Time & Work

Time & Work

Think in rates, not hours. If someone finishes a job in $n$ days, their rate is $\dfrac1n$ of the job per day. Rates add when people work together — that single idea solves almost every question here, including pipes and cisterns.

Core results

• Combined rate: $\dfrac1a+\dfrac1b$; time together $=\dfrac{ab}{a+b}$.
• Work–rate constant: $\dfrac{M_1 D_1 H_1}{W_1}=\dfrac{M_2 D_2 H_2}{W_2}$ (men × days × hours per unit of work).
• Pipes & cisterns: inflow is positive, a leak is negative — just add the signed rates.
• Efficiency ↔ time are inversely proportional: twice as efficient ⇒ half the time.

A worked example

A finishes a job in 12 days, B in 18 days. Working together, how long do they take?

Add the rates:

$\frac1{12}+\frac1{18}=\frac{3}{36}+\frac{2}{36}=\frac{5}{36}\ \text{job/day}\ \Rightarrow\ \text{time}=\frac{36}{5}=\mathbf{7.2\text{ days}}.$

LCM trick: let the total work be $\operatorname{lcm}(12,18)=36$ units. Then A does 3 units/day, B does 2, together 5 units/day, so $36/5=7.2$ days — no fractions at all.

Common traps

• Adding times instead of rates. $12+18$ means nothing; rates add, times don't.
• Forgetting a leak. A draining pipe subtracts from the fill rate.
• Partial work. If A leaves midway, account for the work already done before adding B's contribution.

Checklist

• Convert every worker to a rate (job per day)
• Add rates for simultaneous work; subtract for leaks
• Use the LCM-of-times = total units trick to kill fractions
• Keep efficiency ∝ 1/time in mind for "twice as fast" phrasing