Question

A sum of Rs.$1,200$ becomes Rs.$1,323$ in 2 years at compound interest compounded annually. Find the rate per cent.

Answer 1

Hint:
We are going to solve this problem by using a compound interest formula to get the solution. First we are going to consider the given details and then we are going to substitute the given values in the formula of compound interest. And the formula is listed below.

Formula used:
 ${\text{A = P}}{\left( {{\text{1 + }}\dfrac{{\text{r}}}{{{\text{100}}}}} \right)^{\text{n}}}$
Where ${\text{A}}$ is the total amount, that is ${\text{A}}$ contains Principal amount and interest, ${\text{P}}$ is the principal amount, ${\text{r}}$ is the rate of interest and ${\text{n}}$ is the number of years.

Complete step-by-step answer:
It is given that the principal amount, ${\text{P = Rs}}{\text{.1200}}$
Total amount, ${\text{A = Rs}}{\text{. 1323}}$
And the number of years, ${\text{n = 2}}$
Now we are going to find the rate of the interest ${\text{r}}$ using given details.
To find the rate of the interest ${\text{r}}$, we are going to substitute the given details on ${\text{A = P}}{\left( {{\text{1 + }}\dfrac{{\text{r}}}{{{\text{100}}}}} \right)^{\text{n}}}$
Then we get,
\[1323 = 1200{\left( {1 + \dfrac{{\text{r}}}{{100}}} \right)^2}\]
Now we are going to divide both sides by 1200. Then we get,
$\dfrac{{1323}}{{1200}} = {\left( {1 + \dfrac{{\text{r}}}{{100}}} \right)^2}$
Now we are going to take square root on both sides. Then we get,
 $\sqrt {\dfrac{{1323}}{{1200}}} = \left( {1 + \dfrac{{\text{r}}}{{100}}} \right)$
Now we are going to calculate the value of $\sqrt {\dfrac{{1323}}{{1200}}} $ so that we can simplify and find the solution.
So, $\sqrt {\dfrac{{1323}}{{1200}}} {\text{ }} \approx 1.05$
Now we are going to substitute this value on $\sqrt {\dfrac{{1323}}{{1200}}} = \left( {1 + \dfrac{{\text{r}}}{{100}}} \right)$ to find the value of ${\text{r}}$
$1.05 = \left( {1 + \dfrac{{\text{r}}}{{100}}} \right)$
Now we are going to subtract by 1 on both sides. Then we get,
\[1.05 - 1{\text{ = 1 + }}\dfrac{{\text{r}}}{{100}} - 1\]
Now we can simplify the value on the left hand side term and cancel 1 and -1 on the right hand side term.
Then we get,
\[{\text{0}}{\text{.05 = }}\dfrac{{\text{r}}}{{100}}\]
Now we are going to multiply by 100 on both sides. Then we get,
\[0.05 \times {\text{100 = }}\dfrac{{\text{r}}}{{100}} \times {\text{100 }}\]
Now we are going to cancel 100 on both numerator and denominator terms. Then we get,
\[{\text{r = }}0.05 \times {\text{100 }}\]
This becomes that, \[{\text{r = 5 % }}\]
Hence we finally found that the rate of interest \[{\text{r}}\] is \[{\text{5 % }}\]

Note:
One may confuse the principal amount ${\text{P}}$ with the total amount ${\text{A}}$. Principal amount is the amount that does not include any interest. Principal amount is the initial amount. Total amount is the amount that includes both the principal amount and interest amount. That is, ${\text{A = P + I}}$ where ${\text{I}}$ is the interest.
In this question, they have given a clue that Rs.1323 is the amount that compounded annually. So we can say that Rs.1323 is the total amount and the other amount Rs.1200 is the principal amount.
By seeing the question, we must figure out the principal amount and total amount so that we can substitute the appropriate values on the formula to attain the correct solution.