The short answer
Every work and rate problem on the GRE is built on one relationship: rate × time = amount of work done. Distance problems are the same relationship with distance in place of work.
The second idea does most of the remaining lifting: rates add when things work simultaneously. If one pipe fills a tank at a rate of one-third of a tank per hour and another at one-sixth per hour, together they fill at one-half per hour, and the tank takes two hours.
Almost every difficulty in this topic comes from working with times instead of rates. Times do not add. Rates do. Converting to rates first is the single habit that makes these problems routine.
Set up in rates, not times
If a worker completes a job in six hours, their rate is one job per six hours, or 1/6 of the job per hour. That reciprocal step is the whole setup, and skipping it is the most common source of wrong answers.
Suppose one machine takes 4 hours to complete a job and another takes 6 hours. Their rates are 1/4 and 1/6 of the job per hour. Working together, the combined rate is 1/4 + 1/6 = 3/12 + 2/12 = 5/12 of the job per hour.
Time is then the reciprocal of the combined rate: 12/5 hours, or 2.4 hours. Note that this is less than the faster worker's time alone, which is a useful sanity check — two workers together must finish faster than either one individually, so any answer larger than the smaller of the two times is wrong.
For exactly two workers there is a shortcut worth memorising. If they take a and b hours individually, together they take ab/(a + b) hours. Checking it here: (4 × 6)/(4 + 6) = 24/10 = 2.4. The shortcut only works for two workers, so for three or more, add the rates.
Average speed is not the average of the speeds
This is the most reliably tested trap in the topic. Average speed is total distance divided by total time — always, without exception.
Consider a car that travels 120 miles at 60 miles per hour and returns the same 120 miles at 40 miles per hour. The tempting answer is 50 miles per hour. It is wrong.
The outbound leg takes 2 hours; the return leg takes 3 hours. Total distance is 240 miles, total time is 5 hours, and average speed is 48 miles per hour.
The reason the naive answer fails is that the car spends more time at the slower speed, so the slower speed gets more weight. This generalises: whenever equal distances are covered at different speeds, the average speed is always less than the arithmetic mean of the speeds. That fact alone often lets you eliminate answer choices immediately, and it is a strong tool on quantitative comparison questions where you may not need the exact value at all.
| Problem type | Set-up | Watch for |
|---|---|---|
| Combined work | Add the individual rates | Adding times instead of rates |
| Two workers, find total time | ab/(a + b) | Using it for three or more workers |
| Average speed | Total distance ÷ total time | Averaging the two speeds |
| One worker joins partway | Track work completed in stages | Forgetting the head start |
| Objects approaching | Add the speeds for closing rate | Subtracting when they move apart |
A worked example with a staggered start
Alice can paint a room in 5 hours. Bob can paint the same room in 3 hours. Alice works alone for 2 hours, then Bob joins her. How long does the whole job take?
Rates first. Alice paints at 1/5 of the room per hour, Bob at 1/3.
In her first 2 hours alone, Alice completes 2 × 1/5 = 2/5 of the room. That leaves 3/5 remaining.
Working together their combined rate is 1/5 + 1/3 = 3/15 + 5/15 = 8/15 of the room per hour. Time to finish the remaining 3/5 is (3/5) ÷ (8/15) = (3/5) × (15/8) = 45/40 = 9/8 hours, or 1.125 hours.
Total elapsed time is 2 + 1.125 = 3.125 hours. Note the question's wording matters: "how long does the whole job take" wants total elapsed time, while "how long do they work together" wants only 1.125 hours. GRE answer choices routinely include both.
When to pick numbers instead
Some rate problems are stated entirely in variables — "if m machines produce p parts in h hours, how many parts do n machines produce in k hours?" — and the algebra is more error-prone than the arithmetic.
On those, picking small concrete numbers is usually faster and safer. Let m = 2, p = 10, h = 5. Then two machines make 10 parts in 5 hours, so one machine makes 5 parts in 5 hours, or 1 part per hour. Now let n = 3 and k = 4: three machines at 1 part per hour each for 4 hours give 12 parts.
Substitute your chosen values into each answer choice and keep the one that produces 12. Choose numbers that avoid coincidences — steer clear of 0, 1, and values that make several choices agree, and if two choices both survive, run a second set of numbers.
This is a general GRE technique rather than a rate-specific one, but rate problems are where it pays off most, because the relationships are simple and the variable expressions are ugly.
The common mistake
Adding times is the error that accounts for most misses here. Two workers taking 4 and 6 hours do not take 10 hours together, and they do not take 5. The quantity that combines additively is rate.
The second most common is answering the wrong question. Rate problems frequently ask for something one step removed from what you computed — the time the second worker spent rather than total time, the remaining distance rather than the distance covered, the rate rather than the time. Underline the actual question before you start.
Third is unit mismatch. If a rate is given per minute and a duration in hours, one of them has to be converted, and the answer choices will usually include the version you get if you forget.
Finally, on quantitative comparison, resist computing when you can reason. If the question compares the average speed of a round trip at two different speeds against the arithmetic mean of those speeds, you already know the answer without touching the numbers.
Practice this on Verbloom
Verbloom's GRE quantitative practice includes rate and work problems with step-by-step explanations, including the near-miss answer choices that correspond to specific setup errors.
If your misses in this topic cluster on one thing — usually adding times, or answering the adjacent question — that shows up quickly in review and is straightforward to correct.
Frequently asked questions
What is the formula for two people working together?
If they take a and b hours individually, together they take ab/(a + b) hours. Equivalently, add their rates: 1/a + 1/b, then take the reciprocal. The shortcut applies only to two workers; for three or more, add all the rates and invert the sum.
Why isn't average speed just the average of the two speeds?
Because average speed is total distance divided by total time, and when equal distances are covered at different speeds, more time is spent at the slower speed. That weights the slower speed more heavily, so the average speed is always less than the arithmetic mean of the two speeds.
How do I handle a rate problem where someone starts late?
Break it into stages. Compute how much work the first person completes alone, subtract that from the whole job, then divide the remainder by the combined rate to find the additional time. Read carefully whether the question wants total elapsed time or only the shared portion.
Should I use algebra or pick numbers on rate problems?
Pick numbers when the problem is stated entirely in variables and the answer choices are expressions. Use algebra when the numbers are concrete. If you pick numbers, avoid 0 and 1, and test a second set if two answer choices both survive the first.
Do GRE rate problems require memorising many formulas?
No. Rate × time = work covers essentially all of them, together with the fact that simultaneous rates add. The two-worker shortcut is a convenience rather than a separate concept, and it follows directly from adding rates.
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