The difference between two positive numbers x and y `(x gt y)` is 4 and the difference between their reciprocals is `(4)/(21)`. Find the numbers.

Answer

To solve the problem, we will follow these steps:

### Step 1: Set up the equations
We are given two pieces of information:
1. The difference between two positive numbers \( x \) and \( y \) (where \( x > y \)) is 4.
2. The difference between their reciprocals is \( \frac{4}{21} \).

From the first piece of information, we can write our first equation:
\[ x - y = 4 \quad \text{(Equation 1)} \]

From the second piece of information, we can express the difference between their reciprocals:
\[ \frac{1}{y} - \frac{1}{x} = \frac{4}{21} \quad \text{(Equation 2)} \]

### Step 2: Simplify Equation 2
To simplify Equation 2, we find a common denominator:
\[ \frac{x - y}{xy} = \frac{4}{21} \]

Now, substituting \( x - y \) from Equation 1 into this equation:
\[ \frac{4}{xy} = \frac{4}{21} \]

### Step 3: Cross-multiply to eliminate the fractions
Cross-multiplying gives us:
\[ 4 \cdot 21 = 4 \cdot xy \]
This simplifies to:
\[ 84 = 4xy \]

### Step 4: Solve for \( xy \)
Dividing both sides by 4:
\[ xy = 21 \quad \text{(Equation 3)} \]

### Step 5: Substitute \( x \) in terms of \( y \)
From Equation 1, we can express \( x \) in terms of \( y \):
\[ x = y + 4 \]

### Step 6: Substitute \( x \) into Equation 3
Now substitute \( x \) into Equation 3:
\[ (y + 4)y = 21 \]
Expanding this gives:
\[ y^2 + 4y - 21 = 0 \]

### Step 7: Factor the quadratic equation
We need to factor the quadratic equation:
\[ y^2 + 7y - 3y - 21 = 0 \]
This can be factored as:
\[ (y + 7)(y - 3) = 0 \]

### Step 8: Solve for \( y \)
Setting each factor to zero gives us:
\[ y + 7 = 0 \quad \Rightarrow \quad y = -7 \quad \text{(not valid since \( y \) must be positive)} \]
\[ y - 3 = 0 \quad \Rightarrow \quad y = 3 \]

### Step 9: Find \( x \)
Now, substitute \( y = 3 \) back into the equation for \( x \):
\[ x = y + 4 = 3 + 4 = 7 \]

### Final Answer
The two positive numbers are:
\[ x = 7 \quad \text{and} \quad y = 3 \]