The ratio of a two digit natural number to a number formed by reversing it digit is 4 : 7. Find the sum of all such possible numbers

Options

A. 120
B. 108
C. 330
D. 332

Answer

To solve the problem, we need to find a two-digit natural number and the number formed by reversing its digits, such that the ratio of these two numbers is 4:7. Let's denote the two-digit number as \(10x + y\), where \(x\) is the tens digit and \(y\) is the units digit. The reversed number will then be \(10y + x\).

### Step-by-Step Solution:

1. **Set Up the Equation**:
Given the ratio of the two-digit number to its reverse is \(4:7\), we can write the equation:
\[ \frac{10x + y}{10y + x} = \frac{4}{7} \]

2. **Cross-Multiply**:
Cross-multiplying gives us:
\[ 7(10x + y) = 4(10y + x) \]

3. **Expand Both Sides**:
Expanding both sides results in:
\[ 70x + 7y = 40y + 4x \]

4. **Rearrange the Equation**:
Rearranging the equation to isolate terms involving \(x\) and \(y\):
\[ 70x - 4x = 40y - 7y \]
This simplifies to:
\[ 66x = 33y \]

5. **Simplify the Ratio**:
Dividing both sides by 33 gives:
\[ 2x = y \quad \text{or} \quad \frac{x}{y} = \frac{1}{2} \]

6. **Find Possible Values for \(x\) and \(y\)**:
Since \(y = 2x\), we can determine the possible values for \(x\) (the tens digit) and \(y\) (the units digit):
- If \(x = 1\), then \(y = 2\) → Number = 12
- If \(x = 2\), then \(y = 4\) → Number = 24
- If \(x = 3\), then \(y = 6\) → Number = 36
- If \(x = 4\), then \(y = 8\) → Number = 48
- If \(x = 5\), then \(y = 10\) → Not valid (as \(y\) must be a single digit)

7. **List the Valid Two-Digit Numbers**:
The valid two-digit numbers we found are:
- 12
- 24
- 36
- 48

8. **Sum of All Possible Numbers**:
Now, we sum these valid numbers:
\[ 12 + 24 + 36 + 48 = 120 \]

### Final Answer:
The sum of all such possible numbers is **120**.