This is a mathematical problem involving two variables $a$ and $b$, along with ratios and specific conditions to determine the values of $x$ and $y$, ultimately finding $x + y$.

Here’s how we interpret and solve the problem step by step:

Problem Interpretation:

1. Conditions and Definitions:
   • $a > b > 0$
   • The ratio of the sum ($a + b$) and the difference ($a - b$) is $3 : 2$. $\frac{a + b}{a - b} = \frac{3}{2}$
   • Let $\frac{a}{b} = x$.
   • Another condition is given: $a - 2b = 3a - b$, and you are asked to find $\frac{a - b}{a + b} = y$.
   • Finally, find $x + y$.

Solution Steps:

Step 1: Solve $\frac{a + b}{a - b} = \frac{3}{2}$:

From the ratio: $\frac{a + b}{a - b} = \frac{3}{2}$ Cross-multiply: $2(a + b) = 3(a - b)$ Simplify: $2a + 2b = 3a - 3b$ $a = 5b$

Step 2: Find $x = \frac{a}{b}$:

Since $a = 5b$, $x = \frac{a}{b} = \frac{5b}{b} = 5$

Step 3: Solve for $y = \frac{a - b}{a + b}$:

Using $a = 5b$, $a - b = 5b - b = 4b,$ $a + b = 5b + b = 6b.$ Thus, $y = \frac{a - b}{a + b} = \frac{4b}{6b} = \frac{2}{3}.$

Step 4: Find $x + y$:

$x + y = 5 + \frac{2}{3} = \frac{15}{3} + \frac{2}{3} = \frac{17}{3}.$

Final Answer:

$x + y = \frac{17}{3}$

Would you like further clarification or additional examples on similar problems?

Related Questions:

1. What is the derivation of the condition $a - 2b = 3a - b$?
2. Can you calculate $x + y$ for different ratios, such as $\frac{a+b}{a-b} = \frac{4}{3}$?
3. How do we handle the problem if $a < b$?
4. What happens if the ratio is not given directly but inferred from other conditions?
5. Can $x$ and $y$ be expressed in terms of other variables if more conditions are given?

Tip:

Always simplify ratios and use substitution to solve problems with multiple conditions systematically.