In a survey of 800 investors, it was reported that 479 had invested in stocks, 422 had invested in bonds, and 100 had not invested in either stocks or bonds.
(a) How many investors had invested in both stocks and bonds?

(b) How many investors had invested only in stocks?

To solve this problem, we can use the principle of inclusion-exclusion.

### Given:
- Total number of investors = 800
- Number of investors who invested in stocks = 479
- Number of investors who invested in bonds = 422
- Number of investors who did not invest in either = 100

### Step 1: Find the number of investors who invested in either stocks or bonds (or both).
Since 100 investors did not invest in either stocks or bonds, the number of investors who invested in either stocks or bonds (or both) is:
\[
800 - 100 = 700
\]

### Step 2: Use the formula for inclusion-exclusion to find the number of investors who invested in both stocks and bonds.

Let:
- \( S \) = Number of investors who invested in stocks
- \( B \) = Number of investors who invested in bonds
- \( S \cap B \) = Number of investors who invested in both stocks and bonds

The formula for the number of investors who invested in either stocks or bonds (or both) is given by:
\[
S + B - (S \cap B) = 700
\]
Substituting the values:
\[
479 + 422 - (S \cap B) = 700
\]
Simplifying this equation:
\[
901 - (S \cap B) = 700
\]
\[
(S \cap B) = 901 - 700 = 201
\]

Thus, **201 investors had invested in both stocks and bonds**.

### Step 3: Find the number of investors who invested only in stocks.

The number of investors who invested only in stocks is given by:
\[
S - (S \cap B) = 479 - 201 = 278
\]

Thus, **278 investors had invested only in stocks**.

### Final Answers:
(a) The number of investors who invested in both stocks and bonds is **201**.

(b) The number of investors who invested only in stocks is **278**.

Would you like further details or have any questions about this solution? 

Here are some related questions you might find useful:

1. How many investors invested only in bonds?
2. What percentage of investors invested in both stocks and bonds?
3. What is the total number of investors who invested in at least one of the two options (stocks or bonds)?
4. If 50 investors only invested in bonds, how many invested in both stocks and bonds?
5. How would the solution change if the number of investors who did not invest in either stocks or bonds was different?

**Tip:** When dealing with problems involving sets, always consider using Venn diagrams as a visual aid to organize the data and solve the problem effectively.

In a survey of 800 investors, it was reported that 479 had invested in stocks, 422 had invested in bonds, and 100 had not invested in either stocks or bonds. (a) How many investors had invested in both stocks and bonds? (b) How many investors had invested only in stocks?

In a survey of 800 investors, 479 invested in stocks, 422 in bonds, and 100 in neither. Learn how to calculate the number of investors who invested in both stocks and bonds (201), and those who invested only in stocks (278), using the inclusion-exclusion principle. Perfect for students in Grades 9-12.

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