If `5x + 6y : 8x + 5y = 8 : 9`, find : `x : y`.

Answer

To solve the problem, we need to find the ratio \( x : y \) given the equation:

\[ \frac{5x + 6y}{8x + 5y} = \frac{8}{9} \]

### Step 1: Cross Multiply
We start by cross-multiplying to eliminate the fraction:

\[ 9(5x + 6y) = 8(8x + 5y) \]

### Step 2: Distribute
Next, we distribute the numbers on both sides:

\[ 45x + 54y = 64x + 40y \]

### Step 3: Rearrange the Equation
Now, we rearrange the equation to group the \( x \) terms on one side and the \( y \) terms on the other side:

\[ 45x - 64x = 40y - 54y \]

This simplifies to:

\[ -19x = -14y \]

### Step 4: Simplify the Equation
We can simplify this equation further by dividing both sides by -1:

\[ 19x = 14y \]

### Step 5: Find the Ratio
To find the ratio \( x : y \), we can express it in terms of \( y \):

\[ \frac{x}{y} = \frac{14}{19} \]

Thus, the ratio \( x : y \) is:

\[ x : y = 14 : 19 \]

### Final Answer
The final answer is:

\[ x : y = 14 : 19 \]

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