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.
\(\frac{1}{x}+\frac{1}{y}=\frac{1}{T}\)
Since, they work at the same rate, it can be rewritten as:
\(\frac{1}{x}+\frac{1}{x}=\frac{1}{T}\)
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
\(\frac{1}{5}\)
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