Number System — Find the positive number x that satisfies 80 / x = x / 20. Choose the exact value.

Introduction / Context:
This proportion is a classic “means equals extremes” setup. Cross-multiplication yields a simple quadratic in x^2 with a perfect-square result, so the solution is immediate once the relation is written correctly.

Given Data / Assumptions:

• Equation: 80 / x = x / 20.
• x is taken as positive in typical number system contexts.
• Basic algebraic operations (cross-multiplication) apply.

Concept / Approach:
Cross-multiply to eliminate denominators and solve for x^2. Recognize 1600 as a perfect square to find x without decimals. Always consider the sign context; here, a positive length/value is expected.

Step-by-Step Solution:
1) Cross-multiply: 80 * 20 = x * x.2) Compute left side: 1600 = x^2.3) Take square root: x = sqrt(1600) = 40 (positive solution).4) Therefore, x = 40.

Verification / Alternative check:
Plug back: 80/40 = 2 and 40/20 = 2. Both sides equal, confirming x = 40.

Why Other Options Are Wrong:
400, 800, and 1600 are results of misplacing the square root; 20 yields 80/20 = 4 while 20/20 = 1, not equal.

Common Pitfalls:
Forgetting to take the square root after obtaining x^2; mixing 80 and 20 when cross-multiplying; introducing negative roots where the context implies positivity.

Final Answer:
40