A sum of money amounts to Rs. 6690 after 3 years and to Rs. 10,035 after 6 years on compound interest at the same rate. Find the original principal sum.

Introduction / Context:
This question deals with compound interest when you are given the amounts at two different times (after 3 years and after 6 years) but not the rate or principal explicitly. You must use the relationship between these amounts to determine the original principal. Such problems highlight how compound interest grows exponentially with time and how ratios of amounts can reveal the growth factor without directly knowing the rate.

Given Data / Assumptions:

• Amount after 3 years (A3) = Rs. 6690.

• Amount after 6 years (A6) = Rs. 10,035.

• The rate of interest per annum is the same throughout.

• Interest is compounded annually.

• Principal (P) is unknown and needs to be found.

Concept / Approach:
Let the annual growth factor be (1 + R/100). Then after 3 years, A3 = P * (1 + R/100)^3, and after 6 years, A6 = P * (1 + R/100)^6. Taking the ratio A6 / A3 cancels P and leaves (1 + R/100)^3. From this ratio, we can simplify the relationship and then express P in terms of A3 and the factor. In this specific problem, the numbers are arranged so that the ratio is simple, making the calculations straightforward.

Step-by-Step Solution:
Step 1: Write the expressions for the amounts. Step 2: A3 = P * (1 + R/100)^3 = 6690. Step 3: A6 = P * (1 + R/100)^6 = 10,035. Step 4: Take the ratio A6 / A3 to eliminate P. Step 5: A6 / A3 = [P * (1 + R/100)^6] / [P * (1 + R/100)^3] = (1 + R/100)^3. Step 6: Compute A6 / A3 = 10,035 / 6690. Step 7: 10,035 / 6690 = 1.5. Step 8: Therefore, (1 + R/100)^3 = 1.5. Step 9: From A3 = P * (1 + R/100)^3, we have 6690 = P * 1.5. Step 10: Hence, P = 6690 / 1.5 = Rs. 4460.

Verification / Alternative check:
We can verify by checking the amounts using P = 4460. After 3 years, A3 = 4460 * 1.5 = Rs. 6690, which matches the given value. After 6 years, A6 = A3 * 1.5 = 6690 * 1.5 = Rs. 10,035, also matching the given data. This confirms that the principal must be Rs. 4460.

Why Other Options Are Wrong:
• Rs. 4360, Rs. 4560, and Rs. 4660: None of these values satisfy both conditions simultaneously. If you plug any of them into the compound interest relationship, you will not obtain both A3 = Rs. 6690 and A6 = Rs. 10,035 exactly, so they cannot be the correct principal.

Common Pitfalls:
A frequent mistake is attempting to find the exact interest rate first using fractional powers, which can complicate calculations unnecessarily. The smarter approach is to use the ratio of amounts to determine the growth factor and then directly solve for P. Another error is mis-computing the ratio A6 / A3. Carefully simplifying the ratio and using it to express P makes the problem much easier.

Final Answer:
The original principal sum is Rs. 4460.