Fundamental saturation relationship: Choose the correct fundamental equation linking void ratio (e), specific gravity of solids (G), water content (w), and degree of saturation (S_r).

Introduction / Context:
Phase relationships in soils provide compact formulas to convert between index properties and saturation state. A widely used identity relates the degree of saturation to water content, specific gravity, and void ratio.

Given Data / Assumptions:

• w = water content = Mw / Ms.
• G = specific gravity of solids = rho_s / rho_w.
• e = void ratio = Vv / Vs.
• S_r = degree of saturation = Vw / Vv.

Concept / Approach:
Derive from definitions using masses and volumes. Using Ms = rho_s * Vs and Mw = rho_w * Vw and substituting rho_s = G * rho_w yields S_r = (w * G) / e. This identity is fundamental and allows quick checks on lab data and consistency of reported properties.

Step-by-Step Solution:

Verification / Alternative check:
If a soil is fully saturated (S_r = 1), the relationship reduces to e = w * G; this is a familiar quick check on saturated samples.

Why Other Options Are Wrong:

• (w / e) * G: incorrect placement of e in denominator relative to S_r.
• (e * G) / w and w * e * G: algebraically inconsistent with the basic definitions.

Common Pitfalls:
Mixing e (void ratio) with n (porosity), or using percentages instead of decimals without consistent units.

Final Answer:
S_r = (w * G) / e