If `(3x -4y) : (2x -3y) = (5x -6y) : (4x -5y)`, find : `x : y`.

Answer

To solve the problem `(3x - 4y) : (2x - 3y) = (5x - 6y) : (4x - 5y)`, we will follow these steps:

### Step 1: Write the ratios as fractions
We start by expressing the ratios as fractions:
\[ \frac{3x - 4y}{2x - 3y} = \frac{5x - 6y}{4x - 5y} \]

### Step 2: Apply the component and dividend rule
According to the component and dividend rule, if we have two equal ratios, we can write:
\[ \frac{(3x - 4y) + (2x - 3y)}{(3x - 4y) - (2x - 3y)} = \frac{(5x - 6y) + (4x - 5y)}{(5x - 6y) - (4x - 5y)} \]

### Step 3: Simplify the left side
Now we will simplify the left side:
- Numerator:
\[ (3x - 4y) + (2x - 3y) = 3x + 2x - 4y - 3y = 5x - 7y \]
- Denominator:
\[ (3x - 4y) - (2x - 3y) = 3x - 2x - 4y + 3y = x - y \]
Thus, the left side becomes:
\[ \frac{5x - 7y}{x - y} \]

### Step 4: Simplify the right side
Now we simplify the right side:
- Numerator:
\[ (5x - 6y) + (4x - 5y) = 5x + 4x - 6y - 5y = 9x - 11y \]
- Denominator:
\[ (5x - 6y) - (4x - 5y) = 5x - 4x - 6y + 5y = x - y \]
Thus, the right side becomes:
\[ \frac{9x - 11y}{x - y} \]

### Step 5: Set the two fractions equal to each other
Now we have:
\[ \frac{5x - 7y}{x - y} = \frac{9x - 11y}{x - y} \]

### Step 6: Cross-multiply
Since the denominators are the same, we can cross-multiply:
\[ (5x - 7y) = (9x - 11y) \]

### Step 7: Rearrange the equation
Rearranging gives us:
\[ 5x - 9x = -11y + 7y \]
\[ -4x = -4y \]

### Step 8: Simplify the equation
Dividing both sides by -4 gives:
\[ x = y \]

### Step 9: Write the ratio
Thus, we can express the ratio \( x : y \) as:
\[ x : y = 1 : 1 \]

### Final Answer
The final answer is:
\[ x : y = 1 : 1 \]