A man takes a loan at 5% simple interest for a period of 2 years. He immediately gives this money on loan at 5% compound interest for 2 years. What is the amount of loan he has taken if he makes a profit of ₹ 2,100?

To find the principal amount (P), we calculate the difference between Compound Interest (CI) and Simple Interest (SI) over 2 years.

The formula for Simple Interest is:
SI = (P × R × T) / 100 = (P × 5 × 2) / 100 = 0.10P

The formula for Compound Interest is:
CI = P[(1 + R/100)^T - 1] = P[(1 + 5/100)² - 1] = P[(1.05)² - 1] = P[1.1025 - 1] = 0.1025P

The profit is the difference between the interest earned (CI) and the interest paid (SI):
Profit = CI - SI
2,100 = 0.1025P - 0.10P
2,100 = 0.0025P

Solving for P:
P = 2,100 / 0.0025 = 2,100 / (25/10,000)
P = (2,100 × 10,000) / 25
P = 2,100 × 400 = 8,40,000

Alternatively, using the direct formula for the difference between CI and SI for 2 years:
Difference = P(R/100)²
2,100 = P(5/100)² = P(1/20)² = P/400
P = 2,100 × 400 = 8,40,000.