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# A sum of money is invested at 10% per annum compounded half yearly. If the difference of amount at the end of 6 months and 12 months is Rs.189, find the sum of money invested.

Last updated date: 13th Jun 2024
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Hint: Assuming the sum of money be Rs.y and at rate 10% per annum compounded half yearly. Calculate amount for the first six months, which is the equation of variable y and then calculating amount for first 12 months i.e. 1 year, which again is an equation of variable y. Substituting these two equations would give the value of y, actual sum of money.

Before solving the question, we must first understand the meaning of interest and compound interest. Interest is the money paid regularly at a particular rate for the use of money lent, or for delaying the repayment of a debt.
Compound interest is the addition of interest to the principal sum of a loan or deposit or in other words, interest on interest.
In compound interest, P = Principal, initial amount of money taken.
R = Rate % per annum
n = time in years.
When interest is compounded annually,
$\text{Amount=P}{{\left( 1+\dfrac{R}{100} \right)}^{n}}$
When interest is compounded half yearly,
$\text{Amount=P}{{\left( 1+\dfrac{\dfrac{R}{2}}{100} \right)}^{2n}}$
Now, according to the question:
Let the sum of money be Rs. y and given rate = 10% per annum compounded half yearly.
For first six months (half year, $n=\dfrac{1}{2}$ )
$\text{A=P}{{\left( 1+\dfrac{\dfrac{R}{2}}{100} \right)}^{2n}}$
Here, P = y
$A=y{{\left( 1+\dfrac{\left( \dfrac{10}{2} \right)}{100} \right)}^{2\times \dfrac{1}{2}}}=y\left( \dfrac{21}{20} \right)$
For first 12 months (1 year, n=1)
\begin{align} & \text{A=P}{{\left( 1+\dfrac{\dfrac{R}{2}}{100} \right)}^{2n}}\text{ (Considering P=y)} \\ & \Rightarrow \text{y}{{\left( 1+\dfrac{10}{100} \right)}^{2\times 1}}=y{{\left( \dfrac{21}{20} \right)}^{2}}=y\left( \dfrac{441}{400} \right) \\ \end{align}
Given the difference between the above amounts is Rs.189
\begin{align} & \Rightarrow y\left( \dfrac{441}{400} \right)-y\left( \dfrac{21}{20} \right)=189 \\ & \Rightarrow \left( \dfrac{21}{400} \right)y=189 \\ & \Rightarrow y=\dfrac{189\times 400}{21} \\ & \Rightarrow y=3600 \\ \end{align}
Therefore, the sum of money invested at 10% per annum compounded half yearly is Rs.3600

Note: Students might get confused while keeping the values of n time, which is in year. For half year n should be $\dfrac{1}{2}$ and for 1 year n should be 1. Since, students are much more familiar with simple interest formula i.e. $\text{SI=}\dfrac{\text{P}\times I\times R}{100}$ might use this formula but this would lead to wrong results. Students must use the CI formula as stated here.