Compound Interest – Repeated periods from a known multiplier: A sum becomes 2½ times in 6 years at compound interest. What multiple of the principal will it become in 18 years at the same rate?

Introduction / Context:
Under compound interest, equal time blocks multiply the amount by the same factor each time. If 6 years multiplies by 2.5, then 18 years (three such blocks) multiplies by (2.5)^3.

Given Data / Assumptions:

• Multiplier over 6 years M = 2.5
• 18 years = 3 × 6 years

Concept / Approach:
Overall multiplier over 18 years = M^3 = (2.5)^3 = 15.625.

Step-by-Step Solution:
(2.5)^2 = 6.25(2.5)^3 = 6.25 * 2.5 = 15.62515.625 = 125/8

Verification / Alternative check:
Express 2.5 as 5/2; (5/2)^3 = 125/8 (same result).

Why Other Options Are Wrong:
5/2 and 25/4 reflect 1 and 2 blocks; 625/16 = (2.5)^4 (too many blocks).

Common Pitfalls:
Adding multipliers linearly instead of multiplying them across equal periods.

Final Answer:
125/8