Which vulgar fraction is equal in value to the mixed recurring decimal
0.3125125125…?

Correct option is D

Given:

0. 3125125125... 

Solution:

Separate the decimal into non-repeating and repeating parts.

The decimal 0. 3125125125.... can be written as: 

0.3 + 0.125125125.....

Here: 

0.3 in non-repeating part.

0.125125125..... is repeating part which has a repeating block 125

Represent the repeating part as a fraction.

Let $$x$$ = 0.125125125.....Then:

$$x$$ = $$0.\overline{125}$$​ 

To eliminate the repeating decimal, divide by 999 in the repeating value. (since the repeating block has 3 digits):

$$x$$ = $$\frac{125}{9990}$$ 

Add the repeating and non-repeating part

Total value is:

0.3 + 0.125125125.... = $$\frac{3}{10}$$ + $$\frac{125}{9990}$$ = $$\frac{2997+125}{9990}$$ = $$\frac{3122}{9990}$$ = $$\frac{1561}{4995}$$ 

Thus the correct answer is (D)

x=12.5999=1259990.x = \frac{12.5}{999} = \frac{125}{9990}.