In a survey of 600 investors, it was reported that 380 had invested in stocks, 325 had 

invested in bonds, and 75 had not invested in either stocks or bonds.

a. How many investors had invested in both stocks and bonds?

b. How many investors had only invested in stocks?

Let's break down the problem step by step using set theory.

### Definitions:
- Total number of investors = 600
- Investors who invested in stocks (denoted as \( S \)) = 380
- Investors who invested in bonds (denoted as \( B \)) = 325
- Investors who did not invest in either stocks or bonds = 75

### Step 1: Calculate the number of investors who invested in either stocks or bonds or both.
- The number of investors who invested in either stocks, bonds, or both can be calculated by subtracting those who invested in neither from the total number of investors:
  \[
  \text{Investors in stocks or bonds or both} = 600 - 75 = 525
  \]
  
### Step 2: Use the inclusion-exclusion principle.
The inclusion-exclusion principle states that for two sets \( S \) (stocks) and \( B \) (bonds):
\[
|S \cup B| = |S| + |B| - |S \cap B|
\]
Where:
- \( |S \cup B| \) is the number of investors in either stocks, bonds, or both (525),
- \( |S| \) is the number of investors in stocks (380),
- \( |B| \) is the number of investors in bonds (325),
- \( |S \cap B| \) is the number of investors in both stocks and bonds, which we need to find.

Substitute the known values:
\[
525 = 380 + 325 - |S \cap B|
\]
\[
525 = 705 - |S \cap B|
\]
\[
|S \cap B| = 705 - 525 = 180
\]

### a. **Number of investors who invested in both stocks and bonds**: \( 180 \).

### Step 3: Calculate the number of investors who only invested in stocks.
The number of investors who only invested in stocks is the total number of investors in stocks minus those who invested in both stocks and bonds:
\[
\text{Only in stocks} = |S| - |S \cap B| = 380 - 180 = 200
\]

### b. **Number of investors who only invested in stocks**: \( 200 \).

---

Would you like more details or have any questions on the calculations?

Here are 5 related questions to expand your understanding:
1. How would the result change if 100 investors had not invested in either stocks or bonds?
2. What is the number of investors who only invested in bonds?
3. If 50 additional investors joined who invested in both stocks and bonds, how would that affect the calculations?
4. What percentage of the total investors had invested in both stocks and bonds?
5. How can you represent this problem using a Venn diagram?

**Tip:** When using the inclusion-exclusion principle, remember to subtract the overlap (both) to avoid double-counting in problems with multiple sets.

In a survey of 600 investors, it was reported that 380 had invested in stocks, 325 had invested in bonds, and 75 had not invested in either stocks or bonds. How many investors had invested in both stocks and bonds? How many investors had only invested in stocks?

This problem involves 600 investors, where 380 invested in stocks and 325 in bonds. Learn how to apply the inclusion-exclusion principle to determine how many invested in both or only in stocks, with a clear step-by-step explanation.

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