Phase-relationship identity: Select the correct fundamental relationship linking dry unit weight (gamma_d), bulk unit weight (gamma), and water content (w) for a soil mass.

Introduction / Context:
Routine field and lab work constantly switches between bulk (wet) and dry unit weights using the measured water content. Remembering the equivalent identities avoids algebra mistakes when interpreting compaction and in-situ density results.

Given Data / Assumptions:

• gamma is the total (bulk) unit weight of the moist specimen.
• gamma_d is the unit weight of solids per total volume (i.e., dried basis).
• w is water content expressed as a decimal (not percent).

Concept / Approach:
Start with the mass partition: total mass M = Ms + Mw with Mw = w * Ms by definition of water content. Substituting into unit weight definitions yields a simple proportionality between gamma and gamma_d. Rearrangement then produces equivalent forms useful for conversions in either direction.

Step-by-Step Solution:

Verification / Alternative check:
Example: if gamma = 20 kN/m^3 and w = 0.10, then gamma_d = 20/1.10 ≈ 18.18 kN/m^3. Substituting back: 18.18*(1+0.10) ≈ 20, confirming consistency.

Why Other Options Are Wrong:

• Individually, Options A–C are each correct statements; however, none alone is “the only” relation. Option D correctly recognizes that they are equivalent identities derived from the same definition.

Common Pitfalls:
Using w in percent instead of decimal; mixing mass density with unit weight; rounding too early when converting.

Final Answer:
All of the above are equivalent identities