A sum of money amounts to Rs 6690 after 3 years and to Rs 10035 after 6 years at the same compound interest rate. Find the original principal (sum of money) invested.

Introduction:
This compound interest problem illustrates how knowing the amounts at two different times can help you find both the growth factor and the original principal. You are given the amounts after 3 years and 6 years at the same rate and must work backwards to determine the starting sum.

Given Data / Assumptions:

• Amount after 3 years (A3) = Rs 6690.
• Amount after 6 years (A6) = Rs 10035.
• The same compound interest rate is applied for all years.
• Compounding is assumed to be annual.
• We must find the principal P.

Concept / Approach:
Under compound interest, A = P * (1 + r)^n (taking r as a decimal for simplicity). So A6 = P * (1 + r)^6 and A3 = P * (1 + r)^3. If we divide A6 by A3, the principal cancels, leaving (1 + r)^3. From this ratio we find (1 + r) and then use A3 to compute P.

Step-by-Step Solution:
A3 = P * (1 + r)^3 = 6690A6 = P * (1 + r)^6 = 10035Divide A6 by A3: A6 / A3 = (1 + r)^6 / (1 + r)^3 = (1 + r)^3So (1 + r)^3 = 10035 / 6690 = 1.5Thus 1 + r = (1.5)^(1/3)Now P = A3 / (1 + r)^3 = 6690 / 1.5 = Rs 4460

Verification / Alternative Check:
Check the amounts using P = Rs 4460 and (1 + r)^3 = 1.5. After 3 years: 4460 * 1.5 = 6690, which matches A3. After 6 years: 4460 * (1.5)^2 = 4460 * 2.25 = Rs 10035, which matches A6. Hence, P = Rs 4460 is correct.

Why Other Options Are Wrong:
Rs 4360, Rs 4560, Rs 4660, Rs 4200: When used as principal values, they do not simultaneously produce both 6690 at 3 years and 10035 at 6 years under any single compound rate. They break the ratio relationship between A3 and A6.

Common Pitfalls:
Students sometimes attempt to compute the rate first using complicated logarithms instead of exploiting the ratio trick. Another error is to treat the difference of amounts as simple interest, which is not valid because compound interest grows multiplicatively, not linearly with time.

Final Answer:
The original sum of money invested is Rs 4460.