Sure! Let's go through the problem step-by-step to complete the two-way frequency table, and determine the correct number of students enrolled in either algebra or physics, and fill out the remaining fields.
Given data:
- Total students: 30
- Students enrolled in algebra: 19
- Students enrolled in physics: 12
- Students enrolled in both algebra and physics: 4
First, we will complete the table with step-by-step calculations:
1. Identify the number of students taking only algebra. This is done by subtracting the number of students taking both subjects from those taking algebra:
[tex]\[
\text{Only Algebra} = \text{Algebra} - \text{Both} = 19 - 4 = 15
\][/tex]
2. Identify the number of students taking only physics. This is done by subtracting the number of students taking both subjects from those taking physics:
[tex]\[
\text{Only Physics} = \text{Physics} - \text{Both} = 12 - 4 = 8
\][/tex]
3. Determine the number of students taking neither algebra nor physics. This is done by finding the total students minus those enrolled in either or both subjects:
[tex]\[
\text{Neither} = \text{Total Students} - (\text{Only Algebra} + \text{Only Physics} + \text{Both}) = 30 - (15 + 8 + 4) = 30 - 27 = 3
\][/tex]
4. Now we can fill in the rest of the table with our calculated values:
[tex]\[
\begin{tabular}{|r|l|l|l|}
\hline & Algebra & Not in Algebra & Total \\
\hline Physics & 4 & 8 & 12 \\
\hline Not in Physics & 15 & 3 & 18 \\
\hline Total & 19 & 11 & 30 \\
\hline
\end{tabular}
\][/tex]
- For "Not in Algebra" and "Not in Physics," we found that:
[tex]\[
\text{Not in Algebra, Not in Physics} = \text{Total} - (\text{Algebra} + \text{Only Physics}) = 30 - (19 + 8) = 30 - 27 = 3
\][/tex]
Thus, the table now looks like this:
[tex]\[
\begin{tabular}{|r|l|l|l|}
\hline & Algebra & Not in Algebra & Total \\
\hline Physics & 4 & 8 & 12 \\
\hline Not in Physics & 15 & 3 & 18 \\
\hline Total & 19 & 11 & 30 \\
\hline
\end{tabular}
\][/tex]
Finally, the number of students enrolled in either algebra or physics is calculated by adding those taking only algebra, only physics, and both:
[tex]\[
\text{Either Algebra or Physics} = \text{Only Algebra} + \text{Only Physics} + \text{Both} = 15 + 8 + 4 = 27
\][/tex]
So, 27 students are enrolled in either algebra or physics.
