Question

A sum of Rs. 46,875 was lent out as simple interest and at the end of 1 year 8 months the total amount was Rs. 50,000. Find the rate of interest percent per annum.
(a) 3.5%
(b) 4.5%
(c) 5%
(d) 4%

Answer 1

Hint: To find the rate of interest percent per annum, we have to use the formula $A=P\left( 1+rt \right)$ , where A is the total accrued amount (principal + interest), P is the principal amount, r is the rate of interest per year in decimal and t is the time period involved in months or years. Then, we have to substitute the given values in this formula and solve for r. Then, we have to convert r into percentage by multiplying it by 100.

Complete step by step answer:
We have to find the rate of interest percent per annum. We will use the formula for simple interest which is given by
$\Rightarrow A=P\left( 1+rt \right)...\left( i \right)$
where A is the total accrued amount (principal + interest), P is the principal amount, r is the rate of interest per year in decimal and t is the time period involved in months or years.
We are given that $A=Rs.50,000$ , $P=Rs.46,875$ and $t=1\text{ year }8\text{ months}$ .
We know that 1 year contains 12 months. Let us convert 8 months into years by dividing 8 by 12.
8 months $=\dfrac{8}{12}\text{ years}$
$\begin{align}
  & \Rightarrow \text{8 months}=\dfrac{2}{3}\text{ years} \\
 & \Rightarrow \text{8 months}=0.667\text{ years} \\
\end{align}$
Therefore, we can write t as
$\begin{align}
  & t=\left( 1+0.667 \right)\text{ years} \\
 & \Rightarrow t=1.667\text{ years} \\
\end{align}$
Now, we have to substitute the values in the equation (i).
$\Rightarrow 50000=46875\left( 1+1.667r \right)$
Let us take 46875 to the LHS.
$\begin{align}
  & \Rightarrow \dfrac{50000}{46875}=1+1.667r \\
 & \Rightarrow 1.067=1+1.667r \\
\end{align}$
We have to take 1 from the RHS to the LHS.
$\begin{align}
  & \Rightarrow 1.066=7-1=1.667r \\
 & \Rightarrow 0.067=1.667r \\
\end{align}$
Let us take the coefficient of r to the other side.
\[\begin{align}
  & \Rightarrow \dfrac{0.067}{1.667}=r \\
 & \Rightarrow r=\dfrac{0.067}{1.667} \\
\end{align}\]
We have to multiply the numerator and denominator by 1000.
\[\begin{align}
  & \Rightarrow r=\dfrac{0.067\times 1000}{1.667\times 1000} \\
 & \Rightarrow r=\dfrac{67}{1667} \\
 & \Rightarrow r=0.04 \\
\end{align}\]
We have found the rate as 0.04. We have to convert it into percentage by multiplying it by 100.
$\begin{align}
  & \Rightarrow r=0.04\times 100\% \\
 & \Rightarrow r=4\% \\
\end{align}$

So, the correct answer is “Option d”.

Note: Students should never be confused with the formulas of simple interest and compound interest. The formula for compound interest is given by
$A=P{{\left( 1+\dfrac{r}{n} \right)}^{nt}}$ , where A is the total accrued amount, P is the principal amount, r is the rate of interest per year in decimal, n is the number of times the interest is compounded per year and t is the time period involved in months or years. Students must know to solve equations, convert a number into its percentage and to convert a percentage into its number form.