If `(15x + 16y)/(25x + 4y) = (7)/(6)`, then find x:y.

Answer

To solve the equation \(\frac{15x + 16y}{25x + 4y} = \frac{7}{6}\) and find the ratio \(x:y\), we can follow these steps:

### Step-by-Step Solution:

1. **Cross Multiply**:
Start by cross-multiplying to eliminate the fraction:
\[ 6(15x + 16y) = 7(25x + 4y) \]

2. **Distribute**:
Distribute both sides:
\[ 90x + 96y = 175x + 28y \]

3. **Rearrange the Equation**:
Move all terms involving \(x\) to one side and all terms involving \(y\) to the other side:
\[ 90x - 175x = 28y - 96y \]
This simplifies to:
\[ -85x = -68y \]

4. **Simplify**:
Remove the negative signs:
\[ 85x = 68y \]

5. **Express the Ratio**:
To find the ratio \(x:y\), divide both sides by \(y\) and \(85\):
\[ \frac{x}{y} = \frac{68}{85} \]

6. **Simplify the Ratio**:
To simplify \(\frac{68}{85}\), find the greatest common divisor (GCD) of 68 and 85, which is 17:
\[ \frac{68 \div 17}{85 \div 17} = \frac{4}{5} \]

7. **Final Ratio**:
Thus, the ratio \(x:y\) is:
\[ x:y = 4:5 \]