# \ #
# "One neat way to do this is the following:" #
# "Let:" #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad x = .3777777777... \qquad \qquad \qquad \qquad \qquad \ "(Equation 1)" #
# "Multipy both sides by" \ 10, "and then move the decimal point one place to the right:" #
# \qquad \qquad \qquad \qquad \qquad \qquad \quad \ 10 x = 10 \cdot (.3777777777... ) #
# \qquad \qquad \qquad \qquad \qquad \qquad \quad \ 10 x = 3.777777777... \qquad \qquad \qquad \qquad \qquad \ \ "(Equation 2)" #
# "Subtract (Equation 1) from (Equation 2):" #
# \ 10 x - x = 3.777777777... - .3777777777... \quad \ "(Eqn 2) - (Eqn 1)" #
# \qquad \qquad \ \ \ 9 x = 3.777777777... - .3777777777... #
# \qquad \qquad \qquad \quad \ = \qquad \quad " " \ 3.777777777... #
# \qquad \qquad \qquad \qquad \qquad \quad \ \ \ - \ \ \ \ .3777777777... #
# \qquad \qquad \qquad \quad \ = \qquad " " \ \ \ 3.400000000... #
# \qquad \qquad \qquad \quad \ = \qquad \quad " " \ 3.4 #
# "So we conclude:" #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad\qquad \qquad 9 x = 3.4 #
# "Multiply both sides by 10:" #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad 10 (9 x) = 10 \cdot (3.4) #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ 90 x = 34 #
# "This is easy to solve (!!):" #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad x = 34/90. #
# "Simplify to lowest terms:" #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad x = 17/45. #
# "Remembering what" \ \ x \ \ "is, by (Equation 1), we have:" #
# \qquad \qquad \qquad \qquad \qquad \qquad \quad \quad 3.777777777... = 17/45. #
# "This is our answer." #
# \ #
# "Summarizing:" #
# \qquad \qquad \qquad \qquad \qquad \qquad \quad \quad 3.777777777... = 17/45 \quad. #
