Question

The clan of ₨ \[50,000\] at \[10\% \] p.a. compounded annually for a certain time period is ₨\[10,500\]. What is the time period?

Answer 1

Hint: Here, in the question, we have been given the value of initial amount, compounded interest and the rate of interest. And we are asked to find the time, in years in which the given amount generates that compound interest. We will use the Compound interest formula to find the desired result.
Formula used:
\[CI = P\left[ {{{\left( {1 + \dfrac{R}{{100}}} \right)}^N} - 1} \right]\], where,
\[CI\] is the compound interest, \[P\] is the principal amount, \[R\] is the rate of interest and \[N\] is the time period.

Complete step-by-step solution:
The given values are:
Principal amount, \[P = Rs.50,000\]
Compound Interest, \[CI = Rs.10,500\]
Rate of interest, \[R = 10\% \]
Let \[N\] be the time, in years, in which ₨\[50,000\] generates the interest of ₨ \[10,500\] @ \[10\% \].
Now, using the formula of compound interest,
\[CI = P\left[ {{{\left( {1 + \dfrac{R}{{100}}} \right)}^N} - 1} \right]\] \[\]
Substituting the given values, we get,
\[10,500 = 50,000\left[ {{{\left( {1 + \dfrac{{10}}{{100}}} \right)}^N} - 1} \right]\]
Simplifying it, we get,
\[\dfrac{{10,500}}{{50,000}} = {\left( {1 + \dfrac{1}{{10}}} \right)^N} - 1\]
Now, adding the term inside the bracket, and taking \[1\] on the left hand side, we get,
\[
  \dfrac{{10,500}}{{50,000}} + 1 = {\left( {\dfrac{{11}}{{10}}} \right)^N} \\
   \Rightarrow \dfrac{{60,500}}{{50,000}} = {\left( {\dfrac{{11}}{{10}}} \right)^N} \\
   \Rightarrow \dfrac{{121}}{{100}} = {\left( {\dfrac{{11}}{{10}}} \right)^N} \\
    \\
 \]
Now, we know \[121\] is the square of \[11\] and \[100\] is the square of \[10\],
Therefore, \[{\left( {\dfrac{{11}}{{10}}} \right)^2} = {\left( {\dfrac{{11}}{{10}}} \right)^N}\]
Comparing both the LHS and the RHS, we get,
\[N = 2\]
Hence, the time is \[2\] years, when the clan of ₨\[50,000\] generates the interest of ₨\[10,500\] at the rate of \[10\% \].

Note: Compound Interest means the current interest will be calculated on (say an investment or a loan) the principal amount plus the interest accumulated over previous periods. Or, in other words, it is the result of reinvesting the interest, rather than withdrawing it out.
Compounding was said to be the eighth wonder of the world by a world famous scientist, Albert Einstein.