If `(x)/(y)=9/8`, then the value of `(6/7+(y-x)/(y+x))` is

Options

A. `9/119`
B. `95/119`
C. `19/119`
D. `1 9/119`

Answer

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

\[ \frac{x}{y} = \frac{9}{8} \]

We need to find the value of:

\[ \frac{6}{7} + \frac{y - x}{y + x} \]

### Step 1: Express \(y\) in terms of \(x\)

From the equation \(\frac{x}{y} = \frac{9}{8}\), we can express \(y\) in terms of \(x\):

\[ y = \frac{8}{9}x \]

### Step 2: Substitute \(y\) in the expression

Now, substitute \(y\) into the expression \(\frac{y - x}{y + x}\):

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

### Step 3: Simplify the numerator and denominator

First, simplify the numerator:

\[ \frac{8}{9}x - x = \frac{8}{9}x - \frac{9}{9}x = \frac{8 - 9}{9}x = \frac{-1}{9}x \]

Now simplify the denominator:

\[ \frac{8}{9}x + x = \frac{8}{9}x + \frac{9}{9}x = \frac{8 + 9}{9}x = \frac{17}{9}x \]

### Step 4: Combine the simplified numerator and denominator

Now we can write:

\[ \frac{y - x}{y + x} = \frac{\frac{-1}{9}x}{\frac{17}{9}x} \]

The \(x\) cancels out (assuming \(x \neq 0\)):

\[ \frac{y - x}{y + x} = \frac{-1}{17} \]

### Step 5: Substitute back into the original expression

Now we substitute this back into the original expression:

\[ \frac{6}{7} + \frac{y - x}{y + x} = \frac{6}{7} + \frac{-1}{17} \]

### Step 6: Find a common denominator

To add these fractions, we need a common denominator. The least common multiple of 7 and 17 is 119.

Convert \(\frac{6}{7}\) and \(\frac{-1}{17}\) to have a denominator of 119:

\[ \frac{6}{7} = \frac{6 \times 17}{7 \times 17} = \frac{102}{119} \]

\[ \frac{-1}{17} = \frac{-1 \times 7}{17 \times 7} = \frac{-7}{119} \]

### Step 7: Add the fractions

Now we can add the two fractions:

\[ \frac{102}{119} + \frac{-7}{119} = \frac{102 - 7}{119} = \frac{95}{119} \]

### Final Answer

Thus, the value of \(\frac{6}{7} + \frac{y - x}{y + x}\) is:

\[ \frac{95}{119} \]