The simple interest on a sum at 5% per annum for 8 years is ₹840. For what annual rate of simple interest (in %) would the same sum earn the same interest of ₹840 in 5 years?

Introduction:
This question tests how simple interest scales with time and rate for the same principal. In simple interest, SI = (P * r * t)/100. If the principal and interest are fixed and only the time changes, the rate must adjust inversely with time. A clean way is to compute the principal from the first condition, then compute the new required rate for the new time. An even faster method uses proportionality (r1*t1 = r2*t2 when P and SI are fixed).

Given Data / Assumptions:

• SI = ₹840
• Case 1: r1 = 5% per annum, t1 = 8 years
• Case 2: t2 = 5 years, rate r2 = ?
• Same principal in both cases
• Formula: SI = (P * r * t) / 100

Concept / Approach:
Method 1: Find P from case 1, then solve for r2 from case 2. Method 2 (shortcut): Since SI and P are constant, (r1*t1) must equal (r2*t2). Use r2 = (r1*t1)/t2.

Step-by-Step Solution:
From case 1: 840 = (P * 5 * 8) / 100 840 = (P * 40) / 100 = 0.4P P = 840 / 0.4 = 2100 Now case 2: 840 = (2100 * r2 * 5) / 100 840 = (10500 * r2) / 100 r2 = (840 * 100) / 10500 = 8

Verification / Alternative check:
Shortcut: r2 = (r1*t1)/t2 = (5*8)/5 = 8%. Same result, confirming correctness.

Why Other Options Are Wrong:
7% would produce less interest in 5 years, while 9% and 10% would produce more. 6% is far too low. Only 8% keeps SI at ₹840 for the same principal when time is 5 years.

Common Pitfalls:
Assuming SI is proportional to time only, forgetting the inverse relation between rate and time when SI is fixed, or mixing up years between the two cases.

Final Answer:
The required annual rate is 8%.