The difference of two natural numbers is 5 and the difference of their reciprocals is `(1)/(10)` . Find the numbers.

Answer

To solve the problem, we need to find two natural numbers based on the given conditions. Let's denote the two natural numbers as \( x \) and \( y \).

### Step 1: Set up the equations
According to the problem, we know:
1. The difference of the two numbers is 5:
\[ x - y = 5 \quad \text{(1)} \]
2. The difference of their reciprocals is \( \frac{1}{10} \):
\[ \frac{1}{y} - \frac{1}{x} = \frac{1}{10} \quad \text{(2)} \]

### Step 2: Express \( y \) in terms of \( x \)
From equation (1), we can express \( y \) in terms of \( x \):
\[ y = x - 5 \quad \text{(3)} \]

### Step 3: Substitute \( y \) in equation (2)
Now, substitute equation (3) into equation (2):
\[ \frac{1}{x - 5} - \frac{1}{x} = \frac{1}{10} \]

### Step 4: Find a common denominator
The common denominator for the left side is \( x(x - 5) \):
\[ \frac{x - (x - 5)}{x(x - 5)} = \frac{1}{10} \]
This simplifies to:
\[ \frac{5}{x(x - 5)} = \frac{1}{10} \]

### Step 5: Cross multiply
Cross multiplying gives us:
\[ 5 \cdot 10 = x(x - 5) \]
This simplifies to:
\[ 50 = x^2 - 5x \quad \text{(4)} \]

### Step 6: Rearrange the equation
Rearranging equation (4) gives us:
\[ x^2 - 5x - 50 = 0 \quad \text{(5)} \]

### Step 7: Factor the quadratic equation
We need to factor the quadratic equation (5). We look for two numbers that multiply to \(-50\) and add to \(-5\). These numbers are \(-10\) and \(5\):
\[ (x - 10)(x + 5) = 0 \]

### Step 8: Solve for \( x \)
Setting each factor to zero gives us:
1. \( x - 10 = 0 \) → \( x = 10 \)
2. \( x + 5 = 0 \) → \( x = -5 \)

Since we are looking for natural numbers, we discard \( x = -5 \).

### Step 9: Find \( y \)
Now, substitute \( x = 10 \) back into equation (3) to find \( y \):
\[ y = 10 - 5 = 5 \]

### Conclusion
The two natural numbers are:
\[ \text{First number } (x) = 10, \quad \text{Second number } (y) = 5 \]