The difference between two natural numbers is 196 and the ratio of the two numbers is 9:5. Find the two natural numbers.

Options

A. 441, 245
B. 440, 246
C. 400, 256
D. 410. 235

Answer

To solve the problem step by step, we will use the information given in the question:

1. **Let the two natural numbers be \( x \) and \( y \)**.
- Here, we assume \( x \) is the larger number and \( y \) is the smaller number.

2. **Set up the equations based on the information provided**:
- The difference between the two numbers is given as 196. Therefore, we can write the first equation:
\[ x - y = 196 \quad \text{(Equation 1)} \]
- The ratio of the two numbers is given as 9:5. This can be expressed as:
\[ \frac{x}{y} = \frac{9}{5} \]
- Cross-multiplying gives us the second equation:
\[ 5x = 9y \quad \text{(Equation 2)} \]

3. **Rearranging Equation 1 to express \( x \) in terms of \( y \)**:
- From Equation 1, we can express \( x \) as:
\[ x = y + 196 \]

4. **Substituting \( x \) in Equation 2**:
- Now, we will substitute the expression for \( x \) from Equation 1 into Equation 2:
\[ 5(y + 196) = 9y \]
- Expanding this gives:
\[ 5y + 980 = 9y \]

5. **Rearranging to solve for \( y \)**:
- Now, we will move \( 5y \) to the right side:
\[ 980 = 9y - 5y \]
\[ 980 = 4y \]
- Dividing both sides by 4:
\[ y = \frac{980}{4} = 245 \]

6. **Finding \( x \)**:
- Now that we have \( y \), we can find \( x \) using the expression we found earlier:
\[ x = y + 196 = 245 + 196 = 441 \]

7. **Final answer**:
- The two natural numbers are \( x = 441 \) and \( y = 245 \).

### Summary of the solution:
- The two natural numbers are **441** and **245**.