The combined salaries of three brothers is $90,000 Mr. Big earns twice what Mr. Small earns, and Mr. Middle earns ` 1 1//2` times what Mr. Small earns. What is the smallest salary of the three brother?

Options

A. 20000
B. 22000
C. 25000
D. 30000

Answer

To solve the problem step by step, we can follow these steps:

### Step 1: Define the variables
Let the salary of Mr. Small be \( X \).

### Step 2: Express the salaries of the other two brothers
- Mr. Big earns twice what Mr. Small earns, so his salary is \( 2X \).
- Mr. Middle earns \( 1 \frac{1}{2} \) times what Mr. Small earns. Converting \( 1 \frac{1}{2} \) to an improper fraction gives us \( \frac{3}{2} \). Therefore, Mr. Middle's salary is \( \frac{3}{2}X \).

### Step 3: Write the equation for the combined salary
According to the problem, the combined salaries of the three brothers is $90,000. Thus, we can write the equation:
\[ X + 2X + \frac{3}{2}X = 90,000 \]

### Step 4: Combine the terms
To combine the terms, we first convert all terms to have a common denominator. The common denominator for \( 1X \), \( 2X \), and \( \frac{3}{2}X \) is 2:
\[ \frac{2X}{2} + \frac{4X}{2} + \frac{3X}{2} = 90,000 \]
This simplifies to:
\[ \frac{2X + 4X + 3X}{2} = 90,000 \]
\[ \frac{9X}{2} = 90,000 \]

### Step 5: Solve for \( X \)
To eliminate the fraction, multiply both sides of the equation by 2:
\[ 9X = 180,000 \]
Now, divide both sides by 9:
\[ X = \frac{180,000}{9} = 20,000 \]

### Step 6: Determine the salaries of all brothers
Now that we have \( X \):
- Mr. Small earns \( X = 20,000 \).
- Mr. Big earns \( 2X = 2 \times 20,000 = 40,000 \).
- Mr. Middle earns \( \frac{3}{2}X = \frac{3}{2} \times 20,000 = 30,000 \).

### Step 7: Identify the smallest salary
The salaries of the three brothers are:
- Mr. Small: $20,000
- Mr. Big: $40,000
- Mr. Middle: $30,000

The smallest salary among them is \( 20,000 \).

### Final Answer
The smallest salary of the three brothers is **$20,000**.

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