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

\(\frac{1}{5}\)

.