The largest fraction among the given options is:

To find the largest fraction, let's convert each fraction option into its decimal equivalent form:
* Option A: $\frac{9}{7}\approx 1.2857$
* Option B: $\frac{4}{3}\approx 1.3333$
* Option C: $\frac{3}{2}=1.5000$
* Option D: $\frac{5}{6}\approx 0.8333$

Comparing all decimal values:

$$1.5000>1.3333>1.2857>0.8333$$

Therefore, $\frac{3}{2}$ is the largest fraction.