How many gallons of a solution which is $15\%$ alcohol do we have to mix with a solution that is $35\%$ alcohol to make $250$ gallons of a solution that is $21\%$ alcohol?
1. Let \( x \) be the number of gallons of the solution with \( 15\% \) alcohol.
2. Let \( y \) be the number of gallons of the solution with \( 35\% \) alcohol.
3. We need to mix these to get \( 250 \) gallons of a solution that is \( 21\% \) alcohol.

We can set up the following system of equations based on the given information:
\[
x + y = 250
\]
\[
0.15x + 0.35y = 0.21 \times 250
\]

4. Simplify the second equation:
\[
0.15x + 0.35y = 52.5
\]

5. We now have the system of linear equations:
\[
\begin{cases}
x + y = 250 \\
0.15x + 0.35y = 52.5
\end{cases}
\]

6. Solve the first equation for \( y \):
\[
y = 250 - x
\]

7. Substitute \( y = 250 - x \) into the second equation:
\[
0.15x + 0.35(250 - x) = 52.5
\]

8. Distribute and simplify:
\[
0.15x + 87.5 - 0.35x = 52.5
\]
\[
-0.20x + 87.5 = 52.5
\]

9. Solve for \( x \):
\[
-0.20x = 52.5 - 87.5
\]
\[
-0.20x = -35
\]
\[
x = \frac{-35}{-0.20}
\]
\[
x = 175
\]

10. Substitute \( x = 175 \) back into the equation \( y = 250 - x \):
\[
y = 250 - 175
\]
\[
y = 75
\]

Thus, we need \( 175 \) gallons of the \( 15\% \) alcohol solution and \( 75 \) gallons of the \( 35\% \) alcohol solution.

The final answer is \( \boxed{ 175 } \) gallons of the \( 15\% \) alcohol solution.