Solving problem involving intersection of sets

Question

In a certain office,one-quarter of the staff are left-handed. One-twelfth of them are left-handed and short-sighted; 13 are short sighted while 17 are neither left handed nor short-sighted.Find the number of staff in the office.

Answer 1

n(X) = Staff are left-handed.
n(Y) = Staff are short-sighted.
n(X∪Y) = Number of staff in the office under left-handed and short-sighted.
n(X∩Y) = Both are left-handed and short-sighted in the office.
n(U) = Total number of staff in the office.   
n(X∪Y)' = Neither left-handed and short-sighted in the office.
Then,
n(X∪Y) = n(X)+n(Y)-n(X∩Y)
n(X∪Y) = n(1/4)+n(13)-n(1/12)
n(X∪Y) = n{(1+52)/4}-n(1/12)
n(X∪Y) = n(53/4)-n(1/12)
n(X∪Y) = n{(159-1)/12}
n(X∪Y) = n(158/12)          
n(X∪Y) = n(79/6)                              
Then,
n(X∪Y)' = n(U)-n(X∪Y)
n(17) = n(U)-n(79/6)
n(17)+n(79/6) = n(U)
n{(102+79)/6} = n(U)
n(181/6) = n(U)