If `(r+s)/(x-y)=(3)/(4)`, then `(8r+8s)/(15x-15y)=`

Options

A. `(32)/(45)`
B. `(8)/(15)`
C. `(7)/(16)`
D. `(2)/(5)`

Answer

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

\[ \frac{r+s}{x-y} = \frac{3}{4} \]

We need to find the value of:

\[ \frac{8r + 8s}{15x - 15y} \]

### Step 1: Factor out constants from the numerator and denominator

We can factor out constants from the numerator and denominator:

\[ \frac{8(r+s)}{15(x-y)} \]

### Step 2: Substitute the known value

From the original equation, we know:

\[ \frac{r+s}{x-y} = \frac{3}{4} \]

Now, we can substitute \( \frac{r+s}{x-y} \) into our expression:

\[ \frac{8(r+s)}{15(x-y)} = \frac{8}{15} \cdot \frac{r+s}{x-y} \]

### Step 3: Replace \( \frac{r+s}{x-y} \) with \( \frac{3}{4} \)

Substituting \( \frac{r+s}{x-y} = \frac{3}{4} \):

\[ \frac{8}{15} \cdot \frac{3}{4} \]

### Step 4: Simplify the expression

Now we can simplify the expression:

\[ \frac{8 \cdot 3}{15 \cdot 4} = \frac{24}{60} \]

### Step 5: Reduce the fraction

Now, we can reduce \( \frac{24}{60} \):

\[ \frac{24 \div 12}{60 \div 12} = \frac{2}{5} \]

Thus, the final answer is:

\[ \frac{8r + 8s}{15x - 15y} = \frac{2}{5} \]

### Final Answer

The value of \( \frac{8r + 8s}{15x - 15y} \) is \( \frac{2}{5} \).

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