Question

How much would a sum of $16000$ amount to in $2$ years’ time at $10\% $ per annum compound interest, interest being payable half-yearly $?$

Answer 1

Hint:
Here in this question, they have given the compound interest for a year being payable half-yearly means two times per year, and they are asking us to find how much will be the amount at the end of $2$ years’. So, by using the Amount formula that is $A = P{\left( {1 + \dfrac{R}{{100}}} \right)^n}$ we can find the amount at the end of the $2$ years.

Complete step by step solution:
First by looking at the problem we can say that they asked us to find the amount at the end of $2$ years’. So to find the amount we need to know the formula given below:
 $A = P{\left( {1 + \dfrac{R}{{100}}} \right)^n}$
Where, $A$- Amount at the end of $2$ years’
$P$- Principal amount
$R$-Rate of interest
$n$- Number of compounds
Now, we need to know what and all they have given so that we can find the amount.
They have given the principal amount that is $P = 16000$,
They have given a rate of interest per year as $10\% $ but they have also mentioned that the compound interest is payable half-yearly, therefore the rate of interest for half year becomes $5\% $ that is $R = 5\% $.
The time period given is $2$ years’ but they are also saying that the compound interest is payable at half-year, so we will be having $2$ compounds per year then for $2$ years we will be having $4$ compounds. Therefore $n = 4$.
Substitute the values of $P, R, n$ in the formula of Amount, we get
$A = 16000{\left( {1 + \dfrac{5}{{100}}} \right)^4}$
$ \Rightarrow A = 16000{\left( {1 + 0.05} \right)^4}$
$ \Rightarrow A = 16000{\left( {1.05} \right)^4}$
$ \Rightarrow A = 16000 \times 1.215506$
$ \Rightarrow A = 19,448.1Rs$

Therefore, the sum of $16000$ amount will be $19,448.1$ in $2$ years.

Note:
While taking the compounding period, if the compounding period is not annual, then the rate of interest should be divided in accordance with the compounding period.
The same result can be obtained by using another way is first finding the interest amount that is using the formula $A = P\left[ {{{\left( {1 + \dfrac{{R\% }}{n}} \right)}^{n \times t}}} \right]$here, $t$ is the time period that is $2$ years’, and $n$ will be $2$ as compounded half-yearly(if they will give compounded yearly then it would be $1$), $R\% $will be $\dfrac{{10}}{{100}}$ and principal amount $P = 16000$.