(5) Mrs Li baked cookies for sale at a school event. \( \frac{3}{4} \) of the cookies were chocolate cookies and the rest were vanilla cookies. She sold \( \frac{1}{2} \) of the total number of cookies. \( \frac{5}{6} \) of the cookies sold were chocolate cookies. There were 18 vanilla cookies left. How many chocolate cookies were not sold?

Let the total number of cookies Mrs. Li baked be \( x \). According to the problem, \( \frac{3}{4} \) of the cookies were chocolate cookies, and the rest were vanilla cookies. Therefore, the number of chocolate cookies is: \[ \text{Chocolate cookies} = \frac{3}{4}x \] The number of vanilla cookies is: \[ \text{Vanilla cookies} = x - \frac{3}{4}x = \frac{1}{4}x \] Mrs. Li sold \( \frac{1}{2} \) of the total number of cookies, which is: \[ \text{Cookies sold} = \frac{1}{2}x \] Out of the cookies sold, \( \frac{5}{6} \) were chocolate cookies. Therefore, the number of chocolate cookies sold is: \[ \text{Chocolate cookies sold} = \frac{5}{6} \times \frac{1}{2}x = \frac{5}{12}x \] The number of vanilla cookies sold is then: \[ \text{Vanilla cookies sold} = \text{Cookies sold} - \text{Chocolate cookies sold} = \frac{1}{2}x - \frac{5}{12}x \] To perform this subtraction, we need a common denominator. The common denominator of 2 and 12 is 12. Thus, we convert \( \frac{1}{2}x \) to twelfths: \[ \frac{1}{2}x = \frac{6}{12}x \] Now we can subtract: \[ \text{Vanilla cookies sold} = \frac{6}{12}x - \frac{5}{12}x = \frac{1}{12}x \] We know from the problem that there were 18 vanilla cookies left. The total number of vanilla cookies was \( \frac{1}{4}x \), and the number of vanilla cookies sold was \( \frac{1}{12}x \). Therefore, the number of vanilla cookies left can be expressed as: \[ \text{Vanilla cookies left} = \text{Total vanilla cookies} - \text{Vanilla cookies sold} \] Substituting the expressions we have: \[ 18 = \frac{1}{4}x - \frac{1}{12}x \] To combine these fractions, we need a common denominator, which is 12: \[ \frac{1}{4}x = \frac{3}{12}x \] Now we can rewrite the equation: \[ 18 = \frac{3}{12}x - \frac{1}{12}x = \frac{2}{12}x = \frac{1}{6}x \] To solve for \( x \), we multiply both sides by 6: \[ x = 18 \times 6 = 108 \] Now that we have the total number of cookies, we can find the number of chocolate cookies: \[ \text{Chocolate cookies} = \frac{3}{4}x = \frac{3}{4} \times 108 = 81 \] Next, we find the number of chocolate cookies sold: \[ \text{Chocolate cookies sold} = \frac{5}{12}x = \frac{5}{12} \times 108 = 45 \] Finally, the number of chocolate cookies that were not sold is: \[ \text{Chocolate cookies not sold} = \text{Total chocolate cookies} - \text{Chocolate cookies sold} = 81 - 45 = 36 \] Thus, the number of chocolate cookies that were not sold is: \[ \boxed{36} \]