Introduction / Context:
This problem connects a simple interest statement with a compound interest calculation. We are told that in 3 years at simple interest, a principal grows by 18% of itself. From this we deduce the annual rate. Then, using that rate, we calculate the compound interest on a different principal (Rs. 25,000) over the same 3-year period with annual compounding.
Given Data / Assumptions:
• Under simple interest, in 3 years, the increase is 18% of the principal.
• Therefore, for simple interest: total interest over 3 years = 18% of P.
• Hence, annual simple interest rate R% must satisfy 3R = 18.
• New principal for compound interest calculation is P_new = Rs. 25,000.
• Time for compound interest T = 3 years.
Concept / Approach:
First, we find the annual rate R by using the simple interest relation. Once R is known, we apply the compound interest formula A = P * (1 + R / 100)^T for P = 25000 and T = 3 years. The compound interest is then CI = A - P. Careful calculation is needed to keep decimal places correct.
Step-by-Step Solution:
From simple interest information: in 3 years increase is 18% of principal.
So 3R = 18, which gives R = 18 / 3 = 6% per annum.
Now we use R = 6% to compute compound interest on Rs. 25,000 for 3 years.
Amount A = 25000 * (1 + 6 / 100)^3 = 25000 * (1.06)^3.
Compute (1.06)^2 = 1.1236.
Then (1.06)^3 = 1.1236 * 1.06 = 1.191016.
So A = 25000 * 1.191016 = Rs. 29775.40.
Compound interest CI = A - P = 29775.40 - 25000 = Rs. 4775.40.
Verification / Alternative check:
We can also compute year by year: After 1 year, amount = 25000 * 1.06 = 26500. After 2 years, amount = 26500 * 1.06 = 28090. After 3 years, amount = 28090 * 1.06 = 29775.40. Subtracting the original 25000 yields 4775.40, confirming the compound interest.
Why Other Options Are Wrong:
4557.40 and 5000.00 assume incorrect rate values or mix simple and compound interest ideas. 5575.40 and 5774.40 correspond to higher effective rates than 6% or to extra years. Only Rs. 4775.40 matches the compound interest obtained from a 6% rate over 3 years on Rs. 25,000.
Common Pitfalls:
A common mistake is to treat the 18% as a yearly rate instead of a 3-year cumulative rate, which would incorrectly set R = 18%. Another error is to stop after computing simple interest instead of compound interest, or to miscalculate (1.06)^3. Careful handling of exponents and interpretation of the given statement avoids these issues.
Final Answer:
The compound interest on Rs. 25,000 in 3 years at this rate is
Rs. 4775.40.
Discussion & Comments