In how many years will a sum of Rs. 8,000 amount to Rs. 9,261 at 5% per annum if the interest is compounded annually?

Introduction:
This problem asks us to find the time period required for a principal of Rs. 8,000 to grow to Rs. 9,261 at a compound interest rate of 5% per annum, compounded annually. We have to determine the number of years t that satisfies the compound interest relation.

Given Data / Assumptions:

• Principal (P) = Rs. 8,000.
• Final amount (A) = Rs. 9,261.
• Rate of interest (r) = 5% per annum.
• Interest is compounded annually.
• Time (t) in years is unknown.

Concept / Approach:
With annual compounding, the amount after t years is given by:
A = P * (1 + r/100)^tWe know A, P, and r, and must solve for t. Often, this involves logarithms. However, since the options are small integers or half integers, we can also check powers of (1 + r/100) directly.

Step-by-Step Solution:
Step 1: Substitute the known values into the formula: 9,261 = 8,000 * (1 + 5/100)^t.Step 2: Simplify the base: 1 + 5/100 = 1.05.Step 3: Divide both sides by 8,000: (1.05)^t = 9,261 / 8,000.Step 4: Compute the ratio: 9,261 / 8,000 = 1.157625.Step 5: Now we need t such that (1.05)^t = 1.157625.Step 6: Check integer powers. For t = 2, (1.05)^2 = 1.1025, which is too small. For t = 3, (1.05)^3 = 1.157625, which matches exactly.Step 7: Therefore t = 3 years.Hence, it takes 3 years for Rs. 8,000 to grow to Rs. 9,261 at 5% compound interest.

Verification / Alternative check:
Calculate year by year. Year 1: amount = 8,000 * 1.05 = 8,400. Year 2: amount = 8,400 * 1.05 = 8,820. Year 3: amount = 8,820 * 1.05 = 9,261. This confirms that after 3 full years the amount exactly equals Rs. 9,261.

Why Other Options Are Wrong:
At 2 years, the amount is 8,820, which is less than 9,261. At 3.5 or 4 years, the amount would exceed 9,261. The option of 2 years is clearly too short, while 3.5, 4, and 4.5 years are longer than necessary. Only 3 years matches the given final amount exactly.

Common Pitfalls:
Errors can occur if students attempt to use simple interest formulas, which would not match the compound growth. Others may round intermediate results too much and mistakenly conclude that t is not an integer. Checking exact powers for small integer t is often quicker and more reliable in such problems.

Final Answer:
The sum of Rs. 8,000 will amount to Rs. 9,261 at 5% per annum compounded annually in 3 years.