Incomes of X and Y are in the ratio 15 : 13. Their expenditures are in the ratio 13 : 11. If both save Rs 9600 at the end of the month, then what is the income (in Rs) of X?

Options

A. 62400
B. 52800
C. 80400
D. 72000

Answer

To find the income of X, we can follow these steps:

### Step 1: Set up the ratios
Let the incomes of X and Y be represented as:
- Income of X = 15k
- Income of Y = 13k

Let the expenditures of X and Y be represented as:
- Expenditure of X = 13m
- Expenditure of Y = 11m

### Step 2: Set up the savings equation
According to the problem, both X and Y save Rs 9600 at the end of the month. Therefore, we can write the savings equations as follows:
- Savings of X = Income of X - Expenditure of X
- Savings of Y = Income of Y - Expenditure of Y

This gives us:
- For X: \( 15k - 13m = 9600 \) (1)
- For Y: \( 13k - 11m = 9600 \) (2)

### Step 3: Solve the equations
Now we have two equations:
1. \( 15k - 13m = 9600 \)
2. \( 13k - 11m = 9600 \)

We can solve these equations simultaneously.

From equation (1):
\[ 13m = 15k - 9600 \]
\[ m = \frac{15k - 9600}{13} \] (3)

Substituting (3) into equation (2):
\[ 13k - 11\left(\frac{15k - 9600}{13}\right) = 9600 \]

### Step 4: Clear the fraction
Multiply through by 13 to eliminate the fraction:
\[ 169k - 11(15k - 9600) = 124800 \]
\[ 169k - 165k + 105600 = 124800 \]
\[ 4k + 105600 = 124800 \]
\[ 4k = 124800 - 105600 \]
\[ 4k = 19200 \]
\[ k = 4800 \]

### Step 5: Calculate the income of X
Now that we have the value of k, we can find the income of X:
\[ \text{Income of X} = 15k = 15 \times 4800 = 72000 \]

### Final Answer
The income of X is Rs 72,000.

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