| Steps | Statements | Reasons |
| 1. | 1 grain = 64.798 910 milligrams | by definition |
| 2. | 100 000 000 grains = 6 479 891 000 milligrams | by multiplying both sides by 100,000,000 |
| 3. | 100 000 000 grains = 6 479 891 grams | (right-hand side) by converting milligrams to grams |
| 4. | (28 × 58) grains = (11 × 97 × 6073) grams | by prime factorization |
| 5. | (28 × 58) ---------------- grains = 1 gram (11 × 97 × 6073) | by scaling to grams |
| Steps | Statements | Reasons |
| 1. | a0 ≈ 5.29 × 10-11 m | Definition of Bohr radius |
| 2. | 2a0 ≈ 10.58 × 10-11 m | Double to get diameter |
| 3. | 1.058 × 10-10 m | reduce; width of hydrogen atom |
| 4. | 1.058 × 10-9 m | Multiply by 10 to get width of 10 hydrogren atoms |
| 5. | 1.058 nm | width of 10 hydrogen atoms |
| 6. | ... | ... |
| 7. | 1 mm ≈ 9 451 795.8 hydrogen atoms |
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