A steel component measures 165 BHN. Estimate its tensile strength from the hardness relationship, then determine how many grains per square inch at 100× correspond to an ASTM grain size number of 5.

Start with the thinking

• For steels the empirical relation between hardness and tensile strength is TS ≈ 500 × BHN in psi.
• The ASTM grain size number doubles the grain count for every increment of one.

Step-by-step solution

1. Formula — TS (psi) ≈ 500 × BHN

2. Substituting

   $$TS = 500 \times 165 = 82,500 psi = 82.5 ksi$$

3. Formula

   $$N = 2^(n-1) grains per in^{2} at 100\times$$

4. Substituting

   $$N = 2^(5 - 1) = 16 grains/in^{2}$$

5. Interpretation — finer grains (higher n) raise yield strength through the Hall-Petch effect

Why the other options are there

• 0.33 ksi (ratio inverted)
• 32 grains/in² (exponent off by one)

Reference: FE Reference Handbook — Materials Science → Relationship Between Hardness and Tensile Strength

A steel component measures 391 BHN. Estimate its tensile strength from the hardness relationship, then determine how many grains per square inch at 100× correspond to an ASTM grain size number of 8.

Start with the thinking

• For steels the empirical relation between hardness and tensile strength is TS ≈ 500 × BHN in psi.
• The ASTM grain size number doubles the grain count for every increment of one.

Step-by-step solution

1. Formula — TS (psi) ≈ 500 × BHN

2. Substituting

   $$TS = 500 \times 391 = 195,500 psi = 195.5 ksi$$

3. Formula

   $$N = 2^(n-1) grains per in^{2} at 100\times$$

4. Substituting

   $$N = 2^(8 - 1) = 128 grains/in^{2}$$

5. Interpretation — finer grains (higher n) raise yield strength through the Hall-Petch effect

Why the other options are there

• 0.78 ksi (ratio inverted)
• 256 grains/in² (exponent off by one)

Reference: FE Reference Handbook — Materials Science → Relationship Between Hardness and Tensile Strength

A steel component measures 343 BHN. Estimate its tensile strength from the hardness relationship, then determine how many grains per square inch at 100× correspond to an ASTM grain size number of 6.

Start with the thinking

• For steels the empirical relation between hardness and tensile strength is TS ≈ 500 × BHN in psi.
• The ASTM grain size number doubles the grain count for every increment of one.

Step-by-step solution

1. Formula — TS (psi) ≈ 500 × BHN

2. Substituting

   $$TS = 500 \times 343 = 171,500 psi = 171.5 ksi$$

3. Formula

   $$N = 2^(n-1) grains per in^{2} at 100\times$$

4. Substituting

   $$N = 2^(6 - 1) = 32 grains/in^{2}$$

5. Interpretation — finer grains (higher n) raise yield strength through the Hall-Petch effect

Why the other options are there

• 0.69 ksi (ratio inverted)
• 64 grains/in² (exponent off by one)

Reference: FE Reference Handbook — Materials Science → Relationship Between Hardness and Tensile Strength

A steel component measures 203 BHN. Estimate its tensile strength from the hardness relationship, then determine how many grains per square inch at 100× correspond to an ASTM grain size number of 5.

Start with the thinking

• For steels the empirical relation between hardness and tensile strength is TS ≈ 500 × BHN in psi.
• The ASTM grain size number doubles the grain count for every increment of one.

Step-by-step solution

1. Formula — TS (psi) ≈ 500 × BHN

2. Substituting

   $$TS = 500 \times 203 = 101,500 psi = 101.5 ksi$$

3. Formula

   $$N = 2^(n-1) grains per in^{2} at 100\times$$

4. Substituting

   $$N = 2^(5 - 1) = 16 grains/in^{2}$$

5. Interpretation — finer grains (higher n) raise yield strength through the Hall-Petch effect

Why the other options are there

• 0.41 ksi (ratio inverted)
• 32 grains/in² (exponent off by one)

Reference: FE Reference Handbook — Materials Science → Relationship Between Hardness and Tensile Strength

A steel component measures 183 BHN. Estimate its tensile strength from the hardness relationship, then determine how many grains per square inch at 100× correspond to an ASTM grain size number of 6.

Start with the thinking

• For steels the empirical relation between hardness and tensile strength is TS ≈ 500 × BHN in psi.
• The ASTM grain size number doubles the grain count for every increment of one.

Step-by-step solution

1. Formula — TS (psi) ≈ 500 × BHN

2. Substituting

   $$TS = 500 \times 183 = 91,500 psi = 91.5 ksi$$

3. Formula

   $$N = 2^(n-1) grains per in^{2} at 100\times$$

4. Substituting

   $$N = 2^(6 - 1) = 32 grains/in^{2}$$

5. Interpretation — finer grains (higher n) raise yield strength through the Hall-Petch effect

Why the other options are there

• 0.37 ksi (ratio inverted)
• 64 grains/in² (exponent off by one)

Reference: FE Reference Handbook — Materials Science → Relationship Between Hardness and Tensile Strength

A steel component measures 344 BHN. Estimate its tensile strength from the hardness relationship, then determine how many grains per square inch at 100× correspond to an ASTM grain size number of 6.

