Question

# Sam borrowed Rs.12000 from Arko at some rate of compound interest. After a year, he returned Rs.4000 to Arko. If the compound interest for the second year is Rs.920, find the rate of interest charged to Sam and the amount of debt at the end of the second year.

Verified
91.8k+ views
Hint: Using the formula of calculating the interest at the end of a year find out the amount of interest at the end of the first year, add this amount to the initial principal value and subtract the amount returned to find the new principal amount for the second year. From the formula of calculating the interest at the end of the second year, you can find out the required terms.

Compound Interest to be paid after one year can be calculated by the formula,
$I = \dfrac{{\Pr }}{{100}}$ , where $I =$ interest after one year, $r =$ rate of interest, $P =$ amount borrowed (also called ‘Principal’ and $t =$ time for which amount is borrowed. The time is taken as 1 in this formula.
In the question, we are given that Sam has borrowed Rs.12000 at some rate. Let the rate of interest be x%. So interest to be paid after one year is -
$I = \dfrac{{12000(x)}}{{100}} = 120x$
Find out the interest for the next period using the new principal amount which is the sum of the initial principal amount and the interest at the end of the first year. But Sam also returned Rs.4000. So the new amount will exclude the returned money.
So the new principal amount is
$\Rightarrow P = 12000 + 120x - 4000 \\ P = 8000 + 120x \;$
Interest to be paid at the end of the second year is –
$I = \dfrac{{(8000 + 120x)x}}{{100}} = \dfrac{{800x + 12{x^2}}}{{10}}$
We are given in the question that compound interest for the second year is Rs.920. So,
$\Rightarrow 920 = \dfrac{{800x + 12{x^2}}}{{10}} \\ 12{x^2} + 800x - 9200 = 0 \\ 4(3{x^2} + 200x - 2300) = 0 \\ \Rightarrow 3{x^2} + 200x - 2300 = 0 \;$
Solving the quadratic equation obtained, we get –
$x = \dfrac{{ - 230}}{3}$ and $x = 10$
But the rate of interest cannot be negative so $x = 10$ is the only value possible.
Therefore, the rate of interest charged to Sam is $10\%$ .
Now, after the second year he has to pay back the sum of the principal amount for the second year and the interest to be paid at the end of the second year.
$\Rightarrow debt = 8000 + 120(10) + 920 \\ \Rightarrow debt = 8000 + 1200 + 920 \\ \Rightarrow debt = 10120 \;$
The amount of debt at the end of the second year is Rs.10120.
So, the correct answer is “ Rs.10120.”.

Note: In the case of compound interest, we calculate the interest for the first period and then add it to the amount borrowed, then calculate the interest for the next period using the new amount and so on. Debt is the total amount of money that has to be paid back by a borrower.
Keeping these two definitions in mind and solving the question carefully.