Answer:
[tex]\sf X=\dfrac{2345}{9999}[/tex]
Step-by-step explanation:
Given:
[tex]\sf X=0.234523452345...[/tex]
Therefore, X is a recurring decimal (the decimal numbers 2345 repeat forever).
Converting a recurring decimal to a fraction
Let X equal the recurring decimal:
[tex]\implies \sf X=0.234523452345...[/tex]
Create another number with recurring 2345s by multiplying the above expression by 10000:
[tex]\implies \sf 10000X=2345.23452345...[/tex]
To solve these two equations and write X as a fraction, take away X from 10000X to remove all the recurring decimal places:
[tex]\large\begin{array}{r r l}& \sf 10000X & = \sf 2345.23452345... \\- & \sf X & = \sf \quad \:\:\:0.23452345.... \\\cline{2-3} & \sf 9999X & = \sf 2345\end{array}[/tex]
[tex]\implies \sf 9999X=2345[/tex]
Divide both sides by 9999:
[tex]\implies \sf X=\dfrac{2345}{9999}[/tex]
Therefore, 0.23452345... as a fraction is [tex]\sf \dfrac{2345}{9999}[/tex]
