A sample has moist unit weight 19.2 kN/m³, water content 16% and Gs = 2.68. Find the dry unit weight, void ratio and degree of saturation.

Start with the thinking

• Dry unit weight first — everything else follows.
• Se = wGs closes the saturation calculation.

Step-by-step solution

1. Dry unit weight

   $$\gamma_d = \gamma/(1 + w) = 19.2/1.16 = 16.55 kN/m^{3}$$

2. Void ratio — e = Gsγw/γ_d − 1 = 2.68(9.81)/16.55 − 1

3. Evaluate

   $$e = 26.29/16.55 - 1 = 1.589 - 1 = 0.589$$

4. Saturation

   $$S = wGs/e = 0.16(2.68)/0.589$$

5. Result

   $$\gamma_d = 16.6 kN/m^{3}, e = 0.589, S = 0.728 (72.8$$

Why the other options are there

• γ_d = 22.3 kN/m³ (multiplied by 1 + w)
• S = 100% (assumed saturated)

Reference: FE Reference Handbook — Geotechnical — Phase relationships

A geotechnical problem uses Void ratio–porosity relation. Given porosity (n) = 0.3500, determine the void ratio (e).

Start with the thinking

• The governing relation printed in this handbook section is Void ratio–porosity relation.
• Everything except e is given, so isolate e symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Geotechnical items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$e = n / (1 - n)$$

2. Step 2 — Rearrange the relation so that e stands alone on the left-hand side.

3. Step 3

   $$List the givens: porosity (n) = 0.3500$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$e = 0.5385$$

6. Step 6 — Check: returning e = 0.5385 to

   $$e = n / (1 - n)$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 1.0769 — kept a factor of two that cancels in the correct rearrangement.
• 0.2692 — dropped that same factor in the other direction.
• 0.5923 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Geotechnical → Void ratio

A geotechnical problem uses Void ratio–porosity relation. Given void ratio (e) = 0.6400, determine the porosity (n).

Start with the thinking

• The governing relation printed in this handbook section is Void ratio–porosity relation.
• Everything except n is given, so isolate n symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Geotechnical items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$e = n / (1 - n)$$

2. Step 2 — Rearrange the relation so that n stands alone on the left-hand side.

3. Step 3

   $$List the givens: void ratio (e) = 0.6400$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$n = 0.3902$$

6. Step 6 — Check: returning n = 0.3902 to

   $$e = n / (1 - n)$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 0.7805 — kept a factor of two that cancels in the correct rearrangement.
• 0.1951 — dropped that same factor in the other direction.
• 0.4293 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Geotechnical → Void ratio

A geotechnical problem uses Void ratio–porosity relation. Given porosity (n) = 0.4600, determine the void ratio (e).

Start with the thinking

• The governing relation printed in this handbook section is Void ratio–porosity relation.
• Everything except e is given, so isolate e symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Geotechnical items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$e = n / (1 - n)$$

2. Step 2 — Rearrange the relation so that e stands alone on the left-hand side.

3. Step 3

   $$List the givens: porosity (n) = 0.4600$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$e = 0.8519$$

6. Step 6 — Check: returning e = 0.8519 to

   $$e = n / (1 - n)$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 1.7037 — kept a factor of two that cancels in the correct rearrangement.
• 0.4259 — dropped that same factor in the other direction.
• 0.9370 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Geotechnical → Void ratio

A geotechnical problem uses Void ratio–porosity relation. Given void ratio (e) = 0.6510, determine the porosity (n).

Start with the thinking

• The governing relation printed in this handbook section is Void ratio–porosity relation.
• Everything except n is given, so isolate n symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Geotechnical items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$e = n / (1 - n)$$

2. Step 2 — Rearrange the relation so that n stands alone on the left-hand side.

3. Step 3

   $$List the givens: void ratio (e) = 0.6510$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$n = 0.3943$$

6. Step 6 — Check: returning n = 0.3943 to

   $$e = n / (1 - n)$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 0.7886 — kept a factor of two that cancels in the correct rearrangement.
• 0.1972 — dropped that same factor in the other direction.
• 0.4337 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Geotechnical → Void ratio

A geotechnical problem uses Void ratio–porosity relation. Given porosity (n) = 0.4000, determine the void ratio (e).

