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At a certain rate of simple interest, a certain sum doubles itself in $10$years. It will triple itself in (A)$12$ years (B)$15$ years (C) $20$ years (D) $30$ years

Last updated date: 13th Jun 2024
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Hint: Simple interest is interest calculated on the principal portion of a loan or the original contribution to a saving account. The formula for the simple interest is $S.I. = \dfrac{{P \times R \times T}}{{100}}$. We are going to use the term Amount $A$ in the solution and it stands for sum of simple interest and the principal. It is the amount to which the borrower has to pay after a particular time.

Complete step by step solution:Simple Interest = $S.I.$ (in Rupees)
Amount = $A$ (in Rupees)
Principal =$P$ (in Rupees)
Time = $T$ (in years)
Rate = $R$ (in percentage per annum)
Main goal of the question is to find the time in which the principal will triple to itself. For that we should focus and simplify the given condition.
As given in the question,
In $10$ years principal $P$ becomes $2P$.
In $10$ years,
$A = 2 \times P$
Since Amount $A$ is considered as the sum of principal and the simple interest after $n$ years.
$ \Rightarrow P + S.I. = 2 \times P$
$ \Rightarrow 2P - P = S.I.$
$ \Rightarrow P = S.I.$
As we know the formula for simple interest is $S.I. = \dfrac{{P \times R \times T}}{{100}}$ . On substituting the formula,
$ \Rightarrow P = \dfrac{{P \times R \times T}}{{100}}$
Since it is given that $T = 10$ years.
$ \Rightarrow P = \dfrac{{P \times R \times 10}}{{100}}$
\[ \Rightarrow P = \dfrac{{P \times R}}{{10}}\]
$ \Rightarrow 10 \times P = P \times R$
$ \Rightarrow \dfrac{{10 \times P}}{P} = R$
$ \Rightarrow 10 = R$
Now, by simplifying the given condition we have the rate of interest as $R = 10\% $ per annum.
The time required to triple the principal (say $t$ ) can be found by the formula of Amount and simple interest.
$A = P + S.I.$
$ \Rightarrow 3 \times P = P + S.I.$ [According to question]
$ \Rightarrow 3 \times P - P = S.I.$
$ \Rightarrow 2P = S.I.$
$ \Rightarrow 2 \times P = \dfrac{{P \times R \times t}}{{100}}$
$ \Rightarrow 2 \times P \times 100 = P \times R \times t$
$ \Rightarrow \dfrac{{200 \times P}}{P} = R \times t$
$ \Rightarrow 200 = 10 \times t$
$ \Rightarrow t = 20$

Hence, the time required to triple the principal is $t = 20$ years.

Note: Generally the interest is of two types either simple or compounded. Simple interest is based on the principal amount of a loan or deposit. In contrast, compound interest is based on the principal amount and the interest that accumulates on it in every period.