The Seversk dog sled races involved a total of 315 two-dog and three-dog teams. Within the set time limit, $60\%$ of all two-dog teams and $\frac{1}{3}$ of all three-dog teams reached the finish line, meaning exactly half of all the dogs arrived on time. How many two-dog teams and how many three-dog teams participated in the race?
Let $d$ be the number of two-horse carriages and $t$ be the number of three-horse carriages. From the problem statement, we can set up the system of equations:

$$
\begin{gathered}
d+t=315 \\
0.6 \cdot d \cdot 2+\frac{1}{3} \cdot t \cdot 3=\frac{1}{2}(2 d+3 t)
\end{gathered}
$$

Solving this system, we get $t=90, d=225$.

There were 225 two-horse carriages and 90 three-horse carriages at the start.

[Setting up the equations ... 3 points

solving the system ... 2 points

answer ... 1 point]