Question

Find the amount if $\$72500$ is invested at 6% p.a. for $1\dfrac{1}{2}$ years if compounded half yearly.

Answer 1

Hint: If the principal is P, and we have to invest it for n terms for interest compounded half yearly, the formula to calculate the amount is given by, $A=P{{\left( 1+\dfrac{R}{100} \right)}^{n}}$, where R is the rate of interest for a term and it is to be noted that 1 term = 6 months. So, we will convert the time period given in years to terms. Also, the given rate of interest will be reduced by half. Then, by using the formula, we will get the amount.

Complete step-by-step answer:
We have been given the question that, we have to find the amount if $\$72500$ is invested at 6% p.a. for $1\dfrac{1}{2}$ years if compounded half yearly.
So, from this we get the principal amount, $P=\$72500$, and the rate of interest as 6% per annum. And we know that the formula to find the amount if the interest is compounded half yearly is given as, $A=P{{\left( 1+\dfrac{R}{100} \right)}^{n}}$, where P is the principal amount, n is the term and R is the rate of interest for a term.
So, here R is the rate of interest for half year and not for a year, so 1 term is 6 months or half year. Hence, we will divide the given rate of interest per annum by 2. So, we get,
$R=\dfrac{6}{2}=3\%$
So, we get the rate of interest for half the year as 3%.
Now, the time is given as $1\dfrac{1}{2}$ years. We know that 1 year is equal to 12 months = 6 months + 6 months and $\dfrac{1}{2}$ years is equal to 6 months. Also, we know that 1 term = 6 months for interest compounded half yearly. So, we can write, $1\dfrac{1}{2}$ year = 6 months + 6 months + 6 months = 1 term + 1 term + 1 term = 3 terms, or we can say 3 half years.
Thus, we get, $P=\$72500$, R = 3% and n = 3.
Therefore, on substituting the values of $P=\$72500$, R = 3% and n = 3 in the formula of amount, we get,
$A=72500{{\left( 1+\dfrac{3}{100} \right)}^{3}}$
On taking the LCM of the terms inside the bracket, we get,
$\begin{align}
  & A=72500{{\left( \dfrac{100+3}{100} \right)}^{3}} \\
 & A=72500{{\left( \dfrac{103}{100} \right)}^{3}} \\
\end{align}$
Now, on expanding the power, we get,
$\begin{align}
  & A=72500\times \dfrac{103}{100}\times \dfrac{103}{100}\times \dfrac{103}{100} \\
 & A=\$79222.71\\\end{align}$
Therefore, we get the amount as $\$79222.71$.


Note:The most common mistake that the students make while solving this question is by writing the wrong formula. They may directly put the values of R = 6% and n = 1.5 years, which is wrong. As we know that for interest compounded half yearly, the rate becomes half and also the time period is in terms of months and that 1 term = half year, that is 6 months.