A soccer coach bought 3 times as many T-shirts as soccer balls. Each soccer ball cost $$ \$ 11.50 $$ more than each T-shirt. He paid a total of $$ \$ 166.50 $$. He paid $$ \$ 31.50 $$ more for all the T-shirts than all the soccer balls. How much did the coach pay for a soccer ball?

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Let the number of T-shirts be $$ T $$ and the number of soccer balls be $$ B $$. According to the problem, we have the following relationships:

1. The coach bought 3 times as many T-shirts as soccer balls:
   $$
   T = 3B
   $$

2. Let the cost of each T-shirt be $$ x $$ and the cost of each soccer ball be $$ y $$. We know that:
   $$
   y = x + 11.50
   $$

3. The total cost for T-shirts and soccer balls is:
   $$
   Tx + By = 166.50
   $$

4. The total cost for T-shirts is $$ Tx $$ and for soccer balls is $$ By $$. We also know that:
   $$
   Tx = By + 31.50
   $$

Now, substituting $$ T = 3B $$ into the total cost equation:
$$
3Bx + By = 166.50
$$
Factoring out $$ B $$:
$$
B(3x + y) = 166.50
$$

Next, substituting $$ y = x + 11.50 $$ into the equation:
$$
B(3x + (x + 11.50)) = 166.50
$$
This simplifies to:
$$
B(4x + 11.50) = 166.50
$$

Now, we can express $$ Tx $$ in terms of $$ By $$:
From $$ Tx = By + 31.50 $$:
$$
3Bx = By + 31.50
$$
Substituting $$ y = x + 11.50 $$:
$$
3Bx = B(x + 11.50) + 31.50
$$
Expanding the right side:
$$
3Bx = Bx + 11.50B + 31.50
$$
Rearranging gives:
$$
3Bx - Bx = 11.50B + 31.50
$$
$$
2Bx = 11.50B + 31.50
$$
Dividing both sides by $$ B $$ (assuming $$ B \neq 0 $$):
$$
2x = 11.50 + \frac{31.50}{B}
$$
Thus:
$$
x = \frac{11.50}{2} + \frac{31.50}{2B}
$$
$$
x = 5.75 + \frac{15.75}{B}
$$

Now substituting $$ x $$ back into the total cost equation:
$$
B(4(5.75 + \frac{15.75}{B}) + 11.50) = 166.50
$$
Calculating $$ 4(5.75) + 11.50 $$:
$$
4(5.75) = 23.00
$$
So:
$$
23.00 + 11.50 = 34.50
$$
Thus:
$$
B(34.50 + 63.00/B) = 166.50
$$
This simplifies to:
$$
34.50B + 63.00 = 166.50
$$
Subtracting $$ 63.00 $$ from both sides:
$$
34.50B = 103.50
$$
Dividing by $$ 34.50 $$:
$$
B = 3
$$

Now substituting $$ B = 3 $$ back to find $$ T $$:
$$
T = 3B = 3 \times 3 = 9
$$

Now substituting $$ B = 3 $$ into the equation for $$ y $$:
Using $$ B(4x + 11.50) = 166.50 $$:
$$
3(4x + 11.50) = 166.50
$$
Dividing by 3:
$$
4x + 11.50 = 55.50
$$
Subtracting $$ 11.50 $$:
$$
4x = 44.00
$$
Dividing by 4:
$$
x = 11.00
$$

Now substituting $$ x $$ back to find $$ y $$:
$$
y = x + 11.50 = 11.00 + 11.50 = 22.50
$$

Thus, the coach paid $$ \boxed{22.50} $$ for each soccer ball.
The coach paid $22.50 for each soccer ball.

A soccer coach bought 3 times as many T-shirts as soccer balls. Each soccer ball cost \( \$ 11.50 \) more than each T-shirt. He paid a total of \( \$ 166.50 \). He paid \( \$ 31.50 \) more for all the T-shirts than all the soccer balls. How much did the coach pay for a soccer ball?

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