We are given: $P(A\cup B)=\frac34, \quad P(A\cap B)=\frac14, \quad P(\overline A)=\frac23$

First find $P(A)$: $P(A)=1-P(\overline A)=1-\frac23=\frac13$

Use the formula for union: $P(A\cup B)=P(A)+P(B)-P(A\cap B)$ Substituting the values: $\frac34=\frac13+P(B)-\frac14$

Simplify: $\frac13-\frac14=\frac{4-3}{12}=\frac1{12}$ So, $\frac34=P(B)+\frac1{12}$ $P(B)=\frac34-\frac1{12}=\frac{9-1}{12}=\frac8{12}=\frac23$

Now, $P(\overline A\cap B)=P(B)-P(A\cap B)$ since event $B$ is split into two disjoint parts: $B=(A\cap B)\cup(\overline A\cap B)$

Therefore, $P(\overline A\cap B)=\frac23-\frac14$ $=\frac{8-3}{12}=\frac5{12}$

Hence the correct option is: $\boxed{\frac5{12}}$ which is Option A.