Question

At what rate percent will a sum of Rs. 62500 amounted to Rs. 67600 in 2 years. Interest being compounded annually.

Answer 1

Hint: We have to use the formula for amount when the interest is compounded annually which is given by $A=P{{\left( 1+\dfrac{r}{100} \right)}^{t}}$ , where P is the principal amount, r is the rate of interest and t is the time in years. Now, we have to substitute the given values in this equation such that P will be Rs. 62500, A will be Rs. 67600 and t will be 2 years. We have to solve the equation for r which will be the required answer.

Complete step by step solution:
We have to find the rate percent. We know that when the interest is compounded annually, the amount is given by the formula
$A=P{{\left( 1+\dfrac{r}{100} \right)}^{t}}...\left( i \right)$
where P is the principal amount, r is the rate of interest and t is the time in years.
We are given that principal amount, $P=Rs.62500$ ,the amount, $A=Rs.67600$ at the time, $t=2\text{ years}$ . Let us substitute these values in the formula (i) and find the rate, r.
$\Rightarrow 67600=62500{{\left( 1+\dfrac{r}{100} \right)}^{2}}$
Let us take 62500 to the LHS.
$\Rightarrow \dfrac{67600}{62500}={{\left( 1+\dfrac{r}{100} \right)}^{2}}$
We have to cancel the zeroes from the numerator and denominator.
\[\Rightarrow \dfrac{676\require{cancel}\cancel{0}\require{cancel}\cancel{0}}{625\require{cancel}\cancel{0}\require{cancel}\cancel{0}}={{\left( 1+\dfrac{r}{100} \right)}^{2}}\]
We can write the result of the above cancellation as
\[\Rightarrow \dfrac{676}{625}={{\left( 1+\dfrac{r}{100} \right)}^{2}}\]
Now, we have to take the square root on both sides.
\[\Rightarrow \sqrt{\dfrac{676}{625}}=\left( 1+\dfrac{r}{100} \right)\]
We know that $\sqrt[n]{\dfrac{a}{b}}=\dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}$ . Therefore, we can write the above equation as
\[\Rightarrow \dfrac{\sqrt{676}}{\sqrt{625}}=\left( 1+\dfrac{r}{100} \right)\]
We know that $\sqrt{676}=26$ and $\sqrt{625}=25$ . Let us substitute these values in the above equation.
\[\Rightarrow \dfrac{26}{25}=\left( 1+\dfrac{r}{100} \right)\]
Let us take 1 from the RHS to the LHS.
\[\Rightarrow \dfrac{26}{25}-1=\dfrac{r}{100}\]
We have to take LCM and simplify.
\[\begin{align}
  & \Rightarrow \dfrac{26}{25}-\dfrac{1\times 25}{1\times 25}=\dfrac{r}{100} \\
 & \Rightarrow \dfrac{26}{25}-\dfrac{25}{25}=\dfrac{r}{100} \\
 & \Rightarrow \dfrac{26-25}{25}=\dfrac{r}{100} \\
 & \Rightarrow \dfrac{1}{25}=\dfrac{r}{100} \\
\end{align}\]
Let us take 100 from the denominator of the RHS to the LHS.
\[\begin{align}
  & \Rightarrow \dfrac{100}{25}=r \\
 & \Rightarrow r=\dfrac{100}{25} \\
\end{align}\]
Let us divide 100 by 25.
\[\Rightarrow r=4\]
Hence, the rate percent will be 4%.

Note: Students must be thorough with the formulas of compound interest and simple interest. They can get confused with these formulas. For simple interest, the amount is given by the formula $A=P\left( 1+rt \right)$ . Students must note that in this question, the interest is compounded annually. If the interest was compounded half yearly, we will substitute $n=2$ in the formula $A=P{{\left( 1+\dfrac{r}{n} \right)}^{nt}}$ , where n is the number of times the interest is compounded in a year and r is the rate of the interest (not in percentage).. We can also find the solution to this question using this formula.
Let us substitute the values in the above formula. Here, $n=1$ since the question specifies ‘compounded annually’.
\[\begin{align}
  & \Rightarrow 67600=62500{{\left( 1+\dfrac{r}{1} \right)}^{1\times 2}} \\
 & \Rightarrow \dfrac{67600}{62500}={{\left( 1+r \right)}^{2}} \\
\end{align}\]
On simplifying the LHS, we will get
$\Rightarrow \dfrac{676}{625}={{\left( 1+r \right)}^{2}}$
Now, we have to take the square root on both sides.
$\begin{align}
  & \Rightarrow \sqrt{\dfrac{676}{625}}=\left( 1+r \right) \\
 & \Rightarrow \dfrac{26}{25}=\left( 1+r \right) \\
\end{align}$
Let us take 1 to the LHS.
\[\Rightarrow \dfrac{26}{25}-1=r\]
We have to take LCM and simplify.
\[\begin{align}
  & \Rightarrow \dfrac{26-25}{25}=r \\
 & \Rightarrow \dfrac{1}{25}=r \\
 & \Rightarrow r=0.04 \\
\end{align}\]
We have to convert r into percentage by multiplying r by 100.
$\Rightarrow r=0.04\times 100\%=4\%$
Students may have noticed why the term 100 came on the denominator of the formula (i). This is because we have taken r as a percentage in that equation.