Given the set $\textbf{S}=\{\frac{1}{2},\frac{1}{4},\frac{1}{6},\frac{1}{8},\frac{1}{10},\frac{1}{12},\frac{1}{14}\}$, find a subset such that the sum of all the elements equals to 1
The answer to this question is relatively easy to obtain, however, I am not happy with my approach. Basically my question is, is there an analytical approach to solve this problem?
My approach:
$\frac{1}{2}$ must be an element of the subset since $\frac{1}{4}+\frac{1}{6}+\frac{1}{8}+\frac{1}{10}+\frac{1}{12}+\frac{1}{14}<1$. Using similar logic I also deduced that $\frac{1}{4}$ must be a member of the subset. However, after this, my approach was literally guesswork. I was lucky however and knew that $\frac{1}{6}+\frac{1}{12}=\frac{1}{4}$ which allowed me to solve the problem. However, the latter approach (the guesswork part) was only successful because the given set was small enough for a brute force approach and I was lucky to know $\frac{1}{6}+\frac{1}{12}=\frac{1}{4}$. If the set was bigger however, and lets say the sum of the elements of the subset had to be 99.5 instead of 1 it would take a lot longer to solve.
Is there a more rigorous way to solve this question?