Question

Find the amount and the compound interest on Rs. 6500 for 2 years, compounded annually, the rate of interest being 5 % per annum during the first year and 6 % per annum during the second year.

Answer 1

Hint: To solve this question, we need to know the concept of compound interest. We should also know the formula of compound interest, that is \[A=P{{\left( 1+\dfrac{r}{100} \right)}^{n}}\], where A is the final amount, P is the principal amount, r is the rate of interest and n represents the number of times the interest is applied per time period.

Complete step-by-step answer:
In this question, we have to find the amount and compound interest on Rs. 6500 for 2 years. Now, we have been given that in the first year, the rate of interest is 5 % and in the second year, it is 6 %. So, we will apply the formula of the compound interest for 2 times, first for year 1 for a 5 % rate and then for year 2 for a 6 % rate.
We know that the compound interest is calculated using the formula \[A=P{{\left( 1+\dfrac{r}{100} \right)}^{n}}\], where A is the final amount, P is the principal amount, r is the rate of interest and n is the number of times the interest is applied per time period. So, for the first year, we have P = Rs. 6500, r = 5 % and n = 1. Therefore, we will get
\[A=6500{{\left[ 1+\dfrac{5}{100} \right]}^{1}}\]
Now, we will simplify it to find the value of A’ which is the final amount after the interest of 1 year. So, we get,
\[{{A}^{'}}=6500\left[ \dfrac{100+5}{100} \right]\]
\[{{A}^{'}}=\dfrac{6500\times 105}{100}\]
\[{{A}^{'}}=65\times 105\]
A’ = Rs. 6825
Now, we will apply the formula of compound interest for the second year, for which P = A’ = Rs. 6825, r = 6 % and n = 1. So, we get,
\[A=6825{{\left[ 1+\dfrac{6}{100} \right]}^{1}}\]
Now, we will simplify it further to find the final amount of compound interest after 2 years.
\[A=6825\left[ \dfrac{100+6}{100} \right]\]
\[A=\dfrac{6825\times 106}{100}\]
\[A=Rs.7234.5.....\left( i \right)\]
So, we can say that the final amount after the compound interest of 5 % for the first year and compound interest of 6 % for the second year is Rs. 7234.5. Now, we are asked to find the compound interest for 2 years. So, we will apply the formula for the compound interest, that is
Compound Interest \[=P{{\left( 1+\dfrac{r}{100} \right)}^{n}}-P\]
And we know that \[P{{\left( 1+\dfrac{r}{100} \right)}^{n}}\] is nothing but the final amount after the interest is applied, that is A. So, we can write, Compound Interest = A – P
From equation (i), we can say that A = Rs. 7234.5 and in the question, we have been given that P = Rs. 6500. Therefore, we can write,
Compound Interest = 7234.5 – 6500
Compound Interest = 734.5
Hence, we can say that Rs. 734.5 is the compound interest on Rs. 6500 for 2 years.

Note: In this question, there are high possibilities for calculation mistakes because it has a lot of calculations. Also, when we are asked to find the compound interest, then we have to find the total gain, the amount that is A – P.