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1. The monthly incomes of two persons, Ram and Rahim, are in the ratio [tex]5: 7[/tex], and their monthly expenditures are in the ratio [tex]7: 11[/tex]. If each of them saves ₹ 60 per month, find their monthly incomes.

(Answer: ₹ 200, ₹ 280)



Answer :

GinnyAnswer
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.

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