Relating dry density to bulk density and water content: If a soil's bulk density is denoted by rho (ρ) and its gravimetric water content by w (decimal), what is the correct formula for the dry density ρ_d?

Introduction / Context:
Converting between bulk (wet) density and dry density is routine in compaction control, field density testing, and laboratory reporting. Using a consistent definition of the gravimetric water content w is essential to avoid large interpretation errors.

Given Data / Assumptions:

• Bulk density rho (ρ) = mass of moist soil / total volume.
• Dry density rho_d (ρ_d) = mass of dry solids / total volume.
• Water content w = mass of water / mass of dry solids (decimal form).

Concept / Approach:
The mass of moist soil equals the mass of dry solids plus the mass of water. Because w = Mw/Ms, the moist mass is Ms + Mw = Ms * (1 + w). Dividing by the same total volume, we obtain the density relation: ρ = ρ_d * (1 + w). Rearranging gives ρ_d = ρ / (1 + w). This identity underpins many field conversions, such as nuclear gauge or sand-cone results.

Step-by-Step Solution:

Verification / Alternative check:
Dimensional check confirms that dividing by (1 + w) (dimensionless) yields density units for ρ_d. Example: ρ = 1.9 g/cc, w = 0.10 → ρ_d = 1.9 / 1.10 ≈ 1.727 g/cc.

Why Other Options Are Wrong:

• A: Multiplication by (1 + w) gives bulk density from dry density, not the reverse.
• C and E: Linear subtraction or direct product with w is not dimensionally or conceptually correct.
• D: Division by (1 − w) does not follow from the definitions of w and densities.

Common Pitfalls:
Treating w as a percentage without converting to decimal; using volumetric water content instead of gravimetric water content.

Final Answer:
rho_d = rho / (1 + w)