Two numbers are in ratio 3 : 4. When 3 is subtracted from both the mumbers, the ratio becomes 2 : 3. Find the sum of the numbers.
A. 16
B. 20
C. 21
D. 22

Options

A. C
B. B
C. D
D. A

Answer

To solve the problem step by step, we will follow the information given in the question.

### Step 1: Define the numbers
Let the two numbers be \( x \) and \( y \). According to the problem, the ratio of the two numbers is given as:
\[ \frac{x}{y} = \frac{3}{4} \]
This implies that:
\[ x = \frac{3}{4}y \]

### Step 2: Set up the equation after subtraction
When 3 is subtracted from both numbers, the new ratio becomes:
\[ \frac{x - 3}{y - 3} = \frac{2}{3} \]
Cross-multiplying gives us:
\[ 3(x - 3) = 2(y - 3) \]

### Step 3: Substitute \( x \) in the equation
Substituting \( x = \frac{3}{4}y \) into the equation:
\[ 3\left(\frac{3}{4}y - 3\right) = 2(y - 3) \]
Expanding both sides:
\[ \frac{9}{4}y - 9 = 2y - 6 \]

### Step 4: Eliminate the fraction
To eliminate the fraction, multiply the entire equation by 4:
\[ 9y - 36 = 8y - 24 \]

### Step 5: Solve for \( y \)
Rearranging the equation gives:
\[ 9y - 8y = 36 - 24 \]
\[ y = 12 \]

### Step 6: Find \( x \)
Now that we have \( y \), we can find \( x \):
\[ x = \frac{3}{4}y = \frac{3}{4} \times 12 = 9 \]

### Step 7: Calculate the sum of the numbers
Now, we can find the sum of the two numbers:
\[ x + y = 9 + 12 = 21 \]

### Final Answer
The sum of the numbers is:
\[ \boxed{21} \]