In a survey of 700 investors, it was reported that 440 had invested in stocks, 384 had invested in bonds, and 87 had not invested in either stocks or bonds.

Exercise (a)
How many investors had invested in both stocks and bonds?

Part 1 of 3
Let S = {investors in stock} and B = {investors in bonds}.

Since there were a total of 700 investors and 87 of them had not invested in either stocks or bonds, the number investing in stocks or bonds is
n(S ∪ B)	 = 	700 − 87
Correct: Your answer is correct.
seenKey 87

= 	613
Correct: Your answer is correct.
seenKey 613 .
Part 2 of 3
It is stated in the problem that
n(S) = 440
and
n(B) = 384.

To find the number of investors who had invested in both stocks and bonds, use the Inclusion-Exclusion Principle, given as
n(S ∪ B) = n(S) + n(B) − n(S ∩ B).

Substitute the given information and
n(S ∪ B) = 613
into this formula.
613 = 440 +

Incorrect: Your answer is incorrect.
− n(S ∩ B)

Let's solve this step-by-step using the **Inclusion-Exclusion Principle**.

We are given:
- Total number of investors: \( 700 \)
- Investors who had not invested in either stocks or bonds: \( 87 \)
- Investors who invested in stocks: \( n(S) = 440 \)
- Investors who invested in bonds: \( n(B) = 384 \)
- Investors who invested in either stocks or bonds: \( n(S \cup B) = 700 - 87 = 613 \)

Using the **Inclusion-Exclusion Principle**, we have the formula:
\[
n(S \cup B) = n(S) + n(B) - n(S \cap B)
\]
Substituting the values into this equation:
\[
613 = 440 + 384 - n(S \cap B)
\]
Now, solve for \( n(S \cap B) \):
\[
613 = 824 - n(S \cap B)
\]
\[
n(S \cap B) = 824 - 613
\]
\[
n(S \cap B) = 211
\]

### Conclusion:
The number of investors who had invested in both stocks and bonds is **211**.

Would you like a detailed explanation on any part of this solution? Here are some related questions you might find useful:

1. How does the Inclusion-Exclusion Principle work in set theory?
2. What if some investors had invested in neither, but also in other categories like real estate? How would that affect the calculation?
3. How can we visualize this problem using a Venn diagram?
4. Can the formula be extended for three or more sets, such as adding a third investment type?
5. What if we only knew the number of investors in both categories but not individual counts for stocks or bonds?

**Tip:** The Inclusion-Exclusion Principle is a versatile tool in counting problems, especially when dealing with overlapping sets.

In a survey of 700 investors, it was reported that 440 had invested in stocks, 384 had invested in bonds, and 87 had not invested in either stocks or bonds.

Exercise (a) How many investors had invested in both stocks and bonds?

Let S = {investors in stock} and B = {investors in bonds}.

Since there were a total of 700 investors and 87 of them had not invested in either stocks or bonds, the number investing in stocks or bonds is n(S ∪ B) = 700 − 87 = 613.

It is stated that n(S) = 440 and n(B) = 384. To find the number of investors who had invested in both stocks and bonds, use the Inclusion-Exclusion Principle: n(S ∪ B) = n(S) + n(B) − n(S ∩ B).

Learn how to calculate the number of investors who have invested in both stocks and bonds using the Inclusion-Exclusion Principle. Given 700 total investors, 87 did not invest in either stocks or bonds, 440 invested in stocks, and 384 in bonds. Suitable for students in Grades 9-12.

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