If \(\displaystyle\frac{x}{y}=\frac{3}{5}\) and \(\displaystyle\frac{y}{z}=\frac{5}{2}\), then \(\displaystyle\frac{x^2+z^2}{x^2-z^2}\) would be
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\displaystyle\frac{13}{5}\)
Both ratios contain \(y\) with the same value 5, so they join into one ratio \(x:y:z\), and \(x\) and \(z\) can be written with one common multiplier.
Joining the ratios (\(x:y=3:5\) and \(y:z=5:2\) share \(y=5\)):
\(x:y:z=3:5:2\)
\(x\) and \(z\) are 3 and 2 parts of the same ratio, so take \(x=3k\) and \(z=2k\)
Putting these in (\(k^2\) cancels from top and bottom):
\(\displaystyle \frac{x^2+z^2}{x^2-z^2}=\frac{9k^2+4k^2}{9k^2-4k^2}\)
\(\displaystyle =\frac{13k^2}{5k^2}=\frac{13}{5}\)