Question

At what percent per annum simple interest will double a sum of money in 12 years?

Answer 1

Hint: In this question, we have to find the rate of interest per annum at which simple interest doubles the sum of money in 12 years. For this, we will first suppose the rate of interest at R% per annum and principal amount as P. We will use, Amount = P + Simple interest to calculate the value of the simple interest in terms of P. Here amount will be twice of P. Using the value of SI, P, given time period T in formula $SI=\dfrac{P\times R\times T}{100}$ we will calculate R which is the rate of interest per annum. Here, SI denotes simple interest.

Complete step by step solution:
Here, we are given a time period of 12 years. Therefore, T = 12 years.
Let us suppose the principal amount deposited was Rs.P. After 12 years, the amount doubles itself. Therefore, the amount after the addition of simple interest becomes twice P which is 2P. Hence, A = 2P.
Now, we know, A = P + SI where A is the amount after simple interest, P is the principal amount and SI is the simple interest. Therefore,
\[\begin{align}
  & \Rightarrow 2P=P+SI \\
 & \Rightarrow SI=2P-P \\
 & \Rightarrow SI=P \\
\end{align}\]
Hence, simple interest is Rs.P
As we know, simple interest SI for some principal amount P at the rate of R% per annum for T years is given by $SI=\dfrac{P\times R\times T}{100}$.
So, for this question, supposing the rate of interest as R% per annum and putting all values, we get:
$P=\dfrac{P\times R\times 12}{100}$.
Cancelling P from 12 both sides, we get:
\[1=\dfrac{12R}{100}\].
Taking $\dfrac{12}{100}$ on other side, we get:
$\dfrac{100}{12}=R$.
Now, $\dfrac{100}{12}$ can be written as $8\dfrac{1}{3}$. So, R becomes equal to $8\dfrac{1}{3}$.
Hence, the rate of interest per annum becomes equal to $8\dfrac{1}{3}\%$ per annum.
Hence, option A is the correct answer.

Note: In this question, we had to find the rate of interest per annum. Since the rate of interest is calculated in percentage, students can make mistakes by putting $\dfrac{1}{100}$ on the answer obtained. They should note that we had already included $\dfrac{1}{100}$ in the formula of SI. So we just need numerical value. We have calculated the rate of interest in mixed fractions to match our answer. Students can also convert it into a decimal number.