The problem describes a survey of investors, where we are given the number of investors in stocks, mutual funds, and bonds, as well as the number of investors in various combinations of these investments. The task is to draw a Venn diagram and determine the number of investors in each set. Specifically, we have: Total investors: Not explicitly given, needs to be found. Stocks: 100 Mutual Funds: 60 Bonds: 50 Stocks and Mutual Funds: 35 Mutual Funds and Bonds: 30 Stocks and Bonds: 28 All three: 20

1. Problem Description

The problem describes a survey of investors, where we are given the number of investors in stocks, mutual funds, and bonds, as well as the number of investors in various combinations of these investments. The task is to draw a Venn diagram and determine the number of investors in each set. Specifically, we have:

Total investors: Not explicitly given, needs to be found.

Stocks: 100

Mutual Funds: 60

Bonds: 50

Stocks and Mutual Funds: 35

Mutual Funds and Bonds: 30

Stocks and Bonds: 28

All three: 20

2. Solution Steps

Let $S$ be the set of investors in stocks, $M$ be the set of investors in mutual funds, and $B$ be the set of investors in bonds.

We are given the following information:

$|S| = 100$

$|M| = 60$

$|B| = 50$

$|S \cap M| = 35$

$|M \cap B| = 30$

$|S \cap B| = 28$

$|S \cap M \cap B| = 20$

First, we find the number of investors in only stocks and mutual funds:

$|S \cap M \cap B^c| = |S \cap M| - |S \cap M \cap B| = 35 - 20 = 15$

Next, we find the number of investors in only mutual funds and bonds:

$|M \cap B \cap S^c| = |M \cap B| - |S \cap M \cap B| = 30 - 20 = 10$

Then, we find the number of investors in only stocks and bonds:

$|S \cap B \cap M^c| = |S \cap B| - |S \cap M \cap B| = 28 - 20 = 8$

Now, we find the number of investors only in stocks:

$|S \cap M^c \cap B^c| = |S| - |S \cap M \cap B^c| - |S \cap B \cap M^c| - |S \cap M \cap B| = 100 - 15 - 8 - 20 = 57$

Next, we find the number of investors only in mutual funds:

$|M \cap S^c \cap B^c| = |M| - |S \cap M \cap B^c| - |M \cap B \cap S^c| - |S \cap M \cap B| = 60 - 15 - 10 - 20 = 15$

Then, we find the number of investors only in bonds:

$|B \cap S^c \cap M^c| = |B| - |S \cap B \cap M^c| - |M \cap B \cap S^c| - |S \cap M \cap B| = 50 - 8 - 10 - 20 = 12$

Now, we can calculate the total number of investors in the survey:

$|S \cup M \cup B| = |S \cap M^c \cap B^c| + |M \cap S^c \cap B^c| + |B \cap S^c \cap M^c| + |S \cap M \cap B^c| + |M \cap B \cap S^c| + |S \cap B \cap M^c| + |S \cap M \cap B|$

$|S \cup M \cup B| = 57 + 15 + 12 + 15 + 10 + 8 + 20 = 137$

The total number of investors is

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7. The Venn diagram would have three overlapping circles representing S, M, and B. The number in each region would be:

Only S: 57

Only M: 15

Only B: 12

S and M only: 15

M and B only: 10

S and B only: 8

S and M and B: 20

3. Final Answer

Stocks only: 57

Mutual Funds only: 15

Bonds only: 12

Stocks and Mutual Funds only: 15

Mutual Funds and Bonds only: 10

Stocks and Bonds only: 8

Stocks, Mutual Funds, and Bonds: 20

Total number of investors: 137