Question

If a, b and c are three sums of money such that b is the simple interest on a and c is the simple interest on b for the same time and same rate. Which of the following is correct?

(a) abc = 1
(b) c² = ab
(c) b² = ac
(d) a² = bc

Answer 1

It is given that,
Simple interest (I₁) = b
Sum (P₁) = a
Rate (R₁) = R%
Time (T₁) = T years

Now,
$$\begin{gathered} I_1=\frac{P_1\times R_1\times T_1}{100} \\ \Rightarrow b=\frac{a\times R\times T}{100} \\ \Rightarrow R\times T=\frac{100b}{a}....\left(1\right) \end{gathered}$$

Also,
Simple interest (I₂) = c
Sum (P₂) = b
Rate (R₂) = R%
Time (T₂) = T years

Now,
$$\begin{gathered} I_2=\frac{P_2\times R_2\times T_2}{100} \\ \Rightarrow c=\frac{b\times R\times T}{100} \\ \Rightarrow R\times T=\frac{100c}{b}....\left(2\right) \end{gathered}$$

On equating (1) and (2), we get
$$\begin{gathered} \frac{100b}{a}=\frac{100c}{b} \\ \Rightarrow b^2=ac \end{gathered}$$

Hence, the correct option is (c).