Question

In what rate per annum will a sum of Rs.6750 amount to Rs.8192 in 3years, compounded annually.

Answer 1

Hint: Compound Interest: Compound interest is interest on interest. Addition of the interest in the principal amount. Or reinvesting the interest.
As we know that
$ \Rightarrow C.I. = P{\left( {1 + \dfrac{r}{{100}}} \right)^T} - P$
Here
CI=compound interest
P=principal
r=rate of interest
T=time

Complete step-by-step answer:
Given,
Principal, P=Rs.6750
Amount, A=Rs.8192
Time, T=3yearrs
Rate, R=?
Formula of compound interest,
$ \Rightarrow A = P{\left( {1 + \dfrac{r}{{100}}} \right)^T}$
Put the values in the formula,
$ \Rightarrow 8192 = 6750{\left( {1 + \dfrac{r}{{100}}} \right)^3}$
\[ \Rightarrow \dfrac{{8192}}{{6750}} = {\left( {1 + \dfrac{r}{{100}}} \right)^3}\]
\[ \Rightarrow \dfrac{{4096}}{{3375}} = {\left( {1 + \dfrac{r}{{100}}} \right)^3}\]
\[ \Rightarrow {\left( {\dfrac{{16}}{{15}}} \right)^3} = {\left( {1 + \dfrac{r}{{100}}} \right)^3}\]
The Powers of both sides are the same so the powers are canceled out.
\[ \Rightarrow \left( {\dfrac{{16}}{{15}}} \right) = \left( {1 + \dfrac{r}{{100}}} \right)\]
\[ \Rightarrow \dfrac{r}{{100}} = \dfrac{{16}}{{15}} - 1\]
\[ \Rightarrow \dfrac{r}{{100}} = \dfrac{{16 - 15}}{{15}}\]
\[ \Rightarrow \dfrac{r}{{100}} = \dfrac{1}{{15}}\]
\[ \Rightarrow r = \dfrac{1}{{15}} \times 100\]
\[ \Rightarrow r = \dfrac{{20}}{3}\]
\[ \Rightarrow r = 6\dfrac{2}{3}\% \]
So, the correct answer is “$6\dfrac{2}{3}\%$”.

Note: Compound interest earned or paid on both the principal and previously earned interest. For annually compound interest means “once in a year”, half yearly means “twice in the year”, quarterly means “four times in the year”. Compound interest is always more than simple interest for a given period of time.