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Hint- Ram has given Rs4096 to someone and we have to tell the amount which he will be receiving after 18 months if he lends at some interest rate which is compounded half yearly, thus the principal amount here is Rs4096.

The principal amount P = Rs4096

The duration of time for which he is lending this money is 18 months at $12\dfrac{1}{2}$% per annum and it is compounded half yearly.

Now the interest he will be receiving compounded half yearly in 18 months will be equal to the interest compounded yearly in $18 \times 2 = 36$months.

Here the tenure of interest is 36 months. Now 12 months makes 1 year so 1 month comprises of $\dfrac{1}{{12}}$years and thus 36 months will make $\dfrac{1}{{12}} \times 36 = 3$years. So tenure is of T= $3$years.

Now the rate of interest is given as $12\dfrac{1}{2}$% per annum compounded half yearly.

So we have to change the rate of interest per annum compounded yearly.

Hence that will be $\dfrac{{\dfrac{{25}}{2}}}{2} = \dfrac{{25}}{4}$% per year compounded yearly.

Now the amount that he will be receiving, ${\text{Amount = P}}{\left( {1 + \dfrac{r}{{100}}} \right)^T}$where P is principal amount, R is

Rate of interest, T is the duration of loan given.

${\text{Amount = 4096}}{\left( {1 + \dfrac{{25}}{{4 \times 100}}} \right)^3}$

$

\Rightarrow 4096 \times {\left( {1.0625} \right)^3} \\

\Rightarrow 4096 \times 1.9946 \\

\Rightarrow 4913{\text{ Rs}} \\

$

Thus he will be receiving 4913Rs after 18 months if given at a rate of $12\dfrac{1}{2}$% per annum compounded half yearly.

Note – Whenever we face such a type of problem the key concept that we need to recall is that the amount received is dependent upon the principal amount, rate of interest and the tenure of loan provided, moreover the tenure of loan is always taken in years and not in months while making the calculations.

The principal amount P = Rs4096

The duration of time for which he is lending this money is 18 months at $12\dfrac{1}{2}$% per annum and it is compounded half yearly.

Now the interest he will be receiving compounded half yearly in 18 months will be equal to the interest compounded yearly in $18 \times 2 = 36$months.

Here the tenure of interest is 36 months. Now 12 months makes 1 year so 1 month comprises of $\dfrac{1}{{12}}$years and thus 36 months will make $\dfrac{1}{{12}} \times 36 = 3$years. So tenure is of T= $3$years.

Now the rate of interest is given as $12\dfrac{1}{2}$% per annum compounded half yearly.

So we have to change the rate of interest per annum compounded yearly.

Hence that will be $\dfrac{{\dfrac{{25}}{2}}}{2} = \dfrac{{25}}{4}$% per year compounded yearly.

Now the amount that he will be receiving, ${\text{Amount = P}}{\left( {1 + \dfrac{r}{{100}}} \right)^T}$where P is principal amount, R is

Rate of interest, T is the duration of loan given.

${\text{Amount = 4096}}{\left( {1 + \dfrac{{25}}{{4 \times 100}}} \right)^3}$

$

\Rightarrow 4096 \times {\left( {1.0625} \right)^3} \\

\Rightarrow 4096 \times 1.9946 \\

\Rightarrow 4913{\text{ Rs}} \\

$

Thus he will be receiving 4913Rs after 18 months if given at a rate of $12\dfrac{1}{2}$% per annum compounded half yearly.

Note – Whenever we face such a type of problem the key concept that we need to recall is that the amount received is dependent upon the principal amount, rate of interest and the tenure of loan provided, moreover the tenure of loan is always taken in years and not in months while making the calculations.

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