Question

Mohini lent Rs. 20000 to Sheenu at 12% per annum compounded yearly. Find the interest Sheenu will pay after 2 years and 4 months.

Answer 1

Hint: Use compound interest formula for the calculation of amount $A$, given by: \[A=P{{\left( 1+\dfrac{r}{n} \right)}^{nt}}\]. From this, calculate amount $A$ after $t=\dfrac{7}{3}$ year. Then subtract the principal amount $P$ from the amount $A$ to get the interest.

Complete step-by-step answer:
Compound interest is the addition of interest to the principal sum of a loan or deposit. It is the result of reinvesting interest, rather than paying it out, so the interest in the next period is then earned on the principal sum plus previously accumulated interest.

The total accumulated amount $A$ , on the principal sum \[P\] plus compound interest $I$ is given by the formula \[A=P{{\left( 1+\dfrac{r}{n} \right)}^{nt}}\].

 Here, \[A\] is the amount obtained, $t$ is the number of years, \[r\] is the rate, $P$ is the principal and \[n\] is the number of times the interest is given in a year.

The total compound interest generated is given by: $I=A-P$.

Now, we have been given that:

$P=20000$, $r=12%$, $n=1$ and $t=2+\dfrac{1}{3}=\dfrac{7}{3}$ years.

Substituting these values in the formula for finding amount, we get,

$\begin{align}
  & A=20000\times {{\left( 1+\dfrac{12}{100\times 1} \right)}^{\dfrac{7}{3}}} \\
 & =20000\times {{\left( 1+\dfrac{12}{100} \right)}^{2+\dfrac{1}{3}}} \\
 & =20000\times {{\left( \dfrac{112}{100} \right)}^{2+\dfrac{1}{3}}} \\
\end{align}$

We know that, ${{a}^{m+n}}={{a}^{m}}\times {{a}^{n}}$, therefore, using this formula we get,

$\begin{align}
  & A=20000\times {{\left( \dfrac{112}{100} \right)}^{2}}\times {{\left( \dfrac{112}{100} \right)}^{\dfrac{1}{3}}} \\
 & =2\times {{\left( 112 \right)}^{2}}\times {{\left( 1.12 \right)}^{\dfrac{1}{3}}} \\
 & =2\times 12544\times 1.038 \\
 & =26041.344 \\
\end{align}$

Therefore, $I=A-P=26041.344-20000=6041.344$.

Hence, the interest Sheenu will pay after 2 years and 4 months is Rs. 6041.344


Note: We have used the value of $n$ equal to 1 because it is given in the question that interest is compounded yearly. We have to take the value of $t$ in years, so we have converted the given number of months into a year. Further you may note that we have broken the power term. This is done to simplify the calculation. We have used a calculator to calculate the cube root.