Use the given information to fill in the data missing from the table.

We surveyed 134134 students at PCC. We asked if they were taking either Math or Spanish. Here are the results:

1313 were taking both classes.


6464​ were taking neither class.


Only 2929 were taking Math.

Taking MathNot taking MathTotals

Taking Spanish

a*a*

b*b*

c*c*

Not taking Spanish

d*d*

e*e*

f*f*

Totals

g*g*

h*h*

i*i*
 
Find a*a*​, b*b*​, c*c*​, d*d*​, e*e*​, f*f*​, g*g*​, h*h*​, and i*i*​.

a=*a*=people

Preview




b=*b*=people

Preview




c=*c*=people

Preview




d=*d*=people

Preview




e=*e*=people

Preview




f=*f*=people

Preview




g=*g*=people

Preview




h=*h*=people

Preview




i=*i*=people

Preview
 

To solve the problem, we'll use a Venn diagram approach, assigning the provided data to specific parts of the table. The table represents students at PCC who are either taking Math, Spanish, both, or neither. We’ll find the missing values step by step.

### Given Data:
- Total students surveyed: \( i = 134 \)
- Students taking both Math and Spanish: \( a = 13 \)
- Students taking neither class: \( e = 64 \)
- Students only taking Math: \( d = 29 \)

### Step 1: Fill in the table
We'll first start with the totals for the rows and columns.

1. **Total number of students who are not taking Spanish (including those not taking Math):** 
   \[
   d + e = \text{Not taking Spanish total}
   \]
   Given \( d = 29 \) and \( e = 64 \):
   \[
   f = 29 + 64 = 93
   \]

2. **Total number of students who are taking Math (either taking or not taking Spanish):**
   \[
   a + d = \text{Taking Math total}
   \]
   Since \( a = 13 \) and \( d = 29 \):
   \[
   g = 13 + 29 = 42
   \]

3. **Total number of students who are taking Spanish (either taking or not taking Math):**
   \[
   a + b = \text{Taking Spanish total}
   \]
   Given \( i = 134 \) and knowing that \( e = 64 \), we subtract to find \( c \), the total number of students taking either Math or Spanish:
   \[
   c = 134 - 64 = 70
   \]
   Now, \( c \) is also the total number of students taking Math or Spanish, so:
   \[
   g + b = 70
   \]
   Given \( g = 42 \):
   \[
   b = 70 - 42 = 28
   \]

4. **Fill in remaining totals using the above information:**
   - **\( h = \) Total number of students not taking Math:** 
     \[
     h = 134 - g = 134 - 42 = 92
     \]
   - **\( f = h = 92 \)**: Already calculated.

### Step 2: Calculate the missing individual cell values
Finally, we calculate the values for each individual section:
- **\( a \): Students taking both Math and Spanish:** \( a = 13 \)
- **\( b \): Students only taking Spanish:** \( b = 28 \)
- **\( c = a + b = 13 + 28 = 41 \):** Total taking Spanish
- **\( d \): Students only taking Math:** \( d = 29 \)
- **\( e = 64 \):** Students taking neither Math nor Spanish
- **\( f = 93 \):** Total not taking Spanish (already calculated)
- **\( g = 42 \):** Total taking Math
- **\( h = 92 \):** Total not taking Math
- **\( i = 134 \):** Total students surveyed

### Final Table:
|                   | Taking Math | Not taking Math | Totals    |
|-------------------|-------------|-----------------|-----------|
| **Taking Spanish**    | \( a = 13 \)  | \( b = 28 \)       | \( c = 41 \)  |
| **Not taking Spanish**| \( d = 29 \)  | \( e = 64 \)       | \( f = 93 \)  |
| **Totals**            | \( g = 42 \)  | \( h = 92 \)       | \( i = 134 \) |

Would you like more details, or do you have any questions?

---

1. How would the calculations change if 20 students took both subjects instead of 13?
2. What percentage of students are taking only Spanish?
3. How many students are taking either Math or Spanish but not both?
4. What is the ratio of students taking Math to those taking Spanish?
5. How would the totals be affected if the number of students surveyed increased to 150?

**Tip:** Always cross-check your calculations to avoid errors in data interpretation.

Learn how to solve a set theory problem using Venn diagrams based on a survey of PCC students. Understand concepts like intersection, union, and complement sets. Suitable for high school students.

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