In a company, the average monthly salary of Managers is 43000, that of Executives is 32000 and that of Workers is 24000. The average monthly salary of Managers and Executives combined is 35000 and that of Executives and Workers combined is 26000. What fraction of the workforce is made up of Workers?

To solve this problem, let's denote the number of Managers as \( M \), the number of Executives as \( E \), and the number of Workers as \( W \).

### Step 1: Setting Up the Equations
We know the following:

1. The average monthly salary of Managers is 43,000.
2. The average monthly salary of Executives is 32,000.
3. The average monthly salary of Workers is 24,000.
4. The average monthly salary of Managers and Executives combined is 35,000.
5. The average monthly salary of Executives and Workers combined is 26,000.

From these, we can derive the following equations:

#### Managers and Executives Combined:
\[
\frac{43,000M + 32,000E}{M + E} = 35,000
\]

#### Executives and Workers Combined:
\[
\frac{32,000E + 24,000W}{E + W} = 26,000
\]

### Step 2: Solving the Equations
Let's first solve the equation for Managers and Executives combined:

\[
43,000M + 32,000E = 35,000(M + E)
\]

Expanding and simplifying:

\[
43,000M + 32,000E = 35,000M + 35,000E
\]

\[
43,000M - 35,000M = 35,000E - 32,000E
\]

\[
8,000M = 3,000E
\]

\[
\frac{M}{E} = \frac{3,000}{8,000} = \frac{3}{8}
\]

This means the ratio of Managers to Executives is \( \frac{3}{8} \).

Now, let's solve the equation for Executives and Workers combined:

\[
32,000E + 24,000W = 26,000(E + W)
\]

Expanding and simplifying:

\[
32,000E + 24,000W = 26,000E + 26,000W
\]

\[
32,000E - 26,000E = 26,000W - 24,000W
\]

\[
6,000E = 2,000W
\]

\[
\frac{E}{W} = \frac{2,000}{6,000} = \frac{1}{3}
\]

This means the ratio of Executives to Workers is \( \frac{1}{3} \).

### Step 3: Expressing the Fraction of Workers in the Workforce
Let the total workforce be \( T = M + E + W \).

Using the ratios we found:

- \( M = \frac{3}{8}E \)
- \( W = 3E \)

Substituting these into the total workforce:

\[
T = \frac{3}{8}E + E + 3E = \frac{3E + 8E + 24E}{8} = \frac{35E}{8}
\]

The fraction of the workforce that is made up of Workers \( \frac{W}{T} \) is then:

\[
\frac{W}{T} = \frac{3E}{\frac{35E}{8}} = \frac{3E \times 8}{35E} = \frac{24}{35}
\]

### Final Answer
The fraction of the workforce that is made up of Workers is \( \frac{24}{35} \).

Would you like more details or have any questions?

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Here are five related questions to expand your understanding:

1. What is the fraction of the workforce made up of Managers?
2. What is the fraction of the workforce made up of Executives?
3. If there are 100 Executives, how many Managers and Workers are there?
4. How would the solution change if the average monthly salary of Workers was increased to 26,000?
5. What is the total monthly payroll of the company?

**Tip:** Understanding how to manipulate ratios is crucial in problems involving averages and weighted averages.

Learn how to solve a problem involving average monthly salaries of Managers, Executives, and Workers, and determine the fraction of the workforce made up of Workers. This problem covers concepts like averages, weighted averages, and ratios. Suitable for advanced high school students and college-level students.

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