Certainly! Let's break down the problem and analyze each given expression to determine which one correctly represents Jonah's earnings, given the statement that Jonah earns [tex]$5 more than half of Karen's salary \( k \).
Firstly, let's interpret the statement:
1. Half of Karen's Salary:
If Karen's salary is represented by \( k \), then half of Karen's salary is \( \frac{1}{2}k \).
2. Adding $[/tex]5 to Half of Karen's Salary:
Jonah earns [tex]$5 more than this half. Therefore, we need to add $[/tex]5 to [tex]\( \frac{1}{2}k \)[/tex].
So, the expression for Jonah's earnings becomes:
[tex]\[ \frac{1}{2}k + 5 \][/tex]
Now, let's evaluate each of the given expressions to see if they correspond to the derived expression [tex]\( \frac{1}{2}k + 5 \)[/tex].
1. [tex]\(\frac{1}{2}(k+5)\)[/tex]:
This expression means half of the amount [tex]\( k + 5 \)[/tex]. Let's evaluate it:
[tex]\[ \frac{1}{2}(k+5) = \frac{1}{2}k + \frac{1}{2}\times5 = \frac{1}{2}k + 2.5 \][/tex]
This does not match the desired expression [tex]\( \frac{1}{2}k + 5 \)[/tex].
2. [tex]\(\frac{1}{2}k + 5\)[/tex]:
This expression is the same as our derived expression for Jonah's earnings. Here, we have half of Karen's salary, [tex]\( \frac{1}{2}k \)[/tex], plus [tex]$5.
3. \(\frac{1}{2} + k + 5\):
This expression means adding \( \frac{1}{2} \) to Karen's salary plus an additional $[/tex]5. Let's combine the terms:
[tex]\[ \frac{1}{2} + k + 5 = k + 5.5 \][/tex]
This is not equivalent to our desired expression [tex]\( \frac{1}{2}k + 5 \)[/tex].
4. [tex]\(\frac{1}{2}k > 5\)[/tex]:
This expression is an inequality, stating that half of Karen's salary is more than [tex]$5. This does not represent Jonah's earnings.
Thus, the only correct expression that represents Jonah's earnings, given that he earns $[/tex]5 more than half of Karen's salary [tex]\( k \)[/tex], is:
[tex]\[ \frac{1}{2}k + 5 \][/tex]
Therefore, the correct expression from the given choices is:
[tex]\[ \boxed{\frac{1}{2}k + 5} \][/tex]
