Question

Govind borrows Rs. 18,000 at 10% simple interest. He immediately invests the money borrowed at 10% compound interest compounded half-yearly. How much money does Govind gain in one year?

Answer 1

Hint: In order to solve the problem separately find out the simple interest and the compound interest for the given set of money. The gain for Govind in a year will be the difference between the compound interest and the simple interest.

Complete step-by-step answer:

First let us find out the simple interest that will be paid by Govind.
Principal Amount = Rs.18000
Rate of interest = 10%
Time = 1 year
As we know that:
$SI = \dfrac{{P \times R \times T}}{{100}}$
Where SI represents simple interest, P represents principal amount, R represents rate of interest and T represents Time period.
So let us substitute all the values and find the simple interest
$
   \Rightarrow SI = \dfrac{{P \times R \times T}}{{100}} \\
   \Rightarrow SI = \dfrac{{18000 \times 10 \times 1}}{{100}} \\
   \Rightarrow SI = Rs.1800 \\
 $
Now let us find out the amount to be received by Govind as the compound interest
Principal Amount = Rs.18000
Rate of interest = 10% half yearly rate $ = \dfrac{{10\% }}{2} = 5\% $
Time period = 1 year = 2 half years
As we know that total amount is given by:
$A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T}$
Where A represents total amount i.e. principal + compound interest, P represents principal amount, R represents rate of interest and T represents time period.
So, let us substitute all the values in order to find the total amount
$
   \Rightarrow A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T} \\
   \Rightarrow A = 18000{\left( {1 + \dfrac{5}{{100}}} \right)^2} \\
 $
Now let us simplify the equation to get the total amount.
$
   \Rightarrow A = 18000{\left( {\dfrac{{100 + 5}}{{100}}} \right)^2} \\
   \Rightarrow A = 18000{\left( {\dfrac{{105}}{{100}}} \right)^2} \\
   \Rightarrow A = 18000 \times \dfrac{{105}}{{100}} \times \dfrac{{105}}{{100}} \\
   \Rightarrow A = Rs.19845 \\
 $
Since total amount is the sum of principal amount and the compound interest so
Compound Interest = total amount – principal amount
$
   \Rightarrow CI = A - P \\
   \Rightarrow CI = Rs.19845 - Rs.18000 \\
   \Rightarrow CI = Rs.1845 \\
 $
Now as we know the both simple interest and the compound interest which are same as the money to be paid and received by Govind apart from principal amount so we can easily find out the gain by subtracting them
So, gain = Compound Interest – Simple interest
$
   \Rightarrow {\text{gain}} = Rs.1845 - Rs.1800 \\
   \Rightarrow {\text{gain}} = Rs.45 \\
 $

Hence, Govind gains Rs. 45 in a year.

Note: Students must understand the basic difference between SI and CI. Simple interest is based on the principal amount of a loan or deposit. In contrast, compound interest is based on the principal amount and the interest that accumulates on it in every period. Students must keep in mind that compound interest is always greater than or equal to the simple interest.