Question : If $x, y,$ and $z$ are three sums of money such that y is the simple interest on $x$ and $z$ is the simple interest on $y$ for the same time and at the same rate of interest, then we have:
Option 1: $z^{2}=xy$
Option 2: $xyz=1$
Option 3: $x^{2}=yz$
Option 4: $y^{2}=zx$
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Correct Answer: $y^{2}=zx$
Solution :
Let the time be $t$ years and the rate of interest be $R$.
We know, Simple interest = $\frac{\text{Principal × Rate × Time}}{100}$
According to the question,
Case(I): $y=\frac{x\times R\times t}{100}$-------------(i)
Case(II): $z=\frac{y\times R\times t}{100}$------------(ii)
By dividing equation (i) by equation (ii), we get,
$\frac{y}{z}=\frac{x\times R\times t}{y\times R\times t}$
$⇒y^{2}=zx$
Hence, the correct answer is $y^{2}=zx$.
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