Question

John purchased a black and white T.V. set on credit. If the set costs Rupees \[2400\] and the shopkeeper charges interest at the rate of \[20\% \;\] per annum, find the compound interest that John will have to pay after three years .

Answer 1

Hint: The problem can be solved easily with the concept of compound interest. Compound interest is the interest calculated on the principal and the interest of the previous period. The amount in compound interest to be cumulated depends on the initial principal amount, rate of interest and number of time periods elapsed. The amount A after a certain number of time periods T on a given principal amount P at a specified rate R compounded annually is calculated by the formula: $A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T}$ .

Complete answer:
In the given problem,
Principal $ = P = Rs\,2400$
Rate of interest $ = 20\% $
Time Duration $ = 3\,years$
In the question, the period after which the compound interest is compounded or evaluated is now given. So, we assume that the compound interest is compounded annually by default.
So, Number of time periods$ = n = 3$
Now, The amount A to be paid after a certain number of time periods n on a given principal amount P at a specified rate R compounded annually is calculated by the formula: $A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T}$ .
Hence, Amount $ = A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T}$
Now, substituting all the values that we have with us in the formula $A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T}$, we get,
$A = Rs\,2400{\left( {1 + \dfrac{{20}}{{100}}} \right)^3}$
Taking LCM, we get,
$ \Rightarrow A = Rs\,2400{\left( {\dfrac{{100 + 20}}{{100}}} \right)^3}$
$ \Rightarrow A = Rs\,2400{\left( {\dfrac{6}{5}} \right)^3}$
$ \Rightarrow A = Rs\,2400 \times \left( {\dfrac{6}{5}} \right) \times \left( {\dfrac{6}{5}} \right) \times \left( {\dfrac{6}{5}} \right)$
Cancelling common factors in numerator and denominator, we get,
$ \Rightarrow A = Rs\,96 \times \left( {\dfrac{6}{1}} \right) \times \left( {\dfrac{6}{1}} \right) \times \left( {\dfrac{6}{5}} \right)$
Simplifying the calculations, we get,
$ \Rightarrow A = Rs\,4147.2$
So, John has to pay an amount of $Rs\,4147.2$ after three years.
Now, we can calculate the compound interest by subtracting the principal amount from the total amount payable.
Hence, $Compound\,Interest = Rs\,4147.2 - Rs2400$
$ \Rightarrow Compound\,Interest = Rs\,1747.2$
The compound interest paid by John is $Rs\,1747.2$.

Note:
Time duration is not always equal to the number of time periods. The equality holds only when the compound interest is compounded annually. If the compound interest is compounded half yearly, then the number of time periods doubles in the given time duration and the rate of interest in each time period becomes half of the specified rate of interest. If the time after which interest is compounded is not given, we take it as yearly by default.