If x : y = 3 : 2, then the ratio of 2x² + 3y² : 3x² − 2y² is equal to -

To find the ratio of 2x^2 + 3y^2 : 3x^2 - 2y^2 given that x : y = 3 : 2, follow these steps:

Step-by-Step Solution
Represent the variables: Since x : y = 3 : 2, let x = 3k and y = 2k (where k is a constant).

Substitute into the numerator: The expression 2x^2 + 3y^2 becomes 2(3k)^2 + 3(2k)^2.

This simplifies to 2(9k^2) + 3(4k^2) = 18k^2 + 12k^2 = 30k^2.

Substitute into the denominator: The expression 3x^2 - 2y^2 becomes 3(3k)^2 - 2(2k)^2.

This simplifies to 3(9k^2) - 2(4k^2) = 27k^2 - 8k^2 = 19k^2.

Find the ratio: The ratio is 30k^2 : 19k^2. Canceling out k^2, we get 30 : 19.