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: Set up the two savings equations.
Let the incomes be $9x$ and $7x$, and the expenditures be $4y$ and $3y$. Since each person saves Rs. 2000, \[ 9x - 4y = 2000 \quad \text{and} \quad 7x - 3y = 2000 \]
Step 2: Subtract the equations directly, no LCM needed.
Since both right hand sides equal 2000, subtracting the second equation from the first eliminates the constant right away: \[ (9x - 4y) - (7x - 3y) = 0 \] \[ 2x - y = 0 \quad \Rightarrow \quad y = 2x \]
Step 3: Substitute back into either equation.
Using $7x - 3y = 2000$ with $y = 2x$: \[ 7x - 3(2x) = 2000 \] \[ 7x - 6x = 2000 \quad \Rightarrow \quad x = 2000 \]
Step 4: Find the income difference.
The incomes are $9x$ and $7x$, so their difference is $2x = 2 \times 2000 = 4000$.
\[ \boxed{\text{Rs. } 4000} \]