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John, Robert, and Joseph can complete a job in \(3\frac{1}{3}\) days if they work together and work at the same rate. How much of the job could Robert and Joseph do in one day?

 Jun 14, 2018

Rate of work problems use the formula:
\(\frac{1}{x}+\frac{1}{y}=\frac{1}{T}\); where x = time to complete work for 1st thing, y = time to complete work for 2nd thing, T = time to complete work for the 1st and 2nd thing working together. Also, 1/x = rate of work for thing 1, 1/y = rate of work for thing 2, 1/T = rate of work of thing 1 and 2 together.


Since all of the people work at the same rate and we have the time it takes all of them working together, we can just write:
\(\frac{1}{x}+\frac{1}{x}+\frac{1}{x}=\frac{1}{\frac{10}{3}}\\ \frac{3}{x}=\frac{3}{10}\\ x=10\)

In this case, x=10 means it takes each of them 10 days individually to complete the work (T). It also means, they complete 3/10 of the work in one day (1/T).


To find the rate of only Robert and Joseph, we use the formula I mentioned in the beginning.

Since, they work at the same rate, it can be rewritten as:

We know that it takes each of them 10 days individually to complete the work, so x = 10

\(\frac{1}{10}+\frac{1}{10}=\frac{1}{T}\\ \frac{2}{10}=\frac{1}{T}\\ 2T=10\\ T=5\)

It takes them both 5 days to complete the work.
1/T gives how much they can do in one day, which is

 Jun 14, 2018

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