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Question: Find the compound interest on \(Rs\,16,000\) for \(9\) months at the rate of \(20\% \) per annum com...

Find the compound interest on Rs 16,000Rs\,16,000 for 99 months at the rate of 20%20\% per annum compounded quarterly.

Explanation

Solution

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(1+R100)TA = P{\left( {1 + \dfrac{R}{{100}}} \right)^T} . We are given the values of principal amount, rate of interest and the time period in the question. So, we just switch in the values in the formula and get to the answer straight away.

Complete step by step answer:
In the given problem,
Principal =P=Rs 16,000 = P = Rs\,16,000
Rate of interest =20 = 20 %\% per annum.
Time Duration =9 months = 9\,months
In the question, the period after which the compound interest is compounded or evaluated is given to be a quarter. This means that the interest is compounded quarterly.

So, Number of time periods =n=9 months3 months=3 = n = \dfrac{{9\,months}}{{3\,months}} = 3
Also, the rate of interest given to us is in the per year or per annum format. Hence, we calculate the rate of interest per quarter.
So, we get, rate of interest per quarter as 204%=5%\dfrac{{20}}{4}\% = 5\% .
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(1+R100)TA = P{(1 + \dfrac{R}{{100}})^T} .
Hence, Amount =P(1+R100)T = P{\left( {1 + \dfrac{R}{{100}}} \right)^T}
Now, substituting all the values that we have with us in the formula A=P(1+R100)TA = P{\left( {1 + \dfrac{R}{{100}}} \right)^T}, we get,
⇒A=Rs 16,000(1+5100)3\Rightarrow A = Rs\,16,000{\left( {1 + \dfrac{5}{{100}}} \right)^3}

Taking LCM in the bracket, we get,
⇒A=Rs 16,000(100+5100)3\Rightarrow A = Rs\,16,000{\left( {\dfrac{{100 + 5}}{{100}}} \right)^3}
⇒A=Rs 16,000(105100)3\Rightarrow A = Rs\,16,000{\left( {\dfrac{{105}}{{100}}} \right)^3}
Cancelling out the common factors in numerator and denominator, we get,
⇒A=Rs 16,000(2120)3\Rightarrow A = Rs\,16,000{\left( {\dfrac{{21}}{{20}}} \right)^3}
Now, opening up the cube of the term, we get,
⇒A=Rs 16,000×(2120)×(2120)×(2120)\Rightarrow A = Rs\,16,000 \times \left( {\dfrac{{21}}{{20}}} \right) \times \left( {\dfrac{{21}}{{20}}} \right) \times \left( {\dfrac{{21}}{{20}}} \right)

Cancelling out the common factors in numerator and denominator, we get,
⇒A=Rs 2×21×21×21\Rightarrow A = Rs\,2 \times 21 \times 21 \times 21
Now, computing the product, we get,
⇒A=Rs 42×441\Rightarrow A = Rs\,42 \times 441
⇒A=Rs 18522\Rightarrow A = Rs\,18522
So, the total amount is Rs 18522Rs\,18522.
Now, we know that Compound Interest=Amount−Principal{\text{Compound Interest}} = {\text{Amount}} - {\text{Principal}}.
So, we get the compound interest as,
⇒CI=Rs 18522−Rs16000\Rightarrow CI = Rs\,18522 - Rs16000
∴CI=Rs2522\therefore CI = Rs2522

Therefore, the compound interest on Rs 16,000Rs\,16,000 for 99 months at the rate of 20%20\% per annum compounded quarterly is Rs 2522Rs\,2522.

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 double in the given time duration and the rate of interest in each time period becomes half of the specified rate of interest