Question 469305
Let x = numerator and y = denominator



"the denominator of a certain fraction is six times the numerator." means that {{{y = 6x}}}



"if 3 is added to the numerator and subtracted from the denominator, the resulting fraction is equivalent to 2/5" means that {{{(x+3)/(y-3)=2/5}}}



{{{(x+3)/(y-3)=2/5}}} Start with the second equation.



{{{(x+3)/(6x-3)=2/5}}} Plug in {{{y=6x}}}



{{{5(x+3)=2(6x-3)}}} Cross multiply



{{{5x+15=12x-6}}} Distribute.



{{{5x=12x-6-15}}} Subtract {{{15}}} from both sides.



{{{5x-12x=-6-15}}} Subtract {{{12x}}} from both sides.



{{{-7x=-6-15}}} Combine like terms on the left side.



{{{-7x=-21}}} Combine like terms on the right side.



{{{x=(-21)/(-7)}}} Divide both sides by {{{-7}}} to isolate {{{x}}}.



{{{x=3}}} Reduce.



{{{y=6x}}} Go back to the first equation.



{{{y=6(3)}}} Plug in {{{x=3}}}



{{{y=18}}} Multiply



So the original fraction is {{{3/18}}}