Understanding the Compound Interest Problem

This question asks us to find the number of years (n) it took for an investment to grow to a specific amount under compound interest. We are given the initial investment (principal), the interest rate, and the total extra amount received.

Breaking Down the Given Information

• Principal amount (P) = Rs. 4000
• Annual interest rate (R) = 10%
• Extra amount received = Rs. 1324

The total amount (A) received after n years is the principal plus the extra amount.

Total Amount (A) = Principal + Extra amount

A = Rs. 4000 + Rs. 1324 = Rs. 5324

We need to find the value of n, the number of years.

Applying the Compound Interest Formula

The formula for compound interest is given by:

\(A = P\left(1 + \frac{R}{100}\right)^n\)

Where:

• A is the total amount after n years
• P is the principal amount
• R is the annual interest rate
• n is the number of years

Solving for the Number of Years (n)

Now, let's substitute the known values into the formula:

\(5324 = 4000\left(1 + \frac{10}{100}\right)^n\)

\(5324 = 4000\left(1 + 0.1\right)^n\)

\(5324 = 4000\left(1.1\right)^n\)

To find n, we need to isolate the term \((1.1)^n\). Divide both sides by 4000:

\(\frac{5324}{4000} = (1.1)^n\)

Simplify the fraction on the left side:

\(\frac{5324 \div 4}{4000 \div 4} = \frac{1331}{1000}\)

So, the equation becomes:

\(\frac{1331}{1000} = (1.1)^n\)

Now, we need to express \(\frac{1331}{1000}\) as a power of 1.1. We know that \(1.1 = \frac{11}{10}\).

Let's check small powers of 1.1:

• \((1.1)^1 = 1.1\)
• \((1.1)^2 = 1.1 \times 1.1 = 1.21\)
• \((1.1)^3 = 1.1 \times 1.1 \times 1.1 = 1.21 \times 1.1 = 1.331\)

Also, \(\frac{1331}{1000} = 1.331\).

So, we have:

\(1.331 = (1.1)^n\)

This means:

\((1.1)^3 = (1.1)^n\)

Comparing the exponents on both sides, we find:

\(n = 3\)

Therefore, the value of n is 3 years.

Revision Table: Compound Interest Calculation

Additional Information: Compound Interest Concepts

Compound interest is interest calculated on the initial principal, which also includes all of the accumulated interest from previous periods on a deposit or loan. This means interest earns interest, leading to faster growth than simple interest.

• Compounding Frequency: Interest can be compounded annually, semi-annually, quarterly, monthly, or even daily. The more frequent the compounding, the faster the growth. The formula changes based on frequency: \(A = P(1 + R/(100k))^{nk}\), where k is the number of times interest is compounded per year. In this question, it's annual (k=1).
• Simple Interest vs. Compound Interest: Simple interest is calculated only on the principal amount. \(SI = (P \times R \times n)/100\). Compound interest includes interest on interest. Over longer periods, compound interest yields significantly higher returns than simple interest.
• Applications: Compound interest is used in various financial calculations, including savings accounts, loans, mortgages, and investment growth projections.