Theory:

In the previous topics, we have learnt how to find the solution of the compound variation using the proportion method or multiplicative factor method or formula method.
 
Now, let us learn about the new concept called person days.
 
What do you mean by person days?
 
Person days is the number of persons working per day times the number of days.
Person days \(=\) Number of person \(\times\) Number of days
Also, remember that if \(x\) women or \(y\) men takes \(p\) days to complete the work, then \(a\) women and \(b\) men can complete the same work in \(\frac{xyp}{xb+ya}\) or \(\frac{p}{\frac{a}{x}+\frac{b}{y}}\) days.
Example:
1. In a construction site, if \(20\) women can complete a work in \(7\) days, how long will it take to complete the same work if \(6\) women left the work?
 
Solution:
 
Given that \(20\) women can complete a work in \(7\) days.
Person days \(=\) Number of persons \(\times\) Number of days
Person days \(= 20 \times 7\) \(= 140\)
 
If \(6\) women left the work, then the number of persons become \(20 - 6 = 14\).
 
Now, substituting the number of persons and person days in the formula, we have:
 
\(140 = 14 \times days\)
 
\(\frac{140}{14} = days\)
 
\(10 = days\)
 
Therefore, it takes \(10\) days to complete the same work.
 
 
2. In a company, if \(7\) women or \(3\) men can complete a work in \(90\) days, then how long will it take for \(8\) women and \(6\) men to complete the same work?
 
Solution:
 
Person days method:
 
Let \(M\) denotes men and \(W\) denotes women.
 
Given that \(7\) women or \(3\) men can complete a work in \(90\) days.
Person days \(=\) Number of persons \(\times\) Number of days
Person days \(= 3M \times 90\)
 
Here, \(7W = 3M\)
 
\(W = \frac{3}{7}M\)
 
Now, we have:
 
Number of persons \(= 8W + 6M = 8(\frac{3}{7}M) + 6M\)
 
\(= \frac{24}{7}M + 6M\)
 
\(= \frac{66}{7}M\)
 
Substituting the known values in the above formula, we have:
 
\(3M \times 90 = \frac{66}{7}M \times days\)
 
\(\frac{3M \times 90}{\frac{66}{7}M} = days\)
 
\(3M \times 90 \times \frac{7}{66M} = days\)
 
\(28.64 = days\)
 
Rounding off, we have \(29\) days.
 
Therefore, it takes \(29\) days to complete a work by \(8\) women and \(6\) men.
 
We can also determine the days using the formula method.
 
Formula method:
 
Let us determine the days using the formula \(\frac{xyp}{xb+ya}\).
 
Here, \(x = 7\), \(y = 3\), \(p = 90\), \(a = 8\), \(b = 6\)
 
Number of days \(= \frac{xyp}{xb+ya}\)
 
\(= \frac{7 \times 3 \times 90}{(7 \times 6)+(3 \times 8)}\)
 
\(= \frac{1890}{42 + 24}\)
 
\(= \frac{1890}{66} = 28.64\)
 
Rounding off, we have \(29\) days.
 
(or)
 
Let us determine the days using the formula \(\frac{p}{\frac{a}{x}+\frac{b}{y}}\).
 
Here, \(x = 7\), \(y = 3\), \(p = 90\), \(a = 8\), \(b = 6\)
 
Number of days \(= \frac{p}{\frac{a}{x}+\frac{b}{y}}\)
 
\(= \frac{90}{\frac{8}{7}+\frac{6}{3}}\)
 
\(= \frac{90}{\frac{24 + 42}{21}}\)
 
\(= \frac{90}{\frac{66}{21}}\)
 
\(= 90 \times \frac{21}{66}\)
 
\(= \frac{1890}{66} = 28.64\)
 
Rounding off, we have \(29\) days.
 
Therefore, it takes \(29\) days to complete a work by \(8\) women and \(6\) men.