Let's examine the relationship between Gregory's and Amanda's ages to determine the correct equation.
1. Understand the problem:
- Gregory’s age is given as 5 years greater than one-third of Amanda's age.
- Let Amanda’s age be represented by the variable [tex]\( x \)[/tex].
2. Express Gregory's age in terms of Amanda's age:
- One-third of Amanda’s age is [tex]\( \frac{1}{3} x \)[/tex].
- Gregory's age is 5 years greater than this amount, so Gregory's age can be written as:
[tex]\[
\frac{1}{3} x + 5
\][/tex]
3. Analyze the given equations:
- We are provided with two possible equations:
[tex]\[
A. \quad \frac{1}{3}(x + 5)
\][/tex]
[tex]\[
B. \quad \frac{1}{3} x + 5
\][/tex]
4. Compare the given equations with our expression for Gregory's age:
- Option A: [tex]\(\frac{1}{3}(x + 5)\)[/tex] simplifies to [tex]\(\frac{1}{3} x + \frac{5}{3}\)[/tex], which is not in line with our expression [tex]\(\frac{1}{3} x + 5\)[/tex].
- Option B: [tex]\(\frac{1}{3} x + 5\)[/tex] matches exactly with our expression for Gregory’s age.
5. Conclusion:
- The correct equation that represents Gregory’s age is option B:
[tex]\[
\frac{1}{3} x + 5
\][/tex]
Thus, the correct answer is B.
