To determine how much water Henry will need if he uses [tex]\(1\frac{1}{2}\)[/tex] cups of corn grits, we can follow these steps:
1. Understanding the Ratio:
The recipe specifies that for every [tex]\(\frac{1}{4}\)[/tex] cup of corn grits, [tex]\(\frac{1}{2}\)[/tex] cup of water is required.
2. Corn Grits to be Used:
Henry plans to use [tex]\(1\frac{1}{2}\)[/tex] cups of corn grits. In decimal form, [tex]\(1\frac{1}{2}\)[/tex] cups is [tex]\(1.5\)[/tex] cups.
3. Setting Up a Proportion:
The goal is to find out how much water is needed if 1.5 cups of corn grits are used, keeping in mind the given ratio:
[tex]\[
\frac{\text{corn grits}}{\text{water}} = \frac{1}{4} \text{ cup of corn grits} \Rightarrow \frac{\text{water}}{\text{corn grits}} = \frac{1}{2} \text{ cup of water}
\][/tex]
This relationship indicates that:
[tex]\[
\frac{\frac{1}{2} \text{ cup of water}}{\frac{1}{4} \text{ cup of corn grits}} = \text{constant ratio}
\][/tex]
4. Using the Ratio:
Since 1.5 cups of corn grits are used, and knowing that the water-to-corn grits ratio is constant, we can set up a proportion to find out how much water is required.
[tex]\[
\frac{\frac{1}{2} \text{ cup of water}}{\frac{1}{4} \text{ cup of corn grits}} = \frac{x \text{ cups of water}}{1.5 \text{ cups of corn grits}}
\][/tex]
Simplifying the ratio:
[tex]\[
\frac{1/2}{1/4} = 2
\][/tex]
implies the amount of water is 2 times the amount of corn grits used. Thus:
[tex]\[
x = 1.5 \times 2
\][/tex]
5. Calculate the Amount of Water:
Multiplying through:
[tex]\[
x = 1.5 \times 2 = 3.0
\][/tex]
Therefore, Henry will need [tex]\(\boxed{3.0}\)[/tex] cups of water if he uses [tex]\(1\frac{1}{2}\)[/tex] cups of corn grits.
