Chain Rule

Chain rule is proportion applied several times over. The only decision you make is, for each quantity, whether it is direct or inverse.

• Direct — as one goes up, so does the other. More notebooks, more cost. More machines, more output.
• Inverse — as one goes up, the other goes down. More men, fewer days. More hours per day, fewer days.

The master equation for work problems is

M₁ × D₁ × H₁ / W₁ = M₂ × D₂ × H₂ / W₂

(M = men, D = days, H = hours per day, W = amount of work). Put in what you know, and one unknown falls out.

The single most common mistake is applying a ratio upside down. There is a foolproof test: before multiplying, ask "should the answer get bigger or smaller?" If 15 men take 12 days, then 20 men must take fewer than 12 days — so the multiplier must be 15/20, a fraction less than 1, not 20/15. Check the direction first, then compute.

Shortcut for the standard case: total work = men × days, and that total is fixed. 15 × 12 = 180 man-days is a number you can reuse for any number of men.

M₁D₁H₁/W₁ = M₂D₂H₂/W₂ · men, hours ↔ days are inverse · work, cost, output are direct · total work = men × days

15 men build a wall in 12 days. How long will 20 men take? → Men and days are inverse, so days = 12 × 15/20 = 9 days. Sanity check two ways: more men must mean fewer days, and 9 < 12; and the total work is 15 × 12 = 180 man-days, which 20 men finish in 180 ÷ 20 = 9 days.