Question

What will be the ratio of simple interest earned by a certain amount at the same rate of interest for 6 years and that for 4 years?
(a) 1 : 3
(b) 2 : 3
(c) 1 : 4
(d) 3 : 2

Answer 1

Hint:In this question, we will use a simple interest formula for both time periods and then find the ratio of both the terms.

Complete step-by-step answer:
Let, P be the principal amount that is taken or given on loan. Let, R be the percentage rate of interest per annum on P and let T be the time duration of the loan.
Then, the formula to calculate simple interest is,
Simple interest $=\dfrac{\text{P }\!\!\times\!\!\text{ R }\!\!\times\!\!\text{ T}}{100}$.
Now, in a given question, let the principle amount be P and the rate of interest on which the amount is given on loan be R.
Let, simple interest earned for P, given on rate R for 4 years be ${{S}_{1}}$.
Therefore, using formula of simple interest we have,
${{S}_{1}}=\dfrac{\text{P }\!\!\times\!\!\text{ R }\!\!\times\!\!\text{ 4}}{100}$.
And, let, simple interest earned for P, given on rate R for 6 years be ${{S}_{2}}$.
Therefore, using formula of simple interest we have,
${{S}_{2}}=\dfrac{\text{P }\!\!\times\!\!\text{ R }\!\!\times\!\!\text{ 6}}{100}$.
Now, ratio of ${{S}_{2}}$ to ${{S}_{1}}$is,
$\begin{align}
  & \dfrac{{{S}_{2}}}{{{S}_{1}}}=\dfrac{\dfrac{\text{P }\!\!\times\!\!\text{ R }\!\!\times\!\!\text{ 6}}{100}}{\dfrac{\text{P }\!\!\times\!\!\text{ R }\!\!\times\!\!\text{ 4}}{100}} \\
 & =\dfrac{\text{P }\!\!\times\!\!\text{ R }\!\!\times\!\!\text{ 6}}{100}\div \dfrac{\text{P }\!\!\times\!\!\text{ R }\!\!\times\!\!\text{ 4}}{100} \\
\end{align}$
Changing divide sign to multiply by taking reciprocal, we get,
 $\dfrac{{{S}_{2}}}{{{S}_{1}}}=\dfrac{\text{P }\!\!\times\!\!\text{ R }\!\!\times\!\!\text{ 6}}{100}\times \dfrac{100}{\text{P }\!\!\times\!\!\text{ R }\!\!\times\!\!\text{ 4}}$
Cancelling equal terms, we get,
 $\dfrac{{{S}_{2}}}{{{S}_{1}}}=\dfrac{6}{\text{4}}=\dfrac{3}{2}$.
Hence, the ratio of simple interest earned by a certain amount at the same rate of interest for 6 years and that for 4 years is 3 : 2.

Note: In this type of questions, when all the terms, that is P, R, T are the same except one of them for two different cases, then the ratio of the simple interests of both the cases will be the ratio of unequal terms.