Question & Answer
QUESTION

Calculate the amount and compound interest on Rs. 18,000 for $2\dfrac{1}{2}$ years at 10% per annum compounded annually.

ANSWER Verified Verified
Hint: We will use the formula of compound interest for 2 years and simple interest for the rest \[\dfrac{1}{2}\] year as the time is in fraction and by adding them we will get the total interest on the given principal. The formulae of compound interest and simple interest are as follows:
For compound interest we have, \[amount=p{{\left( 1+\dfrac{R}{100} \right)}^{n}}\] where p= principal, R = rate and n= time in years.
For simple interest we have, \[\text{interest}=\dfrac{P\times R\times T}{100}\] where p= principal, R= rate and T= time in years.

Complete step-by-step answer:
We are given,
Principal = Rs.18,000
Time = $2\dfrac{1}{2}$ years
Rate (R) = 10% per annum.
Since time is in fraction , we will use the formula,
Compound interest for $2\dfrac{1}{2}$ years = compound interest for 2 years + simple interest for \[\dfrac{1}{2}\] year.
Compound interest for 2 years ,
P = Rs.18,000
n = 2
R= 10%
\[\begin{align}
  & \text{Amount= 18000}{{\left( 1+\dfrac{10}{100} \right)}^{2}}\text{ } \\
 & \text{ = }18000{{\left( 1+\dfrac{1}{10} \right)}^{2}} \\
 & \text{ = }18000{{\left( \dfrac{11}{10} \right)}^{2}} \\
 & \text{ = }18000\times \dfrac{11}{10}\times \dfrac{11}{10} \\
 & \text{ = 21,780} \\
\end{align}\]
Since, Amount = principal + interest.
21,780 =18,000 + interest
Interest = 21,780 – 18,000
Interest = 3780
Now, we have interest for two years = Rs. 3780 and amount after two years = Rs. 21780.
For simple interest for the next \[\dfrac{1}{2}\] year we have,
P =Rs. 21,780
R =10%
T= \[\dfrac{1}{2}\] year.
$\begin{align}
  & \text{interest =}\dfrac{21780\times 10\times \dfrac{1}{2}}{100} \\
 & \text{ = }\dfrac{21780\times 10}{2\times 100} \\
 & \text{ =1089} \\
\end{align}$
So, simple interest for next \[\dfrac{1}{2}\] year = Rs. 1089.
Now, interest for $2\dfrac{1}{2}$ years = compound interest for 2 years + simple interest for \[\dfrac{1}{2}\] year.
                                                     = 3780 +1089
                                                     = 4869

Amount = principal + interest
               = 18,000 + 4,869
               = Rs.22,869
Therefore, the amount and interest for the above question are Rs. 22,869 and Rs. 4,869 respectively.

Note: It is important to understand in applying both the compound interest and simple interest formula accordingly. Simple interest is calculated only on the principal, whereas the compound interest is calculated on both the principal and the accumulated interest.