If `p /q = r/s , ` then `((p +q)/( r + s )) = ?`

Options

A. a) `(pq)/(rs) `
B. b) `(pr)/(qs)`
C. c) `((p - q)/(r -s))^2`
D. d) 1

Answer

To solve the problem where \( \frac{p}{q} = \frac{r}{s} \) and we need to find the value of \( \frac{p + q}{r + s} \), we can use the concept of the Compitendo and Dividento rule. Here’s a step-by-step solution:

### Step 1: Start with the given ratio
We know that:
\[ \frac{p}{q} = \frac{r}{s} \]

### Step 2: Apply the Compitendo and Dividento rule
According to the Compitendo and Dividento rule, if \( \frac{a}{b} = \frac{c}{d} \), then:
\[ \frac{a + b}{a - b} = \frac{c + d}{c - d} \]
Applying this to our equation:
\[ \frac{p + q}{p - q} = \frac{r + s}{r - s} \]

### Step 3: Cross-multiply
Now, we cross-multiply to get:
\[ (p + q)(r - s) = (r + s)(p - q) \]

### Step 4: Rearranging the equation
We want to find \( \frac{p + q}{r + s} \). From the previous step, we can express \( p + q \) in terms of \( r + s \):
\[ p + q = \frac{(r + s)(p - q)}{(r - s)} \]

### Step 5: Find \( \frac{p + q}{r + s} \)
Now, we can express \( \frac{p + q}{r + s} \):
\[ \frac{p + q}{r + s} = \frac{(r + s)(p - q)}{(r - s)(r + s)} \]
This simplifies to:
\[ \frac{p + q}{r + s} = \frac{p - q}{r - s} \]

### Final Answer
Thus, the value of \( \frac{p + q}{r + s} \) is:
\[ \frac{p - q}{r - s} \]

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