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 + 700
Incorrect: Your answer is incorrect.
seenKey 384 − n(S ∩ B)
Part 3 of 3
Simplify the above equation and solve for
n(S ∩ B).
613	 =

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

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

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

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

### Part 1:
We are given that the total number of investors is 700, and 87 had not invested in either stocks or bonds. Therefore, the number of investors who invested in stocks or bonds is:
\[
n(S \cup B) = 700 - 87 = 613
\]
This step is correct.

### Part 2:
The Inclusion-Exclusion Principle formula is:
\[
n(S \cup B) = n(S) + n(B) - n(S \cap B)
\]
Where:
- \( n(S \cup B) \) is the number of investors who invested in either stocks or bonds, which is 613.
- \( n(S) \) is the number of investors who invested in stocks, which is 440.
- \( n(B) \) is the number of investors who invested in bonds, which is 384.

Now, we can substitute these values into the Inclusion-Exclusion formula:
\[
613 = 440 + 384 - n(S \cap B)
\]

### Part 3:
Simplify and solve for \( n(S \cap B) \):
\[
613 = 824 - n(S \cap B)
\]
\[
n(S \cap B) = 824 - 613
\]
\[
n(S \cap B) = 211
\]

So, the number of investors who had invested in both stocks and bonds is **211**.

Would you like more details on the Inclusion-Exclusion Principle or any other part of this problem?

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Here are 5 related questions you might find useful:
1. How would the result change if 100 investors had not invested in either stocks or bonds?
2. What if the number of stock investors increased to 500, how would that impact \( n(S \cap B) \)?
3. Can the Inclusion-Exclusion Principle be applied to more than two sets? How?
4. How would you interpret the solution geometrically using Venn diagrams?
5. If a random investor is chosen, what is the probability that they invested in both stocks and bonds?

**Tip:** When using the Inclusion-Exclusion Principle, always ensure you correctly account for the overlap by subtracting the intersection term.

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

Solve how many investors invested in both stocks and bonds using the Inclusion-Exclusion Principle. This problem covers set theory concepts, with 700 total investors, 440 in stocks, 384 in bonds, and 87 in neither. Suitable for Grades 9-12 students.

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