Question

In a simple interest, at what rate percent per annum will a sum of money double in $8$ years?
A.$12.5%$
B.$10.5%$
C.$12.0%$
D.$15.5%$

Answer 1

Hint: Simple interest is one way that interest can be calculated on a loan or investment. The standard formula for simple interest is given as simple interest(S.I) $=\dfrac{P\times R\times T}{100}$.Here, $P$ denotes principal amount of money to be invested in. $R$ denotes the rate percent per annum in the process and $T$ denotes the time period over which the interest is being charged. Here, in simple interest is the same for all periods of the time $T$. And it is based on principal amount only. Now, after a time period $T$, the total amount developed on the basis of principal amount $P$and rate percent per annum $R$ are is the sum of principal amount $P$ and interest.

Complete step-by-step answer:
 Mathematically,
Amount($A$)$=$ Principal($P$) $+$ Interest($I$)
$A=P+\dfrac{P\times R\times T}{100}$
So, taking $P$ common, we get; $A=P\left( 1+\dfrac{RT}{100} \right)$ , where $P=$Principal, $R=$ Rate per annum and $T=$ Time
So, let principal value be $P$ initially and rate per annum be $R$.
 So, accordingly, simple interest (S.I)$=\dfrac{P\times R\times T}{100}$
 We have to find a rate per annum. So, the sum of money doubles in 8 years.
So, amount becomes two times of principal
$A=2P$
$\Rightarrow P\left( 1+\dfrac{R\times 8}{100} \right)=2P$
Rearranging, we get;
 $1+\dfrac{R\times 8}{100}=2$
$\Rightarrow \dfrac{8R}{100}=1$
$\Rightarrow R=\dfrac{25}{2}=12.5%$
So, at a rate of $12.5$percent per annum, the sum of money doubles in $8$ years.
So, the correct option is (A).

Note: Simple interest (S.I) is a linear way of calculating interest. Here, rate percent in per annum and so, time should also be converted in annum. Like, if the rate is $10$ percent per annum and time period is $6$ months, then first calculate time in terms of annum.
Here, $12$ months $=1$annum
So,$1$month$=\dfrac{1}{12}$ annum $\Rightarrow 6$months $=\dfrac{6}{12}$ annum $=\dfrac{1}{2}$ year. Also, one may confuse the $100$ in $\dfrac{P\times R\times T}{100}$ ,that is, if rate is given $10$ percent the one may calculate simple interest as
$S.I=\dfrac{P\times \dfrac{10}{100}\times T}{100}$ which is wrong rather it is calculated as $S.I=\dfrac{P\times 10\times T}{100}$ .