The sum of two positive numbers is 20% of the sum of their squares and 25% of the difference of their squares. If the numbers are x and y the, `(x+y)/(x^(2))` is equal to

Options

A. `(1)/(4)`
B. `(3)/(8)`
C. `(1)/(3)`
D. `(2)/(9)`

Answer

To solve the problem step by step, we will denote the two positive numbers as \( x \) and \( y \).

### Step 1: Set up the equations based on the problem statement.
The problem states that the sum of the two numbers \( x + y \) is equal to:
1. 20% of the sum of their squares:
\[ x + y = 0.2 (x^2 + y^2) \]
This can be rewritten as:
\[ x + y = \frac{1}{5} (x^2 + y^2) \]

2. 25% of the difference of their squares:
\[ x + y = 0.25 (x^2 - y^2) \]
This can be rewritten as:
\[ x + y = \frac{1}{4} (x^2 - y^2) \]

### Step 2: Set the two expressions for \( x + y \) equal to each other.
From the two equations we have:
\[ \frac{1}{5} (x^2 + y^2) = \frac{1}{4} (x^2 - y^2) \]

### Step 3: Clear the fractions by multiplying through by 20 (the least common multiple of 5 and 4).
\[ 4(x^2 + y^2) = 5(x^2 - y^2) \]

### Step 4: Expand and rearrange the equation.
Expanding both sides gives:
\[ 4x^2 + 4y^2 = 5x^2 - 5y^2 \]
Rearranging this leads to:
\[ 4x^2 + 4y^2 + 5y^2 = 5x^2 \]
\[ 4x^2 + 9y^2 = 5x^2 \]
Subtracting \( 4x^2 \) from both sides results in:
\[ 9y^2 = x^2 \]

### Step 5: Solve for the relationship between \( x \) and \( y \).
Taking the square root of both sides gives:
\[ x = 3y \]

### Step 6: Substitute \( x \) back into the expression \( \frac{x+y}{x^2} \).
Now we need to find \( \frac{x+y}{x^2} \):
\[ x + y = 3y + y = 4y \]
\[ x^2 = (3y)^2 = 9y^2 \]
Thus:
\[ \frac{x+y}{x^2} = \frac{4y}{9y^2} \]

### Step 7: Simplify the expression.
This simplifies to:
\[ \frac{4}{9y} \]

### Step 8: Since \( y \) is a positive number, we can express the final result as:
\[ \frac{4}{9} \]

### Final Answer:
Thus, the value of \( \frac{x+y}{x^2} \) is \( \frac{4}{9} \).