Solving for the Principal Amount Using Interest Differences

This problem requires us to find the initial sum of money, also known as the principal amount, given the difference between compound interest and simple interest over a specific period and at a particular interest rate.

Understanding Simple Interest (SI) and Compound Interest (CI)

• Simple Interest (SI): This is calculated only on the initial principal amount. The interest earned remains constant for each time period.
• Compound Interest (CI): This is calculated on the initial principal amount as well as on the accumulated interest from previous periods. This means that interest earns interest, leading to a potentially faster growth of the sum.

Details Provided in the Question

• Interest Rate ($R$): 8% per annum
• Time Period ($T$): 2 years
• Difference between CI and SI ($\text{CI} - \text{SI}$): ₹144
• Principal Amount ($P$): This is what we need to find.

Formula Relating Interest Difference and Principal

The formulas for Simple Interest and Compound Interest are:

• SI = $ \frac{P \times R \times T}{100} $
• CI for 2 years = $ P \left(1 + \frac{R}{100}\right)^2 - P $

For calculations involving the difference between compound interest and simple interest over 2 years, a direct formula is often used:

$$ \text{CI} - \text{SI} = P \left(\frac{R}{100}\right)^2 $$

This formula is derived from the basic SI and CI formulas and is very useful when the difference is given.

Step-by-Step Calculation of the Principal Sum

We are given the difference between CI and SI is ₹144 for 2 years at an 8% annual interest rate. Let's use the derived formula to find the principal amount ($P$).

1. Substitute the known values into the formula:

   We have $\text{CI} - \text{SI} = 144$, $R = 8$, and $T = 2$. Plugging these into the formula $\text{CI} - \text{SI} = P \left(\frac{R}{100}\right)^2$:

   $$ 144 = P \left(\frac{8}{100}\right)^2 $$

2. Simplify the fraction representing the rate:

   The term $ \frac{8}{100} $ simplifies to $ \frac{2}{25} $.

   So the equation becomes:

   $$ 144 = P \left(\frac{2}{25}\right)^2 $$

3. Calculate the square of the simplified rate:

   $$ \left(\frac{2}{25}\right)^2 = \frac{2^2}{25^2} = \frac{4}{625} $$

   Now, substitute this back into the equation:

   $$ 144 = P \times \frac{4}{625} $$

4. Isolate the Principal Amount ($P$):

   To find $P$, we need to rearrange the equation. Multiply both sides by $ \frac{625}{4} $:

   $$ P = 144 \times \frac{625}{4} $$

5. Perform the final calculation:

   First, divide 144 by 4:

   $$ \frac{144}{4} = 36 $$

   Next, multiply the result by 625:

   $$ P = 36 \times 625 $$

   $$ P = 22500 $$

Final Answer Determination

The calculation shows that the principal amount ($P$) is ₹22,500. This is the sum on which the simple and compound interest were calculated, leading to the specified difference of ₹144 over the 2-year period at an 8% interest rate.