Certainly! Let's dissect the given math sentence step by step to write the appropriate equation.
### Analyzing the Given Math Sentence:
"One half times the difference of twenty and a variable is [tex]\(\frac{2}{3}\)[/tex]".
- "One half times": This phrase suggests we will be multiplying by [tex]\(\frac{1}{2}\)[/tex].
- "the difference of twenty and a variable": The order is critical here. "Difference of twenty and a variable" means we should subtract the variable from 20, i.e., [tex]\(20 - \text{variable}\)[/tex].
- "is [tex]\(\frac{2}{3}\)[/tex]": This indicates the resulting expression equals [tex]\(\frac{2}{3}\)[/tex].
### Formulating the Equation:
Putting it all together, we interpret the math sentence into a mathematical equation:
[tex]\[
\frac{1}{2} \left( 20 - \text{variable} \right) = \frac{2}{3}
\][/tex]
We see that the "variable" is typically represented by a letter such as [tex]\(y\)[/tex].
So, the equation becomes:
[tex]\[
\frac{1}{2} (20 - y) = \frac{2}{3}
\][/tex]
### Matching with the Choices:
Let's compare this derived equation with each multiple-choice option provided:
1. [tex]\(\frac{1}{2} (y - 20) = \frac{2}{3}\)[/tex]
- This suggests the difference of [tex]\(y\)[/tex] and 20, which is the opposite of what is needed.
2. [tex]\(\frac{1}{2} (20 - y) = \frac{2}{3}\)[/tex]
- This exactly matches the derived equation.
3. [tex]\(\frac{1}{2}y - 20 = \frac{2}{3}\)[/tex]
- This suggests subtracting 20 from [tex]\(\frac{1}{2}y\)[/tex], which is not the desired interpretation.
4. [tex]\(\frac{1}{2}(20) - y = \frac{2}{3}\)[/tex]
- This suggests half of 20 minus [tex]\(y\)[/tex], which is incorrect as per the given sentence structure.
### Conclusion:
The choice that matches the formulated equation is:
[tex]\[
\frac{1}{2} (20 - y) = \frac{2}{3}
\][/tex]
Thus, the correct equation for the given math sentence is:
[tex]\(
\boxed{\frac{1}{2} (20 - y) = \frac{2}{3}}
\)[/tex]
