If 1/5 : 1/N = 1/N : 1/1.25, find the value of N.

Introduction / Context:
This is a proportion of the form a : b = b : c. When a, b, c are reciprocals or decimals, it is best to translate to an equation in b and solve. Here, a = 1/5, b = 1/N, c = 1/1.25.

Given Data / Assumptions:

• (1/5) : (1/N) = (1/N) : (1/1.25)
• N > 0

Concept / Approach:
For a : b = b : c, we have b^2 = a*c. Hence (1/N)^2 = (1/5) * (1/1.25). Compute the right-hand side and then invert to find N.

Step-by-Step Solution:
(1/N)^2 = (1/5) * (1/1.25) = (1/5) * 0.8 = 0.161/N^2 = 0.16 ⇒ N^2 = 1 / 0.16 = 6.25N = √6.25 = 2.5

Verification / Alternative check:
Plug back: 1/5 : 1/2.5 = 0.2 : 0.4 = 1 : 2; and 1/2.5 : 1/1.25 = 0.4 : 0.8 = 1 : 2. Matches.

Why Other Options Are Wrong:
1.25, 1.5, 2.25, 2.0 do not satisfy b^2 = a*c when substituted.

Common Pitfalls:
Squaring the wrong term or inverting the ratio incorrectly; mishandling 1/1.25 = 0.8.

Final Answer:
2.5