If `x:y::5:2,` then
` (x ^(2) +y)/(x + y ^(2)) =?`

Options

A. 5
B. 4
C. 2
D. Can't be determined

Answer

To solve the problem, we start with the given ratio \( x:y = 5:2 \). This means we can express \( x \) and \( y \) in terms of a common variable \( k \).

### Step 1: Express \( x \) and \( y \) in terms of \( k \)
From the ratio \( x:y = 5:2 \), we can write:
- \( x = 5k \)
- \( y = 2k \)

### Step 2: Substitute \( x \) and \( y \) into the expression
We need to evaluate the expression \( \frac{x^2 + y}{x + y^2} \). Substituting \( x \) and \( y \) gives:
\[ \frac{(5k)^2 + 2k}{5k + (2k)^2} \]

### Step 3: Simplify the numerator
Calculating the numerator:
\[ (5k)^2 + 2k = 25k^2 + 2k \]

### Step 4: Simplify the denominator
Calculating the denominator:
\[ 5k + (2k)^2 = 5k + 4k^2 \]

### Step 5: Write the expression with the simplified numerator and denominator
Now, substituting back, we have:
\[ \frac{25k^2 + 2k}{5k + 4k^2} \]

### Step 6: Factor out \( k \) from both the numerator and denominator
Factoring \( k \) out of both parts:
\[ \frac{k(25k + 2)}{k(5 + 4k)} \]

### Step 7: Cancel \( k \) from numerator and denominator
Assuming \( k \neq 0 \), we can cancel \( k \):
\[ \frac{25k + 2}{5 + 4k} \]

### Step 8: Final expression
Thus, the final expression we need to evaluate is:
\[ \frac{25k + 2}{5 + 4k} \]