Short Answer 
The unit rate is a ratio where one quantity is normalized to 1, and to find a unit rate of 3, certain ratios must be simplified. Valid ratios that yield a unit rate of 3 are 2 cups to 2/3 cups and 15/2 cups to 2 1/2 cups.
Step 1: Understand Unit Rate
A unit rate refers to a ratio where one quantity is normalized to 1. To find the unit rate of a ratio, simplify the fraction to see if it can be expressed as ‘x : 1’. This process helps in identifying how many units of one item correspond to one unit of another.
Step 2: Simplify Given Ratios
To determine which ratios have a unit rate of 3, simplify each provided ratio:
- 2 cups : 2/3 cups simplifies to 3 cups : 1 cup (unit rate of 3).
- 1/2 cups : 5/6 cup simplifies to 3/5 cups : 1 cup (not a unit rate of 3).
- 2/3 cup : 1 cup simplifies to 2/3 cup : 1 cup (not a unit rate of 3).
- 15/2 cups : 2 1/2 cups simplifies to 3 cups : 1 cup (unit rate of 3).
- 1 cup : 1/4 cup simplifies to 4 cups : 1 cup (not a unit rate of 3).
- 3 3/4 cups : 2 cups simplifies to 15/4 cups : 2 cups (not a unit rate of 3).
Step 3: Identify Valid Ratios
By comparing the simplified ratios, we can determine which of them result in a unit rate of 3:
- The valid ratios with a unit rate of 3 are: 2 cups : 2/3 cups.
- Another valid ratio is: 15/2 cups : 2 1/2 cups.
This indicates that both the ratios simplify correctly to reach a unit rate of 3.