Finding the Compound Interest Rate

This problem involves calculating the annual rate of interest when an amount grows from a principal value over a specific period with annual compounding.

We are given the following information:

• Principal amount (P) = Rs. 1000
• Amount after compounding (A) = Rs. 1331
• Time period (n) = 3 years
• Compounding frequency = Annually

We need to find the annual rate of interest (R).

The formula for the amount (A) under compound interest, compounded annually, is given by:

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

Now, we can substitute the given values into this formula:

\( 1331 = 1000\left(1 + \frac{R}{100}\right)^3 \)

To find R, we need to isolate the term containing R. First, divide both sides of the equation by the principal amount, P (1000):

\( \frac{1331}{1000} = \left(1 + \frac{R}{100}\right)^3 \)

Next, we need to find the cube root of both sides of the equation to remove the power of 3. We can rewrite the fraction \(\frac{1331}{1000}\) as a cube. We know that \(11^3 = 1331\) and \(10^3 = 1000\). So, \(\frac{1331}{1000} = \left(\frac{11}{10}\right)^3\).

The equation becomes:

\( \left(\frac{11}{10}\right)^3 = \left(1 + \frac{R}{100}\right)^3 \)

Taking the cube root of both sides:

\( \frac{11}{10} = 1 + \frac{R}{100} \)

Now, we need to isolate \(\frac{R}{100}\). Subtract 1 from both sides:

\( \frac{11}{10} - 1 = \frac{R}{100} \)

Simplify the left side:

\( \frac{11 - 10}{10} = \frac{1}{10} \)

So, we have:

\( \frac{1}{10} = \frac{R}{100} \)

Finally, solve for R by multiplying both sides by 100:

\( R = \frac{1}{10} \times 100 \)

\( R = 10 \)

The rate of interest per annum is 10%.

Verification

Let's check if Rs. 1000 compounded annually at 10% for 3 years amounts to Rs. 1331.

\( A = 1000\left(1 + \frac{10}{100}\right)^3 \)

\( A = 1000\left(1 + 0.1\right)^3 \)

\( A = 1000\left(1.1\right)^3 \)

\( A = 1000 \times 1.331 \)

\( A = 1331 \)

This matches the given amount, confirming our calculated rate is correct.

The rate of interest per annum is 10%.

Revision Table: Compound Interest Terms

Additional Information: Understanding Compound Interest

Compound interest is often called "interest on interest." Unlike simple interest, where interest is calculated only on the initial principal amount, compound interest calculates interest on the initial principal and also on the accumulated interest from previous periods.

Key points about compound interest:

• It leads to faster growth of money compared to simple interest over longer periods.
• The frequency of compounding (annually, semi-annually, etc.) affects the total amount earned. More frequent compounding generally leads to higher returns.
• The formula \(A = P\left(1 + \frac{R/k}{100}\right)^{nk}\) is used when compounding is not annual, where 'k' is the number of times interest is compounded per year. In this problem, compounding is annual, so \(k=1\), and the formula simplifies to \(A = P\left(1 + \frac{R}{100}\right)^n\).
• Understanding compound interest is crucial for investments, loans, and financial planning.