The compound interest earned on a certain principal at 10% per annum for 2 years is Rs 420. At this same rate of interest, what is the principal amount, in rupees, that must have been invested to generate this compound interest?

Introduction / Context:
In this question, the total compound interest over 2 years and the annual rate of interest are given, and we are asked to find the principal amount. This reverses the usual direction of compound interest questions. Instead of computing interest from a known principal, we must work backward from the known interest to determine the original principal. This tests understanding of the compound interest formula and the ability to manipulate it algebraically.

Given Data / Assumptions:

• Total compound interest (CI) for 2 years = Rs 420.
• Rate of interest R = 10% per annum.
• Time T = 2 years.
• Interest is compounded annually.
• Principal P is unknown and must be found.

Concept / Approach:
With annual compounding, amount after T years is:
A = P * (1 + R / 100)^TCompound interest is:
CI = A - PHere, T = 2 and R = 10, so:
A = P * (1 + 10 / 100)^2 = P * (1.1)^2A = P * 1.21Then CI = A - P = P * 1.21 - P = P * (1.21 - 1) = 0.21 * P. We know CI is 420, so 0.21 * P = 420, and we can solve for P.

Step-by-Step Solution:
Step 1: Write the relation between CI and P for 2 years at 10%: CI = P * (1.21 - 1) = 0.21 * P.Step 2: We are told CI = 420, so set up the equation 0.21 * P = 420.Step 3: Solve for P by dividing both sides by 0.21: P = 420 / 0.21.Step 4: Compute 420 / 0.21. Note that 0.21 = 21 / 100. So P = 420 * 100 / 21.Step 5: Simplify: 420 / 21 = 20. Therefore P = 20 * 100 = 2000.Step 6: So the required principal is Rs 2,000.

Verification / Alternative check:
We can verify by computing the compound interest on Rs 2,000 at 10% per annum for 2 years. For year 1, interest = 2000 * 10 / 100 = 200, so amount = 2200. For year 2, interest = 2200 * 10 / 100 = 220, so final amount = 2420. The total compound interest is 2420 - 2000 = 420, which matches the given value. Therefore, P = Rs 2,000 is correct.

Why Other Options Are Wrong:
Rs.1900, Rs.2100, Rs.2150, and Rs.2200 all produce compound interests that are different from Rs 420 when the same formula is applied. For example, taking P = 2100 would give CI = 0.21 * 2100 = 441, which is larger than 420. Similarly, 1900 gives less than 420. Thus none of these meet the stated condition that the compound interest is exactly Rs 420.

Common Pitfalls:
One common mistake is to confuse the simple interest formula with compound interest, for example trying SI = P * R * T / 100 instead of using the 2 year compound interest relation. Another is to compute one year of interest and simply double it, which only works for simple interest. Some candidates also forget that (1.1)^2 equals 1.21, not 1.20, and this small error leads to incorrect results. Working carefully with the formula and double checking the arithmetic avoids these issues.

Final Answer:
The principal that must have been invested to earn a compound interest of Rs 420 in 2 years at 10% per annum is Rs.2000.