Which of the following fractions is greater than $\frac{3}{4}$ and less than $\frac{5}{6}$?

Let's convert the given fractions into decimal values:
* $\frac{3}{4}=0.75$
* $\frac{5}{6}\approx 0.833$

We need to find a fraction whose value lies between $0.75$ and $0.833$:

* Option A: $\frac{2}{3}\approx 0.667$ (less than 0.75)
* Option B: $\frac{1}{2}=0.50$ (less than 0.75)
* Option C: $\frac{4}{5}=0.80$ (lies between 0.75 and 0.833)
* Option D: $\frac{9}{10}=0.90$ (greater than 0.833)

Therefore, $\frac{4}{5}$ satisfies the condition.