In 2012, the arithmetic mean of the annual incomes of Jack and Jill was $3800. The arithmetic mean of the annual incomes of Jill and Jess was $4800, and the arithmetic mean of the annual incomes of Jess and Jack was $5800. What of the incomes of the three?

Options

A. 4000
B. 4200
C. 4400
D. 4800

Answer

To solve the problem step by step, we will use the information given about the arithmetic means of the incomes of Jack, Jill, and Jess.

### Step 1: Set up the equations based on the given means.

1. **Jack and Jill**: The arithmetic mean of Jack (J) and Jill (Ji) is $3800.
\[ \frac{J + Ji}{2} = 3800 \]
Multiplying both sides by 2 gives:
\[ J + Ji = 7600 \quad \text{(Equation 1)} \]

2. **Jill and Jess**: The arithmetic mean of Jill (Ji) and Jess (Je) is $4800.
\[ \frac{Ji + Je}{2} = 4800 \]
Multiplying both sides by 2 gives:
\[ Ji + Je = 9600 \quad \text{(Equation 2)} \]

3. **Jess and Jack**: The arithmetic mean of Jess (Je) and Jack (J) is $5800.
\[ \frac{Je + J}{2} = 5800 \]
Multiplying both sides by 2 gives:
\[ Je + J = 11600 \quad \text{(Equation 3)} \]

### Step 2: Combine the equations to find the total income.

Now we have three equations:
- Equation 1: \( J + Ji = 7600 \)
- Equation 2: \( Ji + Je = 9600 \)
- Equation 3: \( Je + J = 11600 \)

To find the total income of Jack, Jill, and Jess, we can add all three equations together:
\[ (J + Ji) + (Ji + Je) + (Je + J) = 7600 + 9600 + 11600 \]

This simplifies to:
\[ 2J + 2Ji + 2Je = 28800 \]

### Step 3: Solve for the total income of Jack, Jill, and Jess.

Dividing the entire equation by 2 gives:
\[ J + Ji + Je = 14400 \quad \text{(Equation 4)} \]

### Step 4: Calculate the arithmetic mean of their incomes.

The arithmetic mean (AM) of the incomes of Jack, Jill, and Jess is given by:
\[ AM = \frac{J + Ji + Je}{3} \]
Substituting the value from Equation 4:
\[ AM = \frac{14400}{3} = 4800 \]

### Final Answer:
The arithmetic mean of the incomes of Jack, Jill, and Jess is **$4800**.

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