How to Solve Work-Rate Word Problems: Step-by-Step Guide for 2027

Work-rate problems show up under different costumes. Two painters paint a house together. Two pipes fill a pool. One worker takes three hours; another takes five; how long together? The phrasing changes; the math does not. Every work-rate problem in Algebra 1 boils down to one formula and a single equation.

This guide explains the formula, walks through every sub-type with examples, and ends with the four mistakes that trip up most students.

The Universal Work-Rate Formula

If person A takes a hours to do a job and person B takes b hours, then together they take t hours, where:

\[\frac{1}{a} + \frac{1}{b} = \frac{1}{t}\]

The fractions 1/a and 1/b represent rates: the fraction of the job each does per hour. Their rates add when they work together.

That’s the entire formula. Most problems are just plugging in.

Example 1: Two Workers

Anna can paint a fence in 6 hours. Ben can paint the same fence in 4 hours. How long does it take them together?

1/6 + 1/4 = 1/t.

Common denominator: 12. 2/12 + 3/12 = 5/12.

5/12 = 1/t → t = 12/5 = 2.4 hours.

Sanity check: working together should be faster than the faster worker alone (4 hours). 2.4 hours is faster. Good.

Example 2: Three Workers

Three painters can do a job in 6, 9, and 12 hours respectively. How long together?

1/6 + 1/9 + 1/12 = 1/t.

Common denominator: 36. 6/36 + 4/36 + 3/36 = 13/36.

t = 36/13 ≈ 2.77 hours.

The combined time is always less than any individual time, no matter how many workers.

Example 3: Pipes Filling a Tank

Pipe A fills a tank in 3 hours. Pipe B fills the same tank in 5 hours. How long with both pipes open?

1/3 + 1/5 = 1/t. 5/15 + 3/15 = 8/15. t = 15/8 = 1.875 hours = 1 hour 52.5 minutes.

Example 4: A Pipe That Drains

A drain works against fill pipes. Subtract its rate.

Pipe A fills the tank in 4 hours. Drain D empties it in 6 hours. With both open, how long to fill?

1/4 − 1/6 = 1/t. 3/12 − 2/12 = 1/12. t = 12 hours.

The tank fills, but slowly. If the drain were faster than the fill, the tank would never fill.

Example 5: Solving for an Unknown Rate

Together, two workers finish a job in 4 hours. Alone, worker A takes 6 hours. How long does worker B take alone?

1/6 + 1/b = 1/4. 1/b = 1/4 − 1/6. Common denominator: 12. 1/b = 3/12 − 2/12 = 1/12. b = 12 hours.

Example 6: One Worker Stops Early

Maria starts painting a fence; her rate is 1/8 (job per hour). After 2 hours, Sam joins her at a rate of 1/12 per hour. How much longer until the job is done?

In the first 2 hours, Maria completes 2 × 1/8 = 1/4 of the job.

Remaining: 3/4 of the job.

Together their rate: 1/8 + 1/12 = 3/24 + 2/24 = 5/24 per hour.

Time to finish: (3/4) / (5/24) = (3/4) × (24/5) = 72/20 = 3.6 hours.

Total elapsed time: 2 + 3.6 = 5.6 hours.

Example 7: Mixed Units

Watch units. Convert minutes to hours, or rate to “per minute,” consistently.

One photocopier prints 60 pages per minute. Another prints 90 pages per minute. How long to print 1,500 pages together?

Combined rate: 60 + 90 = 150 pages per minute. Time: 1,500 / 150 = 10 minutes.

When rates are given in items per time, you can add them directly. The 1/t formula is for jobs that are “one whole job.”

A Quick Chart for Work-Rate Setup

The Sanity Check Every Time

The combined time t must be less than the slowest worker (and less than every worker, actually). If your t comes out larger than one of the individual times, you set the equation up wrong, probably by adding times instead of rates.

Wrong: “Two workers, 6 and 4 hours each, combined is 6 + 4 = 10 hours.” Adding times is the most common mistake on these problems.

Common Mistakes

1. Adding times instead of rates. 1/6 + 1/4 = 1/t, not 6 + 4 = t.
2. Forgetting to flip the result. After computing the combined rate (say, 5/12), the time is 12/5, not 5/12.
3. Skipping a unit conversion. If one worker is timed in minutes and the other in hours, convert before combining.
4. Treating a drain as a worker. Drains subtract from the rate; they do not add.

A Memorable Pattern

“Rates add. Times do not.”

Tape that under your desk. Almost every error on these problems traces back to that one fact.

Frequently Asked Questions

How do you solve work-rate problems? Turn each worker’s time into a rate, add the rates, then flip the total. If one person takes a hours and another takes b hours, then 1/a + 1/b = 1/t, where t is the time together. For a job one person finishes in 3 hours and another in 6 hours: 1/3 + 1/6 = 1/2, so together they need 2 hours.

What is the work rate formula? The work rate formula is work = rate × time, and for combined work, 1/a + 1/b = 1/t. Each rate is the fraction of the job done per unit of time, so someone who finishes a job in 5 hours works at 1/5 of the job per hour. In 2 hours, that person completes 2 × 1/5 = 2/5 of the job.

What is an example of a work-rate problem? Here is a typical example: Pipe A fills a tank in 3 hours and Pipe B fills it in 5 hours. How long with both open? Add the rates: 1/3 + 1/5 = 5/15 + 3/15 = 8/15 of the tank per hour. Flip it: t = 15/8 = 1.875 hours, or 1 hour 52.5 minutes, which is faster than either pipe alone.

How do you solve for an unknown work rate? Put the unknown time as a variable in the combined-rate equation and solve. Suppose Maya and Leo paint a room together in 3 hours, and Maya alone takes 5 hours. Then 1/5 + 1/x = 1/3, so 1/x = 1/3 − 1/5 = 2/15, and x = 15/2 = 7.5 hours for Leo alone. Check that the answer is longer than the combined time.

How do you solve work-rate problems with a drain? Subtract the rate of anything working against the job, such as a drain or a leak. If a pipe fills a tank in 4 hours and a drain empties it in 6 hours, the net rate is 1/4 − 1/6 = 3/12 − 2/12 = 1/12 of the tank per hour, so filling takes 12 hours. If the drain is faster than the pipe, the tank never fills.

Closing Thought

Work-rate problems are a one-line formula and a four-word rule (rates add, times do not). Use the formula, sanity-check that the combined time is less than every individual time, and the topic stops costing you points. Drill ten problems and the setup becomes automatic.

For more practice, browse our Algebra 1 worksheets and our full Math Topics library. When you are ready for a structured workbook, our Algebra 1 collection covers every word-problem type above.