Discussion
If \(x : y = p : q\), then which of the following is true ?
(a) \(x + y : y = p + q : q\)
(b) \(x - y : y = p - q : q\)
(c) \(x : p = y : q\)
(d) \(x + y : x - y = p - q : p + q\)
Answer: D
The true statements are (a), (b) and (c).
Write \(x = kp\) and \(y = kq\) for some non-zero k. Then:
\[ \begin{aligned} \tfrac{x + y}{y} &= \tfrac{k(p + q)}{kq} = \tfrac{p + q}{q} \\ \tfrac{x - y}{y} &= \tfrac{p - q}{q}, \quad \tfrac{x}{p} = k = \tfrac{y}{q} \end{aligned} \]
These are componendo (a), dividendo (b) and alternendo (c). Statement (d) is wrong: componendo and dividendo give \(x + y : x - y = p + q : p - q\), not \(p - q : p + q\), so any option containing (d) is ruled out.
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