To find the number of carrots needed for 1 cup of water based on the given ratio of 4 carrots for every 2 cups of water, we can set up the proportion.
The ratio of carrots to water can be expressed as:
\[ \frac{4 \text{ carrots}}{2 \text{ cups of water}} \]
To find the number of carrots needed for 1 cup of water, we need to simplify the ratio.
First, we can rewrite the proportion so that we have the carrots per cup of water:
\[ \frac{4 \text{ carrots}}{2 \text{ cups of water}} = \frac{4 \text{ carrots}}{2} \times \frac{1 \text{ cup of water}}{1 \text{ cup of water}} = \frac{4 \text{ carrots}}{2 \text{ cups of water}} \times \frac{1 \text{ cup of water}}{1} \]
This simplifies to:
\[ \frac{4 \text{ carrots}}{2 \text{ cups of water}} = \frac{2 \text{ carrots}}{1 \text{ cup of water}} \]
Thus, to find the expression for the number of carrots needed for 1 cup of water, we can express it as:
\[ \frac{2 \text{ carrots}}{4 \text{ cups of water}} \]
In this format, that would be the third option given which is:
\[ \frac{2 \text{ carrots}}{4 \text{ cups of water}} \]
Therefore, the correct expression is:
\[ \frac{2 \text{ carrots}}{4 \text{ cups of water}} \]