The given proportion is:

We use the first two parts of the proportion:

$$\frac{3}{7} = \frac{27}{A}$$

To solve for A, we can cross-multiply:

$$3 \times A = 27 \times 7$$ $$3A = 189$$

Divide both sides by 3:

$$A = \frac{189}{3}$$ $$A = 63$$

We use the first and third parts of the proportion:

$$\frac{3}{7} = \frac{B}{105}$$

To solve for B, we can cross-multiply:

$$3 \times 105 = 7 \times B$$ $$315 = 7B$$

Divide both sides by 7:

$$B = \frac{315}{7}$$ $$B = 45$$

So, the values are $A = 63$ and $B = 45$.

Comparing this with the given options:
(A) A = 9, B = 45
(B) A = 9, B = 15
(C) A = 63, B = 45
(D) A = 63, B = 15

The correct option is (C).

The final answer is $\boxed{\text{C}}$