Simplify the given expression.

​$$11.45\overline{65} \div 2.\overline{67}$$​​

Correct option is C

Given:

​$$11.45\overline{65} \div 2.\overline{67}$$​​

Formula Used:

​$$(a^2 - b^2) = (a+b)(a-b)$$​​

Solution:

Let 11.45656565 = x

Then 100x = 1145.656565…    …(1)

10000x = 114565.656565…     …(2)

Subtracting eq(1) from eq(2)

10000x -100x = (114565.6565… - 1145.656565…)

9900x= 113420

​$$x= \frac{113420}{9900}$$​​

Now let y = 2.676767…

100y = 267.676767…

100y - y = 267.676767… - 2.676767

99y = 265

​$$y= \frac{265}{99}$$ 

Now, 

​$$11.45\overline{65} \div 2.\overline{67}$$​​

​$$\frac{113420}{9900} \div \frac{265}{99}$$​​

​$$\frac{113420}{9900} \times \frac{99}{265}$$​​

​$$= \frac{107}{25} =4\frac{7}{25}$$​​