If `x: y = 3 : 2`, then the ratio `2x^(2) + 3y^(2) : 3x^(2) - 2y^(2)` is equal to :

Options

A. `12:5`
B. `6:5`
C. `30:19`
D. `5:3`

Answer

To solve the problem, we need to find the ratio \(2x^2 + 3y^2 : 3x^2 - 2y^2\) given that \(x : y = 3 : 2\).

### Step-by-Step Solution:

1. **Express x and y in terms of a common variable:**
Since \(x : y = 3 : 2\), we can let \(x = 3k\) and \(y = 2k\) for some variable \(k\).

2. **Calculate \(x^2\) and \(y^2\):**
\[ x^2 = (3k)^2 = 9k^2 \]
\[ y^2 = (2k)^2 = 4k^2 \]

3. **Substitute \(x^2\) and \(y^2\) into the expressions:**
- For \(2x^2 + 3y^2\):
\[ 2x^2 + 3y^2 = 2(9k^2) + 3(4k^2) = 18k^2 + 12k^2 = 30k^2 \]

- For \(3x^2 - 2y^2\):
\[ 3x^2 - 2y^2 = 3(9k^2) - 2(4k^2) = 27k^2 - 8k^2 = 19k^2 \]

4. **Form the ratio:**
Now we can form the ratio:
\[ 2x^2 + 3y^2 : 3x^2 - 2y^2 = 30k^2 : 19k^2 \]

5. **Simplify the ratio:**
Since \(k^2\) is common in both terms, we can cancel it out:
\[ 30 : 19 \]

### Final Answer:
Thus, the ratio \(2x^2 + 3y^2 : 3x^2 - 2y^2\) is \(30 : 19\).

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