Question

If Kabir pays back Rohan Rs. 6,232.50 for the loan of Rs. 4500 that he had taken at $28\%$ interest p.a., how many months did Kabir take to pay back Rohan’s money?

Answer 1

Hint: Here, first write the given values and then use the formula of amount, A = $\text{P}\,{{\left( 1+\dfrac{\text{r}}{100} \right)}^{\text{n}}}$ and some basic mathematical operations to find the value of n. The n is the time period in the number of years, convert it into months by multiplying it by 12 and we will get the required result.

Complete step-by-step solution:
Here, we have been given the total amount given by Kabir to Rohan, the principal amount Kabir loaned from Rohan at the rate of interest. We need to find the time period, Kabir took to pay back Rohan.
We have,
Amount (A) = Rs. 6,232.50/-, Principal (P) = Rs. 4500/-, Rate of interest (r) = 28 p.c.p.a., number of years (n) =?
We know,
Amount (A) = $\text{P}\,{{\left( 1+\dfrac{\text{r}}{100} \right)}^{\text{n}}}$
Let us substitute the given values in the above formula of calculating the amount and let us find the value of n.
6232.50 = $4500{{\left( 1+\dfrac{28}{100} \right)}^{\text{n}}}$
Now, let us divide by 4500 on both the sides of the equation, we get
$\dfrac{6232.50}{4500}$ = $\dfrac{4500}{4500}{{\left( 1+\dfrac{28}{100} \right)}^{\text{n}}}$
$\Rightarrow 1.385 = {{\left( 1+\dfrac{28}{100} \right)}^{\text{n}}}$
We can write $\dfrac{28}{100}$ as 0.28, we get
$1.385 = {{\left( 1+0.28 \right)}^{\text{n}}}$
$1.385 = {{\left( 1.28 \right)}^{\text{n}}}$
Now, let us take logarithm on both the sides of the equation, we get
$\ln \left( 1.385 \right)=\ln {{\left( 1.28 \right)}^{\text{n}}}$
Since, $\ln {{\left( m \right)}^{n}}=n\times \ln \left( m \right)$, we get
$\ln \left( 1.385 \right)=\text{n}\times \ln \left( 1.28 \right)$
The value of $\ln \left( 1.385 \right)=0.325$ and the value of $\ln \left( 1.28 \right)=0.247$
$0.325 = n \times 0.247$
Divide by 0.247 on both the sides of the equation, we get
$\dfrac{0.325}{0.247}=\text{n}$
$n = 1.315$
The obtained value which is in number of years, to convert it to months, we need to multiply the obtained result by 12 months.
$n = 1.315 \times 12
   = 15.78$
  $\approx $16
Therefore, the total number of months Kabir took to repay Rohan is approximately 16 months.

Note: This question, can also be solved by using two simple formulas of Simple interest, SI = $\dfrac{\text{P}\times \text{N}\times \text{R}}{100}$ and Amount = Principal + Simple Interest, the final value would have been slightly different than what we got in the solution.