Robert O. He holds a Bachelor of Engineering (B.Eng.) degree in Electrical and electronics engineering. He is a career teacher and headed the department of languages and assumed various leadership roles. He writes for Full Potential Learning Academy.