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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?
A. Rs. $ 80 $
b. Rs. $ 50 $
C. Rs. $ 45 $
D. Rs. $ 25 $

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Last updated date: 13th Jun 2024
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Answer
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Hint:
 In this problem, we have two cases, the first case is Govind borrowed money and the second case is Govind investing the same money. Both the cases involve different interest systems, the first case has a simple interest and the second case has a compound interest. So, we will calculate the total amount of both the cases according to the given data by using the known formulas and we will compare both the total amounts to get the result. We will use the formula for the total amount in simple interest is $ A=P\left( 1+\dfrac{TR}{100} \right) $ while the total amount for the compound interest is $ A=P{{\left( 1+\dfrac{R}{100} \right)}^{n}} $ , where $ P $ is the principal amount, $ A $ is the total amount, $ R $ is the rate of interest in percentage, $ T $ is the total time period in simple interest, $ n $ is the number of time compounded yearly.

Complete step by step answer:
Given that,
Govind borrows Rs. $ 18,000 $ at $ 10\% $ simple interest. He immediately invests the money borrowed at $ 10\% $ compound interest compounded half-yearly. As we discussed above the two cases in the given problem are
Case-1: Govind borrows Rs. $ 18,000 $ at $ 10\% $ simple interest.
In this case
Principle amount $ P=18,000 $ ,
Rate of interest in percentage $ R=10 $ ,
Time period $ T=1 $ year.
Now the total amount to be paid by the Govind after one is equal to the total amount and that can be calculated as
 $ A=P\left( 1+\dfrac{TR}{100} \right) $
Substituting all the values we have
 $ \begin{align}
  & \Rightarrow A=18000\left( 1+\dfrac{1\left( 10 \right)}{100} \right) \\
 & \Rightarrow A=18000\left( 1+0.1 \right) \\
 & \Rightarrow A=18000\left( 1.1 \right) \\
 & \Rightarrow A=19800 \\
\end{align} $
Hence the total amount to be paid by Govind is Rs. $ 19,800 $ in this case.
Case-2: Govind invested the same amount at $ 10\% $ compound interest compounded half-yearly.
In this case
Principle amount $ P=18,000 $ ,
Rate of interest in percentage $ R=\dfrac{10}{2}=5 $ since the interest compounds half-yearly,
The interest compounded half-yearly. So, in a year the interest is compounded $ 2 $ times. Then $ n=2 $ . Now the total amount Govind will receive from his investments is
 $ A=P{{\left( 1+\dfrac{R}{100} \right)}^{n}} $
Substituting all the values we have
 $ \begin{align}
  & \Rightarrow A=18000{{\left( 1+\dfrac{5}{100} \right)}^{2}} \\
 & \Rightarrow A=18000{{\left( 1+\dfrac{1}{20} \right)}^{2}} \\
 & \Rightarrow A=18000{{\left( \dfrac{21}{20} \right)}^{2}} \\
 & \Rightarrow A=18000\times \dfrac{441}{400} \\
 & \Rightarrow A=19845 \\
\end{align} $
Hence the total amount received by Govind from his investment is Rs. $ 19,845 $ .
From the above two cases, we can observe that we received more amount than he paid, so he will get profit and the amount of profit is equal to the difference of the total amounts from both the cases.
 $ \begin{align}
  & \therefore P=19845-19800 \\
 & \Rightarrow P=45 \\
\end{align} $
 $ \, therefore, $ Option – C is the correct one.

Note:
 The interest systems play the main roles in business. So, we need to have better clarity on those systems. In this problem, they have asked the profit gain by the Govind from two cases. But in some case, they only ask the profit gained by Govind in simple interest only, then we need to use the formula $ profit=\dfrac{PTR}{100} $ .