To determine the total number of 7th graders who prefer bananas, we need to carefully fill in the missing values in the two-way table using the given information.
Here's the initial table format based on the provided data:
[tex]\[
\begin{array}{|l|c|c|c|c|}
\hline
& \text{Apples} & \text{Bananas} & \text{Oranges} & \text{Totals} \\
\hline
\text{7th graders} & 44 & & & 120 \\
\hline
\text{8th graders} & & 35 & 46 & \\
\hline
\text{Totals} & 72 & & 71 & \\
\hline
\end{array}
\][/tex]
Here are the step-by-step considerations:
1. First, note the totals for each fruit preference.
- Total Apples: 72
- Total Oranges: 71
2. Given that 44 7th graders prefer apples, we can find the number of 8th graders who prefer apples by subtracting:
[tex]\[
72 \, (\text{Total Apples}) - 44 \, (\text{7th graders preferring apples}) = 28 \, (\text{8th graders preferring apples})
\][/tex]
3. Similarly, we can find the number of 7th graders who prefer oranges by subtracting:
[tex]\[
71 \, (\text{Total Oranges}) - 46 \, (\text{8th graders preferring oranges}) = 25 \, (\text{7th graders preferring oranges})
\][/tex]
4. We now know the number of 7th graders who prefer apples (44) and oranges (25). The total number of 7th graders is 120. So, the number of 7th graders who prefer bananas can be calculated by subtracting the number of those who prefer apples and oranges from the total number of 7th graders:
[tex]\[
120 \, (\text{Total 7th graders}) - 44 \, (\text{7th graders preferring apples}) - 25 \, (\text{7th graders preferring oranges}) = 51 \, (\text{7th graders preferring bananas})
\][/tex]
Thus, the total number of 7th graders who prefer bananas is [tex]\( \boxed{51} \)[/tex].
