Question

A sum of money invested at compounded interest amounts to Rs. 800 in 3 years and to Rs. 840 in 4 years. The rate of interest per annum is
A) 3%
B) 4%
C) 5%
D) 6%

Answer 1

Hint:
It is given that the rate of interest should be calculated by using the formula for the compound interest because the sum of money invested at compounded interest amounts to Rs. 840 in 4 years.
Now, the formula to calculate the compound interest is given as
 $ A = {(1 + \dfrac{r}{n})^{nt}} $
Where A is the amount accumulated including interest after t years
P is the principal amount
R is the rate of interest annually
n is the 1 because the rate is being calculated annually
t is the time period.

Complete step by step solution:
Step1
When the time $ {t_1} $ is taken to be 3 years, the formula for compound interest becomes;
 $ {A_1} = P{(1 + \dfrac{r}{n})^{n{t_1}}} $
Step2
But after 3 years, the mounts to be Rs800. Hence the formula becomes;
 $ 800 = P{(1 + \dfrac{r}{1})^{1 \times 3}} $
i.e. $ 800 = P{(1 + r)^3} $
Step 3:
In the case of 4 years, the time $ {t_2} $ becomes 4 and the formula becomes
 $ {A_2} = P{(1 + \dfrac{r}{n})^{n{t_2}}} $
Step4:
It is given that after 4 years, the sum amounts to be Rs 840. Hence, the formula becomes;
 $ 840 = {(1 + \dfrac{r}{1})^{1 \times 4}} $
i.e. $ 840 = {(1 + r)^4} $
Step5
Now, we will divide the equation so formed in step4. By the equation so formed in step2.
 $ \Rightarrow \dfrac{{840 = P{{(1 + r)}^4}}}{{800 = P{{(1 + r)}^3}}} $
Now, the corresponding sides are getting divided by each other
 $ \Rightarrow \dfrac{{840}}{{800}} = \dfrac{{P{{(1 + r)}^3}(1 + r)}}{{P{{(1 + r)}^3}}} $
On solving and rearranging the terms; we get, $ \dfrac{{21}}{{20}} = 1 + r $
 $ \Rightarrow \dfrac{{21}}{{20}} - 1 = r $
 $ \Rightarrow r = \dfrac{{21 - 20}}{{20}} $
 $ \Rightarrow r = \dfrac{1}{{20}} $
To convert the term into a percentage, we multiply it by 100.

So, the rate of interest $ = \dfrac{1}{{20}} \times 100\% = 5\% $ p.a.

Note:
There are two types of interest: simple interest, compound interest.
The formula to calculate the simple interest can be given as $ S.I. = \dfrac{{P \times R \times T}}{{100}} $ .
The formula to calculate the compound interest is $ C.I. = P \times \left[ {{{(1 + \dfrac{r}{n})}^{nt}} - 1} \right] $ .
Also, the formula used in the given problem can also be used in the case of compound interest, when the amount has been mentioned.
I.e. $ A = P{\left( {1 + \dfrac{r}{n}} \right)^{nt}} $
Where P is the principal
R is the rate of interest
N is the number of times interest applied per time period.
t is the number of time periods.
A is the amount accumulated including interest after t years.
Now, in step2 and step3, the equations are having single terms on each side. So, we divided both the equations, and at the end the rate of interest must be in percentage. So, we multiplied 1/20 with 100 and obtained the rate of interest = 5%. Option (C).