Question

The ratio of income of two persons is 9 : 7 and the ratio of their expenditure is 4 : 3. If each of them saves Rs. 2,000 per month, find difference of their monthly income.

• A. Rs. 14,000
• B. Rs. 2,000
• C. Rs. 1,000
• D. Rs. 4,000

Hint:

An elegant ratio method:
Income ratio: \(9 : 7\) (difference is \(2\) units).
Expenditure ratio: \(4 : 3\) (difference is \(1\) unit).
To make the decrease in units equal, multiply the expenditure ratio by the difference of income ratio (\(2\)): new ratio is \(8 : 6\).
Now, compare Income (\(9:7\)) to Expenditure (\(8:6\)):
The change for both is exactly \(1\) unit (\(9-8=1\) and \(7-6=1\)).
This \(1\) unit of saving represents \(\text{Rs. } 2000\).
The difference in income is \(2\) units, which is \(2 \times 2000 = \text{Rs. } 4000\).

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
This question is from the topic of Ratio and Proportion, solved using Linear Equations.
We are given the ratios of incomes and expenditures of two persons and their individual savings.
We need to determine the difference between their monthly incomes.

Step 2: Key Formula or Approach:
The basic relationship between Income, Expenditure, and Savings is:
\[ \text{Income} - \text{Expenditure} = \text{Savings} \]
Let the incomes of the two persons be \(9x\) and \(7x\).
Let their expenditures be \(4y\) and \(3y\).
We set up a system of linear equations using their savings.

Step 3: Detailed Explanation:
Let the income of the first person be \(9x\) and the second person be \(7x\).
Let the expenditure of the first person be \(4y\) and the second person be \(3y\).
Since each person saves \(\text{Rs. } 2000\), we can write:
Equation 1: \(9x - 4y = 2000\)
Equation 2: \(7x - 3y = 2000\)
We can solve this system using the elimination method.
Multiply Equation 1 by 3:
\[ 3 \times (9x - 4y) = 3 \times 2000 \implies 27x - 12y = 6000 \quad \text{(Equation 3)} \]
Multiply Equation 2 by 4:
\[ 4 \times (7x - 3y) = 4 \times 2000 \implies 28x - 12y = 8000 \quad \text{(Equation 4)} \]
Subtract Equation 3 from Equation 4:
\[ (28x - 12y) - (27x - 12y) = 8000 - 6000 \]
\[ 28x - 27x = 2000 \]
\[ x = 2000 \]
Now, we need to find the difference between their monthly incomes.
First person's income = \(9x\)
Second person's income = \(7x\)
Difference in income = \(9x - 7x = 2x\)
Substitute the value of \(x\):
\[ \text{Difference} = 2 \times 2000 = 4000 \]

Step 4: Final Answer:
The difference in their monthly income is \(\text{Rs. } 4,000\).