Understanding the Compound Interest Problem

The question asks us to find the total amount in a bank account after 2 years, given an initial deposit (principal), an annual compound interest rate, and the time period.

Here's what we are given:

• Principal amount (\(P\)): Rs. 10000
• Annual interest rate (\(r\)): 20% per annum
• Time period (\(t\)): 2 years
• Compounding frequency: Annually (which means interest is compounded once a year, so \(n=1\))

We need to calculate the final amount (\(A\)) after 2 years, including the initial principal and the accumulated compound interest.

Compound Interest Formula

To solve this problem, we use the formula for the amount with compound interest:

\(A = P(1 + \frac{r}{n})^{nt}\)

Where:

• \(A\) is the amount after time \(t\)
• \(P\) is the principal amount
• \(r\) is the annual interest rate (expressed as a decimal)
• \(n\) is the number of times that interest is compounded per year
• \(t\) is the time the money is invested for in years

Step-by-Step Calculation of Compound Interest Amount

Let's plug in the given values into the compound interest formula:

• Principal (\(P\)) = 10000
• Annual rate (\(r\)) = 20% = 0.20
• Number of times compounded per year (\(n\)) = 1 (annually)
• Time in years (\(t\)) = 2

Substitute these values into the formula:

\(A = 10000(1 + \frac{0.20}{1})^{1 \times 2}\)

\(A = 10000(1 + 0.20)^2\)

\(A = 10000(1.20)^2\)

Now, calculate \((1.20)^2\):

\((1.20)^2 = 1.20 \times 1.20 = 1.44\)

Substitute this back into the equation for \(A\):

\(A = 10000 \times 1.44\)

\(A = 14400\)

So, the amount in the bank after 2 years will be Rs. 14400.

Analyzing the Calculation

Let's also look at how the compound interest accumulates year by year:

Year 1:

• Interest = Principal \(\times\) Rate = \(10000 \times 0.20 = 2000\)
• Amount at the end of Year 1 = Principal + Interest = \(10000 + 2000 = 12000\)

Year 2:

• For compound interest, the principal for Year 2 is the amount at the end of Year 1, which is Rs. 12000.
• Interest for Year 2 = Amount at start of Year 2 \(\times\) Rate = \(12000 \times 0.20 = 2400\)
• Amount at the end of Year 2 = Amount at start of Year 2 + Interest for Year 2 = \(12000 + 2400 = 14400\)

This step-by-step calculation confirms the result obtained using the compound interest formula.

Final Compound Amount

The amount in the bank after 2 years, with an initial deposit of Rs. 10000 at a 20% annual compound interest rate, is Rs. 14400.

Revision Table: Compound Interest Key Terms

Additional Information: Simple vs. Compound Interest

It's important to understand the difference between simple interest and compound interest.

• Simple Interest: Interest is calculated only on the initial principal amount. The interest earned each period is the same.
• Compound Interest: Interest is calculated on the initial principal AND on the accumulated interest from previous periods. This means the principal grows over time, leading to exponential growth in the total amount.

In this question, since the interest is compounded annually, the interest earned in the first year is added to the principal, and the interest for the second year is calculated on this new, larger amount.

If this were simple interest at 20% per annum:

• Interest per year = \(10000 \times 0.20 = 2000\)
• Total interest after 2 years = \(2000 \times 2 = 4000\)
• Total amount after 2 years = Principal + Total Interest = \(10000 + 4000 = 14000\)

Notice that the simple interest amount (Rs. 14000) is less than the compound interest amount (Rs. 14400). This highlights the power of compounding.