The correct answer is $\frac{{{\text{GS}}}}{{{\text{G}} + {\text{S}}}}$.
A galvanometer is a device that measures electric current. It has a very small resistance, so it can be damaged by large currents. A shunt is a resistor that is connected in parallel with a galvanometer to reduce its resistance. The resistance of the shunt is much smaller than the resistance of the galvanometer, so most of the current flows through the shunt. The current through the galvanometer is then proportional to the current through the shunt.
The resistance of the resulting ammeter is given by the following equation:
$$R = \frac{{{\text{GS}}}}{{{\text{G}} + {\text{S}}}}$$
where $R$ is the resistance of the ammeter, $G$ is the resistance of the galvanometer, and $S$ is the resistance of the shunt.
Option A is the correct answer. Option B is incorrect because the resistance of the ammeter is not equal to the sum of the resistance of the galvanometer and the resistance of the shunt. Option C is incorrect because the resistance of the ammeter is not equal to the difference between the resistance of the galvanometer and the resistance of the shunt. Option D is incorrect because there is a correct answer.