One morning a baker notices that she has $188$ cups of flour and $113$ cups of sugar available. Each loaf of bread that the baker makes takes three cups of flour and a half cup of sugar. Each cake that the baker makes takes two cups of flour and two cups of sugar. The baker decides to make some loaves of bread and some cakes so that she exactly uses up all of her supplies of flour and sugar. Find the number of cakes she should make.
1. Let \( c \) be the number of cakes and \( b \) be the number of loaves of bread.
2. Each loaf of bread requires 3 cups of flour and 0.5 cups of sugar.
3. Each cake requires 2 cups of flour and 2 cups of sugar.
4. We are given that the baker has 188 cups of flour and 113 cups of sugar.

We can set up the following system of linear equations based on the given information:
\[
3b + 2c = 188 \quad \text{(Equation 1: Flour)}
\]
\[
\frac{1}{2}b + 2c = 113 \quad \text{(Equation 2: Sugar)}
\]

5. To eliminate \( c \), we can subtract Equation 2 from Equation 1. First, we need to clear the fraction in Equation 2 by multiplying through by 2:
\[
b + 4c = 226 \quad \text{(Equation 3: Sugar, cleared of fractions)}
\]

6. Now, subtract Equation 1 from Equation 3:
\[
(b + 4c) - (3b + 2c) = 226 - 188
\]
\[
b + 4c - 3b - 2c = 38
\]
\[
-2b + 2c = 38
\]
\[
-2b + 2c = 38 \implies 2c - 2b = 38 \implies c - b = 19 \quad \text{(Equation 4)}
\]

7. From Equation 4, we can express \( c \) in terms of \( b \):
\[
c = b + 19
\]

8. Substitute \( c = b + 19 \) back into Equation 1:
\[
3b + 2(b + 19) = 188
\]
\[
3b + 2b + 38 = 188
\]
\[
5b + 38 = 188
\]
\[
5b = 150
\]
\[
b = 30
\]

9. Substitute \( b = 30 \) back into the expression for \( c \):
\[
c = b + 19
\]
\[
c = 30 + 19
\]
\[
c = 49
\]

The final answer is \( \boxed{49} \).