A sum of money becomes Rs. 13, 380 after 3 years and Rs. 20, 070 after 6 years…

For compound interest, the amount after n years is A = P × (1 + R/100)n, where P is the principal and R is the annual rate. When the amounts at two different time periods are known but R is not given directly, dividing the later-period equation by the earlier-period equation cancels P and leaves an equation purely in the growth factor (1 + R/100) — this ratio, and then the principal, can be found without ever solving for R itself.

1. Write the amount equation for 6 years: A6 = P × (1 + R/100)6 = 20070.

2. Write the amount equation for 3 years: A3 = P × (1 + R/100)3 = 13380.

3. Divide equation (1) by equation (2): (1 + R/100)3 = 20070/13380 = 3/2.

4. Substitute (1 + R/100)3 = 3/2 back into equation (2): P × 3/2 = 13380.

5. Solve for P: P = 13380 × 2/3 = 8920.

Cross-check: since (1 + R/100)6 = [(1 + R/100)3]2 = (3/2)2 = 9/4, the 6-year amount should equal P × 9/4 = 8920 × 9/4 = 20070, which matches the value given in the question exactly — confirming the principal.

So the sum that satisfies both given amounts is 8920 rupees.