Question

Arun deposited $ 5000 $ rupees in a bank which compounds interest half yearly and Mohan deposited the same amount in another bank which compounds interest quarterly. The annual rate of interest is $ 6\% $ at both the banks. How much more would Mohan get after one year?

Answer 1

Hint: The problem can be solved easily with the concept of compound interest. Compound interest is the interest calculated on the principal and the interest of the previous period. The amount in compound interest to be cumulated depends on the initial principal amount, rate of interest and number of time periods elapsed. The amount A after a certain number of time periods T on a given principal amount P at a specified rate R compounded annually is calculated by the formula: $ A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T} $ .

Complete step by step solution:
In the given problem,
Principal amount deposited by both Arun and Mohan is $ 5000 $ rupees.
Rate of interest at both the banks is $ 6\% $ .
The bank at which Arun deposited money compounds the interest annually and bank at which Mohan deposited the amount compounds interest quarterly.
So, in Arun’s case, we have,
Principal $ = P = Rs\,5000 $
Rate of interest $ = 6\% $
Time Duration $ = 1\,year $
The interest is compounded annually.
So, we get the amount after one year as $ A = 5000{\left( {1 + \dfrac{6}{{100}}} \right)^1} $
 $ \Rightarrow A = Rs\,5000\left( {\dfrac{{106}}{{100}}} \right) $
 $ \Rightarrow A = Rs\,50 \times 106 $
 $ \Rightarrow A = Rs\,5300 $
So, the amount with Arun after one year is $ Rs\,5300 $ .
Now, in Mohan’s case, we have,
Principal $ = P = Rs\,5000 $
Rate of interest $ = 6\% $
Time Duration $ = 1\,year $
The interest is compounded quarterly.
Now, in this case, the rate of interest is given on an annual form. So, we divide it by four to get in quarter form. Also, the number of time periods is four.
So, we get the amount after one year as $ A = 5000{\left( {1 + \dfrac{6}{{400}}} \right)^4} $
 $ \Rightarrow A = Rs\,5000{\left( {\dfrac{{406}}{{400}}} \right)^4} $
 $ \Rightarrow A = Rs\,5000{\left( {1.015} \right)^4} $
 $ \Rightarrow A = Rs\,5306.81 $
So, the amount with Mohan after one year is $ Rs\,5306.81 $ .
Hence, Mohan will get $ 6.81 $ rupees after one year more than Arun.
So, the correct answer is “ $ 6.81 $ rupees”.

Note: Time duration is not always equal to the number of time periods. The equality holds only when the compound interest is compounded annually. If the compound interest is compounded half yearly, then the number of time periods doubles in the given time duration and the rate of interest in each time period becomes half of the specified rate of interest.