Question

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 50 students, it is found that 39 are taking algebra, 16 are taking physics, and 7 are enrolled in both. How many students are enrolled in either algebra or physics?


Algebra Not in Algebra Total
Physics 7 16
Not in Physics
Total 39 50
41
2
48
7

Answer 1

To complete the two-way frequency table and find the number of students enrolled in either algebra or physics, we can break down the information provided.

From the sample of 50 students:
- Students taking Algebra (A) = 39
- Students taking Physics (P) = 16
- Students taking both Algebra and Physics (A ∩ P) = 7

### Step 1: Calculate the number of students taking only Algebra and only Physics

- Students taking only Algebra (A only) = Students taking Algebra - Students taking both
\[
A \text{ only} = 39 - 7 = 32
\]

- Students taking only Physics (P only) = Students taking Physics - Students taking both
\[
P \text{ only} = 16 - 7 = 9
\]

### Step 2: Fill in the table

Now we can complete the table with the calculated values:

\[
\begin{array}{|c|c|c|}
\hline
& \textbf{Algebra} & \textbf{Not in Algebra} & \textbf{Total} \\ \hline
\textbf{Physics} & 7 & 9 & 16 \\ \hline
\textbf{Not in Physics} & 32 & 2 & 34 \\ \hline
\textbf{Total} & 39 & 11 & 50 \\ \hline
\end{array}
\]

### Breakdown:
- Total students = 50
- Students "not in Algebra" (total - algebra students) = 50 - 39 = 11
- Students "not in Physics" (total - physics students) = 50 - 16 = 34

The numbers we used for students:
- Algebra and Physics (7)
- Only Physics (9)
- Only Algebra (32)
- Not in Algebra and Not in Physics = 50 - (students in Algebra + students in Physics - students in both) = 50 - (39 + 16 - 7) = 50 - 48 = 2.

### Step 3: Calculate the number of students enrolled in either Algebra or Physics

To find the number of students enrolled in **either** algebra or physics, we can use the principle of inclusion-exclusion:

\[
\text{Total in either Algebra or Physics} = \text{Students in Algebra} + \text{Students in Physics} - \text{Students in both}
\]

So we have:
\[
= 39 + 16 - 7 = 48
\]

Thus, the number of students enrolled in **either** Algebra or Physics is **48**.