Calculating Compound Interest from Simple Interest Rate

This problem requires us to first determine the rate of simple interest from the given information and then use that rate to calculate the compound interest on a different amount over a different period.

Step 1: Finding the Rate of Simple Interest

We are told that an amount increases by 70% in 7 years at simple interest. This means the simple interest earned is 70% of the principal amount.

Let the principal amount be \(P\). The simple interest (\(SI\)) is 70% of \(P\), which is \(0.70 \times P\).

The time period (\(T\)) is 7 years.

The formula for simple interest is:

\(SI = \frac{P \times R \times T}{100}\)

Where \(R\) is the rate of interest per annum.

Substituting the known values:

\(0.70 \times P = \frac{P \times R \times 7}{100}\)

We can cancel \(P\) from both sides (assuming \(P > 0\)):

\(0.70 = \frac{R \times 7}{100}\)

Now, solve for \(R\):

\(0.70 \times 100 = R \times 7\)

\(70 = R \times 7\)

\(R = \frac{70}{7}\)

\(R = 10\%\)

So, the rate of simple interest is 10% per annum. This is the same rate we will use for the compound interest calculation.

Step 2: Calculating the Compound Interest

Now we need to find the compound interest on ₹8,000 after 3 years at a rate of 10% per annum.

Principal amount (\(P\)) = ₹8,000

Rate of interest (\(R\)) = 10% per annum

Time period (\(n\)) = 3 years

The formula for the compound amount (\(A\)) is:

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

Substitute the values:

\(A = 8000 \left(1 + \frac{10}{100}\right)^3\)

\(A = 8000 \left(1 + 0.1\right)^3\)

\(A = 8000 \left(1.1\right)^3\)

Calculate \((1.1)^3\):

\(1.1 \times 1.1 = 1.21\)

\(1.21 \times 1.1 = 1.331\)

So,

\(A = 8000 \times 1.331\)

\(A = 10648\)

The compound amount after 3 years is ₹10,648.

The compound interest (\(CI\)) is the compound amount minus the principal:

\(CI = A - P\)

\(CI = 10648 - 8000\)

\(CI = 2648\)

The compound interest on ₹8,000 after 3 years at 10% per annum is ₹2,648.

Conclusion

First, we found the rate using the simple interest information, which was 10%. Then, we used this rate to calculate the compound interest on ₹8,000 for 3 years, finding it to be ₹2,648.

Revision Table: Interest Calculations

Additional Information: Simple vs. Compound Interest

Understanding the difference between simple interest and compound interest is crucial in finance and mathematics problems.

• Simple Interest: The interest is calculated only on the initial principal amount for the entire duration. The interest earned in each period does not add to the principal for calculating interest in subsequent periods. It's a linear growth.
• Compound Interest: The interest for each period is added to the principal amount before calculating the interest for the next period. This means interest is earned on the principal plus any accumulated interest. This leads to exponential growth over time. The more frequently interest is compounded (annually, half-yearly, quarterly), the faster the amount grows, assuming the same nominal rate.

In this problem, the initial simple interest information helped us find the common rate, which was then applied in a compound interest scenario. The power of compounding becomes evident over longer periods or at higher rates.