Question

In what time will $Rs.1000$ will become $Rs.1331$ at $10\% $ per annum compounded annually?
A. $3$
B. $4$
C. $2$
D. $5$

Answer 1

Hint: First, the given question is all about the compound interest method. The principal amount deposited, the compound interest is given which can be calculated using the amount as $A = P + I$ where A is the amount, P is the principal amount given and I is the interest of them. And then we will use the compound interest formula that $A = P{(1 + \dfrac{R}{{100}})^T}$ where T is the time taken, R is the rate of change per annum.

Complete step by step answer:
Since from the given that we have the principal amount as $P = 1000$ and the rate percent as $R = 10\% $ and also the amount changed is given as $A = 1331$ . Hence, we need to find its time taken T.
Let us assume the time is taken as $n$ then put all the values into the compound interest formula we get $A = P{(1 + \dfrac{R}{{100}})^T} \Rightarrow 1331 = 1000{(1 + \dfrac{{10}}{{100}})^n}$
Now equate them in the right sides we get ${(1 + \dfrac{{10}}{{100}})^n} = \dfrac{{1331}}{{1000}}$
Canceling the common terms and make use of the perfect cube we have ${(1 + \dfrac{{10}}{{100}})^n} = \dfrac{{1331}}{{1000}} \Rightarrow {(\dfrac{{11}}{{10}})^n} = \dfrac{{1331}}{{1000}}$
Hence this will be only possible when the $n = 3$ because ${(\dfrac{{11}}{{10}})^3} = \dfrac{{1331}}{{1000}}$
Thus $3$ years of time will become $Rs.1000$ become $Rs.1331$ at $10\% $ per annum compounded annually.

So, the correct answer is “Option A”.

Note:
Since then ${11^3} = 1331,{10^3} = 1000$ and it can be seen that both the fractions are the same as ${(\dfrac{{11}}{{10}})^n} = \dfrac{{1331}}{{1000}}$ and this implies that the value of their power will also be the same and hence we get $n = 3$
Also, the compound interest of the given problem can be calculated in the same method. This can be obtained using the formula that $A = P{(1 + \dfrac{R}{{100}})^T}$.
Also we need to know about the concept of simple interest. Which is the technique of computing the amount of the interest for a principal amount of money at some rate of interest. Where P refers to the principal amount, T refers to the time taken on the process, R is the rate of the interest percent per annum.
The Simple interest formula is $S.I = \dfrac{{P \times R \times T}}{{100}}$ and note that calculating the amount formula is $A = P + SI$