Question

I borrowed Rs.12000 from Jamshed at 6% per annum simple interest for 2 years. If I had borrowed this sum at 6% per annum compound interest, what extra amount would I have to pay?

Answer 1

Hint: In this question, we are given a principal amount borrowed from a person for a certain time period. We have to compare the simple interest and compound interest on the same amount at the same rate of interest and for the same time period. Using this we have to find an extra amount to be paid in compound interest than simple interest. We will use the formula for simple interest as $SI=\dfrac{P\times R\times T}{100}$ formula for the amount after simple interest as P + SI and formula for calculating amount after compound interest as $A=P{{\left( 1+\dfrac{R}{100} \right)}^{T}}$ where SI is simple interest, P is the principal amount, R is the rate of interest and T is time period.

Complete step-by-step solution:
Here, we are given the principal amount as Rs.12000 and the time period as 2 years. The rate of simple interest is 6% per annum and the rate of compound interest is 6% per annum. We have to compare the amount formed at the end of two years and find out the extra compound amount by amount from simple interest.
Let us calculate the amount of both interests one by one.
Using simple interest:
Principal amount is Rs.12000 therefore, P = 12000. Rate of interest is 6% per annum, therefore R = 6. Time period is given as 2 years, therefore T = 2. Simple interest is given by $SI=\dfrac{P\times R\times T}{100}$ where P is principal amount, R is rate of interest, T is time period. Therefore, simple interest becomes
\[\begin{align}
  & \Rightarrow SI=\dfrac{12000\times 6\times 2}{100} \\
 & \Rightarrow 120\times 12 \\
 & \Rightarrow 1440 \\
\end{align}\]
Hence, simple interest (SI) is Rs.1440. We have to find the total amount. Amount is given as A = P + I where P is principal amount, I is interest. Therefore, amount becomes,
\[\begin{align}
  & \Rightarrow A=12000+1440 \\
 & \Rightarrow 13440 \\
\end{align}\]
Amount by simple interest is Rs.13440 . . . . . . . . . . . . . . . . . (1)
Using compound interest:
P = 12000, R = 6% per annum and T = 2 years.
We know, compound interest can be calculated as $A=P{{\left( 1+\dfrac{R}{100} \right)}^{T}}$ so we get:
\[\Rightarrow A=12000{{\left( 1+\dfrac{6}{100} \right)}^{2}}\]
Taking LCM in the bracket we get:
\[\begin{align}
  & \Rightarrow A=12000{{\left( \dfrac{100+6}{100} \right)}^{2}} \\
 & \Rightarrow A=12000{{\left( \dfrac{106}{100} \right)}^{2}} \\
\end{align}\]
Opening the square, we get:
\[\begin{align}
  & \Rightarrow A=12000\times \dfrac{106\times 106}{100\times 100} \\
 & \Rightarrow A=12000\times \dfrac{11236}{10000} \\
\end{align}\]
Simplifying further, we get:
\[\begin{align}
  & \Rightarrow A=\dfrac{134832}{10} \\
 & \Rightarrow A=Rs.13483.2 \\
\end{align}\]
Therefore, amount after compound interest is equal to Rs.13483.2 . . . . . . . . . . . . . . . . . . . (2)
Now, we need to calculate extra amount using compound interest by the amount using simple interest. Therefore,
Extra amount = amount using compound interest - amount using simple interest.
Using (1) and (2) we get:
\[\begin{align}
  & \Rightarrow \text{Extra amount}=13483.2-13440 \\
 & \Rightarrow Rs.43.2 \\
\end{align}\]
Hence, Rs.43.20 is our required answer.
Hence, I have to pay 43.20 extra if I had the money at 6% per annum compound interest rather than simple interest.

Note: Students should not get confused between simple interest and compound interest. Simple interest is interest calculated on only the principal amount whereas compound interest is the interest calculated on both the principal amount and all previously accumulated interest. Students should keep in mind the formula for calculating the amount and simple interest.