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Assuming the pattern continues, let [tex]x=2.3181818\ldots=2.3\overline{18}[/tex].

Then [tex]10x=23.\overline{18}[/tex] and [tex]1000x=2318.\overline{18}[/tex], and we have

[tex]1000x-10x=990x=2295\implies x=\dfrac{2295}{990}=\dfrac{51}{22}[/tex]

Since [tex]51=2\cdot22+7[/tex], we can write [tex]x[/tex] as the mixed number [tex]2+\dfrac7{22}[/tex].

Answer:

x =  [tex]\frac{51}{22}[/tex].

Step-by-step explanation:

Given  :  2.3181818...

To find : write as a mixed number.

Solution : We have given that

2.3181818...

Let x = 2.3181818...

We can see the number 18 is repeating 18181818......

So, the bar line is over the two digit 18

[tex]x = 2.318.....[/tex].

On multiplying the number both side by 100.

[tex]100x = 100 * 2.318.....[/tex].

100x = 231.81818.....

We can write the number 231.81818..... in term of x

100x = 229 .5 + x

On subtracting both sides by x

100x -x = 229.5

99x = 229.5

On dividing both sides by 99.

x = [tex]\frac{229.5}{99}[/tex].

x =  [tex]\frac{2295}{990}[/tex].

x =  [tex]\frac{459}{198}[/tex].

x =  [tex]\frac{51}{22}[/tex].

Therefore, x =  [tex]\frac{51}{22}[/tex].

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