The incomes of two persons A and B are in the ratio 3 : 4. If each saves Rs.100 per month, the ratio of their expenditures is Rs. 1 : 2. Find their incomes.

Options

A. A) Rs. 100 and Rs.150
B. B) Rs. 150 and Rs.200
C. C) Rs.200 and Rs.250
D. D) Rs.250 and Rs.300

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the incomes of A and B
Let the incomes of A and B be represented as:
- Income of A = 3x
- Income of B = 4x

### Step 2: Define the savings
According to the problem, both A and B save Rs. 100 each month.

### Step 3: Define the expenditures
The expenditure can be calculated using the formula:
- Expenditure = Income - Savings

Thus, we can express their expenditures as:
- Expenditure of A = 3x - 100
- Expenditure of B = 4x - 100

### Step 4: Set up the ratio of expenditures
We know that the ratio of their expenditures is given as 1:2. Therefore, we can write:
\[ \frac{3x - 100}{4x - 100} = \frac{1}{2} \]

### Step 5: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 2(3x - 100) = 1(4x - 100) \]

### Step 6: Expand and simplify the equation
Expanding both sides results in:
\[ 6x - 200 = 4x - 100 \]

### Step 7: Rearranging the equation
Now, let's move all terms involving x to one side and constants to the other side:
\[ 6x - 4x = -100 + 200 \]
\[ 2x = 100 \]

### Step 8: Solve for x
Dividing both sides by 2 gives:
\[ x = 50 \]

### Step 9: Calculate the incomes of A and B
Now that we have the value of x, we can find the incomes:
- Income of A = 3x = 3 * 50 = Rs. 150
- Income of B = 4x = 4 * 50 = Rs. 200

### Final Answer
Thus, the incomes of A and B are Rs. 150 and Rs. 200 respectively.

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