If `(x + y) : (x- y) = 11 : 1`, find the valuee of `((5x + 3y)/(x-2y))`

Options

A. `45/4`
B. `4/25`
C. `(-45)/4`
D. `-4/45`

Answer

To solve the problem, we start with the given ratio:

\[ (x + y) : (x - y) = 11 : 1 \]

### Step 1: Set up the equation
From the ratio, we can write:

\[ \frac{x + y}{x - y} = \frac{11}{1} \]

### Step 2: Cross-multiply
Cross-multiplying gives us:

\[ x + y = 11(x - y) \]

### Step 3: Expand the equation
Expanding the right side:

\[ x + y = 11x - 11y \]

### Step 4: Rearrange the equation
Now, rearranging the terms to isolate \(x\) and \(y\):

\[ x + y + 11y = 11x \]

This simplifies to:

\[ 12y = 10x \]

### Step 5: Simplify the equation
Dividing both sides by 2 gives:

\[ 6y = 5x \]

### Step 6: Express \(x\) in terms of \(y\)
From this, we can express \(x\) in terms of \(y\):

\[ \frac{x}{y} = \frac{6}{5} \implies x = 6k \quad \text{and} \quad y = 5k \]

for some constant \(k\).

### Step 7: Substitute \(x\) and \(y\) into the expression
Now we need to find the value of:

\[ \frac{5x + 3y}{x - 2y} \]

Substituting \(x\) and \(y\):

\[ = \frac{5(6k) + 3(5k)}{6k - 2(5k)} \]

### Step 8: Simplify the numerator and denominator
Calculating the numerator:

\[ 5(6k) + 3(5k) = 30k + 15k = 45k \]

Calculating the denominator:

\[ 6k - 10k = -4k \]

### Step 9: Combine the results
Now substituting back into the fraction:

\[ = \frac{45k}{-4k} \]

### Step 10: Cancel \(k\) and simplify
Cancelling \(k\) (assuming \(k \neq 0\)) gives:

\[ = \frac{45}{-4} = -\frac{45}{4} \]

### Final Answer
Thus, the value of \(\frac{5x + 3y}{x - 2y}\) is:

\[ -\frac{45}{4} \]