Two numbers are in the ratio `2 : 3`. If 2 is subtracted from the first and 2 is added to the second, the ratio becomes `1:2`. The sum of the numbers is :

Options

A. 30
B. 28
C. 24
D. 10

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Define the numbers based on the ratio
Let the two numbers be represented as:
- First number = 2x
- Second number = 3x

### Step 2: Set up the equation based on the condition given
According to the problem, if we subtract 2 from the first number and add 2 to the second number, the new ratio becomes 1:2. We can express this as:
\[ \frac{2x - 2}{3x + 2} = \frac{1}{2} \]

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

### Step 4: Expand both sides
Expanding both sides results in:
\[ 4x - 4 = 3x + 2 \]

### Step 5: Rearrange the equation to isolate x
Now, we can rearrange the equation to solve for x:
\[ 4x - 3x = 2 + 4 \]
\[ x = 6 \]

### Step 6: Find the actual numbers
Now that we have the value of x, we can find the actual numbers:
- First number = 2x = 2(6) = 12
- Second number = 3x = 3(6) = 18

### Step 7: Calculate the sum of the numbers
Finally, we can find the sum of the two numbers:
\[ \text{Sum} = 12 + 18 = 30 \]

### Final Answer
The sum of the numbers is **30**.

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