Let's start by defining the variables and setting up the equation based on the information given:
- Let [tex]\( c \)[/tex] be the total number of pieces of cake.
- Mindy ate 3 pieces of cake.
- Troy ate [tex]\(\frac{1}{4}\)[/tex] of the total cake, which is [tex]\(\frac{1}{4}c\)[/tex] pieces.
We are told that Mindy and Troy combined ate 9 pieces of cake. Thus, we can write the following equation:
[tex]\[ 3 + \frac{1}{4}c = 9 \][/tex]
Now, let's solve this equation step-by-step to find the total number of pieces of cake, [tex]\( c \)[/tex].
1. Start by isolating the term with [tex]\( c \)[/tex]. Subtract 3 from both sides of the equation:
[tex]\[ \frac{1}{4}c = 9 - 3 \][/tex]
2. Simplify the right side:
[tex]\[ \frac{1}{4}c = 6 \][/tex]
3. To isolate [tex]\( c \)[/tex], multiply both sides of the equation by 4:
[tex]\[ c = 6 \times 4 \][/tex]
4. Perform the multiplication:
[tex]\[ c = 24 \][/tex]
So, the total number of pieces of cake is [tex]\( \boxed{24} \)[/tex].