Start with the thinking

• For steels the empirical relation between hardness and tensile strength is TS ≈ 500 × BHN in psi.
• The ASTM grain size number doubles the grain count for every increment of one.

Step-by-step solution

1. Formula — TS (psi) ≈ 500 × BHN

2. Substituting

   $$TS = 500 \times 344 = 172,000 psi = 172.0 ksi$$

3. Formula

   $$N = 2^(n-1) grains per in^{2} at 100\times$$

4. Substituting

   $$N = 2^(6 - 1) = 32 grains/in^{2}$$

5. Interpretation — finer grains (higher n) raise yield strength through the Hall-Petch effect

Why the other options are there

• 0.69 ksi (ratio inverted)
• 64 grains/in² (exponent off by one)

Reference: FE Reference Handbook — Materials Science → Relationship Between Hardness and Tensile Strength

A steel component measures 236 BHN. Estimate its tensile strength from the hardness relationship, then determine how many grains per square inch at 100× correspond to an ASTM grain size number of 8.

Start with the thinking

• For steels the empirical relation between hardness and tensile strength is TS ≈ 500 × BHN in psi.
• The ASTM grain size number doubles the grain count for every increment of one.

Step-by-step solution

1. Formula — TS (psi) ≈ 500 × BHN

2. Substituting

   $$TS = 500 \times 236 = 118,000 psi = 118.0 ksi$$

3. Formula

   $$N = 2^(n-1) grains per in^{2} at 100\times$$

4. Substituting

   $$N = 2^(8 - 1) = 128 grains/in^{2}$$

5. Interpretation — finer grains (higher n) raise yield strength through the Hall-Petch effect

Why the other options are there

• 0.47 ksi (ratio inverted)
• 256 grains/in² (exponent off by one)

Reference: FE Reference Handbook — Materials Science → Relationship Between Hardness and Tensile Strength

A steel component measures 176 BHN. Estimate its tensile strength from the hardness relationship, then determine how many grains per square inch at 100× correspond to an ASTM grain size number of 4.

Start with the thinking

• For steels the empirical relation between hardness and tensile strength is TS ≈ 500 × BHN in psi.
• The ASTM grain size number doubles the grain count for every increment of one.

Step-by-step solution

1. Formula — TS (psi) ≈ 500 × BHN

2. Substituting

   $$TS = 500 \times 176 = 88,000 psi = 88.0 ksi$$

3. Formula

   $$N = 2^(n-1) grains per in^{2} at 100\times$$

4. Substituting

   $$N = 2^(4 - 1) = 8 grains/in^{2}$$

5. Interpretation — finer grains (higher n) raise yield strength through the Hall-Petch effect

Why the other options are there

• 0.35 ksi (ratio inverted)
• 16 grains/in² (exponent off by one)

Reference: FE Reference Handbook — Materials Science → Relationship Between Hardness and Tensile Strength

A steel component measures 220 BHN. Estimate its tensile strength from the hardness relationship, then determine how many grains per square inch at 100× correspond to an ASTM grain size number of 5.

Start with the thinking

• For steels the empirical relation between hardness and tensile strength is TS ≈ 500 × BHN in psi.
• The ASTM grain size number doubles the grain count for every increment of one.

Step-by-step solution

1. Formula — TS (psi) ≈ 500 × BHN

2. Substituting

   $$TS = 500 \times 220 = 110,000 psi = 110.0 ksi$$

3. Formula

   $$N = 2^(n-1) grains per in^{2} at 100\times$$

4. Substituting

   $$N = 2^(5 - 1) = 16 grains/in^{2}$$

5. Interpretation — finer grains (higher n) raise yield strength through the Hall-Petch effect

Why the other options are there

• 0.44 ksi (ratio inverted)
• 32 grains/in² (exponent off by one)

Reference: FE Reference Handbook — Materials Science → Relationship Between Hardness and Tensile Strength

A steel component measures 277 BHN. Estimate its tensile strength from the hardness relationship, then determine how many grains per square inch at 100× correspond to an ASTM grain size number of 6.

Start with the thinking

• For steels the empirical relation between hardness and tensile strength is TS ≈ 500 × BHN in psi.
• The ASTM grain size number doubles the grain count for every increment of one.

Step-by-step solution

1. Formula — TS (psi) ≈ 500 × BHN

2. Substituting

   $$TS = 500 \times 277 = 138,500 psi = 138.5 ksi$$

3. Formula

   $$N = 2^(n-1) grains per in^{2} at 100\times$$

4. Substituting

   $$N = 2^(6 - 1) = 32 grains/in^{2}$$

5. Interpretation — finer grains (higher n) raise yield strength through the Hall-Petch effect

Why the other options are there

• 0.55 ksi (ratio inverted)
• 64 grains/in² (exponent off by one)

Reference: FE Reference Handbook — Materials Science → Relationship Between Hardness and Tensile Strength