$\frac{1}{log_{3}60}+\frac{1}{log_{4}60}+\frac{1}{log_{5}60}$
=$\frac{1}{log_{3}3+log_{3}4+log_{3}5}+\frac{1}{log_{4}3+log_{4}4+log_{4}5}+\frac{1}{log_{5}3+log_{5}4+log_{5}5}$
=$\frac{1}{1+log_{3}4+log_{3}5}+\frac{1}{log_{4}3+1+log_{4}5}+\frac{1}{log_{5}3+log_{5}4+1}$
=$\frac{1}{1+\frac{log4}{log3}+\frac{log5}{log3}}+\frac{1}{\frac{log3}{log4}+1+\frac{log5}{log4}}+\frac{1}{\frac{log3}{log5}+\frac{log4}{log5}+1}$
=$\frac{log3}{log3+log4+log5}+\frac{log4}{log3+log4+log5}+\frac{log5}{log3+log4+log5}$
=$\frac{log3+log4+log5}{log3+log4+log5}$
=$1$
Option B.