If x and y are two numbers such that x / y = 2 / 3, then what is the ratio of (3x + 2y) to (3x − y)?

Introduction / Context:
This algebra question uses a given ratio between two variables x and y and asks for the ratio of two linear expressions in x and y. Instead of trying arbitrary numbers, we can assign convenient values to x and y that preserve the given ratio and then evaluate the expressions. This is a standard technique in ratio problems involving algebraic expressions.

Given Data / Assumptions:

• x / y = 2 / 3.
• x and y are non zero real numbers.
• We need the ratio (3x + 2y) : (3x − y).

Concept / Approach:
From x / y = 2 / 3, we can let x = 2k and y = 3k for some non zero constant k. This representation keeps the ratio intact while making calculations simple. Substituting these into the expressions 3x + 2y and 3x − y converts them into expressions in terms of k alone, which then cancel out in the ratio. Finally we simplify the resulting numeric ratio.

Step-by-Step Solution:
Given x / y = 2 / 3, let x = 2k and y = 3k. Compute 3x + 2y: 3x + 2y = 3 * (2k) + 2 * (3k) = 6k + 6k = 12k. Compute 3x − y: 3x − y = 3 * (2k) − 3k = 6k − 3k = 3k. Required ratio = (3x + 2y) : (3x − y) = 12k : 3k. Divide both parts by 3k: 12k / 3k = 4 and 3k / 3k = 1. Therefore the ratio is 4 : 1.

Verification / Alternative check:
We can choose explicit numbers for x and y that satisfy x / y = 2 / 3. Take x = 2 and y = 3. Then 3x + 2y = 3 * 2 + 2 * 3 = 6 + 6 = 12. Also, 3x − y = 3 * 2 − 3 = 6 − 3 = 3. The ratio 12 : 3 simplifies exactly to 4 : 1, confirming the result. Any other pair (x, y) that satisfies the original ratio, such as (4, 6), will produce the same final ratio.

Why Other Options Are Wrong:
3 : 1 or 2 : 1 would require the numerator or denominator to differ by some factor not supported by the substitution from x / y = 2 / 3. 4 : 3 and 5 : 2 do not match calculations for any consistent value of k and therefore are incorrect.

Common Pitfalls:
A common mistake is to attempt to manipulate the expression (3x + 2y) / (3x − y) symbolically but misapply the given ratio. Another error is to incorrectly set x = 2 and y = 3 without recognizing that they could be multiplied by a common factor, though that does not actually change the outcome if used correctly. The safest method is to express x and y as 2k and 3k and then simplify systematically.

Final Answer:
The ratio of (3x + 2y) to (3x − y) is 4 : 1.