I would like to have a suggestion on how to solve the following problem. I have tried using linear equation with no success. I mainly interested to understand the logic behind the solution.
A man bequeaths to his wife 1/3 of his estate: to his daughter 1/5 of it; to his son, 1/2 of the daughter's share: he divide the renaming equally between a hospital and a public library. What part is received by the hospital?
The solution is: 11/60
So you start with the whole estate: $1$. Now you subtract the portion given to his wife to get $1-1/3$ remaining.
Now you subtract the portion given to his daughter from what was remaining after giving away a share to his wife. $1-1/3-1/5$.
How much was given to the son? Half of what was given to the daughter, so this is $\frac{\frac{1}{5}}{2}=\frac{1}{10}$. Subtracting this from what was leftover after giving away the daughter's share gives $1-1/3-1/5-1/10$ remaining.
Let's compute the fraction that is this remaining share. We rewrite everything so that the denominator is 30 (we use 30 because it is the least common multiple of 1, 3, 5, and 10): $\frac{30}{30}-\frac{10}{30}-\frac{6}{30}-\frac{3}{30}=\frac{30-10-6-3}{30}=\frac{11}{30}$.
Half of this is given to the hospital, so the hospital's share is $\frac{\frac{11}{30}}{2}=\frac{11}{60}$