Question

Calculate the time required for simple interest earned on Rs. 3000 at the rate of \[4\%\] per annum to be equal to the simple interest on Rs. 8000 at the rate of $8\%$ per annum for 3 years.

Answer 1

Hint: We will first find out the simple interest on the principal of 8000 Rs. by the formula for simple interest $S.I.=\dfrac{P\times R\times T}{100}$ , where R=8% and T=3 years. We will then equate it to the simple interest calculated on Rs. 3000 with R=4% and T=t. When we equate we will get the time required as our answer.

Complete step by step answer:
We know that the formula for finding the simple interest is as follows:
$S.I.=\dfrac{P\times R\times T}{100}$ ; Where P = Principal, R= rate of interest, T= time period
Let’s first find out the simple interest on Rs. 8000 at the rate of $8\%$ per annum for 3 years:
Let the simple interest be ${{S}_{1}}$. Now here $P=8000$ , $R=8$ , $T=3$
${{S}_{1}}=\dfrac{8000\times 8\times 3}{100}\Rightarrow {{S}_{1}}=1920\text{ }..........\text{Equation 1}\text{.}$
Now it is given in the question that we have to find the time required for the simple interest earned on Rs. 3000 at the rate of \[4\%\] per annum to be equal to the ${{S}_{1}}$.
Let the simple interest be ${{S}_{2}}$. Now here $P=3000$ , $R=4$ , $T=t$
${{S}_{2}}=\dfrac{3000\times 4\times t}{100}\Rightarrow {{S}_{2}}=120t..........\text{Equation 2}\text{.}$
Now it is given that the simple interest must me equal therefore: ${{S}_{1}}={{S}_{2}}$ , we will put values from equation 1 and equation 2 which will give us:
 $\begin{align}
  & {{S}_{1}}={{S}_{2}}\Rightarrow 1920=120t \\
 & t=\dfrac{1920}{120}\Rightarrow t=16 \\
\end{align}$

Therefore, time required to collect earn the simple interest is $t=16$ years.

Note: Rate per annum is not written as percent in the formula as we wrote 100 in the denominator in its place. You can also equate both the simple interests at initial stage that is when you put values in the formula:
$\begin{align}
  & {{S}_{1}}=\dfrac{8000\times 8\times 3}{100}=\dfrac{3000\times 4\times t}{100}={{S}_{2}} \\
 & \Rightarrow t=16 \\
\end{align}$
This will make the calculation easy and less chances of mistakes will be there.