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Fabina borrows \[Rs.12,500\] at \[12\% \] per annum for 3 years at simple interest and Radha borrows the same amount for the same time period at \[10\% \] per annum, compounded annually. Who pays more interest and by how much?
Answer
Verified
Hint: As we can see the interest mentioned that the amount is compounded yearly for Radha 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, to calculate the amount of Fabina at \[12\% \] per annum is given by the formula of simple interest is given as \[S. I = \dfrac{{P \times R \times T}}{{100}}\], on adding S.I to the principle we get the amount fabina will pay at the end of 3 years. Hence using the above formula, we can obtain our required answer. As we need to compare the final amount of both.
Complete step by step Answer:
As per the given the amount that Fabina borrows is of \[Rs.12,500\] and the rate of interest is \[12\% \], while the time period is \[3\]years.
So, \[P = Rs.12,500\],
\[R = 12\% \] and \[T = 3\]
Hence, the amount that Fabina has to pay is to be
calculated using simple interest formula as,
\[S.I = \dfrac{{P \times R \times T}}{{100}}\]
On substituting the values, we get,
\[
\Rightarrow S.I = \dfrac{{12500 \times 12 \times 3}}{{100}} \\
\Rightarrow S.I = \dfrac{{12500 \times 36}}{{100}} \\
\]
Hence, on further simplification, we get,
\[
\Rightarrow S.I = \dfrac{{450000}}{{100}} \\
\Rightarrow S.I = Rs.4500 \\
\]
Now,
calculating the total amount that Fabina has to pay is,
Amount plus the simple interest incurred at the end of 3 years,
i.e., \[12,500 + 4500 = Rs.17000\]
Now, calculating the amount of Radha it can be given as,
Hence, we can see that the compound interest will be applicable for \[3\]years only at \[10\% \]compound interest for the same principal amount.
So, the amount can be given as,
\[A = P[{(1 + \dfrac{R}{{100}})^n}]\]
On substituting the values we get,
\[
\Rightarrow A = (12500)[{(1 + \dfrac{{10}}{{100}})^3}]\]
Now, rationalize the term inside the bracket as
\[
\Rightarrow A = 12500[{(\dfrac{{110}}{{100}})^3}] \\
= 12500[{(1.1)^3}] \\
\]
Hence, calculating the above value, we get,
\[ \Rightarrow A = Rs.16,637.5\]
Hence, here we can see that Fabina has to pay more amount as compared to Radha and that also by,
\[
= 17,000 - 16,637.5 \\
=
Rs.362.5 \\
\]
Hence, Fabina has to pay more interest and it is also by \[Rs.362.5\].
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. Don’t forget to add the principal amount to the simple interest to get the amount, otherwise, we will reach incorrect conclusions. 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.