Question

Find the approximate rate of compound interest at which Rs. 1000 becomes Rs. 5000 in 4 years.
A) 10%
B) 15%
C) 50%
D) 30%

Answer 1

Hint: We will first of all write the formula for the final amount after compound interest has been applied to the principal amount and then put in all the values given to us already in order to find the rate.

Complete step-by-step solution:
Let us first understand the formula for amount after applying compound interest to it.
The formula for final amount is given by the following expression:-
$ \Rightarrow 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 compound interest and n is the amount of time ( in years ).
Now, since, we are given that P = 1000, A = 5000 and n = 4.
Putting all of these in the above expression, we will then obtain the following expression:-
$ \Rightarrow 5000 = 1000{\left( {1 + \dfrac{R}{{100}}} \right)^4}$
Now, we will take the 1000 from multiplication in RHS to division in LHS to obtain the following expression:-
$ \Rightarrow \dfrac{{5000}}{{1000}} = {\left( {1 + \dfrac{R}{{100}}} \right)^4}$
Simplifying the LHS of the above expression to get the following new expression:-
$ \Rightarrow 5 = {\left( {1 + \dfrac{R}{{100}}} \right)^4}$
Now, we will take the power of $\dfrac{1}{4}$ on both the sides of above expression. We will then obtain:-
$ \Rightarrow {5^{\dfrac{1}{4}}} = {\left[ {{{\left( {1 + \dfrac{R}{{100}}} \right)}^4}} \right]^{\dfrac{1}{4}}}$
Now, since we know that ${\left( {{x^a}} \right)^{\dfrac{1}{a}}} = x$. Using this in the RHS of above expression, we will obtain as follows:-
$ \Rightarrow {5^{\dfrac{1}{4}}} = 1 + \dfrac{R}{{100}}$
Now, putting the value of ${5^{\dfrac{1}{4}}}$ as 1.495 in above expression to get:-
$ \Rightarrow 1.495 = 1 + \dfrac{R}{{100}}$
Now taking the 1 from addition in RHS to subtraction in LHS, we will obtain as follows:-
$ \Rightarrow 1.495 - 1 = \dfrac{R}{{100}}$
Simplifying the LHS to get:-
$ \Rightarrow 0.495 = \dfrac{R}{{100}}$
Taking 100 from division in RHS to multiplication in LHS, we will get:-
$ \Rightarrow R = 49.5$
This is very near to 50.

$\therefore $ The correct option is (C) 50%.

Note: The students must commit to memory the formula for amount after compound interest that is:-
$ \Rightarrow 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 compound interest and n is the amount of time ( in years ).
The students must note that there is a lot of difference between simple interest and compound interest. You must know that generally, if we go for deposition of our money in the bank, we must prefer compound interest because in compound interest, we get interest over our interest in next year which is a win - win situation.