If x, y, z are three sums of money such that y is the simple interest on x, z is the simple interest on y for the same time and at the same rate of interest, then we have.

1x² = yz
2y² = xz
3z² = xy
4xyz = 1

Answer:- 1

Explanation:-

Solution:

Let time be T years and rate be R% p.a.

\begin{aligned} Then, y is the S .I . on x \\ \Rightarrow \frac{x RT}{100} = y . . . . . . (i) \\ And, z is the S .I . on y \\ \Rightarrow \frac{y RT}{100} = z \\ \Rightarrow y = \frac{100 z}{R T} . . . . . . (i i) \\ From (i) and (ii) we have: \\ \frac{x RT}{100} = \frac{100 z}{RT} \\ \Rightarrow \frac{x R^{2} T^{2}}{{(100)}^{2}} = z \\ \Rightarrow \frac{y^{2}}{x} z \\ \Rightarrow y^{2} \\ = x z . \end{aligned}

$\eqalign{ & {\text{Then, }}y{\text{ is the S}}{\text{.I}}{\text{. on x}} \cr & \Rightarrow \frac{{x{\text{RT}}}}{{100}} = y......(i) \cr & {\text{And, }}z{\text{ is the S}}{\text{.I}}{\text{. on y}} \cr & \Rightarrow \frac{{y{\text{RT}}}}{{100}} = z \cr & \Rightarrow y = \frac{{100z}}{{RT}}......(ii) \cr & {\text{From (i) and (ii) we have:}} \cr & \frac{{x{\text{RT}}}}{{100}} = \frac{{100z}}{{{\text{RT}}}} \cr & \Rightarrow \frac{{x{{\text{R}}^2}{{\text{T}}^2}}}{{{{\left( {100} \right)}^2}}} = z \cr & \Rightarrow \frac{{{y^2}}}{x}z \cr & \Rightarrow {y^2} \cr & = xz. \cr}$