Answer:
Let x equal the recurring decimal:
[tex]\sf Equation\:1: \quad x=0.216216216...[/tex]
Create another number with recurring 216s by multiplying the above expression by 1000:
[tex]\sf Equation\:2: \quad 1000x=216.216216...[/tex]
Subtract Equation 1 from Equation 2 to eliminate the recurring digits after the decimal:
[tex]\sf \implies 1000x-x=216.216216...-0.216216...[/tex]
[tex]\sf \implies 999x=216[/tex]
Divide both sides by 999:
[tex]\sf \implies \dfrac{999x}{999}=\dfrac{216}{999}[/tex]
[tex]\sf \implies x=\dfrac{216}{999}[/tex]
Simplify:
[tex]\sf \implies x=\dfrac{216 \div 27}{999 \div 27}=\dfrac{8}{37}[/tex]
