If $(a,b)\ast (c,d)=(a+d)-\frac{c}{b}$, then $\left(\frac{5}{6},\frac{2}{7}\right)\ast \left(\frac{3}{7},\frac{2}{3}\right)$ will be:

The binary operation is defined as:
$$(a,b)\ast (c,d)=(a+d)-\frac{c}{b}$$

Here, the values are:

$$a=\frac{5}{6},b=\frac{2}{7},c=\frac{3}{7},d=\frac{2}{3}$$

Let us compute the components step-by-step:

1. Find $(a+d)$:

$$a+d=\frac{5}{6}+\frac{2}{3}=\frac{5}{6}+\frac{4}{6}=\frac{9}{6}=\frac{3}{2}$$

2. Find $\frac{c}{b}$:

$$\frac{c}{b}=\frac{\frac{3}{7}}{\frac{2}{7}}=\frac{3}{2}$$

3. Substitute these values back into the operation definition:

$$(a,b)\ast (c,d)=(a+d)-\frac{c}{b}=\frac{3}{2}-\frac{3}{2}=0$$