What are work rate problems in math?
Work rate problems ask how long a job takes when people or machines work together. Each worker’s rate is the part of the job done per hour: 1 divided by the time alone. Rates add: 1a + 1b = 1t. If one pump fills a tank in 6 hours and another in 3 hours, together they need 2 hours.
Work Rate Problems in a picture
Formal definition and notation
If worker A completes a job alone in time a and worker B in time b, their rates are 1a and 1b jobs per unit of time. Working together at constant rates, they complete 1a + 1b of the job per unit of time, so the combined time t satisfies 1a + 1b = 1t. A worker that undoes the job (such as a drain) has a negative rate.
| Symbol | Read it as | Meaning |
|---|---|---|
| 1a | one over a | the part of the job done in one unit of time by someone who needs a units alone |
| t | t | the time to finish the job working together |
Key facts
- Turn every time into a rate: 5 hours to finish means 15 of the job per hour.
- Add rates, never times. Adding or averaging the times always gives the wrong answer.
- Together is always faster than the fastest worker alone: if one person needs 4 hours, two people need less than 4.
- A worker who undoes the job, like a drain, gets a minus sign: 15 − 18.
- Work done = rate × time, so after 3 hours at 112 per hour, 312 = 14 of the job is done.
Combined work formula
- a, b
- the times each worker needs alone
- t
- the time needed together
For two workers this gives t = aba + b. It is the same idea as d = rt: work done = rate × time, with the whole job equal to 1.
Work Rate Problems examples, step by step
Example 1: Two painters
Ana can paint a room in 4 hours. Ben needs 12 hours. How long does it take them working together?
- Rates: Ana 14 room per hour, Ben 112 room per hour.
- Together: 14 + 112 = 312 + 112 = 412 = 13 room per hour.
- One third of the room per hour means the whole room takes 3 hours.
Answer: Working together, they take 3 hours.
Example 2: Fill pipe and drain
A pipe fills a pool in 5 hours. A drain empties it in 8 hours. If both are open, how long does it take to fill the empty pool?
- Fill rate 15 per hour; drain rate −18 per hour.
- Net rate: 15 − 18 = 840 − 540 = 340 per hour.
- Time: t = 403 = 1313 hours, which is 13 hours 20 minutes.
Answer: It takes 403 hours, or 13 hours 20 minutes.
Example 3: Find one worker alone
Two machines together pack an order in 6 hours. The first machine alone takes 10 hours. How long would the second machine take alone?
- Let x be the second machine time: 110 + 1x = 16.
- 1x = 16 − 110 = 530 − 330 = 230 = 115.
- So x = 15. Check: 110 + 115 = 330 + 230 = 16.
Answer: The second machine would take 15 hours alone.
Common mistakes
Two people cannot take longer than the faster one would alone. Averaging times ignores that both work at once.
Adding times describes one person working after the other, not both working together.
Anything that undoes the job has a negative rate. Ask whether each worker helps or hurts before you add.
Not to be confused with
d = rt uses a speed. Work problems use the same structure, work = rate × time, with the rate measured in jobs per hour and the whole job equal to 1.
For one fixed job, more workers of equal speed means less time: workers and time are inversely proportional. Work rate problems handle workers with different speeds.
Mixture problems add amount × value; work problems add rates. Both add parts to make a whole.
Quick check: 3 questions
One hose fills a tank in 3 hours and another in 6 hours. How long do both hoses take together?
Answer: C. 13 + 16 = 12 tank per hour, so 2 hours. 4.5 averages the times, 9 adds them, and 1.5 halves the fastest time.Show answer and explanation
Kim mows a lawn in 2 hours and Lee in 3 hours. What part of the lawn do they mow together in 1 hour?
Answer: A. 12 + 13 = 56 of the lawn. 15 adds the times, 25 misuses the formula, and 16 multiplies the rates.Show answer and explanation
Working together, Raj and Mia build a fence in 4 days. Raj alone takes 6 days. How long would Mia take alone?
Answer: D. 14 − 16 = 312 − 212 = 112, so 12 days. 2 subtracts the times, 5 averages them, and 10 adds them.Show answer and explanation
Work Rate Problems: frequently asked questions
What is the formula for work rate problems?
For two workers, 1a + 1b = 1t, where a and b are the times each needs alone and t is the time together. Solved for t, it becomes t = aba + b. For 4 and 12 hours, t = 4816 = 3 hours. More workers just add more fractions.
Why do you add the rates and not the times?
The rates measure how much of the job gets done in each hour, and when two people work at once, their hourly progress adds up. Times measure something else: how long the whole job takes for one person. Adding times would describe working one after the other, which only makes the job take longer.
How can I check a work rate answer?
Two quick tests. First, the time together must be shorter than the fastest worker alone. Second, multiply each rate by your answer and add: the total should be exactly 1 job. For 3 hours with rates 14 and 112, 34 + 312 = 1.
Practice and tools
Sources
- OpenStax, Intermediate Algebra 2e, 7.5 Solve Applications with Rational Equations
- Common Core State Standards, High School Algebra: Creating Equations (A-CED.A.1)
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSA-CED.A.1
Cite this entry
Effortless Math Team. “Work Rate Problems.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/work-rate-problems/
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