Express the following repeating decimal as a fraction in simplest form.0.342 repeating (line over entire decimal)

Answer:The fraction form of given number is [tex]\frac{38}{111}[/tex].Step-by-step explanation:The given repeating decimal number is[tex]0.\overline{342}[/tex]Let [tex]x=0.\overline{342}[/tex]It can be written as[tex]x=0.342342342...[/tex]The digits repeated after 3 decimal places. So multiply both sides by 1000.[tex]1000x=0.342342342...\times 1000[/tex][tex]1000x=342.342342...[/tex][tex]1000x=342+0.342342...[/tex][tex]1000x=342+0.\overline{342}[/tex][tex]1000x=342+x[/tex]Subtract x from both the sides.[tex]1000x-x=342[/tex][tex]999x=342[/tex]Divide both the sides by 999.[tex]x=\frac{342}{999}[/tex][tex]0.\overline{342}=\frac{342}{999}[/tex]Cancel out the common factors.[tex]0.\overline{342}=\frac{38}{111}[/tex]Therefore the fraction form of given number is [tex]\frac{38}{111}[/tex].