To solve this question, follow the steps below.
Step 01: Create an equation such that x equals the decimal number.
[tex]x=0.\bar{210}[/tex]
Let this equation be equation 01.
Step 02: Create a second equation by multiplying both sides of equation 01 by 10³.
We multiply both sides by 10³ since the decimal has 3 repeating numbers.
Since 10³ = 1000:
[tex]1000x=210\bar{.210}[/tex]
Let this equation be equation 02.
Step 03: Subtract equation 01 from equation 02.
[tex]\begin{gathered} 1000x=210.210\ldots \\ -\text{ }x\text{ }=\text{ }0.210... \\ _{------------} \\ 1000x-x=210.210\ldots-0.210\ldots \\ 999x=210 \end{gathered}[/tex]
Step 04: Divide both sides of the equation by 999 to find x.
[tex]\begin{gathered} \frac{999}{999}x=\frac{210}{999} \\ x=\frac{210}{999} \end{gathered}[/tex]
Simplify the fraction by dividing both the numerator and the denominator by 3.
[tex]\begin{gathered} x=\frac{\frac{210}{3}}{\frac{999}{3}} \\ x=\frac{70}{333} \end{gathered}[/tex]
Since
[tex]\begin{gathered} x=0.\bar{210} \\ \text{and} \\ x=\frac{70}{333} \\ \text{Then,} \\ 0.\bar{210}=\frac{70}{333} \end{gathered}[/tex]
Answer:
[tex]0.\bar{210}=\frac{70}{333}[/tex]
