Question

Teju invests Rs. $12,000$ at $5\% $ interest compounded annually. If he receives an amount Rs. $13,230$ at the end, find the period.

Answer 1

Hint: Here we have to put given values in the formula of amount at the end of n years for Principal P and rate of interest R% per annum. On doing some simplification and to find the value of n by solving the equation

Formula used: ${\text{A}} = {\text{P}}{\left( {1 + \dfrac{{\text{R}}}{{100}}} \right)^n}$
Where A = Amount, P = Principal, R = Rate and n = Time period.

Complete step-by-step solution:
It is given that the question stated the value of principal, rate and amount.
Since we called the money invested is called principal.
So, we can write it as the principal = Rs. $12,000$.
Also, rate is the interest paid on Rs $100$ for a specific period.
It is given that the rate = $5\% $ per annum.
Also, it is given that he receives an amount Rs. $13,230$ at the end.
So, Amount = Rs. $13,230$.
Now we have to find the time period using the formula for Amount.
Put these values in ${\text{A}} = {\text{P}}{\left( {1 + \dfrac{{\text{R}}}{{100}}} \right)^n}$, where A = Amount, P = Principal, R = Rate and n = Time period.
$ \Rightarrow 13230 = 12000{\left( {1 + \dfrac{5}{{100}}} \right)^n}$
Take LCM and add the bracket term on RHS, we get
$ \Rightarrow 13230 = 12000{\left( {\dfrac{{105}}{{100}}} \right)^n}$
Divide numerator and denominator by $5$ in bracket on RHS
$ \Rightarrow 13230 = 12000{\left( {\dfrac{{21}}{{20}}} \right)^n}$
Divide both sides of the equation by $12000$.
$ \Rightarrow {\left( {\dfrac{{21}}{{20}}} \right)^n} = \dfrac{{13230}}{{12000}}$
Simplify the fraction in RHS by dividing numerator and denominator by$30$.
$ \Rightarrow {\left( {\dfrac{{21}}{{20}}} \right)^n} = \dfrac{{441}}{{400}}$
We can write $441 = {21^2}$ and $400 = {20^2}$.
$ \Rightarrow {\left( {\dfrac{{21}}{{20}}} \right)^n} = {\left( {\dfrac{{21}}{{20}}} \right)^2}$
Compare the powers of LHS and RHS because the bracket term as same,
$n = 2$
Thus, the period of time is \[2\] years.

Note: Alternative method:
It is given that the question stated the value of principal, rate and amount.
Since we called the money invested is called principal.
So, we can write it as the principal = Rs. $12,000$.
Also, rate is the interest paid on Rs $100$ for a specific period.
It is given that the rate = $5\% $ per annum.
Also, it is given that he receives an amount Rs. $13,230$ at the end.
So, Amount = Rs. $13,230$.
We have to find the time period for which amount would be Rs. $13,230$.
Now we have to find the interest for the first year using the formula for Simple interest
That is we can write it as,$ = \dfrac{{{\text{Principal }} \times {\text{ Rate }} \times {\text{ Time}}}}{{100}}$
Putting the given values and we get,
Interest for the first year = Rs $\dfrac{{12000 \times 5 \times 1}}{{100}}$
On simplification we get
\[ \Rightarrow \]Rs $600$
Also, we have to find the amount at the end of first year using the amount formula,
So we can write it as Amount = Principal + Interest.
Amount at the end of first year = Rs $12000$ + Rs $600$
Let us add the terms and we get
\[ \Rightarrow \]Rs $12600$
Again we can find the interest for the second year using Simple interest
$ \Rightarrow \dfrac{{{\text{Principal }} \times {\text{ Rate }} \times {\text{ Time}}}}{{100}}$.
Use the new Principal amount at the end of first year = Rs $12600$
Time \[ = 1\] year
Putting the value and we get
Interest for the second year = Rs $\dfrac{{12600 \times 5 \times 1}}{{100}}$
On simplification we get
$ \Rightarrow $ Rs $630$
Similarly we can find the amount at the end of second year using Amount = Principal + Interest.
Amount at the end of second year = Rs $12600$ + Rs $630$
$ \Rightarrow $Rs $13230$
This amount is the same as the amount given.
So, at the end of second year he receives an amount Rs. $13,230$.
Thus, the period of time is \[2\] years.