Understanding the Problem: Simple and Compound Interest

The question involves two parts related to interest calculations: first, finding the rate of simple interest based on a given increase in principal over a period, and second, using that rate to calculate the compound interest on a different principal amount over a different period.

Step 1: Calculating the Simple Interest Rate

We are given that a sum of money increases by 45% in 9 years at simple interest. This means the simple interest earned is 45% of the original principal amount. Let the original principal be \(P\).

• Simple Interest (SI) = 45% of \(P\) = \(0.45P\)
• Time (T) = 9 years

The formula for Simple Interest is:

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

Where R is the rate of interest per annum. We can substitute the known values into the formula:

\( 0.45P = \frac{P \times R \times 9}{100} \)

To find the rate (R), we can simplify the equation. Assuming \(P\) is not zero (which it must be for interest to be calculated), we can divide both sides by \(P\):

\( 0.45 = \frac{R \times 9}{100} \)

Now, solve for R:

\( 0.45 \times 100 = R \times 9 \)

\( 45 = 9R \)

\( R = \frac{45}{9} \)

\( R = 5 \)

So, the simple interest rate is 5% per annum.

Step 2: Calculating the Compound Interest

Now we need to calculate the compound interest on Rs. 12,000 after 3 years at the same rate of 5% per annum. The principal for this calculation is Rs. 12,000.

• Principal (P) = Rs. 12,000
• Rate (r) = 5% per annum (same rate found from simple interest)
• Time (n) = 3 years

The formula for the amount (A) after compound interest is:

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

Substitute the values:

\( A = 12000 \left(1 + \frac{5}{100}\right)^3 \)

\( A = 12000 \left(1 + 0.05\right)^3 \)

\( A = 12000 \left(1.05\right)^3 \)

Calculate \((1.05)^3\):

\( (1.05)^1 = 1.05 \)

\( (1.05)^2 = 1.05 \times 1.05 = 1.1025 \)

\( (1.05)^3 = 1.1025 \times 1.05 = 1.157625 \)

Now, substitute this back into the formula for A:

\( A = 12000 \times 1.157625 \)

\( A = 13891.5 \)

This is the amount after 3 years. To find the Compound Interest (CI), subtract the original principal from the amount:

\( \text{CI} = A - P \)

\( \text{CI} = 13891.5 - 12000 \)

\( \text{CI} = 1891.5 \)

The compound interest after 3 years is Rs. 1891.5.

Summary of Calculations

Final Answer

The compound interest of Rs. 12,000 after 3 years at the same rate (5% per annum) is Rs. 1891.5.

Revision Table: Interest Formulas

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 for the entire duration. The interest earned does not get added to the principal for calculating future interest.
• Compound Interest: Interest is calculated on the initial principal and also on the accumulated interest from previous periods. This means the principal amount grows over time, leading to exponential growth in the amount.

In this problem, we first used the simple interest concept to find the consistent annual rate and then applied the compound interest concept using that rate to find the total interest earned over a different period on a different principal.