To construct the two-way table based on the data you provided, we need to complete the missing values.
- We know the totals for each row and column.
- The total number of 7th graders is given as 120.
- The total number of 8th graders can be calculated as the total for all students minus the total for 7th graders.
First, calculate the total number of students.
From the table provided, the totals for each fruit are:
- Apples: 72
- Bananas: (missing)
- Oranges: 71
- 7th graders: 120
- 8th graders: (missing)
Now, from the totals we can fill in the table.
For 7th graders:
- Apples: 44
- Bananas: (unknown)
- Oranges: (unknown)
- Total: 120
For 8th graders:
- Apples: (unknown)
- Bananas: 35
- Oranges: 46
- Total: (unknown)
Let’s denote the missing counts for 7th graders as:
- Bananas: \( x \)
- Oranges: \( y \)
So the equations based on the table are:
- \( 44 + x + y = 120 \) (for 7th graders)
- \( a + 35 + 46 = \text{Total for 8th graders} \) (where \( a \) is the number of 8th graders who prefer apples)
- Total number of students for oranges: \( 71 = y + 46 \) (for oranges)
Now we can solve the equations step by step:
From equation 3: \[ y + 46 = 71 \] \[ y = 71 - 46 \] \[ y = 25 \]
Now substitute \( y \) back into equation 1: \[ 44 + x + 25 = 120 \] \[ x + 69 = 120 \] \[ x = 120 - 69 \] \[ x = 51 \]
Now we can fill the table with the discovered values:
-
For 7th graders:
- Apples: 44
- Bananas: 51
- Oranges: 25
-
For 8th graders, we can find the missing value for apples: Using equation (2): \[ a + 35 + 46 = \text{Total for 8th} \] So total for 8th graders: \[ 35 + 46 = 81 \] Thus, for apples of the 8th graders: \[ a = \text{Total for 8th} - 81 \] Since we do not have a total number of students yet, we'll just leave it in terms of total for now.
The filled table will then look like this:
\[ \begin{array}{|c|c|c|c|c|} \hline & \text{Apples} & \text{Bananas} & \text{Oranges} & \text{Totals} \ \hline \text{7th Graders} & 44 & 51 & 25 & 120 \ \hline \text{8th Graders} & a & 35 & 46 & 81 \ \hline \text{Totals} & 72 & 86 & 71 & 201 \ \hline \end{array} \]
The total number of 7th graders who prefer bananas is 51.