Incomes of X and Y are in the ratio of 5:3. Their expenditures are in the ratio 9:5. If each saves Rs 1600 at the end of month, then what is the income (in Rs) of X?

Options

A. 18000
B. 16000
C. 9600
D. 14000

Answer

To solve the problem step by step, we will break down the information given and use it to find the income of X.

### Step 1: Define the Incomes and Expenditures
Let the incomes of X and Y be represented as:
- Income of X = 5k (where k is a constant)
- Income of Y = 3k

Let the expenditures of X and Y be represented as:
- Expenditure of X = 9m (where m is another constant)
- Expenditure of Y = 5m

### Step 2: Define the Savings
According to the problem, both X and Y save Rs 1600 at the end of the month. We can express savings as:
- Savings of X = Income of X - Expenditure of X
- Savings of Y = Income of Y - Expenditure of Y

Thus, we have:
- Savings of X = 5k - 9m = 1600
- Savings of Y = 3k - 5m = 1600

### Step 3: Set Up the Equations
From the savings equations, we can set up two equations:
1. \( 5k - 9m = 1600 \) (Equation 1)
2. \( 3k - 5m = 1600 \) (Equation 2)

### Step 4: Solve the Equations
We can solve these two equations simultaneously.

First, let's rearrange Equation 1 to express \( m \) in terms of \( k \):
\[ 9m = 5k - 1600 \]
\[ m = \frac{5k - 1600}{9} \] (Equation 3)

Now, substitute Equation 3 into Equation 2:
\[ 3k - 5\left(\frac{5k - 1600}{9}\right) = 1600 \]

Multiply through by 9 to eliminate the fraction:
\[ 27k - 5(5k - 1600) = 14400 \]
\[ 27k - 25k + 8000 = 14400 \]
\[ 2k + 8000 = 14400 \]
\[ 2k = 14400 - 8000 \]
\[ 2k = 6400 \]
\[ k = 3200 \]

### Step 5: Calculate the Income of X
Now that we have the value of \( k \), we can find the income of X:
\[ \text{Income of X} = 5k = 5 \times 3200 = 16000 \]

### Final Answer
The income of X is Rs 16,000.

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