Edvin records pet adoptions at an animal shelter. He notices that for every 3

To determine the relationship between dog adoptions (d) and cat adoptions (c) given the ratio of 3 dog adoptions for every 2 cat adoptions, we can set up the following proportion:

\[
\frac{d}{c} = \frac{3}{2}
\]

From this, we can derive equations that express c in terms of d.

1. **Starting from the proportion:**
\[
2d = 3c
\]
Rearranging gives:
\[
c = \frac{2}{3}d
\]

This means that for every 3 dog adoptions, we have 2 cat adoptions.

Now, let's examine the provided options:

1. **c = 3d**: This implies 3 cat adoptions for every dog adoption and is **incorrect**.
2. **\(\frac{2}{3}d = c\)**: Reflects \(c = \frac{2}{3}d\), which is **correct**.
3. **c = 2d**: This implies 2 cat adoptions for every dog adoption and is **incorrect**.
4. **\(\frac{c}{2} = \frac{d}{3}\)**: This can be rearranged to confirm the ratio (3c = 2d), so it is **correct**.
5. **d = \frac{2}{3}c**: This correctly reflects the inverse relationship based on the original ratio (3 dog adoptions for every 2 cat adoptions), so it is **correct**.

Thus, the correct equations that describe the relationship between cat adoptions (c) and dog adoptions (d) based on the given information are:

- \(\frac{2}{3}d = c\)
- \(\frac{c}{2} = \frac{d}{3}\)
- \(d = \frac{2}{3}c\)