If `(x^(3)+4xy^(2))/(4x^(2)y+y^(3))=15/16` , then find x : y.

Options

A. ` 2 : 3`
B. ` 3 : 2`
C. ` 4 : 3`
D. ` 3 : 4`

Explanation

To solve the equation

\[ \frac{x^3 + 4xy^2}{4x^2y + y^3} = \frac{15}{16}, \]

we will follow these steps:

### Step 1: Cross Multiply
We start by cross-multiplying to eliminate the fraction:

\[ 16(x^3 + 4xy^2) = 15(4x^2y + y^3). \]

### Step 2: Expand Both Sides
Now, we will expand both sides of the equation:

\[ 16x^3 + 64xy^2 = 60x^2y + 15y^3. \]

### Step 3: Rearrange the Equation
Next, we will rearrange the equation to bring all terms to one side:

\[ 16x^3 + 64xy^2 - 60x^2y - 15y^3 = 0. \]

### Step 4: Factor the Equation
Now, we will try to factor the equation. We can group the terms:

\[ 16x^3 - 60x^2y + 64xy^2 - 15y^3 = 0. \]

This can be factored as:

\[ 4x^2(4x - 15y) + y^2(64x - 15y) = 0. \]

### Step 5: Set Each Factor to Zero
We can set each factor to zero:

1. \(4x^2 = 0\) (which gives \(x = 0\), but we will not consider this since we want the ratio \(x:y\)).
2. \(4x - 15y = 0\) (which gives \(x = \frac{15}{4}y\)).
3. \(64x - 15y = 0\) (which gives \(x = \frac{15}{64}y\)).

### Step 6: Find the Ratio
From the equations \(x = \frac{15}{4}y\) and \(x = \frac{15}{64}y\), we can find the ratio \(x:y\).

For \(x = \frac{15}{4}y\):

\[ \frac{x}{y} = \frac{15}{4} \quad \Rightarrow \quad x:y = 15:4. \]

For \(x = \frac{15}{64}y\):

\[ \frac{x}{y} = \frac{15}{64} \quad \Rightarrow \quad x:y = 15:64. \]

Thus, the final ratios are \(15:4\) and \(15:64\).

### Final Answer
The possible ratios of \(x:y\) are \(15:4\) and \(15:64\).

---