Discussion
\(x\) varies directly as \(y\) and inversely as \(z\). When \(y = 1.5\) and \(z = 2\), then \(x = 5\). If \(z = 2.5\) and \(y = 9\), then what is the value of \(x\)?
Answer: A
The value of \(x\) is 24. Direct variation with \(y\) and inverse variation with \(z\) means \(x = \frac{ky}{z}\). From the first case: \[ \begin{aligned} 5 &= \frac{k\times 1.5}{2} \Rightarrow k \\ &= \frac{20}{3} \end{aligned} \] For the second case: \[ \begin{aligned} x &= \frac{20}{3}\times\frac{9}{2.5} \\ &= \frac{20\times 9}{3\times 2.5} \\ &= 24 \end{aligned} \] Because \(x\) falls as \(z\) rises and rises with \(y\), the value must exceed 5 by the factor \(\frac{9}{1.5}\times\frac{2}{2.5} = 4.8\), and \(5\times 4.8 = 24\); 20, 15 and 12 do not satisfy this.
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