Work Problems

Work problems are the “if it takes Ann 4 hours and Bob 6 hours, how long together” questions. They feel like a special trick, but they are really just Distance, Rate, and Time with the word “job” swapped in for “distance” and “rate of work” swapped in for “speed.” The Word Problems five-step method still runs the show.

The one idea to hold onto: rates of work add. A worker who finishes in hours does of the job per hour. Two such workers together do per hour. Multiply a combined rate by the time worked and you get the fraction of the job done — and when the job is finished, those fractions add to .

The Work-Rate Formula

and are the times each worker needs alone, is the time working together, and is the whole job. Each term is the fraction of the job that worker completes in time ; the fractions sum to one finished job.

Building the Equation with a Chart

The distance-rate-time chart carries over, relabeled:

The “Part of job done” column adds to . That sum is the equation.

Two Workers, Full Time Each

Both work the entire time until the job is finished.

Three or More Workers

Every worker adds a term. Three workers finishing alone in , , and hours:

A Worker Who Joins Late or Leaves Early

Give the part-time worker their own time expression. If A works the whole time and B works fewer hours:

Filling While Draining

A drain removes work, so its term is subtracted. A tank filled in hours by a pipe and emptied in hours by an open drain:

This only has a positive solution when — the fill has to outpace the drain, or the tank never fills.

Worked Example A: Two Workers

Ann paints a room in hours. Bob paints the same room in hours. How long does it take them working together?

Chart:

Equation:

Multiply every term by the LCD, :

Answer: hours, which is hours minutes — less than Ann’s hours alone, as it must be.

Worked Example B: Three Pipes

Three pipes fill a pool in , , and hours individually. How long to fill it with all three open?

Equation:

Multiply every term by the LCD, :

Answer: hours, or hours minutes.

Worked Example C: One Worker Leaves Early

A machine can complete an order in hours; a second machine can do it in hours. Both start together, but the first machine breaks down and shuts off hours before the job is finished. How long does the whole job take?

Let be the total time. Machine 1 runs for hours; machine 2 runs the full .

Multiply every term by the LCD, :

Answer: the job takes hours.

Worked Example D: Fill and Drain

A tank fills in hours through an inlet pipe. A drain, left open by mistake, would empty a full tank in hours. With both open, how long does the tank take to fill from empty?

Multiply every term by the LCD, :

Answer: hours, about hours minutes — longer than the -hour solo fill, because the drain is fighting it.

Common Mistakes to Avoid

• Adding the times instead of the rates. ” hours and hours, so hours together” is the classic wrong answer. Rates add; times do not.
• Averaging the solo times. The average of and is , which is also wrong — and larger than one worker’s solo time, which is impossible.
• Forgetting to invert. The combined rate is ; the combined time is the reciprocal of that sum, not the sum itself.
• Using the same time for a part-time worker. A worker who stops early gets , not .
• Adding a drain’s term instead of subtracting it. A drain does negative work.
• Accepting an answer bigger than the fastest solo time (for a “working together” problem). That is a guaranteed sign of an error.

Where This Shows Up Later

• Rational Expressions and rational equations. Every work equation with the variable in a denominator is a rational equation; the LCD-clearing step and the extraneous-solution check both apply.
• Systems of Equations. Problems that give two combined-time facts about three workers are solved as a system.
• Distance, Rate, and Time. The chart and the reciprocal idea are identical; only the labels change.
• Uniform-motion round trips. A round trip with a stated total time produces the same “reciprocals add” structure as a fill-and-drain problem.

Practice Problems

Work each problem yourself before opening the answer.

Problem 1. Carla mows a lawn in hours; Dan mows it in hours. How long together?

Problem 2. One printer prints a batch in minutes; a second does it in minutes. How long with both running?

Problem 3. Three workers finish a task alone in , , and hours. How long together?

Problem 4. Pipe A fills a tank in hours; pipe B fills it in hours. Both are opened. How long to fill the tank?

Problem 5. Ed can roof a house in days. With his apprentice, the job takes days. How long would the apprentice take alone?

Problem 6. A tub fills in minutes with the drain closed. With the drain open it empties a full tub in minutes. If the drain is left open, how long to fill the tub?

Problem 7. Two crews clean a stadium in hours together. Crew A alone takes hours. How long does Crew B take alone?

Problem 8. Two hoses fill a pool in and hours alone. They run together for hours, then the faster hose is shut off. How much longer does the slower hose need to finish?

Problem 9. A machine fills bottles in hours. A newer machine does it in hours. They start together, but the older machine is stopped hour before the job ends. How long is the whole job?

Problem 10. Pipe A fills a tank in hours; pipe B fills it in hours; a drain empties it in hours. All three are open. How long to fill the tank?

Problem 11. Working together, Ravi and Sam assemble a set of shelves in hours. Alone, Ravi is hours faster than Sam. How long does each take alone?

Problem 12. A large pump empties a flooded basement in minutes; a small pump takes minutes. Both run for minutes, then the large pump quits. How much longer does the small pump need?

Quick Reference

Clearing the denominators is the Linear Equations with Fractions step, and when a variable sits in a denominator the equation becomes rational — see Rational Expressions. The chart comes straight from Distance, Rate, and Time, and the overall setup is the Word Problems method. More Algebra lessons are available too.