Complete the two-way frequency table below, which shows the relationship between students who enroll in advanced algebra and physics in a particular high school. From a sample of 30 students, it is found that 19 are taking algebra, 12 are taking physics, and 4 are enrolled in both. How many students are enrolled in either algebra or physics?

\begin{tabular}{|r|l|l|l|}
\hline & Algebra & Not in Algebra & Total \\
\hline Physics & 4 & & 12 \\
\hline Not in Physics & & & \\
\hline Total & 19 & & 30 \\
\hline
\end{tabular}

A. 27
B. 23
C. 3
D. 4



Answer :

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.