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.

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