Certainly! Let's solve the problem step-by-step.
Given:
- The savings of both Ram and Rahim per month is ₹60.
- The ratio of their monthly incomes is 5:7.
- The ratio of their monthly expenditures is 7:11.
Let Ram's monthly income be [tex]\( 5x \)[/tex] and Rahim's monthly income be [tex]\( 7x \)[/tex].
Let Ram's monthly expenditure be [tex]\( 7y \)[/tex] and Rahim's monthly expenditure be [tex]\( 11y \)[/tex].
### Step 1: Establish the savings equations
For Ram:
[tex]\[ \text{Income} - \text{Expenditure} = \text{Savings} \][/tex]
[tex]\[ 5x - 7y = 60 \][/tex]
For Rahim:
[tex]\[ 7x - 11y = 60 \][/tex]
### Step 2: Solve the system of linear equations
We have two equations:
1. [tex]\( 5x - 7y = 60 \)[/tex]
2. [tex]\( 7x - 11y = 60 \)[/tex]
First, we solve for [tex]\( y \)[/tex]:
From equation (2):
[tex]\[ y = \frac{7x - 60}{11} \][/tex]
### Step 3: Substitute [tex]\( y \)[/tex]
Substitute [tex]\( y \)[/tex] from equation (2) into equation (1):
[tex]\[ 5x - 7\left(\frac{7x - 60}{11}\right) = 60 \][/tex]
### Step 4: Solve for [tex]\( x \)[/tex]
Multiply through by 11 to eliminate the fraction:
[tex]\[ 11 \cdot 5x - 11 \cdot \frac{7(7x - 60)}{11} = 11 \cdot 60 \][/tex]
[tex]\[ 55x - 49x + 420 = 660 \][/tex]
[tex]\[ 6x + 420 = 660 \][/tex]
[tex]\[ 6x = 240 \][/tex]
[tex]\[ x = 40 \][/tex]
### Step 5: Calculate their incomes
Now that we have [tex]\( x = 40 \)[/tex]:
- Ram's income [tex]\( = 5x = 5(40) = ₹200 \)[/tex]
- Rahim's income [tex]\( = 7x = 7(40) = ₹280 \)[/tex]
Therefore, Ram's monthly income is ₹200, and Rahim's monthly income is ₹280.
