The compound interest on Rs. 30,000 at 7% per annum is Rs. 4347. For how many years was the money invested at this rate?

Introduction / Context:
This question asks you to determine the time period for which a sum of money was invested under compound interest. You are given the principal, the rate of interest per annum, and the total compound interest earned. Using this information, you can compute the final amount and then match it with powers of (1 + rate) to find the number of years. Such problems are standard in the topic of compound interest and help in understanding how time, rate, and interest are related.

Given Data / Assumptions:

• Principal (P) = Rs. 30,000.

• Rate of interest (R) = 7% per annum.

• Compound interest (CI) = Rs. 4347.

• Compounding is assumed to be annual.

• Time period (n years) is unknown and must be found.

Concept / Approach:
The amount after n years at compound interest is given by A = P * (1 + R/100)^n. Since CI = A − P, we have A = P + CI. Substituting the known values of P and CI will give the amount A. Next, we express A/P as (1 + R/100)^n and compare to known powers of (1.07) to find n. In many exam questions, the numbers are chosen to make this comparison straightforward.

Step-by-Step Solution:
Step 1: Compute the final amount using the given compound interest. Step 2: A = P + CI = 30,000 + 4347 = Rs. 34,347. Step 3: Write A in terms of P and (1 + R/100)^n: A = P * (1.07)^n. Step 4: Therefore, A / P = (1.07)^n = 34,347 / 30,000. Step 5: 34,347 / 30,000 = 1.1449. Step 6: Compute (1.07)^2 = 1.07 * 1.07 = 1.1449. Step 7: Since (1.07)^2 equals 1.1449, we have (1.07)^n = (1.07)^2. Step 8: Therefore, n = 2 years.

Verification / Alternative check:
We can verify directly by calculating the compound amount for 2 years at 7%. After 1 year: amount = 30,000 * 1.07 = Rs. 32,100. After 2 years: new amount = 32,100 * 1.07 = Rs. 34,347. The compound interest over 2 years = 34,347 − 30,000 = Rs. 4,347, which matches the given CI. This confirms that the time period is indeed 2 years.

Why Other Options Are Wrong:
• 2.5 years: This would not give an amount that matches 34,347 when using simple annual compounding and a fractional year, and the CI would differ from the given value.
• 3 years: (1.07)^3 is greater than 1.1449, so the amount would be higher than Rs. 34,347 for the same principal.
• 4 years: Over 4 years, the amount would be much larger and the interest far exceed Rs. 4,347.

Common Pitfalls:
Some students try to use simple interest formulas or approximate the powers without checking the exact values, leading to confusion. Others may forget to add the principal and interest to find the amount A before using the compound interest formula. Remembering that CI = A − P and using A/P to match powers of (1 + R/100) makes such problems straightforward.

Final Answer:
The money was invested for a period of 2 years.