(a) an integer (b) a rational number (c) a natural number (d) an irrational number

We have,

$3 . \overline{27}=3.27272727 \ldots$

Let 

$x=3.27272727 \ldots$

Then,

$100 x=327.272727 \ldots$

$10000 x=32727.2727 \ldots$

Subtract these to get

$9900 x=32400$

$x=\frac{32400}{9900}$

$x=\frac{324}{99}$

Thus, we can also conclude that all infinite repeating decimals are rational numbers.

Hence, the correct choice is (b).