What will be the ratio of simple interest earned by certain amount at the same rate of interest for four years and that for seven years?

Answer

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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.

What will be the ratio of simple interest earned by a certain amount at the same rate of interest for 2 years and for 4 years?

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