Compound Interest: Finding Principal from 2nd Year Interest

This problem requires us to find the original principal sum. We are provided with the amount of interest earned during the second year and the annual interest rate, assuming the interest is compounded annually.

Here is the information given:

• Interest earned in the 2nd year: ₹2,352
• Annual interest rate (r): 5%, which can be written as $0.05$
• Interest compounding: Annually
• We need to find: The principal sum (P)

Method to Calculate Principal Sum

In compound interest, the interest earned each year is calculated on the accumulated amount from the previous year. The interest earned specifically *during* the second year is calculated on the total amount present at the end of the first year.

Let's follow these steps:

1. Determine the Amount After the First Year: The initial principal (P) accrues interest for the first year. The total amount ($A_1$) at the end of the first year is calculated using the formula:

   $ A_1 = P(1 + r) $

   Using the given rate ($r = 0.05$):

   $ A_1 = P(1 + 0.05) = P(1.05) $

2. Calculate the Interest Earned During the Second Year: This interest is calculated on the amount ($A_1$) at the given rate ($r$).

   Interest for the 2nd year = $A_1 \times r$

   By substituting the expression for $A_1$:

   Interest for the 2nd year = $ [P(1.05)] \times 0.05 $

3. Set Up the Equation Using Given Interest: We are told that the interest earned during the second year is ₹2,352. We can equate this to our calculated expression:

   $ [P(1.05)] \times 0.05 = 2352 $

4. Solve for the Principal Sum (P): Now, we rearrange the equation to solve for P.

   $ P \times (1.05 \times 0.05) = 2352 $

   $ P \times 0.0525 = 2352 $

   To find P, divide 2352 by 0.0525:

   $ P = \frac{2352}{0.0525} $

   It's helpful to convert the decimal $0.0525$ into a fraction, which is $\frac{525}{10000}$:

   $ P = \frac{2352}{\frac{525}{10000}} = \frac{2352 \times 10000}{525} $

   Let's simplify this fraction. Divide both the numerator and the denominator by 25:

   $ P = \frac{2352 \times 400}{21} $

   Now, perform the division of 2352 by 21:

   $ \frac{2352}{21} = 112 $

   Finally, multiply the result by 400:

   $ P = 112 \times 400 $

   $ P = 44800 $

Resulting Principal Sum

Based on the calculations, the original principal sum is found to be ₹44,800.