Which simple fraction is equal to 2.(15) ̅ ?

Correct option is C

Solution:

Step 1: Represent the repeating decimal mathematically
Let:

$$x = 2.\overline{15}$$​​

This means:

$$x = 2.151515\ldots$$​​


Step 2: Separate the integer and repeating parts
We rewrite x as:
​$$x = 2 + 0.\overline{15}$$​​

Let:

$$y = 0.\overline{15}$$​​

So:

x = 2 + y

Step 3: Solve for yLet:
​$$y = 0.\overline{15}$$​​

Multiply y by 100 (since the repeating part has 2 digits):

$$100y = 15.\overline{15}$$​​

Subtract the original y from this equation:

$$100y - y = 15.\overline{15} - 0.\overline{15}$$​​

99y = 15

Solve for y:

$$y = \frac{15}{99}$$​​


Simplify the fraction:
​$$y = \frac{5}{33}$$​​


Step 4: Substitute y back into x
​$$x = 2 + y = 2 + \frac{5}{33}$$​​


Convert to a single fraction:
​$$x = \frac{66}{33} + \frac{5}{33} = \frac{71}{33}$$​​