Understanding the Compound Interest Problem
The question asks us to calculate the final amount obtained from an initial investment (principal) after a certain period, considering compound interest that is calculated quarterly.
Here are the details given:
- Principal Amount (\(P\)): INR 15000
- Annual Interest Rate (\(R\)): 10%
- Compounding Frequency: Quarterly
- Time Period (\(T\)): 2 years
Compound Interest Formula Explained
The formula to calculate the future value (\(A\)) of an investment with compound interest is:
\(A = P \left(1 + \frac{r}{n}\right)^{nt}\)
Where:
- \(A\) is the amount of money accumulated after n years, including interest.
- \(P\) is the principal amount (the initial amount of money).
- \(r\) is the annual interest rate (in decimal form).
- \(n\) is the number of times that interest is compounded per year.
- \(t\) is the number of years the money is invested or borrowed for.
Step-by-Step Calculation
First, let's convert the annual interest rate to its decimal form and determine the values for \(n\) and \(nt\).
- Annual interest rate (\(R\)) = 10% = 0.10
- Since interest is compounded quarterly, the number of compounding periods per year (\(n\)) = 4.
- The time period (\(t\)) = 2 years.
Now, let's calculate the interest rate per quarter (\(\frac{r}{n}\)) and the total number of compounding periods (\(nt\)):
- Interest rate per quarter = \(\frac{r}{n} = \frac{0.10}{4} = 0.025\)
- Total number of compounding periods = \(nt = 4 \times 2 = 8\)
Next, we plug these values into the compound interest formula:
\(A = 15000 \left(1 + 0.025\right)^{8}\)
\(A = 15000 \left(1.025\right)^{8}\)
Now, we need to calculate \((1.025)^{8}\):
\((1.025)^{8} \approx 1.2184029\)
Finally, multiply this by the principal amount:
\(A = 15000 \times 1.2184029\)
\(A \approx 18276.0435\)
Determining the Final Yield
The calculated amount after 2 years, compounded quarterly, is approximately INR 18276.04.
Comparing this result with the given options:
- Option 1: INR 18432
- Option 2: INR 18276
- Option 3: INR 18654
- Option 4: INR 18225
The calculated value is closest to Option 2.
Important Questions from Compound Interest
A person borrowed Rs. 10000 on compound interest at the rate of 40 percent per annum. If the interest is compounded half yearly, then what will be the amount to be paid after 1.5 years?
In how many years will a sum of Rs.1875 amount to Rs.2187 at 8 percent p.a. compound interest?
A sum of money has increased by 45% in 9 years at simple interest. What will be the compound interest of Rs. 12,000 after 3 years at the same rate?
At a certain rate of compound interest a certain sum amounts to Rs. 64800 in 4 years and Rs. 93312 in 6 years. What is the compound interest earned in fifth year?
Karuna has an option to invest in the following three schemes. If she wants to invest Rs P, then which scheme will be most beneficial for her?
Scheme that comes with lower interest rates and terms;
I. 28% p.a. interest, compounded every quarter, for one year.
II. 28% p.a. interest, compounded every half year, for one year.
III. 28% p.a. interest, for one year, compounded every year.
