1. **State the problem:** We need to find the coordinates of the point that is $\frac{3}{10}$ of the way from point $A(0,0)$ to point $B(10,9)$.\n\n2. **Formula used:** To find a point that divides a segment between two points $A(x_1,y_1)$ and $B(x_2,y_2)$ in the ratio $t$ (where $t$ is the fraction of the distance from $A$ to $B$), use the formula:\n$$\left(x, y\right) = \left(x_1 + t(x_2 - x_1), y_1 + t(y_2 - y_1)\right)$$\n\n3. **Apply the formula:** Here, $x_1=0$, $y_1=0$, $x_2=10$, $y_2=9$, and $t=\frac{3}{10}$.\nCalculate the $x$-coordinate:\n$$x = 0 + \frac{3}{10}(10 - 0) = \frac{3}{10} \times 10 = 3$$\nCalculate the $y$-coordinate:\n$$y = 0 + \frac{3}{10}(9 - 0) = \frac{3}{10} \times 9 = 2.7$$\n\n4. **Final answer:** The coordinates of the point $\frac{3}{10}$ of the way from $A$ to $B$ are $\boxed{(3, 2.7)}$.