A reliable method
Add positive contributions and subtract a leak or removal rate. Convert completed work into a fraction, then divide the remaining fraction by the active rate.
The common trap
Completion times cannot be added to obtain a joint completion time. A worker joining later does not contribute during the earlier stage.
Check the conclusion
Check that completed fractions sum to exactly one. When machine counts and hours both change, separate output per machine-hour from total output.
Contrast the cases
A completes a job in twelve hours and B in eighteen. If A works alone for three hours, one quarter is finished. The remaining three quarters at joint rate 1/12+1/18=5/36 job per hour require 27/5 additional hours. Returning 36/5 would mistakenly give them a complete new job. A leak is different from a second worker: its rate is subtracted, not added.
Transfer and explain
Transfer: an inlet fills a tank in 6 hours and a leak empties it in 12. The net rate is 1/12 tank per hour, so filling takes 12 hours. Check that the inlet supplies two tankfuls while the leak removes one over that interval.
Three independent printers complete a batch in 6, 9, and 18 hours, respectively. At constant rates, how many hours do they need together?
- The joint rate is 1/6 + 1/9 + 1/18 batch per hour.
- This sum is 1/3.
- The completion time is 3 hours.
Compare the other answer choices
Uses the fastest printer alone.
Averages times instead of summing rates.
Adds completion times instead of rates.
Ignores the third printer.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
GMAT exam contentSiyavula Grade 10 mathematics foundation reference