Simple Interest – Find principal and rate from two amounts at different times: A certain principal amounts to Rs 1,586 in 2 years and Rs 1,729 in 3 years at simple interest. Determine (i) the annual rate of interest and (ii) the original principal (sum).

Introduction / Context:
In simple interest (SI), the interest earned per year is constant. When two amounts are given at two successive times, the yearly interest can be obtained from the difference of the amounts. This allows us to back-calculate both the principal and the rate.

Given Data / Assumptions:

• Amount in 2 years, A2 = Rs 1586
• Amount in 3 years, A3 = Rs 1729
• Simple interest model: A = P * (1 + r * t), with r in decimal per year

Concept / Approach:
The increase from year 2 to year 3 equals one year of simple interest. Thus annual SI = A3 - A2. Then P = A2 - 2 * (annual SI). The rate r = (annual SI / P) * 100 percent per annum.

Step-by-Step Solution:

Verification / Alternative check:

Why Other Options Are Wrong:

• 9% and Rs 1300: rate too low for the observed yearly increase.
• 10% and Rs 1200: would give annual SI of Rs 120, not Rs 143.
• 11% and Rs 1200: inconsistent with A2.
• 10% and Rs 1300: annual SI would be Rs 130, not Rs 143.

Common Pitfalls:

• Using compound interest logic instead of simple interest.
• Subtracting total times instead of recognizing annual increment equals the difference of consecutive amounts.

Final Answer:
Rate = 11% per annum and Principal = Rs 1300.