With distribution:
[tex]\begin{gathered} 6(\frac{1}{2}+\frac{2}{3}+\frac{3}{4}) \\ =(6)\frac{1}{2}+(6)\frac{2}{3}+(6)\frac{3}{4} \\ =\frac{6}{2}+\frac{12}{3}+\frac{18}{4} \\ =\frac{(6\cdot6)+(4\cdot12)+(3\cdot18)}{12} \\ =\frac{36+48+54}{12} \\ =\frac{138}{12} \\ =\frac{23}{2} \end{gathered}[/tex]Without distribution:
[tex]\begin{gathered} 6(\frac{1}{2}+\frac{2}{3}+\frac{3}{4}) \\ =6\lbrack\frac{(6\cdot1)+(4\cdot2)+(3\cdot3)}{12}\rbrack \\ =6(\frac{6+8+9}{12}) \\ =6(\frac{23}{12}) \\ =\frac{6\cdot23}{12} \\ =\frac{23}{2} \end{gathered}[/tex]It's easier to solve it without distribution because we're dealing with smaller numbers if we're not distributing 6 accross the terms. This makes carrying out the math more simple since we're looking at smaller numbers.