A part of monthly expenses of a family is constant and the remaining vary with the number of members in the family. For a family of 4 persons, the total monthly expenses are < 10,400, whereas for a family of 7 persons, the total monthly expenses are 15,800. Find the constant expenses per month and the monthly expenses on each member of a family.

Answer

To solve the problem, we need to set up a system of equations based on the information given about the family's expenses.

### Step 1: Define Variables
Let:
- \( X \) = constant monthly expenses of the family
- \( Y \) = monthly expenses per member of the family

### Step 2: Set Up Equations
From the information provided:
1. For a family of 4 persons, the total monthly expenses are 10,400:
\[ X + 4Y = 10,400 \quad \text{(Equation 1)} \]

2. For a family of 7 persons, the total monthly expenses are 15,800:
\[ X + 7Y = 15,800 \quad \text{(Equation 2)} \]

### Step 3: Subtract Equations
Now, we will subtract Equation 1 from Equation 2 to eliminate \( X \):
\[ (X + 7Y) - (X + 4Y) = 15,800 - 10,400 \]
This simplifies to:
\[ 3Y = 5,400 \]

### Step 4: Solve for \( Y \)
Now, divide both sides by 3 to find \( Y \):
\[ Y = \frac{5,400}{3} = 1,800 \]

### Step 5: Substitute \( Y \) Back to Find \( X \)
Now that we have \( Y \), we can substitute it back into Equation 1 to find \( X \):
\[ X + 4(1,800) = 10,400 \]
This simplifies to:
\[ X + 7,200 = 10,400 \]
Now, subtract 7,200 from both sides:
\[ X = 10,400 - 7,200 = 3,200 \]

### Final Answer
- The constant expenses per month \( X \) is **3,200**.
- The monthly expenses per member \( Y \) is **1,800**.