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The problem you've shared is a classic set theory question involving a survey and Venn diagrams. Let's summarize the given information:

- 208 members enjoy aerobic exercises.
- 145 members enjoy weight training.
- 97 members enjoy both aerobic exercises and weight training.
- 135 members do not enjoy either of these exercises.

We need to find the total number of members surveyed.

### Approach:

We'll use the inclusion-exclusion principle to calculate the number of members who enjoy either aerobic exercises, weight training, or both.

Let:
- \( A \) represent the set of members who enjoy aerobic exercises.
- \( W \) represent the set of members who enjoy weight training.

From the inclusion-exclusion principle:
\[
|A \cup W| = |A| + |W| - |A \cap W|
\]
Where:
- \( |A| = 208 \) (those who enjoy aerobics),
- \( |W| = 145 \) (those who enjoy weight training),
- \( |A \cap W| = 97 \) (those who enjoy both).

Now, substituting the values:
\[
|A \cup W| = 208 + 145 - 97 = 256
\]
Thus, 256 members enjoy either aerobic exercises, weight training, or both.

Finally, adding the 135 members who do not enjoy either type of exercise, we get the total number of members surveyed:
\[
\text{Total members} = 256 + 135 = 391
\]

### Final Answer:
The total number of members surveyed is **391**.

Would you like further details or have any questions?

Here are five related questions to expand your understanding:
1. How would the result change if only 50 members enjoyed both activities?
2. Can you represent this data in a Venn diagram?
3. What if the number of people who enjoy neither exercise was unknown? How would you solve for it?
4. What percentage of members enjoy both types of exercises?
5. How would the formula change if there were a third type of exercise?

**Tip**: Always visualize set problems with a Venn diagram. It helps break down overlapping groups clearly.

208 members indicated that they enjoy aerobic exercises, 145 indicated that they enjoy weight training, 97 indicated that they enjoy both aerobics and weight training, and 135 indicated that they do not enjoy either of these types of exercise. How many members were surveyed?

Solve a set theory problem using the inclusion-exclusion principle. Find how many members were surveyed at a health club where 208 enjoy aerobic exercises, 145 enjoy weight training, and 97 enjoy both. Learn how to apply key formulas and break down overlapping groups with a step-by-step solution.

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