The sum of two natural number is 28 and their product in 192. Find the numbers.

Answer

To solve the problem, we need to find two natural numbers whose sum is 28 and whose product is 192. Let's denote the two numbers as \( x \) and \( y \).

### Step-by-Step Solution:

1. **Set up the equations based on the problem statement:**
- We know that the sum of the two numbers is 28:
\[ x + y = 28 \quad \text{(1)} \]
- We also know that the product of the two numbers is 192:
\[ xy = 192 \quad \text{(2)} \]

2. **Express one variable in terms of the other:**
- From equation (1), we can express \( y \) in terms of \( x \):
\[ y = 28 - x \quad \text{(3)} \]

3. **Substitute equation (3) into equation (2):**
- Substitute \( y \) in equation (2):
\[ x(28 - x) = 192 \]
- Expanding this gives:
\[ 28x - x^2 = 192 \]

4. **Rearrange the equation to standard quadratic form:**
- Rearranging the equation, we get:
\[ x^2 - 28x + 192 = 0 \quad \text{(4)} \]

5. **Factor the quadratic equation:**
- We need to factor the quadratic equation \( x^2 - 28x + 192 = 0 \). We look for two numbers that multiply to 192 and add up to -28.
- The numbers are -12 and -16, so we can write:
\[ (x - 12)(x - 16) = 0 \]

6. **Solve for \( x \):**
- Setting each factor to zero gives us:
\[ x - 12 = 0 \quad \Rightarrow \quad x = 12 \]
\[ x - 16 = 0 \quad \Rightarrow \quad x = 16 \]

7. **Find the corresponding values of \( y \):**
- Using equation (3) to find \( y \):
- If \( x = 12 \):
\[ y = 28 - 12 = 16 \]
- If \( x = 16 \):
\[ y = 28 - 16 = 12 \]

8. **Conclusion:**
- The two natural numbers are 12 and 16.

### Final Answer:
The two natural numbers are 12 and 16.