Three roads, one answer: formula, proportion, analysis
A 5 mL syringe barrel graduated every 0.2 mL beneath the three method columns, plunger seal at the 3 mL graduation, with three tracks descending from the columns and converging on that one graduation.
The order for all three columns: digoxin 0.125 mg by mouth. Supply: scored 0.25 mg tablets.
Column 1 — Desired over have
- Setup: (desired ÷ have) × quantity
- Work: (0.125 mg ÷ 0.25 mg) × 1 tablet = 0.5 × 1
- Answer: 0.5 tablet, and the tablet is scored, so a half is deliverable.
- How it fails: nothing in the expression warns you when desired and have arrive in different units. A milligram measured against a microgram returns a confident thousandfold error with no signal that anything went wrong.
Column 2 — Ratio and proportion
- Setup: have is to quantity as desired is to x
- Work: 0.25 mg : 1 tablet :: 0.125 mg : x. Cross-multiply: 0.25x = 0.125. Divide: x = 0.125 ÷ 0.25.
- Answer: 0.5 tablet
- How it fails: the two ratios can be written in opposite orders and still cross-multiply cleanly. The inverted answer is arithmetically sound and clinically wrong, and the method never objects.
Column 3 — Dimensional analysis
- Work: 0.125 mg × (1 tablet / 0.25 mg) = 0.125 ÷ 0.25
- Answer: 0.5 tablet
- How it fails: it does not fail silently. An illegal chain refuses to cancel, so the setup announces its own error before a number exists.
Second pass, liquid
Order: furosemide 30 mg intravenously. Supply: 10 mg/mL.
- Desired over have: (30 mg ÷ 10 mg) × 1 mL = 3 mL
- Ratio and proportion: 10 mg : 1 mL :: 30 mg : x, so 10x = 30, x = 3 mL
- Dimensional analysis: 30 mg × (1 mL / 10 mg) = 3 mL
Third pass, tablets
Order: levothyroxine 0.075 mg by mouth. Supply: 0.025 mg tablets. 0.075 mg × (1 tablet / 0.025 mg) = 3 tablets.
Watch outpick one road and stay on it under pressure. Switching methods mid-problem is where inverted ratios are born.
Margin notethe drugs here are carriers for arithmetic, nothing more. → BOOK 05 · P7
Takeawayall three roads reach the same answer; only one of them tells you when you have taken a wrong turn.