To solve the problem, we have a system of equations:
- \( q + d = 19 \) (Equation 1)
- \( 0.25q + 0.1d = 4 \) (Equation 2)
We can start by solving Equation 1 for \( d \):
\[ d = 19 - q \]
Next, we substitute \( d \) into Equation 2:
\[ 0.25q + 0.1(19 - q) = 4 \]
Now, distribute \( 0.1 \) in the equation:
\[ 0.25q + 1.9 - 0.1q = 4 \]
Combine like terms:
\[ 0.25q - 0.1q + 1.9 = 4 \ 0.15q + 1.9 = 4 \]
Next, isolate \( q \) by subtracting \( 1.9 \) from both sides:
\[ 0.15q = 4 - 1.9 \ 0.15q = 2.1 \]
Now, divide by \( 0.15 \):
\[ q = \frac{2.1}{0.15} \ q = 14 \]
Thus, Kat has \( \boxed{14} \) quarters.
To confirm, we can find \( d \) using \( d = 19 - q \):
\[ d = 19 - 14 = 5 \]
We can check the total value:
\[ 0.25(14) + 0.1(5) = 3.5 + 0.5 = 4 \]
The values are correct, confirming that Kat has \( \boxed{14} \) quarters.