Start with the thinking

• The governing relation printed in this handbook section is Void ratio–porosity relation.
• Everything except e is given, so isolate e symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Geotechnical items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$e = n / (1 - n)$$

2. Step 2 — Rearrange the relation so that e stands alone on the left-hand side.

3. Step 3

   $$List the givens: porosity (n) = 0.4000$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$e = 0.6667$$

6. Step 6 — Check: returning e = 0.6667 to

   $$e = n / (1 - n)$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 1.3333 — kept a factor of two that cancels in the correct rearrangement.
• 0.3333 — dropped that same factor in the other direction.
• 0.7333 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Geotechnical → Void ratio

A geotechnical problem uses Void ratio–porosity relation. Given void ratio (e) = 0.8690, determine the porosity (n).

Start with the thinking

• The governing relation printed in this handbook section is Void ratio–porosity relation.
• Everything except n is given, so isolate n symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Geotechnical items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$e = n / (1 - n)$$

2. Step 2 — Rearrange the relation so that n stands alone on the left-hand side.

3. Step 3

   $$List the givens: void ratio (e) = 0.8690$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$n = 0.4650$$

6. Step 6 — Check: returning n = 0.4650 to

   $$e = n / (1 - n)$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 0.9299 — kept a factor of two that cancels in the correct rearrangement.
• 0.2325 — dropped that same factor in the other direction.
• 0.5114 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Geotechnical → Void ratio

A geotechnical problem uses Void ratio–porosity relation. Given porosity (n) = 0.2000, determine the void ratio (e).

Start with the thinking

• The governing relation printed in this handbook section is Void ratio–porosity relation.
• Everything except e is given, so isolate e symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Geotechnical items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$e = n / (1 - n)$$

2. Step 2 — Rearrange the relation so that e stands alone on the left-hand side.

3. Step 3

   $$List the givens: porosity (n) = 0.2000$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$e = 0.2500$$

6. Step 6 — Check: returning e = 0.2500 to

   $$e = n / (1 - n)$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 0.5000 — kept a factor of two that cancels in the correct rearrangement.
• 0.1250 — dropped that same factor in the other direction.
• 0.2750 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Geotechnical → Void ratio

A geotechnical problem uses Void ratio–porosity relation. Given void ratio (e) = 0.3030, determine the porosity (n).

Start with the thinking

• The governing relation printed in this handbook section is Void ratio–porosity relation.
• Everything except n is given, so isolate n symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Geotechnical items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$e = n / (1 - n)$$

2. Step 2 — Rearrange the relation so that n stands alone on the left-hand side.

3. Step 3

   $$List the givens: void ratio (e) = 0.3030$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$n = 0.2325$$

6. Step 6 — Check: returning n = 0.2325 to

   $$e = n / (1 - n)$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 0.4651 — kept a factor of two that cancels in the correct rearrangement.
• 0.1163 — dropped that same factor in the other direction.
• 0.2558 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Geotechnical → Void ratio

A geotechnical problem uses Void ratio–porosity relation. Given porosity (n) = 0.4900, determine the void ratio (e).

Start with the thinking

• The governing relation printed in this handbook section is Void ratio–porosity relation.
• Everything except e is given, so isolate e symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Geotechnical items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$e = n / (1 - n)$$

2. Step 2 — Rearrange the relation so that e stands alone on the left-hand side.

3. Step 3

   $$List the givens: porosity (n) = 0.4900$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$e = 0.9608$$

6. Step 6 — Check: returning e = 0.9608 to

   $$e = n / (1 - n)$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 1.9216 — kept a factor of two that cancels in the correct rearrangement.
• 0.4804 — dropped that same factor in the other direction.
• 1.0569 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Geotechnical → Void ratio