Question

A sum of Rs.12,000 deposited at compound interest doubles after 5 years. After 20 years it will become
A. Rs. 1,20,000
B. Rs. 1,92,000
C. Rs. 1,24,000
D. Rs. 96,000

Answer 1

Hint: The amount 12,000 doubles which means it becomes $ 2 \times 12,000 = 24,000 $ after 5 years. This means that when the principal amount is Rs.12,000 and the time period is 5 years, the final amount will be Rs. 24,000. So from this we can calculate the interest rate by using the below formula. With the obtained interest rate, the principal amount Rs.12,000 and the time period 20 years, find the final amount at the end of 20th year.
Compound interest A is calculated by $ A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T} $ , where P is the principal amount, T is the time period and R is the interest rate.

Complete step-by-step answer:
We are given that a sum of Rs.12,000 deposited at compound interest doubles after 5 years.
Twice or double of Rs. 12,000 is $ 2 \times 12,000 = Rs.\;24,000 $ .
So here Principal amount P is Rs.12,000, Time period T is 5 years and the final amount A is Rs.24,000.
Interest rate will be,
 $ A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T} $
 $ \Rightarrow 24,000 = 12,000{\left( {1 + \dfrac{R}{{100}}} \right)^5} $
 $ \Rightarrow {\left( {1 + \dfrac{R}{{100}}} \right)^5} = \dfrac{{24,000}}{{12,000}} = 2 $
 $ \Rightarrow \left( {1 + \dfrac{R}{{100}}} \right) = \sqrt[5]{2} \Rightarrow eq\left( 1 \right) $
The above obtained equation is enough to find the amount after 20 years.
Therefore, the total amount after 20 years will be
 $ A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T} $
 $ \Rightarrow A = 12,000{\left( {1 + \dfrac{R}{{100}}} \right)^{20}} $
We already know from equation 1 that $ \left( {1 + \dfrac{R}{{100}}} \right) = \sqrt[5]{2} $ . Substituting this in the above equation, we get
  $ \Rightarrow A = 12,000{\left( {\sqrt[5]{2}} \right)^{20}} $
 $ \Rightarrow A = 12,000{\left[ {{2^{\left( {\dfrac{1}{5}} \right)}}} \right]^{20}} = 12,000 \times {\left( 2 \right)^{\dfrac{1}{5} \times 20}} = 12,000 \times {2^4} $
 $ \therefore A = 12,000 \times 16 = Rs.1,92,000 $
The amount after 20 years will be Rs. 1,92,000.
So, the correct answer is “Rs. 1,92,000”.

Note: The interest can be either simple or compound. In simple interest, the interest amount does not change till the end of the return period whereas in compound interest, the interest amount gradually changes as the interest is imposed on the principal amount plus the previous accumulated interest combined. Compound interest is much greater than Simple interest.