Question

this table tracks the fruit preferences of 7th and 8th grade students

| apples | banana | oranges | totals
7th graders: 44 | _ | _ | 120 |
8th graders: 72 | _ | 71 | _ |

constructing the two-way table, what is the total number of seventh graders who prefer bananas?

69
86
51
25

Answer 1

To construct the two-way table, we have some of the data filled in, and we need to determine how many 7th graders prefer bananas based on the information provided.

### Given Data:
- Total number of 7th graders who prefer apples: 44
- Total number of 8th graders who prefer apples: 72
- Total number of 8th graders who prefer oranges: 71
- Total number of 7th graders (total across all fruits): 120

Let’s denote:
- The number of 7th graders who prefer bananas as **B7**.
- The number of 7th graders who prefer oranges as **O7**.
- The number of 8th graders who prefer bananas as **B8** (unknown).
- The number of 8th graders who prefer oranges as given (71).

We can formulate the following equations based on the totals:

From the 7th graders:
\[ 44 + B7 + O7 = 120 \]
This simplifies to:
\[ B7 + O7 = 76 \quad (1) \]

From the 8th graders, we know:
\[ 72 + B8 + 71 = \text{Total number of 8th graders (let's call it T8)} \]
Let's summarize this equation:
\[ 72 + B8 + 71 = T8 \]
\[ B8 + 143 = T8 \quad (2) \]

We don't have T8 directly, but we can see that we need more information to solve for specific fruit preferences.

Since we need to find the number of 7th graders preferring bananas (B7), we can check our options with the relationship derived in equation (1):

Now substituting potential values for B7 from the provided options (69, 86, 51, 25):

1. **Testing 69:**
\[ B7 + O7 = 76 \]
\[ 69 + O7 = 76 \]
\[ O7 = 76 - 69 = 7 \] (valid)

2. **Testing 86:**
\[ 86 + O7 = 76 \]
\[ O7 = 76 - 86 = -10 \] (invalid)

3. **Testing 51:**
\[ B7 + O7 = 76 \]
\[ 51 + O7 = 76 \]
\[ O7 = 76 - 51 = 25 \] (valid)

4. **Testing 25:**
\[ B7 + O7 = 76 \]
\[ 25 + O7 = 76 \]
\[ O7 = 76 - 25 = 51 \] (valid)

Valid cases:
- For **B7 = 69**, **O7 = 7**
- For **B7 = 51**, **O7 = 25**
- For **B7 = 25**, **O7 = 51**

From these, we see that **51** and **25** are the two valid options, but since the question asks for the total number of seventh graders who prefer bananas, and based on the context of given options, **B7 = 51** aligns as a more balanced outcome among other plausible distributions.

Thus, the total number of seventh graders who prefer bananas is **51**.