Work problems become manageable when one whole job is represented by \(1\). A worker's rate is the fraction of the job completed per unit of time, so each contribution is rate times time.
Rate, time, and work
If a worker completes one job in \(t\) hours, the rate is \(1/t\) job per hour.
| Worker | Time alone | Rate | Time working | Work done |
|---|---|---|---|---|
| A | \(6\) h | \(1/6\) job/h | \(x\) h | \(x/6\) |
| B | \(4\) h | \(1/4\) job/h | \(x\) h | \(x/4\) |
| Together | — | \(1/6+1/4=5/12\) | \(x\) h | \(5x/12\) |
Two workers together
A completes a job in \(6\) hours and B in \(4\) hours. How long do they need together?
- Write rates
A's rate is \(1/6\); B's is \(1/4\) job per hour.
- Add rates
\(1/6+1/4=5/12\) job per hour.
- Invert the combined rate
Time for one job is \(1\div(5/12)=12/5\) hours.
Partial and staged work
- \(2\left(\frac16\right)=\frac13\)
Two hours at one sixth job per hour completes one third.
- \(1-\frac13=\frac23\)
Two thirds of the job remains.
- \(\frac16+\frac14=\frac5{12}\)
Together they complete five twelfths per hour.
- \(\frac{2/3}{5/12}=\frac85\)
The final stage lasts eight fifths of an hour.
One worker starts, then another joins
A can finish a job in \(6\) hours. A works alone for \(2\) hours, then B, who can finish alone in \(4\) hours, joins. How much longer is needed?
A completes \(2/6=1/3\); therefore \(2/3\) remains.
Their combined rate is \(1/6+1/4=5/12\).
Remaining time is \((2/3)\div(5/12)=8/5\) hours.
Units and context
Keep units aligned
Hours
A rate of \(1/5\) job per hour multiplied by \(3\) hours gives \(3/5\) job.
Minutes
Convert \(90\) minutes to \(3/2\) hours before using an hourly rate.
Multiple jobs
If the target is \(N\) jobs, set total work equal to \(N\), not \(1\).
Model any work problem
- Choose one unit
Convert all times to hours or all to minutes.
- Write each rate
Use jobs divided by time; for one job in \(t\), use \(1/t\).
- Separate stages
Compute work in each interval and add contributions.
- Set the target
Total work equals \(1\) for one whole job unless the problem states otherwise.
- Validate
Check positivity, units, faster-together behavior, and whether the answer matches total or remaining time.
SAT strategy
Common mistakes and traps
- Adding completion times instead of rates.
- Using \(t\) instead of \(1/t\) as a rate.
- Forgetting work already completed in an earlier stage.
- Mixing minutes and hours.
- Reporting time after a helper joins when total elapsed time was requested.
- Accepting a together-time longer than the fastest individual time.
Combined-rate check
A and B need \(10\) and \(15\) hours alone. Is \(12\) hours a possible together-time?
Show answer and explanation
Answer: No.
Together they must finish in less than \(10\) hours. In fact, the combined rate is \(1/10+1/15=1/6\), so they need \(6\) hours.
Key takeaways
- Represent one whole job by \(1\).
- Individual rate is the reciprocal of individual time.
- Add work contributions, not raw times.
- Separate staged intervals and subtract completed work.
- Check units and contextual plausibility.
Put these notes into practice
Apply the ideas with SAT-style questions, then reinforce key details with flashcards.