Question

At what percent annual rate will Rs. 640 amount to 774.40 in 2 years, interest being compounded annually?

Answer 1

Hint: To solve this question, we will apply the formula of compound interest which is as shown:
\[{\rm{A = P}}{\left( {1 + \dfrac{r}{n}} \right)^{nt}}\]
Where A is the final amount, P is the initial Principal balance, r is the rate of interest, n is the number of times interest applied per time period and t is the number of time periods elapsed.

Complete step by step answer:
In the question, we are given that the initial principal balance is Rs. 640, and the total time period is 2 years. Before proceeding to solve the question, we must first know what is compound interest. Compound interest is the addition of interest to the principal sum of a loan or deposit, or in other words, interest on interest. Now, we will apply the formula of compound interest to calculate the interest rate. The formula of compound interest is shown below:
\[{\rm{A = P}}{\left( {1 + \dfrac{r}{n}} \right)^{nt}}\]
In our question, it is given that the value of principal amount is Rs. 640. Also it is given that the total time for which the total amount is calculated is 2 years Also, it is given that the amount is calculated annually, this means that the value of n becomes 1. It is also given that when the 2 years have passed, the total amount we got was Rs.774.40. Now, we will put all these values in the formula of compound interest. After doing this, we will get
\[ \Rightarrow {\rm{774}}{\rm{.40}}\,{\rm{ = }}\,{\rm{640}}{\left( {1 + \dfrac{{\dfrac{r}{{100}}}}{1}} \right)^{1 \times 2}}\]
\[ \Rightarrow 774.40\, = \,640{\left( {1 + \dfrac{r}{{100}}} \right)^2}\]
\[ \Rightarrow \dfrac{{774.40}}{{640}}\, = \,{\left( {1 + \dfrac{r}{{100}}} \right)^2}\]
\[ \Rightarrow \dfrac{{77440}}{{64000}}\, = \,{\left( {1 + \dfrac{r}{{100}}} \right)^2}\]
\[ \Rightarrow \dfrac{{7744}}{{6400}}\, = \,{\left( {1 + \dfrac{r}{{100}}} \right)^2}\]
\[ \Rightarrow \dfrac{{64 \times 121}}{{64 \times 100}}\, = \,{\left( {1 + \dfrac{r}{{100}}} \right)^2}\]
\[ \Rightarrow \dfrac{{121}}{{100}}\, = \,{\left( {1 + \dfrac{r}{{100}}} \right)^2}\]
We will now take square root in both the cases. After doing this we will get following:
\[ \Rightarrow \sqrt {\dfrac{{121}}{{100}}} \, = \,1 + \dfrac{r}{{100}}\]
\[ \Rightarrow \dfrac{{11}}{{10}}\, = \,1 + \dfrac{r}{{100}}\]
\[ \Rightarrow \dfrac{{11}}{{10}} - 1 = \dfrac{r}{{100}}\]
\[ \Rightarrow \dfrac{r}{{100}}\, = \,\dfrac{1}{{10}}\]
\[ \Rightarrow r\, = \,100\, \times \dfrac{1}{{10}}\,\, = \,\,10\% \]

Hence, the interest rate is 10%

Note: Instead of r, we have used it in the formula because we want to calculate the interest rate in percent form. If we had to calculate it in decimal form, we would have omitted the \[\dfrac{1}{{100}}\]. This question is a basic one which requires students to apply the direct formula for amount and then get the rate from it. Just that there might be confusion regarding which value is the amount and principal. So in question, if it says Rs. x will ‘amount to’ Rs. y, then Rs. x will be principal and Rs. y will be amount.