The ratio of incomes of A and B is 9 : 7 and the ratio of their expenditures is 4 : 3. If each of them saves Rs. 4000 per month, then find their monthly incomes. Also, if each of them donates 9% of his income to a charity working for old age destitutes, then find the resulting savings of each. What value is indicated from this action?

Answer

To solve the problem step by step, we will break down the information given and derive the required values systematically.

### Step 1: Define the Variables
Let the incomes of A and B be represented as:
- Income of A = 9x
- Income of B = 7x

Let the expenditures of A and B be represented as:
- Expenditure of A = 4y
- Expenditure of B = 3y

### Step 2: Set Up the Equations
According to the problem, both A and B save Rs. 4000 per month. Therefore, we can write the following equations based on their incomes and expenditures:

1. For A:
\[ \text{Income of A} - \text{Expenditure of A} = \text{Savings of A} \]
\[ 9x - 4y = 4000 \quad \text{(Equation 1)} \]

2. For B:
\[ \text{Income of B} - \text{Expenditure of B} = \text{Savings of B} \]
\[ 7x - 3y = 4000 \quad \text{(Equation 2)} \]

### Step 3: Solve the System of Equations
We have the two equations:
1. \( 9x - 4y = 4000 \)
2. \( 7x - 3y = 4000 \)

To eliminate \(y\), we can multiply Equation 1 by 3 and Equation 2 by 4:

- Multiply Equation 1 by 3:
\[ 27x - 12y = 12000 \quad \text{(Equation 3)} \]

- Multiply Equation 2 by 4:
\[ 28x - 12y = 16000 \quad \text{(Equation 4)} \]

### Step 4: Subtract the Equations
Now, we subtract Equation 3 from Equation 4:
\[ (28x - 12y) - (27x - 12y) = 16000 - 12000 \]
This simplifies to:
\[ x = 4000 \]

### Step 5: Substitute Back to Find \(y\)
Now that we have \(x\), we can substitute it back into either Equation 1 or Equation 2 to find \(y\). Let's use Equation 1:
\[ 9(4000) - 4y = 4000 \]
\[ 36000 - 4y = 4000 \]
\[ -4y = 4000 - 36000 \]
\[ -4y = -32000 \]
\[ y = 8000 \]

### Step 6: Calculate Monthly Incomes
Now we can find the monthly incomes of A and B:
- Income of A:
\[ 9x = 9(4000) = 36000 \text{ Rs.} \]

- Income of B:
\[ 7x = 7(4000) = 28000 \text{ Rs.} \]

### Step 7: Calculate Monthly Expenditures
Now we can find the monthly expenditures of A and B:
- Expenditure of A:
\[ 4y = 4(8000) = 32000 \text{ Rs.} \]

- Expenditure of B:
\[ 3y = 3(8000) = 24000 \text{ Rs.} \]

### Step 8: Calculate Resulting Savings After Donation
Each person donates 9% of their income to charity. We need to calculate their resulting savings after the donation.

1. For A:
- Donation = 9% of Income of A:
\[ \text{Donation} = 0.09 \times 36000 = 3240 \text{ Rs.} \]
- Resulting Savings:
\[ \text{Resulting Savings} = 4000 - 3240 = 760 \text{ Rs.} \]

2. For B:
- Donation = 9% of Income of B:
\[ \text{Donation} = 0.09 \times 28000 = 2520 \text{ Rs.} \]
- Resulting Savings:
\[ \text{Resulting Savings} = 4000 - 2520 = 1480 \text{ Rs.} \]

### Final Results
- Monthly Income of A: Rs. 36,000
- Monthly Income of B: Rs. 28,000
- Resulting Savings of A after donation: Rs. 760
- Resulting Savings of B after donation: Rs. 1480

### Value Indicated
The action of donating to charity indicates a kind and caring attitude towards helping the less fortunate, showcasing social responsibility and empathy.