To find the correct equation that represents the given number sentence "Two more than one-third of a number is 8," let's break down the sentence step-by-step.
1. Identify what the sentence is stating:
- "One-third of a number": Let the number be [tex]\( n \)[/tex]. One-third of [tex]\( n \)[/tex] is [tex]\(\frac{1}{3}n\)[/tex].
- "Two more than one-third of a number": This means we add 2 to [tex]\(\frac{1}{3}n\)[/tex], resulting in [tex]\( 2 + \frac{1}{3}n \)[/tex].
- "is 8": This translates to an equality, with the entire expression on the left being equal to 8.
2. Combine these parts into an equation:
- We have [tex]\( 2 + \frac{1}{3}n = 8 \)[/tex].
Now, let’s compare this resulting equation with the given options:
1. [tex]\( \left(2 + \frac{1}{3}\right)n = 8 \)[/tex]
- This equation suggests multiplying the sum of 2 and [tex]\(\frac{1}{3}\)[/tex] by [tex]\( n\)[/tex], which doesn't match our derived equation.
2. [tex]\( 2 = \frac{1}{3}n + 8 \)[/tex]
- This equation implies that 2 is equal to the result of [tex]\(\frac{1}{3}n\)[/tex] plus 8, which also doesn't match our derived equation.
3. [tex]\( 2 + \frac{1}{3}n + 8 = 0 \)[/tex]
- This equation adds all terms and sets them equal to 0, which again is not equivalent to our derived equation.
4. [tex]\( 2 + \frac{1}{3}n = 8 \)[/tex]
- This exactly matches our derived equation and correctly represents the sentence "Two more than one-third of a number is 8".
Therefore, the correct equation that represents the given number sentence is:
[tex]\[ 2 + \frac{1}{3}n = 8 \][/tex]
The correct answer is:
[tex]\[ \boxed{4} \][/tex]
