Find the third proportional to 0.8 and 0.2. (That is, 0.8 : 0.2 = 0.2 : x. Find x.)

Introduction / Context:
The third proportional to a and b is a number c such that a : b = b : c. This leads to the relation c = b^2 / a for positive a, b. Knowing this formula is a standard ratio-and-proportion skill.

Given Data / Assumptions:

• a = 0.8
• b = 0.2
• Find c such that 0.8 : 0.2 = 0.2 : c

Concept / Approach:
Using a : b = b : c ⇒ c = b^2 / a. Substitute a = 0.8 and b = 0.2 and compute carefully to avoid decimal mistakes.

Step-by-Step Solution:
c = b^2 / a = (0.2 * 0.2) / 0.8= 0.04 / 0.8= 0.05

Verification / Alternative check:
Check the proportion: 0.8 : 0.2 = 4 : 1. With c = 0.05, 0.2 : 0.05 = 4 : 1 as well. Hence the proportion holds.

Why Other Options Are Wrong:
0.4 and 0.8 are too large; 0.032 and 0.02 are too small and do not satisfy 0.2 : c = 4 : 1.

Common Pitfalls:
Using c = a^2 / b (reversing the formula) or mishandling decimals during division.

Final Answer:
0.05