Let's work through this step-by-step:
1. Determine the fractions involved:
- From the problem, [tex]$\frac{3}{4}$[/tex] of the cookies were chocolate cookies.
- Therefore, [tex]$\frac{1}{4}$[/tex] of the cookies were vanilla cookies.
2. Define the variables:
Let [tex]\( T \)[/tex] be the total number of cookies baked.
- Chocolate cookies: [tex]\( \frac{3}{4}T \)[/tex]
- Vanilla cookies: [tex]\( \frac{1}{4}T \)[/tex]
3. Relate the vanilla cookies to the total cookies:
- We're given that 18 vanilla cookies were left.
- Knowing Mrs. Li sold [tex]$\frac{1}{2}$[/tex] of the total cookies, she also kept [tex]$\frac{1}{2}$[/tex] of the total cookies. Thus, the remaining vanilla cookies are half of the originally baked vanilla cookies.
4. Set up the equation for vanilla cookies:
[tex]\[
\text{Vanilla cookies left} = \left( \text{Vanilla cookies originally baked} \right) \times \left(\text{Fraction of unsold cookies}\right)
\][/tex]
Given there were 18 vanilla cookies left:
[tex]\[
18 = \left( \frac{1}{4}T \right) \times \left( \frac{1}{2} \right)
\][/tex]
5. Solve for [tex]\( T \)[/tex]:
[tex]\[
18 = \frac{1}{8}T
\][/tex]
[tex]\[
T = 18 \times 8
\][/tex]
[tex]\[
T = 144
\][/tex]
6. Determine the number of chocolate cookies:
[tex]\[
\text{Chocolate cookies} = \frac{3}{4}T = \frac{3}{4} \times 144 = 108
\][/tex]
7. Determine the number of cookies sold:
[tex]\[
\text{Total cookies sold} = \frac{1}{2}T = \frac{1}{2} \times 144 = 72
\][/tex]
8. Determine how many of the sold cookies were chocolate:
[tex]\[
\text{Chocolate cookies sold} = \frac{5}{6} \times \text{Total cookies sold} = \frac{5}{6} \times 72 = 60
\][/tex]
9. Determine how many chocolate cookies were not sold:
[tex]\[
\text{Chocolate cookies not sold} = \text{Total chocolate cookies} - \text{Chocolate cookies sold}
\][/tex]
[tex]\[
\text{Chocolate cookies not sold} = 108 - 60 = 48
\][/tex]
Therefore, the number of chocolate cookies that were not sold is [tex]\( 48 \)[/tex].
