Weekly incomes of two persons are in the ratio of 7 : 3 and their weekly expenses are in the ratio of 5 : 2. If each of them savesX300 per week, then the weekly income of the first person is

Options

A. X 7500
B. X 4500
C. X 6300
D. X 5400

Answer

To solve the problem step by step, we will follow the information given in the question regarding the incomes and expenses of the two persons.

### Step 1: Define the Variables
Let the common multiple for the incomes be \( x \) and for the expenses be \( y \).
- The income of the first person = \( 7x \)
- The income of the second person = \( 3x \)
- The expenses of the first person = \( 5y \)
- The expenses of the second person = \( 2y \)

### Step 2: Set Up the Savings Equations
According to the question, both persons save \( 300 \) per week. Therefore, we can set up the following equations based on their savings:
1. For the first person:
\[ 7x - 5y = 300 \quad \text{(1)} \]
2. For the second person:
\[ 3x - 2y = 300 \quad \text{(2)} \]

### Step 3: Solve the Equations
We can solve these two equations simultaneously. First, we will manipulate equation (2) to align the coefficients of \( y \).

Multiply equation (2) by \( 5 \) to align it with equation (1):
\[ 5(3x - 2y) = 5(300) \]
This gives us:
\[ 15x - 10y = 1500 \quad \text{(3)} \]

Now, multiply equation (1) by \( 2 \):
\[ 2(7x - 5y) = 2(300) \]
This gives us:
\[ 14x - 10y = 600 \quad \text{(4)} \]

### Step 4: Subtract the Equations
Now, we can subtract equation (4) from equation (3):
\[ (15x - 10y) - (14x - 10y) = 1500 - 600 \]
This simplifies to:
\[ x = 900 \]

### Step 5: Substitute Back to Find \( y \)
Now that we have \( x \), we can substitute it back into one of the original equations to find \( y \). We will use equation (1):
\[ 7(900) - 5y = 300 \]
This simplifies to:
\[ 6300 - 5y = 300 \]
Rearranging gives:
\[ 5y = 6300 - 300 \]
\[ 5y = 6000 \]
\[ y = 1200 \]

### Step 6: Calculate the Weekly Income of the First Person
Now that we have both \( x \) and \( y \), we can find the weekly income of the first person:
\[ \text{Income of the first person} = 7x = 7(900) = 6300 \]

### Final Answer
Thus, the weekly income of the first person is **6300**.

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