This is an example of a "work" problem. Typically there is a task that will be completed in some unit of time. The short-cut here is to think in terms of fractions. So, each person can do a portion (fraction) of work in a portion (fraction) of time. The first step is always the hardest with these problems, but once you master work problems will be forever easier. Let's actually write out the first part of this problem.
Since we're dealing with portions, each sentence will be written as such:
1/L + 1/P + 1/M = 1/10 (Lew, Pau, Mends together complete a job in 10 days)
1/L + 1/P = 1/12 (Lew and Pau finish in 12 days)
1/L + 1/M = 1/20 (Lew and Mends finish in 20 days)
Once we've expressed the problem mathematically, the solution is simple substitution and solving. Since we're solving for Lew (L) alone, we need to rewrite the bottom two equations for Pau and Mends to substitute into the original equation (this will yield all variables in the first equation as a statement of L):
1/P = 1/12 - 1/L and,
1/M = 1/20 - 1/L
With these equations, we substitute into the main equation:
1/L + (1/12 - 1/L) + (1/20 - 1/L) = 1/10 (parentheses there to show substitution)
1/L + 1/12 - 1/L + 1/20 - 1/L = 1/10 (removed parentheses)
1/L + 1/12 - 1/L + 1/20 - 1/L = 1/10
1/12 + 1/20 - 1/L = 1/10
To solve for L, we need to find the common denomenator. With this example, it is 60.
5/60 + 3/60 - 1/L = 6/60 (This equation is equal the previous one)
8/60 - 1/L = 6/60
(-8/60) +8/60 - 1/L = 6/60 (-8/60)
- 1/L = -2/60
1/L = 2/60 (Multiply each side by -1 to have positive numbers)
1/L = 1/30 (Simplify)
So, Lew's portion of the work is that alone it would take 30 days to complete.
*You can rewrite the equation and substitute again to find each person's portion for PRACTICE. So how long would it take each person to complete this work on their own?
I've provided the answers so you can check your work:
Lew = 30 days
Pau = 20 days
Mends = 60 days