To solve this problem, we need to convert the counts in the frequency table to relative frequencies for each preference of biology and chemistry, shown as percentages. We will create a relative frequency table by row, where each cell represents the percentage of students who preferred a specific combination of subjects (Algebra-Biology, Geometry-Biology, Algebra-Chemistry, and Geometry-Chemistry) relative to the total number of students who preferred Biology or Chemistry, respectively.
Starting with Biology:
1. The total number of students who preferred Biology is 92.
2. The number of students who preferred Algebra and Biology is 67.
3. The number of students who preferred Geometry and Biology is 25.
To find the percentage for Algebra-Biology:
[tex]\[ \text{Relative frequency for Algebra-Biology} = \left( \frac{67}{92} \right) \times 100 \approx 72.83\% \][/tex]
To find the percentage for Geometry-Biology:
[tex]\[ \text{Relative frequency for Geometry-Biology} = \left( \frac{25}{92} \right) \times 100 \approx 27.17\% \][/tex]
The total relative frequency for Biology is:
[tex]\[ 72.83\% + 27.17\% = 100.0\% \][/tex]
Next, we calculate the relative frequencies for Chemistry:
1. The total number of students who preferred Chemistry is 66.
2. The number of students who preferred Algebra and Chemistry is 46.
3. The number of students who preferred Geometry and Chemistry is 20.
To find the percentage for Algebra-Chemistry:
[tex]\[ \text{Relative frequency for Algebra-Chemistry} = \left( \frac{46}{66} \right) \times 100 \approx 69.70\% \][/tex]
To find the percentage for Geometry-Chemistry:
[tex]\[ \text{Relative frequency for Geometry-Chemistry} = \left( \frac{20}{66} \right) \times 100 \approx 30.30\% \][/tex]
The total relative frequency for Chemistry is:
[tex]\[ 69.70\% + 30.30\% = 100.0\% \][/tex]
Now we can complete the relative frequency table by row:
\begin{tabular}{|c|c|c|c|}
\hline & Algebra & Geometry & Total \\
\hline Biology & 72.83\% & 27.17\% & 100.0\% \\
\hline Chemistry & 69.70\% & 30.30\% & 100.0\% \\
\hline
\end{tabular}
Let's match the calculated values with the given tiles:
\begin{tabular}{|c|c|c|c|}
\hline 72\% & 27\% & 100\% & 70\% \\
\hline 73\% & 30\% & 100\% & 100\% \\
\hline
\end{tabular}
28\%
The relative frequency table thus becomes:
\begin{tabular}{|c|c|c|c|}
\hline & Algebra & Geometry & Total \\
\hline Biology & 72\% & 28\% & 100\% \\
\hline Chemistry & 70\% & 30\% & 100\% \\
\hline
\end{tabular}
