The monthly pocket money of Ravi and Sanjeev are in the ratio `5 : 7`. Their expenditures are in the ratio `3 : 5`. If each saves `Rs. 80` every month, find their monthly pocket money.

Answer

To solve the problem step by step, we will follow the information provided in the question.

### Step 1: Define the variables for pocket money
Let the monthly pocket money of Ravi be \( 5k \) and that of Sanjeev be \( 7k \), where \( k \) is a common multiplier.

### Step 2: Define the variables for expenditures
Let the expenditures of Ravi and Sanjeev be \( 3m \) and \( 5m \) respectively, where \( m \) is another common multiplier.

### Step 3: Set up the equations based on savings
Since both Ravi and Sanjeev save Rs. 80 each month, we can set up the following equations based on the relationship between pocket money, expenditure, and savings.

For Ravi:
\[ 5k - 3m = 80 \quad \text{(Equation 1)} \]

For Sanjeev:
\[ 7k - 5m = 80 \quad \text{(Equation 2)} \]

### Step 4: Solve the equations simultaneously
We will solve these two equations to find the values of \( k \) and \( m \).

From Equation 1:
\[ 5k - 3m = 80 \quad \Rightarrow \quad 5k = 80 + 3m \quad \Rightarrow \quad k = \frac{80 + 3m}{5} \quad \text{(1)} \]

Now, substitute \( k \) from Equation (1) into Equation 2:
\[ 7\left(\frac{80 + 3m}{5}\right) - 5m = 80 \]

Multiply through by 5 to eliminate the fraction:
\[ 7(80 + 3m) - 25m = 400 \]
\[ 560 + 21m - 25m = 400 \]
\[ 560 - 4m = 400 \]
\[ -4m = 400 - 560 \]
\[ -4m = -160 \quad \Rightarrow \quad m = 40 \]

### Step 5: Substitute back to find \( k \)
Now substitute \( m = 40 \) back into Equation (1) to find \( k \):
\[ k = \frac{80 + 3(40)}{5} \]
\[ k = \frac{80 + 120}{5} = \frac{200}{5} = 40 \]

### Step 6: Calculate the monthly pocket money
Now we can calculate the monthly pocket money for Ravi and Sanjeev:
- For Ravi:
\[ \text{Pocket Money of Ravi} = 5k = 5 \times 40 = 200 \]
- For Sanjeev:
\[ \text{Pocket Money of Sanjeev} = 7k = 7 \times 40 = 280 \]

### Final Answer
Thus, Ravi's monthly pocket money is Rs. 200 and Sanjeev's monthly pocket money is Rs. 280.

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