Question

What sum invested for $1.5{\text{ years}}$ amount to $132651$ in $1\dfrac{1}{2}{\text{ years}}$ compounded half yearly at rate $4\% $ p.a. ?

Answer 1

Hint:Examine the question carefully, this question is talking about compound interest. So, we should be aware of the compound interest formula ${\text{A = P(1 + }}\dfrac{r}{{100}}{)^{2n}}$ . Now to get the invested amount (P), we have to put the given values in the above equation.

Complete step by step answer:
Given, final amount after $1\dfrac{1}{2}{\text{ years = 132651}}$
Rate of interest per annum$ = {{ 4\% p}}{\text{.a}}{\text{.}}$
Rate of interest half yearly $ = {\text{ }}\dfrac{4}{2}{{\% p}}{\text{.a}}{{. = 2\% }}$
Time period elapsed for invested amount $ = {\text{ 1}}{\text{.5 years}}$
In the given question the amount is compounded half yearly.

Formula used in question, compound interest $ \Rightarrow {\text{ A = P(1 + }}\dfrac{r}{{100}}{)^{2n}}$
Here, A is the final amount, P is invested amount, r is rate of interest, n is the time in years.
Now by putting the given values in equation,
${\text{A = P(1 + }}\dfrac{r}{{100}}{)^{2n}}$
$ \Rightarrow {\text{ }}132651{\text{ = P(1 + }}\dfrac{2}{{100}}{)^{(2 \times \dfrac{3}{2})}}$
$ \Rightarrow {\text{ 132651 = P(1}}{\text{.02}}{{\text{)}}^3}$
$ \Rightarrow {\text{ 132651 = P(1}}{\text{.061208)}}$
$ \therefore {\text{ P = 1,25,000}}$

Hence, the principal amount invested is ${\text{Rs}}{\text{. }}1,25,000$ for $1.5{\text{ years}}$ to get an amount of ${\text{Rs}}{\text{. 132651}}$.

Note:Be careful about the time period and rate of interest. Note that the given rate of interest is half yearly or per annum. In compound interest, interest is added to the amount and then interest is applied on that compound amount after a regular interval of time. This means that interest amount will increase after a regular interval of time but this does not happen in simple interest. In simple interest, the interest is applied on the initial amount invested all the time.