Equal sums at 10% — unequal durations, known gap in interest: Two equal sums are lent at simple interest at 10% per annum, one for 4 years and the other for 5 years. If the difference between the interests is ₹ 220, find each sum.

Introduction / Context:
For equal principals but different durations at the same rate, the difference of simple interest directly equals Principal * Rate * (difference in years) / 100.

Given Data / Assumptions:

• Rate r = 10% p.a.
• Times: 4 years and 5 years
• Interest difference ΔI = ₹ 220
• Equal principal P (unknown)

Concept / Approach:
ΔI = P * r * (t2 − t1) / 100 → P * 10 * 1 / 100 = 0.10P. Set 0.10P = 220 to solve for P.

Step-by-Step Solution:
0.10P = 220 → P = ₹ 2200

Verification / Alternative check:
SI for 5 years: 2200 * 0.10 * 5 = ₹ 1100; for 4 years: 2200 * 0.10 * 4 = ₹ 880; difference = ₹ 220 (correct).

Why Other Options Are Wrong:
₹ 880 and ₹ 1100 are actually the interests in those periods, not the principal. ₹ 1640 or ₹ 2000 do not satisfy the ΔI condition.

Common Pitfalls:
Confusing the interest amounts with the principal, or forgetting the /100 factor in the SI formula.

Final Answer:
₹ 2200