Understanding the Difference Between Simple Interest and Compound Interest

The question asks us to find the principal amount given the difference between the simple interest (SI) and the compound interest (CI) for a specific time period and rate. We are given that the difference between CI and SI for 2 years at a rate of 8% per annum is Rs. 16.80. We need to determine the initial principal amount that was invested or borrowed.

Formulas for Simple Interest and Compound Interest

Let's define the terms used in the problem:

• Principal amount (P): The initial sum of money.
• Rate of interest (R): The annual interest rate (as a percentage).
• Time (T): The duration for which the money is invested or borrowed (in years).

The formula for Simple Interest (SI) is:

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

For Compound Interest (CI), compounded annually, the amount (A) after T years is:

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

The Compound Interest (CI) itself is the amount minus the principal:

\(\text{CI} = A - P = P \left(1 + \frac{R}{100}\right)^T - P = P \left[ \left(1 + \frac{R}{100}\right)^T - 1 \right]\)

Calculating the Difference for 2 Years

We are given the time period T = 2 years and the rate R = 8% per annum. The difference between CI and SI is Rs. 16.80.

Let's calculate SI for 2 years:

\(\text{SI} = \frac{P \times 8 \times 2}{100} = \frac{16P}{100} = \frac{4P}{25}\)

Now let's calculate CI for 2 years:

\(\text{CI} = P \left[ \left(1 + \frac{8}{100}\right)^2 - 1 \right]\)

\(\text{CI} = P \left[ \left(1 + \frac{2}{25}\right)^2 - 1 \right]\)

\(\text{CI} = P \left[ \left(\frac{27}{25}\right)^2 - 1 \right]\)

\(\text{CI} = P \left[ \frac{729}{625} - 1 \right]\)

\(\text{CI} = P \left[ \frac{729 - 625}{625} \right]\)

\(\text{CI} = P \left[ \frac{104}{625} \right]\)

The difference between CI and SI is:

\(\text{CI} - \text{SI} = \frac{104P}{625} - \frac{4P}{25}\)

To subtract these fractions, find a common denominator, which is 625:

\(\text{CI} - \text{SI} = \frac{104P}{625} - \frac{4P \times (625/25)}{625} = \frac{104P}{625} - \frac{4P \times 25}{625}\)

\(\text{CI} - \text{SI} = \frac{104P - 100P}{625} = \frac{4P}{625}\)

Using the Direct Formula for CI - SI Difference (2 Years)

Alternatively, there is a direct formula for the difference between CI and SI for 2 years when interest is compounded annually:

\(\text{Difference} = P \left( \frac{R}{100} \right)^2\)

This formula is derived from the general expressions for CI and SI for 2 years. \(\text{CI} = P \left( \left(1 + \frac{R}{100}\right)^2 - 1 \right) = P \left( 1^2 + 2 \times 1 \times \frac{R}{100} + \left(\frac{R}{100}\right)^2 - 1 \right) = P \left( \frac{2R}{100} + \left(\frac{R}{100}\right)^2 \right)\) \(\text{SI} = \frac{P \times R \times 2}{100} = \frac{2PR}{100}\) \(\text{CI} - \text{SI} = P \left( \frac{2R}{100} + \left(\frac{R}{100}\right)^2 \right) - \frac{2PR}{100} = \frac{2PR}{100} + P\left(\frac{R}{100}\right)^2 - \frac{2PR}{100} = P\left(\frac{R}{100}\right)^2\)

Let's use this direct formula as it simplifies the calculation significantly.

Solving for the Principal Amount

We are given the difference is Rs. 16.80 and the rate R = 8%.

Using the formula: \(\text{Difference} = P \left( \frac{R}{100} \right)^2\)

Substitute the values:

\(16.80 = P \left( \frac{8}{100} \right)^2\)

\(16.80 = P \left( \frac{2}{25} \right)^2\)

\(16.80 = P \times \frac{4}{625}\)

Now, solve for P:

\(P = 16.80 \times \frac{625}{4}\)

\(P = \frac{16.80 \times 625}{4}\)

We can perform the calculation:

\(16.80 \div 4 = 4.20\)

So,

\(P = 4.20 \times 625\)

Let's calculate \(4.20 \times 625\):

\(4.2 \times 625 = (4 + 0.2) \times 625 = 4 \times 625 + 0.2 \times 625\)

\(4 \times 625 = 2500\)

\(0.2 \times 625 = \frac{1}{5} \times 625 = \frac{625}{5} = 125\)

\(P = 2500 + 125 = 2625\)

Thus, the principal amount is Rs. 2,625.

Let's check this with the calculated difference formula \(\frac{4P}{625}\):

Difference = \(\frac{4 \times 2625}{625}\)

We know \(2625 = 4.20 \times 625\), so let's use that:

Difference = \(\frac{4 \times (4.20 \times 625)}{625} = 4 \times 4.20 = 16.80\)

This matches the given difference, confirming our principal amount calculation is correct.

Final Answer

The principal amount is Rs. 2,625.

Revision Table: Key Concepts in Interest Calculations

Additional Information on Interest Calculations

Understanding simple interest and compound interest is fundamental in financial mathematics. Simple interest is easier to calculate but earns less over time compared to compound interest, especially over longer periods and at higher rates.

• Impact of Compounding Frequency: The CI formulas assume annual compounding. If interest is compounded semi-annually, quarterly, or monthly, the effective rate per period and the number of periods change, affecting the total CI earned. For example, if the annual rate is R% compounded semi-annually for T years, the rate per period is (R/2)% and the number of periods is 2T.
• Growth Comparison: Simple interest results in linear growth of the principal over time, while compound interest results in exponential growth. This difference becomes substantial over longer durations.
• Applications: Simple interest is often used for short-term loans. Compound interest is standard for savings accounts, fixed deposits, and most loans like mortgages and credit cards. The power of compounding is a key concept in long-term investments.