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:

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.