Question

Kamala borrowed Rs.26,400 from a Bank to buy a scooter at a rate of \[15\% \] p.a. compounded yearly. What amount will she pay at the end of 2 years and 4 months to clear the loan?

Answer 1

Hint: As we can see the interest mentioned above is compounded yearly so using the formula of compound interest as the calculated total amount including the principal amount will be\[A = P[{(1 + \dfrac{R}{{100}})^n}]\], here P is principal amount while R is the rate of interest and n is time period. Similarly, the formula of simple interest is given as \[S.I = \dfrac{{P \times R \times T}}{{100}}\]. We first calculate the amount at the end of 2 years using the compound interest formula, then for the remaining 4 months, we take the amount of 2 years as the principal amount and then on applying the formula of simple interest, we get the interest occurred in 4 months, adding it to the amount we get the amount at the end of 2 years and 4 months.

Complete step by step Answer:

As per the given the principle amount is of \[P = Rs.26,400\] and the rate of interest is \[15\% \], while the time period is 2 years and 4 months.
So, we have
\[P = Rs.26,400\]
\[R = 15\% \]
So, the time period can be simplified to \[(2\dfrac{4}{{12}}) = 2\dfrac{1}{3}\]years as 1 year is equal to 12 months so from the same we can calculate for 4 months.
Hence, we can see that the compound interest will be applicable for 2 years only, and then for \[\dfrac{1}{3}\]year simple interest will be added.
So, compound interest for \[2\dfrac{1}{3}\] is given as the sum of compound interest for 2years and simple interest for \[\dfrac{1}{3}\]year.
Using the formula to find amount for compound interest which is, \[A = P[{(1 + \dfrac{R}{{100}})^n}]\],
So, the amount at compound interest for 2 years is given as,
\[A = (26400)[{(1 + \dfrac{{15}}{{100}})^2}]\]
Now, rationalize the term inside the bracket as
\[A = 26400[{(\dfrac{{115}}{{100}})^2}]\]
Hence, calculating the above value, we get,
\[A = 34914\]
Now, the interest for the next 4 months will be calculated using Simple Interest
Hence, the principal amount for the next session will be the amount calculated for 2 years,
Hence we have our new P as 34914
And T is \[\dfrac{1}{3}\]
Using, \[S.I = \dfrac{{P \times R \times T}}{{100}}\]
On Substituting the values, we get,
\[ \Rightarrow S.I = \dfrac{{34,914 \times 15 \times \dfrac{1}{3}}}{{100}}\]
On dividing 15 by 3 we get,
\[ \Rightarrow S.I = \dfrac{{34,914 \times 5}}{{100}}\]
Hence, on further simplification, we get,
\[
   \Rightarrow S.I = \dfrac{{174570}}{{100}} \\
   \Rightarrow S.I = Rs.1745.70 \\
 \]
Hence, the total amount to be paid at the end of \[2\dfrac{1}{3}\]is given as amount at the end of 2 years plus S.I, i.e.,
\[ = A + S.I\]
Now, substitute the value of both simple interest and amount as ,
\[
   = 34,914 + 1,745.70 \\
   = Rs\;36,659.70 \\
 \]
Hence, Kamala needs to pay an amount Of Rs 36,600 approximately to clear the loan at the end of 2 years and 4 months.

Note: Use the formula of simple interest and compound interest as\[S.I = \dfrac{{P \times R \times T}}{{100}}\]and \[A = P[{(1 + \dfrac{R}{{100}})^n}]\]. And hence identify the terms properly and put them in the above equation.
Compound interest is calculated by multiplying the initial principal amount by one plus the annual interest rate raised to the number of compound periods minus one. The total initial amount of the loan is then subtracted from the resulting value.
Simple interest is a quick and easy method of calculating the interest charge on a loan. Simple interest is determined by multiplying the daily interest rate by the principal by the number of days that elapse between payments.